Group #41034786 — subgroups

Subgroups — C₁₀₇×#8822 #41034786

Order 27392.

354 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 214 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×#1848	C₂	1	2	yes	maximal
2	C₁₀₇×#1566	C₂	1	4	yes	maximal
3	C₂×C₈²×C₁₀₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#108×C₁₀₇	—	2	4	no	—
5	#423×C₁₀₇	C₂²	1	4	yes	—
6	C₂×C₄×C₈×C₁₀₇	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
8	#334×C₁₀₇	—	2	8	no	—
9	C₈²×C₁₀₇	C₄	1	1	yes	characteristic
10	C₈²×C₁₀₇	C₄	1	2	yes	—
11	C₈²×C₁₀₇	C₄	1	1	yes	characteristic
12	C₂×M₁₆×C₁₀₇	—	2	4	no	—
13	C₂²×C₈×C₁₀₇	D₄	1	2	yes	—
14	C₂²×C₈×C₁₀₇	Q₈	1	4	yes	—
15	C₂×C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
16	#108×C₁₀₇	—	4	8	no	—
17	C₄⋊C₈×C₁₀₇	—	2	8	no	—
18	C₄×C₈×C₁₀₇	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₁₀₇	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₁₀₇	D₄	1	4	yes	—
23	C₄×C₈×C₁₀₇	Q₈	1	2	yes	—
24	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₀₇	C₂×Q₈	1	2	yes	—
26	M₁₆×C₁₀₇	—	4	8	no	—
27	C₂×C₈×C₁₀₇	—	2	8	no	—
28	C₂×C₈×C₁₀₇	—	4	4	no	—
29	C₂×C₈×C₁₀₇	C₄⋊C₄	1	4	yes	—
30	C₂×C₈×C₁₀₇	C₄⋊C₄	1	2	yes	—
31	C₂×C₈×C₁₀₇	C₄⋊C₄	1	2	yes	—
32	C₂×C₈×C₁₀₇	C₄⋊C₄	1	4	yes	—
33	C₂×C₈×C₁₀₇	SD₁₆	1	8	yes	—
34	C₄²×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
35	C₄²×C₁₀₇	C₂×D₄	1	2	yes	—
36	C₄²×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
37	C₂³×C₁₀₇	C₄⋊Q₈	1	1	yes	characteristic
38	C₂×C₄×C₁₀₇	—	2	4	no	—
39	C₂×C₄×C₁₀₇	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₇	C₂×C₄⋊C₄	1	2	yes	—
41	C₂×C₄×C₁₀₇	#127	1	1	yes	characteristic
42	C₂×C₄×C₁₀₇	C₄⋊Q₈	1	2	yes	—
43	C₂×C₄×C₁₀₇	C₂×SD₁₆	1	4	yes	—
44	C₈×C₁₀₇	—	2	4	no	—
45	C₈×C₁₀₇	—	8	4	no	—
46	C₈×C₁₀₇	#106	1	4	yes	—
47	C₈×C₁₀₇	#106	1	4	yes	—
48	C₈×C₁₀₇	#108	1	8	yes	complemented
49	C₂²×C₁₀₇	—	2	2	no	—
50	C₂²×C₁₀₇	#384	1	1	yes	characteristic
51	C₂²×C₁₀₇	#498	1	2	yes	—
52	C₄×C₁₀₇	—	2	2	no	—
53	C₄×C₁₀₇	C₂×#106	1	2	yes	—
54	C₄×C₁₀₇	C₂×#108	1	4	yes	—
55	C₄×C₁₀₇	#492	1	2	yes	—
56	#8822	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂×C₁₀₇	—	2	2	no	—
58	C₂×C₁₀₇	#1844	1	1	yes	characteristic
59	C₂×C₁₀₇	#1848	1	2	yes	—
60	#1848	C₂×C₁₀₇	1	2	yes	—
61	#1566	C₂×C₁₀₇	1	4	yes	—
62	C₂×C₈²	C₂×C₁₀₇	1	1	yes	characteristic
63	C₁₀₇	#8822	1	1	yes	characteristic, direct product factor, Sylow
64	C₂×#108	—	2	4	no	—
65	#423	C₂²×C₁₀₇	1	4	yes	—
66	C₂×C₄×C₈	C₂²×C₁₀₇	1	2	yes	—
67	C₂×C₄×C₈	C₂²×C₁₀₇	1	1	yes	characteristic
68	#334	—	2	8	no	—
69	C₈²	C₄×C₁₀₇	1	1	yes	characteristic
70	C₈²	C₄×C₁₀₇	1	2	yes	—
71	C₈²	C₄×C₁₀₇	1	1	yes	characteristic
72	C₂×M₁₆	—	2	4	no	—
73	C₂²×C₈	D₄×C₁₀₇	1	2	yes	—
74	C₂²×C₈	Q₈×C₁₀₇	1	4	yes	—
75	C₂×C₄²	C₂³×C₁₀₇	1	1	yes	characteristic
76	#108	—	4	8	no	—
77	C₄⋊C₈	—	2	8	no	—
78	C₄×C₈	C₂×C₄×C₁₀₇	1	2	yes	—
79	C₄×C₈	C₂×C₄×C₁₀₇	1	1	yes	characteristic
80	C₄×C₈	C₂×C₄×C₁₀₇	1	1	yes	characteristic
81	C₄×C₈	C₂×C₄×C₁₀₇	1	2	yes	—
82	C₄×C₈	D₄×C₁₀₇	1	4	yes	—
83	C₄×C₈	Q₈×C₁₀₇	1	2	yes	—
84	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
85	C₂²×C₄	C₂×Q₈×C₁₀₇	1	2	yes	—
86	M₁₆	—	4	8	no	—
87	C₂×C₈	—	2	8	no	—
88	C₂×C₈	—	4	4	no	—
89	C₂×C₈	C₄⋊C₄×C₁₀₇	1	4	yes	—
90	C₂×C₈	C₄⋊C₄×C₁₀₇	1	2	yes	—
91	C₂×C₈	C₄⋊C₄×C₁₀₇	1	2	yes	—
92	C₂×C₈	C₄⋊C₄×C₁₀₇	1	4	yes	—
93	C₂×C₈	SD₁₆×C₁₀₇	1	8	yes	—
94	C₄²	C₂²×C₄×C₁₀₇	1	1	yes	characteristic
95	C₄²	C₂×D₄×C₁₀₇	1	2	yes	—
96	C₄²	C₂×Q₈×C₁₀₇	1	1	yes	characteristic
97	C₂³	C₄⋊Q₈×C₁₀₇	1	1	yes	characteristic
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	C₂×C₄⋊C₄×C₁₀₇	1	1	yes	characteristic
100	C₂×C₄	C₂×C₄⋊C₄×C₁₀₇	1	2	yes	—
101	C₂×C₄	#127×C₁₀₇	1	1	yes	characteristic
102	C₂×C₄	C₄⋊Q₈×C₁₀₇	1	2	yes	—
103	C₂×C₄	C₂×SD₁₆×C₁₀₇	1	4	yes	—
104	C₈	—	2	4	no	—
105	C₈	—	8	4	no	—
106	C₈	#106×C₁₀₇	1	4	yes	—
107	C₈	#106×C₁₀₇	1	4	yes	—
108	C₈	#108×C₁₀₇	1	8	yes	complemented
109	C₂²	—	2	2	no	—
110	C₂²	#384×C₁₀₇	1	1	yes	characteristic
111	C₂²	#498×C₁₀₇	1	2	yes	—
112	C₄	—	2	2	no	—
113	C₄	C₂×#106×C₁₀₇	1	2	yes	—
114	C₄	C₂×#108×C₁₀₇	1	4	yes	—
115	C₄	#492×C₁₀₇	1	2	yes	—
116	C₂	—	2	2	no	—
117	C₂	C₁₀₇×#1844	1	1	yes	characteristic
118	C₂	C₁₀₇×#1848	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₇×#1848

