Subgroups — #30875806

Order 17152.

636 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562686	C₂	1	1	yes	maximal, characteristic
2	#22561145	C₂	1	1	yes	maximal, characteristic
3	#22561145	C₂	1	1	yes	maximal, characteristic
4	#22560770	C₂	1	2	yes	maximal
5	#22560763	C₂	1	2	yes	maximal
6	#22560353	C₂	1	2	yes	maximal
7	#22560326	C₂	1	1	yes	maximal, characteristic
8	#22560076	C₂	1	1	yes	maximal, characteristic
9	C₈×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic
10	#22559959	C₂	1	2	yes	maximal
11	C₄×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×#14523953	C₂²	1	2	yes	—
16	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
17	#16863302	C₄	1	1	yes	characteristic
18	#16863302	C₄	1	1	yes	characteristic
19	#16863294	C₄	1	1	yes	characteristic
20	#16863294	C₄	1	1	yes	characteristic
21	#16863280	C₄	1	2	yes	—
22	#16863280	C₄	1	2	yes	—
23	C₄×Dic₆₇⋊C₄	C₂²	1	2	yes	—
24	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
25	#16863242	C₂²	1	2	yes	—
26	#16863151	C₂²	1	2	yes	—
27	#16863151	C₂²	1	2	yes	—
28	#16863150	C₂²	1	2	yes	—
29	#16863150	C₂²	1	2	yes	—
30	#16863150	C₂²	1	2	yes	—
31	#16863150	C₂²	1	2	yes	—
32	#16863150	C₂²	1	2	yes	—
33	#16863150	C₂²	1	2	yes	—
34	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
38	C₄³×C₆₇	C₂²	1	1	yes	characteristic
39	#336×C₆₇	C₂²	1	1	yes	characteristic
40	#336×C₆₇	C₂²	1	1	yes	characteristic
41	C₂²×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
43	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
44	C₄×Dic₁₃₄	—	2	2	no	—
45	C₄×Dic₁₃₄	—	2	2	no	—
46	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂×Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
50	C₂×Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
51	C₂×Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
52	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
53	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
54	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
55	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
56	#14523953	—	2	4	no	—
57	C₈×Dic₆₇	—	2	4	no	—
58	C₆₇⋊#95	C₂×C₄	1	2	yes	—
59	C₆₇⋊#95	C₂×C₄	1	2	yes	—
60	C₆₇⋊#95	C₂×C₄	1	2	yes	—
61	C₆₇⋊#95	C₂×C₄	1	2	yes	—
62	C₆₇⋊#95	C₂×C₄	1	2	yes	—
63	C₆₇⋊#95	C₂×C₄	1	2	yes	—
64	C₆₇⋊#95	C₂×C₄	1	2	yes	—
65	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₆₇	—	2	2	no	—
77	C₄⋊C₈×C₆₇	—	2	2	no	—
78	C₄×C₈×C₆₇	—	2	2	no	—
79	C₄×C₈×C₆₇	—	2	2	no	—
80	C₂×Dic₁₃₄	—	2	2	no	—
81	C₂×Dic₁₃₄	—	2	2	no	—
82	C₂×Dic₁₃₄	—	2	2	no	—
83	C₂²×Dic₆₇	C₂²×C₄	1	2	yes	—
84	C₂²×Dic₆₇	C₂²×C₄	1	2	yes	—
85	C₂²×Dic₆₇	C₂²×C₄	1	2	yes	—
86	C₂²×Dic₆₇	C₂²×C₄	1	2	yes	—
87	C₂×C₆₇⋊C₈	—	2	2	no	—
88	C₂×C₆₇⋊C₈	—	2	2	no	—
89	C₂×C₆₇⋊C₈	—	2	2	no	—
90	C₂×C₆₇⋊C₈	—	2	2	no	—
91	C₂×C₆₇⋊C₈	—	2	2	no	—
92	C₂×C₆₇⋊C₈	—	2	2	no	—
93	C₄⋊Dic₆₇	—	2	2	no	—
94	C₄⋊Dic₆₇	—	2	2	no	—
95	C₄⋊Dic₆₇	—	2	2	no	—
96	Dic₆₇⋊C₄	—	2	4	no	—
97	Dic₆₇⋊C₄	—	2	4	no	—
98	Dic₆₇⋊C₄	—	4	2	no	—
99	C₄×Dic₆₇	—	2	2	no	—
100	C₄×Dic₆₇	—	2	2	no	—
101	C₄×Dic₆₇	—	2	2	no	—
102	C₄×Dic₆₇	—	2	2	no	—
103	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
