Group #27141590 — subgroups

Subgroups — #27141590

Order 12416.

292 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 141 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×#15690721	C₂	1	1	yes	maximal, characteristic
3	C₄²×Dic₉₇	C₂	1	1	yes	maximal, characteristic
4	#20260489	C₂	1	2	yes	maximal
5	C₂×#97×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	#15690721	—	2	2	no	—
13	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
15	#97×C₉₇	—	2	2	no	—
16	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
18	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
19	C₂×#11715895	—	2	2	no	—
20	C₂×#11715895	—	2	2	no	—
21	C₂×#11715895	C₂×C₄	1	4	yes	—
22	C₄×Dic₉₇	—	2	2	no	—
23	C₄×Dic₉₇	—	2	4	no	—
24	C₄×Dic₉₇	—	2	4	no	—
25	C₄×Dic₉₇	C₂×C₄	1	4	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	D₄	1	2	yes	—
32	C₄²×C₉₇	Q₈	1	2	yes	—
33	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₉₇	—	2	2	no	—
35	C₂×C₄×C₉₇	—	2	2	no	—
36	C₂×C₄×C₉₇	C₄⋊C₄	1	4	yes	—
37	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₉₇	#41	1	2	yes	—
41	C₂×Dic₉₇	—	2	2	no	—
42	C₂×Dic₉₇	—	2	2	no	—
43	C₂×Dic₉₇	—	2	2	no	—
44	C₂×Dic₉₇	C₄²	1	2	yes	—
45	C₂×Dic₉₇	C₄²	1	2	yes	—
46	C₂×Dic₉₇	M₁₆	1	4	yes	—
47	C₂×Dic₉₇	M₁₆	1	4	yes	—
48	C₈×C₉₇	—	2	4	no	—
49	#11715895	—	2	4	no	—
50	C₂²×C₉₇	C₂×C₄²	1	1	yes	characteristic
51	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₉₇	#117	1	1	yes	characteristic
53	C₂²×C₉₇	C₂×M₁₆	1	2	yes	—
54	C₂²×C₉₇	C₂×M₁₆	1	2	yes	—
55	C₄×C₉₇	—	2	4	no	—
56	C₄×C₉₇	C₄×D₄	1	2	yes	—
57	C₄×C₉₇	C₄×Q₈	1	2	yes	—
58	Dic₉₇	—	2	2	no	—
59	Dic₉₇	—	2	4	no	—
60	Dic₉₇	#97	1	4	yes	complemented
61	C₂×C₉₇	C₄×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₉₇	C₂×#97	1	1	yes	characteristic
63	C₂×C₉₇	C₄×M₁₆	1	1	yes	characteristic
64	C₂×C₉₇	#423	1	2	yes	—
65	C₂×C₉₇	#432	1	2	yes	—
66	#1767	—	97	1	no	maximal, Sylow
67	C₉₇	#1767	1	1	yes	characteristic, complemented, Sylow
68	C₂×C₄⋊C₈	—	97	2	no	—
69	C₂×#97	—	97	1	no	—
70	C₂×#97	D₉₇	1	1	yes	characteristic
71	C₄³	—	97	1	no	—
72	#336	—	97	2	no	—
73	C₂²×C₈	—	97	2	no	—
74	C₂²×C₈	C₂×D₉₇	1	2	yes	—
75	C₂×C₄²	—	97	1	no	—
76	C₂×C₄²	—	97	2	no	—
77	C₂×C₄²	—	97	2	no	—
78	C₂×C₄²	—	97	1	no	—
79	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
80	C₄⋊C₈	—	97	8	no	—
81	#97	—	2	2	no	—
82	#97	—	194	2	no	—
83	C₂²×C₄	—	97	1	no	—
84	C₂²×C₄	—	97	1	no	—
85	C₂²×C₄	—	97	2	no	—
86	C₂²×C₄	C₄×D₉₇	1	2	yes	—
87	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	97	4	no	—
91	C₂×C₈	—	194	2	no	—
92	C₂×C₈	—	194	2	no	—
93	C₂×C₈	C₄×D₉₇	1	4	yes	—
94	C₄²	—	97	4	no	—
95	C₄²	—	194	4	no	—
96	C₄²	—	194	4	no	—
97	C₄²	—	194	2	no	—
98	C₄²	Dic₁₉₄	1	2	yes	—
99	C₄²	C₂²⋊D₉₇	1	2	yes	—
100	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	97	2	no	—
104	C₂×C₄	—	97	4	no	—
105	C₂×C₄	—	97	2	no	—
106	C₂×C₄	—	97	4	no	—
107	C₂×C₄	—	194	2	no	—
108	C₂×C₄	—	194	2	no	—
109	C₂×C₄	—	194	2	no	—
110	C₂×C₄	#13095490	1	4	yes	—
111	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
113	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
114	C₂×C₄	#13095511	1	2	yes	—
115	C₈	—	2	4	no	—
116	C₈	—	194	4	no	—
117	C₂²	C₄²×D₉₇	1	1	yes	characteristic
118	C₂²	C₂×#13095490	1	1	yes	characteristic
119	C₂²	#15690767	1	1	yes	characteristic
120	C₂²	M₁₆×D₉₇	1	2	yes	—
121	C₂²	M₁₆×D₉₇	1	2	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	97	4	no	—
124	C₄	—	194	2	no	—
125	C₄	—	194	4	no	—
126	C₄	C₄×C₂²⋊D₉₇	1	2	yes	—
127	C₄	C₄×Dic₁₉₄	1	2	yes	—
128	C₂	C₄×#13095490	1	1	yes	characteristic
129	C₂	#97×D₉₇	1	1	yes	characteristic
130	C₂	#20260687	1	1	yes	characteristic
131	C₂	#20260758	1	2	yes	—
132	C₂	#20260790	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690739

