Group #27157843 — subgroups

Subgroups — #27157843

Order 12416.

208 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 70 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×#15690880	C₂	1	1	yes	maximal, characteristic
2	#20261742	C₂	1	2	yes	maximal
3	C₂×C₈×#87604	C₂	1	1	yes	maximal, characteristic
4	#20261605	C₂	1	2	yes	maximal
5	#20260770	C₂	1	1	yes	maximal, characteristic
6	C₂×#13095521	C₂²	1	2	yes	—
7	C₂×#13095519	C₂²	1	1	yes	characteristic
8	C₂×C₄×#87604	C₂²	1	1	yes	characteristic
9	#15690880	—	2	2	no	—
10	#15690877	—	2	2	no	—
11	C₄⋊#11715897	—	2	2	no	—
12	C₈×#87604	—	2	2	no	—
13	C₂×C₈×D₉₇	C₂²	1	1	yes	characteristic
14	#15690768	C₂²	1	2	yes	—
15	#15690743	C₄	1	2	yes	complemented
16	#15690742	C₄	1	1	yes	characteristic, complemented
17	#106×C₉₇	C₄	1	1	yes	characteristic, complemented
18	C₂²×#87604	—	2	1	no	—
19	#13095521	—	2	2	no	—
20	#13095521	—	2	2	no	—
21	#13095519	—	2	1	no	—
22	#13095519	—	2	1	no	—
23	C₂×#11715897	—	2	2	no	—
24	C₂×#11715897	D₄	1	2	yes	—
25	C₄×#87604	—	2	1	no	—
26	C₄×#87604	Q₈	1	2	yes	—
27	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
28	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
29	C₈×D₉₇	D₄	1	2	yes	—
30	C₈×D₉₇	Q₈	1	2	yes	—
31	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
32	#13095488	—	2	2	no	—
33	C₄×Dic₉₇	—	2	2	no	—
34	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
35	C₄⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
36	C₂²×D₉₇	#41	1	1	yes	characteristic
37	C₂×#87604	—	2	4	no	—
38	C₂×#87604	—	2	2	no	—
39	C₂×C₄×C₉₇	—	2	2	no	—
40	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×Dic₉₇	—	2	2	no	—
42	C₂×Dic₉₇	C₂×D₄	1	1	yes	characteristic
43	C₄×D₉₇	C₂×D₄	1	1	yes	characteristic
44	C₄×D₉₇	C₂×Q₈	1	1	yes	characteristic
45	C₄×D₉₇	C₂×Q₈	1	1	yes	characteristic
46	C₄×D₉₇	#41	1	1	yes	characteristic
47	#11715897	—	2	2	no	—
48	#11715897	—	4	2	no	—
49	C₈×C₉₇	C₄⋊C₄	1	2	yes	—
50	#11715895	C₄⋊C₄	1	2	yes	—
51	C₂²×C₉₇	C₄×D₄	1	1	yes	characteristic
52	C₂×D₉₇	—	2	2	no	—
53	C₂×D₉₇	C₂²⋊Q₈	1	1	yes	characteristic
54	C₂×D₉₇	#125	1	1	yes	characteristic
55	#87604	—	2	4	no	—
