Subgroups — #30877902

Order 17152.

720 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569158	C₂	1	1	yes	maximal, characteristic, complemented
2	#22568777	C₂	1	1	yes	maximal, characteristic
3	#22561187	C₂	1	1	yes	maximal, characteristic, complemented
4	#22560985	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560958	C₂	1	2	yes	maximal, complemented
6	#22560810	C₂	1	2	yes	maximal, complemented
7	#22560486	C₂	1	2	yes	maximal
8	C₄⋊C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#22560049	C₂	1	2	yes	maximal
10	C₄×#14523970	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#3076	C₂	1	1	yes	maximal, characteristic
12	C₂×#14524050	C₂²	1	2	yes	—
13	#16863527	—	2	1	no	—
14	#16863527	—	2	2	no	—
15	Q₈×C₆₇⋊C₈	—	2	1	no	—
16	C₂×#14523979	C₂²	1	2	yes	—
17	#16863423	—	2	2	no	—
18	#16863423	C₂²	1	2	yes	—
19	#16863423	C₂²	1	2	yes	—
20	#16863420	—	2	1	no	—
21	#16863420	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
23	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
24	C₂×#14523970	C₂²	1	1	yes	characteristic
25	C₂×#14523970	C₂²	1	1	yes	characteristic
26	#16863352	C₂²	1	2	yes	—
27	C₄×#11761612	C₂²	1	1	yes	characteristic
28	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
29	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
30	#16863161	C₂²	1	2	yes	—
31	#16863155	C₂²	1	1	yes	characteristic
32	#16863155	C₂²	1	1	yes	characteristic
33	#16863155	C₂²	1	2	yes	—
34	#16863155	C₂²	1	2	yes	—
35	#16863150	C₂²	1	2	yes	—
36	#533×C₆₇	C₂²	1	1	yes	characteristic
37	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
38	#479×C₆₇	C₄	1	2	yes	complemented
39	#478×C₆₇	C₄	1	2	yes	complemented
40	#477×C₆₇	C₄	1	1	yes	characteristic, complemented
41	#476×C₆₇	C₄	1	1	yes	characteristic, complemented
42	#475×C₆₇	C₄	1	1	yes	characteristic, complemented
43	#474×C₆₇	C₄	1	1	yes	characteristic, complemented
44	#427×C₆₇	C₂²	1	2	yes	—
45	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
46	#416×C₆₇	C₂²	1	2	yes	—
47	#14524050	—	2	4	no	—
48	#14524050	—	4	2	no	—
49	C₂×#11761612	—	2	2	no	—
50	C₂×#11761612	C₂³	1	1	yes	characteristic
51	C₂×#11761612	C₂³	1	2	yes	—
52	C₂×#11761612	C₂³	1	1	yes	characteristic
53	C₂²×C₆₇⋊C₈	—	2	2	no	—
54	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
55	C₂×C₄×Dic₆₇	—	2	1	no	—
56	C₂×C₄×Dic₆₇	—	2	1	no	—
57	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
58	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
59	#14523979	—	4	2	no	—
60	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
61	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
62	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
63	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
64	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
65	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
66	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
