Group #27142884 — subgroups

Subgroups — #27142884

Order 12416.

258 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 87 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×#15690733	C₂	1	2	yes	maximal
2	#20260641	C₂	1	1	yes	maximal, characteristic
3	#20260497	C₂	1	2	yes	maximal
4	#20260490	C₂	1	1	yes	maximal, characteristic
5	C₂×#125×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₉₄	C₂²	1	1	yes	characteristic
7	C₂²×#11715895	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
9	#15690733	—	2	4	no	—
10	#15690717	—	2	2	no	—
11	#125×C₉₇	—	2	4	no	—
12	C₂×C₄⋊C₄×C₉₇	—	2	2	no	—
13	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₉₄	—	2	2	no	—
16	C₂×Dic₁₉₄	—	2	2	no	—
17	C₂×Dic₁₉₄	—	2	1	no	—
18	C₂×Dic₁₉₄	Q₈	1	2	yes	—
19	C₂²×Dic₉₇	—	2	1	no	—
20	C₂²×Dic₉₇	—	2	1	no	—
21	C₂×#11715895	—	2	2	no	—
22	C₂×#11715895	—	2	2	no	—
23	C₂×#11715895	—	2	2	no	—
24	C₄⋊Dic₉₇	—	2	2	no	—
25	C₂²×C₄×C₉₇	—	2	2	no	—
26	C₂²×C₄×C₉₇	D₄	1	2	yes	—
27	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₉₇	—	2	8	no	—
29	C₄⋊C₄×C₉₇	—	2	2	no	—
30	C₄⋊C₄×C₉₇	D₄	1	4	yes	—
31	C₄²×C₉₇	—	2	2	no	—
32	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₉₇	—	2	2	no	—
34	C₂×C₄×C₉₇	—	2	4	no	—
35	C₂×C₄×C₉₇	—	2	2	no	—
36	C₂×C₄×C₉₇	—	2	2	no	—
37	C₂×C₄×C₉₇	—	4	2	no	—
38	C₂×C₄×C₉₇	—	4	2	no	—
39	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₉₇	#41	1	2	yes	—
42	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
43	C₂×Dic₉₇	—	2	2	no	—
44	C₂×Dic₉₇	—	4	1	no	—
45	C₂×Dic₉₇	—	4	1	no	—
46	C₂×Dic₉₇	—	4	1	no	—
47	C₂×Dic₉₇	—	4	2	no	—
48	Dic₁₉₄	—	2	4	no	—
49	Dic₁₉₄	—	4	2	no	—
50	#11715895	—	4	2	no	—
51	C₂²×C₉₇	#120	1	1	yes	characteristic
52	C₂²×C₉₇	C₂²⋊Q₈	1	1	yes	characteristic
53	C₂²×C₉₇	#124	1	1	yes	characteristic
54	C₂²×C₉₇	#135	1	2	yes	—
55	C₂²×C₉₇	#137	1	2	yes	—
56	C₄×C₉₇	—	4	2	no	—
57	C₄×C₉₇	—	4	4	no	—
58	C₄×C₉₇	C₂²⋊Q₈	1	2	yes	—
59	C₄×C₉₇	#124	1	2	yes	—
60	Dic₉₇	—	4	2	no	—
61	Dic₉₇	—	8	1	no	—
62	C₂×C₉₇	#393	1	1	yes	characteristic
63	C₂×C₉₇	#452	1	2	yes	—
64	C₂×C₉₇	#479	1	2	yes	—
65	C₂×C₉₇	#488	1	1	yes	characteristic
66	C₂×C₉₇	#491	1	1	yes	characteristic
67	#2026	—	97	1	no	maximal, Sylow
68	C₉₇	#2026	1	1	yes	characteristic, complemented, Sylow
69	C₂×#125	D₉₇	1	1	yes	characteristic
70	C₂×#103	—	97	2	no	—
71	#391	—	97	1	no	—
72	#340	—	97	2	no	—
73	#336	—	97	1	no	—
74	C₂²×Q₈	—	97	1	no	—
75	C₂²×C₈	—	97	2	no	—
76	#125	—	2	4	no	—
77	C₂×C₄⋊C₄	—	2	2	no	—
78	C₂×C₄⋊C₄	—	97	1	no	—
79	C₂×C₄⋊C₄	C₂×D₉₇	1	2	yes	—
80	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
81	#103	—	194	4	no	—
82	#95	—	194	2	no	—
83	C₂×Q₈	—	97	2	no	—
84	C₂×Q₈	—	194	2	no	—
85	C₂×Q₈	—	194	2	no	—
86	C₂×Q₈	—	194	1	no	—
87	C₂²×C₄	—	2	2	no	—
88	C₂²×C₄	—	194	1	no	—
89	C₂²×C₄	—	194	1	no	—
90	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
91	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
92	C₂×C₈	—	194	2	no	—
93	C₂×C₈	—	194	2	no	—
94	C₂×C₈	—	194	2	no	—
95	C₄⋊C₄	—	2	8	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	194	2	no	—
98	C₄⋊C₄	C₂²⋊D₉₇	1	4	yes	—
99	C₄²	—	2	2	no	—
100	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
101	Q₈	—	388	2	no	—
102	Q₈	—	194	4	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	388	1	no	—
110	C₂×C₄	—	388	2	no	—
111	C₂×C₄	—	388	1	no	—
112	C₂×C₄	—	388	1	no	—
113	C₂×C₄	—	194	2	no	—
114	C₂×C₄	C₂×C₂²⋊D₉₇	1	2	yes	—
115	C₂×C₄	Q₈×D₉₇	1	1	yes	characteristic
116	C₂×C₄	#13095512	1	2	yes	—
117	C₂×C₄	#13095512	1	1	yes	characteristic
118	C₈	—	388	2	no	—
119	C₂²	#15690779	1	1	yes	characteristic
120	C₂²	#15690786	1	1	yes	characteristic
121	C₂²	#15690799	1	1	yes	characteristic
122	C₂²	#15690830	1	2	yes	—
123	C₂²	#15690840	1	2	yes	—
124	C₄	—	4	2	no	—
125	C₄	—	4	4	no	—
126	C₄	—	776	1	no	—
127	C₄	—	388	2	no	—
128	C₄	#15690786	1	2	yes	—
129	C₄	#15690799	1	2	yes	—
130	C₂	#20260646	1	1	yes	characteristic
131	C₂	#20260897	1	2	yes	—
132	C₂	#20261039	1	2	yes	—
133	C₂	#20261092	1	1	yes	characteristic
134	C₂	#20261105	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690733

