Group #22626311 — subgroups

Subgroups — #22626311

Order 9344.

164 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 74 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×#14556665	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#11924031	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊#11924031	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊#11924031	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊#11924031	C₂	1	1	yes	maximal, characteristic, complemented
6	#17989475	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×#10596831	C₂	1	1	yes	maximal, characteristic
8	#14556665	—	2	2	no	—
9	C₂×#11924031	—	2	2	no	—
10	C₂×#11924031	C₂²	1	1	yes	characteristic
11	C₂×#11924031	C₂²	1	1	yes	characteristic
12	C₂×#11924031	C₂²	1	1	yes	characteristic
13	C₂×#11924031	C₂²	1	1	yes	characteristic
14	C₂²×#10596831	C₂²	1	1	yes	characteristic
15	C₂²×#10596831	C₂²	1	1	yes	characteristic
16	C₄⋊#10596831	C₄	1	1	yes	characteristic
17	C₂²⋊#10596831	C₄	1	1	yes	characteristic
18	C₂²⋊#10596831	C₄	1	1	yes	characteristic
19	C₄×#10596831	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₇₃	C₄	1	1	yes	characteristic
21	#11924031	—	2	1	no	—
22	#11924031	—	2	1	no	—
23	#11924031	—	2	1	no	—
24	#11924031	D₄	1	2	yes	complemented
25	C₂×#10596831	—	2	1	no	—
26	C₂×#10596831	—	2	1	no	—
27	C₂×#10596831	—	2	1	no	—
28	C₂×#10596831	—	2	1	no	—
29	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
30	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
31	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
32	C₂×#10596831	C₂³	1	1	yes	characteristic
33	C₂×D₄×C₇₃	C₈	1	1	yes	characteristic
34	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
36	C₄⋊C₄×C₇₃	C₈	1	1	yes	characteristic
37	C₂²⋊C₄×C₇₃	C₈	1	1	yes	characteristic
38	C₂²⋊C₄×C₇₃	C₈	1	1	yes	characteristic
39	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₇₃	C₂×C₈	1	1	yes	characteristic
41	C₂³×C₇₃	C₂×C₈	1	1	yes	characteristic
42	D₄×C₇₃	—	2	2	no	—
43	C₂×C₄×C₇₃	—	2	1	no	—
44	C₂×C₄×C₇₃	—	2	1	no	—
45	C₂×C₄×C₇₃	—	2	1	no	—
46	C₂×C₄×C₇₃	—	2	1	no	—
47	C₂×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
50	C₂×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
52	#10596831	—	2	1	no	—
53	#10596831	—	2	1	no	—
54	#10596831	—	2	1	no	—
55	#10596831	C₂×D₄	1	1	yes	characteristic
56	#10596831	#41	1	1	yes	characteristic
57	C₂²×C₇₃	—	2	1	no	—
58	C₂²×C₇₃	—	2	1	no	—
59	C₂²×C₇₃	—	2	1	no	—
60	C₂²×C₇₃	—	2	1	no	—
61	C₂²×C₇₃	—	2	1	no	—
62	C₂²×C₇₃	M₃₂	1	2	yes	complemented