110	C₂×C₈×C₆₇	—	2	1	no	—
111	C₂×C₈×C₆₇	—	2	1	no	—
112	C₂×C₈×C₆₇	—	2	1	no	—
113	C₂×C₈×C₆₇	—	2	1	no	—
114	C₂×C₈×C₆₇	—	2	1	no	—
115	C₂×C₈×C₆₇	—	2	1	no	—
116	C₂×C₈×C₆₇	—	2	1	no	—
117	C₂×C₈×C₆₇	—	2	1	no	—
118	C₂×C₈×C₆₇	—	2	1	no	—
119	C₂×C₈×C₆₇	—	2	1	no	—
120	C₂×C₈×C₆₇	#41	1	2	yes	—
121	C₂×C₈×C₆₇	#41	1	2	yes	—
122	C₄²×C₆₇	—	2	2	no	—
123	C₄²×C₆₇	—	2	2	no	—
124	C₄²×C₆₇	—	2	2	no	—
125	C₄²×C₆₇	—	2	1	no	—
126	C₄²×C₆₇	—	2	1	no	—
127	C₄²×C₆₇	—	2	1	no	—
128	C₄²×C₆₇	—	2	1	no	—
129	C₄²×C₆₇	—	4	1	no	—
130	C₄²×C₆₇	#41	1	2	yes	—
131	C₄²×C₆₇	#41	1	2	yes	—
132	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₆₇	—	2	1	no	—
134	C₂×C₄×C₆₇	—	2	1	no	—
135	C₂×C₄×C₆₇	—	2	1	no	—
136	C₂×C₄×C₆₇	—	2	1	no	—
137	C₂×C₄×C₆₇	—	2	1	no	—
138	C₂×C₄×C₆₇	—	2	1	no	—
139	C₂×C₄×C₆₇	—	2	1	no	—
140	C₂×C₄×C₆₇	—	2	1	no	—
141	C₂×C₄×C₆₇	—	2	1	no	—
142	C₂×C₄×C₆₇	—	2	1	no	—
143	C₂×C₄×C₆₇	—	2	1	no	—
144	C₂×C₄×C₆₇	—	2	1	no	—
145	C₂×C₄×C₆₇	—	2	1	no	—
146	C₂×C₄×C₆₇	#117	1	2	yes	—
147	C₂×C₄×C₆₇	#117	1	2	yes	—
148	C₂×C₄×C₆₇	#131	1	2	yes	—
149	C₂×C₄×C₆₇	#131	1	2	yes	—
150	C₂×C₄×C₆₇	#131	1	2	yes	—
151	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
154	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
155	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
156	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
157	C₂×Dic₆₇	—	2	2	no	—
158	C₂×Dic₆₇	—	2	2	no	—
159	C₂×Dic₆₇	—	2	2	no	—
160	C₂×Dic₆₇	—	2	2	no	—
161	C₂×Dic₆₇	—	2	2	no	—
162	C₂×Dic₆₇	—	2	2	no	—
163	C₂×Dic₆₇	—	2	2	no	—
164	C₂×Dic₆₇	—	2	2	no	—
165	C₂×Dic₆₇	—	2	2	no	—
166	C₂×Dic₆₇	—	2	2	no	—
167	C₂×Dic₆₇	—	2	2	no	—
168	C₂×Dic₆₇	—	2	2	no	—
169	Dic₁₃₄	—	4	4	no	—
170	C₈×C₆₇	—	2	2	no	—
171	C₈×C₆₇	—	2	2	no	—
172	C₈×C₆₇	—	4	1	no	—
173	C₈×C₆₇	—	4	1	no	—
174	C₆₇⋊C₈	—	4	2	no	—
175	C₆₇⋊C₈	—	4	2	no	—
176	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
177	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
178	C₂²×C₆₇	#519	1	1	yes	characteristic
179	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
180	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
181	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
182	C₂²×C₆₇	#568	1	1	yes	characteristic
183	C₄×C₆₇	—	2	2	no	—
184	C₄×C₆₇	—	2	2	no	—
185	C₄×C₆₇	—	4	1	no	—
186	C₄×C₆₇	—	4	1	no	—
187	C₄×C₆₇	—	4	1	no	—
188	C₄×C₆₇	#433	1	4	yes	—
189	C₄×C₆₇	#552	1	1	yes	characteristic
190	C₄×C₆₇	#552	1	1	yes	characteristic
191	C₄×C₆₇	#556	1	1	yes	characteristic
192	C₄×C₆₇	#556	1	1	yes	characteristic
193	Dic₆₇	—	4	2	no	—
194	Dic₆₇	—	4	2	no	—
195	Dic₆₇	—	4	2	no	—
196	Dic₆₇	—	4	2	no	—
197	#22829	—	67	1	no	maximal, Sylow
198	C₂×C₆₇	#2356	1	1	yes	characteristic
199	C₂×C₆₇	C₂×#433	1	1	yes	characteristic
200	C₂×C₆₇	#2925	1	1	yes	characteristic
201	C₂×C₆₇	#2966	1	1	yes	characteristic
202	C₂×C₆₇	#2969	1	1	yes	characteristic
203	C₂×C₆₇	#2985	1	1	yes	characteristic
204	C₂×C₆₇	#2986	1	1	yes	characteristic
205	#2295	—	67	1	no	—
206	#1984	—	67	1	no	—
207	#1984	—	67	1	no	—
208	#1917	—	67	2	no	—
209	#1916	—	67	2	no	—
210	#1840	—	67	2	no	—
211	#1831	—	67	1	no	—
212	#1770	—	67	1	no	—
