Subgroups — C₂×#22573840 #31763711

Order 17152.

14438 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#22573840	C₂	1	32	yes	maximal, direct product factor
3	C₂×#16864034	C₂	1	8	yes	maximal, complemented
4	C₂×D₄×C₂²⋊D₆₇	C₂	1	8	yes	maximal, complemented
5	C₂×#16863960	C₂	1	2	yes	maximal, complemented
6	C₂×#16863955	C₂	1	4	yes	maximal, complemented
7	C₂×#16863903	C₂	1	2	yes	maximal, complemented
8	C₂²×#14524011	C₂	1	4	yes	maximal, complemented
9	C₂×#16863871	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×H(5,𝔽₂)×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁵×D₆₇	C₂²	1	2	yes	complemented
12	C₂³×D₂₆₈	C₂²	1	1	yes	characteristic, complemented
13	C₂³×C₂²⋊D₆₇	C₂²	1	2	yes	complemented
14	C₂³×C₂²⋊D₆₇	C₂²	1	8	yes	complemented
15	C₂²×D₄×D₆₇	—	2	16	no	—
16	C₂²×D₄×D₆₇	C₂²	1	8	yes	complemented
17	C₂²×D₄×D₆₇	C₂²	1	16	yes	complemented
18	C₂³×C₄×D₆₇	C₂²	1	1	yes	characteristic, complemented
19	#16864034	C₂²	1	128	yes	complemented
20	D₄×C₂²⋊D₆₇	C₂²	1	128	yes	complemented
21	#16863960	C₂²	1	32	yes	complemented
22	#16863955	C₂²	1	64	yes	complemented
23	C₂×#14524039	C₂²	1	4	yes	complemented
24	C₂×#14524032	C₂²	1	4	yes	complemented
25	#16863903	C₂²	1	32	yes	complemented
26	C₂×#14524026	C₂²	1	8	yes	complemented
27	C₂×#14524024	C₂²	1	4	yes	complemented
28	C₂×#14524019	C₂²	1	4	yes	complemented
29	C₂×#14524017	C₂²	1	8	yes	complemented
30	C₂×#14524015	C₂²	1	16	yes	complemented
31	C₂×#14524014	C₂²	1	4	yes	complemented
32	C₂×#14524012	C₂²	1	8	yes	complemented
33	C₂×#14524011	C₂²	1	4	yes	complemented
34	C₂×#14524011	C₂²	1	8	yes	complemented
35	C₂×#14524011	C₂²	1	16	yes	complemented
36	C₂×#14524011	C₂²	1	64	yes	complemented
37	#16863871	C₂²	1	16	yes	complemented
38	C₂×D₄×Dic₆₇	C₂²	1	8	yes	complemented
39	C₂×C₄×C₂²⋊D₆₇	C₂²	1	8	yes	complemented
40	C₂×#14524002	C₂²	1	2	yes	complemented
41	C₂²×C₂²⋊Dic₆₇	C₂²	1	4	yes	complemented
42	C₂²×(C₂×D₆₇)⋊C₄	C₂²	1	8	yes	complemented
43	C₂×H(5,𝔽₂)×C₆₇	—	2	16	no	—
44	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	16	yes	complemented
45	C₂²×#41×C₆₇	C₂²	1	2	yes	complemented
46	C₂²×#41×C₆₇	C₂²	1	4	yes	complemented
47	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
48	C₂³×D₄×C₆₇	C₂²	1	4	yes	complemented
49	C₂³×D₄×C₆₇	C₂²	1	4	yes	complemented
50	C₂⁴×D₆₇	—	2	16	no	—
51	C₂⁴×D₆₇	—	4	16	no	—
52	C₂⁴×D₆₇	C₂³	1	1	yes	characteristic
53	C₂⁴×D₆₇	C₂³	1	8	yes	—
54	C₂⁴×D₆₇	C₂³	1	2	yes	—
55	C₂⁴×D₆₇	C₂³	1	16	yes	complemented
56	C₂²×D₂₆₈	—	2	8	no	—
57	C₂²×D₂₆₈	C₂³	1	4	yes	—
58	C₂²×D₂₆₈	C₂³	1	8	yes	complemented
59	C₂²×C₂²⋊D₆₇	—	2	16	no	—
60	C₂²×C₂²⋊D₆₇	—	4	32	no	—
61	C₂²×C₂²⋊D₆₇	C₂³	1	8	yes	—
62	C₂²×C₂²⋊D₆₇	C₂³	1	8	yes	—
63	C₂²×C₂²⋊D₆₇	C₂³	1	16	yes	complemented
64	C₂²×C₂²⋊D₆₇	C₂³	1	16	yes	complemented
65	C₂²×C₂²⋊D₆₇	C₂³	1	64	yes	complemented
66	C₂×D₄×D₆₇	—	2	64	no	—
67	C₂×D₄×D₆₇	—	2	128	no	—
68	C₂×D₄×D₆₇	C₂³	1	64	yes	complemented
