Group #28335601 — subgroups

Subgroups — #28335601

Order 13696.

206 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 65 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	#20293006	C₂	1	1	yes	maximal, characteristic
2	#20292922	C₂	1	2	yes	maximal
3	#20292376	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊(C₁₀₇⋊C₈)	C₂	1	2	yes	maximal
5	Q₈²×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133808	—	2	2	no	—
7	#16812109	C₂²	1	1	yes	characteristic
8	#16812042	—	2	2	no	—
9	C₄⋊(C₁₀₇⋊C₈)	C₂²	1	2	yes	—
10	#16812033	—	2	2	no	—
11	C₄×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₁₀₇	—	2	2	no	—
13	C₄⋊Q₈×C₁₀₇	—	2	2	no	—
14	C₄⋊Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₁₀₇	—	2	2	no	—
16	C₄×Q₈×C₁₀₇	C₂²	1	2	yes	—
17	C₂×Dic₂₁₄	—	2	1	no	—
18	#13133808	—	4	4	no	—
19	C₂×C₁₀₇⋊C₈	—	2	1	no	—
20	C₂×C₁₀₇⋊C₈	—	2	2	no	—
21	C₄⋊Dic₁₀₇	—	2	2	no	—
22	C₂×Q₈×C₁₀₇	—	2	2	no	—
23	C₂×Q₈×C₁₀₇	D₄	1	2	yes	—
24	C₂×Q₈×C₁₀₇	D₄	1	2	yes	—
25	C₄⋊C₄×C₁₀₇	—	2	4	no	—
26	C₄⋊C₄×C₁₀₇	—	2	2	no	—
27	C₄⋊C₄×C₁₀₇	—	2	2	no	—
28	C₄⋊C₄×C₁₀₇	D₄	1	2	yes	—
29	C₄²×C₁₀₇	—	2	2	no	—
30	C₄²×C₁₀₇	—	2	2	no	—
31	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
32	Q₈×C₁₀₇	—	2	2	no	—
33	Q₈×C₁₀₇	—	4	2	no	—
34	Q₈×C₁₀₇	—	8	2	no	—
35	Q₈×C₁₀₇	Q₁₆	1	4	yes	complemented
36	C₂×C₄×C₁₀₇	—	2	2	no	—
37	C₂×C₄×C₁₀₇	—	2	2	no	—
38	C₂×C₄×C₁₀₇	—	2	2	no	—
39	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
41	C₂×Dic₁₀₇	—	4	1	no	—
42	Dic₂₁₄	—	4	2	no	—
43	C₁₀₇⋊C₈	—	2	2	no	—
44	C₁₀₇⋊C₈	—	4	2	no	—
45	C₂²×C₁₀₇	#120	1	1	yes	characteristic
46	C₄×C₁₀₇	—	2	4	no	—
47	C₄×C₁₀₇	—	2	1	no	—
48	C₄×C₁₀₇	—	4	2	no	—
49	C₄×C₁₀₇	—	4	2	no	—
50	C₄×C₁₀₇	C₂×Q₁₆	1	2	yes	—
51	C₄×C₁₀₇	#137	1	2	yes	—
52	Dic₁₀₇	—	8	1	no	—
53	C₂×C₁₀₇	#451	1	2	yes	—
54	C₂×C₁₀₇	#453	1	1	yes	characteristic
55	#1633	—	107	1	no	maximal, Sylow
56	C₁₀₇	#1633	1	1	yes	characteristic, complemented, Sylow
57	Q₈²	D₁₀₇	1	1	yes	characteristic
58	#494	—	107	1	no	—
59	#477	—	107	2	no	—
60	#333	—	107	1	no	—
61	Q₈⋊C₈	—	107	2	no	—
62	C₂×Q₁₆	—	214	2	no	—
63	C₄⋊Q₈	—	2	2	no	—
64	C₄⋊Q₈	—	2	2	no	—
65	C₄⋊Q₈	—	107	1	no	—
66	C₄⋊Q₈	C₂×D₁₀₇	1	1	yes	characteristic
67	C₄×Q₈	—	2	2	no	—
68	C₄×Q₈	C₂×D₁₀₇	1	2	yes	—
69	#107	—	214	2	no	—
70	C₄⋊C₈	—	107	2	no	—
71	#103	—	214	2	no	—
72	C₄×C₈	—	107	1	no	—
73	C₂×Q₈	—	2	2	no	—
74	C₂×Q₈	—	214	1	no	—
75	C₂×Q₈	C₂²⋊D₁₀₇	1	2	yes	—
76	C₂×Q₈	C₂²⋊D₁₀₇	1	2	yes	—
77	Q₁₆	—	428	4	no	—
78	C₂×C₈	—	214	2	no	—
79	C₂×C₈	—	214	1	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	214	2	no	—
84	C₄⋊C₄	C₂²⋊D₁₀₇	1	2	yes	—
85	C₄²	—	2	2	no	—
86	C₄²	—	2	2	no	—
87	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
88	Q₈	—	2	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	8	2	no	—
91	Q₈	—	428	2	no	—
92	Q₈	#13133808	1	4	yes	complemented
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	428	1	no	—
97	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	2	yes	—
99	C₈	—	428	2	no	—
100	C₈	—	214	2	no	—
101	C₂²	#16812078	1	1	yes	characteristic
102	C₄	—	2	4	no	—
103	C₄	—	2	1	no	—
104	C₄	—	4	2	no	—
105	C₄	—	4	2	no	—
106	C₄	—	856	1	no	—
107	C₄	C₂×#13133808	1	2	yes	—
108	C₄	#16812140	1	2	yes	—
109	C₂	#20292783	1	2	yes	—
110	C₂	#20292795	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20293006

