Group #42130046 — subgroups

Subgroups — C₁₆×Q₁₆×C₁₀₉ #42130046

Order 27904.

218 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 134 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	Q₈×C₁₆×C₁₀₉	C₂	1	2	yes	maximal
2	C₈×Q₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1370	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×Q₈⋊C₁₆	C₂	1	2	yes	maximal
5	C₈×C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₈×Q₈×C₁₀₉	C₂²	1	2	yes	—
7	C₄×Q₁₆×C₁₀₉	C₄	1	1	yes	characteristic
8	C₄⋊C₁₆×C₁₀₉	—	2	2	no	—
9	C₄⋊C₁₆×C₁₀₉	C₂²	1	2	yes	—
10	C₄×C₁₆×C₁₀₉	—	2	2	no	—
11	C₄×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₄×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
13	#335×C₁₀₉	C₄	1	1	yes	characteristic
14	Q₈⋊C₈×C₁₀₉	C₄	1	2	yes	—
15	C₈²×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₁₀₉	C₈	1	1	yes	characteristic
17	C₄×Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
18	C₂×C₁₆×C₁₀₉	—	2	1	no	—
19	C₂×C₁₆×C₁₀₉	—	2	2	no	—
20	C₂×C₁₆×C₁₀₉	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₁₀₉	D₄	1	1	yes	characteristic
22	#107×C₁₀₉	C₈	1	1	yes	characteristic
23	C₄⋊C₈×C₁₀₉	—	2	2	no	—
24	C₄⋊C₈×C₁₀₉	C₂×C₄	1	2	yes	—
25	#103×C₁₀₉	C₈	1	2	yes	—
26	C₄×C₈×C₁₀₉	—	2	2	no	—
27	C₄×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
30	C₂×Q₈×C₁₀₉	C₂×C₈	1	2	yes	—
31	Q₁₆×C₁₀₉	C₁₆	1	4	yes	direct product factor
32	C₂×C₈×C₁₀₉	—	2	1	no	—
33	C₂×C₈×C₁₀₉	—	2	2	no	—
34	C₂×C₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₁₀₉	#41	1	1	yes	characteristic
38	C₄⋊C₄×C₁₀₉	—	2	2	no	—
39	C₄⋊C₄×C₁₀₉	C₂×C₈	1	2	yes	—
40	C₄²×C₁₀₉	—	2	2	no	—
41	C₄²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
42	C₁₆×C₁₀₉	—	2	2	no	—
43	C₁₆×C₁₀₉	—	2	1	no	—
44	C₁₆×C₁₀₉	—	4	2	no	—
45	C₁₆×C₁₀₉	Q₁₆	1	2	yes	direct product factor
46	Q₈×C₁₀₉	—	2	2	no	—
47	Q₈×C₁₀₉	C₂×C₁₆	1	4	yes	—
48	C₂×C₄×C₁₀₉	—	2	2	no	—
49	C₂×C₄×C₁₀₉	—	2	2	no	—
50	C₂×C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
52	C₂×C₄×C₁₀₉	#131	1	1	yes	characteristic
53	C₈×C₁₀₉	—	2	2	no	—
54	C₈×C₁₀₉	—	2	1	no	—
55	C₈×C₁₀₉	—	2	1	no	—
56	C₈×C₁₀₉	—	4	2	no	—
57	C₈×C₁₀₉	C₂×C₁₆	1	2	yes	—
58	C₈×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
59	C₈×C₁₀₉	#135	1	1	yes	characteristic
60	C₂²×C₁₀₉	C₈×D₄	1	1	yes	characteristic
61	C₄×C₁₀₉	—	2	4	no	—
62	C₄×C₁₀₉	—	2	1	no	—
63	C₄×C₁₀₉	—	4	2	no	—
64	C₄×C₁₀₉	C₄×Q₁₆	1	1	yes	characteristic
65	C₄×C₁₀₉	#443	1	1	yes	characteristic
66	C₄×C₁₀₉	C₂²×C₁₆	1	1	yes	characteristic
67	C₄×C₁₀₉	#504	1	1	yes	characteristic
68	C₁₆×Q₁₆	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂×C₁₀₉	C₈×Q₁₆	1	1	yes	characteristic
70	C₂×C₁₀₉	D₄×C₁₆	1	1	yes	characteristic
