Group #22578118 — subgroups

Subgroups — #22578118

Order 8768.

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	#17984088	C₂	1	1	yes	maximal, characteristic, complemented
2	#17984074	C₂	1	4	yes	maximal, complemented
3	#17984062	C₂	1	2	yes	maximal, complemented
4	#17984059	C₂	1	2	yes	maximal, complemented
5	C₄×D₅₄₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#17984047	C₂	1	2	yes	maximal, complemented
7	#17984046	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₂₇₄	C₂²	1	2	yes	—
10	C₂×D₅₄₈	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊D₁₃₇	C₂²	1	4	yes	—
12	C₂×C₄×D₁₃₇	C₂²	1	2	yes	—
13	C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
14	C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
16	C₂²⋊Dic₁₃₇	C₂²	1	4	yes	—
17	#14555487	C₂²	1	4	yes	—
18	#14555487	C₂²	1	2	yes	—
19	C₄×Dic₁₃₇	C₂²	1	4	yes	—
20	C₂×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₁₃₇	C₂²	1	2	yes	—
22	C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
23	C₂²⋊C₄×C₁₃₇	C₂²	1	2	yes	—
24	C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
25	C₂²×D₁₃₇	C₂³	1	2	yes	—
26	Dic₂₇₄	—	4	2	no	—
27	D₅₄₈	—	2	2	no	—
28	C₂²⋊D₁₃₇	—	4	4	no	—
29	C₂×Dic₁₃₇	C₂³	1	2	yes	—
30	C₂×Dic₁₃₇	C₂³	1	4	yes	—
31	C₄×D₁₃₇	—	2	4	no	—
32	C₂³×C₁₃₇	C₂³	1	2	yes	—
33	D₄×C₁₃₇	—	2	2	no	—
34	C₂×C₄×C₁₃₇	—	2	4	no	—
35	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₁₃₇	C₂³	1	2	yes	—
37	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
39	C₂²×C₁₃₇	—	2	2	no	—
40	C₂²×C₁₃₇	—	2	2	no	—
41	C₂²×C₁₃₇	—	2	2	no	—
42	C₂²×C₁₃₇	C₂⁴	1	1	yes	characteristic
43	C₂×D₁₃₇	—	2	2	no	—
44	C₂×D₁₃₇	—	2	2	no	—
45	C₂×D₁₃₇	—	2	2	no	—
46	C₄×C₁₃₇	—	2	2	no	—
47	C₄×C₁₃₇	—	2	1	no	—
48	C₄×C₁₃₇	#41	1	2	yes	—
49	C₄×C₁₃₇	#41	1	2	yes	—
50	Dic₁₃₇	—	2	2	no	—
51	Dic₁₃₇	—	2	4	no	—
52	C₂×C₁₃₇	—	4	2	no	—
53	C₂×C₁₃₇	C₂×#41	1	1	yes	characteristic
54	C₂×C₁₃₇	C₂×#41	1	1	yes	characteristic
55	C₂×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
56	D₁₃₇	—	4	2	no	—
57	C₁₃₇	#555	1	1	yes	characteristic, complemented, Sylow
58	#555	—	137	1	no	maximal, Sylow
59	C₄⋊Q₈	—	137	1	no	—
60	#124	—	137	4	no	—
61	#121	—	137	2	no	—
62	#121	—	137	2	no	—
63	C₄×D₄	—	137	1	no	—
64	C₄×D₄	D₁₃₇	1	1	yes	characteristic, complemented
65	#117	—	137	2	no	—
66	#117	—	137	2	no	—
67	C₂×Q₈	—	137	2	no	—
68	C₂×D₄	—	137	4	no	—
69	C₂×D₄	—	137	1	no	—
70	C₂×D₄	C₂×D₁₃₇	1	1	yes	characteristic
71	C₂²×C₄	—	137	2	no	—
72	C₂²×C₄	C₂×D₁₃₇	1	2	yes	—
73	C₄⋊C₄	—	137	1	no	—
74	C₄⋊C₄	—	137	2	no	—
75	C₄⋊C₄	—	137	2	no	—
76	C₄⋊C₄	C₂×D₁₃₇	1	1	yes	characteristic
77	C₂²⋊C₄	—	137	4	no	—
78	C₂²⋊C₄	—	137	4	no	—
79	C₂²⋊C₄	—	137	2	no	—
80	C₂²⋊C₄	C₂×D₁₃₇	1	2	yes	—
81	C₄²	—	137	4	no	—
82	C₄²	C₂×D₁₃₇	1	1	yes	characteristic
83	C₂³	—	137	2	no	—
84	C₂³	C₂²×D₁₃₇	1	2	yes	—
85	Q₈	—	548	2	no	—
86	D₄	—	2	2	no	—
87	D₄	—	274	2	no	—
88	D₄	—	548	4	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	274	4	no	—
91	C₂×C₄	—	137	2	no	—
92	C₂×C₄	—	137	4	no	—
93	C₂×C₄	C₂²×D₁₃₇	1	2	yes	—
94	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
95	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
96	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
97	C₂²	—	2	2	no	—
98	C₂²	—	2	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	—	274	2	no	—
101	C₂²	—	274	2	no	—
102	C₂²	—	274	2	no	—
103	C₂²	C₂³×D₁₃₇	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	2	1	no	—
106	C₄	—	274	2	no	—
107	C₄	—	274	4	no	—
108	C₄	#14555509	1	2	yes	—
109	C₄	#14555510	1	2	yes	—
110	C₂	—	4	2	no	—
111	C₂	—	548	2	no	—
112	C₂	C₂×#14555509	1	1	yes	characteristic
113	C₂	C₂×#14555510	1	1	yes	characteristic
114	C₂	#17984134	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17984088

