Group #22598999 — subgroups

Subgroups — #22598999

Order 9088.

338 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 125 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	#17986611	C₂	1	2	yes	maximal
2	#17986401	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986379	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986368	C₂	1	2	yes	maximal, complemented
5	#17986257	C₂	1	2	yes	maximal
6	#17986240	C₂	1	2	yes	maximal
7	#17986107	C₂	1	2	yes	maximal, complemented
8	C₄×Dic₂₈₄	C₂	1	2	yes	maximal, complemented
9	#405×C₇₁	C₂	1	1	yes	maximal, characteristic
10	C₂×Dic₂₈₄	C₂²	1	2	yes	—
11	C₂×Dic₂₈₄	C₂²	1	2	yes	—
12	#14556128	C₂²	1	2	yes	—
13	#14556121	—	2	4	no	—
14	#14556120	C₂²	1	2	yes	—
15	#14556118	—	2	4	no	—
16	#14556107	C₂²	1	4	yes	—
17	C₄×Dic₁₄₂	C₂²	1	4	yes	—
18	#14556062	C₂²	1	2	yes	—
19	#14556060	C₂²	1	2	yes	—
20	#14556053	C₂²	1	4	yes	—
21	#14556053	C₂²	1	4	yes	—
22	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
23	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
24	#117×C₇₁	C₂²	1	1	yes	characteristic
25	#97×C₇₁	C₂²	1	1	yes	characteristic
26	#97×C₇₁	C₂²	1	1	yes	characteristic
27	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
28	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
29	C₂×Dic₁₄₂	C₂³	1	4	yes	—
30	Dic₂₈₄	—	2	4	no	—
31	Dic₂₈₄	—	2	4	no	—
32	C₄⋊Dic₇₁	—	2	4	no	—
33	C₄⋊Dic₇₁	C₂³	1	4	yes	—
34	Dic₇₁⋊C₄	—	2	4	no	—
35	Dic₇₁⋊C₄	—	2	4	no	—
36	Dic₇₁⋊C₄	—	2	4	no	—
37	C₂²⋊Dic₇₁	—	2	4	no	—
38	C₄×Dic₇₁	—	2	4	no	—
39	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
40	M₁₆×C₇₁	—	2	2	no	—
41	C₂×C₈×C₇₁	—	2	2	no	—
42	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
45	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₇₁	D₄	1	2	yes	—
47	C₂²⋊C₄×C₇₁	D₄	1	2	yes	—
48	C₄²×C₇₁	C₂³	1	1	yes	characteristic
49	C₄²×C₇₁	C₂³	1	1	yes	characteristic
50	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₁	—	2	1	no	—
52	C₂×C₄×C₇₁	—	2	1	no	—
53	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
57	C₂×Dic₇₁	—	2	4	no	—
58	C₂×Dic₇₁	—	2	4	no	—
59	Dic₁₄₂	—	2	4	no	—
60	Dic₁₄₂	—	2	4	no	—
61	C₈×C₇₁	—	2	1	no	—
62	C₈×C₇₁	—	2	1	no	—
63	C₈×C₇₁	#41	1	2	yes	—
64	C₈×C₇₁	#41	1	2	yes	—
65	C₂²×C₇₁	—	2	1	no	—
66	C₂²×C₇₁	—	2	2	no	—
67	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₇₁	—	2	1	no	—
69	C₄×C₇₁	—	2	2	no	—
70	C₄×C₇₁	—	2	2	no	—
71	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
72	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
73	Dic₇₁	—	4	4	no	—
74	Dic₇₁	—	4	4	no	—
75	C₂×C₇₁	—	4	1	no	—
76	C₂×C₇₁	#532	1	1	yes	characteristic
77	C₂×C₇₁	#578	1	2	yes	—
78	#3177	—	71	1	no	maximal, Sylow
79	C₇₁	#3177	1	1	yes	characteristic, complemented, Sylow
80	#541	—	71	2	no	—
81	#499	—	71	1	no	—
82	#494	—	71	1	no	—
83	#491	—	71	2	no	—
84	#470	—	71	2	no	—
85	#467	—	71	2	no	—
86	#441	—	71	2	no	—
87	C₄×Q₁₆	—	71	2	no	—
88	#405	D₇₁	1	1	yes	characteristic
89	C₂×Q₁₆	—	71	2	no	—
90	C₂×Q₁₆	—	71	2	no	—
91	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
92	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
93	C₄⋊Q₈	—	71	2	no	—
94	#126	—	142	4	no	—
95	#125	—	71	2	no	—
96	#125	—	142	4	no	—
97	C₂²⋊Q₈	—	71	4	no	—
98	C₄×Q₈	—	71	4	no	—
99	#117	C₂×D₇₁	1	1	yes	characteristic
100	#107	—	71	2	no	—
101	#106	—	71	2	no	—
102	#103	—	71	4	no	—
103	#103	—	71	4	no	—
104	#97	C₂×D₇₁	1	1	yes	characteristic
105	#97	C₂×D₇₁	1	1	yes	characteristic
106	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
107	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
108	C₂×Q₈	—	71	4	no	—
109	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
110	Q₁₆	—	142	4	no	—
111	Q₁₆	—	142	4	no	—
112	M₁₆	—	2	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
115	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
116	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
117	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
118	C₄⋊C₄	—	71	4	no	—
119	C₄⋊C₄	—	142	4	no	—
120	C₄⋊C₄	—	142	4	no	—
121	C₄⋊C₄	—	142	4	no	—
122	C₄⋊C₄	—	142	4	no	—
123	C₄⋊C₄	D₂₈₄	1	2	yes	—
124	C₂²⋊C₄	—	142	4	no	—
125	C₂²⋊C₄	D₂₈₄	1	2	yes	—
126	C₄²	—	142	4	no	—
127	C₄²	C₂²×D₇₁	1	1	yes	characteristic
128	C₄²	C₂²×D₇₁	1	1	yes	characteristic
129	C₂³	C₂×D₂₈₄	1	1	yes	characteristic
130	Q₈	—	142	4	no	—
131	Q₈	—	142	4	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	142	4	no	—
135	C₂×C₄	—	142	4	no	—
136	C₂×C₄	C₂×D₂₈₄	1	2	yes	—
137	C₂×C₄	C₂×D₂₈₄	1	2	yes	—
138	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
139	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	#11765865	1	2	yes	—
143	C₈	#11765865	1	2	yes	—
144	C₂²	—	2	2	no	—
145	C₂²	—	2	1	no	—
146	C₂²	C₂²×D₂₈₄	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	2	1	no	—
150	C₄	—	284	4	no	—
151	C₄	—	284	4	no	—
152	C₄	C₂×#11765865	1	1	yes	characteristic
153	C₄	C₂×#11765865	1	1	yes	characteristic
154	C₂	—	4	1	no	—
155	C₂	#17986549	1	1	yes	characteristic
156	C₂	#17986816	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17986611

