Group #28336580 — subgroups

Subgroups — #28336580

Order 13696.

216 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 101 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	#20292473	C₂	1	1	yes	maximal, characteristic
3	#20292381	C₂	1	4	yes	maximal
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	C₁₀₇⋊#95	C₄	1	2	yes	—
9	C₁₀₇⋊#95	C₄	1	2	yes	—
10	C₂²×C₈×C₁₀₇	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₁₀₇	—	2	2	no	—
13	C₂²×Dic₁₀₇	C₂×C₄	1	4	yes	—
14	C₂×C₁₀₇⋊C₈	—	2	2	no	—
15	C₂×C₁₀₇⋊C₈	—	2	4	no	—
16	C₄×Dic₁₀₇	—	2	2	no	—
17	C₄×Dic₁₀₇	—	2	2	no	—
18	C₂²×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
21	C₂×C₈×C₁₀₇	—	2	2	no	—
22	C₂×C₈×C₁₀₇	—	2	4	no	—
23	C₄²×C₁₀₇	D₄	1	2	yes	—
24	C₄²×C₁₀₇	Q₈	1	2	yes	—
25	C₂³×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₄×C₁₀₇	—	2	2	no	—
27	C₂×C₄×C₁₀₇	—	2	2	no	—
28	C₂×C₄×C₁₀₇	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₁₀₇	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄×C₁₀₇	#41	1	2	yes	—
33	C₂×C₄×C₁₀₇	#41	1	2	yes	—
34	C₂×Dic₁₀₇	—	2	4	no	—
35	C₂×Dic₁₀₇	—	2	4	no	—
36	C₂×Dic₁₀₇	—	2	4	no	—
37	C₈×C₁₀₇	—	4	2	no	—
38	C₁₀₇⋊C₈	—	4	2	no	—
39	C₂²×C₁₀₇	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₁₀₇	#117	1	2	yes	—
41	C₂²×C₁₀₇	#131	1	2	yes	—
42	C₂²×C₁₀₇	#131	1	2	yes	—
43	C₄×C₁₀₇	—	2	2	no	—
44	C₄×C₁₀₇	—	2	2	no	—
45	C₄×C₁₀₇	#124	1	1	yes	characteristic
46	C₄×C₁₀₇	#124	1	1	yes	characteristic
47	C₄×C₁₀₇	#125	1	2	yes	—
48	Dic₁₀₇	—	4	4	no	—
49	C₂×C₁₀₇	#382	1	1	yes	characteristic
50	C₂×C₁₀₇	#424	1	1	yes	characteristic
51	C₂×C₁₀₇	#424	1	1	yes	characteristic
52	C₂×C₁₀₇	#433	1	4	yes	—
53	#1832	—	107	1	no	maximal, Sylow
54	C₁₀₇	#1832	1	1	yes	characteristic, complemented, Sylow
55	C₂×C₄⋊C₈	—	107	1	no	—
56	C₂×C₄⋊C₈	D₁₀₇	1	1	yes	characteristic
57	#376	—	107	1	no	—
58	#336	—	107	4	no	—
59	C₂²×C₈	—	107	2	no	—
60	C₂²×C₈	C₂×D₁₀₇	1	2	yes	—
61	C₂×C₄²	—	107	2	no	—
62	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
63	C₄⋊C₈	—	2	2	no	—
64	C₄⋊C₈	—	214	2	no	—
65	#95	—	107	2	no	—
66	#95	—	107	2	no	—
67	C₂²×C₄	—	107	4	no	—
68	C₂²×C₄	C₄×D₁₀₇	1	1	yes	characteristic
69	C₂²×C₄	C₄×D₁₀₇	1	1	yes	characteristic
70	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	4	no	—
73	C₂×C₈	—	214	2	no	—
74	C₂×C₈	—	214	4	no	—
75	C₄²	—	214	2	no	—
76	C₄²	—	214	2	no	—
77	C₄²	Dic₂₁₄	1	2	yes	—
78	C₄²	C₂²⋊D₁₀₇	1	2	yes	—
79	C₂³	C₂×C₄×D₁₀₇	1	1	yes	characteristic
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	214	4	no	—
83	C₂×C₄	—	214	4	no	—
84	C₂×C₄	—	214	4	no	—
85	C₂×C₄	Dic₁₀₇⋊C₄	1	2	yes	—
86	C₂×C₄	Dic₁₀₇⋊C₄	1	2	yes	—
87	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
88	C₂×C₄	C₂×Dic₂₁₄	1	1	yes	characteristic
89	C₂×C₄	#13133817	1	2	yes	—
90	C₂×C₄	#13133819	1	2	yes	—
91	C₈	—	4	2	no	—
92	C₈	—	428	2	no	—
93	C₂²	C₂×Dic₁₀₇⋊C₄	1	1	yes	characteristic
94	C₂²	#16812067	1	2	yes	—
95	C₂²	#16812115	1	2	yes	—
96	C₂²	#16812117	1	2	yes	—
97	C₄	—	2	2	no	—
98	C₄	—	2	2	no	—
99	C₄	—	428	4	no	—
100	C₄	#16812096	1	1	yes	characteristic
101	C₄	#16812098	1	1	yes	characteristic
102	C₄	#16812100	1	2	yes	—
103	C₂	#20292492	1	1	yes	characteristic
104	C₂	#20292653	1	1	yes	characteristic
105	C₂	#20292655	1	1	yes	characteristic
106	C₂	#20292687	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₁₀₇⋊C₈)

