Group #28338281 — subgroups

Subgroups — D₄×C₁₀₇⋊C₁₆ #28338281

Order 13696.

174 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 103 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	C₂²×C₁₀₇⋊C₁₆	C₂	1	2	yes	maximal, complemented
2	C₄⋊(C₁₀₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊(C₁₀₇⋊C₁₆)	C₂	1	2	yes	maximal, complemented
4	C₄×C₁₀₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×D₄×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₀₇⋊C₁₆	—	2	4	no	—
7	C₂×C₁₀₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₁₀₇⋊C₁₆	C₂²	1	2	yes	—
9	C₂×C₁₀₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₁₀₇	C₂²	1	2	yes	—
11	C₄×D₄×C₁₀₇	C₄	1	1	yes	characteristic
12	C₄⋊C₈×C₁₀₇	C₄	1	1	yes	characteristic
13	C₂²⋊C₈×C₁₀₇	C₄	1	2	yes	—
14	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
15	C₁₀₇⋊C₁₆	—	2	2	no	—
16	C₁₀₇⋊C₁₆	—	2	1	no	—
17	C₁₀₇⋊C₁₆	D₄	1	2	yes	direct product factor
18	C₂×D₄×C₁₀₇	C₈	1	1	yes	characteristic
19	C₂²×C₄×C₁₀₇	C₂×C₄	1	2	yes	—
20	C₂×C₈×C₁₀₇	—	2	2	no	—
21	C₂×C₈×C₁₀₇	—	2	2	no	—
22	C₂×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₁₀₇	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₁₀₇	C₈	1	1	yes	characteristic
26	C₂²⋊C₄×C₁₀₇	C₈	1	2	yes	—
27	C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₁₀₇	C₂×C₈	1	2	yes	—
29	D₄×C₁₀₇	C₁₆	1	4	yes	complemented
30	C₂×C₄×C₁₀₇	—	2	2	no	—
31	C₂×C₄×C₁₀₇	—	2	2	no	—
32	C₂×C₄×C₁₀₇	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₁₀₇	C₂×C₈	1	2	yes	—
34	C₂×C₄×C₁₀₇	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
36	C₈×C₁₀₇	—	2	2	no	—
37	C₈×C₁₀₇	—	2	1	no	—
38	C₈×C₁₀₇	C₂×D₄	1	1	yes	characteristic
39	C₈×C₁₀₇	#41	1	1	yes	characteristic
40	C₂²×C₁₀₇	—	2	2	no	—
41	C₂²×C₁₀₇	—	2	2	no	—
42	C₂²×C₁₀₇	C₂×C₁₆	1	4	yes	complemented
43	C₂²×C₁₀₇	C₂²×C₈	1	1	yes	characteristic
44	C₄×C₁₀₇	—	2	2	no	—
45	C₄×C₁₀₇	—	2	1	no	—
46	C₄×C₁₀₇	C₂×C₁₆	1	2	yes	complemented
47	C₄×C₁₀₇	C₄×D₄	1	1	yes	characteristic
48	C₄×C₁₀₇	#131	1	1	yes	characteristic
