Group #27142651 — subgroups

Subgroups — #27142651

Order 12416.

276 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂²×Dic₃₈₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#15690746	C₂	1	1	yes	maximal, characteristic
3	#20260743	C₂	1	2	yes	maximal
4	C₂×#15690728	C₂	1	2	yes	maximal
5	#405×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₉₄	C₂²	1	2	yes	—
7	C₂×Dic₃₈₈	—	2	2	no	—
8	C₂×Dic₃₈₈	—	2	1	no	—
9	C₂×Dic₃₈₈	—	2	1	no	—
10	C₂×Dic₃₈₈	C₄	1	4	yes	complemented
11	C₂×#13095495	C₂²	1	2	yes	—
12	#15690746	—	2	2	no	—
13	#15690733	—	2	4	no	—
14	#15690728	—	2	4	no	—
15	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
17	#117×C₉₇	C₂²	1	1	yes	characteristic
18	#97×C₉₇	—	2	1	no	—
19	C₄×C₈×C₉₇	—	2	1	no	—
20	C₂×Dic₁₉₄	—	2	2	no	—
21	C₂×Dic₁₉₄	—	2	2	no	—
22	C₂×Dic₁₉₄	—	2	2	no	—
23	C₂×Dic₁₉₄	—	2	2	no	—
24	C₂×Dic₁₉₄	—	2	2	no	—
25	C₂×Dic₁₉₄	C₂×C₄	1	4	yes	—
26	C₂²×Dic₉₇	—	2	2	no	—
27	Dic₃₈₈	—	2	4	no	—
28	Dic₃₈₈	—	2	4	no	—
29	#13095495	—	2	2	no	—
30	#13095495	—	2	2	no	—
31	C₂×#11715895	—	2	2	no	—
32	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
33	M₁₆×C₉₇	—	2	1	no	—
34	M₁₆×C₉₇	D₄	1	2	yes	—
35	C₂×C₈×C₉₇	—	2	1	no	—
36	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
37	C₂×C₈×C₉₇	D₄	1	2	yes	—
38	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
39	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
40	C₄⋊C₄×C₉₇	D₄	1	2	yes	—
41	C₂²⋊C₄×C₉₇	—	2	1	no	—
42	C₄²×C₉₇	—	2	1	no	—
43	C₂³×C₉₇	#41	1	1	yes	characteristic
44	C₂×C₄×C₉₇	—	2	2	no	—
45	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
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	1	yes	characteristic
51	C₂×Dic₉₇	—	2	4	no	—
52	C₂×Dic₉₇	—	4	2	no	—
53	C₂×Dic₉₇	—	4	2	no	—
54	Dic₁₉₄	—	2	4	no	—
55	Dic₁₉₄	—	2	4	no	—
56	Dic₁₉₄	—	4	2	no	—
57	Dic₁₉₄	—	4	2	no	—
58	C₈×C₉₇	—	2	2	no	—
59	C₈×C₉₇	C₂²⋊C₄	1	2	yes	—
60	C₈×C₉₇	C₂²⋊C₄	1	2	yes	—
61	#11715895	—	4	2	no	—
62	C₂²×C₉₇	—	2	2	no	—
63	C₂²×C₉₇	C₄×D₄	1	1	yes	characteristic
64	C₂²×C₉₇	#121	1	1	yes	characteristic
65	C₂²×C₉₇	#124	1	1	yes	characteristic
66	C₄×C₉₇	—	4	2	no	—
67	C₄×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₄×C₉₇	C₄×D₄	1	1	yes	characteristic
69	C₄×C₉₇	#121	1	1	yes	characteristic
70	C₄×C₉₇	#127	1	1	yes	characteristic
71	Dic₉₇	—	4	4	no	—
72	C₂×C₉₇	—	2	1	no	—
73	C₂×C₉₇	—	2	1	no	—
74	C₂×C₉₇	#390	1	1	yes	characteristic