63	C₂²×C₇₃	C₂²×C₈	1	1	yes	characteristic
64	C₄×C₇₃	—	2	1	no	—
65	C₄×C₇₃	—	2	1	no	—
66	C₄×C₇₃	—	2	1	no	—
67	C₄×C₇₃	—	2	1	no	—
68	C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
69	C₄×C₇₃	#131	1	1	yes	characteristic
70	C₂×C₇₃	—	2	2	no	—
71	C₂×C₇₃	—	4	1	no	—
72	C₂×C₇₃	C₈×D₄	1	1	yes	characteristic
73	C₂×C₇₃	C₂×M₃₂	1	1	yes	characteristic
74	C₂×C₇₃	#504	1	1	yes	characteristic
75	#2168	—	73	1	no	maximal, Sylow
76	C₇₃	#2168	1	1	yes	characteristic, complemented, Sylow
77	C₂×M₃₂	—	73	1	no	—
78	C₂²×C₁₆	—	73	1	no	—
79	C₈×D₄	—	73	1	no	—
80	C₄⋊C₁₆	—	73	1	no	—
81	C₂²⋊C₁₆	—	73	1	no	—
82	C₂²⋊C₁₆	—	73	1	no	—
83	#346	—	73	1	no	—
84	C₂²×C₈	—	73	1	no	—
85	C₂²×C₈	—	73	1	no	—
86	C₄×D₄	#64408	1	1	yes	characteristic
87	M₃₂	—	146	2	no	—
88	C₂×C₁₆	—	73	1	no	—
89	C₂×C₁₆	—	73	1	no	—
90	C₂×C₁₆	—	73	1	no	—
91	C₂×C₁₆	—	73	1	no	—
92	C₂×C₁₆	—	146	2	no	—
93	C₄⋊C₈	—	73	1	no	—
94	C₂²⋊C₈	—	73	1	no	—
95	C₂²⋊C₈	—	73	1	no	—
96	C₄×C₈	—	73	1	no	—
97	C₂×D₄	#10596833	1	1	yes	characteristic
98	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
99	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
100	C₂×C₈	—	73	1	no	—
101	C₂×C₈	—	73	1	no	—
102	C₂×C₈	—	73	1	no	—
103	C₂×C₈	—	73	1	no	—
104	C₂×C₈	—	146	1	no	—
105	C₂×C₈	—	146	1	no	—
106	C₂×C₈	—	146	1	no	—
107	C₂×C₈	—	146	1	no	—
108	C₄⋊C₄	#10596833	1	1	yes	characteristic
109	C₂²⋊C₄	#10596833	1	1	yes	characteristic
110	C₂²⋊C₄	#10596833	1	1	yes	characteristic
111	C₄²	C₂×#64408	1	1	yes	characteristic
112	C₁₆	—	73	2	no	—
113	C₁₆	—	146	1	no	—
114	C₁₆	—	146	1	no	—
115	C₁₆	—	146	1	no	—
116	C₂³	C₂×#10596833	1	1	yes	characteristic
117	C₂³	C₂×#10596833	1	1	yes	characteristic
118	D₄	—	2	2	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	C₂×#10596833	1	1	yes	characteristic
124	C₂×C₄	C₂×#10596833	1	1	yes	characteristic
125	C₂×C₄	C₂×#10596833	1	1	yes	characteristic
126	C₂×C₄	C₂×#10596833	1	1	yes	characteristic
127	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
128	C₈	—	73	1	no	—
129	C₈	—	73	1	no	—
130	C₈	—	146	1	no	—
131	C₈	—	146	1	no	—
132	C₈	—	146	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	2	1	no	—
138	C₂²	#14556665	1	2	yes	complemented
139	C₂²	C₂²×#10596833	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	D₄×#64408	1	1	yes	characteristic
145	C₄	#14556678	1	1	yes	characteristic
146	C₂	—	2	2	no	—
147	C₂	—	4	1	no	—
148	C₂	D₄×#10596833	1	1	yes	characteristic
149	C₂	C₂×#14556665	1	1	yes	characteristic
150	C₂	#17989645	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556665

