Group #27164582 — subgroups

Subgroups — #27164582

Order 12736.

334 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 111 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	C₂²×D₇₉₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#20263951	C₂	1	2	yes	maximal, complemented
3	#20263944	C₂	1	2	yes	maximal, complemented
4	#20263929	C₂	1	4	yes	maximal, complemented
5	#20263923	C₂	1	4	yes	maximal, complemented
6	#20263914	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#41×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₁₉₉	C₂²	1	2	yes	—
9	C₂×D₇₉₆	—	2	4	no	—
10	C₂×D₇₉₆	C₂²	1	2	yes	—
11	C₂×D₇₉₆	C₂²	1	2	yes	—
12	C₂×C₂²⋊D₁₉₉	C₂²	1	8	yes	complemented
13	C₄⋊Dic₁₉₉	C₂²	1	2	yes	complemented
14	C₂²⋊Dic₁₉₉	C₂²	1	2	yes	—
15	(C₂×D₁₉₉)⋊C₄	C₂²	1	8	yes	complemented
16	C₄×Dic₁₉₉	C₂²	1	2	yes	complemented
17	#41×C₁₉₉	—	2	4	no	—
18	C₂×Q₈×C₁₉₉	C₂²	1	1	yes	characteristic, complemented
19	C₂×D₄×C₁₉₉	C₂²	1	2	yes	—
20	C₂×D₄×C₁₉₉	C₂²	1	1	yes	characteristic, complemented
21	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
22	C₂²×C₄×C₁₉₉	C₂²	1	2	yes	complemented
23	C₂²×D₁₉₉	—	2	4	no	—
24	C₂²×D₁₉₉	—	4	4	no	—
25	C₂²×D₁₉₉	C₂³	1	4	yes	—
26	D₇₉₆	—	2	4	no	—
27	D₇₉₆	—	2	4	no	—
28	C₂²⋊D₁₉₉	—	4	8	no	—
29	C₂×Dic₁₉₉	C₂³	1	4	yes	—
30	C₂³×C₁₉₉	C₂³	1	2	yes	—
31	C₂³×C₁₉₉	C₂³	1	1	yes	characteristic
32	Q₈×C₁₉₉	—	2	2	no	—
33	D₄×C₁₉₉	—	2	4	no	—
34	D₄×C₁₉₉	—	2	2	no	—
35	C₂×C₄×C₁₉₉	—	2	4	no	—
36	C₂×C₄×C₁₉₉	D₄	1	4	yes	complemented
37	C₂×C₄×C₁₉₉	C₂³	1	2	yes	—
38	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
40	C₂²×C₁₉₉	—	2	2	no	—
41	C₂²×C₁₉₉	—	2	2	no	—
42	C₂²×C₁₉₉	—	2	4	no	—
43	C₂²×C₁₉₉	C₂×D₄	1	2	yes	—
44	C₂²×C₁₉₉	C₂⁴	1	1	yes	characteristic
45	C₂×D₁₉₉	—	2	4	no	—
46	C₂×D₁₉₉	—	2	8	no	—
47	C₂×D₁₉₉	—	4	8	no	—
48	C₄×C₁₉₉	—	2	2	no	—
49	C₄×C₁₉₉	C₂×D₄	1	2	yes	—
50	C₄×C₁₉₉	C₂×D₄	1	2	yes	—
51	Dic₁₉₉	—	2	4	no	—
52	C₂×C₁₉₉	—	2	2	no	—
53	C₂×C₁₉₉	—	4	2	no	—
54	C₂×C₁₉₉	C₂²×D₄	1	1	yes	characteristic
55	C₂×C₁₉₉	H(5,𝔽₂)	1	2	yes	—
56	D₁₉₉	—	4	4	no	—
57	C₁₉₉	#535	1	1	yes	characteristic, complemented, Sylow
58	#535	—	199	1	no	maximal, Sylow
59	C₂×#41	D₁₉₉	1	1	yes	characteristic, complemented
60	C₂²×D₄	—	199	1	no	—
61	#127	—	199	2	no	—
62	#124	—	199	2	no	—
63	#121	—	199	4	no	—
64	#120	—	199	4	no	—
65	#117	—	199	1	no	—
66	C₂⁴	—	199	2	no	—
67	#41	—	2	4	no	—
68	C₂×Q₈	C₂×D₁₉₉	1	1	yes	characteristic, complemented
69	C₂×D₄	—	398	4	no	—
70	C₂×D₄	—	199	8	no	—
71	C₂×D₄	—	199	2	no	—
72	C₂×D₄	—	199	2	no	—
73	C₂×D₄	C₂×D₁₉₉	1	2	yes	—
74	C₂×D₄	C₂×D₁₉₉	1	1	yes	characteristic, complemented
75	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
76	C₂²×C₄	C₂×D₁₉₉	1	2	yes	complemented
77	C₄⋊C₄	—	199	2	no	—
78	C₂²⋊C₄	—	199	2	no	—
79	C₂²⋊C₄	—	199	8	no	—
80	C₄²	—	199	2	no	—
81	C₂³	—	796	4	no	—
82	C₂³	—	398	4	no	—
83	C₂³	—	199	4	no	—
84	C₂³	C₂²×D₁₉₉	1	2	yes	—
85	C₂³	C₂²×D₁₉₉	1	1	yes	characteristic
86	Q₈	—	2	2	no	—
87	D₄	—	2	4	no	—
88	D₄	—	2	2	no	—
89	D₄	—	796	8	no	—
90	D₄	—	398	4	no	—
91	D₄	—	398	4	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	199	4	no	—
94	C₂×C₄	C₂²⋊D₁₉₉	1	4	yes	complemented
95	C₂×C₄	C₂²×D₁₉₉	1	2	yes	—
96	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
97	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
98	C₂²	—	2	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	—	2	4	no	—
101	C₂²	—	796	8	no	—
102	C₂²	—	398	4	no	—
103	C₂²	—	398	8	no	—
104	C₂²	C₂×C₂²⋊D₁₉₉	1	2	yes	—
105	C₂²	C₂³×D₁₉₉	1	1	yes	characteristic
106	C₄	—	2	2	no	—
107	C₄	—	398	4	no	—
108	C₄	C₂×C₂²⋊D₁₉₉	1	2	yes	—
109	C₄	C₂×C₂²⋊D₁₉₉	1	2	yes	—
110	C₂	—	2	2	no	—
111	C₂	—	4	2	no	—
112	C₂	—	796	4	no	—
113	C₂	C₂²×C₂²⋊D₁₉₉	1	1	yes	characteristic
114	C₂	#20264001	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₇₉₆