213	C₈×C₄⋊C₄	—	67	1	no	—
214	C₄×C₄⋊C₈	D₆₇	1	1	yes	characteristic
215	#1746	—	67	2	no	—
216	C₆₇	#22829	1	1	yes	characteristic, complemented, Sylow
217	C₂×C₄×Q₈	—	67	1	no	—
218	C₂×C₄⋊C₈	—	67	1	no	—
219	C₂×C₄⋊C₈	—	67	1	no	—
220	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
221	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
222	C₂×#97	—	67	2	no	—
223	C₂×C₄×C₈	—	67	2	no	—
224	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
225	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
226	#391	—	67	1	no	—
227	#391	—	67	1	no	—
228	#389	—	67	1	no	—
229	#389	—	67	1	no	—
230	#387	—	67	2	no	—
231	#387	—	67	2	no	—
232	C₄×C₄⋊C₄	—	67	1	no	—
233	C₄×C₄⋊C₄	—	67	2	no	—
234	#376	—	67	2	no	—
235	C₄³	C₂×D₆₇	1	1	yes	characteristic
236	#336	—	67	2	no	—
237	#336	—	67	2	no	—
238	#336	—	67	2	no	—
239	#336	—	67	2	no	—
240	#336	—	67	2	no	—
241	#336	—	67	2	no	—
242	#336	—	67	2	no	—
243	#336	—	67	2	no	—
244	#336	C₂×D₆₇	1	1	yes	characteristic
245	#336	C₂×D₆₇	1	1	yes	characteristic
246	C₂²×Q₈	—	67	1	no	—
247	C₂²×C₈	—	67	2	no	—
248	C₂²×C₈	—	67	2	no	—
249	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
250	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
251	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
252	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
253	C₄×Q₈	—	134	2	no	—
254	C₄×Q₈	—	134	2	no	—
255	C₂×C₄⋊C₄	—	67	2	no	—
256	C₂×C₄⋊C₄	—	67	2	no	—
257	C₂×C₄⋊C₄	—	67	1	no	—
258	C₂×C₄⋊C₄	—	67	2	no	—
259	C₂×C₄⋊C₄	—	67	1	no	—
260	C₂×C₄⋊C₄	—	67	1	no	—
261	C₂×C₄²	—	67	2	no	—
262	C₂×C₄²	—	67	2	no	—
263	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
264	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
265	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
266	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
267	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
268	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
269	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
270	C₄⋊C₈	—	2	2	no	—
271	C₄⋊C₈	—	2	2	no	—
272	C₄⋊C₈	—	134	2	no	—
273	C₄⋊C₈	—	134	2	no	—
274	#97	—	134	4	no	—
275	C₄×C₈	—	2	2	no	—
276	C₄×C₈	—	2	2	no	—
277	C₄×C₈	—	134	4	no	—
278	#95	—	67	2	no	—
279	#95	—	67	2	no	—
280	#95	—	67	2	no	—
281	#95	—	67	2	no	—
282	#95	—	67	2	no	—
283	#95	—	67	2	no	—
284	#95	—	67	2	no	—
285	C₂×Q₈	—	134	2	no	—
286	C₂×Q₈	—	134	2	no	—
287	C₂×Q₈	—	134	2	no	—
288	C₂²×C₄	—	67	2	no	—
289	C₂²×C₄	—	67	2	no	—
290	C₂²×C₄	—	67	2	no	—
291	C₂²×C₄	—	67	2	no	—
292	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
293	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
294	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
295	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
296	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
297	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
298	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
299	C₂×C₈	—	2	1	no	—
300	C₂×C₈	—	2	1	no	—
301	C₂×C₈	—	2	1	no	—
302	C₂×C₈	—	2	1	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	2	1	no	—
308	C₂×C₈	—	2	1	no	—
309	C₂×C₈	—	134	2	no	—
