Group #27141536 — subgroups

Subgroups — #27141536

Order 12416.

284 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 121 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	#20260719	C₂	1	1	yes	maximal, characteristic
2	#20260709	C₂	1	1	yes	maximal, characteristic
3	#20260690	C₂	1	2	yes	maximal, complemented
4	#20260506	C₂	1	2	yes	maximal, complemented
5	C₂×#131×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13095495	C₂²	1	2	yes	—
7	#15690769	C₂²	1	2	yes	—
8	#15690729	C₄	1	2	yes	complemented
9	#15690727	C₄	1	2	yes	complemented
10	#15690725	C₄	1	4	yes	—
11	#15690720	—	2	2	no	—
12	C₈×Dic₉₇	—	2	2	no	—
13	C₂×#41×C₉₇	C₂²	1	1	yes	characteristic
14	#131×C₉₇	—	2	2	no	—
15	#131×C₉₇	—	2	2	no	—
16	C₂×M₁₆×C₉₇	C₄	1	2	yes	complemented
17	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₉₇	C₄	1	2	yes	complemented
19	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
20	#13095495	—	2	2	no	—
21	#13095495	C₂×C₄	1	4	yes	—
22	C₂×#11715895	—	2	2	no	—
23	C₄⋊Dic₉₇	—	2	2	no	—
24	C₂²⋊Dic₉₇	C₂×C₄	1	4	yes	—
25	C₄×Dic₉₇	—	2	2	no	—
26	#41×C₉₇	—	2	2	no	—
27	#41×C₉₇	—	2	2	no	—
28	C₂×Q₈×C₉₇	C₂×C₄	1	1	yes	characteristic
29	C₂×D₄×C₉₇	C₂×C₄	1	2	yes	—
30	C₂×D₄×C₉₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
33	M₁₆×C₉₇	—	2	2	no	—
34	M₁₆×C₉₇	—	2	4	no	—
35	C₂×C₈×C₉₇	—	2	2	no	—
36	C₂×C₈×C₉₇	—	2	2	no	—
37	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
38	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×C₉₇	D₄	1	2	yes	—
41	C₂×C₈×C₉₇	D₄	1	2	yes	—
42	C₂³×C₉₇	C₄²	1	2	yes	—
43	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
44	Q₈×C₉₇	—	2	2	no	—
45	D₄×C₉₇	—	2	4	no	—
46	D₄×C₉₇	—	2	2	no	—
47	C₂×C₄×C₉₇	—	2	2	no	—
48	C₂×C₄×C₉₇	—	2	2	no	—
49	C₂×C₄×C₉₇	C₄²	1	2	yes	—
50	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₉₇	#41	1	2	yes	—
54	C₂×Dic₉₇	—	2	4	no	—
55	C₈×C₉₇	—	2	2	no	—
56	C₈×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₈×C₉₇	C₂²⋊C₄	1	2	yes	—
58	C₈×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
59	#11715895	—	2	4	no	—
60	C₂²×C₉₇	—	2	2	no	—
61	C₂²×C₉₇	—	4	1	no	—
62	C₂²×C₉₇	—	4	2	no	—
63	C₂²×C₉₇	C₂×C₄²	1	1	yes	characteristic
64	C₂²×C₉₇	C₄×D₄	1	2	yes	—
65	C₄×C₉₇	—	2	2	no	—
66	C₄×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₉₇	#117	1	1	yes	characteristic
68	C₄×C₉₇	C₄×D₄	1	2	yes	—
69	Dic₉₇	—	4	4	no	—
70	C₂×C₉₇	—	2	2	no	—
71	C₂×C₉₇	—	2	1	no	—