56	C₄×C₉₇	—	4	2	no	—
57	C₄×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₄×C₉₇	C₄×Q₈	1	1	yes	characteristic
59	Dic₉₇	—	4	2	no	—
60	Dic₉₇	#121	1	1	yes	characteristic
61	Dic₉₇	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₉₇	#389	1	1	yes	characteristic
63	C₂×C₉₇	#443	1	2	yes	—
64	D₉₇	—	2	1	no	—
65	D₉₇	—	2	1	no	—
66	#1945	—	97	1	no	maximal, Sylow
67	C₉₇	#1945	1	1	yes	characteristic, complemented, Sylow
68	C₂×#108	—	97	1	no	—
69	#427	—	97	1	no	—
70	#424	—	97	2	no	—
71	C₂×C₄×C₈	—	97	1	no	—
72	#339	—	97	2	no	—
73	C₂×M₁₆	—	97	2	no	—
74	C₂²×C₈	—	97	1	no	—
75	C₂²×C₈	—	97	1	no	—
76	#117	—	97	2	no	—
77	C₂×C₄²	—	97	1	no	—
78	#108	—	194	2	no	—
79	#107	—	97	2	no	—
80	#106	—	97	1	no	—
81	#106	#87604	1	1	yes	characteristic, complemented
82	C₄⋊C₈	—	194	2	no	—
83	C₄⋊C₈	—	194	2	no	—
84	C₄×C₈	—	194	2	no	—
85	C₂²×C₄	—	97	1	no	—
86	C₂²×C₄	—	194	1	no	—
87	M₁₆	—	194	2	no	—
88	M₁₆	—	194	2	no	—
89	C₂×C₈	—	97	1	no	—
90	C₂×C₈	—	97	2	no	—
91	C₂×C₈	—	97	2	no	—
92	C₂×C₈	—	97	2	no	—
93	C₂×C₈	—	194	2	no	—
94	C₂×C₈	—	194	1	no	—
95	C₂×C₈	—	194	1	no	—
96	C₂×C₈	C₂×#87604	1	1	yes	characteristic
97	C₄⋊C₄	—	97	2	no	—
98	C₄⋊C₄	C₂×#87604	1	2	yes	—
99	C₂²⋊C₄	—	194	2	no	—
100	C₄²	—	97	2	no	—
101	C₄²	—	194	2	no	—
102	C₄²	—	194	1	no	—
103	C₂³	—	97	1	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	97	1	no	—
106	C₂×C₄	—	97	1	no	—
107	C₂×C₄	—	97	1	no	—
108	C₂×C₄	—	97	1	no	—
109	C₂×C₄	—	97	1	no	—
110	C₂×C₄	—	194	2	no	—
111	C₂×C₄	—	194	2	no	—
112	C₂×C₄	—	194	4	no	—
113	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
114	C₈	—	97	2	no	—
115	C₈	—	388	2	no	—
116	C₈	—	194	2	no	—
117	C₈	C₄⋊#87604	1	2	yes	—
118	C₂²	—	97	1	no	—
119	C₂²	—	97	1	no	—
120	C₂²	—	194	2	no	—
121	C₂²	D₄×#87604	1	1	yes	characteristic
122	C₄	—	4	2	no	—
123	C₄	—	97	1	no	—
124	C₄	—	97	1	no	—
125	C₄	—	388	2	no	—
126	C₄	—	194	4	no	—
127	C₄	C₂×C₄⋊#87604	1	1	yes	characteristic
128	C₄	Q₈×#87604	1	1	yes	characteristic
129	C₂	—	194	1	no	—
130	C₂	—	194	1	no	—
131	C₂	#20261686	1	1	yes	characteristic
132	C₂	#20261778	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690880

