Group #42130501 — subgroups

Subgroups — C₁₀₉×#12507 #42130501

Order 27904.

258 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2254	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#2247	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#2241	C₂	1	2	yes	maximal
4	C₁₀₉×#1707	C₂	1	2	yes	maximal
5	C₁₀₉×#1311	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₃₂×C₁₀₉	—	2	2	no	—
7	#500×C₁₀₉	C₂²	1	2	yes	—
8	#498×C₁₀₉	—	2	2	no	—
9	#494×C₁₀₉	C₂²	1	2	yes	—
10	#487×C₁₀₉	—	2	2	no	—
11	#367×C₁₀₉	—	2	2	no	—
12	#358×C₁₀₉	—	2	4	no	—
13	C₄×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₈²×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₁₀₉	—	2	2	no	—
17	C₂×Q₁₆×C₁₀₉	—	2	2	no	—
18	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
19	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
20	Q₃₂×C₁₀₉	—	4	4	no	—
21	C₂×C₁₆×C₁₀₉	D₄	1	2	yes	—
22	C₂×C₁₆×C₁₀₉	D₄	1	2	yes	—
23	#107×C₁₀₉	—	2	2	no	—
24	#107×C₁₀₉	—	2	2	no	—
25	#106×C₁₀₉	—	4	4	no	—
26	#103×C₁₀₉	—	4	4	no	—
27	C₄×C₈×C₁₀₉	D₄	1	2	yes	—
28	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₁₀₉	—	4	2	no	—
30	C₂×Q₈×C₁₀₉	—	4	2	no	—
31	Q₁₆×C₁₀₉	—	4	4	no	—
32	Q₁₆×C₁₀₉	—	4	4	no	—
33	C₂×C₈×C₁₀₉	D₈	1	4	yes	—
34	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₁₀₉	—	4	2	no	—
37	C₄⋊C₄×C₁₀₉	—	4	2	no	—
38	C₄⋊C₄×C₁₀₉	—	4	2	no	—
39	C₄⋊C₄×C₁₀₉	—	4	2	no	—
40	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₁₀₉	—	2	2	no	—
42	C₁₆×C₁₀₉	SD₁₆	1	4	yes	—
43	Q₈×C₁₀₉	—	8	2	no	—
44	Q₈×C₁₀₉	—	8	2	no	—
45	Q₈×C₁₀₉	—	8	4	no	—
46	C₂×C₄×C₁₀₉	—	8	2	no	—
47	C₂×C₄×C₁₀₉	—	8	2	no	—
48	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
49	C₂×C₄×C₁₀₉	C₂×D₈	1	2	yes	—
50	C₈×C₁₀₉	—	2	4	no	—
51	C₈×C₁₀₉	C₂×SD₁₆	1	2	yes	—
52	C₈×C₁₀₉	#135	1	2	yes	—
53	C₂²×C₁₀₉	#493	1	1	yes	characteristic
54	C₄×C₁₀₉	—	16	2	no	—
55	C₄×C₁₀₉	—	16	2	no	—
56	C₄×C₁₀₉	#492	1	1	yes	characteristic
57	C₄×C₁₀₉	#495	1	1	yes	characteristic
58	C₄×C₁₀₉	#510	1	4	yes	—
59	#12507	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂×C₁₀₉	#1706	1	1	yes	characteristic
61	C₂×C₁₀₉	#2251	1	2	yes	—
62	#2254	C₂×C₁₀₉	1	1	yes	characteristic
63	#2247	C₂×C₁₀₉	1	1	yes	characteristic
64	#2241	C₂×C₁₀₉	1	2	yes	—
65	#1707	C₂×C₁₀₉	1	2	yes	—
66	#1311	C₂×C₁₀₉	1	1	yes	characteristic
67	C₁₀₉	#12507	1	1	yes	characteristic, direct product factor, Sylow
68	C₂×Q₃₂	—	2	2	no	—
69	#500	C₂²×C₁₀₉	1	2	yes	—
70	#498	—	2	2	no	—
71	#494	C₂²×C₁₀₉	1	2	yes	—
72	#487	—	2	2	no	—
73	#367	—	2	2	no	—
74	#358	—	2	4	no	—
75	C₄×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
76	C₄×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
77	C₈²	C₂²×C₁₀₉	1	1	yes	characteristic
78	C₂×Q₁₆	—	2	2	no	—
79	C₂×Q₁₆	—	2	2	no	—
80	C₄⋊Q₈	—	2	2	no	—
81	C₄⋊Q₈	—	2	2	no	—
82	Q₃₂	—	4	4	no	—
83	C₂×C₁₆	D₄×C₁₀₉	1	2	yes	—
84	C₂×C₁₆	D₄×C₁₀₉	1	2	yes	—
85	#107	—	2	2	no	—
86	#107	—	2	2	no	—
87	#106	—	4	4	no	—
88	#103	—	4	4	no	—
89	C₄×C₈	D₄×C₁₀₉	1	2	yes	—
90	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
91	C₂×Q₈	—	4	2	no	—
92	C₂×Q₈	—	4	2	no	—
93	Q₁₆	—	4	4	no	—
94	Q₁₆	—	4	4	no	—
95	C₂×C₈	D₈×C₁₀₉	1	4	yes	—
96	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
97	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
98	C₄⋊C₄	—	4	2	no	—
99	C₄⋊C₄	—	4	2	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	4	2	no	—
102	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
103	C₁₆	—	2	2	no	—
104	C₁₆	SD₁₆×C₁₀₉	1	4	yes	—
105	Q₈	—	8	4	no	—
106	Q₈	—	8	2	no	—
107	Q₈	—	8	2	no	—
108	C₂×C₄	—	8	2	no	—
109	C₂×C₄	—	8	2	no	—
110	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
111	C₂×C₄	C₂×D₈×C₁₀₉	1	2	yes	—
112	C₈	—	2	4	no	—
113	C₈	C₂×SD₁₆×C₁₀₉	1	2	yes	—
114	C₈	#135×C₁₀₉	1	2	yes	—
115	C₂²	#493×C₁₀₉	1	1	yes	characteristic
116	C₄	—	16	2	no	—
117	C₄	—	16	2	no	—
118	C₄	#492×C₁₀₉	1	1	yes	characteristic
119	C₄	#495×C₁₀₉	1	1	yes	characteristic
120	C₄	#510×C₁₀₉	1	4	yes	—
121	C₂	C₁₀₉×#1706	1	1	yes	characteristic
122	C₂	C₁₀₉×#2251	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉×#2254

