Subgroups — #30875730

Order 17152.

656 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562687	C₂	1	1	yes	maximal, characteristic
2	#22561146	C₂	1	1	yes	maximal, characteristic
3	#22560774	C₂	1	2	yes	maximal
4	#22560765	C₂	1	2	yes	maximal
5	#22560360	C₂	1	1	yes	maximal, characteristic
6	#22560327	C₂	1	2	yes	maximal
7	#22560079	C₂	1	1	yes	maximal, characteristic
8	C₈×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic
9	#22560065	C₂	1	2	yes	maximal
10	C₄×Dic₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1984	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×Dic₆₇⋊C₈	C₂²	1	2	yes	—
15	C₂×Dic₆₇⋊C₈	C₂²	1	2	yes	—
16	C₂×Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×#14523953	C₂²	1	2	yes	—
18	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
20	#16863303	C₄	1	1	yes	characteristic
21	#16863303	C₄	1	1	yes	characteristic
22	#16863295	C₄	1	1	yes	characteristic
23	#16863295	C₄	1	1	yes	characteristic
24	#16863282	C₄	1	2	yes	—
25	#16863282	C₄	1	2	yes	—
26	C₄×C₄⋊Dic₆₇	C₂²	1	2	yes	—
27	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
28	#16863242	C₂²	1	2	yes	—
29	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
30	#16863151	C₂²	1	2	yes	—
31	#16863151	C₂²	1	2	yes	—
32	#16863150	C₂²	1	1	yes	characteristic
33	#16863150	C₂²	1	2	yes	—
34	#16863150	C₂²	1	1	yes	characteristic
35	#16863150	C₂²	1	2	yes	—
36	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
39	#336×C₆₇	C₂²	1	2	yes	—
40	#336×C₆₇	C₂²	1	2	yes	—
41	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
42	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
44	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
45	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
48	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
53	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
54	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
55	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
56	Dic₆₇⋊C₈	—	2	2	no	—
57	Dic₆₇⋊C₈	—	2	4	no	—
58	Dic₆₇⋊C₈	—	4	2	no	—
59	Dic₆₇⋊C₈	Q₈	1	4	yes	complemented
60	#14523953	—	4	2	no	—
61	C₄×C₆₇⋊C₈	—	2	2	no	—
62	C₈×Dic₆₇	—	4	2	no	—
63	C₆₇⋊#95	C₂×C₄	1	2	yes	—
64	C₆₇⋊#95	C₂×C₄	1	2	yes	—
65	C₆₇⋊#95	C₂×C₄	1	2	yes	—
66	C₆₇⋊#95	C₂×C₄	1	2	yes	—
67	C₆₇⋊#95	C₂×C₄	1	2	yes	—
68	C₆₇⋊#95	C₂×C₄	1	2	yes	—
69	C₆₇⋊#95	C₂×C₄	1	2	yes	—
70	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
71	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
72	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
74	C₄×Q₈×C₆₇	—	2	2	no	—
75	C₄×Q₈×C₆₇	—	2	2	no	—
