Group #42128352 — subgroups

Subgroups — C₁₀₉×#10358 #42128352

Order 27904.

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

normal subgroups only

This group has 484 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×#2844	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₀₉×#1961	C₂	1	3	yes	maximal
3	C₁₀₉×#1295	C₂	1	3	yes	maximal, complemented
4	C₂×#127×C₁₀₉	C₂²	1	1	yes	characteristic
5	C₂×#102×C₁₀₉	—	2	3	no	—
6	C₂×#102×C₁₀₉	—	2	3	no	—
7	C₂×#97×C₁₀₉	C₂²	1	3	yes	—
8	#400×C₁₀₉	—	2	3	no	—
9	#395×C₁₀₉	—	2	3	no	—
10	#382×C₁₀₉	C₂²	1	3	yes	—
11	C₂²×Q₈×C₁₀₉	—	2	1	no	—
12	C₂²×D₄×C₁₀₉	—	2	3	no	—
13	C₂²×C₈×C₁₀₉	—	2	3	no	—
14	#127×C₁₀₉	—	2	3	no	—
15	#127×C₁₀₉	Q₈	1	2	yes	complemented
16	C₂×C₄⋊C₄×C₁₀₉	—	2	3	no	—
17	C₂×C₄⋊C₄×C₁₀₉	D₄	1	3	yes	—
18	C₂×C₄⋊C₄×C₁₀₉	D₄	1	3	yes	—
19	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
20	#102×C₁₀₉	—	4	6	no	—
21	#102×C₁₀₉	—	4	6	no	—
22	#97×C₁₀₉	—	2	6	no	—
23	#95×C₁₀₉	—	2	3	no	—
24	#95×C₁₀₉	—	2	1	no	—
25	#95×C₁₀₉	—	2	6	no	—
26	C₂⁴×C₁₀₉	—	4	1	no	—
27	C₂×Q₈×C₁₀₉	—	8	3	no	—
28	C₂×D₄×C₁₀₉	—	2	6	no	—
29	C₂×D₄×C₁₀₉	—	4	3	no	—
30	C₂×D₄×C₁₀₉	—	4	3	no	—
31	C₂×D₄×C₁₀₉	—	4	3	no	—
32	C₂×D₄×C₁₀₉	—	4	3	no	—
33	C₂×D₄×C₁₀₉	—	4	3	no	—
34	C₂²×C₄×C₁₀₉	—	2	3	no	—
35	C₂²×C₄×C₁₀₉	—	2	3	no	—
36	C₂²×C₄×C₁₀₉	C₂×D₄	1	3	yes	—
37	C₂×C₈×C₁₀₉	—	2	6	no	—
38	C₂×C₈×C₁₀₉	—	4	3	no	—
39	C₂×C₈×C₁₀₉	—	4	3	no	—
40	C₄⋊C₄×C₁₀₉	—	4	3	no	—
41	C₄⋊C₄×C₁₀₉	—	4	3	no	—
42	C₄⋊C₄×C₁₀₉	—	8	3	no	—
43	C₄²×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
44	C₄²×C₁₀₉	#41	1	3	yes	—
45	C₂³×C₁₀₉	—	4	2	no	—
46	C₂³×C₁₀₉	—	8	3	no	—
47	C₂³×C₁₀₉	—	8	3	no	—
48	C₂³×C₁₀₉	#120	1	1	yes	characteristic
49	Q₈×C₁₀₉	—	8	4	no	—
50	D₄×C₁₀₉	—	4	6	no	—
51	D₄×C₁₀₉	—	4	6	no	—
52	D₄×C₁₀₉	—	8	3	no	—
53	D₄×C₁₀₉	—	8	3	no	—
54	C₂×C₄×C₁₀₉	—	2	3	no	—
55	C₂×C₄×C₁₀₉	—	2	3	no	—
56	C₂×C₄×C₁₀₉	—	4	3	no	—
57	C₂×C₄×C₁₀₉	—	4	3	no	—
58	C₂×C₄×C₁₀₉	—	4	6	no	—
59	C₂×C₄×C₁₀₉	—	4	6	no	—
60	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	3	yes	—
61	C₂×C₄×C₁₀₉	#124	1	3	yes	—
62	C₈×C₁₀₉	—	4	6	no	—
63	C₂²×C₁₀₉	—	8	6	no	—
64	C₂²×C₁₀₉	—	16	1	no	—
65	C₂²×C₁₀₉	—	16	3	no	—
66	C₂²×C₁₀₉	#393	1	1	yes	characteristic
67	C₂²×C₁₀₉	#453	1	3	yes	—
68	C₂²×C₁₀₉	#454	1	3	yes	—
69	C₄×C₁₀₉	—	2	3	no	—
70	C₄×C₁₀₉	—	2	3	no	—
71	C₄×C₁₀₉	—	8	3	no	—
