Group #27141785 — subgroups

Subgroups — #27141785

Order 12416.

252 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 81 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×#15690818	C₂	1	1	yes	maximal, characteristic
2	C₂×#15690745	C₂	1	2	yes	maximal, complemented
3	#20260717	C₂	1	2	yes	maximal
4	#20260493	C₂	1	1	yes	maximal, characteristic, complemented
5	#338×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13095511	C₂²	1	1	yes	characteristic
7	#15690818	—	2	4	no	—
8	C₂×#13095495	C₂²	1	2	yes	—
9	C₂×#13095494	C₄	1	2	yes	—
10	C₂²×#11715895	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₉₇	C₄	1	2	yes	—
12	#15690745	—	2	4	no	—
13	#15690728	—	2	2	no	—
14	#15690726	—	2	2	no	—
15	C₂×M₁₆×C₉₇	C₂²	1	2	yes	—
16	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
17	#13095511	—	2	2	no	—
18	#13095511	—	2	2	no	—
19	C₂×Dic₁₉₄	D₄	1	1	yes	characteristic
20	C₂×D₃₈₈	D₄	1	1	yes	characteristic
21	C₂×C₂²⋊D₉₇	Q₈	1	2	yes	—
22	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	—
23	#13095495	—	2	2	no	—
24	#13095495	—	2	2	no	—
25	#13095494	—	2	4	no	—
26	C₂×#11715895	—	2	2	no	—
27	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
28	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
29	C₂×#11715895	D₄	1	4	yes	—
30	C₈×D₉₇	—	2	4	no	—
31	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
32	M₁₆×C₉₇	—	2	2	no	—
33	M₁₆×C₉₇	—	2	2	no	—
34	M₁₆×C₉₇	—	2	2	no	—
35	C₂×C₈×C₉₇	—	2	2	no	—
36	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
37	C₂³×C₉₇	C₂×Q₈	1	1	yes	characteristic
38	C₂²×D₉₇	C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₉₇	#41	1	2	yes	—
43	C₂²⋊D₉₇	—	2	4	no	—
44	C₂×Dic₉₇	C₄⋊C₄	1	2	yes	—
45	D₃₈₈	—	2	2	no	—
46	C₄×D₉₇	—	2	2	no	—
47	C₄×D₉₇	—	2	2	no	—
48	Dic₁₉₄	—	2	2	no	—
49	C₈×C₉₇	—	2	2	no	—
50	C₈×C₉₇	—	4	2	no	—
51	#11715895	—	4	2	no	—
52	#11715895	C₂²⋊C₄	1	2	yes	—
53	#11715895	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₉₇	—	4	1	no	—
55	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₉₇	C₂²⋊Q₈	1	2	yes	—
57	C₂×D₉₇	—	2	2	no	—
58	C₂×D₉₇	—	4	2	no	—
59	C₄×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₉₇	#117	1	1	yes	characteristic
61	C₄×C₉₇	#121	1	2	yes	—
62	Dic₉₇	—	2	2	no	—
63	C₂×C₉₇	—	2	1	no	—
64	C₂×C₉₇	—	2	2	no	—
65	C₂×C₉₇	#380	1	1	yes	characteristic
66	D₉₇	—	4	2	no	—
67	#1813	—	97	1	no	maximal, Sylow
68	C₉₇	#1813	1	1	yes	characteristic, complemented, Sylow
69	C₂×#131	—	97	1	no	—
70	C₂×#108	—	97	2	no	—
71	#413	—	97	2	no	—
72	#338	—	97	1	no	—
73	#338	D₉₇	1	1	yes	characteristic, complemented
74	C₂×#41	—	97	1	no	—
75	#131	—	194	4	no	—
76	C₂×M₁₆	—	97	2	no	—
77	C₂×M₁₆	—	97	2	no	—
78	C₂×M₁₆	C₂×D₉₇	1	2	yes	—
79	C₂×M₁₆	C₂×D₉₇	1	1	yes	characteristic
80	C₂²×C₈	—	97	1	no	—
81	C₂²×C₈	—	97	2	no	—
82	#108	—	194	4	no	—
83	#101	—	194	2	no	—
84	#100	—	194	2	no	—
85	#41	—	194	2	no	—
86	#41	—	194	2	no	—
87	C₂×Q₈	—	97	1	no	—
88	C₂×D₄	—	97	2	no	—
89	C₂×D₄	—	97	1	no	—
90	C₂²×C₄	—	97	2	no	—
91	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
92	M₁₆	—	2	2	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	2	no	—
95	M₁₆	—	194	4	no	—
96	M₁₆	—	194	2	no	—
97	M₁₆	—	194	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	97	1	no	—
100	C₂×C₈	—	97	1	no	—
101	C₂×C₈	—	97	4	no	—
102	C₂×C₈	—	194	4	no	—
103	C₂×C₈	—	194	2	no	—
104	C₂×C₈	C₄×D₉₇	1	2	yes	—
105	C₂³	—	97	2	no	—
106	C₂³	Q₈×D₉₇	1	1	yes	characteristic
107	Q₈	—	194	2	no	—
108	D₄	—	194	4	no	—
109	D₄	—	194	2	no	—
110	C₂×C₄	—	97	2	no	—
111	C₂×C₄	—	194	2	no	—
112	C₂×C₄	—	194	2	no	—
113	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
114	C₂×C₄	D₄×D₉₇	1	2	yes	—
115	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
116	C₂×C₄	#13095511	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	4	2	no	—
119	C₈	—	97	2	no	—
120	C₈	—	97	2	no	—
121	C₈	—	388	2	no	—
122	C₂²	—	4	1	no	—
123	C₂²	—	388	2	no	—
124	C₂²	—	194	2	no	—
125	C₂²	C₄⋊C₄×D₉₇	1	1	yes	characteristic
126	C₂²	#15690788	1	2	yes	—
127	C₄	—	194	2	no	—
128	C₄	C₂²⋊C₄×D₉₇	1	1	yes	characteristic
129	C₄	#15690767	1	1	yes	characteristic
130	C₄	#15690783	1	2	yes	—
131	C₂	—	2	1	no	—
132	C₂	—	2	2	no	—
133	C₂	—	388	2	no	—
134	C₂	#20260594	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690818