C₂

yes

maximal

C₂×#15690721

C₂

yes

maximal, characteristic

C₄²×Dic₉₇

C₂

yes

maximal, characteristic

#20260489

C₂

yes

maximal

C₂×#97×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

#15690721

—

C₂²×C₈×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

#97×C₉₇

—

C₂²×Dic₉₇

C₂×C₄

yes

characteristic

C₂²×Dic₉₇

C₂×C₄

yes

—

C₂²×Dic₉₇

C₂×C₄

yes

—

—

C₂×C₄

yes

—

C₄×Dic₉₇

—

C₄×Dic₉₇

—

C₄×Dic₉₇

—

C₄×Dic₉₇

C₂×C₄

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

D₄

yes

—

C₄²×C₉₇

Q₈

yes

—

C₂³×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₄⋊C₄

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

—

C₂×Dic₉₇

—

C₂×Dic₉₇

C₄²

yes

—

C₂×Dic₉₇

C₄²

yes

—

C₂×Dic₉₇

M₁₆

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

C₂×M₁₆

yes

—

C₂²×C₉₇

C₂×M₁₆

yes

—

C₄×C₉₇

—

C₄×C₉₇

C₄×D₄

yes

—

C₄×C₉₇

C₄×Q₈

yes

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

#97

yes

complemented

C₂×C₉₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₉₇

C₂×#97

yes

characteristic

C₂×C₉₇

C₄×M₁₆

yes

characteristic

C₂×C₉₇

#423

yes

—

C₂×C₉₇

#432

yes

—

#1767

—

maximal, Sylow

C₉₇

#1767

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×#97

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

—

#97

—

#97

—

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

—

194

—

C₄²

—

194

—

C₄²

—

194

—

C₄²

Dic₁₉₄

yes

—

C₄²

C₂²⋊D₉₇

yes

—

100

C₂³

C₂×C₄×D₉₇

yes

characteristic

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

—

194

—

109

C₂×C₄

—

194

—

110

C₂×C₄

#13095490

yes

—

111

C₂×C₄

C₂×C₄×D₉₇

yes

—

112

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

113

C₂×C₄

C₂×Dic₁₉₄

yes

characteristic

114

C₂×C₄

#13095511

yes

—

115

C₈

—

116

C₈

—

194

—

117

C₂²

C₄²×D₉₇

yes

characteristic

118

C₂²

C₂×#13095490

yes

characteristic

119

C₂²

#15690767

yes

characteristic

120

C₂²

M₁₆×D₉₇

yes

—

121

C₂²

M₁₆×D₉₇

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

194

—

125

C₄

—

194

—

126

C₄

C₄×C₂²⋊D₉₇

yes

—

127

C₄

C₄×Dic₁₉₄

yes

—

128

C₂

C₄×#13095490

yes

characteristic

129

C₂

#97×D₉₇

yes

characteristic

130

C₂

#20260687

yes

characteristic

131

C₂

#20260758

yes

—

132

C₂

#20260790

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₉₇
commutator	C₂²×C₉₇	C₂×C₄²
Fitting	C₂×#97×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂²×C₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂×#97	D₉₇
hypocenter	C₉₇	#1767
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₉₇

commutator

C₂²×C₉₇

C₂×C₄²

Fitting

C₂×#97×C₉₇

C₂

socle, prime radical

C₂³×C₉₇

C₂²×C₄

cosocle

C₂²×C₄×C₉₇

C₂³

hypercenter

C₂×#97

D₉₇

C₉₇

G ▷ C₂²×C₉₇ ▷ C₉₇

C₁ ◁ C₂²×C₄ ◁ C₂×#97

N	H ≅ G/N	type
11. #15690739	C₄	semidirect
60. Dic₉₇	#97	semidirect
67. C₉₇	#1767	semidirect

H ≅ G/N

type

11. #15690739

C₄

semidirect

60. Dic₉₇

#97

semidirect

67. C₉₇

#1767

semidirect

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

subgroup	index
C₂×#15690739	2
C₂×#15690721	2
C₄²×Dic₉₇	2
#20260489	2
C₂×#97×C₉₇	2
#1767	97

subgroup

index

C₄²×Dic₉₇

C₂×#97×C₉₇

Subgroups — #27141590

Conjugacy classes of subgroups

Complements

Maximal subgroups