Subgroups — #37575112

Order 24448.

880 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20259281	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄⋊Dic₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#27132570	C₂	1	2	yes	maximal, complemented
4	#27132566	C₂	1	2	yes	maximal, complemented
5	#27132543	C₂	1	4	yes	maximal, complemented
6	#27132523	C₂	1	4	yes	maximal, complemented
7	C₂²×#41×C₁₉₁	C₂	1	1	yes	maximal, characteristic
8	C₂³×Dic₁₉₁	C₂²	1	2	yes	—
9	#20259281	—	2	4	no	—
10	C₂×C₄⋊Dic₁₉₁	—	2	4	no	—
11	C₂×C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
14	C₂×C₂²⋊Dic₁₉₁	C₂²	1	2	yes	—
15	C₂×C₂²⋊Dic₁₉₁	C₂²	1	8	yes	—
16	C₂×C₄×Dic₁₉₁	C₂²	1	2	yes	—
17	C₁₉₁⋊#95	C₂²	1	8	yes	—
18	C₂×#41×C₁₉₁	—	2	4	no	—
19	C₂×#41×C₁₉₁	—	2	4	no	—
20	C₂×#41×C₁₉₁	C₄	1	8	yes	complemented
21	C₂²×Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
22	C₂²×D₄×C₁₉₁	C₂²	1	2	yes	—
23	C₂²×D₄×C₁₉₁	C₂²	1	1	yes	characteristic
24	C₂³×C₄×C₁₉₁	C₂²	1	2	yes	—
25	C₂³×C₄×C₁₉₁	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₉₁	—	2	4	no	—
27	C₂²×Dic₁₉₁	—	4	4	no	—
28	C₂²×Dic₁₉₁	C₂³	1	4	yes	—
29	C₂²×Dic₁₉₁	C₂³	1	4	yes	—
30	C₄⋊Dic₁₉₁	—	2	4	no	—
31	C₄⋊Dic₁₉₁	—	2	4	no	—
32	C₄⋊Dic₁₉₁	—	2	4	no	—
33	C₂²⋊Dic₁₉₁	—	2	4	no	—
34	C₂²⋊Dic₁₉₁	—	4	8	no	—
35	C₄×Dic₁₉₁	—	2	4	no	—
36	C₂⁴×C₁₉₁	C₂³	1	2	yes	—
37	C₂⁴×C₁₉₁	C₂³	1	1	yes	characteristic
38	#41×C₁₉₁	—	2	32	no	—
39	C₂×Q₈×C₁₉₁	—	2	2	no	—
40	C₂×Q₈×C₁₉₁	—	2	2	no	—
41	C₂×Q₈×C₁₉₁	C₂×C₄	1	4	yes	complemented
42	C₂×D₄×C₁₉₁	—	2	4	no	—
43	C₂×D₄×C₁₉₁	—	2	4	no	—
44	C₂×D₄×C₁₉₁	—	2	2	no	—
45	C₂×D₄×C₁₉₁	—	2	2	no	—
46	C₂×D₄×C₁₉₁	C₂×C₄	1	8	yes	—
47	C₂×D₄×C₁₉₁	C₂×C₄	1	4	yes	complemented
48	C₂²×C₄×C₁₉₁	—	2	4	no	—
49	C₂²×C₄×C₁₉₁	—	2	4	no	—
50	C₂²×C₄×C₁₉₁	C₂×C₄	1	4	yes	—
51	C₂²×C₄×C₁₉₁	C₂×C₄	1	8	yes	complemented
52	C₂²×C₄×C₁₉₁	D₄	1	4	yes	—
53	C₂²×C₄×C₁₉₁	D₄	1	4	yes	—
54	C₂²×C₄×C₁₉₁	C₂³	1	2	yes	—
55	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
57	C₂×Dic₁₉₁	—	2	4	no	—
58	C₂×Dic₁₉₁	—	2	4	no	—
59	C₂×Dic₁₉₁	—	2	4	no	—
60	C₂×Dic₁₉₁	—	2	8	no	—
61	C₂×Dic₁₉₁	—	2	4	no	—
62	C₂×Dic₁₉₁	—	4	8	no	—
63	C₂³×C₁₉₁	—	2	2	no	—
64	C₂³×C₁₉₁	—	2	2	no	—
65	C₂³×C₁₉₁	—	2	2	no	—
66	C₂³×C₁₉₁	—	2	2	no	—
67	C₂³×C₁₉₁	—	2	4	no	—
68	C₂³×C₁₉₁	—	2	4	no	—
69	C₂³×C₁₉₁	C₂²×C₄	1	4	yes	—
70	C₂³×C₁₉₁	C₂²×C₄	1	2	yes	—
71	C₂³×C₁₉₁	C₂×D₄	1	2	yes	—
72	C₂³×C₁₉₁	C₂×D₄	1	2	yes	—
73	C₂³×C₁₉₁	C₂⁴	1	1	yes	characteristic
74	Q₈×C₁₉₁	—	2	8	no	—
75	D₄×C₁₉₁	—	2	16	no	—
76	D₄×C₁₉₁	—	2	8	no	—
77	C₂×C₄×C₁₉₁	—	2	2	no	—
78	C₂×C₄×C₁₉₁	—	2	2	no	—
79	C₂×C₄×C₁₉₁	—	2	16	no	—
80	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	16	yes	complemented
81	C₂×C₄×C₁₉₁	C₂²×C₄	1	4	yes	—
