Group #27143023 — subgroups

Subgroups — #27143023

Order 12416.

266 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 65 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	#20261535	C₂	1	1	yes	maximal, characteristic
2	C₂×#15690738	C₂	1	2	yes	maximal, complemented
3	#20260717	C₂	1	2	yes	maximal, complemented
4	#20260493	C₂	1	1	yes	maximal, characteristic, complemented
5	#423×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13095511	C₂²	1	2	yes	—
7	#15690843	—	2	2	no	—
8	#15690838	—	2	2	no	—
9	#15690832	—	2	2	no	—
10	C₂×#13095495	C₂²	1	2	yes	—
11	#15690738	—	2	2	no	—
12	#15690738	—	2	2	no	—
13	#15690728	—	2	2	no	—
14	#15690726	—	2	2	no	—
15	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
18	C₄⋊C₈×C₉₇	—	2	2	no	—
19	#13095511	—	2	2	no	—
20	#13095511	—	2	2	no	—
21	#13095511	—	2	2	no	—
22	#13095511	—	2	2	no	—
23	C₂×Dic₁₉₄	D₄	1	2	yes	—
24	C₂×D₃₈₈	D₄	1	2	yes	—
25	C₂×C₂²⋊D₉₇	—	2	2	no	—
26	C₂×C₄×D₉₇	—	2	2	no	—
27	Dic₃₈₈	—	2	2	no	—
28	#13095500	—	2	4	no	—
29	D₇₇₆	—	2	2	no	—
30	#13095495	—	2	2	no	—
31	#13095495	—	2	2	no	—
32	C₂×#11715895	—	2	2	no	—
33	C₂²×C₄×C₉₇	—	2	1	no	—
34	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
35	M₁₆×C₉₇	—	2	2	no	—
36	M₁₆×C₉₇	—	2	2	no	—
37	C₂×C₈×C₉₇	—	2	1	no	—
38	C₂×C₈×C₉₇	D₄	1	2	yes	—
39	C₄²×C₉₇	—	2	1	no	—
40	C₄²×C₉₇	D₄	1	1	yes	characteristic
41	C₄²×C₉₇	D₄	1	1	yes	characteristic
42	C₂³×C₉₇	#41	1	1	yes	characteristic
43	C₂²×D₉₇	—	2	2	no	—
44	C₂×C₄×C₉₇	—	2	1	no	—
45	C₂×C₄×C₉₇	—	2	1	no	—
46	C₂×C₄×C₉₇	—	4	2	no	—
47	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₉₇	#41	1	2	yes	—
51	C₂²⋊D₉₇	—	4	2	no	—
52	C₂²⋊D₉₇	—	4	2	no	—
53	C₂×Dic₉₇	—	2	2	no	—
54	D₃₈₈	—	2	2	no	—
55	D₃₈₈	—	2	2	no	—
56	C₄×D₉₇	—	4	2	no	—
57	C₄×D₉₇	—	4	2	no	—
58	Dic₁₉₄	—	2	2	no	—
59	Dic₁₉₄	—	2	2	no	—
60	C₈×C₉₇	—	2	2	no	—
61	C₈×C₉₇	—	4	1	no	—
62	#11715895	—	4	2	no	—
63	C₂²×C₉₇	—	4	1	no	—
64	C₂²×C₉₇	#121	1	1	yes	characteristic
65	C₂²×C₉₇	#123	1	2	yes	—
66	C₂×D₉₇	—	4	2	no	—
67	C₂×D₉₇	—	8	2	no	—
68	C₄×C₉₇	—	2	2	no	—
69	C₄×C₉₇	—	4	1	no	—
70	C₄×C₉₇	#120	1	1	yes	characteristic
71	C₄×C₉₇	#121	1	2	yes	—
72	C₄×C₉₇	#124	1	1	yes	characteristic
73	Dic₉₇	—	4	2	no	—
74	C₂×C₉₇	—	2	1	no	—
75	C₂×C₉₇	—	2	2	no	—
