Group #42126133 — subgroups

Subgroups — C₁₀₉×#8139 #42126133

Order 27904.

330 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 174 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×#1726	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1452	C₂	1	2	yes	maximal
3	C₁₀₉×#1275	C₂	1	2	yes	maximal
4	C₁₀₉×#1274	C₂	1	2	yes	maximal
5	#431×C₁₀₉	C₂²	1	4	yes	—
6	C₂×#97×C₁₀₉	C₄	1	2	yes	—
7	C₂×#97×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₀₉	C₄	1	2	yes	—
9	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
10	#376×C₁₀₉	C₂²	1	1	yes	characteristic
11	#322×C₁₀₉	C₄	1	8	yes	complemented
12	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
17	C₄⋊C₈×C₁₀₉	—	2	4	no	—
18	C₂²⋊C₈×C₁₀₉	—	2	4	no	—
19	#97×C₁₀₉	—	2	4	no	—
20	#97×C₁₀₉	—	2	2	no	—
21	C₄×C₈×C₁₀₉	—	2	4	no	—
22	C₄×C₈×C₁₀₉	C₂×C₄	1	8	yes	—
23	C₄×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
24	#95×C₁₀₉	—	2	2	no	—
25	C₂²×C₄×C₁₀₉	—	2	2	no	—
26	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
28	C₂×C₈×C₁₀₉	—	2	2	no	—
29	C₂×C₈×C₁₀₉	—	2	2	no	—
30	C₂×C₈×C₁₀₉	—	2	2	no	—
31	C₂×C₈×C₁₀₉	—	2	2	no	—
32	C₂×C₈×C₁₀₉	—	2	8	no	—
33	C₂×C₈×C₁₀₉	C₄²	1	2	yes	—
34	C₂×C₈×C₁₀₉	C₄²	1	2	yes	—
35	C₂×C₈×C₁₀₉	M₁₆	1	4	yes	—
36	C₄²×C₁₀₉	—	2	2	no	—
37	C₄²×C₁₀₉	M₁₆	1	4	yes	—
38	C₄²×C₁₀₉	C₂²×C₄	1	2	yes	—
39	C₄²×C₁₀₉	#41	1	2	yes	—
40	C₂³×C₁₀₉	#117	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	—	2	4	no	—
42	C₂×C₄×C₁₀₉	—	4	2	no	—
43	C₂×C₄×C₁₀₉	—	4	4	no	—
44	C₂×C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
45	C₂×C₄×C₁₀₉	#117	1	4	yes	—
46	C₂×C₄×C₁₀₉	#117	1	1	yes	characteristic
47	C₂×C₄×C₁₀₉	C₂×M₁₆	1	2	yes	—
48	C₂×C₄×C₁₀₉	C₂×M₁₆	1	2	yes	—
49	C₈×C₁₀₉	—	2	2	no	—
50	C₈×C₁₀₉	—	2	2	no	—
51	C₈×C₁₀₉	—	2	4	no	—
52	C₈×C₁₀₉	—	4	8	no	—
53	C₂²×C₁₀₉	—	2	2	no	—
54	C₂²×C₁₀₉	#376	1	1	yes	characteristic
55	C₂²×C₁₀₉	#432	1	2	yes	—
56	C₄×C₁₀₉	—	2	4	no	—
57	C₄×C₁₀₉	—	4	4	no	—
58	C₄×C₁₀₉	C₄×M₁₆	1	1	yes	characteristic
59	C₄×C₁₀₉	C₄×M₁₆	1	1	yes	characteristic
60	C₄×C₁₀₉	#432	1	2	yes	—
61	#8139	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂×C₁₀₉	—	2	2	no	—
