Group #27196701 — subgroups

Subgroups — #27196701

Order 13184.

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

normal subgroups only

This group has 214 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	#20267689	C₂	1	1	yes	maximal, characteristic
2	#20267553	C₂	1	2	yes	maximal
3	#20267553	C₂	1	2	yes	maximal
4	#20267546	C₂	1	1	yes	maximal, characteristic
5	#384×C₁₀₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₀₃⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₀₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₀₃	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₀₃⋊C₄	—	2	1	no	—
10	C₁₀₃⋊#95	—	2	1	no	—
11	C₂×C₄⋊C₄×C₁₀₃	—	2	1	no	—
12	C₂×C₄⋊C₄×C₁₀₃	—	2	1	no	—
13	C₂×C₄⋊C₄×C₁₀₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₀₃	—	2	1	no	—
16	C₂²×Dic₁₀₃	—	2	1	no	—
17	C₂×C₁₀₃⋊C₈	—	2	2	no	—
18	C₂×C₁₀₃⋊C₈	—	2	2	no	—
19	C₂×C₁₀₃⋊C₈	—	2	2	no	—
20	C₄⋊Dic₁₀₃	—	2	1	no	—
21	C₄⋊Dic₁₀₃	—	2	2	no	—
22	C₄⋊Dic₁₀₃	Q₈	1	2	yes	—
23	Dic₁₀₃⋊C₄	—	4	2	no	—
24	C₂²×C₄×C₁₀₃	—	2	2	no	—
25	C₂²×C₄×C₁₀₃	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₀₃	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₁₀₃	—	2	4	no	—
29	C₄⋊C₄×C₁₀₃	—	4	2	no	—
30	C₄⋊C₄×C₁₀₃	—	4	2	no	—
31	C₄²×C₁₀₃	—	2	2	no	—
32	C₂³×C₁₀₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₃	—	2	1	no	—
34	C₂×C₄×C₁₀₃	—	2	1	no	—
35	C₂×C₄×C₁₀₃	—	2	1	no	—
36	C₂×C₄×C₁₀₃	—	2	1	no	—
37	C₂×C₄×C₁₀₃	—	4	2	no	—
38	C₂×C₄×C₁₀₃	—	4	2	no	—
39	C₂×C₄×C₁₀₃	—	4	2	no	—
40	C₂×C₄×C₁₀₃	D₈	1	2	yes	—
41	C₂×C₄×C₁₀₃	SD₁₆	1	2	yes	—
42	C₂×C₄×C₁₀₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₁₀₃	#41	1	2	yes	—
44	C₂×C₄×C₁₀₃	#41	1	2	yes	—
45	C₂×C₄×C₁₀₃	#41	1	1	yes	characteristic
46	C₂×Dic₁₀₃	—	2	2	no	—
47	C₂×Dic₁₀₃	—	4	1	no	—
48	C₂×Dic₁₀₃	—	4	1	no	—
49	C₂×Dic₁₀₃	—	4	1	no	—
50	C₂×Dic₁₀₃	—	4	1	no	—
51	C₂×Dic₁₀₃	—	4	1	no	—
52	C₁₀₃⋊C₈	—	4	2	no	—
53	C₂²×C₁₀₃	C₂²⋊Q₈	1	1	yes	characteristic
54	C₂²×C₁₀₃	#123	1	1	yes	characteristic
55	C₂²×C₁₀₃	#124	1	1	yes	characteristic
56	C₂²×C₁₀₃	C₂×D₈	1	1	yes	characteristic
57	C₂²×C₁₀₃	C₂×SD₁₆	1	1	yes	characteristic
58	C₂²×C₁₀₃	#137	1	1	yes	characteristic
59	C₂²×C₁₀₃	#137	1	1	yes	characteristic
60	C₄×C₁₀₃	—	2	2	no	—
61	C₄×C₁₀₃	—	2	2	no	—
62	C₄×C₁₀₃	—	8	2	no	—
63	C₄×C₁₀₃	#125	1	2	yes	—
64	C₄×C₁₀₃	#126	1	2	yes	—
65	Dic₁₀₃	—	4	2	no	—
66	Dic₁₀₃	—	8	1	no	—
67	C₂×C₁₀₃	#400	1	1	yes	characteristic
