Subgroups — C₂×#22574366 #31762954

Order 17152.

9542 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22574366	C₂	1	32	yes	maximal, direct product factor
2	C₂×#16864135	C₂	1	3	yes	maximal, complemented
3	C₂×#16864076	C₂	1	3	yes	maximal, complemented
4	C₂×#16864039	C₂	1	6	yes	maximal, complemented
5	C₂×#16864028	C₂	1	3	yes	maximal, complemented
6	C₂×#16863980	C₂	1	6	yes	maximal, complemented
7	C₂×#16863964	C₂	1	3	yes	maximal, complemented
8	C₂×#16863956	C₂	1	3	yes	maximal, complemented
9	C₂×#120×D₆₇	C₂	1	3	yes	maximal, complemented
10	C₂×H(3,𝔽₄)×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁵×D₆₇	C₂²	1	1	yes	characteristic, complemented
12	C₂²×#11761629	C₂²	1	3	yes	complemented
13	C₂³×C₂²⋊D₆₇	C₂²	1	3	yes	complemented
14	C₂²×D₄×D₆₇	C₂²	1	6	yes	complemented
15	C₂²×D₄×D₆₇	C₂²	1	3	yes	complemented
16	#16864135	C₂²	1	48	yes	complemented
17	#16864076	C₂²	1	48	yes	complemented
18	#16864039	C₂²	1	96	yes	complemented
19	#16864028	C₂²	1	48	yes	complemented
20	#16863980	C₂²	1	96	yes	complemented
21	#16863964	C₂²	1	48	yes	complemented
22	#16863956	C₂²	1	48	yes	complemented
23	C₂×#14524040	C₂²	1	1	yes	characteristic, complemented
24	C₂×#14524039	C₂²	1	3	yes	complemented
25	C₂×#14524037	C₂²	1	2	yes	complemented
26	C₂×#14524036	C₂²	1	6	yes	complemented
27	C₂×#14524032	C₂²	1	3	yes	complemented
28	C₂×#14524027	C₂²	1	3	yes	complemented
29	C₂×#14524025	C₂²	1	6	yes	complemented
30	C₂×#14524025	C₂²	1	6	yes	complemented
31	C₂×#14524024	C₂²	1	3	yes	complemented
32	C₂×#14524019	C₂²	1	6	yes	complemented
33	C₂×#14524016	C₂²	1	6	yes	complemented
34	C₂×#14524016	C₂²	1	6	yes	complemented
35	C₂×#14524016	C₂²	1	6	yes	complemented
36	C₂×#14524015	C₂²	1	6	yes	complemented
37	C₂×#14524014	C₂²	1	6	yes	complemented
38	C₂×#14524013	C₂²	1	3	yes	complemented
39	C₂×#14524013	C₂²	1	6	yes	complemented
40	C₂×#14524013	C₂²	1	6	yes	complemented
41	C₂×#14524012	C₂²	1	3	yes	complemented
42	C₂×#14524012	C₂²	1	1	yes	characteristic, complemented
43	C₂×#14524011	C₂²	1	3	yes	—
44	C₂×#14524011	C₂²	1	6	yes	complemented
45	#120×D₆₇	C₂²	1	48	yes	complemented
46	C₂×C₄×D₂₆₈	C₂²	1	6	yes	complemented
47	C₂×(C₂²⋊D₆₇)⋊C₄	C₂²	1	6	yes	complemented
48	C₂×#14524000	C₂²	1	3	yes	complemented
49	C₂×C₂²⋊C₄×D₆₇	C₂²	1	6	yes	complemented
50	C₂×C₂²⋊C₄×D₆₇	C₂²	1	3	yes	complemented
51	C₂×C₂²⋊C₄×D₆₇	C₂²	1	3	yes	complemented
52	H(3,𝔽₄)×C₆₇	C₂²	1	16	yes	complemented
53	C₂×#124×C₆₇	C₂²	1	6	yes	complemented
54	C₂×#124×C₆₇	C₂²	1	3	yes	complemented
55	C₂×#120×C₆₇	C₂²	1	3	yes	complemented
56	C₂×#120×C₆₇	C₂²	1	3	yes	complemented
57	C₂⁴×D₆₇	—	4	12	no	—
58	C₂⁴×D₆₇	C₂³	1	3	yes	—
59	C₂⁴×D₆₇	C₂³	1	3	yes	—
60	C₂⁴×D₆₇	C₂³	1	3	yes	—
61	C₂⁴×D₆₇	C₂³	1	8	yes	complemented
62	C₂×#11761629	—	4	12	no	—
63	C₂×#11761629	C₂³	1	24	yes	complemented
64	C₂²×D₂₆₈	C₂³	1	6	yes	—
65	C₂²×D₂₆₈	C₂³	1	3	yes	—
66	C₂²×D₂₆₈	C₂³	1	3	yes	complemented
67	C₂²×C₂²⋊D₆₇	—	4	12	no	—
