Group #27167178 — subgroups

Subgroups — C₁₀₁×#1824 #27167178

Order 12928.

344 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 136 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	C₂²×C₄⋊C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊C₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic
3	#380×C₁₀₁	C₂	1	1	yes	maximal, characteristic
4	#340×C₁₀₁	C₂	1	4	yes	maximal, complemented
5	C₂³×C₄×C₁₀₁	—	2	1	no	—
6	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₁₀₁	C₂²	1	2	yes	—
8	C₂×C₄⋊C₄×C₁₀₁	—	2	1	no	—
9	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
10	C₂×C₄⋊C₄×C₁₀₁	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
13	C₂×C₂²⋊C₄×C₁₀₁	—	2	1	no	—
14	C₂²⋊C₈×C₁₀₁	—	2	2	no	—
15	#95×C₁₀₁	—	2	1	no	—
16	C₂⁴×C₁₀₁	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₁₀₁	—	2	4	no	—
18	C₂²×C₄×C₁₀₁	—	2	4	no	—
19	C₂²×C₄×C₁₀₁	—	2	2	no	—
20	C₂²×C₄×C₁₀₁	—	2	2	no	—
21	C₂²×C₄×C₁₀₁	—	4	4	no	—
22	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
25	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₁₀₁	—	2	2	no	—
27	C₂×C₈×C₁₀₁	—	2	4	no	—
28	C₄⋊C₄×C₁₀₁	—	2	4	no	—
29	C₄⋊C₄×C₁₀₁	—	2	2	no	—
30	C₄⋊C₄×C₁₀₁	—	2	4	no	—
31	C₄⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	—
32	C₂²⋊C₄×C₁₀₁	—	4	2	no	—
33	C₂³×C₁₀₁	—	2	2	no	—
34	C₂³×C₁₀₁	—	2	2	no	—
35	C₂³×C₁₀₁	C₂²⋊C₄	1	2	yes	—
36	C₂³×C₁₀₁	SD₁₆	1	2	yes	—
37	C₂³×C₁₀₁	Q₁₆	1	2	yes	—
38	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₁	—	2	2	no	—
40	C₂×C₄×C₁₀₁	—	2	4	no	—
41	C₂×C₄×C₁₀₁	—	2	4	no	—
42	C₂×C₄×C₁₀₁	—	2	4	no	—
43	C₂×C₄×C₁₀₁	—	4	8	no	—
44	C₂×C₄×C₁₀₁	—	4	4	no	—
45	C₂×C₄×C₁₀₁	—	4	2	no	—
46	C₂×C₄×C₁₀₁	—	4	2	no	—
47	C₂×C₄×C₁₀₁	—	4	2	no	—
48	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
49	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₀₁	#41	1	4	yes	—
52	C₈×C₁₀₁	—	4	2	no	—
53	C₂²×C₁₀₁	—	2	2	no	—
54	C₂²×C₁₀₁	—	2	2	no	—
55	C₂²×C₁₀₁	—	2	4	no	—
56	C₂²×C₁₀₁	—	2	4	no	—
57	C₂²×C₁₀₁	#103	1	4	yes	—
58	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₁₀₁	#123	1	2	yes	—
60	C₂²×C₁₀₁	C₂×SD₁₆	1	1	yes	characteristic
61	C₂²×C₁₀₁	C₂×Q₁₆	1	1	yes	characteristic
62	C₂²×C₁₀₁	AGL(1,ℤ/8)	1	2	yes	—
63	C₄×C₁₀₁	—	2	2	no	—
64	C₄×C₁₀₁	—	4	4	no	—
65	C₄×C₁₀₁	—	8	2	no	—
66	C₄×C₁₀₁	#117	1	2	yes	—
67	C₄×C₁₀₁	#123	1	2	yes	—
68	C₂×C₁₀₁	—	2	4	no	—
69	C₂×C₁₀₁	#381	1	1	yes	characteristic
70	C₂×C₁₀₁	C₂×#103	1	1	yes	characteristic
71	C₂×C₁₀₁	#418	1	1	yes	characteristic
72	C₂×C₁₀₁	#481	1	2	yes	—
73	C₂×C₁₀₁	#484	1	2	yes	—
74	#1824	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₁₀₁	#1824	1	1	yes	characteristic, direct product factor, Sylow
76	C₂²×C₄⋊C₄	C₂×C₁₀₁	1	1	yes	characteristic
77	C₂×C₂²⋊C₈	C₂×C₁₀₁	1	1	yes	characteristic
78	#380	C₂×C₁₀₁	1	1	yes	characteristic
79	#340	C₂×C₁₀₁	1	4	yes	complemented
80	C₂³×C₄	—	2	1	no	—
81	C₂³×C₄	C₂²×C₁₀₁	1	1	yes	characteristic
82	C₂²×C₈	C₂²×C₁₀₁	1	2	yes	—
83	C₂×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄⋊C₄	—	2	2	no	—
85	C₂×C₄⋊C₄	C₄×C₁₀₁	1	4	yes	complemented
86	C₂×C₄⋊C₄	C₂²×C₁₀₁	1	2	yes	—
87	C₂×C₄⋊C₄	C₂²×C₁₀₁	1	2	yes	—
88	C₂×C₂²⋊C₄	—	2	1	no	—
89	C₂²⋊C₈	—	2	2	no	—
90	#95	—	2	1	no	—
91	C₂⁴	D₄×C₁₀₁	1	1	yes	characteristic
92	C₂²×C₄	—	2	4	no	—
93	C₂²×C₄	—	2	4	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	4	4	no	—
97	C₂²×C₄	C₂×C₄×C₁₀₁	1	2	yes	—
98	C₂²×C₄	D₄×C₁₀₁	1	1	yes	characteristic
99	C₂²×C₄	D₄×C₁₀₁	1	2	yes	—
100	C₂²×C₄	C₂³×C₁₀₁	1	1	yes	characteristic
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	4	no	—
103	C₄⋊C₄	—	2	4	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	2	4	no	—
106	C₄⋊C₄	C₂×C₄×C₁₀₁	1	4	yes	—
107	C₂²⋊C₄	—	4	2	no	—
108	C₂³	—	2	2	no	—
109	C₂³	—	2	2	no	—
110	C₂³	C₂²⋊C₄×C₁₀₁	1	2	yes	—
111	C₂³	SD₁₆×C₁₀₁	1	2	yes	—
112	C₂³	Q₁₆×C₁₀₁	1	2	yes	—
113	C₂³	C₂×D₄×C₁₀₁	1	1	yes	characteristic
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	4	8	no	—
119	C₂×C₄	—	4	4	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	C₂²⋊C₄×C₁₀₁	1	2	yes	—
124	C₂×C₄	C₂²×C₄×C₁₀₁	1	1	yes	characteristic
125	C₂×C₄	C₂×D₄×C₁₀₁	1	1	yes	characteristic
126	C₂×C₄	#41×C₁₀₁	1	4	yes	—
127	C₈	—	4	2	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	4	no	—
130	C₂²	—	2	4	no	—
131	C₂²	—	2	2	no	—
132	C₂²	#103×C₁₀₁	1	4	yes	—
133	C₂²	C₂×C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
134	C₂²	#123×C₁₀₁	1	2	yes	—
135	C₂²	C₂×SD₁₆×C₁₀₁	1	1	yes	characteristic
136	C₂²	C₂×Q₁₆×C₁₀₁	1	1	yes	characteristic
137	C₂²	AGL(1,ℤ/8)×C₁₀₁	1	2	yes	—
138	C₄	—	2	2	no	—
139	C₄	—	4	4	no	—
140	C₄	—	8	2	no	—
141	C₄	#117×C₁₀₁	1	2	yes	—
142	C₄	#123×C₁₀₁	1	2	yes	—
143	C₂	—	2	4	no	—
144	C₂	#381×C₁₀₁	1	1	yes	characteristic
145	C₂	C₂×#103×C₁₀₁	1	1	yes	characteristic
146	C₂	#418×C₁₀₁	1	1	yes	characteristic
147	C₂	#481×C₁₀₁	1	2	yes	—
148	C₂	#484×C₁₀₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊C₄×C₁₀₁

