Group #27150008 — subgroups

Subgroups — #27150008

Order 12416.

356 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 133 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	#20260677	C₂	1	4	yes	maximal
2	#20260674	C₂	1	2	yes	maximal
3	#20260656	C₂	1	4	yes	maximal
4	#20260655	C₂	1	2	yes	maximal
5	#20260601	C₂	1	2	yes	maximal
6	#382×C₉₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₉₄	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
11	C₂×#13095490	C₂²	1	4	yes	—
12	#15690717	C₂²	1	4	yes	—
13	#15690717	C₂²	1	2	yes	—
14	#15690717	C₂²	1	4	yes	—
15	#15690717	C₂²	1	4	yes	—
16	#15690717	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
20	#95×C₉₇	C₂²	1	4	yes	—
21	C₂×Dic₁₉₄	—	4	4	no	—
22	C₂×Dic₁₉₄	—	4	2	no	—
23	C₂²×Dic₉₇	C₂³	1	4	yes	—
24	C₂²×Dic₉₇	C₂³	1	4	yes	—
25	C₄⋊Dic₉₇	—	4	2	no	—
26	C₄⋊Dic₉₇	—	4	2	no	—
27	C₄⋊Dic₉₇	—	4	2	no	—
28	#13095490	—	4	4	no	—
29	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
30	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
31	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
32	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₉₇	—	4	1	no	—
34	C₄⋊C₄×C₉₇	—	4	1	no	—
35	C₄²×C₉₇	Q₈	1	4	yes	complemented
36	C₂³×C₉₇	C₂⁴	1	1	yes	characteristic
37	C₂×C₄×C₉₇	—	2	1	no	—
38	C₂×C₄×C₉₇	—	2	2	no	—
39	C₂×C₄×C₉₇	—	2	2	no	—
40	C₂×C₄×C₉₇	—	2	1	no	—
41	C₂×C₄×C₉₇	—	2	2	no	—
42	C₂×C₄×C₉₇	—	2	2	no	—
43	C₂×C₄×C₉₇	—	2	4	no	—
44	C₂×C₄×C₉₇	—	2	4	no	—
45	C₂×C₄×C₉₇	C₂×Q₈	1	4	yes	—
46	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
47	C₂×Dic₉₇	—	2	4	no	—
48	C₂×Dic₉₇	—	2	4	no	—
49	C₂×Dic₉₇	—	2	4	no	—
50	C₂×Dic₉₇	—	2	4	no	—
51	C₂×Dic₉₇	—	2	4	no	—
52	C₂×Dic₉₇	—	2	4	no	—
53	Dic₁₉₄	—	4	8	no	—
54	C₂²×C₉₇	C₂²×Q₈	1	1	yes	characteristic
55	C₂²×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
56	C₂²×C₉₇	H(5,𝔽₂)	1	2	yes	—
57	C₂²×C₉₇	2₋¹⁺⁴	1	2	yes	—
58	C₂²×C₉₇	2₋¹⁺⁴	1	1	yes	characteristic
59	C₄×C₉₇	—	2	4	no	—
60	C₄×C₉₇	—	2	2	no	—
61	C₄×C₉₇	—	4	2	no	—
62	C₄×C₉₇	—	4	2	no	—
63	Dic₉₇	—	4	4	no	—
64	Dic₉₇	—	4	4	no	—
65	C₂×C₉₇	#544	1	2	yes	—
66	C₂×C₉₇	#544	1	1	yes	characteristic
67	C₂×C₉₇	H(3,𝔽₄)	1	1	yes	characteristic
68	C₂×C₉₇	#563	1	1	yes	characteristic
69	C₂×C₉₇	#563	1	2	yes	—
70	#2844	—	97	1	no	maximal, Sylow
71	C₉₇	#2844	1	1	yes	characteristic, complemented, Sylow
72	#400	—	97	2	no	—
73	#400	—	97	4	no	—
74	#395	—	97	2	no	—
75	#395	—	97	4	no	—
76	#382	—	97	2	no	—
77	#382	D₉₇	1	1	yes	characteristic
78	C₂²×Q₈	—	97	2	no	—
79	C₂×C₄⋊C₄	—	97	4	no	—
80	C₂×C₄⋊C₄	—	97	2	no	—
81	C₂×C₄⋊C₄	—	97	2	no	—
82	C₂×C₄⋊C₄	—	97	2	no	—
83	C₂×C₄⋊C₄	C₂×D₉₇	1	1	yes	characteristic
84	C₂×C₄⋊C₄	C₂×D₉₇	1	1	yes	characteristic
85	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
86	#95	—	97	4	no	—
87	#95	—	97	4	no	—
88	#95	—	97	2	no	—
89	#95	—	97	4	no	—
90	#95	—	97	2	no	—
91	#95	C₂×D₉₇	1	4	yes	—
92	C₂×Q₈	—	388	2	no	—
93	C₂×Q₈	—	388	4	no	—
94	C₂²×C₄	—	97	4	no	—
95	C₂²×C₄	—	97	4	no	—
96	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
97	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
98	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
99	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
100	C₄⋊C₄	—	4	1	no	—
101	C₄⋊C₄	—	4	1	no	—
102	C₄⋊C₄	—	388	4	no	—
103	C₄⋊C₄	—	388	2	no	—
104	C₄⋊C₄	—	388	2	no	—
105	C₄⋊C₄	—	388	2	no	—
106	C₄²	Dic₁₉₄	1	4	yes	complemented
107	C₂³	C₂³×D₉₇	1	1	yes	characteristic
108	Q₈	—	388	8	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	194	4	no	—
118	C₂×C₄	—	194	4	no	—
119	C₂×C₄	—	194	4	no	—
120	C₂×C₄	—	194	4	no	—
121	C₂×C₄	—	194	4	no	—
122	C₂×C₄	—	194	4	no	—
123	C₂×C₄	C₂×Dic₁₉₄	1	4	yes	—
124	C₂×C₄	C₂×Dic₁₉₄	1	2	yes	—
125	C₂²	C₂²×Dic₁₉₄	1	1	yes	characteristic
126	C₂²	#15690855	1	1	yes	characteristic
127	C₂²	#15690856	1	2	yes	—
128	C₂²	#15690857	1	2	yes	—
129	C₂²	#15690858	1	1	yes	characteristic
130	C₄	—	2	4	no	—
131	C₄	—	2	2	no	—
132	C₄	—	4	2	no	—
133	C₄	—	4	2	no	—
134	C₄	—	388	4	no	—
135	C₄	—	388	4	no	—
136	C₂	#20261363	1	2	yes	—
137	C₂	#20261366	1	1	yes	characteristic
138	C₂	#20261477	1	1	yes	characteristic
139	C₂	#20261485	1	1	yes	characteristic
140	C₂	#20261489	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20260677

