Subgroups — #27133391

Order 12224.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259322	C₂	1	1	yes	maximal, characteristic
2	#20259320	C₂	1	2	yes	maximal
3	#20259314	C₂	1	2	yes	maximal
4	#20259312	C₂	1	4	yes	maximal
5	C₄×Dic₃₈₂	C₂	1	2	yes	maximal
6	C₄×Dic₃₈₂	C₂	1	1	yes	maximal, characteristic
7	#20259287	C₂	1	2	yes	maximal
8	C₄×Q₈×C₁₉₁	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₃₈₂	C₂²	1	2	yes	—
10	C₂×Dic₃₈₂	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
13	C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
14	Dic₁₉₁⋊C₄	C₂²	1	4	yes	—
15	Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
16	Dic₁₉₁⋊C₄	C₂²	1	4	yes	—
17	Dic₁₉₁⋊C₄	C₂²	1	4	yes	—
18	C₄×Dic₁₉₁	C₂²	1	4	yes	—
19	C₄×Dic₁₉₁	C₂²	1	2	yes	—
20	C₂×Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×C₁₉₁	C₂²	1	2	yes	—
23	C₄²×C₁₉₁	C₂²	1	2	yes	—
24	C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
25	Dic₃₈₂	—	2	4	no	—
26	Dic₃₈₂	Q₈	1	4	yes	complemented
27	C₂×Dic₁₉₁	C₂³	1	2	yes	—
28	C₂×Dic₁₉₁	C₂³	1	4	yes	—
29	C₂×Dic₁₉₁	C₂³	1	2	yes	—
30	Q₈×C₁₉₁	—	2	2	no	—
31	C₂×C₄×C₁₉₁	C₂³	1	2	yes	—
32	C₂×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₁₉₁	C₂³	1	2	yes	—
34	C₂×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
36	C₂²×C₁₉₁	C₂⁴	1	1	yes	characteristic
37	C₄×C₁₉₁	—	2	2	no	—
38	C₄×C₁₉₁	—	2	2	no	—
39	C₄×C₁₉₁	—	2	1	no	—
40	C₄×C₁₉₁	C₂×Q₈	1	2	yes	—
41	C₄×C₁₉₁	#41	1	2	yes	—
42	Dic₁₉₁	—	2	4	no	—
43	Dic₁₉₁	—	2	2	no	—
44	Dic₁₉₁	C₂×Q₈	1	4	yes	—
45	C₂×C₁₉₁	C₂²×Q₈	1	1	yes	characteristic
46	C₂×C₁₉₁	C₂×#41	1	1	yes	characteristic
47	C₂×C₁₉₁	2₋¹⁺⁴	1	1	yes	characteristic
48	C₁₉₁	#557	1	1	yes	characteristic, complemented, Sylow
49	#557	—	191	1	no	maximal, Sylow
50	C₄⋊Q₈	—	191	2	no	—
51	C₄⋊Q₈	—	191	1	no	—
52	#125	—	191	4	no	—
53	#125	—	191	2	no	—
54	C₄×Q₈	—	191	2	no	—
55	C₄×Q₈	—	191	2	no	—
56	C₄×Q₈	—	191	1	no	—
57	C₄×Q₈	D₁₉₁	1	1	yes	characteristic
58	C₂×Q₈	—	191	2	no	—
59	C₂×Q₈	—	191	1	no	—
60	C₂×Q₈	C₂×D₁₉₁	1	1	yes	characteristic
61	C₄⋊C₄	—	191	4	no	—
62	C₄⋊C₄	—	191	2	no	—
63	C₄⋊C₄	—	191	4	no	—
64	C₄⋊C₄	—	191	4	no	—
65	C₄⋊C₄	—	191	1	no	—
66	C₄⋊C₄	—	191	2	no	—
67	C₄⋊C₄	—	191	2	no	—
68	C₄⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
69	C₄⋊C₄	C₂×D₁₉₁	1	2	yes	—
70	C₄²	—	191	4	no	—
71	C₄²	—	191	2	no	—
72	C₄²	C₂×D₁₉₁	1	2	yes	—
73	C₄²	C₂×D₁₉₁	1	1	yes	characteristic
74	Q₈	—	2	2	no	—
75	Q₈	—	382	4	no	—
76	Q₈	—	191	4	no	—
77	C₂×C₄	—	191	2	no	—
78	C₂×C₄	—	191	4	no	—
79	C₂×C₄	—	191	2	no	—
80	C₂×C₄	C₂²×D₁₉₁	1	2	yes	—
81	C₂×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
82	C₂×C₄	C₂²×D₁₉₁	1	2	yes	—
83	C₂×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
84	C₂×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
85	C₂²	C₂³×D₁₉₁	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	2	2	no	—
88	C₄	—	2	1	no	—
89	C₄	—	382	4	no	—
90	C₄	—	382	2	no	—
91	C₄	—	191	4	no	—
92	C₄	Q₈×D₁₉₁	1	2	yes	—
93	C₄	#15690287	1	2	yes	—
94	C₂	C₂×Q₈×D₁₉₁	1	1	yes	characteristic
95	C₂	C₂×#15690287	1	1	yes	characteristic
96	C₂	#20259369	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₉₁
commutator, socle, prime radical, cosocle	C₂²×C₁₉₁	C₂⁴
Fitting	C₄×Q₈×C₁₉₁	C₂
hypercenter	C₄×Q₈	D₁₉₁
hypocenter	C₁₉₁	#557
LCS	G ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
26. Dic₃₈₂	Q₈	semidirect
48. C₁₉₁	#557	semidirect

Maximal subgroups

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

subgroup	index
#20259322	2
#20259320	2
#20259314	2
#20259312	2
C₄×Dic₃₈₂	2
#20259287	2
C₄×Q₈×C₁₉₁	2
#557	191