Group #16840428 — subgroups

Subgroups — D₄×Dic₁₂₂ #16840428

Order 3904.

280 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 123 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×Dic₁₂₂	C₂	1	2	yes	maximal, complemented
2	#14519777	C₂	1	1	yes	maximal, characteristic, complemented
3	#14519776	C₂	1	2	yes	maximal, complemented
4	#14519756	C₂	1	2	yes	maximal, complemented
5	#14519754	C₂	1	4	yes	maximal, complemented
6	C₄×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×Dic₆₁	C₂	1	2	yes	maximal
8	C₄×D₄×C₆₁	C₂	1	1	yes	maximal, characteristic
9	C₂×D₄×C₆₁	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₆₁	C₂²	1	2	yes	—
11	C₂²×Dic₆₁	C₂²	1	4	yes	—
12	C₂×Dic₁₂₂	—	2	8	no	—
13	C₂×Dic₁₂₂	C₂²	1	4	yes	—
14	C₂×Dic₁₂₂	C₂²	1	2	yes	—
15	C₂×Dic₁₂₂	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
17	C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
18	C₄²×C₆₁	C₂²	1	1	yes	characteristic
19	C₂²⋊Dic₆₁	C₂²	1	4	yes	—
20	C₄⋊Dic₆₁	C₂²	1	2	yes	—
21	C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
22	C₄⋊Dic₆₁	C₂²	1	2	yes	—
23	#11758196	C₂²	1	2	yes	—
24	#11758196	C₂²	1	4	yes	—
25	C₄×Dic₆₁	C₂²	1	2	yes	—
26	C₂³×C₆₁	C₂³	1	2	yes	—
27	D₄×C₆₁	Q₈	1	4	yes	complemented
28	C₂×C₄×C₆₁	—	2	4	no	—
29	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₆₁	C₂³	1	2	yes	—
31	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
33	C₂×Dic₆₁	—	2	8	no	—
34	C₂×Dic₆₁	C₂³	1	2	yes	—
35	C₂×Dic₆₁	C₂³	1	4	yes	—
36	C₂×Dic₆₁	C₂³	1	2	yes	—
37	Dic₁₂₂	—	2	4	no	—
38	Dic₁₂₂	—	2	8	no	—
39	Dic₁₂₂	D₄	1	4	yes	direct product factor
40	C₂²×C₆₁	—	2	2	no	—
41	C₂²×C₆₁	—	2	2	no	—
42	C₂²×C₆₁	C₂×Q₈	1	4	yes	complemented
43	C₂²×C₆₁	C₂⁴	1	1	yes	characteristic
44	C₄×C₆₁	—	2	2	no	—
45	C₄×C₆₁	—	2	1	no	—
46	C₄×C₆₁	C₂×D₄	1	2	yes	—
47	C₄×C₆₁	C₂×Q₈	1	2	yes	complemented
48	Dic₆₁	—	2	4	no	—
49	Dic₆₁	—	2	2	no	—
50	Dic₆₁	C₂×D₄	1	4	yes	—
51	C₂×C₆₁	—	2	4	no	—
52	C₂×C₆₁	C₂²×D₄	1	1	yes	characteristic
53	C₂×C₆₁	C₂²×Q₈	1	1	yes	characteristic
54	C₂×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
55	D₄×Q₈	—	61	1	no	maximal, Sylow
56	C₆₁	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
57	C₂²×Q₈	—	61	2	no	—
58	C₄⋊Q₈	—	61	2	no	—
59	C₄⋊Q₈	—	61	1	no	—
60	C₂²⋊Q₈	—	61	2	no	—
61	C₂²⋊Q₈	—	61	4	no	—
62	C₄×Q₈	—	61	1	no	—
63	C₄×D₄	—	61	2	no	—
64	C₄×D₄	D₆₁	1	1	yes	characteristic
65	C₂×Q₈	—	61	1	no	—
66	C₂×Q₈	—	61	4	no	—
67	C₂×Q₈	—	61	2	no	—
68	C₂×Q₈	—	122	8	no	—
69	C₂×D₄	C₂×D₆₁	1	1	yes	characteristic
70	C₂²×C₄	—	61	4	no	—
71	C₂²×C₄	C₂×D₆₁	1	2	yes	—
72	C₄⋊C₄	—	61	2	no	—
73	C₄⋊C₄	—	61	4	no	—
74	C₄⋊C₄	—	61	2	no	—
75	C₄⋊C₄	—	61	1	no	—
76	C₄⋊C₄	—	61	2	no	—
77	C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
78	C₂²⋊C₄	—	61	4	no	—
79	C₂²⋊C₄	C₂×D₆₁	1	2	yes	—
80	C₄²	—	61	2	no	—
81	C₄²	C₂×D₆₁	1	1	yes	characteristic
82	C₂³	C₂²×D₆₁	1	2	yes	—
83	Q₈	—	61	4	no	—
84	Q₈	—	122	8	no	—
85	Q₈	—	122	4	no	—
86	D₄	Dic₁₂₂	1	4	yes	direct product factor
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	61	2	no	—
89	C₂×C₄	—	61	2	no	—
90	C₂×C₄	—	61	4	no	—
91	C₂×C₄	—	122	8	no	—
92	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
93	C₂×C₄	C₂²×D₆₁	1	2	yes	—
94	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
95	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
96	C₂²	—	2	2	no	—
97	C₂²	—	2	2	no	—
98	C₂²	C₂×Dic₁₂₂	1	4	yes	complemented
99	C₂²	C₂³×D₆₁	1	1	yes	characteristic
100	C₄	—	2	2	no	—
101	C₄	—	2	1	no	—
102	C₄	—	61	4	no	—
103	C₄	—	122	4	no	—
104	C₄	—	122	2	no	—
105	C₄	C₂×Dic₁₂₂	1	2	yes	complemented
106	C₄	D₄×D₆₁	1	2	yes	—
107	C₂	—	2	4	no	—
108	C₂	C₂×D₄×D₆₁	1	1	yes	characteristic
109	C₂	C₂²×Dic₁₂₂	1	1	yes	characteristic
110	C₂	#14519824	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₁₂₂

