Group #27150093 — subgroups

Subgroups — #27150093

Order 12416.

752 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 213 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, complemented
2	C₂×#15690807	C₂	1	1	yes	maximal, characteristic
3	C₂×#15690798	C₂	1	4	yes	maximal, complemented
4	#20260637	C₂	1	8	yes	maximal, complemented
5	C₄³×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Dic₁₉₄	C₂²	1	2	yes	—
7	C₂²×D₃₈₈	C₂²	1	4	yes	—
8	C₂²×D₃₈₈	C₂²	1	2	yes	—
9	#15690810	—	2	4	no	—
10	#15690807	—	2	4	no	—
11	#15690798	—	2	16	no	—
12	C₂×C₄⋊Dic₉₇	C₂²	1	4	yes	—
13	C₂×#13095488	C₂²	1	16	yes	—
14	C₂×C₄²×C₉₇	C₂²	1	4	yes	—
15	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₉₇	C₂²	1	2	yes	—
17	C₂³×D₉₇	C₂³	1	4	yes	—
18	C₂×Dic₁₉₄	—	2	8	no	—
19	C₂×Dic₁₉₄	—	2	4	no	—
20	C₂×D₃₈₈	—	2	8	no	—
21	C₂×D₃₈₈	—	2	8	no	—
22	C₂×D₃₈₈	—	2	4	no	—
23	C₂×D₃₈₈	—	2	16	no	—
24	C₂²×Dic₉₇	C₂³	1	4	yes	—
25	C₄⋊Dic₉₇	—	2	8	no	—
26	#13095488	—	2	32	no	—
27	C₂²×C₄×C₉₇	C₂³	1	4	yes	—
28	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
30	C₄²×C₉₇	—	2	8	no	—
31	C₄²×C₉₇	D₄	1	4	yes	—
32	C₄²×C₉₇	D₄	1	8	yes	complemented
33	C₂³×C₉₇	C₂⁴	1	1	yes	characteristic
34	C₂²×D₉₇	—	2	8	no	—
35	C₂²×D₉₇	—	2	4	no	—
36	C₂²×D₉₇	—	2	16	no	—
37	C₂×C₄×C₉₇	—	2	8	no	—
38	C₂×C₄×C₉₇	C₂×D₄	1	8	yes	—
39	C₂×C₄×C₉₇	C₂×D₄	1	4	yes	—
40	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₉₇	C₂×D₄	1	4	yes	—
42	C₂×C₄×C₉₇	#41	1	8	yes	—
43	C₂×Dic₉₇	—	2	8	no	—
44	C₂×Dic₉₇	—	2	4	no	—
45	D₃₈₈	—	4	16	no	—
46	D₃₈₈	—	4	8	no	—
47	Dic₁₉₄	—	4	8	no	—
48	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
49	C₂²×C₉₇	C₂²×D₄	1	2	yes	—
50	C₂²×C₉₇	C₂×#41	1	4	yes	—
51	C₂×D₉₇	—	4	8	no	—
52	C₂×D₉₇	—	4	4	no	—
53	C₂×D₉₇	—	4	16	no	—
54	C₄×C₉₇	—	2	8	no	—
55	C₄×C₉₇	#124	1	8	yes	complemented
56	C₄×C₉₇	#127	1	4	yes	—
57	Dic₉₇	—	4	4	no	—
58	C₂×C₉₇	C₂×#124	1	2	yes	—
