Group #27142671 — subgroups

Subgroups — #27142671

Order 12416.

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

normal subgroups only

This group has 292 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×Dic₁₉₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11715895	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×Dic₁₉₄	C₄	1	1	yes	characteristic
6	C₂²×#11715895	C₂²	1	2	yes	—
7	C₄×Dic₁₉₄	—	2	2	no	—
8	C₄×Dic₁₉₄	—	2	2	no	—
9	C₂×C₄⋊Dic₉₇	C₄	1	1	yes	characteristic
10	C₂×#13095490	C₄	1	2	yes	—
11	C₂×C₄×Dic₉₇	C₂²	1	2	yes	—
12	C₄×#11715895	—	2	2	no	—
13	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₉₇	—	2	2	no	—
16	C₂×Dic₁₉₄	—	2	2	no	—
17	C₂×Dic₁₉₄	—	2	2	no	—
18	C₂×Dic₁₉₄	C₈	1	4	yes	complemented
19	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
20	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
21	C₂×#11715895	—	2	2	no	—
22	C₂×#11715895	—	2	2	no	—
23	C₂×#11715895	Q₈	1	4	yes	complemented
24	C₄⋊Dic₉₇	—	2	2	no	—
25	#13095490	—	2	4	no	—
26	C₄×Dic₉₇	—	2	2	no	—
27	C₄×Dic₉₇	—	2	2	no	—
28	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₉₇	—	2	2	no	—
32	C₂×C₈×C₉₇	—	2	4	no	—
33	C₄²×C₉₇	D₄	1	2	yes	—
34	C₄²×C₉₇	D₄	1	2	yes	—
35	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₉₇	—	2	2	no	—
37	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₉₇	C₂×C₈	1	2	yes	—
40	C₂×C₄×C₉₇	M₁₆	1	2	yes	—
41	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₉₇	#41	1	2	yes	—
45	C₂×Dic₉₇	—	2	2	no	—
46	C₂×Dic₉₇	—	2	2	no	—
47	C₂×Dic₉₇	—	2	2	no	—
48	C₂×Dic₉₇	—	2	4	no	—
49	C₂×Dic₉₇	C₂×C₈	1	4	yes	—
50	Dic₁₉₄	—	2	8	no	—
51	C₈×C₉₇	—	4	2	no	—
52	#11715895	—	2	4	no	—
53	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₉₇	C₄×Q₈	1	2	yes	—
55	C₂²×C₉₇	C₂²×C₈	1	1	yes	characteristic
56	C₂²×C₉₇	C₂×M₁₆	1	1	yes	characteristic
57	C₂²×C₉₇	#131	1	2	yes	—
58	C₄×C₉₇	—	2	2	no	—
59	C₄×C₉₇	C₂²⋊C₈	1	4	yes	—
60	C₄×C₉₇	C₂²⋊Q₈	1	2	yes	—
61	C₄×C₉₇	#124	1	1	yes	characteristic
62	C₄×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
63	Dic₉₇	—	2	4	no	—
64	Dic₉₇	—	4	2	no	—
65	C₂×C₉₇	#391	1	1	yes	characteristic
66	C₂×C₉₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
67	C₂×C₉₇	#408	1	1	yes	characteristic
68	C₂×C₉₇	C₈×Q₈	1	2	yes	—