C₂

yes

maximal

C₁₀₇×#1566

C₂

yes

maximal

C₂×C₈²×C₁₀₇

C₂

yes

maximal, characteristic

C₂×#108×C₁₀₇

—

#423×C₁₀₇

C₂²

yes

—

C₂×C₄×C₈×C₁₀₇

C₂²

yes

—

C₂×C₄×C₈×C₁₀₇

C₂²

yes

characteristic

#334×C₁₀₇

—

C₈²×C₁₀₇

C₄

yes

characteristic

C₈²×C₁₀₇

C₄

yes

—

C₈²×C₁₀₇

C₄

yes

characteristic

C₂×M₁₆×C₁₀₇

—

C₂²×C₈×C₁₀₇

D₄

yes

—

C₂²×C₈×C₁₀₇

Q₈

yes

—

C₂×C₄²×C₁₀₇

C₂³

yes

characteristic

#108×C₁₀₇

—

C₄⋊C₈×C₁₀₇

—

C₄×C₈×C₁₀₇

C₂×C₄

yes

—

C₄×C₈×C₁₀₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₀₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₀₇

C₂×C₄

yes

—

C₄×C₈×C₁₀₇

D₄

yes

—

C₄×C₈×C₁₀₇

Q₈

yes

—

C₂²×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₀₇

C₂×Q₈

yes

—

M₁₆×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₀₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₀₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₀₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₀₇

