Group #22578072 — subgroups

Subgroups — #22578072

Order 8768.

312 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 113 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×#14555511	C₂	1	1	yes	maximal, characteristic, complemented
2	#17984084	C₂	1	3	yes	maximal, complemented
3	#17984062	C₂	1	3	yes	maximal, complemented
4	#17984059	C₂	1	3	yes	maximal, complemented
5	Q₈×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₂²⋊D₁₃₇	C₂	1	3	yes	maximal, complemented
7	C₂×#41×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14555511	—	2	4	no	—
9	C₂×D₅₄₈	C₂²	1	3	yes	—
10	C₂×C₂²⋊D₁₃₇	C₂²	1	3	yes	—
11	C₂×C₂²⋊D₁₃₇	C₂²	1	6	yes	—
12	C₂×C₄×D₁₃₇	C₂²	1	3	yes	—
13	C₄⋊Dic₁₃₇	C₂²	1	3	yes	—
14	#14555489	C₂²	1	1	yes	characteristic
15	C₂²⋊Dic₁₃₇	C₂²	1	3	yes	—
16	#14555487	C₂²	1	3	yes	—
17	C₄×Dic₁₃₇	C₂²	1	3	yes	—
18	#41×C₁₃₇	—	2	4	no	—
19	C₂×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₂×D₄×C₁₃₇	C₂²	1	3	yes	—
21	C₂²×C₄×C₁₃₇	C₂²	1	3	yes	—
22	C₂²×D₁₃₇	C₂³	1	3	yes	—
23	D₅₄₈	—	2	6	no	—
24	C₂²⋊D₁₃₇	—	2	6	no	—
25	C₂²⋊D₁₃₇	—	4	6	no	—
26	C₂×Dic₁₃₇	C₂³	1	3	yes	—
27	C₂×Dic₁₃₇	C₂³	1	1	yes	characteristic
28	C₂×Dic₁₃₇	C₂³	1	1	yes	characteristic
29	C₄×D₁₃₇	—	2	6	no	—
30	C₂³×C₁₃₇	C₂³	1	3	yes	—
31	Q₈×C₁₃₇	D₄	1	4	yes	complemented
32	D₄×C₁₃₇	—	2	6	no	—
33	C₂×C₄×C₁₃₇	—	2	6	no	—
34	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₁₃₇	C₂³	1	3	yes	—
36	C₂²×C₁₃₇	—	2	3	no	—
37	C₂²×C₁₃₇	—	2	3	no	—
38	C₂²×C₁₃₇	—	2	3	no	—
39	C₂²×C₁₃₇	C₂⁴	1	1	yes	characteristic
40	C₂×D₁₃₇	—	2	3	no	—
41	C₂×D₁₃₇	—	2	3	no	—
42	C₂×D₁₃₇	—	2	3	no	—
43	C₄×C₁₃₇	—	2	1	no	—
44	C₄×C₁₃₇	C₂×D₄	1	6	yes	—
45	Dic₁₃₇	—	2	1	no	—
46	Dic₁₃₇	—	2	3	no	—
47	Dic₁₃₇	#41	1	2	yes	complemented
48	C₂×C₁₃₇	—	4	3	no	—
49	C₂×C₁₃₇	C₂²×D₄	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	—	4	3	no	—
53	C₁₃₇	#550	1	1	yes	characteristic, complemented, Sylow
54	#550	—	137	1	no	maximal, Sylow
55	C₂×#41	—	137	1	no	—
56	C₂×#41	D₁₃₇	1	1	yes	characteristic, complemented
57	#127	—	137	3	no	—
58	#121	—	137	3	no	—
59	#121	—	137	3	no	—
60	C₄×Q₈	—	137	1	no	—
61	C₄×D₄	—	137	3	no	—
62	#41	—	2	4	no	—
63	#41	—	274	4	no	—
64	C₂×Q₈	C₂×D₁₃₇	1	1	yes	characteristic
65	C₂×D₄	—	137	3	no	—
66	C₂×D₄	—	137	6	no	—
67	C₂×D₄	—	137	3	no	—
68	C₂×D₄	C₂×D₁₃₇	1	3	yes	—
69	C₂²×C₄	—	137	3	no	—
70	C₂²×C₄	C₂×D₁₃₇	1	3	yes	—
71	C₄⋊C₄	—	137	1	no	—
72	C₄⋊C₄	—	137	3	no	—
73	C₂²⋊C₄	—	137	3	no	—
74	C₂²⋊C₄	—	137	3	no	—
75	C₄²	—	137	3	no	—
76	C₂³	—	137	3	no	—
77	C₂³	C₂²×D₁₃₇	1	3	yes	—
78	Q₈	C₂²⋊D₁₃₇	1	4	yes	complemented
79	D₄	—	2	6	no	—
80	D₄	—	274	6	no	—
81	D₄	—	274	6	no	—
82	D₄	—	548	6	no	—
83	C₂×C₄	—	2	6	no	—
84	C₂×C₄	—	274	6	no	—
85	C₂×C₄	—	137	3	no	—
86	C₂×C₄	—	137	1	no	—
87	C₂×C₄	—	137	1	no	—
88	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
89	C₂×C₄	C₂²×D₁₃₇	1	3	yes	—
90	C₂²	—	2	3	no	—
91	C₂²	—	2	3	no	—
92	C₂²	—	2	3	no	—
93	C₂²	—	274	3	no	—
94	C₂²	—	274	3	no	—
95	C₂²	—	274	3	no	—
96	C₂²	C₂³×D₁₃₇	1	1	yes	characteristic
97	C₄	—	2	1	no	—
98	C₄	—	274	1	no	—
99	C₄	—	274	3	no	—
100	C₄	—	137	2	no	—
101	C₄	C₂×C₂²⋊D₁₃₇	1	6	yes	—
102	C₂	—	4	3	no	—
103	C₂	—	548	3	no	—
104	C₂	C₂²×C₂²⋊D₁₃₇	1	1	yes	characteristic
105	C₂	#41×D₁₃₇	1	1	yes	characteristic
106	C₂	#17984134	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14555511

