Group #22578345 — subgroups

Subgroups — #22578345

Order 8768.

234 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 70 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×C₄⋊#10587739	C₂	1	2	yes	maximal, complemented
2	C₂×C₂²⋊#10587739	C₂	1	2	yes	maximal
3	(C₂×#10587739)⋊C₄	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄×D₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10587739	C₂²	1	4	yes	—
6	C₄⋊#10587739	—	2	4	no	—
7	C₂²⋊#10587739	—	2	4	no	—
8	C₂³×D₁₃₇	C₂²	1	1	yes	characteristic
9	C₂²×Dic₁₃₇	C₄	1	1	yes	characteristic, complemented
10	C₂×C₄×D₁₃₇	—	2	4	no	—
11	C₂×C₄×D₁₃₇	C₄	1	1	yes	characteristic, complemented
12	C₂×C₄×D₁₃₇	C₄	1	1	yes	characteristic, complemented
13	C₂×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₁₃₇	C₄	1	1	yes	characteristic, complemented
16	C₂×#10587739	—	2	4	no	—
17	C₂×#10587739	—	2	4	no	—
18	C₂×#10587739	—	2	4	no	—
19	C₂²×D₁₃₇	—	2	2	no	—
20	C₂²×D₁₃₇	—	2	2	no	—
21	C₂²×D₁₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×D₁₃₇	D₄	1	2	yes	—
23	C₂²×D₁₃₇	Q₈	1	2	yes	—
24	C₂²×D₁₃₇	C₂³	1	1	yes	characteristic
25	C₂×Dic₁₃₇	—	2	2	no	—
26	C₂×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
27	C₂×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
28	C₄×D₁₃₇	—	2	2	no	—
29	C₄×D₁₃₇	—	2	2	no	—
30	C₄×D₁₃₇	—	2	2	no	—
31	C₄×D₁₃₇	D₄	1	4	yes	—
32	C₂³×C₁₃₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₃₇	—	2	2	no	—
34	C₂×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₁₃₇	—	2	1	no	—
37	C₂²×C₁₃₇	—	2	1	no	—
38	C₂²×C₁₃₇	C₄⋊C₄	1	2	yes	complemented
39	C₂²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×D₁₃₇	—	2	1	no	—
41	C₂×D₁₃₇	—	2	1	no	—
42	C₂×D₁₃₇	—	2	4	no	—
43	C₂×D₁₃₇	—	2	4	no	—
44	C₂×D₁₃₇	C₄⋊C₄	1	2	yes	complemented
45	C₂×D₁₃₇	C₂×D₄	1	1	yes	characteristic
46	C₂×D₁₃₇	C₂×D₄	1	2	yes	—
47	C₂×D₁₃₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×D₁₃₇	#41	1	2	yes	—
49	#10587739	—	4	4	no	—
50	C₄×C₁₃₇	—	2	1	no	—
51	C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	complemented
52	Dic₁₃₇	—	2	1	no	—
53	Dic₁₃₇	C₂²⋊C₄	1	2	yes	complemented
54	C₂×C₁₃₇	—	2	2	no	—
55	C₂×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₁₃₇	#117	1	1	yes	characteristic
58	D₁₃₇	—	2	2	no	—
59	D₁₃₇	#121	1	2	yes	—
60	D₁₃₇	C₂²⋊Q₈	1	2	yes	—
61	C₁₃₇	#380	1	1	yes	characteristic, complemented, Sylow
62	#380	—	137	1	no	maximal, Sylow
63	C₂³×C₄	—	137	1	no	—
64	C₂×C₄⋊C₄	—	137	2	no	—
65	C₂×C₂²⋊C₄	—	137	2	no	—
66	#95	—	137	2	no	—
67	C₂⁴	—	137	1	no	—
68	C₂²×C₄	—	274	4	no	—
69	C₂²×C₄	—	137	4	no	—
70	C₂²×C₄	—	137	1	no	—
71	C₂²×C₄	—	137	1	no	—
72	C₂²×C₄	—	137	1	no	—
73	C₂²×C₄	—	137	1	no	—
74	C₂²×C₄	—	137	1	no	—
75	C₂²×C₄	#10587739	1	1	yes	characteristic, complemented
76	C₄⋊C₄	—	274	4	no	—
77	C₂²⋊C₄	—	274	4	no	—
78	C₂³	—	274	2	no	—
79	C₂³	—	274	2	no	—
80	C₂³	—	137	2	no	—
81	C₂³	—	137	2	no	—
82	C₂³	—	137	1	no	—
83	C₂³	—	137	1	no	—
84	C₂³	C₂×#10587739	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	274	2	no	—
87	C₂×C₄	—	274	2	no	—
88	C₂×C₄	—	274	2	no	—
89	C₂×C₄	—	274	2	no	—
90	C₂×C₄	—	274	4	no	—
91	C₂×C₄	—	274	4	no	—
92	C₂×C₄	—	274	4	no	—
93	C₂×C₄	—	137	1	no	—
94	C₂×C₄	—	137	1	no	—
95	C₂×C₄	—	137	4	no	—
96	C₂×C₄	C₂×#10587739	1	1	yes	characteristic
97	C₂×C₄	C₂×#10587739	1	1	yes	characteristic
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	—	274	4	no	—
101	C₂²	—	274	4	no	—
102	C₂²	—	274	1	no	—
103	C₂²	—	274	1	no	—
104	C₂²	—	137	2	no	—
105	C₂²	—	137	2	no	—
106	C₂²	—	137	1	no	—
107	C₂²	—	137	1	no	—
108	C₂²	—	137	2	no	—
109	C₂²	C₄⋊#10587739	1	2	yes	complemented
110	C₂²	C₂²×#10587739	1	1	yes	characteristic
111	C₄	—	2	1	no	—
112	C₄	—	274	1	no	—
113	C₄	—	548	4	no	—
114	C₄	—	137	2	no	—
115	C₄	C₂²⋊#10587739	1	2	yes	complemented
116	C₂	—	2	2	no	—
117	C₂	—	274	2	no	—
118	C₂	—	137	2	no	—
119	C₂	—	137	2	no	—
120	C₂	C₂×C₂²⋊#10587739	1	1	yes	characteristic
121	C₂	C₂×C₄⋊#10587739	1	1	yes	characteristic
122	C₂	#17984167	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#10587739

