Subgroups — C₂×#22574003 #31764304

Order 17152.

11534 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14524089	C₂	1	1	yes	maximal, characteristic, complemented
2	#22574003	C₂	1	32	yes	maximal, direct product factor
3	C₂×#16864057	C₂	1	4	yes	maximal, complemented
4	C₂×#16864034	C₂	1	12	yes	maximal, complemented
5	C₂×#16863970	C₂	1	3	yes	maximal, complemented
6	C₂×#16863960	C₂	1	3	yes	maximal, complemented
7	C₂×#16863942	C₂	1	3	yes	maximal, complemented
8	C₂²×#14524032	C₂	1	3	yes	maximal, complemented
9	C₂×#41×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×2₋¹⁺⁴×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#14524089	—	2	16	no	—
12	C₂×#14524089	C₂²	1	16	yes	complemented
13	C₂²×#11761629	C₂²	1	2	yes	—
14	C₂²×#11761628	C₂²	1	3	yes	—
15	C₂³×D₂₆₈	C₂²	1	3	yes	—
16	C₂²×D₄×D₆₇	C₂²	1	6	yes	—
17	#16864057	C₂²	1	64	yes	complemented
18	#16864034	C₂²	1	192	yes	complemented
19	#16863970	C₂²	1	48	yes	complemented
20	#16863960	C₂²	1	48	yes	complemented
21	#16863942	C₂²	1	48	yes	complemented
22	C₂×#14524041	C₂²	1	3	yes	complemented
23	C₂×#14524039	C₂²	1	3	yes	complemented
24	C₂×#14524032	C₂²	1	12	yes	complemented
25	C₂×#14524032	C₂²	1	3	yes	complemented
26	C₂×#14524032	C₂²	1	6	yes	complemented
27	C₂×#14524032	C₂²	1	48	yes	complemented
28	C₂×#14524030	C₂²	1	1	yes	characteristic, complemented
29	C₂×#14524026	C₂²	1	12	yes	complemented
30	C₂×#14524019	C₂²	1	12	yes	complemented
31	C₂×#14524017	C₂²	1	12	yes	complemented
32	C₂×#14524015	C₂²	1	12	yes	complemented
33	C₂×#14524011	C₂²	1	12	yes	complemented
34	#41×Dic₆₇	C₂²	1	16	yes	complemented
35	C₂×Q₈×Dic₆₇	C₂²	1	2	yes	complemented
36	C₂×D₄×Dic₆₇	C₂²	1	6	yes	complemented
37	C₂×C₄×C₂²⋊D₆₇	C₂²	1	12	yes	complemented
38	C₂×#14524002	C₂²	1	3	yes	complemented
39	C₂²×(C₂×D₆₇)⋊C₄	C₂²	1	12	yes	complemented
40	C₂²×C₄×Dic₆₇	C₂²	1	3	yes	complemented
41	C₂×2₋¹⁺⁴×C₆₇	—	2	16	no	—
42	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	16	yes	complemented
43	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
44	C₂²×#41×C₆₇	C₂²	1	6	yes	complemented
45	C₂²×#41×C₆₇	C₂²	1	3	yes	complemented
46	C₂³×Q₈×C₆₇	C₂²	1	2	yes	complemented
47	C₂³×Q₈×C₆₇	C₂²	1	3	yes	complemented
48	C₂⁴×D₆₇	C₂³	1	6	yes	—
