Group #22578476 — subgroups

Subgroups — C₂×D₁₃₇⋊M₁₆ #22578476

Order 8768.

298 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 148 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×#14555520	C₂	1	2	yes	maximal, complemented
2	D₁₃₇⋊M₁₆	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14555519	C₂	1	2	yes	maximal, complemented
4	C₂×#14555518	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×D₁₃₇	C₂	1	1	yes	maximal, characteristic
6	#14555520	C₂²	1	8	yes	complemented
7	#14555519	C₂²	1	8	yes	complemented
8	#14555518	C₂²	1	8	yes	complemented
9	C₂×#11764047	C₂²	1	4	yes	complemented
10	C₂³×D₁₃₇	C₄	1	1	yes	characteristic
11	C₂²×Dic₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₁₃₇	C₄	1	4	yes	—
13	C₂×C₄×D₁₃₇	C₄	1	1	yes	characteristic
14	C₂×C₄×D₁₃₇	C₄	1	1	yes	characteristic
15	C₂×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₁₃₇	C₂²	1	4	yes	—
17	C₂×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₁₃₇	C₄	1	1	yes	characteristic
19	#11764047	C₂³	1	8	yes	complemented
20	C₂²×D₁₃₇	—	2	4	no	—
21	C₂²×D₁₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×D₁₃₇	C₂×C₄	1	4	yes	—
23	C₂²×D₁₃₇	C₂×C₄	1	1	yes	characteristic
24	C₂×Dic₁₃₇	C₂×C₄	1	2	yes	—
25	C₂×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
26	C₂×Dic₁₃₇	C₂³	1	1	yes	characteristic
27	C₂×Dic₁₃₇	C₂³	1	2	yes	—
28	C₄×D₁₃₇	C₂×C₄	1	4	yes	—
29	C₄×D₁₃₇	C₂×C₄	1	2	yes	—
30	C₄×D₁₃₇	C₂×C₄	1	2	yes	—
31	C₄×D₁₃₇	C₂×C₄	1	4	yes	—
32	C₄×D₁₃₇	C₂³	1	2	yes	—
33	C₄×D₁₃₇	C₂³	1	2	yes	—
34	C₂³×C₁₃₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₃₇	C₂×C₄	1	4	yes	—
36	C₂×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₁₃₇	—	2	2	no	—
39	C₂²×C₁₃₇	C₂²×C₄	1	2	yes	—
40	C₂²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×D₁₃₇	—	2	2	no	—
42	C₂×D₁₃₇	—	2	8	no	—
43	C₂×D₁₃₇	M₁₆	1	4	yes	complemented
44	C₂×D₁₃₇	C₂²×C₄	1	2	yes	—
45	C₂×D₁₃₇	C₂²×C₄	1	2	yes	—
46	C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
47	C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
48	Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
49	Dic₁₃₇	C₂²×C₄	1	2	yes	—
50	Dic₁₃₇	C₂⁴	1	1	yes	characteristic
51	C₂×C₁₃₇	—	2	2	no	—
52	C₂×C₁₃₇	C₂×M₁₆	1	2	yes	complemented
53	C₂×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
54	D₁₃₇	—	2	2	no	—
55	D₁₃₇	C₂×M₁₆	1	4	yes	complemented
56	C₁₃₇	C₂²×M₁₆	1	1	yes	characteristic, complemented, Sylow
57	C₂²×M₁₆	—	137	1	no	maximal, Sylow
58	C₂³×C₄	—	137	1	no	—
59	C₂×M₁₆	—	137	2	no	—
60	C₂×M₁₆	—	137	8	no	—
61	C₂×M₁₆	—	137	2	no	—
62	C₂²×C₈	—	137	2	no	—
63	C₂⁴	—	137	1	no	—
64	C₂²×C₄	—	137	1	no	—
65	C₂²×C₄	—	137	4	no	—
66	C₂²×C₄	—	137	1	no	—
67	C₂²×C₄	—	137	1	no	—
68	C₂²×C₄	—	137	4	no	—
69	C₂²×C₄	—	137	1	no	—
70	C₂²×C₄	—	137	1	no	—
71	C₂²×C₄	#10587739	1	1	yes	characteristic
72	M₁₆	—	137	8	no	—
73	M₁₆	—	137	8	no	—
74	C₂×C₈	—	137	4	no	—
75	C₂×C₈	—	137	8	no	—
76	C₂³	—	274	4	no	—
77	C₂³	—	137	4	no	—
78	C₂³	—	137	1	no	—
79	C₂³	—	137	1	no	—
80	C₂³	C₂×#10587739	1	1	yes	characteristic
81	C₂×C₄	—	137	1	no	—
82	C₂×C₄	—	137	2	no	—
83	C₂×C₄	—	137	2	no	—
84	C₂×C₄	—	137	1	no	—
85	C₂×C₄	—	137	2	no	—
86	C₂×C₄	—	137	2	no	—
87	C₂×C₄	—	137	2	no	—
88	C₂×C₄	—	137	4	no	—
89	C₂×C₄	—	137	4	no	—
90	C₂×C₄	—	137	2	no	—
91	C₂×C₄	C₂×#10587739	1	1	yes	characteristic
92	C₂×C₄	C₂×#10587739	1	4	yes	—
93	C₂×C₄	C₂×#10587739	1	1	yes	characteristic
94	C₈	—	137	8	no	—
95	C₂²	—	2	2	no	—
96	C₂²	—	274	8	no	—
97	C₂²	—	274	2	no	—
98	C₂²	—	137	2	no	—
99	C₂²	—	137	4	no	—
100	C₂²	—	137	2	no	—
101	C₂²	C₂²×#10587739	1	2	yes	—
102	C₂²	C₂²×#10587739	1	1	yes	characteristic
103	C₄	—	137	1	no	—
104	C₄	—	137	1	no	—
105	C₄	—	137	2	no	—
106	C₄	C₂²×#10587739	1	2	yes	—
107	C₄	C₂²×#10587739	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	—	274	2	no	—
110	C₂	—	137	4	no	—
111	C₂	D₁₃₇⋊M₁₆	1	2	yes	direct product factor
112	C₂	C₂³×#10587739	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14555520