76	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
77	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
79	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
80	C₄⋊C₈×C₆₇	—	2	2	no	—
81	C₄×C₈×C₆₇	—	2	2	no	—
82	C₂²×Dic₆₇	C₂²×C₄	1	2	yes	—
83	C₂²×Dic₆₇	C₂²×C₄	1	2	yes	—
84	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
86	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₆₇⋊C₈	—	2	1	no	—
89	C₂×C₆₇⋊C₈	—	2	1	no	—
90	C₂×C₆₇⋊C₈	—	2	1	no	—
91	C₂×C₆₇⋊C₈	—	2	1	no	—
92	C₂×C₆₇⋊C₈	—	2	2	no	—
93	C₂×C₆₇⋊C₈	—	2	2	no	—
94	C₂×C₆₇⋊C₈	—	2	2	no	—
95	C₂×C₆₇⋊C₈	—	2	1	no	—
96	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
97	C₄⋊Dic₆₇	—	2	2	no	—
98	C₄⋊Dic₆₇	—	2	2	no	—
99	C₄⋊Dic₆₇	—	2	4	no	—
100	C₄⋊Dic₆₇	—	4	2	no	—
101	C₄×Dic₆₇	—	2	2	no	—
102	C₄×Dic₆₇	—	2	2	no	—
103	C₄×Dic₆₇	—	2	2	no	—
104	C₄×Dic₆₇	—	2	1	no	—
105	C₄×Dic₆₇	—	2	1	no	—
106	C₄×Dic₆₇	—	2	2	no	—
107	C₄×Dic₆₇	—	2	2	no	—
108	C₄×Dic₆₇	—	4	1	no	—
109	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
110	C₄×Dic₆₇	#41	1	2	yes	—
111	C₂×Q₈×C₆₇	—	2	2	no	—
112	C₂×Q₈×C₆₇	—	2	2	no	—
113	C₂×Q₈×C₆₇	—	2	2	no	—
114	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
115	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
116	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
117	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
119	C₂×C₈×C₆₇	—	2	1	no	—
120	C₂×C₈×C₆₇	—	2	1	no	—
121	C₂×C₈×C₆₇	—	2	2	no	—
122	C₂×C₈×C₆₇	—	2	1	no	—
123	C₂×C₈×C₆₇	—	2	2	no	—
124	C₂×C₈×C₆₇	—	2	1	no	—
125	C₂×C₈×C₆₇	—	2	2	no	—
126	C₂×C₈×C₆₇	—	2	1	no	—
127	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
128	C₄⋊C₄×C₆₇	—	2	2	no	—
129	C₄⋊C₄×C₆₇	—	2	4	no	—
130	C₄²×C₆₇	—	2	2	no	—
131	C₄²×C₆₇	—	2	2	no	—
132	C₄²×C₆₇	#41	1	2	yes	—
133	C₄²×C₆₇	#41	1	2	yes	—
134	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
135	Q₈×C₆₇	—	4	4	no	—
136	C₂×C₄×C₆₇	—	2	1	no	—
137	C₂×C₄×C₆₇	—	2	1	no	—
138	C₂×C₄×C₆₇	—	2	2	no	—
139	C₂×C₄×C₆₇	—	2	2	no	—
140	C₂×C₄×C₆₇	—	2	2	no	—
141	C₂×C₄×C₆₇	—	2	2	no	—
142	C₂×C₄×C₆₇	—	2	2	no	—
143	C₂×C₄×C₆₇	—	2	2	no	—
144	C₂×C₄×C₆₇	#117	1	2	yes	—
145	C₂×C₄×C₆₇	#117	1	2	yes	—
146	C₂×C₄×C₆₇	#131	1	2	yes	—
147	C₂×C₄×C₆₇	#131	1	2	yes	—
148	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
149	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
151	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
153	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
154	C₂×Dic₆₇	—	2	1	no	—
155	C₂×Dic₆₇	—	2	1	no	—
156	C₂×Dic₆₇	—	2	1	no	—
157	C₂×Dic₆₇	—	2	1	no	—
158	C₂×Dic₆₇	—	2	1	no	—
159	C₂×Dic₆₇	—	2	1	no	—
160	C₂×Dic₆₇	—	2	2	no	—
161	C₂×Dic₆₇	—	2	1	no	—