69	C₂³×Dic₆₇	C₂³	1	1	yes	characteristic
70	C₂³×Dic₆₇	C₂³	1	8	yes	complemented
71	C₂²×C₄×D₆₇	—	2	8	no	—
72	C₂²×C₄×D₆₇	C₂³	1	4	yes	—
73	C₂²×C₄×D₆₇	C₂³	1	8	yes	complemented
74	#14524039	C₂³	1	32	yes	complemented
75	#14524032	C₂³	1	32	yes	complemented
76	#14524026	C₂³	1	64	yes	complemented
77	#14524024	C₂³	1	32	yes	complemented
78	#14524019	C₂³	1	32	yes	complemented
79	#14524017	C₂³	1	64	yes	complemented
80	#14524015	C₂³	1	128	yes	complemented
81	#14524014	C₂³	1	32	yes	complemented
82	#14524012	C₂³	1	64	yes	complemented
83	#14524011	C₂³	1	32	yes	complemented
84	#14524011	C₂³	1	64	yes	complemented
85	#14524011	C₂³	1	128	yes	complemented
86	D₄×Dic₆₇	C₂³	1	64	yes	complemented
87	C₄×C₂²⋊D₆₇	C₂³	1	64	yes	complemented
88	#14524002	C₂³	1	16	yes	complemented
89	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	complemented
90	C₂×Dic₆₇⋊C₄	C₂³	1	8	yes	complemented
91	C₂×C₂²⋊Dic₆₇	C₂³	1	4	yes	—
92	C₂×C₂²⋊Dic₆₇	C₂³	1	32	yes	complemented
93	C₂×C₂²⋊Dic₆₇	C₂³	1	8	yes	complemented
94	C₂×C₂²⋊Dic₆₇	C₂³	1	8	yes	complemented
95	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	8	yes	—
96	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	16	yes	complemented
97	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	64	yes	complemented
98	C₂×C₄×Dic₆₇	C₂³	1	4	yes	complemented
99	C₂⁵×C₆₇	C₂³	1	2	yes	—
100	C₂⁵×C₆₇	C₂³	1	4	yes	complemented
101	H(5,𝔽₂)×C₆₇	—	2	128	no	—
102	C₂×#41×C₆₇	—	2	32	no	—
103	C₂×#41×C₆₇	—	2	16	no	—
104	C₂×#41×C₆₇	C₂³	1	16	yes	complemented
105	C₂×#41×C₆₇	C₂³	1	32	yes	complemented
106	C₂²×Q₈×C₆₇	C₂³	1	2	yes	complemented
107	C₂²×D₄×C₆₇	—	2	32	no	—
108	C₂²×D₄×C₆₇	—	2	32	no	—
109	C₂²×D₄×C₆₇	D₄	1	16	yes	complemented
110	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
111	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
112	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
113	C₂²×D₄×C₆₇	C₂³	1	8	yes	complemented
114	C₂²×D₄×C₆₇	C₂³	1	2	yes	complemented
115	C₂²×D₄×C₆₇	C₂³	1	32	yes	complemented
116	C₂²×D₄×C₆₇	C₂³	1	32	yes	complemented
117	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
118	C₂³×C₄×C₆₇	C₂³	1	4	yes	complemented
119	C₂³×C₄×C₆₇	C₂³	1	4	yes	complemented
120	C₂³×D₆₇	—	2	4	no	—
121	C₂³×D₆₇	—	2	8	no	—
122	C₂³×D₆₇	—	2	32	no	—
123	C₂³×D₆₇	—	2	8	no	—
124	C₂³×D₆₇	—	2	64	no	—
125	C₂³×D₆₇	—	2	32	no	—
126	C₂³×D₆₇	—	4	64	no	—
127	C₂³×D₆₇	—	4	16	no	—
128	C₂³×D₆₇	—	4	8	no	—
129	C₂³×D₆₇	—	4	128	no	—
130	C₂³×D₆₇	C₂⁴	1	4	yes	—
131	C₂³×D₆₇	C₂⁴	1	4	yes	—
132	C₂³×D₆₇	C₂⁴	1	32	yes	—
133	C₂³×D₆₇	C₂⁴	1	2	yes	—
134	C₂³×D₆₇	C₂⁴	1	8	yes	—
135	C₂×D₂₆₈	—	2	16	no	—
136	C₂×D₂₆₈	—	2	32	no	—
137	C₂×D₂₆₈	C₂⁴	1	16	yes	—
138	C₂×C₂²⋊D₆₇	—	2	32	no	—
139	C₂×C₂²⋊D₆₇	—	2	64	no	—
140	C₂×C₂²⋊D₆₇	—	2	32	no	—
141	C₂×C₂²⋊D₆₇	—	4	32	no	—
142	C₂×C₂²⋊D₆₇	—	4	128	no	—
143	C₂×C₂²⋊D₆₇	C₂⁴	1	32	yes	—
144	C₂×C₂²⋊D₆₇	C₂⁴	1	32	yes	—
145	C₂×C₂²⋊D₆₇	C₂⁴	1	64	yes	complemented
