Group #41034746 — subgroups

Subgroups — C₁₀₇×#8782 #41034746

Order 27392.

262 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 126 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	C₁₀₇×#1746	C₂	1	1	yes	maximal, characteristic
2	C₁₀₇×#1564	C₂	1	2	yes	maximal
3	C₁₀₇×#1281	C₂	1	2	yes	maximal
4	C₁₀₇×#1274	C₂	1	2	yes	maximal
5	#431×C₁₀₇	C₂²	1	4	yes	—
6	C₂×#97×C₁₀₇	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
8	#376×C₁₀₇	C₂²	1	1	yes	characteristic
9	#336×C₁₀₇	C₄	1	4	yes	—
10	#335×C₁₀₇	—	2	2	no	—
11	#334×C₁₀₇	—	2	2	no	—
12	C₂²×C₈×C₁₀₇	C₂×C₄	1	2	yes	—
13	C₂²×C₈×C₁₀₇	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₁₀₇	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
16	C₄⋊C₈×C₁₀₇	—	2	4	no	—
17	C₂²⋊C₈×C₁₀₇	—	2	4	no	—
18	#97×C₁₀₇	—	2	2	no	—
19	C₄×C₈×C₁₀₇	—	2	4	no	—
20	C₄×C₈×C₁₀₇	D₄	1	1	yes	characteristic
21	C₄×C₈×C₁₀₇	D₄	1	1	yes	characteristic
22	C₄×C₈×C₁₀₇	Q₈	1	2	yes	—
23	#95×C₁₀₇	—	2	2	no	—
24	C₂²×C₄×C₁₀₇	—	2	2	no	—
25	C₂²×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₀₇	#41	1	2	yes	—
27	C₂×C₈×C₁₀₇	—	2	4	no	—
28	C₂×C₈×C₁₀₇	—	2	2	no	—
29	C₂×C₈×C₁₀₇	—	2	4	no	—
30	C₂×C₈×C₁₀₇	—	4	4	no	—
31	C₂×C₈×C₁₀₇	C₄⋊C₄	1	2	yes	—
32	C₂×C₈×C₁₀₇	C₄⋊C₄	1	2	yes	—
33	C₄²×C₁₀₇	—	2	4	no	—
34	C₄²×C₁₀₇	C₂×D₄	1	1	yes	characteristic
35	C₄²×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
36	C₄²×C₁₀₇	#41	1	2	yes	—
37	C₂³×C₁₀₇	#117	1	1	yes	characteristic
38	C₂×C₄×C₁₀₇	—	4	2	no	—
39	C₂×C₄×C₁₀₇	—	4	4	no	—
40	C₂×C₄×C₁₀₇	—	4	2	no	—
41	C₂×C₄×C₁₀₇	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₀₇	#117	1	1	yes	characteristic
43	C₂×C₄×C₁₀₇	#124	1	2	yes	—
44	C₂×C₄×C₁₀₇	#125	1	2	yes	—
45	C₂×C₄×C₁₀₇	#131	1	4	yes	—
46	C₈×C₁₀₇	—	2	2	no	—
47	C₈×C₁₀₇	—	2	2	no	—
48	C₈×C₁₀₇	—	4	2	no	—
49	C₈×C₁₀₇	—	8	4	no	—
50	C₂²×C₁₀₇	—	2	2	no	—
51	C₂²×C₁₀₇	#382	1	1	yes	characteristic
52	C₂²×C₁₀₇	#433	1	2	yes	—
53	C₄×C₁₀₇	—	2	2	no	—
54	C₄×C₁₀₇	—	2	2	no	—
55	C₄×C₁₀₇	—	8	2	no	—
56	C₄×C₁₀₇	#424	1	2	yes	—
57	C₄×C₁₀₇	#433	1	2	yes	—
58	#8782	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂×C₁₀₇	—	2	2	no	—
