Group #42128313 — subgroups

Subgroups — C₁₀₉×#10319 #42128313

Order 27904.

538 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 162 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	C₂×#487×C₁₀₉	C₂	1	3	yes	maximal
2	C₁₀₉×#1990	C₂	1	3	yes	maximal
3	C₁₀₉×#1273	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	3	yes	—
5	#487×C₁₀₉	—	2	6	no	—
6	#487×C₁₀₉	—	2	6	no	—
7	C₂×#103×C₁₀₉	—	2	6	no	—
8	C₂×C₄×C₈×C₁₀₉	C₂²	1	3	yes	—
9	#384×C₁₀₉	C₂²	1	1	yes	characteristic
10	#340×C₁₀₉	—	2	6	no	—
11	C₂²×Q₈×C₁₀₉	—	2	3	no	—
12	C₂²×C₈×C₁₀₉	D₄	1	6	yes	—
13	C₄⋊Q₈×C₁₀₉	—	2	3	no	—
14	C₄⋊Q₈×C₁₀₉	—	2	3	no	—
15	C₄⋊Q₈×C₁₀₉	—	2	3	no	—
16	C₄⋊Q₈×C₁₀₉	D₄	1	6	yes	—
17	C₂×C₄⋊C₄×C₁₀₉	—	2	3	no	—
18	C₂×C₄⋊C₄×C₁₀₉	—	2	3	no	—
19	C₂×C₄⋊C₄×C₁₀₉	—	2	3	no	—
20	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
21	#103×C₁₀₉	—	2	24	no	—
22	C₄×C₈×C₁₀₉	—	2	3	no	—
23	C₄×C₈×C₁₀₉	—	2	3	no	—
24	C₂×Q₈×C₁₀₉	—	2	6	no	—
25	C₂×Q₈×C₁₀₉	—	4	3	no	—
26	C₂×Q₈×C₁₀₉	—	4	6	no	—
27	C₂×Q₈×C₁₀₉	—	4	6	no	—
28	C₂²×C₄×C₁₀₉	—	4	3	no	—
29	C₂²×C₄×C₁₀₉	—	4	1	no	—
30	C₂²×C₄×C₁₀₉	C₂×D₄	1	3	yes	—
31	C₂×C₈×C₁₀₉	—	2	6	no	—
32	C₂×C₈×C₁₀₉	—	2	6	no	—
33	C₂×C₈×C₁₀₉	—	2	6	no	—
34	C₄⋊C₄×C₁₀₉	—	2	6	no	—
35	C₄⋊C₄×C₁₀₉	—	2	6	no	—
36	C₄⋊C₄×C₁₀₉	—	4	3	no	—
37	C₄⋊C₄×C₁₀₉	—	4	3	no	—
38	C₄⋊C₄×C₁₀₉	—	4	6	no	—
39	C₄²×C₁₀₉	C₂×D₄	1	3	yes	—
40	C₄²×C₁₀₉	#41	1	1	yes	characteristic
41	C₂³×C₁₀₉	#127	1	1	yes	characteristic
42	Q₈×C₁₀₉	—	4	12	no	—
43	Q₈×C₁₀₉	—	8	6	no	—
44	C₂×C₄×C₁₀₉	—	4	6	no	—
45	C₂×C₄×C₁₀₉	—	8	3	no	—
46	C₂×C₄×C₁₀₉	—	8	3	no	—
47	C₂×C₄×C₁₀₉	—	8	3	no	—
48	C₂×C₄×C₁₀₉	#120	1	3	yes	—
49	C₂×C₄×C₁₀₉	#121	1	3	yes	—
50	C₂×C₄×C₁₀₉	#135	1	6	yes	—
51	C₂×C₄×C₁₀₉	#137	1	6	yes	—
52	C₈×C₁₀₉	—	4	6	no	—
53	C₂²×C₁₀₉	#392	1	1	yes	characteristic
54	C₂²×C₁₀₉	#495	1	3	yes	—
55	C₂²×C₁₀₉	#497	1	3	yes	—
56	C₄×C₁₀₉	—	8	6	no	—
57	C₄×C₁₀₉	—	16	1	no	—
58	C₄×C₁₀₉	#452	1	6	yes	—
59	C₄×C₁₀₉	#464	1	6	yes	—
