Group #27142887 — subgroups

Subgroups — #27142887

Order 12416.

258 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 85 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×#15690802	C₂	1	1	yes	maximal, characteristic
2	C₂×#15690734	C₂	1	2	yes	maximal
3	#20260497	C₂	1	2	yes	maximal
4	#20260490	C₂	1	1	yes	maximal, characteristic
5	#391×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11715895	C₂²	1	2	yes	—
7	#15690802	—	2	4	no	—
8	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
9	C₂×#13095490	—	2	2	no	—
10	#15690734	—	2	4	no	—
11	C₂²×Q₈×C₉₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
14	#95×C₉₇	—	2	2	no	—
15	C₂²×Dic₉₇	—	2	2	no	—
16	C₂×#11715895	—	2	2	no	—
17	C₂×#11715895	—	2	2	no	—
18	C₂×#11715895	—	2	2	no	—
19	C₄⋊Dic₉₇	—	2	2	no	—
20	C₄⋊Dic₉₇	D₄	1	4	yes	—
21	#13095490	—	2	8	no	—
22	C₂×Q₈×C₉₇	—	2	1	no	—
23	C₂×Q₈×C₉₇	—	2	2	no	—
24	C₂×Q₈×C₉₇	—	2	2	no	—
25	C₂×Q₈×C₉₇	Q₈	1	2	yes	—
26	C₂²×C₄×C₉₇	—	2	1	no	—
27	C₂²×C₄×C₉₇	—	2	1	no	—
28	C₂²×C₄×C₉₇	D₄	1	2	yes	—
29	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₉₇	—	2	2	no	—
31	C₄²×C₉₇	—	2	2	no	—
32	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
33	Q₈×C₉₇	—	2	4	no	—
34	Q₈×C₉₇	—	4	2	no	—
35	C₂×C₄×C₉₇	—	2	2	no	—
36	C₂×C₄×C₉₇	—	2	2	no	—
37	C₂×C₄×C₉₇	—	2	2	no	—
38	C₂×C₄×C₉₇	—	2	2	no	—
39	C₂×C₄×C₉₇	—	4	1	no	—
40	C₂×C₄×C₉₇	—	4	1	no	—
41	C₂×C₄×C₉₇	—	4	1	no	—
42	C₂×C₄×C₉₇	—	4	2	no	—
43	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₉₇	#41	1	2	yes	—
46	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
47	C₂×Dic₉₇	—	2	4	no	—
48	C₂×Dic₉₇	—	4	2	no	—
49	C₂×Dic₉₇	—	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	2	no	—
58	C₄×C₉₇	—	8	1	no	—
59	C₄×C₉₇	C₂²⋊Q₈	1	2	yes	—
60	C₄×C₉₇	#124	1	2	yes	—
61	Dic₉₇	—	4	4	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	—	97	1	no	—
70	C₂×#103	—	97	2	no	—
71	#391	D₉₇	1	1	yes	characteristic
72	#340	—	97	2	no	—
73	#336	—	97	1	no	—
74	C₂²×Q₈	C₂×D₉₇	1	1	yes	characteristic
75	C₂²×C₈	—	97	2	no	—
76	#125	—	194	4	no	—
77	C₂×C₄⋊C₄	—	97	2	no	—
78	C₂×C₄⋊C₄	—	194	2	no	—
79	C₂×C₄⋊C₄	C₂×D₉₇	1	1	yes	characteristic
80	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
81	#103	—	194	4	no	—
82	#95	—	2	2	no	—
83	C₂×Q₈	—	2	1	no	—
84	C₂×Q₈	—	2	2	no	—
85	C₂×Q₈	—	2	2	no	—
86	C₂×Q₈	Dic₁₉₄	1	2	yes	—
87	C₂²×C₄	—	2	1	no	—
88	C₂²×C₄	—	2	1	no	—
89	C₂²×C₄	—	194	2	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	2	no	—
96	C₄⋊C₄	—	97	4	no	—
97	C₄⋊C₄	—	194	8	no	—
98	C₄⋊C₄	—	194	2	no	—
99	C₄²	—	2	2	no	—
100	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
101	Q₈	—	2	4	no	—
102	Q₈	—	4	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	4	1	no	—
109	C₂×C₄	—	4	1	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	388	2	no	—
112	C₂×C₄	—	388	2	no	—
113	C₂×C₄	—	194	4	no	—
114	C₂×C₄	D₄×D₉₇	1	2	yes	—
115	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
116	C₂×C₄	#13095510	1	2	yes	—
117	C₂×C₄	#13095511	1	1	yes	characteristic
118	C₈	—	388	2	no	—
119	C₂²	#15690778	1	1	yes	characteristic
120	C₂²	#15690789	1	1	yes	characteristic
121	C₂²	#15690797	1	1	yes	characteristic
122	C₂²	#15690830	1	2	yes	—
123	C₂²	#15690841	1	2	yes	—
124	C₄	—	4	2	no	—
125	C₄	—	4	2	no	—
126	C₄	—	8	1	no	—
127	C₄	—	388	4	no	—
128	C₄	#15690787	1	2	yes	—
129	C₄	#15690796	1	2	yes	—
130	C₂	#20260647	1	1	yes	characteristic
131	C₂	#20260895	1	2	yes	—
132	C₂	#20261042	1	2	yes	—
133	C₂	#20261089	1	1	yes	characteristic
134	C₂	#20261103	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690802