146	D₄×D₆₇	—	2	256	no	—
147	C₂²×Dic₆₇	—	2	4	no	—
148	C₂²×Dic₆₇	—	4	16	no	—
149	C₂²×Dic₆₇	C₂⁴	1	8	yes	—
150	C₂²×Dic₆₇	C₂⁴	1	2	yes	—
151	C₂²×Dic₆₇	C₂⁴	1	4	yes	—
152	C₂²×Dic₆₇	C₂⁴	1	32	yes	complemented
153	C₂×C₄×D₆₇	—	2	16	no	—
154	C₂×C₄×D₆₇	—	2	32	no	—
155	C₂×C₄×D₆₇	C₂⁴	1	16	yes	—
156	C₄⋊Dic₆₇	C₂⁴	1	16	yes	complemented
157	Dic₆₇⋊C₄	C₂⁴	1	32	yes	complemented
158	C₂²⋊Dic₆₇	C₂⁴	1	16	yes	—
159	C₂²⋊Dic₆₇	C₂⁴	1	32	yes	complemented
160	C₂²⋊Dic₆₇	C₂⁴	1	32	yes	complemented
161	(C₂×D₆₇)⋊C₄	C₂⁴	1	32	yes	—
162	(C₂×D₆₇)⋊C₄	C₂⁴	1	64	yes	complemented
163	C₄×Dic₆₇	C₂⁴	1	16	yes	complemented
164	C₂⁴×C₆₇	—	2	16	no	—
165	C₂⁴×C₆₇	—	4	8	no	—
166	C₂⁴×C₆₇	—	4	16	no	—
167	C₂⁴×C₆₇	C₂×D₄	1	16	yes	complemented
168	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
169	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
170	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
171	C₂⁴×C₆₇	C₂⁴	1	8	yes	—
172	C₂⁴×C₆₇	C₂⁴	1	16	yes	complemented
173	#41×C₆₇	—	2	128	no	—
174	#41×C₆₇	—	2	64	no	—
175	C₂×Q₈×C₆₇	—	2	8	no	—
176	C₂×Q₈×C₆₇	C₂⁴	1	8	yes	complemented
177	C₂×D₄×C₆₇	—	2	16	no	—
178	C₂×D₄×C₆₇	—	2	32	no	—
179	C₂×D₄×C₆₇	—	2	128	no	—
180	C₂×D₄×C₆₇	—	2	128	no	—
181	C₂×D₄×C₆₇	—	2	8	no	—
182	C₂×D₄×C₆₇	C₂×D₄	1	32	yes	complemented
183	C₂×D₄×C₆₇	C₂×D₄	1	64	yes	complemented
184	C₂×D₄×C₆₇	C₂⁴	1	16	yes	—
185	C₂×D₄×C₆₇	C₂⁴	1	16	yes	—
186	C₂×D₄×C₆₇	C₂⁴	1	32	yes	complemented
187	C₂×D₄×C₆₇	C₂⁴	1	8	yes	complemented
188	C₂²×C₄×C₆₇	—	2	16	no	—
189	C₂²×C₄×C₆₇	—	2	16	no	—
190	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	complemented
191	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
192	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
193	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
194	C₂²×C₄×C₆₇	C₂⁴	1	16	yes	complemented
195	C₂²×C₄×C₆₇	C₂⁴	1	16	yes	complemented
196	C₂³×C₆₇	—	2	16	no	—
197	C₂³×C₆₇	—	2	4	no	—
198	C₂³×C₆₇	—	2	8	no	—
199	C₂³×C₆₇	—	2	16	no	—
200	C₂³×C₆₇	—	2	32	no	—
201	C₂³×C₆₇	—	4	32	no	—
202	C₂³×C₆₇	—	4	16	no	—
203	C₂³×C₆₇	—	4	32	no	—
204	C₂³×C₆₇	—	8	32	no	—
205	C₂³×C₆₇	C₂²×D₄	1	16	yes	—
206	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
207	C₂³×C₆₇	C₂²×D₄	1	32	yes	complemented
208	C₂³×C₆₇	C₂⁵	1	8	yes	—
209	C₂³×C₆₇	C₂⁵	1	2	yes	—
210	C₂³×C₆₇	C₂⁵	1	8	yes	—
211	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
212	C₂²×D₆₇	—	2	32	no	—
213	C₂²×D₆₇	—	2	8	no	—
214	C₂²×D₆₇	—	2	4	no	—
215	C₂²×D₆₇	—	2	4	no	—
216	C₂²×D₆₇	—	2	16	no	—
217	C₂²×D₆₇	—	2	64	no	—
218	C₂²×D₆₇	—	2	16	no	—
219	C₂²×D₆₇	—	2	8	no	—
220	C₂²×D₆₇	—	2	16	no	—
221	C₂²×D₆₇	—	2	64	no	—
222	C₂²×D₆₇	—	4	32	no	—
223	C₂²×D₆₇	—	4	16	no	—
224	C₂²×D₆₇	—	4	128	no	—
225	C₂²×D₆₇	—	4	256	no	—
226	C₂²×D₆₇	—	4	32	no	—
227	C₂²×D₆₇	H(5,𝔽₂)	1	8	yes	complemented
228	C₂²×D₆₇	C₂⁵	1	8	yes	—
229	C₂²×D₆₇	C₂⁵	1	4	yes	—
230	Q₈×C₆₇	—	2	16	no	—
231	D₄×C₆₇	—	2	32	no	—
232	D₄×C₆₇	—	2	64	no	—
233	D₄×C₆₇	—	2	16	no	—
