Group #42128205 — subgroups

Subgroups — C₁₀₉×#10211 #42128205

Order 27904.

406 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 94 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×#3237	C₂	1	3	yes	maximal
2	C₁₀₉×#1515	C₂	1	3	yes	maximal
3	C₁₀₉×#1308	C₂	1	1	yes	maximal, characteristic
4	#533×C₁₀₉	C₂²	1	3	yes	—
5	#501×C₁₀₉	—	2	3	no	—
6	#491×C₁₀₉	—	2	3	no	—
7	#485×C₁₀₉	—	2	6	no	—
8	#477×C₁₀₉	—	2	3	no	—
9	#475×C₁₀₉	—	2	3	no	—
10	#432×C₁₀₉	C₂²	1	3	yes	—
11	#383×C₁₀₉	C₂²	1	1	yes	characteristic
12	#328×C₁₀₉	—	2	6	no	—
13	C₄⋊Q₈×C₁₀₉	D₄	1	3	yes	—
14	C₄⋊Q₈×C₁₀₉	D₄	1	3	yes	—
15	#125×C₁₀₉	—	2	3	no	—
16	C₂²⋊Q₈×C₁₀₉	—	2	3	no	—
17	C₂²⋊Q₈×C₁₀₉	—	2	3	no	—
18	C₄×Q₈×C₁₀₉	—	2	3	no	—
19	C₄×Q₈×C₁₀₉	—	2	3	no	—
20	#117×C₁₀₉	—	2	3	no	—
21	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
22	#107×C₁₀₉	—	2	6	no	—
23	#106×C₁₀₉	—	2	6	no	—
24	C₄⋊C₈×C₁₀₉	—	2	3	no	—
25	#103×C₁₀₉	—	2	6	no	—
26	#103×C₁₀₉	—	2	6	no	—
27	C₂²⋊C₈×C₁₀₉	D₄	1	6	yes	—
28	#97×C₁₀₉	—	2	3	no	—
29	#95×C₁₀₉	—	2	3	no	—
30	C₂×Q₈×C₁₀₉	—	2	3	no	—
31	C₂×Q₈×C₁₀₉	—	2	3	no	—
32	C₂²×C₄×C₁₀₉	—	4	1	no	—
33	C₂²×C₄×C₁₀₉	C₂×D₄	1	3	yes	—
34	C₂×C₈×C₁₀₉	—	2	6	no	—
35	C₄⋊C₄×C₁₀₉	—	2	3	no	—
36	C₄⋊C₄×C₁₀₉	—	2	3	no	—
37	C₄⋊C₄×C₁₀₉	—	2	3	no	—
38	C₄⋊C₄×C₁₀₉	—	2	3	no	—
39	C₄⋊C₄×C₁₀₉	—	4	3	no	—
40	C₄⋊C₄×C₁₀₉	—	4	3	no	—
41	C₂²⋊C₄×C₁₀₉	—	4	3	no	—
42	C₄²×C₁₀₉	—	4	3	no	—
43	C₄²×C₁₀₉	C₂×D₄	1	3	yes	—
44	C₄²×C₁₀₉	#41	1	1	yes	characteristic
45	C₂³×C₁₀₉	#127	1	1	yes	characteristic
46	Q₈×C₁₀₉	—	4	6	no	—
47	Q₈×C₁₀₉	—	8	3	no	—
48	C₂×C₄×C₁₀₉	—	4	3	no	—
49	C₂×C₄×C₁₀₉	—	4	3	no	—
50	C₂×C₄×C₁₀₉	—	4	3	no	—
51	C₂×C₄×C₁₀₉	—	8	3	no	—
52	C₂×C₄×C₁₀₉	#120	1	3	yes	—
53	C₂×C₄×C₁₀₉	#121	1	3	yes	—
54	C₈×C₁₀₉	—	4	6	no	—
55	C₂²×C₁₀₉	—	2	3	no	—
56	C₂²×C₁₀₉	#392	1	1	yes	characteristic
57	C₄×C₁₀₉	—	2	3	no	—
58	C₄×C₁₀₉	—	2	3	no	—
59	C₄×C₁₀₉	—	8	3	no	—
60	C₄×C₁₀₉	—	8	3	no	—
61	C₄×C₁₀₉	—	16	1	no	—
62	#10211	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂×C₁₀₉	—	4	1	no	—
64	C₂×C₁₀₉	#2009	1	3	yes	—
65	#3237	C₂×C₁₀₉	1	3	yes	—
66	#1515	C₂×C₁₀₉	1	3	yes	—
67	#1308	C₂×C₁₀₉	1	1	yes	characteristic
68	C₁₀₉	#10211	1	1	yes	characteristic, direct product factor, Sylow
69	#533	C₂²×C₁₀₉	1	3	yes	—
70	#501	—	2	3	no	—
71	#491	—	2	3	no	—
72	#485	—	2	6	no	—
73	#477	—	2	3	no	—
74	#475	—	2	3	no	—
75	#432	C₂²×C₁₀₉	1	3	yes	—
76	#383	C₂²×C₁₀₉	1	1	yes	characteristic
77	#328	—	2	6	no	—
78	C₄⋊Q₈	D₄×C₁₀₉	1	3	yes	—
79	C₄⋊Q₈	D₄×C₁₀₉	1	3	yes	—
80	#125	—	2	3	no	—
81	C₂²⋊Q₈	—	2	3	no	—
82	C₂²⋊Q₈	—	2	3	no	—
83	C₄×Q₈	—	2	3	no	—
84	C₄×Q₈	—	2	3	no	—
85	#117	—	2	3	no	—
86	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
87	#107	—	2	6	no	—
88	#106	—	2	6	no	—
89	C₄⋊C₈	—	2	3	no	—
90	#103	—	2	6	no	—
91	#103	—	2	6	no	—
92	C₂²⋊C₈	D₄×C₁₀₉	1	6	yes	—
93	#97	—	2	3	no	—
94	#95	—	2	3	no	—
95	C₂×Q₈	—	2	3	no	—
96	C₂×Q₈	—	2	3	no	—
97	C₂²×C₄	—	4	1	no	—
98	C₂²×C₄	C₂×D₄×C₁₀₉	1	3	yes	—
99	C₂×C₈	—	2	6	no	—
100	C₄⋊C₄	—	2	3	no	—
101	C₄⋊C₄	—	2	3	no	—
102	C₄⋊C₄	—	2	3	no	—
103	C₄⋊C₄	—	2	3	no	—
104	C₄⋊C₄	—	4	3	no	—
105	C₄⋊C₄	—	4	3	no	—
106	C₂²⋊C₄	—	4	3	no	—
107	C₄²	—	4	3	no	—
108	C₄²	C₂×D₄×C₁₀₉	1	3	yes	—
109	C₄²	#41×C₁₀₉	1	1	yes	characteristic
110	C₂³	#127×C₁₀₉	1	1	yes	characteristic
111	Q₈	—	4	6	no	—
112	Q₈	—	8	3	no	—
113	C₂×C₄	—	4	3	no	—
114	C₂×C₄	—	4	3	no	—
115	C₂×C₄	—	4	3	no	—
116	C₂×C₄	—	8	3	no	—
117	C₂×C₄	#120×C₁₀₉	1	3	yes	—
118	C₂×C₄	#121×C₁₀₉	1	3	yes	—
119	C₈	—	4	6	no	—
120	C₂²	—	2	3	no	—
121	C₂²	#392×C₁₀₉	1	1	yes	characteristic
122	C₄	—	2	3	no	—
123	C₄	—	2	3	no	—
124	C₄	—	8	3	no	—
125	C₄	—	8	3	no	—
126	C₄	—	16	1	no	—
127	C₂	—	4	1	no	—
128	C₂	C₁₀₉×#2009	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉×#3237