71	C₂×C₁₀₉	#2170	1	1	yes	characteristic
72	Q₈×C₁₆	C₂×C₁₀₉	1	2	yes	—
73	C₈×Q₁₆	C₂×C₁₀₉	1	1	yes	characteristic
74	#1370	C₂×C₁₀₉	1	1	yes	characteristic
75	Q₈⋊C₁₆	C₂×C₁₀₉	1	2	yes	—
76	C₈×C₁₆	C₂×C₁₀₉	1	1	yes	characteristic
77	C₁₀₉	C₁₆×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
78	C₈×Q₈	C₂²×C₁₀₉	1	2	yes	—
79	C₄×Q₁₆	C₄×C₁₀₉	1	1	yes	characteristic
80	C₄⋊C₁₆	—	2	2	no	—
81	C₄⋊C₁₆	C₂²×C₁₀₉	1	2	yes	—
82	C₄×C₁₆	—	2	2	no	—
83	C₄×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
84	C₄×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
85	#335	C₄×C₁₀₉	1	1	yes	characteristic
86	Q₈⋊C₈	C₄×C₁₀₉	1	2	yes	—
87	C₈²	C₂²×C₁₀₉	1	1	yes	characteristic
88	C₂×Q₁₆	C₈×C₁₀₉	1	1	yes	characteristic
89	C₄×Q₈	C₂×C₄×C₁₀₉	1	2	yes	—
90	C₂×C₁₆	—	2	1	no	—
91	C₂×C₁₆	—	2	2	no	—
92	C₂×C₁₆	D₄×C₁₀₉	1	1	yes	characteristic
93	C₂×C₁₆	D₄×C₁₀₉	1	1	yes	characteristic
94	#107	C₈×C₁₀₉	1	1	yes	characteristic
95	C₄⋊C₈	—	2	2	no	—
96	C₄⋊C₈	C₂×C₄×C₁₀₉	1	2	yes	—
97	#103	C₈×C₁₀₉	1	2	yes	—
98	C₄×C₈	—	2	2	no	—
99	C₄×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
100	C₄×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
101	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
102	C₂×Q₈	C₂×C₈×C₁₀₉	1	2	yes	—
103	Q₁₆	C₁₆×C₁₀₉	1	4	yes	direct product factor
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	C₂×C₈×C₁₀₉	1	1	yes	characteristic
107	C₂×C₈	C₂×C₈×C₁₀₉	1	1	yes	characteristic
108	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
109	C₂×C₈	#41×C₁₀₉	1	1	yes	characteristic
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	C₂×C₈×C₁₀₉	1	2	yes	—
112	C₄²	—	2	2	no	—
113	C₄²	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
114	C₁₆	—	2	2	no	—
115	C₁₆	—	2	1	no	—
116	C₁₆	—	4	2	no	—
117	C₁₆	Q₁₆×C₁₀₉	1	2	yes	direct product factor
118	Q₈	—	2	2	no	—
119	Q₈	C₂×C₁₆×C₁₀₉	1	4	yes	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	C₄×D₄×C₁₀₉	1	1	yes	characteristic
123	C₂×C₄	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
124	C₂×C₄	#131×C₁₀₉	1	1	yes	characteristic
125	C₈	—	2	2	no	—
126	C₈	—	2	1	no	—
127	C₈	—	2	1	no	—
128	C₈	—	4	2	no	—
129	C₈	C₂×C₁₆×C₁₀₉	1	2	yes	—
130	C₈	C₂×Q₁₆×C₁₀₉	1	1	yes	characteristic
131	C₈	#135×C₁₀₉	1	1	yes	characteristic
132	C₂²	C₈×D₄×C₁₀₉	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	4	no	—
135	C₄	—	4	2	no	—
136	C₄	C₄×Q₁₆×C₁₀₉	1	1	yes	characteristic
137	C₄	#443×C₁₀₉	1	1	yes	characteristic
138	C₄	C₂²×C₁₆×C₁₀₉	1	1	yes	characteristic
139	C₄	#504×C₁₀₉	1	1	yes	characteristic
140	C₂	C₈×Q₁₆×C₁₀₉	1	1	yes	characteristic
141	C₂	D₄×C₁₆×C₁₀₉	1	1	yes	characteristic
142	C₂	C₁₀₉×#2170	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×C₁₆×C₁₀₉

