Group #24923409 — subgroups

Subgroups — #24923409

Order 10688.

158 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 45 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×Q₈×D₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14581939	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14581937	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128807	C₂	1	1	yes	maximal, characteristic
5	#19128755	C₂	1	1	yes	maximal, characteristic, complemented
6	(C₂×D₁₆₇)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#103×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₃₃₄	—	2	1	no	—
9	C₂×Dic₃₃₄	C₂²	1	1	yes	characteristic
10	Q₈×D₁₆₇	—	2	2	no	—
11	Q₈×D₁₆₇	—	2	2	no	—
12	C₂×D₆₆₈	—	2	1	no	—
13	C₂×D₆₆₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₁₆₇	—	2	1	no	—
15	C₂×C₄×D₁₆₇	C₂²	1	1	yes	characteristic
16	#14581939	—	2	2	no	—
17	#14581937	—	2	2	no	—
18	C₂×C₁₆₇⋊C₈	C₂²	1	1	yes	characteristic
19	(C₂×D₁₆₇)⋊C₄	—	2	1	no	—
20	C₂×Q₈×C₁₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×C₁₆₇	C₂²	1	1	yes	characteristic
23	C₂²×D₁₆₇	—	2	1	no	—
24	C₂²×D₁₆₇	D₄	1	1	yes	characteristic
25	Dic₃₃₄	—	2	4	no	—
26	Dic₃₃₄	—	2	1	no	—
27	Dic₃₃₄	D₄	1	2	yes	complemented
28	D₆₆₈	—	2	1	no	—
29	D₆₆₈	—	2	1	no	—
30	D₆₆₈	—	4	2	no	—
31	C₂×Dic₁₆₇	—	2	1	no	—
32	C₂×Dic₁₆₇	D₄	1	1	yes	characteristic
33	C₄×D₁₆₇	—	2	4	no	—
34	C₄×D₁₆₇	—	2	1	no	—
35	C₄×D₁₆₇	—	2	1	no	—
36	C₁₆₇⋊C₈	—	2	1	no	—
37	Q₈×C₁₆₇	—	2	1	no	—
38	Q₈×C₁₆₇	D₄	1	2	yes	complemented
39	C₂×C₄×C₁₆₇	—	2	1	no	—
40	C₂×C₄×C₁₆₇	—	2	1	no	—
41	C₂×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
42	C₈×C₁₆₇	—	2	1	no	—
43	C₂²×C₁₆₇	C₂×D₄	1	1	yes	characteristic
44	C₂×D₁₆₇	—	2	1	no	—
45	C₂×D₁₆₇	—	2	1	no	—
46	C₂×D₁₆₇	—	4	1	no	—
47	C₂×D₁₆₇	—	4	1	no	—
48	C₂×D₁₆₇	—	4	1	no	—
49	C₂×D₁₆₇	SD₁₆	1	2	yes	complemented
50	C₄×C₁₆₇	—	2	2	no	—
51	C₄×C₁₆₇	—	4	1	no	—
52	C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
53	C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
54	Dic₁₆₇	—	2	2	no	—
55	Dic₁₆₇	—	2	1	no	—
56	C₂×C₁₆₇	#120	1	1	yes	characteristic
57	C₂×C₁₆₇	C₂×SD₁₆	1	1	yes	characteristic
58	C₂×C₁₆₇	#137	1	1	yes	characteristic
59	D₁₆₇	—	2	2	no	—
60	D₁₆₇	—	8	1	no	—
61	C₁₆₇	#448	1	1	yes	characteristic, complemented, Sylow
62	#448	—	167	1	no	maximal, Sylow
63	C₂²×Q₈	—	167	1	no	—
64	C₂×SD₁₆	—	167	1	no	—
65	C₂×SD₁₆	—	167	1	no	—
66	#121	—	167	1	no	—
67	#103	—	167	1	no	—
68	#103	D₁₆₇	1	1	yes	characteristic, complemented
69	C₂²⋊C₈	—	167	1	no	—
70	C₂×Q₈	—	334	1	no	—
71	C₂×Q₈	—	334	2	no	—
72	C₂×Q₈	—	334	2	no	—
73	C₂×Q₈	—	167	1	no	—
74	C₂×Q₈	C₂×D₁₆₇	1	1	yes	characteristic
75	C₂×D₄	—	334	1	no	—
76	C₂×D₄	—	167	1	no	—
77	C₂²×C₄	—	334	1	no	—
78	C₂²×C₄	—	167	1	no	—
79	SD₁₆	—	334	2	no	—
80	SD₁₆	—	334	2	no	—
81	C₂×C₈	—	167	1	no	—
82	C₂×C₈	C₂×D₁₆₇	1	1	yes	characteristic
83	C₄⋊C₄	C₂×D₁₆₇	1	1	yes	characteristic
84	C₂²⋊C₄	—	334	1	no	—
85	C₂³	—	334	1	no	—
86	C₂³	—	167	1	no	—
87	Q₈	—	2	1	no	—
88	Q₈	—	334	4	no	—
89	Q₈	—	334	1	no	—
90	Q₈	—	167	2	no	—
91	Q₈	D₆₆₈	1	2	yes	complemented
92	D₄	—	334	1	no	—
93	D₄	—	334	1	no	—
94	D₄	—	668	2	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	334	4	no	—
98	C₂×C₄	—	334	1	no	—
99	C₂×C₄	—	334	1	no	—
100	C₂×C₄	—	334	1	no	—
101	C₂×C₄	—	167	1	no	—
102	C₂×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
103	C₈	—	2	1	no	—
104	C₈	—	334	1	no	—
105	C₂²	—	334	1	no	—
106	C₂²	—	334	1	no	—
107	C₂²	—	668	1	no	—
108	C₂²	—	668	1	no	—
109	C₂²	—	668	1	no	—
110	C₂²	—	167	2	no	—
111	C₂²	D₄×D₁₆₇	1	1	yes	characteristic
112	C₄	—	2	2	no	—
113	C₄	—	4	1	no	—
114	C₄	—	334	2	no	—
115	C₄	—	334	1	no	—
116	C₄	D₄×D₁₆₇	1	1	yes	characteristic
117	C₄	C₂×D₆₆₈	1	1	yes	characteristic
118	C₂	—	1336	1	no	—
119	C₂	—	334	2	no	—
120	C₂	#19128802	1	1	yes	characteristic
121	C₂	SD₁₆×D₁₆₇	1	1	yes	characteristic
122	C₂	#19128864	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×D₁₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#14581939