68	C₂²×C₂²⋊D₆₇	C₂³	1	6	yes	—
69	C₂²×C₂²⋊D₆₇	C₂³	1	3	yes	—
70	C₂²×C₂²⋊D₆₇	C₂³	1	3	yes	—
71	C₂²×C₂²⋊D₆₇	C₂³	1	6	yes	—
72	C₂²×C₂²⋊D₆₇	C₂³	1	6	yes	complemented
73	C₂²×C₂²⋊D₆₇	C₂³	1	24	yes	complemented
74	C₂×D₄×D₆₇	—	4	12	no	—
75	C₂×D₄×D₆₇	—	4	24	no	—
76	C₂×D₄×D₆₇	C₂³	1	48	yes	complemented
77	C₂×D₄×D₆₇	C₂³	1	24	yes	complemented
78	C₂³×Dic₆₇	C₂³	1	3	yes	complemented
79	C₂²×C₄×D₆₇	C₂³	1	3	yes	—
80	C₂²×C₄×D₆₇	C₂³	1	6	yes	—
81	C₂²×C₄×D₆₇	C₂³	1	6	yes	complemented
82	#14524040	C₂³	1	8	yes	complemented
83	#14524039	C₂³	1	24	yes	complemented
84	#14524037	C₂³	1	16	yes	complemented
85	#14524036	C₂³	1	48	yes	complemented
86	#14524032	C₂³	1	24	yes	complemented
87	#14524027	C₂³	1	24	yes	complemented
88	#14524025	C₂³	1	48	yes	complemented
89	#14524025	C₂³	1	48	yes	complemented
90	#14524024	C₂³	1	24	yes	complemented
91	#14524019	C₂³	1	48	yes	complemented
92	#14524016	C₂³	1	48	yes	complemented
93	#14524016	C₂³	1	48	yes	complemented
94	#14524016	C₂³	1	48	yes	complemented
95	#14524015	C₂³	1	48	yes	complemented
96	#14524014	C₂³	1	48	yes	complemented
97	#14524013	C₂³	1	24	yes	complemented
98	#14524013	C₂³	1	48	yes	complemented
99	#14524013	C₂³	1	48	yes	complemented
100	#14524012	C₂³	1	24	yes	complemented
101	#14524012	C₂³	1	8	yes	complemented
102	#14524011	C₂³	1	24	yes	—
103	#14524011	C₂³	1	48	yes	complemented
104	C₄×D₂₆₈	C₂³	1	48	yes	complemented
105	(C₂²⋊D₆₇)⋊C₄	C₂³	1	48	yes	complemented
106	#14524000	C₂³	1	24	yes	complemented
107	C₂×C₄⋊Dic₆₇	C₂³	1	6	yes	complemented
108	C₂×Dic₆₇⋊C₄	C₂³	1	6	yes	complemented
109	C₂×Dic₆₇⋊C₄	C₂³	1	6	yes	complemented
110	C₂×C₂²⋊Dic₆₇	C₂³	1	3	yes	—
111	C₂×C₂²⋊Dic₆₇	C₂³	1	6	yes	—
112	C₂×C₂²⋊Dic₆₇	C₂³	1	3	yes	complemented
113	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	6	yes	—
114	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	6	yes	—
115	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	6	yes	—
116	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	6	yes	complemented
117	C₂²⋊C₄×D₆₇	C₂³	1	48	yes	complemented
118	C₂²⋊C₄×D₆₇	C₂³	1	24	yes	complemented
119	C₂²⋊C₄×D₆₇	C₂³	1	24	yes	complemented
120	C₂×C₄×Dic₆₇	C₂³	1	3	yes	complemented
121	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
122	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic, complemented
123	C₂²×Q₈×C₆₇	C₂³	1	3	yes	complemented
124	C₂²×D₄×C₆₇	C₂³	1	3	yes	—
125	C₂²×D₄×C₆₇	C₂³	1	6	yes	—
126	#124×C₆₇	C₂³	1	48	yes	complemented
127	#124×C₆₇	C₂³	1	24	yes	complemented
128	#120×C₆₇	C₂³	1	24	yes	complemented
129	#120×C₆₇	C₂³	1	24	yes	complemented
130	C₂×C₂²⋊C₄×C₆₇	C₂³	1	6	yes	—
131	C₂×C₂²⋊C₄×C₆₇	C₂³	1	3	yes	—
132	C₂×C₂²⋊C₄×C₆₇	C₂³	1	3	yes	complemented
133	C₂×C₂²⋊C₄×C₆₇	C₂³	1	6	yes	complemented
134	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic, complemented
135	C₂×C₄²×C₆₇	C₂³	1	2	yes	complemented
136	C₂³×D₆₇	—	2	24	no	—
137	C₂³×D₆₇	—	2	12	no	—
138	C₂³×D₆₇	—	2	12	no	—
