Group #27142992 — subgroups

Subgroups — #27142992

Order 12416.

296 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 69 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×D₇₇₆	C₂	1	1	yes	maximal, characteristic
2	#20260739	C₂	1	2	yes	maximal, complemented
3	C₂×#15690726	C₂	1	2	yes	maximal, complemented
4	#20260500	C₂	1	1	yes	maximal, characteristic, complemented
5	#424×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₈₈	C₂²	1	2	yes	—
7	C₂×D₇₇₆	—	2	4	no	—
8	C₂×D₇₇₆	—	2	2	no	—
9	C₂×#13095495	C₂²	1	2	yes	—
10	#15690730	—	2	4	no	—
11	#15690726	—	2	4	no	—
12	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
14	#117×C₉₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₉₇	—	2	2	no	—
16	C₂³×D₉₇	—	2	2	no	—
17	C₂×D₃₈₈	—	2	2	no	—
18	C₂×D₃₈₈	—	2	2	no	—
19	C₂×D₃₈₈	—	2	2	no	—
20	C₂×D₃₈₈	—	2	2	no	—
21	C₂×D₃₈₈	—	2	2	no	—
22	C₂×D₃₈₈	D₄	1	4	yes	—
23	D₇₇₆	—	2	8	no	—
24	#13095495	—	2	2	no	—
25	#13095495	—	2	2	no	—
26	C₂×#11715895	—	2	2	no	—
27	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
28	M₁₆×C₉₇	—	2	2	no	—
29	C₂×C₈×C₉₇	—	2	2	no	—
30	C₂×C₈×C₉₇	—	2	1	no	—
31	C₂×C₈×C₉₇	D₄	1	2	yes	—
32	C₄⋊C₄×C₉₇	D₄	1	2	yes	—
33	C₂²⋊C₄×C₉₇	—	2	1	no	—
34	C₄²×C₉₇	—	2	1	no	—
35	C₂³×C₉₇	#41	1	1	yes	characteristic
36	C₂²×D₉₇	—	2	4	no	—
37	C₂²×D₉₇	—	4	2	no	—
38	C₂²×D₉₇	—	4	2	no	—
39	C₂²×D₉₇	—	8	4	no	—
40	C₂×C₄×C₉₇	—	2	2	no	—
41	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
42	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₉₇	#41	1	2	yes	—
45	D₃₈₈	—	2	4	no	—
46	D₃₈₈	—	2	4	no	—
47	D₃₈₈	—	4	2	no	—
48	D₃₈₈	—	4	2	no	—
49	C₈×C₉₇	—	2	2	no	—
50	C₈×C₉₇	—	4	1	no	—
51	#11715895	—	4	2	no	—
52	C₂²×C₉₇	—	2	2	no	—
53	C₂²×C₉₇	#121	1	1	yes	characteristic
54	C₂²×C₉₇	#123	1	2	yes	—
55	C₂×D₉₇	—	4	4	no	—
56	C₂×D₉₇	—	8	4	no	—
57	C₂×D₉₇	—	8	4	no	—
58	C₂×D₉₇	—	8	4	no	—
59	C₄×C₉₇	—	4	2	no	—
60	C₄×C₉₇	#120	1	1	yes	characteristic
61	C₄×C₉₇	#121	1	2	yes	—
62	C₄×C₉₇	#124	1	1	yes	characteristic
63	C₂×C₉₇	—	2	2	no	—
64	C₂×C₉₇	#394	1	1	yes	characteristic