C₂

yes

maximal, complemented

#14519777

C₂

yes

maximal, characteristic, complemented

#14519776

C₂

yes

maximal, complemented

#14519756

C₂

yes

maximal, complemented

#14519754

C₂

yes

maximal, complemented

C₄×Dic₁₂₂

C₂

yes

maximal, characteristic, complemented

D₄×Dic₆₁

C₂

yes

maximal

C₄×D₄×C₆₁

C₂

yes

maximal, characteristic

C₂×D₄×C₆₁

C₂²

yes

characteristic

C₂²×C₄×C₆₁

C₂²

yes

—

C₂²×Dic₆₁

C₂²

yes

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

C₂²

yes

—

C₂×Dic₁₂₂

C₂²

yes

—

C₂×Dic₁₂₂

C₂²

yes

characteristic

C₄⋊C₄×C₆₁

C₂²

yes

characteristic

C₂²⋊C₄×C₆₁

C₂²

yes

—

C₄²×C₆₁

C₂²

yes

characteristic

C₂²⋊Dic₆₁

C₂²

yes

—

C₄⋊Dic₆₁

C₂²

yes

—

C₄⋊Dic₆₁

C₂²

yes

characteristic

C₄⋊Dic₆₁

C₂²

yes

—

#11758196

C₂²

yes

—

#11758196

C₂²

yes

—

C₄×Dic₆₁

C₂²

yes

—

C₂³×C₆₁

C₂³

yes

—

D₄×C₆₁

Q₈

yes

complemented

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂³

yes

characteristic

C₂×C₄×C₆₁

C₂³

yes

—

C₂×C₄×C₆₁

C₂³

yes

characteristic

C₂×C₄×C₆₁

C₂³

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂³

yes

—

C₂×Dic₆₁

C₂³

yes

—

C₂×Dic₆₁

C₂³

yes

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

D₄

yes

direct product factor

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×Q₈

yes

complemented

C₂²×C₆₁

C₂⁴

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×D₄

yes

—

C₄×C₆₁

C₂×Q₈

yes

complemented

Dic₆₁

—

Dic₆₁

—

Dic₆₁

C₂×D₄

yes

—

C₂×C₆₁

—

C₂×C₆₁

C₂²×D₄

yes

characteristic

C₂×C₆₁

C₂²×Q₈

yes

characteristic

C₂×C₆₁

2₋¹⁺⁴

yes

characteristic

D₄×Q₈

—

maximal, Sylow

C₆₁

D₄×Q₈

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

D₆₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂×D₄

C₂×D₆₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×D₆₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₆₁

