Group #27143300 — subgroups

Subgroups — #27143300

Order 12416.

214 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 87 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	#20260633	C₂	1	1	yes	maximal, characteristic
2	#20260605	C₂	1	1	yes	maximal, characteristic
3	#20260498	C₂	1	2	yes	maximal
4	#20260498	C₂	1	2	yes	maximal
5	#336×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₉₇	—	2	1	no	—
7	C₂×C₄⋊Dic₉₇	—	2	1	no	—
8	C₂×C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
11	C₂×#13095490	—	2	1	no	—
12	#15690717	—	2	1	no	—
13	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₉₇	—	2	1	no	—
16	C₂²×Dic₉₇	—	2	1	no	—
17	C₂²×Dic₉₇	—	2	2	no	—
18	C₄⋊Dic₉₇	—	2	1	no	—
19	C₄⋊Dic₉₇	—	2	2	no	—
20	C₄⋊Dic₉₇	—	2	4	no	—
21	C₄⋊Dic₉₇	—	4	2	no	—
22	C₄⋊Dic₉₇	—	4	2	no	—
23	C₄⋊Dic₉₇	Q₈	1	2	yes	—
24	#13095490	—	4	2	no	—
25	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₉₇	—	2	2	no	—
29	C₂×C₈×C₉₇	—	2	2	no	—
30	C₂×C₈×C₉₇	—	2	2	no	—
31	C₄²×C₉₇	—	2	2	no	—
32	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₉₇	—	2	1	no	—
34	C₂×C₄×C₉₇	—	2	1	no	—
35	C₂×C₄×C₉₇	—	2	1	no	—
36	C₂×C₄×C₉₇	—	2	1	no	—
37	C₂×C₄×C₉₇	SD₁₆	1	2	yes	—
38	C₂×C₄×C₉₇	Q₁₆	1	2	yes	—
39	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₉₇	#41	1	2	yes	—
41	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
42	C₂×C₄×C₉₇	#41	1	2	yes	—
43	C₂×Dic₉₇	—	2	2	no	—
44	C₂×Dic₉₇	—	4	1	no	—
45	C₂×Dic₉₇	—	4	1	no	—
46	C₂×Dic₉₇	—	4	1	no	—
47	C₂×Dic₉₇	—	4	2	no	—
48	C₂×Dic₉₇	—	4	1	no	—
49	C₂×Dic₉₇	—	4	2	no	—
50	C₂×Dic₉₇	—	4	2	no	—
51	C₂×Dic₉₇	—	4	1	no	—
52	C₈×C₉₇	—	4	2	no	—
53	C₂²×C₉₇	C₂²⋊Q₈	1	1	yes	characteristic
54	C₂²×C₉₇	#123	1	1	yes	characteristic
55	C₂²×C₉₇	#124	1	1	yes	characteristic
56	C₂²×C₉₇	C₂×SD₁₆	1	1	yes	characteristic
57	C₂²×C₉₇	C₂×Q₁₆	1	1	yes	characteristic
58	C₂²×C₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
59	C₂²×C₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₄×C₉₇	—	2	2	no	—
61	C₄×C₉₇	—	2	2	no	—
62	C₄×C₉₇	#125	1	2	yes	—
63	C₄×C₉₇	#126	1	2	yes	—
64	Dic₉₇	—	4	2	no	—
65	Dic₉₇	—	8	1	no	—
66	Dic₉₇	—	8	2	no	—
67	C₂×C₉₇	#400	1	1	yes	characteristic
68	C₂×C₉₇	#476	1	1	yes	characteristic
69	C₂×C₉₇	#477	1	1	yes	characteristic
70	C₂×C₉₇	#481	1	1	yes	characteristic
71	C₂×C₉₇	#484	1	1	yes	characteristic
