Group #27141781 — subgroups

Subgroups — #27141781

Order 12416.

260 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 81 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	C₂×#15690818	C₂	1	1	yes	maximal, characteristic
2	#20260771	C₂	1	2	yes	maximal, complemented
3	#20260708	C₂	1	2	yes	maximal
4	#20260492	C₂	1	1	yes	maximal, characteristic, complemented
5	H(3,ℤ/4)×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13095511	C₂²	1	1	yes	characteristic
7	#15690818	—	2	2	no	—
8	#15690818	—	2	2	no	—
9	C₂×#13095494	C₄	1	2	yes	complemented
10	C₂²×#11715895	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₉₇	C₄	1	2	yes	complemented
12	#15690769	C₂²	1	2	yes	—
13	#15690743	—	2	2	no	—
14	#15690741	—	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	C₂×Dic₁₉₄	Q₈	1	1	yes	characteristic
21	C₂×D₃₈₈	Q₈	1	1	yes	characteristic
22	C₂×C₂²⋊D₉₇	D₄	1	2	yes	—
23	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	—
24	#13095494	—	2	4	no	—
25	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
26	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
27	C₂×#11715895	D₄	1	2	yes	—
28	C₂×#11715895	D₄	1	2	yes	—
29	C₈×D₉₇	—	2	2	no	—
30	C₈×D₉₇	—	2	2	no	—
31	C₄⋊Dic₉₇	—	2	2	no	—
32	C₂²⋊Dic₉₇	—	2	2	no	—
33	C₄×Dic₉₇	—	2	2	no	—
34	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
35	M₁₆×C₉₇	—	2	2	no	—
36	C₂×C₈×C₉₇	C₂×C₄	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	C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₉₇	—	4	2	no	—
43	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₉₇	#41	1	2	yes	—
47	C₂²⋊D₉₇	—	2	4	no	—
48	C₂×Dic₉₇	—	4	2	no	—
49	C₂×Dic₉₇	C₄⋊C₄	1	2	yes	—
50	D₃₈₈	—	2	2	no	—
51	C₄×D₉₇	—	2	2	no	—
52	C₄×D₉₇	—	2	2	no	—
53	Dic₁₉₄	—	2	2	no	—
54	C₈×C₉₇	—	2	2	no	—
55	#11715895	C₂²⋊C₄	1	1	yes	characteristic
56	#11715895	C₂²⋊C₄	1	1	yes	characteristic
57	#11715895	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₉₇	—	4	1	no	—
59	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₉₇	#121	1	2	yes	—
61	C₂×D₉₇	—	2	2	no	—
62	C₂×D₉₇	—	4	2	no	—
63	C₄×C₉₇	—	8	2	no	—
64	C₄×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₄×C₉₇	#117	1	1	yes	characteristic
66	C₄×C₉₇	C₂²⋊Q₈	1	2	yes	—
67	Dic₉₇	—	2	2	no	—
68	Dic₉₇	—	8	2	no	—
69	C₂×C₉₇	—	2	2	no	—
70	C₂×C₉₇	—	2	1	no	—
71	C₂×C₉₇	#380	1	1	yes	characteristic
72	D₉₇	—	4	2	no	—
73	#1812	—	97	1	no	maximal, Sylow
74	C₉₇	#1812	1	1	yes	characteristic, complemented, Sylow
75	C₂×#131	—	97	1	no	—
76	#427	—	97	2	no	—
77	#410	—	97	2	no	—
78	H(3,ℤ/4)	—	97	1	no	—
79	H(3,ℤ/4)	D₉₇	1	1	yes	characteristic, complemented
80	C₂×#41	—	97	1	no	—
81	#131	—	194	2	no	—
82	#131	—	194	2	no	—
83	C₂×M₁₆	—	97	2	no	—
84	C₂×M₁₆	C₂×D₉₇	1	1	yes	characteristic
85	C₂²×C₈	—	97	1	no	—
86	C₂²×C₈	—	97	2	no	—
87	#117	—	97	2	no	—
88	#117	C₂×D₉₇	1	2	yes	—
89	#107	—	194	2	no	—
90	#106	—	194	2	no	—
91	#99	—	194	4	no	—
92	#41	—	194	2	no	—
93	#41	—	194	2	no	—
94	C₂×Q₈	—	97	1	no	—
95	C₂×D₄	—	97	2	no	—
96	C₂×D₄	—	97	1	no	—
97	C₂²×C₄	—	97	2	no	—
98	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
99	M₁₆	—	2	2	no	—
100	M₁₆	—	194	4	no	—
101	C₂×C₈	—	97	1	no	—
102	C₂×C₈	—	97	1	no	—
103	C₂×C₈	—	97	2	no	—
104	C₂×C₈	—	97	2	no	—
105	C₂×C₈	—	194	2	no	—
106	C₂×C₈	—	194	2	no	—
107	C₂×C₈	C₄×D₉₇	1	2	yes	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	194	2	no	—
110	C₂²⋊C₄	—	2	2	no	—
111	C₂²⋊C₄	—	194	2	no	—
112	C₄²	—	2	2	no	—
113	C₄²	—	194	2	no	—
114	C₂³	—	97	2	no	—
115	C₂³	D₄×D₉₇	1	1	yes	characteristic
116	Q₈	—	194	2	no	—
117	D₄	—	194	4	no	—
118	D₄	—	194	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	97	2	no	—
121	C₂×C₄	—	388	2	no	—
122	C₂×C₄	—	194	2	no	—
123	C₂×C₄	—	194	2	no	—
124	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
125	C₂×C₄	D₄×D₉₇	1	2	yes	—
126	C₂×C₄	Q₈×D₉₇	1	1	yes	characteristic
127	C₂×C₄	#13095511	1	2	yes	—
128	C₈	—	2	2	no	—
129	C₈	—	97	1	no	—
130	C₈	—	97	1	no	—
131	C₈	—	97	2	no	—
132	C₂²	—	4	1	no	—
133	C₂²	—	388	2	no	—
134	C₂²	—	194	2	no	—
135	C₂²	C₄⋊C₄×D₉₇	1	1	yes	characteristic
136	C₂²	#15690783	1	2	yes	—
137	C₄	—	8	2	no	—
138	C₄	—	776	2	no	—
139	C₄	—	194	2	no	—
140	C₄	C₂²⋊C₄×D₉₇	1	1	yes	characteristic
141	C₄	#15690767	1	1	yes	characteristic
142	C₄	#15690788	1	2	yes	—
143	C₂	—	2	2	no	—
144	C₂	—	2	1	no	—
145	C₂	—	388	2	no	—
146	C₂	#20260594	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690818

