Group #27142685 — subgroups

Subgroups — #27142685

Order 12416.

292 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 129 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×C₄⋊#11715895	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₉₇	C₂	1	1	yes	maximal, characteristic
3	#20260497	C₂	1	4	yes	maximal
4	C₂×C₄⋊Q₈×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11715895	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
7	C₂×C₄×Dic₉₇	—	2	1	no	—
8	C₄⋊#11715895	—	2	2	no	—
9	#15690717	—	2	1	no	—
10	C₂²×Q₈×C₉₇	—	2	1	no	—
11	C₄⋊Q₈×C₉₇	—	2	2	no	—
12	C₄⋊Q₈×C₉₇	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₉₇	—	2	1	no	—
14	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
16	C₂²×Dic₉₇	—	2	2	no	—
17	C₂×#11715895	—	2	2	no	—
18	C₂×#11715895	—	2	4	no	—
19	C₄⋊Dic₉₇	—	2	2	no	—
20	C₄⋊Dic₉₇	Q₈	1	4	yes	complemented
21	C₄×Dic₉₇	—	4	2	no	—
22	C₂×Q₈×C₉₇	—	2	4	no	—
23	C₂×Q₈×C₉₇	—	4	4	no	—
24	C₂²×C₄×C₉₇	—	2	2	no	—
25	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₉₇	—	2	4	no	—
29	C₄⋊C₄×C₉₇	—	2	2	no	—
30	C₄⋊C₄×C₉₇	C₂×C₄	1	4	yes	—
31	C₄²×C₉₇	C₂×C₄	1	2	yes	—
32	C₄²×C₉₇	D₄	1	2	yes	—
33	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
34	Q₈×C₉₇	—	4	8	no	—
35	C₂×C₄×C₉₇	—	2	2	no	—
36	C₂×C₄×C₉₇	—	2	4	no	—
37	C₂×C₄×C₉₇	—	4	4	no	—
38	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₉₇	SD₁₆	1	2	yes	—
41	C₂×C₄×C₉₇	Q₁₆	1	2	yes	—
42	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₉₇	#41	1	2	yes	—
46	C₂×Dic₉₇	—	2	4	no	—
47	C₂×Dic₉₇	—	4	2	no	—
48	C₂×Dic₉₇	—	4	2	no	—
49	#11715895	—	4	2	no	—
50	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₉₇	C₂²⋊Q₈	1	2	yes	—
52	C₂²×C₉₇	C₂×SD₁₆	1	1	yes	characteristic
53	C₂²×C₉₇	C₂×Q₁₆	1	1	yes	characteristic
54	C₂²×C₉₇	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₉₇	—	2	2	no	—
56	C₄×C₉₇	—	4	4	no	—
57	C₄×C₉₇	#103	1	4	yes	—
58	C₄×C₉₇	C₄×Q₈	1	2	yes	—
59	C₄×C₉₇	#124	1	1	yes	characteristic
60	C₄×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
61	Dic₉₇	—	4	4	no	—
62	C₂×C₉₇	#391	1	1	yes	characteristic
63	C₂×C₉₇	C₂×#103	1	1	yes	characteristic
64	C₂×C₉₇	#418	1	1	yes	characteristic
65	C₂×C₉₇	#476	1	2	yes	—
66	C₂×C₉₇	#477	1	2	yes	—
67	#1987	—	97	1	no	maximal, Sylow
68	C₉₇	#1987	1	1	yes	characteristic, complemented, Sylow
69	C₂×C₄⋊Q₈	D₉₇	1	1	yes	characteristic
70	C₂×C₄⋊C₈	—	97	1	no	—
71	C₄×C₄⋊C₄	—	97	1	no	—
72	#340	—	97	4	no	—
73	C₂²×Q₈	—	2	1	no	—
74	C₂²×C₈	—	97	2	no	—
75	C₄⋊Q₈	—	2	2	no	—
76	C₄⋊Q₈	Dic₉₇	1	4	yes	complemented
77	C₂×C₄⋊C₄	—	2	1	no	—
78	C₂×C₄⋊C₄	—	97	2	no	—
79	C₂×C₄⋊C₄	C₂×D₉₇	1	2	yes	—
80	C₂×C₄²	—	194	1	no	—
81	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
82	C₄⋊C₈	—	194	2	no	—
83	#95	—	194	1	no	—
84	C₂×Q₈	—	2	4	no	—
85	C₂×Q₈	—	4	4	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	—	194	2	no	—
88	C₂²×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
89	C₂²×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
90	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
91	C₂×C₈	—	194	2	no	—
92	C₂×C₈	—	194	4	no	—
93	C₄⋊C₄	—	2	4	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	97	4	no	—
96	C₄⋊C₄	—	194	2	no	—
97	C₄⋊C₄	C₂×Dic₉₇	1	4	yes	—
98	C₄²	—	388	2	no	—
99	C₄²	C₂×Dic₉₇	1	2	yes	—
100	C₄²	C₂²⋊D₉₇	1	2	yes	—
101	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
102	Q₈	—	4	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	4	4	no	—
106	C₂×C₄	—	388	2	no	—
107	C₂×C₄	—	388	2	no	—
108	C₂×C₄	—	194	4	no	—
109	C₂×C₄	C₂²⋊Dic₉₇	1	2	yes	—
110	C₂×C₄	C₂²⋊Dic₉₇	1	2	yes	—
111	C₂×C₄	#13095498	1	2	yes	—
112	C₂×C₄	#13095501	1	2	yes	—
113	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
114	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
115	C₂×C₄	Q₈×D₉₇	1	2	yes	—
116	C₂×C₄	#13095512	1	2	yes	—
117	C₈	—	388	2	no	—
118	C₂²	C₂×C₂²⋊Dic₉₇	1	1	yes	characteristic
119	C₂²	#15690786	1	2	yes	—
120	C₂²	C₂×#13095498	1	1	yes	characteristic
121	C₂²	C₂×#13095501	1	1	yes	characteristic
122	C₂²	#15690836	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	—	4	4	no	—
125	C₄	—	388	4	no	—
126	C₄	#15690734	1	4	yes	—
127	C₄	Q₈×Dic₉₇	1	2	yes	—
128	C₄	#15690799	1	1	yes	characteristic
129	C₄	#15690808	1	1	yes	characteristic
130	C₂	#20260642	1	1	yes	characteristic
131	C₂	C₂×#15690734	1	1	yes	characteristic
132	C₂	#20260740	1	1	yes	characteristic
133	C₂	#20261022	1	2	yes	—
134	C₂	#20261027	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11715895