82	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
87	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
88	C₂²×C₁₉₁	—	2	4	no	—
89	C₂²×C₁₉₁	—	2	2	no	—
90	C₂²×C₁₉₁	—	2	2	no	—
91	C₂²×C₁₉₁	—	2	4	no	—
92	C₂²×C₁₉₁	—	2	4	no	—
93	C₂²×C₁₉₁	—	2	8	no	—
94	C₂²×C₁₉₁	—	4	2	no	—
95	C₂²×C₁₉₁	—	4	2	no	—
96	C₂²×C₁₉₁	—	4	4	no	—
97	C₂²×C₁₉₁	C₂×C₂²⋊C₄	1	4	yes	—
98	C₂²×C₁₉₁	C₂³×C₄	1	1	yes	characteristic
99	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
100	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
101	C₂²×C₁₉₁	H(5,𝔽₂)	1	2	yes	—
102	C₂²×C₁₉₁	2₋¹⁺⁴	1	2	yes	—
103	C₄×C₁₉₁	—	2	4	no	—
104	C₄×C₁₉₁	C₂×C₂²⋊C₄	1	4	yes	—
105	C₄×C₁₉₁	C₂×C₂²⋊C₄	1	4	yes	—
106	Dic₁₉₁	—	4	4	no	—
107	Dic₁₉₁	—	4	4	no	—
108	C₂×C₁₉₁	—	2	4	no	—
109	C₂×C₁₉₁	—	4	4	no	—
110	C₂×C₁₉₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₁₉₁	#520	1	2	yes	—
112	C₂×C₁₉₁	#535	1	1	yes	characteristic
113	C₂×C₁₉₁	#536	1	1	yes	characteristic
114	C₂×C₁₉₁	#537	1	2	yes	—
115	C₁₉₁	#2311	1	1	yes	characteristic, complemented, Sylow
116	#2311	—	191	1	no	maximal, Sylow
117	C₂²×#41	D₁₉₁	1	1	yes	characteristic
118	C₂×#117	—	191	1	no	—
119	C₂²×C₄⋊C₄	—	191	1	no	—
120	#391	—	191	2	no	—
121	#390	—	191	2	no	—
122	#386	—	191	4	no	—
123	#380	—	191	4	no	—
124	C₂×#41	—	2	4	no	—
125	C₂×#41	—	2	4	no	—
126	C₂×#41	Dic₁₉₁	1	8	yes	complemented
127	C₂²×Q₈	C₂×D₁₉₁	1	1	yes	characteristic
128	C₂²×D₄	C₂×D₁₉₁	1	2	yes	—
129	C₂²×D₄	C₂×D₁₉₁	1	1	yes	characteristic
130	C₂³×C₄	—	191	2	no	—
131	C₂³×C₄	C₂×D₁₉₁	1	2	yes	—
132	C₂³×C₄	C₂×D₁₉₁	1	1	yes	characteristic
133	#117	—	382	4	no	—
134	C₂×C₄⋊C₄	—	382	4	no	—
135	C₂×C₄⋊C₄	—	191	2	no	—
136	C₂×C₄⋊C₄	—	191	2	no	—
137	C₂×C₄⋊C₄	—	191	2	no	—
138	C₂×C₂²⋊C₄	—	191	2	no	—
139	C₂×C₂²⋊C₄	—	191	8	no	—
140	C₂×C₄²	—	191	2	no	—
141	#95	—	191	8	no	—
142	C₂⁴	C₂²×D₁₉₁	1	2	yes	—
143	C₂⁴	C₂²×D₁₉₁	1	1	yes	characteristic
144	#41	—	2	32	no	—
145	C₂×Q₈	—	2	2	no	—
146	C₂×Q₈	—	2	2	no	—
147	C₂×Q₈	C₂×Dic₁₉₁	1	4	yes	complemented
148	C₂×D₄	—	2	4	no	—
149	C₂×D₄	—	2	4	no	—
150	C₂×D₄	—	2	2	no	—
151	C₂×D₄	—	2	2	no	—
152	C₂×D₄	C₂×Dic₁₉₁	1	8	yes	—
153	C₂×D₄	C₂×Dic₁₉₁	1	4	yes	complemented
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	382	4	no	—
157	C₂²×C₄	—	191	4	no	—
158	C₂²×C₄	—	191	4	no	—
159	C₂²×C₄	—	764	4	no	—
160	C₂²×C₄	C₂×Dic₁₉₁	1	4	yes	—
161	C₂²×C₄	C₂×Dic₁₉₁	1	8	yes	complemented
162	C₂²×C₄	C₂²⋊D₁₉₁	1	4	yes	—
163	C₂²×C₄	C₂²⋊D₁₉₁	1	4	yes	—
164	C₂²×C₄	C₂²×D₁₉₁	1	2	yes	—
165	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
166	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
167	C₄⋊C₄	—	382	4	no	—
168	C₄⋊C₄	—	382	4	no	—
169	C₄⋊C₄	—	382	4	no	—
170	C₂²⋊C₄	—	382	4	no	—
171	C₂²⋊C₄	—	764	8	no	—
172	C₄²	—	382	4	no	—