49	#14524089	—	2	128	no	—
50	C₂×#11761629	—	2	16	no	—
51	C₂×#11761629	C₂³	1	16	yes	—
52	C₂×#11761628	—	2	24	no	—
53	C₂×#11761628	C₂³	1	24	yes	—
54	C₂²×Dic₁₃₄	C₂³	1	1	yes	characteristic
55	C₂²×D₂₆₈	—	2	24	no	—
56	C₂²×D₂₆₈	C₂³	1	6	yes	—
57	C₂²×D₂₆₈	C₂³	1	3	yes	—
58	C₂²×D₂₆₈	C₂³	1	24	yes	—
59	C₂²×C₂²⋊D₆₇	C₂³	1	6	yes	—
60	C₂²×C₂²⋊D₆₇	C₂³	1	12	yes	—
61	C₂×D₄×D₆₇	—	2	48	no	—
62	C₂×D₄×D₆₇	C₂³	1	48	yes	—
63	C₂³×Dic₆₇	C₂³	1	6	yes	—
64	C₂²×C₄×D₆₇	C₂³	1	6	yes	—
65	#14524041	C₂³	1	24	yes	complemented
66	#14524039	C₂³	1	24	yes	complemented
67	#14524032	C₂³	1	96	yes	complemented
68	#14524032	C₂³	1	24	yes	complemented
69	#14524032	C₂³	1	48	yes	complemented
70	#14524030	C₂³	1	8	yes	complemented
71	#14524026	C₂³	1	96	yes	complemented
72	#14524019	C₂³	1	96	yes	complemented
73	#14524017	C₂³	1	96	yes	complemented
74	#14524015	C₂³	1	96	yes	complemented
75	#14524011	C₂³	1	96	yes	complemented
76	Q₈×Dic₆₇	C₂³	1	16	yes	complemented
77	D₄×Dic₆₇	C₂³	1	48	yes	complemented
78	C₄×C₂²⋊D₆₇	C₂³	1	96	yes	complemented
79	#14524002	C₂³	1	24	yes	complemented
80	C₂×C₄⋊Dic₆₇	C₂³	1	6	yes	—
81	C₂×Dic₆₇⋊C₄	C₂³	1	12	yes	—
82	C₂×C₂²⋊Dic₆₇	C₂³	1	6	yes	—
83	C₂×C₂²⋊Dic₆₇	C₂³	1	4	yes	—
84	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	12	yes	—
85	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	24	yes	—
86	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	96	yes	complemented
87	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
88	C₂×C₄×Dic₆₇	C₂³	1	3	yes	—
89	C₂×C₄×Dic₆₇	C₂³	1	6	yes	—
90	C₂×C₄×Dic₆₇	C₂³	1	24	yes	complemented
91	2₋¹⁺⁴×C₆₇	—	2	128	no	—
92	C₂×#41×C₆₇	—	2	48	no	—
93	C₂×#41×C₆₇	—	2	24	no	—
94	C₂×#41×C₆₇	D₄	1	16	yes	complemented
95	C₂×#41×C₆₇	C₂³	1	8	yes	—
96	C₂×#41×C₆₇	C₂³	1	48	yes	complemented
97	C₂×#41×C₆₇	C₂³	1	24	yes	complemented
98	C₂²×Q₈×C₆₇	—	2	16	no	—
99	C₂²×Q₈×C₆₇	—	2	24	no	—
100	C₂²×Q₈×C₆₇	C₂³	1	6	yes	—
101	C₂²×Q₈×C₆₇	C₂³	1	3	yes	—
102	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
103	C₂²×Q₈×C₆₇	C₂³	1	16	yes	complemented
104	C₂²×Q₈×C₆₇	C₂³	1	24	yes	complemented
105	C₂²×D₄×C₆₇	C₂³	1	6	yes	—
106	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
107	C₂²×D₄×C₆₇	C₂³	1	3	yes	—
108	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
