Group #27140060 — subgroups

Subgroups — #27140060

Order 12416.

248 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 113 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	#20260788	C₂	1	2	yes	maximal
2	#20260461	C₂	1	4	yes	maximal
3	C₂×C₄×Q₈×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11715895	—	2	2	no	—
5	C₂²⋊#11715895	C₂²	1	4	yes	—
6	C₄×#11715895	—	2	2	no	—
7	C₂²×Q₈×C₉₇	C₄	1	1	yes	characteristic
8	C₄×Q₈×C₉₇	—	2	2	no	—
9	C₄×Q₈×C₉₇	—	2	2	no	—
10	C₂×C₄⋊C₄×C₉₇	C₄	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
12	C₂×C₄²×C₉₇	C₄	1	2	yes	—
13	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
14	C₂×#11715895	—	2	4	no	—
15	C₂×Q₈×C₉₇	—	2	2	no	—
16	C₂×Q₈×C₉₇	—	2	2	no	—
17	C₂×Q₈×C₉₇	C₈	1	4	yes	complemented
18	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₉₇	—	2	2	no	—
24	C₄⋊C₄×C₉₇	—	2	4	no	—
25	C₄²×C₉₇	—	2	2	no	—
26	C₄²×C₉₇	—	2	2	no	—
27	C₄²×C₉₇	D₄	1	2	yes	—
28	C₄²×C₉₇	D₄	1	2	yes	—
29	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
30	Q₈×C₉₇	—	2	8	no	—
31	C₂×C₄×C₉₇	—	2	2	no	—
32	C₂×C₄×C₉₇	—	2	2	no	—
33	C₂×C₄×C₉₇	—	2	2	no	—
34	C₂×C₄×C₉₇	—	2	2	no	—
35	C₂×C₄×C₉₇	—	4	1	no	—
36	C₂×C₄×C₉₇	—	4	2	no	—
37	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₉₇	C₂×C₈	1	4	yes	—
41	C₂×C₄×C₉₇	C₂×C₈	1	2	yes	—
42	C₂×C₄×C₉₇	M₁₆	1	2	yes	—
43	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
44	#11715895	—	4	4	no	—
45	C₂²×C₉₇	—	2	1	no	—
46	C₂²×C₉₇	—	2	1	no	—
47	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₉₇	C₂²×C₈	1	1	yes	characteristic
49	C₂²×C₉₇	C₂×M₁₆	1	1	yes	characteristic
50	C₄×C₉₇	—	2	4	no	—
51	C₄×C₉₇	—	2	2	no	—
52	C₄×C₉₇	—	2	1	no	—
53	C₄×C₉₇	—	4	2	no	—
54	C₄×C₉₇	C₂²⋊C₈	1	4	yes	—
55	C₄×C₉₇	#101	1	2	yes	—
56	C₂×C₉₇	—	2	1	no	—
57	C₂×C₉₇	—	2	1	no	—
58	C₂×C₉₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
59	C₂×C₉₇	#410	1	1	yes	characteristic
60	C₂×C₉₇	C₂×#101	1	1	yes	characteristic
61	#1461	—	97	1	no	maximal, Sylow
62	C₉₇	#1461	1	1	yes	characteristic, complemented, Sylow
63	C₂×C₄×Q₈	D₉₇	1	1	yes	characteristic
64	#431	—	97	2	no	—
65	#324	—	97	4	no	—
66	C₂²×Q₈	Dic₉₇	1	1	yes	characteristic
67	C₄×Q₈	—	2	2	no	—
68	C₄×Q₈	—	2	2	no	—
69	C₂×C₄⋊C₄	Dic₉₇	1	1	yes	characteristic
70	C₂×C₄⋊C₄	C₂×D₉₇	1	2	yes	—
71	C₂×C₄²	Dic₉₇	1	2	yes	—
72	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
73	C₄⋊C₈	—	194	2	no	—
74	C₂²⋊C₈	—	97	4	no	—
75	C₄×C₈	—	194	2	no	—
76	C₂×Q₈	—	2	2	no	—
77	C₂×Q₈	—	2	2	no	—
78	C₂×Q₈	#11715895	1	4	yes	complemented
79	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
80	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
81	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
82	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
83	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
84	C₂×C₈	—	194	4	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	2	4	no	—
87	C₄²	—	2	2	no	—
88	C₄²	—	2	2	no	—
89	C₄²	C₂²⋊D₉₇	1	2	yes	—
90	C₄²	C₂²⋊D₉₇	1	2	yes	—
91	C₂³	C₂²×Dic₉₇	1	1	yes	characteristic
92	Q₈	—	2	8	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	1	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
100	C₂×C₄	C₂²⋊Dic₉₇	1	2	yes	—
101	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
102	C₂×C₄	C₂×#11715895	1	4	yes	—
103	C₂×C₄	C₂×#11715895	1	2	yes	—
104	C₂×C₄	#13095495	1	2	yes	—
105	C₂×C₄	C₂×C₂²⋊D₉₇	1	2	yes	—
106	C₈	—	388	4	no	—
107	C₂²	—	2	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	C₂×C₂²⋊Dic₉₇	1	1	yes	characteristic
110	C₂²	C₂²×#11715895	1	1	yes	characteristic
111	C₂²	C₂×#13095495	1	1	yes	characteristic
112	C₄	—	2	4	no	—
113	C₄	—	2	2	no	—
114	C₄	—	2	1	no	—
115	C₄	—	4	2	no	—
116	C₄	C₂²⋊#11715895	1	4	yes	—
117	C₄	#15690729	1	2	yes	—
118	C₂	—	2	1	no	—
119	C₂	—	2	1	no	—
120	C₂	C₂×C₂²⋊#11715895	1	1	yes	characteristic
121	C₂	#20260709	1	1	yes	characteristic
122	C₂	C₂×#15690729	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20260788