76	C₂×C₉₇	#394	1	1	yes	characteristic
77	D₉₇	—	8	2	no	—
78	#2050	—	97	1	no	maximal, Sylow
79	C₉₇	#2050	1	1	yes	characteristic, complemented, Sylow
80	#575	—	97	1	no	—
81	#423	D₉₇	1	1	yes	characteristic, complemented
82	C₂×#104	—	97	2	no	—
83	#413	—	97	2	no	—
84	#338	—	97	1	no	—
85	C₂×#41	—	97	2	no	—
86	#137	—	194	2	no	—
87	AGL(1,ℤ/8)	—	194	2	no	—
88	#135	—	194	2	no	—
89	C₂×M₁₆	—	97	2	no	—
90	C₂×M₁₆	C₂×D₉₇	1	1	yes	characteristic
91	C₂×M₁₆	C₂×D₉₇	1	1	yes	characteristic
92	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
93	C₄⋊C₈	—	2	2	no	—
94	#104	—	194	2	no	—
95	#104	—	194	2	no	—
96	#101	—	194	2	no	—
97	#100	—	194	2	no	—
98	#41	—	194	2	no	—
99	#41	—	194	2	no	—
100	#41	—	194	2	no	—
101	#41	—	194	2	no	—
102	C₂×Q₈	—	97	2	no	—
103	C₂×D₄	—	97	2	no	—
104	C₂×D₄	—	194	2	no	—
105	C₂²×C₄	—	2	1	no	—
106	C₂²×C₄	—	194	2	no	—
107	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
108	Q₁₆	—	194	2	no	—
109	SD₁₆	—	194	4	no	—
110	D₈	—	194	2	no	—
111	M₁₆	—	2	2	no	—
112	M₁₆	—	2	2	no	—
113	M₁₆	—	194	2	no	—
114	M₁₆	—	194	2	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	194	2	no	—
117	C₂×C₈	C₂²⋊D₉₇	1	2	yes	—
118	C₄²	—	2	1	no	—
119	C₄²	D₃₈₈	1	1	yes	characteristic
120	C₄²	D₃₈₈	1	1	yes	characteristic
121	C₂³	—	194	2	no	—
122	C₂³	#13095511	1	1	yes	characteristic
123	Q₈	—	194	2	no	—
124	Q₈	—	194	2	no	—
125	D₄	—	388	2	no	—
126	D₄	—	388	2	no	—
127	D₄	—	194	2	no	—
128	D₄	—	194	2	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	388	2	no	—
133	C₂×C₄	—	388	2	no	—
134	C₂×C₄	—	194	2	no	—
135	C₂×C₄	D₄×D₉₇	1	2	yes	—
136	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
137	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
138	C₂×C₄	#13095512	1	2	yes	—
139	C₈	—	2	2	no	—
140	C₈	—	4	1	no	—
141	C₈	—	388	2	no	—
142	C₂²	—	4	1	no	—
143	C₂²	—	776	2	no	—
144	C₂²	—	388	2	no	—
145	C₂²	#15690784	1	1	yes	characteristic
146	C₂²	#15690795	1	2	yes	—
147	C₄	—	2	2	no	—
148	C₄	—	4	1	no	—
149	C₄	—	388	2	no	—
150	C₄	#15690778	1	1	yes	characteristic
151	C₄	#15690785	1	2	yes	—
152	C₄	#15690799	1	1	yes	characteristic
153	C₂	—	2	1	no	—
154	C₂	—	2	2	no	—
155	C₂	—	776	2	no	—
156	C₂	#20260652	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20261535

