Group #27142569 — subgroups

Subgroups — #27142569

Order 12416.

320 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 97 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	#20260759	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15690737	C₂	1	4	yes	maximal, complemented
3	#20260581	C₂	1	1	yes	maximal, characteristic, complemented
4	#532×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13095495	C₂²	1	2	yes	—
6	C₂×C₄×Dic₉₇	C₂²	1	2	yes	—
7	C₄⋊#11715895	—	2	2	no	—
8	#15690737	—	2	8	no	—
9	#15690717	—	2	2	no	—
10	C₂×#41×C₉₇	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₉₇	C₄	1	1	yes	characteristic, complemented
12	#127×C₉₇	C₄	1	1	yes	characteristic, complemented
13	#124×C₉₇	C₄	1	2	yes	complemented
14	#121×C₉₇	—	2	2	no	—
15	C₄×D₄×C₉₇	—	2	2	no	—
16	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
17	C₂²×Dic₉₇	—	2	2	no	—
18	#13095495	—	2	4	no	—
19	C₂×#11715895	—	2	2	no	—
20	C₄×Dic₉₇	—	2	2	no	—
21	C₄×Dic₉₇	D₄	1	4	yes	complemented
22	#41×C₉₇	—	2	4	no	—
23	#41×C₉₇	—	2	4	no	—
24	C₂×Q₈×C₉₇	C₂×C₄	1	2	yes	—
25	C₂×D₄×C₉₇	—	2	2	no	—
26	C₂×D₄×C₉₇	—	2	2	no	—
27	C₂×D₄×C₉₇	C₂×C₄	1	2	yes	—
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	C₄⋊C₄×C₉₇	—	2	2	no	—
32	C₂²⋊C₄×C₉₇	—	2	4	no	—
33	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₉₇	D₄	1	2	yes	—
36	C₂³×C₉₇	—	2	2	no	—
37	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
38	Q₈×C₉₇	—	2	4	no	—
39	D₄×C₉₇	—	2	4	no	—
40	D₄×C₉₇	—	4	4	no	—
41	D₄×C₉₇	—	4	4	no	—
42	C₂×C₄×C₉₇	—	2	2	no	—
43	C₂×C₄×C₉₇	—	4	2	no	—
44	C₂×C₄×C₉₇	—	4	2	no	—
45	C₂×C₄×C₉₇	—	4	2	no	—
46	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₉₇	#41	1	2	yes	—
52	C₂×Dic₉₇	—	2	4	no	—
53	C₂×Dic₉₇	—	4	4	no	—
54	#11715895	—	4	2	no	—
55	C₂²×C₉₇	—	2	2	no	—
56	C₂²×C₉₇	—	4	2	no	—
57	C₂²×C₉₇	—	4	4	no	—
58	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₉₇	#121	1	2	yes	—
60	C₄×C₉₇	—	2	2	no	—
61	C₄×C₉₇	—	2	2	no	—
62	C₄×C₉₇	—	4	2	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	4	no	—
67	C₂×C₉₇	—	2	2	no	—
68	C₂×C₉₇	—	8	2	no	—
69	C₂×C₉₇	#390	1	1	yes	characteristic
70	C₂×C₉₇	#421	1	2	yes	—
71	#1963	—	97	1	no	maximal, Sylow
72	C₉₇	#1963	1	1	yes	characteristic, complemented, Sylow
73	#532	D₉₇	1	1	yes	characteristic
74	#423	—	97	1	no	—
75	C₂×#104	—	97	4	no	—
76	#376	—	97	1	no	—
77	C₂×#41	C₂×D₉₇	1	2	yes	—
78	C₂×M₁₆	—	97	2	no	—
79	C₄⋊Q₈	Dic₉₇	1	1	yes	characteristic, complemented
80	#127	Dic₉₇	1	1	yes	characteristic, complemented
81	#124	Dic₉₇	1	2	yes	complemented
82	#121	—	2	2	no	—
83	C₄×D₄	—	2	2	no	—
84	C₂×C₄²	—	97	2	no	—
85	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
86	C₄⋊C₈	—	194	2	no	—
87	#104	—	194	8	no	—
88	#95	—	194	2	no	—
89	#41	—	2	4	no	—
90	#41	—	2	4	no	—
91	C₂×Q₈	C₂×Dic₉₇	1	2	yes	—
92	C₂×D₄	—	2	2	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	C₂×Dic₉₇	1	2	yes	—
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	—	2	2	no	—
102	C₂²⋊C₄	—	2	4	no	—
103	C₄²	—	97	4	no	—
104	C₄²	—	194	2	no	—
105	C₄²	C₂×Dic₉₇	1	1	yes	characteristic
106	C₄²	C₂×Dic₉₇	1	1	yes	characteristic
107	C₄²	C₂²⋊D₉₇	1	2	yes	—
108	C₂³	—	2	2	no	—
109	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
110	Q₈	—	2	4	no	—
111	D₄	—	2	4	no	—
112	D₄	—	4	4	no	—
113	D₄	—	4	4	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	388	4	no	—
119	C₂×C₄	—	194	4	no	—
120	C₂×C₄	C₂²⋊Dic₉₇	1	2	yes	—
121	C₂×C₄	C₂²⋊Dic₉₇	1	2	yes	—
122	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
123	C₂×C₄	D₄×D₉₇	1	2	yes	—
124	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
125	C₂×C₄	#13095510	1	2	yes	—
126	C₈	—	388	2	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	4	2	no	—
129	C₂²	—	4	4	no	—
130	C₂²	C₂×C₂²⋊Dic₉₇	1	1	yes	characteristic
131	C₂²	#15690781	1	2	yes	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	4	2	no	—
135	C₄	—	388	4	no	—
136	C₄	D₄×Dic₉₇	1	2	yes	—
137	C₄	#15690797	1	1	yes	characteristic
138	C₄	#15690806	1	1	yes	characteristic
139	C₂	—	2	2	no	—
140	C₂	—	8	2	no	—
141	C₂	#20260638	1	1	yes	characteristic
142	C₂	#20260752	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20260759

