Group #27133378 — subgroups

Subgroups — #27133378

Order 12224.

236 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259322	C₂	1	1	yes	maximal, characteristic, complemented
2	#20259308	C₂	1	4	yes	maximal, complemented
3	#20259296	C₂	1	2	yes	maximal, complemented
4	#20259293	C₂	1	2	yes	maximal, complemented
5	C₄×D₇₆₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#20259281	C₂	1	2	yes	maximal, complemented
7	#20259280	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₃₈₂	C₂²	1	2	yes	—
10	C₂×D₇₆₄	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊D₁₉₁	C₂²	1	4	yes	—
12	C₂×C₄×D₁₉₁	C₂²	1	2	yes	—
13	C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
14	C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
16	C₂²⋊Dic₁₉₁	C₂²	1	4	yes	—
17	(C₂×D₁₉₁)⋊C₄	C₂²	1	4	yes	—
18	(C₂×D₁₉₁)⋊C₄	C₂²	1	2	yes	—
19	C₄×Dic₁₉₁	C₂²	1	4	yes	—
20	C₂×D₄×C₁₉₁	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₁₉₁	C₂²	1	2	yes	—
22	C₄⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
23	C₂²⋊C₄×C₁₉₁	C₂²	1	2	yes	—
24	C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
25	C₂²×D₁₉₁	C₂³	1	2	yes	—
26	Dic₃₈₂	—	4	2	no	—
27	D₇₆₄	—	2	2	no	—
28	C₂²⋊D₁₉₁	—	4	4	no	—
29	C₂×Dic₁₉₁	C₂³	1	2	yes	—
30	C₂×Dic₁₉₁	C₂³	1	4	yes	—
31	C₄×D₁₉₁	—	2	4	no	—
32	C₂³×C₁₉₁	C₂³	1	2	yes	—
33	D₄×C₁₉₁	—	2	2	no	—
34	C₂×C₄×C₁₉₁	—	2	4	no	—
35	C₂×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₁₉₁	C₂³	1	2	yes	—
37	C₂×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
39	C₂²×C₁₉₁	—	2	2	no	—
40	C₂²×C₁₉₁	—	2	2	no	—
41	C₂²×C₁₉₁	—	2	2	no	—
42	C₂²×C₁₉₁	C₂⁴	1	1	yes	characteristic
43	C₂×D₁₉₁	—	2	2	no	—
44	C₂×D₁₉₁	—	2	2	no	—
45	C₂×D₁₉₁	—	2	2	no	—
46	C₄×C₁₉₁	—	2	2	no	—
47	C₄×C₁₉₁	—	2	1	no	—
48	C₄×C₁₉₁	#41	1	2	yes	—
49	C₄×C₁₉₁	#41	1	2	yes	—
50	Dic₁₉₁	—	2	2	no	—
51	Dic₁₉₁	—	2	4	no	—
52	C₂×C₁₉₁	—	4	2	no	—
53	C₂×C₁₉₁	C₂×#41	1	1	yes	characteristic
54	C₂×C₁₉₁	C₂×#41	1	1	yes	characteristic
55	C₂×C₁₉₁	H(5,𝔽₂)	1	1	yes	characteristic
56	D₁₉₁	—	4	2	no	—
57	C₁₉₁	#555	1	1	yes	characteristic, complemented, Sylow
58	#555	—	191	1	no	maximal, Sylow
59	C₄⋊Q₈	—	191	1	no	—
60	#124	—	191	4	no	—
61	#121	—	191	2	no	—
62	#121	—	191	2	no	—
63	C₄×D₄	—	191	1	no	—
64	C₄×D₄	D₁₉₁	1	1	yes	characteristic, complemented
65	#117	—	191	2	no	—
66	#117	—	191	2	no	—
67	C₂×Q₈	—	191	2	no	—
68	C₂×D₄	—	191	4	no	—
69	C₂×D₄	—	191	1	no	—
70	C₂×D₄	C₂×D₁₉₁	1	1	yes	characteristic
71	C₂²×C₄	—	191	2	no	—
72	C₂²×C₄	C₂×D₁₉₁	1	2	yes	—
73	C₄⋊C₄	—	191	1	no	—
74	C₄⋊C₄	—	191	2	no	—
75	C₄⋊C₄	—	191	2	no	—
76	C₄⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
77	C₂²⋊C₄	—	191	4	no	—
78	C₂²⋊C₄	—	191	4	no	—
79	C₂²⋊C₄	—	191	2	no	—
80	C₂²⋊C₄	C₂×D₁₉₁	1	2	yes	—
81	C₄²	—	191	4	no	—
82	C₄²	C₂×D₁₉₁	1	1	yes	characteristic
83	C₂³	—	191	2	no	—
84	C₂³	C₂²×D₁₉₁	1	2	yes	—
85	Q₈	—	764	2	no	—
86	D₄	—	2	2	no	—
87	D₄	—	382	2	no	—
88	D₄	—	764	4	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	382	4	no	—
91	C₂×C₄	—	191	2	no	—
92	C₂×C₄	—	191	4	no	—
93	C₂×C₄	C₂²×D₁₉₁	1	2	yes	—
94	C₂×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
95	C₂×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
96	C₂×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
97	C₂²	—	2	2	no	—
98	C₂²	—	2	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	—	382	2	no	—
101	C₂²	—	382	2	no	—
102	C₂²	—	382	2	no	—
103	C₂²	C₂³×D₁₉₁	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	2	1	no	—
106	C₄	—	382	2	no	—
107	C₄	—	382	4	no	—
108	C₄	#15690286	1	2	yes	—
109	C₄	#15690287	1	2	yes	—
110	C₂	—	4	2	no	—
111	C₂	—	764	2	no	—
112	C₂	C₂×#15690286	1	1	yes	characteristic
113	C₂	C₂×#15690287	1	1	yes	characteristic
114	C₂	#20259368	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20259322

