Group #22626845 — subgroups

Subgroups — #22626845

Order 9344.

314 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 144 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	#17989639	C₂	1	4	yes	maximal, complemented
2	#17989638	C₂	1	4	yes	maximal, complemented
3	#17989612	C₂	1	4	yes	maximal
4	#17989610	C₂	1	1	yes	maximal, characteristic
5	#17989610	C₂	1	1	yes	maximal, characteristic
6	#17989142	C₂	1	1	yes	maximal, characteristic
7	#14556658	C₂²	1	2	yes	—
8	#14556658	C₂²	1	4	yes	—
9	#14556658	C₂²	1	2	yes	—
10	C₄⋊#10596833	C₂²	1	4	yes	—
11	#14556648	C₂²	1	2	yes	—
12	#14556648	C₂²	1	2	yes	—
13	#14556646	C₂²	1	2	yes	—
14	#14556646	C₂²	1	4	yes	—
15	#14556646	C₂²	1	2	yes	—
16	C₄×#10596833	C₂²	1	4	yes	—
17	#14556591	C₄	1	1	yes	characteristic
18	#14556582	C₄	1	2	yes	—
19	#14556571	C₄	1	2	yes	—
20	#14556567	C₄	1	2	yes	—
21	Q₈×Dic₇₃	C₂²	1	2	yes	—
22	#14556556	C₂²	1	2	yes	—
23	#14556549	C₂²	1	2	yes	—
24	C₄²×D₇₃	C₂²	1	1	yes	characteristic
25	C₄⋊Q₈×C₇₃	C₄	1	1	yes	characteristic
26	C₂×#10596833	C₂³	1	4	yes	—
27	C₂×#10596833	C₂³	1	4	yes	—
28	C₂×Dic₁₄₆	C₂×C₄	1	2	yes	—
29	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
32	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
33	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
34	C₄⋊Dic₇₃	C₂×C₄	1	4	yes	—
35	#11924007	C₂×C₄	1	4	yes	—
36	#11924005	C₂×C₄	1	4	yes	—
37	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
38	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
39	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
40	C₄×Dic₇₃	C₂³	1	4	yes	—
41	C₂×Q₈×C₇₃	C₂×C₄	1	2	yes	—
42	C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
43	C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
44	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×D₇₃	C₂²×C₄	1	1	yes	characteristic
46	Q₈×C₇₃	—	4	2	no	—
47	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₇₃	C₂²×C₄	1	4	yes	—
51	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×Dic₇₃	C₂²×C₄	1	4	yes	—
54	C₂×Dic₇₃	C₂⁴	1	1	yes	characteristic
55	C₄×D₇₃	—	2	2	no	—
56	C₄×D₇₃	—	4	1	no	—
57	C₄×D₇₃	—	4	1	no	—
58	Dic₁₄₆	—	4	2	no	—
59	#10596833	—	2	4	no	—
60	#10596833	—	2	4	no	—
61	C₂²×C₇₃	C₂³×C₄	1	1	yes	characteristic
62	C₂×D₇₃	—	2	1	no	—
63	C₂×D₇₃	—	2	1	no	—
64	C₂×D₇₃	—	2	1	no	—
65	C₄×C₇₃	—	2	1	no	—