75	C₂×C₉₇	#473	1	2	yes	—
76	#1979	—	97	1	no	maximal, Sylow
77	C₉₇	#1979	1	1	yes	characteristic, complemented, Sylow
78	C₂²×Q₁₆	—	97	1	no	—
79	C₂×#108	—	97	1	no	—
80	#419	—	97	2	no	—
81	C₂×#101	—	97	2	no	—
82	#405	D₉₇	1	1	yes	characteristic
83	C₂²×Q₈	—	97	2	no	—
84	C₂×Q₁₆	—	97	4	no	—
85	C₂×Q₁₆	—	194	2	no	—
86	C₂×Q₁₆	—	194	1	no	—
87	C₂×Q₁₆	—	194	1	no	—
88	C₂×M₁₆	—	97	2	no	—
89	C₂×M₁₆	C₂×D₉₇	1	1	yes	characteristic
90	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
91	#117	C₂×D₉₇	1	1	yes	characteristic
92	#108	—	194	2	no	—
93	#103	—	194	4	no	—
94	#101	—	194	4	no	—
95	#97	—	2	1	no	—
96	C₄×C₈	—	2	1	no	—
97	C₂×Q₈	—	97	4	no	—
98	C₂×Q₈	—	194	2	no	—
99	C₂×Q₈	—	194	2	no	—
100	C₂×Q₈	—	194	2	no	—
101	C₂×Q₈	—	194	2	no	—
102	C₂×Q₈	—	194	2	no	—
103	C₂²×C₄	—	194	2	no	—
104	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
105	Q₁₆	—	194	4	no	—
106	Q₁₆	—	194	4	no	—
107	M₁₆	—	2	1	no	—
108	M₁₆	—	194	2	no	—
109	M₁₆	—	194	2	no	—
110	M₁₆	D₃₈₈	1	2	yes	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	194	2	no	—
113	C₂×C₈	C₄×D₉₇	1	2	yes	—
114	C₂×C₈	D₃₈₈	1	2	yes	—
115	C₂×C₈	C₂²⋊D₉₇	1	1	yes	characteristic
116	C₂×C₈	C₂²⋊D₉₇	1	1	yes	characteristic
117	C₄⋊C₄	D₃₈₈	1	2	yes	—
118	C₂²⋊C₄	—	2	1	no	—
119	C₄²	—	2	1	no	—
120	C₂³	#13095511	1	1	yes	characteristic
121	Q₈	—	388	2	no	—
122	Q₈	—	388	2	no	—
123	Q₈	—	194	4	no	—
124	Q₈	—	194	4	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	388	2	no	—
127	C₂×C₄	—	388	2	no	—
128	C₂×C₄	—	194	4	no	—
129	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
131	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
132	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
133	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
134	C₂×C₄	#13095511	1	1	yes	characteristic
135	C₈	—	2	2	no	—
136	C₈	—	388	2	no	—
137	C₈	#13095488	1	2	yes	—
138	C₈	#13095488	1	2	yes	—
139	C₂²	—	2	2	no	—
140	C₂²	C₄×D₃₈₈	1	1	yes	characteristic
141	C₂²	#15690784	1	1	yes	characteristic
142	C₂²	#15690798	1	1	yes	characteristic
143	C₄	—	4	2	no	—
144	C₄	—	388	4	no	—
145	C₄	C₂×#13095488	1	1	yes	characteristic
146	C₄	C₄×D₃₈₈	1	1	yes	characteristic
147	C₄	#15690784	1	1	yes	characteristic
148	C₄	#15690807	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	#20260637	1	1	yes	characteristic
152	C₂	#20261010	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₃₈₈

