Subgroups — #27132941

Order 12224.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259321	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₁₉₁	C₂	1	1	yes	maximal, characteristic
3	#20259256	C₂	1	1	yes	maximal, characteristic, complemented
4	#20259255	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₁₉₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#20259243	C₂	1	1	yes	maximal, characteristic
7	#102×C₁₉₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₃₈₂	—	2	1	no	—
9	C₂²×Dic₁₉₁	—	2	1	no	—
10	C₂×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
13	Dic₁₉₁⋊C₄	—	2	1	no	—
14	C₂²⋊Dic₁₉₁	—	2	1	no	—
15	C₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
16	C₂×D₄×C₁₉₁	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
19	Dic₃₈₂	—	4	2	no	—
20	C₂×Dic₁₉₁	—	2	1	no	—
21	C₂×Dic₁₉₁	—	2	1	no	—
22	C₂×Dic₁₉₁	—	4	2	no	—
23	C₂×Dic₁₉₁	D₄	1	1	yes	characteristic
24	C₂×Dic₁₉₁	D₄	1	1	yes	characteristic
25	C₁₉₁⋊C₈	—	2	1	no	—
26	C₂³×C₁₉₁	—	2	1	no	—
27	D₄×C₁₉₁	—	2	1	no	—
28	D₄×C₁₉₁	Q₈	1	2	yes	complemented
29	C₂×C₄×C₁₉₁	—	2	1	no	—
30	C₂×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
31	C₈×C₁₉₁	—	2	1	no	—
32	C₂²×C₁₉₁	—	2	2	no	—
33	C₂²×C₁₉₁	—	4	1	no	—
34	C₂²×C₁₉₁	—	4	1	no	—
35	C₂²×C₁₉₁	C₂×D₄	1	1	yes	characteristic
36	C₄×C₁₉₁	—	4	1	no	—
37	C₄×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
38	C₄×C₁₉₁	#41	1	1	yes	characteristic
39	Dic₁₉₁	—	2	1	no	—
40	Dic₁₉₁	—	4	1	no	—
41	Dic₁₉₁	—	4	1	no	—
42	Dic₁₉₁	D₈	1	2	yes	—
43	C₂×C₁₉₁	—	4	2	no	—
44	C₂×C₁₉₁	C₂²⋊Q₈	1	1	yes	characteristic
45	C₂×C₁₉₁	C₂×D₈	1	1	yes	characteristic
46	C₂×C₁₉₁	#137	1	1	yes	characteristic
47	C₁₉₁	#474	1	1	yes	characteristic, complemented, Sylow
48	#474	—	191	1	no	maximal, Sylow
49	C₄⋊Q₈	—	191	1	no	—
50	C₄×D₄	—	191	1	no	—
51	#107	—	191	1	no	—
52	#107	—	191	1	no	—
53	C₄⋊C₈	—	191	1	no	—
54	#102	—	191	1	no	—
55	#102	D₁₉₁	1	1	yes	characteristic
56	C₂×Q₈	—	382	1	no	—
57	C₂×D₄	C₂×D₁₉₁	1	1	yes	characteristic
58	C₂²×C₄	—	382	1	no	—
59	C₂×C₈	—	191	1	no	—
60	C₂×C₈	C₂×D₁₉₁	1	1	yes	characteristic
61	C₄⋊C₄	—	382	1	no	—
62	C₄⋊C₄	—	191	1	no	—
63	C₄⋊C₄	—	191	1	no	—
64	C₄⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
65	C₂²⋊C₄	—	382	1	no	—
66	C₄²	—	191	1	no	—
67	C₂³	—	2	1	no	—
68	Q₈	—	764	2	no	—
69	D₄	—	2	1	no	—
70	D₄	Dic₃₈₂	1	2	yes	complemented
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	382	1	no	—
73	C₂×C₄	—	382	1	no	—
74	C₂×C₄	—	191	1	no	—
75	C₂×C₄	—	191	1	no	—
76	C₂×C₄	—	764	2	no	—
77	C₂×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	382	1	no	—
80	C₂²	—	2	2	no	—
81	C₂²	—	4	1	no	—
82	C₂²	—	4	1	no	—
83	C₂²	D₄×D₁₉₁	1	1	yes	characteristic
84	C₄	—	4	1	no	—
85	C₄	—	382	1	no	—
86	C₄	—	191	2	no	—
87	C₄	—	764	1	no	—
88	C₄	—	764	1	no	—
89	C₄	C₂×Dic₃₈₂	1	1	yes	characteristic
90	C₄	#15690286	1	1	yes	characteristic
91	C₂	—	4	2	no	—
92	C₂	#20259299	1	1	yes	characteristic
93	C₂	D₈×D₁₉₁	1	1	yes	characteristic
94	C₂	#20259351	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₁
Fitting	#102×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂×D₄
prime radical	C₂×D₄×C₁₉₁	C₂²
hypercenter	#102	D₁₉₁
hypocenter	C₁₉₁	#474
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

Complements

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

N	H ≅ G/N	type
1. #20259321	C₂	semidirect
3. #20259256	C₂	semidirect
4. #20259255	C₂	semidirect
5. Dic₁₉₁⋊C₈	C₂	semidirect
28. D₄×C₁₉₁	Q₈	semidirect
47. C₁₉₁	#474	semidirect
70. D₄	Dic₃₈₂	semidirect

Maximal subgroups

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

subgroup	index
#20259321	2
D₄×Dic₁₉₁	2
#20259256	2
#20259255	2
Dic₁₉₁⋊C₈	2
#20259243	2
#102×C₁₉₁	2
#474	191