Group #22625558 — subgroups

Subgroups — #22625558

Order 9344.

166 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 56 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	Q₈×#10596833	C₂	1	1	yes	maximal, characteristic, complemented
2	#17989621	C₂	1	1	yes	maximal, characteristic
3	#17989450	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊#10596833	C₂	1	1	yes	maximal, characteristic, complemented
5	#17989435	C₂	1	1	yes	maximal, characteristic
6	#17989429	C₂	1	1	yes	maximal, characteristic, complemented
7	#17988672	C₂	1	1	yes	maximal, characteristic
8	C₂×#11924038	—	2	1	no	—
9	C₄⋊#10596833	—	2	1	no	—
10	C₄⋊#10596833	C₂²	1	1	yes	characteristic
11	C₄⋊#10596833	C₂²	1	1	yes	characteristic
12	#14556648	—	2	1	no	—
13	C₄×#10596833	—	2	1	no	—
14	C₄×#10596833	C₂²	1	1	yes	characteristic
15	C₄×#10596833	C₂²	1	1	yes	characteristic
16	C₂×#11924015	C₄	1	1	yes	characteristic
17	Q₈×Dic₇₃	C₂²	1	1	yes	characteristic
18	#14556553	C₂²	1	1	yes	characteristic
19	#14556522	C₄	1	1	yes	characteristic
20	#14556513	C₄	1	1	yes	characteristic
21	C₈×Dic₇₃	C₂²	1	1	yes	characteristic
22	#103×C₇₃	C₄	1	1	yes	characteristic
23	#11924038	—	4	2	no	—
24	C₂×#10596833	—	2	1	no	—
25	C₂×#10596833	—	2	1	no	—
26	C₂×#10596833	—	2	1	no	—
27	C₂×#10596833	D₄	1	1	yes	characteristic
28	C₂×#10596833	D₄	1	1	yes	characteristic
29	C₂×D₂₉₂	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₇₃	—	2	1	no	—
31	#11924015	—	2	2	no	—
32	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
33	C₄⋊Dic₇₃	—	2	1	no	—
34	C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
35	#11924005	—	2	1	no	—
36	C₄×Dic₇₃	—	2	1	no	—
37	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
38	C₂×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
40	C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×D₇₃	—	2	1	no	—
42	Q₈×C₇₃	—	2	1	no	—
43	Q₈×C₇₃	M₁₆	1	2	yes	complemented
44	C₂×C₄×C₇₃	—	2	1	no	—
45	C₂×C₄×C₇₃	—	2	1	no	—
46	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×Dic₇₃	—	2	1	no	—
48	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₇₃	#41	1	1	yes	characteristic
50	D₂₉₂	—	2	1	no	—
51	D₂₉₂	—	2	1	no	—
52	C₄×D₇₃	—	4	1	no	—
53	C₄×D₇₃	—	4	1	no	—
54	#10596833	—	2	2	no	—
55	#10596833	—	2	1	no	—
56	#10596833	—	2	1	no	—
57	#10596833	—	4	1	no	—
58	#10596833	—	4	1	no	—
59	C₈×C₇₃	—	2	1	no	—
60	#10596831	—	2	1	no	—
61	C₂²×C₇₃	C₄×D₄	1	1	yes	characteristic
62	C₂×D₇₃	—	4	1	no	—
