Group #27141584 — subgroups

Subgroups — #27141584

Order 12416.

316 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 161 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×#15690739	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₉₇	C₂	1	2	yes	maximal
3	C₄²×Dic₉₇	C₂	1	1	yes	maximal, characteristic
4	#20260490	C₂	1	1	yes	maximal, characteristic
5	#336×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11715895	C₂²	1	2	yes	—
7	C₂×C₄×Dic₉₇	C₄	1	2	yes	—
8	C₂×C₄×Dic₉₇	C₄	1	2	yes	—
9	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
11	#15690739	C₄	1	8	yes	complemented
12	C₈×Dic₉₇	—	2	4	no	—
13	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
18	C₂×#11715895	—	2	2	no	—
19	C₂×#11715895	—	2	2	no	—
20	C₂×#11715895	C₂×C₄	1	4	yes	—
21	C₄×Dic₉₇	—	2	4	no	—
22	C₄×Dic₉₇	C₈	1	8	yes	complemented
23	C₄×Dic₉₇	C₂×C₄	1	4	yes	—
24	C₄×Dic₉₇	D₄	1	2	yes	—
25	C₄×Dic₉₇	Q₈	1	2	yes	—
26	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₉₇	—	2	2	no	—
29	C₂×C₈×C₉₇	—	2	2	no	—
30	C₂×C₈×C₉₇	C₂×C₄	1	4	yes	—
31	C₄²×C₉₇	—	2	2	no	—
32	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₉₇	—	2	2	no	—
34	C₂×C₄×C₉₇	—	2	2	no	—
35	C₂×C₄×C₉₇	C₂×C₈	1	4	yes	—
36	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
37	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₉₇	#41	1	2	yes	—
40	C₂×Dic₉₇	—	2	2	no	—
41	C₂×Dic₉₇	C₄²	1	2	yes	—
42	C₂×Dic₉₇	C₄²	1	2	yes	—
43	C₂×Dic₉₇	C₄⋊C₄	1	4	yes	—
44	C₂×Dic₉₇	C₂×C₈	1	4	yes	—
45	C₂×Dic₉₇	C₂×C₈	1	4	yes	—
46	C₂×Dic₉₇	M₁₆	1	4	yes	—
47	C₈×C₉₇	—	2	4	no	—
48	#11715895	—	2	4	no	—
49	C₂²×C₉₇	C₂×C₄²	1	1	yes	characteristic
50	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₉₇	#117	1	1	yes	characteristic
52	C₂²×C₉₇	C₂²×C₈	1	2	yes	—
53	C₂²×C₉₇	C₂×M₁₆	1	2	yes	—
54	C₄×C₉₇	—	2	4	no	—
55	C₄×C₉₇	C₄×D₄	1	2	yes	—
56	C₄×C₉₇	C₄×Q₈	1	2	yes	—
57	Dic₉₇	—	2	2	no	—
58	Dic₉₇	C₄×C₈	1	4	yes	complemented
59	Dic₉₇	C₄⋊C₈	1	8	yes	complemented
60	C₂×C₉₇	C₄×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₉₇	C₂×C₄×C₈	1	1	yes	characteristic