C₂

yes

maximal, characteristic

#20260771

C₂

yes

maximal, complemented

#20260708

C₂

yes

maximal

#20260492

C₂

yes

maximal, characteristic, complemented

H(3,ℤ/4)×C₉₇

C₂

yes

maximal, characteristic, complemented

C₂×#13095511

C₂²

yes

—

—

C₄

yes

complemented

C₂²×#11715895

C₂²

yes

characteristic

C₂×C₈×D₉₇

C₄

yes

complemented

#15690769

C₂²

yes

—

#15690743

—

#15690741

—

#15690724

—

C₂×M₁₆×C₉₇

C₂²

yes

characteristic

#117×C₉₇

C₂²

yes

—

#13095511

—

#13095511

—

C₂×Dic₁₉₄

Q₈

yes

characteristic

C₂×D₃₈₈

Q₈

yes

characteristic

C₂×C₂²⋊D₉₇

D₄

yes

—

C₂×C₄×D₉₇

C₂×C₄

yes

—

#13095494

—

C₂×#11715895

C₂×C₄

yes

characteristic

C₂×#11715895

C₂×C₄

yes

characteristic

C₂×#11715895

D₄

yes

—

C₂×#11715895

D₄

yes

—

C₈×D₉₇

—

C₈×D₉₇

—

C₄⋊Dic₉₇

—

C₂²⋊Dic₉₇

—

C₄×Dic₉₇

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

M₁₆×C₉₇

—

C₂×C₈×C₉₇

C₂×C₄

yes

—

C₄⋊C₄×C₉₇

—

C₂²⋊C₄×C₉₇

—

C₄²×C₉₇

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

C₂²×D₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂²⋊D₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

