Subgroups — #22570701

Order 8576.

532 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14524075	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14524072	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14524064	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₁₆×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863971	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863966	C₂	1	1	yes	maximal, characteristic, complemented
7	#16863790	C₂	1	1	yes	maximal, characteristic, complemented
8	#16863789	C₂	1	1	yes	maximal, characteristic, complemented
9	#16863787	C₂	1	1	yes	maximal, characteristic, complemented
10	#16863778	C₂	1	1	yes	maximal, characteristic, complemented
11	#16863777	C₂	1	1	yes	maximal, characteristic, complemented
12	#16863775	C₂	1	1	yes	maximal, characteristic, complemented
13	#16863774	C₂	1	1	yes	maximal, characteristic, complemented
14	#16863353	C₂	1	1	yes	maximal, characteristic, complemented
15	#497×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11761629	C₂²	1	1	yes	characteristic
17	C₂×#11761627	C₂²	1	1	yes	characteristic
18	C₂×Q₈×D₆₇	C₂²	1	1	yes	characteristic
19	C₂×Q₈×D₆₇	C₂²	1	1	yes	characteristic
20	#14524075	—	2	4	no	—
21	#14524072	—	2	4	no	—
22	#14524064	—	2	4	no	—
23	C₂×Dic₂₆₈	C₂²	1	1	yes	characteristic
24	C₂×Dic₂₆₈	C₂²	1	1	yes	characteristic
25	C₂×#11761618	C₂²	1	1	yes	characteristic
26	C₂×#11761618	C₂²	1	1	yes	characteristic
27	C₂×#11761618	C₂²	1	1	yes	characteristic
28	C₂×#11761618	C₂²	1	1	yes	characteristic
29	C₂×#11761618	C₂²	1	1	yes	characteristic
30	Q₁₆×D₆₇	—	2	4	no	—
31	C₂×#11761617	C₂²	1	1	yes	characteristic
32	C₂×#11761617	C₂²	1	1	yes	characteristic
33	C₂×#11761616	C₂²	1	1	yes	characteristic
34	C₂×#11761615	C₂²	1	1	yes	characteristic
35	C₂×#11761613	C₂²	1	1	yes	characteristic
36	C₂×#11761611	C₂²	1	1	yes	characteristic
37	C₂×C₈×D₆₇	C₂²	1	1	yes	characteristic
38	#14524043	C₂²	1	1	yes	characteristic
39	#14524041	C₂²	1	1	yes	characteristic
40	#14524041	C₂²	1	1	yes	characteristic
41	#14524032	C₂²	1	1	yes	characteristic
42	#14524031	C₂²	1	1	yes	characteristic
43	#14524030	C₂²	1	1	yes	characteristic
44	#14524028	—	2	2	no	—
45	#14524025	—	2	2	no	—
46	#14524021	—	2	2	no	—
47	#14524020	—	2	2	no	—
48	#14524000	C₂²	1	1	yes	characteristic
49	#14523953	C₂²	1	1	yes	characteristic
