Group #27157972 — subgroups

Subgroups — #27157972

Order 12416.

220 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 86 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×#15690884	C₂	1	2	yes	maximal, complemented
2	#20261631	C₂	1	4	yes	maximal, complemented
3	C₂×#15690817	C₂	1	1	yes	maximal, characteristic
4	#15690884	—	2	2	no	—
5	#15690884	C₂²	1	4	yes	—
6	C₂×#13095514	—	2	2	no	—
7	#15690817	—	2	2	no	—
8	#15690817	—	2	2	no	—
9	C₂×#13095495	C₄	1	1	yes	characteristic
10	C₂×#13095495	C₂²	1	2	yes	—
11	C₂²×#11715895	C₄	1	2	yes	—
12	C₂²×#11715895	C₂²	1	1	yes	characteristic
13	C₂×#41×C₉₇	C₄	1	1	yes	characteristic
14	#13095514	—	2	4	no	—
15	#13095495	—	2	2	no	—
16	#13095495	—	2	4	no	—
17	C₂×#11715895	—	2	2	no	—
18	C₂×#11715895	—	2	2	no	—
19	C₂×#11715895	C₂×C₄	1	2	yes	—
20	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
21	C₂×#11715895	D₄	1	2	yes	—
22	C₂×#11715895	D₄	1	2	yes	—
23	C₂×#11715895	C₂³	1	1	yes	characteristic
24	#41×C₉₇	—	2	2	no	—
25	#41×C₉₇	—	2	2	no	—
26	C₂×Q₈×C₉₇	C₈	1	1	yes	characteristic
27	C₂×D₄×C₉₇	C₈	1	2	yes	—
28	C₂×D₄×C₉₇	C₈	1	1	yes	characteristic
29	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₉₇	C₂×C₈	1	2	yes	—
32	C₂³×C₉₇	C₂×C₈	1	1	yes	characteristic
33	Q₈×C₉₇	—	2	2	no	—
34	D₄×C₉₇	—	2	4	no	—
35	D₄×C₉₇	—	2	2	no	—
36	C₂×C₄×C₉₇	—	2	2	no	—
37	C₂×C₄×C₉₇	—	2	2	no	—
38	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₉₇	C₂×C₈	1	2	yes	—
40	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
41	C₂×C₄×C₉₇	M₁₆	1	2	yes	—
42	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
43	#11715895	—	2	2	no	—
44	#11715895	C₂²⋊C₄	1	1	yes	characteristic
45	#11715895	C₂²⋊C₄	1	1	yes	characteristic
46	#11715895	C₂×D₄	1	2	yes	—
47	C₂²×C₉₇	—	2	2	no	—
48	C₂²×C₉₇	—	4	1	no	—
49	C₂²×C₉₇	—	4	2	no	—
50	C₂²×C₉₇	C₂²⋊C₈	1	2	yes	—
51	C₂²×C₉₇	C₂²×C₈	1	1	yes	characteristic
52	C₄×C₉₇	—	2	2	no	—
53	C₄×C₉₇	C₂²⋊C₈	1	2	yes	—
54	C₄×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₄×C₉₇	C₂×M₁₆	1	1	yes	characteristic
56	C₂×C₉₇	—	2	1	no	—
57	C₂×C₉₇	—	2	2	no	—
58	C₂×C₉₇	—	4	2	no	—
59	C₂×C₉₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
60	#2115	—	97	1	no	maximal, Sylow
61	C₉₇	#2115	1	1	yes	characteristic, complemented, Sylow
62	C₂×#131	—	97	1	no	—
63	C₂×M₃₂	—	97	2	no	—
64	#349	—	97	4	no	—
65	C₂×#41	#87604	1	1	yes	characteristic
66	#131	—	194	2	no	—
67	#131	—	194	2	no	—
68	C₂×M₁₆	—	97	2	no	—
69	C₂×M₁₆	—	97	1	no	—
70	C₂²×C₈	—	97	1	no	—
71	C₂²×C₈	—	97	2	no	—
72	M₃₂	—	97	4	no	—
73	M₃₂	—	194	2	no	—
74	C₂×C₁₆	—	194	2	no	—
75	#41	—	2	2	no	—
76	#41	—	2	2	no	—
77	C₂×Q₈	#11715897	1	1	yes	characteristic
78	C₂×D₄	#11715897	1	2	yes	—
79	C₂×D₄	#11715897	1	1	yes	characteristic
80	C₂²×C₄	C₂×#87604	1	2	yes	—
81	C₂²×C₄	C₂×#87604	1	1	yes	characteristic
82	M₁₆	—	194	4	no	—
83	M₁₆	—	194	2	no	—
84	C₂×C₈	—	97	2	no	—
85	C₂×C₈	—	97	2	no	—
86	C₂×C₈	—	97	1	no	—
87	C₂×C₈	—	97	1	no	—
88	C₂×C₈	—	97	2	no	—
89	C₂×C₈	—	194	2	no	—
90	C₂×C₈	—	194	2	no	—
91	C₁₆	—	194	4	no	—
92	C₂³	C₂×#11715897	1	2	yes	—
93	C₂³	C₂×#11715897	1	1	yes	characteristic
94	Q₈	—	2	2	no	—
95	D₄	—	2	4	no	—
96	D₄	—	2	2	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	C₂²⋊#87604	1	2	yes	—
100	C₂×C₄	C₂×#11715897	1	2	yes	—
101	C₂×C₄	C₂×#11715897	1	1	yes	characteristic
102	C₂×C₄	#13095520	1	2	yes	—
103	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
104	C₈	—	97	2	no	—
105	C₈	—	97	1	no	—
106	C₈	—	97	1	no	—
107	C₈	—	194	2	no	—
108	C₂²	—	2	2	no	—
109	C₂²	—	4	1	no	—
110	C₂²	—	4	2	no	—
111	C₂²	C₂²⋊#11715897	1	2	yes	—
112	C₂²	C₂²×#11715897	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	C₂²⋊#11715897	1	2	yes	—
115	C₄	C₂×C₂²⋊#87604	1	1	yes	characteristic
116	C₄	C₂×#13095520	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	2	2	no	—
119	C₂	—	4	2	no	—
120	C₂	C₂×C₂²⋊#11715897	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690884