67	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
68	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
69	#14523970	—	2	2	no	—
70	#14523970	Q₈	1	4	yes	complemented
71	#14523954	—	2	2	no	—
72	#14523954	—	2	1	no	—
73	C₄×C₆₇⋊C₈	—	2	2	no	—
74	C₄×C₆₇⋊C₈	—	2	1	no	—
75	C₈×Dic₆₇	—	4	2	no	—
76	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
78	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
79	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
80	#125×C₆₇	C₂×C₄	1	2	yes	—
81	C₂²⋊Q₈×C₆₇	—	2	1	no	—
82	C₂²⋊Q₈×C₆₇	—	2	1	no	—
83	C₄×Q₈×C₆₇	—	2	1	no	—
84	C₄×Q₈×C₆₇	—	2	1	no	—
85	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
86	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
87	C₄×D₄×C₆₇	—	2	1	no	—
88	C₄×D₄×C₆₇	—	2	1	no	—
89	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
90	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
91	#117×C₆₇	—	2	1	no	—
92	#117×C₆₇	C₂³	1	1	yes	characteristic
93	#117×C₆₇	C₂³	1	2	yes	—
94	C₂×C₄²×C₆₇	—	2	1	no	—
95	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
96	#107×C₆₇	C₂×C₄	1	2	yes	—
97	#107×C₆₇	C₂×C₄	1	2	yes	—
98	#106×C₆₇	C₂×C₄	1	2	yes	—
99	#106×C₆₇	C₂×C₄	1	2	yes	—
100	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
101	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
102	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
103	#103×C₆₇	C₂×C₄	1	2	yes	—
104	#103×C₆₇	C₂×C₄	1	2	yes	—
105	#102×C₆₇	C₂×C₄	1	2	yes	—
106	#102×C₆₇	C₂×C₄	1	2	yes	—
107	C₂²×Dic₆₇	—	2	1	no	—
108	C₂²×Dic₆₇	—	2	1	no	—
109	#11761612	—	2	1	no	—
110	#11761612	—	2	1	no	—
111	#11761612	—	2	2	no	—
112	#11761612	—	2	1	no	—
113	#11761612	—	2	2	no	—
114	#11761612	—	4	4	no	—
115	#11761612	C₂×Q₈	1	2	yes	—
116	C₂×C₆₇⋊C₈	—	2	1	no	—
117	C₂×C₆₇⋊C₈	—	2	2	no	—
118	C₂×C₆₇⋊C₈	—	2	2	no	—
119	C₂×C₆₇⋊C₈	—	2	1	no	—
120	C₂×C₆₇⋊C₈	—	2	2	no	—
121	C₂×C₆₇⋊C₈	—	4	4	no	—
122	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
123	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
124	C₄×Dic₆₇	—	2	4	no	—
125	C₄×Dic₆₇	—	2	1	no	—
126	C₄×Dic₆₇	—	2	1	no	—
127	C₄×Dic₆₇	—	2	1	no	—
128	C₄×Dic₆₇	—	4	2	no	—
129	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
130	#41×C₆₇	—	2	1	no	—
131	#41×C₆₇	—	2	1	no	—
132	#41×C₆₇	C₂×Q₈	1	2	yes	—
133	#41×C₆₇	#41	1	2	yes	—
134	C₂×Q₈×C₆₇	—	2	1	no	—
135	C₂×Q₈×C₆₇	—	2	1	no	—
136	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
137	C₂×D₄×C₆₇	—	2	1	no	—
138	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
139	C₂²×C₄×C₆₇	—	2	1	no	—
140	C₂²×C₄×C₆₇	—	2	1	no	—
141	C₂²×C₄×C₆₇	—	2	1	no	—
142	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
143	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
144	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
145	C₂×C₈×C₆₇	—	2	2	no	—
146	C₂×C₈×C₆₇	—	2	2	no	—
147	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
