Group #27141852 — subgroups

Subgroups — #27141852

Order 12416.

260 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 129 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×C₄⋊#11715895	C₂	1	1	yes	maximal, characteristic
2	C₄²×Dic₉₇	C₂	1	1	yes	maximal, characteristic
3	#20260489	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11715895	C₂²	1	2	yes	—
6	C₂×C₄×Dic₉₇	C₄	1	2	yes	—
7	C₂×C₄×Dic₉₇	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₉₇	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₉₇	C₂²	1	2	yes	—
10	C₄⋊#11715895	—	2	2	no	—
11	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
12	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₉₇	—	2	2	no	—
14	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
15	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
16	C₂×#11715895	—	2	2	no	—
17	C₂×#11715895	—	2	4	no	—
18	C₄×Dic₉₇	—	2	2	no	—
19	C₄×Dic₉₇	—	2	2	no	—
20	C₄×Dic₉₇	—	2	2	no	—
21	C₄×Dic₉₇	—	2	4	no	—
22	C₄×Dic₉₇	C₈	1	4	yes	complemented
23	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₉₇	—	2	2	no	—
27	C₂×C₈×C₉₇	—	2	4	no	—
28	C₄²×C₉₇	D₄	1	2	yes	—
29	C₄²×C₉₇	Q₈	1	2	yes	—
30	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₉₇	—	2	2	no	—
32	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
33	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
34	C₂×C₄×C₉₇	C₂×C₈	1	2	yes	—
35	C₂×C₄×C₉₇	M₁₆	1	2	yes	—
36	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₉₇	#41	1	2	yes	—
39	C₂×C₄×C₉₇	#41	1	2	yes	—
40	C₂×Dic₉₇	—	2	2	no	—
41	C₂×Dic₉₇	—	2	2	no	—
42	C₂×Dic₉₇	—	2	2	no	—
43	C₂×Dic₉₇	—	2	2	no	—
44	C₂×Dic₉₇	C₂×C₈	1	4	yes	—
45	C₂×Dic₉₇	M₁₆	1	4	yes	—
46	C₈×C₉₇	—	4	2	no	—
47	#11715895	—	4	2	no	—
48	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₉₇	#117	1	2	yes	—
50	C₂²×C₉₇	C₂²×C₈	1	1	yes	characteristic
51	C₂²×C₉₇	C₂×M₁₆	1	2	yes	—
52	C₂²×C₉₇	C₂×M₁₆	1	1	yes	characteristic
53	C₄×C₉₇	—	2	2	no	—
54	C₄×C₉₇	C₄⋊C₈	1	4	yes	—
55	C₄×C₉₇	#124	1	1	yes	characteristic
56	C₄×C₉₇	#124	1	1	yes	characteristic
57	C₄×C₉₇	#125	1	2	yes	—
58	Dic₉₇	—	2	4	no	—
59	Dic₉₇	—	2	4	no	—
60	C₂×C₉₇	#382	1	1	yes	characteristic
61	C₂×C₉₇	C₂×C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₉₇	#423	1	1	yes	characteristic
63	C₂×C₉₇	#431	1	2	yes	—
64	C₂×C₉₇	#432	1	2	yes	—
65	#1831	—	97	1	no	maximal, Sylow
66	C₉₇	#1831	1	1	yes	characteristic, complemented, Sylow
67	C₂×C₄⋊C₈	—	97	1	no	—
68	C₂×C₄⋊C₈	D₉₇	1	1	yes	characteristic
69	C₄³	—	97	1	no	—
70	#336	—	97	4	no	—
71	C₂²×C₈	—	97	2	no	—
72	C₂²×C₈	C₂×D₉₇	1	2	yes	—
73	C₂×C₄²	—	97	1	no	—
74	C₂×C₄²	—	97	2	no	—
75	C₂×C₄²	—	97	1	no	—
76	C₂×C₄²	—	97	2	no	—
77	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
78	C₄⋊C₈	—	2	2	no	—
79	C₄⋊C₈	—	194	2	no	—
80	C₂²×C₄	—	97	2	no	—
81	C₂²×C₄	—	97	2	no	—
82	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
83	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
84	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	4	no	—
87	C₂×C₈	—	194	2	no	—
88	C₂×C₈	—	194	4	no	—
89	C₄²	—	97	4	no	—
90	C₄²	—	194	4	no	—
91	C₄²	—	194	2	no	—
92	C₄²	—	194	2	no	—
93	C₄²	—	194	2	no	—
94	C₄²	Dic₁₉₄	1	2	yes	—
95	C₄²	C₂²⋊D₉₇	1	2	yes	—
96	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	97	4	no	—
99	C₂×C₄	—	97	4	no	—
100	C₂×C₄	—	194	2	no	—
101	C₂×C₄	—	194	2	no	—
102	C₂×C₄	—	194	2	no	—
103	C₂×C₄	—	194	2	no	—
104	C₂×C₄	#13095490	1	2	yes	—
105	C₂×C₄	#13095490	1	2	yes	—
106	C₂×C₄	C₈×D₉₇	1	2	yes	—
107	C₂×C₄	#13095494	1	2	yes	—
108	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
109	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
110	C₂×C₄	#13095510	1	2	yes	—
111	C₂×C₄	#13095512	1	2	yes	—
112	C₈	—	4	2	no	—
113	C₈	—	388	2	no	—
114	C₂²	C₂×#13095490	1	1	yes	characteristic
115	C₂²	#15690768	1	2	yes	—
116	C₂²	C₂×C₈×D₉₇	1	1	yes	characteristic
117	C₂²	M₁₆×D₉₇	1	2	yes	—
118	C₂²	C₂×#13095494	1	1	yes	characteristic
119	C₄	—	2	2	no	—
120	C₄	—	194	4	no	—
121	C₄	—	194	4	no	—
122	C₄	#15690739	1	4	yes	—
123	C₄	#15690797	1	1	yes	characteristic
124	C₄	#15690799	1	1	yes	characteristic
125	C₄	#15690801	1	2	yes	—
126	C₂	#20260600	1	1	yes	characteristic
127	C₂	C₂×#15690739	1	1	yes	characteristic
128	C₂	#20260758	1	1	yes	characteristic
129	C₂	#20260787	1	2	yes	—
130	C₂	#20260791	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11715895

