Group #22626067 — subgroups

Subgroups — #22626067

Order 9344.

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

normal subgroups only

This group has 320 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	#17989592	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14556656	C₂	1	4	yes	maximal, complemented
3	#17989523	C₂	1	1	yes	maximal, characteristic, complemented
4	#17989138	C₂	1	1	yes	maximal, characteristic
5	C₂×#11924038	C₂²	1	2	yes	—
6	C₂×C₄×#64408	C₂²	1	2	yes	—
7	#14556658	—	2	2	no	—
8	#14556656	—	2	8	no	—
9	(C₂×#64408)⋊C₄	—	2	2	no	—
10	C₂×#11924029	C₂²	1	2	yes	—
11	#14556588	C₄	1	1	yes	characteristic, complemented
12	#14556580	C₄	1	2	yes	complemented
13	#14556566	—	2	2	no	—
14	#14556553	—	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	#11924038	—	2	4	no	—
19	C₂×#10596833	—	2	2	no	—
20	C₄×#64408	—	2	2	no	—
21	C₄×#64408	D₄	1	4	yes	complemented
22	#11924029	—	2	4	no	—
23	#11924029	—	2	4	no	—
24	C₂×D₂₉₂	—	2	2	no	—
25	C₂×D₂₉₂	—	2	2	no	—
26	C₂×D₂₉₂	C₂×C₄	1	2	yes	—
27	C₂×C₄×D₇₃	—	2	2	no	—
28	C₂×C₄×D₇₃	D₄	1	2	yes	—
29	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
30	#11924005	—	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	—	2	2	no	—
37	C₂²×D₇₃	C₂×D₄	1	1	yes	characteristic
38	C₂×#64408	—	2	4	no	—
39	C₂×#64408	—	4	4	no	—
40	Q₈×C₇₃	—	2	4	no	—
41	C₂×C₄×C₇₃	—	2	2	no	—
42	C₂×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×Dic₇₃	C₂²⋊C₄	1	2	yes	—
45	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
46	D₂₉₂	—	2	4	no	—
47	D₂₉₂	—	4	4	no	—
48	D₂₉₂	—	4	4	no	—
49	C₄×D₇₃	—	4	2	no	—
50	C₄×D₇₃	—	4	2	no	—
51	C₄×D₇₃	—	4	2	no	—
52	C₄×D₇₃	C₂×D₄	1	2	yes	—
53	C₄×D₇₃	#41	1	2	yes	—
54	#10596833	—	4	2	no	—
55	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×D₇₃	—	2	2	no	—
57	C₂×D₇₃	—	4	2	no	—
58	C₂×D₇₃	—	4	4	no	—
59	C₂×D₇₃	#121	1	2	yes	—
60	#64408	—	4	4	no	—
61	C₄×C₇₃	—	2	2	no	—
62	C₄×C₇₃	—	4	2	no	—
63	C₄×C₇₃	C₄×D₄	1	2	yes	—
64	Dic₇₃	—	2	2	no	—
65	Dic₇₃	#124	1	1	yes	characteristic
66	Dic₇₃	#127	1	1	yes	characteristic
67	C₂×C₇₃	#390	1	1	yes	characteristic
68	C₂×C₇₃	#421	1	2	yes	—
69	D₇₃	—	2	2	no	—
70	D₇₃	—	8	2	no	—
71	#1963	—	73	1	no	maximal, Sylow
72	C₇₃	#1963	1	1	yes	characteristic, complemented, Sylow
73	#532	—	73	1	no	—
74	#423	—	73	1	no	—
75	C₂×#104	—	73	4	no	—
76	#376	—	73	1	no	—
77	C₂×#41	—	73	2	no	—
78	C₂×M₁₆	—	73	2	no	—
79	C₄⋊Q₈	#64408	1	1	yes	characteristic, complemented
80	#127	—	73	1	no	—
81	#124	—	73	2	no	—
82	#121	—	146	2	no	—
83	C₄×D₄	—	146	2	no	—
84	C₂×C₄²	—	73	1	no	—
85	C₂×C₄²	—	73	2	no	—
86	C₄⋊C₈	—	146	2	no	—
87	#104	—	146	8	no	—
88	#95	—	146	2	no	—
89	#41	—	146	4	no	—
90	#41	—	146	4	no	—
91	C₂×Q₈	C₂×#64408	1	2	yes	—
92	C₂×D₄	—	73	2	no	—
93	C₂×D₄	—	146	2	no	—
94	C₂×D₄	—	146	2	no	—
95	C₂²×C₄	—	73	2	no	—
96	C₂²×C₄	—	73	1	no	—
97	C₂²×C₄	—	146	2	no	—
98	C₂²×C₄	—	146	2	no	—
99	M₁₆	—	146	4	no	—
100	C₂×C₈	—	146	2	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₂²⋊C₄	—	146	4	no	—
103	C₄²	—	73	2	no	—
104	C₄²	—	73	1	no	—
105	C₄²	—	73	4	no	—
106	C₄²	—	146	2	no	—
107	C₄²	C₂×#64408	1	1	yes	characteristic
108	C₂³	—	73	1	no	—
109	C₂³	—	146	2	no	—
110	Q₈	—	2	4	no	—
111	D₄	—	292	4	no	—
112	D₄	—	292	4	no	—
113	D₄	—	146	4	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	292	2	no	—
116	C₂×C₄	—	292	2	no	—
117	C₂×C₄	—	292	2	no	—
118	C₂×C₄	—	292	4	no	—
119	C₂×C₄	—	73	1	no	—
120	C₂×C₄	—	73	2	no	—
121	C₂×C₄	—	73	2	no	—
122	C₂×C₄	—	73	2	no	—
123	C₂×C₄	—	146	4	no	—
124	C₂×C₄	C₂²⋊#64408	1	2	yes	—
125	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
126	C₈	—	292	2	no	—
127	C₂²	—	292	2	no	—
128	C₂²	—	292	4	no	—
129	C₂²	—	73	2	no	—
130	C₂²	—	146	2	no	—
131	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	4	2	no	—
134	C₄	—	292	4	no	—
135	C₄	—	73	1	no	—
136	C₄	—	73	1	no	—
137	C₄	—	146	2	no	—
138	C₄	D₄×#64408	1	2	yes	—
139	C₂	—	584	2	no	—
140	C₂	—	146	2	no	—
141	C₂	#17989539	1	1	yes	characteristic
142	C₂	#17989586	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17989592

