Group #22588384 — subgroups

Subgroups — #22588384

Order 9088.

260 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 89 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×#14556135	C₂	1	1	yes	maximal, characteristic
2	#17986035	C₂	1	2	yes	maximal, complemented
3	#17985973	C₂	1	1	yes	maximal, characteristic
4	#17985755	C₂	1	2	yes	maximal, complemented
5	#410×C₇₁	C₂	1	1	yes	maximal, characteristic
6	#14556135	—	2	2	no	—
7	#14556135	—	2	2	no	—
8	C₂×#11765849	C₄	1	2	yes	complemented
9	C₂×#11765849	C₂²	1	1	yes	characteristic
10	C₂²×C₇₁⋊C₈	C₄	1	2	yes	complemented
11	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
12	#14556087	C₂²	1	2	yes	—
13	#14556061	—	2	2	no	—
14	#14556059	—	2	2	no	—
15	#14556043	—	2	2	no	—
16	C₂×#41×C₇₁	C₂²	1	1	yes	characteristic
17	#117×C₇₁	C₂²	1	2	yes	—
18	#99×C₇₁	—	2	2	no	—
19	#11765849	—	2	2	no	—
20	#11765849	—	2	4	no	—
21	C₂×C₇₁⋊C₈	—	2	2	no	—
22	C₂×C₇₁⋊C₈	—	2	2	no	—
23	C₂×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
24	C₂×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₇₁⋊C₈	D₄	1	2	yes	—
27	C₂×C₇₁⋊C₈	D₄	1	2	yes	—
28	C₄⋊Dic₇₁	—	2	2	no	—
29	C₂²⋊Dic₇₁	—	2	2	no	—
30	C₄×Dic₇₁	—	2	2	no	—
31	#41×C₇₁	—	2	2	no	—
32	#41×C₇₁	—	2	2	no	—
33	C₂×Q₈×C₇₁	Q₈	1	1	yes	characteristic
34	C₂×D₄×C₇₁	D₄	1	2	yes	—
35	C₂×D₄×C₇₁	Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₇₁	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₇₁	—	2	2	no	—
39	C₂²⋊C₄×C₇₁	—	2	2	no	—
40	C₄²×C₇₁	—	2	2	no	—
41	C₂³×C₇₁	C₄⋊C₄	1	2	yes	—
42	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₇₁	—	2	2	no	—
44	D₄×C₇₁	—	2	4	no	—
45	D₄×C₇₁	—	2	2	no	—
46	C₂×C₄×C₇₁	—	2	2	no	—
47	C₂×C₄×C₇₁	—	2	2	no	—
48	C₂×C₄×C₇₁	—	4	2	no	—
49	C₂×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
50	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₇₁	#41	1	2	yes	—
54	C₂×Dic₇₁	—	4	2	no	—
55	C₇₁⋊C₈	—	2	2	no	—
56	C₇₁⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
57	C₇₁⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
58	C₇₁⋊C₈	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₇₁	—	2	2	no	—
60	C₂²×C₇₁	—	4	1	no	—
61	C₂²×C₇₁	—	4	2	no	—
62	C₂²×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₇₁	#121	1	2	yes	—
64	C₄×C₇₁	—	2	2	no	—
65	C₄×C₇₁	—	8	2	no	—
66	C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₇₁	#117	1	1	yes	characteristic
68	C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
69	Dic₇₁	—	8	2	no	—
70	C₂×C₇₁	—	2	2	no	—
71	C₂×C₇₁	—	2	1	no	—
72	C₂×C₇₁	—	4	2	no	—
73	C₂×C₇₁	#380	1	1	yes	characteristic
74	#1812	—	71	1	no	maximal, Sylow
75	C₇₁	#1812	1	1	yes	characteristic, complemented, Sylow
76	C₂×#131	—	71	1	no	—
77	#427	—	71	2	no	—
78	#410	—	71	1	no	—
79	#410	D₇₁	1	1	yes	characteristic
80	H(3,ℤ/4)	—	71	2	no	—
81	C₂×#41	C₂×D₇₁	1	1	yes	characteristic
82	#131	—	142	2	no	—
83	#131	—	142	2	no	—
84	C₂×M₁₆	—	71	2	no	—
85	C₂×M₁₆	—	71	1	no	—
86	C₂²×C₈	—	71	1	no	—
87	C₂²×C₈	—	71	2	no	—
88	#117	—	71	2	no	—
89	#117	C₂×D₇₁	1	2	yes	—
90	#107	—	142	2	no	—
91	#106	—	142	2	no	—
92	#99	—	2	2	no	—
93	#99	—	142	2	no	—
94	#41	—	2	2	no	—
95	#41	—	2	2	no	—
96	C₂×Q₈	Dic₁₄₂	1	1	yes	characteristic
97	C₂×D₄	Dic₁₄₂	1	1	yes	characteristic
98	C₂×D₄	C₂²⋊D₇₁	1	2	yes	—
99	C₂²×C₄	C₄×D₇₁	1	2	yes	—
100	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
101	M₁₆	—	142	4	no	—
102	M₁₆	—	142	2	no	—
103	C₂×C₈	—	71	1	no	—
104	C₂×C₈	—	71	1	no	—
105	C₂×C₈	—	71	2	no	—
106	C₂×C₈	—	71	2	no	—
107	C₂×C₈	—	71	2	no	—
108	C₂×C₈	—	142	2	no	—
109	C₂×C₈	—	142	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	142	2	no	—
112	C₂²⋊C₄	—	2	2	no	—
113	C₂²⋊C₄	—	142	2	no	—
114	C₄²	—	2	2	no	—
115	C₄²	—	142	2	no	—
116	C₂³	Dic₇₁⋊C₄	1	2	yes	—
117	C₂³	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
118	Q₈	—	2	2	no	—
119	D₄	—	2	4	no	—
120	D₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	284	2	no	—
125	C₂×C₄	Dic₇₁⋊C₄	1	2	yes	—
126	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
127	C₂×C₄	D₄×D₇₁	1	2	yes	—
128	C₂×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
129	C₂×C₄	#11765864	1	2	yes	—
130	C₈	—	71	1	no	—
131	C₈	—	71	1	no	—
132	C₈	—	71	2	no	—
133	C₈	—	142	2	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	4	1	no	—
136	C₂²	—	4	2	no	—
137	C₂²	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
138	C₂²	#14556100	1	2	yes	—
139	C₄	—	2	2	no	—
140	C₄	—	8	2	no	—
141	C₄	—	568	2	no	—
142	C₄	C₂²⋊C₄×D₇₁	1	1	yes	characteristic
143	C₄	#14556084	1	1	yes	characteristic
144	C₄	#14556105	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	2	1	no	—
147	C₂	—	4	2	no	—
148	C₂	#17985857	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556135