63	C₂×C₁₀₉	#1745	1	1	yes	characteristic
64	C₂×C₁₀₉	#1747	1	2	yes	—
65	#1726	C₂×C₁₀₉	1	1	yes	characteristic
66	#1452	C₂×C₁₀₉	1	2	yes	—
67	#1275	C₂×C₁₀₉	1	2	yes	—
68	#1274	C₂×C₁₀₉	1	2	yes	—
69	C₁₀₉	#8139	1	1	yes	characteristic, direct product factor, Sylow
70	#431	C₂²×C₁₀₉	1	4	yes	—
71	C₂×#97	C₄×C₁₀₉	1	2	yes	—
72	C₂×#97	C₂²×C₁₀₉	1	1	yes	characteristic
73	C₂×C₄×C₈	C₄×C₁₀₉	1	2	yes	—
74	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
75	#376	C₂²×C₁₀₉	1	1	yes	characteristic
76	#322	C₄×C₁₀₉	1	8	yes	complemented
77	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
78	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
79	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
80	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
81	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
82	C₄⋊C₈	—	2	4	no	—
83	C₂²⋊C₈	—	2	4	no	—
84	#97	—	2	4	no	—
85	#97	—	2	2	no	—
86	C₄×C₈	—	2	4	no	—
87	C₄×C₈	C₂×C₄×C₁₀₉	1	8	yes	—
88	C₄×C₈	C₂×C₄×C₁₀₉	1	4	yes	—
89	#95	—	2	2	no	—
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
92	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	8	no	—
98	C₂×C₈	C₄²×C₁₀₉	1	2	yes	—
99	C₂×C₈	C₄²×C₁₀₉	1	2	yes	—
100	C₂×C₈	M₁₆×C₁₀₉	1	4	yes	—
101	C₄²	—	2	2	no	—
102	C₄²	M₁₆×C₁₀₉	1	4	yes	—
103	C₄²	C₂²×C₄×C₁₀₉	1	2	yes	—
104	C₄²	#41×C₁₀₉	1	2	yes	—
105	C₂³	#117×C₁₀₉	1	1	yes	characteristic
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	4	no	—
109	C₂×C₄	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
110	C₂×C₄	#117×C₁₀₉	1	4	yes	—
111	C₂×C₄	#117×C₁₀₉	1	1	yes	characteristic
112	C₂×C₄	C₂×M₁₆×C₁₀₉	1	2	yes	—
113	C₂×C₄	C₂×M₁₆×C₁₀₉	1	2	yes	—
114	C₈	—	2	2	no	—
115	C₈	—	2	2	no	—
116	C₈	—	2	4	no	—
117	C₈	—	4	8	no	—
118	C₂²	—	2	2	no	—
119	C₂²	#376×C₁₀₉	1	1	yes	characteristic
120	C₂²	#432×C₁₀₉	1	2	yes	—
121	C₄	—	2	4	no	—
122	C₄	—	4	4	no	—
123	C₄	C₄×M₁₆×C₁₀₉	1	1	yes	characteristic
124	C₄	C₄×M₁₆×C₁₀₉	1	1	yes	characteristic
125	C₄	#432×C₁₀₉	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	C₁₀₉×#1745	1	1	yes	characteristic
128	C₂	C₁₀₉×#1747	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉×#1726