66	C₄×C₇₃	—	2	1	no	—
67	C₄×C₇₃	—	2	1	no	—
68	C₄×C₇₃	—	2	4	no	—
69	Dic₇₃	—	2	4	no	—
70	Dic₇₃	—	2	1	no	—
71	Dic₇₃	—	2	1	no	—
72	Dic₇₃	2₋¹⁺⁴	1	2	yes	—
73	C₂×C₇₃	#519	1	1	yes	characteristic
74	C₂×C₇₃	#568	1	1	yes	characteristic
75	C₂×C₇₃	#568	1	1	yes	characteristic
76	D₇₃	—	4	1	no	—
77	#2987	—	73	1	no	maximal, Sylow
78	C₇₃	#2987	1	1	yes	characteristic, complemented, Sylow
79	#533	—	73	1	no	—
80	#446	—	73	4	no	—
81	#446	—	73	4	no	—
82	#433	—	73	4	no	—
83	#432	—	73	1	no	—
84	#432	—	73	1	no	—
85	C₄⋊Q₈	—	73	1	no	—
86	C₄⋊Q₈	#64408	1	1	yes	characteristic
87	#125	—	73	2	no	—
88	C₂²⋊Q₈	—	73	2	no	—
89	C₂²⋊Q₈	—	73	2	no	—
90	C₄×Q₈	—	73	2	no	—
91	C₄×Q₈	—	73	2	no	—
92	#117	—	73	2	no	—
93	C₂×C₄²	—	73	1	no	—
94	C₄⋊C₈	—	73	2	no	—
95	C₄⋊C₈	—	73	2	no	—
96	C₄⋊C₈	—	73	4	no	—
97	C₄⋊C₈	—	73	4	no	—
98	C₂²⋊C₈	—	73	2	no	—
99	C₂²⋊C₈	—	73	2	no	—
100	#97	—	73	2	no	—
101	#97	—	73	2	no	—
102	#97	—	73	4	no	—
103	C₄×C₈	—	73	4	no	—
104	C₂×Q₈	—	73	2	no	—
105	C₂×Q₈	C₂×#64408	1	2	yes	—
106	C₂²×C₄	—	73	1	no	—
107	C₂²×C₄	—	73	1	no	—
108	C₂²×C₄	—	73	1	no	—
109	C₂×C₈	—	73	4	no	—
110	C₂×C₈	—	73	4	no	—
111	C₄⋊C₄	—	73	4	no	—
112	C₄⋊C₄	—	73	2	no	—
113	C₄⋊C₄	—	73	4	no	—
114	C₄⋊C₄	—	73	2	no	—
115	C₄⋊C₄	C₂×#64408	1	2	yes	—
116	C₄⋊C₄	C₂×#64408	1	2	yes	—
117	C₂²⋊C₄	—	73	4	no	—
118	C₄²	—	73	1	no	—
119	C₄²	—	73	4	no	—
120	C₄²	—	73	1	no	—
121	C₄²	—	73	1	no	—
122	C₄²	C₂×#64408	1	1	yes	characteristic
123	C₂³	—	73	1	no	—
124	Q₈	—	4	2	no	—
125	Q₈	—	292	2	no	—
126	C₂×C₄	—	292	1	no	—
127	C₂×C₄	—	292	1	no	—
128	C₂×C₄	—	73	1	no	—
129	C₂×C₄	—	73	1	no	—
130	C₂×C₄	—	73	1	no	—
131	C₂×C₄	—	73	4	no	—
132	C₂×C₄	—	146	2	no	—
133	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
134	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
135	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
136	C₂×C₄	C₂²×#64408	1	4	yes	—
137	C₈	—	146	4	no	—
138	C₈	—	146	4	no	—
139	C₂²	—	146	1	no	—
140	C₂²	—	146	1	no	—
141	C₂²	—	146	1	no	—
142	C₂²	C₂³×#64408	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	2	4	no	—
146	C₄	—	2	1	no	—
147	C₄	—	73	2	no	—
148	C₄	—	146	1	no	—
149	C₄	—	146	1	no	—
150	C₄	—	146	4	no	—
151	C₂	—	292	1	no	—
152	C₂	#17989655	1	1	yes	characteristic
153	C₂	#17989665	1	1	yes	characteristic
154	C₂	#17989667	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989639

