Group #22578113 — subgroups

Subgroups — #22578113

Order 8768.

228 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 107 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	#17984080	C₂	1	2	yes	maximal, complemented
3	#17984066	C₂	1	4	yes	maximal, complemented
4	#17984064	C₂	1	2	yes	maximal, complemented
5	C₄×D₅₄₈	C₂	1	2	yes	maximal
6	C₄×D₅₄₈	C₂	1	1	yes	maximal, characteristic
7	C₄⋊C₄×D₁₃₇	C₂	1	2	yes	maximal, complemented
8	C₄×Q₈×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×C₄×D₁₃₇	C₂²	1	2	yes	—
12	C₂×C₄×D₁₃₇	C₂²	1	4	yes	—
13	C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
14	C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
16	#14555489	C₂²	1	4	yes	—
17	#14555489	C₂²	1	4	yes	—
18	#14555487	C₂²	1	4	yes	—
19	#14555487	C₂²	1	2	yes	—
20	C₂×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
23	C₄²×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	Q₈	1	4	yes	complemented
28	C₂×Dic₁₃₇	C₂³	1	2	yes	—
29	C₂×Dic₁₃₇	C₂³	1	4	yes	—
30	C₄×D₁₃₇	—	2	4	no	—
31	C₄×D₁₃₇	—	4	4	no	—
32	Q₈×C₁₃₇	—	2	2	no	—
33	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
34	C₂×C₄×C₁₃₇	C₂³	1	2	yes	—
35	C₂×C₄×C₁₃₇	C₂³	1	2	yes	—
36	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
38	C₂²×C₁₃₇	C₂⁴	1	1	yes	characteristic
39	C₂×D₁₃₇	—	2	2	no	—
40	C₂×D₁₃₇	—	2	2	no	—
41	C₂×D₁₃₇	C₂×Q₈	1	4	yes	—
42	C₄×C₁₃₇	—	2	2	no	—
43	C₄×C₁₃₇	—	2	2	no	—
44	C₄×C₁₃₇	—	2	1	no	—
45	C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
46	C₄×C₁₃₇	#41	1	2	yes	—
47	Dic₁₃₇	—	2	2	no	—
48	Dic₁₃₇	—	2	4	no	—
49	C₂×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
50	C₂×C₁₃₇	C₂×#41	1	1	yes	characteristic
51	C₂×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
52	D₁₃₇	—	2	4	no	—
53	C₁₃₇	#554	1	1	yes	characteristic, complemented, Sylow
54	#554	—	137	1	no	maximal, Sylow
55	C₄⋊Q₈	—	137	1	no	—
56	#125	—	137	2	no	—
57	C₂²⋊Q₈	—	137	2	no	—
58	C₂²⋊Q₈	—	137	4	no	—
59	C₄×Q₈	D₁₃₇	1	1	yes	characteristic, complemented
60	C₄×D₄	—	137	2	no	—
61	C₄×D₄	—	137	1	no	—
62	C₂×C₄⋊C₄	—	137	2	no	—
63	C₂×Q₈	—	137	2	no	—
64	C₂×Q₈	C₂×D₁₃₇	1	1	yes	characteristic
65	C₂×D₄	—	137	1	no	—
66	C₂²×C₄	—	137	2	no	—
67	C₂²×C₄	—	137	4	no	—
68	C₄⋊C₄	—	137	4	no	—
69	C₄⋊C₄	—	137	4	no	—
70	C₄⋊C₄	—	137	2	no	—
71	C₄⋊C₄	—	137	1	no	—
72	C₄⋊C₄	—	137	2	no	—
73	C₄⋊C₄	C₂×D₁₃₇	1	1	yes	characteristic
74	C₄⋊C₄	C₂×D₁₃₇	1	2	yes	—
75	C₂²⋊C₄	—	137	2	no	—
76	C₂²⋊C₄	—	137	4	no	—
77	C₄²	C₂×D₁₃₇	1	2	yes	—
78	C₄²	C₂×D₁₃₇	1	1	yes	characteristic
79	C₂³	—	137	2	no	—
80	Q₈	—	2	2	no	—
81	Q₈	—	548	2	no	—
82	D₄	—	137	4	no	—
83	C₂×C₄	—	274	4	no	—
84	C₂×C₄	—	548	4	no	—
85	C₂×C₄	—	137	2	no	—
86	C₂×C₄	—	137	4	no	—
87	C₂×C₄	C₂²×D₁₃₇	1	2	yes	—
88	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
89	C₂×C₄	C₂²×D₁₃₇	1	2	yes	—
90	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
91	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
92	C₂²	—	274	2	no	—
93	C₂²	—	274	2	no	—
94	C₂²	—	137	4	no	—
95	C₂²	C₂³×D₁₃₇	1	1	yes	characteristic
96	C₄	—	2	2	no	—
97	C₄	—	2	2	no	—
98	C₄	—	2	1	no	—
99	C₄	—	274	2	no	—
100	C₄	—	274	4	no	—
101	C₄	Q₈×D₁₃₇	1	2	yes	—
102	C₄	#14555510	1	2	yes	—
103	C₂	—	274	4	no	—
104	C₂	C₂×Q₈×D₁₃₇	1	1	yes	characteristic
105	C₂	C₂×#14555510	1	1	yes	characteristic
106	C₂	#17984134	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17984088