139	C₂³×D₆₇	—	4	48	no	—
140	C₂³×D₆₇	—	4	6	no	—
141	C₂³×D₆₇	—	4	12	no	—
142	C₂³×D₆₇	—	4	12	no	—
143	C₂³×D₆₇	—	4	64	no	—
144	C₂³×D₆₇	C₂⁴	1	1	yes	characteristic
145	C₂³×D₆₇	C₂⁴	1	3	yes	—
146	C₂³×D₆₇	C₂⁴	1	12	yes	—
147	C₂³×D₆₇	C₂⁴	1	3	yes	—
148	C₂³×D₆₇	C₂⁴	1	12	yes	—
149	C₂³×D₆₇	C₂⁴	1	12	yes	—
150	#11761629	—	4	48	no	—
151	C₂×D₂₆₈	—	2	24	no	—
152	C₂×D₂₆₈	—	4	6	no	—
153	C₂×D₂₆₈	—	4	6	no	—
154	C₂×D₂₆₈	C₂⁴	1	24	yes	—
155	C₂×D₂₆₈	C₂⁴	1	12	yes	—
156	C₂×D₂₆₈	C₂⁴	1	12	yes	complemented
157	C₂×C₂²⋊D₆₇	—	2	24	no	—
158	C₂×C₂²⋊D₆₇	—	4	6	no	—
159	C₂×C₂²⋊D₆₇	—	4	6	no	—
160	C₂×C₂²⋊D₆₇	—	4	12	no	—
161	C₂×C₂²⋊D₆₇	—	4	12	no	—
162	C₂×C₂²⋊D₆₇	—	4	48	no	—
163	C₂×C₂²⋊D₆₇	C₂⁴	1	24	yes	—
164	C₂×C₂²⋊D₆₇	C₂⁴	1	12	yes	—
165	C₂×C₂²⋊D₆₇	C₂⁴	1	12	yes	—
166	C₂×C₂²⋊D₆₇	C₂⁴	1	24	yes	—
167	C₂×C₂²⋊D₆₇	C₂⁴	1	24	yes	complemented
168	D₄×D₆₇	—	4	48	no	—
169	D₄×D₆₇	—	4	96	no	—
170	C₂²×Dic₆₇	—	4	6	no	—
171	C₂²×Dic₆₇	C₂⁴	1	6	yes	—
172	C₂²×Dic₆₇	C₂⁴	1	3	yes	—
173	C₂²×Dic₆₇	C₂⁴	1	12	yes	complemented
174	C₂×C₄×D₆₇	—	2	12	no	—
175	C₂×C₄×D₆₇	—	4	12	no	—
176	C₂×C₄×D₆₇	—	4	12	no	—
177	C₂×C₄×D₆₇	C₂⁴	1	12	yes	—
178	C₂×C₄×D₆₇	C₂⁴	1	24	yes	—
179	C₂×C₄×D₆₇	C₂⁴	1	24	yes	complemented
180	C₄⋊Dic₆₇	C₂⁴	1	24	yes	complemented
181	Dic₆₇⋊C₄	C₂⁴	1	24	yes	complemented
182	Dic₆₇⋊C₄	C₂⁴	1	24	yes	complemented
183	C₂²⋊Dic₆₇	C₂⁴	1	12	yes	—
184	C₂²⋊Dic₆₇	C₂⁴	1	24	yes	—
185	C₂²⋊Dic₆₇	C₂⁴	1	12	yes	complemented
186	(C₂×D₆₇)⋊C₄	C₂⁴	1	24	yes	—
187	(C₂×D₆₇)⋊C₄	C₂⁴	1	24	yes	—
188	(C₂×D₆₇)⋊C₄	C₂⁴	1	24	yes	—
189	(C₂×D₆₇)⋊C₄	C₂⁴	1	24	yes	complemented
190	C₄×Dic₆₇	C₂⁴	1	12	yes	complemented
191	C₂⁴×C₆₇	—	4	6	no	—
192	C₂⁴×C₆₇	—	4	6	no	—
193	C₂⁴×C₆₇	C₂⁴	1	3	yes	—
194	C₂⁴×C₆₇	C₂⁴	1	3	yes	—
195	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
196	C₂⁴×C₆₇	C₂⁴	1	4	yes	complemented
197	C₂×Q₈×C₆₇	—	4	6	no	—
198	C₂×Q₈×C₆₇	C₂⁴	1	12	yes	complemented
199	C₂×D₄×C₆₇	—	4	6	no	—
200	C₂×D₄×C₆₇	—	4	12	no	—
201	C₂×D₄×C₆₇	C₂⁴	1	12	yes	—
202	C₂×D₄×C₆₇	C₂⁴	1	24	yes	—
203	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
204	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
205	C₂²⋊C₄×C₆₇	C₂⁴	1	24	yes	—
206	C₂²⋊C₄×C₆₇	C₂⁴	1	12	yes	—
207	C₂²⋊C₄×C₆₇	C₂⁴	1	12	yes	complemented
208	C₂²⋊C₄×C₆₇	C₂⁴	1	24	yes	complemented
209	C₄²×C₆₇	C₂⁴	1	4	yes	complemented
210	C₄²×C₆₇	C₂⁴	1	8	yes	complemented
211	C₂³×C₆₇	—	2	6	no	—
212	C₂³×C₆₇	—	2	12	no	—
213	C₂³×C₆₇	—	2	6	no	—
214	C₂³×C₆₇	—	2	12	no	—
215	C₂³×C₆₇	—	4	12	no	—
216	C₂³×C₆₇	—	4	12	no	—
217	C₂³×C₆₇	—	4	16	no	—
218	C₂³×C₆₇	—	8	8	no	—
219	C₂³×C₆₇	C₂⁵	1	6	yes	—
220	C₂³×C₆₇	C₂⁵	1	6	yes	—
221	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
222	C₂²×D₆₇	—	2	12	no	—
223	C₂²×D₆₇	—	2	12	no	—
224	C₂²×D₆₇	—	2	6	no	—
225	C₂²×D₆₇	—	2	6	no	—
226	C₂²×D₆₇	—	2	48	no	—
227	C₂²×D₆₇	—	2	24	no	—
228	C₂²×D₆₇	—	2	24	no	—
229	C₂²×D₆₇	—	4	12	no	—
230	C₂²×D₆₇	—	4	24	no	—
231	C₂²×D₆₇	—	4	48	no	—
232	C₂²×D₆₇	—	4	24	no	—
233	C₂²×D₆₇	—	4	48	no	—
