Subgroups — #37550519

Order 23168.

356 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26042331	C₂	1	2	yes	maximal, complemented
2	#26042331	C₂	1	2	yes	maximal, complemented
3	#26042329	C₂	1	2	yes	maximal, complemented
4	#26042328	C₂	1	2	yes	maximal, complemented
5	#26042286	C₂	1	1	yes	maximal, characteristic
6	#26042286	C₂	1	1	yes	maximal, characteristic
7	D₁₈₁⋊(C₄⋊C₈)	C₂	1	1	yes	maximal, characteristic
8	#26042244	C₂	1	1	yes	maximal, characteristic
9	#26042244	C₂	1	1	yes	maximal, characteristic
10	C₄×#15668459	C₂	1	1	yes	maximal, characteristic
11	#26041835	C₂	1	1	yes	maximal, characteristic
12	C₂×#15668459	C₂²	1	1	yes	characteristic
13	C₂×#15668459	C₂²	1	1	yes	characteristic
14	C₂×D₁₈₁⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×D₁₈₁⋊C₈	C₂²	1	1	yes	characteristic
16	#19153883	C₂²	1	1	yes	characteristic
17	#19153883	C₂²	1	1	yes	characteristic
18	#19153883	C₂²	1	2	yes	—
19	#19153883	C₂²	1	2	yes	—
20	C₄⋊#13089685	C₂²	1	1	yes	characteristic
21	C₄⋊#13089685	C₂²	1	1	yes	characteristic
22	C₄⋊#13089685	C₂²	1	2	yes	—
23	C₄⋊#13089685	C₂²	1	2	yes	—
24	#19153871	C₂²	1	2	yes	—
25	#19153871	C₂²	1	2	yes	—
26	#19153871	C₂²	1	2	yes	—
27	C₄×#13089685	C₂²	1	2	yes	—
28	C₄×#13089685	C₂²	1	2	yes	—
29	C₄×#13089685	C₂²	1	2	yes	—
30	#19153815	C₄	1	1	yes	characteristic
31	#19153815	C₄	1	1	yes	characteristic
32	#19153808	C₄	1	1	yes	characteristic
33	#19153794	C₄	1	2	yes	—
34	#19153794	C₄	1	2	yes	—
35	#19153781	C₂²	1	2	yes	—
36	#19153781	C₂²	1	2	yes	—
37	#19153774	C₂²	1	1	yes	characteristic
38	#19153774	C₂²	1	1	yes	characteristic
39	C₄²×D₁₈₁	C₂²	1	1	yes	characteristic
40	#125×C₁₈₁	C₄	1	1	yes	characteristic
41	#15668459	—	2	2	no	—
42	#15668459	Q₈	1	4	yes	complemented
43	D₁₈₁⋊C₈	—	2	2	no	—
44	D₁₈₁⋊C₈	—	2	2	no	—
45	C₂×#13089685	C₂³	1	1	yes	characteristic
46	C₂×#13089685	C₂³	1	1	yes	characteristic
47	C₂×#13089685	C₂³	1	2	yes	—
48	C₂×#13089685	C₂³	1	1	yes	characteristic
49	C₂×#13089685	C₂³	1	1	yes	characteristic
50	C₂×#13089685	C₂³	1	2	yes	—
51	C₂×Dic₃₆₂	C₂×C₄	1	2	yes	—
52	C₂×Dic₃₆₂	C₂×C₄	1	2	yes	—
53	C₂×C₄×D₁₈₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₁₈₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₁₈₁	C₂³	1	1	yes	characteristic
56	C₄⋊Dic₁₈₁	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Dic₁₈₁	C₂×C₄	1	1	yes	characteristic
58	#15668428	C₂×C₄	1	2	yes	—
59	#15668428	C₂×C₄	1	2	yes	—
60	#15668428	C₂×C₄	1	2	yes	—
61	#15668428	C₂×C₄	1	2	yes	—
62	#15668426	C₂×C₄	1	2	yes	—
63	#15668426	C₂×C₄	1	2	yes	—
64	C₄×Dic₁₈₁	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₈₁	C₂³	1	1	yes	characteristic