50	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₆₇	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
54	C₂×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
55	C₂×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
56	C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
57	#124×C₆₇	C₂²	1	1	yes	characteristic
58	#97×C₆₇	C₂²	1	1	yes	characteristic
59	#11761629	—	2	2	no	—
60	#11761629	—	2	2	no	—
61	#11761627	—	2	2	no	—
62	#11761627	—	2	2	no	—
63	C₂×Dic₁₃₄	—	2	1	no	—
64	C₂×Dic₁₃₄	—	2	1	no	—
65	C₂×Dic₁₃₄	C₂³	1	1	yes	characteristic
66	C₂×Dic₁₃₄	C₂³	1	1	yes	characteristic
67	C₂×Dic₁₃₄	C₂³	1	1	yes	characteristic
68	Q₈×D₆₇	—	2	2	no	—
69	Q₈×D₆₇	—	2	2	no	—
70	Q₈×D₆₇	—	2	2	no	—
71	Q₈×D₆₇	—	2	2	no	—
72	C₂×D₂₆₈	—	2	1	no	—
73	C₂×D₂₆₈	C₂³	1	1	yes	characteristic
74	C₂×C₂²⋊D₆₇	—	2	1	no	—
75	C₂²×Dic₆₇	—	2	1	no	—
76	C₂×C₄×D₆₇	—	2	1	no	—
77	C₂×C₄×D₆₇	—	2	1	no	—
78	C₂×C₄×D₆₇	—	2	1	no	—
79	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
80	Dic₂₆₈	—	2	2	no	—
81	Dic₂₆₈	—	2	2	no	—
82	#11761618	—	2	2	no	—
83	#11761618	—	2	2	no	—
84	#11761618	—	2	2	no	—
85	#11761618	—	2	2	no	—
86	#11761618	—	2	2	no	—
87	#11761617	—	2	2	no	—
88	#11761617	—	2	2	no	—
89	#11761616	—	2	2	no	—
90	#11761615	—	2	2	no	—
91	#11761613	—	2	2	no	—
92	#11761611	D₄	1	4	yes	—
93	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
94	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
95	C₈×D₆₇	D₄	1	4	yes	—
96	C₄⋊Dic₆₇	—	2	2	no	—
97	Dic₆₇⋊C₄	—	2	2	no	—
98	Dic₆₇⋊C₄	—	2	2	no	—
99	Dic₆₇⋊C₄	D₄	1	2	yes	—
100	C₂²⋊Dic₆₇	—	2	2	no	—
101	(C₂×D₆₇)⋊C₄	—	2	2	no	—
102	(C₂×D₆₇)⋊C₄	—	2	2	no	—
103	(C₂×D₆₇)⋊C₄	D₄	1	2	yes	—
104	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
105	C₂×Q₈×C₆₇	C₂³	1	1	yes	characteristic
106	C₂×Q₈×C₆₇	C₂³	1	1	yes	characteristic
107	C₂×Q₈×C₆₇	C₂³	1	1	yes	characteristic
108	C₂×D₄×C₆₇	C₂³	1	1	yes	characteristic
109	Q₁₆×C₆₇	—	2	2	no	—
110	Q₁₆×C₆₇	—	2	2	no	—
111	SD₁₆×C₆₇	—	2	2	no	—
112	SD₁₆×C₆₇	—	2	2	no	—
113	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
114	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
115	C₄⋊C₄×C₆₇	—	2	2	no	—
116	C₂²⋊C₄×C₆₇	—	2	2	no	—
117	C₄²×C₆₇	C₂³	1	1	yes	characteristic
118	C₂³×C₆₇	—	2	1	no	—
119	C₂²×D₆₇	—	2	1	no	—
120	C₂²×D₆₇	C₂×D₄	1	1	yes	characteristic
121	Q₈×C₆₇	—	2	1	no	—
122	Q₈×C₆₇	—	2	1	no	—
123	Q₈×C₆₇	—	2	1	no	—
