Subgroups — #37545766

Order 23168.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×Dic₃₆₂	C₂	1	1	yes	maximal, characteristic
2	#26041649	C₂	1	1	yes	maximal, characteristic, complemented
3	#26041620	C₂	1	1	yes	maximal, characteristic, complemented
4	#26041609	C₂	1	1	yes	maximal, characteristic, complemented
5	#26041583	C₂	1	1	yes	maximal, characteristic, complemented
6	#26041535	C₂	1	1	yes	maximal, characteristic, complemented
7	#26041518	C₂	1	1	yes	maximal, characteristic, complemented
8	#26041489	C₂	1	1	yes	maximal, characteristic, complemented
9	#26041434	C₂	1	1	yes	maximal, characteristic, complemented
10	#26041434	C₂	1	1	yes	maximal, characteristic, complemented
11	#26041384	C₂	1	1	yes	maximal, characteristic, complemented
12	#26041351	C₂	1	1	yes	maximal, characteristic
13	#26041287	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#19153739	C₂	1	1	yes	maximal, characteristic, complemented
15	#552×C₁₈₁	C₂	1	1	yes	maximal, characteristic
16	C₂²×Dic₃₆₂	C₂²	1	1	yes	characteristic
17	C₂²×Dic₃₆₂	C₂²	1	1	yes	characteristic
18	C₂×#15668439	C₂²	1	1	yes	characteristic
19	C₂×#15668439	C₂²	1	1	yes	characteristic
20	C₂×#15668439	C₂²	1	1	yes	characteristic
21	C₂×#15668433	C₂²	1	1	yes	characteristic
22	C₂²×#13089678	C₂²	1	1	yes	characteristic
23	#19153816	C₂²	1	1	yes	characteristic
24	#19153815	—	2	1	no	—
25	#19153795	C₂²	1	1	yes	characteristic
26	#19153795	C₂²	1	1	yes	characteristic
27	#19153793	—	2	1	no	—
28	#19153793	—	2	1	no	—
29	C₄×Dic₃₆₂	C₂²	1	1	yes	characteristic
30	D₄×Dic₁₈₁	—	2	1	no	—
31	#19153749	C₂²	1	1	yes	characteristic
32	#19153747	C₂²	1	1	yes	characteristic
33	#19153747	C₂²	1	1	yes	characteristic
34	C₄⋊#13089678	C₂²	1	1	yes	characteristic
35	#19153740	C₂²	1	1	yes	characteristic
36	#19153739	C₂²	1	1	yes	characteristic
37	#19153739	C₂²	1	1	yes	characteristic
38	#19153739	C₂²	1	1	yes	characteristic
39	#19153739	C₂²	1	1	yes	characteristic
40	#19153739	C₂²	1	1	yes	characteristic
41	#19153739	C₂²	1	1	yes	characteristic
42	#19153739	C₂²	1	1	yes	characteristic
43	#19153739	C₂²	1	1	yes	characteristic
44	#19153739	C₂²	1	1	yes	characteristic
45	C₂²⋊#13089678	C₂²	1	1	yes	characteristic
46	C₂²⋊#13089678	C₂²	1	1	yes	characteristic