69	C₂×C₉₇	#446	1	2	yes	—
70	#1984	—	97	1	no	maximal, Sylow
71	C₉₇	#1984	1	1	yes	characteristic, complemented, Sylow
72	C₂×C₄×Q₈	—	97	1	no	—
73	C₂×C₄⋊C₈	D₉₇	1	1	yes	characteristic
74	C₂×C₄×C₈	—	97	1	no	—
75	#336	—	97	4	no	—
76	C₂²×Q₈	—	97	1	no	—
77	C₂²×C₈	—	97	2	no	—
78	C₂²×C₈	C₂×D₉₇	1	2	yes	—
79	C₄×Q₈	—	194	2	no	—
80	C₄×Q₈	—	194	2	no	—
81	C₂×C₄⋊C₄	—	97	2	no	—
82	C₂×C₄⋊C₄	—	97	1	no	—
83	C₂×C₄²	—	97	2	no	—
84	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
85	C₄⋊C₈	—	2	2	no	—
86	C₄×C₈	—	194	2	no	—
87	C₂×Q₈	—	97	4	no	—
88	C₂×Q₈	—	194	2	no	—
89	C₂×Q₈	—	194	2	no	—
90	C₂²×C₄	—	97	2	no	—
91	C₂²×C₄	—	97	2	no	—
92	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
93	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
94	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	4	no	—
97	C₂×C₈	—	97	4	no	—
98	C₂×C₈	—	194	2	no	—
99	C₂×C₈	—	194	2	no	—
100	C₄⋊C₄	—	194	4	no	—
101	C₄⋊C₄	—	194	2	no	—
102	C₄²	—	194	2	no	—
103	C₄²	—	194	2	no	—
104	C₄²	D₃₈₈	1	2	yes	—
105	C₄²	C₂²⋊D₉₇	1	2	yes	—
106	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
107	Q₈	—	194	8	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	97	4	no	—
110	C₂×C₄	—	194	2	no	—
111	C₂×C₄	—	194	4	no	—
112	C₂×C₄	—	194	2	no	—
113	C₂×C₄	—	194	2	no	—
114	C₂×C₄	#13095488	1	2	yes	—
115	C₂×C₄	#13095488	1	2	yes	—
116	C₂×C₄	C₈×D₉₇	1	2	yes	—
117	C₂×C₄	#13095494	1	2	yes	—
118	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
119	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
120	C₂×C₄	Q₈×D₉₇	1	2	yes	—
121	C₂×C₄	#13095510	1	2	yes	—
122	C₈	—	4	2	no	—
123	C₈	—	194	4	no	—
124	C₂²	C₂×#13095488	1	1	yes	characteristic
125	C₂²	#15690775	1	2	yes	—
126	C₂²	C₂×C₈×D₉₇	1	1	yes	characteristic
127	C₂²	C₂×#13095494	1	1	yes	characteristic
128	C₂²	#15690818	1	2	yes	—
129	C₄	—	2	2	no	—
130	C₄	—	388	2	no	—
131	C₄	—	194	4	no	—
132	C₄	#15690722	1	4	yes	—
133	C₄	#15690790	1	2	yes	—
134	C₄	#15690797	1	1	yes	characteristic
135	C₄	#15690808	1	1	yes	characteristic
136	C₂	#20260640	1	1	yes	characteristic
137	C₂	C₂×#15690722	1	1	yes	characteristic
138	C₂	#20260703	1	1	yes	characteristic
139	C₂	#20260859	1	2	yes	—
140	C₂	#20260862	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₉₄

