Group #27142460 — subgroups

Subgroups — #27142460

Order 12416.

248 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 121 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×C₄⋊#11715895	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11715895	C₂	1	2	yes	maximal
3	C₂×#15690721	C₂	1	1	yes	maximal, characteristic
4	#20260490	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11715895	C₂²	1	2	yes	—
7	C₂²×#11715895	C₂²	1	1	yes	characteristic
8	C₂²×#11715895	C₂²	1	1	yes	characteristic
9	C₄⋊#11715895	—	2	4	no	—
10	C₄⋊#11715895	—	2	2	no	—
11	#15690721	—	2	2	no	—
12	C₂×C₄⋊C₄×C₉₇	C₄	1	2	yes	—
13	C₂×C₄²×C₉₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
15	#95×C₉₇	C₄	1	1	yes	characteristic
16	#95×C₉₇	C₄	1	1	yes	characteristic
17	C₂×#11715895	—	2	1	no	—
18	C₂×#11715895	—	2	1	no	—
19	C₂×#11715895	—	2	1	no	—
20	C₂×#11715895	—	2	1	no	—
21	C₂×#11715895	—	2	2	no	—
22	C₂×#11715895	—	2	2	no	—
23	C₂×#11715895	—	2	2	no	—
24	C₂×#11715895	D₄	1	2	yes	—
25	C₂×#11715895	Q₈	1	2	yes	—
26	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₉₇	—	2	4	no	—
32	C₄²×C₉₇	—	2	2	no	—
33	C₄²×C₉₇	—	2	2	no	—
34	C₄²×C₉₇	D₄	1	2	yes	—
35	C₄²×C₉₇	Q₈	1	2	yes	—
36	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₉₇	—	2	2	no	—
38	C₂×C₄×C₉₇	—	2	2	no	—
39	C₂×C₄×C₉₇	—	2	2	no	—
40	C₂×C₄×C₉₇	—	2	2	no	—
41	C₂×C₄×C₉₇	—	2	2	no	—
42	C₂×C₄×C₉₇	—	2	2	no	—
43	C₂×C₄×C₉₇	—	2	2	no	—
44	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
45	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₉₇	M₁₆	1	4	yes	—
47	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
52	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
53	#11715895	—	2	2	no	—
54	#11715895	—	2	2	no	—
55	#11715895	—	4	2	no	—
56	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₉₇	C₄×D₄	1	1	yes	characteristic
58	C₂²×C₉₇	C₄×Q₈	1	1	yes	characteristic
59	C₂²×C₉₇	C₂×M₁₆	1	2	yes	—
60	C₂²×C₉₇	#131	1	2	yes	—
61	C₄×C₉₇	—	2	2	no	—
62	C₄×C₉₇	—	2	2	no	—
63	C₄×C₉₇	—	2	4	no	—
64	C₄×C₉₇	—	4	2	no	—
65	C₄×C₉₇	#121	1	1	yes	characteristic
66	C₄×C₉₇	C₂²⋊Q₈	1	1	yes	characteristic
67	C₄×C₉₇	#125	1	1	yes	characteristic
68	C₄×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₉₇	#389	1	1	yes	characteristic
70	C₂×C₉₇	#423	1	1	yes	characteristic
71	C₂×C₉₇	#424	1	1	yes	characteristic
72	C₂×C₉₇	#435	1	2	yes	—
73	C₂×C₉₇	#446	1	2	yes	—
