Group #20293255 — subgroups

Subgroups — #20293255

Order 6848.

206 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 111 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	#16812109	C₂	1	2	yes	maximal, complemented
2	#16812108	C₂	1	2	yes	maximal, complemented
3	#16812101	C₂	1	2	yes	maximal, complemented
4	#16812100	C₂	1	2	yes	maximal, complemented
5	#16812086	C₂	1	4	yes	maximal
6	#16812068	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊Dic₁₀₇	C₂	1	1	yes	maximal, characteristic
8	#117×C₁₀₇	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₂₁₄	C₂²	1	4	yes	—
10	C₂²×Dic₁₀₇	C₂²	1	2	yes	—
11	C₄⋊Dic₁₀₇	C₂²	1	2	yes	—
12	C₄⋊Dic₁₀₇	C₂²	1	2	yes	—
13	C₄⋊Dic₁₀₇	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₁₀₇	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₁₀₇	C₂²	1	4	yes	—
16	Dic₁₀₇⋊C₄	C₂²	1	4	yes	—
17	Dic₁₀₇⋊C₄	C₂²	1	4	yes	—
18	C₂²⋊Dic₁₀₇	C₂²	1	2	yes	—
19	C₄×Dic₁₀₇	C₂²	1	2	yes	—
20	C₂²×C₄×C₁₀₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₁₀₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×C₁₀₇	C₂²	1	1	yes	characteristic
23	C₂²⋊C₄×C₁₀₇	C₂²	1	2	yes	—
24	C₄²×C₁₀₇	C₂²	1	2	yes	—
25	C₂³×C₁₀₇	C₂³	1	1	yes	characteristic
26	C₂×C₄×C₁₀₇	Q₈	1	4	yes	complemented
27	C₂×C₄×C₁₀₇	C₂³	1	2	yes	—
28	C₂×C₄×C₁₀₇	C₂³	1	2	yes	—
29	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₀₇	—	4	2	no	—
32	C₂×Dic₁₀₇	C₂³	1	2	yes	—
33	C₂×Dic₁₀₇	C₂³	1	2	yes	—
34	C₂×Dic₁₀₇	C₂³	1	4	yes	—
35	Dic₂₁₄	—	4	4	no	—
36	C₂²×C₁₀₇	—	2	1	no	—
37	C₂²×C₁₀₇	—	2	1	no	—
38	C₂²×C₁₀₇	C₂×Q₈	1	2	yes	—
39	C₂²×C₁₀₇	C₂⁴	1	1	yes	characteristic
40	C₄×C₁₀₇	—	2	2	no	—
41	C₄×C₁₀₇	—	2	2	no	—
42	C₄×C₁₀₇	C₂×Q₈	1	2	yes	—
43	C₄×C₁₀₇	C₂×Q₈	1	2	yes	—
44	Dic₁₀₇	—	2	2	no	—
