Group #22577972 — subgroups

Subgroups — #22577972

Order 8768.

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	C₂×#14555510	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₅₄₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#17984085	C₂	1	2	yes	maximal, complemented
4	#17984076	C₂	1	2	yes	maximal, complemented
5	#17984063	C₂	1	4	yes	maximal, complemented
6	#17984058	C₂	1	4	yes	maximal, complemented
7	#117×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₁₃₇	C₂²	1	2	yes	—
9	#14555510	—	2	4	no	—
10	C₂×Dic₂₇₄	C₂²	1	1	yes	characteristic, complemented
11	C₂×D₅₄₈	—	2	4	no	—
12	C₂×D₅₄₈	C₂²	1	2	yes	—
13	C₂×D₅₄₈	C₂²	1	2	yes	—
14	C₂×D₅₄₈	C₂²	1	1	yes	characteristic, complemented
15	C₂×D₅₄₈	C₂²	1	8	yes	complemented
16	C₂×C₂²⋊D₁₃₇	C₂²	1	2	yes	—
17	C₂×C₄×D₁₃₇	C₂²	1	2	yes	complemented
18	#14555487	C₂²	1	8	yes	complemented
19	C₂²×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	complemented
21	C₂²⋊C₄×C₁₃₇	C₂²	1	2	yes	—
22	C₄²×C₁₃₇	C₂²	1	2	yes	complemented
23	C₂²×D₁₃₇	—	2	4	no	—
24	C₂²×D₁₃₇	—	4	4	no	—
25	C₂²×D₁₃₇	C₂³	1	4	yes	—
26	C₂²×D₁₃₇	C₂³	1	2	yes	—
27	Dic₂₇₄	—	2	2	no	—
28	D₅₄₈	—	2	4	no	—
29	D₅₄₈	—	2	4	no	—
30	D₅₄₈	—	2	2	no	—
31	D₅₄₈	—	4	8	no	—
32	C₂²⋊D₁₃₇	—	2	4	no	—
33	C₂×Dic₁₃₇	C₂³	1	2	yes	—
34	C₄×D₁₃₇	—	2	4	no	—
35	C₂³×C₁₃₇	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₁₃₇	D₄	1	4	yes	complemented
37	C₂×C₄×C₁₃₇	C₂³	1	4	yes	—
38	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
40	C₂²×C₁₃₇	—	2	2	no	—
41	C₂²×C₁₃₇	C₂×D₄	1	2	yes	—
42	C₂²×C₁₃₇	C₂⁴	1	1	yes	characteristic
43	C₂×D₁₃₇	—	2	2	no	—
44	C₂×D₁₃₇	—	2	4	no	—
45	C₂×D₁₃₇	—	2	4	no	—
46	C₂×D₁₃₇	—	2	8	no	—
47	C₂×D₁₃₇	—	4	8	no	—
48	C₄×C₁₃₇	—	2	4	no	—
49	C₄×C₁₃₇	C₂×D₄	1	2	yes	—
50	C₄×C₁₃₇	C₂×D₄	1	2	yes	—
51	Dic₁₃₇	—	2	2	no	—
52	C₂×C₁₃₇	—	2	2	no	—
53	C₂×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
54	C₂×C₁₃₇	H(5,𝔽₂)	1	2	yes	—
55	D₁₃₇	—	4	2	no	—
56	D₁₃₇	—	4	4	no	—
57	C₁₃₇	#535	1	1	yes	characteristic, complemented, Sylow
58	#535	—	137	1	no	maximal, Sylow
59	C₂×#41	—	137	1	no	—
60	C₂²×D₄	—	137	1	no	—
61	#127	—	137	2	no	—
62	#124	—	137	2	no	—
63	#121	—	137	4	no	—
64	#120	—	137	4	no	—
65	#117	D₁₃₇	1	1	yes	characteristic, complemented
66	C₂⁴	—	137	2	no	—
67	#41	—	274	4	no	—
68	C₂×Q₈	—	137	1	no	—
69	C₂×D₄	—	274	4	no	—
70	C₂×D₄	—	137	2	no	—
71	C₂×D₄	—	137	2	no	—
72	C₂×D₄	—	137	2	no	—
73	C₂×D₄	—	137	1	no	—
74	C₂×D₄	—	137	8	no	—
75	C₂²×C₄	—	137	2	no	—
76	C₂²×C₄	C₂×D₁₃₇	1	1	yes	characteristic
77	C₄⋊C₄	C₂×D₁₃₇	1	2	yes	complemented
78	C₂²⋊C₄	—	137	8	no	—
79	C₂²⋊C₄	C₂×D₁₃₇	1	2	yes	—
80	C₄²	C₂×D₁₃₇	1	2	yes	complemented
81	C₂³	—	274	4	no	—
82	C₂³	—	548	4	no	—
83	C₂³	—	137	4	no	—
84	C₂³	—	137	2	no	—
85	C₂³	C₂²×D₁₃₇	1	1	yes	characteristic
86	Q₈	—	274	2	no	—
87	D₄	—	274	4	no	—
88	D₄	—	274	4	no	—
89	D₄	—	274	4	no	—
90	D₄	—	274	2	no	—
91	D₄	—	548	8	no	—
92	C₂×C₄	—	274	4	no	—
93	C₂×C₄	—	137	2	no	—
94	C₂×C₄	D₅₄₈	1	4	yes	complemented
95	C₂×C₄	C₂²×D₁₃₇	1	4	yes	—
96	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
97	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
98	C₂²	—	2	2	no	—
99	C₂²	—	274	2	no	—
100	C₂²	—	274	4	no	—
101	C₂²	—	274	4	no	—
102	C₂²	—	274	8	no	—
103	C₂²	—	548	8	no	—
104	C₂²	C₂×D₅₄₈	1	2	yes	—
105	C₂²	C₂³×D₁₃₇	1	1	yes	characteristic
106	C₄	—	2	4	no	—
107	C₄	—	274	2	no	—
108	C₄	C₂×D₅₄₈	1	2	yes	—
109	C₄	C₂×D₅₄₈	1	2	yes	—
110	C₂	—	2	2	no	—
111	C₂	—	548	4	no	—
112	C₂	—	548	2	no	—
113	C₂	C₂²×D₅₄₈	1	1	yes	characteristic
114	C₂	#17984134	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14555510