65	C₂×C₉₇	#472	1	2	yes	—
66	D₉₇	—	8	4	no	—
67	#2045	—	97	1	no	maximal, Sylow
68	C₉₇	#2045	1	1	yes	characteristic, complemented, Sylow
69	C₂²×D₈	—	97	1	no	—
70	#424	D₉₇	1	1	yes	characteristic, complemented
71	#418	—	97	2	no	—
72	C₂×#100	—	97	2	no	—
73	#341	—	97	1	no	—
74	C₂²×D₄	—	97	2	no	—
75	C₂×D₈	—	194	4	no	—
76	C₂×D₈	—	194	2	no	—
77	C₂×M₁₆	—	97	2	no	—
78	C₂×M₁₆	C₂×D₉₇	1	1	yes	characteristic
79	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
80	#117	C₂×D₉₇	1	1	yes	characteristic
81	C₄⋊C₈	—	2	2	no	—
82	#102	—	194	4	no	—
83	#100	—	194	4	no	—
84	C₂⁴	—	194	2	no	—
85	C₂×D₄	—	97	4	no	—
86	C₂×D₄	—	194	2	no	—
87	C₂×D₄	—	194	2	no	—
88	C₂×D₄	—	194	2	no	—
89	C₂×D₄	—	194	2	no	—
90	C₂×D₄	—	194	2	no	—
91	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
92	D₈	—	194	8	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	194	2	no	—
95	M₁₆	—	194	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	194	2	no	—
99	C₂×C₈	C₂²⋊D₉₇	1	2	yes	—
100	C₄⋊C₄	D₃₈₈	1	2	yes	—
101	C₂²⋊C₄	—	2	1	no	—
102	C₄²	—	2	1	no	—
103	C₂³	—	776	4	no	—
104	C₂³	—	388	2	no	—
105	C₂³	—	388	2	no	—
106	C₂³	—	194	4	no	—
107	C₂³	#13095511	1	1	yes	characteristic
108	D₄	—	388	2	no	—
109	D₄	—	388	2	no	—
110	D₄	—	194	4	no	—
111	D₄	—	194	4	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	D₄×D₉₇	1	2	yes	—
114	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
115	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
116	C₂×C₄	#13095512	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	4	1	no	—
119	C₈	—	388	2	no	—
120	C₂²	—	2	2	no	—
121	C₂²	—	776	4	no	—
122	C₂²	—	776	4	no	—
123	C₂²	—	776	4	no	—
124	C₂²	—	388	4	no	—
125	C₂²	#15690784	1	1	yes	characteristic
126	C₂²	#15690795	1	2	yes	—
127	C₄	—	4	2	no	—
128	C₄	#15690778	1	1	yes	characteristic
129	C₄	#15690785	1	2	yes	—
130	C₄	#15690799	1	1	yes	characteristic
131	C₂	—	2	2	no	—
132	C₂	—	776	4	no	—
133	C₂	#20260652	1	1	yes	characteristic
134	C₂	#20261005	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₇₇₆