234	D₄×C₆₇	C₂²×D₄	1	64	yes	complemented
235	C₂×C₄×C₆₇	—	2	8	no	—
236	C₂×C₄×C₆₇	—	2	32	no	—
237	C₂×C₄×C₆₇	—	2	32	no	—
238	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
239	C₂×C₄×C₆₇	C₂²×D₄	1	16	yes	complemented
240	C₂×C₄×C₆₇	C₂⁵	1	8	yes	—
241	C₂×C₄×C₆₇	C₂⁵	1	4	yes	—
242	C₂²⋊D₆₇	—	2	64	no	—
243	C₂²⋊D₆₇	—	2	64	no	—
244	C₂²⋊D₆₇	—	4	64	no	—
245	C₂×Dic₆₇	—	2	8	no	—
246	C₂×Dic₆₇	—	2	4	no	—
247	C₂×Dic₆₇	—	2	16	no	—
248	C₂×Dic₆₇	—	4	32	no	—
249	C₂×Dic₆₇	C₂⁵	1	16	yes	—
250	C₂×Dic₆₇	C₂⁵	1	4	yes	—
251	D₂₆₈	—	2	32	no	—
252	C₄×D₆₇	—	2	32	no	—
253	C₂²×C₆₇	—	2	16	no	—
254	C₂²×C₆₇	—	2	4	no	—
255	C₂²×C₆₇	—	2	8	no	—
256	C₂²×C₆₇	—	2	32	no	—
257	C₂²×C₆₇	—	2	8	no	—
258	C₂²×C₆₇	—	2	16	no	—
259	C₂²×C₆₇	—	4	32	no	—
260	C₂²×C₆₇	—	4	16	no	—
261	C₂²×C₆₇	—	8	32	no	—
262	C₂²×C₆₇	C₂³×D₄	1	4	yes	—
263	C₂²×C₆₇	C₂³×D₄	1	16	yes	—
264	C₂²×C₆₇	C₂³×D₄	1	2	yes	—
265	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	4	yes	—
266	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
267	C₂×D₆₇	—	2	32	no	—
268	C₂×D₆₇	—	2	4	no	—
269	C₂×D₆₇	—	2	4	no	—
270	C₂×D₆₇	—	2	16	no	—
271	C₂×D₆₇	—	2	8	no	—
272	C₂×D₆₇	—	2	16	no	—
273	C₂×D₆₇	—	4	128	no	—
274	C₂×D₆₇	—	4	32	no	—
275	C₂×D₆₇	—	4	16	no	—
276	C₂×D₆₇	C₂×H(5,𝔽₂)	1	8	yes	complemented
277	C₄×C₆₇	—	2	8	no	—
278	C₄×C₆₇	C₂³×D₄	1	8	yes	—
279	Dic₆₇	—	2	4	no	—
280	Dic₆₇	—	2	16	no	—
281	C₂×#3484	—	67	1	no	maximal, Sylow
282	C₂×C₆₇	—	2	4	no	—
283	C₂×C₆₇	—	2	16	no	—
284	C₂×C₆₇	—	4	8	no	—
285	C₂×C₆₇	#3484	1	4	yes	complemented
286	C₂×C₆₇	C₂⁴×D₄	1	1	yes	characteristic
287	C₂×C₆₇	C₂²×H(5,𝔽₂)	1	2	yes	—
288	D₆₇	—	2	8	no	—
289	D₆₇	—	4	8	no	—
290	C₂²×H(5,𝔽₂)	D₆₇	1	1	yes	characteristic, complemented
291	C₂⁴×D₄	—	67	1	no	—
292	#3484	—	67	32	no	—
293	C₂×#546	—	67	8	no	—
294	C₂×D₄²	—	67	8	no	—
295	C₂×#535	—	67	2	no	—
296	C₂×#534	—	67	4	no	—
297	C₂×#525	—	67	2	no	—
298	C₂²×#120	—	67	4	no	—
299	C₂×#518	—	67	1	no	—
300	C₆₇	C₂×#3484	1	1	yes	characteristic, complemented, Sylow
301	C₂⁶	—	67	2	no	—
302	C₂×H(5,𝔽₂)	—	2	16	no	—
303	C₂×H(5,𝔽₂)	C₂×D₆₇	1	16	yes	complemented
304	C₂²×#41	C₂×D₆₇	1	2	yes	complemented
305	C₂²×#41	C₂×D₆₇	1	4	yes	complemented
306	C₂³×D₄	—	67	8	no	—
307	C₂³×D₄	—	67	2	no	—
308	C₂³×D₄	—	67	8	no	—
309	C₂³×D₄	—	67	16	no	—
310	C₂³×D₄	—	67	1	no	—
311	C₂³×D₄	—	134	16	no	—
312	C₂³×D₄	C₂×D₆₇	1	1	yes	characteristic
313	C₂³×D₄	C₂×D₆₇	1	4	yes	complemented
314	C₂³×D₄	C₂×D₆₇	1	4	yes	complemented
315	C₂⁴×C₄	—	67	1	no	—
316	#546	—	67	128	no	—
317	D₄²	—	67	128	no	—
318	#535	—	67	32	no	—
319	#534	—	67	64	no	—
320	C₂×#127	—	67	4	no	—
321	C₂×#124	—	67	4	no	—
322	#525	—	67	32	no	—
323	C₂×#123	—	67	4	no	—
324	C₂×#123	—	67	8	no	—
325	C₂×C₂²⋊Q₈	—	67	4	no	—
326	C₂×#121	—	67	4	no	—
327	C₂×#121	—	67	16	no	—
328	C₂×#121	—	67	8	no	—
329	C₂×#120	—	67	8	no	—
330	C₂×#120	—	67	4	no	—
331	C₂×#120	—	67	8	no	—
332	C₂×#120	—	67	16	no	—
333	C₂×#120	—	67	64	no	—
334	#518	—	67	16	no	—