C₂

yes

maximal, characteristic

#20292473

C₂

yes

maximal, characteristic

#20292381

C₂

yes

maximal

C₂×C₄⋊C₈×C₁₀₇

C₂

yes

maximal, characteristic

C₂²×C₁₀₇⋊C₈

C₂²

yes

—

C₂×C₄×Dic₁₀₇

C₂²

yes

—

C₄⋊(C₁₀₇⋊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

—

C₄×Dic₁₀₇

—

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

characteristic

C₂×C₄×C₁₀₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₇

#41

yes

—

C₂×C₄×C₁₀₇

#41

yes

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₈×C₁₀₇

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₇

#117

yes

—

C₂²×C₁₀₇

#131

yes

—

C₂²×C₁₀₇

#131

yes

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

yes

C₄×C₁₀₇

yes

C₄×C₁₀₇

yes

—

Dic₁₀₇

—

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

—

#1832

—

107

maximal, Sylow

C₁₀₇

#1832

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

107

—

C₂×C₄⋊C₈

D₁₀₇

yes

characteristic

#376

—

107

—

#336

—

107

—

C₂²×C₈

—

107

—

C₂²×C₈

C₂×D₁₀₇

yes

—

C₂×C₄²

—

107

—

C₂×C₄²

C₂×D₁₀₇

yes

characteristic

C₄⋊C₈

—

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

—

214

—

C₄²

—

214

—

C₄²

Dic₂₁₄

yes

—

C₄²

C₂²⋊D₁₀₇

yes

—

C₂³

C₂×C₄×D₁₀₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

214

—

C₂×C₄

—

214

—

C₂×C₄

—

214

—

C₂×C₄

Dic₁₀₇⋊C₄

yes

—

C₂×C₄

Dic₁₀₇⋊C₄

yes

—

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

characteristic

C₂×C₄

C₂×Dic₂₁₄

yes

characteristic

C₂×C₄

#13133817

yes

—

C₂×C₄

#13133819

yes

—

C₈

—

C₈

—

428

—

C₂²

C₂×Dic₁₀₇⋊C₄

yes

characteristic

C₂²

#16812067

yes

—

C₂²

#16812115

yes

—

C₂²

#16812117

yes

—

C₄

—

C₄

—

C₄

—

428

—

100

C₄

#16812096

yes

characteristic

101

C₄

#16812098

yes

characteristic

102

C₄

#16812100

yes

—

103

C₂

#20292492

yes

characteristic

104

C₂

#20292653

yes

characteristic

105

C₂

#20292655

yes

characteristic

106

C₂

#20292687

yes

—

	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	#1832
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
54. C₁₀₇	#1832	semidirect

H ≅ G/N

type

54. C₁₀₇

#1832

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
#20292473	2
#20292381	2
C₂×C₄⋊C₈×C₁₀₇	2
#1832	107

subgroup

index

C₂×C₄⋊(C₁₀₇⋊C₈)

#20292473

#20292381

C₂×C₄⋊C₈×C₁₀₇

#1832

107

Subgroups — #28336580

Conjugacy classes of subgroups

Complements

Maximal subgroups