Subgroups — C₂×#22560260 #30855490

Order 17152.

1330 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14524049	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14523979	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16863414	C₂	1	2	yes	maximal
4	#22560260	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16863166	C₂	1	2	yes	maximal, complemented
6	C₂×#427×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11761628	C₂²	1	1	yes	characteristic
8	C₂×#14524049	—	2	2	no	—
9	C₂×#14524049	—	2	1	no	—
10	C₂×#14524049	—	2	1	no	—
11	C₂×#14524049	—	2	2	no	—
12	C₂×#14524049	C₄	1	2	yes	complemented
13	C₂×#14524049	C₄	1	2	yes	complemented
14	C₂×#14524049	C₄	1	4	yes	complemented
15	C₂×#14524049	C₂²	1	4	yes	—
16	C₂²×#11761612	C₂²	1	2	yes	—
17	C₂²×#11761612	C₂²	1	1	yes	characteristic
18	C₂²×#11761611	—	2	1	no	—
19	C₂²×C₈×D₆₇	—	2	1	no	—
20	C₂×#14523979	—	2	1	no	—
21	C₂×#14523979	—	2	1	no	—
22	C₂×#14523979	—	2	2	no	—
23	C₂×#14523979	C₂²	1	4	yes	complemented
24	#16863414	—	2	4	no	—
25	#16863414	C₂²	1	8	yes	—
26	C₂×#14523971	—	2	4	no	—
27	#16863166	C₂²	1	8	yes	complemented
28	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#117×C₆₇	C₂²	1	2	yes	—
30	#427×C₆₇	—	2	2	no	—
31	#427×C₆₇	C₂²	1	4	yes	complemented
32	C₂×#107×C₆₇	—	2	1	no	—
33	C₂×#106×C₆₇	—	2	1	no	—
34	C₂×#11761628	—	2	1	no	—
35	C₂×#11761628	—	2	1	no	—
36	C₂×#11761628	—	2	1	no	—
37	C₂×#11761628	—	2	1	no	—
38	C₂×#11761628	—	2	1	no	—
39	C₂×#11761628	—	2	1	no	—
40	C₂×#11761628	C₂×C₄	1	2	yes	—
41	C₂×#11761628	C₂×C₄	1	2	yes	—
42	C₂×#11761628	D₄	1	2	yes	—
43	C₂×#11761628	D₄	1	2	yes	—
44	C₂×#11761628	Q₈	1	2	yes	—
45	C₂×#11761628	C₂³	1	2	yes	—
46	C₂²×Dic₁₃₄	Q₈	1	1	yes	characteristic
47	C₂²×D₂₆₈	Q₈	1	1	yes	characteristic
48	C₂²×C₂²⋊D₆₇	—	2	1	no	—
49	C₂²×C₄×D₆₇	—	2	1	no	—
50	#14524049	—	2	4	no	—
51	#14524049	—	2	4	no	—
52	#14524049	—	2	4	no	—
53	#14524049	—	2	4	no	—
54	#14524049	—	2	4	no	—
55	#14524049	—	2	4	no	—
56	#14524049	C₂×C₄	1	8	yes	complemented
57	#14524049	C₂×C₄	1	8	yes	complemented
58	C₂×#11761612	—	2	1	no	—
59	C₂×#11761612	—	2	1	no	—
60	C₂×#11761612	—	2	1	no	—
61	C₂×#11761612	—	2	2	no	—
62	C₂×#11761612	—	2	2	no	—
63	C₂×#11761612	—	2	2	no	—
64	C₂×#11761612	—	2	2	no	—
65	C₂×#11761612	—	2	2	no	—
66	C₂×#11761612	C₂×C₄	1	2	yes	—
67	C₂×#11761612	C₂×C₄	1	2	yes	—
68	C₂×#11761612	C₂³	1	4	yes	—
69	C₂×#11761612	C₂³	1	2	yes	—
70	C₂×#11761611	—	2	2	no	—
71	C₂×#11761611	—	2	2	no	—
72	C₂×#11761611	—	2	4	no	—
73	C₂×#11761611	—	2	4	no	—
74	C₂²×C₆₇⋊C₈	—	2	1	no	—
75	C₂²×C₆₇⋊C₈	—	2	2	no	—
76	C₂×C₈×D₆₇	—	2	2	no	—
77	C₂×C₈×D₆₇	—	2	2	no	—
78	C₂×C₈×D₆₇	—	2	4	no	—
79	C₂×C₈×D₆₇	—	2	4	no	—
80	#14523979	—	2	4	no	—
81	#14523979	—	2	4	no	—
82	#14523971	—	2	16	no	—
83	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
84	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
85	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
86	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
87	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
88	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
89	C₂²×C₈×C₆₇	D₄	1	2	yes	—
90	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