148	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
149	C₄⋊C₄×C₆₇	—	2	1	no	—
150	C₄⋊C₄×C₆₇	—	2	1	no	—
151	C₄⋊C₄×C₆₇	—	2	2	no	—
152	C₄⋊C₄×C₆₇	—	2	2	no	—
153	C₄⋊C₄×C₆₇	—	2	1	no	—
154	C₄⋊C₄×C₆₇	—	2	1	no	—
155	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
156	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
157	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
158	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
159	C₂²⋊C₄×C₆₇	—	2	1	no	—
160	C₂²⋊C₄×C₆₇	—	2	2	no	—
161	C₂²⋊C₄×C₆₇	—	2	1	no	—
162	C₂²⋊C₄×C₆₇	—	2	1	no	—
163	C₄²×C₆₇	—	2	1	no	—
164	C₄²×C₆₇	—	2	1	no	—
165	C₄²×C₆₇	—	2	1	no	—
166	C₄²×C₆₇	—	2	2	no	—
167	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
168	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
169	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
170	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
171	C₂³×C₆₇	—	2	1	no	—
172	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
173	Q₈×C₆₇	—	2	1	no	—
174	Q₈×C₆₇	—	4	2	no	—
175	Q₈×C₆₇	—	8	1	no	—
176	Q₈×C₆₇	C₄×Q₈	1	2	yes	—
177	D₄×C₆₇	—	2	2	no	—
178	D₄×C₆₇	—	2	2	no	—
179	D₄×C₆₇	—	2	1	no	—
180	D₄×C₆₇	C₄×Q₈	1	2	yes	—
181	C₂×C₄×C₆₇	—	2	1	no	—
182	C₂×C₄×C₆₇	—	2	2	no	—
183	C₂×C₄×C₆₇	—	2	2	no	—
184	C₂×C₄×C₆₇	—	2	2	no	—
185	C₂×C₄×C₆₇	—	2	1	no	—
186	C₂×C₄×C₆₇	—	2	1	no	—
187	C₂×C₄×C₆₇	—	2	2	no	—
188	C₂×C₄×C₆₇	—	2	2	no	—
189	C₂×C₄×C₆₇	—	4	1	no	—
190	C₂×C₄×C₆₇	—	4	2	no	—
191	C₂×C₄×C₆₇	—	8	1	no	—
192	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
193	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
194	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
195	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
196	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
197	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
198	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
199	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
200	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
201	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
202	C₂×Dic₆₇	—	2	2	no	—
203	C₂×Dic₆₇	—	2	2	no	—
204	C₂×Dic₆₇	—	2	2	no	—
205	C₂×Dic₆₇	—	2	2	no	—
206	C₂×Dic₆₇	—	4	1	no	—
207	C₂×Dic₆₇	—	4	1	no	—
208	C₈×C₆₇	—	2	2	no	—
209	C₈×C₆₇	—	2	2	no	—
210	C₆₇⋊C₈	—	2	2	no	—
211	C₆₇⋊C₈	—	2	2	no	—
212	C₆₇⋊C₈	—	2	2	no	—
213	C₆₇⋊C₈	—	4	2	no	—
214	C₆₇⋊C₈	—	4	1	no	—
215	C₂²×C₆₇	—	2	1	no	—
216	C₂²×C₆₇	—	2	1	no	—
217	C₂²×C₆₇	—	2	2	no	—
218	C₂²×C₆₇	—	4	1	no	—
219	C₂²×C₆₇	—	4	1	no	—
220	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
221	C₂²×C₆₇	#552	1	1	yes	characteristic
222	C₂²×C₆₇	#554	1	1	yes	characteristic
223	C₄×C₆₇	—	2	2	no	—
224	C₄×C₆₇	—	2	1	no	—
225	C₄×C₆₇	—	2	1	no	—
226	C₄×C₆₇	—	2	1	no	—
227	C₄×C₆₇	—	4	1	no	—
228	C₄×C₆₇	—	4	2	no	—
229	C₄×C₆₇	—	4	1	no	—
230	C₄×C₆₇	—	4	2	no	—
231	C₄×C₆₇	#443	1	2	yes	—