C₂

yes

maximal, characteristic

C₁₀₉×#2247

C₂

yes

maximal, characteristic

C₁₀₉×#2241

C₂

yes

maximal

C₁₀₉×#1707

C₂

yes

maximal

C₁₀₉×#1311

C₂

yes

maximal, characteristic

C₂×Q₃₂×C₁₀₉

—

#500×C₁₀₉

C₂²

yes

—

#498×C₁₀₉

—

#494×C₁₀₉

C₂²

yes

—

#487×C₁₀₉

—

#367×C₁₀₉

—

#358×C₁₀₉

—

C₄×C₁₆×C₁₀₉

C₂²

yes

characteristic

C₄×C₁₆×C₁₀₉

C₂²

yes

characteristic

C₈²×C₁₀₉

C₂²

yes

characteristic

C₂×Q₁₆×C₁₀₉

—

C₂×Q₁₆×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

Q₃₂×C₁₀₉

—

C₂×C₁₆×C₁₀₉

D₄

yes

—

C₂×C₁₆×C₁₀₉

D₄

yes

—

#107×C₁₀₉

—

#107×C₁₀₉

—

#106×C₁₀₉

—

#103×C₁₀₉

—

C₄×C₈×C₁₀₉

D₄

yes

—

C₄×C₈×C₁₀₉

C₂³

yes

characteristic

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

Q₁₆×C₁₀₉

—

Q₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

D₈

yes

—

C₂×C₈×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂×D₄

yes

characteristic

C₁₆×C₁₀₉

—

C₁₆×C₁₀₉

SD₁₆

yes

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

#127

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₈

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₂×SD₁₆

yes

—

C₈×C₁₀₉

#135

yes

—

C₂²×C₁₀₉

#493

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

—

#12507

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

#1706

yes

characteristic

C₂×C₁₀₉

#2251

yes

—

#2254

C₂×C₁₀₉

yes

characteristic

#2247

C₂×C₁₀₉

yes

characteristic

#2241

C₂×C₁₀₉

yes

—

#1707

C₂×C₁₀₉

yes

—

#1311

C₂×C₁₀₉

yes

characteristic

C₁₀₉

#12507

yes

characteristic, direct product factor, Sylow

C₂×Q₃₂

—

#500

C₂²×C₁₀₉

yes

—

#498

—

#494

C₂²×C₁₀₉

yes

—

#487

—

#367

—

#358

—

C₄×C₁₆

C₂²×C₁₀₉

yes

characteristic

C₄×C₁₆

C₂²×C₁₀₉

yes

characteristic

C₈²

C₂²×C₁₀₉

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

C₂×C₁₆

D₄×C₁₀₉

yes

—

C₂×C₁₆

D₄×C₁₀₉

yes

—

—

—

—

—

C₄×C₈

D₄×C₁₀₉

yes

—

C₄×C₈

C₂³×C₁₀₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

D₈×C₁₀₉

yes

—

C₂×C₈

C₂×D₄×C₁₀₉

yes

characteristic

C₂×C₈

C₂×D₄×C₁₀₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄²

C₂×D₄×C₁₀₉

yes

characteristic

103

C₁₆

—

104

C₁₆

SD₁₆×C₁₀₉

yes

—

105

Q₈

—

106

Q₈

—

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

#127×C₁₀₉

yes

characteristic

111

C₂×C₄

C₂×D₈×C₁₀₉

yes

—

112

C₈

—

113

C₈

C₂×SD₁₆×C₁₀₉

yes

—

114

C₈

#135×C₁₀₉

yes

—

115

C₂²

#493×C₁₀₉

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

#492×C₁₀₉

yes

characteristic

119

C₄

#495×C₁₀₉

yes

characteristic

120

C₄

#510×C₁₀₉

yes

—

121

C₂

C₁₀₉×#1706

yes

characteristic

122

C₂

C₁₀₉×#2251

yes

—

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₀₉	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₀₉
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₄²×C₁₀₉ ◁ C₈²×C₁₀₉ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₁₀₉

#493

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₁₀₉

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₉ ◁ C₄²×C₁₀₉ ◁ C₈²×C₁₀₉ ◁ G

N	H ≅ G/N	type
59. #12507	C₁₀₉	direct (N × H)
67. C₁₀₉	#12507	direct (N × H)

H ≅ G/N

type

59. #12507

C₁₀₉

direct (N × H)

67. C₁₀₉

#12507

direct (N × H)

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

subgroup	index
C₁₀₉×#2254	2
C₁₀₉×#2247	2
C₁₀₉×#2241	2
C₁₀₉×#1707	2
C₁₀₉×#1311	2
#12507	109

subgroup

index

C₁₀₉×#2254

C₁₀₉×#2247

C₁₀₉×#2241

C₁₀₉×#1707

C₁₀₉×#1311

#12507

109

Subgroups — C₁₀₉×#12507 #42130501

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#12507 #42130501

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12507 #42130501