Group #27142718 — subgroups

Subgroups — #27142718

Order 12416.

244 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 97 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	#20260759	C₂	1	1	yes	maximal, characteristic
2	#20260581	C₂	1	1	yes	maximal, characteristic
3	#20260494	C₂	1	4	yes	maximal
4	#533×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13095495	C₂²	1	2	yes	—
6	C₂×C₄×Dic₉₇	C₂²	1	2	yes	—
7	C₄⋊#11715895	—	2	2	no	—
8	#15690717	—	2	2	no	—
9	C₄⋊Q₈×C₉₇	C₄	1	2	yes	complemented
10	#125×C₉₇	C₄	1	2	yes	complemented
11	C₂²⋊Q₈×C₉₇	—	2	2	no	—
12	C₄×Q₈×C₉₇	—	2	2	no	—
13	#117×C₉₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₉₇	—	2	2	no	—
16	#13095495	—	2	4	no	—
17	C₂×#11715895	—	2	2	no	—
18	C₄×Dic₉₇	—	2	2	no	—
19	C₄×Dic₉₇	Q₈	1	4	yes	complemented
20	C₂×Q₈×C₉₇	—	2	2	no	—
21	C₂²×C₄×C₉₇	D₄	1	2	yes	—
22	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₉₇	—	2	4	no	—
24	C₄⋊C₄×C₉₇	—	2	2	no	—
25	C₄⋊C₄×C₉₇	C₂×C₄	1	4	yes	—
26	C₂²⋊C₄×C₉₇	—	2	2	no	—
27	C₄²×C₉₇	—	2	2	no	—
28	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₉₇	D₄	1	2	yes	—
31	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
32	Q₈×C₉₇	—	4	4	no	—
33	C₂×C₄×C₉₇	—	2	4	no	—
34	C₂×C₄×C₉₇	—	4	2	no	—
35	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₉₇	#41	1	2	yes	—
41	C₂×Dic₉₇	—	2	4	no	—
42	C₂×Dic₉₇	—	4	4	no	—
43	#11715895	—	4	2	no	—
44	C₂²×C₉₇	—	2	2	no	—
45	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₉₇	C₂²⋊Q₈	1	2	yes	—
47	C₄×C₉₇	—	2	2	no	—
48	C₄×C₉₇	—	2	2	no	—
49	C₄×C₉₇	—	4	4	no	—
50	C₄×C₉₇	C₄×Q₈	1	2	yes	—
51	C₄×C₉₇	#124	1	1	yes	characteristic
52	C₄×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
53	Dic₉₇	—	4	4	no	—
54	C₂×C₉₇	—	2	2	no	—
55	C₂×C₉₇	#391	1	1	yes	characteristic
56	C₂×C₉₇	#421	1	2	yes	—
57	#1995	—	97	1	no	maximal, Sylow
58	C₉₇	#1995	1	1	yes	characteristic, complemented, Sylow
59	#533	D₉₇	1	1	yes	characteristic
60	#423	—	97	1	no	—
61	#376	—	97	1	no	—
62	#339	—	97	4	no	—
63	C₂×M₁₆	—	97	2	no	—
64	C₄⋊Q₈	Dic₉₇	1	2	yes	complemented
65	#125	Dic₉₇	1	2	yes	complemented
66	C₂²⋊Q₈	—	2	2	no	—
67	C₄×Q₈	—	2	2	no	—
68	#117	C₂×D₉₇	1	2	yes	—
69	C₂×C₄²	—	97	2	no	—
70	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
71	C₄⋊C₈	—	194	2	no	—
72	#95	—	194	2	no	—
73	C₂×Q₈	—	2	2	no	—
74	C₂²×C₄	—	194	2	no	—
75	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
76	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
77	M₁₆	—	194	4	no	—
78	C₂×C₈	—	194	2	no	—
79	C₄⋊C₄	—	2	4	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	C₂×Dic₉₇	1	4	yes	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₄²	—	2	2	no	—
84	C₄²	—	97	4	no	—
85	C₄²	—	194	2	no	—
86	C₄²	C₂×Dic₉₇	1	1	yes	characteristic
87	C₄²	C₂×Dic₉₇	1	1	yes	characteristic
88	C₄²	C₂²⋊D₉₇	1	2	yes	—
89	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
90	Q₈	—	4	4	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	—	388	4	no	—
94	C₂×C₄	—	194	4	no	—
95	C₂×C₄	C₂²⋊Dic₉₇	1	2	yes	—
96	C₂×C₄	C₂²⋊Dic₉₇	1	2	yes	—
97	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
99	C₂×C₄	Q₈×D₉₇	1	2	yes	—
100	C₂×C₄	#13095512	1	2	yes	—
101	C₈	—	388	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	C₂×C₂²⋊Dic₉₇	1	1	yes	characteristic
104	C₂²	#15690786	1	2	yes	—
105	C₄	—	2	2	no	—
106	C₄	—	2	2	no	—
107	C₄	—	4	4	no	—
108	C₄	—	388	4	no	—
109	C₄	Q₈×Dic₉₇	1	2	yes	—
110	C₄	#15690799	1	1	yes	characteristic
111	C₄	#15690808	1	1	yes	characteristic
112	C₂	—	2	2	no	—
113	C₂	#20260642	1	1	yes	characteristic
114	C₂	#20260752	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20260759