310	C₂×C₈	—	134	2	no	—
311	C₂×C₈	—	134	2	no	—
312	C₂×C₈	—	134	2	no	—
313	C₂×C₈	—	134	2	no	—
314	C₂×C₈	—	134	2	no	—
315	C₂×C₈	#11761628	1	2	yes	—
316	C₂×C₈	#11761628	1	2	yes	—
317	C₄⋊C₄	—	268	2	no	—
318	C₄⋊C₄	—	134	4	no	—
319	C₄⋊C₄	—	134	2	no	—
320	C₄⋊C₄	—	134	4	no	—
321	C₄⋊C₄	—	134	2	no	—
322	C₄⋊C₄	—	134	2	no	—
323	C₄²	—	2	2	no	—
324	C₄²	—	2	2	no	—
325	C₄²	—	2	2	no	—
326	C₄²	—	2	1	no	—
327	C₄²	—	2	1	no	—
328	C₄²	—	2	1	no	—
329	C₄²	—	2	1	no	—
330	C₄²	—	4	1	no	—
331	C₄²	—	134	2	no	—
332	C₄²	—	134	2	no	—
333	C₄²	—	134	2	no	—
334	C₄²	—	134	2	no	—
335	C₄²	#11761627	1	2	yes	—
336	C₄²	#11761629	1	2	yes	—
337	C₂³	C₂²×C₄×D₆₇	1	1	yes	characteristic
338	Q₈	—	268	4	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	134	2	no	—
353	C₂×C₄	—	134	2	no	—
354	C₂×C₄	—	134	2	no	—
355	C₂×C₄	—	134	2	no	—
356	C₂×C₄	—	134	2	no	—
357	C₂×C₄	—	134	2	no	—
358	C₂×C₄	—	134	2	no	—
359	C₂×C₄	—	134	2	no	—
360	C₂×C₄	—	134	2	no	—
361	C₂×C₄	—	134	2	no	—
362	C₂×C₄	—	134	2	no	—
363	C₂×C₄	—	134	2	no	—
364	C₂×C₄	#14524001	1	2	yes	—
365	C₂×C₄	#14524001	1	2	yes	—
366	C₂×C₄	#14524049	1	2	yes	—
367	C₂×C₄	#14524049	1	2	yes	—
368	C₂×C₄	#14524051	1	2	yes	—
369	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
370	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
371	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
372	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
373	C₂×C₄	#14524090	1	1	yes	characteristic
374	C₂×C₄	#14524090	1	1	yes	characteristic
375	C₈	—	2	2	no	—
376	C₈	—	2	2	no	—
377	C₈	—	4	1	no	—
378	C₈	—	4	1	no	—
379	C₈	—	268	2	no	—
380	C₈	—	268	2	no	—
381	C₂²	C₂×#14524001	1	1	yes	characteristic
382	C₂²	C₄×#11761628	1	1	yes	characteristic
383	C₂²	#16863873	1	1	yes	characteristic
384	C₂²	C₂×#14524049	1	1	yes	characteristic
385	C₂²	C₂×#14524049	1	1	yes	characteristic
386	C₂²	C₂×#14524051	1	1	yes	characteristic
387	C₂²	#16864162	1	1	yes	characteristic
388	C₄	—	2	2	no	—
389	C₄	—	2	2	no	—
390	C₄	—	4	1	no	—
391	C₄	—	4	1	no	—
392	C₄	—	4	1	no	—
393	C₄	—	268	2	no	—
394	C₄	—	268	2	no	—
395	C₄	—	268	2	no	—
396	C₄	—	268	2	no	—
397	C₄	#16863456	1	4	yes	—
398	C₄	#16864086	1	1	yes	characteristic
399	C₄	#16864086	1	1	yes	characteristic
400	C₄	#16864119	1	1	yes	characteristic
401	C₄	#16864119	1	1	yes	characteristic
402	C₂	#22563172	1	1	yes	characteristic
403	C₂	C₂×#16863456	1	1	yes	characteristic
404	C₂	#22568874	1	1	yes	characteristic
405	C₂	#22569184	1	1	yes	characteristic
406	C₂	#22569213	1	1	yes	characteristic
407	C₂	#22569303	1	1	yes	characteristic
408	C₂	#22569310	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle, prime radical	C₂³×C₆₇	C₂³×C₄
Fitting	C₄×C₄⋊C₈×C₆₇	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄×C₄⋊C₈	D₆₇
hypocenter	C₆₇	#22829
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
216. C₆₇	#22829	semidirect

Maximal subgroups

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

subgroup	index
#22562686	2
#22561145	2
#22560770	2
#22560763	2
#22560353	2
#22560326	2
#22560076	2
C₈×C₄⋊Dic₆₇	2
#22559959	2
C₄×C₄⋊C₈×C₆₇	2
#22829	67