59	C₂×C₉₇	C₂×#127	1	1	yes	characteristic
60	C₂×C₉₇	#532	1	4	yes	—
61	D₉₇	—	8	4	no	—
62	#2858	—	97	1	no	maximal, Sylow
63	C₉₇	#2858	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄⋊Q₈	—	97	1	no	—
65	C₂×#127	—	97	1	no	—
66	C₂×#124	—	97	4	no	—
67	#390	—	97	8	no	—
68	C₄³	D₉₇	1	1	yes	characteristic, complemented
69	C₂²×Q₈	—	97	2	no	—
70	C₂²×D₄	—	97	2	no	—
71	C₂²×D₄	—	97	4	no	—
72	C₄⋊Q₈	—	194	4	no	—
73	#127	—	194	4	no	—
74	#124	—	194	16	no	—
75	C₂×C₄⋊C₄	—	97	4	no	—
76	C₂×C₂²⋊C₄	—	97	16	no	—
77	C₂×C₄²	C₂×D₉₇	1	4	yes	—
78	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
79	C₂×C₄²	C₂×D₉₇	1	2	yes	—
80	C₂⁴	—	97	4	no	—
81	C₂×Q₈	—	194	8	no	—
82	C₂×Q₈	—	194	4	no	—
83	C₂×D₄	—	194	8	no	—
84	C₂×D₄	—	194	8	no	—
85	C₂×D₄	—	194	4	no	—
86	C₂×D₄	—	194	16	no	—
87	C₂²×C₄	—	97	4	no	—
88	C₂²×C₄	C₂²×D₉₇	1	4	yes	—
89	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
90	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
91	C₄⋊C₄	—	194	8	no	—
92	C₂²⋊C₄	—	194	32	no	—
93	C₄²	—	2	8	no	—
94	C₄²	D₃₈₈	1	4	yes	—
95	C₄²	D₃₈₈	1	8	yes	complemented
96	C₂³	—	194	8	no	—
97	C₂³	—	194	4	no	—
98	C₂³	—	194	16	no	—
99	C₂³	C₂³×D₉₇	1	1	yes	characteristic
100	Q₈	—	388	8	no	—
101	D₄	—	388	16	no	—
102	D₄	—	388	8	no	—
103	C₂×C₄	—	2	8	no	—
104	C₂×C₄	—	194	8	no	—
105	C₂×C₄	—	194	4	no	—
106	C₂×C₄	C₂×D₃₈₈	1	8	yes	—
107	C₂×C₄	C₂×D₃₈₈	1	4	yes	—
108	C₂×C₄	C₂×D₃₈₈	1	2	yes	—
109	C₂×C₄	C₂×D₃₈₈	1	4	yes	—
110	C₂×C₄	#13095511	1	8	yes	—
111	C₂²	—	388	8	no	—
112	C₂²	—	388	4	no	—
113	C₂²	—	388	16	no	—
114	C₂²	C₂²×D₃₈₈	1	1	yes	characteristic
115	C₂²	C₂²×D₃₈₈	1	2	yes	—
116	C₂²	C₂×#13095511	1	4	yes	—
117	C₄	—	2	8	no	—
118	C₄	—	388	4	no	—
119	C₄	#15690798	1	8	yes	complemented
120	C₄	#15690807	1	4	yes	—
121	C₂	—	776	4	no	—
122	C₂	C₂×#15690798	1	2	yes	—
123	C₂	C₂×#15690807	1	1	yes	characteristic
124	C₂	#20261285	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690810

