Group #27143043 — subgroups

Subgroups — #27143043

Order 12416.

264 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 63 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	#20260763	C₂	1	1	yes	maximal, characteristic, complemented
3	#20260729	C₂	1	2	yes	maximal, complemented
4	#20260708	C₂	1	2	yes	maximal, complemented
5	H(3,ℤ/4)×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	C₂²×#11715895	C₂²	1	1	yes	characteristic
11	#15690769	C₂²	1	1	yes	characteristic
12	#15690739	—	2	2	no	—
13	#15690733	—	2	2	no	—
14	#15690730	—	2	2	no	—
15	#15690724	—	2	4	no	—
16	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
17	#117×C₉₇	C₂²	1	2	yes	—
18	#13095511	—	2	2	no	—
19	#13095511	—	2	2	no	—
20	#13095511	—	2	2	no	—
21	#13095511	—	2	2	no	—
22	C₂×Dic₁₉₄	D₄	1	2	yes	—
23	C₂×D₃₈₈	D₄	1	2	yes	—
24	C₂×C₂²⋊D₉₇	—	2	2	no	—
25	C₂×C₄×D₉₇	—	2	2	no	—
26	Dic₃₈₈	—	2	2	no	—
27	#13095500	—	2	4	no	—
28	D₇₇₆	—	2	2	no	—
29	C₂×#11715895	—	2	2	no	—
30	C₂×#11715895	—	2	1	no	—
31	C₄⋊Dic₉₇	—	2	1	no	—
32	C₂²⋊Dic₉₇	D₄	1	2	yes	—
33	C₄×Dic₉₇	—	2	1	no	—
34	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
35	M₁₆×C₉₇	—	2	2	no	—
36	C₂×C₈×C₉₇	D₄	1	2	yes	—
37	C₄⋊C₄×C₉₇	—	2	2	no	—
38	C₂²⋊C₄×C₉₇	—	2	2	no	—
39	C₄²×C₉₇	—	2	2	no	—
40	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×D₉₇	—	2	2	no	—
42	C₂×C₄×C₉₇	—	4	2	no	—
43	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₉₇	#41	1	2	yes	—
46	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
47	C₂²⋊D₉₇	—	4	2	no	—
48	C₂²⋊D₉₇	—	4	2	no	—
49	C₂×Dic₉₇	—	2	2	no	—
50	C₂×Dic₉₇	—	2	2	no	—
51	D₃₈₈	—	2	2	no	—
52	D₃₈₈	—	2	2	no	—
53	C₄×D₉₇	—	4	2	no	—
54	C₄×D₉₇	—	4	2	no	—
55	Dic₁₉₄	—	2	2	no	—
56	Dic₁₉₄	—	2	2	no	—
57	C₈×C₉₇	—	2	2	no	—
58	#11715895	—	4	1	no	—
59	C₂²×C₉₇	—	4	1	no	—
60	C₂²×C₉₇	#121	1	2	yes	—
61	C₂²×C₉₇	#121	1	1	yes	characteristic
62	C₂×D₉₇	—	4	2	no	—
63	C₂×D₉₇	—	8	2	no	—
64	C₄×C₉₇	—	8	2	no	—
65	C₄×C₉₇	#120	1	1	yes	characteristic
66	C₄×C₉₇	#123	1	2	yes	—
67	C₄×C₉₇	#124	1	1	yes	characteristic
68	Dic₉₇	—	4	2	no	—
69	Dic₉₇	—	4	2	no	—
70	C₂×C₉₇	—	2	1	no	—
71	C₂×C₉₇	—	2	2	no	—