C₂

yes

maximal, characteristic

#20292922

C₂

yes

maximal

#20292376

C₂

yes

maximal, characteristic

Q₈⋊(C₁₀₇⋊C₈)

C₂

yes

maximal

Q₈²×C₁₀₇

C₂

yes

maximal, characteristic

C₂×#13133808

—

#16812109

C₂²

yes

characteristic

#16812042

—

C₄⋊(C₁₀₇⋊C₈)

C₂²

yes

—

#16812033

—

C₄×C₁₀₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₇

—

C₄⋊Q₈×C₁₀₇

—

C₄⋊Q₈×C₁₀₇

C₂²

yes

characteristic

C₄×Q₈×C₁₀₇

—

C₄×Q₈×C₁₀₇

C₂²

yes

—

C₂×Dic₂₁₄

—

#13133808

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

—

C₄⋊Dic₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂×Q₈×C₁₀₇

D₄

yes

—

C₂×Q₈×C₁₀₇

D₄

yes

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

D₄

yes

—

C₄²×C₁₀₇

—

C₄²×C₁₀₇

—

C₄²×C₁₀₇

C₂³

yes

characteristic

Q₈×C₁₀₇

—

Q₈×C₁₀₇

—

Q₈×C₁₀₇

—

Q₈×C₁₀₇

Q₁₆

yes

complemented

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×Dic₁₀₇

—

Dic₂₁₄

—

C₁₀₇⋊C₈

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

#120

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂×Q₁₆

yes

—

C₄×C₁₀₇

#137

yes

—

Dic₁₀₇

—

C₂×C₁₀₇

#451

yes

—

C₂×C₁₀₇

#453

yes

characteristic

#1633

—

107

maximal, Sylow

C₁₀₇

#1633

yes

characteristic, complemented, Sylow

Q₈²

D₁₀₇

yes

characteristic

#494

—

107

—

#477

—

107

—

#333

—

107

—

Q₈⋊C₈

—

107

—

C₂×Q₁₆

—

214

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

107

—

C₄⋊Q₈

C₂×D₁₀₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

C₂×D₁₀₇

yes

—

#107

—

214

—

C₄⋊C₈

—

107

—

#103

—

214

—

C₄×C₈

—

107

—

C₂×Q₈

—

C₂×Q₈

—

214

—

C₂×Q₈

C₂²⋊D₁₀₇

yes

—

C₂×Q₈

C₂²⋊D₁₀₇

yes

—

Q₁₆

—

428

—

C₂×C₈

—

214

—

C₂×C₈

—

214

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

214

—

C₄⋊C₄

C₂²⋊D₁₀₇

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×D₁₀₇

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

428

—

Q₈

#13133808

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

428

—

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

—

C₈

—

428

—

100

C₈

—

214

—

101

C₂²

#16812078

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

856

—

107

C₄

C₂×#13133808

yes

—

108

C₄

#16812140

yes

—

109

C₂

#20292783

yes

—

110

C₂

#20292795

yes

characteristic

	subgroup	G/H
center	C₂²	#16812078
commutator, cosocle	C₄²×C₁₀₇	C₂³
Frattini	C₄²	C₂²×D₁₀₇
Fitting	Q₈²×C₁₀₇	C₂
socle, prime radical	C₂²×C₁₀₇	#120
hypercenter	Q₈²	D₁₀₇
hypocenter	C₁₀₇	#1633
LCS	G ▷ C₄²×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

subgroup

G/H

center

C₂²

#16812078

commutator, cosocle

C₄²×C₁₀₇

C₂³

Frattini

C₄²

C₂²×D₁₀₇

Fitting

Q₈²×C₁₀₇

C₂

socle, prime radical

C₂²×C₁₀₇

#120

hypercenter

Q₈²

D₁₀₇

hypocenter

C₁₀₇

#1633

LCS

G ▷ C₄²×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

N	H ≅ G/N	type
35. Q₈×C₁₀₇	Q₁₆	semidirect
56. C₁₀₇	#1633	semidirect
92. Q₈	#13133808	semidirect

H ≅ G/N

type

35. Q₈×C₁₀₇

Q₁₆

semidirect

56. C₁₀₇

#1633

semidirect

92. Q₈

#13133808

semidirect

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

subgroup	index
#20293006	2
#20292922	2
#20292376	2
Q₈⋊(C₁₀₇⋊C₈)	2
Q₈²×C₁₀₇	2
#1633	107

subgroup

index

#20293006

#20292922

#20292376

Q₈⋊(C₁₀₇⋊C₈)

Q₈²×C₁₀₇

#1633

107

Subgroups — #28335601

Conjugacy classes of subgroups

Complements

Maximal subgroups