Group #28336614 — subgroups

Subgroups — #28336614

Order 13696.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂×C₄⋊(C₁₀₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₀₇⋊C₈	C₂	1	4	yes	maximal
3	#20292473	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₀₇⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄×Dic₁₀₇	C₂²	1	2	yes	—
7	C₄⋊(C₁₀₇⋊C₈)	—	2	2	no	—
8	Dic₁₀₇⋊C₈	—	2	8	no	—
9	C₁₀₇⋊#95	C₄	1	2	yes	—
10	C₁₀₇⋊#95	C₄	1	2	yes	—
11	C₂²×C₈×C₁₀₇	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₀₇	—	2	2	no	—
14	C₂²×Dic₁₀₇	C₂×C₄	1	4	yes	—
15	C₂×C₁₀₇⋊C₈	—	2	4	no	—
16	C₂×C₁₀₇⋊C₈	—	2	2	no	—
17	C₄×Dic₁₀₇	D₄	1	4	yes	—
18	C₄×Dic₁₀₇	Q₈	1	4	yes	—
19	C₂²×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₁₀₇	—	2	4	no	—
23	C₂×C₈×C₁₀₇	—	2	2	no	—
24	C₄²×C₁₀₇	D₄	1	2	yes	—
25	C₄²×C₁₀₇	Q₈	1	2	yes	—
26	C₂³×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₀₇	—	2	2	no	—
28	C₂×C₄×C₁₀₇	—	2	2	no	—
29	C₂×C₄×C₁₀₇	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₁₀₇	C₄⋊C₄	1	2	yes	—
31	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
32	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₇	C₂×Q₈	1	2	yes	—
34	C₂×C₄×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
35	C₂×Dic₁₀₇	—	2	4	no	—
36	C₂×Dic₁₀₇	—	2	4	no	—
37	C₂×Dic₁₀₇	C₄⋊C₄	1	8	yes	—
38	C₈×C₁₀₇	—	4	2	no	—
39	C₁₀₇⋊C₈	—	4	2	no	—
40	C₂²×C₁₀₇	C₂×C₄⋊C₄	1	2	yes	—
41	C₂²×C₁₀₇	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₁₀₇	#131	1	2	yes	—
43	C₂²×C₁₀₇	#131	1	2	yes	—
44	C₄×C₁₀₇	—	2	2	no	—
45	C₄×C₁₀₇	—	2	2	no	—
46	C₄×C₁₀₇	#127	1	1	yes	characteristic
47	C₄×C₁₀₇	C₄⋊Q₈	1	1	yes	characteristic
48	C₄×C₁₀₇	C₄⋊Q₈	1	2	yes	—
49	Dic₁₀₇	—	2	8	no	—
50	C₂×C₁₀₇	#384	1	1	yes	characteristic
51	C₂×C₁₀₇	#424	1	4	yes	—
52	C₂×C₁₀₇	#424	1	1	yes	characteristic
53	C₂×C₁₀₇	#424	1	1	yes	characteristic
54	#1843	—	107	1	no	maximal, Sylow
55	C₁₀₇	#1843	1	1	yes	characteristic, complemented, Sylow
56	C₂×C₄⋊C₈	—	107	4	no	—
57	C₂×C₄⋊C₈	—	107	1	no	—
58	C₂×C₄⋊C₈	D₁₀₇	1	1	yes	characteristic
59	#376	—	107	1	no	—
60	C₂²×C₈	—	107	2	no	—
61	C₂²×C₈	C₂×D₁₀₇	1	2	yes	—
62	C₂×C₄²	—	107	2	no	—
63	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
64	C₄⋊C₈	—	2	2	no	—
65	C₄⋊C₈	—	214	8	no	—
66	C₄⋊C₈	—	214	2	no	—
67	#95	—	107	2	no	—
68	#95	—	107	2	no	—
69	C₂²×C₄	—	107	4	no	—
70	C₂²×C₄	C₄×D₁₀₇	1	1	yes	characteristic
71	C₂²×C₄	C₄×D₁₀₇	1	1	yes	characteristic
72	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
73	C₂×C₈	—	2	4	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	214	4	no	—
76	C₂×C₈	—	214	2	no	—
77	C₄²	—	107	4	no	—
78	C₄²	—	107	4	no	—
79	C₄²	Dic₂₁₄	1	2	yes	—
80	C₄²	C₂²⋊D₁₀₇	1	2	yes	—
81	C₂³	C₂×C₄×D₁₀₇	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	107	8	no	—
85	C₂×C₄	—	214	4	no	—
86	C₂×C₄	—	214	4	no	—
87	C₂×C₄	Dic₁₀₇⋊C₄	1	2	yes	—
88	C₂×C₄	Dic₁₀₇⋊C₄	1	2	yes	—
89	C₂×C₄	D₄×D₁₀₇	1	2	yes	—
90	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
91	C₂×C₄	Q₈×D₁₀₇	1	2	yes	—
92	C₂×C₄	C₂×Dic₂₁₄	1	1	yes	characteristic
93	C₈	—	4	2	no	—
94	C₈	—	428	2	no	—
95	C₂²	C₄⋊C₄×D₁₀₇	1	2	yes	—
96	C₂²	C₂×Dic₁₀₇⋊C₄	1	1	yes	characteristic
97	C₂²	#16812115	1	2	yes	—
98	C₂²	#16812117	1	2	yes	—
99	C₄	—	2	2	no	—
100	C₄	—	2	2	no	—
101	C₄	—	214	8	no	—
102	C₄	#16812105	1	1	yes	characteristic
103	C₄	#16812107	1	1	yes	characteristic
104	C₄	#16812108	1	2	yes	—
105	C₂	#20292496	1	1	yes	characteristic
106	C₂	#20292652	1	4	yes	—
107	C₂	#20292653	1	1	yes	characteristic
108	C₂	#20292655	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₁₀₇⋊C₈)