91	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
92	C₂²×C₈×C₆₇	D₄	1	2	yes	—
93	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
94	#117×C₆₇	—	2	2	no	—
95	#117×C₆₇	—	2	2	no	—
96	#117×C₆₇	—	2	2	no	—
97	#117×C₆₇	C₂³	1	4	yes	—
98	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
99	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
100	C₂×C₄²×C₆₇	—	2	2	no	—
101	#107×C₆₇	—	2	4	no	—
102	#106×C₆₇	—	2	4	no	—
103	C₂³×D₆₇	—	2	1	no	—
104	#11761628	—	2	2	no	—
105	#11761628	—	2	2	no	—
106	#11761628	—	2	2	no	—
107	#11761628	—	2	2	no	—
108	#11761628	—	2	2	no	—
109	#11761628	—	2	2	no	—
110	#11761628	—	2	2	no	—
111	#11761628	—	2	2	no	—
112	#11761628	—	2	2	no	—
113	#11761628	—	2	2	no	—
114	#11761628	—	2	2	no	—
115	#11761628	—	2	2	no	—
116	#11761628	C₄⋊C₄	1	4	yes	—
117	#11761628	C₄⋊C₄	1	4	yes	—
118	#11761628	C₂²×C₄	1	4	yes	—
119	#11761628	C₂×D₄	1	4	yes	—
120	C₂×Dic₁₃₄	—	2	1	no	—
121	C₂×Dic₁₃₄	—	2	1	no	—
122	C₂×Dic₁₃₄	—	2	1	no	—
123	C₂×Dic₁₃₄	C₄⋊C₄	1	2	yes	—
124	C₂×Dic₁₃₄	C₄⋊C₄	1	2	yes	—
125	C₂×Dic₁₃₄	C₂×Q₈	1	2	yes	—
126	C₂×D₂₆₈	—	2	1	no	—
127	C₂×D₂₆₈	—	2	1	no	—
128	C₂×D₂₆₈	—	2	1	no	—
129	C₂×D₂₆₈	C₄⋊C₄	1	2	yes	—
130	C₂×D₂₆₈	C₄⋊C₄	1	2	yes	—
131	C₂×D₂₆₈	C₂×Q₈	1	2	yes	—
132	C₂×C₂²⋊D₆₇	—	2	2	no	—
133	C₂×C₂²⋊D₆₇	—	2	2	no	—
134	C₂×C₂²⋊D₆₇	—	2	2	no	—
135	C₂×C₂²⋊D₆₇	—	2	2	no	—
136	C₂×C₂²⋊D₆₇	—	2	2	no	—
137	C₂×C₂²⋊D₆₇	—	2	2	no	—
138	C₂²×Dic₆₇	—	2	1	no	—
139	C₂×C₄×D₆₇	—	2	2	no	—
140	C₂×C₄×D₆₇	—	2	2	no	—
141	C₂×C₄×D₆₇	—	2	2	no	—
142	C₂×C₄×D₆₇	—	2	2	no	—
143	C₂×C₄×D₆₇	—	2	2	no	—
144	C₂×C₄×D₆₇	—	2	2	no	—
145	#11761612	—	2	2	no	—
146	#11761612	—	2	2	no	—
147	#11761612	—	2	2	no	—
148	#11761612	—	2	4	no	—
149	#11761612	—	2	4	no	—
150	#11761612	—	4	2	no	—
151	#11761612	—	4	2	no	—
152	#11761612	C₂²×C₄	1	4	yes	—
153	#11761611	—	2	8	no	—
154	#11761611	—	2	8	no	—
155	C₂×C₆₇⋊C₈	—	2	2	no	—
156	C₂×C₆₇⋊C₈	—	2	2	no	—
157	C₂×C₆₇⋊C₈	—	2	2	no	—
158	C₂×C₆₇⋊C₈	—	2	4	no	—
159	C₂×C₆₇⋊C₈	—	4	2	no	—
160	C₂×C₆₇⋊C₈	—	4	2	no	—
161	C₈×D₆₇	—	2	8	no	—
162	C₈×D₆₇	—	2	8	no	—
163	C₂⁴×C₆₇	#41	1	1	yes	characteristic
164	C₂²×C₄×C₆₇	—	4	2	no	—
165	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
166	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
167	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
174	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
175	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
176	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
177	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
178	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
179	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
180	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
181	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
182	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
183	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
184	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
185	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
186	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
187	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
