Group #27142522 — subgroups

Subgroups — #27142522

Order 12416.

196 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 97 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	#20260777	C₂	1	2	yes	maximal
2	C₂×#15690744	C₂	1	1	yes	maximal, characteristic
3	#20260690	C₂	1	1	yes	maximal, characteristic
4	#20260503	C₂	1	2	yes	maximal
5	C₂×#108×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13095495	C₂²	1	1	yes	characteristic
7	#15690769	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
9	#15690744	—	2	2	no	—
10	#15690744	—	2	1	no	—
11	#15690744	—	2	1	no	—
12	#15690742	—	2	2	no	—
13	#15690720	—	2	1	no	—
14	C₈×Dic₉₇	—	2	1	no	—
15	C₂×M₁₆×C₉₇	C₂²	1	2	yes	—
16	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
17	#108×C₉₇	C₄	1	4	yes	complemented
18	C₂²×Dic₉₇	—	2	2	no	—
19	#13095495	—	2	1	no	—
20	#13095495	Q₈	1	2	yes	—
21	C₂×#11715895	—	2	1	no	—
22	C₄⋊Dic₉₇	—	2	2	no	—
23	C₄⋊Dic₉₇	—	2	2	no	—
24	C₄⋊Dic₉₇	D₄	1	1	yes	characteristic
25	C₄⋊Dic₉₇	D₄	1	1	yes	characteristic
26	C₂²⋊Dic₉₇	—	2	1	no	—
27	C₄×Dic₉₇	—	2	1	no	—
28	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
29	M₁₆×C₉₇	—	2	2	no	—
30	M₁₆×C₉₇	C₂×C₄	1	4	yes	—
31	C₂×C₈×C₉₇	—	2	2	no	—
32	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
33	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
34	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
35	C₂×C₈×C₉₇	Q₈	1	2	yes	—
36	C₂³×C₉₇	#41	1	1	yes	characteristic
37	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
43	C₂×Dic₉₇	—	2	1	no	—
44	C₂×Dic₉₇	—	2	1	no	—
45	C₂×Dic₉₇	—	4	2	no	—
46	C₂×Dic₉₇	—	8	2	no	—
47	C₈×C₉₇	—	2	4	no	—
48	C₈×C₉₇	C₄⋊C₄	1	2	yes	—
49	C₈×C₉₇	C₄⋊C₄	1	2	yes	—
50	#11715895	—	2	2	no	—
51	C₂²×C₉₇	—	2	1	no	—
52	C₂²×C₉₇	—	2	1	no	—
53	C₂²×C₉₇	C₄×Q₈	1	1	yes	characteristic
54	C₂²×C₉₇	#121	1	1	yes	characteristic
55	C₂²×C₉₇	#125	1	1	yes	characteristic
56	C₄×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₄×C₉₇	C₄×D₄	1	1	yes	characteristic
58	C₄×C₉₇	C₂²⋊Q₈	1	1	yes	characteristic
59	C₄×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
60	Dic₉₇	—	4	1	no	—
61	Dic₉₇	—	4	1	no	—
62	Dic₉₇	—	8	2	no	—
63	C₂×C₉₇	—	2	1	no	—
64	C₂×C₉₇	—	2	1	no	—
65	C₂×C₉₇	#389	1	1	yes	characteristic
66	C₂×C₉₇	#472	1	1	yes	characteristic
67	C₂×C₉₇	#473	1	1	yes	characteristic
68	#1952	—	97	1	no	maximal, Sylow
69	C₉₇	#1952	1	1	yes	characteristic, complemented, Sylow
70	C₂×#108	D₉₇	1	1	yes	characteristic
71	#428	—	97	2	no	—
72	C₂×#107	—	97	1	no	—
73	#405	—	97	1	no	—
74	#343	—	97	2	no	—
75	C₂×M₁₆	—	97	1	no	—
76	C₂×M₁₆	C₂×D₉₇	1	2	yes	—
77	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
78	#117	—	97	1	no	—
79	C₂×C₄⋊C₄	—	97	2	no	—
80	#108	Dic₉₇	1	4	yes	complemented
81	#107	—	194	2	no	—
82	#107	—	194	1	no	—
83	#107	—	194	1	no	—
84	#106	—	194	2	no	—
85	#97	—	194	1	no	—
86	C₄×C₈	—	194	1	no	—
87	C₂²×C₄	—	194	2	no	—
88	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
89	M₁₆	—	2	2	no	—
90	M₁₆	—	97	2	no	—
91	M₁₆	—	194	1	no	—
92	M₁₆	C₂×Dic₉₇	1	4	yes	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	194	1	no	—
95	C₂×C₈	Dic₁₉₄	1	2	yes	—
96	C₂×C₈	D₃₈₈	1	1	yes	characteristic
97	C₂×C₈	D₃₈₈	1	1	yes	characteristic
98	C₂×C₈	C₂×Dic₉₇	1	2	yes	—
99	C₄⋊C₄	—	97	1	no	—
100	C₄⋊C₄	—	97	1	no	—
101	C₄⋊C₄	—	194	2	no	—
102	C₄⋊C₄	—	194	2	no	—
103	C₂²⋊C₄	—	194	1	no	—
104	C₄²	—	194	1	no	—
105	C₂³	#13095512	1	1	yes	characteristic
106	C₂×C₄	—	776	2	no	—
107	C₂×C₄	—	388	2	no	—
108	C₂×C₄	—	194	1	no	—
109	C₂×C₄	—	194	1	no	—
110	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
111	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
112	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
113	C₂×C₄	Q₈×D₉₇	1	1	yes	characteristic
114	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
115	C₂×C₄	#13095510	1	1	yes	characteristic
116	C₈	—	2	4	no	—
117	C₈	—	194	2	no	—
118	C₈	C₄⋊Dic₉₇	1	2	yes	—
119	C₈	C₄⋊Dic₉₇	1	2	yes	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	Q₈×Dic₉₇	1	1	yes	characteristic
123	C₂²	#15690785	1	1	yes	characteristic
124	C₂²	#15690801	1	1	yes	characteristic
125	C₄	—	776	2	no	—
126	C₄	—	388	1	no	—
127	C₄	—	388	1	no	—
128	C₄	C₂×C₄⋊Dic₉₇	1	1	yes	characteristic
129	C₄	D₄×Dic₉₇	1	1	yes	characteristic
130	C₄	#15690790	1	1	yes	characteristic
131	C₄	#15690809	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	2	1	no	—
134	C₂	#20260634	1	1	yes	characteristic
135	C₂	#20261006	1	1	yes	characteristic
136	C₂	#20261011	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20260777

