Group #41033102 — subgroups

Subgroups — C₁₀₇×#7138 #41033102

Order 27392.

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

normal subgroups only

This group has 278 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×C₈²×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	C₈²×C₁₀₇	—	2	2	no	—
9	C₂²×C₈×C₁₀₇	—	2	2	no	—
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	C₈	1	4	yes	—
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	M₁₆	1	2	yes	—
22	C₂×C₈×C₁₀₇	—	2	8	no	—
23	C₂×C₈×C₁₀₇	—	2	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	C₂×C₈	1	2	yes	—
31	C₁₆×C₁₀₇	—	2	8	no	—
32	C₂³×C₁₀₇	#97	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	1	yes	characteristic
36	C₂×C₄×C₁₀₇	C₄×C₈	1	2	yes	—
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	—	2	8	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	#322	1	2	yes	—
45	C₂²×C₁₀₇	#336	1	1	yes	characteristic
46	C₄×C₁₀₇	—	2	2	no	—
47	C₄×C₁₀₇	C₈²	1	1	yes	characteristic
48	C₄×C₁₀₇	#322	1	1	yes	characteristic
49	C₄×C₁₀₇	#336	1	2	yes	—
50	C₄×C₁₀₇	#364	1	4	yes	—
51	#7138	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂×C₁₀₇	—	2	2	no	—
53	C₂×C₁₀₇	#1273	1	1	yes	characteristic
54	C₂×C₁₀₇	#1376	1	2	yes	—
55	C₄×M₃₂	C₂×C₁₀₇	1	2	yes	—
56	C₂×C₈²	C₂×C₁₀₇	1	1	yes	characteristic
57	C₁₀₇	#7138	1	1	yes	characteristic, direct product factor, Sylow
58	C₂×M₃₂	C₄×C₁₀₇	1	4	yes	—
59	C₂×C₄×C₈	C₄×C₁₀₇	1	2	yes	—
60	C₂×C₄×C₈	C₂²×C₁₀₇	1	1	yes	characteristic
61	#346	—	2	2	no	—
62	C₄×C₁₆	—	2	2	no	—
63	C₈²	—	2	2	no	—
64	C₂²×C₈	—	2	2	no	—
65	C₂²×C₈	C₂×C₄×C₁₀₇	1	2	yes	—
66	C₂×C₄²	C₂×C₄×C₁₀₇	1	1	yes	characteristic
67	M₃₂	—	2	4	no	—
68	M₃₂	C₈×C₁₀₇	1	8	yes	complemented
69	C₂×C₁₆	—	2	4	no	—
70	C₄×C₈	—	2	2	no	—
71	C₄×C₈	C₈×C₁₀₇	1	4	yes	—
72	C₄×C₈	D₄×C₁₀₇	1	2	yes	—
73	C₄×C₈	D₄×C₁₀₇	1	1	yes	characteristic
74	C₄×C₈	Q₈×C₁₀₇	1	1	yes	characteristic
75	C₂²×C₄	C₄²×C₁₀₇	1	1	yes	characteristic
76	C₂²×C₄	M₁₆×C₁₀₇	1	2	yes	—
77	C₂×C₈	—	2	8	no	—
78	C₂×C₈	—	2	4	no	—
79	C₂×C₈	C₂²⋊C₄×C₁₀₇	1	2	yes	—
80	C₂×C₈	C₄⋊C₄×C₁₀₇	1	2	yes	—
81	C₂×C₈	C₂×C₈×C₁₀₇	1	4	yes	—
82	C₂×C₈	M₁₆×C₁₀₇	1	4	yes	—
83	C₄²	C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
84	C₄²	C₄⋊C₄×C₁₀₇	1	1	yes	characteristic
85	C₄²	C₂×C₈×C₁₀₇	1	2	yes	—
86	C₁₆	—	2	8	no	—
87	C₂³	#97×C₁₀₇	1	1	yes	characteristic
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	#95×C₁₀₇	1	1	yes	characteristic
90	C₂×C₄	C₄×C₈×C₁₀₇	1	1	yes	characteristic
91	C₂×C₄	C₄×C₈×C₁₀₇	1	2	yes	—
92	C₂×C₄	#97×C₁₀₇	1	2	yes	—
93	C₂×C₄	C₂²⋊C₈×C₁₀₇	1	2	yes	—
94	C₂×C₄	C₄⋊C₈×C₁₀₇	1	2	yes	—
95	C₈	—	2	8	no	—
96	C₈	C₂²⋊C₈×C₁₀₇	1	4	yes	—
97	C₈	C₄⋊C₈×C₁₀₇	1	4	yes	—
98	C₂²	—	2	2	no	—
99	C₂²	#322×C₁₀₇	1	2	yes	—
100	C₂²	#336×C₁₀₇	1	1	yes	characteristic
101	C₄	—	2	2	no	—
102	C₄	C₈²×C₁₀₇	1	1	yes	characteristic
103	C₄	#322×C₁₀₇	1	1	yes	characteristic
104	C₄	#336×C₁₀₇	1	2	yes	—
105	C₄	#364×C₁₀₇	1	4	yes	—
106	C₂	—	2	2	no	—
107	C₂	C₁₀₇×#1273	1	1	yes	characteristic
108	C₂	C₁₀₇×#1376	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×M₃₂×C₁₀₇

