Group #28359055 — subgroups

Subgroups — C₁₀₉×#1801 #28359055

Order 13952.

272 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 124 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×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	#423×C₁₀₉	C₂	1	2	yes	maximal
3	C₂×#101×C₁₀₉	C₂	1	2	yes	maximal
4	#343×C₁₀₉	C₂	1	2	yes	maximal
5	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
6	C₂×M₁₆×C₁₀₉	C₂²	1	4	yes	—
7	C₄×Q₈×C₁₀₉	—	2	2	no	—
8	C₄×Q₈×C₁₀₉	—	2	2	no	—
9	C₂×C₄⋊C₄×C₁₀₉	C₄	1	2	yes	—
10	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₁₀₉	C₄	1	2	yes	—
12	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₀₉	—	2	2	no	—
14	C₄⋊C₈×C₁₀₉	—	2	2	no	—
15	#101×C₁₀₉	—	2	4	no	—
16	C₂×Q₈×C₁₀₉	—	2	2	no	—
17	C₂×Q₈×C₁₀₉	—	2	2	no	—
18	C₂×Q₈×C₁₀₉	D₄	1	2	yes	—
19	C₂×Q₈×C₁₀₉	Q₈	1	2	yes	—
20	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
25	M₁₆×C₁₀₉	—	2	4	no	—
26	M₁₆×C₁₀₉	—	2	4	no	—
27	C₂×C₈×C₁₀₉	—	2	4	no	—
28	C₄⋊C₄×C₁₀₉	—	2	4	no	—
29	C₄⋊C₄×C₁₀₉	—	2	2	no	—
30	C₄²×C₁₀₉	—	2	2	no	—
31	C₄²×C₁₀₉	—	2	2	no	—
32	C₄²×C₁₀₉	D₄	1	2	yes	—
33	C₄²×C₁₀₉	D₄	1	2	yes	—
34	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
35	Q₈×C₁₀₉	—	2	4	no	—
36	Q₈×C₁₀₉	—	2	4	no	—
37	C₂×C₄×C₁₀₉	—	2	2	no	—
38	C₂×C₄×C₁₀₉	—	2	2	no	—
39	C₂×C₄×C₁₀₉	—	2	2	no	—
40	C₂×C₄×C₁₀₉	—	2	2	no	—
41	C₂×C₄×C₁₀₉	—	4	1	no	—
42	C₂×C₄×C₁₀₉	—	4	2	no	—
43	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
47	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₁₀₉	#41	1	2	yes	—
51	C₈×C₁₀₉	—	4	4	no	—
52	C₂²×C₁₀₉	—	2	1	no	—
53	C₂²×C₁₀₉	—	2	1	no	—
54	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₀₉	#117	1	1	yes	characteristic
57	C₄×C₁₀₉	—	2	4	no	—
58	C₄×C₁₀₉	—	2	1	no	—
59	C₄×C₁₀₉	—	2	2	no	—
60	C₄×C₁₀₉	—	4	2	no	—
61	C₄×C₁₀₉	#101	1	2	yes	complemented
62	C₄×C₁₀₉	#121	1	2	yes	—
63	C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
64	C₂×C₁₀₉	—	2	1	no	—
65	C₂×C₁₀₉	—	2	1	no	—
66	C₂×C₁₀₉	#380	1	1	yes	characteristic
67	C₂×C₁₀₉	C₂×#101	1	1	yes	characteristic
68	C₂×C₁₀₉	#413	1	1	yes	characteristic
69	#1801	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₁₀₉	#1801	1	1	yes	characteristic, direct product factor, Sylow
71	C₂×C₄×Q₈	C₂×C₁₀₉	1	1	yes	characteristic
72	#423	C₂×C₁₀₉	1	2	yes	—
73	C₂×#101	C₂×C₁₀₉	1	2	yes	—
74	#343	C₂×C₁₀₉	1	2	yes	—
75	C₂²×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
76	C₂×M₁₆	C₂²×C₁₀₉	1	4	yes	—
77	C₄×Q₈	—	2	2	no	—
78	C₄×Q₈	—	2	2	no	—
79	C₂×C₄⋊C₄	C₄×C₁₀₉	1	2	yes	—
80	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
81	C₂×C₄²	C₄×C₁₀₉	1	2	yes	—
82	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
83	C₄⋊C₈	—	2	2	no	—
84	C₄⋊C₈	—	2	2	no	—
85	#101	—	2	4	no	—
86	C₂×Q₈	—	2	2	no	—
87	C₂×Q₈	—	2	2	no	—
88	C₂×Q₈	D₄×C₁₀₉	1	2	yes	—
89	C₂×Q₈	Q₈×C₁₀₉	1	2	yes	—
90	C₂²×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
91	C₂²×C₄	C₂×C₄×C₁₀₉	1	2	yes	—
92	C₂²×C₄	C₂×C₄×C₁₀₉	1	2	yes	—
93	C₂²×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
94	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
95	M₁₆	—	2	4	no	—
96	M₁₆	—	2	4	no	—
97	C₂×C₈	—	2	4	no	—
98	C₄⋊C₄	—	2	4	no	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄²	—	2	2	no	—
101	C₄²	—	2	2	no	—
102	C₄²	D₄×C₁₀₉	1	2	yes	—
103	C₄²	D₄×C₁₀₉	1	2	yes	—
104	C₂³	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
105	Q₈	—	2	4	no	—
106	Q₈	—	2	4	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
114	C₂×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
115	C₂×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
116	C₂×C₄	C₄⋊C₄×C₁₀₉	1	4	yes	—
117	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
118	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
119	C₂×C₄	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
120	C₂×C₄	#41×C₁₀₉	1	2	yes	—
121	C₈	—	4	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
125	C₂²	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
126	C₂²	#117×C₁₀₉	1	1	yes	characteristic
127	C₄	—	2	4	no	—
128	C₄	—	2	1	no	—
129	C₄	—	2	2	no	—
130	C₄	—	4	2	no	—
131	C₄	#101×C₁₀₉	1	2	yes	complemented
132	C₄	#121×C₁₀₉	1	2	yes	—
133	C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
134	C₂	—	2	1	no	—
135	C₂	—	2	1	no	—
136	C₂	#380×C₁₀₉	1	1	yes	characteristic
137	C₂	C₂×#101×C₁₀₉	1	1	yes	characteristic
138	C₂	#413×C₁₀₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Q₈×C₁₀₉