C₂

yes

maximal, characteristic, complemented

C₂×#14581937

C₂

yes

maximal, characteristic, complemented

#19128807

C₂

yes

maximal, characteristic

#19128755

C₂

yes

maximal, characteristic, complemented

(C₂×D₁₆₇)⋊C₈

C₂

yes

maximal, characteristic, complemented

#103×C₁₆₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₃₃₄

—

C₂×Dic₃₃₄

C₂²

yes

characteristic

Q₈×D₁₆₇

—

Q₈×D₁₆₇

—

C₂×D₆₆₈

—

C₂×D₆₆₈

C₂²

yes

characteristic

C₂×C₄×D₁₆₇

—

C₂×C₄×D₁₆₇

C₂²

yes

characteristic

#14581939

—

#14581937

—

C₂×C₁₆₇⋊C₈

C₂²

yes

characteristic

(C₂×D₁₆₇)⋊C₄

—

C₂×Q₈×C₁₆₇

C₂²

yes

characteristic

C₂×C₈×C₁₆₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₆₇

C₂²

yes

characteristic

C₂²×D₁₆₇

—

C₂²×D₁₆₇

D₄

yes

characteristic

Dic₃₃₄

—

Dic₃₃₄

—

Dic₃₃₄

D₄

yes

complemented

D₆₆₈

—

D₆₆₈

—

D₆₆₈

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

D₄

yes

characteristic

C₄×D₁₆₇

—

C₄×D₁₆₇

—

C₄×D₁₆₇

—

C₁₆₇⋊C₈

—

Q₈×C₁₆₇

—

Q₈×C₁₆₇

D₄

yes

complemented

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

C₂³

yes

characteristic

C₈×C₁₆₇

—

C₂²×C₁₆₇

C₂×D₄

yes

characteristic

C₂×D₁₆₇

—

C₂×D₁₆₇

—

C₂×D₁₆₇

—

C₂×D₁₆₇

—

C₂×D₁₆₇

—

C₂×D₁₆₇

SD₁₆

yes

complemented

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₄×C₁₆₇

C₂×D₄

yes

characteristic

Dic₁₆₇

—

Dic₁₆₇

—

C₂×C₁₆₇

#120

yes

characteristic

C₂×C₁₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₁₆₇

#137

yes

characteristic

D₁₆₇

—

D₁₆₇

—

C₁₆₇

#448

yes

characteristic, complemented, Sylow

#448

—

167

maximal, Sylow

C₂²×Q₈

—

167

—

C₂×SD₁₆

—

167

—

C₂×SD₁₆

—

167

—

#121

—

167

—

#103

—

167

—

#103

D₁₆₇

yes

characteristic, complemented

C₂²⋊C₈

—

167

—

C₂×Q₈

—

334

—

C₂×Q₈

—

334

—

C₂×Q₈

—

334

—

C₂×Q₈

—

167

—

C₂×Q₈

C₂×D₁₆₇

yes

characteristic

C₂×D₄

—

334

—

C₂×D₄

—

167

—

C₂²×C₄

—

334

—

C₂²×C₄

—

167

—

SD₁₆

—

334

—

SD₁₆

—

334

—

C₂×C₈

—

167

—

C₂×C₈

C₂×D₁₆₇

yes

characteristic

C₄⋊C₄

C₂×D₁₆₇

yes

characteristic

C₂²⋊C₄

—

334

—

C₂³

—

334

—

C₂³

—

167

—

Q₈

