Group #27152075 — subgroups

Subgroups — C₂×#20261103 #27152075

Order 12416.

384 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 149 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×#15690802	C₂	1	1	yes	maximal, characteristic
2	#20261103	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15690734	C₂	1	4	yes	maximal
4	C₂×#15690721	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11715895	C₂²	1	2	yes	—
7	#15690802	—	2	2	no	—
8	#15690802	C₂²	1	4	yes	—
9	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
10	C₂×#13095490	—	2	2	no	—
11	#15690734	C₂²	1	16	yes	—
12	#15690721	C₂²	1	4	yes	—
13	C₂²×Q₈×C₉₇	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₉₇	—	2	2	no	—
15	C₄⋊Q₈×C₉₇	C₂²	1	4	yes	—
16	C₂×C₄⋊C₄×C₉₇	—	2	2	no	—
17	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
18	C₂²×Dic₉₇	—	2	2	no	—
19	C₂×#11715895	—	2	4	no	—
20	C₂×#11715895	C₂³	1	4	yes	—
21	C₄⋊Dic₉₇	—	2	2	no	—
22	C₄⋊Dic₉₇	C₂³	1	4	yes	—
23	#13095490	—	2	8	no	—
24	C₂×Q₈×C₉₇	—	2	2	no	—
25	C₂×Q₈×C₉₇	—	2	8	no	—
26	C₂×Q₈×C₉₇	C₂³	1	4	yes	—
27	C₂²×C₄×C₉₇	—	2	2	no	—
28	C₂²×C₄×C₉₇	D₄	1	2	yes	—
29	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₉₇	—	2	8	no	—
31	C₄²×C₉₇	D₄	1	2	yes	—
32	C₄²×C₉₇	C₂³	1	2	yes	—
33	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
34	Q₈×C₉₇	—	2	8	no	—
35	Q₈×C₉₇	—	4	4	no	—
36	C₂×C₄×C₉₇	—	2	4	no	—
37	C₂×C₄×C₉₇	—	2	4	no	—
38	C₂×C₄×C₉₇	—	4	4	no	—
39	C₂×C₄×C₉₇	C₂×D₄	1	4	yes	—
40	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₉₇	#41	1	4	yes	—
42	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
43	C₂×Dic₉₇	—	2	4	no	—
44	C₂×Dic₉₇	—	4	4	no	—
45	#11715895	—	2	4	no	—
46	C₂²×C₉₇	#124	1	2	yes	—
47	C₂²×C₉₇	#137	1	4	yes	—
48	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
49	C₄×C₉₇	—	2	4	no	—
50	C₄×C₉₇	—	4	4	no	—
51	C₄×C₉₇	#124	1	2	yes	—
52	C₄×C₉₇	C₂×#41	1	2	yes	—
53	Dic₉₇	—	4	4	no	—
54	C₂×C₉₇	#491	1	4	yes	complemented
55	C₂×C₉₇	C₂×#124	1	1	yes	characteristic
56	C₂×C₉₇	C₂×#137	1	2	yes	—
57	C₂×#491	—	97	1	no	maximal, Sylow
58	C₉₇	C₂×#491	1	1	yes	characteristic, complemented, Sylow
59	C₂×C₄⋊Q₈	D₉₇	1	1	yes	characteristic
60	C₂×#125	—	97	1	no	—
61	#491	—	97	8	no	—
62	C₂×#103	—	97	4	no	—
63	C₂×#97	—	97	1	no	—
64	C₂²×Q₈	C₂×D₉₇	1	2	yes	—
65	C₂²×C₈	—	97	2	no	—
66	C₄⋊Q₈	—	2	2	no	—
67	C₄⋊Q₈	C₂×D₉₇	1	4	yes	—
68	#125	—	97	4	no	—
69	#125	—	194	2	no	—
70	C₂×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄⋊C₄	—	97	2	no	—
72	C₂×C₄⋊C₄	—	194	2	no	—
73	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
74	#103	—	97	16	no	—
75	#97	—	97	4	no	—
76	C₂×Q₈	—	2	2	no	—
77	C₂×Q₈	—	2	8	no	—
78	C₂×Q₈	C₂²×D₉₇	1	4	yes	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	194	2	no	—
81	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
82	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
83	C₂×C₈	—	97	4	no	—
84	C₂×C₈	—	194	4	no	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	97	4	no	—
87	C₄⋊C₄	—	194	8	no	—
88	C₄⋊C₄	—	194	2	no	—
89	C₄²	C₂²⋊D₉₇	1	2	yes	—
90	C₄²	C₂²×D₉₇	1	2	yes	—
91	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
92	Q₈	—	2	8	no	—
93	Q₈	—	4	4	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	4	4	no	—
97	C₂×C₄	—	388	4	no	—
98	C₂×C₄	—	194	4	no	—
99	C₂×C₄	C₂×C₂²⋊D₉₇	1	4	yes	—
100	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
101	C₂×C₄	#13095510	1	4	yes	—
102	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
103	C₈	—	194	4	no	—
104	C₂²	#15690797	1	2	yes	—
105	C₂²	#15690841	1	4	yes	—
106	C₂²	C₂²×C₂²⋊D₉₇	1	1	yes	characteristic
107	C₄	—	2	4	no	—
108	C₄	—	4	4	no	—
109	C₄	—	388	4	no	—
110	C₄	#15690797	1	2	yes	—
111	C₄	C₂×#13095510	1	2	yes	—
112	C₂	#20261103	1	4	yes	direct product factor
113	C₂	C₂×#15690797	1	1	yes	characteristic
114	C₂	C₂×#15690841	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690802