232	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
233	C₄×C₆₇	#520	1	1	yes	characteristic
234	C₄×C₆₇	#546	1	1	yes	characteristic
235	C₄×C₆₇	D₄×Q₈	1	1	yes	characteristic
236	Dic₆₇	—	2	4	no	—
237	Dic₆₇	—	4	2	no	—
238	#22982	—	67	1	no	maximal, Sylow
239	C₂×C₆₇	—	2	1	no	—
240	C₂×C₆₇	—	2	1	no	—
241	C₂×C₆₇	—	4	2	no	—
242	C₂×C₆₇	#2360	1	1	yes	characteristic
243	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
244	C₂×C₆₇	#2956	1	1	yes	characteristic
245	#3076	D₆₇	1	1	yes	characteristic
246	#2962	—	67	1	no	—
247	#2906	—	67	1	no	—
248	#1994	—	67	1	no	—
249	#1950	—	67	1	no	—
250	#1945	—	67	2	no	—
251	#1922	—	67	2	no	—
252	#1866	—	67	2	no	—
253	C₄×C₄⋊C₈	—	67	1	no	—
254	#1764	—	67	2	no	—
255	C₄×#104	—	67	1	no	—
256	C₆₇	#22982	1	1	yes	characteristic, complemented, Sylow
257	C₂×#131	—	67	2	no	—
258	#533	C₂×D₆₇	1	1	yes	characteristic
259	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
260	#479	Dic₆₇	1	2	yes	complemented
261	#478	Dic₆₇	1	2	yes	complemented
262	#477	Dic₆₇	1	1	yes	characteristic, complemented
263	#476	Dic₆₇	1	1	yes	characteristic, complemented
264	#475	Dic₆₇	1	1	yes	characteristic, complemented
265	#474	Dic₆₇	1	1	yes	characteristic, complemented
266	#446	—	134	2	no	—
267	#446	—	134	1	no	—
268	C₈×Q₈	—	134	1	no	—
269	C₂×#108	—	67	2	no	—
270	#427	C₂×D₆₇	1	2	yes	—
271	#424	—	67	2	no	—
272	#424	—	67	2	no	—
273	#424	—	134	2	no	—
274	#423	—	67	1	no	—
275	#423	—	134	1	no	—
276	C₂×C₄⋊C₈	—	67	1	no	—
277	C₂×C₄⋊C₈	—	134	1	no	—
278	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
279	C₂×#104	—	67	1	no	—
280	C₂×#104	—	67	1	no	—
281	#416	C₂×D₆₇	1	2	yes	—
282	#405	—	67	2	no	—
283	C₄×M₁₆	—	67	1	no	—
284	C₂×C₄×C₈	—	67	2	no	—
285	C₄³	—	67	1	no	—
286	#341	—	67	2	no	—
287	#339	—	67	2	no	—
288	#339	—	67	1	no	—
289	#339	—	67	1	no	—
290	#339	—	67	2	no	—
291	#336	—	67	2	no	—
292	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
293	#131	—	268	2	no	—
294	#131	—	134	4	no	—
295	C₂×M₁₆	—	67	1	no	—
296	C₂×M₁₆	—	67	2	no	—
297	C₂×M₁₆	—	67	1	no	—
298	C₂×M₁₆	—	134	2	no	—
299	C₂²×C₈	—	67	2	no	—
300	C₂²×C₈	—	134	2	no	—
301	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
302	C₄⋊Q₈	C₂×Dic₆₇	1	1	yes	characteristic
303	C₄⋊Q₈	C₂×Dic₆₇	1	1	yes	characteristic
304	#125	C₂×Dic₆₇	1	2	yes	—
305	C₂²⋊Q₈	—	2	1	no	—
306	C₂²⋊Q₈	—	2	1	no	—
307	C₄×Q₈	—	2	1	no	—
308	C₄×Q₈	—	2	1	no	—
309	C₄×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
310	C₄×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
311	C₄×D₄	—	2	1	no	—
312	C₄×D₄	—	2	1	no	—
313	C₄×D₄	C₂×Dic₆₇	1	1	yes	characteristic
314	C₄×D₄	C₂×Dic₆₇	1	1	yes	characteristic
315	#117	—	2	1	no	—
316	#117	C₂²×D₆₇	1	1	yes	characteristic
317	#117	C₂²×D₆₇	1	2	yes	—
318	C₂×C₄²	—	2	1	no	—
319	C₂×C₄²	—	67	1	no	—
320	C₂×C₄²	—	67	1	no	—
321	C₂×C₄²	—	134	1	no	—
322	C₂×C₄²	—	134	1	no	—
323	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