C₂

yes

maximal, characteristic, complemented

#17984084

C₂

yes

maximal, complemented

#17984062

C₂

yes

maximal, complemented

#17984059

C₂

yes

maximal, complemented

Q₈×Dic₁₃₇

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₁₃₇

C₂

yes

maximal, complemented

C₂×#41×C₁₃₇

C₂

yes

maximal, characteristic, complemented

#14555511

—

C₂×D₅₄₈

C₂²

yes

—

C₂×C₂²⋊D₁₃₇

C₂²

yes

—

C₂×C₂²⋊D₁₃₇

C₂²

yes

—

C₂×C₄×D₁₃₇

C₂²

yes

—

C₄⋊Dic₁₃₇

C₂²

yes

—

#14555489

C₂²

yes

characteristic

C₂²⋊Dic₁₃₇

C₂²

yes

—

#14555487

C₂²

yes

—

C₄×Dic₁₃₇

C₂²

yes

—

#41×C₁₃₇

—

C₂×Q₈×C₁₃₇

C₂²

yes

characteristic

C₂×D₄×C₁₃₇

C₂²

yes

—

C₂²×C₄×C₁₃₇

C₂²

yes

—

C₂²×D₁₃₇

C₂³

yes

—

D₅₄₈

—

C₂²⋊D₁₃₇

—

C₂²⋊D₁₃₇

—

C₂×Dic₁₃₇

C₂³

yes

—

C₂×Dic₁₃₇

C₂³

yes

characteristic

C₂×Dic₁₃₇

C₂³

yes

characteristic

C₄×D₁₃₇

—

C₂³×C₁₃₇

C₂³

yes

—

Q₈×C₁₃₇

D₄

yes

complemented

D₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂³

yes

characteristic

C₂×C₄×C₁₃₇

C₂³

yes

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂⁴

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×D₄

yes

—

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

#41

yes

complemented

C₂×C₁₃₇

—

C₂×C₁₃₇

C₂²×D₄

yes

characteristic

C₂×C₁₃₇

C₂×#41

yes

characteristic

C₂×C₁₃₇

H(5,𝔽₂)

