Group #22625355 — subgroups

Subgroups — #22625355

Order 9344.

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

normal subgroups only

This group has 208 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	#17989610	C₂	1	1	yes	maximal, characteristic
2	#17989532	C₂	1	1	yes	maximal, characteristic
3	#17989441	C₂	1	4	yes	maximal
4	#17989142	C₂	1	1	yes	maximal, characteristic
5	#14556658	—	2	1	no	—
6	#14556648	C₂²	1	2	yes	—
7	#14556646	—	2	1	no	—
8	(C₂×#64408)⋊C₄	—	2	1	no	—
9	(C₂×#64408)⋊C₄	—	2	1	no	—
10	(C₂×#64408)⋊C₄	C₂²	1	2	yes	—
11	#14556591	C₄	1	1	yes	characteristic, complemented
12	#14556582	—	2	1	no	—
13	#14556571	—	2	1	no	—
14	#14556567	C₂²	1	2	yes	—
15	Q₈×Dic₇₃	C₄	1	2	yes	complemented
16	#14556556	—	2	1	no	—
17	#14556549	—	2	1	no	—
18	C₄²×D₇₃	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₇₃	C₄	1	1	yes	characteristic, complemented
20	C₂²×#64408	—	2	2	no	—
21	C₂×#10596833	—	2	2	no	—
22	C₂×Dic₁₄₆	—	2	1	no	—
23	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
24	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
25	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
26	C₄⋊Dic₇₃	—	2	1	no	—
27	C₄⋊Dic₇₃	—	2	2	no	—
28	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
29	#11924007	—	2	2	no	—
30	#11924005	—	2	2	no	—
31	C₄×Dic₇₃	—	2	2	no	—
32	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₇₃	D₄	1	1	yes	characteristic
34	C₄×Dic₇₃	D₄	1	1	yes	characteristic
35	C₂×Q₈×C₇₃	C₂×C₄	1	2	yes	—
36	C₄⋊C₄×C₇₃	—	2	1	no	—
37	C₄⋊C₄×C₇₃	—	2	1	no	—
38	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×D₇₃	C₂×D₄	1	1	yes	characteristic
40	C₂×#64408	—	4	2	no	—
41	C₂×#64408	—	4	2	no	—
42	C₂×#64408	—	4	2	no	—
43	Q₈×C₇₃	—	4	2	no	—
44	C₂×C₄×C₇₃	—	2	2	no	—
45	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×Dic₇₃	—	2	2	no	—
49	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
52	C₄×D₇₃	—	2	2	no	—
53	C₄×D₇₃	—	4	1	no	—
54	C₄×D₇₃	—	4	1	no	—
55	Dic₁₄₆	—	4	2	no	—
56	#10596833	—	4	2	no	—
57	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×D₇₃	—	2	1	no	—
59	C₂×D₇₃	—	2	1	no	—
60	C₂×D₇₃	#137	1	2	yes	—
61	#64408	—	8	2	no	—
62	C₄×C₇₃	—	2	1	no	—
63	C₄×C₇₃	—	2	1	no	—
64	C₄×C₇₃	—	2	1	no	—
65	C₄×C₇₃	—	4	2	no	—
66	Dic₇₃	—	2	1	no	—
67	Dic₇₃	—	2	1	no	—
68	Dic₇₃	—	2	1	no	—
69	Dic₇₃	—	4	2	no	—
70	C₂×C₇₃	#410	1	1	yes	characteristic
71	C₂×C₇₃	#419	1	1	yes	characteristic
72	C₂×C₇₃	#421	1	1	yes	characteristic
73	D₇₃	—	2	2	no	—
74	#1515	—	73	1	no	maximal, Sylow
75	C₇₃	#1515	1	1	yes	characteristic, complemented, Sylow
76	#533	—	73	1	no	—
77	#432	—	73	1	no	—
78	#383	—	73	1	no	—
79	#328	—	73	4	no	—
80	C₄⋊Q₈	—	73	1	no	—
81	C₄⋊Q₈	#64408	1	1	yes	characteristic, complemented
82	#125	—	146	1	no	—
83	C₂²⋊Q₈	—	73	2	no	—
84	C₂²⋊Q₈	—	146	1	no	—
85	C₄×Q₈	—	73	2	no	—
86	C₄×Q₈	—	146	1	no	—
87	#117	—	146	1	no	—
88	C₂×C₄²	—	73	1	no	—
89	C₄⋊C₈	—	146	1	no	—
90	C₂²⋊C₈	—	73	2	no	—
91	#97	—	146	1	no	—
92	#95	—	73	2	no	—
93	#95	—	146	1	no	—
94	#95	—	146	1	no	—
95	C₂×Q₈	—	146	1	no	—
96	C₂×Q₈	C₂×#64408	1	2	yes	—
97	C₂²×C₄	—	73	1	no	—
98	C₂²×C₄	—	73	1	no	—
99	C₂²×C₄	—	73	1	no	—
100	C₂²×C₄	—	146	2	no	—
101	C₂×C₈	—	146	2	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	73	2	no	—
105	C₄⋊C₄	—	146	2	no	—
106	C₄⋊C₄	—	146	2	no	—
107	C₄⋊C₄	—	146	1	no	—
108	C₂²⋊C₄	—	146	2	no	—
109	C₄²	—	73	1	no	—
110	C₄²	—	73	1	no	—
111	C₄²	—	73	1	no	—
112	C₄²	—	146	2	no	—
113	C₄²	C₂×#64408	1	1	yes	characteristic
114	C₂³	—	73	1	no	—
115	Q₈	—	4	2	no	—
116	Q₈	—	292	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	292	1	no	—
119	C₂×C₄	—	292	1	no	—
120	C₂×C₄	—	292	2	no	—
121	C₂×C₄	—	292	2	no	—
122	C₂×C₄	—	292	2	no	—
123	C₂×C₄	—	73	1	no	—
124	C₂×C₄	—	73	1	no	—
125	C₂×C₄	—	73	1	no	—
126	C₂×C₄	—	146	2	no	—
127	C₂×C₄	—	146	2	no	—
128	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
129	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
130	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
131	C₈	—	292	2	no	—
132	C₂²	—	73	2	no	—
133	C₂²	—	146	1	no	—
134	C₂²	—	146	1	no	—
135	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	4	2	no	—
140	C₄	—	292	2	no	—
141	C₄	—	584	2	no	—
142	C₄	—	146	1	no	—
143	C₄	—	146	1	no	—
144	C₄	—	146	1	no	—
145	C₂	—	146	2	no	—
146	C₂	#17989567	1	1	yes	characteristic
147	C₂	#17989580	1	1	yes	characteristic
148	C₂	#17989586	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989610