SD₁₆

yes

—

C₄²×C₁₀₇

C₂²×C₄

yes

characteristic

C₄²×C₁₀₇

C₂×D₄

yes

—

C₄²×C₁₀₇

C₂×Q₈

yes

characteristic

C₂³×C₁₀₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₁₀₇

#127

yes

characteristic

C₂×C₄×C₁₀₇

C₄⋊Q₈

yes

—

C₂×C₄×C₁₀₇

C₂×SD₁₆

yes

—

C₈×C₁₀₇

—

C₈×C₁₀₇

—

C₈×C₁₀₇

#106

yes

—

C₈×C₁₀₇

#106

yes

—

C₈×C₁₀₇

#108

yes

complemented

C₂²×C₁₀₇

—

C₂²×C₁₀₇

#384

yes

characteristic

C₂²×C₁₀₇

#498

yes

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂×#106

yes

—

C₄×C₁₀₇

C₂×#108

yes

—

C₄×C₁₀₇

#492

yes

—

#8822

C₁₀₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₇

—

C₂×C₁₀₇

#1844

yes

characteristic

C₂×C₁₀₇

#1848

yes

—

#1848

C₂×C₁₀₇

yes

—

#1566

C₂×C₁₀₇

yes

—

C₂×C₈²

C₂×C₁₀₇

yes

characteristic

C₁₀₇

#8822

yes

characteristic, direct product factor, Sylow

C₂×#108

—

#423

C₂²×C₁₀₇

yes

—

C₂×C₄×C₈

C₂²×C₁₀₇

yes

—

C₂×C₄×C₈

C₂²×C₁₀₇

yes

characteristic

#334

—

C₈²

C₄×C₁₀₇

yes

characteristic

C₈²

C₄×C₁₀₇

yes

—

C₈²

C₄×C₁₀₇

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

D₄×C₁₀₇

yes

—

C₂²×C₈

Q₈×C₁₀₇

yes

—

C₂×C₄²

C₂³×C₁₀₇

yes

characteristic

#108

—

C₄⋊C₈

—

C₄×C₈

C₂×C₄×C₁₀₇

yes

—

C₄×C₈

C₂×C₄×C₁₀₇

yes

characteristic

C₄×C₈

C₂×C₄×C₁₀₇

yes

characteristic

C₄×C₈

C₂×C₄×C₁₀₇

yes

—

C₄×C₈

D₄×C₁₀₇

yes

—

C₄×C₈

Q₈×C₁₀₇

yes

—

C₂²×C₄

C₂×D₄×C₁₀₇

yes

characteristic

C₂²×C₄

C₂×Q₈×C₁₀₇

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₁₀₇

yes

—

C₂×C₈

C₄⋊C₄×C₁₀₇

yes

—

C₂×C₈

C₄⋊C₄×C₁₀₇

yes

—

C₂×C₈

C₄⋊C₄×C₁₀₇

yes

—

C₂×C₈

SD₁₆×C₁₀₇

yes

—

C₄²

C₂²×C₄×C₁₀₇

yes

characteristic

C₄²

C₂×D₄×C₁₀₇

yes

—

C₄²

C₂×Q₈×C₁₀₇

yes

characteristic

C₂³

C₄⋊Q₈×C₁₀₇

yes

characteristic

C₂×C₄

—

C₂×C₄

C₂×C₄⋊C₄×C₁₀₇

yes

characteristic

100

C₂×C₄

C₂×C₄⋊C₄×C₁₀₇

yes

—

101

C₂×C₄

#127×C₁₀₇

yes

characteristic

102

C₂×C₄

C₄⋊Q₈×C₁₀₇

yes

—

103

C₂×C₄

C₂×SD₁₆×C₁₀₇

yes

—

104

C₈

—

105

C₈

—

106

C₈

#106×C₁₀₇

yes

—

107

C₈

#106×C₁₀₇

yes

—

108

C₈

#108×C₁₀₇

yes

complemented

109

C₂²

—

110

C₂²

#384×C₁₀₇

yes

characteristic

111

C₂²

#498×C₁₀₇

yes

—

112

C₄

—

113

C₄

C₂×#106×C₁₀₇

yes

—

114

C₄

C₂×#108×C₁₀₇

yes

—

115

C₄

#492×C₁₀₇

yes

—

116

C₂

—

117

C₂

C₁₀₇×#1844

yes

characteristic

118

C₂

C₁₀₇×#1848

yes

—

	subgroup	G/H
center	C₂×C₄×C₁₀₇	#127
commutator	C₄²	C₂²×C₄×C₁₀₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₇
socle	C₂²×C₁₀₇	#384
prime radical	C₂³×C₁₀₇	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₇ ◁ C₂×C₄²×C₁₀₇ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₀₇

#127

commutator

C₄²

C₂²×C₄×C₁₀₇

Frattini, cosocle

C₂×C₄²

C₂³×C₁₀₇

socle

C₂²×C₁₀₇

#384

prime radical

C₂³×C₁₀₇

C₄⋊Q₈

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₁₀₇ ◁ C₂×C₄²×C₁₀₇ ◁ G

N	H ≅ G/N	type
48. C₈×C₁₀₇	#108	semidirect
56. #8822	C₁₀₇	direct (N × H)
63. C₁₀₇	#8822	direct (N × H)
108. C₈	#108×C₁₀₇	semidirect

H ≅ G/N

type

48. C₈×C₁₀₇

#108

semidirect

56. #8822

C₁₀₇

direct (N × H)

63. C₁₀₇

#8822

direct (N × H)

108. C₈

#108×C₁₀₇

semidirect

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

subgroup	index
C₁₀₇×#1848	2
C₁₀₇×#1566	2
C₂×C₈²×C₁₀₇	2
#8822	107

subgroup

index

C₁₀₇×#1848

C₁₀₇×#1566

C₂×C₈²×C₁₀₇

#8822

107

Subgroups — C₁₀₇×#8822 #41034786

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C107×#8822 #41034786

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#8822 #41034786