173	C₂³	—	2	2	no	—
174	C₂³	—	2	2	no	—
175	C₂³	—	2	2	no	—
176	C₂³	—	2	2	no	—
177	C₂³	—	2	4	no	—
178	C₂³	—	2	4	no	—
179	C₂³	C₂²×Dic₁₉₁	1	4	yes	—
180	C₂³	C₂²×Dic₁₉₁	1	2	yes	—
181	C₂³	C₂×C₂²⋊D₁₉₁	1	2	yes	—
182	C₂³	C₂×C₂²⋊D₁₉₁	1	2	yes	—
183	C₂³	C₂³×D₁₉₁	1	1	yes	characteristic
184	Q₈	—	2	8	no	—
185	D₄	—	2	16	no	—
186	D₄	—	2	8	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	16	no	—
190	C₂×C₄	—	382	4	no	—
191	C₂×C₄	—	382	4	no	—
192	C₂×C₄	—	382	4	no	—
193	C₂×C₄	—	382	8	no	—
194	C₂×C₄	—	382	4	no	—
195	C₂×C₄	—	764	8	no	—
196	C₂×C₄	C₂²⋊Dic₁₉₁	1	16	yes	complemented
197	C₂×C₄	C₂²×Dic₁₉₁	1	4	yes	—
198	C₂×C₄	C₂²×Dic₁₉₁	1	2	yes	—
199	C₂×C₄	C₂²×Dic₁₉₁	1	2	yes	—
200	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	2	yes	—
201	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	2	yes	—
202	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	2	yes	—
203	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	2	yes	—
204	C₂²	—	2	2	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	4	no	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	4	no	—
209	C₂²	—	2	8	no	—
210	C₂²	—	4	2	no	—
211	C₂²	—	4	2	no	—
212	C₂²	—	4	4	no	—
213	C₂²	C₂×C₂²⋊Dic₁₉₁	1	4	yes	—
214	C₂²	C₂³×Dic₁₉₁	1	1	yes	characteristic
215	C₂²	C₂²×C₂²⋊D₁₉₁	1	1	yes	characteristic
216	C₂²	C₂²×C₂²⋊D₁₉₁	1	1	yes	characteristic
217	C₂²	#20259368	1	2	yes	—
218	C₂²	#20259369	1	2	yes	—
219	C₄	—	2	4	no	—
220	C₄	—	764	4	no	—
221	C₄	—	764	4	no	—
222	C₄	C₂×C₂²⋊Dic₁₉₁	1	4	yes	—
223	C₄	C₂×C₂²⋊Dic₁₉₁	1	4	yes	—
224	C₂	—	2	4	no	—
225	C₂	—	4	4	no	—
226	C₂	C₂²×C₂²⋊Dic₁₉₁	1	1	yes	characteristic
227	C₂	#27133146	1	2	yes	—
228	C₂	#27133227	1	1	yes	characteristic
229	C₂	#27133234	1	1	yes	characteristic
230	C₂	#27133241	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉₁
commutator	C₂²×C₁₉₁	C₂³×C₄
Fitting, prime radical	C₂²×#41×C₁₉₁	C₂
socle, cosocle	C₂³×C₁₉₁	C₂⁴
hypercenter	C₂²×#41	D₁₉₁
hypocenter	C₁₉₁	#2311
LCS	G ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#20259281	C₂	semidirect
2. C₂²×C₄⋊Dic₁₉₁	C₂	semidirect
3. #27132570	C₂	semidirect
4. #27132566	C₂	semidirect
5. #27132543	C₂	semidirect
6. #27132523	C₂	semidirect
20. C₂×#41×C₁₉₁	C₄	semidirect
41. C₂×Q₈×C₁₉₁	C₂×C₄	semidirect
47. C₂×D₄×C₁₉₁	C₂×C₄	semidirect
51. C₂²×C₄×C₁₉₁	C₂×C₄	semidirect
80. C₂×C₄×C₁₉₁	C₂²⋊C₄	semidirect
115. C₁₉₁	#2311	semidirect
126. C₂×#41	Dic₁₉₁	semidirect
147. C₂×Q₈	C₂×Dic₁₉₁	semidirect
153. C₂×D₄	C₂×Dic₁₉₁	semidirect
161. C₂²×C₄	C₂×Dic₁₉₁	semidirect
196. C₂×C₄	C₂²⋊Dic₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20259281	2
C₂²×C₄⋊Dic₁₉₁	2
#27132570	2
#27132566	2
#27132543	2
#27132523	2
C₂²×#41×C₁₉₁	2
#2311	191