C₂

yes

maximal, characteristic

C₂×C₂²⋊C₈×C₁₀₁

C₂

yes

maximal, characteristic

#380×C₁₀₁

C₂

yes

maximal, characteristic

#340×C₁₀₁

C₂

yes

maximal, complemented

C₂³×C₄×C₁₀₁

—

C₂³×C₄×C₁₀₁

C₂²

yes

characteristic

C₂²×C₈×C₁₀₁

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₁

—

C₂×C₄⋊C₄×C₁₀₁

—

C₂×C₄⋊C₄×C₁₀₁

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₀₁

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₁

C₂²

yes

—

C₂×C₂²⋊C₄×C₁₀₁

—

C₂²⋊C₈×C₁₀₁

—

#95×C₁₀₁

—

C₂⁴×C₁₀₁

D₄

yes

characteristic

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

D₄

yes

characteristic

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

C₂×C₄

yes

—

C₂²⋊C₄×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₂²⋊C₄

yes

—

C₂³×C₁₀₁

SD₁₆

yes

—

C₂³×C₁₀₁

Q₁₆

yes

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₈×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

#103

yes

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₁

#123

yes

—

C₂²×C₁₀₁

C₂×SD₁₆

yes

characteristic

C₂²×C₁₀₁

C₂×Q₁₆

yes

characteristic

C₂²×C₁₀₁

AGL(1,ℤ/8)