C₂

yes

maximal, characteristic

#17989532

C₂

yes

maximal, characteristic

#17989441

C₂

yes

maximal

#17989142

C₂

yes

maximal, characteristic

#14556658

—

#14556648

C₂²

yes

—

—

—

—

C₂²

yes

—

#14556591

C₄

yes

characteristic, complemented

#14556582

—

#14556571

—

#14556567

C₂²

yes

—

Q₈×Dic₇₃

C₄

yes

complemented

#14556556

—

#14556549

—

C₄²×D₇₃

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₄

yes

characteristic, complemented

C₂²×#64408

—

C₂×#10596833

—

C₂×Dic₁₄₆

—

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

C₂³

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

#11924007

—

#11924005

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₂×Q₈×C₇₃

C₂×C₄

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂²×D₇₃

C₂×D₄

yes

—

—

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

#10596833

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

#137

yes

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1515

—

maximal, Sylow

C₇₃

#1515

yes

characteristic, complemented, Sylow

—

—

—

—

C₄⋊Q₈

—

C₄⋊Q₈

#64408

yes

characteristic, complemented

#125

—

146

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

146

—

C₄×Q₈

—

C₄×Q₈

—

146

—

#117

—

146

—

C₂×C₄²

—

C₄⋊C₈

—

146

—

C₂²⋊C₈

—

#97

—

146

—

#95

—

#95

—

146

—

#95

—

146

—

C₂×Q₈

—

146

—

C₂×Q₈

C₂×#64408

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

146

—

101

C₂×C₈

—

146

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

146

—

106

C₄⋊C₄

—

146

—

107

C₄⋊C₄

—

146

—

108

C₂²⋊C₄

—

146

—

109

C₄²

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

146

—

113

C₄²

C₂×#64408

yes

characteristic

114

C₂³

—

115

Q₈

—

116

Q₈

—

292

—

117

C₂×C₄

—

118

C₂×C₄

—

292

—

119

C₂×C₄

—

292

—

120

C₂×C₄

—

292

—

121

C₂×C₄

—

292

—

122

C₂×C₄

—

292

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

146

—

127

C₂×C₄

—

146

—

128

C₂×C₄

C₂²⋊#64408

yes

characteristic

129

C₂×C₄

C₂²⋊#64408

yes

characteristic

130

C₂×C₄

C₂²×#64408

yes

characteristic

131

C₈

—

292

—

132

C₂²

—

133

C₂²

—

146

—

134

C₂²

—

146

—

135

C₂²

C₂×C₂²⋊#64408

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

292

—

141

C₄

—

584

—

142

C₄

—

146

—

143

C₄

—

146

—

144

C₄

—

146

—

145

C₂

—

146

—

146

C₂

#17989567

yes

characteristic

147

C₂

#17989580

yes

characteristic

148

C₂

#17989586

yes

characteristic

	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	#1515
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×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 ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
11. #14556591	C₄	semidirect
15. Q₈×Dic₇₃	C₄	semidirect
19. C₄⋊Q₈×C₇₃	C₄	semidirect
75. C₇₃	#1515	semidirect
81. C₄⋊Q₈	#64408	semidirect

H ≅ G/N

type

11. #14556591

C₄

semidirect

15. Q₈×Dic₇₃

C₄

semidirect

19. C₄⋊Q₈×C₇₃

C₄

semidirect

75. C₇₃

#1515

semidirect

81. C₄⋊Q₈

#64408

semidirect

Subgroups — #22625355

Conjugacy classes of subgroups

Complements