Group #27152246 — subgroups

Subgroups — #27152246

Order 12416.

356 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 157 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	#20261292	C₂	1	1	yes	maximal, characteristic
2	#20261138	C₂	1	2	yes	maximal, complemented
3	#20261134	C₂	1	4	yes	maximal, complemented
4	#20261131	C₂	1	2	yes	maximal, complemented
5	#20260771	C₂	1	4	yes	maximal
6	C₂×C₄×#11715895	C₂	1	1	yes	maximal, characteristic
7	#533×C₉₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11715895	C₂²	1	2	yes	—
9	#15690810	C₂²	1	2	yes	—
10	#15690802	C₂²	1	2	yes	—
11	#15690789	—	2	2	no	—
12	C₄×Dic₁₉₄	—	2	2	no	—
13	#15690769	C₂²	1	2	yes	—
14	#15690743	C₂²	1	8	yes	—
15	#15690741	C₂²	1	8	yes	—
16	C₄×#11715895	C₂²	1	1	yes	characteristic
17	C₄×#11715895	C₂²	1	2	yes	—
18	C₄×#11715895	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₉₇	C₂²	1	2	yes	—
20	#125×C₉₇	C₂²	1	2	yes	—
21	C₂²⋊Q₈×C₉₇	—	2	2	no	—
22	C₄×Q₈×C₉₇	—	2	2	no	—
23	#117×C₉₇	C₂²	1	2	yes	—
24	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
25	C₂×Dic₁₉₄	—	2	2	no	—
26	C₂×#11715895	Q₈	1	8	yes	complemented
27	C₂×#11715895	C₂³	1	2	yes	—
28	C₂×#11715895	C₂³	1	2	yes	—
29	C₄⋊Dic₉₇	—	2	2	no	—
30	C₄⋊Dic₉₇	C₂³	1	4	yes	—
31	#13095490	—	2	4	no	—
32	C₂²⋊Dic₉₇	—	2	2	no	—
33	C₄×Dic₉₇	—	2	2	no	—
34	C₂×Q₈×C₉₇	—	2	2	no	—
35	C₂²×C₄×C₉₇	D₄	1	2	yes	—
36	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₉₇	—	2	4	no	—
38	C₄⋊C₄×C₉₇	—	2	2	no	—
39	C₄⋊C₄×C₉₇	C₂³	1	4	yes	—
40	C₂²⋊C₄×C₉₇	—	2	2	no	—
41	C₄²×C₉₇	—	2	2	no	—
42	C₄²×C₉₇	D₄	1	2	yes	—
43	C₄²×C₉₇	C₂³	1	1	yes	characteristic
44	C₄²×C₉₇	C₂³	1	1	yes	characteristic
45	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₉₇	—	4	4	no	—
47	C₂×C₄×C₉₇	—	2	4	no	—
48	C₂×C₄×C₉₇	—	2	4	no	—
49	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₉₇	C₂×Q₈	1	4	yes	—
53	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₉₇	—	2	4	no	—
55	Dic₁₉₄	—	4	4	no	—
56	#11715895	C₂×Q₈	1	4	yes	—
57	#11715895	C₂×Q₈	1	4	yes	—
58	C₂²×C₉₇	—	2	2	no	—
59	C₂²×C₉₇	C₄⋊Q₈	1	2	yes	—
60	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
61	C₄×C₉₇	—	2	2	no	—
62	C₄×C₉₇	—	4	4	no	—
63	C₄×C₉₇	C₄⋊Q₈	1	2	yes	—
64	C₄×C₉₇	#135	1	4	yes	—
65	C₄×C₉₇	C₂²×Q₈	1	2	yes	—