74	#1940	—	97	1	no	maximal, Sylow
75	C₉₇	#1940	1	1	yes	characteristic, complemented, Sylow
76	C₂×C₄⋊C₈	—	97	2	no	—
77	C₂×C₄⋊C₈	—	97	1	no	—
78	C₂×#97	—	97	1	no	—
79	C₄×C₄⋊C₄	D₉₇	1	1	yes	characteristic
80	#336	—	97	2	no	—
81	C₂²×C₈	—	97	1	no	—
82	C₂²×C₈	—	97	1	no	—
83	C₂²×C₈	—	97	2	no	—
84	C₂×C₄⋊C₄	Dic₉₇	1	2	yes	—
85	C₂×C₄²	C₂×D₉₇	1	2	yes	—
86	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
87	C₄⋊C₈	—	194	4	no	—
88	C₄⋊C₈	—	194	2	no	—
89	#97	—	194	2	no	—
90	#95	Dic₉₇	1	1	yes	characteristic
91	#95	Dic₉₇	1	1	yes	characteristic
92	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
93	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
94	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
95	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
96	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
97	C₂×C₈	—	97	2	no	—
98	C₂×C₈	—	97	2	no	—
99	C₂×C₈	—	194	2	no	—
100	C₂×C₈	—	194	2	no	—
101	C₂×C₈	—	194	2	no	—
102	C₂×C₈	—	194	1	no	—
103	C₂×C₈	—	194	1	no	—
104	C₂×C₈	—	194	1	no	—
105	C₂×C₈	—	194	1	no	—
106	C₄⋊C₄	—	2	4	no	—
107	C₄²	—	2	2	no	—
108	C₄²	—	2	2	no	—
109	C₄²	Dic₁₉₄	1	2	yes	—
110	C₄²	D₃₈₈	1	2	yes	—
111	C₂³	C₂²×Dic₉₇	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	C₄⋊Dic₉₇	1	2	yes	—
120	C₂×C₄	C₄⋊Dic₉₇	1	2	yes	—
121	C₂×C₄	#13095495	1	4	yes	—
122	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
123	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
124	C₂×C₄	Q₈×D₉₇	1	1	yes	characteristic
125	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
126	C₂×C₄	#13095510	1	1	yes	characteristic
127	C₂×C₄	#13095512	1	1	yes	characteristic
128	C₈	—	388	2	no	—
129	C₈	—	194	2	no	—
130	C₈	—	194	2	no	—
131	C₂²	C₂×C₄⋊Dic₉₇	1	1	yes	characteristic
132	C₂²	D₄×Dic₉₇	1	1	yes	characteristic
133	C₂²	Q₈×Dic₉₇	1	1	yes	characteristic
134	C₂²	C₂×#13095495	1	2	yes	—
135	C₂²	#15690817	1	2	yes	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	2	4	no	—
139	C₄	—	4	2	no	—
140	C₄	#15690785	1	1	yes	characteristic
141	C₄	#15690790	1	1	yes	characteristic
142	C₄	#15690801	1	1	yes	characteristic
143	C₄	#15690809	1	1	yes	characteristic
144	C₂	#20260634	1	1	yes	characteristic
145	C₂	#20260759	1	1	yes	characteristic
146	C₂	#20260762	1	1	yes	characteristic
147	C₂	#20260807	1	2	yes	—
148	C₂	#20260866	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11715895