72	C₂×C₉₇	#487	1	1	yes	characteristic
73	C₂×C₉₇	#490	1	1	yes	characteristic
74	#2100	—	97	1	no	maximal, Sylow
75	C₉₇	#2100	1	1	yes	characteristic, complemented, Sylow
76	#389	—	97	1	no	—
77	#384	—	97	1	no	—
78	#340	—	97	2	no	—
79	#340	—	97	2	no	—
80	#336	D₉₇	1	1	yes	characteristic
81	C₂²×C₈	C₂×D₉₇	1	2	yes	—
82	C₂×C₄⋊C₄	—	97	1	no	—
83	C₂×C₄⋊C₄	—	97	2	no	—
84	C₂×C₄⋊C₄	—	97	1	no	—
85	C₂×C₄⋊C₄	—	194	1	no	—
86	C₂×C₄⋊C₄	—	194	1	no	—
87	C₂×C₄⋊C₄	—	194	1	no	—
88	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
89	#95	—	194	1	no	—
90	C₂²×C₄	—	194	1	no	—
91	C₂²×C₄	—	194	2	no	—
92	C₂²×C₄	—	194	1	no	—
93	C₂²×C₄	D₃₈₈	1	1	yes	characteristic
94	C₂²×C₄	D₃₈₈	1	1	yes	characteristic
95	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₄⋊C₄	—	97	2	no	—
100	C₄⋊C₄	—	388	2	no	—
101	C₄⋊C₄	—	388	2	no	—
102	C₄⋊C₄	—	388	2	no	—
103	C₄⋊C₄	—	194	2	no	—
104	C₄⋊C₄	—	194	4	no	—
105	C₄⋊C₄	—	194	1	no	—
106	C₄²	—	2	2	no	—
107	C₂³	C₂×D₃₈₈	1	1	yes	characteristic
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	388	1	no	—
113	C₂×C₄	—	388	2	no	—
114	C₂×C₄	—	388	1	no	—
115	C₂×C₄	—	388	2	no	—
116	C₂×C₄	—	388	2	no	—
117	C₂×C₄	—	388	1	no	—
118	C₂×C₄	—	388	1	no	—
119	C₂×C₄	—	388	1	no	—
120	C₂×C₄	—	194	2	no	—
121	C₂×C₄	#13095500	1	2	yes	—
122	C₂×C₄	Dic₃₈₈	1	2	yes	—
123	C₂×C₄	Q₈×D₉₇	1	1	yes	characteristic
124	C₂×C₄	#13095510	1	2	yes	—
125	C₂×C₄	#13095510	1	1	yes	characteristic
126	C₂×C₄	#13095511	1	2	yes	—
127	C₈	—	4	2	no	—
128	C₂²	#15690790	1	1	yes	characteristic
129	C₂²	#15690794	1	1	yes	characteristic
130	C₂²	#15690798	1	1	yes	characteristic
131	C₂²	C₂×#13095500	1	1	yes	characteristic
132	C₂²	C₂×Dic₃₈₈	1	1	yes	characteristic
133	C₂²	#15690838	1	1	yes	characteristic
134	C₂²	#15690838	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	2	2	no	—
137	C₄	—	776	1	no	—
138	C₄	—	776	2	no	—
139	C₄	—	388	2	no	—
140	C₄	#15690800	1	2	yes	—
141	C₄	#15690803	1	2	yes	—
142	C₂	#20260675	1	1	yes	characteristic
143	C₂	#20261026	1	1	yes	characteristic
144	C₂	#20261031	1	1	yes	characteristic
145	C₂	#20261054	1	1	yes	characteristic
146	C₂	#20261071	1	1	yes	characteristic
147	C₂	#20261086	1	1	yes	characteristic
148	C₂	#20261100	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20260633

