Group #31884666 — subgroups

Subgroups — #31884666

Order 18176.

298 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 75 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	C₄×#14556047	C₂	1	2	yes	maximal
2	C₇₁×#1985	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556047	—	2	2	no	—
4	C₄×#11765849	—	2	2	no	—
5	#17985768	C₄	1	4	yes	—
6	C₂×C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₈×C₇₁	C₄	1	1	yes	characteristic
8	C₂×#97×C₇₁	C₄	1	1	yes	characteristic
9	#336×C₇₁	—	2	2	no	—
10	C₂×#11765849	—	2	4	no	—
11	#14556047	—	2	8	no	—
12	#14556039	—	4	2	no	—
13	C₄×C₇₁⋊C₈	—	4	2	no	—
14	C₂²×Q₈×C₇₁	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₇₁	—	2	1	no	—
16	C₂²×C₈×C₇₁	—	2	1	no	—
17	C₄×Q₈×C₇₁	—	4	1	no	—
18	C₄×Q₈×C₇₁	—	4	1	no	—
19	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
20	C₂×C₄⋊C₄×C₇₁	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₇₁	D₄	1	2	yes	—
23	C₄⋊C₈×C₇₁	—	4	1	no	—
24	#97×C₇₁	—	4	1	no	—
25	#11765849	—	2	8	no	—
26	#11765849	—	4	4	no	—
27	C₂×C₇₁⋊C₈	—	4	4	no	—
28	C₂×Q₈×C₇₁	—	2	4	no	—
29	C₂×Q₈×C₇₁	—	4	1	no	—
30	C₂²×C₄×C₇₁	C₄²	1	1	yes	characteristic
31	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
32	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₇₁	—	2	2	no	—
36	C₂×C₈×C₇₁	—	4	1	no	—
37	C₂×C₈×C₇₁	—	4	1	no	—
38	C₂×C₈×C₇₁	—	4	2	no	—
39	C₂×C₈×C₇₁	—	4	1	no	—
40	C₄⋊C₄×C₇₁	—	4	2	no	—
41	C₄⋊C₄×C₇₁	—	4	1	no	—
42	C₄²×C₇₁	—	2	1	no	—
43	C₄²×C₇₁	—	2	1	no	—
44	C₄²×C₇₁	—	2	2	no	—
45	C₄²×C₇₁	—	2	2	no	—
46	C₂³×C₇₁	#95	1	1	yes	characteristic
47	Q₈×C₇₁	—	4	4	no	—
48	C₂×C₄×C₇₁	—	2	2	no	—
49	C₂×C₄×C₇₁	—	2	2	no	—
50	C₂×C₄×C₇₁	—	2	1	no	—
51	C₂×C₄×C₇₁	—	2	1	no	—
52	C₂×C₄×C₇₁	—	2	2	no	—
53	C₂×C₄×C₇₁	—	2	1	no	—
54	C₂×C₄×C₇₁	—	2	2	no	—
55	C₂×C₄×C₇₁	—	2	1	no	—
56	C₂×C₄×C₇₁	—	4	1	no	—
57	C₂×C₄×C₇₁	—	4	2	no	—
58	C₂×C₄×C₇₁	#99	1	1	yes	characteristic
59	C₂×C₄×C₇₁	#99	1	1	yes	characteristic
60	C₂×C₄×C₇₁	#104	1	4	yes	—
61	C₈×C₇₁	—	4	2	no	—
62	C₈×C₇₁	—	8	1	no	—
63	C₇₁⋊C₈	—	4	8	no	—
64	C₂²×C₇₁	—	2	1	no	—
65	C₂²×C₇₁	—	2	1	no	—
66	C₂²×C₇₁	#339	1	2	yes	—
67	C₂²×C₇₁	#342	1	1	yes	characteristic
68	C₄×C₇₁	—	2	4	no	—
69	C₄×C₇₁	—	2	1	no	—
70	C₄×C₇₁	—	4	1	no	—
71	C₄×C₇₁	—	4	1	no	—
72	C₄×C₇₁	—	4	2	no	—
73	C₄×C₇₁	#355	1	1	yes	characteristic
74	C₄×C₇₁	#356	1	1	yes	characteristic