yes

characteristic

D₁₃₇

—

C₁₃₇

#550

yes

characteristic, complemented, Sylow

#550

—

137

maximal, Sylow

C₂×#41

—

137

—

C₂×#41

D₁₃₇

yes

characteristic, complemented

#127

—

137

—

#121

—

137

—

#121

—

137

—

C₄×Q₈

—

137

—

C₄×D₄

—

137

—

#41

—

#41

—

274

—

C₂×Q₈

C₂×D₁₃₇

yes

characteristic

C₂×D₄

—

137

—

C₂×D₄

—

137

—

C₂×D₄

—

137

—

C₂×D₄

C₂×D₁₃₇

yes

—

C₂²×C₄

—

137

—

C₂²×C₄

C₂×D₁₃₇

yes

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₂²⋊C₄

—

137

—

C₂²⋊C₄

—

137

—

C₄²

—

137

—

C₂³

—

137

—

C₂³

C₂²×D₁₃₇

yes

—

Q₈

C₂²⋊D₁₃₇

yes

complemented

D₄

—

D₄

—

274

—

D₄

—

274

—

D₄

—

548

—

C₂×C₄

—

C₂×C₄

—

274

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

C₂²×D₁₃₇

yes

characteristic

C₂×C₄

C₂²×D₁₃₇

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

274

—

C₂²

—

274

—

C₂²

—

274

—

C₂²

C₂³×D₁₃₇

yes

characteristic

C₄

—

C₄

—

274

—

C₄

—

274

—

100

C₄

—

137

—

101

C₄

C₂×C₂²⋊D₁₃₇

yes

—

102

C₂

—

103

C₂

—

548

—

104

C₂

C₂²×C₂²⋊D₁₃₇

yes

characteristic

105

C₂

#41×D₁₃₇

yes

characteristic

106

C₂

#17984134

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂⁴
Fitting	C₂×#41×C₁₃₇	C₂
hypercenter	C₂×#41	D₁₃₇
hypocenter	C₁₃₇	#550
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×#41

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₃₇

commutator, socle

C₂²×C₁₃₇

C₂⁴

Fitting

C₂×#41×C₁₃₇

C₂

hypercenter

C₂×#41

D₁₃₇

C₁₃₇

G ▷ C₂²×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂² ◁ C₂×#41

N	H ≅ G/N	type
1. C₂×#14555511	C₂	semidirect
2. #17984084	C₂	semidirect
3. #17984062	C₂	semidirect
4. #17984059	C₂	semidirect
5. Q₈×Dic₁₃₇	C₂	semidirect
6. C₄×C₂²⋊D₁₃₇	C₂	semidirect
7. C₂×#41×C₁₃₇	C₂	semidirect
31. Q₈×C₁₃₇	D₄	semidirect
47. Dic₁₃₇	#41	semidirect
53. C₁₃₇	#550	semidirect
56. C₂×#41	D₁₃₇	semidirect
78. Q₈	C₂²⋊D₁₃₇	semidirect

H ≅ G/N

type

1. C₂×#14555511

C₂

semidirect

2. #17984084

C₂

semidirect

3. #17984062

C₂

semidirect

4. #17984059

C₂

semidirect

5. Q₈×Dic₁₃₇

C₂

semidirect

6. C₄×C₂²⋊D₁₃₇

C₂

semidirect

7. C₂×#41×C₁₃₇

C₂

semidirect

31. Q₈×C₁₃₇

D₄

semidirect

47. Dic₁₃₇

#41

semidirect

53. C₁₃₇

#550

semidirect

56. C₂×#41

D₁₃₇

semidirect

78. Q₈

C₂²⋊D₁₃₇

semidirect

Subgroups — #22578072

Conjugacy classes of subgroups

Complements