C₂

yes

maximal, characteristic, complemented

C₂×#15690807

C₂

yes

maximal, characteristic

C₂×#15690798

C₂

yes

maximal, complemented

#20260637

C₂

yes

maximal, complemented

C₄³×C₉₇

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₁₉₄

C₂²

yes

—

C₂²×D₃₈₈

C₂²

yes

—

C₂²×D₃₈₈

C₂²

yes

—

#15690810

—

#15690807

—

#15690798

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

C₂×#13095488

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂×C₄²×C₉₇

C₂²

yes

—

C₂³×D₉₇

C₂³

yes

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×D₃₈₈

—

C₂×D₃₈₈

—

C₂×D₃₈₈

—

C₂×D₃₈₈

—

C₂²×Dic₉₇

C₂³

yes

—

C₄⋊Dic₉₇

—

#13095488

—

C₂²×C₄×C₉₇

C₂³

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₂²×C₄×C₉₇

C₂³

yes

—

C₄²×C₉₇

—

C₄²×C₉₇

D₄

yes

—

C₄²×C₉₇

D₄

yes

complemented

C₂³×C₉₇

C₂⁴

yes

characteristic

C₂²×D₉₇

—

C₂²×D₉₇

—

C₂²×D₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

#41

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

D₃₈₈

—

D₃₈₈

—

Dic₁₉₄

—

C₂²×C₉₇

C₂²×D₄

yes

characteristic

C₂²×C₉₇

C₂²×D₄

yes

—

C₂²×C₉₇

C₂×#41

yes

—

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#124

yes

complemented

C₄×C₉₇

#127

yes

—

Dic₉₇

—

C₂×C₉₇

C₂×#124

yes

—

C₂×C₉₇

C₂×#127

yes

characteristic

C₂×C₉₇

#532

yes

—

D₉₇

—

#2858

—

maximal, Sylow

C₉₇

#2858

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×#127

—

C₂×#124

—

#390

—

C₄³

D₉₇

yes

characteristic, complemented

C₂²×Q₈

—

C₂²×D₄

—

C₂²×D₄

—

C₄⋊Q₈

—

194

—

#127

—

194

—

#124

—

194

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₂×C₄²

C₂×D₉₇

yes

—

C₂⁴

—

C₂×Q₈

—

194

—

C₂×Q₈

—

194

—

C₂×D₄

—

194

—

C₂×D₄

—

194

—

C₂×D₄

—

194

—

C₂×D₄

—

194

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂²×C₄

C₂²×D₉₇

yes

—

C₄⋊C₄

—

194

—

C₂²⋊C₄

—

194

—

C₄²

—

C₄²

D₃₈₈

yes

—

C₄²

D₃₈₈

yes

complemented

C₂³

—

194

—

C₂³

—

194

—

C₂³

—

194

—

C₂³

C₂³×D₉₇

yes

characteristic

100

Q₈

—

388

—

101

D₄

—

388

—

102

D₄

—

388

—

103

C₂×C₄

—

104

C₂×C₄

—

194

—

105

C₂×C₄

—

194

—

106

C₂×C₄

C₂×D₃₈₈

yes

—

107

C₂×C₄

C₂×D₃₈₈

yes

—

108

C₂×C₄

C₂×D₃₈₈

yes

—

109

C₂×C₄

C₂×D₃₈₈

yes

—

110

C₂×C₄

#13095511

yes

—

111

C₂²

—

388

—

112

C₂²

—

388

—

113

C₂²

—

388

—

114

C₂²

C₂²×D₃₈₈

yes

characteristic

115

C₂²

C₂²×D₃₈₈

yes

—

116

C₂²

C₂×#13095511

yes

—

117

C₄

—

118

C₄

—

388

—

119

C₄

#15690798

yes

complemented

120

C₄

#15690807

yes

—

121

C₂

—

776

—

122

C₂

C₂×#15690798

yes

—

123

C₂

C₂×#15690807

yes

characteristic

124

C₂

#20261285

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₉₇
commutator, socle, cosocle	C₂³×C₉₇	C₂⁴
Fitting	C₄³×C₉₇	C₂
prime radical	C₂×#15690807	C₂
hypercenter	C₄³	D₉₇
hypocenter	C₉₇	#2858
LCS	G ▷ C₂³×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₄³

subgroup

G/H

center, Frattini

C₂³

C₂³×D₉₇

commutator, socle, cosocle

C₂³×C₉₇

C₂⁴

Fitting

C₄³×C₉₇

C₂

prime radical

C₂×#15690807

C₂

hypercenter

C₄³

D₉₇

C₉₇

G ▷ C₂³×C₉₇ ▷ C₉₇

C₁ ◁ C₂³ ◁ C₄³

N	H ≅ G/N	type
1. C₂×#15690810	C₂	semidirect
3. C₂×#15690798	C₂	semidirect
4. #20260637	C₂	semidirect
5. C₄³×C₉₇	C₂	semidirect
32. C₄²×C₉₇	D₄	semidirect
55. C₄×C₉₇	#124	semidirect
63. C₉₇	#2858	semidirect
68. C₄³	D₉₇	semidirect
95. C₄²	D₃₈₈	semidirect
119. C₄	#15690798	semidirect

H ≅ G/N

type

1. C₂×#15690810

C₂

semidirect

3. C₂×#15690798

C₂

semidirect

4. #20260637

C₂

semidirect

5. C₄³×C₉₇

C₂

semidirect

32. C₄²×C₉₇

D₄

semidirect

55. C₄×C₉₇

#124

semidirect

63. C₉₇

#2858

semidirect

68. C₄³

D₉₇

semidirect

95. C₄²

D₃₈₈

semidirect

119. C₄

#15690798

semidirect

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

subgroup	index
C₂×#15690810	2
C₂×#15690807	2
C₂×#15690798	2
#20260637	2
C₄³×C₉₇	2
#2858	97

subgroup

index

C₄³×C₉₇

Subgroups — #27150093

Conjugacy classes of subgroups

Complements

Maximal subgroups