Group #27141780 — subgroups

Subgroups — #27141780

Order 12416.

260 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 89 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×#15690817	C₂	1	1	yes	maximal, characteristic
2	#20260771	C₂	1	2	yes	maximal, complemented
3	#20260709	C₂	1	1	yes	maximal, characteristic
4	#20260491	C₂	1	2	yes	maximal, complemented
5	#410×C₉₇	C₂	1	1	yes	maximal, characteristic
6	#15690817	—	2	2	no	—
7	#15690817	—	2	2	no	—
8	C₂×#13095495	C₄	1	2	yes	complemented
9	C₂×#13095495	C₂²	1	1	yes	characteristic
10	C₂²×#11715895	C₄	1	2	yes	complemented
11	C₂²×#11715895	C₂²	1	1	yes	characteristic
12	#15690769	C₂²	1	2	yes	—
13	#15690743	—	2	2	no	—
14	#15690741	—	2	2	no	—
15	#15690725	—	2	2	no	—
16	C₂×#41×C₉₇	C₂²	1	1	yes	characteristic
17	#117×C₉₇	C₂²	1	2	yes	—
18	#99×C₉₇	—	2	2	no	—
19	#13095495	—	2	2	no	—
20	#13095495	—	2	4	no	—
21	C₂×#11715895	—	2	2	no	—
22	C₂×#11715895	—	2	2	no	—
23	C₂×#11715895	C₂×C₄	1	2	yes	—
24	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
25	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
26	C₂×#11715895	D₄	1	2	yes	—
27	C₂×#11715895	D₄	1	2	yes	—
28	C₄⋊Dic₉₇	—	2	2	no	—
29	C₂²⋊Dic₉₇	—	2	2	no	—
30	C₄×Dic₉₇	—	2	2	no	—
31	#41×C₉₇	—	2	2	no	—
32	#41×C₉₇	—	2	2	no	—
33	C₂×Q₈×C₉₇	Q₈	1	1	yes	characteristic
34	C₂×D₄×C₉₇	D₄	1	2	yes	—
35	C₂×D₄×C₉₇	Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₉₇	—	2	2	no	—
39	C₂²⋊C₄×C₉₇	—	2	2	no	—
40	C₄²×C₉₇	—	2	2	no	—
41	C₂³×C₉₇	C₄⋊C₄	1	2	yes	—
42	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₉₇	—	2	2	no	—
44	D₄×C₉₇	—	2	4	no	—
45	D₄×C₉₇	—	2	2	no	—
46	C₂×C₄×C₉₇	—	2	2	no	—
47	C₂×C₄×C₉₇	—	2	2	no	—
48	C₂×C₄×C₉₇	—	4	2	no	—
49	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
50	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₉₇	#41	1	2	yes	—
54	C₂×Dic₉₇	—	4	2	no	—
55	#11715895	—	2	2	no	—
56	#11715895	C₂²⋊C₄	1	1	yes	characteristic
57	#11715895	C₂²⋊C₄	1	1	yes	characteristic
58	#11715895	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₉₇	—	2	2	no	—
60	C₂²×C₉₇	—	4	1	no	—
61	C₂²×C₉₇	—	4	2	no	—
62	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₉₇	#121	1	2	yes	—
64	C₄×C₉₇	—	2	2	no	—
65	C₄×C₉₇	—	8	2	no	—
66	C₄×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₉₇	#117	1	1	yes	characteristic
