Group #27143785 — subgroups

Subgroups — #27143785

Order 12416.

154 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 49 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×Dic₇₇₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#15690756	C₂	1	1	yes	maximal, characteristic
3	#20261113	C₂	1	1	yes	maximal, characteristic
4	#20260829	C₂	1	1	yes	maximal, characteristic
5	#20260553	C₂	1	1	yes	maximal, characteristic
6	#20260538	C₂	1	1	yes	maximal, characteristic
7	#358×C₉₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₃₈₈	C₂²	1	1	yes	characteristic
9	C₂×Dic₃₈₈	C₂²	1	1	yes	characteristic
10	C₂×#13095501	—	2	1	no	—
11	#15690808	—	2	1	no	—
12	#15690775	—	2	1	no	—
13	Dic₇₇₆	—	2	2	no	—
14	#15690756	—	2	2	no	—
15	C₂×#13095486	C₂²	1	1	yes	characteristic
16	#15690733	—	2	1	no	—
17	C₈×Dic₉₇	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₉₇	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×C₉₇	C₂²	1	1	yes	characteristic
20	#107×C₉₇	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₉₄	—	2	1	no	—
22	C₂×Dic₁₉₄	—	2	1	no	—
23	Dic₃₈₈	—	2	1	no	—
24	Dic₃₈₈	—	2	1	no	—
25	Dic₃₈₈	—	2	1	no	—
26	Dic₃₈₈	D₄	1	2	yes	—
27	#13095501	—	4	2	no	—
28	C₂×#11715895	D₄	1	1	yes	characteristic
29	#13095490	—	4	1	no	—
30	#13095490	—	4	1	no	—
31	#13095490	—	4	1	no	—
32	C₄×Dic₉₇	—	4	1	no	—
33	C₄×Dic₉₇	D₄	1	1	yes	characteristic
34	#13095486	—	2	1	no	—
35	C₂×Q₈×C₉₇	—	2	1	no	—
36	Q₁₆×C₉₇	—	2	1	no	—
37	Q₁₆×C₉₇	—	2	1	no	—
38	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₉₇	—	2	1	no	—
40	C₁₆×C₉₇	—	2	1	no	—
41	Q₈×C₉₇	—	4	1	no	—
42	Q₈×C₉₇	—	4	1	no	—
43	C₂×C₄×C₉₇	—	4	1	no	—
44	C₂×C₄×C₉₇	—	4	1	no	—
45	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
46	C₂×Dic₉₇	—	4	1	no	—
47	C₂×Dic₉₇	—	4	1	no	—
48	C₂×Dic₉₇	D₈	1	1	yes	characteristic
49	C₂×Dic₉₇	D₈	1	1	yes	characteristic
50	Dic₁₉₄	—	2	2	no	—
51	Dic₁₉₄	—	4	1	no	—
52	Dic₁₉₄	—	4	1	no	—
53	Dic₁₉₄	—	4	1	no	—
54	C₈×C₉₇	C₂×D₄	1	1	yes	characteristic
55	C₈×C₉₇	#41	1	1	yes	characteristic
56	#11715895	—	2	1	no	—
57	C₂²×C₉₇	C₂×D₈	1	1	yes	characteristic
58	C₄×C₉₇	—	8	1	no	—
59	C₄×C₉₇	—	8	1	no	—
60	C₄×C₉₇	#121	1	1	yes	characteristic