72	C₂×C₉₇	—	4	2	no	—
73	C₂×C₉₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
74	#1757	—	97	1	no	maximal, Sylow
75	C₉₇	#1757	1	1	yes	characteristic, complemented, Sylow
76	C₂×#131	D₉₇	1	1	yes	characteristic
77	#413	—	97	1	no	—
78	#410	—	97	1	no	—
79	#405	—	97	2	no	—
80	#344	—	97	2	no	—
81	C₂×#41	C₂×D₉₇	1	1	yes	characteristic
82	#131	—	2	2	no	—
83	#131	—	2	2	no	—
84	C₂×M₁₆	—	97	2	no	—
85	C₂×M₁₆	Dic₉₇	1	2	yes	complemented
86	C₂×M₁₆	C₂×D₉₇	1	1	yes	characteristic
87	C₂²×C₈	Dic₉₇	1	2	yes	complemented
88	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
89	#117	—	97	2	no	—
90	#101	—	97	2	no	—
91	#100	—	97	2	no	—
92	#99	—	97	4	no	—
93	#97	—	194	2	no	—
94	C₄×C₈	—	194	2	no	—
95	#41	—	2	2	no	—
96	#41	—	2	2	no	—
97	C₂×Q₈	C₄×D₉₇	1	1	yes	characteristic
98	C₂×D₄	C₄×D₉₇	1	2	yes	—
99	C₂×D₄	C₄×D₉₇	1	1	yes	characteristic
100	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
101	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
102	M₁₆	—	2	2	no	—
103	M₁₆	—	2	4	no	—
104	M₁₆	—	97	4	no	—
105	M₁₆	—	194	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	194	2	no	—
109	C₂×C₈	C₂×Dic₉₇	1	2	yes	—
110	C₂×C₈	C₂×Dic₉₇	1	1	yes	characteristic
111	C₂×C₈	C₂×Dic₉₇	1	1	yes	characteristic
112	C₂×C₈	C₂²⋊D₉₇	1	2	yes	—
113	C₂×C₈	C₂²⋊D₉₇	1	2	yes	—
114	C₄⋊C₄	—	194	2	no	—
115	C₂²⋊C₄	—	97	4	no	—
116	C₄²	—	194	2	no	—
117	C₂³	C₄×Dic₉₇	1	2	yes	—
118	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
119	Q₈	—	2	2	no	—
120	D₄	—	2	4	no	—
121	D₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	194	4	no	—
125	C₂×C₄	C₄×Dic₉₇	1	2	yes	—
126	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
127	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
128	C₂×C₄	C₂×C₂²⋊D₉₇	1	2	yes	—
129	C₂×C₄	#13095511	1	2	yes	—
130	C₈	—	2	2	no	—
131	C₈	—	194	4	no	—
132	C₈	C₂²⋊Dic₉₇	1	1	yes	characteristic
133	C₈	C₂²⋊Dic₉₇	1	2	yes	—
134	C₈	C₂²⋊Dic₉₇	1	1	yes	characteristic
135	C₂²	—	2	2	no	—
136	C₂²	—	4	1	no	—
137	C₂²	—	4	2	no	—
138	C₂²	C₂×C₄×Dic₉₇	1	1	yes	characteristic
139	C₂²	C₄×C₂²⋊D₉₇	1	2	yes	—
140	C₄	—	2	2	no	—
141	C₄	—	388	4	no	—
142	C₄	C₂×C₂²⋊Dic₉₇	1	1	yes	characteristic
143	C₄	#15690769	1	1	yes	characteristic
144	C₄	C₄×C₂²⋊D₉₇	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	2	1	no	—
147	C₂	—	4	2	no	—
148	C₂	C₄×C₂²⋊Dic₉₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20260719

