Group #28359239 — subgroups

Subgroups — C₁₀₉×#1985 #28359239

Order 13952.

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

normal subgroups only

This group has 280 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	C₂×C₄×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#97×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#336×C₁₀₉	C₂	1	4	yes	maximal
5	C₂²×Q₈×C₁₀₉	C₄	1	1	yes	characteristic
6	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
7	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
8	C₄×Q₈×C₁₀₉	—	2	2	no	—
9	C₄×Q₈×C₁₀₉	—	2	2	no	—
10	C₂×C₄⋊C₄×C₁₀₉	C₄	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₁₀₉	C₄	1	2	yes	—
12	C₂×C₄²×C₁₀₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₁₀₉	—	2	2	no	—
15	#97×C₁₀₉	—	2	2	no	—
16	C₂×Q₈×C₁₀₉	—	2	2	no	—
17	C₂×Q₈×C₁₀₉	—	2	4	no	—
18	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₁₀₉	—	2	2	no	—
24	C₂×C₈×C₁₀₉	—	2	2	no	—
25	C₂×C₈×C₁₀₉	—	2	2	no	—
26	C₂×C₈×C₁₀₉	—	2	4	no	—
27	C₂×C₈×C₁₀₉	Q₈	1	4	yes	complemented
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	8	no	—
36	C₂×C₄×C₁₀₉	—	2	2	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	—	2	4	no	—
42	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₀₉	M₁₆	1	4	yes	—
45	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₁₀₉	#41	1	2	yes	—
49	C₈×C₁₀₉	—	2	4	no	—
50	C₈×C₁₀₉	—	4	2	no	—
51	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₁₀₉	C₄×Q₈	1	2	yes	—
53	C₂²×C₁₀₉	C₂×M₁₆	1	2	yes	—
54	C₂²×C₁₀₉	#131	1	2	yes	—
55	C₄×C₁₀₉	—	2	2	no	—
56	C₄×C₁₀₉	—	2	4	no	—
57	C₄×C₁₀₉	—	2	2	no	—
58	C₄×C₁₀₉	—	4	2	no	—
59	C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
60	C₄×C₁₀₉	#124	1	1	yes	characteristic
61	C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₁₀₉	#391	1	1	yes	characteristic
63	C₂×C₁₀₉	#407	1	1	yes	characteristic
64	C₂×C₁₀₉	#408	1	1	yes	characteristic
65	C₂×C₁₀₉	#446	1	4	yes	—
66	#1985	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₁₀₉	#1985	1	1	yes	characteristic, direct product factor, Sylow
68	C₂×C₄×Q₈	C₂×C₁₀₉	1	1	yes	characteristic
69	C₂×C₄⋊C₈	C₂×C₁₀₉	1	1	yes	characteristic
70	C₂×#97	C₂×C₁₀₉	1	1	yes	characteristic
71	#336	C₂×C₁₀₉	1	4	yes	—
72	C₂²×Q₈	C₄×C₁₀₉	1	1	yes	characteristic
73	C₂²×C₈	C₂²×C₁₀₉	1	2	yes	—
74	C₂²×C₈	C₂²×C₁₀₉	1	2	yes	—
75	C₄×Q₈	—	2	2	no	—
76	C₄×Q₈	—	2	2	no	—
77	C₂×C₄⋊C₄	C₄×C₁₀₉	1	1	yes	characteristic
78	C₂×C₄⋊C₄	C₄×C₁₀₉	1	2	yes	—
79	C₂×C₄²	C₂²×C₁₀₉	1	2	yes	—
80	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
81	C₄⋊C₈	—	2	2	no	—
82	#97	—	2	2	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂×Q₈	—	2	4	no	—
85	C₂²×C₄	C₂×C₄×C₁₀₉	1	2	yes	—
86	C₂²×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
87	C₂²×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
88	C₂²×C₄	C₂×C₄×C₁₀₉	1	2	yes	—
89	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	4	no	—
94	C₂×C₈	Q₈×C₁₀₉	1	4	yes	complemented
95	C₄⋊C₄	—	2	4	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄²	—	2	2	no	—
98	C₄²	—	2	2	no	—
99	C₄²	D₄×C₁₀₉	1	2	yes	—
100	C₄²	D₄×C₁₀₉	1	2	yes	—
101	C₂³	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
102	Q₈	—	2	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
110	C₂×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
111	C₂×C₄	M₁₆×C₁₀₉	1	4	yes	—
112	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
113	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
114	C₂×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
115	C₂×C₄	#41×C₁₀₉	1	2	yes	—
116	C₈	—	2	4	no	—
117	C₈	—	4	2	no	—
118	C₂²	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
119	C₂²	C₄×Q₈×C₁₀₉	1	2	yes	—
120	C₂²	C₂×M₁₆×C₁₀₉	1	2	yes	—
121	C₂²	#131×C₁₀₉	1	2	yes	—
122	C₄	—	2	2	no	—
123	C₄	—	2	4	no	—
124	C₄	—	2	2	no	—
125	C₄	—	4	2	no	—
126	C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
127	C₄	#124×C₁₀₉	1	1	yes	characteristic
128	C₄	C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
129	C₂	#391×C₁₀₉	1	1	yes	characteristic
130	C₂	#407×C₁₀₉	1	1	yes	characteristic
131	C₂	#408×C₁₀₉	1	1	yes	characteristic
132	C₂	#446×C₁₀₉	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Q₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#97×C₁₀₉

