Group #27142713 — subgroups

Subgroups — #27142713

Order 12416.

284 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 101 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	#20261292	C₂	1	1	yes	maximal, characteristic
2	#20260759	C₂	1	1	yes	maximal, characteristic
3	#20260495	C₂	1	4	yes	maximal
4	C₄³×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13095495	C₂²	1	2	yes	—
6	#15690810	C₄	1	2	yes	complemented
7	#15690802	C₄	1	2	yes	complemented
8	#15690789	—	2	2	no	—
9	C₄×Dic₁₉₄	—	2	2	no	—
10	#15690769	C₂²	1	2	yes	—
11	C₄⋊#11715895	—	2	2	no	—
12	C₂×C₄²×C₉₇	—	2	2	no	—
13	C₂×C₄²×C₉₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₉₄	—	2	2	no	—
16	#13095495	—	2	4	no	—
17	C₂×#11715895	—	2	2	no	—
18	C₄⋊Dic₉₇	—	2	2	no	—
19	C₄⋊Dic₉₇	C₂×C₄	1	4	yes	—
20	#13095490	—	2	4	no	—
21	C₂²⋊Dic₉₇	—	2	2	no	—
22	C₄×Dic₉₇	—	2	2	no	—
23	C₂²×C₄×C₉₇	—	2	2	no	—
24	C₂²×C₄×C₉₇	D₄	1	2	yes	—
25	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
26	C₄²×C₉₇	—	2	8	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	C₄²×C₉₇	Q₈	1	4	yes	complemented
32	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₉₇	—	2	8	no	—
34	C₂×C₄×C₉₇	—	2	4	no	—
35	C₂×C₄×C₉₇	—	2	4	no	—
36	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₉₇	#41	1	2	yes	—
42	C₂×Dic₉₇	—	2	4	no	—
43	Dic₁₉₄	—	4	4	no	—
44	#11715895	—	4	2	no	—
45	C₂²×C₉₇	—	2	2	no	—
46	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₉₇	C₂²⋊Q₈	1	2	yes	—
48	C₄×C₉₇	—	2	8	no	—
49	C₄×C₉₇	—	2	2	no	—
50	C₄×C₉₇	#104	1	4	yes	—
51	C₄×C₉₇	C₄×Q₈	1	2	yes	—
52	C₄×C₉₇	#124	1	1	yes	characteristic
53	C₄×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
54	Dic₉₇	—	4	4	no	—
55	C₂×C₉₇	—	2	2	no	—
56	C₂×C₉₇	#391	1	1	yes	characteristic
57	C₂×C₉₇	C₂×#104	1	2	yes	—
58	#1994	—	97	1	no	maximal, Sylow
59	C₉₇	#1994	1	1	yes	characteristic, complemented, Sylow
60	#533	—	97	1	no	—
61	#423	—	97	1	no	—
62	C₄³	D₉₇	1	1	yes	characteristic
63	#339	—	97	4	no	—
64	C₂×M₁₆	—	97	2	no	—
65	C₄⋊Q₈	—	97	2	no	—
66	#125	—	97	2	no	—
67	C₂²⋊Q₈	—	194	2	no	—
68	C₄×Q₈	—	194	2	no	—
69	#117	—	97	2	no	—
70	C₂×C₄²	—	2	2	no	—
71	C₂×C₄²	C₂×D₉₇	1	2	yes	—
72	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
73	C₄⋊C₈	—	194	2	no	—
74	C₂×Q₈	—	194	2	no	—
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
77	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
78	M₁₆	—	194	4	no	—
79	C₂×C₈	—	194	2	no	—
80	C₄⋊C₄	—	97	4	no	—
81	C₄⋊C₄	—	194	4	no	—
82	C₄⋊C₄	—	194	2	no	—
83	C₂²⋊C₄	—	194	2	no	—
84	C₄²	—	2	8	no	—
85	C₄²	—	2	2	no	—
86	C₄²	—	194	2	no	—
87	C₄²	Dic₁₉₄	1	4	yes	complemented
88	C₄²	C₄×D₉₇	1	1	yes	characteristic
89	C₄²	C₄×D₉₇	1	1	yes	characteristic
90	C₄²	D₃₈₈	1	2	yes	—
91	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
92	Q₈	—	388	4	no	—
93	C₂×C₄	—	2	8	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	194	4	no	—
97	C₂×C₄	#13095488	1	2	yes	—
98	C₂×C₄	#13095488	1	2	yes	—
99	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
100	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
101	C₂×C₄	C₂×Dic₁₉₄	1	2	yes	—
102	C₂×C₄	#13095511	1	2	yes	—
103	C₈	—	388	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	C₂×#13095488	1	1	yes	characteristic
106	C₂²	#15690789	1	2	yes	—
107	C₄	—	2	8	no	—
108	C₄	—	2	2	no	—
109	C₄	—	388	4	no	—
110	C₄	#15690738	1	4	yes	—
111	C₄	C₄×Dic₁₉₄	1	2	yes	—
112	C₄	#15690798	1	1	yes	characteristic
113	C₄	#15690810	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	#20260641	1	1	yes	characteristic
116	C₂	C₂×#15690738	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20261292