C₂

yes

maximal, characteristic

#20260709

C₂

yes

maximal, characteristic

#20260690

C₂

yes

maximal, complemented

#20260506

C₂

yes

maximal, complemented

C₂×#131×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

C₂²

yes

—

#15690769

C₂²

yes

—

#15690729

C₄

yes

complemented

#15690727

C₄

yes

complemented

#15690725

C₄

yes

—

#15690720

—

C₈×Dic₉₇

—

C₂×#41×C₉₇

C₂²

yes

characteristic

#131×C₉₇

—

#131×C₉₇

—

C₂×M₁₆×C₉₇

C₄

yes

complemented

C₂×M₁₆×C₉₇

C₂²

yes

characteristic

C₂²×C₈×C₉₇

C₄

yes

complemented

C₂²×C₈×C₉₇

C₂²

yes

characteristic

#13095495

—

#13095495

C₂×C₄

yes

—

C₂×#11715895

—

C₄⋊Dic₉₇

—

C₂²⋊Dic₉₇

C₂×C₄

yes

—

C₄×Dic₉₇

—

#41×C₉₇

—

#41×C₉₇

—

C₂×Q₈×C₉₇

C₂×C₄

yes

characteristic

C₂×D₄×C₉₇

C₂×C₄

yes

—

C₂×D₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

M₁₆×C₉₇

—

M₁₆×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

C₂×C₄

yes

—

C₂×C₈×C₉₇

C₂×C₄

yes

characteristic

C₂×C₈×C₉₇

C₂×C₄

yes

characteristic

C₂×C₈×C₉₇

D₄

yes

—

C₂×C₈×C₉₇

D₄

yes

—

C₂³×C₉₇

C₄²

yes

—

C₂³×C₉₇

C₂²×C₄

yes

characteristic

Q₈×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₄²

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

#41

yes

—

C₂×Dic₉₇

—

C₈×C₉₇

—

C₈×C₉₇

C₂²⋊C₄

yes

characteristic

C₈×C₉₇

C₂²⋊C₄

yes

—

C₈×C₉₇

C₂²⋊C₄

yes

characteristic

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×C₄²

yes

characteristic

C₂²×C₉₇

C₄×D₄

yes

—

C₄×C₉₇

—

C₄×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₉₇

#117

yes

characteristic

C₄×C₉₇

C₄×D₄

yes

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

C₄×C₂²⋊C₄

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₂×#131

D₉₇

yes

—

—

—

—

C₂×#41

C₂×D₉₇

yes

characteristic

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

Dic₉₇

yes

complemented

C₂×M₁₆

C₂×D₉₇

yes

characteristic

C₂²×C₈

Dic₉₇

yes

complemented

C₂²×C₈

C₂×D₉₇

yes

—

—

—

—

—

194

—

C₄×C₈

—

194

—

#41

—

#41

—

C₂×Q₈

C₄×D₉₇

yes

characteristic

C₂×D₄

C₄×D₉₇

yes

—

C₂×D₄

C₄×D₉₇

yes

characteristic

100

C₂²×C₄

C₂×Dic₉₇

yes

—

101

C₂²×C₄

C₂²×D₉₇

yes

characteristic

102

M₁₆

—

103

M₁₆

—

104

M₁₆

—

105

M₁₆

—

194

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

194

—

109

C₂×C₈

C₂×Dic₉₇

yes

—

110

C₂×C₈

C₂×Dic₉₇

yes

characteristic

111

C₂×C₈

C₂×Dic₉₇

yes

characteristic

112

C₂×C₈

C₂²⋊D₉₇

yes

—

113

C₂×C₈

C₂²⋊D₉₇

yes

—

114

C₄⋊C₄

—

194

—

115

C₂²⋊C₄

—

116

C₄²

—

194

—

117

C₂³

C₄×Dic₉₇

yes

—

118

C₂³

C₂×C₄×D₉₇

yes

characteristic

119

Q₈

—

120

D₄

—

121

D₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

194

—

125

C₂×C₄

C₄×Dic₉₇

yes

—

126

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

127

C₂×C₄

C₂²×Dic₉₇

yes

characteristic

128

C₂×C₄

C₂×C₂²⋊D₉₇

yes

—

129

C₂×C₄

#13095511

yes

—

130

C₈

—

131

C₈

—

194

—

132

C₈

C₂²⋊Dic₉₇

yes

characteristic

133

C₈

C₂²⋊Dic₉₇

yes

—

134

C₈

C₂²⋊Dic₉₇

yes

characteristic

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

C₂×C₄×Dic₉₇

yes

characteristic

139

C₂²

C₄×C₂²⋊D₉₇

yes

—

140

C₄

—

141

C₄

—

388

—

142

C₄

C₂×C₂²⋊Dic₉₇

yes

characteristic

143

C₄

#15690769

yes

characteristic

144

C₄

C₄×C₂²⋊D₉₇

yes

—

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

C₄×C₂²⋊Dic₉₇

yes

characteristic

	subgroup	G/H
center	C₈	C₂²⋊Dic₉₇
commutator	C₂²×C₉₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	C₂×#131×C₉₇	C₂
socle	C₂×C₉₇	C₄×C₂²⋊C₄
prime radical	C₂×#41×C₉₇	C₂²
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂×#131	D₉₇
hypocenter	C₉₇	#1757
LCS	G ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#131

subgroup

G/H

center

C₈

C₂²⋊Dic₉₇

commutator

C₂²×C₉₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

C₂×#131×C₉₇

C₂

socle

C₂×C₉₇

C₄×C₂²⋊C₄

prime radical

C₂×#41×C₉₇

C₂²

cosocle

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#131

N	H ≅ G/N	type
3. #20260690	C₂	semidirect
4. #20260506	C₂	semidirect
8. #15690729	C₄	semidirect
9. #15690727	C₄	semidirect
16. C₂×M₁₆×C₉₇	C₄	semidirect
18. C₂²×C₈×C₉₇	C₄	semidirect
75. C₉₇	#1757	semidirect
85. C₂×M₁₆	Dic₉₇	semidirect
87. C₂²×C₈	Dic₉₇	semidirect

H ≅ G/N

type

3. #20260690

C₂

semidirect

4. #20260506

C₂

semidirect

8. #15690729

C₄

semidirect

9. #15690727

C₄

semidirect

16. C₂×M₁₆×C₉₇

C₄

semidirect

18. C₂²×C₈×C₉₇

C₄

semidirect

75. C₉₇

#1757

semidirect

85. C₂×M₁₆

Dic₉₇

semidirect

87. C₂²×C₈

Dic₉₇

semidirect

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

subgroup	index
#20260719	2
#20260709	2
#20260690	2
#20260506	2
C₂×#131×C₉₇	2
#1757	97

subgroup

index

C₂×#131×C₉₇

Subgroups — #27141536

Conjugacy classes of subgroups

Complements

Maximal subgroups