61	C₄×C₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
62	Dic₉₇	—	2	1	no	—
63	Dic₉₇	—	4	2	no	—
64	Dic₉₇	—	8	1	no	—
65	Dic₉₇	Q₃₂	1	2	yes	complemented
66	C₂×C₉₇	#459	1	1	yes	characteristic
67	C₂×C₉₇	C₂×Q₃₂	1	1	yes	characteristic
68	C₂×C₉₇	#510	1	1	yes	characteristic
69	#2209	—	97	1	no	maximal, Sylow
70	C₉₇	#2209	1	1	yes	characteristic, complemented, Sylow
71	C₂×Q₃₂	—	97	1	no	—
72	C₂×Q₃₂	—	97	1	no	—
73	#494	—	97	1	no	—
74	C₄×Q₁₆	—	97	1	no	—
75	C₄⋊C₁₆	—	97	1	no	—
76	#358	—	97	1	no	—
77	#358	D₉₇	1	1	yes	characteristic
78	C₂×Q₁₆	—	97	1	no	—
79	C₂×Q₁₆	—	97	1	no	—
80	C₂×Q₁₆	—	194	1	no	—
81	C₂×Q₁₆	C₂×D₉₇	1	1	yes	characteristic
82	C₄⋊Q₈	—	194	1	no	—
83	C₄×Q₈	—	194	1	no	—
84	Q₃₂	—	194	2	no	—
85	Q₃₂	—	194	2	no	—
86	C₂×C₁₆	—	97	1	no	—
87	C₂×C₁₆	C₂×D₉₇	1	1	yes	characteristic
88	#107	C₂×D₉₇	1	1	yes	characteristic
89	#103	—	194	1	no	—
90	C₄×C₈	—	97	1	no	—
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	194	1	no	—
93	C₂×Q₈	—	194	1	no	—
94	Q₁₆	—	2	1	no	—
95	Q₁₆	—	2	1	no	—
96	Q₁₆	—	97	2	no	—
97	Q₁₆	—	388	2	no	—
98	Q₁₆	—	194	1	no	—
99	Q₁₆	—	194	1	no	—
100	Q₁₆	—	194	1	no	—
101	C₂×C₈	—	97	1	no	—
102	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	388	1	no	—
105	C₄⋊C₄	—	388	1	no	—
106	C₄⋊C₄	—	388	1	no	—
107	C₄²	—	97	1	no	—
108	C₄²	—	388	1	no	—
109	C₁₆	—	2	1	no	—
110	C₁₆	—	194	1	no	—
111	Q₈	—	4	1	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	388	1	no	—
114	Q₈	—	388	1	no	—
115	Q₈	—	388	1	no	—
116	Q₈	—	194	2	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	97	1	no	—
120	C₂×C₄	—	97	1	no	—
121	C₂×C₄	—	388	1	no	—
122	C₂×C₄	—	388	1	no	—
123	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
124	C₈	—	194	1	no	—
125	C₈	D₄×D₉₇	1	1	yes	characteristic
126	C₈	#13095511	1	1	yes	characteristic
127	C₂²	D₈×D₉₇	1	1	yes	characteristic
128	C₄	—	8	1	no	—
129	C₄	—	8	1	no	—
130	C₄	—	776	1	no	—
131	C₄	—	97	2	no	—
132	C₄	—	388	2	no	—
133	C₄	—	194	1	no	—
134	C₄	#15690783	1	1	yes	characteristic
135	C₄	#15690837	1	1	yes	characteristic
136	C₂	#20260929	1	1	yes	characteristic
137	C₂	Q₃₂×D₉₇	1	1	yes	characteristic
138	C₂	#20261174	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₇₇₆

