Group #42128294 — subgroups

Subgroups — C₁₀₉×#10300 #42128294

Order 27904.

592 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 106 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	C₁₀₉×#3253	C₂	1	3	yes	maximal, complemented
2	C₁₀₉×#1962	C₂	1	3	yes	maximal, complemented
3	C₁₀₉×#1309	C₂	1	1	yes	maximal, characteristic, complemented
4	#532×C₁₀₉	C₂²	1	3	yes	—
5	#476×C₁₀₉	—	2	6	no	—
6	#469×C₁₀₉	—	2	6	no	—
7	#459×C₁₀₉	—	2	6	no	—
8	#423×C₁₀₉	C₂²	1	3	yes	—
9	C₂×#104×C₁₀₉	—	2	6	no	—
10	C₄³×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×#41×C₁₀₉	—	2	3	no	—
12	C₂×D₈×C₁₀₉	—	2	6	no	—
13	C₂×M₁₆×C₁₀₉	D₄	1	6	yes	—
14	C₄⋊Q₈×C₁₀₉	D₄	1	3	yes	—
15	#127×C₁₀₉	D₄	1	3	yes	—
16	#124×C₁₀₉	—	2	3	no	—
17	#121×C₁₀₉	—	2	6	no	—
18	C₄×D₄×C₁₀₉	—	2	6	no	—
19	C₂×C₄²×C₁₀₉	—	2	3	no	—
20	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
21	#106×C₁₀₉	—	2	6	no	—
22	C₄⋊C₈×C₁₀₉	—	2	6	no	—
23	#104×C₁₀₉	—	4	12	no	—
24	#102×C₁₀₉	—	2	12	no	—
25	#41×C₁₀₉	—	4	6	no	—
26	#41×C₁₀₉	—	4	6	no	—
27	C₂×Q₈×C₁₀₉	—	2	3	no	—
28	C₂×D₄×C₁₀₉	—	2	6	no	—
29	C₂×D₄×C₁₀₉	—	2	3	no	—
30	C₂×D₄×C₁₀₉	—	4	3	no	—
31	C₂²×C₄×C₁₀₉	—	4	1	no	—
32	C₂²×C₄×C₁₀₉	—	4	3	no	—
33	C₂²×C₄×C₁₀₉	C₂×D₄	1	3	yes	—
34	D₈×C₁₀₉	—	8	6	no	—
35	M₁₆×C₁₀₉	—	4	6	no	—
36	C₂×C₈×C₁₀₉	—	2	6	no	—
37	C₄⋊C₄×C₁₀₉	—	2	6	no	—
38	C₂²⋊C₄×C₁₀₉	—	4	6	no	—
39	C₄²×C₁₀₉	—	2	6	no	—
40	C₄²×C₁₀₉	—	4	3	no	—
41	C₄²×C₁₀₉	C₂×D₄	1	3	yes	—
42	C₄²×C₁₀₉	#41	1	1	yes	characteristic
43	C₂³×C₁₀₉	—	4	3	no	—
44	C₂³×C₁₀₉	#127	1	1	yes	characteristic
45	Q₈×C₁₀₉	—	4	6	no	—
46	D₄×C₁₀₉	—	4	6	no	—
47	D₄×C₁₀₉	—	4	6	no	—
48	D₄×C₁₀₉	—	4	6	no	—
49	D₄×C₁₀₉	—	8	6	no	—
50	C₂×C₄×C₁₀₉	—	2	3	no	—
51	C₂×C₄×C₁₀₉	—	2	3	no	—
52	C₂×C₄×C₁₀₉	—	4	6	no	—
53	C₂×C₄×C₁₀₉	—	4	3	no	—
54	C₂×C₄×C₁₀₉	—	8	3	no	—
55	C₂×C₄×C₁₀₉	—	8	3	no	—
56	C₂×C₄×C₁₀₉	#120	1	3	yes	—
57	C₂×C₄×C₁₀₉	#121	1	3	yes	—
58	C₈×C₁₀₉	—	4	6	no	—
59	C₂²×C₁₀₉	—	2	3	no	—
60	C₂²×C₁₀₉	—	8	3	no	—
61	C₂²×C₁₀₉	—	8	6	no	—
62	C₂²×C₁₀₉	#392	1	1	yes	characteristic
63	C₄×C₁₀₉	—	2	3	no	—
64	C₄×C₁₀₉	—	4	4	no	—
65	C₄×C₁₀₉	—	8	3	no	—