335	C₂×C₄×D₄	—	67	8	no	—
336	C₂×C₄×D₄	—	67	8	no	—
337	C₂×#117	—	67	2	no	—
338	C₂²×C₂²⋊C₄	—	67	4	no	—
339	C₂²×C₂²⋊C₄	—	67	8	no	—
340	C₂⁵	—	268	16	no	—
341	C₂⁵	—	67	2	no	—
342	C₂⁵	—	67	16	no	—
343	C₂⁵	—	67	8	no	—
344	C₂⁵	—	67	1	no	—
345	C₂⁵	—	134	16	no	—
346	C₂⁵	C₂²×D₆₇	1	2	yes	—
347	C₂⁵	C₂²×D₆₇	1	4	yes	complemented
348	H(5,𝔽₂)	—	2	128	no	—
349	C₂×#41	—	2	32	no	—
350	C₂×#41	—	2	16	no	—
351	C₂×#41	C₂²×D₆₇	1	16	yes	complemented
352	C₂×#41	C₂²×D₆₇	1	32	yes	complemented
353	C₂²×Q₈	C₂²×D₆₇	1	2	yes	complemented
354	C₂²×D₄	—	2	32	no	—
355	C₂²×D₄	—	2	32	no	—
356	C₂²×D₄	—	268	32	no	—
357	C₂²×D₄	—	67	16	no	—
358	C₂²×D₄	—	67	64	no	—
359	C₂²×D₄	—	67	8	no	—
360	C₂²×D₄	—	67	8	no	—
361	C₂²×D₄	—	67	16	no	—
362	C₂²×D₄	—	67	64	no	—
363	C₂²×D₄	—	67	4	no	—
364	C₂²×D₄	—	67	8	no	—
365	C₂²×D₄	—	134	16	no	—
366	C₂²×D₄	—	134	64	no	—
367	C₂²×D₄	—	134	128	no	—
368	C₂²×D₄	—	134	8	no	—
369	C₂²×D₄	C₂²⋊D₆₇	1	16	yes	complemented
370	C₂²×D₄	C₂²×D₆₇	1	4	yes	—
371	C₂²×D₄	C₂²×D₆₇	1	4	yes	—
372	C₂²×D₄	C₂²×D₆₇	1	8	yes	—
373	C₂²×D₄	C₂²×D₆₇	1	8	yes	complemented
374	C₂²×D₄	C₂²×D₆₇	1	2	yes	complemented
375	C₂²×D₄	C₂²×D₆₇	1	32	yes	complemented
376	C₂²×D₄	C₂²×D₆₇	1	32	yes	complemented
377	C₂³×C₄	—	67	8	no	—
378	C₂³×C₄	—	67	1	no	—
379	C₂³×C₄	—	67	4	no	—
380	C₂³×C₄	—	67	8	no	—
381	C₂³×C₄	—	134	8	no	—
382	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
383	C₂³×C₄	C₂²×D₆₇	1	4	yes	complemented
384	C₂³×C₄	C₂²×D₆₇	1	4	yes	complemented
385	#127	—	67	32	no	—
386	#124	—	67	32	no	—
387	#123	—	67	32	no	—
388	#123	—	67	64	no	—
389	C₂²⋊Q₈	—	67	32	no	—
390	#121	—	67	32	no	—
391	#121	—	67	128	no	—
392	#121	—	67	64	no	—
393	#120	—	67	64	no	—
394	#120	—	67	32	no	—
395	#120	—	67	64	no	—
396	#120	—	67	128	no	—
397	C₄×D₄	—	67	64	no	—
398	C₄×D₄	—	67	64	no	—
399	#117	—	67	16	no	—
400	C₂×C₄⋊C₄	—	67	4	no	—
401	C₂×C₄⋊C₄	—	67	8	no	—
402	C₂×C₂²⋊C₄	—	67	4	no	—
403	C₂×C₂²⋊C₄	—	67	8	no	—
404	C₂×C₂²⋊C₄	—	67	32	no	—
405	C₂×C₂²⋊C₄	—	67	8	no	—
406	C₂×C₂²⋊C₄	—	67	8	no	—
407	C₂×C₂²⋊C₄	—	67	16	no	—
408	C₂×C₂²⋊C₄	—	67	64	no	—
409	C₂×C₄²	—	67	4	no	—
410	C₂⁴	—	2	16	no	—
411	C₂⁴	—	4	8	no	—
412	C₂⁴	—	4	16	no	—
413	C₂⁴	—	268	16	no	—
414	C₂⁴	—	268	8	no	—
415	C₂⁴	—	268	128	no	—
416	C₂⁴	—	268	64	no	—
417	C₂⁴	—	67	8	no	—
418	C₂⁴	—	67	32	no	—
419	C₂⁴	—	67	2	no	—
420	C₂⁴	—	67	4	no	—
421	C₂⁴	—	67	4	no	—
422	C₂⁴	—	134	32	no	—
423	C₂⁴	—	134	8	no	—
424	C₂⁴	—	134	8	no	—
425	C₂⁴	—	134	32	no	—
426	C₂⁴	—	134	64	no	—
427	C₂⁴	—	134	4	no	—
428	C₂⁴	C₂×C₂²⋊D₆₇	1	16	yes	complemented
429	C₂⁴	C₂³×D₆₇	1	4	yes	—
430	C₂⁴	C₂³×D₆₇	1	1	yes	characteristic
431	C₂⁴	C₂³×D₆₇	1	4	yes	—
432	C₂⁴	C₂³×D₆₇	1	8	yes	—
433	C₂⁴	C₂³×D₆₇	1	16	yes	complemented
434	#41	—	2	128	no	—
435	#41	—	2	64	no	—
436	C₂×Q₈	—	2	8	no	—
437	C₂×Q₈	C₂³×D₆₇	1	8	yes	complemented
438	C₂×D₄	—	2	16	no	—
439	C₂×D₄	—	2	32	no	—
440	C₂×D₄	—	2	128	no	—
441	C₂×D₄	—	2	128	no	—
442	C₂×D₄	—	2	8	no	—
