Group #37554930 — subgroups

Subgroups — #37554930

Order 23488.

220 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 91 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	#26044253	C₂	1	1	yes	maximal, characteristic, complemented
2	#26044249	C₂	1	1	yes	maximal, characteristic, complemented
3	#26044245	C₂	1	2	yes	maximal, complemented
4	#26044243	C₂	1	2	yes	maximal, complemented
5	#26044238	C₂	1	2	yes	maximal, complemented
6	#26044237	C₂	1	2	yes	maximal, complemented
7	#26044234	C₂	1	2	yes	maximal, complemented
8	#26044233	C₂	1	2	yes	maximal, complemented
9	#124×C₃₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₇₃₄	C₂²	1	1	yes	characteristic
11	C₂×D₁₄₆₈	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₃₆₇	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₃₆₇	C₂²	1	2	yes	—
14	C₂²×Dic₃₆₇	C₂²	1	2	yes	—
15	C₂×C₄×D₃₆₇	C₂²	1	2	yes	—
16	C₄⋊Dic₃₆₇	C₂²	1	2	yes	—
17	C₄⋊Dic₃₆₇	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₃₆₇	C₂²	1	1	yes	characteristic
19	Dic₃₆₇⋊C₄	C₂²	1	2	yes	—
20	Dic₃₆₇⋊C₄	C₂²	1	2	yes	—
21	Dic₃₆₇⋊C₄	C₂²	1	2	yes	—
22	C₂²⋊Dic₃₆₇	C₂²	1	2	yes	—
23	(C₂×D₃₆₇)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₃₆₇)⋊C₄	C₂²	1	2	yes	—
25	(C₂×D₃₆₇)⋊C₄	C₂²	1	2	yes	—
26	C₂×Q₈×C₃₆₇	C₂²	1	1	yes	characteristic
27	C₂×D₄×C₃₆₇	C₂²	1	1	yes	characteristic
28	C₂²⋊C₄×C₃₆₇	C₂²	1	2	yes	—
29	C₂²⋊C₄×C₃₆₇	C₂²	1	2	yes	—
30	C₄²×C₃₆₇	C₂²	1	1	yes	characteristic
31	C₂²×D₃₆₇	C₂³	1	2	yes	—
32	Dic₇₃₄	—	4	1	no	—
33	D₁₄₆₈	—	4	1	no	—
34	C₂²⋊D₃₆₇	—	4	2	no	—
35	C₂²⋊D₃₆₇	—	4	2	no	—
36	C₂×Dic₃₆₇	—	4	2	no	—
37	C₂×Dic₃₆₇	C₂³	1	2	yes	—
38	C₂×Dic₃₆₇	C₂³	1	2	yes	—
39	C₂×Dic₃₆₇	C₂³	1	2	yes	—
40	C₄×D₃₆₇	—	4	2	no	—
41	C₂³×C₃₆₇	C₂³	1	2	yes	—
42	Q₈×C₃₆₇	—	4	1	no	—
43	D₄×C₃₆₇	—	4	1	no	—
44	C₂×C₄×C₃₆₇	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₃₆₇	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₃₆₇	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₃₆₇	C₂³	1	2	yes	—
48	C₂²×C₃₆₇	—	2	2	no	—
49	C₂²×C₃₆₇	—	2	2	no	—
50	C₂²×C₃₆₇	—	2	2	no	—
51	C₂²×C₃₆₇	C₂⁴	1	1	yes	characteristic
52	C₂×D₃₆₇	—	2	2	no	—
53	C₂×D₃₆₇	—	2	2	no	—
