Subgroups — #37546623

Order 23168.

374 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26041985	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₃₆₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#26041689	C₂	1	1	yes	maximal, characteristic, complemented
4	#26041687	C₂	1	1	yes	maximal, characteristic, complemented
5	#26041676	C₂	1	1	yes	maximal, characteristic, complemented
6	#26041581	C₂	1	1	yes	maximal, characteristic, complemented
7	#26041570	C₂	1	1	yes	maximal, characteristic, complemented
8	#26041568	C₂	1	1	yes	maximal, characteristic, complemented
9	#26041567	C₂	1	1	yes	maximal, characteristic, complemented
10	#26041558	C₂	1	1	yes	maximal, characteristic, complemented
11	#26041408	C₂	1	1	yes	maximal, characteristic, complemented
12	#26041389	C₂	1	1	yes	maximal, characteristic
13	#26041388	C₂	1	1	yes	maximal, characteristic
14	#26041359	C₂	1	1	yes	maximal, characteristic
15	#478×C₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Dic₃₆₂	—	2	1	no	—
17	C₂×#15668434	C₂²	1	1	yes	characteristic
18	#19153816	C₂²	1	1	yes	characteristic
19	#19153815	C₂²	1	1	yes	characteristic
20	#19153815	C₂²	1	1	yes	characteristic
21	#19153815	C₂²	1	1	yes	characteristic
22	#19153807	C₂²	1	1	yes	characteristic
23	#19153796	—	2	1	no	—
24	#19153795	—	2	1	no	—
25	#19153794	—	2	1	no	—
26	#19153794	—	2	1	no	—
27	#19153793	—	2	1	no	—
28	#19153793	—	2	1	no	—
29	C₄×Dic₃₆₂	C₂²	1	1	yes	characteristic
30	C₄×D₇₂₄	C₂²	1	1	yes	characteristic
31	D₄×Dic₁₈₁	C₂²	1	1	yes	characteristic
32	D₄×Dic₁₈₁	C₂²	1	1	yes	characteristic
33	#19153778	C₂²	1	1	yes	characteristic
34	#19153778	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₁₈₁	—	2	1	no	—
36	#19153749	C₂²	1	1	yes	characteristic
37	#19153749	C₂²	1	1	yes	characteristic
38	#19153748	C₂²	1	1	yes	characteristic
39	#19153748	C₂²	1	1	yes	characteristic
40	#19153747	C₂²	1	1	yes	characteristic
41	#19153747	C₂²	1	1	yes	characteristic
42	#19153747	C₂²	1	1	yes	characteristic
43	#19153745	C₂²	1	1	yes	characteristic
44	#19153745	C₂²	1	1	yes	characteristic
45	#19153738	C₂²	1	1	yes	characteristic
46	#19153738	C₂²	1	1	yes	characteristic
47	#19153737	C₂²	1	1	yes	characteristic
48	#19153736	C₂²	1	1	yes	characteristic
49	#19153727	C₂²	1	1	yes	characteristic
50	#19153726	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₈₁	C₂²	1	1	yes	characteristic
52	#125×C₁₈₁	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₁₈₁	C₂²	1	1	yes	characteristic
54	#107×C₁₈₁	C₂²	1	1	yes	characteristic
55	#106×C₁₈₁	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₈₁	C₂²	1	1	yes	characteristic
57	#102×C₁₈₁	C₂²	1	1	yes	characteristic
58	#102×C₁₈₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₃₆₂	—	2	1	no	—
