Group #27141865 — subgroups

Subgroups — #27141865

Order 12416.

216 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 93 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×C₈×Dic₉₇	C₂	1	1	yes	maximal, characteristic
2	#20260600	C₂	1	2	yes	maximal
3	#20260498	C₂	1	1	yes	maximal, characteristic
4	#20260497	C₂	1	2	yes	maximal
5	#340×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11715895	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
8	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
9	C₈×Dic₉₇	C₄	1	4	yes	complemented
10	#15690717	—	2	2	no	—
11	#15690717	—	2	2	no	—
12	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
14	C₂²×Dic₉₇	—	2	2	no	—
15	C₂²×Dic₉₇	D₄	1	1	yes	characteristic
16	C₂²×Dic₉₇	D₄	1	1	yes	characteristic
17	C₂×#11715895	—	2	2	no	—
18	C₂×#11715895	C₂×C₄	1	2	yes	—
19	C₄⋊Dic₉₇	—	2	2	no	—
20	C₄⋊Dic₉₇	—	2	2	no	—
21	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
22	C₄×Dic₉₇	Q₈	1	2	yes	—
23	C₂²×C₄×C₉₇	—	2	2	no	—
24	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₉₇	—	2	2	no	—
26	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
27	C₄⋊C₄×C₉₇	—	2	2	no	—
28	C₄⋊C₄×C₉₇	—	2	2	no	—
29	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₉₇	—	4	2	no	—
31	C₂×C₄×C₉₇	—	4	2	no	—
32	C₂×C₄×C₉₇	—	4	2	no	—
33	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₉₇	#41	1	2	yes	—
36	C₂×C₄×C₉₇	#41	1	2	yes	—
37	C₂×Dic₉₇	—	2	1	no	—
38	C₂×Dic₉₇	—	2	2	no	—
39	C₂×Dic₉₇	—	2	1	no	—
40	C₂×Dic₉₇	—	4	2	no	—
41	C₂×Dic₉₇	—	4	2	no	—
42	C₂×Dic₉₇	—	4	2	no	—
43	C₂×Dic₉₇	C₄⋊C₄	1	2	yes	—
44	C₂×Dic₉₇	C₄⋊C₄	1	2	yes	—
45	C₈×C₉₇	—	2	2	no	—
46	#11715895	—	2	2	no	—
47	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₉₇	#124	1	2	yes	—
49	C₂²×C₉₇	#135	1	1	yes	characteristic
50	C₂²×C₉₇	#135	1	2	yes	—
51	C₂²×C₉₇	#135	1	1	yes	characteristic
52	C₄×C₉₇	—	8	2	no	—
53	C₄×C₉₇	#117	1	1	yes	characteristic
54	C₄×C₉₇	#117	1	1	yes	characteristic
55	C₄×C₉₇	#125	1	2	yes	—
56	Dic₉₇	—	2	2	no	—
57	Dic₉₇	—	2	2	no	—
58	Dic₉₇	—	8	2	no	—
59	C₂×C₉₇	#382	1	1	yes	characteristic
60	C₂×C₉₇	#427	1	1	yes	characteristic
61	C₂×C₉₇	#427	1	1	yes	characteristic
62	C₂×C₉₇	#488	1	2	yes	—
63	C₂×C₉₇	#488	1	2	yes	—
64	#1835	—	97	1	no	maximal, Sylow
65	C₉₇	#1835	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₄×C₈	—	97	1	no	—
67	#382	—	97	2	no	—
68	#340	—	97	1	no	—
69	#340	—	97	2	no	—
70	#340	D₉₇	1	1	yes	characteristic
71	C₂²×C₈	—	97	1	no	—
72	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
73	C₂×C₄⋊C₄	—	97	2	no	—
74	C₂×C₄⋊C₄	C₂×D₉₇	1	2	yes	—
75	C₂×C₄²	—	97	1	no	—
76	C₄×C₈	—	97	4	no	—
77	#95	—	194	2	no	—
78	#95	—	194	2	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	97	1	no	—
81	C₂²×C₄	—	97	1	no	—
82	C₂²×C₄	—	194	2	no	—
83	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	97	2	no	—
86	C₂×C₈	—	194	2	no	—
87	C₂×C₈	C₄×D₉₇	1	2	yes	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	—	194	2	no	—
91	C₄⋊C₄	—	194	2	no	—
92	C₄²	—	97	2	no	—
93	C₄²	—	97	2	no	—
94	C₂³	D₄×D₉₇	1	1	yes	characteristic
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	97	2	no	—
99	C₂×C₄	—	97	2	no	—
100	C₂×C₄	—	388	2	no	—
101	C₂×C₄	—	388	2	no	—
102	C₂×C₄	—	388	2	no	—
103	C₂×C₄	—	194	1	no	—
104	C₂×C₄	—	194	2	no	—
105	C₂×C₄	—	194	1	no	—
106	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
107	C₂×C₄	Q₈×D₉₇	1	1	yes	characteristic
108	C₂×C₄	#13095510	1	2	yes	—
109	C₂×C₄	#13095511	1	2	yes	—
110	C₈	—	2	2	no	—
111	C₈	—	194	2	no	—
112	C₂²	C₄⋊C₄×D₉₇	1	1	yes	characteristic
113	C₂²	#15690796	1	2	yes	—
114	C₂²	#15690829	1	1	yes	characteristic
115	C₂²	#15690831	1	2	yes	—
116	C₂²	#15690833	1	1	yes	characteristic
117	C₄	—	8	2	no	—
118	C₄	—	776	2	no	—
119	C₄	—	194	2	no	—
120	C₄	—	194	2	no	—
121	C₄	#15690766	1	1	yes	characteristic
122	C₄	#15690767	1	1	yes	characteristic
123	C₄	#15690800	1	2	yes	—
124	C₂	#20260599	1	1	yes	characteristic
125	C₂	#20260770	1	1	yes	characteristic
126	C₂	#20260772	1	1	yes	characteristic
127	C₂	#20261088	1	2	yes	—
128	C₂	#20261091	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₉₇

