Group #27142684 — subgroups

Subgroups — #27142684

Order 12416.

292 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 125 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×#15690810	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11715895	C₂	1	1	yes	maximal, characteristic
3	#20260497	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₄×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₉₄	—	2	1	no	—
6	C₂²×#11715895	C₂²	1	2	yes	—
7	#15690810	—	2	2	no	—
8	#15690810	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₉₇	—	2	1	no	—
10	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
11	C₄⋊#11715895	—	2	2	no	—
12	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₉₇	—	2	1	no	—
14	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
15	#95×C₉₇	—	2	1	no	—
16	C₂×Dic₁₉₄	—	2	4	no	—
17	C₂×Dic₁₉₄	—	4	4	no	—
18	C₂²×Dic₉₇	—	2	2	no	—
19	C₂×#11715895	—	2	2	no	—
20	C₂×#11715895	—	2	4	no	—
21	C₄⋊Dic₉₇	—	2	4	no	—
22	C₄⋊Dic₉₇	—	2	2	no	—
23	C₄⋊Dic₉₇	C₂×C₄	1	4	yes	—
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	2	no	—
29	C₄⋊C₄×C₉₇	Q₈	1	4	yes	complemented
30	C₄²×C₉₇	—	4	2	no	—
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	C₂×C₄×C₉₇	—	2	2	no	—
35	C₂×C₄×C₉₇	—	2	4	no	—
36	C₂×C₄×C₉₇	—	4	2	no	—
37	C₂×C₄×C₉₇	—	4	2	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	4	no	—
48	Dic₁₉₄	—	4	8	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	—	97	1	no	—
70	C₂×C₄⋊C₈	—	97	1	no	—
71	C₄×C₄⋊C₄	D₉₇	1	1	yes	characteristic
72	#340	—	97	4	no	—
73	C₂²×Q₈	—	194	1	no	—
74	C₂²×C₈	—	97	2	no	—
75	C₄⋊Q₈	—	97	4	no	—
76	C₄⋊Q₈	—	194	2	no	—
77	C₂×C₄⋊C₄	—	97	2	no	—
78	C₂×C₄⋊C₄	—	194	1	no	—
79	C₂×C₄⋊C₄	C₂×D₉₇	1	2	yes	—
80	C₂×C₄²	—	2	1	no	—
81	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
82	C₄⋊C₈	—	194	2	no	—
83	#95	—	2	1	no	—
84	C₂×Q₈	—	388	4	no	—
85	C₂×Q₈	—	194	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	2	no	—
94	C₄⋊C₄	—	97	4	no	—
95	C₄⋊C₄	—	194	4	no	—
96	C₄⋊C₄	—	194	2	no	—
97	C₄⋊C₄	Dic₁₉₄	1	4	yes	complemented
98	C₄²	—	4	2	no	—
99	C₄²	C₄×D₉₇	1	2	yes	—
100	C₄²	D₃₈₈	1	2	yes	—
101	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
102	Q₈	—	388	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	388	4	no	—
108	C₂×C₄	—	194	4	no	—
109	C₂×C₄	#13095488	1	2	yes	—
110	C₂×C₄	#13095488	1	2	yes	—
111	C₂×C₄	#13095499	1	2	yes	—
112	C₂×C₄	#13095501	1	2	yes	—
113	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
114	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
115	C₂×C₄	C₂×Dic₁₉₄	1	2	yes	—
116	C₂×C₄	#13095511	1	2	yes	—
117	C₈	—	388	2	no	—
118	C₂²	C₂×#13095488	1	1	yes	characteristic
119	C₂²	#15690789	1	2	yes	—
120	C₂²	C₂×#13095499	1	1	yes	characteristic
121	C₂²	C₂×#13095501	1	1	yes	characteristic
122	C₂²	#15690835	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	—	4	4	no	—
125	C₄	—	388	4	no	—
126	C₄	#15690733	1	4	yes	—
127	C₄	C₄×Dic₁₉₄	1	2	yes	—
128	C₄	#15690798	1	1	yes	characteristic
129	C₄	#15690810	1	1	yes	characteristic
130	C₂	#20260641	1	1	yes	characteristic
131	C₂	C₂×#15690733	1	1	yes	characteristic
132	C₂	#20260739	1	1	yes	characteristic
133	C₂	#20261025	1	2	yes	—
134	C₂	#20261030	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690810