72	C₂×C₉₇	#394	1	1	yes	characteristic
73	D₉₇	—	8	2	no	—
74	#2053	—	97	1	no	maximal, Sylow
75	C₉₇	#2053	1	1	yes	characteristic, complemented, Sylow
76	#575	—	97	1	no	—
77	#424	—	97	1	no	—
78	#416	—	97	2	no	—
79	#410	—	97	2	no	—
80	H(3,ℤ/4)	D₉₇	1	1	yes	characteristic, complemented
81	C₂×#41	—	97	2	no	—
82	#137	—	194	2	no	—
83	AGL(1,ℤ/8)	—	194	2	no	—
84	#135	—	194	2	no	—
85	C₂×M₁₆	C₂×D₉₇	1	1	yes	characteristic
86	C₂²×C₈	—	97	1	no	—
87	#117	—	97	1	no	—
88	#117	C₂×D₉₇	1	2	yes	—
89	C₄⋊C₈	—	194	2	no	—
90	#103	—	194	2	no	—
91	#102	—	194	2	no	—
92	#99	—	194	4	no	—
93	#41	—	194	2	no	—
94	#41	—	194	2	no	—
95	#41	—	194	2	no	—
96	#41	—	194	2	no	—
97	C₂×Q₈	—	97	2	no	—
98	C₂×D₄	—	97	2	no	—
99	C₂×D₄	—	194	2	no	—
100	C₂²×C₄	—	194	2	no	—
101	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
102	Q₁₆	—	194	2	no	—
103	SD₁₆	—	194	4	no	—
104	D₈	—	194	2	no	—
105	M₁₆	—	2	2	no	—
106	C₂×C₈	—	194	2	no	—
107	C₂×C₈	—	194	1	no	—
108	C₂×C₈	D₃₈₈	1	2	yes	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	194	1	no	—
111	C₂²⋊C₄	—	2	2	no	—
112	C₂²⋊C₄	—	97	2	no	—
113	C₄²	—	2	2	no	—
114	C₄²	—	194	1	no	—
115	C₂³	—	194	2	no	—
116	C₂³	D₄×D₉₇	1	1	yes	characteristic
117	Q₈	—	194	2	no	—
118	Q₈	—	194	2	no	—
119	D₄	—	388	2	no	—
120	D₄	—	388	2	no	—
121	D₄	—	194	2	no	—
122	D₄	—	194	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	388	2	no	—
125	C₂×C₄	—	388	2	no	—
126	C₂×C₄	—	194	2	no	—
127	C₂×C₄	—	194	2	no	—
128	C₂×C₄	D₄×D₉₇	1	2	yes	—
129	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
130	C₂×C₄	#13095511	1	2	yes	—
131	C₂×C₄	#13095512	1	1	yes	characteristic
132	C₈	—	2	2	no	—
133	C₈	—	388	1	no	—
134	C₂²	—	4	1	no	—
135	C₂²	—	776	2	no	—
136	C₂²	—	388	2	no	—
137	C₂²	#15690783	1	2	yes	—
138	C₂²	#15690785	1	1	yes	characteristic
139	C₄	—	8	2	no	—
140	C₄	—	388	2	no	—
141	C₄	—	388	2	no	—
142	C₄	#15690780	1	1	yes	characteristic
143	C₄	#15690795	1	2	yes	—
144	C₄	#15690798	1	1	yes	characteristic
145	C₂	—	2	2	no	—
146	C₂	—	2	1	no	—
147	C₂	—	776	2	no	—
148	C₂	#20260653	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20261535