C₂

yes

maximal, characteristic, complemented

#20259308

C₂

yes

maximal, complemented

#20259296

C₂

yes

maximal, complemented

#20259293

C₂

yes

maximal, complemented

C₄×D₇₆₄

C₂

yes

maximal, characteristic, complemented

#20259281

C₂

yes

maximal, complemented

#20259280

C₂

yes

maximal, complemented

C₄×D₄×C₁₉₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₃₈₂

C₂²

yes

—

C₂×D₇₆₄

C₂²

yes

characteristic

C₂×C₂²⋊D₁₉₁

C₂²

yes

—

C₂×C₄×D₁₉₁

C₂²

yes

—

C₄⋊Dic₁₉₁

C₂²

yes

—

C₄⋊Dic₁₉₁

C₂²

yes

characteristic

C₄⋊Dic₁₉₁

C₂²

yes

—

C₂²⋊Dic₁₉₁

C₂²

yes

—

(C₂×D₁₉₁)⋊C₄

C₂²

yes

—

(C₂×D₁₉₁)⋊C₄

C₂²

yes

—

C₄×Dic₁₉₁

C₂²

yes

—

C₂×D₄×C₁₉₁

C₂²

yes

characteristic

C₂²×C₄×C₁₉₁

C₂²

yes

—

C₄⋊C₄×C₁₉₁

C₂²

yes

characteristic

C₂²⋊C₄×C₁₉₁

C₂²

yes

—

C₄²×C₁₉₁

C₂²

yes

characteristic

C₂²×D₁₉₁

C₂³

yes

—

Dic₃₈₂

—

D₇₆₄

—

C₂²⋊D₁₉₁

—

C₂×Dic₁₉₁

C₂³

yes

—

C₂×Dic₁₉₁

C₂³

yes

—

C₄×D₁₉₁

—

C₂³×C₁₉₁

C₂³

yes

—

D₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂³

yes

characteristic

C₂×C₄×C₁₉₁

C₂³

yes

—

C₂×C₄×C₁₉₁

C₂³

yes

characteristic

C₂×C₄×C₁₉₁

C₂³

yes

characteristic

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₂⁴

yes

characteristic

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#41

yes

—

C₄×C₁₉₁

#41

yes

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

C₂×#41

yes

characteristic

C₂×C₁₉₁

C₂×#41

yes

characteristic

C₂×C₁₉₁

H(5,𝔽₂)