162	C₂×Dic₆₇	—	2	2	no	—
163	C₂×Dic₆₇	—	2	1	no	—
164	C₂×Dic₆₇	—	2	2	no	—
165	C₂×Dic₆₇	—	2	2	no	—
166	C₂×Dic₆₇	—	2	2	no	—
167	C₂×Dic₆₇	—	2	1	no	—
168	C₂×Dic₆₇	—	2	2	no	—
169	C₂×Dic₆₇	C₄×Q₈	1	2	yes	—
170	C₂×Dic₆₇	C₄×Q₈	1	2	yes	—
171	C₂×Dic₆₇	#131	1	2	yes	—
172	C₈×C₆₇	—	2	2	no	—
173	C₈×C₆₇	—	4	1	no	—
174	C₈×C₆₇	—	4	2	no	—
175	C₆₇⋊C₈	—	2	2	no	—
176	C₆₇⋊C₈	—	4	1	no	—
177	C₆₇⋊C₈	—	4	2	no	—
178	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
179	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
180	C₂²×C₆₇	#520	1	1	yes	characteristic
181	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
182	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
183	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
184	C₂²×C₆₇	#568	1	1	yes	characteristic
185	C₄×C₆₇	—	2	2	no	—
186	C₄×C₆₇	—	4	2	no	—
187	C₄×C₆₇	—	4	2	no	—
188	C₄×C₆₇	#433	1	4	yes	—
189	C₄×C₆₇	#554	1	1	yes	characteristic
190	C₄×C₆₇	#556	1	1	yes	characteristic
191	C₄×C₆₇	#557	1	1	yes	characteristic
192	C₄×C₆₇	#559	1	1	yes	characteristic
193	Dic₆₇	—	2	2	no	—
194	Dic₆₇	—	2	2	no	—
195	Dic₆₇	—	2	2	no	—
196	Dic₆₇	—	4	1	no	—
197	Dic₆₇	—	4	2	no	—
198	Dic₆₇	—	4	2	no	—
199	#22822	—	67	1	no	maximal, Sylow
200	C₂×C₆₇	#2355	1	1	yes	characteristic
201	C₂×C₆₇	C₂×#433	1	1	yes	characteristic
202	C₂×C₆₇	#2925	1	1	yes	characteristic
203	C₂×C₆₇	#2960	1	1	yes	characteristic
204	C₂×C₆₇	#2962	1	1	yes	characteristic
205	C₂×C₆₇	#2993	1	1	yes	characteristic
206	C₂×C₆₇	#2994	1	1	yes	characteristic
207	#2295	—	67	1	no	—
208	#1984	—	67	1	no	—
209	#1984	D₆₇	1	1	yes	characteristic
210	#1917	—	67	2	no	—
211	#1916	—	67	2	no	—
212	#1842	—	67	1	no	—
213	#1832	—	67	2	no	—
214	#1770	—	67	1	no	—
215	C₈×C₄⋊C₄	—	67	1	no	—
216	#1768	—	67	2	no	—
217	C₄×C₄⋊C₈	—	67	1	no	—
218	C₆₇	#22822	1	1	yes	characteristic, complemented, Sylow
219	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
220	C₂×C₄⋊C₈	—	67	2	no	—
221	C₂×C₄⋊C₈	—	67	1	no	—
222	C₂×C₄⋊C₈	—	67	1	no	—
223	C₂×C₄⋊C₈	—	67	2	no	—
224	C₂×C₄⋊C₈	—	67	1	no	—
225	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
226	C₂×#97	—	67	2	no	—
227	C₂×C₄×C₈	—	67	2	no	—
228	C₂×C₄×C₈	—	67	1	no	—
229	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
230	#391	—	67	1	no	—
231	#391	—	67	1	no	—
232	#389	—	67	1	no	—
233	#389	—	67	1	no	—
234	#387	—	67	2	no	—
235	#387	—	67	2	no	—
236	C₄×C₄⋊C₄	—	67	2	no	—
237	C₄×C₄⋊C₄	—	67	1	no	—
238	#376	—	67	2	no	—
239	C₄³	—	67	1	no	—
240	#336	—	67	2	no	—
241	#336	—	67	2	no	—
242	#336	—	67	1	no	—
243	#336	—	67	1	no	—
244	#336	—	67	2	no	—
245	#336	—	67	2	no	—
246	#336	C₂×D₆₇	1	2	yes	—