72	C₄×C₁₀₉	—	8	3	no	—
73	#10358	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂×C₁₀₉	—	16	2	no	—
75	C₂×C₁₀₉	#1655	1	1	yes	characteristic
76	C₂×C₁₀₉	#1657	1	3	yes	—
77	C₂×C₁₀₉	#2027	1	3	yes	—
78	#2844	C₂×C₁₀₉	1	1	yes	characteristic, complemented
79	#1961	C₂×C₁₀₉	1	3	yes	—
80	#1295	C₂×C₁₀₉	1	3	yes	complemented
81	C₁₀₉	#10358	1	1	yes	characteristic, direct product factor, Sylow
82	C₂×#127	C₂²×C₁₀₉	1	1	yes	characteristic
83	C₂×#102	—	2	3	no	—
84	C₂×#102	—	2	3	no	—
85	C₂×#97	C₂²×C₁₀₉	1	3	yes	—
86	#400	—	2	3	no	—
87	#395	—	2	3	no	—
88	#382	C₂²×C₁₀₉	1	3	yes	—
89	C₂²×Q₈	—	2	1	no	—
90	C₂²×D₄	—	2	3	no	—
91	C₂²×C₈	—	2	3	no	—
92	#127	—	2	3	no	—
93	#127	Q₈×C₁₀₉	1	2	yes	complemented
94	C₂×C₄⋊C₄	—	2	3	no	—
95	C₂×C₄⋊C₄	D₄×C₁₀₉	1	3	yes	—
96	C₂×C₄⋊C₄	D₄×C₁₀₉	1	3	yes	—
97	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
98	#102	—	4	6	no	—
99	#102	—	4	6	no	—
100	#97	—	2	6	no	—
101	#95	—	2	3	no	—
102	#95	—	2	1	no	—
103	#95	—	2	6	no	—
104	C₂⁴	—	4	1	no	—
105	C₂×Q₈	—	8	3	no	—
106	C₂×D₄	—	2	6	no	—
107	C₂×D₄	—	4	3	no	—
108	C₂×D₄	—	4	3	no	—
109	C₂×D₄	—	4	3	no	—
110	C₂×D₄	—	4	3	no	—
111	C₂×D₄	—	4	3	no	—
112	C₂²×C₄	—	2	3	no	—
113	C₂²×C₄	—	2	3	no	—
114	C₂²×C₄	C₂×D₄×C₁₀₉	1	3	yes	—
115	C₂×C₈	—	2	6	no	—
116	C₂×C₈	—	4	3	no	—
117	C₂×C₈	—	4	3	no	—
118	C₄⋊C₄	—	4	3	no	—
119	C₄⋊C₄	—	4	3	no	—
120	C₄⋊C₄	—	8	3	no	—
121	C₄²	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
122	C₄²	#41×C₁₀₉	1	3	yes	—
123	C₂³	—	4	2	no	—
124	C₂³	—	8	3	no	—
125	C₂³	—	8	3	no	—
126	C₂³	#120×C₁₀₉	1	1	yes	characteristic
127	Q₈	—	8	4	no	—
128	D₄	—	4	6	no	—
129	D₄	—	4	6	no	—
130	D₄	—	8	3	no	—
131	D₄	—	8	3	no	—
132	C₂×C₄	—	2	3	no	—
133	C₂×C₄	—	2	3	no	—
134	C₂×C₄	—	4	3	no	—
135	C₂×C₄	—	4	3	no	—
136	C₂×C₄	—	4	6	no	—
137	C₂×C₄	—	4	6	no	—
138	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	3	yes	—
139	C₂×C₄	#124×C₁₀₉	1	3	yes	—
140	C₈	—	4	6	no	—
141	C₂²	—	8	6	no	—
142	C₂²	—	16	1	no	—
143	C₂²	—	16	3	no	—
144	C₂²	#393×C₁₀₉	1	1	yes	characteristic
145	C₂²	#453×C₁₀₉	1	3	yes	—
146	C₂²	#454×C₁₀₉	1	3	yes	—
147	C₄	—	2	3	no	—
148	C₄	—	2	3	no	—
149	C₄	—	8	3	no	—
150	C₄	—	8	3	no	—
151	C₂	—	16	2	no	—
152	C₂	C₁₀₉×#1655	1	1	yes	characteristic
153	C₂	C₁₀₉×#1657	1	3	yes	—
154	C₂	C₁₀₉×#2027	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉×#2844