C₂

yes

maximal, characteristic, complemented

#17984080

C₂

yes

maximal, complemented

#17984066

C₂

yes

maximal, complemented

#17984064

C₂

yes

maximal, complemented

C₄×D₅₄₈

C₂

yes

maximal

C₄×D₅₄₈

C₂

yes

maximal, characteristic

C₄⋊C₄×D₁₃₇

C₂

yes

maximal, complemented

C₄×Q₈×C₁₃₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₇₄

C₂²

yes

—

C₂×D₅₄₈

C₂²

yes

characteristic

C₂×C₄×D₁₃₇

C₂²

yes

—

C₂×C₄×D₁₃₇

C₂²

yes

—

C₄⋊Dic₁₃₇

C₂²

yes

—

C₄⋊Dic₁₃₇

C₂²

yes

characteristic

C₄⋊Dic₁₃₇

C₂²

yes

—

#14555489

C₂²

yes

—

#14555489

C₂²

yes

—

#14555487

C₂²

yes

—

#14555487

C₂²

yes

—

C₂×Q₈×C₁₃₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₇

C₂²

yes

—

C₄²×C₁₃₇

C₂²

yes

—

C₄²×C₁₃₇

C₂²

yes

characteristic

C₂²×D₁₃₇

C₂³

yes

—

Dic₂₇₄

—

D₅₄₈

Q₈

yes

complemented

C₂×Dic₁₃₇

C₂³

yes

—

C₂×Dic₁₃₇

C₂³

yes

—

C₄×D₁₃₇

—

C₄×D₁₃₇

—

Q₈×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂³

yes

characteristic

C₂×C₄×C₁₃₇

C₂³

yes

—

C₂×C₄×C₁₃₇

C₂³

yes

—

C₂×C₄×C₁₃₇

C₂³

yes

characteristic

C₂×C₄×C₁₃₇

C₂³

yes

characteristic

C₂²×C₁₃₇

C₂⁴

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

C₂×Q₈

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×Q₈

yes

—

C₄×C₁₃₇

#41

yes

—

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

C₂²×Q₈

yes

characteristic

C₂×C₁₃₇

C₂×#41

yes

characteristic

C₂×C₁₃₇

H(5,𝔽₂)

yes

characteristic

D₁₃₇

—

C₁₃₇

#554

yes

characteristic, complemented, Sylow

#554

—

137

maximal, Sylow

C₄⋊Q₈

—

137

—

#125

—

137

—

C₂²⋊Q₈

—

137

—

C₂²⋊Q₈

—

137

—

C₄×Q₈

D₁₃₇

yes

characteristic, complemented

C₄×D₄

—

137

—

C₄×D₄

—

137

—

C₂×C₄⋊C₄

—

137

—

C₂×Q₈

—

137

—

C₂×Q₈

C₂×D₁₃₇

yes

characteristic

C₂×D₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₄⋊C₄

C₂×D₁₃₇

yes

characteristic

C₄⋊C₄

C₂×D₁₃₇

yes

—

C₂²⋊C₄

—

137

—

C₂²⋊C₄

—

137

—

C₄²

C₂×D₁₃₇

yes

—

C₄²

C₂×D₁₃₇

yes

characteristic

C₂³

—

137

—

Q₈

—

Q₈

—

548

—

D₄

—

137

—

C₂×C₄

—

274

—

C₂×C₄

—

548

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

C₂²×D₁₃₇

yes

—

C₂×C₄

C₂²×D₁₃₇

yes

characteristic

C₂×C₄

C₂²×D₁₃₇

yes

—

C₂×C₄

C₂²×D₁₃₇

yes

characteristic

C₂×C₄

C₂²×D₁₃₇

yes

characteristic

C₂²

—

274

—

C₂²

—

274

—

C₂²

—

137

—

C₂²

C₂³×D₁₃₇

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

274

—

100

C₄

—

274

—

101

C₄

Q₈×D₁₃₇

yes

—

102

C₄

#14555510

yes

—

103

C₂

—

274

—

104

C₂

C₂×Q₈×D₁₃₇

yes

characteristic

105

C₂

C₂×#14555510

yes

characteristic

106

C₂

#17984134

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂⁴
Fitting	C₄×Q₈×C₁₃₇	C₂
hypercenter	C₄×Q₈	D₁₃₇
hypocenter	C₁₃₇	#554
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₃₇

commutator, socle

C₂²×C₁₃₇

C₂⁴

Fitting

C₄×Q₈×C₁₃₇

C₂

hypercenter

C₄×Q₈

D₁₃₇

C₁₃₇

G ▷ C₂²×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
1. #17984088	C₂	semidirect
2. #17984080	C₂	semidirect
3. #17984066	C₂	semidirect
4. #17984064	C₂	semidirect
7. C₄⋊C₄×D₁₃₇	C₂	semidirect
8. C₄×Q₈×C₁₃₇	C₂	semidirect
27. D₅₄₈	Q₈	semidirect
53. C₁₃₇	#554	semidirect
59. C₄×Q₈	D₁₃₇	semidirect

H ≅ G/N

type

1. #17984088

C₂

semidirect

2. #17984080

C₂

semidirect

3. #17984066

C₂

semidirect

4. #17984064

C₂

semidirect

7. C₄⋊C₄×D₁₃₇

C₂

semidirect

8. C₄×Q₈×C₁₃₇

C₂

semidirect

27. D₅₄₈

Q₈

semidirect

53. C₁₃₇

#554

semidirect

59. C₄×Q₈

D₁₃₇

semidirect

Subgroups — #22578113

Conjugacy classes of subgroups

Complements