C₂

yes

maximal, characteristic

#20261103

C₂

yes

maximal, direct product factor

C₂×#15690734

C₂

yes

maximal

C₂×#15690721

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₉₇

C₂

yes

maximal, characteristic

C₂²×#11715895

C₂²

yes

—

#15690802

—

#15690802

C₂²

yes

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

C₂×#13095490

—

#15690734

C₂²

yes

—

#15690721

C₂²

yes

—

C₂²×Q₈×C₉₇

C₂²

yes

—

C₄⋊Q₈×C₉₇

—

C₄⋊Q₈×C₉₇

C₂²

yes

—

C₂×C₄⋊C₄×C₉₇

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂²×Dic₉₇

—

C₂×#11715895

—

C₂×#11715895

C₂³

yes

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

C₂³

yes

—

#13095490

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

C₂³

yes

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

D₄

yes

—

C₄²×C₉₇

C₂³

yes

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

Q₈×C₉₇

—

Q₈×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂×C₄×C₉₇

C₂⁴

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

—

#11715895

—

C₂²×C₉₇

#124

yes

—

C₂²×C₉₇

#137

yes

—

C₂²×C₉₇

C₂²×D₄

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#124

yes

—

C₄×C₉₇

C₂×#41

yes

—

Dic₉₇

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₂×#491

—

maximal, Sylow

C₉₇

C₂×#491

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₉₇

yes

—

—

—

C₂×#97

—

C₂²×Q₈

C₂×D₉₇

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₉₇

yes

—

#125

—

#125

—

194

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

#103

—

#97

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₉₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

194

—

C₂²×C₄

C₂²⋊D₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

194

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

194

—

C₄⋊C₄

—

194

—

C₄²

C₂²⋊D₉₇

yes

—

C₄²

C₂²×D₉₇

yes

—

C₂³

C₂×C₂²⋊D₉₇

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

388

—

C₂×C₄

—

194

—

C₂×C₄

C₂×C₂²⋊D₉₇

yes

—

100

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

101

C₂×C₄

#13095510

yes

—

102

C₂×C₄

C₂³×D₉₇

yes

characteristic

103

C₈

—

194

—

104

C₂²

#15690797

yes

—

105

C₂²

#15690841

yes

—

106

C₂²

C₂²×C₂²⋊D₉₇

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

388

—

110

C₄

#15690797

yes

—

111

C₄

C₂×#13095510

yes

—

112

C₂

#20261103

yes

direct product factor

113

C₂

C₂×#15690797

yes

characteristic

114

C₂

C₂×#15690841

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₉₇
commutator, cosocle	C₂×C₄×C₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	C₂×C₄⋊Q₈×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₉₇
hypocenter	C₉₇	C₂×#491
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₉₇

commutator, cosocle

C₂×C₄×C₉₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₉₇

Fitting

C₂×C₄⋊Q₈×C₉₇

C₂

socle, prime radical

C₂³×C₉₇

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₉₇

hypocenter

C₉₇

C₂×#491

LCS

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #20261103	C₂	direct (N × H)
54. C₂×C₉₇	#491	semidirect
58. C₉₇	C₂×#491	semidirect
112. C₂	#20261103	direct (N × H)

H ≅ G/N

type

2. #20261103

C₂

direct (N × H)

54. C₂×C₉₇

#491

semidirect

58. C₉₇

C₂×#491

semidirect

112. C₂

#20261103

direct (N × H)

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

subgroup	index
C₂×#15690802	2
#20261103	2
C₂×#15690734	2
C₂×#15690721	2
C₂×C₄⋊Q₈×C₉₇	2
C₂×#491	97

subgroup

index

C₂×C₄⋊Q₈×C₉₇

Subgroups — C₂×#20261103 #27152075

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20261103 #27152075

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20261103 #27152075