Group #22626062 — subgroups

Subgroups — #22626062

Order 9344.

308 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 78 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×#14556654	C₂	1	4	yes	maximal
2	#17989546	C₂	1	1	yes	maximal, characteristic
3	#17989531	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11924036	C₂²	1	2	yes	—
6	C₂×C₄⋊#64408	C₂²	1	2	yes	—
7	#14556654	—	2	8	no	—
8	#14556646	—	2	2	no	—
9	(C₂×#64408)⋊C₄	—	2	2	no	—
10	C₂×Q₈×D₇₃	C₂²	1	2	yes	—
11	#14556591	C₄	1	1	yes	characteristic, complemented
12	#14556590	—	2	2	no	—
13	#14556589	C₄	1	2	yes	complemented
14	C₄⋊C₄×D₇₃	—	2	2	no	—
15	C₄²×D₇₃	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈×C₇₃	C₄	1	1	yes	characteristic, complemented
17	C₂²×#64408	—	2	2	no	—
18	#11924036	—	2	2	no	—
19	#11924036	D₄	1	4	yes	—
20	C₂×#10596833	—	2	2	no	—
21	C₄⋊#64408	—	2	4	no	—
22	C₂×Dic₁₄₆	—	2	2	no	—
23	C₂×Dic₁₄₆	C₂×C₄	1	2	yes	—
24	Q₈×D₇₃	—	2	4	no	—
25	Q₈×D₇₃	—	2	4	no	—
26	C₂×C₄×D₇₃	—	2	2	no	—
27	C₂×C₄×D₇₃	D₄	1	2	yes	—
28	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₇₃	—	2	2	no	—
30	#11924007	—	2	4	no	—
31	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
32	C₄×Dic₇₃	D₄	1	2	yes	—
33	C₂×Q₈×C₇₃	C₂×C₄	1	2	yes	—
34	C₄⋊C₄×C₇₃	—	2	2	no	—
35	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×D₇₃	C₂×D₄	1	1	yes	characteristic
37	C₂×#64408	—	4	2	no	—
38	C₂×#64408	—	4	4	no	—
39	Q₈×C₇₃	—	2	4	no	—
40	C₂×C₄×C₇₃	—	2	2	no	—
41	C₂×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×Dic₇₃	—	2	2	no	—
44	C₂×Dic₇₃	C₂²⋊C₄	1	2	yes	—
45	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
46	C₄×D₇₃	—	2	4	no	—
47	C₄×D₇₃	—	4	2	no	—
48	C₄×D₇₃	—	4	2	no	—
49	C₄×D₇₃	C₂×D₄	1	2	yes	—
50	C₄×D₇₃	#41	1	2	yes	—
51	Dic₁₄₆	—	2	4	no	—
52	Dic₁₄₆	—	4	4	no	—
53	#10596833	—	2	4	no	—
54	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂×D₇₃	#124	1	1	yes	characteristic
56	C₂×D₇₃	#127	1	1	yes	characteristic
57	C₂×D₇₃	#137	1	4	yes	—
58	#64408	—	8	2	no	—
59	C₄×C₇₃	—	2	2	no	—
60	C₄×C₇₃	—	4	2	no	—
61	C₄×C₇₃	C₄×D₄	1	2	yes	—
62	Dic₇₃	—	2	2	no	—
63	Dic₇₃	—	4	2	no	—
64	Dic₇₃	#121	1	2	yes	—
65	C₂×C₇₃	#390	1	1	yes	characteristic
66	C₂×C₇₃	#419	1	2	yes	—
67	D₇₃	#491	1	2	yes	—
68	D₇₃	#497	1	2	yes	—
69	#1960	—	73	1	no	maximal, Sylow
70	C₇₃	#1960	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₄⋊Q₈	—	73	1	no	—
72	C₂×#103	—	73	4	no	—
73	C₂×#97	—	73	1	no	—
74	#382	—	73	1	no	—
75	C₂²×Q₈	—	73	2	no	—
76	C₂²×C₈	—	73	2	no	—
77	C₄⋊Q₈	—	73	2	no	—
78	C₄⋊Q₈	—	73	1	no	—
79	C₄⋊Q₈	—	146	2	no	—
80	C₄⋊Q₈	#64408	1	1	yes	characteristic, complemented
81	C₂×C₄⋊C₄	—	73	2	no	—
82	C₂×C₄⋊C₄	—	146	2	no	—
83	C₂×C₄²	—	73	1	no	—
84	#103	—	146	8	no	—
85	#97	—	146	2	no	—
86	#95	—	146	2	no	—
87	C₂×Q₈	—	73	2	no	—
88	C₂×Q₈	—	146	2	no	—
89	C₂×Q₈	—	146	4	no	—
90	C₂×Q₈	—	146	4	no	—
91	C₂×Q₈	C₂×#64408	1	2	yes	—
92	C₂²×C₄	—	73	1	no	—
93	C₂²×C₄	—	73	2	no	—
94	C₂²×C₄	—	146	2	no	—
95	C₂²×C₄	—	146	2	no	—
96	C₂×C₈	—	73	4	no	—
97	C₂×C₈	—	146	2	no	—
98	C₂×C₈	—	146	2	no	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	146	4	no	—
101	C₄⋊C₄	—	146	4	no	—
102	C₄⋊C₄	—	146	2	no	—
103	C₄²	—	73	2	no	—
104	C₄²	—	73	1	no	—
105	C₄²	C₂×#64408	1	1	yes	characteristic
106	C₂³	—	73	1	no	—
107	Q₈	—	2	4	no	—
108	Q₈	—	292	4	no	—
109	Q₈	—	146	4	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	292	2	no	—
112	C₂×C₄	—	292	2	no	—
113	C₂×C₄	—	292	2	no	—
114	C₂×C₄	—	292	4	no	—
115	C₂×C₄	—	73	1	no	—
116	C₂×C₄	—	73	2	no	—
117	C₂×C₄	—	73	2	no	—
118	C₂×C₄	—	73	2	no	—
119	C₂×C₄	—	146	4	no	—
120	C₂×C₄	—	146	2	no	—
121	C₂×C₄	C₂²⋊#64408	1	2	yes	—
122	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
123	C₈	—	146	4	no	—
124	C₂²	—	73	1	no	—
125	C₂²	—	73	1	no	—
126	C₂²	—	73	4	no	—
127	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
128	C₄	—	2	2	no	—
129	C₄	—	4	2	no	—
130	C₄	—	292	2	no	—
131	C₄	—	584	2	no	—
132	C₄	—	73	2	no	—
133	C₄	—	146	2	no	—
134	C₄	D₄×#64408	1	2	yes	—
135	C₂	—	73	2	no	—
136	C₂	—	73	2	no	—
137	C₂	#17989539	1	1	yes	characteristic
138	C₂	#17989580	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556654