C₂

yes

maximal, complemented

D₁₃₇⋊M₁₆

C₂

yes

maximal, direct product factor

C₂×#14555519

C₂

yes

maximal, complemented

C₂×#14555518

C₂

yes

maximal, complemented

C₂²×C₄×D₁₃₇

C₂

yes

maximal, characteristic

#14555520

C₂²

yes

complemented

#14555519

C₂²

yes

complemented

#14555518

C₂²

yes

complemented

C₂×#11764047

C₂²

yes

complemented

C₂³×D₁₃₇

C₄

yes

characteristic

C₂²×Dic₁₃₇

C₂²

yes

characteristic

C₂×C₄×D₁₃₇

C₄

yes

—

C₂×C₄×D₁₃₇

C₄

yes

characteristic

C₂×C₄×D₁₃₇

C₄

yes

characteristic

C₂×C₄×D₁₃₇

C₂²

yes

characteristic

C₂×C₄×D₁₃₇

C₂²

yes

—

C₂×C₄×D₁₃₇

C₂²

yes

characteristic

C₂²×C₄×C₁₃₇

C₄

yes

characteristic

#11764047

C₂³

yes

complemented

C₂²×D₁₃₇

—

C₂²×D₁₃₇

C₂×C₄

yes

characteristic

C₂²×D₁₃₇

C₂×C₄

yes

—

C₂²×D₁₃₇

C₂×C₄

yes

characteristic

C₂×Dic₁₃₇

C₂×C₄

yes

—

C₂×Dic₁₃₇

C₂×C₄

yes

characteristic

C₂×Dic₁₃₇

C₂³

yes

characteristic

C₂×Dic₁₃₇

C₂³

yes

—

C₄×D₁₃₇

C₂×C₄

yes

—

C₄×D₁₃₇

C₂×C₄

yes

—

C₄×D₁₃₇

C₂×C₄

yes

—

C₄×D₁₃₇

C₂×C₄

yes

—

C₄×D₁₃₇

C₂³

yes

—

C₄×D₁₃₇

C₂³

yes

—

C₂³×C₁₃₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×C₄

yes

—

C₂×C₄×C₁₃₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×C₄

yes

characteristic

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂²×C₄

yes

—

C₂²×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

M₁₆

yes

complemented

C₂×D₁₃₇

C₂²×C₄

yes

—

C₂×D₁₃₇

C₂²×C₄

yes

—

C₄×C₁₃₇

C₂²×C₄

yes

—

C₄×C₁₃₇

C₂²×C₄

yes

—

Dic₁₃₇

C₂²×C₄

yes

characteristic

Dic₁₃₇

C₂²×C₄

yes

—

Dic₁₃₇

C₂⁴

yes

characteristic

C₂×C₁₃₇

—

C₂×C₁₃₇

C₂×M₁₆

yes

complemented

C₂×C₁₃₇

C₂³×C₄

yes

characteristic

D₁₃₇

—

D₁₃₇

C₂×M₁₆

yes

complemented

C₁₃₇

C₂²×M₁₆

yes

characteristic, complemented, Sylow

C₂²×M₁₆

—

137

maximal, Sylow

C₂³×C₄

—

137

—

C₂×M₁₆

—

137

—

C₂×M₁₆

—

137

—

C₂×M₁₆

—

137

—

C₂²×C₈

—

137

—

C₂⁴

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

#10587739

yes

characteristic

M₁₆

—

137

—

M₁₆

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂³

—

274

—

C₂³

—

137

—

C₂³

—

137

—

C₂³

—

137

—

C₂³

C₂×#10587739

yes

characteristic

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