188	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
189	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
190	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
191	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
192	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
193	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
194	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
195	C₄⋊C₄×C₆₇	—	2	4	no	—
196	C₄⋊C₄×C₆₇	—	4	2	no	—
197	C₂²⋊C₄×C₆₇	—	2	4	no	—
198	C₂²⋊C₄×C₆₇	—	4	2	no	—
199	C₄²×C₆₇	—	2	4	no	—
200	C₄²×C₆₇	—	4	2	no	—
201	C₂³×C₆₇	—	4	2	no	—
202	C₂³×C₆₇	#117	1	1	yes	characteristic
203	C₂³×C₆₇	#117	1	1	yes	characteristic
204	C₂³×C₆₇	#121	1	1	yes	characteristic
205	C₂³×C₆₇	#121	1	1	yes	characteristic
206	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
207	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
208	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
209	C₂²×D₆₇	—	2	2	no	—
210	C₂²×D₆₇	—	2	2	no	—
211	C₂²×D₆₇	—	2	2	no	—
212	C₂²×D₆₇	—	4	4	no	—
213	C₂×C₄×C₆₇	—	4	4	no	—
214	C₂×C₄×C₆₇	—	8	4	no	—
215	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
216	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
217	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
218	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
219	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
220	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
221	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
222	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
223	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
224	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
225	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
226	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
227	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
228	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
229	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
230	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
231	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
232	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
233	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
234	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
235	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
236	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
237	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
238	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
239	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
240	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
242	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
243	C₂²⋊D₆₇	—	2	4	no	—
244	C₂²⋊D₆₇	—	2	4	no	—
245	C₂²⋊D₆₇	—	2	4	no	—
246	C₂²⋊D₆₇	—	2	4	no	—
247	C₂×Dic₆₇	—	2	2	no	—
248	C₂×Dic₆₇	—	2	2	no	—
249	C₂×Dic₆₇	—	2	2	no	—
250	D₂₆₈	—	2	2	no	—
251	D₂₆₈	—	2	2	no	—
252	D₂₆₈	—	2	2	no	—
253	D₂₆₈	C₂×C₄⋊C₄	1	4	yes	—
254	C₄×D₆₇	—	2	4	no	—
255	C₄×D₆₇	—	2	4	no	—
256	C₄×D₆₇	—	2	4	no	—
257	C₄×D₆₇	—	2	4	no	—
258	Dic₁₃₄	—	2	2	no	—
259	Dic₁₃₄	—	2	2	no	—
260	Dic₁₃₄	—	2	2	no	—
261	Dic₁₃₄	C₂×C₄⋊C₄	1	4	yes	—