C₂

yes

maximal, characteristic

C₂×C₄⋊#11715895

C₂

yes

maximal

C₂×#15690721

C₂

yes

maximal, characteristic

#20260490

C₂

yes

maximal

C₄×C₄⋊C₄×C₉₇

C₂

yes

maximal, characteristic

C₂²×#11715895

C₂²

yes

—

C₂²×#11715895

C₂²

yes

characteristic

C₂²×#11715895

C₂²

yes

—

—

—

C₂×C₄⋊C₄×C₉₇

C₄

yes

—

C₂×C₄²×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

#95×C₉₇

C₄

yes

characteristic

#95×C₉₇

C₄

yes

—

—

—

—

—

—

—

D₄

yes

—

C₂×#11715895

Q₈

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

—

C₄²×C₉₇

—

C₄²×C₉₇

D₄

yes

—

C₄²×C₉₇

Q₈

yes

—

C₂³×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₉₇

M₁₆

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

C₂×C₄×C₉₇

yes

C₂×C₄×C₉₇

yes

—

—

—

C₂²×C₉₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₉₇

C₄×D₄

yes

characteristic

C₂²×C₉₇

C₄×Q₈

yes

characteristic

C₂²×C₉₇

C₂×M₁₆

yes

—

C₂²×C₉₇

#131

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#121

yes

characteristic

C₄×C₉₇

C₂²⋊Q₈

yes

characteristic

C₄×C₉₇

#125

yes

characteristic

C₄×C₉₇

C₄⋊Q₈

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₂×C₉₇

#446

yes

—

#1940

—

maximal, Sylow

C₉₇

#1940

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₄×C₄⋊C₄

D₉₇

yes

characteristic

#336

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

Dic₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

194

—

C₄⋊C₈

—

194

—

#97

—

194

—

#95

Dic₉₇

yes

characteristic

#95

Dic₉₇

yes

characteristic

C₂²×C₄

C₂×Dic₉₇

yes

—

C₂²×C₄

C₂×Dic₉₇

yes

characteristic

C₂²×C₄

C₂×Dic₉₇

yes

characteristic

C₂²×C₄

C₂×Dic₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

194

—

100

C₂×C₈

—

194

—

101

C₂×C₈

—

194

—

102

C₂×C₈

—

194

—

103

C₂×C₈

—

194

—

104

C₂×C₈

—

194

—

105

C₂×C₈

—

194

—

106

C₄⋊C₄

—

107

C₄²

—

108

C₄²

—

109

C₄²

Dic₁₉₄

yes

—

110

C₄²

D₃₈₈

yes

—

111

C₂³

C₂²×Dic₉₇

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₄⋊Dic₉₇

yes

—

120

C₂×C₄

C₄⋊Dic₉₇

yes

—

121

C₂×C₄

#13095495

yes

—

122

C₂×C₄

D₄×D₉₇

yes

characteristic

123

C₂×C₄

C₂×D₃₈₈

yes

characteristic

124

C₂×C₄

Q₈×D₉₇

yes

characteristic

125

C₂×C₄

C₂×Dic₁₉₄

yes

characteristic

126

C₂×C₄

#13095510

yes

characteristic

127

C₂×C₄

#13095512

yes

characteristic

128

C₈

—

388

—

129

C₈

—

194

—

130

C₈

—

194

—

131

C₂²

C₂×C₄⋊Dic₉₇

yes

characteristic

132

C₂²

D₄×Dic₉₇

yes

characteristic

133

C₂²

Q₈×Dic₉₇

yes

characteristic

134

C₂²

C₂×#13095495

yes

—

135

C₂²

#15690817

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

#15690785

yes

characteristic

141

C₄

#15690790

yes

characteristic

142

C₄

#15690801

yes

characteristic

143

C₄

#15690809

yes

characteristic

144

C₂

#20260634

yes

characteristic

145

C₂

#20260759

yes

characteristic

146

C₂

#20260762

yes

characteristic

147

C₂

#20260807

yes

—

148

C₂

#20260866

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₉₇
commutator, socle, prime radical	C₂³×C₉₇	C₂²×C₄
Fitting	C₄×C₄⋊C₄×C₉₇	C₂
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₄×C₄⋊C₄	D₉₇
hypocenter	C₉₇	#1940
LCS	G ▷ C₂³×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₉₇

commutator, socle, prime radical

C₂³×C₉₇

C₂²×C₄

Fitting

C₄×C₄⋊C₄×C₉₇

C₂

cosocle

C₂²×C₄×C₉₇

C₂³

hypercenter

C₄×C₄⋊C₄

D₉₇

C₉₇

G ▷ C₂³×C₉₇ ▷ C₉₇

C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
75. C₉₇	#1940	semidirect

H ≅ G/N

type

75. C₉₇

#1940

semidirect

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

subgroup	index
C₂×C₄⋊#11715895	2
C₂×#15690721	2
#20260490	2
C₄×C₄⋊C₄×C₉₇	2
#1940	97

subgroup

index

C₄×C₄⋊C₄×C₉₇

Subgroups — #27142460

Conjugacy classes of subgroups

Complements

Maximal subgroups