C₂

yes

maximal, characteristic, complemented

C₁₀₉×#1961

C₂

yes

maximal

C₁₀₉×#1295

C₂

yes

maximal, complemented

C₂×#127×C₁₀₉

C₂²

yes

characteristic

C₂×#102×C₁₀₉

—

C₂×#102×C₁₀₉

—

C₂×#97×C₁₀₉

C₂²

yes

—

#400×C₁₀₉

—

#395×C₁₀₉

—

#382×C₁₀₉

C₂²

yes

—

C₂²×Q₈×C₁₀₉

—

C₂²×D₄×C₁₀₉

—

C₂²×C₈×C₁₀₉

—

#127×C₁₀₉

—

#127×C₁₀₉

Q₈

yes

complemented

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

D₄

yes

—

C₂×C₄⋊C₄×C₁₀₉

D₄

yes

—

C₂×C₄²×C₁₀₉

C₂³

yes

characteristic

#102×C₁₀₉

—

#102×C₁₀₉

—

#97×C₁₀₉

—

#95×C₁₀₉

—

#95×C₁₀₉

—

#95×C₁₀₉

—

C₂⁴×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂×Q₈

yes

characteristic

C₄²×C₁₀₉

#41

yes

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

#120

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

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₀₉

#124

yes

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#393

yes

characteristic

C₂²×C₁₀₉

#453

yes

—

C₂²×C₁₀₉

#454

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

#10358

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

—

C₂×C₁₀₉

#1655

yes

characteristic

C₂×C₁₀₉

#1657

yes

—

C₂×C₁₀₉

#2027

yes

—

#2844

C₂×C₁₀₉

yes

characteristic, complemented

#1961

C₂×C₁₀₉

yes

—

#1295

C₂×C₁₀₉

yes

complemented

C₁₀₉

#10358

yes

characteristic, direct product factor, Sylow

C₂×#127

C₂²×C₁₀₉

yes

characteristic

C₂×#102

—

C₂×#102

—

C₂×#97

C₂²×C₁₀₉

yes

—

#400

—

#395

—

#382

C₂²×C₁₀₉

yes

—

C₂²×Q₈

—

C₂²×D₄

—

C₂²×C₈

—

#127

—

#127

Q₈×C₁₀₉

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄×C₁₀₉

yes

—

C₂×C₄⋊C₄

D₄×C₁₀₉

yes

—

C₂×C₄²

C₂³×C₁₀₉

yes

characteristic

#102

—

#102

—

100

#97

—

101

#95

—

102

#95

—

103

#95

—

104

C₂⁴

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂×D₄×C₁₀₉

yes

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄²

C₂×Q₈×C₁₀₉

yes

characteristic

122

C₄²

#41×C₁₀₉

yes

—

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

#120×C₁₀₉

yes

characteristic

127

Q₈

—

128

D₄

—

129

D₄

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

C₂²⋊Q₈×C₁₀₉

yes

—

139

C₂×C₄

#124×C₁₀₉

yes

—

140

C₈

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

#393×C₁₀₉

yes

characteristic

145

C₂²

#453×C₁₀₉

yes

—

146

C₂²

#454×C₁₀₉

yes

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₂

—

152

C₂

C₁₀₉×#1655

yes

characteristic

153

C₂

C₁₀₉×#1657

yes

—

154

C₂

C₁₀₉×#2027

yes

—

	subgroup	G/H
center, socle	C₂³×C₁₀₉	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
prime radical	C₂×#127×C₁₀₉	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

subgroup

G/H

center, socle

C₂³×C₁₀₉

#120

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₁₀₉

prime radical

C₂×#127×C₁₀₉

C₂²

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

N	H ≅ G/N	type
1. C₁₀₉×#2844	C₂	semidirect
3. C₁₀₉×#1295	C₂	semidirect
15. #127×C₁₀₉	Q₈	semidirect
73. #10358	C₁₀₉	direct (N × H)
78. #2844	C₂×C₁₀₉	semidirect
80. #1295	C₂×C₁₀₉	semidirect
81. C₁₀₉	#10358	direct (N × H)
93. #127	Q₈×C₁₀₉	semidirect

H ≅ G/N

type

1. C₁₀₉×#2844

C₂

semidirect

3. C₁₀₉×#1295

C₂

semidirect

15. #127×C₁₀₉

Q₈

semidirect

73. #10358

C₁₀₉

direct (N × H)

78. #2844

C₂×C₁₀₉

semidirect

80. #1295

C₂×C₁₀₉

semidirect

81. C₁₀₉

#10358

direct (N × H)

93. #127

Q₈×C₁₀₉

semidirect

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

subgroup	index
C₁₀₉×#2844	2
C₁₀₉×#1961	2
C₁₀₉×#1295	2
#10358	109

subgroup

index

C₁₀₉×#2844

C₁₀₉×#1961

C₁₀₉×#1295

#10358

109

Subgroups — C₁₀₉×#10358 #42128352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#10358 #42128352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#10358 #42128352