C₂

yes

maximal, characteristic

C₂×#15690738

C₂

yes

maximal, complemented

#20260717

C₂

yes

maximal, complemented

#20260493

C₂

yes

maximal, characteristic, complemented

#423×C₉₇

C₂

yes

maximal, characteristic, complemented

C₂×#13095511

C₂²

yes

—

—

—

—

C₂²

yes

—

—

—

—

—

C₂×M₁₆×C₉₇

C₂²

yes

characteristic

C₂×M₁₆×C₉₇

C₂²

yes

characteristic

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₉₇

—

—

—

—

—

C₂×Dic₁₉₄

D₄

yes

—

C₂×D₃₈₈

D₄

yes

—

C₂×C₂²⋊D₉₇

—

C₂×C₄×D₉₇

—

Dic₃₈₈

—

#13095500

—

D₇₇₆

—

#13095495

—

#13095495

—

C₂×#11715895

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

M₁₆×C₉₇

—

M₁₆×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

D₄

yes

—

C₄²×C₉₇

—

C₄²×C₉₇

D₄

yes

characteristic

C₄²×C₉₇

D₄

yes

characteristic

C₂³×C₉₇

#41

yes

characteristic

C₂²×D₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂²⋊D₉₇

—

C₂²⋊D₉₇

—

C₂×Dic₉₇

—

D₃₈₈

—

D₃₈₈

—

C₄×D₉₇

—

C₄×D₉₇

—

Dic₁₉₄

—

Dic₁₉₄

—

C₈×C₉₇

—

C₈×C₉₇

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

#121

yes

characteristic

C₂²×C₉₇

#123

yes

—

C₂×D₉₇

—

C₂×D₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#120

yes

characteristic

C₄×C₉₇

#121

yes

—

C₄×C₉₇

#124

yes

characteristic

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#394

yes

characteristic

D₉₇

—

#2050

—

maximal, Sylow

C₉₇

#2050

yes

characteristic, complemented, Sylow

#575

—

#423

D₉₇

yes

characteristic, complemented

C₂×#104

—

#413

—

#338

—

C₂×#41

—

#137

—

194

—

AGL(1,ℤ/8)

—

194

—

#135

—

194

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₉₇

yes

characteristic

C₂×M₁₆

C₂×D₉₇

yes

characteristic

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

#104

—

194

—

#104

—

194

—

#101

—

194

—

#100

—

194

—

#41

—

194

—

#41

—

194

—

100

#41

—

194

—

101

#41

—

194

—

102

C₂×Q₈

—

103

C₂×D₄

—

104

C₂×D₄

—

194

—

105

C₂²×C₄

—

106

C₂²×C₄

—

194

—

107

C₂²×C₄

C₂²×D₉₇

yes

characteristic

108

Q₁₆

—

194

—

109

SD₁₆

—

194

—

110

D₈

—

194

—

111

M₁₆

—

112

M₁₆

—

113

M₁₆

—

194

—

114

M₁₆

—

194

—

115

C₂×C₈

—

116

C₂×C₈

—

194

—

117

C₂×C₈

C₂²⋊D₉₇

yes

—

118

C₄²

—

119

C₄²

D₃₈₈

yes

characteristic

120

C₄²

D₃₈₈

yes

characteristic

121

C₂³

—

194

—

122

C₂³

#13095511

yes

characteristic

123

Q₈

—

194

—

124

Q₈

—

194

—

125

D₄

—

388

—

126

D₄

—

388

—

127

D₄

—

194

—

128

D₄

—

194

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

388

—

133

C₂×C₄

—

388

—

134

C₂×C₄

—

194

—

135

C₂×C₄

D₄×D₉₇

yes

—

136

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

137

C₂×C₄

C₂×D₃₈₈

yes

characteristic

138

C₂×C₄

#13095512

yes

—

139

C₈

—

140

C₈

—

141

C₈

—

388

—

142

C₂²

—

143

C₂²

—

776

—

144

C₂²

—

388

—

145

C₂²

#15690784

yes

characteristic

146

C₂²

#15690795

yes

—

147

C₄

—

148

C₄

—

149

C₄

—

388

—

150

C₄

#15690778

yes

characteristic

151

C₄

#15690785

yes

—

152

C₄

#15690799

yes

characteristic

153

C₂

—

154

C₂

—

155

C₂

—

776

—

156

C₂

#20260652

yes

characteristic

	subgroup	G/H
center	C₄	#15690799
commutator, cosocle	C₂²×C₄×C₉₇	C₂³
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#423×C₉₇	C₂
socle	C₂×C₉₇	#394
prime radical	#20261535	C₂
hypercenter	#423	D₉₇
hypocenter	C₉₇	#2050
LCS	G ▷ C₂²×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #423

subgroup

G/H

center

C₄

#15690799

commutator, cosocle

C₂²×C₄×C₉₇

C₂³

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

#423×C₉₇

C₂

C₂×C₉₇

C₂

D₉₇

C₉₇

G ▷ C₂²×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #423

N	H ≅ G/N	type
2. C₂×#15690738	C₂	semidirect
3. #20260717	C₂	semidirect
4. #20260493	C₂	semidirect
5. #423×C₉₇	C₂	semidirect
79. C₉₇	#2050	semidirect
81. #423	D₉₇	semidirect

H ≅ G/N

type

2. C₂×#15690738

C₂

semidirect

3. #20260717

C₂

semidirect

4. #20260493

C₂

semidirect

5. #423×C₉₇

C₂

semidirect

79. C₉₇

#2050

semidirect

81. #423

D₉₇

semidirect

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

subgroup	index
#20261535	2
C₂×#15690738	2
#20260717	2
#20260493	2
#423×C₉₇	2
#2050	97

subgroup

index

#423×C₉₇

Subgroups — #27143023

Conjugacy classes of subgroups

Complements

Maximal subgroups