C₂

yes

maximal, characteristic

C₂×#15690745

C₂

yes

maximal, complemented

#20260717

C₂

yes

maximal

#20260493

C₂

yes

maximal, characteristic, complemented

#338×C₉₇

C₂

yes

maximal, characteristic, complemented

C₂×#13095511

C₂²

yes

characteristic

#15690818

—

C₂×#13095495

C₂²

yes

—

C₂×#13095494

C₄

yes

—

C₂²×#11715895

C₂²

yes

characteristic

C₂×C₈×D₉₇

C₄

yes

—

#15690745

—

#15690728

—

#15690726

—

C₂×M₁₆×C₉₇

C₂²

yes

—

C₂×M₁₆×C₉₇

C₂²

yes

characteristic

#13095511

—

#13095511

—

C₂×Dic₁₉₄

D₄

yes

characteristic

C₂×D₃₈₈

D₄

yes

characteristic

C₂×C₂²⋊D₉₇

Q₈

yes

—

C₂×C₄×D₉₇

C₂×C₄

yes

—

—

—

—

—

C₂×C₄

yes

characteristic

C₂×#11715895

C₂×C₄

yes

characteristic

C₂×#11715895

D₄

yes

—

C₈×D₉₇

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

M₁₆×C₉₇

—

M₁₆×C₉₇

—

M₁₆×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

C₂×C₄

yes

—

C₂³×C₉₇

C₂×Q₈

yes

characteristic

C₂²×D₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂²⋊D₉₇

—

C₂×Dic₉₇

C₄⋊C₄

yes

—

D₃₈₈

—

C₄×D₉₇

—

C₄×D₉₇

—

Dic₁₉₄

—

C₈×C₉₇

—

C₈×C₉₇

—

#11715895

—

#11715895

C₂²⋊C₄

yes

—

#11715895

C₂²⋊C₄

yes

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₉₇

C₂²⋊Q₈

yes

—

C₂×D₉₇

—

C₂×D₉₇

—

C₄×C₉₇

C₂×C₂²⋊C₄

yes

C₄×C₉₇

yes

C₄×C₉₇

yes

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#380

yes

characteristic

D₉₇

—

#1813

—

maximal, Sylow

C₉₇

#1813

yes

characteristic, complemented, Sylow

—

—

—

—

D₉₇

yes

characteristic, complemented

C₂×#41

—

#131

—

194

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₉₇

yes

—

C₂×M₁₆

C₂×D₉₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

#108

—

194

—

#101

—

194

—

#100

—

194

—

#41

—

194

—

#41

—

194

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

194

—

M₁₆

—

194

—

M₁₆

—

194

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

194

—

103

C₂×C₈

—

194

—

104

C₂×C₈

C₄×D₉₇

yes

—

105

C₂³

—

106

C₂³

Q₈×D₉₇

yes

characteristic

107

Q₈

—

194

—

108

D₄

—

194

—

109

D₄

—

194

—

110

C₂×C₄

—

111

C₂×C₄

—

194

—

112

C₂×C₄

—

194

—

113

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

114

C₂×C₄

D₄×D₉₇

yes

—

115

C₂×C₄

D₄×D₉₇

yes

characteristic

116

C₂×C₄

#13095511

yes

—

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

—

388

—

122

C₂²

—

123

C₂²

—

388

—

124

C₂²

—

194

—

125

C₂²

C₄⋊C₄×D₉₇

yes

characteristic

126

C₂²

#15690788

yes

—

127

C₄

—

194

—

128

C₄

C₂²⋊C₄×D₉₇

yes

characteristic

129

C₄

#15690767

yes

characteristic

130

C₄

#15690783

yes

—

131

C₂

—

132

C₂

—

133

C₂

—

388

—

134

C₂

#20260594

yes

characteristic

	subgroup	G/H
center	C₄	#15690767
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#338×C₉₇	C₂
socle	C₂×C₉₇	#380
prime radical	C₂×#13095511	C₂²
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#338	D₉₇
hypocenter	C₉₇	#1813
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#15690767

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

#338×C₉₇

C₂

C₂×C₉₇

C₂²

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
2. C₂×#15690745	C₂	semidirect
4. #20260493	C₂	semidirect
5. #338×C₉₇	C₂	semidirect
68. C₉₇	#1813	semidirect
73. #338	D₉₇	semidirect

H ≅ G/N

type

2. C₂×#15690745

C₂

semidirect

4. #20260493

C₂

semidirect

5. #338×C₉₇

C₂

semidirect

68. C₉₇

#1813

semidirect

73. #338

D₉₇

semidirect

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

subgroup	index
C₂×#15690818	2
C₂×#15690745	2
#20260717	2
#20260493	2
#338×C₉₇	2
#1813	97

subgroup

index

#338×C₉₇

Subgroups — #27141785

Conjugacy classes of subgroups

Complements

Maximal subgroups