C₂

yes

maximal, characteristic

C₄²×Dic₉₇

C₂

yes

maximal, characteristic

#20260489

C₂

yes

maximal

C₂×C₄⋊C₈×C₉₇

C₂

yes

maximal, characteristic

C₂²×#11715895

C₂²

yes

—

C₂×C₄×Dic₉₇

C₄

yes

—

C₂×C₄×Dic₉₇

C₄

yes

characteristic

C₂×C₄×Dic₉₇

C₄

yes

characteristic

C₂×C₄×Dic₉₇

C₂²

yes

—

C₄⋊#11715895

—

C₂²×C₈×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₉₇

—

C₂²×Dic₉₇

C₂×C₄

yes

—

C₂²×Dic₉₇

C₂×C₄

yes

—

C₂×#11715895

—

C₂×#11715895

—

C₄×Dic₉₇

—

C₄×Dic₉₇

—

C₄×Dic₉₇

—

C₄×Dic₉₇

—

C₄×Dic₉₇

C₈

yes

complemented

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₄²×C₉₇

D₄

yes

—

C₄²×C₉₇

Q₈

yes

—

C₂³×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₉₇

C₂×C₈

yes

—

C₂×C₄×C₉₇

M₁₆

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂×C₄×C₉₇

#41

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

C₂×C₈

yes

—

C₂×Dic₉₇

M₁₆

yes

—

C₈×C₉₇

—

#11715895

—

C₂²×C₉₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₉₇

#117

yes

—

C₂²×C₉₇

C₂²×C₈

yes

characteristic

C₂²×C₉₇

C₂×M₁₆

yes

—

C₂²×C₉₇

C₂×M₁₆

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

C₄⋊C₈

yes

—

C₄×C₉₇

yes

C₄×C₉₇

yes

C₄×C₉₇

yes

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

#382

yes

characteristic

C₂×C₉₇

C₂×C₄⋊C₈

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₂×C₉₇

#432

yes

—

#1831

—

maximal, Sylow

C₉₇

#1831

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₉₇

yes

characteristic

C₄³

—

#336

—

C₂²×C₈

—

C₂²×C₈

C₂×D₉₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

194

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₉₇

yes

characteristic

C₂²×C₄

C₄×D₉₇

yes

characteristic

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₄²

—

C₄²

—

194

—

C₄²

—

194

—

C₄²

—

194

—

C₄²

—

194

—

C₄²

Dic₁₉₄

yes

—

C₄²

C₂²⋊D₉₇

yes

—

C₂³

C₂×C₄×D₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

194

—

101

C₂×C₄

—

194

—

102

C₂×C₄

—

194

—

103

C₂×C₄

—

194

—

104

C₂×C₄

#13095490

yes

—

105

C₂×C₄

#13095490

yes

—

106

C₂×C₄

C₈×D₉₇

yes

—

107

C₂×C₄

#13095494

yes

—

108

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

109

C₂×C₄

C₂×Dic₁₉₄

yes

characteristic

110

C₂×C₄

#13095510

yes

—

111

C₂×C₄

#13095512

yes

—

112

C₈

—

113

C₈

—

388

—

114

C₂²

C₂×#13095490

yes

characteristic

115

C₂²

#15690768

yes

—

116

C₂²

C₂×C₈×D₉₇

yes

characteristic

117

C₂²

M₁₆×D₉₇

yes

—

118

C₂²

C₂×#13095494

yes

characteristic

119

C₄

—

120

C₄

—

194

—

121

C₄

—

194

—

122

C₄

#15690739

yes

—

123

C₄

#15690797

yes

characteristic

124

C₄

#15690799

yes

characteristic

125

C₄

#15690801

yes

—

126

C₂

#20260600

yes

characteristic

127

C₂

C₂×#15690739

yes

characteristic

128

C₂

#20260758

yes

characteristic

129

C₂

#20260787

yes

—

130

C₂

#20260791

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₉₇
commutator	C₂²×C₉₇	C₂²×C₈
Fitting	C₂×C₄⋊C₈×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂²×C₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂×C₄⋊C₈	D₉₇
hypocenter	C₉₇	#1831
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₉₇

commutator

C₂²×C₉₇

C₂²×C₈

Fitting

C₂×C₄⋊C₈×C₉₇

C₂

socle, prime radical

C₂³×C₉₇

C₂²×C₄

cosocle

C₂²×C₄×C₉₇

C₂³

hypercenter

C₂×C₄⋊C₈

D₉₇

C₉₇

G ▷ C₂²×C₉₇ ▷ C₉₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
22. C₄×Dic₉₇	C₈	semidirect
66. C₉₇	#1831	semidirect

H ≅ G/N

type

22. C₄×Dic₉₇

C₈

semidirect

66. C₉₇

#1831

semidirect

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

subgroup	index
C₂×C₄⋊#11715895	2
C₄²×Dic₉₇	2
#20260489	2
C₂×C₄⋊C₈×C₉₇	2
#1831	97

subgroup

index

C₂×C₄⋊#11715895

C₄²×Dic₉₇

#20260489

C₂×C₄⋊C₈×C₉₇

#1831

Subgroups — #27141852

Conjugacy classes of subgroups

Complements

Maximal subgroups