68	C₂×C₁₀₃	#474	1	1	yes	characteristic
69	C₂×C₁₀₃	#475	1	1	yes	characteristic
70	C₂×C₁₀₃	#480	1	1	yes	characteristic
71	C₂×C₁₀₃	#483	1	1	yes	characteristic
72	C₂×C₁₀₃	#486	1	1	yes	characteristic
73	C₂×C₁₀₃	#491	1	1	yes	characteristic
74	#2099	—	103	1	no	maximal, Sylow
75	C₁₀₃	#2099	1	1	yes	characteristic, complemented, Sylow
76	#389	—	103	1	no	—
77	#384	D₁₀₃	1	1	yes	characteristic
78	#340	—	103	2	no	—
79	#340	—	103	2	no	—
80	#336	—	103	1	no	—
81	C₂²×C₈	—	103	2	no	—
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄⋊C₄	—	103	1	no	—
85	C₂×C₄⋊C₄	—	103	1	no	—
86	C₂×C₄⋊C₄	—	206	1	no	—
87	C₂×C₄⋊C₄	C₂×D₁₀₃	1	2	yes	—
88	C₂×C₄²	C₂×D₁₀₃	1	1	yes	characteristic
89	#95	—	206	1	no	—
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	—	206	1	no	—
92	C₂²×C₄	—	206	1	no	—
93	C₂²×C₄	C₂²⋊D₁₀₃	1	1	yes	characteristic
94	C₂²×C₄	C₂²⋊D₁₀₃	1	1	yes	characteristic
95	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
96	C₂×C₈	—	206	2	no	—
97	C₂×C₈	—	206	2	no	—
98	C₂×C₈	—	206	2	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	4	2	no	—
102	C₄⋊C₄	—	103	2	no	—
103	C₄⋊C₄	—	412	2	no	—
104	C₄⋊C₄	—	206	2	no	—
105	C₄⋊C₄	—	206	1	no	—
106	C₄²	—	2	2	no	—
107	C₂³	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	412	1	no	—
116	C₂×C₄	—	412	1	no	—
117	C₂×C₄	—	412	1	no	—
118	C₂×C₄	—	412	1	no	—
119	C₂×C₄	—	412	1	no	—
120	C₂×C₄	—	206	2	no	—
121	C₂×C₄	#13110090	1	2	yes	—
122	C₂×C₄	#13110093	1	2	yes	—
123	C₂×C₄	Q₈×D₁₀₃	1	1	yes	characteristic
124	C₂×C₄	#13110104	1	2	yes	—
125	C₂×C₄	#13110105	1	2	yes	—
126	C₂×C₄	#13110106	1	1	yes	characteristic
127	C₈	—	412	2	no	—
128	C₂²	#15692997	1	1	yes	characteristic
129	C₂²	#15693004	1	1	yes	characteristic
130	C₂²	#15693008	1	1	yes	characteristic
131	C₂²	C₂×#13110090	1	1	yes	characteristic
132	C₂²	C₂×#13110093	1	1	yes	characteristic
133	C₂²	#15693052	1	1	yes	characteristic
134	C₂²	#15693052	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	2	2	no	—
137	C₄	—	8	2	no	—
138	C₄	—	824	1	no	—
139	C₄	—	412	2	no	—
140	C₄	#15693011	1	2	yes	—
141	C₄	#15693016	1	2	yes	—
142	C₂	#20267729	1	1	yes	characteristic
143	C₂	#20268068	1	1	yes	characteristic
144	C₂	#20268073	1	1	yes	characteristic
145	C₂	#20268104	1	1	yes	characteristic
146	C₂	#20268121	1	1	yes	characteristic
147	C₂	#20268137	1	1	yes	characteristic
148	C₂	#20268159	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20267689