109	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
110	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
111	C₂³×D₆₇	—	2	24	no	—
112	C₂³×D₆₇	—	4	24	no	—
113	C₂³×D₆₇	C₂⁴	1	6	yes	—
114	C₂³×D₆₇	C₂⁴	1	24	yes	—
115	#11761629	—	2	64	no	—
116	#11761628	—	2	96	no	—
117	C₂×Dic₁₃₄	—	2	4	no	—
118	C₂×Dic₁₃₄	C₂⁴	1	4	yes	—
119	C₂×D₂₆₈	—	2	24	no	—
120	C₂×D₂₆₈	—	2	12	no	—
121	C₂×D₂₆₈	—	2	96	no	—
122	C₂×D₂₆₈	C₂⁴	1	24	yes	—
123	C₂×D₂₆₈	C₂⁴	1	12	yes	—
124	C₂×C₂²⋊D₆₇	—	2	24	no	—
125	C₂×C₂²⋊D₆₇	—	4	24	no	—
126	C₂×C₂²⋊D₆₇	C₂⁴	1	24	yes	—
127	C₂×C₂²⋊D₆₇	C₂⁴	1	48	yes	—
128	D₄×D₆₇	—	2	192	no	—
129	C₂²×Dic₆₇	—	4	12	no	—
130	C₂²×Dic₆₇	C₂⁴	1	2	yes	—
131	C₂²×Dic₆₇	C₂⁴	1	6	yes	—
132	C₂²×Dic₆₇	C₂⁴	1	2	yes	—
133	C₂²×Dic₆₇	C₂⁴	1	24	yes	—
134	C₂×C₄×D₆₇	—	2	24	no	—
135	C₂×C₄×D₆₇	C₂⁴	1	24	yes	—
136	C₄⋊Dic₆₇	C₂⁴	1	24	yes	—
137	Dic₆₇⋊C₄	C₂⁴	1	48	yes	—
138	C₂²⋊Dic₆₇	C₂⁴	1	24	yes	—
139	C₂²⋊Dic₆₇	C₂⁴	1	16	yes	—
140	(C₂×D₆₇)⋊C₄	C₂⁴	1	48	yes	—
141	(C₂×D₆₇)⋊C₄	C₂⁴	1	96	yes	—
142	C₄×Dic₆₇	C₂⁴	1	4	yes	—
143	C₄×Dic₆₇	C₂⁴	1	12	yes	—
144	C₄×Dic₆₇	C₂⁴	1	24	yes	—
145	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
146	C₂⁴×C₆₇	C₂⁴	1	3	yes	—
147	#41×C₆₇	—	2	192	no	—
148	#41×C₆₇	—	2	96	no	—
149	#41×C₆₇	C₂×D₄	1	64	yes	complemented
150	C₂×Q₈×C₆₇	—	2	64	no	—
151	C₂×Q₈×C₆₇	—	2	24	no	—
152	C₂×Q₈×C₆₇	—	2	12	no	—
153	C₂×Q₈×C₆₇	—	2	96	no	—
154	C₂×Q₈×C₆₇	C₂×D₄	1	8	yes	—
155	C₂×Q₈×C₆₇	C₂⁴	1	24	yes	—
156	C₂×Q₈×C₆₇	C₂⁴	1	12	yes	—
157	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
158	C₂×D₄×C₆₇	—	2	24	no	—
159	C₂×D₄×C₆₇	—	2	4	no	—
160	C₂×D₄×C₆₇	C₂×D₄	1	24	yes	—
161	C₂×D₄×C₆₇	C₂⁴	1	24	yes	—
162	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
163	C₂×D₄×C₆₇	C₂⁴	1	12	yes	—
164	C₂²×C₄×C₆₇	—	2	24	no	—
165	C₂²×C₄×C₆₇	—	2	24	no	—
166	C₂²×C₄×C₆₇	C₂×D₄	1	24	yes	—
167	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
168	C₂²×C₄×C₆₇	C₂⁴	1	24	yes	—
169	C₂²×C₄×C₆₇	C₂⁴	1	24	yes	—
170	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
171	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
172	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
173	C₂³×C₆₇	—	2	12	no	—
174	C₂³×C₆₇	—	2	4	no	—
175	C₂³×C₆₇	—	2	8	no	—
176	C₂³×C₆₇	C₂²×D₄	1	12	yes	—
177	C₂³×C₆₇	C₂⁵	1	4	yes	—