60	#10319	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂×C₁₀₉	#1711	1	1	yes	characteristic
62	C₂×C₁₀₉	#1718	1	3	yes	—
63	C₂×C₁₀₉	#2015	1	3	yes	—
64	C₂×#487	C₂×C₁₀₉	1	3	yes	—
65	#1990	C₂×C₁₀₉	1	3	yes	—
66	#1273	C₂×C₁₀₉	1	1	yes	characteristic
67	C₁₀₉	#10319	1	1	yes	characteristic, direct product factor, Sylow
68	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	3	yes	—
69	#487	—	2	6	no	—
70	#487	—	2	6	no	—
71	C₂×#103	—	2	6	no	—
72	C₂×C₄×C₈	C₂²×C₁₀₉	1	3	yes	—
73	#384	C₂²×C₁₀₉	1	1	yes	characteristic
74	#340	—	2	6	no	—
75	C₂²×Q₈	—	2	3	no	—
76	C₂²×C₈	D₄×C₁₀₉	1	6	yes	—
77	C₄⋊Q₈	—	2	3	no	—
78	C₄⋊Q₈	—	2	3	no	—
79	C₄⋊Q₈	—	2	3	no	—
80	C₄⋊Q₈	D₄×C₁₀₉	1	6	yes	—
81	C₂×C₄⋊C₄	—	2	3	no	—
82	C₂×C₄⋊C₄	—	2	3	no	—
83	C₂×C₄⋊C₄	—	2	3	no	—
84	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
85	#103	—	2	24	no	—
86	C₄×C₈	—	2	3	no	—
87	C₄×C₈	—	2	3	no	—
88	C₂×Q₈	—	2	6	no	—
89	C₂×Q₈	—	4	3	no	—
90	C₂×Q₈	—	4	6	no	—
91	C₂×Q₈	—	4	6	no	—
92	C₂²×C₄	—	4	3	no	—
93	C₂²×C₄	—	4	1	no	—
94	C₂²×C₄	C₂×D₄×C₁₀₉	1	3	yes	—
95	C₂×C₈	—	2	6	no	—
96	C₂×C₈	—	2	6	no	—
97	C₂×C₈	—	2	6	no	—
98	C₄⋊C₄	—	2	6	no	—
99	C₄⋊C₄	—	2	6	no	—
100	C₄⋊C₄	—	4	3	no	—
101	C₄⋊C₄	—	4	3	no	—
102	C₄⋊C₄	—	4	6	no	—
103	C₄²	C₂×D₄×C₁₀₉	1	3	yes	—
104	C₄²	#41×C₁₀₉	1	1	yes	characteristic
105	C₂³	#127×C₁₀₉	1	1	yes	characteristic
106	Q₈	—	4	12	no	—
107	Q₈	—	8	6	no	—
108	C₂×C₄	—	4	6	no	—
109	C₂×C₄	—	8	3	no	—
110	C₂×C₄	—	8	3	no	—
111	C₂×C₄	—	8	3	no	—
112	C₂×C₄	#120×C₁₀₉	1	3	yes	—
113	C₂×C₄	#121×C₁₀₉	1	3	yes	—
114	C₂×C₄	#135×C₁₀₉	1	6	yes	—
115	C₂×C₄	#137×C₁₀₉	1	6	yes	—
116	C₈	—	4	6	no	—
117	C₂²	#392×C₁₀₉	1	1	yes	characteristic
118	C₂²	#495×C₁₀₉	1	3	yes	—
119	C₂²	#497×C₁₀₉	1	3	yes	—
120	C₄	—	8	6	no	—
121	C₄	—	16	1	no	—
122	C₄	#452×C₁₀₉	1	6	yes	—
123	C₄	#464×C₁₀₉	1	6	yes	—
124	C₂	C₁₀₉×#1711	1	1	yes	characteristic
125	C₂	C₁₀₉×#1718	1	3	yes	—
126	C₂	C₁₀₉×#2015	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#487×C₁₀₉