C₂

yes

maximal

#17989546

C₂

yes

maximal, characteristic

#17989531

C₂

yes

maximal, characteristic

C₄⋊Q₈×D₇₃

C₂

yes

maximal, characteristic

C₂×#11924036

C₂²

yes

—

C₂×C₄⋊#64408

C₂²

yes

—

#14556654

—

#14556646

—

(C₂×#64408)⋊C₄

—

C₂×Q₈×D₇₃

C₂²

yes

—

#14556591

C₄

yes

characteristic, complemented

#14556590

—

#14556589

C₄

yes

complemented

C₄⋊C₄×D₇₃

—

C₄²×D₇₃

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₄

yes

characteristic, complemented

C₂²×#64408

—

#11924036

—

#11924036

D₄

yes

—

C₂×#10596833

—

C₄⋊#64408

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

C₂×C₄

yes

—

Q₈×D₇₃

—

Q₈×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

D₄

yes

—

C₂×C₄×D₇₃

C₂³

yes

characteristic

C₄⋊Dic₇₃

—

#11924007

—

C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

—

C₂×Q₈×C₇₃

C₂×C₄

yes

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂²×D₇₃

C₂×D₄

yes

characteristic

C₂×#64408

—

C₂×#64408

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂²⋊C₄

yes

—

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₂×D₄

yes

—

C₄×D₇₃

#41

yes

—

Dic₁₄₆

—

Dic₁₄₆

—

#10596833

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

C₂×D₇₃

yes

C₂×D₇₃

yes

C₂×D₇₃

yes

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₄×D₄

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#121

yes

—

C₂×C₇₃

#390

yes

characteristic

C₂×C₇₃

#419

yes

—

D₇₃

#491

yes

—

D₇₃

#497

yes

—

#1960

—

maximal, Sylow

C₇₃

#1960

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×#103

—

C₂×#97

—

#382

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

146

—

C₄⋊Q₈

#64408

yes

characteristic, complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄²

—

#103

—

146

—

#97

—

146

—

#95

—

146

—

C₂×Q₈

—

C₂×Q₈

—

146

—

C₂×Q₈

—

146

—

C₂×Q₈

—

146

—

C₂×Q₈

C₂×#64408

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

146

—

C₂²×C₄

—

146

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

C₄⋊C₄

—

100

C₄⋊C₄

—

146

—

101

C₄⋊C₄

—

146

—

102

C₄⋊C₄

—

146

—

103

C₄²

—

104

C₄²

—

105

C₄²

C₂×#64408

yes

characteristic

106

C₂³

—

107

Q₈

—

108

Q₈

—

292

—

109

Q₈

—

146

—

110

C₂×C₄

—

111

C₂×C₄

—

292

—

112

C₂×C₄

—

292

—

113

C₂×C₄

—

292

—

114

C₂×C₄

—

292

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

146

—

120

C₂×C₄

—

146

—

121

C₂×C₄

C₂²⋊#64408

yes

—

122

C₂×C₄

C₂²×#64408

yes

characteristic

123

C₈

—

146

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

C₂×C₂²⋊#64408

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

292

—

131

C₄

—

584

—

132

C₄

—

133

C₄

—

146

—

134

C₄

D₄×#64408

yes

—

135

C₂

—

136

C₂

—

137

C₂

#17989539

yes

characteristic

138

C₂

#17989580

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#64408
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#64408
Fitting	C₄⋊Q₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	C₄⋊Q₈	#64408
hypocenter	C₇₃	#1960
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#64408

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#64408

Fitting

C₄⋊Q₈×C₇₃

C₄

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

C₄⋊Q₈

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
11. #14556591	C₄	semidirect
13. #14556589	C₄	semidirect
16. C₄⋊Q₈×C₇₃	C₄	semidirect
70. C₇₃	#1960	semidirect
80. C₄⋊Q₈	#64408	semidirect

H ≅ G/N

type

11. #14556591

C₄

semidirect

13. #14556589

C₄

semidirect

16. C₄⋊Q₈×C₇₃

C₄

semidirect

70. C₇₃

#1960

semidirect

80. C₄⋊Q₈

#64408

semidirect

Subgroups — #22626062

Conjugacy classes of subgroups

Complements