262	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
263	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
264	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
265	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
266	C₆₇⋊C₈	—	2	4	no	—
267	C₆₇⋊C₈	—	4	4	no	—
268	C₂²×C₆₇	—	2	2	no	—
269	C₂²×C₆₇	—	2	2	no	—
270	C₂²×C₆₇	—	2	2	no	—
271	C₂²×C₆₇	—	4	1	no	—
272	C₂²×C₆₇	—	4	1	no	—
273	C₂²×C₆₇	—	4	1	no	—
274	C₂²×C₆₇	—	4	1	no	—
275	C₂²×C₆₇	#380	1	1	yes	characteristic
276	C₂²×C₆₇	#380	1	1	yes	characteristic
277	C₂²×C₆₇	#380	1	1	yes	characteristic
278	C₂²×C₆₇	#380	1	1	yes	characteristic
279	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
280	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
281	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
282	C₂×D₆₇	—	2	4	no	—
283	C₂×D₆₇	—	4	4	no	—
284	C₂×D₆₇	—	4	4	no	—
285	C₂×D₆₇	—	4	4	no	—
286	C₄×C₆₇	—	8	4	no	—
287	C₄×C₆₇	#380	1	1	yes	characteristic
288	C₄×C₆₇	#380	1	1	yes	characteristic
289	C₄×C₆₇	#380	1	1	yes	characteristic
290	C₄×C₆₇	#380	1	1	yes	characteristic
291	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
292	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
293	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
294	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
295	Dic₆₇	—	2	4	no	—
296	C₂×#1814	—	67	1	no	maximal, Sylow
297	C₂×C₆₇	—	2	1	no	—
298	C₂×C₆₇	—	2	1	no	—
299	C₂×C₆₇	—	2	1	no	—
300	C₂×C₆₇	—	2	1	no	—
301	C₂×C₆₇	—	2	1	no	—
302	C₂×C₆₇	—	2	1	no	—
303	C₂×C₆₇	#1814	1	2	yes	complemented
304	C₂×C₆₇	C₂×#380	1	1	yes	characteristic
305	D₆₇	—	4	4	no	—
306	C₂²×#131	—	67	1	no	—
307	C₂²×#108	—	67	1	no	—
308	C₂×#427	D₆₇	1	1	yes	characteristic, complemented
309	C₂×#421	—	67	2	no	—
310	#1814	—	67	8	no	—
311	C₂×#344	—	67	2	no	—
312	C₆₇	C₂×#1814	1	1	yes	characteristic, complemented, Sylow
313	C₂²×#41	—	67	1	no	—
314	C₂×#131	—	67	2	no	—
315	C₂×#131	—	67	2	no	—
316	C₂×#131	—	67	4	no	—
317	C₂×#131	—	67	4	no	—
318	C₂×#131	—	134	2	no	—
319	C₂×#131	—	134	1	no	—
320	C₂×#131	—	134	1	no	—
321	C₂×#131	—	134	2	no	—
322	C₂²×M₁₆	—	67	1	no	—
323	C₂²×M₁₆	—	67	2	no	—
324	C₂²×M₁₆	—	134	1	no	—
325	C₂³×C₈	—	134	1	no	—
326	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
327	C₂×#117	C₂×D₆₇	1	2	yes	—
328	C₂×#108	—	67	4	no	—
329	C₂×#108	—	134	1	no	—
330	C₂×#108	—	134	1	no	—
331	C₂×#108	—	134	2	no	—
332	#427	—	2	2	no	—
333	#427	C₂×D₆₇	1	4	yes	complemented
334	C₂×#107	—	2	1	no	—
335	C₂×#106	—	2	1	no	—
336	#421	—	67	8	no	—
337	#421	—	134	4	no	—
338	C₂×#104	—	134	4	no	—
339	#344	—	67	8	no	—
340	C₂×#41	—	67	2	no	—
341	C₂×#41	—	67	2	no	—
342	C₂×#41	—	67	2	no	—
343	C₂×#41	—	67	2	no	—
344	C₂×#41	—	67	2	no	—
345	C₂×#41	—	67	2	no	—
346	C₂×#41	—	134	1	no	—
347	C₂×#41	—	134	1	no	—
348	C₂×#41	—	134	1	no	—
349	C₂×#41	—	134	1	no	—
350	C₂×#41	—	134	1	no	—
351	C₂×#41	—	134	1	no	—
352	C₂²×Q₈	—	67	1	no	—
353	C₂²×D₄	—	67	1	no	—
354	C₂²×D₄	—	134	1	no	—
355	C₂³×C₄	—	134	1	no	—
356	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
357	#131	—	67	8	no	—
358	#131	—	67	8	no	—
359	#131	—	134	4	no	—
360	#131	—	134	4	no	—
361	#131	—	134	4	no	—
362	#131	—	134	4	no	—
363	#131	—	134	4	no	—