66	Dic₉₇	—	4	4	no	—
67	C₂×C₉₇	—	2	2	no	—
68	C₂×C₉₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₉₇	C₂×#135	1	2	yes	—
70	#3160	—	97	1	no	maximal, Sylow
71	C₉₇	#3160	1	1	yes	characteristic, complemented, Sylow
72	#533	—	97	1	no	—
73	#533	D₉₇	1	1	yes	characteristic
74	#500	—	97	2	no	—
75	#499	—	97	4	no	—
76	#498	—	97	2	no	—
77	#427	—	97	4	no	—
78	C₂×C₄×C₈	—	97	1	no	—
79	C₂²×C₈	—	97	2	no	—
80	C₄⋊Q₈	—	97	2	no	—
81	C₄⋊Q₈	C₂×D₉₇	1	2	yes	—
82	#125	—	97	2	no	—
83	#125	C₂×D₉₇	1	2	yes	—
84	C₂²⋊Q₈	—	2	2	no	—
85	C₂²⋊Q₈	—	194	2	no	—
86	C₄×Q₈	—	2	2	no	—
87	C₄×Q₈	—	194	2	no	—
88	#117	—	97	2	no	—
89	#117	C₂×D₉₇	1	2	yes	—
90	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
91	#107	—	97	8	no	—
92	#106	—	97	8	no	—
93	C₄×C₈	—	97	1	no	—
94	C₄×C₈	—	97	2	no	—
95	C₄×C₈	—	97	1	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	194	2	no	—
98	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
99	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
100	C₂×C₈	—	97	8	no	—
101	C₂×C₈	—	97	2	no	—
102	C₂×C₈	—	97	2	no	—
103	C₄⋊C₄	—	2	4	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	97	4	no	—
106	C₄⋊C₄	—	194	4	no	—
107	C₄⋊C₄	—	194	2	no	—
108	C₄⋊C₄	C₂²×D₉₇	1	4	yes	—
109	C₂²⋊C₄	—	2	2	no	—
110	C₂²⋊C₄	—	194	2	no	—
111	C₄²	—	2	2	no	—
112	C₄²	—	194	2	no	—
113	C₄²	C₂²⋊D₉₇	1	2	yes	—
114	C₄²	C₂²×D₉₇	1	1	yes	characteristic
115	C₄²	C₂²×D₉₇	1	1	yes	characteristic
116	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
117	Q₈	—	4	4	no	—
118	Q₈	—	388	4	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	194	4	no	—
122	C₂×C₄	C₂×C₂²⋊D₉₇	1	2	yes	—
123	C₂×C₄	C₂×C₂²⋊D₉₇	1	2	yes	—
124	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
125	C₂×C₄	Q₈×D₉₇	1	4	yes	—
126	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
127	C₈	—	97	4	no	—
128	C₈	—	97	4	no	—
129	C₂²	—	2	2	no	—
130	C₂²	#15690808	1	2	yes	—
131	C₂²	C₂²×C₂²⋊D₉₇	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	4	4	no	—
134	C₄	—	388	4	no	—
135	C₄	#15690808	1	2	yes	—
136	C₄	#15690830	1	4	yes	—
137	C₄	C₂×Q₈×D₉₇	1	2	yes	—
138	C₂	—	2	2	no	—
139	C₂	C₂×#15690808	1	1	yes	characteristic
140	C₂	C₂×#15690830	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20261292