49	C₂×C₁₀₇	—	2	4	no	—
50	C₂×C₁₀₇	C₈×D₄	1	1	yes	characteristic
51	C₂×C₁₀₇	C₂²×C₁₆	1	1	yes	characteristic
52	C₂×C₁₀₇	#504	1	1	yes	characteristic
53	D₄×C₁₆	—	107	1	no	maximal, Sylow
54	C₁₀₇	D₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
55	C₂²×C₁₆	—	107	2	no	—
56	C₈×D₄	D₁₀₇	1	1	yes	characteristic
57	C₄⋊C₁₆	—	107	1	no	—
58	C₂²⋊C₁₆	—	107	2	no	—
59	C₄×C₁₆	—	107	1	no	—
60	C₂²×C₈	C₂×D₁₀₇	1	2	yes	—
61	C₄×D₄	Dic₁₀₇	1	1	yes	characteristic
62	C₂×C₁₆	—	107	1	no	—
63	C₂×C₁₆	—	107	1	no	—
64	C₂×C₁₆	—	107	2	no	—
65	C₂×C₁₆	—	214	4	no	—
66	C₄⋊C₈	Dic₁₀₇	1	1	yes	characteristic
67	C₂²⋊C₈	Dic₁₀₇	1	2	yes	—
68	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic
69	C₂×D₄	C₁₀₇⋊C₈	1	1	yes	characteristic
70	C₂²×C₄	C₂×Dic₁₀₇	1	2	yes	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	C₂×Dic₁₀₇	1	1	yes	characteristic
74	C₂×C₈	C₂×Dic₁₀₇	1	2	yes	—
75	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
76	C₄⋊C₄	C₁₀₇⋊C₈	1	1	yes	characteristic
77	C₂²⋊C₄	C₁₀₇⋊C₈	1	2	yes	—
78	C₄²	C₂×Dic₁₀₇	1	1	yes	characteristic
79	C₁₆	—	107	2	no	—
80	C₁₆	—	214	2	no	—
81	C₁₆	—	214	1	no	—
82	C₂³	C₂×C₁₀₇⋊C₈	1	2	yes	—
83	D₄	C₁₀₇⋊C₁₆	1	4	yes	direct product factor
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	C₂×C₁₀₇⋊C₈	1	1	yes	characteristic
87	C₂×C₄	C₂×C₁₀₇⋊C₈	1	2	yes	—
88	C₂×C₄	C₂×C₁₀₇⋊C₈	1	1	yes	characteristic
89	C₂×C₄	C₂²×Dic₁₀₇	1	1	yes	characteristic
90	C₈	—	2	2	no	—
91	C₈	—	2	1	no	—
92	C₈	D₄×D₁₀₇	1	1	yes	characteristic
93	C₈	#13133817	1	1	yes	characteristic
94	C₂²	—	2	2	no	—
95	C₂²	—	2	2	no	—
96	C₂²	C₂×C₁₀₇⋊C₁₆	1	4	yes	complemented
97	C₂²	C₂²×C₁₀₇⋊C₈	1	1	yes	characteristic
98	C₄	—	2	2	no	—
99	C₄	—	2	1	no	—
100	C₄	C₂×C₁₀₇⋊C₁₆	1	2	yes	complemented
101	C₄	D₄×Dic₁₀₇	1	1	yes	characteristic
102	C₄	#16812116	1	1	yes	characteristic
103	C₂	—	2	4	no	—
104	C₂	D₄×C₁₀₇⋊C₈	1	1	yes	characteristic
105	C₂	C₂²×C₁₀₇⋊C₁₆	1	1	yes	characteristic
106	C₂	#20293044	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₁₀₇⋊C₁₆