C₂

yes

maximal, characteristic

C₁₀₉×#1452

C₂

yes

maximal

C₁₀₉×#1275

C₂

yes

maximal

C₁₀₉×#1274

C₂

yes

maximal

#431×C₁₀₉

C₂²

yes

—

C₂×#97×C₁₀₉

C₄

yes

—

C₂×#97×C₁₀₉

C₂²

yes

characteristic

C₂×C₄×C₈×C₁₀₉

C₄

yes

—

C₂×C₄×C₈×C₁₀₉

C₂²

yes

characteristic

#376×C₁₀₉

C₂²

yes

characteristic

#322×C₁₀₉

C₄

yes

complemented

C₂²×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₄²×C₁₀₉

C₂×C₄

yes

—

C₂×C₄²×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₈×C₁₀₉

—

C₂²⋊C₈×C₁₀₉

—

#97×C₁₀₉

—

#97×C₁₀₉

—

C₄×C₈×C₁₀₉

—

C₄×C₈×C₁₀₉

C₂×C₄

yes

—

C₄×C₈×C₁₀₉

C₂×C₄

yes

—

#95×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

#41

yes

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₄²

yes

—

C₂×C₈×C₁₀₉

C₄²

yes

—

C₂×C₈×C₁₀₉

M₁₆

yes

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

M₁₆

yes

—

C₄²×C₁₀₉

C₂²×C₄

yes

—

C₄²×C₁₀₉

#41

yes

—

C₂³×C₁₀₉

#117

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×C₄²

yes

characteristic

C₂×C₄×C₁₀₉

#117

yes

—

C₂×C₄×C₁₀₉

#117

yes

characteristic

C₂×C₄×C₁₀₉

C₂×M₁₆

yes

—

C₂×C₄×C₁₀₉

C₂×M₁₆

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#376

yes

characteristic

C₂²×C₁₀₉

#432

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×M₁₆

yes

characteristic

C₄×C₁₀₉

C₄×M₁₆

yes

characteristic

C₄×C₁₀₉

#432

yes

—

#8139

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

—

C₂×C₁₀₉

#1745

yes

characteristic

C₂×C₁₀₉

#1747

yes

—

#1726

C₂×C₁₀₉

yes

characteristic

#1452

C₂×C₁₀₉

yes

—

#1275

C₂×C₁₀₉

yes

—

#1274

C₂×C₁₀₉

yes

—

C₁₀₉

#8139

yes

characteristic, direct product factor, Sylow

#431

C₂²×C₁₀₉

yes

—

C₂×#97

C₄×C₁₀₉

yes

—

C₂×#97

C₂²×C₁₀₉

yes

characteristic

C₂×C₄×C₈

C₄×C₁₀₉

yes

—

C₂×C₄×C₈

C₂²×C₁₀₉

yes

characteristic

#376

C₂²×C₁₀₉

yes

characteristic

#322

C₄×C₁₀₉

yes

complemented

C₂²×C₈

C₂×C₄×C₁₀₉

yes

characteristic

C₂²×C₈

C₂×C₄×C₁₀₉

yes

characteristic

C₂²×C₈

C₂×C₄×C₁₀₉

yes

—

C₂×C₄²

C₂×C₄×C₁₀₉

yes

—

C₂×C₄²

C₂³×C₁₀₉

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×C₄×C₁₀₉

yes

—

C₄×C₈

C₂×C₄×C₁₀₉

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

C₂²×C₄×C₁₀₉

yes

characteristic

C₂²×C₄

#41×C₁₀₉

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²×C₁₀₉

yes

—

C₂×C₈

C₄²×C₁₀₉

yes

—

100

C₂×C₈

M₁₆×C₁₀₉

yes

—

101

C₄²

—

102

C₄²

M₁₆×C₁₀₉

yes

—

103

C₄²

C₂²×C₄×C₁₀₉

yes

—

104

C₄²

#41×C₁₀₉

yes

—

105

C₂³

#117×C₁₀₉

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂×C₄²×C₁₀₉

yes

characteristic

110

C₂×C₄

#117×C₁₀₉

yes

—

111

C₂×C₄

#117×C₁₀₉

yes

characteristic

112

C₂×C₄

C₂×M₁₆×C₁₀₉

yes

—

113

C₂×C₄

C₂×M₁₆×C₁₀₉

yes

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

118

C₂²

—

119

C₂²

#376×C₁₀₉

yes

characteristic

120

C₂²

#432×C₁₀₉

yes

—

121

C₄

—

122

C₄

—

123

C₄

C₄×M₁₆×C₁₀₉

yes

characteristic

124

C₄

C₄×M₁₆×C₁₀₉

yes

characteristic

125

C₄

#432×C₁₀₉

yes

—

126

C₂

—

127

C₂

C₁₀₉×#1745

yes

characteristic

128

C₂

C₁₀₉×#1747

yes

—

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	#376
prime radical	C₂³×C₁₀₉	#117
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₀₉

C₂×C₄²

commutator

C₂×C₄

C₂×C₄²×C₁₀₉

Frattini, cosocle

C₂×C₄²

C₂³×C₁₀₉

C₂²×C₁₀₉

C₂³×C₁₀₉

G ▷ C₂×C₄ ▷ C₁

C₁ ◁ C₂×C₄×C₁₀₉ ◁ G

N	H ≅ G/N	type
11. #322×C₁₀₉	C₄	semidirect
61. #8139	C₁₀₉	direct (N × H)
69. C₁₀₉	#8139	direct (N × H)
76. #322	C₄×C₁₀₉	semidirect

H ≅ G/N

type

11. #322×C₁₀₉

C₄

semidirect

61. #8139

C₁₀₉

direct (N × H)

69. C₁₀₉

#8139

direct (N × H)

76. #322

C₄×C₁₀₉

semidirect

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

subgroup	index
C₁₀₉×#1726	2
C₁₀₉×#1452	2
C₁₀₉×#1275	2
C₁₀₉×#1274	2
#8139	109

subgroup

index

C₁₀₉×#1726

C₁₀₉×#1452

C₁₀₉×#1275

C₁₀₉×#1274

#8139

109

Subgroups — C₁₀₉×#8139 #42126133

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#8139 #42126133

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#8139 #42126133