45	Dic₁₀₇	—	2	4	no	—
46	Dic₁₀₇	—	2	2	no	—
47	C₂×C₁₀₇	—	2	2	no	—
48	C₂×C₁₀₇	C₂²×Q₈	1	1	yes	characteristic
49	C₂×C₁₀₇	H(5,𝔽₂)	1	1	yes	characteristic
50	C₂×C₁₀₇	2₋¹⁺⁴	1	1	yes	characteristic
51	C₁₀₇	#544	1	1	yes	characteristic, complemented, Sylow
52	#544	—	107	1	no	maximal, Sylow
53	C₄⋊Q₈	—	107	2	no	—
54	C₄⋊Q₈	—	107	2	no	—
55	#125	—	107	2	no	—
56	#125	—	107	2	no	—
57	C₂²⋊Q₈	—	107	4	no	—
58	#117	—	107	1	no	—
59	#117	D₁₀₇	1	1	yes	characteristic
60	C₂×C₄⋊C₄	—	107	1	no	—
61	C₂×Q₈	—	107	4	no	—
62	C₂²×C₄	—	107	2	no	—
63	C₂²×C₄	C₂×D₁₀₇	1	1	yes	characteristic
64	C₄⋊C₄	—	107	2	no	—
65	C₄⋊C₄	—	107	2	no	—
66	C₄⋊C₄	—	107	1	no	—
67	C₄⋊C₄	—	107	1	no	—
68	C₄⋊C₄	—	107	4	no	—
69	C₄⋊C₄	—	107	4	no	—
70	C₄⋊C₄	—	107	4	no	—
71	C₄⋊C₄	C₂×D₁₀₇	1	1	yes	characteristic
72	C₄⋊C₄	C₂×D₁₀₇	1	1	yes	characteristic
73	C₂²⋊C₄	—	107	2	no	—
74	C₂²⋊C₄	C₂×D₁₀₇	1	2	yes	—
75	C₄²	—	107	2	no	—
76	C₄²	C₂×D₁₀₇	1	2	yes	—
77	C₂³	C₂²×D₁₀₇	1	1	yes	characteristic
78	Q₈	—	428	4	no	—
79	C₂×C₄	—	107	2	no	—
80	C₂×C₄	—	107	2	no	—
81	C₂×C₄	—	107	4	no	—
82	C₂×C₄	—	428	2	no	—
83	C₂×C₄	Dic₂₁₄	1	4	yes	complemented
84	C₂×C₄	C₂²×D₁₀₇	1	2	yes	—
85	C₂×C₄	C₂²×D₁₀₇	1	2	yes	—
86	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
87	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
88	C₂²	—	2	1	no	—
89	C₂²	—	2	1	no	—
90	C₂²	C₂×Dic₂₁₄	1	2	yes	—
91	C₂²	C₂³×D₁₀₇	1	1	yes	characteristic
92	C₄	—	2	2	no	—
93	C₄	—	2	2	no	—
94	C₄	—	214	4	no	—
95	C₄	—	214	2	no	—
96	C₄	—	214	2	no	—
97	C₄	C₂×Dic₂₁₄	1	2	yes	—
98	C₄	C₂×Dic₂₁₄	1	2	yes	—
99	C₂	—	2	2	no	—
100	C₂	C₂²×Dic₂₁₄	1	1	yes	characteristic
101	C₂	#16812155	1	1	yes	characteristic
102	C₂	#16812156	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16812109