443	C₂×D₄	—	268	128	no	—
444	C₂×D₄	—	268	32	no	—
445	C₂×D₄	—	67	64	no	—
446	C₂×D₄	—	67	32	no	—
447	C₂×D₄	—	67	32	no	—
448	C₂×D₄	—	67	16	no	—
449	C₂×D₄	—	134	32	no	—
450	C₂×D₄	—	134	32	no	—
451	C₂×D₄	—	134	64	no	—
452	C₂×D₄	—	134	256	no	—
453	C₂×D₄	—	134	16	no	—
454	C₂×D₄	—	134	32	no	—
455	C₂×D₄	C₂×C₂²⋊D₆₇	1	32	yes	complemented
456	C₂×D₄	C₂×C₂²⋊D₆₇	1	64	yes	complemented
457	C₂×D₄	C₂³×D₆₇	1	16	yes	—
458	C₂×D₄	C₂³×D₆₇	1	16	yes	—
459	C₂×D₄	C₂³×D₆₇	1	32	yes	complemented
460	C₂×D₄	C₂³×D₆₇	1	8	yes	complemented
461	C₂²×C₄	—	2	16	no	—
462	C₂²×C₄	—	2	16	no	—
463	C₂²×C₄	—	268	16	no	—
464	C₂²×C₄	—	67	32	no	—
465	C₂²×C₄	—	67	4	no	—
466	C₂²×C₄	—	67	8	no	—
467	C₂²×C₄	—	67	2	no	—
468	C₂²×C₄	—	67	16	no	—
469	C₂²×C₄	—	134	4	no	—
470	C₂²×C₄	—	134	16	no	—
471	C₂²×C₄	—	134	32	no	—
472	C₂²×C₄	C₂×C₂²⋊D₆₇	1	8	yes	complemented
473	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
474	C₂²×C₄	C₂³×D₆₇	1	2	yes	—
475	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
476	C₂²×C₄	C₂³×D₆₇	1	16	yes	complemented
477	C₂²×C₄	C₂³×D₆₇	1	16	yes	complemented
478	C₄⋊C₄	—	67	16	no	—
479	C₄⋊C₄	—	67	32	no	—
480	C₂²⋊C₄	—	67	16	no	—
481	C₂²⋊C₄	—	67	32	no	—
482	C₂²⋊C₄	—	67	32	no	—
483	C₂²⋊C₄	—	67	32	no	—
484	C₂²⋊C₄	—	67	64	no	—
485	C₄²	—	67	16	no	—
486	C₂³	—	2	16	no	—
487	C₂³	—	2	4	no	—
488	C₂³	—	2	8	no	—
489	C₂³	—	2	16	no	—
490	C₂³	—	2	32	no	—
491	C₂³	—	4	32	no	—
492	C₂³	—	4	16	no	—
493	C₂³	—	4	32	no	—
494	C₂³	—	8	32	no	—
495	C₂³	—	268	32	no	—
496	C₂³	—	268	128	no	—
497	C₂³	—	268	32	no	—
498	C₂³	—	268	16	no	—
499	C₂³	—	268	256	no	—
500	C₂³	—	67	8	no	—
501	C₂³	—	67	4	no	—
502	C₂³	—	67	8	no	—
503	C₂³	—	134	64	no	—
504	C₂³	—	134	16	no	—
505	C₂³	—	134	32	no	—
506	C₂³	—	134	16	no	—
507	C₂³	—	134	64	no	—
508	C₂³	—	134	8	no	—
509	C₂³	—	134	16	no	—
510	C₂³	—	134	8	no	—
511	C₂³	—	134	4	no	—
512	C₂³	—	134	4	no	—
513	C₂³	C₂²×C₂²⋊D₆₇	1	16	yes	—
514	C₂³	C₂²×C₂²⋊D₆₇	1	4	yes	—
515	C₂³	C₂²×C₂²⋊D₆₇	1	32	yes	complemented
516	C₂³	C₂⁴×D₆₇	1	8	yes	—
517	C₂³	C₂⁴×D₆₇	1	2	yes	—
518	C₂³	C₂⁴×D₆₇	1	8	yes	—
519	C₂³	C₂⁴×D₆₇	1	1	yes	characteristic
520	Q₈	—	2	16	no	—
521	D₄	—	2	32	no	—
522	D₄	—	2	64	no	—
523	D₄	—	2	16	no	—
524	D₄	—	268	64	no	—
525	D₄	—	134	64	no	—
526	D₄	—	134	64	no	—
527	D₄	—	134	32	no	—
528	D₄	C₂²×C₂²⋊D₆₇	1	64	yes	complemented
529	C₂×C₄	—	2	8	no	—
530	C₂×C₄	—	2	32	no	—
531	C₂×C₄	—	2	32	no	—
532	C₂×C₄	—	268	32	no	—
533	C₂×C₄	—	67	16	no	—
534	C₂×C₄	—	67	4	no	—
535	C₂×C₄	—	134	8	no	—
536	C₂×C₄	—	134	4	no	—
537	C₂×C₄	—	134	16	no	—
538	C₂×C₄	—	134	32	no	—
539	C₂×C₄	C₂²×C₂²⋊D₆₇	1	8	yes	—
540	C₂×C₄	C₂²×C₂²⋊D₆₇	1	16	yes	complemented
541	C₂×C₄	C₂⁴×D₆₇	1	8	yes	—
542	C₂×C₄	C₂⁴×D₆₇	1	4	yes	—
543	C₂²	—	2	8	no	—
544	C₂²	—	2	32	no	—
545	C₂²	—	2	16	no	—
546	C₂²	—	2	4	no	—
547	C₂²	—	2	8	no	—
548	C₂²	—	2	16	no	—
549	C₂²	—	4	32	no	—
550	C₂²	—	4	16	no	—
551	C₂²	—	8	32	no	—
552	C₂²	—	268	32	no	—
553	C₂²	—	268	128	no	—