C₂

yes

maximal

#20260461

C₂

yes

maximal

C₂×C₄×Q₈×C₉₇

C₂

yes

maximal, characteristic

C₄⋊#11715895

—

C₂²⋊#11715895

C₂²

yes

—

C₄×#11715895

—

C₂²×Q₈×C₉₇

C₄

yes

characteristic

C₄×Q₈×C₉₇

—

C₄×Q₈×C₉₇

—

C₂×C₄⋊C₄×C₉₇

C₄

yes

characteristic

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₄

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂×#11715895

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

C₈

yes

complemented

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

—

C₄²×C₉₇

—

C₄²×C₉₇

D₄

yes

—

C₄²×C₉₇

D₄

yes

—

C₂³×C₉₇

C₂²×C₄

yes

characteristic

Q₈×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×C₈

yes

—

C₂×C₄×C₉₇

C₂×C₈

yes

—

C₂×C₄×C₉₇

M₁₆

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₉₇

C₂²×C₈

yes

characteristic

C₂²×C₉₇

C₂×M₁₆

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂²⋊C₈

yes

—

C₄×C₉₇

#101

yes

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

C₂×C₂²⋊C₈

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

D₉₇

yes

characteristic

#431

—

#324

—

C₂²×Q₈

Dic₉₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

Dic₉₇

yes

characteristic

C₂×C₄⋊C₄

C₂×D₉₇

yes

—

C₂×C₄²

Dic₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

194

—

C₂²⋊C₈

—

C₄×C₈

—

194

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

#11715895

yes

complemented

C₂²×C₄

C₂×Dic₉₇

yes

characteristic

C₂²×C₄

C₂×Dic₉₇

yes

—

C₂²×C₄

C₂×Dic₉₇

yes

—

C₂²×C₄

C₂×Dic₉₇

yes

characteristic

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₈

—

194

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²⋊D₉₇

yes

—

C₄²

C₂²⋊D₉₇

yes

—

C₂³

C₂²×Dic₉₇

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₉₇

yes

characteristic

100

C₂×C₄

C₂²⋊Dic₉₇

yes

—

101

C₂×C₄

C₂²⋊Dic₉₇

yes

characteristic

102

C₂×C₄

C₂×#11715895

yes

—

103

C₂×C₄

C₂×#11715895

yes

—

104

C₂×C₄

#13095495

yes

—

105

C₂×C₄

C₂×C₂²⋊D₉₇

yes

—

106

C₈

—

388

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₂×C₂²⋊Dic₉₇

yes

characteristic

110

C₂²

C₂²×#11715895

yes

characteristic

111

C₂²

C₂×#13095495

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

C₂²⋊#11715895

yes

—

117

C₄

#15690729

yes

—

118

C₂

—

119

C₂

—

120

C₂

C₂×C₂²⋊#11715895

yes

characteristic

121

C₂

#20260709

yes

characteristic

122

C₂

C₂×#15690729

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₉₇
commutator	C₂²×C₉₇	C₂²×C₈
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	C₂×C₄×Q₈×C₉₇	C₂
socle	C₂²×C₉₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₉₇	C₂²×C₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂×C₄×Q₈	D₉₇
hypocenter	C₉₇	#1461
LCS	G ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂×C₄

C₂²⋊Dic₉₇

commutator

C₂²×C₉₇

C₂²×C₈

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

C₂×C₄×Q₈×C₉₇

C₂

socle

C₂²×C₉₇

C₂×C₂²⋊C₄

prime radical

C₂³×C₉₇

C₂²×C₄

cosocle

C₂²×C₄×C₉₇

C₂³

hypercenter

C₂×C₄×Q₈

D₉₇

hypocenter

C₉₇

#1461

LCS

G ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
17. C₂×Q₈×C₉₇	C₈	semidirect
62. C₉₇	#1461	semidirect
78. C₂×Q₈	#11715895	semidirect

H ≅ G/N

type

17. C₂×Q₈×C₉₇

C₈

semidirect

62. C₉₇

#1461

semidirect

78. C₂×Q₈

#11715895

semidirect

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

subgroup	index
#20260788	2
#20260461	2
C₂×C₄×Q₈×C₉₇	2
#1461	97

subgroup

index

#20260788

#20260461

C₂×C₄×Q₈×C₉₇

#1461

Subgroups — #27140060

Conjugacy classes of subgroups

Complements

Maximal subgroups