C₂

yes

maximal, characteristic, complemented

C₂²×#11924031

C₂

yes

maximal, characteristic, complemented

C₄⋊#11924031

C₂

yes

maximal, characteristic, complemented

C₂²⋊#11924031

C₂

yes

maximal, characteristic, complemented

C₂²⋊#11924031

C₂

yes

maximal, characteristic, complemented

#17989475

C₂

yes

maximal, characteristic, complemented

D₄×#10596831

C₂

yes

maximal, characteristic

#14556665

—

C₂×#11924031

—

C₂×#11924031

C₂²

yes

characteristic

C₂×#11924031

C₂²

yes

characteristic

C₂×#11924031

C₂²

yes

characteristic

C₂×#11924031

C₂²

yes

characteristic

C₂²×#10596831

C₂²

yes

characteristic

C₂²×#10596831

C₂²

yes

characteristic

C₄⋊#10596831

C₄

yes

characteristic

C₂²⋊#10596831

C₄

yes

characteristic

C₂²⋊#10596831

C₄

yes

characteristic

C₄×#10596831

C₂²

yes

characteristic

C₄×D₄×C₇₃

C₄

yes

—

—

—

D₄

yes

—

—

—

—

C₂×C₄

yes

characteristic

C₂×#10596831

C₂×C₄

yes

characteristic

C₂×#10596831

C₂×C₄

yes

characteristic

C₂×#10596831

C₂³

yes

characteristic

C₂×D₄×C₇₃

C₈

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₇₃

C₈

yes

characteristic

C₂²⋊C₄×C₇₃

C₈

yes

characteristic

C₂²⋊C₄×C₇₃

C₈

yes

characteristic

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂³×C₇₃

C₂×C₈

yes

characteristic

C₂³×C₇₃

C₂×C₈

yes

characteristic

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

—

—

—

C₂×D₄

yes

yes

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

M₃₂

yes

complemented

C₂²×C₇₃

C₂²×C₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₄×D₄

yes

characteristic

C₄×C₇₃

#131

yes

characteristic

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

C₈×D₄

yes

characteristic

C₂×C₇₃

C₂×M₃₂

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

—

C₈×D₄

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

C₂²×C₈

—

C₂²×C₈

—

C₄×D₄

#64408

yes

characteristic

M₃₂

—

146

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂×D₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

146

—

105

C₂×C₈

—

146

—

106

C₂×C₈

—

146

—

107

C₂×C₈

—

146

—

108

C₄⋊C₄

#10596833

yes

characteristic

109

C₂²⋊C₄

#10596833

yes

characteristic

110

C₂²⋊C₄

#10596833

yes

characteristic

111

C₄²

C₂×#64408

yes

characteristic

112

C₁₆

—

113

C₁₆

—

146

—

114

C₁₆

—

146

—

115

C₁₆

—

146

—

116

C₂³

C₂×#10596833

yes

characteristic

117

C₂³

C₂×#10596833

yes

characteristic

118

D₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

C₂×#10596833

yes

characteristic

124

C₂×C₄

C₂×#10596833

yes

characteristic

125

C₂×C₄

C₂×#10596833

yes

characteristic

126

C₂×C₄

C₂×#10596833

yes

characteristic

127

C₂×C₄

C₂²×#64408

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

146

—

131

C₈

—

146

—

132

C₈

—

146

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#14556665

yes

complemented

139

C₂²

C₂²×#10596833

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

D₄×#64408

yes

characteristic

145

C₄

#14556678

yes

characteristic

146

C₂

—

147

C₂

—

148

C₂

D₄×#10596833

yes

characteristic

149

C₂

C₂×#14556665

yes

characteristic

150

C₂

#17989645

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#64408
commutator, socle	C₂²×C₇₃	C₂²×C₈
Fitting	C₄×D₄×C₇₃	C₄
hypercenter	C₄×D₄	#64408
hypocenter	C₇₃	#2168
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#64408

commutator, socle

C₂²×C₇₃

C₂²×C₈

Fitting

C₄×D₄×C₇₃

C₄

C₄×D₄

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₄ ◁ C₄×D₄

N	H ≅ G/N	type
1. C₂×#14556665	C₂	semidirect
2. C₂²×#11924031	C₂	semidirect
3. C₄⋊#11924031	C₂	semidirect
4. C₂²⋊#11924031	C₂	semidirect
5. C₂²⋊#11924031	C₂	semidirect
6. #17989475	C₂	semidirect
24. #11924031	D₄	semidirect
62. C₂²×C₇₃	M₃₂	semidirect
76. C₇₃	#2168	semidirect
138. C₂²	#14556665	semidirect

H ≅ G/N

type

1. C₂×#14556665

C₂

semidirect

2. C₂²×#11924031

C₂

semidirect

3. C₄⋊#11924031

C₂

semidirect

4. C₂²⋊#11924031

C₂

semidirect

5. C₂²⋊#11924031

C₂

semidirect

6. #17989475

C₂

semidirect

24. #11924031

D₄

semidirect

62. C₂²×C₇₃

M₃₂

semidirect

76. C₇₃

#2168

semidirect

138. C₂²

#14556665

semidirect

Subgroups — #22626311

Conjugacy classes of subgroups

Complements