47	#19153727	C₂²	1	1	yes	characteristic
48	#126×C₁₈₁	—	2	1	no	—
49	#125×C₁₈₁	—	2	1	no	—
50	#125×C₁₈₁	C₂²	1	1	yes	characteristic
51	#123×C₁₈₁	—	2	1	no	—
52	C₂²⋊Q₈×C₁₈₁	C₂²	1	1	yes	characteristic
53	C₂²⋊Q₈×C₁₈₁	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₁₈₁	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₁₈₁	C₂²	1	1	yes	characteristic
56	#117×C₁₈₁	—	2	1	no	—
57	#117×C₁₈₁	C₂²	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₁₈₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₃₆₂	—	2	2	no	—
60	C₂×Dic₃₆₂	—	2	2	no	—
61	C₂×Dic₃₆₂	—	2	1	no	—
62	C₂×Dic₃₆₂	—	2	1	no	—
63	C₂×Dic₃₆₂	—	4	1	no	—
64	C₂×Dic₃₆₂	—	4	1	no	—
65	C₂×Dic₃₆₂	C₂³	1	1	yes	characteristic
66	C₂×Dic₃₆₂	C₂³	1	1	yes	characteristic
67	C₂×Dic₃₆₂	C₂³	1	1	yes	characteristic
68	C₂²×Dic₁₈₁	—	2	1	no	—
69	C₂²×Dic₁₈₁	—	2	1	no	—
70	#15668439	—	2	2	no	—
71	#15668439	—	4	1	no	—
72	#15668439	—	4	1	no	—
73	#15668433	—	4	1	no	—
74	C₂×#13089678	—	4	1	no	—
75	C₂×#13089678	C₂³	1	1	yes	characteristic
76	C₂×#13089678	C₂³	1	1	yes	characteristic
77	C₂×#13089678	C₂³	1	1	yes	characteristic
78	C₂×#13089678	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₁₈₁	—	2	1	no	—
80	C₄⋊Dic₁₈₁	—	2	1	no	—
81	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
82	#15668428	—	2	1	no	—
83	#15668428	—	2	1	no	—
84	#15668428	—	2	1	no	—
85	C₂²⋊Dic₁₈₁	—	2	1	no	—
86	C₂²⋊Dic₁₈₁	—	2	1	no	—
87	C₄×Dic₁₈₁	—	2	1	no	—
88	C₂×Q₈×C₁₈₁	C₂³	1	1	yes	characteristic
89	C₂×D₄×C₁₈₁	D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₈₁	—	2	1	no	—
91	C₂²×C₄×C₁₈₁	C₂³	1	1	yes	characteristic
92	C₂²×C₄×C₁₈₁	C₂³	1	1	yes	characteristic
93	C₄⋊C₄×C₁₈₁	—	2	1	no	—
94	C₄⋊C₄×C₁₈₁	—	2	1	no	—
95	C₄⋊C₄×C₁₈₁	—	2	1	no	—
96	C₄⋊C₄×C₁₈₁	—	2	1	no	—
97	C₄⋊C₄×C₁₈₁	—	2	1	no	—
98	C₄⋊C₄×C₁₈₁	—	2	1	no	—
99	C₄⋊C₄×C₁₈₁	D₄	1	1	yes	characteristic
100	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
103	C₂²⋊C₄×C₁₈₁	—	2	1	no	—
104	C₂²⋊C₄×C₁₈₁	—	2	1	no	—
105	C₂²⋊C₄×C₁₈₁	—	2	1	no	—
106	C₂²⋊C₄×C₁₈₁	D₄	1	1	yes	characteristic
107	C₂²⋊C₄×C₁₈₁	D₄	1	1	yes	characteristic
108	C₄²×C₁₈₁	—	2	1	no	—
109	C₄²×C₁₈₁	—	2	1	no	—
110	C₄²×C₁₈₁	C₂³	1	1	yes	characteristic
111	Dic₃₆₂	—	2	1	no	—
112	Dic₃₆₂	—	2	1	no	—
113	Dic₃₆₂	—	2	1	no	—
114	Dic₃₆₂	—	2	1	no	—
115	Dic₃₆₂	—	2	1	no	—
116	Dic₃₆₂	—	4	2	no	—