54	C₂×D₃₆₇	—	2	2	no	—
55	C₄×C₃₆₇	—	2	1	no	—
56	C₄×C₃₆₇	—	2	2	no	—
57	C₄×C₃₆₇	—	2	1	no	—
58	C₄×C₃₆₇	—	2	1	no	—
59	Dic₃₆₇	—	2	2	no	—
60	Dic₃₆₇	—	2	2	no	—
61	Dic₃₆₇	—	2	2	no	—
62	C₂×C₃₆₇	—	4	2	no	—
63	C₂×C₃₆₇	H(5,𝔽₂)	1	1	yes	characteristic
64	C₂×C₃₆₇	H(5,𝔽₂)	1	1	yes	characteristic
65	C₂×C₃₆₇	2₋¹⁺⁴	1	1	yes	characteristic
66	D₃₆₇	—	4	2	no	—
67	C₃₆₇	#562	1	1	yes	characteristic, complemented, Sylow
68	#562	—	367	1	no	maximal, Sylow
69	#125	—	367	1	no	—
70	#124	—	367	1	no	—
71	#124	D₃₆₇	1	1	yes	characteristic, complemented
72	#123	—	367	2	no	—
73	#123	—	367	2	no	—
74	C₂²⋊Q₈	—	367	2	no	—
75	C₂²⋊Q₈	—	367	2	no	—
76	#121	—	367	2	no	—
77	#121	—	367	2	no	—
78	C₂×Q₈	—	367	1	no	—
79	C₂×Q₈	C₂×D₃₆₇	1	1	yes	characteristic
80	C₂×D₄	—	367	2	no	—
81	C₂×D₄	—	367	2	no	—
82	C₂×D₄	—	367	1	no	—
83	C₂×D₄	C₂×D₃₆₇	1	1	yes	characteristic
84	C₂²×C₄	—	367	2	no	—
85	C₂²×C₄	—	367	2	no	—
86	C₄⋊C₄	—	367	2	no	—
87	C₄⋊C₄	—	367	2	no	—
88	C₄⋊C₄	—	367	2	no	—
89	C₄⋊C₄	—	367	2	no	—
90	C₄⋊C₄	—	367	1	no	—
91	C₄⋊C₄	—	367	1	no	—
92	C₂²⋊C₄	—	367	2	no	—
93	C₂²⋊C₄	—	367	2	no	—
94	C₂²⋊C₄	—	367	2	no	—
95	C₂²⋊C₄	—	367	2	no	—
96	C₂²⋊C₄	C₂×D₃₆₇	1	2	yes	—
97	C₂²⋊C₄	C₂×D₃₆₇	1	2	yes	—
98	C₄²	C₂×D₃₆₇	1	1	yes	characteristic
99	C₂³	—	367	2	no	—
100	C₂³	C₂²×D₃₆₇	1	2	yes	—
101	Q₈	—	4	1	no	—
102	Q₈	—	1468	1	no	—
103	D₄	—	4	1	no	—
104	D₄	—	1468	1	no	—
105	D₄	—	1468	2	no	—
106	D₄	—	1468	2	no	—
107	C₂×C₄	—	367	2	no	—
108	C₂×C₄	—	367	2	no	—
109	C₂×C₄	—	367	2	no	—
110	C₂×C₄	—	1468	2	no	—
111	C₂×C₄	—	1468	2	no	—
112	C₂×C₄	C₂²×D₃₆₇	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₃₆₇	1	1	yes	characteristic
114	C₂×C₄	C₂²×D₃₆₇	1	1	yes	characteristic
115	C₂×C₄	C₂²×D₃₆₇	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	734	2	no	—
120	C₂²	—	734	2	no	—
121	C₂²	—	734	2	no	—
122	C₂²	C₂³×D₃₆₇	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	734	2	no	—
128	C₄	—	734	2	no	—
129	C₄	—	734	2	no	—
130	C₂	—	4	2	no	—
131	C₂	—	1468	2	no	—
132	C₂	#26044306	1	1	yes	characteristic
133	C₂	#26044307	1	1	yes	characteristic
134	C₂	#26044308	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26044253