C₂

yes

maximal, characteristic

#20260581

C₂

yes

maximal, characteristic

#20260494

C₂

yes

maximal

#533×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

C₂²

yes

—

C₂×C₄×Dic₉₇

C₂²

yes

—

C₄⋊#11715895

—

#15690717

—

C₄⋊Q₈×C₉₇

C₄

yes

complemented

#125×C₉₇

C₄

yes

complemented

C₂²⋊Q₈×C₉₇

—

C₄×Q₈×C₉₇

—

#117×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂²×Dic₉₇

—

#13095495

—

C₂×#11715895

—

C₄×Dic₉₇

—

C₄×Dic₉₇

Q₈

yes

complemented

C₂×Q₈×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

C₂×C₄

yes

—

C₂²⋊C₄×C₉₇

—

C₄²×C₉₇

—

C₄²×C₉₇

C₂×C₄

yes

characteristic

C₄²×C₉₇

C₂×C₄

yes

characteristic

C₄²×C₉₇

D₄

yes

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

Q₈×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

—

C₂×C₄×C₉₇

#41

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₉₇

C₂²⋊Q₈

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₄×Q₈

yes

—

C₄×C₉₇

#124

yes

characteristic

C₄×C₉₇

C₄⋊Q₈

yes

characteristic

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#391

yes

characteristic

C₂×C₉₇

#421

yes

—

#1995

—

maximal, Sylow

C₉₇

#1995

yes

characteristic, complemented, Sylow

#533

D₉₇

yes

—

—

—

C₂×M₁₆

—

C₄⋊Q₈

Dic₉₇

yes

complemented

#125

Dic₉₇

yes

complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₉₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

194

—

#95

—

194

—

C₂×Q₈

—

C₂²×C₄

—

194

—

C₂²×C₄

C₂²⋊D₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

M₁₆

—

194

—

C₂×C₈

—

194

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₉₇

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

194

—

C₄²

C₂×Dic₉₇

yes

characteristic

C₄²

C₂×Dic₉₇

yes

characteristic

C₄²

C₂²⋊D₉₇

yes

—

C₂³

C₂×C₂²⋊D₉₇

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

388

—

C₂×C₄

—

194

—

C₂×C₄

C₂²⋊Dic₉₇

yes

—

C₂×C₄

C₂²⋊Dic₉₇

yes

—

C₂×C₄

C₂²×Dic₉₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

C₂×C₄

Q₈×D₉₇

yes

—

100

C₂×C₄

#13095512

yes

—

101

C₈

—

388

—

102

C₂²

—

103

C₂²

C₂×C₂²⋊Dic₉₇

yes

characteristic

104

C₂²

#15690786

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

388

—

109

C₄

Q₈×Dic₉₇

yes

—

110

C₄

#15690799

yes

characteristic

111

C₄

#15690808

yes

characteristic

112

C₂

—

113

C₂

#20260642

yes

characteristic

114

C₂

#20260752

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₉₇
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#533×C₉₇	C₂
socle	C₂²×C₉₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₉₇	C₂×D₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#533	D₉₇
hypocenter	C₉₇	#1995
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₉₇

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

#533×C₉₇

C₂

socle

C₂²×C₉₇

C₂×C₂²⋊C₄

prime radical

C₂³×C₉₇

C₂×D₄

cosocle

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
9. C₄⋊Q₈×C₉₇	C₄	semidirect
10. #125×C₉₇	C₄	semidirect
19. C₄×Dic₉₇	Q₈	semidirect
58. C₉₇	#1995	semidirect
64. C₄⋊Q₈	Dic₉₇	semidirect
65. #125	Dic₉₇	semidirect

H ≅ G/N

type

9. C₄⋊Q₈×C₉₇

C₄

semidirect

10. #125×C₉₇

C₄

semidirect

19. C₄×Dic₉₇

Q₈

semidirect

58. C₉₇

#1995

semidirect

64. C₄⋊Q₈

Dic₉₇

semidirect

65. #125

Dic₉₇

semidirect

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

subgroup	index
#20260759	2
#20260581	2
#20260494	2
#533×C₉₇	2
#1995	97

subgroup

index

#533×C₉₇

Subgroups — #27142718

Conjugacy classes of subgroups

Complements

Maximal subgroups