66	C₄×C₁₀₉	#453	1	3	yes	—
67	C₄×C₁₀₉	#454	1	3	yes	—
68	#10300	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂×C₁₀₉	—	4	1	no	—
70	C₂×C₁₀₉	—	16	3	no	—
71	C₂×C₁₀₉	#2006	1	3	yes	—
72	#3253	C₂×C₁₀₉	1	3	yes	complemented
73	#1962	C₂×C₁₀₉	1	3	yes	complemented
74	#1309	C₂×C₁₀₉	1	1	yes	characteristic, complemented
75	C₁₀₉	#10300	1	1	yes	characteristic, direct product factor, Sylow
76	#532	C₂²×C₁₀₉	1	3	yes	—
77	#476	—	2	6	no	—
78	#469	—	2	6	no	—
79	#459	—	2	6	no	—
80	#423	C₂²×C₁₀₉	1	3	yes	—
81	C₂×#104	—	2	6	no	—
82	C₄³	C₂²×C₁₀₉	1	1	yes	characteristic
83	C₂×#41	—	2	3	no	—
84	C₂×D₈	—	2	6	no	—
85	C₂×M₁₆	D₄×C₁₀₉	1	6	yes	—
86	C₄⋊Q₈	D₄×C₁₀₉	1	3	yes	—
87	#127	D₄×C₁₀₉	1	3	yes	—
88	#124	—	2	3	no	—
89	#121	—	2	6	no	—
90	C₄×D₄	—	2	6	no	—
91	C₂×C₄²	—	2	3	no	—
92	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
93	#106	—	2	6	no	—
94	C₄⋊C₈	—	2	6	no	—
95	#104	—	4	12	no	—
96	#102	—	2	12	no	—
97	#41	—	4	6	no	—
98	#41	—	4	6	no	—
99	C₂×Q₈	—	2	3	no	—
100	C₂×D₄	—	2	6	no	—
101	C₂×D₄	—	2	3	no	—
102	C₂×D₄	—	4	3	no	—
103	C₂²×C₄	—	4	1	no	—
104	C₂²×C₄	—	4	3	no	—
105	C₂²×C₄	C₂×D₄×C₁₀₉	1	3	yes	—
106	D₈	—	8	6	no	—
107	M₁₆	—	4	6	no	—
108	C₂×C₈	—	2	6	no	—
109	C₄⋊C₄	—	2	6	no	—
110	C₂²⋊C₄	—	4	6	no	—
111	C₄²	—	2	6	no	—
112	C₄²	—	4	3	no	—
113	C₄²	C₂×D₄×C₁₀₉	1	3	yes	—
114	C₄²	#41×C₁₀₉	1	1	yes	characteristic
115	C₂³	—	4	3	no	—
116	C₂³	#127×C₁₀₉	1	1	yes	characteristic
117	Q₈	—	4	6	no	—
118	D₄	—	4	6	no	—
119	D₄	—	4	6	no	—
120	D₄	—	4	6	no	—
121	D₄	—	8	6	no	—
122	C₂×C₄	—	2	3	no	—
123	C₂×C₄	—	2	3	no	—
124	C₂×C₄	—	4	6	no	—
125	C₂×C₄	—	4	3	no	—
126	C₂×C₄	—	8	3	no	—
127	C₂×C₄	—	8	3	no	—
128	C₂×C₄	#120×C₁₀₉	1	3	yes	—
129	C₂×C₄	#121×C₁₀₉	1	3	yes	—
130	C₈	—	4	6	no	—
131	C₂²	—	2	3	no	—
132	C₂²	—	8	3	no	—
133	C₂²	—	8	6	no	—
134	C₂²	#392×C₁₀₉	1	1	yes	characteristic
135	C₄	—	2	3	no	—
136	C₄	—	4	4	no	—
137	C₄	—	8	3	no	—
138	C₄	#453×C₁₀₉	1	3	yes	—
139	C₄	#454×C₁₀₉	1	3	yes	—
140	C₂	—	4	1	no	—
141	C₂	—	16	3	no	—
142	C₂	C₁₀₉×#2006	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉×#3253