C₂

yes

maximal, characteristic

#20261138

C₂

yes

maximal, complemented

#20261134

C₂

yes

maximal, complemented

#20261131

C₂

yes

maximal, complemented

#20260771

C₂

yes

maximal

C₂×C₄×#11715895

C₂

yes

maximal, characteristic

#533×C₉₇

C₂

yes

maximal, characteristic

C₂²×#11715895

C₂²

yes

—

#15690810

C₂²

yes

—

#15690802

C₂²

yes

—

#15690789

—

C₄×Dic₁₉₄

—

#15690769

C₂²

yes

—

#15690743

C₂²

yes

—

#15690741

C₂²

yes

—

C₄×#11715895

C₂²

yes

characteristic

C₄×#11715895

C₂²

yes

—

C₄×#11715895

C₂²

yes

characteristic

C₄⋊Q₈×C₉₇

C₂²

yes

—

#125×C₉₇

C₂²

yes

—

C₂²⋊Q₈×C₉₇

—

C₄×Q₈×C₉₇

—

#117×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂×Dic₁₉₄

—

C₂×#11715895

Q₈

yes

complemented

C₂×#11715895

C₂³

yes

—

C₂×#11715895

C₂³

yes

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

C₂³

yes

—

#13095490

—

C₂²⋊Dic₉₇

—

C₄×Dic₉₇

—

C₂×Q₈×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

C₂³

yes

—

C₂²⋊C₄×C₉₇

—

C₄²×C₉₇

—

C₄²×C₉₇

D₄

yes

—

C₄²×C₉₇

C₂³

yes

characteristic

C₄²×C₉₇

C₂³

yes

characteristic

C₂³×C₉₇

C₂×D₄

yes

characteristic

Q₈×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

—

C₂×C₄×C₉₇

C₂⁴

yes

characteristic

C₂×Dic₉₇

—

Dic₁₉₄

—

#11715895

C₂×Q₈

yes

—

#11715895

C₂×Q₈

yes

—

C₂²×C₉₇

—

C₂²×C₉₇

C₄⋊Q₈

yes

—

C₂²×C₉₇

C₂²×D₄

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₄⋊Q₈

yes

—

C₄×C₉₇

#135

yes

—

C₄×C₉₇

C₂²×Q₈

yes

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₉₇

C₂×#135

yes

—

#3160

—

maximal, Sylow

C₉₇

#3160

yes

characteristic, complemented, Sylow

#533

—

#533

D₉₇

yes

—

—

—

—

C₂×C₄×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₉₇

yes

—

#125

—

#125

C₂×D₉₇

yes

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

194

—

C₄×Q₈

—

C₄×Q₈

—

194

—

#117

—

#117

C₂×D₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

#107

—

#106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

194

—

C₂²×C₄

C₂²⋊D₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

194

—

107

C₄⋊C₄

—

194

—

108

C₄⋊C₄

C₂²×D₉₇

yes

—

109

C₂²⋊C₄

—

110

C₂²⋊C₄

—

194

—

111

C₄²

—

112

C₄²

—

194

—

113

C₄²

C₂²⋊D₉₇

yes

—

114

C₄²

C₂²×D₉₇

yes

characteristic

115

C₄²

C₂²×D₉₇

yes

characteristic

116

C₂³

C₂×C₂²⋊D₉₇

yes

characteristic

117

Q₈

—

118

Q₈

—

388

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

194

—

122

C₂×C₄

C₂×C₂²⋊D₉₇

yes

—

123

C₂×C₄

C₂×C₂²⋊D₉₇

yes

—

124

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

125

C₂×C₄

Q₈×D₉₇

yes

—

126

C₂×C₄

C₂³×D₉₇

yes

characteristic

127

C₈

—

128

C₈

—

129

C₂²

—

130

C₂²

#15690808

yes

—

131

C₂²

C₂²×C₂²⋊D₉₇

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

388

—

135

C₄

#15690808

yes

—

136

C₄

#15690830

yes

—

137

C₄

C₂×Q₈×D₉₇

yes

—

138

C₂

—

139

C₂

C₂×#15690808

yes

characteristic

140

C₂

C₂×#15690830

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₉₇
commutator, cosocle	C₂×C₄×C₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	#533×C₉₇	C₂
socle	C₂²×C₉₇	C₂²×D₄
prime radical	C₂³×C₉₇	C₂×D₄
hypercenter	#533	D₉₇
hypocenter	C₉₇	#3160
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₉₇

commutator, cosocle

C₂×C₄×C₉₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₉₇

Fitting

#533×C₉₇

C₂

socle

C₂²×C₉₇

C₂²×D₄

prime radical

C₂³×C₉₇

C₂×D₄

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #20261138	C₂	semidirect
3. #20261134	C₂	semidirect
4. #20261131	C₂	semidirect
26. C₂×#11715895	Q₈	semidirect
71. C₉₇	#3160	semidirect

H ≅ G/N

type

2. #20261138

C₂

semidirect

3. #20261134

C₂

semidirect

4. #20261131

C₂

semidirect

26. C₂×#11715895

Q₈

semidirect

71. C₉₇

#3160

semidirect

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

subgroup	index
#20261292	2
#20261138	2
#20261134	2
#20261131	2
#20260771	2
C₂×C₄×#11715895	2
#533×C₉₇	2
#3160	97

subgroup

index

#533×C₉₇

Subgroups — #27152246

Conjugacy classes of subgroups

Complements

Maximal subgroups