C₂

yes

maximal, characteristic

C₂×C₄⋊#11715895

C₂

yes

maximal, characteristic

#20260497

C₂

yes

maximal

C₄×C₄⋊C₄×C₉₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₉₄

—

C₂²×#11715895

C₂²

yes

—

#15690810

—

#15690810

C₄

yes

complemented

C₂×C₄⋊Dic₉₇

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

C₄⋊#11715895

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

#95×C₉₇

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂²×Dic₉₇

—

C₂×#11715895

—

C₂×#11715895

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

C₂×C₄

yes

—

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

Q₈

yes

complemented

C₄²×C₉₇

—

C₄²×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

D₄

yes

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×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

—

Dic₁₉₄

—

#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

—

C₂×C₄⋊C₈

—

C₄×C₄⋊C₄

D₉₇

yes

characteristic

#340

—

C₂²×Q₈

—

194

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

194

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄⋊C₄

C₂×D₉₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

194

—

#95

—

C₂×Q₈

—

388

—

C₂×Q₈

—

194

—

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

—

194

—

C₄⋊C₄

—

194

—

C₄⋊C₄

Dic₁₉₄

yes

complemented

C₄²

—

C₄²

C₄×D₉₇

yes

—

100

C₄²

D₃₈₈

yes

—

101

C₂³

C₂×C₂²⋊D₉₇

yes

characteristic

102

Q₈

—

388

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

388

—

108

C₂×C₄

—

194

—

109

C₂×C₄

#13095488

yes

—

110

C₂×C₄

#13095488

yes

—

111

C₂×C₄

#13095499

yes

—

112

C₂×C₄

#13095501

yes

—

113

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

114

C₂×C₄

C₂×D₃₈₈

yes

characteristic

115

C₂×C₄

C₂×Dic₁₉₄

yes

—

116

C₂×C₄

#13095511

yes

—

117

C₈

—

388

—

118

C₂²

C₂×#13095488

yes

characteristic

119

C₂²

#15690789

yes

—

120

C₂²

C₂×#13095499

yes

characteristic

121

C₂²

C₂×#13095501

yes

characteristic

122

C₂²

#15690835

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

388

—

126

C₄

#15690733

yes

—

127

C₄

C₄×Dic₁₉₄

yes

—

128

C₄

#15690798

yes

characteristic

129

C₄

#15690810

yes

characteristic

130

C₂

#20260641

yes

characteristic

131

C₂

C₂×#15690733

yes

characteristic

132

C₂

#20260739

yes

characteristic

133

C₂

#20261025

yes

—

134

C₂

#20261030

yes

—

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

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₉₇

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

C₄×C₄⋊C₄×C₉₇

C₂

socle, prime radical

C₂³×C₉₇

C₂×D₄

cosocle

C₂²×C₄×C₉₇

C₂³

hypercenter

C₄×C₄⋊C₄

D₉₇

hypocenter

C₉₇

#1987

LCS

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
8. #15690810	C₄	semidirect
29. C₄⋊C₄×C₉₇	Q₈	semidirect
68. C₉₇	#1987	semidirect
97. C₄⋊C₄	Dic₁₉₄	semidirect

H ≅ G/N

type

8. #15690810

C₄

semidirect

29. C₄⋊C₄×C₉₇

Q₈

semidirect

68. C₉₇

#1987

semidirect

97. C₄⋊C₄

Dic₁₉₄

semidirect

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

subgroup	index
C₂×#15690810	2
C₂×C₄⋊#11715895	2
#20260497	2
C₄×C₄⋊C₄×C₉₇	2
#1987	97

subgroup

index

C₄×C₄⋊C₄×C₉₇

Subgroups — #27142684

Conjugacy classes of subgroups

Complements

Maximal subgroups