C₂

yes

maximal

C₂×#15690744

C₂

yes

maximal, characteristic

#20260690

C₂

yes

maximal, characteristic

#20260503

C₂

yes

maximal

C₂×#108×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

C₂²

yes

characteristic

#15690769

C₂²

yes

characteristic

C₂×C₄⋊Dic₉₇

C₂²

yes

—

—

—

—

—

—

C₈×Dic₉₇

—

C₂×M₁₆×C₉₇

C₂²

yes

—

C₂²×C₈×C₉₇

C₂²

yes

characteristic

#108×C₉₇

C₄

yes

complemented

C₂²×Dic₉₇

—

#13095495

—

#13095495

Q₈

yes

—

C₂×#11715895

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

D₄

yes

characteristic

C₄⋊Dic₉₇

D₄

yes

characteristic

C₂²⋊Dic₉₇

—

C₄×Dic₉₇

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

M₁₆×C₉₇

—

M₁₆×C₉₇

C₂×C₄

yes

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

C₂×C₄

yes

—

C₂×C₈×C₉₇

D₄

yes

characteristic

C₂×C₈×C₉₇

D₄

yes

characteristic

C₂×C₈×C₉₇

Q₈

yes

—

C₂³×C₉₇

#41

yes

characteristic

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₉₇

#41

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₈×C₉₇

—

C₈×C₉₇

C₄⋊C₄

yes

—

C₈×C₉₇

C₄⋊C₄

yes

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₄×Q₈

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₄×C₉₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₉₇

