Group #27142568 — subgroups

Subgroups — #27142568

Order 12416.

320 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 93 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	#20261285	C₂	1	1	yes	maximal, characteristic
2	#20260759	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15690738	C₂	1	4	yes	maximal, complemented
4	#376×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13095511	C₂²	1	2	yes	—
6	C₂×#13095495	C₂²	1	2	yes	—
7	#15690810	C₄	1	1	yes	characteristic, complemented
8	#15690807	C₄	1	1	yes	characteristic, complemented
9	#15690798	C₄	1	2	yes	complemented
10	#15690784	—	2	2	no	—
11	C₄×D₃₈₈	—	2	2	no	—
12	C₄⋊#11715895	—	2	2	no	—
13	#15690738	—	2	8	no	—
14	C₂×C₄²×C₉₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
16	#95×C₉₇	—	2	2	no	—
17	#13095511	—	2	4	no	—
18	#13095511	—	2	4	no	—
19	C₂×Dic₁₉₄	C₂×C₄	1	2	yes	—
20	C₂×D₃₈₈	—	2	2	no	—
21	C₂×D₃₈₈	C₂×C₄	1	2	yes	—
22	C₂×C₂²⋊D₉₇	—	2	2	no	—
23	C₂×C₄×D₉₇	—	2	2	no	—
24	#13095495	—	2	4	no	—
25	C₂×#11715895	—	2	2	no	—
26	C₄⋊Dic₉₇	—	2	2	no	—
27	#13095488	—	2	4	no	—
28	C₂²×C₄×C₉₇	—	2	2	no	—
29	C₂²×C₄×C₉₇	D₄	1	2	yes	—
30	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
31	C₄²×C₉₇	—	2	2	no	—
32	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₉₇	D₄	1	2	yes	—
35	C₄²×C₉₇	D₄	1	4	yes	complemented
36	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×D₉₇	—	2	2	no	—
38	C₂×C₄×C₉₇	—	2	4	no	—
39	C₂×C₄×C₉₇	—	4	2	no	—
40	C₂×C₄×C₉₇	—	4	4	no	—
41	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₉₇	#41	1	2	yes	—
47	C₂²⋊D₉₇	—	4	4	no	—
48	C₂×Dic₉₇	—	2	2	no	—
49	D₃₈₈	—	2	4	no	—
50	D₃₈₈	—	4	4	no	—
51	C₄×D₉₇	—	4	2	no	—
52	C₄×D₉₇	—	4	2	no	—
53	Dic₁₉₄	—	2	4	no	—
54	#11715895	—	4	2	no	—
55	C₂²×C₉₇	—	2	2	no	—
56	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₉₇	#121	1	2	yes	—
58	C₂×D₉₇	—	4	2	no	—
59	C₂×D₉₇	—	4	4	no	—
60	C₄×C₉₇	—	2	2	no	—
61	C₄×C₉₇	—	2	2	no	—
62	C₄×C₉₇	—	4	4	no	—
63	C₄×C₉₇	C₄×D₄	1	2	yes	—
64	C₄×C₉₇	#124	1	1	yes	characteristic
65	C₄×C₉₇	#127	1	1	yes	characteristic
66	Dic₉₇	—	4	2	no	—
67	C₂×C₉₇	—	2	2	no	—
68	C₂×C₉₇	#390	1	1	yes	characteristic
69	C₂×C₉₇	#421	1	2	yes	—