C₂

yes

maximal, complemented

#16812108

C₂

yes

maximal, complemented

#16812101

C₂

yes

maximal, complemented

#16812100

C₂

yes

maximal, complemented

#16812086

C₂

yes

maximal

#16812068

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₁₀₇

C₂

yes

maximal, characteristic

#117×C₁₀₇

C₂

yes

maximal, characteristic

C₂×Dic₂₁₄

C₂²

yes

—

C₂²×Dic₁₀₇

C₂²

yes

—

C₄⋊Dic₁₀₇

C₂²

yes

—

C₄⋊Dic₁₀₇

C₂²

yes

—

C₄⋊Dic₁₀₇

C₂²

yes

characteristic

C₄⋊Dic₁₀₇

C₂²

yes

characteristic

C₄⋊Dic₁₀₇

C₂²

yes

—

Dic₁₀₇⋊C₄

C₂²

yes

—

Dic₁₀₇⋊C₄

C₂²

yes

—

C₂²⋊Dic₁₀₇

C₂²

yes

—

C₄×Dic₁₀₇

C₂²

yes

—

C₂²×C₄×C₁₀₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₇

C₂²

yes

characteristic

C₂²⋊C₄×C₁₀₇

C₂²

yes

—

C₄²×C₁₀₇

C₂²

yes

—

C₂³×C₁₀₇

C₂³

yes

characteristic

C₂×C₄×C₁₀₇

Q₈

yes

complemented

C₂×C₄×C₁₀₇

C₂³

yes

—

C₂×C₄×C₁₀₇

C₂³

yes

—

C₂×C₄×C₁₀₇

C₂³

yes

characteristic

C₂×C₄×C₁₀₇

C₂³

yes

characteristic

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

C₂³

yes

—

C₂×Dic₁₀₇

C₂³

yes

—

C₂×Dic₁₀₇

C₂³

yes

—

Dic₂₁₄

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₂×Q₈

yes

—

C₂²×C₁₀₇

C₂⁴

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂×Q₈

yes

—

C₄×C₁₀₇

C₂×Q₈

yes

—

Dic₁₀₇

—

Dic₁₀₇

—

Dic₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

C₂²×Q₈

yes

characteristic

C₂×C₁₀₇

H(5,𝔽₂)

yes

characteristic

C₂×C₁₀₇

2₋¹⁺⁴

yes

characteristic

C₁₀₇

#544

yes

characteristic, complemented, Sylow

#544

—

107

maximal, Sylow

C₄⋊Q₈

—

107

—

C₄⋊Q₈

—

107

—

#125

—

107

—

#125

—

107

—

C₂²⋊Q₈

—

107

—

#117

—

107

—

#117

D₁₀₇

yes

characteristic

C₂×C₄⋊C₄

—

107

—

C₂×Q₈

—

107

—

C₂²×C₄

—

107

—

C₂²×C₄

C₂×D₁₀₇

yes

characteristic

C₄⋊C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

C₂×D₁₀₇

yes

characteristic

C₄⋊C₄

C₂×D₁₀₇

yes

characteristic

C₂²⋊C₄

—

107

—

C₂²⋊C₄

C₂×D₁₀₇

yes

—

C₄²

—

107

—

C₄²

C₂×D₁₀₇

yes

—

C₂³

C₂²×D₁₀₇

yes

characteristic

Q₈

—

428

—

C₂×C₄

—

107

—

C₂×C₄

—

107

—

C₂×C₄

—

107

—

C₂×C₄

—

428

—

C₂×C₄

Dic₂₁₄

yes

complemented

C₂×C₄

C₂²×D₁₀₇

yes

—

C₂×C₄

C₂²×D₁₀₇

yes

—

C₂×C₄

C₂²×D₁₀₇

yes

characteristic

C₂×C₄

C₂²×D₁₀₇

yes

characteristic

C₂²

—

C₂²

—

C₂²

C₂×Dic₂₁₄

yes

—

C₂²

C₂³×D₁₀₇

yes

characteristic

C₄

—

C₄

—

C₄

—

214

—

C₄

—

214

—

C₄

—

214

—

C₄

C₂×Dic₂₁₄

yes

—

C₄

C₂×Dic₂₁₄

yes

—

C₂

—

100

C₂

C₂²×Dic₂₁₄

yes

characteristic

101

C₂

#16812155

yes

characteristic

102

C₂

#16812156

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₇
commutator, socle	C₂²×C₁₀₇	C₂⁴
Fitting	#117×C₁₀₇	C₂
hypercenter	#117	D₁₀₇
hypocenter	C₁₀₇	#544
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₀₇

commutator, socle

C₂²×C₁₀₇

C₂⁴

Fitting

#117×C₁₀₇

C₂

D₁₀₇

C₁₀₇

G ▷ C₂²×C₁₀₇ ▷ C₁₀₇

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
1. #16812109	C₂	semidirect
2. #16812108	C₂	semidirect
3. #16812101	C₂	semidirect
4. #16812100	C₂	semidirect
26. C₂×C₄×C₁₀₇	Q₈	semidirect
51. C₁₀₇	#544	semidirect
83. C₂×C₄	Dic₂₁₄	semidirect

H ≅ G/N

type

1. #16812109

C₂

semidirect

2. #16812108

C₂

semidirect

3. #16812101

C₂

semidirect

4. #16812100

C₂

semidirect

26. C₂×C₄×C₁₀₇

Q₈

semidirect

51. C₁₀₇

#544

semidirect

83. C₂×C₄

Dic₂₁₄

semidirect

Subgroups — #20293255

Conjugacy classes of subgroups

Complements