Group #22577977 — subgroups

Subgroups — #22577977

Order 8768.

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

normal subgroups only

This group has 270 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
2	#17984086	C₂	1	2	yes	maximal, complemented
3	#17984077	C₂	1	2	yes	maximal, complemented
4	#17984071	C₂	1	4	yes	maximal, complemented
5	#17984064	C₂	1	4	yes	maximal, complemented
6	#17984047	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14555510	—	2	4	no	—
9	C₂×Dic₂₇₄	C₂²	1	1	yes	characteristic
10	C₂×D₅₄₈	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊D₁₃₇	C₂²	1	2	yes	—
12	C₂×C₄×D₁₃₇	C₂²	1	2	yes	—
13	C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
14	#14555489	C₂²	1	8	yes	—
15	C₂²⋊Dic₁₃₇	C₂²	1	2	yes	—
16	#14555487	C₂²	1	8	yes	—
17	C₄×Dic₁₃₇	C₂²	1	2	yes	—
18	C₂×Q₈×C₁₃₇	—	2	4	no	—
19	C₂×Q₈×C₁₃₇	C₂²	1	2	yes	—
20	C₂×Q₈×C₁₃₇	C₂²	1	2	yes	—
21	C₂²×C₄×C₁₃₇	C₂²	1	2	yes	—
22	C₂²×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
23	C₂²×D₁₃₇	C₂³	1	2	yes	—
24	Dic₂₇₄	—	2	2	no	—
25	D₅₄₈	—	2	2	no	—
26	C₂²⋊D₁₃₇	—	2	4	no	—
27	C₂×Dic₁₃₇	C₂³	1	4	yes	—
28	C₂×Dic₁₃₇	C₂³	1	2	yes	—
29	C₄×D₁₃₇	—	2	4	no	—
30	C₂³×C₁₃₇	C₂³	1	1	yes	characteristic
31	Q₈×C₁₃₇	—	2	4	no	—
32	Q₈×C₁₃₇	—	2	4	no	—
33	C₂×C₄×C₁₃₇	—	2	4	no	—
34	C₂×C₄×C₁₃₇	D₄	1	4	yes	—
35	C₂×C₄×C₁₃₇	C₂³	1	4	yes	—
36	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
38	C₂²×C₁₃₇	—	2	2	no	—
39	C₂²×C₁₃₇	C₂×D₄	1	2	yes	—
40	C₂²×C₁₃₇	C₂⁴	1	1	yes	characteristic
41	C₂×D₁₃₇	—	2	2	no	—
42	C₂×D₁₃₇	—	2	4	no	—
43	C₄×C₁₃₇	—	2	4	no	—
44	C₄×C₁₃₇	C₂×D₄	1	2	yes	—
45	C₄×C₁₃₇	C₂×D₄	1	2	yes	—
46	Dic₁₃₇	—	2	2	no	—
47	Dic₁₃₇	—	2	4	no	—
48	C₂×C₁₃₇	—	2	2	no	—
49	C₂×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
50	C₂×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
51	D₁₃₇	—	4	2	no	—
52	C₁₃₇	#536	1	1	yes	characteristic, complemented, Sylow
53	#536	—	137	1	no	maximal, Sylow
54	C₂×#41	—	137	1	no	—
55	C₂²×Q₈	D₁₃₇	1	1	yes	characteristic, complemented
56	C₄⋊Q₈	—	137	2	no	—
57	#124	—	137	2	no	—
58	#123	—	137	4	no	—
59	C₂²⋊Q₈	—	137	4	no	—
60	#117	—	137	1	no	—
61	#41	—	274	4	no	—
62	C₂×Q₈	—	2	4	no	—
63	C₂×Q₈	—	137	1	no	—
64	C₂×Q₈	C₂×D₁₃₇	1	2	yes	—
65	C₂×Q₈	C₂×D₁₃₇	1	2	yes	—
66	C₂×D₄	—	137	2	no	—
67	C₂×D₄	—	137	1	no	—
68	C₂²×C₄	—	137	2	no	—
69	C₂²×C₄	C₂×D₁₃₇	1	2	yes	—
70	C₂²×C₄	C₂×D₁₃₇	1	1	yes	characteristic
71	C₄⋊C₄	—	137	8	no	—
72	C₄⋊C₄	—	137	2	no	—
73	C₂²⋊C₄	—	137	2	no	—
74	C₂²⋊C₄	—	137	8	no	—
75	C₄²	—	137	2	no	—
76	C₂³	—	137	2	no	—
77	C₂³	C₂²×D₁₃₇	1	1	yes	characteristic
78	Q₈	—	2	4	no	—
79	Q₈	—	2	4	no	—
80	Q₈	—	274	2	no	—
81	D₄	—	274	4	no	—
82	D₄	—	274	2	no	—
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	—	274	4	no	—
85	C₂×C₄	—	137	4	no	—
86	C₂×C₄	—	137	2	no	—
87	C₂×C₄	C₂²⋊D₁₃₇	1	4	yes	—
88	C₂×C₄	C₂²×D₁₃₇	1	4	yes	—
89	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
90	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
91	C₂²	—	2	2	no	—
92	C₂²	—	274	2	no	—
93	C₂²	—	274	4	no	—
94	C₂²	C₂×C₂²⋊D₁₃₇	1	2	yes	—
95	C₂²	C₂³×D₁₃₇	1	1	yes	characteristic
96	C₄	—	2	4	no	—
97	C₄	—	274	2	no	—
98	C₄	—	274	4	no	—
99	C₄	C₂×C₂²⋊D₁₃₇	1	2	yes	—
100	C₄	C₂×C₂²⋊D₁₃₇	1	2	yes	—
101	C₂	—	2	2	no	—
102	C₂	—	548	2	no	—
103	C₂	C₂²×C₂²⋊D₁₃₇	1	1	yes	characteristic
104	C₂	#17984136	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14555510