C₂

yes

maximal, characteristic

#20260739

C₂

yes

maximal, complemented

C₂×#15690726

C₂

yes

maximal, complemented

#20260500

C₂

yes

maximal, characteristic, complemented

#424×C₉₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₈₈

C₂²

yes

—

C₂×D₇₇₆

—

C₂×D₇₇₆

—

C₂×#13095495

C₂²

yes

—

#15690730

—

#15690726

—

C₂×M₁₆×C₉₇

C₂²

yes

characteristic

C₂²×C₈×C₉₇

C₂²

yes

characteristic

#117×C₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₉₇

—

C₂³×D₉₇

—

C₂×D₃₈₈

—

C₂×D₃₈₈

—

C₂×D₃₈₈

—

C₂×D₃₈₈

—

C₂×D₃₈₈

—

C₂×D₃₈₈

D₄

yes

—

D₇₇₆

—

#13095495

—

#13095495

—

C₂×#11715895

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

M₁₆×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

D₄

yes

—

C₄⋊C₄×C₉₇

D₄

yes

—

C₂²⋊C₄×C₉₇

—

C₄²×C₉₇

—

C₂³×C₉₇

#41

yes

characteristic

C₂²×D₉₇

—

C₂²×D₉₇

—

C₂²×D₉₇

—

C₂²×D₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

D₃₈₈

—

D₃₈₈

—

D₃₈₈

—

D₃₈₈

—

C₈×C₉₇

—

C₈×C₉₇

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

#121

yes

characteristic

C₂²×C₉₇

#123

yes

—

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#120

yes

characteristic

C₄×C₉₇

#121

yes

—

C₄×C₉₇

#124

yes

characteristic

C₂×C₉₇

—

C₂×C₉₇

#394

yes

characteristic

C₂×C₉₇

#472

yes

—

D₉₇

—

#2045

—

maximal, Sylow

C₉₇

#2045

yes

characteristic, complemented, Sylow

C₂²×D₈

—

#424

D₉₇

yes

characteristic, complemented

#418

—

C₂×#100

—

#341

—

C₂²×D₄

—

C₂×D₈

—

194

—

C₂×D₈

—

194

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₉₇

yes

characteristic

C₂²×C₈

C₂×D₉₇

yes

characteristic

#117

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

#102

—

194

—

#100

—

194

—

C₂⁴

—

194

—

C₂×D₄

—

C₂×D₄

—

194

—

C₂×D₄

—

194

—

C₂×D₄

—

194

—

C₂×D₄

—

194

—

C₂×D₄

—

194

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

D₈

—

194

—

M₁₆

—

M₁₆

—

194

—

M₁₆

—

194

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

C₂²⋊D₉₇

yes

—

100

C₄⋊C₄

D₃₈₈

yes

—

101

C₂²⋊C₄

—

102

C₄²

—

103

C₂³

—

776

—

104

C₂³

—

388

—

105

C₂³

—

388

—

106

C₂³

—

194

—

107

C₂³

#13095511

yes

characteristic

108

D₄

—

388

—

109

D₄

—

388

—

110

D₄

—

194

—

111

D₄

—

194

—

112

C₂×C₄

—

113

C₂×C₄

D₄×D₉₇

yes

—

114

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

115

C₂×C₄

C₂×D₃₈₈

yes

characteristic

116

C₂×C₄

#13095512

yes

—

117

C₈

—

118

C₈

—

119

C₈

—

388

—

120

C₂²

—

121

C₂²

—

776

—

122

C₂²

—

776

—

123

C₂²

—

776

—

124

C₂²

—

388

—

125

C₂²

#15690784

yes

characteristic

126

C₂²

#15690795

yes

—

127

C₄

—

128

C₄

#15690778

yes

characteristic

129

C₄

#15690785

yes

—

130

C₄

#15690799

yes

characteristic

131

C₂

—

132

C₂

—

776

—

133

C₂

#20260652

yes

characteristic

134

C₂

#20261005

yes

—

	subgroup	G/H
center	C₂²	#15690784
commutator, cosocle	C₂²×C₄×C₉₇	C₂³
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#424×C₉₇	C₂
socle	C₂²×C₉₇	#121
prime radical	C₂²×D₇₇₆	C₂
hypercenter	#424	D₉₇
hypocenter	C₉₇	#2045
LCS	G ▷ C₂²×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#15690784

commutator, cosocle

C₂²×C₄×C₉₇

C₂³

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

#424×C₉₇

C₂

socle

C₂²×C₉₇

#121

prime radical

C₂²×D₇₇₆

C₂

D₉₇

C₉₇

G ▷ C₂²×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
2. #20260739	C₂	semidirect
3. C₂×#15690726	C₂	semidirect
4. #20260500	C₂	semidirect
5. #424×C₉₇	C₂	semidirect
68. C₉₇	#2045	semidirect
70. #424	D₉₇	semidirect

H ≅ G/N

type

2. #20260739

C₂

semidirect

3. C₂×#15690726

C₂

semidirect

4. #20260500

C₂

semidirect

5. #424×C₉₇

C₂

semidirect

68. C₉₇

#2045

semidirect

70. #424

D₉₇

semidirect

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

subgroup	index
C₂²×D₇₇₆	2
#20260739	2
C₂×#15690726	2
#20260500	2
#424×C₉₇	2
#2045	97

subgroup

index

C₂²×D₇₇₆

#424×C₉₇

Subgroups — #27142992

Conjugacy classes of subgroups

Complements

Maximal subgroups