C₂

yes

maximal, characteristic

#20260759

C₂

yes

maximal, characteristic

#20260495

C₂

yes

maximal

C₄³×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

C₂²

yes

—

#15690810

C₄

yes

complemented

#15690802

C₄

yes

complemented

#15690789

—

C₄×Dic₁₉₄

—

#15690769

C₂²

yes

—

C₄⋊#11715895

—

C₂×C₄²×C₉₇

—

C₂×C₄²×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂×Dic₁₉₄

—

#13095495

—

C₂×#11715895

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

C₂×C₄

yes

—

#13095490

—

C₂²⋊Dic₉₇

—

C₄×Dic₉₇

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄²×C₉₇

—

C₄²×C₉₇

—

C₄²×C₉₇

C₂×C₄

yes

characteristic

C₄²×C₉₇

C₂×C₄

yes

characteristic

C₄²×C₉₇

D₄

yes

—

C₄²×C₉₇

Q₈

yes

complemented

C₂³×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×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

—

Dic₁₉₄

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₉₇

C₂²⋊Q₈

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#104

yes

—

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

C₂×#104

yes

—

#1994

—

maximal, Sylow

C₉₇

#1994

yes

characteristic, complemented, Sylow

#533

—

#423

—

C₄³

D₉₇

yes

characteristic

#339

—

C₂×M₁₆

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

194

—

C₄×Q₈

—

194

—

#117

—

C₂×C₄²

—

C₂×C₄²

C₂×D₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

194

—

C₂×Q₈

—

194

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

M₁₆

—

194

—

C₂×C₈

—

194

—

C₄⋊C₄

—

C₄⋊C₄

—

194

—

C₄⋊C₄

—

194

—

C₂²⋊C₄

—

194

—

C₄²

—

C₄²

—

C₄²

—

194

—

C₄²

Dic₁₉₄

yes

complemented

C₄²

C₄×D₉₇

yes

characteristic

C₄²

C₄×D₉₇

yes

characteristic

C₄²

D₃₈₈

yes

—

C₂³

C₂×C₂²⋊D₉₇

yes

characteristic

Q₈

—

388

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

194

—

C₂×C₄

#13095488

yes

—

C₂×C₄

#13095488

yes

—

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

100

C₂×C₄

C₂×D₃₈₈

yes

characteristic

101

C₂×C₄

C₂×Dic₁₉₄

yes

—

102

C₂×C₄

#13095511

yes

—

103

C₈

—

388

—

104

C₂²

—

105

C₂²

C₂×#13095488

yes

characteristic

106

C₂²

#15690789

yes

—

107

C₄

—

108

C₄

—

109

C₄

—

388

—

110

C₄

#15690738

yes

—

111

C₄

C₄×Dic₁₉₄

yes

—

112

C₄

#15690798

yes

characteristic

113

C₄

#15690810

yes

characteristic

114

C₂

—

115

C₂

#20260641

yes

characteristic

116

C₂

C₂×#15690738

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₃₈₈
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	C₄³×C₉₇	C₂
socle	C₂²×C₉₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₉₇	C₂×D₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₄³	D₉₇
hypocenter	C₉₇	#1994
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄³

subgroup

G/H

center

C₂×C₄

C₂×D₃₈₈

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

C₄³×C₉₇

C₂

socle

C₂²×C₉₇

C₂×C₂²⋊C₄

prime radical

C₂³×C₉₇

C₂×D₄

cosocle

C₂²×C₄×C₉₇

C₂³

hypercenter

C₄³

D₉₇

hypocenter

C₉₇

#1994

LCS

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄³

N	H ≅ G/N	type
6. #15690810	C₄	semidirect
7. #15690802	C₄	semidirect
31. C₄²×C₉₇	Q₈	semidirect
59. C₉₇	#1994	semidirect
87. C₄²	Dic₁₉₄	semidirect

H ≅ G/N

type

6. #15690810

C₄

semidirect

7. #15690802

C₄

semidirect

31. C₄²×C₉₇

Q₈

semidirect

59. C₉₇

#1994

semidirect

87. C₄²

Dic₁₉₄

semidirect

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

subgroup	index
#20261292	2
#20260759	2
#20260495	2
C₄³×C₉₇	2
#1994	97

subgroup

index

C₄³×C₉₇

Subgroups — #27142713

Conjugacy classes of subgroups

Complements

Maximal subgroups