C₂×#10587739

yes

characteristic

C₂×C₄

C₂×#10587739

yes

—

C₂×C₄

C₂×#10587739

yes

characteristic

C₈

—

137

—

C₂²

—

C₂²

—

274

—

C₂²

—

274

—

C₂²

—

137

—

C₂²

—

137

—

100

C₂²

—

137

—

101

C₂²

C₂²×#10587739

yes

—

102

C₂²

C₂²×#10587739

yes

characteristic

103

C₄

—

137

—

104

C₄

—

137

—

105

C₄

—

137

—

106

C₄

C₂²×#10587739

yes

—

107

C₄

C₂²×#10587739

yes

—

108

C₂

—

109

C₂

—

274

—

110

C₂

—

137

—

111

C₂

D₁₃₇⋊M₁₆

yes

direct product factor

112

C₂

C₂³×#10587739

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#10587739
commutator	C₂×C₁₃₇	C₂³×C₄
Frattini	C₂	C₂³×#10587739
Fitting	C₂²×C₄×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂²×C₄
hypercenter	C₂²×C₄	#10587739
hypocenter	C₁₃₇	C₂²×M₁₆
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

subgroup

G/H

center

C₂×C₄

C₂×#10587739

commutator

C₂×C₁₃₇

C₂³×C₄

Frattini

C₂

C₂³×#10587739

Fitting

C₂²×C₄×C₁₃₇

C₄

socle

C₂²×C₁₃₇

C₂²×C₄

hypercenter

C₂²×C₄

#10587739

hypocenter

C₁₃₇

C₂²×M₁₆

LCS

G ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄

N	H ≅ G/N	type
1. C₂×#14555520	C₂	semidirect
2. D₁₃₇⋊M₁₆	C₂	direct (N × H)
3. C₂×#14555519	C₂	semidirect
4. C₂×#14555518	C₂	semidirect
6. #14555520	C₂²	semidirect
7. #14555519	C₂²	semidirect
8. #14555518	C₂²	semidirect
9. C₂×#11764047	C₂²	semidirect
19. #11764047	C₂³	semidirect
43. C₂×D₁₃₇	M₁₆	semidirect
52. C₂×C₁₃₇	C₂×M₁₆	semidirect
55. D₁₃₇	C₂×M₁₆	semidirect
56. C₁₃₇	C₂²×M₁₆	semidirect
111. C₂	D₁₃₇⋊M₁₆	direct (N × H)

H ≅ G/N

type

1. C₂×#14555520

C₂

semidirect

2. D₁₃₇⋊M₁₆

C₂

direct (N × H)

3. C₂×#14555519

C₂

semidirect

4. C₂×#14555518

C₂

semidirect

6. #14555520

C₂²

semidirect

7. #14555519

C₂²

semidirect

8. #14555518

C₂²

semidirect

9. C₂×#11764047

C₂²

semidirect

19. #11764047

C₂³

semidirect

43. C₂×D₁₃₇

M₁₆

semidirect

52. C₂×C₁₃₇

C₂×M₁₆

semidirect

55. D₁₃₇

C₂×M₁₆

semidirect

56. C₁₃₇

C₂²×M₁₆

semidirect

111. C₂

D₁₃₇⋊M₁₆

direct (N × H)

Subgroups — C₂×D₁₃₇⋊M₁₆ #22578476

Conjugacy classes of subgroups

Complements

Subgroups — C2×D137⋊M16 #22578476

Conjugacy classes of subgroups

Complements

Subgroups — C₂×D₁₃₇⋊M₁₆ #22578476