62	C₂×C₉₇	C₄×M₁₆	1	1	yes	characteristic
63	C₂×C₉₇	C₂×C₄⋊C₈	1	2	yes	—
64	C₂×C₉₇	#431	1	2	yes	—
65	C₄×C₄⋊C₈	—	97	1	no	maximal, Sylow
66	C₉₇	C₄×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
67	C₂×C₄⋊C₈	—	97	2	no	—
68	C₂×C₄×C₈	—	97	2	no	—
69	C₄³	—	97	1	no	—
70	#336	—	97	1	no	—
71	#336	D₉₇	1	1	yes	characteristic
72	C₂²×C₈	—	97	2	no	—
73	C₂²×C₈	C₂×D₉₇	1	2	yes	—
74	C₂×C₄²	—	97	2	no	—
75	C₂×C₄²	—	97	1	no	—
76	C₂×C₄²	—	97	1	no	—
77	C₂×C₄²	—	97	2	no	—
78	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
79	C₄⋊C₈	—	97	8	no	—
80	C₄×C₈	—	194	4	no	—
81	C₂²×C₄	—	97	1	no	—
82	C₂²×C₄	—	97	2	no	—
83	C₂²×C₄	—	97	1	no	—
84	C₂²×C₄	C₄×D₉₇	1	2	yes	—
85	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	97	4	no	—
89	C₂×C₈	—	194	2	no	—
90	C₂×C₈	—	194	2	no	—
91	C₂×C₈	C₄×D₉₇	1	4	yes	—
92	C₄²	—	2	2	no	—
93	C₄²	—	97	8	no	—
94	C₄²	—	97	4	no	—
95	C₄²	—	97	2	no	—
96	C₄²	—	97	2	no	—
97	C₄²	—	194	4	no	—
98	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	97	2	no	—
102	C₂×C₄	—	97	4	no	—
103	C₂×C₄	—	97	4	no	—
104	C₂×C₄	—	97	4	no	—
105	C₂×C₄	—	97	4	no	—
106	C₂×C₄	—	97	2	no	—
107	C₂×C₄	—	194	2	no	—
108	C₂×C₄	C₈×D₉₇	1	4	yes	—
109	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
110	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
111	C₂×C₄	Q₈×D₉₇	1	1	yes	characteristic
112	C₂×C₄	#13095510	1	2	yes	—
113	C₈	—	2	4	no	—
114	C₈	—	194	4	no	—
115	C₂²	C₄²×D₉₇	1	1	yes	characteristic
116	C₂²	C₄⋊C₄×D₉₇	1	1	yes	characteristic
117	C₂²	#15690766	1	1	yes	characteristic
118	C₂²	C₂×C₈×D₉₇	1	2	yes	—
119	C₂²	M₁₆×D₉₇	1	2	yes	—
120	C₄	—	2	4	no	—
121	C₄	—	97	4	no	—
122	C₄	—	97	8	no	—
123	C₄	—	194	2	no	—
124	C₄	#15690770	1	2	yes	—
125	C₄	#15690775	1	2	yes	—
126	C₂	#20260586	1	1	yes	characteristic
127	C₂	C₄×C₈×D₉₇	1	1	yes	characteristic
128	C₂	#20260687	1	1	yes	characteristic
129	C₂	C₄⋊C₈×D₉₇	1	2	yes	—
130	C₂	#20260785	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690739