364	#131	—	134	4	no	—
365	C₂×M₁₆	—	67	2	no	—
366	C₂×M₁₆	—	67	2	no	—
367	C₂×M₁₆	—	67	2	no	—
368	C₂×M₁₆	—	67	4	no	—
369	C₂×M₁₆	—	134	2	no	—
370	C₂×M₁₆	—	134	2	no	—
371	C₂×M₁₆	—	134	2	no	—
372	C₂×M₁₆	—	134	2	no	—
373	C₂×M₁₆	—	134	2	no	—
374	C₂×M₁₆	—	134	1	no	—
375	C₂×M₁₆	—	134	1	no	—
376	C₂×M₁₆	—	134	1	no	—
377	C₂×M₁₆	—	134	2	no	—
378	C₂×M₁₆	—	134	2	no	—
379	C₂×M₁₆	—	134	4	no	—
380	C₂×M₁₆	—	134	4	no	—
381	C₂²×C₈	—	134	2	no	—
382	C₂²×C₈	—	134	1	no	—
383	C₂²×C₈	—	134	2	no	—
384	C₂²×C₈	—	134	2	no	—
385	C₂²×C₈	—	134	4	no	—
386	C₂²×C₈	—	134	4	no	—
387	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
388	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
389	C₂²×C₈	C₄×D₆₇	1	2	yes	—
390	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
391	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
392	C₂²×C₈	D₂₆₈	1	2	yes	—
393	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
394	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
395	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
396	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
397	#117	—	2	2	no	—
398	#117	—	2	2	no	—
399	#117	—	2	2	no	—
400	#117	C₂²×D₆₇	1	4	yes	—
401	C₂×C₄⋊C₄	—	2	2	no	—
402	C₂×C₂²⋊C₄	—	2	2	no	—
403	C₂×C₄²	—	2	2	no	—
404	#108	—	134	4	no	—
405	#108	—	134	4	no	—
406	#107	—	2	4	no	—
407	#106	—	2	4	no	—
408	#104	—	134	16	no	—
409	C₂⁴	—	134	1	no	—
410	C₂⁴	#11761627	1	1	yes	characteristic
411	#41	—	67	4	no	—
412	#41	—	67	4	no	—
413	#41	—	67	4	no	—
414	#41	—	67	4	no	—
415	#41	—	134	2	no	—
416	#41	—	134	2	no	—
417	#41	—	134	2	no	—
418	#41	—	134	2	no	—
419	#41	—	134	2	no	—
420	#41	—	134	2	no	—
421	#41	—	134	2	no	—
422	#41	—	134	2	no	—
423	#41	—	134	2	no	—
424	#41	—	134	2	no	—
425	#41	—	134	2	no	—
426	#41	—	134	2	no	—
427	C₂×Q₈	—	67	2	no	—
428	C₂×Q₈	—	67	2	no	—
429	C₂×Q₈	—	67	2	no	—
430	C₂×Q₈	—	134	1	no	—
431	C₂×Q₈	—	134	1	no	—
432	C₂×Q₈	—	134	1	no	—
433	C₂×D₄	—	67	2	no	—
434	C₂×D₄	—	67	2	no	—
435	C₂×D₄	—	67	2	no	—
436	C₂×D₄	—	134	2	no	—
437	C₂×D₄	—	134	2	no	—
438	C₂×D₄	—	134	2	no	—
439	C₂×D₄	—	134	2	no	—
440	C₂×D₄	—	134	2	no	—
441	C₂×D₄	—	134	2	no	—
442	C₂×D₄	—	134	1	no	—
443	C₂×D₄	—	134	1	no	—
444	C₂×D₄	—	134	1	no	—
445	C₂²×C₄	—	4	2	no	—
446	C₂²×C₄	—	134	1	no	—
447	C₂²×C₄	—	134	2	no	—
448	C₂²×C₄	—	134	2	no	—
449	C₂²×C₄	—	134	2	no	—
450	C₂²×C₄	—	134	2	no	—
451	C₂²×C₄	—	134	2	no	—
452	C₂²×C₄	—	134	2	no	—
453	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
454	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
455	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
456	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
457	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
458	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
459	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
460	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
461	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
462	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
463	C₂²×C₄	#11761627	1	1	yes	characteristic
464	C₂²×C₄	#11761629	1	1	yes	characteristic
465	C₂²×C₄	#11761629	1	1	yes	characteristic
