Group #27141773 — subgroups

Subgroups — #27141773

Order 12416.

292 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 89 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×C₈×D₉₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#15690745	C₂	1	2	yes	maximal, complemented
3	#20260698	C₂	1	2	yes	maximal
4	#20260500	C₂	1	1	yes	maximal, characteristic, complemented
5	#341×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₉₇	C₂²	1	1	yes	characteristic
7	C₂×#13095495	C₂²	1	2	yes	—
8	C₂²×#11715895	C₂²	1	1	yes	characteristic
9	C₂×C₈×D₉₇	—	2	4	no	—
10	C₂×C₈×D₉₇	C₄	1	4	yes	—
11	#15690745	—	2	4	no	—
12	#15690722	—	2	4	no	—
13	C₂×M₁₆×C₉₇	C₂²	1	2	yes	—
14	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
15	C₂³×D₉₇	Q₈	1	1	yes	characteristic
16	C₂²×Dic₉₇	Q₈	1	1	yes	characteristic
17	C₂×C₄×D₉₇	—	2	2	no	—
18	C₂×C₄×D₉₇	—	2	2	no	—
19	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	—
20	C₂×C₄×D₉₇	D₄	1	2	yes	—
21	#13095495	—	2	2	no	—
22	#13095495	—	2	2	no	—
23	C₂×#11715895	—	2	2	no	—
24	C₂×#11715895	C₂×C₄	1	2	yes	—
25	C₂×#11715895	D₄	1	4	yes	—
26	C₈×D₉₇	—	2	8	no	—
27	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
28	M₁₆×C₉₇	—	2	2	no	—
29	M₁₆×C₉₇	—	2	2	no	—
30	C₂×C₈×C₉₇	—	2	2	no	—
31	C₂×C₈×C₉₇	—	2	2	no	—
32	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
33	C₂³×C₉₇	C₂×Q₈	1	1	yes	characteristic
34	C₂²×D₉₇	—	2	2	no	—
35	C₂²×D₉₇	—	2	4	no	—
36	C₂²×D₉₇	C₄⋊C₄	1	2	yes	—
37	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₉₇	#41	1	2	yes	—
41	C₂×Dic₉₇	—	2	2	no	—
42	C₂×Dic₉₇	C₄⋊C₄	1	2	yes	—
43	C₄×D₉₇	—	2	2	no	—
44	C₄×D₉₇	—	2	2	no	—
45	C₄×D₉₇	—	2	4	no	—
46	C₈×C₉₇	—	2	2	no	—
47	C₈×C₉₇	—	4	2	no	—
48	#11715895	—	4	2	no	—
49	#11715895	C₂²⋊C₄	1	4	yes	—
50	C₂²×C₉₇	—	2	2	no	—
51	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₉₇	C₂²⋊Q₈	1	2	yes	—
53	C₂×D₉₇	—	2	8	no	—
54	C₂×D₉₇	—	2	2	no	—
55	C₂×D₉₇	—	2	2	no	—
56	C₂×D₉₇	#108	1	4	yes	complemented
57	C₄×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₄×C₉₇	#117	1	1	yes	characteristic
59	C₄×C₉₇	#121	1	2	yes	—
60	Dic₉₇	—	2	2	no	—
61	C₂×C₉₇	—	2	2	no	—
62	C₂×C₉₇	#380	1	1	yes	characteristic
63	C₂×C₉₇	C₂×#108	1	2	yes	—
64	D₉₇	—	2	4	no	—
65	#1810	—	97	1	no	maximal, Sylow
66	C₉₇	#1810	1	1	yes	characteristic, complemented, Sylow
67	C₂³×C₈	—	97	1	no	—
68	C₂×#108	—	97	2	no	—
69	#407	—	97	2	no	—
70	#341	—	97	1	no	—
71	#341	D₉₇	1	1	yes	characteristic, complemented
72	C₂³×C₄	—	97	1	no	—
73	C₂×M₁₆	—	97	2	no	—
74	C₂×M₁₆	C₂×D₉₇	1	2	yes	—
75	C₂²×C₈	—	97	1	no	—
76	C₂²×C₈	—	97	4	no	—
77	C₂²×C₈	—	194	4	no	—
78	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
79	#108	—	194	4	no	—
80	C₂²⋊C₈	—	194	4	no	—
81	C₂⁴	—	97	1	no	—
82	C₂²×C₄	—	97	1	no	—
83	C₂²×C₄	—	97	2	no	—
84	C₂²×C₄	—	97	2	no	—
85	C₂²×C₄	—	194	2	no	—
86	C₂²×C₄	—	194	2	no	—
87	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
88	M₁₆	—	2	2	no	—
89	M₁₆	—	2	2	no	—
90	M₁₆	—	194	2	no	—
91	M₁₆	—	194	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	97	2	no	—
95	C₂×C₈	—	97	4	no	—
96	C₂×C₈	—	194	8	no	—
97	C₂×C₈	—	194	2	no	—
98	C₂×C₈	C₄×D₉₇	1	2	yes	—
99	C₂³	—	97	2	no	—
100	C₂³	—	194	4	no	—
101	C₂³	—	194	2	no	—
102	C₂³	Q₈×D₉₇	1	1	yes	characteristic
103	C₂×C₄	—	97	2	no	—
104	C₂×C₄	—	194	2	no	—
105	C₂×C₄	—	194	2	no	—
106	C₂×C₄	—	194	2	no	—
107	C₂×C₄	—	194	4	no	—
108	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
109	C₂×C₄	D₄×D₉₇	1	2	yes	—
110	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
111	C₂×C₄	#13095511	1	2	yes	—
112	C₈	—	2	2	no	—
113	C₈	—	4	2	no	—
114	C₈	—	97	4	no	—
115	C₈	—	388	2	no	—
116	C₂²	—	2	2	no	—
117	C₂²	—	97	4	no	—
118	C₂²	—	194	8	no	—
119	C₂²	—	194	2	no	—
120	C₂²	—	194	2	no	—
121	C₂²	C₄⋊C₄×D₉₇	1	1	yes	characteristic
122	C₂²	#15690788	1	2	yes	—
123	C₄	—	194	2	no	—
124	C₄	C₂²⋊C₄×D₉₇	1	1	yes	characteristic
125	C₄	#15690767	1	1	yes	characteristic
126	C₄	#15690783	1	2	yes	—
127	C₂	—	2	2	no	—
128	C₂	—	194	4	no	—
129	C₂	#20260594	1	1	yes	characteristic
130	C₂	#108×D₉₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₈×D₉₇