117	Dic₃₆₂	—	4	2	no	—
118	Dic₃₆₂	#41	1	2	yes	—
119	C₂×Dic₁₈₁	—	2	1	no	—
120	C₂×Dic₁₈₁	—	2	1	no	—
121	C₂×Dic₁₈₁	—	2	1	no	—
122	C₂×Dic₁₈₁	—	2	1	no	—
123	C₂×Dic₁₈₁	—	4	2	no	—
124	C₂×Dic₁₈₁	—	4	2	no	—
125	#13089678	—	2	1	no	—
126	#13089678	—	2	1	no	—
127	#13089678	—	2	1	no	—
128	#13089678	—	2	1	no	—
129	C₂³×C₁₈₁	C₂×D₄	1	1	yes	characteristic
130	C₂³×C₁₈₁	C₂×D₄	1	1	yes	characteristic
131	Q₈×C₁₈₁	—	2	1	no	—
132	Q₈×C₁₈₁	—	2	1	no	—
133	D₄×C₁₈₁	—	2	2	no	—
134	C₂×C₄×C₁₈₁	—	2	1	no	—
135	C₂×C₄×C₁₈₁	—	2	1	no	—
136	C₂×C₄×C₁₈₁	—	2	1	no	—
137	C₂×C₄×C₁₈₁	—	2	1	no	—
138	C₂×C₄×C₁₈₁	—	2	1	no	—
139	C₂×C₄×C₁₈₁	—	2	1	no	—
140	C₂×C₄×C₁₈₁	—	2	1	no	—
141	C₂×C₄×C₁₈₁	—	2	1	no	—
142	C₂×C₄×C₁₈₁	—	4	2	no	—
143	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
146	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
147	C₂×C₄×C₁₈₁	C₂⁴	1	1	yes	characteristic
148	C₂²×C₁₈₁	—	2	1	no	—
149	C₂²×C₁₈₁	—	2	1	no	—
150	C₂²×C₁₈₁	—	2	1	no	—
151	C₂²×C₁₈₁	—	2	1	no	—
152	C₂²×C₁₈₁	—	2	1	no	—
153	C₂²×C₁₈₁	#137	1	2	yes	—
154	C₂²×C₁₈₁	C₂²×D₄	1	1	yes	characteristic
155	C₄×C₁₈₁	—	2	1	no	—
156	C₄×C₁₈₁	—	2	1	no	—
157	C₄×C₁₈₁	—	2	1	no	—
158	C₄×C₁₈₁	—	2	1	no	—
159	C₄×C₁₈₁	—	4	1	no	—
160	C₄×C₁₈₁	—	4	1	no	—
161	C₄×C₁₈₁	—	4	1	no	—
162	C₄×C₁₈₁	—	4	1	no	—
163	C₄×C₁₈₁	C₂×#41	1	1	yes	characteristic
164	C₄×C₁₈₁	H(5,𝔽₂)	1	1	yes	characteristic
165	Dic₁₈₁	—	2	2	no	—
166	Dic₁₈₁	—	4	1	no	—
167	Dic₁₈₁	—	4	1	no	—
168	Dic₁₈₁	—	4	1	no	—
169	C₂×C₁₈₁	—	2	2	no	—
170	C₂×C₁₈₁	—	4	1	no	—
171	C₂×C₁₈₁	#546	1	1	yes	characteristic
172	C₂×C₁₈₁	C₂×#137	1	1	yes	characteristic
173	C₂×C₁₈₁	#578	1	1	yes	characteristic
174	C₁₈₁	#3312	1	1	yes	characteristic, complemented, Sylow
175	#3312	—	181	1	no	maximal, Sylow
176	#552	D₁₈₁	1	1	yes	characteristic
177	D₄×Q₈	—	181	1	no	—
178	#491	—	181	1	no	—
179	#485	—	181	1	no	—
180	#483	—	181	1	no	—
181	#479	—	181	1	no	—
182	#470	—	181	1	no	—
183	#467	—	181	1	no	—
184	#462	—	181	1	no	—
185	#451	—	181	1	no	—
186	#451	—	181	1	no	—
187	#441	—	181	1	no	—
188	#435	—	181	1	no	—
189	#419	—	181	1	no	—
190	C₂×#103	—	181	1	no	—
191	C₂²×Q₈	—	181	1	no	—
192	C₂²×Q₈	—	181	1	no	—
193	C₂×Q₁₆	—	181	1	no	—
194	C₂×Q₁₆	—	181	1	no	—
195	C₂×Q₁₆	—	181	1	no	—
196	C₂×M₁₆	—	181	1	no	—
197	C₂²×C₈	—	181	1	no	—
