Group #27142579 — subgroups

Subgroups — #27142579

Order 12416.

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

normal subgroups only

This group has 292 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	C₂×#15690810	C₂	1	1	yes	maximal, characteristic
2	C₂×#15690728	C₂	1	4	yes	maximal
3	C₄×#13095495	C₂	1	1	yes	maximal, characteristic
4	#423×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₉₄	C₂²	1	2	yes	—
6	C₂×#13095495	C₂²	1	2	yes	—
7	#15690810	—	2	1	no	—
8	#15690810	—	2	1	no	—
9	#15690810	—	2	2	no	—
10	C₂×C₄⋊Dic₉₇	C₄	1	4	yes	—
11	#15690728	—	2	8	no	—
12	#15690721	—	2	1	no	—
13	C₄×#11715895	—	2	1	no	—
14	C₂×M₁₆×C₉₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₉₇	—	2	2	no	—
17	C₂×Dic₁₉₄	—	2	2	no	—
18	C₂×Dic₁₉₄	—	2	2	no	—
19	C₂×Dic₁₉₄	—	2	4	no	—
20	C₂×Dic₁₉₄	—	2	4	no	—
21	C₂²×Dic₉₇	C₂×C₄	1	4	yes	—
22	#13095495	—	2	2	no	—
23	#13095495	D₄	1	4	yes	—
24	C₂×#11715895	—	2	2	no	—
25	C₄⋊Dic₉₇	—	2	4	no	—
26	C₄⋊Dic₉₇	—	2	4	no	—
27	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
29	M₁₆×C₉₇	—	2	4	no	—
30	C₂×C₈×C₉₇	—	2	2	no	—
31	C₄²×C₉₇	D₄	1	1	yes	characteristic
32	C₄²×C₉₇	D₄	1	1	yes	characteristic
33	C₄²×C₉₇	D₄	1	2	yes	—
34	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
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	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₉₇	#41	1	2	yes	—
42	C₂×Dic₉₇	—	2	4	no	—
43	C₂×Dic₉₇	—	2	8	no	—
44	Dic₁₉₄	—	4	4	no	—
45	Dic₁₉₄	—	4	2	no	—
46	Dic₁₉₄	—	4	2	no	—
47	C₈×C₉₇	—	4	2	no	—
48	#11715895	—	2	4	no	—
49	C₂²×C₉₇	—	2	2	no	—
50	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₉₇	C₄×D₄	1	2	yes	—
52	C₄×C₉₇	—	2	2	no	—
53	C₄×C₉₇	#101	1	4	yes	complemented
54	C₄×C₉₇	#121	1	2	yes	—
55	C₄×C₉₇	#124	1	1	yes	characteristic
56	C₄×C₉₇	#127	1	1	yes	characteristic
57	Dic₉₇	—	4	4	no	—
58	C₂×C₉₇	—	2	2	no	—
59	C₂×C₉₇	#390	1	1	yes	characteristic
60	C₂×C₉₇	C₂×#101	1	2	yes	—
61	#1966	—	97	1	no	maximal, Sylow
62	C₉₇	#1966	1	1	yes	characteristic, complemented, Sylow
63	C₂×C₄⋊Q₈	—	97	1	no	—
64	#423	D₉₇	1	1	yes	characteristic
65	C₂×#101	—	97	4	no	—
66	C₄×M₁₆	—	97	1	no	—
67	C₂²×Q₈	—	97	2	no	—
68	C₂×M₁₆	—	97	2	no	—
69	C₂×M₁₆	C₂×D₉₇	1	2	yes	—
70	C₄⋊Q₈	—	194	1	no	—
71	C₄⋊Q₈	—	194	1	no	—
72	C₄⋊Q₈	—	194	2	no	—
73	C₂×C₄⋊C₄	—	97	4	no	—
74	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
75	C₄⋊C₈	—	2	2	no	—
76	#101	—	194	8	no	—
77	#97	—	194	1	no	—
78	C₄×C₈	—	194	1	no	—
79	C₂×Q₈	—	194	4	no	—
80	C₂×Q₈	—	194	2	no	—
81	C₂×Q₈	—	194	2	no	—
82	C₂×Q₈	—	194	4	no	—
83	C₂²×C₄	—	97	4	no	—
84	C₂²×C₄	C₄×D₉₇	1	2	yes	—
85	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
86	M₁₆	—	2	4	no	—
87	M₁₆	—	97	4	no	—
88	M₁₆	—	194	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	194	2	no	—
91	C₄⋊C₄	—	194	4	no	—
92	C₄⋊C₄	—	194	4	no	—
93	C₄²	D₃₈₈	1	1	yes	characteristic
94	C₄²	D₃₈₈	1	1	yes	characteristic
95	C₄²	C₂²⋊D₉₇	1	2	yes	—
96	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
97	Q₈	—	388	4	no	—
98	Q₈	—	388	2	no	—
99	Q₈	—	388	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	194	4	no	—
102	C₂×C₄	—	194	8	no	—
103	C₂×C₄	#13095488	1	2	yes	—
104	C₂×C₄	#13095488	1	2	yes	—
105	C₂×C₄	D₄×D₉₇	1	2	yes	—
106	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
107	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
108	C₂×C₄	#13095512	1	2	yes	—
109	C₈	—	4	2	no	—
110	C₈	—	194	4	no	—
111	C₂²	—	2	2	no	—
112	C₂²	C₂×#13095488	1	1	yes	characteristic
113	C₂²	#15690772	1	2	yes	—
114	C₄	—	2	2	no	—
115	C₄	—	388	4	no	—
116	C₄	#15690728	1	4	yes	complemented
117	C₄	#15690785	1	2	yes	—
118	C₄	#15690799	1	1	yes	characteristic
119	C₄	#15690806	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	#20260636	1	1	yes	characteristic
122	C₂	C₂×#15690728	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690810