63	C₂×D₇₃	—	4	1	no	—
64	C₂×D₇₃	—	4	1	no	—
65	C₄×C₇₃	—	2	2	no	—
66	C₄×C₇₃	—	4	1	no	—
67	C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
68	C₄×C₇₃	#131	1	1	yes	characteristic
69	Dic₇₃	—	2	1	no	—
70	Dic₇₃	—	4	1	no	—
71	Dic₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
72	Dic₇₃	#137	1	1	yes	characteristic
73	C₂×C₇₃	#435	1	1	yes	characteristic
74	C₂×C₇₃	#440	1	1	yes	characteristic
75	C₂×C₇₃	#443	1	1	yes	characteristic
76	D₇₃	—	8	1	no	—
77	#1588	—	73	1	no	maximal, Sylow
78	C₇₃	#1588	1	1	yes	characteristic, complemented, Sylow
79	C₈×Q₈	—	73	1	no	—
80	C₄×SD₁₆	—	73	1	no	—
81	#436	—	73	1	no	—
82	#335	—	73	1	no	—
83	Q₈⋊C₈	—	73	1	no	—
84	D₄⋊C₈	—	73	1	no	—
85	#322	—	73	1	no	—
86	C₂×SD₁₆	—	73	1	no	—
87	C₂×M₁₆	—	146	1	no	—
88	C₄×Q₈	—	73	1	no	—
89	C₄×D₄	—	73	1	no	—
90	#106	—	73	1	no	—
91	C₄⋊C₈	—	73	1	no	—
92	C₄⋊C₈	—	73	1	no	—
93	C₄⋊C₈	—	146	1	no	—
94	#103	#64408	1	1	yes	characteristic
95	#102	—	73	1	no	—
96	C₂²⋊C₈	—	146	1	no	—
97	C₄×C₈	—	73	1	no	—
98	C₄×C₈	—	73	1	no	—
99	C₄×C₈	—	73	1	no	—
100	C₄×C₈	—	146	1	no	—
101	C₂×Q₈	C₂×#64408	1	1	yes	characteristic
102	C₂×D₄	—	73	1	no	—
103	C₂²×C₄	—	146	1	no	—
104	SD₁₆	—	146	2	no	—
105	M₁₆	—	292	2	no	—
106	C₂×C₈	—	73	1	no	—
107	C₂×C₈	—	73	1	no	—
108	C₂×C₈	—	73	1	no	—
109	C₂×C₈	—	146	1	no	—
110	C₂×C₈	—	146	1	no	—
111	C₂×C₈	—	146	1	no	—
112	C₂×C₈	C₂×#64408	1	1	yes	characteristic
113	C₄⋊C₄	—	73	1	no	—
114	C₄⋊C₄	—	146	1	no	—
115	C₄⋊C₄	C₂×#64408	1	1	yes	characteristic
116	C₂²⋊C₄	—	146	1	no	—
117	C₄²	—	73	1	no	—
118	C₄²	—	146	1	no	—
119	C₂³	—	146	1	no	—
120	Q₈	—	2	1	no	—
121	Q₈	#11924038	1	2	yes	complemented
122	D₄	—	146	1	no	—
123	D₄	—	146	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	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	—	146	1	no	—
131	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	292	1	no	—
134	C₈	—	292	1	no	—
135	C₈	—	146	1	no	—
136	C₈	—	146	1	no	—
137	C₈	—	146	2	no	—
138	C₈	—	146	1	no	—
139	C₂²	—	292	1	no	—
140	C₂²	—	292	1	no	—
141	C₂²	—	292	1	no	—
142	C₂²	D₄×#64408	1	1	yes	characteristic
143	C₄	—	2	2	no	—
144	C₄	—	4	1	no	—
145	C₄	—	292	1	no	—
146	C₄	—	73	1	no	—
147	C₄	—	73	1	no	—
148	C₄	—	146	1	no	—
149	C₄	C₂×#11924038	1	1	yes	characteristic
150	C₄	#14556678	1	1	yes	characteristic
151	C₂	—	584	1	no	—
152	C₂	#17989616	1	1	yes	characteristic
153	C₂	#17989626	1	1	yes	characteristic
154	C₂	#17989631	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#10596833