466	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
467	M₁₆	—	268	2	no	—
468	M₁₆	—	268	2	no	—
469	M₁₆	—	67	4	no	—
470	M₁₆	—	134	4	no	—
471	M₁₆	—	134	4	no	—
472	M₁₆	—	134	2	no	—
473	M₁₆	—	134	2	no	—
474	M₁₆	—	134	2	no	—
475	M₁₆	—	134	8	no	—
476	M₁₆	—	134	8	no	—
477	C₂×C₈	—	268	2	no	—
478	C₂×C₈	—	268	2	no	—
479	C₂×C₈	—	134	4	no	—
480	C₂×C₈	—	134	2	no	—
481	C₂×C₈	—	134	2	no	—
482	C₂×C₈	—	134	2	no	—
483	C₂×C₈	—	134	8	no	—
484	C₂×C₈	—	134	8	no	—
485	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
486	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
487	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
488	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
489	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
490	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
491	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
492	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
493	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
494	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
495	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
496	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
497	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
498	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
499	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
500	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
501	C₄⋊C₄	—	2	4	no	—
502	C₄⋊C₄	—	4	2	no	—
503	C₂²⋊C₄	—	2	4	no	—
504	C₂²⋊C₄	—	4	2	no	—
505	C₄²	—	2	4	no	—
506	C₄²	—	4	2	no	—
507	C₂³	—	4	2	no	—
508	C₂³	—	268	4	no	—
509	C₂³	—	134	2	no	—
510	C₂³	—	134	2	no	—
511	C₂³	—	134	2	no	—
512	C₂³	#14524001	1	1	yes	characteristic
513	C₂³	#14524001	1	1	yes	characteristic
514	C₂³	#14524014	1	1	yes	characteristic
515	C₂³	#14524014	1	1	yes	characteristic
516	C₂³	#14524023	1	1	yes	characteristic
517	C₂³	#14524023	1	1	yes	characteristic
518	C₂³	C₂×#11761627	1	1	yes	characteristic
519	Q₈	—	67	4	no	—
520	Q₈	—	134	2	no	—
521	Q₈	—	134	2	no	—
522	Q₈	—	134	2	no	—
523	D₄	—	67	4	no	—
524	D₄	—	134	4	no	—
525	D₄	—	134	4	no	—
526	D₄	—	134	4	no	—
527	D₄	—	134	4	no	—
528	D₄	—	134	2	no	—
529	D₄	—	134	2	no	—
530	D₄	—	134	2	no	—
531	C₂×C₄	—	4	4	no	—
532	C₂×C₄	—	8	4	no	—
533	C₂×C₄	—	134	2	no	—
534	C₂×C₄	—	134	2	no	—
535	C₂×C₄	—	134	2	no	—
536	C₂×C₄	—	134	4	no	—
537	C₂×C₄	—	134	4	no	—
538	C₂×C₄	—	134	4	no	—
539	C₂×C₄	—	134	4	no	—
540	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
541	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
542	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
543	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
544	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
545	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
546	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
547	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
548	C₂×C₄	#14524001	1	1	yes	characteristic
549	C₂×C₄	#14524001	1	1	yes	characteristic
550	C₂×C₄	#14524014	1	1	yes	characteristic
551	C₂×C₄	#14524014	1	1	yes	characteristic
552	C₂×C₄	#14524018	1	1	yes	characteristic
553	C₂×C₄	#14524018	1	1	yes	characteristic