C₂

yes

maximal

C₁₀₉×#1515

C₂

yes

maximal

C₁₀₉×#1308

C₂

yes

maximal, characteristic

#533×C₁₀₉

C₂²

yes

—

#501×C₁₀₉

—

#491×C₁₀₉

—

#485×C₁₀₉

—

#477×C₁₀₉

—

#475×C₁₀₉

—

#432×C₁₀₉

C₂²

yes

—

#383×C₁₀₉

C₂²

yes

characteristic

#328×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

D₄

yes

—

C₄⋊Q₈×C₁₀₉

D₄

yes

—

#125×C₁₀₉

—

C₂²⋊Q₈×C₁₀₉

—

C₂²⋊Q₈×C₁₀₉

—

C₄×Q₈×C₁₀₉

—

C₄×Q₈×C₁₀₉

—

#117×C₁₀₉

—

C₂×C₄²×C₁₀₉

C₂³

yes

characteristic

#107×C₁₀₉

—

#106×C₁₀₉

—

C₄⋊C₈×C₁₀₉

—

#103×C₁₀₉

—

#103×C₁₀₉

—

C₂²⋊C₈×C₁₀₉

D₄

yes

—

#97×C₁₀₉

—

#95×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₈×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

C₂×D₄

yes

—

C₄²×C₁₀₉

yes

C₂³×C₁₀₉

yes

Q₈×C₁₀₉

—

Q₈×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

#392

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

#10211

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

—

C₂×C₁₀₉

#2009

yes

—

#3237

C₂×C₁₀₉

yes

—

#1515

C₂×C₁₀₉

yes

—

#1308

C₂×C₁₀₉

yes

characteristic

C₁₀₉

#10211

yes

characteristic, direct product factor, Sylow

#533

C₂²×C₁₀₉

yes

—

—

—

—

—

—

C₂²×C₁₀₉

yes

—

#383

C₂²×C₁₀₉

yes

characteristic

#328

—

C₄⋊Q₈

D₄×C₁₀₉

yes

—

C₄⋊Q₈

D₄×C₁₀₉

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂³×C₁₀₉

yes

characteristic

#107

—

#106

—

C₄⋊C₈

—

#103

—

#103

—

C₂²⋊C₈

D₄×C₁₀₉

yes

—

#97

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄×C₁₀₉

yes

—

C₂×C₈

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₂²⋊C₄

—

107

C₄²

—

108

C₄²

C₂×D₄×C₁₀₉

yes

—

109

C₄²

#41×C₁₀₉

yes

characteristic

110

C₂³

#127×C₁₀₉

yes

characteristic

111

Q₈

—

112

Q₈

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

#120×C₁₀₉

yes

—

118

C₂×C₄

#121×C₁₀₉

yes

—

119

C₈

—

120

C₂²

—

121

C₂²

#392×C₁₀₉

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₂

—

128

C₂

C₁₀₉×#2009

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
prime radical	C₂³×C₁₀₉	#127
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

prime radical

C₂³×C₁₀₉

#127

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

N	H ≅ G/N	type
62. #10211	C₁₀₉	direct (N × H)
68. C₁₀₉	#10211	direct (N × H)

H ≅ G/N

type

62. #10211

C₁₀₉

direct (N × H)

68. C₁₀₉

#10211

direct (N × H)

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

subgroup	index
C₁₀₉×#3237	2
C₁₀₉×#1515	2
C₁₀₉×#1308	2
#10211	109

subgroup

index

C₁₀₉×#3237

C₁₀₉×#1515

C₁₀₉×#1308

#10211

109

Subgroups — C₁₀₉×#10211 #42128205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#10211 #42128205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#10211 #42128205