178	C₂³×C₆₇	C₂⁵	1	6	yes	—
179	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
180	C₂²×D₆₇	—	2	12	no	—
181	C₂²×D₆₇	—	2	24	no	—
182	C₂²×D₆₇	—	2	48	no	—
183	C₂²×D₆₇	—	4	48	no	—
184	C₂²×D₆₇	—	8	48	no	—
185	C₂²×D₆₇	C₂⁵	1	12	yes	—
186	Q₈×C₆₇	—	2	24	no	—
187	Q₈×C₆₇	—	2	48	no	—
188	Q₈×C₆₇	C₂²×D₄	1	16	yes	—
189	D₄×C₆₇	—	2	48	no	—
190	D₄×C₆₇	—	2	8	no	—
191	D₄×C₆₇	C₂²×D₄	1	48	yes	—
192	C₂×C₄×C₆₇	—	2	12	no	—
193	C₂×C₄×C₆₇	—	2	48	no	—
194	C₂×C₄×C₆₇	—	2	48	no	—
195	C₂×C₄×C₆₇	C₂²×D₄	1	12	yes	—
196	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
197	C₂×C₄×C₆₇	C₂²×D₄	1	48	yes	—
198	C₂×C₄×C₆₇	C₂⁵	1	12	yes	—
199	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
200	C₂×C₄×C₆₇	C₂⁵	1	6	yes	—
201	C₂²⋊D₆₇	—	2	48	no	—
202	C₂²⋊D₆₇	—	4	48	no	—
203	C₂×Dic₆₇	—	2	4	no	—
204	C₂×Dic₆₇	—	2	4	no	—
205	C₂×Dic₆₇	—	2	12	no	—
206	C₂×Dic₆₇	—	4	24	no	—
207	C₂×Dic₆₇	C₂⁵	1	4	yes	—
208	C₂×Dic₆₇	C₂⁵	1	12	yes	—
209	C₂×Dic₆₇	C₂⁵	1	4	yes	—
210	D₂₆₈	—	2	48	no	—
211	D₂₆₈	—	2	24	no	—
212	C₄×D₆₇	—	2	48	no	—
213	Dic₁₃₄	—	2	8	no	—
214	C₂²×C₆₇	—	2	12	no	—
215	C₂²×C₆₇	—	2	24	no	—
216	C₂²×C₆₇	—	2	4	no	—
217	C₂²×C₆₇	—	2	8	no	—
218	C₂²×C₆₇	—	4	8	no	—
219	C₂²×C₆₇	C₂³×D₄	1	12	yes	—
220	C₂²×C₆₇	C₂³×D₄	1	2	yes	—
221	C₂²×C₆₇	#585	1	4	yes	—
222	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
223	C₂×D₆₇	—	2	12	no	—
224	C₂×D₆₇	—	2	24	no	—
225	C₂×D₆₇	—	4	24	no	—
226	C₂×D₆₇	—	8	48	no	—
227	C₄×C₆₇	—	2	12	no	—
228	C₄×C₆₇	C₂³×D₄	1	4	yes	—
229	C₄×C₆₇	C₂³×D₄	1	12	yes	—
230	Dic₆₇	—	2	4	no	—
231	Dic₆₇	—	2	4	no	—
232	Dic₆₇	—	2	12	no	—
233	C₂×#3500	—	67	1	no	maximal, Sylow
234	C₂×C₆₇	—	2	12	no	—
235	C₂×C₆₇	—	4	4	no	—
236	C₂×C₆₇	#3500	1	4	yes	complemented
237	C₂×C₆₇	C₂⁴×D₄	1	1	yes	characteristic
238	C₂×C₆₇	C₂×#585	1	2	yes	—
239	D₆₇	—	4	12	no	—
240	C₂²×2₋¹⁺⁴	D₆₇	1	1	yes	characteristic, complemented
241	C₂²×H(5,𝔽₂)	—	67	1	no	—
242	#3500	—	67	32	no	—
243	C₂×#548	—	67	4	no	—
244	C₂×#546	—	67	12	no	—
245	C₂×#536	—	67	3	no	—
246	C₂×#535	—	67	3	no	—
247	C₂×#532	—	67	3	no	—
248	C₂²×#124	—	67	3	no	—
249	C₂×C₄×#41	—	67	1	no	—
250	C₆₇	C₂×#3500	1	1	yes	characteristic, complemented, Sylow
251	C₂×2₋¹⁺⁴	—	2	16	no	—
252	C₂×2₋¹⁺⁴	C₂×D₆₇	1	16	yes	complemented
253	C₂×H(5,𝔽₂)	—	67	16	no	—
254	C₂×H(5,𝔽₂)	—	134	16	no	—