C₂

yes

maximal, characteristic

#20267553

C₂

yes

maximal

#20267553

C₂

yes

maximal

#20267546

C₂

yes

maximal, characteristic

#384×C₁₀₃

C₂

yes

maximal, characteristic

C₂²×C₁₀₃⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₀₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₀₃

C₂²

yes

characteristic

C₂×Dic₁₀₃⋊C₄

—

C₁₀₃⋊#95

—

C₂×C₄⋊C₄×C₁₀₃

—

C₂×C₄⋊C₄×C₁₀₃

—

C₂×C₄⋊C₄×C₁₀₃

C₂²

yes

—

C₂×C₄²×C₁₀₃

C₂²

yes

characteristic

C₂²×Dic₁₀₃

—

C₂²×Dic₁₀₃

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

Q₈

yes

—

Dic₁₀₃⋊C₄

—

C₂²×C₄×C₁₀₃

—

C₂²×C₄×C₁₀₃

D₄

yes

characteristic

C₂²×C₄×C₁₀₃

D₄

yes

characteristic

C₂²×C₄×C₁₀₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₃

—

C₄⋊C₄×C₁₀₃

—

C₄⋊C₄×C₁₀₃

—

C₄²×C₁₀₃

—

C₂³×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

D₈

yes

—

C₂×C₄×C₁₀₃

SD₁₆

yes

—

C₂×C₄×C₁₀₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₃

#41

yes

—

C₂×C₄×C₁₀₃

#41

yes

—

C₂×C₄×C₁₀₃

#41

yes

characteristic

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

C₂²⋊Q₈

yes

C₂²×C₁₀₃

yes

C₂²×C₁₀₃

yes

C₂²×C₁₀₃

C₂×D₈

yes

characteristic

C₂²×C₁₀₃

C₂×SD₁₆

yes

C₂²×C₁₀₃

yes

C₂²×C₁₀₃

yes

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

#125

yes

—

C₄×C₁₀₃

#126

yes

—

Dic₁₀₃

—

Dic₁₀₃

—

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

—

103

maximal, Sylow

C₁₀₃

#2099

yes

characteristic, complemented, Sylow

#389

—

103

—

#384

D₁₀₃

yes

characteristic

#340

—

103

—

#340

—

103

—

#336

—

103

—

C₂²×C₈

—

103

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

103

—

C₂×C₄⋊C₄

—

103

—

C₂×C₄⋊C₄

—

206

—

C₂×C₄⋊C₄

C₂×D₁₀₃

yes

—

C₂×C₄²

C₂×D₁₀₃

yes

characteristic

#95

—

206

—

C₂²×C₄

—

C₂²×C₄

—

206

—

C₂²×C₄

—

206

—

C₂²×C₄

C₂²⋊D₁₀₃

yes

characteristic

C₂²×C₄

C₂²⋊D₁₀₃

yes

characteristic

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

C₂×C₈

—

206

—

C₂×C₈

—

206

—

C₂×C₈

—

206

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

—

103

C₄⋊C₄

—

412

—

104

C₄⋊C₄

—

206

—

105

C₄⋊C₄

—

206

—

106

C₄²

—

107

C₂³

C₂×C₂²⋊D₁₀₃

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

412

—

116

C₂×C₄

—

412

—

117

C₂×C₄

—

412

—

118

C₂×C₄

—

412

—

119

C₂×C₄

—

412

—

120

C₂×C₄

—

206

—

121

C₂×C₄

#13110090

yes

—

122

C₂×C₄

#13110093

yes

—

123

C₂×C₄

Q₈×D₁₀₃

yes

characteristic

124

C₂×C₄

#13110104

yes

—

125

C₂×C₄

#13110105

yes

—

126

C₂×C₄

#13110106

yes

characteristic

127

C₈

—

412

—

128

C₂²

#15692997

yes

characteristic

129

C₂²

#15693004

yes

characteristic

130

C₂²

#15693008

yes

characteristic

131

C₂²

C₂×#13110090

yes

characteristic

132

C₂²

C₂×#13110093

yes

characteristic

133

C₂²

#15693052

yes

characteristic

134

C₂²

#15693052

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

824

—

139

C₄

—

412

—

140

C₄

#15693011

yes

—

141

C₄

#15693016

yes

—

142

C₂

#20267729

yes

characteristic

143

C₂

#20268068

yes

characteristic

144

C₂

#20268073

yes

characteristic

145

C₂

#20268104

yes

characteristic

146

C₂

#20268121

yes

characteristic

147

C₂

#20268137

yes

characteristic

148

C₂

#20268159

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₃
commutator, cosocle	C₂²×C₄×C₁₀₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₃
Fitting	#384×C₁₀₃	C₂
socle, prime radical	C₂³×C₁₀₃	C₂×D₄
hypercenter	#384	D₁₀₃
hypocenter	C₁₀₃	#2099
LCS	G ▷ C₂²×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₃

commutator, cosocle

C₂²×C₄×C₁₀₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₃

Fitting

#384×C₁₀₃

C₂

socle, prime radical

C₂³×C₁₀₃

C₂×D₄

D₁₀₃

C₁₀₃

G ▷ C₂²×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
75. C₁₀₃	#2099	semidirect

H ≅ G/N

type

75. C₁₀₃

#2099

semidirect

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

subgroup	index
#20267689	2
#20267553	2
#20267546	2
#384×C₁₀₃	2
#2099	103

subgroup

index

#384×C₁₀₃

103

Subgroups — #27196701

Conjugacy classes of subgroups

Complements

Maximal subgroups