68	C₄×C₉₇	C₂²⋊Q₈	1	2	yes	—
69	Dic₉₇	—	8	2	no	—
70	C₂×C₉₇	—	2	2	no	—
71	C₂×C₉₇	—	2	1	no	—
72	C₂×C₉₇	—	4	2	no	—
73	C₂×C₉₇	#380	1	1	yes	characteristic
74	#1812	—	97	1	no	maximal, Sylow
75	C₉₇	#1812	1	1	yes	characteristic, complemented, Sylow
76	C₂×#131	—	97	1	no	—
77	#427	—	97	2	no	—
78	#410	—	97	1	no	—
79	#410	D₉₇	1	1	yes	characteristic
80	H(3,ℤ/4)	—	97	2	no	—
81	C₂×#41	C₂×D₉₇	1	1	yes	characteristic
82	#131	—	194	2	no	—
83	#131	—	194	2	no	—
84	C₂×M₁₆	—	97	2	no	—
85	C₂×M₁₆	—	97	1	no	—
86	C₂²×C₈	—	97	1	no	—
87	C₂²×C₈	—	97	2	no	—
88	#117	—	97	2	no	—
89	#117	C₂×D₉₇	1	2	yes	—
90	#107	—	194	2	no	—
91	#106	—	194	2	no	—
92	#99	—	2	2	no	—
93	#99	—	194	2	no	—
94	#41	—	2	2	no	—
95	#41	—	2	2	no	—
96	C₂×Q₈	Dic₁₉₄	1	1	yes	characteristic
97	C₂×D₄	Dic₁₉₄	1	1	yes	characteristic
98	C₂×D₄	C₂²⋊D₉₇	1	2	yes	—
99	C₂²×C₄	C₄×D₉₇	1	2	yes	—
100	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
101	M₁₆	—	194	4	no	—
102	M₁₆	—	194	2	no	—
103	C₂×C₈	—	97	1	no	—
104	C₂×C₈	—	97	1	no	—
105	C₂×C₈	—	97	2	no	—
106	C₂×C₈	—	97	2	no	—
107	C₂×C₈	—	97	2	no	—
108	C₂×C₈	—	194	2	no	—
109	C₂×C₈	—	194	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	194	2	no	—
112	C₂²⋊C₄	—	2	2	no	—
113	C₂²⋊C₄	—	194	2	no	—
114	C₄²	—	2	2	no	—
115	C₄²	—	194	2	no	—
116	C₂³	#13095490	1	2	yes	—
117	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
118	Q₈	—	2	2	no	—
119	D₄	—	2	4	no	—
120	D₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	388	2	no	—
125	C₂×C₄	#13095490	1	2	yes	—
126	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
127	C₂×C₄	D₄×D₉₇	1	2	yes	—
128	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
129	C₂×C₄	#13095510	1	2	yes	—
130	C₈	—	97	1	no	—
131	C₈	—	97	1	no	—
132	C₈	—	97	2	no	—
133	C₈	—	194	2	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	4	1	no	—
136	C₂²	—	4	2	no	—
137	C₂²	C₂×#13095490	1	1	yes	characteristic
138	C₂²	#15690782	1	2	yes	—
139	C₄	—	2	2	no	—
140	C₄	—	8	2	no	—
141	C₄	—	776	2	no	—
142	C₄	C₂²⋊C₄×D₉₇	1	1	yes	characteristic
143	C₄	#15690766	1	1	yes	characteristic
144	C₄	#15690787	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	2	1	no	—
147	C₂	—	4	2	no	—
148	C₂	#20260593	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690817