234	C₂²×D₆₇	—	4	128	no	—
235	C₂²×D₆₇	—	8	24	no	—
236	C₂²×D₆₇	H(5,𝔽₂)	1	12	yes	—
237	C₂²×D₆₇	C₂⁵	1	2	yes	—
238	C₂²×D₆₇	C₂⁵	1	6	yes	—
239	C₂²×D₆₇	C₂⁵	1	6	yes	—
240	Q₈×C₆₇	—	4	12	no	—
241	D₄×C₆₇	—	4	12	no	—
242	D₄×C₆₇	—	4	24	no	—
243	C₂×C₄×C₆₇	—	2	6	no	—
244	C₂×C₄×C₆₇	—	2	12	no	—
245	C₂×C₄×C₆₇	C₂⁵	1	6	yes	—
246	C₂×C₄×C₆₇	C₂⁵	1	12	yes	—
247	C₂²⋊D₆₇	—	2	48	no	—
248	C₂²⋊D₆₇	—	4	12	no	—
249	C₂²⋊D₆₇	—	4	12	no	—
250	C₂²⋊D₆₇	—	4	24	no	—
251	C₂²⋊D₆₇	—	4	24	no	—
252	C₂×Dic₆₇	—	2	6	no	—
253	C₂×Dic₆₇	—	2	12	no	—
254	C₂×Dic₆₇	—	4	12	no	—
255	C₂×Dic₆₇	C₂⁵	1	12	yes	—
256	C₂×Dic₆₇	C₂⁵	1	6	yes	—
257	D₂₆₈	—	2	48	no	—
258	D₂₆₈	—	4	12	no	—
259	D₂₆₈	—	4	12	no	—
260	C₄×D₆₇	—	2	24	no	—
261	C₄×D₆₇	—	4	24	no	—
262	C₄×D₆₇	—	4	24	no	—
263	C₂²×C₆₇	—	2	6	no	—
264	C₂²×C₆₇	—	2	12	no	—
265	C₂²×C₆₇	—	2	6	no	—
266	C₂²×C₆₇	—	2	12	no	—
267	C₂²×C₆₇	—	4	16	no	—
268	C₂²×C₆₇	—	4	12	no	—
269	C₂²×C₆₇	—	4	12	no	—
270	C₂²×C₆₇	—	8	8	no	—
271	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	6	yes	—
272	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
273	C₂×D₆₇	—	2	12	no	—
274	C₂×D₆₇	—	2	24	no	—
275	C₂×D₆₇	—	2	12	no	—
276	C₂×D₆₇	—	2	6	no	—
277	C₂×D₆₇	—	2	6	no	—
278	C₂×D₆₇	—	4	48	no	—
279	C₂×D₆₇	—	4	48	no	—
280	C₂×D₆₇	—	4	4	no	—
281	C₂×D₆₇	—	4	12	no	—
282	C₂×D₆₇	—	8	24	no	—
283	C₂×D₆₇	C₂×H(5,𝔽₂)	1	12	yes	—
284	C₄×C₆₇	—	2	6	no	—
285	C₄×C₆₇	—	2	12	no	—
286	Dic₆₇	—	2	6	no	—
287	Dic₆₇	—	2	12	no	—
288	C₂×#3532	—	67	1	no	maximal, Sylow
289	C₂×C₆₇	—	4	6	no	—
290	C₂×C₆₇	—	4	6	no	—
291	C₂×C₆₇	#3532	1	4	yes	complemented
292	C₂×C₆₇	C₂²×H(5,𝔽₂)	1	3	yes	—
293	D₆₇	—	2	12	no	—
294	D₆₇	—	4	2	no	—
295	D₆₇	—	4	6	no	—
296	#3532	—	67	32	no	—
297	C₂×H(3,𝔽₄)	D₆₇	1	1	yes	characteristic, complemented
298	C₂×#560	—	67	3	no	—
299	C₂×#551	—	67	3	no	—
300	C₂×#546	—	67	6	no	—
301	C₂×D₄²	—	67	3	no	—
302	C₂×#538	—	67	6	no	—
303	C₂×#535	—	67	3	no	—
304	C₂×#534	—	67	3	no	—
305	C₂²×#120	—	67	3	no	—
306	C₆₇	C₂×#3532	1	1	yes	characteristic, complemented, Sylow
307	C₂⁶	—	67	1	no	—
308	C₂²×#41	—	67	3	no	—
309	C₂³×D₄	—	67	3	no	—
310	C₂³×D₄	—	67	3	no	—
311	C₂³×D₄	—	67	6	no	—
312	H(3,𝔽₄)	C₂×D₆₇	1	16	yes	complemented
313	#560	—	67	48	no	—
314	#551	—	67	48	no	—
315	#546	—	67	96	no	—
316	D₄²	—	67	48	no	—
317	#538	—	67	96	no	—
318	#535	—	67	48	no	—
319	#534	—	67	48	no	—
320	C₂×#127	—	67	3	no	—
321	C₂×#127	—	67	1	no	—
322	C₂×#126	—	67	6	no	—
323	C₂×#126	—	67	2	no	—
324	C₂×#124	—	67	3	no	—
325	C₂×#124	C₂×D₆₇	1	3	yes	complemented
326	C₂×#124	C₂×D₆₇	1	6	yes	complemented
327	C₂×#123	—	67	3	no	—
328	C₂×#123	—	67	6	no	—
329	C₂×#123	—	67	3	no	—
330	C₂×#123	—	67	6	no	—
331	C₂×C₂²⋊Q₈	—	67	6	no	—
332	C₂×#121	—	67	6	no	—
333	C₂×#121	—	67	6	no	—
334	C₂×#121	—	67	6	no	—
335	C₂×#121	—	67	6	no	—
336	C₂×#121	—	67	6	no	—
337	C₂×#120	—	67	3	no	—
338	C₂×#120	—	67	1	no	—
339	C₂×#120	—	67	6	no	—
340	C₂×#120	—	67	3	no	—
341	C₂×#120	—	67	48	no	—
342	C₂×#120	—	67	6	no	—