C₂

yes

maximal

C₈×Q₁₆×C₁₀₉

C₂

yes

maximal, characteristic

C₁₀₉×#1370

C₂

yes

maximal, characteristic

C₁₀₉×Q₈⋊C₁₆

C₂

yes

maximal

C₈×C₁₆×C₁₀₉

C₂

yes

maximal, characteristic

C₈×Q₈×C₁₀₉

C₂²

yes

—

C₄×Q₁₆×C₁₀₉

C₄

yes

characteristic

C₄⋊C₁₆×C₁₀₉

—

C₄⋊C₁₆×C₁₀₉

C₂²

yes

—

C₄×C₁₆×C₁₀₉

—

C₄×C₁₆×C₁₀₉

C₂²

yes

characteristic

C₄×C₁₆×C₁₀₉

C₂²

yes

characteristic

#335×C₁₀₉

C₄

yes

characteristic

Q₈⋊C₈×C₁₀₉

C₄

yes

—

C₈²×C₁₀₉

C₂²

yes

characteristic

C₂×Q₁₆×C₁₀₉

C₈

yes

characteristic

C₄×Q₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₁₆×C₁₀₉

—

C₂×C₁₆×C₁₀₉

—

C₂×C₁₆×C₁₀₉

D₄

yes

characteristic

C₂×C₁₆×C₁₀₉

D₄

yes

characteristic

#107×C₁₀₉

C₈

yes

characteristic

C₄⋊C₈×C₁₀₉

—

C₄⋊C₈×C₁₀₉

C₂×C₄

yes

—

#103×C₁₀₉

C₈

yes

—

C₄×C₈×C₁₀₉

—

C₄×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₄×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₄×C₈×C₁₀₉

C₂³

yes

characteristic

C₂×Q₈×C₁₀₉

C₂×C₈

yes

—

Q₁₆×C₁₀₉

C₁₆

yes

direct product factor

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₈×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₈×C₁₀₉