247	#336	C₂×D₆₇	1	2	yes	—
248	C₂²×Q₈	C₄×D₆₇	1	1	yes	characteristic
249	C₂²×C₈	—	67	1	no	—
250	C₂²×C₈	—	67	1	no	—
251	C₂²×C₈	—	67	2	no	—
252	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
253	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
254	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
255	C₄×Q₈	—	2	2	no	—
256	C₄×Q₈	—	2	2	no	—
257	C₂×C₄⋊C₄	—	67	2	no	—
258	C₂×C₄⋊C₄	—	67	2	no	—
259	C₂×C₄⋊C₄	—	67	1	no	—
260	C₂×C₄⋊C₄	—	67	1	no	—
261	C₂×C₄⋊C₄	C₄×D₆₇	1	2	yes	—
262	C₂×C₄⋊C₄	C₄×D₆₇	1	1	yes	characteristic
263	C₂×C₄²	—	67	1	no	—
264	C₂×C₄²	—	67	1	no	—
265	C₂×C₄²	—	67	1	no	—
266	C₂×C₄²	—	67	1	no	—
267	C₂×C₄²	—	67	1	no	—
268	C₂×C₄²	—	67	1	no	—
269	C₂×C₄²	—	67	2	no	—
270	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
271	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
272	C₄⋊C₈	—	2	2	no	—
273	C₄⋊C₈	—	268	2	no	—
274	C₄⋊C₈	—	67	4	no	—
275	C₄⋊C₈	—	134	2	no	—
276	C₄⋊C₈	—	134	4	no	—
277	C₄⋊C₈	—	134	2	no	—
278	#97	—	268	2	no	—
279	C₄×C₈	—	2	2	no	—
280	C₄×C₈	—	268	2	no	—
281	C₄×C₈	—	134	2	no	—
282	#95	—	67	2	no	—
283	#95	—	67	2	no	—
284	#95	—	67	2	no	—
285	#95	—	67	2	no	—
286	#95	—	67	2	no	—
287	#95	—	67	2	no	—
288	#95	—	67	2	no	—
289	C₂×Q₈	—	2	2	no	—
290	C₂×Q₈	—	2	2	no	—
291	C₂×Q₈	—	2	2	no	—
292	C₂²×C₄	—	67	1	no	—
293	C₂²×C₄	—	67	1	no	—
294	C₂²×C₄	—	67	1	no	—
295	C₂²×C₄	—	67	2	no	—
296	C₂²×C₄	—	67	2	no	—
297	C₂²×C₄	—	67	1	no	—
298	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
299	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
300	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
301	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
302	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	2	1	no	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	2	2	no	—
310	C₂×C₈	—	2	1	no	—
311	C₂×C₈	—	67	2	no	—
312	C₂×C₈	—	134	1	no	—
313	C₂×C₈	—	134	2	no	—
314	C₂×C₈	—	134	1	no	—
315	C₂×C₈	—	134	2	no	—
316	C₂×C₈	—	134	2	no	—
317	C₂×C₈	—	134	1	no	—
318	C₂×C₈	—	134	1	no	—
319	C₂×C₈	—	134	1	no	—
320	C₂×C₈	Q₈×D₆₇	1	2	yes	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	4	no	—
323	C₄⋊C₄	—	268	2	no	—
324	C₄⋊C₄	—	134	4	no	—
325	C₄⋊C₄	—	134	2	no	—
326	C₄⋊C₄	—	134	2	no	—
327	C₄²	—	2	2	no	—
328	C₄²	—	2	2	no	—
329	C₄²	—	268	1	no	—
330	C₄²	—	67	2	no	—
331	C₄²	—	67	2	no	—
332	C₄²	—	134	2	no	—
333	C₄²	—	134	2	no	—
334	C₄²	—	134	2	no	—
335	C₄²	—	134	2	no	—
336	C₄²	—	134	1	no	—
337	C₄²	—	134	2	no	—
338	C₄²	—	134	1	no	—
339	C₄²	#11761627	1	2	yes	—
340	C₄²	#11761627	1	2	yes	—
341	C₂³	C₂²×C₄×D₆₇	1	1	yes	characteristic
342	Q₈	—	4	4	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	67	2	no	—
