Group #27141854 — subgroups

Subgroups — #27141854

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×#15690739	C₂	1	2	yes	maximal
2	#20260581	C₂	1	1	yes	maximal, characteristic
3	#20260490	C₂	1	1	yes	maximal, characteristic
4	#20260489	C₂	1	2	yes	maximal
5	#336×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11715895	C₂²	1	2	yes	—
7	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
9	#15690739	—	2	4	no	—
10	#15690717	C₄	1	2	yes	—
11	#15690717	C₄	1	2	yes	—
12	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
17	C₂×#11715895	—	2	2	no	—
18	C₂×#11715895	—	2	2	no	—
19	C₂×#11715895	—	2	2	no	—
20	C₄×Dic₉₇	—	2	2	no	—
21	C₄×Dic₉₇	D₄	1	2	yes	—
22	C₄×Dic₉₇	Q₈	1	2	yes	—
23	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₉₇	—	2	2	no	—
26	C₂×C₈×C₉₇	—	2	2	no	—
27	C₂×C₈×C₉₇	—	2	2	no	—
28	C₄²×C₉₇	—	2	2	no	—
29	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₉₇	—	2	2	no	—
31	C₂×C₄×C₉₇	—	2	2	no	—
32	C₂×C₄×C₉₇	—	2	2	no	—
33	C₂×C₄×C₉₇	C₂×D₄	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	2	no	—
38	C₂×Dic₉₇	—	2	2	no	—
39	C₂×Dic₉₇	—	2	1	no	—
40	C₂×Dic₉₇	—	2	1	no	—
41	C₂×Dic₉₇	—	2	2	no	—
42	C₂×Dic₉₇	—	2	2	no	—
43	C₂×Dic₉₇	C₄⋊C₄	1	4	yes	—
44	C₈×C₉₇	—	4	2	no	—
45	#11715895	—	4	2	no	—
46	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₉₇	#117	1	1	yes	characteristic
48	C₂²×C₉₇	#117	1	1	yes	characteristic
49	C₂²×C₉₇	#131	1	2	yes	—
50	C₂²×C₉₇	#131	1	2	yes	—
51	C₄×C₉₇	—	4	2	no	—
52	C₄×C₉₇	#124	1	2	yes	—
53	C₄×C₉₇	#125	1	2	yes	—
54	Dic₉₇	—	2	4	no	—
55	Dic₉₇	—	4	1	no	—
56	Dic₉₇	—	4	1	no	—
57	C₂×C₉₇	#382	1	1	yes	characteristic
58	C₂×C₉₇	#424	1	2	yes	—
59	C₂×C₉₇	#433	1	2	yes	—
60	C₂×C₉₇	#433	1	1	yes	characteristic
61	C₂×C₉₇	#433	1	1	yes	characteristic
62	#1832	—	97	1	no	maximal, Sylow
63	C₉₇	#1832	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄⋊C₈	—	97	2	no	—
65	#376	—	97	1	no	—
66	#336	—	97	2	no	—
67	#336	—	97	1	no	—
68	#336	D₉₇	1	1	yes	characteristic
69	C₂²×C₈	—	97	2	no	—
70	C₂²×C₈	C₂×D₉₇	1	2	yes	—
71	C₂×C₄²	—	97	1	no	—
72	C₂×C₄²	—	97	1	no	—
73	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
74	C₄⋊C₈	—	194	4	no	—
75	#95	—	97	2	no	—
76	#95	—	97	2	no	—
77	C₂²×C₄	—	97	2	no	—
78	C₂²×C₄	—	97	1	no	—
79	C₂²×C₄	—	97	1	no	—
80	C₂²×C₄	C₄×D₉₇	1	2	yes	—
81	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	194	2	no	—
86	C₂×C₈	—	194	2	no	—
87	C₂×C₈	—	194	2	no	—
88	C₄²	—	2	2	no	—
89	C₄²	—	97	2	no	—
90	C₄²	—	97	2	no	—
91	C₄²	—	194	2	no	—
92	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	97	4	no	—
97	C₂×C₄	—	194	1	no	—
98	C₂×C₄	—	194	1	no	—
99	C₂×C₄	—	194	2	no	—
100	C₂×C₄	—	194	2	no	—
101	C₂×C₄	—	194	2	no	—
102	C₂×C₄	—	194	2	no	—
103	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
104	C₂×C₄	Q₈×D₉₇	1	1	yes	characteristic
105	C₂×C₄	#13095510	1	2	yes	—
106	C₂×C₄	#13095511	1	2	yes	—
107	C₈	—	4	2	no	—
108	C₈	—	388	2	no	—
109	C₂²	C₄⋊C₄×D₉₇	1	1	yes	characteristic
110	C₂²	#15690766	1	1	yes	characteristic
111	C₂²	#15690767	1	1	yes	characteristic
112	C₂²	#15690816	1	2	yes	—
113	C₂²	#15690818	1	2	yes	—
114	C₄	—	4	2	no	—
115	C₄	—	388	1	no	—
116	C₄	—	388	1	no	—
117	C₄	—	194	4	no	—
118	C₄	#15690796	1	2	yes	—
119	C₄	#15690800	1	2	yes	—
120	C₂	#20260599	1	1	yes	characteristic
121	C₂	#20260760	1	2	yes	—
122	C₂	#20260793	1	2	yes	—
123	C₂	#20260794	1	1	yes	characteristic
124	C₂	#20260797	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690739