#41

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂×C₈

yes

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

C₂²×C₄

yes

characteristic

C₁₆×C₁₀₉

—

C₁₆×C₁₀₉

—

C₁₆×C₁₀₉

—

C₁₆×C₁₀₉

Q₁₆

yes

direct product factor

Q₈×C₁₀₉

—

Q₈×C₁₀₉

C₂×C₁₆

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₀₉

#131

yes

characteristic

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₂×C₁₆

yes

—

C₈×C₁₀₉

C₂×Q₁₆

yes

characteristic

C₈×C₁₀₉

#135

yes

characteristic

C₂²×C₁₀₉

C₈×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×Q₁₆

yes

characteristic

C₄×C₁₀₉

#443

yes

characteristic

C₄×C₁₀₉

C₂²×C₁₆

yes

characteristic

C₄×C₁₀₉

#504

yes

characteristic

C₁₆×Q₁₆

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

C₈×Q₁₆

yes

characteristic

C₂×C₁₀₉

D₄×C₁₆

yes

characteristic

C₂×C₁₀₉

#2170

yes

characteristic

Q₈×C₁₆

C₂×C₁₀₉

yes

—

C₈×Q₁₆

C₂×C₁₀₉

yes

characteristic

#1370

C₂×C₁₀₉

yes

characteristic

Q₈⋊C₁₆

C₂×C₁₀₉

yes

—

C₈×C₁₆

C₂×C₁₀₉

yes

characteristic

C₁₀₉

C₁₆×Q₁₆

yes

characteristic, direct product factor, Sylow

C₈×Q₈

C₂²×C₁₀₉

yes

—

C₄×Q₁₆

C₄×C₁₀₉

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂²×C₁₀₉

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂²×C₁₀₉

yes

characteristic

C₄×C₁₆

C₂²×C₁₀₉

yes

characteristic

#335

C₄×C₁₀₉

yes

characteristic

Q₈⋊C₈

C₄×C₁₀₉

yes

—

C₈²

C₂²×C₁₀₉

yes

characteristic

C₂×Q₁₆

C₈×C₁₀₉

yes

characteristic

C₄×Q₈

C₂×C₄×C₁₀₉

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₄×C₁₀₉

yes

characteristic

C₂×C₁₆

D₄×C₁₀₉

yes

characteristic

#107

C₈×C₁₀₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×C₄×C₁₀₉

yes

—

#103

C₈×C₁₀₉

yes

—

C₄×C₈

—

C₄×C₈

C₂×C₄×C₁₀₉

yes

characteristic

100

C₄×C₈

C₂×C₄×C₁₀₉

yes

characteristic

101

C₄×C₈

C₂³×C₁₀₉

yes

characteristic

102

C₂×Q₈

C₂×C₈×C₁₀₉

yes

—

103

Q₁₆

C₁₆×C₁₀₉

yes

direct product factor

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₂×C₈×C₁₀₉

yes

characteristic

107

C₂×C₈

C₂×C₈×C₁₀₉

yes

characteristic

108

C₂×C₈

C₂×D₄×C₁₀₉

yes

characteristic

109

C₂×C₈

#41×C₁₀₉

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

C₂×C₈×C₁₀₉

yes

—

112

C₄²

—

113

C₄²

C₂²×C₄×C₁₀₉

yes

characteristic

114

C₁₆

—

115

C₁₆

—

116

C₁₆

—

117

C₁₆

Q₁₆×C₁₀₉

yes

direct product factor

118

Q₈

—

119

Q₈

C₂×C₁₆×C₁₀₉

yes

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₄×D₄×C₁₀₉

yes

characteristic

123

C₂×C₄

C₂²×C₈×C₁₀₉

yes

characteristic

124

C₂×C₄

#131×C₁₀₉

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

C₂×C₁₆×C₁₀₉

yes

—

130

C₈

C₂×Q₁₆×C₁₀₉

yes

characteristic

131

C₈

#135×C₁₀₉

yes

characteristic

132

C₂²

C₈×D₄×C₁₀₉

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

C₄×Q₁₆×C₁₀₉

yes

characteristic

137

C₄

#443×C₁₀₉

yes

characteristic

138

C₄

C₂²×C₁₆×C₁₀₉

yes

characteristic

139

C₄

#504×C₁₀₉

yes

characteristic

140

C₂

C₈×Q₁₆×C₁₀₉

yes

characteristic

141

C₂

D₄×C₁₆×C₁₀₉

yes

characteristic

142

C₂

C₁₀₉×#2170

yes

characteristic

	subgroup	G/H
center	C₂×C₁₆×C₁₀₉	D₄
commutator	C₄	C₂²×C₁₆×C₁₀₉
Frattini, cosocle	C₄×C₈	C₂³×C₁₀₉
socle, prime radical	C₂²×C₁₀₉	C₈×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₁₀₉ ◁ C₄×C₁₆×C₁₀₉ ◁ G

subgroup

G/H

center

C₂×C₁₆×C₁₀₉

D₄

commutator

C₄

C₂²×C₁₆×C₁₀₉

Frattini, cosocle

C₄×C₈

C₂³×C₁₀₉

socle, prime radical

C₂²×C₁₀₉

C₈×D₄

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₁₆×C₁₀₉ ◁ C₄×C₁₆×C₁₀₉ ◁ G

N	H ≅ G/N	type
31. Q₁₆×C₁₀₉	C₁₆	direct (N × H)
45. C₁₆×C₁₀₉	Q₁₆	direct (N × H)
68. C₁₆×Q₁₆	C₁₀₉	direct (N × H)
77. C₁₀₉	C₁₆×Q₁₆	direct (N × H)
103. Q₁₆	C₁₆×C₁₀₉	direct (N × H)
117. C₁₆	Q₁₆×C₁₀₉	direct (N × H)

H ≅ G/N

type

31. Q₁₆×C₁₀₉

C₁₆

direct (N × H)

45. C₁₆×C₁₀₉

Q₁₆

direct (N × H)

68. C₁₆×Q₁₆

C₁₀₉

direct (N × H)

77. C₁₀₉

C₁₆×Q₁₆

direct (N × H)

103. Q₁₆

C₁₆×C₁₀₉

direct (N × H)

117. C₁₆

Q₁₆×C₁₀₉

direct (N × H)

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

subgroup	index
Q₈×C₁₆×C₁₀₉	2
C₈×Q₁₆×C₁₀₉	2
C₁₀₉×#1370	2
C₁₀₉×Q₈⋊C₁₆	2
C₈×C₁₆×C₁₀₉	2
C₁₆×Q₁₆	109

subgroup

index

Q₈×C₁₆×C₁₀₉

C₈×Q₁₆×C₁₀₉

C₁₀₉×#1370

C₁₀₉×Q₈⋊C₁₆

C₈×C₁₆×C₁₀₉

C₁₆×Q₁₆

109

Subgroups — C₁₆×Q₁₆×C₁₀₉ #42130046

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C16×Q16×C109 #42130046

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₆×Q₁₆×C₁₀₉ #42130046