352	C₂×C₄	—	67	2	no	—
353	C₂×C₄	—	67	2	no	—
354	C₂×C₄	—	134	2	no	—
355	C₂×C₄	—	134	1	no	—
356	C₂×C₄	—	134	2	no	—
357	C₂×C₄	—	134	2	no	—
358	C₂×C₄	—	134	1	no	—
359	C₂×C₄	—	134	2	no	—
360	C₂×C₄	—	134	2	no	—
361	C₂×C₄	—	134	1	no	—
362	C₂×C₄	—	134	2	no	—
363	C₂×C₄	—	134	1	no	—
364	C₂×C₄	—	134	1	no	—
365	C₂×C₄	—	134	1	no	—
366	C₂×C₄	—	134	1	no	—
367	C₂×C₄	—	134	1	no	—
368	C₂×C₄	—	134	1	no	—
369	C₂×C₄	#14523999	1	2	yes	—
370	C₂×C₄	#14523999	1	2	yes	—
371	C₂×C₄	#14524049	1	2	yes	—
372	C₂×C₄	#14524051	1	2	yes	—
373	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
374	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
375	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
376	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
377	C₂×C₄	#14524089	1	1	yes	characteristic
378	C₂×C₄	#14524091	1	1	yes	characteristic
379	C₈	—	2	2	no	—
380	C₈	—	4	1	no	—
381	C₈	—	4	2	no	—
382	C₈	—	268	1	no	—
383	C₈	—	268	2	no	—
384	C₈	—	134	2	no	—
385	C₂²	C₂×#14523999	1	1	yes	characteristic
386	C₂²	C₄×Q₈×D₆₇	1	1	yes	characteristic
387	C₂²	#16863878	1	1	yes	characteristic
388	C₂²	#131×D₆₇	1	1	yes	characteristic
389	C₂²	C₂×#14524049	1	1	yes	characteristic
390	C₂²	C₂×#14524051	1	1	yes	characteristic
391	C₂²	#16864161	1	1	yes	characteristic
392	C₄	—	2	2	no	—
393	C₄	—	4	2	no	—
394	C₄	—	4	2	no	—
395	C₄	—	268	1	no	—
396	C₄	—	268	2	no	—
397	C₄	—	268	2	no	—
398	C₄	—	134	2	no	—
399	C₄	—	134	2	no	—
400	C₄	—	134	2	no	—
401	C₄	#16863454	1	4	yes	—
402	C₄	#16864104	1	1	yes	characteristic
403	C₄	#16864117	1	1	yes	characteristic
404	C₄	#16864121	1	1	yes	characteristic
405	C₄	#16864128	1	1	yes	characteristic
406	C₂	#22563165	1	1	yes	characteristic
407	C₂	C₂×#16863454	1	1	yes	characteristic
408	C₂	#22568871	1	1	yes	characteristic
409	C₂	#22569138	1	1	yes	characteristic
410	C₂	#22569157	1	1	yes	characteristic
411	C₂	#22569357	1	1	yes	characteristic
412	C₂	#22569372	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle, prime radical	C₂³×C₆₇	C₂³×C₄
Fitting	C₆₇×#1984	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1984	D₆₇
hypocenter	C₆₇	#22822
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1984

Complements

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

N	H ≅ G/N	type
59. Dic₆₇⋊C₈	Q₈	semidirect
218. C₆₇	#22822	semidirect

Maximal subgroups

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

subgroup	index
#22562687	2
#22561146	2
#22560774	2
#22560765	2
#22560360	2
#22560327	2
#22560079	2
C₈×C₄⋊Dic₆₇	2
#22560065	2
C₄×Dic₆₇⋊C₈	2
C₆₇×#1984	2
#22822	67