C₂

yes

maximal, complemented

C₂×C₂²⋊#10587739

C₂

yes

maximal

(C₂×#10587739)⋊C₄

C₂

yes

maximal, complemented

C₂²×C₄×D₁₃₇

C₂

yes

maximal, characteristic

C₂²×#10587739

C₂²

yes

—

C₄⋊#10587739

—

C₂²⋊#10587739

—

C₂³×D₁₃₇

C₂²

yes

characteristic

C₂²×Dic₁₃₇

C₄

yes

characteristic, complemented

C₂×C₄×D₁₃₇

—

C₂×C₄×D₁₃₇

C₄

yes

characteristic, complemented

C₂×C₄×D₁₃₇

C₄

yes

characteristic, complemented

C₂×C₄×D₁₃₇

C₂²

yes

characteristic

C₂×C₄×D₁₃₇

C₂²

yes

characteristic

C₂²×C₄×C₁₃₇

C₄

yes

characteristic, complemented

C₂×#10587739

—

C₂×#10587739

—

C₂×#10587739

—

C₂²×D₁₃₇

—

C₂²×D₁₃₇

—

C₂²×D₁₃₇

C₂×C₄

yes

characteristic

C₂²×D₁₃₇

D₄

yes

—

C₂²×D₁₃₇

Q₈

yes

—

C₂²×D₁₃₇

C₂³

yes

characteristic

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

C₂×C₄

yes

characteristic

C₂×Dic₁₃₇

C₂×C₄

yes

characteristic

C₄×D₁₃₇

—

C₄×D₁₃₇

—

C₄×D₁₃₇

—

C₄×D₁₃₇

D₄

yes

—

C₂³×C₁₃₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×C₄

yes

characteristic

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₄⋊C₄

yes

complemented

C₂²×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

C₄⋊C₄

yes

complemented

C₂×D₁₃₇

C₂×D₄

yes

characteristic

C₂×D₁₃₇

C₂×D₄

yes

—

C₂×D₁₃₇

C₂×Q₈

yes

characteristic

C₂×D₁₃₇

#41

yes

—

#10587739

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂²⋊C₄

yes

complemented

Dic₁₃₇

—

Dic₁₃₇

C₂²⋊C₄

yes

complemented

C₂×C₁₃₇

—

C₂×C₁₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₁₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₁₃₇