C₂

yes

maximal, characteristic

C₂×#15690745

C₂

yes

maximal, complemented

#20260698

C₂

yes

maximal

#20260500

C₂

yes

maximal, characteristic, complemented

#341×C₉₇

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₉₇

C₂²

yes

characteristic

C₂×#13095495

C₂²

yes

—

C₂²×#11715895

C₂²

yes

characteristic

C₂×C₈×D₉₇

—

C₂×C₈×D₉₇

C₄

yes

—

#15690745

—

#15690722

—

C₂×M₁₆×C₉₇

C₂²

yes

—

C₂²×C₈×C₉₇

C₂²

yes

characteristic

C₂³×D₉₇

Q₈

yes

characteristic

C₂²×Dic₉₇

Q₈

yes

characteristic

C₂×C₄×D₉₇

—

C₂×C₄×D₉₇

—

C₂×C₄×D₉₇

C₂×C₄

yes

—

C₂×C₄×D₉₇

D₄

yes

—

—

—

—

C₂×C₄

yes

—

C₂×#11715895

D₄

yes

—

C₈×D₉₇

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

M₁₆×C₉₇

—

M₁₆×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

C₂×C₄

yes

—

C₂³×C₉₇

C₂×Q₈

yes

characteristic

C₂²×D₉₇

—

C₂²×D₉₇

—

C₂²×D₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

C₄⋊C₄

yes

—

C₄×D₉₇

—

C₄×D₉₇

—

C₄×D₉₇

—

C₈×C₉₇

—

C₈×C₉₇

—

#11715895

—

#11715895

C₂²⋊C₄

yes

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₉₇

C₂²⋊Q₈

yes

—

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

#108

yes

complemented

C₄×C₉₇

C₂×C₂²⋊C₄

yes

C₄×C₉₇

yes

C₄×C₉₇

yes

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#380

yes

characteristic

C₂×C₉₇

C₂×#108

yes

—

D₉₇

—

#1810

—

maximal, Sylow

C₉₇

#1810

yes

characteristic, complemented, Sylow

C₂³×C₈

—

—

—

—

D₉₇

yes

characteristic, complemented

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₉₇

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

194

—

C₂²×C₈

C₂×D₉₇

yes

characteristic

#108

—

194

—

C₂²⋊C₈

—

194

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

194

—

C₂²×C₄

—

194

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

194

—

M₁₆

—

194

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₂×C₈

C₄×D₉₇

yes

—

C₂³

—

100

C₂³

—

194

—

101

C₂³

—

194

—

102

C₂³

Q₈×D₉₇

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

194

—

105

C₂×C₄

—

194

—

106

C₂×C₄

—

194

—

107

C₂×C₄

—

194

—

108

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

109

C₂×C₄

D₄×D₉₇

yes

—

110

C₂×C₄

D₄×D₉₇

yes

characteristic

111

C₂×C₄

#13095511

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

388

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

194

—

119

C₂²

—

194

—

120

C₂²

—

194

—

121

C₂²

C₄⋊C₄×D₉₇

yes

characteristic

122

C₂²

#15690788

yes

—

123

C₄

—

194

—

124

C₄

C₂²⋊C₄×D₉₇

yes

characteristic

125

C₄

#15690767

yes

characteristic

126

C₄

#15690783

yes

—

127

C₂

—

128

C₂

—

194

—

129

C₂

#20260594

yes

characteristic

130

C₂

#108×D₉₇

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₉₇
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#341×C₉₇	C₂
socle	C₂²×C₉₇	C₂×C₄⋊C₄
prime radical	C₂³×D₉₇	Q₈
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#341	D₉₇
hypocenter	C₉₇	#1810
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₉₇

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

#341×C₉₇

C₂

socle

C₂²×C₉₇

C₂×C₄⋊C₄

prime radical

C₂³×D₉₇

Q₈

cosocle

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
2. C₂×#15690745	C₂	semidirect
4. #20260500	C₂	semidirect
5. #341×C₉₇	C₂	semidirect
56. C₂×D₉₇	#108	semidirect
66. C₉₇	#1810	semidirect
71. #341	D₉₇	semidirect

H ≅ G/N

type

2. C₂×#15690745

C₂

semidirect

4. #20260500

C₂

semidirect

5. #341×C₉₇

C₂

semidirect

56. C₂×D₉₇

#108

semidirect

66. C₉₇

#1810

semidirect

71. #341

D₉₇

semidirect

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

subgroup	index
C₂²×C₈×D₉₇	2
C₂×#15690745	2
#20260698	2
#20260500	2
#341×C₉₇	2
#1810	97

subgroup

index

C₂²×C₈×D₉₇

#341×C₉₇

Subgroups — #27141773

Conjugacy classes of subgroups

Complements

Maximal subgroups