C₂

yes

maximal, characteristic, complemented

C₂²×D₅₄₈

C₂

yes

maximal, characteristic, complemented

#17984085

C₂

yes

maximal, complemented

#17984076

C₂

yes

maximal, complemented

#17984063

C₂

yes

maximal, complemented

#17984058

C₂

yes

maximal, complemented

#117×C₁₃₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₃₇

C₂²

yes

—

#14555510

—

C₂×Dic₂₇₄

C₂²

yes

characteristic, complemented

C₂×D₅₄₈

—

C₂×D₅₄₈

C₂²

yes

—

C₂×D₅₄₈

C₂²

yes

—

C₂×D₅₄₈

C₂²

yes

characteristic, complemented

C₂×D₅₄₈

C₂²

yes

complemented

C₂×C₂²⋊D₁₃₇

C₂²

yes

—

C₂×C₄×D₁₃₇

C₂²

yes

complemented

#14555487

C₂²

yes

complemented

C₂²×C₄×C₁₃₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₇

C₂²

yes

complemented

C₂²⋊C₄×C₁₃₇

C₂²

yes

—

C₄²×C₁₃₇

C₂²

yes

complemented

C₂²×D₁₃₇

—

C₂²×D₁₃₇

—

C₂²×D₁₃₇

C₂³

yes

—

C₂²×D₁₃₇

C₂³

yes

—

Dic₂₇₄

—

D₅₄₈

—

D₅₄₈

—

D₅₄₈

—

D₅₄₈

—

C₂²⋊D₁₃₇

—

C₂×Dic₁₃₇

C₂³

yes

—

C₄×D₁₃₇

—

C₂³×C₁₃₇

C₂³

yes

characteristic

C₂×C₄×C₁₃₇

D₄

yes

complemented

C₂×C₄×C₁₃₇

C₂³

yes

—

C₂×C₄×C₁₃₇

C₂³

yes

characteristic

C₂×C₄×C₁₃₇

C₂³

yes

characteristic

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂×D₄

yes

—

C₂²×C₁₃₇

C₂⁴

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×D₄

yes

—

C₄×C₁₃₇

C₂×D₄

yes

—

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

C₂²×D₄

yes

characteristic

C₂×C₁₃₇

H(5,𝔽₂)