C₂

yes

maximal

C₂×C₈×Dic₉₇

C₂

yes

maximal

C₄²×Dic₉₇

C₂

yes

maximal, characteristic

#20260490

C₂

yes

maximal, characteristic

#336×C₉₇

C₂

yes

maximal, characteristic

C₂²×#11715895

C₂²

yes

—

C₂×C₄×Dic₉₇

C₄

yes

—

C₂×C₄×Dic₉₇

C₄

yes

—

C₂×C₄×Dic₉₇

C₂²

yes

characteristic

C₂×C₄×Dic₉₇

C₂²

yes

characteristic

#15690739

C₄

yes

complemented

C₈×Dic₉₇

—

C₂²×C₈×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂²×Dic₉₇

C₂×C₄

yes

characteristic

C₂²×Dic₉₇

C₂×C₄

yes

characteristic

C₂²×Dic₉₇

C₂×C₄

yes

—

C₂×#11715895

—

C₂×#11715895

—

C₂×#11715895

C₂×C₄

yes

—

C₄×Dic₉₇

—

C₄×Dic₉₇

C₈

yes

complemented

C₄×Dic₉₇

C₂×C₄

yes

—

C₄×Dic₉₇

D₄

yes

—

C₄×Dic₉₇

Q₈

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

—

C₂³×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×C₈

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

C₄²

yes

—

C₂×Dic₉₇

C₄²

yes

—

C₂×Dic₉₇

C₄⋊C₄

yes

—

C₂×Dic₉₇

C₂×C₈

yes

—

C₂×Dic₉₇

C₂×C₈

yes

—

C₂×Dic₉₇

M₁₆

yes

—

C₈×C₉₇

—

#11715895

—

C₂²×C₉₇

C₂×C₄²

yes

characteristic

C₂²×C₉₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₉₇

#117

yes

characteristic

C₂²×C₉₇

C₂²×C₈

yes

—

C₂²×C₉₇

C₂×M₁₆

yes

—

C₄×C₉₇

—

C₄×C₉₇

C₄×D₄

yes

—

C₄×C₉₇

C₄×Q₈

yes

—

Dic₉₇

—

Dic₉₇

C₄×C₈

yes

complemented

Dic₉₇

C₄⋊C₈

yes

complemented

C₂×C₉₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₉₇

C₂×C₄×C₈

yes

characteristic

C₂×C₉₇

C₄×M₁₆

yes

characteristic

C₂×C₉₇

C₂×C₄⋊C₈

yes

—

C₂×C₉₇

#431

yes

—

C₄×C₄⋊C₈

—

maximal, Sylow

C₉₇

C₄×C₄⋊C₈

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₄³

—

#336

—

#336

D₉₇

yes

characteristic

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

—

C₂²×C₄

C₄×D₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₂×C₈

C₄×D₉₇

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

194

—

C₂³

C₂×C₄×D₉₇

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

194

—

108

C₂×C₄

C₈×D₉₇

yes

—

109

C₂×C₄

C₂×C₄×D₉₇

yes

—

110

C₂×C₄

D₄×D₉₇

yes

characteristic

111

C₂×C₄

Q₈×D₉₇

yes

characteristic

112

C₂×C₄

#13095510

yes

—

113

C₈

—

114

C₈

—

194

—

115

C₂²

C₄²×D₉₇

yes

characteristic

116

C₂²

C₄⋊C₄×D₉₇

yes

characteristic

117

C₂²

#15690766

yes

characteristic

118

C₂²

C₂×C₈×D₉₇

yes

—

119

C₂²

M₁₆×D₉₇

yes

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

194

—

124

C₄

#15690770

yes

—

125

C₄

#15690775

yes

—

126

C₂

#20260586

yes

characteristic

127

C₂

C₄×C₈×D₉₇

yes

characteristic

128

C₂

#20260687

yes

characteristic

129

C₂

C₄⋊C₈×D₉₇

yes

—

130

C₂

#20260785

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₉₇
commutator	C₂×C₉₇	C₂×C₄×C₈
Fitting	#336×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂²×C₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#336	D₉₇
hypocenter	C₉₇	C₄×C₄⋊C₈
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₉₇

commutator

C₂×C₉₇

C₂×C₄×C₈

Fitting

#336×C₉₇

C₂

socle, prime radical

C₂³×C₉₇

C₂²×C₄

cosocle

C₂²×C₄×C₉₇

C₂³

hypercenter

#336

D₉₇

hypocenter

C₉₇

C₄×C₄⋊C₈

LCS

G ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
11. #15690739	C₄	semidirect
22. C₄×Dic₉₇	C₈	semidirect
58. Dic₉₇	C₄×C₈	semidirect
59. Dic₉₇	C₄⋊C₈	semidirect
66. C₉₇	C₄×C₄⋊C₈	semidirect

H ≅ G/N

type

11. #15690739

C₄

semidirect

22. C₄×Dic₉₇

C₈

semidirect

58. Dic₉₇

C₄×C₈

semidirect

59. Dic₉₇

C₄⋊C₈

semidirect

66. C₉₇

C₄×C₄⋊C₈

semidirect

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

subgroup	index
C₂×#15690739	2
C₂×C₈×Dic₉₇	2
C₄²×Dic₉₇	2
#20260490	2
#336×C₉₇	2
C₄×C₄⋊C₈	97

subgroup

index

C₂×#15690739

C₂×C₈×Dic₉₇

C₄²×Dic₉₇

#20260490

#336×C₉₇

C₄×C₄⋊C₈

Subgroups — #27141584

Conjugacy classes of subgroups

Complements

Maximal subgroups