C₂

yes

maximal, characteristic

#423×C₁₀₉

C₂

yes

maximal

C₂×#101×C₁₀₉

C₂

yes

maximal

#343×C₁₀₉

C₂

yes

maximal

C₂²×Q₈×C₁₀₉

C₂²

yes

characteristic

C₂×M₁₆×C₁₀₉

C₂²

yes

—

C₄×Q₈×C₁₀₉

—

C₄×Q₈×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

C₄

yes

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

characteristic

C₂×C₄²×C₁₀₉

C₄

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₉

—

C₄⋊C₈×C₁₀₉

—

#101×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

D₄

yes

—

C₂×Q₈×C₁₀₉

Q₈

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

#117

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#101

yes

complemented

C₄×C₁₀₉

#121

yes

—

C₄×C₁₀₉

C₂²⋊Q₈

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₁₀₉

#1801

yes

characteristic, direct product factor, Sylow

C₂×C₄×Q₈

C₂×C₁₀₉

yes

characteristic

#423

C₂×C₁₀₉

yes

—

C₂×#101

C₂×C₁₀₉

yes

—

#343

C₂×C₁₀₉

yes

—

C₂²×Q₈

C₂²×C₁₀₉

yes

characteristic

C₂×M₁₆

C₂²×C₁₀₉

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

C₄×C₁₀₉

yes

—

C₂×C₄⋊C₄

C₂²×C₁₀₉

yes

characteristic

C₂×C₄²

C₄×C₁₀₉

yes

—

C₂×C₄²

C₂²×C₁₀₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#101

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₄×C₁₀₉

yes

—

C₂×Q₈

Q₈×C₁₀₉

yes

—

C₂²×C₄

C₂×C₄×C₁₀₉

yes

characteristic

C₂²×C₄

C₂×C₄×C₁₀₉

yes

—

C₂²×C₄

C₂×C₄×C₁₀₉

yes

—

C₂²×C₄

C₂×C₄×C₁₀₉

yes

characteristic

C₂²×C₄

C₂³×C₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

—

101

C₄²

—

102

C₄²

D₄×C₁₀₉

yes

—

103

C₄²

D₄×C₁₀₉

yes

—

104

C₂³

C₂²×C₄×C₁₀₉

yes

characteristic

105

Q₈

—

106

Q₈

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²⋊C₄×C₁₀₉

yes

characteristic

114

C₂×C₄

C₂²⋊C₄×C₁₀₉

yes

—

115

C₂×C₄

C₂²⋊C₄×C₁₀₉

yes

characteristic

116

C₂×C₄

C₄⋊C₄×C₁₀₉

yes

—

117

C₂×C₄

C₂×D₄×C₁₀₉

yes

characteristic

118

C₂×C₄

C₂×D₄×C₁₀₉

yes

—

119

C₂×C₄

C₂×Q₈×C₁₀₉

yes

characteristic

120

C₂×C₄

#41×C₁₀₉

yes

—

121

C₈

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂×C₂²⋊C₄×C₁₀₉

yes

characteristic

125

C₂²

C₂×C₄⋊C₄×C₁₀₉

yes

characteristic

126

C₂²

#117×C₁₀₉

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

#101×C₁₀₉

yes

complemented

132

C₄

#121×C₁₀₉

yes

—

133

C₄

C₂²⋊Q₈×C₁₀₉

yes

—

134

C₂

—

135

C₂

—

136

C₂

#380×C₁₀₉

yes

characteristic

137

C₂

C₂×#101×C₁₀₉

yes

characteristic

138

C₂

#413×C₁₀₉

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₉
prime radical	C₂³×C₁₀₉	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₀₉

C₂×C₂²⋊C₄

commutator

C₂³

C₂²×C₄×C₁₀₉

Frattini, cosocle

C₂²×C₄

C₂³×C₁₀₉

prime radical

C₂³×C₁₀₉

C₂²×C₄

LCS

G ▷ C₂³ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

N	H ≅ G/N	type
61. C₄×C₁₀₉	#101	semidirect
69. #1801	C₁₀₉	direct (N × H)
70. C₁₀₉	#1801	direct (N × H)
131. C₄	#101×C₁₀₉	semidirect

H ≅ G/N

type

61. C₄×C₁₀₉

#101

semidirect

69. #1801

C₁₀₉

direct (N × H)

70. C₁₀₉

#1801

direct (N × H)

131. C₄

#101×C₁₀₉

semidirect

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

subgroup	index
C₂×C₄×Q₈×C₁₀₉	2
#423×C₁₀₉	2
C₂×#101×C₁₀₉	2
#343×C₁₀₉	2
#1801	109

subgroup

index

C₂×C₄×Q₈×C₁₀₉

#423×C₁₀₉

C₂×#101×C₁₀₉

#343×C₁₀₉

#1801

109

Subgroups — C₁₀₉×#1801 #28359055

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#1801 #28359055

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#1801 #28359055