554	C₂²	—	268	16	no	—
555	C₂²	—	67	8	no	—
556	C₂²	—	134	32	no	—
557	C₂²	—	134	8	no	—
558	C₂²	—	134	16	no	—
559	C₂²	—	134	16	no	—
560	C₂²	—	134	4	no	—
561	C₂²	—	134	4	no	—
562	C₂²	C₂³×C₂²⋊D₆₇	1	16	yes	—
563	C₂²	C₂³×C₂²⋊D₆₇	1	4	yes	—
564	C₂²	C₂³×C₂²⋊D₆₇	1	2	yes	—
565	C₂²	H(5,𝔽₂)×D₆₇	1	4	yes	—
566	C₂²	C₂⁵×D₆₇	1	1	yes	characteristic
567	C₄	—	2	8	no	—
568	C₄	—	134	4	no	—
569	C₄	—	134	16	no	—
570	C₄	C₂³×C₂²⋊D₆₇	1	8	yes	—
571	C₂	—	2	4	no	—
572	C₂	—	2	16	no	—
573	C₂	—	4	8	no	—
574	C₂	—	268	8	no	—
575	C₂	—	134	8	no	—
576	C₂	#22573840	1	4	yes	direct product factor
577	C₂	C₂⁴×C₂²⋊D₆₇	1	1	yes	characteristic
578	C₂	C₂×H(5,𝔽₂)×D₆₇	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₆₇
commutator, cosocle	C₂²×C₆₇	C₂⁶
Frattini	C₂²	C₂⁵×D₆₇
Fitting	C₂²×H(5,𝔽₂)×C₆₇	C₂
socle	C₂³×C₆₇	C₂⁵
hypercenter	C₂²×H(5,𝔽₂)	D₆₇
hypocenter	C₆₇	C₂×#3484
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₂³×D₄×D₆₇	C₂	semidirect
2. #22573840	C₂	direct (N × H)
3. C₂×#16864034	C₂	semidirect
4. C₂×D₄×C₂²⋊D₆₇	C₂	semidirect
5. C₂×#16863960	C₂	semidirect
6. C₂×#16863955	C₂	semidirect
7. C₂×#16863903	C₂	semidirect
8. C₂²×#14524011	C₂	semidirect
9. C₂×#16863871	C₂	semidirect
10. C₂²×H(5,𝔽₂)×C₆₇	C₂	semidirect
11. C₂⁵×D₆₇	C₂²	semidirect
12. C₂³×D₂₆₈	C₂²	semidirect
13. C₂³×C₂²⋊D₆₇	C₂²	semidirect
14. C₂³×C₂²⋊D₆₇	C₂²	semidirect
16. C₂²×D₄×D₆₇	C₂²	semidirect
17. C₂²×D₄×D₆₇	C₂²	semidirect
18. C₂³×C₄×D₆₇	C₂²	semidirect
19. #16864034	C₂²	semidirect
20. D₄×C₂²⋊D₆₇	C₂²	semidirect
21. #16863960	C₂²	semidirect
22. #16863955	C₂²	semidirect
23. C₂×#14524039	C₂²	semidirect
24. C₂×#14524032	C₂²	semidirect
25. #16863903	C₂²	semidirect
26. C₂×#14524026	C₂²	semidirect
27. C₂×#14524024	C₂²	semidirect
28. C₂×#14524019	C₂²	semidirect
29. C₂×#14524017	C₂²	semidirect
30. C₂×#14524015	C₂²	semidirect
31. C₂×#14524014	C₂²	semidirect
32. C₂×#14524012	C₂²	semidirect
33. C₂×#14524011	C₂²	semidirect
34. C₂×#14524011	C₂²	semidirect
35. C₂×#14524011	C₂²	semidirect
36. C₂×#14524011	C₂²	semidirect
37. #16863871	C₂²	semidirect
38. C₂×D₄×Dic₆₇	C₂²	semidirect
39. C₂×C₄×C₂²⋊D₆₇	C₂²	semidirect
40. C₂×#14524002	C₂²	semidirect
41. C₂²×C₂²⋊Dic₆₇	C₂²	semidirect
42. C₂²×(C₂×D₆₇)⋊C₄	C₂²	semidirect
44. C₂×H(5,𝔽₂)×C₆₇	C₂²	semidirect
45. C₂²×#41×C₆₇	C₂²	semidirect
46. C₂²×#41×C₆₇	C₂²	semidirect
48. C₂³×D₄×C₆₇	C₂²	semidirect
49. C₂³×D₄×C₆₇	C₂²	semidirect
55. C₂⁴×D₆₇	C₂³	semidirect
58. C₂²×D₂₆₈	C₂³	semidirect
63. C₂²×C₂²⋊D₆₇	C₂³	semidirect
64. C₂²×C₂²⋊D₆₇	C₂³	semidirect
65. C₂²×C₂²⋊D₆₇	C₂³	semidirect
68. C₂×D₄×D₆₇	C₂³	semidirect
70. C₂³×Dic₆₇	C₂³	semidirect
73. C₂²×C₄×D₆₇	C₂³	semidirect
74. #14524039	C₂³	semidirect
75. #14524032	C₂³	semidirect
76. #14524026	C₂³	semidirect
77. #14524024	C₂³	semidirect
78. #14524019	C₂³	semidirect
79. #14524017	C₂³	semidirect
80. #14524015	C₂³	semidirect
81. #14524014	C₂³	semidirect
82. #14524012	C₂³	semidirect