C₂

yes

maximal, characteristic

#17986035

C₂

yes

maximal, complemented

#17985973

C₂

yes

maximal, characteristic

#17985755

C₂

yes

maximal, complemented

#410×C₇₁

C₂

yes

maximal, characteristic

#14556135

—

#14556135

—

C₂×#11765849

C₄

yes

complemented

C₂×#11765849

C₂²

yes

characteristic

C₂²×C₇₁⋊C₈

C₄

yes

complemented

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

#14556087

C₂²

yes

—

#14556061

—

#14556059

—

#14556043

—

C₂×#41×C₇₁

C₂²

yes

characteristic

#117×C₇₁

C₂²

yes

—

#99×C₇₁

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₈

D₄

yes

—

C₂×C₇₁⋊C₈

D₄

yes

—

C₄⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

#41×C₇₁

—

#41×C₇₁

—

C₂×Q₈×C₇₁

Q₈

yes

characteristic

C₂×D₄×C₇₁

D₄

yes

—

C₂×D₄×C₇₁

Q₈

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₂³×C₇₁

C₄⋊C₄

yes

—

C₂³×C₇₁

C₂×D₄

yes

characteristic

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄⋊C₄

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₁

#41

yes

—

C₂×Dic₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

C₂²⋊C₄

yes

characteristic

C₇₁⋊C₈

C₂²⋊C₄

yes

characteristic

C₇₁⋊C₈

C₂²⋊C₄

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₁

#121

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₁

#117

yes

characteristic

C₄×C₇₁

C₂²⋊Q₈

yes

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

—

—

—

D₇₁

yes

characteristic

H(3,ℤ/4)

—

C₂×#41

C₂×D₇₁

yes

characteristic

#131

—

142

—

#131

—

142

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

#117

—

#117

C₂×D₇₁

yes

—

#107

—

142

—

#106

—

142

—

#99

—

#99

—

142

—

#41

—

#41

—

C₂×Q₈

Dic₁₄₂

yes

characteristic

C₂×D₄

Dic₁₄₂

yes

characteristic

C₂×D₄

C₂²⋊D₇₁

yes

—

C₂²×C₄

C₄×D₇₁

yes

—

100

C₂²×C₄

C₂²×D₇₁

yes

characteristic

101

M₁₆

—

142

—

102

M₁₆

—

142

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

142

—

109

C₂×C₈

—

142

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

142

—

112

C₂²⋊C₄

—

113

C₂²⋊C₄

—

142

—

114

C₄²

—

115

C₄²

—

142

—

116

C₂³

Dic₇₁⋊C₄

yes

—

117

C₂³

C₂×C₂²⋊D₇₁

yes

characteristic

118

Q₈

—

119

D₄

—

120

D₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

284

—

125

C₂×C₄

Dic₇₁⋊C₄

yes

—

126

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

127

C₂×C₄

D₄×D₇₁

yes

—

128

C₂×C₄

C₂×Dic₁₄₂

yes

characteristic

129

C₂×C₄

#11765864

yes

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

142

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂×Dic₇₁⋊C₄

yes

characteristic

138

C₂²

#14556100

yes

—

139

C₄

—

140

C₄

—

141

C₄

—

568

—

142

C₄

C₂²⋊C₄×D₇₁

yes

characteristic

143

C₄

#14556084

yes

characteristic

144

C₄

#14556105

yes

—

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

#17985857

yes

characteristic

	subgroup	G/H
center	C₄	#14556084
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	#410×C₇₁	C₂
socle	C₂×C₇₁	#380
hypercenter	#410	D₇₁
hypocenter	C₇₁	#1812
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

subgroup

G/H

center

C₄

#14556084

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

#410×C₇₁

C₂

C₂×C₇₁

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

N	H ≅ G/N	type
2. #17986035	C₂	semidirect
4. #17985755	C₂	semidirect
8. C₂×#11765849	C₄	semidirect
10. C₂²×C₇₁⋊C₈	C₄	semidirect
75. C₇₁	#1812	semidirect

H ≅ G/N

type

2. #17986035

C₂

semidirect

4. #17985755

C₂

semidirect

8. C₂×#11765849

C₄

semidirect

10. C₂²×C₇₁⋊C₈

C₄

semidirect

75. C₇₁

#1812

semidirect

Subgroups — #22588384

Conjugacy classes of subgroups

Complements