70	D₉₇	—	8	2	no	—
71	#1963	—	97	1	no	maximal, Sylow
72	C₉₇	#1963	1	1	yes	characteristic, complemented, Sylow
73	#532	—	97	1	no	—
74	#423	—	97	1	no	—
75	C₂×#104	—	97	4	no	—
76	#376	D₉₇	1	1	yes	characteristic, complemented
77	C₂×#41	—	97	2	no	—
78	C₂×M₁₆	—	97	2	no	—
79	C₄⋊Q₈	—	97	1	no	—
80	#127	—	97	1	no	—
81	#124	—	97	2	no	—
82	#121	—	194	2	no	—
83	C₄×D₄	—	194	2	no	—
84	C₂×C₄²	C₂×D₉₇	1	2	yes	—
85	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
86	C₄⋊C₈	—	194	2	no	—
87	#104	—	194	8	no	—
88	#95	—	2	2	no	—
89	#41	—	194	4	no	—
90	#41	—	194	4	no	—
91	C₂×Q₈	—	97	2	no	—
92	C₂×D₄	—	97	2	no	—
93	C₂×D₄	—	194	2	no	—
94	C₂×D₄	—	194	2	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	194	2	no	—
97	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
98	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
99	M₁₆	—	194	4	no	—
100	C₂×C₈	—	194	2	no	—
101	C₄⋊C₄	—	194	2	no	—
102	C₂²⋊C₄	—	194	4	no	—
103	C₄²	—	2	2	no	—
104	C₄²	C₄×D₉₇	1	1	yes	characteristic
105	C₄²	C₄×D₉₇	1	1	yes	characteristic
106	C₄²	D₃₈₈	1	2	yes	—
107	C₄²	D₃₈₈	1	4	yes	complemented
108	C₂³	—	194	2	no	—
109	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
110	Q₈	—	194	4	no	—
111	D₄	—	388	4	no	—
112	D₄	—	388	4	no	—
113	D₄	—	194	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	—	388	2	no	—
118	C₂×C₄	—	388	2	no	—
119	C₂×C₄	—	194	2	no	—
120	C₂×C₄	#13095488	1	2	yes	—
121	C₂×C₄	#13095488	1	2	yes	—
122	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
123	C₂×C₄	C₂×D₃₈₈	1	2	yes	—
124	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
125	C₂×C₄	#13095511	1	2	yes	—
126	C₈	—	388	2	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	388	2	no	—
129	C₂²	—	388	4	no	—
130	C₂²	C₂×#13095488	1	1	yes	characteristic
131	C₂²	#15690784	1	2	yes	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	4	4	no	—
135	C₄	—	388	2	no	—
136	C₄	C₄×D₃₈₈	1	2	yes	—
137	C₄	#15690798	1	1	yes	characteristic
138	C₄	#15690807	1	1	yes	characteristic
139	C₂	—	2	2	no	—
140	C₂	—	776	2	no	—
141	C₂	#20260637	1	1	yes	characteristic
142	C₂	#20260753	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20261285