C₂

yes

maximal

#17986401

C₂

yes

maximal, characteristic, complemented

#17986379

C₂

yes

maximal, characteristic, complemented

#17986368

C₂

yes

maximal, complemented

#17986257

C₂

yes

maximal

#17986240

C₂

yes

maximal

#17986107

C₂

yes

maximal, complemented

C₄×Dic₂₈₄

C₂

yes

maximal, complemented

#405×C₇₁

C₂

yes

maximal, characteristic

C₂×Dic₂₈₄

C₂²

yes

—

C₂×Dic₂₈₄

C₂²

yes

—

#14556128

C₂²

yes

—

#14556121

—

#14556120

C₂²

yes

—

#14556118

—

#14556107

C₂²

yes

—

C₄×Dic₁₄₂

C₂²

yes

—

#14556062

C₂²

yes

—

#14556060

C₂²

yes

—

#14556053

C₂²

yes

—

#14556053

C₂²

yes

—

C₂×M₁₆×C₇₁

C₂²

yes

characteristic

C₂²×C₈×C₇₁

C₂²

yes

characteristic

#117×C₇₁

C₂²

yes

characteristic

#97×C₇₁

C₂²

yes

characteristic

#97×C₇₁

C₂²

yes

characteristic

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

C₂³

yes

—

Dic₂₈₄

—

Dic₂₈₄

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

C₂³

yes

—

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

D₄

yes

—

C₂²⋊C₄×C₇₁

D₄

yes

—

C₄²×C₇₁

C₂³

yes

characteristic

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂⁴

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

#41

yes

—

C₈×C₇₁

#41

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂²×D₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×#41