255	C₂²×#41	—	67	2	no	—
256	C₂²×#41	—	67	3	no	—
257	C₂²×#41	C₂×D₆₇	1	1	yes	characteristic
258	C₂²×#41	C₂×D₆₇	1	3	yes	complemented
259	C₂²×#41	C₂×D₆₇	1	6	yes	complemented
260	C₂³×Q₈	C₂×D₆₇	1	3	yes	complemented
261	C₂³×Q₈	C₂×D₆₇	1	2	yes	complemented
262	C₂³×D₄	—	67	6	no	—
263	C₂³×D₄	—	67	3	no	—
264	#548	—	67	64	no	—
265	#546	—	67	192	no	—
266	#536	—	67	48	no	—
267	#535	—	67	48	no	—
268	#532	—	67	48	no	—
269	C₂×C₄⋊Q₈	—	67	3	no	—
270	C₂×#127	—	67	3	no	—
271	C₂×#124	—	67	3	no	—
272	C₂×#124	—	67	6	no	—
273	C₂×#124	—	67	12	no	—
274	C₂×#124	—	67	48	no	—
275	C₂×#124	—	67	1	no	—
276	C₂×#123	—	67	12	no	—
277	C₂×C₂²⋊Q₈	—	67	12	no	—
278	C₂×#121	—	67	12	no	—
279	C₂×#121	—	67	12	no	—
280	C₂×#120	—	67	12	no	—
281	C₄×#41	—	67	16	no	—
282	C₂×C₄×Q₈	—	67	2	no	—
283	C₂×C₄×D₄	—	67	6	no	—
284	C₂×C₄×D₄	—	67	12	no	—
285	C₂×#117	—	67	3	no	—
286	C₂²×C₂²⋊C₄	—	67	12	no	—
287	C₂²×C₄²	—	67	3	no	—
288	C₂⁵	—	67	6	no	—
289	2₋¹⁺⁴	—	2	128	no	—
290	H(5,𝔽₂)	—	134	128	no	—
291	C₂×#41	—	2	48	no	—
292	C₂×#41	—	2	24	no	—
293	C₂×#41	—	67	16	no	—
294	C₂×#41	—	67	24	no	—
295	C₂×#41	—	134	16	no	—
296	C₂×#41	—	134	24	no	—
297	C₂×#41	C₂²⋊D₆₇	1	16	yes	complemented
298	C₂×#41	C₂²×D₆₇	1	8	yes	—
299	C₂×#41	C₂²×D₆₇	1	24	yes	complemented
300	C₂×#41	C₂²×D₆₇	1	48	yes	complemented
301	C₂²×Q₈	—	2	16	no	—
302	C₂²×Q₈	—	2	24	no	—
303	C₂²×Q₈	—	67	1	no	—
304	C₂²×Q₈	C₂²×D₆₇	1	6	yes	—
305	C₂²×Q₈	C₂²×D₆₇	1	3	yes	—
306	C₂²×Q₈	C₂²×D₆₇	1	1	yes	characteristic
307	C₂²×Q₈	C₂²×D₆₇	1	24	yes	complemented
308	C₂²×Q₈	C₂²×D₆₇	1	16	yes	complemented
309	C₂²×D₄	—	67	12	no	—
310	C₂²×D₄	—	67	6	no	—
311	C₂²×D₄	—	67	48	no	—
312	C₂²×D₄	—	67	6	no	—
313	C₂²×D₄	—	67	3	no	—
314	C₂²×D₄	—	67	24	no	—
315	C₂²×D₄	—	134	48	no	—
316	C₂²×D₄	—	134	24	no	—
317	C₂²×D₄	C₂²×D₆₇	1	6	yes	—
318	C₂²×D₄	C₂²×D₆₇	1	1	yes	characteristic
319	C₂²×D₄	C₂²×D₆₇	1	3	yes	—
320	C₂³×C₄	—	67	6	no	—
321	C₂³×C₄	—	67	6	no	—
322	C₂³×C₄	C₂²×D₆₇	1	6	yes	—
323	C₂³×C₄	C₂²×D₆₇	1	6	yes	—
324	C₂³×C₄	C₂²×D₆₇	1	3	yes	—
325	C₄⋊Q₈	—	67	24	no	—
326	#127	—	67	24	no	—
327	#124	—	67	24	no	—
328	#124	—	67	48	no	—
329	#124	—	67	96	no	—
330	#124	—	67	8	no	—
331	#123	—	67	96	no	—
332	C₂²⋊Q₈	—	67	96	no	—
333	#121	—	67	96	no	—
334	#121	—	67	96	no	—
335	#120	—	67	96	no	—