66	C₄×Dic₁₈₁	C₂³	1	1	yes	characteristic
67	C₄×Dic₁₈₁	C₂³	1	2	yes	—
68	C₄×Dic₁₈₁	C₂³	1	2	yes	—
69	C₄⋊C₄×C₁₈₁	C₂×C₄	1	1	yes	characteristic
70	C₄⋊C₄×C₁₈₁	C₂×C₄	1	1	yes	characteristic
71	C₄⋊C₄×C₁₈₁	C₂×C₄	1	2	yes	—
72	C₄⋊C₄×C₁₈₁	C₂×C₄	1	2	yes	—
73	C₄²×C₁₈₁	C₂×C₄	1	1	yes	characteristic
74	#13089685	—	2	1	no	—
75	#13089685	—	2	1	no	—
76	#13089685	—	2	1	no	—
77	#13089685	—	2	1	no	—
78	#13089685	—	2	2	no	—
79	#13089685	C₂×Q₈	1	4	yes	—
80	C₂²×D₁₈₁	C₂²×C₄	1	1	yes	characteristic
81	Dic₃₆₂	—	2	4	no	—
82	Dic₃₆₂	—	4	2	no	—
83	C₂×Dic₁₈₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₁₈₁	C₂²×C₄	1	2	yes	—
85	C₂×Dic₁₈₁	C₂²×C₄	1	2	yes	—
86	C₂×Dic₁₈₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₁₈₁	C₂⁴	1	1	yes	characteristic
88	C₄×D₁₈₁	—	2	2	no	—
89	C₄×D₁₈₁	—	4	1	no	—
90	C₄×D₁₈₁	C₂×Q₈	1	2	yes	—
91	C₄×D₁₈₁	#41	1	2	yes	—
92	C₂×C₄×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₈₁	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₁₈₁	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₁₈₁	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₁₈₁	—	2	1	no	—
99	C₂×D₁₈₁	—	2	1	no	—
100	C₂×D₁₈₁	C₄×Q₈	1	2	yes	—
101	C₄×C₁₈₁	—	2	1	no	—
102	C₄×C₁₈₁	—	2	2	no	—
103	C₄×C₁₈₁	—	2	1	no	—
104	C₄×C₁₈₁	—	2	2	no	—
105	C₄×C₁₈₁	C₄×Q₈	1	2	yes	—
106	Dic₁₈₁	—	2	2	no	—
107	Dic₁₈₁	—	2	1	no	—
108	Dic₁₈₁	—	2	2	no	—
109	Dic₁₈₁	#131	1	2	yes	—
110	Dic₁₈₁	C₂²×Q₈	1	1	yes	characteristic
111	Dic₁₈₁	C₂×#41	1	1	yes	characteristic
112	C₂×C₁₈₁	C₂×C₄×Q₈	1	1	yes	characteristic
113	C₂×C₁₈₁	C₂×#131	1	1	yes	characteristic
114	C₂×C₁₈₁	#568	1	1	yes	characteristic
115	D₁₈₁	—	2	2	no	—
116	C₁₈₁	#2962	1	1	yes	characteristic, complemented, Sylow
117	#2962	—	181	1	no	maximal, Sylow
118	#533	—	181	1	no	—
119	#446	—	181	2	no	—
120	#446	—	181	2	no	—
121	#446	—	181	2	no	—
122	C₈×Q₈	—	181	2	no	—
123	#424	—	181	1	no	—
124	#424	—	181	1	no	—
125	C₂×C₄⋊C₈	—	181	1	no	—
126	#405	—	181	1	no	—
127	#405	—	181	1	no	—
128	C₄×M₁₆	—	181	1	no	—
129	C₂×M₁₆	—	181	1	no	—
130	C₂×M₁₆	—	181	1	no	—
131	C₂²×C₈	—	181	1	no	—
132	C₂²×C₈	—	181	1	no	—
133	C₄⋊Q₈	—	181	1	no	—
134	C₄⋊Q₈	—	181	1	no	—
135	#125	—	181	1	no	—
136	#125	#10624382	1	1	yes	characteristic
137	C₂²⋊Q₈	—	181	2	no	—
138	C₂²⋊Q₈	—	181	2	no	—
139	C₄×Q₈	—	181	2	no	—
140	C₄×Q₈	—	181	2	no	—
141	#117	—	181	1	no	—
142	#117	—	181	1	no	—
143	C₂×C₄²	—	181	1	no	—
144	C₄⋊C₈	—	181	1	no	—
145	C₄⋊C₈	—	181	1	no	—
146	C₄⋊C₈	—	181	1	no	—
147	C₄⋊C₈	—	181	1	no	—
148	C₄⋊C₈	—	181	2	no	—
149	C₄⋊C₈	—	181	2	no	—
150	C₄⋊C₈	—	181	2	no	—
151	C₄⋊C₈	—	181	2	no	—
152	#97	—	181	2	no	—
153	#97	—	181	2	no	—
154	#97	—	181	2	no	—
155	C₄×C₈	—	181	2	no	—
156	C₄×C₈	—	181	2	no	—