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₉₇

C₂

yes

maximal, characteristic

#20260497

C₂

yes

maximal

C₂×C₄⋊Q₈×C₉₇

C₂

yes

maximal, characteristic

C₂²×#11715895

C₂²

yes

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

C₂×C₄×Dic₉₇

—

C₄⋊#11715895

—

#15690717

—

C₂²×Q₈×C₉₇

—

C₄⋊Q₈×C₉₇

—

C₄⋊Q₈×C₉₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₉₇

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂²×Dic₉₇

—

C₂×#11715895

—

C₂×#11715895

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

Q₈

yes

complemented

C₄×Dic₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

D₄

yes

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

Q₈×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

SD₁₆

yes

—

C₂×C₄×C₉₇

Q₁₆

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

—

C₂×C₄×C₉₇

#41

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

#11715895

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₉₇

C₂²⋊Q₈

yes

—

C₂²×C₉₇

C₂×SD₁₆

yes

characteristic

C₂²×C₉₇

C₂×Q₁₆

yes

characteristic

C₂²×C₉₇

AGL(1,ℤ/8)

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#103

yes

—

C₄×C₉₇

C₄×Q₈

yes

—

C₄×C₉₇

#124

yes

characteristic

C₄×C₉₇

C₄⋊Q₈

yes

characteristic

Dic₉₇

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₂×C₉₇

#477

yes

—

#1987

—

maximal, Sylow

C₉₇

#1987

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₉₇

yes

characteristic

C₂×C₄⋊C₈

—

C₄×C₄⋊C₄

—

#340

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₉₇

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₉₇

yes

—

C₂×C₄²

—

194

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

194

—

#95

—

194

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

194

—

C₂²×C₄

C₂²⋊D₉₇

yes

characteristic

C₂²×C₄

C₂²⋊D₉₇

yes

characteristic

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

194

—

C₄⋊C₄

C₂×Dic₉₇

yes

—

C₄²

—

388

—

C₄²

C₂×Dic₉₇

yes

—

100

C₄²

C₂²⋊D₉₇

yes

—

101

C₂³

C₂×C₂²⋊D₉₇

yes

characteristic

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

388

—

107

C₂×C₄

—

388

—

108

C₂×C₄

—

194

—

109

C₂×C₄

C₂²⋊Dic₉₇

yes

—

110

C₂×C₄

C₂²⋊Dic₉₇

yes

—

111

C₂×C₄

#13095498

yes

—

112

C₂×C₄

#13095501

yes

—

113

C₂×C₄

C₂²×Dic₉₇

yes

characteristic

114

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

115

C₂×C₄

Q₈×D₉₇

yes

—

116

C₂×C₄

#13095512

yes

—

117

C₈

—

388

—

118

C₂²

C₂×C₂²⋊Dic₉₇

yes

characteristic

119

C₂²

#15690786

yes

—

120

C₂²

C₂×#13095498

yes

characteristic

121

C₂²

C₂×#13095501

yes

characteristic

122

C₂²

#15690836

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

388

—

126

C₄

#15690734

yes

—

127

C₄

Q₈×Dic₉₇

yes

—

128

C₄

#15690799

yes

characteristic

129

C₄

#15690808

yes

characteristic

130

C₂

#20260642

yes

characteristic

131

C₂

C₂×#15690734

yes

characteristic

132

C₂

#20260740

yes

characteristic

133

C₂

#20261022

yes

—

134

C₂

#20261027

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₉₇
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	C₂×C₄⋊Q₈×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂×D₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂×C₄⋊Q₈	D₉₇
hypocenter	C₉₇	#1987
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₉₇

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

C₂×C₄⋊Q₈×C₉₇

C₂

socle, prime radical

C₂³×C₉₇

C₂×D₄

cosocle

C₂²×C₄×C₉₇

C₂³

hypercenter

C₂×C₄⋊Q₈

D₉₇

hypocenter

C₉₇

#1987

LCS

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₉₇	C₄	semidirect
20. C₄⋊Dic₉₇	Q₈	semidirect
68. C₉₇	#1987	semidirect
76. C₄⋊Q₈	Dic₉₇	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₉₇

C₄

semidirect

20. C₄⋊Dic₉₇

Q₈

semidirect

68. C₉₇

#1987

semidirect

76. C₄⋊Q₈

Dic₉₇

semidirect

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

subgroup	index
C₂×C₄⋊#11715895	2
C₄×C₄⋊Dic₉₇	2
#20260497	2
C₂×C₄⋊Q₈×C₉₇	2
#1987	97

subgroup

index

C₂×C₄⋊#11715895

C₄×C₄⋊Dic₉₇

#20260497

C₂×C₄⋊Q₈×C₉₇

#1987

Subgroups — #27142685

Conjugacy classes of subgroups

Complements

Maximal subgroups