75	#7284	—	71	1	no	maximal, Sylow
76	C₂×C₇₁	—	2	2	no	—
77	C₂×C₇₁	#1297	1	1	yes	characteristic
78	C₂×C₇₁	#1392	1	1	yes	characteristic
79	C₂×C₇₁	#1394	1	1	yes	characteristic
80	#1985	D₇₁	1	1	yes	characteristic
81	C₄×#101	—	71	2	no	—
82	C₇₁	#7284	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
84	C₂×C₄⋊C₈	Dic₇₁	1	1	yes	characteristic
85	C₂×#101	—	142	2	no	—
86	C₄×M₁₆	—	142	2	no	—
87	C₂×#97	Dic₇₁	1	1	yes	characteristic
88	#343	—	71	4	no	—
89	#336	—	2	2	no	—
90	C₂²×Q₈	D₂₈₄	1	1	yes	characteristic
91	C₂×M₁₆	—	142	4	no	—
92	C₂²×C₈	—	2	1	no	—
93	C₂²×C₈	—	2	1	no	—
94	C₄×Q₈	—	4	1	no	—
95	C₄×Q₈	—	4	1	no	—
96	C₂×C₄⋊C₄	Dic₁₄₂	1	1	yes	characteristic
97	C₂×C₄⋊C₄	C₄×D₇₁	1	2	yes	—
98	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
99	C₂×C₄²	C₂²⋊D₇₁	1	2	yes	—
100	C₄⋊C₈	—	4	1	no	—
101	#101	—	142	8	no	—
102	#97	—	4	1	no	—
103	#97	—	284	2	no	—
104	C₄×C₈	—	284	2	no	—
105	C₂×Q₈	—	2	4	no	—
106	C₂×Q₈	—	4	1	no	—
107	C₂²×C₄	C₄×Dic₇₁	1	1	yes	characteristic
108	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	2	yes	—
109	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
110	C₂²×C₄	Dic₇₁⋊C₄	1	2	yes	—
111	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
112	M₁₆	—	284	4	no	—
113	M₁₆	—	142	8	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	4	1	no	—
116	C₂×C₈	—	4	1	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	4	1	no	—
119	C₂×C₈	—	284	4	no	—
120	C₄⋊C₄	—	4	1	no	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄²	—	2	1	no	—
123	C₄²	—	2	2	no	—
124	C₄²	—	2	2	no	—
125	C₄²	—	2	1	no	—
126	C₂³	C₇₁⋊#95	1	1	yes	characteristic
127	Q₈	—	4	4	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	#14556043	1	1	yes	characteristic
139	C₂×C₄	#14556043	1	1	yes	characteristic
140	C₂×C₄	#14556054	1	4	yes	—
141	C₈	—	4	2	no	—
142	C₈	—	8	1	no	—
143	C₈	—	284	8	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	#17985760	1	2	yes	—
147	C₂²	#17985766	1	1	yes	characteristic
148	C₄	—	2	1	no	—
149	C₄	—	2	4	no	—
150	C₄	—	4	1	no	—
151	C₄	—	4	2	no	—
152	C₄	—	4	1	no	—
153	C₄	#17985796	1	1	yes	characteristic
154	C₄	#17985798	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	#22586225	1	1	yes	characteristic
157	C₂	#22586463	1	1	yes	characteristic
158	C₂	#22586467	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14556047

