Group #31884875 — subgroups

Subgroups — #31884875

Order 18176.

148 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 39 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	#22588514	C₂	1	1	yes	maximal, characteristic
2	#22586291	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#1327	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556063	—	2	1	no	—
5	C₂×#14556063	—	2	1	no	—
6	#17986023	C₄	1	1	yes	characteristic
7	C₂×#14556039	C₄	1	1	yes	characteristic
8	#17985781	—	2	1	no	—
9	#423×C₇₁	C₂²	1	1	yes	characteristic
10	#349×C₇₁	—	2	1	no	—
11	C₂×#11765849	—	2	1	no	—
12	C₂²×C₇₁⋊C₈	—	2	1	no	—
13	#14556068	—	4	1	no	—
14	#14556063	—	4	2	no	—
15	#14556063	—	4	2	no	—
16	C₄⋊(C₇₁⋊C₈)	—	4	1	no	—
17	#14556039	—	2	1	no	—
18	#14556039	C₈	1	2	yes	—
19	C₂×M₁₆×C₇₁	D₄	1	1	yes	characteristic
20	C₂×M₁₆×C₇₁	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
22	M₃₂×C₇₁	—	4	1	no	—
23	C₄⋊C₈×C₇₁	—	4	1	no	—
24	#11765849	—	4	1	no	—
25	#11765849	—	4	1	no	—
26	C₂×C₇₁⋊C₈	—	2	2	no	—
27	C₂×C₇₁⋊C₈	—	2	2	no	—
28	C₂×C₇₁⋊C₈	—	2	2	no	—
29	C₂×C₇₁⋊C₈	—	4	1	no	—
30	C₇₁⋊C₁₆	—	8	1	no	—
31	C₂²×C₄×C₇₁	—	2	1	no	—
32	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
33	M₁₆×C₇₁	—	4	1	no	—
34	M₁₆×C₇₁	—	4	1	no	—
35	C₂×C₈×C₇₁	—	2	1	no	—
36	C₂×C₈×C₇₁	—	2	1	no	—
37	C₄²×C₇₁	—	2	1	no	—
38	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
39	C₄²×C₇₁	M₁₆	1	1	yes	characteristic
40	C₁₆×C₇₁	—	8	1	no	—
41	C₂³×C₇₁	#101	1	1	yes	characteristic
42	C₂×C₄×C₇₁	—	2	1	no	—
43	C₂×C₄×C₇₁	—	2	1	no	—
44	C₂×C₄×C₇₁	—	2	1	no	—
45	C₂×C₄×C₇₁	—	2	1	no	—
46	C₂×C₄×C₇₁	—	8	1	no	—
47	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
48	C₂×C₄×C₇₁	#99	1	1	yes	characteristic
49	C₈×C₇₁	—	4	1	no	—
50	C₈×C₇₁	—	4	1	no	—
51	C₇₁⋊C₈	—	2	4	no	—
52	C₇₁⋊C₈	—	8	1	no	—
53	C₂²×C₇₁	—	2	1	no	—
54	C₂²×C₇₁	—	4	1	no	—
55	C₂²×C₇₁	#324	1	1	yes	characteristic
56	C₄×C₇₁	—	2	1	no	—
57	C₄×C₇₁	—	4	1	no	—
58	C₄×C₇₁	—	4	1	no	—
59	C₄×C₇₁	#353	1	1	yes	characteristic
60	C₄×C₇₁	#356	1	1	yes	characteristic
61	#7374	—	71	1	no	maximal, Sylow
62	C₂×C₇₁	—	2	1	no	—
63	C₂×C₇₁	—	4	1	no	—
64	C₂×C₇₁	#1325	1	1	yes	characteristic
65	#1849	—	71	1	no	—
66	#1327	—	71	1	no	—
67	#1327	D₇₁	1	1	yes	characteristic
68	C₇₁	#7374	1	1	yes	characteristic, complemented, Sylow
69	C₂×#108	—	142	1	no	—
70	C₂×#108	—	142	1	no	—
71	#423	—	71	1	no	—
72	#423	C₂×D₇₁	1	1	yes	characteristic
73	C₂×#97	—	71	1	no	—
74	#349	—	2	1	no	—
75	#349	—	142	1	no	—
76	C₂×M₁₆	—	142	1	no	—
77	C₂×M₁₆	D₂₈₄	1	1	yes	characteristic
78	C₂×M₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
79	C₂²×C₈	—	142	1	no	—
80	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
81	M₃₂	—	4	1	no	—
82	M₃₂	—	284	1	no	—
83	#108	—	284	2	no	—
84	#108	—	284	2	no	—
85	C₄⋊C₈	—	4	1	no	—
86	C₄⋊C₈	—	284	1	no	—
87	#97	—	71	2	no	—
88	#97	—	142	1	no	—
89	C₂²×C₄	—	2	1	no	—
90	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
91	M₁₆	—	4	1	no	—
92	M₁₆	—	4	1	no	—
93	M₁₆	—	284	1	no	—
94	M₁₆	—	284	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	284	1	no	—
98	C₂×C₈	—	142	2	no	—
99	C₂×C₈	—	142	2	no	—
100	C₂×C₈	—	142	2	no	—
101	C₄²	—	2	1	no	—
102	C₄²	C₈×D₇₁	1	1	yes	characteristic
103	C₄²	#11765848	1	1	yes	characteristic
104	C₁₆	—	8	1	no	—
105	C₁₆	—	568	1	no	—
106	C₂³	#14556046	1	1	yes	characteristic
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	8	1	no	—
112	C₂×C₄	(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
113	C₂×C₄	#14556042	1	1	yes	characteristic
114	C₈	—	4	1	no	—
115	C₈	—	4	1	no	—
116	C₈	—	568	1	no	—
117	C₈	—	142	4	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	4	1	no	—
120	C₂²	#17985724	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	4	1	no	—
123	C₄	—	4	1	no	—
124	C₄	#17985791	1	1	yes	characteristic
125	C₄	#17985797	1	1	yes	characteristic
126	C₂	—	2	1	no	—
127	C₂	—	4	1	no	—
128	C₂	#22586285	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22588514