C₂

yes

maximal, characteristic, complemented

C₂×#14556656

C₂

yes

maximal, complemented

#17989523

C₂

yes

maximal, characteristic, complemented

#17989138

C₂

yes

maximal, characteristic

C₂×#11924038

C₂²

yes

—

C₂×C₄×#64408

C₂²

yes

—

—

—

—

C₂²

yes

—

#14556588

C₄

yes

characteristic, complemented

#14556580

C₄

yes

complemented

#14556566

—

#14556553

—

C₄²×D₇₃

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₄

yes

characteristic, complemented

C₂²×#64408

—

—

—

—

D₄

yes

complemented

#11924029

—

#11924029

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

C₂×C₄

yes

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

D₄

yes

—

C₂×C₄×D₇₃

C₂³

yes

characteristic

#11924005

—

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

—

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

C₂²⋊C₄

yes

—

C₂×Dic₇₃

C₂×D₄

yes

characteristic

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₂×D₄

yes

—

C₄×D₇₃

#41

yes

—

#10596833

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

#121

yes

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₄×D₄

yes

—

Dic₇₃

—

Dic₇₃

yes

Dic₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

—

D₇₃

—

#1963

—

maximal, Sylow

C₇₃

#1963

yes

characteristic, complemented, Sylow

—

—

—

—

C₂×#41

—

C₂×M₁₆

—

C₄⋊Q₈

#64408

yes

characteristic, complemented

#127

—

#124

—

#121

—

146

—

C₄×D₄

—

146

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

146

—

#104

—

146

—

#95

—

146

—

#41

—

146

—

#41

—

146

—

C₂×Q₈

C₂×#64408

yes

—

C₂×D₄

—

C₂×D₄

—

146

—

C₂×D₄

—

146

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

146

—

C₂²×C₄

—

146

—

M₁₆

—

146

—

100

C₂×C₈

—

146

—

101

C₄⋊C₄

—

102

C₂²⋊C₄

—

146

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₄²

—

146

—

107

C₄²

C₂×#64408

yes

characteristic

108

C₂³

—

109

C₂³

—

146

—

110

Q₈

—

111

D₄

—

292

—

112

D₄

—

292

—

113

D₄

—

146

—

114

C₂×C₄

—

115

C₂×C₄

—

292

—

116

C₂×C₄

—

292

—

117

C₂×C₄

—

292

—

118

C₂×C₄

—

292

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

146

—

124

C₂×C₄

C₂²⋊#64408

yes

—

125

C₂×C₄

C₂²×#64408

yes

characteristic

126

C₈

—

292

—

127

C₂²

—

292

—

128

C₂²

—

292

—

129

C₂²

—

130

C₂²

—

146

—

131

C₂²

C₂×C₂²⋊#64408

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

292

—

135

C₄

—

136

C₄

—

137

C₄

—

146

—

138

C₄

D₄×#64408

yes

—

139

C₂

—

584

—

140

C₂

—

146

—

141

C₂

#17989539

yes

characteristic

142

C₂

#17989586

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	#1963
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
1. #17989592	C₂	semidirect
2. C₂×#14556656	C₂	semidirect
3. #17989523	C₂	semidirect
11. #14556588	C₄	semidirect
12. #14556580	C₄	semidirect
16. C₄⋊Q₈×C₇₃	C₄	semidirect
21. C₄×#64408	D₄	semidirect
72. C₇₃	#1963	semidirect
79. C₄⋊Q₈	#64408	semidirect

H ≅ G/N

type

1. #17989592

C₂

semidirect

2. C₂×#14556656

C₂

semidirect

3. #17989523

C₂

semidirect

11. #14556588

C₄

semidirect

12. #14556580

C₄

semidirect

16. C₄⋊Q₈×C₇₃

C₄

semidirect

21. C₄×#64408

D₄

semidirect

72. C₇₃

#1963

semidirect

79. C₄⋊Q₈

#64408

semidirect

Subgroups — #22626067

Conjugacy classes of subgroups

Complements