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₆₁

yes

—

C₄²

—

C₄²

C₂×D₆₁

yes

characteristic

C₂³

C₂²×D₆₁

yes

—

Q₈

—

Q₈

—

122

—

Q₈

—

122

—

D₄

Dic₁₂₂

yes

direct product factor

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

122

—

C₂×C₄

C₂²×D₆₁

yes

characteristic

C₂×C₄

C₂²×D₆₁

yes

—

C₂×C₄

C₂²×D₆₁

yes

characteristic

C₂×C₄

C₂²×D₆₁

yes

characteristic

C₂²

—

C₂²

—

C₂²

C₂×Dic₁₂₂

yes

complemented

C₂²

C₂³×D₆₁

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

122

—

104

C₄

—

122

—

105

C₄

C₂×Dic₁₂₂

yes

complemented

106

C₄

D₄×D₆₁

yes

—

107

C₂

—

108

C₂

C₂×D₄×D₆₁

yes

characteristic

109

C₂

C₂²×Dic₁₂₂

yes

characteristic

110

C₂

#14519824

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₆₁
commutator, socle, cosocle	C₂²×C₆₁	C₂⁴
Fitting	C₄×D₄×C₆₁	C₂
prime radical	C₂×D₄×C₆₁	C₂²
hypercenter	C₄×D₄	D₆₁
hypocenter	C₆₁	D₄×Q₈
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₆₁

commutator, socle, cosocle

C₂²×C₆₁

C₂⁴

Fitting

C₄×D₄×C₆₁

C₂

prime radical

C₂×D₄×C₆₁

C₂²

hypercenter

C₄×D₄

D₆₁

hypocenter

C₆₁

D₄×Q₈

LCS

G ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. C₂²×Dic₁₂₂	C₂	semidirect
2. #14519777	C₂	semidirect
3. #14519776	C₂	semidirect
4. #14519756	C₂	semidirect
5. #14519754	C₂	semidirect
6. C₄×Dic₁₂₂	C₂	semidirect
27. D₄×C₆₁	Q₈	semidirect
39. Dic₁₂₂	D₄	direct (N × H)
42. C₂²×C₆₁	C₂×Q₈	semidirect
47. C₄×C₆₁	C₂×Q₈	semidirect
56. C₆₁	D₄×Q₈	semidirect
86. D₄	Dic₁₂₂	direct (N × H)
98. C₂²	C₂×Dic₁₂₂	semidirect
105. C₄	C₂×Dic₁₂₂	semidirect

H ≅ G/N

type

1. C₂²×Dic₁₂₂

C₂

semidirect

2. #14519777

C₂

semidirect

3. #14519776

C₂

semidirect

4. #14519756

C₂

semidirect

5. #14519754

C₂

semidirect

6. C₄×Dic₁₂₂

C₂

semidirect

27. D₄×C₆₁

Q₈

semidirect

39. Dic₁₂₂

D₄

direct (N × H)

42. C₂²×C₆₁

C₂×Q₈

semidirect

47. C₄×C₆₁

C₂×Q₈

semidirect

56. C₆₁

D₄×Q₈

semidirect

86. D₄

Dic₁₂₂

direct (N × H)

98. C₂²

C₂×Dic₁₂₂

semidirect

105. C₄

C₂×Dic₁₂₂

semidirect

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

subgroup	index
C₂²×Dic₁₂₂	2
#14519777	2
#14519776	2
#14519756	2
#14519754	2
C₄×Dic₁₂₂	2
D₄×Dic₆₁	2
C₄×D₄×C₆₁	2
D₄×Q₈	61

subgroup

index

C₂²×Dic₁₂₂

C₄×Dic₁₂₂

D₄×Dic₆₁

C₄×D₄×C₆₁

D₄×Q₈

Subgroups — D₄×Dic₁₂₂ #16840428

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D4×Dic122 #16840428

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×Dic₁₂₂ #16840428