C₂

yes

maximal, characteristic, complemented

#20263951

C₂

yes

maximal, complemented

#20263944

C₂

yes

maximal, complemented

#20263929

C₂

yes

maximal, complemented

#20263923

C₂

yes

maximal, complemented

#20263914

C₂

yes

maximal, characteristic, complemented

C₂×#41×C₁₉₉

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₉₉

C₂²

yes

—

C₂×D₇₉₆

—

C₂×D₇₉₆

C₂²

yes

—

C₂×D₇₉₆

C₂²

yes

—

C₂×C₂²⋊D₁₉₉

C₂²

yes

complemented

C₄⋊Dic₁₉₉

C₂²

yes

complemented

C₂²⋊Dic₁₉₉

C₂²

yes

—

(C₂×D₁₉₉)⋊C₄

C₂²

yes

complemented

C₄×Dic₁₉₉

C₂²

yes

complemented

#41×C₁₉₉

—

C₂×Q₈×C₁₉₉

C₂²

yes

characteristic, complemented

C₂×D₄×C₁₉₉

C₂²

yes

—

C₂×D₄×C₁₉₉

C₂²

yes

characteristic, complemented

C₂²×C₄×C₁₉₉

C₂²

yes

characteristic

C₂²×C₄×C₁₉₉

C₂²

yes

complemented

C₂²×D₁₉₉

—

C₂²×D₁₉₉

—

C₂²×D₁₉₉

C₂³

yes

—

D₇₉₆

—

D₇₉₆

—

C₂²⋊D₁₉₉

—

C₂×Dic₁₉₉

C₂³

yes

—

C₂³×C₁₉₉

C₂³

yes

—

C₂³×C₁₉₉

C₂³

yes

characteristic

Q₈×C₁₉₉

—

D₄×C₁₉₉

—

D₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

D₄

yes

complemented

C₂×C₄×C₁₉₉

C₂³

yes

—

C₂×C₄×C₁₉₉

C₂³

yes

characteristic

C₂×C₄×C₁₉₉

C₂³

yes

characteristic

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

C₂×D₄

yes

—

C₂²×C₁₉₉

C₂⁴

yes

characteristic

C₂×D₁₉₉

—

C₂×D₁₉₉

—

C₂×D₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

C₂×D₄

yes

—

C₄×C₁₉₉

C₂×D₄

yes

—

Dic₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

C₂²×D₄

yes

characteristic

C₂×C₁₉₉

H(5,𝔽₂)