554	C₂×C₄	#14524018	1	1	yes	characteristic
555	C₂×C₄	#14524018	1	1	yes	characteristic
556	C₂×C₄	#14524019	1	1	yes	characteristic
557	C₂×C₄	#14524019	1	1	yes	characteristic
558	C₂×C₄	#14524019	1	1	yes	characteristic
559	C₂×C₄	#14524019	1	1	yes	characteristic
560	C₂×C₄	#14524023	1	1	yes	characteristic
561	C₂×C₄	#14524023	1	1	yes	characteristic
562	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
563	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
564	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
565	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
566	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
567	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
568	C₈	—	268	4	no	—
569	C₈	—	134	4	no	—
570	C₈	C₂×(C₂×D₆₇)⋊C₄	1	2	yes	—
571	C₈	C₂×(C₂×D₆₇)⋊C₄	1	2	yes	—
572	C₈	C₂×(C₂×D₆₇)⋊C₄	1	2	yes	—
573	C₈	C₂×(C₂×D₆₇)⋊C₄	1	2	yes	—
574	C₂²	—	2	2	no	—
575	C₂²	—	2	2	no	—
576	C₂²	—	2	2	no	—
577	C₂²	—	4	1	no	—
578	C₂²	—	4	1	no	—
579	C₂²	—	4	1	no	—
580	C₂²	—	4	1	no	—
581	C₂²	—	268	4	no	—
582	C₂²	—	268	4	no	—
583	C₂²	—	268	4	no	—
584	C₂²	—	134	4	no	—
585	C₂²	#16863253	1	1	yes	characteristic
586	C₂²	#16863253	1	1	yes	characteristic
587	C₂²	#16863253	1	1	yes	characteristic
588	C₂²	#16863253	1	1	yes	characteristic
589	C₂²	C₂×#14524001	1	1	yes	characteristic
590	C₂²	C₂×#14524014	1	1	yes	characteristic
591	C₂²	C₂×#14524023	1	1	yes	characteristic
592	C₄	—	8	4	no	—
593	C₄	—	134	4	no	—
594	C₄	#16863253	1	1	yes	characteristic
595	C₄	#16863253	1	1	yes	characteristic
596	C₄	#16863253	1	1	yes	characteristic
597	C₄	#16863253	1	1	yes	characteristic
598	C₄	C₂²×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
599	C₄	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
600	C₄	C₂×#14524018	1	1	yes	characteristic
601	C₄	C₂×#14524019	1	1	yes	characteristic
602	C₂	—	2	1	no	—
603	C₂	—	2	1	no	—
604	C₂	—	2	1	no	—
605	C₂	—	2	1	no	—
606	C₂	—	2	1	no	—
607	C₂	—	2	1	no	—
608	C₂	—	268	4	no	—
609	C₂	#22560260	1	2	yes	direct product factor
610	C₂	C₂×#16863253	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14524018
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#427×C₆₇	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂²×#11761628	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#427	D₆₇
hypocenter	C₆₇	C₂×#1814
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
2. C₂²×#14523979	C₂	semidirect
4. #22560260	C₂	direct (N × H)
5. C₂×#16863166	C₂	semidirect
6. C₂×#427×C₆₇	C₂	semidirect
12. C₂×#14524049	C₄	semidirect
13. C₂×#14524049	C₄	semidirect
14. C₂×#14524049	C₄	semidirect
23. C₂×#14523979	C₂²	semidirect
27. #16863166	C₂²	semidirect
31. #427×C₆₇	C₂²	semidirect
56. #14524049	C₂×C₄	semidirect
57. #14524049	C₂×C₄	semidirect
303. C₂×C₆₇	#1814	semidirect
308. C₂×#427	D₆₇	semidirect
312. C₆₇	C₂×#1814	semidirect
333. #427	C₂×D₆₇	semidirect
609. C₂	#22560260	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×#14524049	2
C₂²×#14523979	2
C₂×#16863414	2
#22560260	2
C₂×#16863166	2
C₂×#427×C₆₇	2
C₂×#1814	67

Subgroups — C2×#22560260 #30855490

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22560260 #30855490