C₂

yes

maximal, characteristic

#22586291

C₂

yes

maximal, characteristic

C₇₁×#1327

C₂

yes

maximal, characteristic

C₂×#14556063

—

C₂×#14556063

—

#17986023

C₄

yes

characteristic

C₂×#14556039

C₄

yes

characteristic

#17985781

—

#423×C₇₁

C₂²

yes

characteristic

#349×C₇₁

—

C₂×#11765849

—

C₂²×C₇₁⋊C₈

—

#14556068

—

#14556063

—

#14556063

—

C₄⋊(C₇₁⋊C₈)

—

#14556039

—

#14556039

C₈

yes

—

C₂×M₁₆×C₇₁

D₄

yes

characteristic

C₂×M₁₆×C₇₁

D₄

yes

characteristic

C₂×C₄²×C₇₁

C₂×C₄

yes

characteristic

M₃₂×C₇₁

—

C₄⋊C₈×C₇₁

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂×C₈

yes

characteristic

C₄²×C₇₁

M₁₆

yes

characteristic

C₁₆×C₇₁

—

C₂³×C₇₁

#101

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₁

#99

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#324

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

—

—

D₇₁

yes

characteristic

C₇₁

#7374

yes

characteristic, complemented, Sylow

C₂×#108

—

142

—

C₂×#108

—

142

—

#423

—

#423

C₂×D₇₁

yes

characteristic

C₂×#97

—

#349

—

#349

—

142

—

C₂×M₁₆

—

142

—

C₂×M₁₆

D₂₈₄

yes

characteristic

C₂×M₁₆

C₂²⋊D₇₁

yes

characteristic

C₂²×C₈

—

142

—

C₂×C₄²

C₄×D₇₁

yes

characteristic

M₃₂

—

M₃₂

—

284

—

#108

—

284

—

#108

—

284

—

C₄⋊C₈

—

C₄⋊C₈

—

284

—

#97

—

#97

—

142

—

C₂²×C₄

—

C₂²×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

284

—

M₁₆

—

284

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

284

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

100

C₂×C₈

—

142

—

101

C₄²

—

102

C₄²

C₈×D₇₁

yes

characteristic

103

C₄²

#11765848

yes

characteristic

104

C₁₆

—

105

C₁₆

—

568

—

106

C₂³

#14556046

yes

characteristic

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

(C₂×D₇₁)⋊C₈

yes

characteristic

113

C₂×C₄

#14556042

yes

characteristic

114

C₈

—

115

C₈

—

116

C₈

—

568

—

117

C₈

—

142

—

118

C₂²

—

119

C₂²

—

120

C₂²

#17985724

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

#17985791

yes

characteristic

125

C₄

#17985797

yes

characteristic

126

C₂

—

127

C₂

—

128

C₂

#22586285

yes

characteristic

	subgroup	G/H
center	C₄	#17985791
commutator	C₄²×C₇₁	C₂×C₈
Frattini	#423	C₂×D₇₁
Fitting	C₇₁×#1327	C₂
socle	C₂×C₇₁	#1325
prime radical	C₂³×C₇₁	#101
cosocle	#423×C₇₁	C₂²
hypercenter	#1327	D₇₁
hypocenter	C₇₁	#7374
LCS	G ▷ C₄²×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #423 ◁ #1327

subgroup

G/H

center

C₄

#17985791

commutator

C₄²×C₇₁

C₂×C₈

Frattini

#423

C₂×D₇₁

Fitting

C₇₁×#1327

C₂

C₂×C₇₁

C₂³×C₇₁

#423×C₇₁

C₂²

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #423 ◁ #1327

N	H ≅ G/N	type
68. C₇₁	#7374	semidirect

H ≅ G/N

type

68. C₇₁

#7374

semidirect

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

subgroup	index
#22588514	2
#22586291	2
C₇₁×#1327	2
#7374	71

subgroup

index

#22588514

#22586291

C₇₁×#1327

#7374

Subgroups — #31884875

Conjugacy classes of subgroups

Complements

Maximal subgroups