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#117

yes

—

C₄×C₁₀₁

#123

yes

—

C₂×C₁₀₁

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

C₂×C₁₀₁

#484

yes

—

#1824

C₁₀₁

yes

maximal, characteristic, direct product factor, Sylow

C₁₀₁

#1824

yes

characteristic, direct product factor, Sylow

C₂²×C₄⋊C₄

C₂×C₁₀₁

yes

characteristic

C₂×C₂²⋊C₈

C₂×C₁₀₁

yes

characteristic

#380

C₂×C₁₀₁

yes

characteristic

#340

C₂×C₁₀₁

yes

complemented

C₂³×C₄

—

C₂³×C₄

C₂²×C₁₀₁

yes

characteristic

C₂²×C₈

C₂²×C₁₀₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₄×C₁₀₁

yes

complemented

C₂×C₄⋊C₄

C₂²×C₁₀₁

yes

—

C₂×C₄⋊C₄

C₂²×C₁₀₁

yes

—

C₂×C₂²⋊C₄

—

C₂²⋊C₈

—

#95

—

C₂⁴

D₄×C₁₀₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×C₁₀₁

yes

—

C₂²×C₄

D₄×C₁₀₁

yes

characteristic

C₂²×C₄

D₄×C₁₀₁

yes

—

100

C₂²×C₄

C₂³×C₁₀₁

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

C₂×C₄×C₁₀₁

yes

—

107

C₂²⋊C₄

—

108

C₂³

—

109

C₂³

—

110

C₂³

C₂²⋊C₄×C₁₀₁

yes

—

111

C₂³

SD₁₆×C₁₀₁

yes

—

112

C₂³

Q₁₆×C₁₀₁

yes

—

113

C₂³

C₂×D₄×C₁₀₁

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

C₂²⋊C₄×C₁₀₁

yes

—

124

C₂×C₄

C₂²×C₄×C₁₀₁

yes

characteristic

125

C₂×C₄

C₂×D₄×C₁₀₁

yes

characteristic

126

C₂×C₄

#41×C₁₀₁

yes

—

127

C₈

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

#103×C₁₀₁

yes

—

133

C₂²

C₂×C₂²⋊C₄×C₁₀₁

yes

characteristic

134

C₂²

#123×C₁₀₁

yes

—

135

C₂²

C₂×SD₁₆×C₁₀₁

yes

characteristic

136

C₂²

C₂×Q₁₆×C₁₀₁

yes

characteristic

137

C₂²

AGL(1,ℤ/8)×C₁₀₁

yes

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#117×C₁₀₁

yes

—

142

C₄

#123×C₁₀₁

yes

—

143

C₂

—

144

C₂

#381×C₁₀₁

yes

characteristic

145

C₂

C₂×#103×C₁₀₁

yes

characteristic

146

C₂

#418×C₁₀₁

yes

characteristic

147

C₂

#481×C₁₀₁

yes

—

148

C₂

#484×C₁₀₁

yes

—

	subgroup	G/H
center, socle	C₂³×C₁₀₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₁₀₁
Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₁
prime radical	C₂⁴×C₁₀₁	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₁ ◁ C₂³×C₄×C₁₀₁ ◁ G

subgroup

G/H

center, socle

C₂³×C₁₀₁

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₁₀₁

Frattini, cosocle

C₂²×C₄

C₂³×C₁₀₁

prime radical

C₂⁴×C₁₀₁

D₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₀₁ ◁ C₂³×C₄×C₁₀₁ ◁ G

N	H ≅ G/N	type
4. #340×C₁₀₁	C₂	semidirect
10. C₂×C₄⋊C₄×C₁₀₁	C₄	semidirect
74. #1824	C₁₀₁	direct (N × H)
75. C₁₀₁	#1824	direct (N × H)
79. #340	C₂×C₁₀₁	semidirect
85. C₂×C₄⋊C₄	C₄×C₁₀₁	semidirect

H ≅ G/N

type

4. #340×C₁₀₁

C₂

semidirect

10. C₂×C₄⋊C₄×C₁₀₁

C₄

semidirect

74. #1824

C₁₀₁

direct (N × H)

75. C₁₀₁

#1824

direct (N × H)

79. #340

C₂×C₁₀₁

semidirect

85. C₂×C₄⋊C₄

C₄×C₁₀₁

semidirect

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

subgroup	index
C₂²×C₄⋊C₄×C₁₀₁	2
C₂×C₂²⋊C₈×C₁₀₁	2
#380×C₁₀₁	2
#340×C₁₀₁	2
#1824	101

subgroup

index

C₂²×C₄⋊C₄×C₁₀₁

C₂×C₂²⋊C₈×C₁₀₁

#380×C₁₀₁

#340×C₁₀₁

#1824

101

Subgroups — C₁₀₁×#1824 #27167178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C101×#1824 #27167178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#1824 #27167178