Group #28337441 — subgroups

Subgroups — #28337441

Order 13696.

244 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 89 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	#20293184	C₂	1	1	yes	maximal, characteristic
2	#20292473	C₂	1	1	yes	maximal, characteristic
3	#20292388	C₂	1	4	yes	maximal
4	#423×C₁₀₇	C₂	1	1	yes	maximal, characteristic
5	#16812109	C₄	1	2	yes	complemented
6	#16812101	C₄	1	2	yes	complemented
7	#16812088	—	2	2	no	—
8	C₄×Dic₂₁₄	—	2	2	no	—
9	#16812068	C₂²	1	2	yes	—
10	C₂×C₄×Dic₁₀₇	C₂²	1	2	yes	—
11	C₁₀₇⋊#95	—	2	2	no	—
12	C₂×M₁₆×C₁₀₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₁₀₇	—	2	2	no	—
15	C₂×Dic₂₁₄	—	2	2	no	—
16	C₂²×Dic₁₀₇	—	2	2	no	—
17	C₄⋊Dic₁₀₇	—	2	2	no	—
18	C₄⋊Dic₁₀₇	C₂×C₄	1	4	yes	—
19	Dic₁₀₇⋊C₄	—	2	4	no	—
20	C₂²⋊Dic₁₀₇	—	2	2	no	—
21	C₄×Dic₁₀₇	—	2	2	no	—
22	C₄×Dic₁₀₇	—	2	2	no	—
23	C₄×Dic₁₀₇	Q₈	1	4	yes	complemented
24	C₂²×C₄×C₁₀₇	D₄	1	2	yes	—
25	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₀₇	—	2	4	no	—
27	C₂×C₈×C₁₀₇	—	2	2	no	—
28	C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₁₀₇	D₄	1	2	yes	—
31	C₂³×C₁₀₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₇	—	4	2	no	—
33	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	2	yes	—
34	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	2	yes	—
35	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₇	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₁₀₇	#41	1	2	yes	—
39	C₂×Dic₁₀₇	—	2	4	no	—
40	C₂×Dic₁₀₇	—	2	4	no	—
41	C₂×Dic₁₀₇	—	4	4	no	—
42	Dic₂₁₄	—	4	4	no	—
43	C₈×C₁₀₇	—	4	2	no	—
44	C₂²×C₁₀₇	—	2	2	no	—
45	C₂²×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₁₀₇	C₂²⋊Q₈	1	2	yes	—
47	C₄×C₁₀₇	—	2	2	no	—
48	C₄×C₁₀₇	—	2	2	no	—
49	C₄×C₁₀₇	C₄×Q₈	1	2	yes	—
50	C₄×C₁₀₇	#124	1	1	yes	characteristic
51	C₄×C₁₀₇	C₄⋊Q₈	1	1	yes	characteristic
52	Dic₁₀₇	—	4	4	no	—
53	Dic₁₀₇	—	4	4	no	—
54	C₂×C₁₀₇	—	2	2	no	—
55	C₂×C₁₀₇	#391	1	1	yes	characteristic
56	C₂×C₁₀₇	#421	1	2	yes	—
57	#1995	—	107	1	no	maximal, Sylow
58	C₁₀₇	#1995	1	1	yes	characteristic, complemented, Sylow
59	#533	—	107	1	no	—
60	#423	D₁₀₇	1	1	yes	characteristic
61	#376	—	107	1	no	—
62	#339	—	107	4	no	—
63	C₂×M₁₆	C₂×D₁₀₇	1	2	yes	—
64	C₄⋊Q₈	—	107	2	no	—
65	#125	—	107	2	no	—
66	C₂²⋊Q₈	—	214	2	no	—
67	C₄×Q₈	—	214	2	no	—
68	#117	—	107	2	no	—
69	C₂×C₄²	—	107	2	no	—
70	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
71	C₄⋊C₈	—	2	2	no	—
72	#95	—	214	2	no	—
73	C₂×Q₈	—	214	2	no	—
74	C₂²×C₄	—	214	2	no	—
75	C₂²×C₄	D₄₂₈	1	2	yes	—
76	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
77	M₁₆	—	2	4	no	—
78	C₂×C₈	—	2	2	no	—
79	C₄⋊C₄	—	107	4	no	—
80	C₄⋊C₄	—	214	4	no	—
81	C₄⋊C₄	—	214	2	no	—
82	C₂²⋊C₄	—	214	2	no	—
83	C₄²	—	107	4	no	—
84	C₄²	—	214	2	no	—
85	C₄²	—	214	2	no	—
86	C₄²	C₄×D₁₀₇	1	1	yes	characteristic
87	C₄²	C₄×D₁₀₇	1	1	yes	characteristic
88	C₄²	C₂²⋊D₁₀₇	1	2	yes	—
89	C₂³	C₂×D₄₂₈	1	1	yes	characteristic
90	Q₈	—	428	4	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	428	4	no	—
93	C₂×C₄	—	214	4	no	—
94	C₂×C₄	—	214	4	no	—
95	C₂×C₄	(C₂×D₁₀₇)⋊C₄	1	2	yes	—
96	C₂×C₄	(C₂×D₁₀₇)⋊C₄	1	2	yes	—
97	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
99	C₂×C₄	Q₈×D₁₀₇	1	2	yes	—
100	C₂×C₄	#13133817	1	2	yes	—
101	C₈	—	4	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	C₂×(C₂×D₁₀₇)⋊C₄	1	1	yes	characteristic
104	C₂²	#16812089	1	2	yes	—
105	C₄	—	2	2	no	—
106	C₄	—	2	2	no	—
107	C₄	—	428	4	no	—
108	C₄	—	428	4	no	—
109	C₄	#16812074	1	2	yes	—
110	C₄	#16812096	1	1	yes	characteristic
111	C₄	#16812107	1	1	yes	characteristic
112	C₂	—	2	2	no	—
113	C₂	#20292532	1	1	yes	characteristic
114	C₂	#20292643	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20293184