—

Q₈

—

334

—

Q₈

—

334

—

Q₈

—

167

—

Q₈

D₆₆₈

yes

complemented

D₄

—

334

—

D₄

—

334

—

D₄

—

668

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

334

—

C₂×C₄

—

334

—

C₂×C₄

—

334

—

100

C₂×C₄

—

334

—

101

C₂×C₄

—

167

—

102

C₂×C₄

C₂²×D₁₆₇

yes

characteristic

103

C₈

—

104

C₈

—

334

—

105

C₂²

—

334

—

106

C₂²

—

334

—

107

C₂²

—

668

—

108

C₂²

—

668

—

109

C₂²

—

668

—

110

C₂²

—

167

—

111

C₂²

D₄×D₁₆₇

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

334

—

115

C₄

—

334

—

116

C₄

D₄×D₁₆₇

yes

characteristic

117

C₄

C₂×D₆₆₈

yes

characteristic

118

C₂

—

1336

—

119

C₂

—

334

—

120

C₂

#19128802

yes

characteristic

121

C₂

SD₁₆×D₁₆₇

yes

characteristic

122

C₂

#19128864

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₆₇
commutator, cosocle	C₂×C₄×C₁₆₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₆₇
Fitting	#103×C₁₆₇	C₂
socle	C₂²×C₁₆₇	C₂×D₄
prime radical	#19128807	C₂
hypercenter	#103	D₁₆₇
hypocenter	C₁₆₇	#448
LCS	G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

subgroup

G/H

center

C₂²

D₄×D₁₆₇

commutator, cosocle

C₂×C₄×C₁₆₇

C₂³

Frattini

C₂×C₄

C₂²×D₁₆₇

Fitting

#103×C₁₆₇

C₂

socle

C₂²×C₁₆₇

C₂×D₄

C₂

D₁₆₇

C₁₆₇

G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

N	H ≅ G/N	type
1. C₂×Q₈×D₁₆₇	C₂	semidirect
2. C₂×#14581939	C₂	semidirect
3. C₂×#14581937	C₂	semidirect
5. #19128755	C₂	semidirect
6. (C₂×D₁₆₇)⋊C₈	C₂	semidirect
7. #103×C₁₆₇	C₂	semidirect
27. Dic₃₃₄	D₄	semidirect
38. Q₈×C₁₆₇	D₄	semidirect
49. C₂×D₁₆₇	SD₁₆	semidirect
61. C₁₆₇	#448	semidirect
68. #103	D₁₆₇	semidirect
91. Q₈	D₆₆₈	semidirect

H ≅ G/N

type

1. C₂×Q₈×D₁₆₇

C₂

semidirect

2. C₂×#14581939

C₂

semidirect

3. C₂×#14581937

C₂

semidirect

5. #19128755

C₂

semidirect

6. (C₂×D₁₆₇)⋊C₈

C₂

semidirect

7. #103×C₁₆₇

C₂

semidirect

27. Dic₃₃₄

D₄

semidirect

38. Q₈×C₁₆₇

D₄

semidirect

49. C₂×D₁₆₇

SD₁₆

semidirect

61. C₁₆₇

#448

semidirect

68. #103

D₁₆₇

semidirect

91. Q₈

D₆₆₈

semidirect

Subgroups — #24923409

Conjugacy classes of subgroups

Complements