198	C₄⋊Q₈	—	362	1	no	—
199	C₄⋊Q₈	—	181	1	no	—
200	#126	—	2	1	no	—
201	#125	—	2	1	no	—
202	#125	C₂×D₁₈₁	1	1	yes	characteristic
203	#123	—	2	1	no	—
204	C₂²⋊Q₈	—	362	1	no	—
205	C₂²⋊Q₈	—	362	1	no	—
206	C₂²⋊Q₈	—	181	1	no	—
207	C₂²⋊Q₈	—	181	1	no	—
208	C₂²⋊Q₈	C₂×D₁₈₁	1	1	yes	characteristic
209	C₂²⋊Q₈	C₂×D₁₈₁	1	1	yes	characteristic
210	C₄×Q₈	—	181	1	no	—
211	C₄×Q₈	C₂×D₁₈₁	1	1	yes	characteristic
212	C₄×D₄	—	362	1	no	—
213	C₄×D₄	C₂×D₁₈₁	1	1	yes	characteristic
214	#117	—	2	1	no	—
215	#117	C₂×D₁₈₁	1	1	yes	characteristic
216	C₂×C₄⋊C₄	C₂×D₁₈₁	1	1	yes	characteristic
217	#107	—	181	1	no	—
218	#106	—	181	1	no	—
219	#106	—	181	1	no	—
220	C₄⋊C₈	—	181	1	no	—
221	#103	—	181	1	no	—
222	#103	—	181	1	no	—
223	#103	—	181	1	no	—
224	#103	—	181	1	no	—
225	#103	—	181	1	no	—
226	#103	—	181	1	no	—
227	#103	—	181	1	no	—
228	#103	—	181	1	no	—
229	#103	—	181	1	no	—
230	#103	—	181	1	no	—
231	C₂²⋊C₈	—	181	1	no	—
232	C₂²⋊C₈	—	181	1	no	—
233	#97	—	181	1	no	—
234	C₂×Q₈	—	362	2	no	—
235	C₂×Q₈	—	362	2	no	—
236	C₂×Q₈	—	362	1	no	—
237	C₂×Q₈	—	362	1	no	—
238	C₂×Q₈	—	181	1	no	—
239	C₂×Q₈	—	181	1	no	—
240	C₂×Q₈	—	181	1	no	—
241	C₂×Q₈	—	724	1	no	—
242	C₂×Q₈	—	724	1	no	—
243	C₂×Q₈	C₂²×D₁₈₁	1	1	yes	characteristic
244	C₂×D₄	C₂²⋊D₁₈₁	1	1	yes	characteristic
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	362	1	no	—
247	C₂²×C₄	—	362	1	no	—
248	C₂²×C₄	C₂²×D₁₈₁	1	1	yes	characteristic
249	C₂²×C₄	C₂²×D₁₈₁	1	1	yes	characteristic
250	Q₁₆	—	362	2	no	—
251	Q₁₆	—	724	1	no	—
252	Q₁₆	—	724	1	no	—
253	M₁₆	—	724	1	no	—
254	C₂×C₈	—	181	1	no	—
255	C₂×C₈	—	181	1	no	—
256	C₂×C₈	—	181	1	no	—
257	C₂×C₈	—	181	1	no	—
258	C₂×C₈	—	724	1	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	362	1	no	—
266	C₄⋊C₄	—	362	1	no	—
267	C₄⋊C₄	—	362	1	no	—
268	C₄⋊C₄	—	362	1	no	—
269	C₄⋊C₄	—	362	1	no	—
270	C₄⋊C₄	—	181	1	no	—
271	C₄⋊C₄	C₂²⋊D₁₈₁	1	1	yes	characteristic
272	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
273	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	362	1	no	—
279	C₂²⋊C₄	—	362	1	no	—
280	C₂²⋊C₄	C₂²⋊D₁₈₁	1	1	yes	characteristic
281	C₂²⋊C₄	C₂²⋊D₁₈₁	1	1	yes	characteristic
282	C₄²	—	2	1	no	—
283	C₄²	—	2	1	no	—
284	C₄²	—	362	1	no	—
285	C₄²	C₂²×D₁₈₁	1	1	yes	characteristic
286	C₂³	C₂×C₂²⋊D₁₈₁	1	1	yes	characteristic