C₂

yes

maximal, characteristic, complemented

#26044249

C₂

yes

maximal, characteristic, complemented

#26044245

C₂

yes

maximal, complemented

#26044243

C₂

yes

maximal, complemented

#26044238

C₂

yes

maximal, complemented

#26044237

C₂

yes

maximal, complemented

#26044234

C₂

yes

maximal, complemented

#26044233

C₂

yes

maximal, complemented

#124×C₃₆₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₇₃₄

C₂²

yes

characteristic

C₂×D₁₄₆₈

C₂²

yes

characteristic

C₂×C₂²⋊D₃₆₇

C₂²

yes

—

C₂×C₂²⋊D₃₆₇

C₂²

yes

—

C₂²×Dic₃₆₇

C₂²

yes

—

C₂×C₄×D₃₆₇

C₂²

yes

—

C₄⋊Dic₃₆₇

C₂²

yes

—

C₄⋊Dic₃₆₇

C₂²

yes

characteristic

C₄⋊Dic₃₆₇

C₂²

yes

characteristic

Dic₃₆₇⋊C₄

C₂²

yes

—

Dic₃₆₇⋊C₄

C₂²

yes

—

Dic₃₆₇⋊C₄

C₂²

yes

—

C₂²⋊Dic₃₆₇

C₂²

yes

—

(C₂×D₃₆₇)⋊C₄

C₂²

yes

—

(C₂×D₃₆₇)⋊C₄

C₂²

yes

—

(C₂×D₃₆₇)⋊C₄

C₂²

yes

—

C₂×Q₈×C₃₆₇

C₂²

yes

characteristic

C₂×D₄×C₃₆₇

C₂²

yes

characteristic

C₂²⋊C₄×C₃₆₇

C₂²

yes

—

C₂²⋊C₄×C₃₆₇

C₂²

yes

—

C₄²×C₃₆₇

C₂²

yes

characteristic

C₂²×D₃₆₇

C₂³

yes

—

Dic₇₃₄

—

D₁₄₆₈

—

C₂²⋊D₃₆₇

—

C₂²⋊D₃₆₇

—

C₂×Dic₃₆₇

—

C₂×Dic₃₆₇

C₂³

yes

—

C₂×Dic₃₆₇

C₂³

yes

—

C₂×Dic₃₆₇

C₂³

yes

—

C₄×D₃₆₇

—

C₂³×C₃₆₇

C₂³

yes

—

Q₈×C₃₆₇

—

D₄×C₃₆₇

—

C₂×C₄×C₃₆₇

C₂³

yes

characteristic

C₂×C₄×C₃₆₇

C₂³

yes

characteristic

C₂×C₄×C₃₆₇

C₂³

yes

characteristic

C₂×C₄×C₃₆₇

C₂³

yes

—

C₂²×C₃₆₇

—

C₂²×C₃₆₇

—

C₂²×C₃₆₇

—

C₂²×C₃₆₇

C₂⁴

yes

characteristic

C₂×D₃₆₇

—

C₂×D₃₆₇

—

C₂×D₃₆₇

—

C₄×C₃₆₇

—

C₄×C₃₆₇

—

C₄×C₃₆₇

—

C₄×C₃₆₇

—

Dic₃₆₇

—

Dic₃₆₇

—

Dic₃₆₇

—

C₂×C₃₆₇

—

C₂×C₃₆₇

yes

C₂×C₃₆₇

yes

C₂×C₃₆₇

2₋¹⁺⁴

yes

characteristic

D₃₆₇

—

C₃₆₇

#562

yes

characteristic, complemented, Sylow

#562

—

367

maximal, Sylow

#125

—

367

—

#124

—

367

—

#124

D₃₆₇

yes

characteristic, complemented

#123

—

367

—

#123

—

367

—

C₂²⋊Q₈

—

367

—

C₂²⋊Q₈

—

367

—

#121

—

367

—

#121

—

367

—

C₂×Q₈

—

367

—

C₂×Q₈

C₂×D₃₆₇

yes

characteristic

C₂×D₄

—

367

—

C₂×D₄

—

367

—

C₂×D₄

—

367

—

C₂×D₄

C₂×D₃₆₇

yes

characteristic

C₂²×C₄

—

367

—

C₂²×C₄

—

367

—

C₄⋊C₄

—

367

—

C₄⋊C₄

—

367

—

C₄⋊C₄

—

367

—

C₄⋊C₄

—

367

—

C₄⋊C₄

—

367

—

C₄⋊C₄

—

367

—

C₂²⋊C₄