C₂

yes

maximal, complemented

C₄⋊(C₁₀₇⋊C₁₆)

C₂

yes

maximal, characteristic, complemented

C₂²⋊(C₁₀₇⋊C₁₆)

C₂

yes

maximal, complemented

C₄×C₁₀₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₈×D₄×C₁₀₇

C₂

yes

maximal, characteristic

C₂×C₁₀₇⋊C₁₆

—

C₂×C₁₀₇⋊C₁₆

C₂²

yes

characteristic

C₂×C₁₀₇⋊C₁₆

C₂²

yes

—

C₂×C₁₀₇⋊C₁₆

C₂²

yes

characteristic

C₂²×C₈×C₁₀₇

C₂²

yes

—

C₄×D₄×C₁₀₇

C₄

yes

characteristic

C₄⋊C₈×C₁₀₇

C₄

yes

characteristic

C₂²⋊C₈×C₁₀₇

C₄

yes

—

C₄×C₈×C₁₀₇

C₂²

yes

characteristic

C₁₀₇⋊C₁₆

—

C₁₀₇⋊C₁₆

—

C₁₀₇⋊C₁₆

D₄

yes

direct product factor

C₂×D₄×C₁₀₇

C₈

yes

characteristic

C₂²×C₄×C₁₀₇

C₂×C₄

yes

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₇

C₂×C₄

yes

—

C₂×C₈×C₁₀₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₇

C₈

yes

characteristic

C₂²⋊C₄×C₁₀₇

C₈

yes

—

C₄²×C₁₀₇

C₂×C₄

yes

characteristic

C₂³×C₁₀₇

C₂×C₈

yes

—

D₄×C₁₀₇

C₁₆

yes

complemented

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₇

C₂×C₈

yes

—

C₂×C₄×C₁₀₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₇

C₂²×C₄

yes

characteristic

C₈×C₁₀₇

—

C₈×C₁₀₇

—

C₈×C₁₀₇

C₂×D₄

yes

characteristic

C₈×C₁₀₇

#41

yes

characteristic

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₂×C₁₆

yes

complemented

C₂²×C₁₀₇

C₂²×C₈

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂×C₁₆

yes

complemented

C₄×C₁₀₇

C₄×D₄

yes

characteristic

C₄×C₁₀₇

#131

yes

characteristic

C₂×C₁₀₇

—

C₂×C₁₀₇

C₈×D₄

yes

characteristic

C₂×C₁₀₇

C₂²×C₁₆

yes

characteristic

C₂×C₁₀₇

#504

yes

characteristic

D₄×C₁₆

—

107

maximal, Sylow

C₁₀₇

D₄×C₁₆

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

107

—

C₈×D₄

D₁₀₇

yes

characteristic

C₄⋊C₁₆

—

107

—

C₂²⋊C₁₆

—

107

—

C₄×C₁₆

—

107

—

C₂²×C₈

C₂×D₁₀₇

yes

—

C₄×D₄

Dic₁₀₇

yes

characteristic

C₂×C₁₆

—

107

—

C₂×C₁₆

—

107

—

C₂×C₁₆

—

107

—

C₂×C₁₆

—

214

—

C₄⋊C₈

Dic₁₀₇

yes

characteristic

C₂²⋊C₈

Dic₁₀₇

yes

—

C₄×C₈

C₂×D₁₀₇

yes

characteristic

C₂×D₄

C₁₀₇⋊C₈

yes

characteristic

C₂²×C₄

C₂×Dic₁₀₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₁₀₇

yes

characteristic

C₂×C₈

C₂×Dic₁₀₇

yes

—

C₂×C₈

C₂²×D₁₀₇

yes

characteristic

C₄⋊C₄

C₁₀₇⋊C₈

yes

characteristic

C₂²⋊C₄

C₁₀₇⋊C₈

yes

—

C₄²

C₂×Dic₁₀₇

yes

characteristic

C₁₆

—

107

—

C₁₆

—

214

—

C₁₆

—

214

—

C₂³

C₂×C₁₀₇⋊C₈

yes

—

D₄

C₁₀₇⋊C₁₆

yes

direct product factor

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₁₀₇⋊C₈