yes

characteristic

D₁₉₁

—

C₁₉₁

#555

yes

characteristic, complemented, Sylow

#555

—

191

maximal, Sylow

C₄⋊Q₈

—

191

—

#124

—

191

—

#121

—

191

—

#121

—

191

—

C₄×D₄

—

191

—

C₄×D₄

D₁₉₁

yes

characteristic, complemented

#117

—

191

—

#117

—

191

—

C₂×Q₈

—

191

—

C₂×D₄

—

191

—

C₂×D₄

—

191

—

C₂×D₄

C₂×D₁₉₁

yes

characteristic

C₂²×C₄

—

191

—

C₂²×C₄

C₂×D₁₉₁

yes

—

C₄⋊C₄

—

191

—

C₄⋊C₄

—

191

—

C₄⋊C₄

—

191

—

C₄⋊C₄

C₂×D₁₉₁

yes

characteristic

C₂²⋊C₄

—

191

—

C₂²⋊C₄

—

191

—

C₂²⋊C₄

—

191

—

C₂²⋊C₄

C₂×D₁₉₁

yes

—

C₄²

—

191

—

C₄²

C₂×D₁₉₁

yes

characteristic

C₂³

—

191

—

C₂³

C₂²×D₁₉₁

yes

—

Q₈

—

764

—

D₄

—

D₄

—

382

—

D₄

—

764

—

C₂×C₄

—

C₂×C₄

—

382

—

C₂×C₄

—

191

—

C₂×C₄

—

191

—

C₂×C₄

C₂²×D₁₉₁

yes

—

C₂×C₄

C₂²×D₁₉₁

yes

characteristic

C₂×C₄

C₂²×D₁₉₁

yes

characteristic

C₂×C₄

C₂²×D₁₉₁

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

382

—

101

C₂²

—

382

—

102

C₂²

—

382

—

103

C₂²

C₂³×D₁₉₁

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

382

—

107

C₄

—

382

—

108

C₄

#15690286

yes

—

109

C₄

#15690287

yes

—

110

C₂

—

111

C₂

—

764

—

112

C₂

C₂×#15690286

yes

characteristic

113

C₂

C₂×#15690287

yes

characteristic

114

C₂

#20259368

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₉₁
commutator, socle, cosocle	C₂²×C₁₉₁	C₂⁴
Fitting	C₄×D₄×C₁₉₁	C₂
hypercenter	C₄×D₄	D₁₉₁
hypocenter	C₁₉₁	#555
LCS	G ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₉₁

commutator, socle, cosocle

C₂²×C₁₉₁

C₂⁴

Fitting

C₄×D₄×C₁₉₁

C₂

hypercenter

C₄×D₄

D₁₉₁

C₁₉₁

G ▷ C₂²×C₁₉₁ ▷ C₁₉₁

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. #20259322	C₂	semidirect
2. #20259308	C₂	semidirect
3. #20259296	C₂	semidirect
4. #20259293	C₂	semidirect
5. C₄×D₇₆₄	C₂	semidirect
6. #20259281	C₂	semidirect
7. #20259280	C₂	semidirect
8. C₄×D₄×C₁₉₁	C₂	semidirect
57. C₁₉₁	#555	semidirect
64. C₄×D₄	D₁₉₁	semidirect

H ≅ G/N

type

1. #20259322

C₂

semidirect

2. #20259308

C₂

semidirect

3. #20259296

C₂

semidirect

4. #20259293

C₂

semidirect

5. C₄×D₇₆₄

C₂

semidirect

6. #20259281

C₂

semidirect

7. #20259280

C₂

semidirect

8. C₄×D₄×C₁₉₁

C₂

semidirect

57. C₁₉₁

#555

semidirect

64. C₄×D₄

D₁₉₁

semidirect

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

subgroup	index
#20259322	2
#20259308	2
#20259296	2
#20259293	2
C₄×D₇₆₄	2
#20259281	2
#20259280	2
C₄×D₄×C₁₉₁	2
#555	191

subgroup

index

C₄×D₇₆₄

C₄×D₄×C₁₉₁

191

Subgroups — #27133378

Conjugacy classes of subgroups

Complements

Maximal subgroups