343	C₂×#120	—	67	3	no	—
344	C₂×#120	—	67	6	no	—
345	C₂×#120	C₂×D₆₇	1	3	yes	complemented
346	C₂×#120	C₂×D₆₇	1	3	yes	complemented
347	C₂×C₄×D₄	—	67	6	no	—
348	C₂×C₄×D₄	—	67	6	no	—
349	C₂×#117	—	67	3	no	—
350	C₂²×C₂²⋊C₄	—	67	3	no	—
351	C₂²×C₂²⋊C₄	—	67	6	no	—
352	C₂²×C₂²⋊C₄	—	67	3	no	—
353	C₂⁵	—	268	12	no	—
354	C₂⁵	—	67	8	no	—
355	C₂⁵	—	67	3	no	—
356	C₂⁵	—	67	3	no	—
357	C₂⁵	—	67	3	no	—
358	C₂⁵	C₂²×D₆₇	1	1	yes	characteristic
359	C₂⁵	C₂²×D₆₇	1	1	yes	characteristic, complemented
360	C₂×#41	—	268	12	no	—
361	C₂×#41	—	67	24	no	—
362	C₂²×Q₈	C₂²×D₆₇	1	3	yes	complemented
363	C₂²×D₄	—	268	24	no	—
364	C₂²×D₄	—	268	12	no	—
365	C₂²×D₄	—	268	12	no	—
366	C₂²×D₄	—	67	3	no	—
367	C₂²×D₄	—	67	3	no	—
368	C₂²×D₄	—	67	6	no	—
369	C₂²×D₄	—	67	6	no	—
370	C₂²×D₄	—	67	24	no	—
371	C₂²×D₄	—	67	24	no	—
372	C₂²×D₄	—	67	48	no	—
373	C₂²×D₄	—	67	6	no	—
374	C₂²×D₄	—	67	3	no	—
375	C₂²×D₄	—	67	3	no	—
376	C₂²×D₄	—	67	6	no	—
377	C₂²×D₄	C₂²×D₆₇	1	6	yes	—
378	C₂²×D₄	C₂²×D₆₇	1	3	yes	—
379	C₂³×C₄	—	67	3	no	—
380	C₂³×C₄	—	67	6	no	—
381	C₂³×C₄	—	67	6	no	—
382	C₂³×C₄	—	67	3	no	—
383	#127	—	67	24	no	—
384	#127	—	67	8	no	—
385	#126	—	67	48	no	—
386	#126	—	67	16	no	—
387	#124	—	67	24	no	—
388	#124	C₂²×D₆₇	1	24	yes	complemented
389	#124	C₂²×D₆₇	1	48	yes	complemented
390	#123	—	67	24	no	—
391	#123	—	67	48	no	—
392	#123	—	67	24	no	—
393	#123	—	67	48	no	—
394	C₂²⋊Q₈	—	67	48	no	—
395	#121	—	67	48	no	—
396	#121	—	67	48	no	—
397	#121	—	67	48	no	—
398	#121	—	67	48	no	—
399	#121	—	67	48	no	—
400	#120	—	67	24	no	—
401	#120	—	67	8	no	—
402	#120	—	67	48	no	—
403	#120	—	67	24	no	—
404	#120	—	67	48	no	—
405	#120	—	67	24	no	—
406	#120	—	67	48	no	—
407	#120	C₂²×D₆₇	1	24	yes	complemented
408	#120	C₂²×D₆₇	1	24	yes	complemented
409	C₄×D₄	—	67	48	no	—
410	C₄×D₄	—	67	48	no	—
411	#117	—	67	24	no	—
412	C₂×C₄⋊C₄	—	67	6	no	—
413	C₂×C₄⋊C₄	—	67	6	no	—
414	C₂×C₄⋊C₄	—	67	6	no	—
415	C₂×C₂²⋊C₄	—	67	3	no	—
416	C₂×C₂²⋊C₄	—	67	6	no	—
417	C₂×C₂²⋊C₄	—	67	3	no	—
418	C₂×C₂²⋊C₄	—	67	6	no	—
419	C₂×C₂²⋊C₄	—	67	6	no	—
420	C₂×C₂²⋊C₄	—	67	6	no	—
421	C₂×C₂²⋊C₄	—	67	6	no	—
422	C₂×C₂²⋊C₄	—	67	24	no	—
423	C₂×C₂²⋊C₄	—	67	48	no	—
424	C₂×C₂²⋊C₄	—	67	24	no	—
425	C₂×C₂²⋊C₄	C₂²×D₆₇	1	6	yes	—
426	C₂×C₂²⋊C₄	C₂²×D₆₇	1	3	yes	—
427	C₂×C₂²⋊C₄	C₂²×D₆₇	1	3	yes	complemented
428	C₂×C₂²⋊C₄	C₂²×D₆₇	1	6	yes	complemented
429	C₂×C₄²	—	67	3	no	—
430	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic, complemented
431	C₂×C₄²	C₂²×D₆₇	1	2	yes	complemented
432	C₂⁴	—	4	6	no	—
433	C₂⁴	—	4	6	no	—
434	C₂⁴	—	268	12	no	—
435	C₂⁴	—	268	6	no	—
436	C₂⁴	—	268	12	no	—
437	C₂⁴	—	268	64	no	—
438	C₂⁴	—	268	48	no	—
439	C₂⁴	—	67	12	no	—
440	C₂⁴	—	67	12	no	—
441	C₂⁴	—	67	12	no	—
442	C₂⁴	—	67	3	no	—
443	C₂⁴	—	67	1	no	—
444	C₂⁴	—	67	3	no	—
445	C₂⁴	—	134	12	no	—
446	C₂⁴	—	134	24	no	—
447	C₂⁴	—	134	12	no	—
448	C₂⁴	C₂³×D₆₇	1	3	yes	—
449	C₂⁴	C₂³×D₆₇	1	3	yes	—
450	C₂⁴	C₂³×D₆₇	1	4	yes	—
