Group #41033103 — subgroups

Subgroups — C₁₀₇×#7139 #41033103

Order 27392.

262 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 166 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	C₄×M₃₂×C₁₀₇	C₂	1	2	yes	maximal
2	C₂×#322×C₁₀₇	C₂	1	1	yes	maximal, characteristic
3	C₂×M₃₂×C₁₀₇	C₄	1	4	yes	—
4	C₂×C₄×C₈×C₁₀₇	C₄	1	2	yes	—
5	C₂×C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
6	#346×C₁₀₇	—	2	2	no	—
7	C₄×C₁₆×C₁₀₇	—	2	2	no	—
8	#322×C₁₀₇	—	2	2	no	—
9	C₂²×C₈×C₁₀₇	C₈	1	4	yes	—
10	C₂²×C₈×C₁₀₇	C₂×C₄	1	2	yes	—
11	C₂×C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
12	M₃₂×C₁₀₇	—	2	4	no	—
13	M₃₂×C₁₀₇	C₈	1	8	yes	complemented
14	C₂×C₁₆×C₁₀₇	—	2	4	no	—
15	C₄×C₈×C₁₀₇	—	2	2	no	—
16	C₄×C₈×C₁₀₇	—	2	2	no	—
17	C₄×C₈×C₁₀₇	D₄	1	2	yes	—
18	C₄×C₈×C₁₀₇	D₄	1	1	yes	characteristic
19	C₄×C₈×C₁₀₇	Q₈	1	1	yes	characteristic
20	C₂²×C₄×C₁₀₇	C₄²	1	1	yes	characteristic
21	C₂²×C₄×C₁₀₇	C₂×C₈	1	2	yes	—
22	C₂×C₈×C₁₀₇	—	2	4	no	—
23	C₂×C₈×C₁₀₇	—	4	4	no	—
24	C₂×C₈×C₁₀₇	C₂²⋊C₄	1	2	yes	—
25	C₂×C₈×C₁₀₇	C₄⋊C₄	1	2	yes	—
26	C₂×C₈×C₁₀₇	C₂×C₈	1	4	yes	—
27	C₂×C₈×C₁₀₇	M₁₆	1	4	yes	—
28	C₄²×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₄²×C₁₀₇	C₄⋊C₄	1	1	yes	characteristic
30	C₄²×C₁₀₇	M₁₆	1	2	yes	—
31	C₁₆×C₁₀₇	—	2	8	no	—
32	C₂³×C₁₀₇	C₄×C₈	1	1	yes	characteristic
33	C₂×C₄×C₁₀₇	—	2	4	no	—
34	C₂×C₄×C₁₀₇	#95	1	1	yes	characteristic
35	C₂×C₄×C₁₀₇	C₄×C₈	1	2	yes	—
36	C₂×C₄×C₁₀₇	#97	1	1	yes	characteristic
37	C₂×C₄×C₁₀₇	#97	1	2	yes	—
38	C₂×C₄×C₁₀₇	C₂²⋊C₈	1	2	yes	—
39	C₂×C₄×C₁₀₇	C₄⋊C₈	1	2	yes	—
40	C₈×C₁₀₇	—	4	4	no	—
41	C₈×C₁₀₇	C₂²⋊C₈	1	4	yes	—
42	C₈×C₁₀₇	C₄⋊C₈	1	4	yes	—
43	C₂²×C₁₀₇	—	2	2	no	—
44	C₂²×C₁₀₇	C₈²	1	1	yes	characteristic
45	C₂²×C₁₀₇	#322	1	1	yes	characteristic
46	C₂²×C₁₀₇	#336	1	1	yes	characteristic
47	C₄×C₁₀₇	—	2	2	no	—
48	C₄×C₁₀₇	#322	1	2	yes	—
49	C₄×C₁₀₇	#336	1	2	yes	—
50	C₄×C₁₀₇	#349	1	4	yes	—
51	#7139	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂×C₁₀₇	—	2	1	no	—