C₂

yes

maximal

C₁₀₉×#1990

C₂

yes

maximal

C₁₀₉×#1273

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₀₉

C₂²

yes

—

#487×C₁₀₉

—

#487×C₁₀₉

—

C₂×#103×C₁₀₉

—

C₂×C₄×C₈×C₁₀₉

C₂²

yes

—

#384×C₁₀₉

C₂²

yes

characteristic

#340×C₁₀₉

—

C₂²×Q₈×C₁₀₉

—

C₂²×C₈×C₁₀₉

D₄

yes

—

C₄⋊Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

D₄

yes

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄²×C₁₀₉

C₂³

yes

characteristic

#103×C₁₀₉

—

C₄×C₈×C₁₀₉

—

C₄×C₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×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

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×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

—

C₂×C₄×C₁₀₉

#135

yes

—

C₂×C₄×C₁₀₉

#137

yes

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

#392

yes

characteristic

C₂²×C₁₀₉

#495

yes

—

C₂²×C₁₀₉

#497

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#452

yes

—

C₄×C₁₀₉

#464

yes

—

#10319

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

#1711

yes

characteristic

C₂×C₁₀₉

#1718

yes

—

C₂×C₁₀₉

#2015

yes

—

C₂×#487

C₂×C₁₀₉

yes

—

#1990

C₂×C₁₀₉

yes

—

#1273

C₂×C₁₀₉

yes

characteristic

C₁₀₉

#10319

yes

characteristic, direct product factor, Sylow

C₂×C₄⋊Q₈

C₂²×C₁₀₉

yes

—

#487

—

#487

—

C₂×#103

—

C₂×C₄×C₈

C₂²×C₁₀₉

yes

—

#384

C₂²×C₁₀₉

yes

characteristic

#340

—

C₂²×Q₈

—

C₂²×C₈

D₄×C₁₀₉

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₄×C₁₀₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³×C₁₀₉

yes

characteristic

#103

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄×C₁₀₉

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄²

C₂×D₄×C₁₀₉

yes

—

104

C₄²

#41×C₁₀₉

yes

characteristic

105

C₂³

#127×C₁₀₉

yes

characteristic

106

Q₈

—

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

#120×C₁₀₉

yes

—

113

C₂×C₄

#121×C₁₀₉

yes

—

114

C₂×C₄

#135×C₁₀₉

yes

—

115

C₂×C₄

#137×C₁₀₉

yes

—

116

C₈

—

117

C₂²

#392×C₁₀₉

yes

characteristic

118

C₂²

#495×C₁₀₉

yes

—

119

C₂²

#497×C₁₀₉

yes

—

120

C₄

—

121

C₄

—

122

C₄

#452×C₁₀₉

yes

—

123

C₄

#464×C₁₀₉

yes

—

124

C₂

C₁₀₉×#1711

yes

characteristic

125

C₂

C₁₀₉×#1718

yes

—

126

C₂

C₁₀₉×#2015

yes

—

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

subgroup

G/H

center, socle, prime radical

C₂³×C₁₀₉

#127

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₁₀₉

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

N	H ≅ G/N	type
60. #10319	C₁₀₉	direct (N × H)
67. C₁₀₉	#10319	direct (N × H)

H ≅ G/N

type

60. #10319

C₁₀₉

direct (N × H)

67. C₁₀₉

#10319

direct (N × H)

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

subgroup	index
C₂×#487×C₁₀₉	2
C₁₀₉×#1990	2
C₁₀₉×#1273	2
#10319	109

subgroup

index

C₂×#487×C₁₀₉

C₁₀₉×#1990

C₁₀₉×#1273

#10319

109

Subgroups — C₁₀₉×#10319 #42128313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#10319 #42128313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#10319 #42128313