451	C₂⁴	C₂³×D₆₇	1	4	yes	complemented
452	#41	—	268	48	no	—
453	C₂×Q₈	—	4	6	no	—
454	C₂×Q₈	C₂³×D₆₇	1	12	yes	complemented
455	C₂×D₄	—	4	12	no	—
456	C₂×D₄	—	4	6	no	—
457	C₂×D₄	—	268	6	no	—
458	C₂×D₄	—	268	6	no	—
459	C₂×D₄	—	268	96	no	—
460	C₂×D₄	—	268	48	no	—
461	C₂×D₄	—	268	48	no	—
462	C₂×D₄	—	268	6	no	—
463	C₂×D₄	—	268	6	no	—
464	C₂×D₄	—	268	12	no	—
465	C₂×D₄	—	268	12	no	—
466	C₂×D₄	—	67	12	no	—
467	C₂×D₄	—	67	12	no	—
468	C₂×D₄	—	67	24	no	—
469	C₂×D₄	—	67	24	no	—
470	C₂×D₄	—	67	24	no	—
471	C₂×D₄	—	67	12	no	—
472	C₂×D₄	—	67	12	no	—
473	C₂×D₄	—	67	24	no	—
474	C₂×D₄	—	134	24	no	—
475	C₂×D₄	—	134	24	no	—
476	C₂×D₄	C₂³×D₆₇	1	24	yes	—
477	C₂×D₄	C₂³×D₆₇	1	12	yes	—
478	C₂²×C₄	—	268	12	no	—
479	C₂²×C₄	—	268	12	no	—
480	C₂²×C₄	—	268	6	no	—
481	C₂²×C₄	—	67	12	no	—
482	C₂²×C₄	—	67	24	no	—
483	C₂²×C₄	—	67	24	no	—
484	C₂²×C₄	—	67	6	no	—
485	C₂²×C₄	—	67	3	no	—
486	C₂²×C₄	—	67	12	no	—
487	C₂²×C₄	—	134	12	no	—
488	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
489	C₂²×C₄	C₂³×D₆₇	1	6	yes	—
490	C₄⋊C₄	—	67	24	no	—
491	C₄⋊C₄	—	67	24	no	—
492	C₄⋊C₄	—	67	24	no	—
493	C₂²⋊C₄	—	67	12	no	—
494	C₂²⋊C₄	—	67	24	no	—
495	C₂²⋊C₄	—	67	12	no	—
496	C₂²⋊C₄	—	67	24	no	—
497	C₂²⋊C₄	—	67	24	no	—
498	C₂²⋊C₄	—	67	24	no	—
499	C₂²⋊C₄	—	67	24	no	—
500	C₂²⋊C₄	C₂³×D₆₇	1	24	yes	—
501	C₂²⋊C₄	C₂³×D₆₇	1	12	yes	—
502	C₂²⋊C₄	C₂³×D₆₇	1	12	yes	complemented
503	C₂²⋊C₄	C₂³×D₆₇	1	24	yes	complemented
504	C₄²	—	67	12	no	—
505	C₄²	C₂³×D₆₇	1	4	yes	complemented
506	C₄²	C₂³×D₆₇	1	8	yes	complemented
507	C₂³	—	2	6	no	—
508	C₂³	—	2	12	no	—
509	C₂³	—	2	12	no	—
510	C₂³	—	2	6	no	—
511	C₂³	—	4	12	no	—
512	C₂³	—	4	12	no	—
513	C₂³	—	4	16	no	—
514	C₂³	—	8	8	no	—
515	C₂³	—	268	24	no	—
516	C₂³	—	268	12	no	—
517	C₂³	—	268	24	no	—
518	C₂³	—	268	128	no	—
519	C₂³	—	268	48	no	—
520	C₂³	—	268	48	no	—
521	C₂³	—	536	24	no	—
522	C₂³	—	67	12	no	—
523	C₂³	—	67	6	no	—
524	C₂³	—	67	2	no	—
525	C₂³	—	67	6	no	—
526	C₂³	—	134	24	no	—
527	C₂³	—	134	48	no	—
528	C₂³	—	134	24	no	—
529	C₂³	—	134	12	no	—
530	C₂³	—	134	6	no	—
531	C₂³	—	134	6	no	—
532	C₂³	—	134	12	no	—
533	C₂³	C₂⁴×D₆₇	1	6	yes	—
534	C₂³	C₂⁴×D₆₇	1	6	yes	—
535	C₂³	C₂⁴×D₆₇	1	1	yes	characteristic
536	Q₈	—	4	12	no	—
537	D₄	—	4	24	no	—
538	D₄	—	4	12	no	—
539	D₄	—	268	12	no	—
540	D₄	—	268	12	no	—
541	D₄	—	268	12	no	—
542	D₄	—	268	12	no	—
543	D₄	—	268	24	no	—
544	D₄	—	268	24	no	—
545	D₄	—	134	48	no	—
546	D₄	—	134	48	no	—
547	C₂×C₄	—	2	12	no	—
548	C₂×C₄	—	2	6	no	—
549	C₂×C₄	—	268	24	no	—
550	C₂×C₄	—	268	24	no	—
551	C₂×C₄	—	268	12	no	—
552	C₂×C₄	—	67	12	no	—
553	C₂×C₄	—	67	6	no	—
554	C₂×C₄	—	134	24	no	—
555	C₂×C₄	—	134	6	no	—
556	C₂×C₄	—	134	12	no	—
557	C₂×C₄	C₂⁴×D₆₇	1	6	yes	—
558	C₂×C₄	C₂⁴×D₆₇	1	12	yes	—
559	C₂²	—	2	6	no	—
560	C₂²	—	2	12	no	—
561	C₂²	—	2	12	no	—
562	C₂²	—	2	6	no	—
563	C₂²	—	4	16	no	—
564	C₂²	—	4	12	no	—
565	C₂²	—	4	12	no	—
566	C₂²	—	8	8	no	—
567	C₂²	—	268	48	no	—
568	C₂²	—	268	48	no	—