83. #14524011	C₂³	semidirect
84. #14524011	C₂³	semidirect
85. #14524011	C₂³	semidirect
86. D₄×Dic₆₇	C₂³	semidirect
87. C₄×C₂²⋊D₆₇	C₂³	semidirect
88. #14524002	C₂³	semidirect
89. C₂×C₄⋊Dic₆₇	C₂³	semidirect
90. C₂×Dic₆₇⋊C₄	C₂³	semidirect
92. C₂×C₂²⋊Dic₆₇	C₂³	semidirect
93. C₂×C₂²⋊Dic₆₇	C₂³	semidirect
94. C₂×C₂²⋊Dic₆₇	C₂³	semidirect
96. C₂×(C₂×D₆₇)⋊C₄	C₂³	semidirect
97. C₂×(C₂×D₆₇)⋊C₄	C₂³	semidirect
98. C₂×C₄×Dic₆₇	C₂³	semidirect
100. C₂⁵×C₆₇	C₂³	semidirect
104. C₂×#41×C₆₇	C₂³	semidirect
105. C₂×#41×C₆₇	C₂³	semidirect
106. C₂²×Q₈×C₆₇	C₂³	semidirect
109. C₂²×D₄×C₆₇	D₄	semidirect
113. C₂²×D₄×C₆₇	C₂³	semidirect
114. C₂²×D₄×C₆₇	C₂³	semidirect
115. C₂²×D₄×C₆₇	C₂³	semidirect
116. C₂²×D₄×C₆₇	C₂³	semidirect
118. C₂³×C₄×C₆₇	C₂³	semidirect
119. C₂³×C₄×C₆₇	C₂³	semidirect
145. C₂×C₂²⋊D₆₇	C₂⁴	semidirect
152. C₂²×Dic₆₇	C₂⁴	semidirect
156. C₄⋊Dic₆₇	C₂⁴	semidirect
157. Dic₆₇⋊C₄	C₂⁴	semidirect
159. C₂²⋊Dic₆₇	C₂⁴	semidirect
160. C₂²⋊Dic₆₇	C₂⁴	semidirect
162. (C₂×D₆₇)⋊C₄	C₂⁴	semidirect
163. C₄×Dic₆₇	C₂⁴	semidirect
167. C₂⁴×C₆₇	C₂×D₄	semidirect
172. C₂⁴×C₆₇	C₂⁴	semidirect
176. C₂×Q₈×C₆₇	C₂⁴	semidirect
182. C₂×D₄×C₆₇	C₂×D₄	semidirect
183. C₂×D₄×C₆₇	C₂×D₄	semidirect
186. C₂×D₄×C₆₇	C₂⁴	semidirect
187. C₂×D₄×C₆₇	C₂⁴	semidirect
190. C₂²×C₄×C₆₇	C₂×D₄	semidirect
194. C₂²×C₄×C₆₇	C₂⁴	semidirect
195. C₂²×C₄×C₆₇	C₂⁴	semidirect
207. C₂³×C₆₇	C₂²×D₄	semidirect
227. C₂²×D₆₇	H(5,𝔽₂)	semidirect
234. D₄×C₆₇	C₂²×D₄	semidirect
239. C₂×C₄×C₆₇	C₂²×D₄	semidirect
276. C₂×D₆₇	C₂×H(5,𝔽₂)	semidirect
285. C₂×C₆₇	#3484	semidirect
290. C₂²×H(5,𝔽₂)	D₆₇	semidirect
300. C₆₇	C₂×#3484	semidirect
303. C₂×H(5,𝔽₂)	C₂×D₆₇	semidirect
304. C₂²×#41	C₂×D₆₇	semidirect
305. C₂²×#41	C₂×D₆₇	semidirect
313. C₂³×D₄	C₂×D₆₇	semidirect
314. C₂³×D₄	C₂×D₆₇	semidirect
347. C₂⁵	C₂²×D₆₇	semidirect
351. C₂×#41	C₂²×D₆₇	semidirect
352. C₂×#41	C₂²×D₆₇	semidirect
353. C₂²×Q₈	C₂²×D₆₇	semidirect
369. C₂²×D₄	C₂²⋊D₆₇	semidirect
373. C₂²×D₄	C₂²×D₆₇	semidirect
374. C₂²×D₄	C₂²×D₆₇	semidirect
375. C₂²×D₄	C₂²×D₆₇	semidirect
376. C₂²×D₄	C₂²×D₆₇	semidirect
383. C₂³×C₄	C₂²×D₆₇	semidirect
384. C₂³×C₄	C₂²×D₆₇	semidirect
428. C₂⁴	C₂×C₂²⋊D₆₇	semidirect
433. C₂⁴	C₂³×D₆₇	semidirect
437. C₂×Q₈	C₂³×D₆₇	semidirect
455. C₂×D₄	C₂×C₂²⋊D₆₇	semidirect
456. C₂×D₄	C₂×C₂²⋊D₆₇	semidirect
459. C₂×D₄	C₂³×D₆₇	semidirect
460. C₂×D₄	C₂³×D₆₇	semidirect
472. C₂²×C₄	C₂×C₂²⋊D₆₇	semidirect
476. C₂²×C₄	C₂³×D₆₇	semidirect
477. C₂²×C₄	C₂³×D₆₇	semidirect
515. C₂³	C₂²×C₂²⋊D₆₇	semidirect
528. D₄	C₂²×C₂²⋊D₆₇	semidirect
540. C₂×C₄	C₂²×C₂²⋊D₆₇	semidirect
576. C₂	#22573840	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₄×D₆₇	2
#22573840	2
C₂×#16864034	2
C₂×D₄×C₂²⋊D₆₇	2
C₂×#16863960	2
C₂×#16863955	2
C₂×#16863903	2
C₂²×#14524011	2
C₂×#16863871	2
C₂²×H(5,𝔽₂)×C₆₇	2
C₂×#3484	67

Subgroups — C2×#22573840 #31763711

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22573840 #31763711