yes

characteristic

C₄×C₇₁

C₂×#41

yes

characteristic

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#532

yes

characteristic

C₂×C₇₁

#578

yes

—

#3177

—

maximal, Sylow

C₇₁

#3177

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

—

C₄×Q₁₆

—

#405

D₇₁

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

C₂×D₇₁

yes

characteristic

C₂²×C₈

C₂×D₇₁

yes

characteristic

C₄⋊Q₈

—

#126

—

142

—

#125

—

#125

—

142

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₇₁

yes

characteristic

100

#107

—

101

#106

—

102

#103

—

103

#103

—

104

#97

C₂×D₇₁

yes

characteristic

105

#97

C₂×D₇₁

yes

characteristic

106

C₄×C₈

C₂×D₇₁

yes

characteristic

107

C₄×C₈

C₂×D₇₁

yes

characteristic

108

C₂×Q₈

—

109

C₂²×C₄

C₂²×D₇₁

yes

characteristic

110

Q₁₆

—

142

—

111

Q₁₆

—

142

—

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

C₂²×D₇₁

yes

characteristic

115

C₂×C₈

C₂²×D₇₁

yes

characteristic

116

C₂×C₈

C₂²×D₇₁

yes

characteristic

117

C₂×C₈

C₂²×D₇₁

yes

characteristic

118

C₄⋊C₄

—

119

C₄⋊C₄

—

142

—

120

C₄⋊C₄

—

142

—

121

C₄⋊C₄

—

142

—

122

C₄⋊C₄

—

142

—

123

C₄⋊C₄

D₂₈₄

yes

—

124

C₂²⋊C₄

—

142

—

125

C₂²⋊C₄

D₂₈₄

yes

—

126

C₄²

—

142

—

127

C₄²

C₂²×D₇₁

yes

characteristic

128

C₄²

C₂²×D₇₁

yes

characteristic

129

C₂³

C₂×D₂₈₄

yes

characteristic

130

Q₈

—

142

—

131

Q₈

—

142

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

142

—

135

C₂×C₄

—

142

—

136

C₂×C₄

C₂×D₂₈₄

yes

—

137

C₂×C₄

C₂×D₂₈₄

yes

—

138

C₂×C₄

C₂×D₂₈₄

yes

characteristic

139

C₂×C₄

C₂³×D₇₁

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

#11765865

yes

—

143

C₈

#11765865

yes

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂²×D₂₈₄

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

284

—

151

C₄

—

284

—

152

C₄

C₂×#11765865

yes

characteristic

153

C₄

C₂×#11765865

yes

characteristic

154

C₂

—

155

C₂

#17986549

yes

characteristic

156

C₂

#17986816

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#405×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#405	D₇₁
hypocenter	C₇₁	#3177
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

subgroup

G/H

center

C₂²

C₂²×D₂₈₄

commutator

C₂×C₄×C₇₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₁

Fitting

#405×C₇₁

C₂

socle

C₂²×C₇₁

C₂²×D₄

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ #117 ◁ #405

N	H ≅ G/N	type
2. #17986401	C₂	semidirect
3. #17986379	C₂	semidirect
4. #17986368	C₂	semidirect
7. #17986107	C₂	semidirect
8. C₄×Dic₂₈₄	C₂	semidirect
79. C₇₁	#3177	semidirect

H ≅ G/N

type

2. #17986401

C₂

semidirect

3. #17986379

C₂

semidirect

4. #17986368

C₂

semidirect

7. #17986107

C₂

semidirect

8. C₄×Dic₂₈₄

C₂

semidirect

79. C₇₁

#3177

semidirect

Subgroups — #22598999

Conjugacy classes of subgroups

Complements