yes

—

D₁₉₉

—

C₁₉₉

#535

yes

characteristic, complemented, Sylow

#535

—

199

maximal, Sylow

C₂×#41

D₁₉₉

yes

characteristic, complemented

C₂²×D₄

—

199

—

#127

—

199

—

#124

—

199

—

#121

—

199

—

#120

—

199

—

#117

—

199

—

C₂⁴

—

199

—

#41

—

C₂×Q₈

C₂×D₁₉₉

yes

characteristic, complemented

C₂×D₄

—

398

—

C₂×D₄

—

199

—

C₂×D₄

—

199

—

C₂×D₄

—

199

—

C₂×D₄

C₂×D₁₉₉

yes

—

C₂×D₄

C₂×D₁₉₉

yes

characteristic, complemented

C₂²×C₄

C₂×D₁₉₉

yes

characteristic

C₂²×C₄

C₂×D₁₉₉

yes

complemented

C₄⋊C₄

—

199

—

C₂²⋊C₄

—

199

—

C₂²⋊C₄

—

199

—

C₄²

—

199

—

C₂³

—

796

—

C₂³

—

398

—

C₂³

—

199

—

C₂³

C₂²×D₁₉₉

yes

—

C₂³

C₂²×D₁₉₉

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

796

—

D₄

—

398

—

D₄

—

398

—

C₂×C₄

—

C₂×C₄

—

199

—

C₂×C₄

C₂²⋊D₁₉₉

yes

complemented

C₂×C₄

C₂²×D₁₉₉

yes

—

C₂×C₄

C₂²×D₁₉₉

yes

characteristic

C₂×C₄

C₂²×D₁₉₉

yes

characteristic

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

—

796

—

102

C₂²

—

398

—

103

C₂²

—

398

—

104

C₂²

C₂×C₂²⋊D₁₉₉

yes

—

105

C₂²

C₂³×D₁₉₉

yes

characteristic

106

C₄

—

107

C₄

—

398

—

108

C₄

C₂×C₂²⋊D₁₉₉

yes

—

109

C₄

C₂×C₂²⋊D₁₉₉

yes

—

110

C₂

—

111

C₂

—

112

C₂

—

796

—

113

C₂

C₂²×C₂²⋊D₁₉₉

yes

characteristic

114

C₂

#20264001

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₉₉
commutator, socle, cosocle	C₂²×C₁₉₉	C₂⁴
Fitting	C₂×#41×C₁₉₉	C₂
hypercenter	C₂×#41	D₁₉₉
hypocenter	C₁₉₉	#535
LCS	G ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂×#41

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₉₉

commutator, socle, cosocle

C₂²×C₁₉₉

C₂⁴

Fitting

C₂×#41×C₁₉₉

C₂

hypercenter

C₂×#41

D₁₉₉

C₁₉₉

G ▷ C₂²×C₁₉₉ ▷ C₁₉₉

C₁ ◁ C₂² ◁ C₂×#41

N	H ≅ G/N	type
1. C₂²×D₇₉₆	C₂	semidirect
2. #20263951	C₂	semidirect
3. #20263944	C₂	semidirect
4. #20263929	C₂	semidirect
5. #20263923	C₂	semidirect
6. #20263914	C₂	semidirect
7. C₂×#41×C₁₉₉	C₂	semidirect
12. C₂×C₂²⋊D₁₉₉	C₂²	semidirect
13. C₄⋊Dic₁₉₉	C₂²	semidirect
15. (C₂×D₁₉₉)⋊C₄	C₂²	semidirect
16. C₄×Dic₁₉₉	C₂²	semidirect
18. C₂×Q₈×C₁₉₉	C₂²	semidirect
20. C₂×D₄×C₁₉₉	C₂²	semidirect
22. C₂²×C₄×C₁₉₉	C₂²	semidirect
36. C₂×C₄×C₁₉₉	D₄	semidirect
57. C₁₉₉	#535	semidirect
59. C₂×#41	D₁₉₉	semidirect
68. C₂×Q₈	C₂×D₁₉₉	semidirect
74. C₂×D₄	C₂×D₁₉₉	semidirect
76. C₂²×C₄	C₂×D₁₉₉	semidirect
94. C₂×C₄	C₂²⋊D₁₉₉	semidirect

H ≅ G/N

type

1. C₂²×D₇₉₆

C₂

semidirect

2. #20263951

C₂

semidirect

3. #20263944

C₂

semidirect

4. #20263929

C₂

semidirect

5. #20263923

C₂

semidirect

6. #20263914

C₂

semidirect

7. C₂×#41×C₁₉₉

C₂

semidirect

12. C₂×C₂²⋊D₁₉₉

C₂²

semidirect

13. C₄⋊Dic₁₉₉

C₂²

semidirect

15. (C₂×D₁₉₉)⋊C₄

C₂²

semidirect

16. C₄×Dic₁₉₉

C₂²

semidirect

18. C₂×Q₈×C₁₉₉

C₂²

semidirect

20. C₂×D₄×C₁₉₉

C₂²

semidirect

22. C₂²×C₄×C₁₉₉

C₂²

semidirect

36. C₂×C₄×C₁₉₉

D₄

semidirect

57. C₁₉₉

#535

semidirect

59. C₂×#41

D₁₉₉

semidirect

68. C₂×Q₈

C₂×D₁₉₉

semidirect

74. C₂×D₄

C₂×D₁₉₉

semidirect

76. C₂²×C₄

C₂×D₁₉₉

semidirect

94. C₂×C₄

C₂²⋊D₁₉₉

semidirect

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

subgroup	index
C₂²×D₇₉₆	2
#20263951	2
#20263944	2
#20263929	2
#20263923	2
#20263914	2
C₂×#41×C₁₉₉	2
#535	199

subgroup

index

C₂²×D₇₉₆

C₂×#41×C₁₉₉

199

Subgroups — #27164582

Conjugacy classes of subgroups

Complements

Maximal subgroups