C₂

yes

maximal, characteristic

C₂×Dic₁₀₇⋊C₈

C₂

yes

maximal

#20292473

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₁₀₇

C₂

yes

maximal, characteristic

C₂²×C₁₀₇⋊C₈

C₂²

yes

—

C₂×C₄×Dic₁₀₇

C₂²

yes

—

C₄⋊(C₁₀₇⋊C₈)

—

Dic₁₀₇⋊C₈

—

C₁₀₇⋊#95

C₄

yes

—

C₁₀₇⋊#95

C₄

yes

—

C₂²×C₈×C₁₀₇

C₂²

yes

—

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₇

—

C₂²×Dic₁₀₇

C₂×C₄

yes

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

—

C₄×Dic₁₀₇

D₄

yes

—

C₄×Dic₁₀₇

Q₈

yes

—

C₂²×C₄×C₁₀₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₄²×C₁₀₇

D₄

yes

—

C₄²×C₁₀₇

Q₈

yes

—

C₂³×C₁₀₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×Q₈

yes

—

C₂×C₄×C₁₀₇

C₂×Q₈

yes

characteristic

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

C₄⋊C₄

yes

—

C₈×C₁₀₇

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

C₂×C₄⋊C₄

yes

—

C₂²×C₁₀₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₇

#131

yes

—

C₂²×C₁₀₇

#131

yes

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

#127

yes

characteristic

C₄×C₁₀₇

C₄⋊Q₈

yes

characteristic

C₄×C₁₀₇

C₄⋊Q₈

yes

—

Dic₁₀₇

—

C₂×C₁₀₇

#384

yes

characteristic

C₂×C₁₀₇

#424

yes

—

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

—

107

maximal, Sylow

C₁₀₇

#1843

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

107

—

C₂×C₄⋊C₈

—

107

—

C₂×C₄⋊C₈

D₁₀₇

yes

characteristic

#376

—

107

—

C₂²×C₈

—

107

—

C₂²×C₈

C₂×D₁₀₇

yes

—

C₂×C₄²

—

107

—

C₂×C₄²

C₂×D₁₀₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

214

—

C₄⋊C₈

—

214

—

#95

—

107

—

#95

—

107

—

C₂²×C₄

—

107

—

C₂²×C₄

C₄×D₁₀₇

yes

characteristic

C₂²×C₄

C₄×D₁₀₇

yes

characteristic

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

214

—

C₂×C₈

—

214

—

C₄²

—

107

—

C₄²

—

107

—

C₄²

Dic₂₁₄

yes

—

C₄²

C₂²⋊D₁₀₇

yes

—

C₂³

C₂×C₄×D₁₀₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

107

—

C₂×C₄

—

214

—

C₂×C₄

—

214

—

C₂×C₄

Dic₁₀₇⋊C₄

yes

—

C₂×C₄

Dic₁₀₇⋊C₄

yes

—

C₂×C₄

D₄×D₁₀₇

yes

—

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

characteristic

C₂×C₄

Q₈×D₁₀₇

yes

—

C₂×C₄

C₂×Dic₂₁₄

yes

characteristic

C₈

—

C₈

—

428

—

C₂²

C₄⋊C₄×D₁₀₇

yes

—

C₂²

C₂×Dic₁₀₇⋊C₄

yes

characteristic

C₂²

#16812115

yes

—

C₂²

#16812117

yes

—

C₄

—

100

C₄

—

101

C₄

—

214

—

102

C₄

#16812105

yes

characteristic

103

C₄

#16812107

yes

characteristic

104

C₄

#16812108

yes

—

105

C₂

#20292496

yes

characteristic

106

C₂

#20292652

yes

—

107

C₂

#20292653

yes

characteristic

108

C₂

#20292655

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₀₇
commutator, socle, prime radical	C₂³×C₁₀₇	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₁₀₇	C₂
cosocle	C₂²×C₄×C₁₀₇	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₀₇
hypocenter	C₁₀₇	#1843
LCS	G ▷ C₂³×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₀₇

commutator, socle, prime radical

C₂³×C₁₀₇

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₁₀₇

C₂

cosocle

C₂²×C₄×C₁₀₇

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₀₇

C₁₀₇

G ▷ C₂³×C₁₀₇ ▷ C₁₀₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
55. C₁₀₇	#1843	semidirect

H ≅ G/N

type

55. C₁₀₇

#1843

semidirect

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

subgroup	index
C₂×C₄⋊(C₁₀₇⋊C₈)	2
C₂×Dic₁₀₇⋊C₈	2
#20292473	2
C₂×C₄⋊C₈×C₁₀₇	2
#1843	107

subgroup

index

C₂×C₄⋊(C₁₀₇⋊C₈)

C₂×Dic₁₀₇⋊C₈

#20292473

C₂×C₄⋊C₈×C₁₀₇

#1843

107

Subgroups — #28336614

Conjugacy classes of subgroups

Complements

Maximal subgroups