yes

characteristic

C₂×C₄

C₂×C₁₀₇⋊C₈

yes

—

C₂×C₄

C₂×C₁₀₇⋊C₈

yes

characteristic

C₂×C₄

C₂²×Dic₁₀₇

yes

characteristic

C₈

—

C₈

—

C₈

D₄×D₁₀₇

yes

characteristic

C₈

#13133817

yes

characteristic

C₂²

—

C₂²

—

C₂²

C₂×C₁₀₇⋊C₁₆

yes

complemented

C₂²

C₂²×C₁₀₇⋊C₈

yes

characteristic

C₄

—

C₄

—

100

C₄

C₂×C₁₀₇⋊C₁₆

yes

complemented

101

C₄

D₄×Dic₁₀₇

yes

characteristic

102

C₄

#16812116

yes

characteristic

103

C₂

—

104

C₂

D₄×C₁₀₇⋊C₈

yes

characteristic

105

C₂

C₂²×C₁₀₇⋊C₁₆

yes

characteristic

106

C₂

#20293044

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₀₇
commutator	C₂×C₁₀₇	C₂²×C₁₆
Fitting	C₈×D₄×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂²×C₈
prime radical	C₂×D₄×C₁₀₇	C₈
cosocle	C₂×C₈×C₁₀₇	C₂³
hypercenter	C₈×D₄	D₁₀₇
hypocenter	C₁₀₇	D₄×C₁₆
LCS	G ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂×C₈ ◁ C₈×D₄

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₀₇

commutator

C₂×C₁₀₇

C₂²×C₁₆

Fitting

C₈×D₄×C₁₀₇

C₂

socle

C₂²×C₁₀₇

C₂²×C₈

prime radical

C₂×D₄×C₁₀₇

C₈

cosocle

C₂×C₈×C₁₀₇

C₂³

hypercenter

C₈×D₄

D₁₀₇

hypocenter

C₁₀₇

D₄×C₁₆

LCS

G ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂×C₈ ◁ C₈×D₄

N	H ≅ G/N	type
1. C₂²×C₁₀₇⋊C₁₆	C₂	semidirect
2. C₄⋊(C₁₀₇⋊C₁₆)	C₂	semidirect
3. C₂²⋊(C₁₀₇⋊C₁₆)	C₂	semidirect
4. C₄×C₁₀₇⋊C₁₆	C₂	semidirect
17. C₁₀₇⋊C₁₆	D₄	direct (N × H)
29. D₄×C₁₀₇	C₁₆	semidirect
42. C₂²×C₁₀₇	C₂×C₁₆	semidirect
46. C₄×C₁₀₇	C₂×C₁₆	semidirect
54. C₁₀₇	D₄×C₁₆	semidirect
83. D₄	C₁₀₇⋊C₁₆	direct (N × H)
96. C₂²	C₂×C₁₀₇⋊C₁₆	semidirect
100. C₄	C₂×C₁₀₇⋊C₁₆	semidirect

H ≅ G/N

type

1. C₂²×C₁₀₇⋊C₁₆

C₂

semidirect

2. C₄⋊(C₁₀₇⋊C₁₆)

C₂

semidirect

3. C₂²⋊(C₁₀₇⋊C₁₆)

C₂

semidirect

4. C₄×C₁₀₇⋊C₁₆

C₂

semidirect

17. C₁₀₇⋊C₁₆

D₄

direct (N × H)

29. D₄×C₁₀₇

C₁₆

semidirect

42. C₂²×C₁₀₇

C₂×C₁₆

semidirect

46. C₄×C₁₀₇

C₂×C₁₆

semidirect

54. C₁₀₇

D₄×C₁₆

semidirect

83. D₄

C₁₀₇⋊C₁₆

direct (N × H)

96. C₂²

C₂×C₁₀₇⋊C₁₆

semidirect

100. C₄

C₂×C₁₀₇⋊C₁₆

semidirect

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

subgroup	index
C₂²×C₁₀₇⋊C₁₆	2
C₄⋊(C₁₀₇⋊C₁₆)	2
C₂²⋊(C₁₀₇⋊C₁₆)	2
C₄×C₁₀₇⋊C₁₆	2
C₈×D₄×C₁₀₇	2
D₄×C₁₆	107

subgroup

index

C₂²×C₁₀₇⋊C₁₆

C₄⋊(C₁₀₇⋊C₁₆)

C₂²⋊(C₁₀₇⋊C₁₆)

C₄×C₁₀₇⋊C₁₆

C₈×D₄×C₁₀₇

D₄×C₁₆

107

Subgroups — D₄×C₁₀₇⋊C₁₆ #28338281

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D4×C107⋊C16 #28338281

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×C₁₀₇⋊C₁₆ #28338281