C₂

yes

maximal, characteristic

#20260771

C₂

yes

maximal, complemented

#20260709

C₂

yes

maximal, characteristic

#20260491

C₂

yes

maximal, complemented

#410×C₉₇

C₂

yes

maximal, characteristic

#15690817

—

#15690817

—

C₂×#13095495

C₄

yes

complemented

C₂×#13095495

C₂²

yes

characteristic

C₂²×#11715895

C₄

yes

complemented

C₂²×#11715895

C₂²

yes

characteristic

#15690769

C₂²

yes

—

#15690743

—

#15690741

—

#15690725

—

C₂×#41×C₉₇

C₂²

yes

characteristic

#117×C₉₇

C₂²

yes

—

#99×C₉₇

—

—

—

—

—

C₂×C₄

yes

—

C₂×#11715895

C₂×C₄

yes

characteristic

C₂×#11715895

C₂×C₄

yes

characteristic

C₂×#11715895

D₄

yes

—

C₂×#11715895

D₄

yes

—

C₄⋊Dic₉₇

—

C₂²⋊Dic₉₇

—

C₄×Dic₉₇

—

#41×C₉₇

—

#41×C₉₇

—

C₂×Q₈×C₉₇

Q₈

yes

characteristic

C₂×D₄×C₉₇

D₄

yes

—

C₂×D₄×C₉₇

Q₈

yes

characteristic

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₂²⋊C₄×C₉₇

—

C₄²×C₉₇

—

C₂³×C₉₇

C₄⋊C₄

yes

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

Q₈×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂×Dic₉₇

—

#11715895

—

#11715895

C₂²⋊C₄

yes

characteristic

#11715895

C₂²⋊C₄

yes

characteristic

#11715895

C₂²⋊C₄

yes

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₉₇

#121

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₉₇

#117

yes

characteristic

C₄×C₉₇

C₂²⋊Q₈

yes

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

—

—

—

D₉₇

yes

characteristic

H(3,ℤ/4)

—

C₂×#41

C₂×D₉₇

yes

characteristic

#131

—

194

—

#131

—

194

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

#117

—

#117

C₂×D₉₇

yes

—

#107

—

194

—

#106

—

194

—

#99

—

#99

—

194

—

#41

—

#41

—

C₂×Q₈

Dic₁₉₄

yes

characteristic

C₂×D₄

Dic₁₉₄

yes

characteristic

C₂×D₄

C₂²⋊D₉₇

yes

—

C₂²×C₄

C₄×D₉₇

yes

—

100

C₂²×C₄

C₂²×D₉₇

yes

characteristic

101

M₁₆

—

194

—

102

M₁₆

—

194

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

194

—

109

C₂×C₈

—

194

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

194

—

112

C₂²⋊C₄

—

113

C₂²⋊C₄

—

194

—

114

C₄²

—

115

C₄²

—

194

—

116

C₂³

#13095490

yes

—

117

C₂³

C₂×C₂²⋊D₉₇

yes

characteristic

118

Q₈

—

119

D₄

—

120

D₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

388

—

125

C₂×C₄

#13095490

yes

—

126

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

127

C₂×C₄

D₄×D₉₇

yes

—

128

C₂×C₄

C₂×Dic₁₉₄

yes

characteristic

129

C₂×C₄

#13095510

yes

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

194

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂×#13095490

yes

characteristic

138

C₂²

#15690782

yes

—

139

C₄

—

140

C₄

—

141

C₄

—

776

—

142

C₄

C₂²⋊C₄×D₉₇

yes

characteristic

143

C₄

#15690766

yes

characteristic

144

C₄

#15690787

yes

—

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

#20260593

yes

characteristic

	subgroup	G/H
center	C₄	#15690766
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#410×C₉₇	C₂
socle	C₂×C₉₇	#380
prime radical	C₂×#41×C₉₇	C₂²
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#410	D₉₇
hypocenter	C₉₇	#1812
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

subgroup

G/H

center

C₄

#15690766

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

#410×C₉₇

C₂

socle

C₂×C₉₇

#380

prime radical

C₂×#41×C₉₇

C₂²

cosocle

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

N	H ≅ G/N	type
2. #20260771	C₂	semidirect
4. #20260491	C₂	semidirect
8. C₂×#13095495	C₄	semidirect
10. C₂²×#11715895	C₄	semidirect
75. C₉₇	#1812	semidirect

H ≅ G/N

type

2. #20260771

C₂

semidirect

4. #20260491

C₂

semidirect

8. C₂×#13095495

C₄

semidirect

10. C₂²×#11715895

C₄

semidirect

75. C₉₇

#1812

semidirect

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

subgroup	index
C₂×#15690817	2
#20260771	2
#20260709	2
#20260491	2
#410×C₉₇	2
#1812	97

subgroup

index

#410×C₉₇

Subgroups — #27141780

Conjugacy classes of subgroups

Complements

Maximal subgroups