C₄⋊C₄

yes

—

D₃₈₈

—

C₄×D₉₇

—

C₄×D₉₇

—

Dic₁₉₄

—

C₈×C₉₇

—

#11715895

C₂²⋊C₄

yes

characteristic

#11715895

C₂²⋊C₄

yes

characteristic

#11715895

C₂²⋊C₄

yes

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₉₇

#121

yes

—

C₂×D₉₇

—

C₂×D₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₉₇

#117

yes

characteristic

C₄×C₉₇

C₂²⋊Q₈

yes

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#380

yes

characteristic

D₉₇

—

#1812

—

maximal, Sylow

C₉₇

#1812

yes

characteristic, complemented, Sylow

—

—

—

—

D₉₇

yes

characteristic, complemented

C₂×#41

—

#131

—

194

—

#131

—

194

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₉₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

#117

—

#117

C₂×D₉₇

yes

—

#107

—

194

—

#106

—

194

—

#99

—

194

—

#41

—

194

—

#41

—

194

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

M₁₆

—

100

M₁₆

—

194

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

194

—

106

C₂×C₈

—

194

—

107

C₂×C₈

C₄×D₉₇

yes

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

194

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

—

194

—

112

C₄²

—

113

C₄²

—

194

—

114

C₂³

—

115

C₂³

D₄×D₉₇

yes

characteristic

116

Q₈

—

194

—

117

D₄

—

194

—

118

D₄

—

194

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

388

—

122

C₂×C₄

—

194

—

123

C₂×C₄

—

194

—

124

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

125

C₂×C₄

D₄×D₉₇

yes

—

126

C₂×C₄

Q₈×D₉₇

yes

characteristic

127

C₂×C₄

#13095511

yes

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₂²

—

133

C₂²

—

388

—

134

C₂²

—

194

—

135

C₂²

C₄⋊C₄×D₉₇

yes

characteristic

136

C₂²

#15690783

yes

—

137

C₄

—

138

C₄

—

776

—

139

C₄

—

194

—

140

C₄

C₂²⋊C₄×D₉₇

yes

characteristic

141

C₄

#15690767

yes

characteristic

142

C₄

#15690788

yes

—

143

C₂

—

144

C₂

—

145

C₂

—

388

—

146

C₂

#20260594

yes

characteristic

	subgroup	G/H
center	C₄	#15690767
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	H(3,ℤ/4)×C₉₇	C₂
socle	C₂×C₉₇	#380
prime radical	C₂×#13095511	C₂²
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	H(3,ℤ/4)	D₉₇
hypocenter	C₉₇	#1812
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#15690767

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

H(3,ℤ/4)×C₉₇

C₂

C₂×C₉₇

C₂²

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
2. #20260771	C₂	semidirect
4. #20260492	C₂	semidirect
5. H(3,ℤ/4)×C₉₇	C₂	semidirect
9. C₂×#13095494	C₄	semidirect
11. C₂×C₈×D₉₇	C₄	semidirect
74. C₉₇	#1812	semidirect
79. H(3,ℤ/4)	D₉₇	semidirect

H ≅ G/N

type

2. #20260771

C₂

semidirect

4. #20260492

C₂

semidirect

5. H(3,ℤ/4)×C₉₇

C₂

semidirect

9. C₂×#13095494

C₄

semidirect

11. C₂×C₈×D₉₇

C₄

semidirect

74. C₉₇

#1812

semidirect

79. 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
C₂×#15690818	2
#20260771	2
#20260708	2
#20260492	2
H(3,ℤ/4)×C₉₇	2
#1812	97

subgroup

index

H(3,ℤ/4)×C₉₇

Subgroups — #27141781

Conjugacy classes of subgroups

Complements

Maximal subgroups