60	C₂×C₁₀₇	#1832	1	1	yes	characteristic
61	C₂×C₁₀₇	#1833	1	2	yes	—
62	#1746	C₂×C₁₀₇	1	1	yes	characteristic
63	#1564	C₂×C₁₀₇	1	2	yes	—
64	#1281	C₂×C₁₀₇	1	2	yes	—
65	#1274	C₂×C₁₀₇	1	2	yes	—
66	C₁₀₇	#8782	1	1	yes	characteristic, direct product factor, Sylow
67	#431	C₂²×C₁₀₇	1	4	yes	—
68	C₂×#97	C₂²×C₁₀₇	1	1	yes	characteristic
69	C₂×C₄×C₈	C₂²×C₁₀₇	1	1	yes	characteristic
70	#376	C₂²×C₁₀₇	1	1	yes	characteristic
71	#336	C₄×C₁₀₇	1	4	yes	—
72	#335	—	2	2	no	—
73	#334	—	2	2	no	—
74	C₂²×C₈	C₂×C₄×C₁₀₇	1	2	yes	—
75	C₂²×C₈	C₂×C₄×C₁₀₇	1	2	yes	—
76	C₂×C₄²	C₂×C₄×C₁₀₇	1	2	yes	—
77	C₂×C₄²	C₂³×C₁₀₇	1	1	yes	characteristic
78	C₄⋊C₈	—	2	4	no	—
79	C₂²⋊C₈	—	2	4	no	—
80	#97	—	2	2	no	—
81	C₄×C₈	—	2	4	no	—
82	C₄×C₈	D₄×C₁₀₇	1	1	yes	characteristic
83	C₄×C₈	D₄×C₁₀₇	1	1	yes	characteristic
84	C₄×C₈	Q₈×C₁₀₇	1	2	yes	—
85	#95	—	2	2	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	C₂²×C₄×C₁₀₇	1	1	yes	characteristic
88	C₂²×C₄	#41×C₁₀₇	1	2	yes	—
89	C₂×C₈	—	2	4	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	4	no	—
92	C₂×C₈	—	4	4	no	—
93	C₂×C₈	C₄⋊C₄×C₁₀₇	1	2	yes	—
94	C₂×C₈	C₄⋊C₄×C₁₀₇	1	2	yes	—
95	C₄²	—	2	4	no	—
96	C₄²	C₂×D₄×C₁₀₇	1	1	yes	characteristic
97	C₄²	C₂×Q₈×C₁₀₇	1	1	yes	characteristic
98	C₄²	#41×C₁₀₇	1	2	yes	—
99	C₂³	#117×C₁₀₇	1	1	yes	characteristic
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	4	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	C₂×C₄⋊C₄×C₁₀₇	1	1	yes	characteristic
104	C₂×C₄	#117×C₁₀₇	1	1	yes	characteristic
105	C₂×C₄	#124×C₁₀₇	1	2	yes	—
106	C₂×C₄	#125×C₁₀₇	1	2	yes	—
107	C₂×C₄	#131×C₁₀₇	1	4	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	2	2	no	—
110	C₈	—	4	2	no	—
111	C₈	—	8	4	no	—
112	C₂²	—	2	2	no	—
113	C₂²	#382×C₁₀₇	1	1	yes	characteristic
114	C₂²	#433×C₁₀₇	1	2	yes	—
115	C₄	—	2	2	no	—
116	C₄	—	2	2	no	—
117	C₄	—	8	2	no	—
118	C₄	#424×C₁₀₇	1	2	yes	—
119	C₄	#433×C₁₀₇	1	2	yes	—
120	C₂	—	2	2	no	—
121	C₂	C₁₀₇×#1832	1	1	yes	characteristic
122	C₂	C₁₀₇×#1833	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₇×#1746