C₂

yes

maximal, characteristic

#20260763

C₂

yes

maximal, characteristic, complemented

#20260729

C₂

yes

maximal, complemented

#20260708

C₂

yes

maximal, complemented

H(3,ℤ/4)×C₉₇

C₂

yes

maximal, characteristic, complemented

C₂×#13095511

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

#15690769

C₂²

yes

—

—

—

—

C₂×M₁₆×C₉₇

C₂²

yes

characteristic

#117×C₉₇

C₂²

yes

—

—

—

—

—

C₂×Dic₁₉₄

D₄

yes

—

C₂×D₃₈₈

D₄

yes

—

C₂×C₂²⋊D₉₇

—

C₂×C₄×D₉₇

—

Dic₃₈₈

—

#13095500

—

D₇₇₆

—

C₂×#11715895

—

C₂×#11715895

—

C₄⋊Dic₉₇

—

C₂²⋊Dic₉₇

D₄

yes

—

C₄×Dic₉₇

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

M₁₆×C₉₇

—

C₂×C₈×C₉₇

D₄

yes

—

C₄⋊C₄×C₉₇

—

C₂²⋊C₄×C₉₇

—

C₄²×C₉₇

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

C₂²×D₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂×C₄×C₉₇

#41

yes

characteristic

C₂²⋊D₉₇

—

C₂²⋊D₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

D₃₈₈

—

D₃₈₈

—

C₄×D₉₇

—

C₄×D₉₇

—

Dic₁₉₄

—

Dic₁₉₄

—

C₈×C₉₇

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

#121

yes

—

C₂²×C₉₇

#121

yes

characteristic

C₂×D₉₇

—

C₂×D₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#120

yes

characteristic

C₄×C₉₇

#123

yes

—

C₄×C₉₇

#124

yes

characteristic

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#394

yes

characteristic

D₉₇

—

#2053

—

maximal, Sylow

C₉₇

#2053

yes

characteristic, complemented, Sylow

—

—

—

—

D₉₇

yes

characteristic, complemented

C₂×#41

—

#137

—

194

—

AGL(1,ℤ/8)

—

194

—

#135

—

194

—

C₂×M₁₆

C₂×D₉₇

yes

characteristic

C₂²×C₈

—

#117

—

#117

C₂×D₉₇

yes

—

C₄⋊C₈

—

194

—

#103

—

194

—

#102

—

194

—

#99

—

194

—

#41

—

194

—

#41

—

194

—

#41

—

194

—

#41

—

194

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

194

—

100

C₂²×C₄

—

194

—

101

C₂²×C₄

C₂²×D₉₇

yes

characteristic

102

Q₁₆

—

194

—

103

SD₁₆

—

194

—

104

D₈

—

194

—

105

M₁₆

—

106

C₂×C₈

—

194

—

107

C₂×C₈

—

194

—

108

C₂×C₈

D₃₈₈

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

194

—

111

C₂²⋊C₄

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

—

194

—

115

C₂³

—

194

—

116

C₂³

D₄×D₉₇

yes

characteristic

117

Q₈

—

194

—

118

Q₈

—

194

—

119

D₄

—

388

—

120

D₄

—

388

—

121

D₄

—

194

—

122

D₄

—

194

—

123

C₂×C₄

—

124

C₂×C₄

—

388

—

125

C₂×C₄

—

388

—

126

C₂×C₄

—

194

—

127

C₂×C₄

—

194

—

128

C₂×C₄

D₄×D₉₇

yes

—

129

C₂×C₄

C₂×D₃₈₈

yes

characteristic

130

C₂×C₄

#13095511

yes

—

131

C₂×C₄

#13095512

yes

characteristic

132

C₈

—

133

C₈

—

388

—

134

C₂²

—

135

C₂²

—

776

—

136

C₂²

—

388

—

137

C₂²

#15690783

yes

—

138

C₂²

#15690785

yes

characteristic

139

C₄

—

140

C₄

—

388

—

141

C₄

—

388

—

142

C₄

#15690780

yes

characteristic

143

C₄

#15690795

yes

—

144

C₄

#15690798

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

776

—

148

C₂

#20260653

yes

characteristic

	subgroup	G/H
center	C₄	#15690798
commutator, cosocle	C₂²×C₄×C₉₇	C₂³
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	H(3,ℤ/4)×C₉₇	C₂
socle	C₂×C₉₇	#394
prime radical	#20261535	C₂
hypercenter	H(3,ℤ/4)	D₉₇
hypocenter	C₉₇	#2053
LCS	G ▷ C₂²×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#15690798

commutator, cosocle

C₂²×C₄×C₉₇

C₂³

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

H(3,ℤ/4)×C₉₇

C₂

C₂×C₉₇

C₂

D₉₇

C₉₇

G ▷ C₂²×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
2. #20260763	C₂	semidirect
3. #20260729	C₂	semidirect
4. #20260708	C₂	semidirect
5. H(3,ℤ/4)×C₉₇	C₂	semidirect
75. C₉₇	#2053	semidirect
80. H(3,ℤ/4)	D₉₇	semidirect

H ≅ G/N

type

2. #20260763

C₂

semidirect

3. #20260729

C₂

semidirect

4. #20260708

C₂

semidirect

5. H(3,ℤ/4)×C₉₇

C₂

semidirect

75. C₉₇

#2053

semidirect

80. H(3,ℤ/4)

D₉₇

semidirect

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

subgroup	index
#20261535	2
#20260763	2
#20260729	2
#20260708	2
H(3,ℤ/4)×C₉₇	2
#2053	97

subgroup

index

H(3,ℤ/4)×C₉₇

Subgroups — #27143043

Conjugacy classes of subgroups

Complements

Maximal subgroups