124	Q₈×C₆₇	—	2	1	no	—
125	Q₈×C₆₇	—	2	1	no	—
126	Q₈×C₆₇	—	2	1	no	—
127	D₄×C₆₇	—	2	1	no	—
128	D₄×C₆₇	—	2	1	no	—
129	C₂×C₄×C₆₇	—	2	1	no	—
130	C₂×C₄×C₆₇	—	2	1	no	—
131	C₂×C₄×C₆₇	—	2	1	no	—
132	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
133	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
134	C₂²⋊D₆₇	—	4	2	no	—
135	C₂×Dic₆₇	—	2	1	no	—
136	C₂×Dic₆₇	—	2	1	no	—
137	C₂×Dic₆₇	—	2	1	no	—
138	C₂×Dic₆₇	—	4	2	no	—
139	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
140	C₂×Dic₆₇	C₂×D₄	1	1	yes	characteristic
141	D₂₆₈	—	2	1	no	—
142	D₂₆₈	—	2	1	no	—
143	D₂₆₈	—	4	2	no	—
144	C₄×D₆₇	—	4	2	no	—
145	C₄×D₆₇	—	4	2	no	—
146	C₄×D₆₇	—	4	2	no	—
147	C₄×D₆₇	C₂×D₄	1	2	yes	—
148	C₄×D₆₇	C₂×D₄	1	2	yes	—
149	Dic₁₃₄	—	2	1	no	—
150	Dic₁₃₄	—	2	1	no	—
151	Dic₁₃₄	—	2	1	no	—
152	Dic₁₃₄	—	2	1	no	—
153	Dic₁₃₄	—	2	1	no	—
154	Dic₁₃₄	—	2	1	no	—
155	Dic₁₃₄	—	4	2	no	—
156	Dic₁₃₄	—	4	2	no	—
157	C₈×C₆₇	C₂×D₄	1	2	yes	—
158	C₈×C₆₇	C₂×D₄	1	2	yes	—
159	C₆₇⋊C₈	C₂×D₄	1	2	yes	—
160	C₆₇⋊C₈	C₂×D₄	1	2	yes	—
161	C₂²×C₆₇	—	4	1	no	—
162	C₂²×C₆₇	—	4	2	no	—
163	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
164	C₂×D₆₇	—	2	2	no	—
165	C₂×D₆₇	—	4	1	no	—
166	C₂×D₆₇	—	4	2	no	—
167	C₂×D₆₇	#127	1	2	yes	—
168	C₄×C₆₇	—	2	2	no	—
169	C₄×C₆₇	—	4	1	no	—
170	C₄×C₆₇	—	4	1	no	—
171	C₄×C₆₇	—	4	1	no	—
172	C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
173	C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
174	Dic₆₇	—	2	2	no	—
175	Dic₆₇	—	4	1	no	—
176	Dic₆₇	—	4	1	no	—
177	Dic₆₇	—	4	1	no	—
178	Dic₆₇	#127	1	2	yes	—
179	C₂×C₆₇	—	8	1	no	—
180	C₂×C₆₇	C₂×#127	1	1	yes	characteristic
181	C₂×C₆₇	#578	1	2	yes	—
182	D₆₇	—	2	2	no	—
183	D₆₇	—	8	1	no	—
184	#3156	—	67	1	no	maximal, Sylow
185	C₆₇	#3156	1	1	yes	characteristic, complemented, Sylow
186	C₂×#137	—	67	1	no	—
187	C₂×#137	—	67	1	no	—
188	C₂×#135	—	67	1	no	—
189	C₂²×Q₁₆	—	67	1	no	—
190	#536	—	67	1	no	—
191	#536	—	67	1	no	—
192	#497	—	67	1	no	—
193	#497	—	67	1	no	—
194	#497	—	67	1	no	—
195	#497	D₆₇	1	1	yes	characteristic, complemented
196	#495	—	67	1	no	—
197	#495	—	67	1	no	—
198	#494	—	67	1	no	—
199	#494	—	67	1	no	—
200	#405	—	67	1	no	—
201	C₂×#41	—	67	1	no	—
202	C₂×#41	—	67	1	no	—
203	C₂²×Q₈	—	67	1	no	—
204	C₂²×Q₈	—	67	1	no	—
205	#137	—	134	4	no	—
206	#137	—	134	4	no	—
207	#135	—	134	4	no	—
208	C₂×Q₁₆	—	67	1	no	—
209	C₂×Q₁₆	—	67	1	no	—