C₂

yes

maximal, characteristic

C₂×#15690756

C₂

yes

maximal, characteristic

#20261113

C₂

yes

maximal, characteristic

#20260829

C₂

yes

maximal, characteristic

#20260553

C₂

yes

maximal, characteristic

#20260538

C₂

yes

maximal, characteristic

#358×C₉₇

C₂

yes

maximal, characteristic

C₂×Dic₃₈₈

C₂²

yes

characteristic

C₂×Dic₃₈₈

C₂²

yes

—

—

—

Dic₇₇₆

—

#15690756

—

C₂×#13095486

C₂²

yes

characteristic

#15690733

—

C₈×Dic₉₇

C₂²

yes

characteristic

C₂×Q₁₆×C₉₇

C₂²

yes

characteristic

C₂×C₁₆×C₉₇

C₂²

yes

characteristic

#107×C₉₇

C₂²

yes

characteristic

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

Dic₃₈₈

—

Dic₃₈₈

—

Dic₃₈₈

—

Dic₃₈₈

D₄

yes

—

#13095501

—

C₂×#11715895

D₄

yes

—

—

—

C₄×Dic₉₇

—

C₄×Dic₉₇

D₄

yes

characteristic

#13095486

—

C₂×Q₈×C₉₇

—

Q₁₆×C₉₇

—

Q₁₆×C₉₇

—

C₂×C₈×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₁₆×C₉₇

—

Q₈×C₉₇

—

Q₈×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

D₈

yes

characteristic

C₂×Dic₉₇

D₈

yes

characteristic

Dic₁₉₄

—

Dic₁₉₄

—

Dic₁₉₄

—

Dic₁₉₄

—

C₈×C₉₇

C₂×D₄

yes

C₈×C₉₇

yes

—

C₂²×C₉₇

C₂×D₈

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

yes

C₄×C₉₇

yes

Dic₉₇

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

Q₃₂

yes

complemented

C₂×C₉₇

#459

yes

characteristic

C₂×C₉₇

C₂×Q₃₂

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

#494

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

#358

—

#358

D₉₇

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

194

—

C₂×Q₁₆

C₂×D₉₇

yes

characteristic

C₄⋊Q₈

—

194

—

C₄×Q₈

—

194

—

Q₃₂

—

194

—

Q₃₂

—

194

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₉₇

yes

characteristic

#107

C₂×D₉₇

yes

characteristic

#103

—

194

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

194

—

C₂×Q₈

—

194

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

388

—

Q₁₆

—

194

—

Q₁₆

—

194

—

100

Q₁₆

—

194

—

101

C₂×C₈

—

102

C₂×C₈

C₂²×D₉₇

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

388

—

105

C₄⋊C₄

—

388

—

106

C₄⋊C₄

—

388

—

107

C₄²

—

108

C₄²

—

388

—

109

C₁₆

—

110

C₁₆

—

194

—

111

Q₈

—

112

Q₈

—

113

Q₈

—

388

—

114

Q₈

—

388

—

115

Q₈

—

388

—

116

Q₈

—

194

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

388

—

122

C₂×C₄

—

388

—

123

C₂×C₄

D₄×D₉₇

yes

characteristic

124

C₈

—

194

—

125

C₈

D₄×D₉₇

yes

characteristic

126

C₈

#13095511

yes

characteristic

127

C₂²

D₈×D₉₇

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

776

—

131

C₄

—

132

C₄

—

388

—

133

C₄

—

194

—

134

C₄

#15690783

yes

characteristic

135

C₄

#15690837

yes

characteristic

136

C₂

#20260929

yes

characteristic

137

C₂

Q₃₂×D₉₇

yes

characteristic

138

C₂

#20261174

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₉₇
commutator, cosocle	C₂×C₈×C₉₇	C₂³
Frattini	C₂×C₈	C₂²×D₉₇
Fitting	#358×C₉₇	C₂
socle, prime radical	C₂²×C₉₇	C₂×D₈
hypercenter	#358	D₉₇
hypocenter	C₉₇	#2209
LCS	G ▷ C₂×C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₉₇

commutator, cosocle

C₂×C₈×C₉₇

C₂³

Frattini

C₂×C₈

C₂²×D₉₇

Fitting

#358×C₉₇

C₂

socle, prime radical

C₂²×C₉₇

C₂×D₈

D₉₇

C₉₇

G ▷ C₂×C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
65. Dic₉₇	Q₃₂	semidirect
70. C₉₇	#2209	semidirect

H ≅ G/N

type

65. Dic₉₇

Q₃₂

semidirect

70. C₉₇

#2209

semidirect

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

subgroup	index
C₂×Dic₇₇₆	2
C₂×#15690756	2
#20261113	2
#20260829	2
#20260553	2
#20260538	2
#358×C₉₇	2
#2209	97

subgroup

index

C₂×Dic₇₇₆

#358×C₉₇

Subgroups — #27143785

Conjugacy classes of subgroups

Complements

Maximal subgroups