569	C₂²	—	268	4	no	—
570	C₂²	—	268	12	no	—
571	C₂²	—	536	24	no	—
572	C₂²	—	67	12	no	—
573	C₂²	—	134	12	no	—
574	C₂²	—	134	6	no	—
575	C₂²	—	134	6	no	—
576	C₂²	—	134	24	no	—
577	C₂²	—	134	12	no	—
578	C₂²	H(5,𝔽₂)×D₆₇	1	6	yes	—
579	C₂²	C₂⁵×D₆₇	1	1	yes	characteristic
580	C₄	—	2	12	no	—
581	C₄	—	2	6	no	—
582	C₄	—	134	6	no	—
583	C₄	—	134	12	no	—
584	C₂	—	4	6	no	—
585	C₂	—	4	6	no	—
586	C₂	—	268	2	no	—
587	C₂	—	268	6	no	—
588	C₂	—	134	12	no	—
589	C₂	#22574366	1	4	yes	direct product factor
590	C₂	C₂×H(5,𝔽₂)×D₆₇	1	3	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₆₇
commutator, cosocle	C₂²×C₆₇	C₂⁶
Frattini	C₂²	C₂⁵×D₆₇
Fitting	C₂×H(3,𝔽₄)×C₆₇	C₂
socle	C₂³×C₆₇	C₂⁵
hypercenter	C₂×H(3,𝔽₄)	D₆₇
hypocenter	C₆₇	C₂×#3532
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×H(3,𝔽₄)

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #22574366	C₂	direct (N × H)
2. C₂×#16864135	C₂	semidirect
3. C₂×#16864076	C₂	semidirect
4. C₂×#16864039	C₂	semidirect
5. C₂×#16864028	C₂	semidirect
6. C₂×#16863980	C₂	semidirect
7. C₂×#16863964	C₂	semidirect
8. C₂×#16863956	C₂	semidirect
9. C₂×#120×D₆₇	C₂	semidirect
10. C₂×H(3,𝔽₄)×C₆₇	C₂	semidirect
11. C₂⁵×D₆₇	C₂²	semidirect
12. C₂²×#11761629	C₂²	semidirect
13. C₂³×C₂²⋊D₆₇	C₂²	semidirect
14. C₂²×D₄×D₆₇	C₂²	semidirect
15. C₂²×D₄×D₆₇	C₂²	semidirect
16. #16864135	C₂²	semidirect
17. #16864076	C₂²	semidirect
18. #16864039	C₂²	semidirect
19. #16864028	C₂²	semidirect
20. #16863980	C₂²	semidirect
21. #16863964	C₂²	semidirect
22. #16863956	C₂²	semidirect
23. C₂×#14524040	C₂²	semidirect
24. C₂×#14524039	C₂²	semidirect
25. C₂×#14524037	C₂²	semidirect
26. C₂×#14524036	C₂²	semidirect
27. C₂×#14524032	C₂²	semidirect
28. C₂×#14524027	C₂²	semidirect
29. C₂×#14524025	C₂²	semidirect
30. C₂×#14524025	C₂²	semidirect
31. C₂×#14524024	C₂²	semidirect
32. C₂×#14524019	C₂²	semidirect
33. C₂×#14524016	C₂²	semidirect
34. C₂×#14524016	C₂²	semidirect
35. C₂×#14524016	C₂²	semidirect
36. C₂×#14524015	C₂²	semidirect
37. C₂×#14524014	C₂²	semidirect
38. C₂×#14524013	C₂²	semidirect
39. C₂×#14524013	C₂²	semidirect
40. C₂×#14524013	C₂²	semidirect
41. C₂×#14524012	C₂²	semidirect
42. C₂×#14524012	C₂²	semidirect
44. C₂×#14524011	C₂²	semidirect
45. #120×D₆₇	C₂²	semidirect
46. C₂×C₄×D₂₆₈	C₂²	semidirect
47. C₂×(C₂²⋊D₆₇)⋊C₄	C₂²	semidirect
48. C₂×#14524000	C₂²	semidirect
49. C₂×C₂²⋊C₄×D₆₇	C₂²	semidirect
50. C₂×C₂²⋊C₄×D₆₇	C₂²	semidirect
51. C₂×C₂²⋊C₄×D₆₇	C₂²	semidirect
52. H(3,𝔽₄)×C₆₇	C₂²	semidirect
53. C₂×#124×C₆₇	C₂²	semidirect
54. C₂×#124×C₆₇	C₂²	semidirect
55. C₂×#120×C₆₇	C₂²	semidirect
56. C₂×#120×C₆₇	C₂²	semidirect
61. C₂⁴×D₆₇	C₂³	semidirect
63. C₂×#11761629	C₂³	semidirect
66. C₂²×D₂₆₈	C₂³	semidirect
72. C₂²×C₂²⋊D₆₇	C₂³	semidirect
73. C₂²×C₂²⋊D₆₇	C₂³	semidirect
76. C₂×D₄×D₆₇	C₂³	semidirect
77. C₂×D₄×D₆₇	C₂³	semidirect
78. C₂³×Dic₆₇	C₂³	semidirect
81. C₂²×C₄×D₆₇	C₂³	semidirect
82. #14524040	C₂³	semidirect