336	C₄×Q₈	—	67	16	no	—
337	C₄×D₄	—	67	48	no	—
338	C₄×D₄	—	67	96	no	—
339	#117	—	67	24	no	—
340	C₂×C₄⋊C₄	—	67	6	no	—
341	C₂×C₄⋊C₄	—	67	12	no	—
342	C₂×C₂²⋊C₄	—	67	6	no	—
343	C₂×C₂²⋊C₄	—	67	4	no	—
344	C₂×C₂²⋊C₄	—	67	12	no	—
345	C₂×C₂²⋊C₄	—	67	24	no	—
346	C₂×C₂²⋊C₄	—	67	96	no	—
347	C₂×C₄²	—	67	3	no	—
348	C₂×C₄²	—	67	6	no	—
349	C₂×C₄²	—	67	24	no	—
350	C₂×C₄²	—	67	1	no	—
351	C₂⁴	—	268	24	no	—
352	C₂⁴	—	67	24	no	—
353	C₂⁴	—	67	6	no	—
354	C₂⁴	—	134	24	no	—
355	C₂⁴	C₂³×D₆₇	1	2	yes	—
356	C₂⁴	C₂³×D₆₇	1	3	yes	—
357	#41	—	2	192	no	—
358	#41	—	2	96	no	—
359	#41	—	134	64	no	—
360	#41	—	134	96	no	—
361	#41	C₂×C₂²⋊D₆₇	1	64	yes	complemented
362	C₂×Q₈	—	2	64	no	—
363	C₂×Q₈	—	2	24	no	—
364	C₂×Q₈	—	2	12	no	—
365	C₂×Q₈	—	2	96	no	—
366	C₂×Q₈	—	67	4	no	—
367	C₂×Q₈	—	134	4	no	—
368	C₂×Q₈	C₂×C₂²⋊D₆₇	1	8	yes	—
369	C₂×Q₈	C₂³×D₆₇	1	24	yes	—
370	C₂×Q₈	C₂³×D₆₇	1	12	yes	—
371	C₂×Q₈	C₂³×D₆₇	1	4	yes	—
372	C₂×D₄	—	2	24	no	—
373	C₂×D₄	—	2	4	no	—
374	C₂×D₄	—	268	24	no	—
375	C₂×D₄	—	67	48	no	—
376	C₂×D₄	—	67	24	no	—
377	C₂×D₄	—	67	24	no	—
378	C₂×D₄	—	67	12	no	—
379	C₂×D₄	—	134	24	no	—
380	C₂×D₄	—	134	192	no	—
381	C₂×D₄	—	134	24	no	—
382	C₂×D₄	—	134	12	no	—
383	C₂×D₄	—	134	96	no	—
384	C₂×D₄	C₂×C₂²⋊D₆₇	1	24	yes	—
385	C₂×D₄	C₂³×D₆₇	1	24	yes	—
386	C₂×D₄	C₂³×D₆₇	1	4	yes	—
387	C₂×D₄	C₂³×D₆₇	1	12	yes	—
388	C₂²×C₄	—	2	24	no	—
389	C₂²×C₄	—	2	24	no	—
390	C₂²×C₄	—	268	12	no	—
391	C₂²×C₄	—	67	24	no	—
392	C₂²×C₄	—	67	2	no	—
393	C₂²×C₄	—	67	6	no	—
394	C₂²×C₄	—	67	2	no	—
395	C₂²×C₄	—	67	24	no	—
396	C₂²×C₄	—	134	24	no	—
397	C₂²×C₄	C₂×C₂²⋊D₆₇	1	24	yes	—
398	C₂²×C₄	C₂³×D₆₇	1	24	yes	—
399	C₂²×C₄	C₂³×D₆₇	1	24	yes	—
400	C₂²×C₄	C₂³×D₆₇	1	6	yes	—
401	C₂²×C₄	C₂³×D₆₇	1	12	yes	—
402	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
403	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
404	C₄⋊C₄	—	67	24	no	—
405	C₄⋊C₄	—	67	48	no	—
406	C₂²⋊C₄	—	67	24	no	—
407	C₂²⋊C₄	—	67	16	no	—
408	C₂²⋊C₄	—	67	48	no	—
409	C₂²⋊C₄	—	67	96	no	—
410	C₄²	—	67	4	no	—
411	C₄²	—	67	12	no	—
412	C₄²	—	67	24	no	—
413	C₂³	—	2	12	no	—
414	C₂³	—	2	4	no	—
415	C₂³	—	2	8	no	—
416	C₂³	—	268	48	no	—
417	C₂³	—	536	48	no	—
418	C₂³	—	67	12	no	—
419	C₂³	—	134	12	no	—
420	C₂³	—	134	24	no	—
421	C₂³	—	134	48	no	—
422	C₂³	C₂²×C₂²⋊D₆₇	1	12	yes	—