324	#108	—	268	2	no	—
325	#107	C₂×Dic₆₇	1	2	yes	—
326	#107	C₂×Dic₆₇	1	2	yes	—
327	#106	C₂×Dic₆₇	1	2	yes	—
328	#106	C₂×Dic₆₇	1	2	yes	—
329	C₄⋊C₈	—	67	1	no	—
330	C₄⋊C₈	—	67	1	no	—
331	C₄⋊C₈	—	67	2	no	—
332	C₄⋊C₈	—	134	2	no	—
333	C₄⋊C₈	—	134	2	no	—
334	C₄⋊C₈	—	134	2	no	—
335	C₄⋊C₈	—	134	2	no	—
336	C₄⋊C₈	—	134	1	no	—
337	C₄⋊C₈	—	134	1	no	—
338	C₄⋊C₈	C₂×Dic₆₇	1	1	yes	characteristic
339	C₄⋊C₈	C₂×Dic₆₇	1	1	yes	characteristic
340	C₄⋊C₈	C₂×Dic₆₇	1	2	yes	—
341	#104	—	67	4	no	—
342	#104	—	134	2	no	—
343	#103	C₂×Dic₆₇	1	2	yes	—
344	#103	C₂×Dic₆₇	1	2	yes	—
345	#102	C₂×Dic₆₇	1	2	yes	—
346	#102	C₂×Dic₆₇	1	2	yes	—
347	#97	—	134	2	no	—
348	#97	—	134	1	no	—
349	C₄×C₈	—	268	2	no	—
350	C₄×C₈	—	134	2	no	—
351	C₄×C₈	—	134	1	no	—
352	#41	—	2	1	no	—
353	#41	—	2	1	no	—
354	#41	Q₈×D₆₇	1	2	yes	—
355	#41	#11761629	1	2	yes	—
356	C₂×Q₈	—	2	1	no	—
357	C₂×Q₈	—	2	1	no	—
358	C₂×Q₈	C₂²×Dic₆₇	1	1	yes	characteristic
359	C₂×D₄	—	2	1	no	—
360	C₂×D₄	C₂²×Dic₆₇	1	1	yes	characteristic
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	1	no	—
364	C₂²×C₄	—	134	1	no	—
365	C₂²×C₄	—	134	1	no	—
366	C₂²×C₄	#11761627	1	1	yes	characteristic
367	C₂²×C₄	#11761627	1	1	yes	characteristic
368	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
369	M₁₆	—	268	4	no	—
370	M₁₆	—	67	2	no	—
371	M₁₆	—	134	1	no	—
372	M₁₆	—	134	2	no	—
373	M₁₆	—	134	1	no	—
374	M₁₆	—	134	2	no	—
375	M₁₆	—	134	1	no	—
376	C₂×C₈	—	2	2	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	268	4	no	—
379	C₂×C₈	—	67	2	no	—
380	C₂×C₈	—	67	2	no	—
381	C₂×C₈	—	134	1	no	—
382	C₂×C₈	—	134	2	no	—
383	C₂×C₈	—	134	1	no	—
384	C₂×C₈	—	134	2	no	—
385	C₂×C₈	—	134	2	no	—
386	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
387	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
388	C₄⋊C₄	—	2	1	no	—
389	C₄⋊C₄	—	2	1	no	—
390	C₄⋊C₄	—	2	2	no	—
391	C₄⋊C₄	—	2	2	no	—
392	C₄⋊C₄	—	2	1	no	—
393	C₄⋊C₄	—	2	1	no	—
394	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
395	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
396	C₄⋊C₄	C₂²×Dic₆₇	1	1	yes	characteristic
397	C₄⋊C₄	C₂²×Dic₆₇	1	1	yes	characteristic
398	C₂²⋊C₄	—	2	1	no	—
399	C₂²⋊C₄	—	2	2	no	—
400	C₂²⋊C₄	—	2	1	no	—
401	C₂²⋊C₄	—	2	1	no	—
402	C₄²	—	2	1	no	—
403	C₄²	—	2	1	no	—
404	C₄²	—	2	2	no	—
405	C₄²	—	2	1	no	—
406	C₄²	—	268	2	no	—
407	C₄²	—	67	2	no	—
408	C₄²	—	134	4	no	—
409	C₄²	—	134	1	no	—
410	C₄²	—	134	1	no	—
411	C₄²	—	134	1	no	—
412	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
413	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
414	C₄²	D₄×D₆₇	1	1	yes	characteristic
415	C₄²	D₄×D₆₇	1	1	yes	characteristic
416	C₂³	—	2	1	no	—
417	C₂³	C₂×#11761627	1	1	yes	characteristic
418	Q₈	—	2	1	no	—
419	Q₈	—	4	2	no	—
420	Q₈	—	8	1	no	—
421	Q₈	Q₈×Dic₆₇	1	2	yes	—
422	D₄	—	2	2	no	—
423	D₄	—	2	2	no	—
424	D₄	—	2	1	no	—