C₂

yes

maximal, characteristic

#336×C₁₀₉

C₂

yes

maximal

C₂²×Q₈×C₁₀₉

C₄

yes

characteristic

C₂²×C₈×C₁₀₉

C₂²

yes

—

C₂²×C₈×C₁₀₉

C₂²

yes

—

C₄×Q₈×C₁₀₉

—

C₄×Q₈×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

C₄

yes

characteristic

C₂×C₄⋊C₄×C₁₀₉

C₄

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₉

—

#97×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

Q₈

yes

complemented

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

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

M₁₆

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₄×Q₈

yes

—

C₂²×C₁₀₉

C₂×M₁₆

yes

—

C₂²×C₁₀₉

#131

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²⋊Q₈

yes

—

C₄×C₁₀₉

#124

yes

characteristic

C₄×C₁₀₉

C₄⋊Q₈

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

#1985

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₁₀₉

#1985

yes

characteristic, direct product factor, Sylow

C₂×C₄×Q₈

C₂×C₁₀₉

yes

characteristic

C₂×C₄⋊C₈

C₂×C₁₀₉

yes

characteristic

C₂×#97

C₂×C₁₀₉

yes

characteristic

#336

C₂×C₁₀₉

yes

—

C₂²×Q₈

C₄×C₁₀₉

yes

characteristic

C₂²×C₈

C₂²×C₁₀₉

yes

—

C₂²×C₈

C₂²×C₁₀₉

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

C₄×C₁₀₉

yes

characteristic

C₂×C₄⋊C₄

C₄×C₁₀₉

yes

—

C₂×C₄²

C₂²×C₁₀₉

yes

—

C₂×C₄²

C₂²×C₁₀₉

yes

characteristic

C₄⋊C₈

—

#97

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×C₄×C₁₀₉

yes

—

C₂²×C₄

C₂×C₄×C₁₀₉

yes

characteristic

C₂²×C₄

C₂×C₄×C₁₀₉

yes

characteristic

C₂²×C₄

C₂×C₄×C₁₀₉

yes

—

C₂²×C₄

C₂³×C₁₀₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Q₈×C₁₀₉

yes

complemented

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

D₄×C₁₀₉

yes

—

100

C₄²

D₄×C₁₀₉

yes

—

101

C₂³

C₂²×C₄×C₁₀₉

yes

characteristic

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂²⋊C₄×C₁₀₉

yes

—

110

C₂×C₄

C₂²⋊C₄×C₁₀₉

yes

—

111

C₂×C₄

M₁₆×C₁₀₉

yes

—

112

C₂×C₄

C₂×D₄×C₁₀₉

yes

characteristic

113

C₂×C₄

C₂×D₄×C₁₀₉

yes

characteristic

114

C₂×C₄

C₂×Q₈×C₁₀₉

yes

—

115

C₂×C₄

#41×C₁₀₉

yes

—

116

C₈

—

117

C₈

—

118

C₂²

C₂×C₂²⋊C₄×C₁₀₉

yes

characteristic

119

C₂²

C₄×Q₈×C₁₀₉

yes

—

120

C₂²

C₂×M₁₆×C₁₀₉

yes

—

121

C₂²

#131×C₁₀₉

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

C₂²⋊Q₈×C₁₀₉

yes

—

127

C₄

#124×C₁₀₉

yes

characteristic

128

C₄

C₄⋊Q₈×C₁₀₉

yes

characteristic

129

C₂

#391×C₁₀₉

yes

characteristic

130

C₂

#407×C₁₀₉

yes

characteristic

131

C₂

#408×C₁₀₉

yes

characteristic

132

C₂

#446×C₁₀₉

yes

—

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

subgroup

G/H

center

C₂²×C₄×C₁₀₉

C₂³

commutator

C₂³

C₂²×C₄×C₁₀₉

Frattini, cosocle

C₂²×C₄

C₂³×C₁₀₉

socle, prime radical

C₂³×C₁₀₉

C₂²×C₄

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂²×C₄×C₁₀₉ ◁ G

N	H ≅ G/N	type
27. C₂×C₈×C₁₀₉	Q₈	semidirect
66. #1985	C₁₀₉	direct (N × H)
67. C₁₀₉	#1985	direct (N × H)
94. C₂×C₈	Q₈×C₁₀₉	semidirect

H ≅ G/N

type

27. C₂×C₈×C₁₀₉

Q₈

semidirect

66. #1985

C₁₀₉

direct (N × H)

67. C₁₀₉

#1985

direct (N × H)

94. C₂×C₈

Q₈×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
C₂×C₄⋊C₈×C₁₀₉	2
C₂×#97×C₁₀₉	2
#336×C₁₀₉	2
#1985	109

subgroup

index

C₂×C₄×Q₈×C₁₀₉

C₂×C₄⋊C₈×C₁₀₉

C₂×#97×C₁₀₉

#336×C₁₀₉

#1985

109

Subgroups — C₁₀₉×#1985 #28359239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#1985 #28359239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#1985 #28359239