60	C₂×Dic₃₆₂	—	2	1	no	—
61	C₂×Dic₃₆₂	—	2	1	no	—
62	C₂×Dic₃₆₂	—	2	1	no	—
63	C₂×Dic₃₆₂	—	4	4	no	—
64	C₂×Dic₃₆₂	D₄	1	1	yes	characteristic
65	C₂×D₇₂₄	C₂³	1	1	yes	characteristic
66	C₂²×Dic₁₈₁	—	2	1	no	—
67	C₂²×Dic₁₈₁	—	2	1	no	—
68	C₂×C₄×D₁₈₁	—	2	1	no	—
69	C₂×C₄×D₁₈₁	—	2	1	no	—
70	C₂×C₄×D₁₈₁	—	2	1	no	—
71	#15668434	Q₈	1	4	yes	complemented
72	C₂×#13089678	C₂³	1	1	yes	characteristic
73	C₂×#13089678	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₈₁	—	2	1	no	—
75	C₄⋊Dic₁₈₁	—	2	1	no	—
76	C₄⋊Dic₁₈₁	D₄	1	1	yes	characteristic
77	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
80	#15668428	—	2	1	no	—
81	#15668428	—	2	1	no	—
82	#15668428	—	2	1	no	—
83	#15668428	—	2	1	no	—
84	#15668428	D₄	1	1	yes	characteristic
85	#15668428	D₄	1	1	yes	characteristic
86	C₂²⋊Dic₁₈₁	—	2	1	no	—
87	C₂²⋊Dic₁₈₁	—	2	1	no	—
88	#15668426	—	2	1	no	—
89	#15668426	—	2	1	no	—
90	#15668426	—	2	1	no	—
91	C₄×Dic₁₈₁	C₂³	1	1	yes	characteristic
92	C₄×Dic₁₈₁	C₂³	1	1	yes	characteristic
93	C₂×D₄×C₁₈₁	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₁₈₁	—	2	1	no	—
95	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
96	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
97	C₄⋊C₄×C₁₈₁	—	2	1	no	—
98	C₄⋊C₄×C₁₈₁	—	2	1	no	—
99	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
102	C₂²⋊C₄×C₁₈₁	—	2	1	no	—
103	C₄²×C₁₈₁	C₂³	1	1	yes	characteristic
104	C₂²×D₁₈₁	—	2	1	no	—
105	Dic₃₆₂	—	2	2	no	—
106	Dic₃₆₂	—	4	2	no	—
107	Dic₃₆₂	—	4	2	no	—
108	Dic₃₆₂	—	4	2	no	—
109	Dic₃₆₂	—	8	1	no	—
110	D₇₂₄	—	2	1	no	—
111	D₇₂₄	C₂×Q₈	1	2	yes	complemented
112	C₂×Dic₁₈₁	—	2	1	no	—
113	C₂×Dic₁₈₁	—	2	1	no	—
114	C₂×Dic₁₈₁	—	2	1	no	—
115	C₂×Dic₁₈₁	—	4	2	no	—
116	C₂×Dic₁₈₁	—	4	2	no	—
117	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
118	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
121	C₄×D₁₈₁	—	4	2	no	—
122	C₄×D₁₈₁	—	8	1	no	—
123	C₄×D₁₈₁	—	8	1	no	—
124	#13089678	—	2	1	no	—
125	#13089678	C₂×Q₈	1	2	yes	complemented
126	C₂³×C₁₈₁	—	2	1	no	—
127	D₄×C₁₈₁	—	2	1	no	—
128	D₄×C₁₈₁	C₂×Q₈	1	2	yes	—
129	C₂×C₄×C₁₈₁	—	2	1	no	—
130	C₂×C₄×C₁₈₁	—	2	1	no	—
131	C₂×C₄×C₁₈₁	—	2	1	no	—
132	C₂×C₄×C₁₈₁	—	4	2	no	—
133	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₈₁	C₂⁴	1	1	yes	characteristic
136	C₈×C₁₈₁	—	2	1	no	—
137	C₈×C₁₈₁	—	2	1	no	—
138	C₂²×C₁₈₁	—	2	2	no	—
139	C₂²×C₁₈₁	—	4	1	no	—
140	C₂²×C₁₈₁	—	4	1	no	—
141	C₂²×C₁₈₁	C₂²×D₄	1	1	yes	characteristic
142	C₂×D₁₈₁	—	2	2	no	—
143	C₂×D₁₈₁	—	4	1	no	—
144	C₂×D₁₈₁	—	4	1	no	—
145	C₄×C₁₈₁	—	2	1	no	—
146	C₄×C₁₈₁	—	2	1	no	—
147	C₄×C₁₈₁	—	4	1	no	—
148	C₄×C₁₈₁	—	4	1	no	—
149	C₄×C₁₈₁	—	4	1	no	—
150	C₄×C₁₈₁	C₂²×Q₈	1	1	yes	characteristic
151	C₄×C₁₈₁	2₋¹⁺⁴	1	1	yes	characteristic
152	Dic₁₈₁	—	2	1	no	—
153	Dic₁₈₁	—	2	1	no	—
154	Dic₁₈₁	—	2	1	no	—