210	C₂×Q₁₆	—	67	1	no	—
211	C₂×Q₁₆	—	67	1	no	—
212	C₂×Q₁₆	—	67	1	no	—
213	C₂×Q₁₆	—	67	1	no	—
214	C₂×Q₁₆	—	67	1	no	—
215	C₂×Q₁₆	—	134	4	no	—
216	C₂×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
217	C₂×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
218	C₂×SD₁₆	—	67	1	no	—
219	C₂×SD₁₆	—	67	1	no	—
220	C₂×SD₁₆	—	67	1	no	—
221	C₂×SD₁₆	—	67	1	no	—
222	C₂×SD₁₆	C₂×D₆₇	1	1	yes	characteristic
223	C₂×SD₁₆	C₂×D₆₇	1	1	yes	characteristic
224	C₂×D₈	—	67	1	no	—
225	C₂×M₁₆	—	67	1	no	—
226	C₂²×C₈	—	67	1	no	—
227	C₄⋊Q₈	—	67	1	no	—
228	C₄⋊Q₈	—	67	1	no	—
229	C₄⋊Q₈	—	67	1	no	—
230	C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
231	#124	—	67	1	no	—
232	#124	—	67	1	no	—
233	#124	—	67	1	no	—
234	#124	C₂×D₆₇	1	1	yes	characteristic
235	#123	—	134	2	no	—
236	#123	—	134	2	no	—
237	C₂²⋊Q₈	—	134	2	no	—
238	C₂²⋊Q₈	—	134	2	no	—
239	#117	—	67	1	no	—
240	#97	—	67	1	no	—
241	#97	C₂×D₆₇	1	1	yes	characteristic
242	C₄×C₈	—	67	1	no	—
243	C₄×C₈	—	67	1	no	—
244	#41	—	134	2	no	—
245	#41	—	134	2	no	—
246	#41	—	134	2	no	—
247	#41	—	134	2	no	—
248	C₂×Q₈	—	67	1	no	—
249	C₂×Q₈	—	67	1	no	—
250	C₂×Q₈	—	67	1	no	—
251	C₂×Q₈	—	134	1	no	—
252	C₂×Q₈	—	134	1	no	—
253	C₂×Q₈	—	134	2	no	—
254	C₂×Q₈	—	134	2	no	—
255	C₂×Q₈	—	134	2	no	—
256	C₂×Q₈	—	134	2	no	—
257	C₂×Q₈	C₂²×D₆₇	1	1	yes	characteristic
258	C₂×Q₈	C₂²×D₆₇	1	1	yes	characteristic
259	C₂×Q₈	C₂²×D₆₇	1	1	yes	characteristic
260	C₂×D₄	—	67	1	no	—
261	C₂×D₄	—	134	1	no	—
262	C₂×D₄	—	134	1	no	—
263	C₂×D₄	C₂²×D₆₇	1	1	yes	characteristic
264	C₂²×C₄	—	67	1	no	—
265	C₂²×C₄	—	134	1	no	—
266	C₂²×C₄	—	134	1	no	—
267	C₂²×C₄	—	134	1	no	—
268	C₂²×C₄	—	134	1	no	—
269	Q₁₆	—	2	2	no	—
270	Q₁₆	—	2	2	no	—
271	Q₁₆	—	134	2	no	—
272	Q₁₆	—	134	2	no	—
273	Q₁₆	—	134	2	no	—
274	Q₁₆	—	134	2	no	—
275	Q₁₆	—	134	2	no	—
276	Q₁₆	—	134	2	no	—
277	Q₁₆	—	134	2	no	—
278	SD₁₆	—	2	2	no	—
279	SD₁₆	—	2	2	no	—
280	SD₁₆	—	134	2	no	—
281	SD₁₆	—	134	2	no	—
282	SD₁₆	—	134	2	no	—
283	SD₁₆	—	134	2	no	—
284	D₈	—	134	2	no	—
285	M₁₆	—	67	4	no	—
286	C₂×C₈	—	67	1	no	—
287	C₂×C₈	—	67	1	no	—
288	C₂×C₈	—	67	4	no	—
289	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
290	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	67	2	no	—
293	C₄⋊C₄	—	134	2	no	—
294	C₄⋊C₄	—	134	2	no	—
295	C₄⋊C₄	—	134	2	no	—
296	C₂²⋊C₄	—	2	2	no	—
297	C₂²⋊C₄	—	67	2	no	—
298	C₂²⋊C₄	—	134	2	no	—