C₂

yes

maximal

C₇₁×#1985

C₂

yes

maximal, characteristic

C₂×#14556047

—

C₄×#11765849

—

#17985768

C₄

yes

—

C₂×C₄×Q₈×C₇₁

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₇₁

C₄

yes

characteristic

C₂×#97×C₇₁

C₄

yes

characteristic

#336×C₇₁

—

C₂×#11765849

—

#14556047

—

#14556039

—

C₄×C₇₁⋊C₈

—

C₂²×Q₈×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

—

C₂²×C₈×C₇₁

—

C₄×Q₈×C₇₁

—

C₄×Q₈×C₇₁

—

C₂×C₄⋊C₄×C₇₁

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₇₁

Q₈

yes

characteristic

C₂×C₄²×C₇₁

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₁

D₄

yes

—

C₄⋊C₈×C₇₁

—

#97×C₇₁

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂²×C₄×C₇₁

C₄²

yes

characteristic

C₂²×C₄×C₇₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₁

C₄⋊C₄

yes

—

C₂²×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₂³×C₇₁

#95

yes

characteristic

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#339

yes

—

C₂²×C₇₁

#342

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

yes

characteristic

C₄×#101

—

C₇₁

#7284

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

C₂×D₇₁

yes

characteristic

C₂×C₄⋊C₈

Dic₇₁

yes

characteristic

C₂×#101

—

142

—

C₄×M₁₆

—

142

—

C₂×#97

Dic₇₁

yes

characteristic

#343

—

#336

—

C₂²×Q₈

D₂₈₄

yes

characteristic

C₂×M₁₆

—

142

—

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

Dic₁₄₂

yes

characteristic

C₂×C₄⋊C₄

C₄×D₇₁

yes

—

C₂×C₄²

C₂×Dic₇₁

yes

characteristic

C₂×C₄²

C₂²⋊D₇₁

yes

—

100

C₄⋊C₈

—

101

#101

—

142

—

102

#97

—

103

#97

—

284

—

104

C₄×C₈

—

284

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂²×C₄

C₄×Dic₇₁

yes

characteristic

108

C₂²×C₄

(C₂×D₇₁)⋊C₄

yes

—

109

C₂²×C₄

C₂²⋊Dic₇₁

yes

characteristic

110

C₂²×C₄

Dic₇₁⋊C₄

yes

—

111

C₂²×C₄

C₄⋊Dic₇₁

yes

characteristic

112

M₁₆

—

284

—

113

M₁₆

—

142

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

284

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₂³

C₇₁⋊#95

yes

characteristic

127

Q₈

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

#14556043

yes

characteristic

139

C₂×C₄

#14556043

yes

characteristic

140

C₂×C₄

#14556054

yes

—

141

C₈

—

142

C₈

—

143

C₈

—

284

—

144

C₂²

—

145

C₂²

—

146

C₂²

#17985760

yes

—

147

C₂²

#17985766

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

#17985796

yes

characteristic

154

C₄

#17985798

yes

characteristic

155

C₂

—

156

C₂

#22586225

yes

characteristic

157

C₂

#22586463

yes

characteristic

158

C₂

#22586467

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556043
commutator	C₂²×C₄×C₇₁	C₄²
Frattini	C₂×C₄×Q₈	C₂×D₇₁
Fitting	C₇₁×#1985	C₂
socle	C₂²×C₇₁	#342
prime radical	C₂³×C₇₁	#95
cosocle	C₂×C₄×Q₈×C₇₁	C₂²
hypercenter	#1985	D₇₁
hypocenter	C₇₁	#7284
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂³×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ #1985

subgroup

G/H

center

C₂×C₄

#14556043

commutator

C₂²×C₄×C₇₁

C₄²

Frattini

C₂×C₄×Q₈

C₂×D₇₁

Fitting

C₇₁×#1985

C₂

C₂²×C₇₁

C₂³×C₇₁

C₂×C₄×Q₈×C₇₁

C₂²

D₇₁

C₇₁

G ▷ C₂²×C₄×C₇₁ ▷ C₂³×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ #1985

N	H ≅ G/N	type
82. C₇₁	#7284	semidirect

H ≅ G/N

type

82. C₇₁

#7284

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×#14556047	2
C₇₁×#1985	2
#7284	71

subgroup

index

C₄×#14556047

C₇₁×#1985

#7284

Subgroups — #31884666

Conjugacy classes of subgroups

Complements

Maximal subgroups