C₂

yes

maximal, complemented

C₁₀₉×#1962

C₂

yes

maximal, complemented

C₁₀₉×#1309

C₂

yes

maximal, characteristic, complemented

#532×C₁₀₉

C₂²

yes

—

#476×C₁₀₉

—

#469×C₁₀₉

—

#459×C₁₀₉

—

#423×C₁₀₉

C₂²

yes

—

C₂×#104×C₁₀₉

—

C₄³×C₁₀₉

C₂²

yes

characteristic

C₂×#41×C₁₀₉

—

C₂×D₈×C₁₀₉

—

C₂×M₁₆×C₁₀₉

D₄

yes

—

C₄⋊Q₈×C₁₀₉

D₄

yes

—

#127×C₁₀₉

D₄

yes

—

#124×C₁₀₉

—

#121×C₁₀₉

—

C₄×D₄×C₁₀₉

—

C₂×C₄²×C₁₀₉

—

C₂×C₄²×C₁₀₉

C₂³

yes

characteristic

#106×C₁₀₉

—

C₄⋊C₈×C₁₀₉

—

#104×C₁₀₉

—

#102×C₁₀₉

—

#41×C₁₀₉

—

#41×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×D₄

yes

—

D₈×C₁₀₉

—

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

C₂×D₄

yes

—

C₄²×C₁₀₉

#41

yes

characteristic

C₂³×C₁₀₉

—

C₂³×C₁₀₉

#127

yes

characteristic

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

#120

yes

—

C₂×C₄×C₁₀₉

#121

yes

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#392

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#453

yes

—

C₄×C₁₀₉

#454

yes

—

#10300

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#2006

yes

—

#3253

C₂×C₁₀₉

yes

complemented

#1962

C₂×C₁₀₉

yes

complemented

#1309

C₂×C₁₀₉

yes

characteristic, complemented

C₁₀₉

#10300

yes

characteristic, direct product factor, Sylow

#532

C₂²×C₁₀₉

yes

—

—

—

—

C₂²×C₁₀₉

yes

—

C₂×#104

—

C₄³

C₂²×C₁₀₉

yes

characteristic

C₂×#41

—

C₂×D₈

—

C₂×M₁₆

D₄×C₁₀₉

yes

—

C₄⋊Q₈

D₄×C₁₀₉

yes

—

#127

D₄×C₁₀₉

yes

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

—

C₂×C₄²

C₂³×C₁₀₉

yes

characteristic

#106

—

C₄⋊C₈

—

—

—

—

—

C₂×Q₈

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

C₂×D₄×C₁₀₉

yes

—

106

D₈

—

107

M₁₆

—

108

C₂×C₈

—

109

C₄⋊C₄

—

110

C₂²⋊C₄

—

111

C₄²

—

112

C₄²

—

113

C₄²

C₂×D₄×C₁₀₉

yes

—

114

C₄²

#41×C₁₀₉

yes

characteristic

115

C₂³

—

116

C₂³

#127×C₁₀₉

yes

characteristic

117

Q₈

—

118

D₄

—

119

D₄

—

120

D₄

—

121

D₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

#120×C₁₀₉

yes

—

129

C₂×C₄

#121×C₁₀₉

yes

—

130

C₈

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

#392×C₁₀₉

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#453×C₁₀₉

yes

—

139

C₄

#454×C₁₀₉

yes

—

140

C₂

—

141

C₂

—

142

C₂

C₁₀₉×#2006

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₀₉

#392

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₁₀₉

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

N	H ≅ G/N	type
1. C₁₀₉×#3253	C₂	semidirect
2. C₁₀₉×#1962	C₂	semidirect
3. C₁₀₉×#1309	C₂	semidirect
68. #10300	C₁₀₉	direct (N × H)
72. #3253	C₂×C₁₀₉	semidirect
73. #1962	C₂×C₁₀₉	semidirect
74. #1309	C₂×C₁₀₉	semidirect
75. C₁₀₉	#10300	direct (N × H)

H ≅ G/N

type

1. C₁₀₉×#3253

C₂

semidirect

2. C₁₀₉×#1962

C₂

semidirect

3. C₁₀₉×#1309

C₂

semidirect

68. #10300

C₁₀₉

direct (N × H)

72. #3253

C₂×C₁₀₉

semidirect

73. #1962

C₂×C₁₀₉

semidirect

74. #1309

C₂×C₁₀₉

semidirect

75. C₁₀₉

#10300

direct (N × H)

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

subgroup	index
C₁₀₉×#3253	2
C₁₀₉×#1962	2
C₁₀₉×#1309	2
#10300	109

subgroup

index

C₁₀₉×#3253

C₁₀₉×#1962

C₁₀₉×#1309

#10300

109

Subgroups — C₁₀₉×#10300 #42128294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#10300 #42128294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#10300 #42128294