Group #27142485 — subgroups

Subgroups — #27142485

Order 12416.

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

normal subgroups only

This group has 208 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×#15690746	C₂	1	1	yes	maximal, characteristic
2	#20260762	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₉₇	C₂	1	1	yes	maximal, characteristic
4	#20260494	C₂	1	2	yes	maximal
5	#427×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13095495	C₂²	1	2	yes	—
7	C₂²×#11715895	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
9	#15690746	—	2	2	no	—
10	C₄⋊#11715895	—	2	4	no	—
11	C₈×Dic₉₇	—	2	2	no	—
12	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
13	#117×C₉₇	C₂²	1	2	yes	—
14	#107×C₉₇	C₄	1	2	yes	complemented
15	#106×C₉₇	C₄	1	2	yes	complemented
16	C₂²×Dic₉₇	—	2	1	no	—
17	#13095495	—	2	2	no	—
18	#13095495	—	2	2	no	—
19	C₂×#11715895	—	2	1	no	—
20	C₂×#11715895	—	2	1	no	—
21	C₂×#11715895	—	2	2	no	—
22	C₂×#11715895	D₄	1	2	yes	—
23	C₄×Dic₉₇	—	2	1	no	—
24	C₄×Dic₉₇	Q₈	1	2	yes	—
25	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₉₇	D₄	1	2	yes	—
29	C₂×C₈×C₉₇	Q₈	1	2	yes	—
30	C₄⋊C₄×C₉₇	C₂×C₄	1	4	yes	—
31	C₂²⋊C₄×C₉₇	—	2	2	no	—
32	C₄²×C₉₇	—	2	2	no	—
33	C₂³×C₉₇	#41	1	1	yes	characteristic
34	C₂×C₄×C₉₇	—	2	4	no	—
35	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
41	C₂×Dic₉₇	—	2	4	no	—
42	C₂×Dic₉₇	—	2	2	no	—
43	C₈×C₉₇	C₄⋊C₄	1	2	yes	—
44	C₈×C₉₇	C₄⋊C₄	1	2	yes	—
45	#11715895	—	2	2	no	—
46	#11715895	—	4	2	no	—
47	C₂²×C₉₇	—	2	2	no	—
48	C₂²×C₉₇	C₄×D₄	1	1	yes	characteristic
49	C₂²×C₉₇	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₉₇	#125	1	1	yes	characteristic
51	C₄×C₉₇	—	4	4	no	—
52	C₄×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₉₇	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₉₇	#121	1	1	yes	characteristic
55	C₄×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
56	Dic₉₇	—	2	4	no	—
57	C₂×C₉₇	—	2	1	no	—
58	C₂×C₉₇	—	2	1	no	—
59	C₂×C₉₇	#389	1	1	yes	characteristic
60	C₂×C₉₇	#443	1	2	yes	—
61	#1945	—	97	1	no	maximal, Sylow
62	C₉₇	#1945	1	1	yes	characteristic, complemented, Sylow
63	C₂×#108	—	97	1	no	—
64	#427	D₉₇	1	1	yes	characteristic
65	#424	—	97	2	no	—
66	C₂×C₄×C₈	—	97	1	no	—
67	#339	—	97	2	no	—
68	C₂×M₁₆	—	97	2	no	—
69	C₂²×C₈	—	97	1	no	—
70	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
71	#117	C₂×D₉₇	1	2	yes	—
72	C₂×C₄²	—	97	1	no	—
73	#108	—	194	2	no	—
74	#107	Dic₉₇	1	2	yes	complemented
75	#106	Dic₉₇	1	2	yes	complemented
76	C₄⋊C₈	—	194	4	no	—
77	C₄×C₈	—	194	2	no	—
78	C₂²×C₄	—	194	1	no	—
79	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
80	M₁₆	—	194	2	no	—
81	M₁₆	—	194	2	no	—
82	C₂×C₈	—	97	2	no	—
83	C₂×C₈	—	194	2	no	—
84	C₂×C₈	—	194	1	no	—
85	C₂×C₈	—	194	1	no	—
86	C₂×C₈	Dic₁₉₄	1	2	yes	—
87	C₂×C₈	D₃₈₈	1	2	yes	—
88	C₂×C₈	C₂×Dic₉₇	1	1	yes	characteristic
89	C₂×C₈	C₂×Dic₉₇	1	1	yes	characteristic
90	C₄⋊C₄	C₂×Dic₉₇	1	4	yes	—
91	C₂²⋊C₄	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	—	97	2	no	—
94	C₄²	—	194	1	no	—
95	C₂³	#13095510	1	1	yes	characteristic
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	194	4	no	—
98	C₂×C₄	—	194	2	no	—
99	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
100	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
101	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
102	C₂×C₄	Q₈×D₉₇	1	1	yes	characteristic
103	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
104	C₂×C₄	#13095512	1	1	yes	characteristic
105	C₈	—	388	2	no	—
106	C₈	—	194	2	no	—
107	C₈	C₄⋊Dic₉₇	1	2	yes	—
108	C₈	C₄⋊Dic₉₇	1	2	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	D₄×Dic₉₇	1	1	yes	characteristic
111	C₂²	#15690790	1	1	yes	characteristic
112	C₂²	#15690801	1	1	yes	characteristic
113	C₄	—	4	4	no	—
114	C₄	—	194	4	no	—
115	C₄	C₂×C₄⋊Dic₉₇	1	1	yes	characteristic
116	C₄	Q₈×Dic₉₇	1	1	yes	characteristic
117	C₄	#15690785	1	1	yes	characteristic
118	C₄	#15690809	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#20260634	1	1	yes	characteristic
122	C₂	#20260852	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690746