423	C₂³	C₂⁴×D₆₇	1	4	yes	—
424	C₂³	C₂⁴×D₆₇	1	6	yes	—
425	C₂³	C₂⁴×D₆₇	1	1	yes	characteristic
426	Q₈	—	2	48	no	—
427	Q₈	—	2	24	no	—
428	Q₈	—	134	8	no	—
429	Q₈	C₂²×C₂²⋊D₆₇	1	16	yes	—
430	D₄	—	2	48	no	—
431	D₄	—	2	8	no	—
432	D₄	—	268	48	no	—
433	D₄	—	134	48	no	—
434	D₄	—	134	48	no	—
435	D₄	—	134	24	no	—
436	D₄	C₂²×C₂²⋊D₆₇	1	48	yes	—
437	C₂×C₄	—	2	12	no	—
438	C₂×C₄	—	2	48	no	—
439	C₂×C₄	—	2	48	no	—
440	C₂×C₄	—	268	24	no	—
441	C₂×C₄	—	67	4	no	—
442	C₂×C₄	—	67	12	no	—
443	C₂×C₄	—	67	4	no	—
444	C₂×C₄	—	134	4	no	—
445	C₂×C₄	—	134	4	no	—
446	C₂×C₄	—	134	12	no	—
447	C₂×C₄	—	134	48	no	—
448	C₂×C₄	C₂²×C₂²⋊D₆₇	1	12	yes	—
449	C₂×C₄	C₂²×C₂²⋊D₆₇	1	4	yes	—
450	C₂×C₄	C₂²×C₂²⋊D₆₇	1	48	yes	—
451	C₂×C₄	C₂⁴×D₆₇	1	12	yes	—
452	C₂×C₄	C₂⁴×D₆₇	1	2	yes	—
453	C₂×C₄	C₂⁴×D₆₇	1	6	yes	—
454	C₂²	—	2	12	no	—
455	C₂²	—	2	24	no	—
456	C₂²	—	2	4	no	—
457	C₂²	—	2	8	no	—
458	C₂²	—	4	8	no	—
459	C₂²	—	268	24	no	—
460	C₂²	—	536	48	no	—
461	C₂²	—	134	12	no	—
462	C₂²	—	134	24	no	—
463	C₂²	C₂³×C₂²⋊D₆₇	1	12	yes	—
464	C₂²	C₂³×C₂²⋊D₆₇	1	2	yes	—
465	C₂²	#16864233	1	4	yes	—
466	C₂²	C₂⁵×D₆₇	1	1	yes	characteristic
467	C₄	—	2	12	no	—
468	C₄	—	134	4	no	—
469	C₄	—	134	4	no	—
470	C₄	—	134	12	no	—
471	C₄	C₂³×C₂²⋊D₆₇	1	12	yes	—
472	C₄	C₂³×C₂²⋊D₆₇	1	4	yes	—
473	C₂	—	2	12	no	—
474	C₂	—	4	4	no	—
475	C₂	—	268	12	no	—
476	C₂	#22574003	1	4	yes	direct product factor
477	C₂	C₂⁴×C₂²⋊D₆₇	1	1	yes	characteristic
478	C₂	C₂×#16864233	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₆₇
commutator, cosocle	C₂²×C₆₇	C₂⁶
Frattini	C₂²	C₂⁵×D₆₇
Fitting	C₂²×2₋¹⁺⁴×C₆₇	C₂
socle	C₂³×C₆₇	C₂⁵
hypercenter	C₂²×2₋¹⁺⁴	D₆₇
hypocenter	C₆₇	C₂×#3500
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. C₂²×#14524089	C₂	semidirect
2. #22574003	C₂	direct (N × H)
3. C₂×#16864057	C₂	semidirect
4. C₂×#16864034	C₂	semidirect
5. C₂×#16863970	C₂	semidirect
6. C₂×#16863960	C₂	semidirect
7. C₂×#16863942	C₂	semidirect
8. C₂²×#14524032	C₂	semidirect
9. C₂×#41×Dic₆₇	C₂	semidirect
10. C₂²×2₋¹⁺⁴×C₆₇	C₂	semidirect
12. C₂×#14524089	C₂²	semidirect
17. #16864057	C₂²	semidirect
18. #16864034	C₂²	semidirect
19. #16863970	C₂²	semidirect
20. #16863960	C₂²	semidirect
21. #16863942	C₂²	semidirect
22. C₂×#14524041	C₂²	semidirect