425	D₄	Q₈×Dic₆₇	1	2	yes	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	4	1	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	8	1	no	—
437	C₂×C₄	—	268	1	no	—
438	C₂×C₄	—	268	1	no	—
439	C₂×C₄	—	134	2	no	—
440	C₂×C₄	—	134	2	no	—
441	C₂×C₄	—	134	2	no	—
442	C₂×C₄	—	134	2	no	—
443	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
444	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
445	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
446	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
447	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
448	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
449	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
450	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
451	C₂×C₄	#14524089	1	1	yes	characteristic
452	C₂×C₄	#14524090	1	1	yes	characteristic
453	C₈	—	2	2	no	—
454	C₈	—	2	2	no	—
455	C₈	—	268	2	no	—
456	C₈	—	268	1	no	—
457	C₈	—	134	2	no	—
458	C₈	—	134	2	no	—
459	C₈	—	134	2	no	—
460	C₂²	—	2	1	no	—
461	C₂²	—	2	1	no	—
462	C₂²	—	2	2	no	—
463	C₂²	—	4	1	no	—
464	C₂²	—	4	1	no	—
465	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
466	C₂²	#16864088	1	1	yes	characteristic
467	C₂²	#16864104	1	1	yes	characteristic
468	C₄	—	2	1	no	—
469	C₄	—	2	2	no	—
470	C₄	—	2	1	no	—
471	C₄	—	2	1	no	—
472	C₄	—	4	1	no	—
473	C₄	—	4	2	no	—
474	C₄	—	4	1	no	—
475	C₄	—	4	2	no	—
476	C₄	—	268	2	no	—
477	C₄	—	134	4	no	—
478	C₄	#16863513	1	2	yes	—
479	C₄	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
480	C₄	#16863879	1	1	yes	characteristic
481	C₄	#16864037	1	1	yes	characteristic
482	C₄	#16864059	1	1	yes	characteristic
483	C₂	—	2	1	no	—
484	C₂	—	2	1	no	—
485	C₂	—	4	2	no	—
486	C₂	#22563218	1	1	yes	characteristic
487	C₂	C₂×#16863513	1	1	yes	characteristic
488	C₂	#22569119	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#3076	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#41×C₆₇	C₂³
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#3076	D₆₇
hypocenter	C₆₇	#22982
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3076

Complements

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

N	H ≅ G/N	type
1. #22569158	C₂	semidirect
3. #22561187	C₂	semidirect
4. #22560985	C₂	semidirect
5. #22560958	C₂	semidirect
6. #22560810	C₂	semidirect
8. C₄⋊C₈×Dic₆₇	C₂	semidirect
38. #479×C₆₇	C₄	semidirect
39. #478×C₆₇	C₄	semidirect
40. #477×C₆₇	C₄	semidirect
41. #476×C₆₇	C₄	semidirect
42. #475×C₆₇	C₄	semidirect
43. #474×C₆₇	C₄	semidirect
70. #14523970	Q₈	semidirect
256. C₆₇	#22982	semidirect
260. #479	Dic₆₇	semidirect
261. #478	Dic₆₇	semidirect
262. #477	Dic₆₇	semidirect
263. #476	Dic₆₇	semidirect
264. #475	Dic₆₇	semidirect
265. #474	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22569158	2
#22568777	2
#22561187	2
#22560985	2
#22560958	2
#22560810	2
#22560486	2
C₄⋊C₈×Dic₆₇	2
#22560049	2
C₄×#14523970	2
C₆₇×#3076	2
#22982	67