C₂

yes

maximal

#20260674

C₂

yes

maximal

#20260656

C₂

yes

maximal

#20260655

C₂

yes

maximal

#20260601

C₂

yes

maximal

#382×C₉₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₉₄

C₂²

yes

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

C₂×#13095490

C₂²

yes

—

#15690717

C₂²

yes

—

#15690717

C₂²

yes

—

#15690717

C₂²

yes

—

#15690717

C₂²

yes

—

#15690717

C₂²

yes

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₉₇

C₂²

yes

characteristic

C₂×C₄²×C₉₇

C₂²

yes

characteristic

#95×C₉₇

C₂²

yes

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂²×Dic₉₇

C₂³

yes

—

C₂²×Dic₉₇

C₂³

yes

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

#13095490

—

C₂²×C₄×C₉₇

C₂³

yes

—

C₂²×C₄×C₉₇

C₂³

yes

—

C₂²×C₄×C₉₇

C₂³

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

Q₈

yes

complemented

C₂³×C₉₇

C₂⁴

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×Q₈

yes

—

C₂×C₄×C₉₇

C₂×Q₈

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

Dic₁₉₄

—

C₂²×C₉₇

C₂²×Q₈

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

—

C₂²×C₉₇

2₋¹⁺⁴

yes

—

C₂²×C₉₇

2₋¹⁺⁴

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

#544

yes

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

#2844

—

maximal, Sylow

C₉₇

#2844

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₉₇

yes

characteristic

C₂²×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₉₇

yes

characteristic

C₂×C₄⋊C₄

C₂×D₉₇

yes

characteristic

C₂×C₄²

C₂×D₉₇

yes

—

—

—

—

—

C₂×D₉₇

yes

—

C₂×Q₈

—

388

—

C₂×Q₈

—

388

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂²×C₄

C₂²×D₉₇

yes

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

388

—

103

C₄⋊C₄

—

388

—

104

C₄⋊C₄

—

388

—

105

C₄⋊C₄

—

388

—

106

C₄²

Dic₁₉₄

yes

complemented

107

C₂³

C₂³×D₉₇

yes

characteristic

108

Q₈

—

388

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

194

—

118

C₂×C₄

—

194

—

119

C₂×C₄

—

194

—

120

C₂×C₄

—

194

—

121

C₂×C₄

—

194

—

122

C₂×C₄

—

194

—

123

C₂×C₄

C₂×Dic₁₉₄

yes

—

124

C₂×C₄

C₂×Dic₁₉₄

yes

—

125

C₂²

C₂²×Dic₁₉₄

yes

characteristic

126

C₂²

#15690855

yes

characteristic

127

C₂²

#15690856

yes

—

128

C₂²

#15690857

yes

—

129

C₂²

#15690858

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

388

—

135

C₄

—

388

—

136

C₂

#20261363

yes

—

137

C₂

#20261366

yes

characteristic

138

C₂

#20261477

yes

characteristic

139

C₂

#20261485

yes

characteristic

140

C₂

#20261489

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₉₇
commutator, socle, prime radical, cosocle	C₂³×C₉₇	C₂⁴
Fitting	#382×C₉₇	C₂
hypercenter	#382	D₉₇
hypocenter	C₉₇	#2844
LCS	G ▷ C₂³×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ #382

subgroup

G/H

center, Frattini

C₂³

C₂³×D₉₇

commutator, socle, prime radical, cosocle

C₂³×C₉₇

C₂⁴

Fitting

#382×C₉₇

C₂

D₉₇

C₉₇

G ▷ C₂³×C₉₇ ▷ C₉₇

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
35. C₄²×C₉₇	Q₈	semidirect
71. C₉₇	#2844	semidirect
106. C₄²	Dic₁₉₄	semidirect

H ≅ G/N

type

35. C₄²×C₉₇

Q₈

semidirect

71. C₉₇

#2844

semidirect

106. C₄²

Dic₁₉₄

semidirect

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

subgroup	index
#20260677	2
#20260674	2
#20260656	2
#20260655	2
#20260601	2
#382×C₉₇	2
#2844	97

subgroup

index

#382×C₉₇

Subgroups — #27150008

Conjugacy classes of subgroups

Complements

Maximal subgroups