#117

yes

characteristic

D₁₃₇

—

D₁₃₇

#121

yes

—

D₁₃₇

C₂²⋊Q₈

yes

—

C₁₃₇

#380

yes

characteristic, complemented, Sylow

#380

—

137

maximal, Sylow

C₂³×C₄

—

137

—

C₂×C₄⋊C₄

—

137

—

C₂×C₂²⋊C₄

—

137

—

#95

—

137

—

C₂⁴

—

137

—

C₂²×C₄

—

274

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

#10587739

yes

characteristic, complemented

C₄⋊C₄

—

274

—

C₂²⋊C₄

—

274

—

C₂³

—

274

—

C₂³

—

274

—

C₂³

—

137

—

C₂³

—

137

—

C₂³

—

137

—

C₂³

—

137

—

C₂³

C₂×#10587739

yes

characteristic

C₂×C₄

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

yes

C₂×C₄

yes

C₂²

—

C₂²

—

100

C₂²

—

274

—

101

C₂²

—

274

—

102

C₂²

—

274

—

103

C₂²

—

274

—

104

C₂²

—

137

—

105

C₂²

—

137

—

106

C₂²

—

137

—

107

C₂²

—

137

—

108

C₂²

—

137

—

109

C₂²

C₄⋊#10587739

yes

complemented

110

C₂²

C₂²×#10587739

yes

characteristic

111

C₄

—

112

C₄

—

274

—

113

C₄

—

548

—

114

C₄

—

137

—

115

C₄

C₂²⋊#10587739

yes

complemented

116

C₂

—

117

C₂

—

274

—

118

C₂

—

137

—

119

C₂

—

137

—

120

C₂

C₂×C₂²⋊#10587739

yes

characteristic

121

C₂

C₂×C₄⋊#10587739

yes

characteristic

122

C₂

#17984167

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#10587739
commutator, socle	C₂²×C₁₃₇	C₂²×C₄
Fitting	C₂²×C₄×C₁₃₇	C₄
hypercenter	C₂²×C₄	#10587739
hypocenter	C₁₃₇	#380
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

subgroup

G/H

center, Frattini

C₂²

C₂²×#10587739

commutator, socle

C₂²×C₁₃₇

C₂²×C₄

Fitting

C₂²×C₄×C₁₃₇

C₄

C₂²×C₄

C₁₃₇

G ▷ C₂²×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂² ◁ C₂²×C₄

N	H ≅ G/N	type
1. C₂×C₄⋊#10587739	C₂	semidirect
3. (C₂×#10587739)⋊C₄	C₂	semidirect
9. C₂²×Dic₁₃₇	C₄	semidirect
11. C₂×C₄×D₁₃₇	C₄	semidirect
12. C₂×C₄×D₁₃₇	C₄	semidirect
15. C₂²×C₄×C₁₃₇	C₄	semidirect
38. C₂²×C₁₃₇	C₄⋊C₄	semidirect
44. C₂×D₁₃₇	C₄⋊C₄	semidirect
51. C₄×C₁₃₇	C₂²⋊C₄	semidirect
53. Dic₁₃₇	C₂²⋊C₄	semidirect
61. C₁₃₇	#380	semidirect
75. C₂²×C₄	#10587739	semidirect
109. C₂²	C₄⋊#10587739	semidirect
115. C₄	C₂²⋊#10587739	semidirect

H ≅ G/N

type

1. C₂×C₄⋊#10587739

C₂

semidirect

3. (C₂×#10587739)⋊C₄

C₂

semidirect

9. C₂²×Dic₁₃₇

C₄

semidirect

11. C₂×C₄×D₁₃₇

C₄

semidirect

12. C₂×C₄×D₁₃₇

C₄

semidirect

15. C₂²×C₄×C₁₃₇

C₄

semidirect

38. C₂²×C₁₃₇

C₄⋊C₄

semidirect

44. C₂×D₁₃₇

C₄⋊C₄

semidirect

51. C₄×C₁₃₇

C₂²⋊C₄

semidirect

53. Dic₁₃₇

C₂²⋊C₄

semidirect

61. C₁₃₇

#380

semidirect

75. C₂²×C₄

#10587739

semidirect

109. C₂²

C₄⋊#10587739

semidirect

115. C₄

C₂²⋊#10587739

semidirect

Subgroups — #22578345

Conjugacy classes of subgroups

Complements