C₂

yes

maximal, characteristic

C₁₀₇×#1564

C₂

yes

maximal

C₁₀₇×#1281

C₂

yes

maximal

C₁₀₇×#1274

C₂

yes

maximal

#431×C₁₀₇

C₂²

yes

—

C₂×#97×C₁₀₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₁₀₇

C₂²

yes

characteristic

#376×C₁₀₇

C₂²

yes

characteristic

#336×C₁₀₇

C₄

yes

—

#335×C₁₀₇

—

#334×C₁₀₇

—

C₂²×C₈×C₁₀₇

C₂×C₄

yes

—

C₂²×C₈×C₁₀₇

C₂×C₄

yes

—

C₂×C₄²×C₁₀₇

C₂×C₄

yes

—

C₂×C₄²×C₁₀₇

C₂³

yes

characteristic

C₄⋊C₈×C₁₀₇

—

C₂²⋊C₈×C₁₀₇

—

#97×C₁₀₇

—

C₄×C₈×C₁₀₇

—

C₄×C₈×C₁₀₇

D₄

yes

characteristic

C₄×C₈×C₁₀₇

D₄

yes

characteristic

C₄×C₈×C₁₀₇

Q₈

yes

—

#95×C₁₀₇

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₀₇

#41

yes

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₀₇

C₄⋊C₄

yes

—

C₄²×C₁₀₇

—

C₄²×C₁₀₇

C₂×D₄

yes

characteristic

C₄²×C₁₀₇

C₂×Q₈

yes

characteristic

C₄²×C₁₀₇

#41

yes

—

C₂³×C₁₀₇

#117

yes

characteristic

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂×C₄⋊C₄

yes

C₂×C₄×C₁₀₇

yes

C₂×C₄×C₁₀₇

yes

—

C₂×C₄×C₁₀₇

#125

yes

—

C₂×C₄×C₁₀₇

#131

yes

—

C₈×C₁₀₇

—

C₈×C₁₀₇

—

C₈×C₁₀₇

—

C₈×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

#382

yes

characteristic

C₂²×C₁₀₇

#433

yes

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

#424

yes

—

C₄×C₁₀₇

#433

yes

—

#8782

C₁₀₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₇

—

C₂×C₁₀₇

#1832

yes

characteristic

C₂×C₁₀₇

#1833

yes

—

#1746

C₂×C₁₀₇

yes

characteristic

#1564

C₂×C₁₀₇

yes

—

#1281

C₂×C₁₀₇

yes

—

#1274

C₂×C₁₀₇

yes

—

C₁₀₇

#8782

yes

characteristic, direct product factor, Sylow

#431

C₂²×C₁₀₇

yes

—

C₂×#97

C₂²×C₁₀₇

yes

characteristic

C₂×C₄×C₈

C₂²×C₁₀₇

yes

characteristic

#376

C₂²×C₁₀₇

yes

characteristic

#336

C₄×C₁₀₇

yes

—

#335

—

#334

—

C₂²×C₈

C₂×C₄×C₁₀₇

yes

—

C₂²×C₈

C₂×C₄×C₁₀₇

yes

—

C₂×C₄²

C₂×C₄×C₁₀₇

yes

—

C₂×C₄²

C₂³×C₁₀₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

D₄×C₁₀₇

yes

characteristic

C₄×C₈

D₄×C₁₀₇

yes

characteristic

C₄×C₈

Q₈×C₁₀₇

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

C₂²×C₄×C₁₀₇

yes

characteristic

C₂²×C₄

#41×C₁₀₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₁₀₇

yes

—

C₂×C₈

C₄⋊C₄×C₁₀₇

yes

—

C₄²

—

C₄²

C₂×D₄×C₁₀₇

yes

characteristic

C₄²

C₂×Q₈×C₁₀₇

yes

characteristic

C₄²

#41×C₁₀₇

yes

—

C₂³

#117×C₁₀₇

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂×C₄⋊C₄×C₁₀₇

yes

characteristic

104

C₂×C₄

#117×C₁₀₇

yes

characteristic

105

C₂×C₄

#124×C₁₀₇

yes

—

106

C₂×C₄

#125×C₁₀₇

yes

—

107

C₂×C₄

#131×C₁₀₇

yes

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₂²

—

113

C₂²

#382×C₁₀₇

yes

characteristic

114

C₂²

#433×C₁₀₇

yes

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

#424×C₁₀₇

yes

—

119

C₄

#433×C₁₀₇

yes

—

120

C₂

—

121

C₂

C₁₀₇×#1832

yes

characteristic

122

C₂

C₁₀₇×#1833

yes

—

	subgroup	G/H
center	C₂×C₄×C₁₀₇	#117
commutator	C₂²×C₄	C₂²×C₄×C₁₀₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₇
socle	C₂²×C₁₀₇	#382
prime radical	C₂³×C₁₀₇	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₇ ◁ #376×C₁₀₇ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₀₇

#117

commutator

C₂²×C₄

C₂²×C₄×C₁₀₇

Frattini, cosocle

C₂×C₄²

C₂³×C₁₀₇

C₂²×C₁₀₇

C₂³×C₁₀₇

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂×C₄×C₁₀₇ ◁ #376×C₁₀₇ ◁ G

N	H ≅ G/N	type
58. #8782	C₁₀₇	direct (N × H)
66. C₁₀₇	#8782	direct (N × H)

H ≅ G/N

type

58. #8782

C₁₀₇

direct (N × H)

66. C₁₀₇

#8782

direct (N × H)

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

subgroup	index
C₁₀₇×#1746	2
C₁₀₇×#1564	2
C₁₀₇×#1281	2
C₁₀₇×#1274	2
#8782	107

subgroup

index

C₁₀₇×#1746

C₁₀₇×#1564

C₁₀₇×#1281

C₁₀₇×#1274

#8782

107

Subgroups — C₁₀₇×#8782 #41034746

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C107×#8782 #41034746

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#8782 #41034746