C₂

yes

maximal, characteristic, complemented

#17984074

C₂

yes

maximal, complemented

#17984062

C₂

yes

maximal, complemented

#17984059

C₂

yes

maximal, complemented

C₄×D₅₄₈

C₂

yes

maximal, characteristic, complemented

#17984047

C₂

yes

maximal, complemented

#17984046

C₂

yes

maximal, complemented

C₄×D₄×C₁₃₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₇₄

C₂²

yes

—

C₂×D₅₄₈

C₂²

yes

characteristic

C₂×C₂²⋊D₁₃₇

C₂²

yes

—

C₂×C₄×D₁₃₇

C₂²

yes

—

C₄⋊Dic₁₃₇

C₂²

yes

—

C₄⋊Dic₁₃₇

C₂²

yes

characteristic

C₄⋊Dic₁₃₇

C₂²

yes

—

C₂²⋊Dic₁₃₇

C₂²

yes

—

#14555487

C₂²

yes

—

#14555487

C₂²

yes

—

C₄×Dic₁₃₇

C₂²

yes

—

C₂×D₄×C₁₃₇

C₂²

yes

characteristic

C₂²×C₄×C₁₃₇

C₂²

yes

—

C₄⋊C₄×C₁₃₇

C₂²

yes

characteristic

C₂²⋊C₄×C₁₃₇

C₂²

yes

—

C₄²×C₁₃₇

C₂²

yes

characteristic

C₂²×D₁₃₇

C₂³

yes

—

Dic₂₇₄

—

D₅₄₈

—

C₂²⋊D₁₃₇

—

C₂×Dic₁₃₇

C₂³

yes

—

C₂×Dic₁₃₇

C₂³

yes

—

C₄×D₁₃₇

—

C₂³×C₁₃₇

C₂³

yes

—

D₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂³

yes

characteristic

C₂×C₄×C₁₃₇

C₂³

yes

—

C₂×C₄×C₁₃₇

C₂³

yes

characteristic

C₂×C₄×C₁₃₇

C₂³

yes

characteristic

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂⁴

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#41

yes

—

C₄×C₁₃₇

#41

yes

—

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

C₂×#41

yes

characteristic

C₂×C₁₃₇

C₂×#41

yes

characteristic

C₂×C₁₃₇

H(5,𝔽₂)

yes

characteristic

D₁₃₇

—

C₁₃₇

#555

yes

characteristic, complemented, Sylow

#555

—

137

maximal, Sylow

C₄⋊Q₈

—

137

—

#124

—

137

—

#121

—

137

—

#121

—

137

—

C₄×D₄

—

137

—

C₄×D₄

D₁₃₇

yes

characteristic, complemented

#117

—

137

—

#117

—

137

—

C₂×Q₈

—

137

—

C₂×D₄

—

137

—

C₂×D₄

—

137

—

C₂×D₄

C₂×D₁₃₇

yes

characteristic

C₂²×C₄

—

137

—

C₂²×C₄

C₂×D₁₃₇

yes

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₄⋊C₄

C₂×D₁₃₇

yes

characteristic

C₂²⋊C₄

—

137

—

C₂²⋊C₄

—

137

—

C₂²⋊C₄

—

137

—

C₂²⋊C₄

C₂×D₁₃₇

yes

—

C₄²

—

137

—

C₄²

C₂×D₁₃₇

yes

characteristic

C₂³

—

137

—

C₂³

C₂²×D₁₃₇

yes

—

Q₈

—

548

—

D₄

—

D₄

—

274

—

D₄

—

548

—

C₂×C₄

—

C₂×C₄

—

274

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

C₂²×D₁₃₇

yes

—

C₂×C₄

C₂²×D₁₃₇

yes

characteristic

C₂×C₄

C₂²×D₁₃₇

yes

characteristic

C₂×C₄

C₂²×D₁₃₇

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

274

—

101

C₂²

—

274

—

102

C₂²

—

274

—

103

C₂²

C₂³×D₁₃₇

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

274

—

107

C₄

—

274

—

108

C₄

#14555509

yes

—

109

C₄

#14555510

yes

—

110

C₂

—

111

C₂

—

548

—

112

C₂

C₂×#14555509

yes

characteristic

113

C₂

C₂×#14555510

yes

characteristic

114

C₂

#17984134

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂⁴
Fitting	C₄×D₄×C₁₃₇	C₂
hypercenter	C₄×D₄	D₁₃₇
hypocenter	C₁₃₇	#555
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₃₇

commutator, socle

C₂²×C₁₃₇

C₂⁴

Fitting

C₄×D₄×C₁₃₇

C₂

hypercenter

C₄×D₄

D₁₃₇

C₁₃₇

G ▷ C₂²×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. #17984088	C₂	semidirect
2. #17984074	C₂	semidirect
3. #17984062	C₂	semidirect
4. #17984059	C₂	semidirect
5. C₄×D₅₄₈	C₂	semidirect
6. #17984047	C₂	semidirect
7. #17984046	C₂	semidirect
8. C₄×D₄×C₁₃₇	C₂	semidirect
57. C₁₃₇	#555	semidirect
64. C₄×D₄	D₁₃₇	semidirect

H ≅ G/N

type

1. #17984088

C₂

semidirect

2. #17984074

C₂

semidirect

3. #17984062

C₂

semidirect

4. #17984059

C₂

semidirect

5. C₄×D₅₄₈

C₂

semidirect

6. #17984047

C₂

semidirect

7. #17984046

C₂

semidirect

8. C₄×D₄×C₁₃₇

C₂

semidirect

57. C₁₃₇

#555

semidirect

64. C₄×D₄

D₁₃₇

semidirect

Subgroups — #22578118

Conjugacy classes of subgroups

Complements