287	C₂³	C₂×C₂²⋊D₁₈₁	1	1	yes	characteristic
288	Q₈	—	2	1	no	—
289	Q₈	—	2	1	no	—
290	Q₈	—	362	1	no	—
291	Q₈	—	362	1	no	—
292	Q₈	—	362	1	no	—
293	Q₈	—	362	1	no	—
294	Q₈	—	362	1	no	—
295	Q₈	—	181	2	no	—
296	Q₈	—	724	2	no	—
297	Q₈	—	724	2	no	—
298	D₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	362	1	no	—
309	C₂×C₄	—	362	1	no	—
310	C₂×C₄	—	362	1	no	—
311	C₂×C₄	—	362	1	no	—
312	C₂×C₄	—	724	2	no	—
313	C₂×C₄	—	724	2	no	—
314	C₂×C₄	C₂×C₂²⋊D₁₈₁	1	1	yes	characteristic
315	C₂×C₄	C₂×C₂²⋊D₁₈₁	1	1	yes	characteristic
316	C₂×C₄	C₂×C₂²⋊D₁₈₁	1	1	yes	characteristic
317	C₂×C₄	C₂×C₂²⋊D₁₈₁	1	1	yes	characteristic
318	C₂×C₄	C₂³×D₁₈₁	1	1	yes	characteristic
319	C₈	—	362	1	no	—
320	C₈	—	362	1	no	—
321	C₈	—	362	1	no	—
322	C₈	—	362	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	#19153846	1	2	yes	—
329	C₂²	C₂²×C₂²⋊D₁₈₁	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	362	2	no	—
339	C₄	—	724	1	no	—
340	C₄	—	724	1	no	—
341	C₄	—	724	1	no	—
342	C₄	#41×D₁₈₁	1	1	yes	characteristic
343	C₄	#19153862	1	1	yes	characteristic
344	C₂	—	2	2	no	—
345	C₂	—	4	1	no	—
346	C₂	#26041917	1	1	yes	characteristic
347	C₂	C₂×#19153846	1	1	yes	characteristic
348	C₂	#26042094	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₈₁
commutator, cosocle	C₂×C₄×C₁₈₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₈₁
Fitting	#552×C₁₈₁	C₂
socle	C₂²×C₁₈₁	C₂²×D₄
prime radical	C₂×D₄×C₁₈₁	D₄
hypercenter	#552	D₁₈₁
hypocenter	C₁₈₁	#3312
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #552

Complements

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

N	H ≅ G/N	type
2. #26041649	C₂	semidirect
3. #26041620	C₂	semidirect
4. #26041609	C₂	semidirect
5. #26041583	C₂	semidirect
6. #26041535	C₂	semidirect
7. #26041518	C₂	semidirect
8. #26041489	C₂	semidirect
9. #26041434	C₂	semidirect
10. #26041434	C₂	semidirect
11. #26041384	C₂	semidirect
13. #26041287	C₂	semidirect
14. C₂×#19153739	C₂	semidirect
174. C₁₈₁	#3312	semidirect

Maximal subgroups

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

subgroup	index
D₄×Dic₃₆₂	2
#26041649	2
#26041620	2
#26041609	2
#26041583	2
#26041535	2
#26041518	2
#26041489	2
#26041434	2
#26041384	2
#26041351	2
#26041287	2
C₂×#19153739	2
#552×C₁₈₁	2
#3312	181