C₂

yes

maximal, characteristic

C₂×C₄×#11715895

C₂

yes

maximal, characteristic

#20260489

C₂

yes

maximal

C₂×C₄⋊C₈×C₉₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₉₄

C₄

yes

characteristic

C₂²×#11715895

C₂²

yes

—

C₄×Dic₁₉₄

—

C₄×Dic₁₉₄

—

C₂×C₄⋊Dic₉₇

C₄

yes

characteristic

C₂×#13095490

C₄

yes

—

C₂×C₄×Dic₉₇

C₂²

yes

—

C₄×#11715895

—

C₂²×C₈×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₉₇

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

C₈

yes

complemented

C₂²×Dic₉₇

C₂×C₄

yes

—

C₂²×Dic₉₇

C₂×C₄

yes

—

C₂×#11715895

—

C₂×#11715895

—

C₂×#11715895

Q₈

yes

complemented

C₄⋊Dic₉₇

—

#13095490

—

C₄×Dic₉₇

—

C₄×Dic₉₇

—

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

D₄

yes

—

C₂³×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂×C₈

yes

—

C₂×C₄×C₉₇

M₁₆

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

—

C₂×C₄×C₉₇

#41

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

C₂×C₈

yes

—

Dic₁₉₄

—

C₈×C₉₇

—

#11715895

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₉₇

C₄×Q₈

yes

—

C₂²×C₉₇

C₂²×C₈

yes

characteristic

C₂²×C₉₇

C₂×M₁₆

yes

characteristic

C₂²×C₉₇

#131

yes

—

C₄×C₉₇

—

C₄×C₉₇

C₂²⋊C₈

yes

—

C₄×C₉₇

C₂²⋊Q₈

yes

—

C₄×C₉₇

#124

yes

characteristic

C₄×C₉₇

C₄⋊Q₈

yes

characteristic

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

#391

yes

characteristic

C₂×C₉₇

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₉₇

#408

yes

characteristic

C₂×C₉₇

C₈×Q₈

yes

—

C₂×C₉₇

#446

yes

—

#1984

—

maximal, Sylow

C₉₇

#1984

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

D₉₇

yes

characteristic

C₂×C₄×C₈

—

#336

—

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₉₇

yes

—

C₄×Q₈

—

194

—

C₄×Q₈

—

194

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

C₄×C₈

—

194

—

C₂×Q₈

—

C₂×Q₈

—

194

—

C₂×Q₈

—

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

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

100

C₄⋊C₄

—

194

—

101

C₄⋊C₄

—

194

—

102

C₄²

—

194

—

103

C₄²

—

194

—

104

C₄²

D₃₈₈

yes

—

105

C₄²

C₂²⋊D₉₇

yes

—

106

C₂³

C₂×C₄×D₉₇

yes

characteristic

107

Q₈

—

194

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

194

—

111

C₂×C₄

—

194

—

112

C₂×C₄

—

194

—

113

C₂×C₄

—

194

—

114

C₂×C₄

#13095488

yes

—

115

C₂×C₄

#13095488

yes

—

116

C₂×C₄

C₈×D₉₇

yes

—

117

C₂×C₄

#13095494

yes

—

118

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

119

C₂×C₄

C₂×D₃₈₈

yes

characteristic

120

C₂×C₄

Q₈×D₉₇

yes

—

121

C₂×C₄

#13095510

yes

—

122

C₈

—

123

C₈

—

194

—

124

C₂²

C₂×#13095488

yes

characteristic

125

C₂²

#15690775

yes

—

126

C₂²

C₂×C₈×D₉₇

yes

characteristic

127

C₂²

C₂×#13095494

yes

characteristic

128

C₂²

#15690818

yes

—

129

C₄

—

130

C₄

—

388

—

131

C₄

—

194

—

132

C₄

#15690722

yes

—

133

C₄

#15690790

yes

—

134

C₄

#15690797

yes

characteristic

135

C₄

#15690808

yes

characteristic

136

C₂

#20260640

yes

characteristic

137

C₂

C₂×#15690722

yes

characteristic

138

C₂

#20260703

yes

characteristic

139

C₂

#20260859

yes

—

140

C₂

#20260862

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	#1984
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
18. C₂×Dic₁₉₄	C₈	semidirect
23. C₂×#11715895	Q₈	semidirect
71. C₉₇	#1984	semidirect

H ≅ G/N

type

18. C₂×Dic₁₉₄

C₈

semidirect

23. C₂×#11715895

Q₈

semidirect

71. C₉₇

#1984

semidirect

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

subgroup	index
C₂×C₄×Dic₁₉₄	2
C₂×C₄×#11715895	2
#20260489	2
C₂×C₄⋊C₈×C₉₇	2
#1984	97

subgroup

index

C₂×C₄×Dic₁₉₄

C₂×C₄×#11715895

#20260489

C₂×C₄⋊C₈×C₉₇

#1984

Subgroups — #27142671

Conjugacy classes of subgroups

Complements

Maximal subgroups