C₂

yes

maximal, characteristic, complemented

#17989621

C₂

yes

maximal, characteristic

#17989450

C₂

yes

maximal, characteristic, complemented

Q₈⋊#10596833

C₂

yes

maximal, characteristic, complemented

#17989435

C₂

yes

maximal, characteristic

#17989429

C₂

yes

maximal, characteristic, complemented

#17988672

C₂

yes

maximal, characteristic

C₂×#11924038

—

C₄⋊#10596833

—

C₄⋊#10596833

C₂²

yes

characteristic

C₄⋊#10596833

C₂²

yes

—

—

C₂²

yes

characteristic

C₄×#10596833

C₂²

yes

characteristic

C₂×#11924015

C₄

yes

characteristic

Q₈×Dic₇₃

C₂²

yes

characteristic

#14556553

C₂²

yes

characteristic

#14556522

C₄

yes

characteristic

#14556513

C₄

yes

characteristic

C₈×Dic₇₃

C₂²

yes

characteristic

#103×C₇₃

C₄

yes

—

—

—

—

D₄

yes

characteristic

C₂×#10596833

D₄

yes

characteristic

C₂×D₂₉₂

C₂×C₄

yes

characteristic

C₂×C₄×D₇₃

—

#11924015

—

C₂×#10596831

C₂×C₄

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂×C₄

yes

characteristic

#11924005

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₂³

yes

characteristic

C₂×Q₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×D₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

M₁₆

yes

complemented

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

#41

yes

characteristic

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

—

—

—

—

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

C₄×D₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×M₁₆

yes

characteristic

C₄×C₇₃

#131

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

yes

Dic₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1588

—

maximal, Sylow

C₇₃

#1588

yes

characteristic, complemented, Sylow

C₈×Q₈

—

C₄×SD₁₆

—

#436

—

#335

—

Q₈⋊C₈

—

D₄⋊C₈

—

#322

—

C₂×SD₁₆

—

C₂×M₁₆

—

146

—

C₄×Q₈

—

C₄×D₄

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

146

—

yes

—

C₂²⋊C₈

—

146

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

—

146

—

101

C₂×Q₈

C₂×#64408

yes

characteristic

102

C₂×D₄

—

103

C₂²×C₄

—

146

—

104

SD₁₆

—

146

—

105

M₁₆

—

292

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

146

—

110

C₂×C₈

—

146

—

111

C₂×C₈

—

146

—

112

C₂×C₈

C₂×#64408

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

146

—

115

C₄⋊C₄

C₂×#64408

yes

characteristic

116

C₂²⋊C₄

—

146

—

117

C₄²

—

118

C₄²

—

146

—

119

C₂³

—

146

—

120

Q₈

—

121

Q₈

#11924038

yes

complemented

122

D₄

—

146

—

123

D₄

—

146

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

292

—

127

C₂×C₄

—

292

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

146

—

131

C₂×C₄

C₂²×#64408

yes

characteristic

132

C₈

—

133

C₈

—

292

—

134

C₈

—

292

—

135

C₈

—

146

—

136

C₈

—

146

—

137

C₈

—

146

—

138

C₈

—

146

—

139

C₂²

—

292

—

140

C₂²

—

292

—

141

C₂²

—

292

—

142

C₂²

D₄×#64408

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

292

—

146

C₄

—

147

C₄

—

148

C₄

—

146

—

149

C₄

C₂×#11924038

yes

characteristic

150

C₄

#14556678

yes

characteristic

151

C₂

—

584

—

152

C₂

#17989616

yes

characteristic

153

C₂

#17989626

yes

characteristic

154

C₂

#17989631

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#64408
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#64408
Fitting	#103×C₇₃	C₄
socle	C₂²×C₇₃	C₄×D₄
hypercenter	#103	#64408
hypocenter	C₇₃	#1588
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

subgroup

G/H

center

C₂²

D₄×#64408

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#64408

Fitting

#103×C₇₃

C₄

socle

C₂²×C₇₃

C₄×D₄

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

N	H ≅ G/N	type
1. Q₈×#10596833	C₂	semidirect
3. #17989450	C₂	semidirect
4. Q₈⋊#10596833	C₂	semidirect
6. #17989429	C₂	semidirect
43. Q₈×C₇₃	M₁₆	semidirect
78. C₇₃	#1588	semidirect
121. Q₈	#11924038	semidirect

H ≅ G/N

type

1. Q₈×#10596833

C₂

semidirect

3. #17989450

C₂

semidirect

4. Q₈⋊#10596833

C₂

semidirect

6. #17989429

C₂

semidirect

43. Q₈×C₇₃

M₁₆

semidirect

78. C₇₃

#1588

semidirect

121. Q₈

#11924038

semidirect

Subgroups — #22625558

Conjugacy classes of subgroups

Complements