155	Dic₁₈₁	—	4	1	no	—
156	Dic₁₈₁	—	4	1	no	—
157	Dic₁₈₁	—	4	1	no	—
158	Dic₁₈₁	AGL(1,ℤ/8)	1	2	yes	—
159	C₂×C₁₈₁	—	4	2	no	—
160	C₂×C₁₈₁	D₄×Q₈	1	1	yes	characteristic
161	C₂×C₁₈₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
162	C₂×C₁₈₁	#577	1	1	yes	characteristic
163	D₁₈₁	—	4	2	no	—
164	C₁₈₁	#3384	1	1	yes	characteristic, complemented, Sylow
165	#3384	—	181	1	no	maximal, Sylow
166	#554	—	181	1	no	—
167	D₄×Q₈	—	181	1	no	—
168	#501	—	181	1	no	—
169	#501	—	181	1	no	—
170	#498	—	181	1	no	—
171	#478	—	181	1	no	—
172	#478	D₁₈₁	1	1	yes	characteristic, complemented
173	#476	—	181	1	no	—
174	#476	—	181	1	no	—
175	#476	—	181	1	no	—
176	#474	—	181	1	no	—
177	#446	—	181	1	no	—
178	#442	—	181	1	no	—
179	#442	—	181	1	no	—
180	C₄×D₈	—	181	1	no	—
181	C₂²×Q₈	—	362	1	no	—
182	C₂×D₈	—	181	1	no	—
183	C₄⋊Q₈	—	181	1	no	—
184	C₄⋊Q₈	—	181	1	no	—
185	C₄⋊Q₈	—	181	1	no	—
186	C₄⋊Q₈	—	181	1	no	—
187	#125	—	181	1	no	—
188	#125	C₂×D₁₈₁	1	1	yes	characteristic
189	C₂²⋊Q₈	—	362	1	no	—
190	C₂²⋊Q₈	—	362	1	no	—
191	C₂²⋊Q₈	—	362	1	no	—
192	C₂²⋊Q₈	—	362	1	no	—
193	C₂²⋊Q₈	—	362	1	no	—
194	C₂²⋊Q₈	—	362	1	no	—
195	C₄×Q₈	—	181	1	no	—
196	C₄×D₄	—	181	1	no	—
197	C₄×D₄	—	181	1	no	—
198	C₄×D₄	—	181	1	no	—
199	C₄×D₄	—	181	1	no	—
200	C₄×D₄	—	181	1	no	—
201	C₄×D₄	C₂×D₁₈₁	1	1	yes	characteristic
202	C₂×C₄⋊C₄	—	362	1	no	—
203	#107	—	181	1	no	—
204	#107	—	181	1	no	—
205	#107	C₂×D₁₈₁	1	1	yes	characteristic
206	#106	—	181	1	no	—
207	#106	—	181	1	no	—
208	#106	—	181	1	no	—
209	#106	—	181	1	no	—
210	#106	—	181	1	no	—
211	#106	C₂×D₁₈₁	1	1	yes	characteristic
212	C₄⋊C₈	—	181	1	no	—
213	C₄⋊C₈	—	181	1	no	—
214	C₄⋊C₈	C₂×D₁₈₁	1	1	yes	characteristic
215	#102	—	181	1	no	—
216	#102	—	181	1	no	—
217	#102	—	181	1	no	—
218	#102	—	181	1	no	—
219	#102	C₂×D₁₈₁	1	1	yes	characteristic
220	#102	C₂×D₁₈₁	1	1	yes	characteristic
221	#97	—	181	1	no	—
222	#97	—	181	1	no	—
223	C₄×C₈	—	181	1	no	—
224	C₂×Q₈	—	362	1	no	—
225	C₂×Q₈	—	362	1	no	—
226	C₂×Q₈	—	362	1	no	—
227	C₂×Q₈	—	362	1	no	—
228	C₂×Q₈	—	181	1	no	—
229	C₂×Q₈	—	724	4	no	—
230	C₂×D₄	—	181	1	no	—
231	C₂×D₄	C₂²×D₁₈₁	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	362	1	no	—
234	C₂²×C₄	—	362	1	no	—
235	C₂²×C₄	—	362	1	no	—
236	C₂²×C₄	—	362	1	no	—
237	C₂²×C₄	—	362	1	no	—
238	D₈	—	181	4	no	—
239	C₂×C₈	—	181	1	no	—
240	C₂×C₈	—	181	1	no	—
241	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
242	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	362	1	no	—
246	C₄⋊C₄	—	362	1	no	—
247	C₄⋊C₄	—	362	1	no	—
248	C₄⋊C₄	—	362	1	no	—
249	C₄⋊C₄	—	362	1	no	—
250	C₄⋊C₄	—	362	1	no	—
251	C₄⋊C₄	—	181	1	no	—
252	C₄⋊C₄	—	181	1	no	—
253	C₄⋊C₄	—	181	1	no	—
254	C₄⋊C₄	—	181	1	no	—
255	C₄⋊C₄	—	181	1	no	—
256	C₄⋊C₄	—	181	1	no	—
257	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