C₂

yes

maximal, characteristic

#20292473

C₂

yes

maximal, characteristic

#20292388

C₂

yes

maximal

#423×C₁₀₇

C₂

yes

maximal, characteristic

#16812109

C₄

yes

complemented

#16812101

C₄

yes

complemented

#16812088

—

C₄×Dic₂₁₄

—

#16812068

C₂²

yes

—

C₂×C₄×Dic₁₀₇

C₂²

yes

—

C₁₀₇⋊#95

—

C₂×M₁₆×C₁₀₇

C₂²

yes

—

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₇

—

C₂×Dic₂₁₄

—

C₂²×Dic₁₀₇

—

C₄⋊Dic₁₀₇

—

C₄⋊Dic₁₀₇

C₂×C₄

yes

—

Dic₁₀₇⋊C₄

—

C₂²⋊Dic₁₀₇

—

C₄×Dic₁₀₇

—

C₄×Dic₁₀₇

—

C₄×Dic₁₀₇

Q₈

yes

complemented

C₂²×C₄×C₁₀₇

D₄

yes

—

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

M₁₆×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₄²×C₁₀₇

C₂×C₄

yes

characteristic

C₄²×C₁₀₇

C₂×C₄

yes

characteristic

C₄²×C₁₀₇

D₄

yes

—

C₂³×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×Q₈

yes

—

C₂×C₄×C₁₀₇

#41

yes

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

Dic₂₁₄

—

C₈×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₇

C₂²⋊Q₈

yes

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₄×Q₈

yes

—

C₄×C₁₀₇

#124

yes

characteristic

C₄×C₁₀₇

C₄⋊Q₈

yes

characteristic

Dic₁₀₇

—

Dic₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

#391

yes

characteristic

C₂×C₁₀₇

#421

yes

—

#1995

—

107

maximal, Sylow

C₁₀₇

#1995

yes

characteristic, complemented, Sylow

#533

—

107

—

#423

D₁₀₇

yes

characteristic

#376

—

107

—

#339

—

107

—

C₂×M₁₆

C₂×D₁₀₇

yes

—

C₄⋊Q₈

—

107

—

#125

—

107

—

C₂²⋊Q₈

—

214

—

C₄×Q₈

—

214

—

#117

—

107

—

C₂×C₄²

—

107

—

C₂×C₄²

C₂×D₁₀₇

yes

characteristic

C₄⋊C₈

—

#95

—

214

—

C₂×Q₈

—

214

—

C₂²×C₄

—

214

—

C₂²×C₄

D₄₂₈

yes

—

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

107

—

C₄⋊C₄

—

214

—

C₄⋊C₄

—

214

—

C₂²⋊C₄

—

214

—

C₄²

—

107

—

C₄²

—

214

—

C₄²

—

214

—

C₄²

C₄×D₁₀₇

yes

characteristic

C₄²

C₄×D₁₀₇

yes

characteristic

C₄²

C₂²⋊D₁₀₇

yes

—

C₂³

C₂×D₄₂₈

yes

characteristic

Q₈

—

428

—

C₂×C₄

—

C₂×C₄

—

428

—

C₂×C₄

—

214

—

C₂×C₄

—

214

—

C₂×C₄

(C₂×D₁₀₇)⋊C₄

yes

—

C₂×C₄

(C₂×D₁₀₇)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₁₀₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

characteristic

C₂×C₄

Q₈×D₁₀₇

yes

—

100

C₂×C₄

#13133817

yes

—

101

C₈

—

102

C₂²

—

103

C₂²

C₂×(C₂×D₁₀₇)⋊C₄

yes

characteristic

104

C₂²

#16812089

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

428

—

108

C₄

—

428

—

109

C₄

#16812074

yes

—

110

C₄

#16812096

yes

characteristic

111

C₄

#16812107

yes

characteristic

112

C₂

—

113

C₂

#20292532

yes

characteristic

114

C₂

#20292643

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₀₇
commutator	C₂×C₄×C₁₀₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₇
Fitting	#423×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₀₇	C₂×D₄
cosocle	C₂²×C₄×C₁₀₇	C₂³
hypercenter	#423	D₁₀₇
hypocenter	C₁₀₇	#1995
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₀₇

commutator

C₂×C₄×C₁₀₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₇

Fitting

#423×C₁₀₇

C₂

socle

C₂²×C₁₀₇

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₀₇

C₂×D₄

cosocle

C₂²×C₄×C₁₀₇

C₂³

D₁₀₇

C₁₀₇

G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
5. #16812109	C₄	semidirect
6. #16812101	C₄	semidirect
23. C₄×Dic₁₀₇	Q₈	semidirect
58. C₁₀₇	#1995	semidirect

H ≅ G/N

type

5. #16812109

C₄

semidirect

6. #16812101

C₄

semidirect

23. C₄×Dic₁₀₇

Q₈

semidirect

58. C₁₀₇

#1995

semidirect

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

subgroup	index
#20293184	2
#20292473	2
#20292388	2
#423×C₁₀₇	2
#1995	107

subgroup

index

#423×C₁₀₇

107

Subgroups — #28337441

Conjugacy classes of subgroups

Complements

Maximal subgroups