C₂

yes

maximal, characteristic

#20260605

C₂

yes

maximal, characteristic

#20260498

C₂

yes

maximal

#20260498

C₂

yes

maximal

#336×C₉₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₉₇

—

C₂×C₄⋊Dic₉₇

—

C₂×C₄⋊Dic₉₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₉₇

C₂²

yes

—

C₂×C₄⋊Dic₉₇

C₂²

yes

characteristic

C₂×#13095490

—

#15690717

—

C₂²×C₈×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂²×Dic₉₇

—

C₂²×Dic₉₇

—

C₂²×Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

Q₈

yes

—

#13095490

—

C₂²×C₄×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₄²×C₉₇

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

SD₁₆

yes

—

C₂×C₄×C₉₇

Q₁₆

yes

—

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂×C₄×C₉₇

#41

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₈×C₉₇

—

C₂²×C₉₇

C₂²⋊Q₈

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₂²×C₉₇

C₂×SD₁₆

yes

characteristic

C₂²×C₉₇

C₂×Q₁₆

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#125

yes

—

C₄×C₉₇

#126

yes

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

—

—

—

—

D₉₇

yes

characteristic

C₂²×C₈

C₂×D₉₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

#95

—

194

—

C₂²×C₄

—

194

—

C₂²×C₄

—

194

—

C₂²×C₄

—

194

—

C₂²×C₄

D₃₈₈

yes

characteristic

C₂²×C₄

D₃₈₈

yes

characteristic

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

100

C₄⋊C₄

—

388

—

101

C₄⋊C₄

—

388

—

102

C₄⋊C₄

—

388

—

103

C₄⋊C₄

—

194

—

104

C₄⋊C₄

—

194

—

105

C₄⋊C₄

—

194

—

106

C₄²

—

107

C₂³

C₂×D₃₈₈

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

388

—

113

C₂×C₄

—

388

—

114

C₂×C₄

—

388

—

115

C₂×C₄

—

388

—

116

C₂×C₄

—

388

—

117

C₂×C₄

—

388

—

118

C₂×C₄

—

388

—

119

C₂×C₄

—

388

—

120

C₂×C₄

—

194

—

121

C₂×C₄

#13095500

yes

—

122

C₂×C₄

Dic₃₈₈

yes

—

123

C₂×C₄

Q₈×D₉₇

yes

characteristic

124

C₂×C₄

#13095510

yes

—

125

C₂×C₄

#13095510

yes

characteristic

126

C₂×C₄

#13095511

yes

—

127

C₈

—

128

C₂²

#15690790

yes

characteristic

129

C₂²

#15690794

yes

characteristic

130

C₂²

#15690798

yes

characteristic

131

C₂²

C₂×#13095500

yes

characteristic

132

C₂²

C₂×Dic₃₈₈

yes

characteristic

133

C₂²

#15690838

yes

characteristic

134

C₂²

#15690838

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

776

—

138

C₄

—

776

—

139

C₄

—

388

—

140

C₄

#15690800

yes

—

141

C₄

#15690803

yes

—

142

C₂

#20260675

yes

characteristic

143

C₂

#20261026

yes

characteristic

144

C₂

#20261031

yes

characteristic

145

C₂

#20261054

yes

characteristic

146

C₂

#20261071

yes

characteristic

147

C₂

#20261086

yes

characteristic

148

C₂

#20261100

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₈₈
commutator, cosocle	C₂²×C₄×C₉₇	C₂³
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#336×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂×D₄
hypercenter	#336	D₉₇
hypocenter	C₉₇	#2100
LCS	G ▷ C₂²×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₃₈₈

commutator, cosocle

C₂²×C₄×C₉₇

C₂³

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

#336×C₉₇

C₂

socle, prime radical

C₂³×C₉₇

C₂×D₄

D₉₇

C₉₇

G ▷ C₂²×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
75. C₉₇	#2100	semidirect

H ≅ G/N

type

75. C₉₇

#2100

semidirect

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

subgroup	index
#20260633	2
#20260605	2
#20260498	2
#336×C₉₇	2
#2100	97

subgroup

index

#336×C₉₇

Subgroups — #27143300

Conjugacy classes of subgroups

Complements

Maximal subgroups