C₂

yes

maximal, characteristic

C₂×#15690746

C₂

yes

maximal, characteristic

#20260743

C₂

yes

maximal

C₂×#15690728

C₂

yes

maximal

#405×C₉₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₉₄

C₂²

yes

—

C₂×Dic₃₈₈

—

C₂×Dic₃₈₈

—

C₂×Dic₃₈₈

—

C₂×Dic₃₈₈

C₄

yes

complemented

C₂×#13095495

C₂²

yes

—

#15690746

—

#15690733

—

#15690728

—

C₂×M₁₆×C₉₇

C₂²

yes

characteristic

C₂²×C₈×C₉₇

C₂²

yes

characteristic

#117×C₉₇

C₂²

yes

characteristic

#97×C₉₇

—

C₄×C₈×C₉₇

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

C₂×C₄

yes

—

C₂²×Dic₉₇

—

Dic₃₈₈

—

Dic₃₈₈

—

#13095495

—

#13095495

—

C₂×#11715895

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

M₁₆×C₉₇

—

M₁₆×C₉₇

D₄

yes

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

C₂×C₄

yes

—

C₂×C₈×C₉₇

D₄

yes

—

C₂×C₈×C₉₇

D₄

yes

characteristic

C₂×C₈×C₉₇

D₄

yes

characteristic

C₄⋊C₄×C₉₇

D₄

yes

—

C₂²⋊C₄×C₉₇

—

C₄²×C₉₇

—

C₂³×C₉₇

#41

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

#41

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

Dic₁₉₄

—

Dic₁₉₄

—

Dic₁₉₄

—

Dic₁₉₄

—

C₈×C₉₇

—

C₈×C₉₇

C₂²⋊C₄

yes

—

C₈×C₉₇

C₂²⋊C₄

yes

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

C₄×D₄

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₄×C₉₇

—

C₄×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₉₇

C₄×D₄

yes

C₄×C₉₇

yes

C₄×C₉₇

yes

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#390

yes

characteristic

C₂×C₉₇

#473

yes

—

#1979

—

maximal, Sylow

C₉₇

#1979

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

—

—

—

D₉₇

yes

characteristic

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

194

—

C₂×Q₁₆

—

194

—

C₂×Q₁₆

—

194

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₉₇

yes

characteristic

C₂²×C₈

C₂×D₉₇

yes

characteristic

#117

C₂×D₉₇

yes

characteristic

#108

—

194

—

#103

—

194

—

#101

—

194

—

#97

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

194

—

C₂×Q₈

—

194

—

100

C₂×Q₈

—

194

—

101

C₂×Q₈

—

194

—

102

C₂×Q₈

—

194

—

103

C₂²×C₄

—

194

—

104

C₂²×C₄

C₂²×D₉₇

yes

characteristic

105

Q₁₆

—

194

—

106

Q₁₆

—

194

—

107

M₁₆

—

108

M₁₆

—

194

—

109

M₁₆

—

194

—

110

M₁₆

D₃₈₈

yes

—

111

C₂×C₈

—

112

C₂×C₈

—

194

—

113

C₂×C₈

C₄×D₉₇

yes

—

114

C₂×C₈

D₃₈₈

yes

—

115

C₂×C₈

C₂²⋊D₉₇

yes

characteristic

116

C₂×C₈

C₂²⋊D₉₇

yes

characteristic

117

C₄⋊C₄

D₃₈₈

yes

—

118

C₂²⋊C₄

—

119

C₄²

—

120

C₂³

#13095511

yes

characteristic

121

Q₈

—

388

—

122

Q₈

—

388

—

123

Q₈

—

194

—

124

Q₈

—

194

—

125

C₂×C₄

—

126

C₂×C₄

—

388

—

127

C₂×C₄

—

388

—

128

C₂×C₄

—

194

—

129

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

131

C₂×C₄

C₂×D₃₈₈

yes

characteristic

132

C₂×C₄

C₂×D₃₈₈

yes

characteristic

133

C₂×C₄

C₂×D₃₈₈

yes

characteristic

134

C₂×C₄

#13095511

yes

characteristic

135

C₈

—

136

C₈

—

388

—

137

C₈

#13095488

yes

—

138

C₈

#13095488

yes

—

139

C₂²

—

140

C₂²

C₄×D₃₈₈

yes

characteristic

141

C₂²

#15690784

yes

characteristic

142

C₂²

#15690798

yes

characteristic

143

C₄

—

144

C₄

—

388

—

145

C₄

C₂×#13095488

yes

characteristic

146

C₄

C₄×D₃₈₈

yes

characteristic

147

C₄

#15690784

yes

characteristic

148

C₄

#15690807

yes

characteristic

149

C₂

—

150

C₂

—

151

C₂

#20260637

yes

characteristic

152

C₂

#20261010

yes

—

	subgroup	G/H
center	C₂²	#15690784
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#405×C₉₇	C₂
socle	C₂²×C₉₇	#121
prime radical	C₂³×C₉₇	#41
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#405	D₉₇
hypocenter	C₉₇	#1979
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

subgroup

G/H

center

C₂²

#15690784

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

#405×C₉₇

C₂

C₂²×C₉₇

C₂³×C₉₇

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

N	H ≅ G/N	type
10. C₂×Dic₃₈₈	C₄	semidirect
77. C₉₇	#1979	semidirect

H ≅ G/N

type

10. C₂×Dic₃₈₈

C₄

semidirect

77. C₉₇

#1979

semidirect

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

subgroup	index
C₂²×Dic₃₈₈	2
C₂×#15690746	2
#20260743	2
C₂×#15690728	2
#405×C₉₇	2
#1979	97

subgroup

index

C₂²×Dic₃₈₈

#405×C₉₇

Subgroups — #27142651

Conjugacy classes of subgroups

Complements

Maximal subgroups