299	C₂²⋊C₄	—	134	2	no	—
300	C₂²⋊C₄	—	134	2	no	—
301	C₄²	—	67	1	no	—
302	C₄²	C₂²×D₆₇	1	1	yes	characteristic
303	C₂³	—	2	1	no	—
304	C₂³	—	67	1	no	—
305	C₂³	—	134	1	no	—
306	Q₈	—	2	1	no	—
307	Q₈	—	2	1	no	—
308	Q₈	—	2	1	no	—
309	Q₈	—	2	1	no	—
310	Q₈	—	2	1	no	—
311	Q₈	—	2	1	no	—
312	Q₈	—	268	2	no	—
313	Q₈	—	268	2	no	—
314	Q₈	—	134	1	no	—
315	Q₈	—	134	1	no	—
316	Q₈	—	134	1	no	—
317	Q₈	—	134	1	no	—
318	Q₈	—	134	1	no	—
319	Q₈	—	134	1	no	—
320	D₄	—	2	1	no	—
321	D₄	—	2	1	no	—
322	D₄	—	268	2	no	—
323	D₄	—	268	2	no	—
324	D₄	—	134	1	no	—
325	D₄	—	134	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	268	2	no	—
330	C₂×C₄	—	268	2	no	—
331	C₂×C₄	—	268	2	no	—
332	C₂×C₄	—	268	2	no	—
333	C₂×C₄	—	67	1	no	—
334	C₂×C₄	—	67	2	no	—
335	C₂×C₄	—	67	2	no	—
336	C₂×C₄	—	67	2	no	—
337	C₂×C₄	—	134	1	no	—
338	C₂×C₄	—	134	1	no	—
339	C₂×C₄	—	134	1	no	—
340	C₂×C₄	D₄×D₆₇	1	2	yes	—
341	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
342	C₈	—	67	2	no	—
343	C₈	—	67	2	no	—
344	C₈	D₄×D₆₇	1	2	yes	—
345	C₈	D₄×D₆₇	1	2	yes	—
346	C₂²	—	4	1	no	—
347	C₂²	—	4	2	no	—
348	C₂²	—	268	1	no	—
349	C₂²	—	268	2	no	—
350	C₂²	—	67	2	no	—
351	C₂²	—	134	2	no	—
352	C₂²	C₂×D₄×D₆₇	1	1	yes	characteristic
353	C₄	—	2	2	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	268	1	no	—
358	C₄	—	268	1	no	—
359	C₄	—	268	1	no	—
360	C₄	—	67	2	no	—
361	C₄	—	134	2	no	—
362	C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
363	C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
364	C₂	—	8	1	no	—
365	C₂	—	536	1	no	—
366	C₂	—	134	2	no	—
367	C₂	#127×D₆₇	1	1	yes	characteristic
368	C₂	#16864212	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	#497×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	#497	D₆₇
hypocenter	C₆₇	#3156
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #497

Complements

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

N	H ≅ G/N	type
1. C₂×#14524075	C₂	semidirect
2. C₂×#14524072	C₂	semidirect
4. C₂×Q₁₆×D₆₇	C₂	semidirect
5. #16863971	C₂	semidirect
6. #16863966	C₂	semidirect
7. #16863790	C₂	semidirect
8. #16863789	C₂	semidirect
9. #16863787	C₂	semidirect
10. #16863778	C₂	semidirect
11. #16863777	C₂	semidirect
12. #16863775	C₂	semidirect
13. #16863774	C₂	semidirect
14. #16863353	C₂	semidirect
15. #497×C₆₇	C₂	semidirect
185. C₆₇	#3156	semidirect
195. #497	D₆₇	semidirect