C₂

yes

maximal, characteristic

C₂×#15690734

C₂

yes

maximal

#20260497

C₂

yes

maximal

#20260490

C₂

yes

maximal, characteristic

#391×C₉₇

C₂

yes

maximal, characteristic

C₂²×#11715895

C₂²

yes

—

#15690802

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

C₂×#13095490

—

#15690734

—

C₂²×Q₈×C₉₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₉₇

C₂²

yes

characteristic

C₂×C₄²×C₉₇

C₂²

yes

characteristic

#95×C₉₇

—

C₂²×Dic₉₇

—

C₂×#11715895

—

C₂×#11715895

—

C₂×#11715895

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

D₄

yes

—

#13095490

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

Q₈

yes

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

Q₈×C₉₇

—

Q₈×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

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

—

#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

—

C₄×C₉₇

C₂²⋊Q₈

yes

—

C₄×C₉₇

#124

yes

—

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

—

C₂×#103

—

#391

D₉₇

yes

characteristic

#340

—

#336

—

C₂²×Q₈

C₂×D₉₇

yes

characteristic

C₂²×C₈

—

#125

—

194

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄⋊C₄

C₂×D₉₇

yes

characteristic

C₂×C₄²

C₂×D₉₇

yes

characteristic

#103

—

194

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₁₉₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

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

—

194

—

C₄²

—

100

C₂³

C₂×C₂²⋊D₉₇

yes

characteristic

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

388

—

112

C₂×C₄

—

388

—

113

C₂×C₄

—

194

—

114

C₂×C₄

D₄×D₉₇

yes

—

115

C₂×C₄

C₂×Dic₁₉₄

yes

characteristic

116

C₂×C₄

#13095510

yes

—

117

C₂×C₄

#13095511

yes

characteristic

118

C₈

—

388

—

119

C₂²

#15690778

yes

characteristic

120

C₂²

#15690789

yes

characteristic

121

C₂²

#15690797

yes

characteristic

122

C₂²

#15690830

yes

—

123

C₂²

#15690841

yes

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

388

—

128

C₄

#15690787

yes

—

129

C₄

#15690796

yes

—

130

C₂

#20260647

yes

characteristic

131

C₂

#20260895

yes

—

132

C₂

#20261042

yes

—

133

C₂

#20261089

yes

characteristic

134

C₂

#20261103

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₉₇
commutator, cosocle	C₂²×C₄×C₉₇	C₂³
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#391×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂×D₄
hypercenter	#391	D₉₇
hypocenter	C₉₇	#2026
LCS	G ▷ C₂²×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₉₇

commutator, cosocle

C₂²×C₄×C₉₇

C₂³

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

#391×C₉₇

C₂

socle, prime radical

C₂³×C₉₇

C₂×D₄

D₉₇

C₉₇

G ▷ C₂²×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

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×#15690802	2
C₂×#15690734	2
#20260497	2
#20260490	2
#391×C₉₇	2
#2026	97

subgroup

index

#391×C₉₇

Subgroups — #27142887

Conjugacy classes of subgroups

Complements

Maximal subgroups