C₂

yes

maximal, characteristic

#17984086

C₂

yes

maximal, complemented

#17984077

C₂

yes

maximal, complemented

#17984071

C₂

yes

maximal, complemented

#17984064

C₂

yes

maximal, complemented

#17984047

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₁₃₇

C₂

yes

maximal, characteristic, complemented

#14555510

—

C₂×Dic₂₇₄

C₂²

yes

characteristic

C₂×D₅₄₈

C₂²

yes

characteristic

C₂×C₂²⋊D₁₃₇

C₂²

yes

—

C₂×C₄×D₁₃₇

C₂²

yes

—

C₄⋊Dic₁₃₇

C₂²

yes

—

#14555489

C₂²

yes

—

C₂²⋊Dic₁₃₇

C₂²

yes

—

#14555487

C₂²

yes

—

C₄×Dic₁₃₇

C₂²

yes

—

C₂×Q₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

C₂²

yes

—

C₂×Q₈×C₁₃₇

C₂²

yes

—

C₂²×C₄×C₁₃₇

C₂²

yes

—

C₂²×C₄×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

characteristic

Q₈×C₁₃₇

—

Q₈×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

D₄

yes

—

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

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×D₄

yes

—

C₄×C₁₃₇

C₂×D₄

yes

—

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

C₂²×D₄

yes

characteristic

C₂×C₁₃₇

2₋¹⁺⁴

yes

—

D₁₃₇

—

C₁₃₇

#536

yes

characteristic, complemented, Sylow

#536

—

137

maximal, Sylow

C₂×#41

—

137

—

C₂²×Q₈

D₁₃₇

yes

characteristic, complemented

C₄⋊Q₈

—

137

—

#124

—

137

—

#123

—

137

—

C₂²⋊Q₈

—

137

—

#117

—

137

—

#41

—

274

—

C₂×Q₈

—

C₂×Q₈

—

137

—

C₂×Q₈

C₂×D₁₃₇

yes

—

C₂×Q₈

C₂×D₁₃₇

yes

—

C₂×D₄

—

137

—

C₂×D₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

C₂×D₁₃₇

yes

—

C₂²×C₄

C₂×D₁₃₇

yes

characteristic

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₂²⋊C₄

—

137

—

C₂²⋊C₄

—

137

—

C₄²

—

137

—

C₂³

—

137

—

C₂³

C₂²×D₁₃₇

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

274

—

D₄

—

274

—

D₄

—

274

—

C₂×C₄

—

C₂×C₄

—

274

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

C₂²⋊D₁₃₇

yes

—

C₂×C₄

C₂²×D₁₃₇

yes

—

C₂×C₄

C₂²×D₁₃₇

yes

characteristic

C₂×C₄

C₂²×D₁₃₇

yes

characteristic

C₂²

—

C₂²

—

274

—

C₂²

—

274

—

C₂²

C₂×C₂²⋊D₁₃₇

yes

—

C₂²

C₂³×D₁₃₇

yes

characteristic

C₄

—

C₄

—

274

—

C₄

—

274

—

C₄

C₂×C₂²⋊D₁₃₇

yes

—

100

C₄

C₂×C₂²⋊D₁₃₇

yes

—

101

C₂

—

102

C₂

—

548

—

103

C₂

C₂²×C₂²⋊D₁₃₇

yes

characteristic

104

C₂

#17984136

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂⁴
Fitting	C₂²×Q₈×C₁₃₇	C₂
hypercenter	C₂²×Q₈	D₁₃₇
hypocenter	C₁₃₇	#536
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₃₇

commutator, socle

C₂²×C₁₃₇

C₂⁴

Fitting

C₂²×Q₈×C₁₃₇

C₂

hypercenter

C₂²×Q₈

D₁₃₇

C₁₃₇

G ▷ C₂²×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂² ◁ C₂²×Q₈

N	H ≅ G/N	type
2. #17984086	C₂	semidirect
3. #17984077	C₂	semidirect
4. #17984071	C₂	semidirect
5. #17984064	C₂	semidirect
6. #17984047	C₂	semidirect
7. C₂²×Q₈×C₁₃₇	C₂	semidirect
52. C₁₃₇	#536	semidirect
55. C₂²×Q₈	D₁₃₇	semidirect

H ≅ G/N

type

2. #17984086

C₂

semidirect

3. #17984077

C₂

semidirect

4. #17984071

C₂

semidirect

5. #17984064

C₂

semidirect

6. #17984047

C₂

semidirect

7. C₂²×Q₈×C₁₃₇

C₂

semidirect

52. C₁₃₇

#536

semidirect

55. C₂²×Q₈

D₁₃₇

semidirect

Subgroups — #22577977

Conjugacy classes of subgroups

Complements