C₂

yes

maximal, characteristic, complemented

C₂×#15690737

C₂

yes

maximal, complemented

#20260581

C₂

yes

maximal, characteristic, complemented

#532×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

C₂²

yes

—

C₂×C₄×Dic₉₇

C₂²

yes

—

C₄⋊#11715895

—

#15690737

—

#15690717

—

C₂×#41×C₉₇

C₂²

yes

—

C₄⋊Q₈×C₉₇

C₄

yes

characteristic, complemented

#127×C₉₇

C₄

yes

characteristic, complemented

#124×C₉₇

C₄

yes

complemented

#121×C₉₇

—

C₄×D₄×C₉₇

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂²×Dic₉₇

—

#13095495

—

C₂×#11715895

—

C₄×Dic₉₇

—

C₄×Dic₉₇

D₄

yes

complemented

#41×C₉₇

—

#41×C₉₇

—

C₂×Q₈×C₉₇

C₂×C₄

yes

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₂²⋊C₄×C₉₇

—

C₄²×C₉₇

C₂×C₄

yes

characteristic

C₄²×C₉₇

C₂×C₄

yes

characteristic

C₄²×C₉₇

D₄

yes

—

C₂³×C₉₇

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

Q₈×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

C₂×C₄×C₉₇

—

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

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₉₇

#121

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₄×D₄

yes

—

C₄×C₉₇

yes

C₄×C₉₇

yes

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#390

yes

characteristic

C₂×C₉₇

#421

yes

—

#1963

—

maximal, Sylow

C₉₇

#1963

yes

characteristic, complemented, Sylow

#532

D₉₇

yes

—

—

—

C₂×#41

C₂×D₉₇

yes

—

C₂×M₁₆

—

C₄⋊Q₈

Dic₉₇

yes

characteristic, complemented

#127

Dic₉₇

yes

characteristic, complemented

#124

Dic₉₇

yes

complemented

#121

—

C₄×D₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

194

—

#104

—

194

—

#95

—

194

—

#41

—

#41

—

C₂×Q₈

C₂×Dic₉₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₉₇

yes

—

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

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

—

194

—

105

C₄²

C₂×Dic₉₇

yes

characteristic

106

C₄²

C₂×Dic₉₇

yes

characteristic

107

C₄²

C₂²⋊D₉₇

yes

—

108

C₂³

—

109

C₂³

C₂×C₂²⋊D₉₇

yes

characteristic

110

Q₈

—

111

D₄

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

388

—

119

C₂×C₄

—

194

—

120

C₂×C₄

C₂²⋊Dic₉₇

yes

—

121

C₂×C₄

C₂²⋊Dic₉₇

yes

—

122

C₂×C₄

C₂²×Dic₉₇

yes

characteristic

123

C₂×C₄

D₄×D₉₇

yes

—

124

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

125

C₂×C₄

#13095510

yes

—

126

C₈

—

388

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

C₂×C₂²⋊Dic₉₇

yes

characteristic

131

C₂²

#15690781

yes

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

388

—

136

C₄

D₄×Dic₉₇

yes

—

137

C₄

#15690797

yes

characteristic

138

C₄

#15690806

yes

characteristic

139

C₂

—

140

C₂

—

141

C₂

#20260638

yes

characteristic

142

C₂

#20260752

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₉₇
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting, prime radical	#532×C₉₇	C₂
socle	C₂²×C₉₇	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#532	D₉₇
hypocenter	C₉₇	#1963
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₉₇

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting, prime radical

#532×C₉₇

C₂

socle

C₂²×C₉₇

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #20260759	C₂	semidirect
2. C₂×#15690737	C₂	semidirect
3. #20260581	C₂	semidirect
11. C₄⋊Q₈×C₉₇	C₄	semidirect
12. #127×C₉₇	C₄	semidirect
13. #124×C₉₇	C₄	semidirect
21. C₄×Dic₉₇	D₄	semidirect
72. C₉₇	#1963	semidirect
79. C₄⋊Q₈	Dic₉₇	semidirect
80. #127	Dic₉₇	semidirect
81. #124	Dic₉₇	semidirect

H ≅ G/N

type

1. #20260759

C₂

semidirect

2. C₂×#15690737

C₂

semidirect

3. #20260581

C₂

semidirect

11. C₄⋊Q₈×C₉₇

C₄

semidirect

12. #127×C₉₇

C₄

semidirect

13. #124×C₉₇

C₄

semidirect

21. C₄×Dic₉₇

D₄

semidirect

72. C₉₇

#1963

semidirect

79. C₄⋊Q₈

Dic₉₇

semidirect

80. #127

Dic₉₇

semidirect

81. #124

Dic₉₇

semidirect

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

subgroup	index
#20260759	2
C₂×#15690737	2
#20260581	2
#532×C₉₇	2
#1963	97

subgroup

index

#532×C₉₇

Subgroups — #27142569

Conjugacy classes of subgroups

Complements

Maximal subgroups