C₂

yes

maximal, characteristic

#20260762

C₂

yes

maximal

C₂×C₈×Dic₉₇

C₂

yes

maximal, characteristic

#20260494

C₂

yes

maximal

#427×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

C₂²

yes

—

C₂²×#11715895

C₂²

yes

characteristic

C₂×C₄×Dic₉₇

C₂²

yes

characteristic

#15690746

—

C₄⋊#11715895

—

C₈×Dic₉₇

—

C₂²×C₈×C₉₇

C₂²

yes

characteristic

#117×C₉₇

C₂²

yes

—

#107×C₉₇

C₄

yes

complemented

#106×C₉₇

C₄

yes

complemented

C₂²×Dic₉₇

—

—

—

—

—

—

D₄

yes

—

C₄×Dic₉₇

—

C₄×Dic₉₇

Q₈

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₂×C₈×C₉₇

C₂×C₄

yes

characteristic

C₂×C₈×C₉₇

C₂×C₄

yes

characteristic

C₂×C₈×C₉₇

D₄

yes

—

C₂×C₈×C₉₇

Q₈

yes

—

C₄⋊C₄×C₉₇

C₂×C₄

yes

—

C₂²⋊C₄×C₉₇

—

C₄²×C₉₇

—

C₂³×C₉₇

#41

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₉₇

#41

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₈×C₉₇

C₄⋊C₄

yes

—

C₈×C₉₇

C₄⋊C₄

yes

—

#11715895

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

C₄×D₄

yes

characteristic

C₂²×C₉₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₉₇

#125

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₉₇

C₄×Q₈

yes

characteristic

C₄×C₉₇

#121

yes

characteristic

C₄×C₉₇

C₄⋊Q₈

yes

characteristic

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#389

yes

characteristic

C₂×C₉₇

#443

yes

—

#1945

—

maximal, Sylow

C₉₇

#1945

yes

characteristic, complemented, Sylow

C₂×#108

—

#427

D₉₇

yes

characteristic

#424

—

C₂×C₄×C₈

—

#339

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₉₇

yes

characteristic

#117

C₂×D₉₇

yes

—

C₂×C₄²

—

#108

—

194

—

#107

Dic₉₇

yes

complemented

#106

Dic₉₇

yes

complemented

C₄⋊C₈

—

194

—

C₄×C₈

—

194

—

C₂²×C₄

—

194

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

M₁₆

—

194

—

M₁₆

—

194

—

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₂×C₈

Dic₁₉₄

yes

—

C₂×C₈

D₃₈₈

yes

—

C₂×C₈

C₂×Dic₉₇

yes

characteristic

C₂×C₈

C₂×Dic₉₇

yes

characteristic

C₄⋊C₄

C₂×Dic₉₇

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

194

—

C₂³

#13095510

yes

characteristic

C₂×C₄

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

C₂×C₄

C₂²×Dic₉₇

yes

characteristic

100

C₂×C₄

D₄×D₉₇

yes

characteristic

101

C₂×C₄

C₂×D₃₈₈

yes

characteristic

102

C₂×C₄

Q₈×D₉₇

yes

characteristic

103

C₂×C₄

C₂×Dic₁₉₄

yes

characteristic

104

C₂×C₄

#13095512

yes

characteristic

105

C₈

—

388

—

106

C₈

—

194

—

107

C₈

C₄⋊Dic₉₇

yes

—

108

C₈

C₄⋊Dic₉₇

yes

—

109

C₂²

—

110

C₂²

D₄×Dic₉₇

yes

characteristic

111

C₂²

#15690790

yes

characteristic

112

C₂²

#15690801

yes

characteristic

113

C₄

—

114

C₄

—

194

—

115

C₄

C₂×C₄⋊Dic₉₇

yes

characteristic

116

C₄

Q₈×Dic₉₇

yes

characteristic

117

C₄

#15690785

yes

characteristic

118

C₄

#15690809

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#20260634

yes

characteristic

122

C₂

#20260852

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₉₇
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#427×C₉₇	C₂
socle	C₂²×C₉₇	C₄×D₄
prime radical	C₂³×C₉₇	#41
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#427	D₉₇
hypocenter	C₉₇	#1945
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₉₇

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

#427×C₉₇

C₂

socle

C₂²×C₉₇

C₄×D₄

prime radical

C₂³×C₉₇

#41

cosocle

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
14. #107×C₉₇	C₄	semidirect
15. #106×C₉₇	C₄	semidirect
62. C₉₇	#1945	semidirect
74. #107	Dic₉₇	semidirect
75. #106	Dic₉₇	semidirect

H ≅ G/N

type

14. #107×C₉₇

C₄

semidirect

15. #106×C₉₇

C₄

semidirect

62. C₉₇

#1945

semidirect

74. #107

Dic₉₇

semidirect

75. #106

Dic₉₇

semidirect

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

subgroup	index
C₂×#15690746	2
#20260762	2
C₂×C₈×Dic₉₇	2
#20260494	2
#427×C₉₇	2
#1945	97

subgroup

index

C₂×C₈×Dic₉₇

#427×C₉₇

Subgroups — #27142485

Conjugacy classes of subgroups

Complements

Maximal subgroups