C₂

yes

maximal, characteristic

#20261742

C₂

yes

maximal

C₂×C₈×#87604

C₂

yes

maximal, characteristic

#20261605

C₂

yes

maximal

#20260770

C₂

yes

maximal, characteristic

C₂×#13095521

C₂²

yes

—

C₂×#13095519

C₂²

yes

characteristic

C₂×C₄×#87604

C₂²

yes

—

—

—

—

C₂×C₈×D₉₇

C₂²

yes

characteristic

#15690768

C₂²

yes

—

#15690743

C₄

yes

complemented

#15690742

C₄

yes

characteristic, complemented

#106×C₉₇

C₄

yes

characteristic, complemented

C₂²×#87604

—

—

—

—

—

—

D₄

yes

—

C₄×#87604

—

C₄×#87604

Q₈

yes

—

C₂×C₄×D₉₇

C₂³

yes

characteristic

C₂×#11715895

C₂×C₄

yes

characteristic

C₈×D₉₇

D₄

yes

—

C₈×D₉₇

Q₈

yes

—

C₄⋊Dic₉₇

C₂×C₄

yes

—

#13095488

—

C₄×Dic₉₇

—

C₂×C₈×C₉₇

C₂×C₄

yes

characteristic

C₄⋊C₄×C₉₇

C₂×C₄

yes

—

C₂²×D₉₇

yes

—

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

C₂×D₄

yes

characteristic

C₄×D₉₇

C₂×D₄

yes

characteristic

C₄×D₉₇

C₂×Q₈

yes

characteristic

C₄×D₉₇

C₂×Q₈

yes

C₄×D₉₇

yes

—

—

C₈×C₉₇

C₄⋊C₄

yes

—

#11715895

C₄⋊C₄

yes

—

C₂²×C₉₇

C₄×D₄

yes

characteristic

C₂×D₉₇

—

C₂×D₉₇

C₂²⋊Q₈

yes

C₂×D₉₇

yes

—

C₄×C₉₇

—

C₄×C₉₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₉₇

C₄×Q₈

yes

characteristic

Dic₉₇

—

Dic₉₇

#121

yes

characteristic

Dic₉₇

C₄⋊Q₈

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

D₉₇

—

D₉₇

—

#1945

—

maximal, Sylow

C₉₇

#1945

yes

characteristic, complemented, Sylow

C₂×#108

—

#427

—

#424

—

C₂×C₄×C₈

—

#339

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

#117

—

C₂×C₄²

—

#108

—

194

—

—

—

yes

characteristic, complemented

C₄⋊C₈

—

194

—

C₄⋊C₈

—

194

—

C₄×C₈

—

194

—

C₂²×C₄

—

C₂²×C₄

—

194

—

M₁₆

—

194

—

M₁₆

—

194

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₂×C₈

C₂×#87604

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

C₂×#87604

yes

—

C₂²⋊C₄

—

194

—

100

C₄²

—

101

C₄²

—

194

—

102

C₄²

—

194

—

103

C₂³

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

194

—

111

C₂×C₄

—

194

—

112

C₂×C₄

—

194

—

113

C₂×C₄

C₂²×#87604

yes

characteristic

114

C₈

—

115

C₈

—

388

—

116

C₈

—

194

—

117

C₈

C₄⋊#87604

yes

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

194

—

121

C₂²

D₄×#87604

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

388

—

126

C₄

—

194

—

127

C₄

C₂×C₄⋊#87604

yes

characteristic

128

C₄

Q₈×#87604

yes

characteristic

129

C₂

—

194

—

130

C₂

—

194

—

131

C₂

#20261686

yes

characteristic

132

C₂

#20261778

yes

—

	subgroup	G/H
center	C₂²	D₄×#87604
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#87604
Fitting	#106×C₉₇	C₄
socle	C₂²×C₉₇	C₄×D₄
prime radical	C₂²×D₉₇	#41
cosocle	C₂×C₄×D₉₇	C₂³
hypercenter	#106	#87604
hypocenter	C₉₇	#1945
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

subgroup

G/H

center

C₂²

D₄×#87604

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#87604

Fitting

#106×C₉₇

C₄

socle

C₂²×C₉₇

C₄×D₄

prime radical

C₂²×D₉₇

#41

cosocle

C₂×C₄×D₉₇

C₂³

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

N	H ≅ G/N	type
15. #15690743	C₄	semidirect
16. #15690742	C₄	semidirect
17. #106×C₉₇	C₄	semidirect
67. C₉₇	#1945	semidirect
81. #106	#87604	semidirect

H ≅ G/N

type

15. #15690743

C₄

semidirect

16. #15690742

C₄

semidirect

17. #106×C₉₇

C₄

semidirect

67. C₉₇

#1945

semidirect

81. #106

#87604

semidirect

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

subgroup	index
C₂×#15690880	2
#20261742	2
C₂×C₈×#87604	2
#20261605	2
#20260770	2
#1945	97

subgroup

index

C₂×C₈×#87604

Subgroups — #27157843

Conjugacy classes of subgroups

Complements

Maximal subgroups