—

367

—

C₂²⋊C₄

—

367

—

C₂²⋊C₄

—

367

—

C₂²⋊C₄

—

367

—

C₂²⋊C₄

C₂×D₃₆₇

yes

—

C₂²⋊C₄

C₂×D₃₆₇

yes

—

C₄²

C₂×D₃₆₇

yes

characteristic

C₂³

—

367

—

100

C₂³

C₂²×D₃₆₇

yes

—

101

Q₈

—

102

Q₈

—

1468

—

103

D₄

—

104

D₄

—

1468

—

105

D₄

—

1468

—

106

D₄

—

1468

—

107

C₂×C₄

—

367

—

108

C₂×C₄

—

367

—

109

C₂×C₄

—

367

—

110

C₂×C₄

—

1468

—

111

C₂×C₄

—

1468

—

112

C₂×C₄

C₂²×D₃₆₇

yes

characteristic

113

C₂×C₄

C₂²×D₃₆₇

yes

characteristic

114

C₂×C₄

C₂²×D₃₆₇

yes

characteristic

115

C₂×C₄

C₂²×D₃₆₇

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

734

—

120

C₂²

—

734

—

121

C₂²

—

734

—

122

C₂²

C₂³×D₃₆₇

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

734

—

128

C₄

—

734

—

129

C₄

—

734

—

130

C₂

—

131

C₂

—

1468

—

132

C₂

#26044306

yes

characteristic

133

C₂

#26044307

yes

characteristic

134

C₂

#26044308

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₆₇
commutator, socle, cosocle	C₂²×C₃₆₇	C₂⁴
Fitting	#124×C₃₆₇	C₂
hypercenter	#124	D₃₆₇
hypocenter	C₃₆₇	#562
LCS	G ▷ C₂²×C₃₆₇ ▷ C₃₆₇
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₆₇

commutator, socle, cosocle

C₂²×C₃₆₇

C₂⁴

Fitting

#124×C₃₆₇

C₂

D₃₆₇

C₃₆₇

G ▷ C₂²×C₃₆₇ ▷ C₃₆₇

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. #26044253	C₂	semidirect
2. #26044249	C₂	semidirect
3. #26044245	C₂	semidirect
4. #26044243	C₂	semidirect
5. #26044238	C₂	semidirect
6. #26044237	C₂	semidirect
7. #26044234	C₂	semidirect
8. #26044233	C₂	semidirect
9. #124×C₃₆₇	C₂	semidirect
67. C₃₆₇	#562	semidirect
71. #124	D₃₆₇	semidirect

H ≅ G/N

type

1. #26044253

C₂

semidirect

2. #26044249

C₂

semidirect

3. #26044245

C₂

semidirect

4. #26044243

C₂

semidirect

5. #26044238

C₂

semidirect

6. #26044237

C₂

semidirect

7. #26044234

C₂

semidirect

8. #26044233

C₂

semidirect

9. #124×C₃₆₇

C₂

semidirect

67. C₃₆₇

#562

semidirect

71. #124

D₃₆₇

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#26044253	2
#26044249	2
#26044245	2
#26044243	2
#26044238	2
#26044237	2
#26044234	2
#26044233	2
#124×C₃₆₇	2
#562	367

subgroup

index

#124×C₃₆₇

367

Subgroups — #37554930

Conjugacy classes of subgroups

Complements

Maximal subgroups