53	C₂×C₁₀₇	—	2	1	no	—
54	C₂×C₁₀₇	#1273	1	1	yes	characteristic
55	C₂×C₁₀₇	#1377	1	2	yes	—
56	C₄×M₃₂	C₂×C₁₀₇	1	2	yes	—
57	C₂×#322	C₂×C₁₀₇	1	1	yes	characteristic
58	C₁₀₇	#7139	1	1	yes	characteristic, direct product factor, Sylow
59	C₂×M₃₂	C₄×C₁₀₇	1	4	yes	—
60	C₂×C₄×C₈	C₄×C₁₀₇	1	2	yes	—
61	C₂×C₄×C₈	C₂²×C₁₀₇	1	1	yes	characteristic
62	#346	—	2	2	no	—
63	C₄×C₁₆	—	2	2	no	—
64	#322	—	2	2	no	—
65	C₂²×C₈	C₈×C₁₀₇	1	4	yes	—
66	C₂²×C₈	C₂×C₄×C₁₀₇	1	2	yes	—
67	C₂×C₄²	C₂×C₄×C₁₀₇	1	1	yes	characteristic
68	M₃₂	—	2	4	no	—
69	M₃₂	C₈×C₁₀₇	1	8	yes	complemented
70	C₂×C₁₆	—	2	4	no	—
71	C₄×C₈	—	2	2	no	—
72	C₄×C₈	—	2	2	no	—
73	C₄×C₈	D₄×C₁₀₇	1	2	yes	—
74	C₄×C₈	D₄×C₁₀₇	1	1	yes	characteristic
75	C₄×C₈	Q₈×C₁₀₇	1	1	yes	characteristic
76	C₂²×C₄	C₄²×C₁₀₇	1	1	yes	characteristic
77	C₂²×C₄	C₂×C₈×C₁₀₇	1	2	yes	—
78	C₂×C₈	—	2	4	no	—
79	C₂×C₈	—	4	4	no	—
80	C₂×C₈	C₂²⋊C₄×C₁₀₇	1	2	yes	—
81	C₂×C₈	C₄⋊C₄×C₁₀₇	1	2	yes	—
82	C₂×C₈	C₂×C₈×C₁₀₇	1	4	yes	—
83	C₂×C₈	M₁₆×C₁₀₇	1	4	yes	—
84	C₄²	C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
85	C₄²	C₄⋊C₄×C₁₀₇	1	1	yes	characteristic
86	C₄²	M₁₆×C₁₀₇	1	2	yes	—
87	C₁₆	—	2	8	no	—
88	C₂³	C₄×C₈×C₁₀₇	1	1	yes	characteristic
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	#95×C₁₀₇	1	1	yes	characteristic
91	C₂×C₄	C₄×C₈×C₁₀₇	1	2	yes	—
92	C₂×C₄	#97×C₁₀₇	1	1	yes	characteristic
93	C₂×C₄	#97×C₁₀₇	1	2	yes	—
94	C₂×C₄	C₂²⋊C₈×C₁₀₇	1	2	yes	—
95	C₂×C₄	C₄⋊C₈×C₁₀₇	1	2	yes	—
96	C₈	—	4	4	no	—
97	C₈	C₂²⋊C₈×C₁₀₇	1	4	yes	—
98	C₈	C₄⋊C₈×C₁₀₇	1	4	yes	—
99	C₂²	—	2	2	no	—
100	C₂²	C₈²×C₁₀₇	1	1	yes	characteristic
101	C₂²	#322×C₁₀₇	1	1	yes	characteristic
102	C₂²	#336×C₁₀₇	1	1	yes	characteristic
103	C₄	—	2	2	no	—
104	C₄	#322×C₁₀₇	1	2	yes	—
105	C₄	#336×C₁₀₇	1	2	yes	—
106	C₄	#349×C₁₀₇	1	4	yes	—
107	C₂	—	2	1	no	—
108	C₂	—	2	1	no	—
109	C₂	C₁₀₇×#1273	1	1	yes	characteristic
110	C₂	C₁₀₇×#1377	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×M₃₂×C₁₀₇