C₄×D₄

yes

characteristic

C₄×C₉₇

C₂²⋊Q₈

yes

characteristic

C₄×C₉₇

C₄⋊Q₈

yes

characteristic

Dic₉₇

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₂×#108

D₉₇

yes

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₉₇

yes

—

C₂²×C₈

C₂×D₉₇

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

#108

Dic₉₇

yes

complemented

#107

—

194

—

#107

—

194

—

#107

—

194

—

#106

—

194

—

#97

—

194

—

C₄×C₈

—

194

—

C₂²×C₄

—

194

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

194

—

M₁₆

C₂×Dic₉₇

yes

—

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

Dic₁₉₄

yes

—

C₂×C₈

D₃₈₈

yes

characteristic

C₂×C₈

D₃₈₈

yes

characteristic

C₂×C₈

C₂×Dic₉₇

yes

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

194

—

102

C₄⋊C₄

—

194

—

103

C₂²⋊C₄

—

194

—

104

C₄²

—

194

—

105

C₂³

#13095512

yes

characteristic

106

C₂×C₄

—

776

—

107

C₂×C₄

—

388

—

108

C₂×C₄

—

194

—

109

C₂×C₄

—

194

—

110

C₂×C₄

C₂²×Dic₉₇

yes

characteristic

111

C₂×C₄

D₄×D₉₇

yes

characteristic

112

C₂×C₄

C₂×D₃₈₈

yes

characteristic

113

C₂×C₄

Q₈×D₉₇

yes

characteristic

114

C₂×C₄

C₂×Dic₁₉₄

yes

characteristic

115

C₂×C₄

#13095510

yes

characteristic

116

C₈

—

117

C₈

—

194

—

118

C₈

C₄⋊Dic₉₇

yes

—

119

C₈

C₄⋊Dic₉₇

yes

—

120

C₂²

—

121

C₂²

—

122

C₂²

Q₈×Dic₉₇

yes

characteristic

123

C₂²

#15690785

yes

characteristic

124

C₂²

#15690801

yes

characteristic

125

C₄

—

776

—

126

C₄

—

388

—

127

C₄

—

388

—

128

C₄

C₂×C₄⋊Dic₉₇

yes

characteristic

129

C₄

D₄×Dic₉₇

yes

characteristic

130

C₄

#15690790

yes

characteristic

131

C₄

#15690809

yes

characteristic

132

C₂

—

133

C₂

—

134

C₂

#20260634

yes

characteristic

135

C₂

#20261006

yes

characteristic

136

C₂

#20261011

yes

characteristic

	subgroup	G/H
center	C₂²	#15690785
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	C₂×#108×C₉₇	C₂
socle	C₂²×C₉₇	#121
prime radical	C₂³×C₉₇	#41
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂×#108	D₉₇
hypocenter	C₉₇	#1952
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂²

#15690785

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

C₂×#108×C₉₇

C₂

C₂²×C₉₇

C₂³×C₉₇

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
17. #108×C₉₇	C₄	semidirect
69. C₉₇	#1952	semidirect
80. #108	Dic₉₇	semidirect

H ≅ G/N

type

17. #108×C₉₇

C₄

semidirect

69. C₉₇

#1952

semidirect

80. #108

Dic₉₇

semidirect

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

subgroup	index
#20260777	2
C₂×#15690744	2
#20260690	2
#20260503	2
C₂×#108×C₉₇	2
#1952	97

subgroup

index

C₂×#108×C₉₇

Subgroups — #27142522

Conjugacy classes of subgroups

Complements

Maximal subgroups