83. #14524039	C₂³	semidirect
84. #14524037	C₂³	semidirect
85. #14524036	C₂³	semidirect
86. #14524032	C₂³	semidirect
87. #14524027	C₂³	semidirect
88. #14524025	C₂³	semidirect
89. #14524025	C₂³	semidirect
90. #14524024	C₂³	semidirect
91. #14524019	C₂³	semidirect
92. #14524016	C₂³	semidirect
93. #14524016	C₂³	semidirect
94. #14524016	C₂³	semidirect
95. #14524015	C₂³	semidirect
96. #14524014	C₂³	semidirect
97. #14524013	C₂³	semidirect
98. #14524013	C₂³	semidirect
99. #14524013	C₂³	semidirect
100. #14524012	C₂³	semidirect
101. #14524012	C₂³	semidirect
103. #14524011	C₂³	semidirect
104. C₄×D₂₆₈	C₂³	semidirect
105. (C₂²⋊D₆₇)⋊C₄	C₂³	semidirect
106. #14524000	C₂³	semidirect
107. C₂×C₄⋊Dic₆₇	C₂³	semidirect
108. C₂×Dic₆₇⋊C₄	C₂³	semidirect
109. C₂×Dic₆₇⋊C₄	C₂³	semidirect
112. C₂×C₂²⋊Dic₆₇	C₂³	semidirect
116. C₂×(C₂×D₆₇)⋊C₄	C₂³	semidirect
117. C₂²⋊C₄×D₆₇	C₂³	semidirect
118. C₂²⋊C₄×D₆₇	C₂³	semidirect
119. C₂²⋊C₄×D₆₇	C₂³	semidirect
120. C₂×C₄×Dic₆₇	C₂³	semidirect
122. C₂⁵×C₆₇	C₂³	semidirect
123. C₂²×Q₈×C₆₇	C₂³	semidirect
126. #124×C₆₇	C₂³	semidirect
127. #124×C₆₇	C₂³	semidirect
128. #120×C₆₇	C₂³	semidirect
129. #120×C₆₇	C₂³	semidirect
132. C₂×C₂²⋊C₄×C₆₇	C₂³	semidirect
133. C₂×C₂²⋊C₄×C₆₇	C₂³	semidirect
134. C₂×C₄²×C₆₇	C₂³	semidirect
135. C₂×C₄²×C₆₇	C₂³	semidirect
156. C₂×D₂₆₈	C₂⁴	semidirect
167. C₂×C₂²⋊D₆₇	C₂⁴	semidirect
173. C₂²×Dic₆₇	C₂⁴	semidirect
179. C₂×C₄×D₆₇	C₂⁴	semidirect
180. C₄⋊Dic₆₇	C₂⁴	semidirect
181. Dic₆₇⋊C₄	C₂⁴	semidirect
182. Dic₆₇⋊C₄	C₂⁴	semidirect
185. C₂²⋊Dic₆₇	C₂⁴	semidirect
189. (C₂×D₆₇)⋊C₄	C₂⁴	semidirect
190. C₄×Dic₆₇	C₂⁴	semidirect
196. C₂⁴×C₆₇	C₂⁴	semidirect
198. C₂×Q₈×C₆₇	C₂⁴	semidirect
207. C₂²⋊C₄×C₆₇	C₂⁴	semidirect
208. C₂²⋊C₄×C₆₇	C₂⁴	semidirect
209. C₄²×C₆₇	C₂⁴	semidirect
210. C₄²×C₆₇	C₂⁴	semidirect
291. C₂×C₆₇	#3532	semidirect
297. C₂×H(3,𝔽₄)	D₆₇	semidirect
306. C₆₇	C₂×#3532	semidirect
312. H(3,𝔽₄)	C₂×D₆₇	semidirect
325. C₂×#124	C₂×D₆₇	semidirect
326. C₂×#124	C₂×D₆₇	semidirect
345. C₂×#120	C₂×D₆₇	semidirect
346. C₂×#120	C₂×D₆₇	semidirect
359. C₂⁵	C₂²×D₆₇	semidirect
362. C₂²×Q₈	C₂²×D₆₇	semidirect
388. #124	C₂²×D₆₇	semidirect
389. #124	C₂²×D₆₇	semidirect
407. #120	C₂²×D₆₇	semidirect
408. #120	C₂²×D₆₇	semidirect
427. C₂×C₂²⋊C₄	C₂²×D₆₇	semidirect
428. C₂×C₂²⋊C₄	C₂²×D₆₇	semidirect
430. C₂×C₄²	C₂²×D₆₇	semidirect
431. C₂×C₄²	C₂²×D₆₇	semidirect
451. C₂⁴	C₂³×D₆₇	semidirect
454. C₂×Q₈	C₂³×D₆₇	semidirect
502. C₂²⋊C₄	C₂³×D₆₇	semidirect
503. C₂²⋊C₄	C₂³×D₆₇	semidirect
505. C₄²	C₂³×D₆₇	semidirect
506. C₄²	C₂³×D₆₇	semidirect
589. C₂	#22574366	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22574366	2
C₂×#16864135	2
C₂×#16864076	2
C₂×#16864039	2
C₂×#16864028	2
C₂×#16863980	2
C₂×#16863964	2
C₂×#16863956	2
C₂×#120×D₆₇	2
C₂×H(3,𝔽₄)×C₆₇	2
C₂×#3532	67

Subgroups — C2×#22574366 #31762954

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22574366 #31762954