C₂

yes

maximal, complemented

#20261631

C₂

yes

maximal, complemented

C₂×#15690817

C₂

yes

maximal, characteristic

#15690884

—

#15690884

C₂²

yes

—

—

—

—

C₄

yes

characteristic

C₂×#13095495

C₂²

yes

—

C₂²×#11715895

C₄

yes

—

C₂²×#11715895

C₂²

yes

characteristic

C₂×#41×C₉₇

C₄

yes

—

—

—

—

—

C₂×C₄

yes

—

C₂×#11715895

C₂×C₄

yes

characteristic

C₂×#11715895

D₄

yes

—

C₂×#11715895

D₄

yes

—

C₂×#11715895

C₂³

yes

characteristic

#41×C₉₇

—

#41×C₉₇

—

C₂×Q₈×C₉₇

C₈

yes

characteristic

C₂×D₄×C₉₇

C₈

yes

—

C₂×D₄×C₉₇

C₈

yes

characteristic

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂³×C₉₇

C₂×C₈

yes

—

C₂³×C₉₇

C₂×C₈

yes

characteristic

Q₈×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂×C₈

yes

—

C₂×C₄×C₉₇

C₂×C₈

yes

characteristic

C₂×C₄×C₉₇

M₁₆

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

#11715895

—

#11715895

C₂²⋊C₄

yes

characteristic

#11715895

C₂²⋊C₄

yes

characteristic

#11715895

C₂×D₄

yes

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂²⋊C₈

yes

—

C₂²×C₉₇

C₂²×C₈

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

C₂²⋊C₈

yes

—

C₄×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₉₇

C₂×M₁₆

yes

characteristic

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

C₂×C₂²⋊C₈

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₂×#131

—

C₂×M₃₂

—

#349

—

C₂×#41

#87604

yes

characteristic

#131

—

194

—

#131

—

194

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

M₃₂

—

M₃₂

—

194

—

C₂×C₁₆

—

194

—

#41

—

#41

—

C₂×Q₈

#11715897

yes

characteristic

C₂×D₄

#11715897

yes

—

C₂×D₄

#11715897

yes

characteristic

C₂²×C₄

C₂×#87604

yes

—

C₂²×C₄

C₂×#87604

yes

characteristic

M₁₆

—

194

—

M₁₆

—

194

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₁₆

—

194

—

C₂³

C₂×#11715897

yes

—

C₂³

C₂×#11715897

yes

characteristic

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊#87604

yes

—

100

C₂×C₄

C₂×#11715897

yes

—

101

C₂×C₄

C₂×#11715897

yes

characteristic

102

C₂×C₄

#13095520

yes

—

103

C₂×C₄

C₂²×#87604

yes

characteristic

104

C₈

—

105

C₈

—

106

C₈

—

107

C₈

—

194

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

C₂²⋊#11715897

yes

—

112

C₂²

C₂²×#11715897

yes

characteristic

113

C₄

—

114

C₄

C₂²⋊#11715897

yes

—

115

C₄

C₂×C₂²⋊#87604

yes

characteristic

116

C₄

C₂×#13095520

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

—

120

C₂

C₂×C₂²⋊#11715897

yes

characteristic

	subgroup	G/H
center	C₄	C₂×C₂²⋊#87604
commutator	C₂²×C₉₇	C₂²×C₈
Frattini	C₂×C₄	C₂²×#87604
Fitting, prime radical	C₂×#41×C₉₇	C₄
socle	C₂×C₉₇	C₂×C₂²⋊C₈
cosocle	C₂×#11715895	C₂³
hypercenter	C₂×#41	#87604
hypocenter	C₉₇	#2115
LCS	G ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#41

subgroup

G/H

center

C₄

C₂×C₂²⋊#87604

commutator

C₂²×C₉₇

C₂²×C₈

Frattini

C₂×C₄

C₂²×#87604

Fitting, prime radical

C₂×#41×C₉₇

C₄

socle

C₂×C₉₇

C₂×C₂²⋊C₈

C₂³

C₂×#41

C₉₇

G ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#41

N	H ≅ G/N	type
1. C₂×#15690884	C₂	semidirect
2. #20261631	C₂	semidirect
61. C₉₇	#2115	semidirect

H ≅ G/N

type

1. C₂×#15690884

C₂

semidirect

2. #20261631

C₂

semidirect

61. C₉₇

#2115

semidirect

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

subgroup	index
C₂×#15690884	2
#20261631	2
C₂×#15690817	2
#2115	97

subgroup

index

Subgroups — #27157972

Conjugacy classes of subgroups

Complements

Maximal subgroups