157	C₄×C₈	—	181	2	no	—
158	C₂×Q₈	—	181	2	no	—
159	C₂×Q₈	—	181	2	no	—
160	C₂²×C₄	—	181	1	no	—
161	C₂²×C₄	—	181	1	no	—
162	C₂²×C₄	—	181	1	no	—
163	M₁₆	—	362	2	no	—
164	M₁₆	—	181	4	no	—
165	C₂×C₈	—	362	2	no	—
166	C₂×C₈	—	362	2	no	—
167	C₂×C₈	—	181	1	no	—
168	C₂×C₈	—	181	1	no	—
169	C₂×C₈	—	181	2	no	—
170	C₂×C₈	—	181	1	no	—
171	C₂×C₈	—	181	1	no	—
172	C₂×C₈	—	181	2	no	—
173	C₄⋊C₄	—	181	1	no	—
174	C₄⋊C₄	—	181	1	no	—
175	C₄⋊C₄	—	181	2	no	—
176	C₄⋊C₄	—	181	2	no	—
177	C₄⋊C₄	—	181	2	no	—
178	C₄⋊C₄	—	181	2	no	—
179	C₄⋊C₄	C₂×#10624382	1	1	yes	characteristic
180	C₄⋊C₄	C₂×#10624382	1	1	yes	characteristic
181	C₄⋊C₄	C₂×#10624382	1	2	yes	—
182	C₄⋊C₄	C₂×#10624382	1	2	yes	—
183	C₂²⋊C₄	—	181	2	no	—
184	C₂²⋊C₄	—	181	2	no	—
185	C₄²	—	181	1	no	—
186	C₄²	—	181	2	no	—
187	C₄²	—	181	2	no	—
188	C₄²	—	181	1	no	—
189	C₄²	—	181	1	no	—
190	C₄²	C₂×#10624382	1	1	yes	characteristic
191	C₂³	—	181	1	no	—
192	Q₈	—	362	4	no	—
193	Q₈	—	724	2	no	—
194	C₂×C₄	—	362	2	no	—
195	C₂×C₄	—	181	1	no	—
196	C₂×C₄	—	181	2	no	—
197	C₂×C₄	—	181	2	no	—
198	C₂×C₄	—	181	1	no	—
199	C₂×C₄	—	181	1	no	—
200	C₂×C₄	—	181	2	no	—
201	C₂×C₄	—	181	2	no	—
202	C₂×C₄	—	724	1	no	—
203	C₂×C₄	C₂²×#10624382	1	1	yes	characteristic
204	C₂×C₄	C₂²×#10624382	1	1	yes	characteristic
205	C₂×C₄	C₂²×#10624382	1	2	yes	—
206	C₂×C₄	C₂²×#10624382	1	2	yes	—
207	C₂×C₄	C₂²×#10624382	1	1	yes	characteristic
208	C₈	—	362	1	no	—
209	C₈	—	362	1	no	—
210	C₈	—	362	1	no	—
211	C₈	—	362	1	no	—
212	C₈	—	362	2	no	—
213	C₈	—	181	4	no	—
214	C₂²	—	362	1	no	—
215	C₂²	—	362	1	no	—
216	C₂²	—	181	2	no	—
217	C₂²	C₂³×#10624382	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	1	no	—
221	C₄	—	2	2	no	—
222	C₄	—	362	2	no	—
223	C₄	—	362	1	no	—
224	C₄	—	362	2	no	—
225	C₄	—	181	1	no	—
226	C₄	—	181	1	no	—
227	C₄	—	181	2	no	—
228	C₄	Q₈×#10624382	1	2	yes	—
229	C₂	—	362	2	no	—
230	C₂	C₂×Q₈×#10624382	1	1	yes	characteristic
231	C₂	D₁₈₁⋊#131	1	1	yes	characteristic
232	C₂	#26042358	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10624382
commutator, socle	C₂²×C₁₈₁	C₂³×C₄
Fitting	#125×C₁₈₁	C₄
prime radical	C₂²×D₁₈₁	C₂²×C₄
cosocle	C₂×Dic₁₈₁	C₂⁴
hypercenter	#125	#10624382
hypocenter	C₁₈₁	#2962
LCS	G ▷ C₂²×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. #26042331	C₂	semidirect
2. #26042331	C₂	semidirect
3. #26042329	C₂	semidirect
4. #26042328	C₂	semidirect
42. #15668459	Q₈	semidirect
116. C₁₈₁	#2962	semidirect

Maximal subgroups

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

subgroup	index
#26042331	2
#26042329	2
#26042328	2
#26042286	2
D₁₈₁⋊(C₄⋊C₈)	2
#26042244	2
C₄×#15668459	2
#26041835	2
#2962	181