23. C₂×#14524039	C₂²	semidirect
24. C₂×#14524032	C₂²	semidirect
25. C₂×#14524032	C₂²	semidirect
26. C₂×#14524032	C₂²	semidirect
27. C₂×#14524032	C₂²	semidirect
28. C₂×#14524030	C₂²	semidirect
29. C₂×#14524026	C₂²	semidirect
30. C₂×#14524019	C₂²	semidirect
31. C₂×#14524017	C₂²	semidirect
32. C₂×#14524015	C₂²	semidirect
33. C₂×#14524011	C₂²	semidirect
34. #41×Dic₆₇	C₂²	semidirect
35. C₂×Q₈×Dic₆₇	C₂²	semidirect
36. C₂×D₄×Dic₆₇	C₂²	semidirect
37. C₂×C₄×C₂²⋊D₆₇	C₂²	semidirect
38. C₂×#14524002	C₂²	semidirect
39. C₂²×(C₂×D₆₇)⋊C₄	C₂²	semidirect
40. C₂²×C₄×Dic₆₇	C₂²	semidirect
42. C₂×2₋¹⁺⁴×C₆₇	C₂²	semidirect
44. C₂²×#41×C₆₇	C₂²	semidirect
45. C₂²×#41×C₆₇	C₂²	semidirect
46. C₂³×Q₈×C₆₇	C₂²	semidirect
47. C₂³×Q₈×C₆₇	C₂²	semidirect
65. #14524041	C₂³	semidirect
66. #14524039	C₂³	semidirect
67. #14524032	C₂³	semidirect
68. #14524032	C₂³	semidirect
69. #14524032	C₂³	semidirect
70. #14524030	C₂³	semidirect
71. #14524026	C₂³	semidirect
72. #14524019	C₂³	semidirect
73. #14524017	C₂³	semidirect
74. #14524015	C₂³	semidirect
75. #14524011	C₂³	semidirect
76. Q₈×Dic₆₇	C₂³	semidirect
77. D₄×Dic₆₇	C₂³	semidirect
78. C₄×C₂²⋊D₆₇	C₂³	semidirect
79. #14524002	C₂³	semidirect
86. C₂×(C₂×D₆₇)⋊C₄	C₂³	semidirect
90. C₂×C₄×Dic₆₇	C₂³	semidirect
94. C₂×#41×C₆₇	D₄	semidirect
96. C₂×#41×C₆₇	C₂³	semidirect
97. C₂×#41×C₆₇	C₂³	semidirect
103. C₂²×Q₈×C₆₇	C₂³	semidirect
104. C₂²×Q₈×C₆₇	C₂³	semidirect
149. #41×C₆₇	C₂×D₄	semidirect
236. C₂×C₆₇	#3500	semidirect
240. C₂²×2₋¹⁺⁴	D₆₇	semidirect
250. C₆₇	C₂×#3500	semidirect
252. C₂×2₋¹⁺⁴	C₂×D₆₇	semidirect
258. C₂²×#41	C₂×D₆₇	semidirect
259. C₂²×#41	C₂×D₆₇	semidirect
260. C₂³×Q₈	C₂×D₆₇	semidirect
261. C₂³×Q₈	C₂×D₆₇	semidirect
297. C₂×#41	C₂²⋊D₆₇	semidirect
299. C₂×#41	C₂²×D₆₇	semidirect
300. C₂×#41	C₂²×D₆₇	semidirect
307. C₂²×Q₈	C₂²×D₆₇	semidirect
308. C₂²×Q₈	C₂²×D₆₇	semidirect
361. #41	C₂×C₂²⋊D₆₇	semidirect
476. C₂	#22574003	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14524089	2
#22574003	2
C₂×#16864057	2
C₂×#16864034	2
C₂×#16863970	2
C₂×#16863960	2
C₂×#16863942	2
C₂²×#14524032	2
C₂×#41×Dic₆₇	2
C₂²×2₋¹⁺⁴×C₆₇	2
C₂×#3500	67

Subgroups — C2×#22574003 #31764304

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22574003 #31764304