C₂

yes

maximal

C₂×#322×C₁₀₇

C₂

yes

maximal, characteristic

C₂×M₃₂×C₁₀₇

C₄

yes

—

C₂×C₄×C₈×C₁₀₇

C₄

yes

—

C₂×C₄×C₈×C₁₀₇

C₂²

yes

characteristic

#346×C₁₀₇

—

C₄×C₁₆×C₁₀₇

—

#322×C₁₀₇

—

C₂²×C₈×C₁₀₇

C₈

yes

—

C₂²×C₈×C₁₀₇

C₂×C₄

yes

—

C₂×C₄²×C₁₀₇

C₂×C₄

yes

characteristic

M₃₂×C₁₀₇

—

M₃₂×C₁₀₇

C₈

yes

complemented

C₂×C₁₆×C₁₀₇

—

C₄×C₈×C₁₀₇

—

C₄×C₈×C₁₀₇

—

C₄×C₈×C₁₀₇

D₄

yes

—

C₄×C₈×C₁₀₇

D₄

yes

characteristic

C₄×C₈×C₁₀₇

Q₈

yes

characteristic

C₂²×C₄×C₁₀₇

C₄²

yes

characteristic

C₂²×C₄×C₁₀₇

C₂×C₈

yes

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

C₂²⋊C₄

yes

—

C₂×C₈×C₁₀₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₀₇

C₂×C₈

yes

—

C₂×C₈×C₁₀₇

M₁₆

yes

—

C₄²×C₁₀₇

C₂²⋊C₄

yes

characteristic

C₄²×C₁₀₇

C₄⋊C₄

yes

characteristic

C₄²×C₁₀₇

M₁₆

yes

—

C₁₆×C₁₀₇

—

C₂³×C₁₀₇

C₄×C₈

yes

characteristic

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

#95

yes

characteristic

C₂×C₄×C₁₀₇

C₄×C₈

yes

—

C₂×C₄×C₁₀₇

#97

yes

characteristic

C₂×C₄×C₁₀₇

#97

yes

—

C₂×C₄×C₁₀₇

C₂²⋊C₈

yes

—

C₂×C₄×C₁₀₇

C₄⋊C₈

yes

—

C₈×C₁₀₇

—

C₈×C₁₀₇

C₂²⋊C₈

yes

—

C₈×C₁₀₇

C₄⋊C₈

yes

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₈²

yes

C₂²×C₁₀₇

yes

C₂²×C₁₀₇

yes

C₄×C₁₀₇

—

C₄×C₁₀₇

#322

yes

—

C₄×C₁₀₇

#336

yes

—

C₄×C₁₀₇

#349

yes

—

#7139

C₁₀₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

#1273

yes

characteristic

C₂×C₁₀₇

#1377

yes

—

C₄×M₃₂

C₂×C₁₀₇

yes

—

C₂×#322

C₂×C₁₀₇

yes

characteristic

C₁₀₇

#7139

yes

characteristic, direct product factor, Sylow

C₂×M₃₂

C₄×C₁₀₇

yes

—

C₂×C₄×C₈

C₄×C₁₀₇

yes

—

C₂×C₄×C₈

C₂²×C₁₀₇

yes

characteristic

#346

—

C₄×C₁₆

—

#322

—

C₂²×C₈

C₈×C₁₀₇

yes

—

C₂²×C₈

C₂×C₄×C₁₀₇

yes

—

C₂×C₄²

C₂×C₄×C₁₀₇

yes

characteristic

M₃₂

—

M₃₂

C₈×C₁₀₇

yes

complemented

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₄×C₁₀₇

yes

—

C₄×C₈

D₄×C₁₀₇

yes

characteristic

C₄×C₈

Q₈×C₁₀₇

yes

characteristic

C₂²×C₄

C₄²×C₁₀₇

yes

characteristic

C₂²×C₄

C₂×C₈×C₁₀₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄×C₁₀₇

yes

—

C₂×C₈

C₄⋊C₄×C₁₀₇

yes

—

C₂×C₈

C₂×C₈×C₁₀₇

yes

—

C₂×C₈

M₁₆×C₁₀₇

yes

—

C₄²

C₂²⋊C₄×C₁₀₇

yes

characteristic

C₄²

C₄⋊C₄×C₁₀₇

yes

characteristic

C₄²

M₁₆×C₁₀₇

yes

—

C₁₆

—

C₂³

C₄×C₈×C₁₀₇

yes

characteristic

C₂×C₄

—

C₂×C₄

#95×C₁₀₇

yes

characteristic

C₂×C₄

C₄×C₈×C₁₀₇

yes

—

C₂×C₄

#97×C₁₀₇

yes

characteristic

C₂×C₄

#97×C₁₀₇

yes

—

C₂×C₄

C₂²⋊C₈×C₁₀₇

yes

—

C₂×C₄

C₄⋊C₈×C₁₀₇

yes

—

C₈

—

C₈

C₂²⋊C₈×C₁₀₇

yes

—

C₈

C₄⋊C₈×C₁₀₇

yes

—

C₂²

—

100

C₂²

C₈²×C₁₀₇

yes

characteristic

101

C₂²

#322×C₁₀₇

yes

characteristic

102

C₂²

#336×C₁₀₇

yes

characteristic

103

C₄

—

104

C₄

#322×C₁₀₇

yes

—

105

C₄

#336×C₁₀₇

yes

—

106

C₄

#349×C₁₀₇

yes

—

107

C₂

—

108

C₂

—

109

C₂

C₁₀₇×#1273

yes

characteristic

110

C₂

C₁₀₇×#1377

yes

—

	subgroup	G/H
center	C₄²×C₁₀₇	C₂²⋊C₄
commutator	C₂²	C₈²×C₁₀₇
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₀₇
socle	C₂²×C₁₀₇	#336
prime radical	C₂³×C₁₀₇	C₄×C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₁₀₇ ◁ C₂×C₄×C₈×C₁₀₇ ◁ G

subgroup

G/H

center

C₄²×C₁₀₇

C₂²⋊C₄

commutator

C₂²

C₈²×C₁₀₇

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₁₀₇

socle

C₂²×C₁₀₇

#336

prime radical

C₂³×C₁₀₇

C₄×C₈

LCS

G ▷ C₂² ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄²×C₁₀₇ ◁ C₂×C₄×C₈×C₁₀₇ ◁ G

N	H ≅ G/N	type
13. M₃₂×C₁₀₇	C₈	semidirect
51. #7139	C₁₀₇	direct (N × H)
58. C₁₀₇	#7139	direct (N × H)
69. M₃₂	C₈×C₁₀₇	semidirect

H ≅ G/N

type

13. M₃₂×C₁₀₇

C₈

semidirect

51. #7139

C₁₀₇

direct (N × H)

58. C₁₀₇

#7139

direct (N × H)

69. M₃₂

C₈×C₁₀₇

semidirect

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

subgroup	index
C₄×M₃₂×C₁₀₇	2
C₂×#322×C₁₀₇	2
#7139	107

subgroup

index

C₄×M₃₂×C₁₀₇

C₂×#322×C₁₀₇

#7139

107

Subgroups — C₁₀₇×#7139 #41033103

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C107×#7139 #41033103

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#7139 #41033103