C₂

yes

maximal

C₂×C₈²×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

—

C₈²×C₁₀₇

—

C₂²×C₈×C₁₀₇

—

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

C₈

yes

—

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

M₁₆

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

C₂×C₈

yes

—

C₁₆×C₁₀₇

—

C₂³×C₁₀₇

#97

yes

characteristic

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

#95

yes

characteristic

C₂×C₄×C₁₀₇

C₄×C₈

yes

characteristic

C₂×C₄×C₁₀₇

C₄×C₈

yes

—

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

#322

yes

—

C₂²×C₁₀₇

#336

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

C₈²

yes

C₄×C₁₀₇

yes

C₄×C₁₀₇

yes

—

C₄×C₁₀₇

#364

yes

—

#7138

C₁₀₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₇

—

C₂×C₁₀₇

#1273

yes

characteristic

C₂×C₁₀₇

#1376

yes

—

C₄×M₃₂

C₂×C₁₀₇

yes

—

C₂×C₈²

C₂×C₁₀₇

yes

characteristic

C₁₀₇

#7138

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

—

C₈²

—

C₂²×C₈

—

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

C₈×C₁₀₇

yes

—

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

M₁₆×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

C₂×C₈×C₁₀₇

yes

—

C₁₆

—

C₂³

#97×C₁₀₇

yes

characteristic

C₂×C₄

—

C₂×C₄

#95×C₁₀₇

yes

characteristic

C₂×C₄

C₄×C₈×C₁₀₇

yes

characteristic

C₂×C₄

C₄×C₈×C₁₀₇

yes

—

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

—

C₂²

#322×C₁₀₇

yes

—

100

C₂²

#336×C₁₀₇

yes

characteristic

101

C₄

—

102

C₄

C₈²×C₁₀₇

yes

characteristic

103

C₄

#322×C₁₀₇

yes

characteristic

104

C₄

#336×C₁₀₇

yes

—

105

C₄

#364×C₁₀₇

yes

—

106

C₂

—

107

C₂

C₁₀₇×#1273

yes

characteristic

108

C₂

C₁₀₇×#1376

yes

—

	subgroup	G/H
center	C₄×C₈×C₁₀₇	D₄
commutator	C₄	C₈²×C₁₀₇
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₀₇
socle	C₂²×C₁₀₇	#336
prime radical	C₂³×C₁₀₇	#97
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈×C₁₀₇ ◁ C₂×C₄×C₈×C₁₀₇ ◁ G

subgroup

G/H

center

C₄×C₈×C₁₀₇

D₄

commutator

C₄

C₈²×C₁₀₇

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₁₀₇

C₂²×C₁₀₇

C₂³×C₁₀₇

G ▷ C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₄×C₈×C₁₀₇ ◁ C₂×C₄×C₈×C₁₀₇ ◁ G

N	H ≅ G/N	type
13. M₃₂×C₁₀₇	C₈	semidirect
51. #7138	C₁₀₇	direct (N × H)
57. C₁₀₇	#7138	direct (N × H)
68. M₃₂	C₈×C₁₀₇	semidirect

H ≅ G/N

type

13. M₃₂×C₁₀₇

C₈

semidirect

51. #7138

C₁₀₇

direct (N × H)

57. C₁₀₇

#7138

direct (N × H)

68. 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×C₈²×C₁₀₇	2
#7138	107

subgroup

index

C₄×M₃₂×C₁₀₇

C₂×C₈²×C₁₀₇

#7138

107

Subgroups — C₁₀₇×#7138 #41033102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C107×#7138 #41033102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#7138 #41033102