C₂

yes

maximal

#20260641

C₂

yes

maximal, characteristic

#20260497

C₂

yes

maximal

#20260490

C₂

yes

maximal, characteristic

C₂×#125×C₉₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₉₄

C₂²

yes

characteristic

C₂²×#11715895

C₂²

yes

—

C₂×C₄⋊Dic₉₇

C₂²

yes

characteristic

#15690733

—

#15690717

—

#125×C₉₇

—

C₂×C₄⋊C₄×C₉₇

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

Q₈

yes

—

C₂²×Dic₉₇

—

C₂²×Dic₉₇

—

C₂×#11715895

—

C₂×#11715895

—

C₂×#11715895

—

C₄⋊Dic₉₇

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

D₄

yes

—

C₄²×C₉₇

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂×C₄×C₉₇

#41

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

Dic₁₉₄

—

Dic₁₉₄

—

#11715895

—

C₂²×C₉₇

#120

yes

characteristic

C₂²×C₉₇

C₂²⋊Q₈

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

—

C₂²×C₉₇

#137

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂²⋊Q₈

yes

—

C₄×C₉₇

#124

yes

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

#393

yes

characteristic

C₂×C₉₇

#452

yes

—

C₂×C₉₇

#479

yes

—

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₂×#125

D₉₇

yes

—

—

—

—

C₂²×Q₈

—

C₂²×C₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

#103

—

194

—

#95

—

194

—

C₂×Q₈

—

C₂×Q₈

—

194

—

C₂×Q₈

—

194

—

C₂×Q₈

—

194

—

C₂²×C₄

—

C₂²×C₄

—

194

—

C₂²×C₄

—

194

—

C₂²×C₄

C₂²⋊D₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

194

—

C₄⋊C₄

C₂²⋊D₉₇

yes

—

C₄²

—

100

C₂³

C₂×C₂²⋊D₉₇

yes

characteristic

101

Q₈

—

388

—

102

Q₈

—

194

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

388

—

110

C₂×C₄

—

388

—

111

C₂×C₄

—

388

—

112

C₂×C₄

—

388

—

113

C₂×C₄

—

194

—

114

C₂×C₄

C₂×C₂²⋊D₉₇

yes

—

115

C₂×C₄

Q₈×D₉₇

yes

characteristic

116

C₂×C₄

#13095512

yes

—

117

C₂×C₄

#13095512

yes

characteristic

118

C₈

—

388

—

119

C₂²

#15690779

yes

characteristic

120

C₂²

#15690786

yes

characteristic

121

C₂²

#15690799

yes

characteristic

122

C₂²

#15690830

yes

—

123

C₂²

#15690840

yes

—

124

C₄

—

125

C₄

—

126

C₄

—

776

—

127

C₄

—

388

—

128

C₄

#15690786

yes

—

129

C₄

#15690799

yes

—

130

C₂

#20260646

yes

characteristic

131

C₂

#20260897

yes

—

132

C₂

#20261039

yes

—

133

C₂

#20261092

yes

characteristic

134

C₂

#20261105

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₉₇
commutator, cosocle	C₂²×C₄×C₉₇	C₂³
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	C₂×#125×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂×D₄
hypercenter	C₂×#125	D₉₇
hypocenter	C₉₇	#2026
LCS	G ▷ C₂²×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₉₇

commutator, cosocle

C₂²×C₄×C₉₇

C₂³

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

C₂×#125×C₉₇

C₂

socle, prime radical

C₂³×C₉₇

C₂×D₄

D₉₇

C₉₇

G ▷ C₂²×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
68. C₉₇	#2026	semidirect

H ≅ G/N

type

68. C₉₇

#2026

semidirect

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

subgroup	index
C₂×#15690733	2
#20260641	2
#20260497	2
#20260490	2
C₂×#125×C₉₇	2
#2026	97

subgroup

index

C₂×#125×C₉₇

Subgroups — #27142884

Conjugacy classes of subgroups

Complements

Maximal subgroups