C₂

yes

maximal, characteristic

#20260759

C₂

yes

maximal, characteristic, complemented

C₂×#15690738

C₂

yes

maximal, complemented

#376×C₉₇

C₂

yes

maximal, characteristic, complemented

C₂×#13095511

C₂²

yes

—

C₂×#13095495

C₂²

yes

—

#15690810

C₄

yes

characteristic, complemented

#15690807

C₄

yes

characteristic, complemented

#15690798

C₄

yes

complemented

#15690784

—

C₄×D₃₈₈

—

C₄⋊#11715895

—

#15690738

—

C₂×C₄²×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

#95×C₉₇

—

#13095511

—

#13095511

—

C₂×Dic₁₉₄

C₂×C₄

yes

—

C₂×D₃₈₈

—

C₂×D₃₈₈

C₂×C₄

yes

—

C₂×C₂²⋊D₉₇

—

C₂×C₄×D₉₇

—

#13095495

—

C₂×#11715895

—

C₄⋊Dic₉₇

—

#13095488

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄²×C₉₇

—

C₄²×C₉₇

C₂×C₄

yes

characteristic

C₄²×C₉₇

C₂×C₄

yes

characteristic

C₄²×C₉₇

D₄

yes

—

C₄²×C₉₇

D₄

yes

complemented

C₂³×C₉₇

C₂×D₄

yes

characteristic

C₂²×D₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂²⋊D₉₇

—

C₂×Dic₉₇

—

D₃₈₈

—

D₃₈₈

—

C₄×D₉₇

—

C₄×D₉₇

—

Dic₁₉₄

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₉₇

#121

yes

—

C₂×D₉₇

—

C₂×D₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₄×D₄

yes

—

C₄×C₉₇

yes

C₄×C₉₇

yes

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#390

yes

characteristic

C₂×C₉₇

#421

yes

—

D₉₇

—

#1963

—

maximal, Sylow

C₉₇

#1963

yes

characteristic, complemented, Sylow

—

—

—

D₉₇

yes

characteristic, complemented

C₂×#41

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

—

#124

—

#121

—

194

—

C₄×D₄

—

194

—

C₂×C₄²

C₂×D₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

194

—

#104

—

194

—

#95

—

#41

—

194

—

#41

—

194

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

194

—

C₂×D₄

—

194

—

C₂²×C₄

—

C₂²×C₄

—

194

—

C₂²×C₄

C₂²⋊D₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

M₁₆

—

194

—

100

C₂×C₈

—

194

—

101

C₄⋊C₄

—

194

—

102

C₂²⋊C₄

—

194

—

103

C₄²

—

104

C₄²

C₄×D₉₇

yes

characteristic

105

C₄²

C₄×D₉₇

yes

characteristic

106

C₄²

D₃₈₈

yes

—

107

C₄²

D₃₈₈

yes

complemented

108

C₂³

—

194

—

109

C₂³

C₂×C₂²⋊D₉₇

yes

characteristic

110

Q₈

—

194

—

111

D₄

—

388

—

112

D₄

—

388

—

113

D₄

—

194

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

388

—

118

C₂×C₄

—

388

—

119

C₂×C₄

—

194

—

120

C₂×C₄

#13095488

yes

—

121

C₂×C₄

#13095488

yes

—

122

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

123

C₂×C₄

C₂×D₃₈₈

yes

—

124

C₂×C₄

C₂×D₃₈₈

yes

characteristic

125

C₂×C₄

#13095511

yes

—

126

C₈

—

388

—

127

C₂²

—

128

C₂²

—

388

—

129

C₂²

—

388

—

130

C₂²

C₂×#13095488

yes

characteristic

131

C₂²

#15690784

yes

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

388

—

136

C₄

C₄×D₃₈₈

yes

—

137

C₄

#15690798

yes

characteristic

138

C₄

#15690807

yes

characteristic

139

C₂

—

140

C₂

—

776

—

141

C₂

#20260637

yes

characteristic

142

C₂

#20260753

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₃₈₈
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#376×C₉₇	C₂
socle	C₂²×C₉₇	C₂×C₂²⋊C₄
prime radical	#20261285	C₂
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#376	D₉₇
hypocenter	C₉₇	#1963
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

subgroup

G/H

center

C₂×C₄

C₂×D₃₈₈

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

#376×C₉₇

C₂

socle

C₂²×C₉₇

C₂×C₂²⋊C₄

prime radical

#20261285

C₂

cosocle

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

N	H ≅ G/N	type
2. #20260759	C₂	semidirect
3. C₂×#15690738	C₂	semidirect
4. #376×C₉₇	C₂	semidirect
7. #15690810	C₄	semidirect
8. #15690807	C₄	semidirect
9. #15690798	C₄	semidirect
35. C₄²×C₉₇	D₄	semidirect
72. C₉₇	#1963	semidirect
76. #376	D₉₇	semidirect
107. C₄²	D₃₈₈	semidirect

H ≅ G/N

type

2. #20260759

C₂

semidirect

3. C₂×#15690738

C₂

semidirect

4. #376×C₉₇

C₂

semidirect

7. #15690810

C₄

semidirect

8. #15690807

C₄

semidirect

9. #15690798

C₄

semidirect

35. C₄²×C₉₇

D₄

semidirect

72. C₉₇

#1963

semidirect

76. #376

D₉₇

semidirect

107. C₄²

D₃₈₈

semidirect

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

subgroup	index
#20261285	2
#20260759	2
C₂×#15690738	2
#376×C₉₇	2
#1963	97

subgroup

index

#376×C₉₇

Subgroups — #27142568

Conjugacy classes of subgroups

Complements

Maximal subgroups