yes

—

D₁₃₇

—

D₁₃₇

—

C₁₃₇

#535

yes

characteristic, complemented, Sylow

#535

—

137

maximal, Sylow

C₂×#41

—

137

—

C₂²×D₄

—

137

—

#127

—

137

—

#124

—

137

—

#121

—

137

—

#120

—

137

—

#117

D₁₃₇

yes

characteristic, complemented

C₂⁴

—

137

—

#41

—

274

—

C₂×Q₈

—

137

—

C₂×D₄

—

274

—

C₂×D₄

—

137

—

C₂×D₄

—

137

—

C₂×D₄

—

137

—

C₂×D₄

—

137

—

C₂×D₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

C₂×D₁₃₇

yes

characteristic

C₄⋊C₄

C₂×D₁₃₇

yes

complemented

C₂²⋊C₄

—

137

—

C₂²⋊C₄

C₂×D₁₃₇

yes

—

C₄²

C₂×D₁₃₇

yes

complemented

C₂³

—

274

—

C₂³

—

548

—

C₂³

—

137

—

C₂³

—

137

—

C₂³

C₂²×D₁₃₇

yes

characteristic

Q₈

—

274

—

D₄

—

274

—

D₄

—

274

—

D₄

—

274

—

D₄

—

274

—

D₄

—

548

—

C₂×C₄

—

274

—

C₂×C₄

—

137

—

C₂×C₄

D₅₄₈

yes

complemented

C₂×C₄

C₂²×D₁₃₇

yes

—

C₂×C₄

C₂²×D₁₃₇

yes

characteristic

C₂×C₄

C₂²×D₁₃₇

yes

characteristic

C₂²

—

C₂²

—

274

—

100

C₂²

—

274

—

101

C₂²

—

274

—

102

C₂²

—

274

—

103

C₂²

—

548

—

104

C₂²

C₂×D₅₄₈

yes

—

105

C₂²

C₂³×D₁₃₇

yes

characteristic

106

C₄

—

107

C₄

—

274

—

108

C₄

C₂×D₅₄₈

yes

—

109

C₄

C₂×D₅₄₈

yes

—

110

C₂

—

111

C₂

—

548

—

112

C₂

—

548

—

113

C₂

C₂²×D₅₄₈

yes

characteristic

114

C₂

#17984134

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂⁴
Fitting	#117×C₁₃₇	C₂
hypercenter	#117	D₁₃₇
hypocenter	C₁₃₇	#535
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₃₇

commutator, socle

C₂²×C₁₃₇

C₂⁴

Fitting

#117×C₁₃₇

C₂

D₁₃₇

C₁₃₇

G ▷ C₂²×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
1. C₂×#14555510	C₂	semidirect
2. C₂²×D₅₄₈	C₂	semidirect
3. #17984085	C₂	semidirect
4. #17984076	C₂	semidirect
5. #17984063	C₂	semidirect
6. #17984058	C₂	semidirect
7. #117×C₁₃₇	C₂	semidirect
10. C₂×Dic₂₇₄	C₂²	semidirect
14. C₂×D₅₄₈	C₂²	semidirect
15. C₂×D₅₄₈	C₂²	semidirect
17. C₂×C₄×D₁₃₇	C₂²	semidirect
18. #14555487	C₂²	semidirect
20. C₄⋊C₄×C₁₃₇	C₂²	semidirect
22. C₄²×C₁₃₇	C₂²	semidirect
36. C₂×C₄×C₁₃₇	D₄	semidirect
57. C₁₃₇	#535	semidirect
65. #117	D₁₃₇	semidirect
77. C₄⋊C₄	C₂×D₁₃₇	semidirect
80. C₄²	C₂×D₁₃₇	semidirect
94. C₂×C₄	D₅₄₈	semidirect

H ≅ G/N

type

1. C₂×#14555510

C₂

semidirect

2. C₂²×D₅₄₈

C₂

semidirect

3. #17984085

C₂

semidirect

4. #17984076

C₂

semidirect

5. #17984063

C₂

semidirect

6. #17984058

C₂

semidirect

7. #117×C₁₃₇

C₂

semidirect

10. C₂×Dic₂₇₄

C₂²

semidirect

14. C₂×D₅₄₈

C₂²

semidirect

15. C₂×D₅₄₈

C₂²

semidirect

17. C₂×C₄×D₁₃₇

C₂²

semidirect

18. #14555487

C₂²

semidirect

20. C₄⋊C₄×C₁₃₇

C₂²

semidirect

22. C₄²×C₁₃₇

C₂²

semidirect

36. C₂×C₄×C₁₃₇

D₄

semidirect

57. C₁₃₇

#535

semidirect

65. #117

D₁₃₇

semidirect

77. C₄⋊C₄

C₂×D₁₃₇

semidirect

80. C₄²

C₂×D₁₃₇

semidirect

94. C₂×C₄

D₅₄₈

semidirect

Subgroups — #22577972

Conjugacy classes of subgroups

Complements