258	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
259	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	362	1	no	—
262	C₂²⋊C₄	—	362	1	no	—
263	C₂²⋊C₄	—	362	1	no	—
264	C₂²⋊C₄	—	362	1	no	—
265	C₂²⋊C₄	—	362	1	no	—
266	C₄²	—	181	1	no	—
267	C₄²	—	181	1	no	—
268	C₄²	C₂²×D₁₈₁	1	1	yes	characteristic
269	C₂³	—	2	1	no	—
270	C₂³	—	362	1	no	—
271	Q₈	—	362	2	no	—
272	Q₈	—	1448	1	no	—
273	Q₈	—	724	2	no	—
274	Q₈	—	724	2	no	—
275	Q₈	—	724	2	no	—
276	D₄	—	2	1	no	—
277	D₄	—	362	1	no	—
278	D₄	—	181	2	no	—
279	D₄	Q₈×D₁₈₁	1	2	yes	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	362	1	no	—
285	C₂×C₄	—	362	1	no	—
286	C₂×C₄	—	362	1	no	—
287	C₂×C₄	—	1448	1	no	—
288	C₂×C₄	—	1448	1	no	—
289	C₂×C₄	—	181	1	no	—
290	C₂×C₄	—	181	1	no	—
291	C₂×C₄	—	181	1	no	—
292	C₂×C₄	—	181	1	no	—
293	C₂×C₄	—	724	2	no	—
294	C₂×C₄	—	724	2	no	—
295	C₂×C₄	—	724	2	no	—
296	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
297	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₁₈₁	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	362	1	no	—
302	C₈	—	181	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	362	2	no	—
307	C₂²	—	724	1	no	—
308	C₂²	—	724	1	no	—
309	C₂²	C₂×D₄×D₁₈₁	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	362	1	no	—
316	C₄	—	362	1	no	—
317	C₄	—	362	1	no	—
318	C₄	—	181	2	no	—
319	C₄	—	724	1	no	—
320	C₄	—	724	1	no	—
321	C₄	—	724	1	no	—
322	C₄	C₂×Q₈×D₁₈₁	1	1	yes	characteristic
323	C₄	#19153863	1	1	yes	characteristic
324	C₂	—	4	2	no	—
325	C₂	—	724	2	no	—
326	C₂	#26041942	1	1	yes	characteristic
327	C₂	AGL(1,ℤ/8)×D₁₈₁	1	1	yes	characteristic
328	C₂	#26042086	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₈₁
commutator, cosocle	C₂×C₄×C₁₈₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₈₁
Fitting	#478×C₁₈₁	C₂
socle	C₂²×C₁₈₁	C₂²×D₄
prime radical	C₂×#15668434	C₂²
hypercenter	#478	D₁₈₁
hypocenter	C₁₈₁	#3384
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #26041985	C₂	semidirect
2. D₄×Dic₃₆₂	C₂	semidirect
3. #26041689	C₂	semidirect
4. #26041687	C₂	semidirect
5. #26041676	C₂	semidirect
6. #26041581	C₂	semidirect
7. #26041570	C₂	semidirect
8. #26041568	C₂	semidirect
9. #26041567	C₂	semidirect
10. #26041558	C₂	semidirect
11. #26041408	C₂	semidirect
15. #478×C₁₈₁	C₂	semidirect
71. #15668434	Q₈	semidirect
111. D₇₂₄	C₂×Q₈	semidirect
125. #13089678	C₂×Q₈	semidirect
164. C₁₈₁	#3384	semidirect
172. #478	D₁₈₁	semidirect

Maximal subgroups

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

subgroup	index
#26041985	2
D₄×Dic₃₆₂	2
#26041689	2
#26041687	2
#26041676	2
#26041581	2
#26041570	2
#26041568	2
#26041567	2
#26041558	2
#26041408	2
#26041389	2
#26041388	2
#26041359	2
#478×C₁₈₁	2
#3384	181