C₂

yes

maximal, complemented

#17989638

C₂

yes

maximal, complemented

#17989612

C₂

yes

maximal

#17989610

C₂

yes

maximal, characteristic

#17989610

C₂

yes

maximal, characteristic

#17989142

C₂

yes

maximal, characteristic

#14556658

C₂²

yes

—

#14556658

C₂²

yes

—

#14556658

C₂²

yes

—

C₄⋊#10596833

C₂²

yes

—

#14556648

C₂²

yes

—

#14556648

C₂²

yes

—

#14556646

C₂²

yes

—

#14556646

C₂²

yes

—

#14556646

C₂²

yes

—

C₄×#10596833

C₂²

yes

—

#14556591

C₄

yes

characteristic

#14556582

C₄

yes

—

#14556571

C₄

yes

—

#14556567

C₄

yes

—

Q₈×Dic₇₃

C₂²

yes

—

#14556556

C₂²

yes

—

#14556549

C₂²

yes

—

C₄²×D₇₃

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₄

yes

characteristic

C₂×#10596833

C₂³

yes

—

C₂×#10596833

C₂³

yes

—

C₂×Dic₁₄₆

C₂×C₄

yes

—

C₂×C₄×D₇₃

C₂×C₄

yes

characteristic

C₂×C₄×D₇₃

C₂×C₄

yes

characteristic

C₂×C₄×D₇₃

C₂³

yes

characteristic

C₄⋊Dic₇₃

C₂×C₄

yes

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

#11924007

C₂×C₄

yes

—

#11924005

C₂×C₄

yes

—

C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₄×Dic₇₃

C₂³

yes

characteristic

C₄×Dic₇₃

C₂³

yes

characteristic

C₄×Dic₇₃

C₂³

yes

—

C₂×Q₈×C₇₃

C₂×C₄

yes

—

C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂²×D₇₃

C₂²×C₄

yes

characteristic

Q₈×C₇₃

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

—

C₂×Dic₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

C₂²×C₄

yes

—

C₂×Dic₇₃

C₂⁴

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

#10596833

—

#10596833

—

C₂²×C₇₃

C₂³×C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

2₋¹⁺⁴

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#2987

—

maximal, Sylow

C₇₃

#2987

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

C₄⋊Q₈

—

C₄⋊Q₈

#64408

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

100

#97

—

101

#97

—

102

#97

—

103

C₄×C₈

—

104

C₂×Q₈

—

105

C₂×Q₈

C₂×#64408

yes

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

C₂×#64408

yes

—

116

C₄⋊C₄

C₂×#64408

yes

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₄²

C₂×#64408

yes

characteristic

123

C₂³

—

124

Q₈

—

125

Q₈

—

292

—

126

C₂×C₄

—

292

—

127

C₂×C₄

—

292

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

146

—

133

C₂×C₄

C₂²×#64408

yes

characteristic

134

C₂×C₄

C₂²×#64408

yes

characteristic

135

C₂×C₄

C₂²×#64408

yes

characteristic

136

C₂×C₄

C₂²×#64408

yes

—

137

C₈

—

146

—

138

C₈

—

146

—

139

C₂²

—

146

—

140

C₂²

—

146

—

141

C₂²

—

146

—

142

C₂²

C₂³×#64408

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

146

—

149

C₄

—

146

—

150

C₄

—

146

—

151

C₂

—

292

—

152

C₂

#17989655

yes

characteristic

153

C₂

#17989665

yes

characteristic

154

C₂

#17989667

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₄
Fitting	C₄⋊Q₈×C₇₃	C₄
hypercenter	C₄⋊Q₈	#64408
hypocenter	C₇₃	#2987
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#64408

commutator, socle

C₂²×C₇₃

C₂³×C₄

Fitting

C₄⋊Q₈×C₇₃

C₄

C₄⋊Q₈

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. #17989639	C₂	semidirect
2. #17989638	C₂	semidirect
78. C₇₃	#2987	semidirect

H ≅ G/N

type

1. #17989639

C₂

semidirect

2. #17989638

C₂

semidirect

78. C₇₃

#2987

semidirect

Subgroups — #22626845

Conjugacy classes of subgroups

Complements