C₂

yes

maximal, characteristic

C₂×#15690728

C₂

yes

maximal

C₄×#13095495

C₂

yes

maximal, characteristic

#423×C₉₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₉₄

C₂²

yes

—

C₂×#13095495

C₂²

yes

—

#15690810

—

#15690810

—

#15690810

—

C₂×C₄⋊Dic₉₇

C₄

yes

—

#15690728

—

#15690721

—

C₄×#11715895

—

C₂×M₁₆×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₉₇

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂²×Dic₉₇

C₂×C₄

yes

—

#13095495

—

#13095495

D₄

yes

—

C₂×#11715895

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

M₁₆×C₉₇

—

C₂×C₈×C₉₇

—

C₄²×C₉₇

D₄

yes

characteristic

C₄²×C₉₇

D₄

yes

characteristic

C₄²×C₉₇

D₄

yes

—

C₂³×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

Dic₁₉₄

—

Dic₁₉₄

—

Dic₁₉₄

—

C₈×C₉₇

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₉₇

C₄×D₄

yes

—

C₄×C₉₇

—

C₄×C₉₇

#101

yes

complemented

C₄×C₉₇

#121

yes

—

C₄×C₉₇

yes

C₄×C₉₇

yes

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#390

yes

characteristic

C₂×C₉₇

C₂×#101

yes

—

#1966

—

maximal, Sylow

C₉₇

#1966

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

#423

D₉₇

yes

characteristic

C₂×#101

—

C₄×M₁₆

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₉₇

yes

—

C₄⋊Q₈

—

194

—

C₄⋊Q₈

—

194

—

C₄⋊Q₈

—

194

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

#101

—

194

—

#97

—

194

—

C₄×C₈

—

194

—

C₂×Q₈

—

194

—

C₂×Q₈

—

194

—

C₂×Q₈

—

194

—

C₂×Q₈

—

194

—

C₂²×C₄

—

C₂²×C₄

C₄×D₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

194

—

C₂×C₈

—

C₂×C₈

—

194

—

C₄⋊C₄

—

194

—

C₄⋊C₄

—

194

—

C₄²

D₃₈₈

yes

characteristic

C₄²

D₃₈₈

yes

characteristic

C₄²

C₂²⋊D₉₇

yes

—

C₂³

C₂×C₄×D₉₇

yes

characteristic

Q₈

—

388

—

Q₈

—

388

—

Q₈

—

388

—

100

C₂×C₄

—

101

C₂×C₄

—

194

—

102

C₂×C₄

—

194

—

103

C₂×C₄

#13095488

yes

—

104

C₂×C₄

#13095488

yes

—

105

C₂×C₄

D₄×D₉₇

yes

—

106

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

107

C₂×C₄

C₂×D₃₈₈

yes

characteristic

108

C₂×C₄

#13095512

yes

—

109

C₈

—

110

C₈

—

194

—

111

C₂²

—

112

C₂²

C₂×#13095488

yes

characteristic

113

C₂²

#15690772

yes

—

114

C₄

—

115

C₄

—

388

—

116

C₄

#15690728

yes

complemented

117

C₄

#15690785

yes

—

118

C₄

#15690799

yes

characteristic

119

C₄

#15690806

yes

characteristic

120

C₂

—

121

C₂

#20260636

yes

characteristic

122

C₂

C₂×#15690728

yes

—

	subgroup	G/H
center	C₂²	C₂×#13095488
commutator, prime radical	C₂³×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#423×C₉₇	C₂
socle	C₂²×C₉₇	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#423	D₉₇
hypocenter	C₉₇	#1966
LCS	G ▷ C₂³×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×#13095488

commutator, prime radical

C₂³×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

#423×C₉₇

C₂

socle

C₂²×C₉₇

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂³×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
53. C₄×C₉₇	#101	semidirect
62. C₉₇	#1966	semidirect
116. C₄	#15690728	semidirect

H ≅ G/N

type

53. C₄×C₉₇

#101

semidirect

62. C₉₇

#1966

semidirect

116. C₄

#15690728

semidirect

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

subgroup	index
C₂×#15690810	2
C₂×#15690728	2
C₄×#13095495	2
#423×C₉₇	2
#1966	97

subgroup

index

#423×C₉₇

Subgroups — #27142579

Conjugacy classes of subgroups

Complements

Maximal subgroups