C₂

yes

maximal, characteristic

#20260600

C₂

yes

maximal

#20260498

C₂

yes

maximal, characteristic

#20260497

C₂

yes

maximal

#340×C₉₇

C₂

yes

maximal, characteristic

C₂²×#11715895

C₂²

yes

characteristic

C₂×C₄⋊Dic₉₇

C₂²

yes

—

C₂×C₄×Dic₉₇

C₂²

yes

characteristic

C₈×Dic₉₇

C₄

yes

complemented

#15690717

—

#15690717

—

C₂²×C₈×C₉₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

C₂²×Dic₉₇

—

C₂²×Dic₉₇

D₄

yes

characteristic

C₂²×Dic₉₇

D₄

yes

characteristic

C₂×#11715895

—

C₂×#11715895

C₂×C₄

yes

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₄×Dic₉₇

C₂×C₄

yes

—

C₄×Dic₉₇

Q₈

yes

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

C₂×C₄

yes

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂×C₄×C₉₇

#41

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

C₄⋊C₄

yes

—

C₂×Dic₉₇

C₄⋊C₄

yes

—

C₈×C₉₇

—

#11715895

—

C₂²×C₉₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₉₇

#124

yes

—

C₂²×C₉₇

#135

yes

characteristic

C₂²×C₉₇

#135

yes

—

C₂²×C₉₇

#135

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

yes

C₄×C₉₇

yes

C₄×C₉₇

yes

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₂×C₉₇

#488

yes

—

#1835

—

maximal, Sylow

C₉₇

#1835

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

—

—

—

D₉₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×D₉₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₉₇

yes

—

C₂×C₄²

—

C₄×C₈

—

#95

—

194

—

#95

—

194

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

194

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

C₄×D₉₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

194

—

C₄⋊C₄

—

194

—

C₄²

—

C₄²

—

C₂³

D₄×D₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

388

—

101

C₂×C₄

—

388

—

102

C₂×C₄

—

388

—

103

C₂×C₄

—

194

—

104

C₂×C₄

—

194

—

105

C₂×C₄

—

194

—

106

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

107

C₂×C₄

Q₈×D₉₇

yes

characteristic

108

C₂×C₄

#13095510

yes

—

109

C₂×C₄

#13095511

yes

—

110

C₈

—

111

C₈

—

194

—

112

C₂²

C₄⋊C₄×D₉₇

yes

characteristic

113

C₂²

#15690796

yes

—

114

C₂²

#15690829

yes

characteristic

115

C₂²

#15690831

yes

—

116

C₂²

#15690833

yes

characteristic

117

C₄

—

118

C₄

—

776

—

119

C₄

—

194

—

120

C₄

—

194

—

121

C₄

#15690766

yes

characteristic

122

C₄

#15690767

yes

characteristic

123

C₄

#15690800

yes

—

124

C₂

#20260599

yes

characteristic

125

C₂

#20260770

yes

characteristic

126

C₂

#20260772

yes

characteristic

127

C₂

#20261088

yes

—

128

C₂

#20261091

yes

—

	subgroup	G/H
center	C₂³	D₄×D₉₇
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#340×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂×D₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#340	D₉₇
hypocenter	C₉₇	#1835
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₉₇

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

#340×C₉₇

C₂

socle, prime radical

C₂³×C₉₇

C₂×D₄

cosocle

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
9. C₈×Dic₉₇	C₄	semidirect
65. C₉₇	#1835	semidirect

H ≅ G/N

type

9. C₈×Dic₉₇

C₄

semidirect

65. C₉₇

#1835

semidirect

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

subgroup	index
C₂×C₈×Dic₉₇	2
#20260600	2
#20260498	2
#20260497	2
#340×C₉₇	2
#1835	97

subgroup

index

C₂×C₈×Dic₉₇

#340×C₉₇

Subgroups — #27141865

Conjugacy classes of subgroups

Complements

Maximal subgroups