C₂

yes

maximal

#20260581

C₂

yes

maximal, characteristic

#20260490

C₂

yes

maximal, characteristic

#20260489

C₂

yes

maximal

#336×C₉₇

C₂

yes

maximal, characteristic

C₂²×#11715895

C₂²

yes

—

C₂×C₄×Dic₉₇

C₂²

yes

characteristic

C₂×C₄×Dic₉₇

C₂²

yes

characteristic

#15690739

—

#15690717

C₄

yes

—

#15690717

C₄

yes

—

C₂²×C₈×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂²×Dic₉₇

C₂×C₄

yes

characteristic

C₂²×Dic₉₇

C₂×C₄

yes

characteristic

C₂²×Dic₉₇

C₂×C₄

yes

—

C₂×#11715895

—

C₂×#11715895

—

C₂×#11715895

—

C₄×Dic₉₇

—

C₄×Dic₉₇

D₄

yes

—

C₄×Dic₉₇

Q₈

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₄²×C₉₇

—

C₂³×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

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

—

C₈×C₉₇

—

#11715895

—

C₂²×C₉₇

C₂×C₄⋊C₄

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

—

C₂²×C₉₇

#131

yes

—

C₄×C₉₇

—

C₄×C₉₇

#124

yes

—

C₄×C₉₇

#125

yes

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

#382

yes

characteristic

C₂×C₉₇

#424

yes

—

C₂×C₉₇

#433

yes

—

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

—

—

—

D₉₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×D₉₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

194

—

#95

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

194

—

C₂³

C₂×C₄×D₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

100

C₂×C₄

—

194

—

101

C₂×C₄

—

194

—

102

C₂×C₄

—

194

—

103

C₂×C₄

D₄×D₉₇

yes

characteristic

104

C₂×C₄

Q₈×D₉₇

yes

characteristic

105

C₂×C₄

#13095510

yes

—

106

C₂×C₄

#13095511

yes

—

107

C₈

—

108

C₈

—

388

—

109

C₂²

C₄⋊C₄×D₉₇

yes

characteristic

110

C₂²

#15690766

yes

characteristic

111

C₂²

#15690767

yes

characteristic

112

C₂²

#15690816

yes

—

113

C₂²

#15690818

yes

—

114

C₄

—

115

C₄

—

388

—

116

C₄

—

388

—

117

C₄

—

194

—

118

C₄

#15690796

yes

—

119

C₄

#15690800

yes

—

120

C₂

#20260599

yes

characteristic

121

C₂

#20260760

yes

—

122

C₂

#20260793

yes

—

123

C₂

#20260794

yes

characteristic

124

C₂

#20260797

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₉₇
commutator, socle, prime radical	C₂³×C₉₇	C₂²×C₄
Fitting	#336×C₉₇	C₂
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#336	D₉₇
hypocenter	C₉₇	#1832
LCS	G ▷ C₂³×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₉₇

commutator, socle, prime radical

C₂³×C₉₇

C₂²×C₄

Fitting

#336×C₉₇

C₂

cosocle

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂³×C₉₇ ▷ C₉₇

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
63. C₉₇	#1832	semidirect

H ≅ G/N

type

63. C₉₇

#1832

semidirect

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

subgroup	index
C₂×#15690739	2
#20260581	2
#20260490	2
#20260489	2
#336×C₉₇	2
#1832	97

subgroup

index

#336×C₉₇

Subgroups — #27141854

Conjugacy classes of subgroups

Complements

Maximal subgroups