Group #37535541 — subgroups

Subgroups — D₄×#15668424 #37535541

Order 23168.

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×#15668424	C₂	1	2	yes	maximal, complemented
2	C₄⋊#15668424	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊#15668424	C₂	1	2	yes	maximal, complemented
4	C₄×#15668424	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×D₄×C₁₈₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668424	—	2	4	no	—
7	C₂×#15668424	C₂²	1	2	yes	—
8	C₂×#15668424	C₂²	1	1	yes	characteristic
9	C₂×#15668424	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₁₈₁	C₂²	1	2	yes	—
11	C₄×D₄×C₁₈₁	C₄	1	1	yes	characteristic
12	C₄⋊C₈×C₁₈₁	C₄	1	1	yes	characteristic
13	C₂²⋊C₈×C₁₈₁	C₄	1	2	yes	—
14	C₄×C₈×C₁₈₁	C₂²	1	1	yes	characteristic
15	#15668424	—	2	2	no	—
16	#15668424	—	2	1	no	—
17	#15668424	D₄	1	2	yes	direct product factor
18	C₂×D₄×C₁₈₁	C₈	1	1	yes	characteristic
19	C₂²×C₄×C₁₈₁	C₂×C₄	1	2	yes	—
20	C₂×C₈×C₁₈₁	—	2	2	no	—
21	C₂×C₈×C₁₈₁	—	2	2	no	—
22	C₂×C₈×C₁₈₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₁₈₁	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₁₈₁	C₈	1	1	yes	characteristic
26	C₂²⋊C₄×C₁₈₁	C₈	1	2	yes	—
27	C₄²×C₁₈₁	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₁₈₁	C₂×C₈	1	2	yes	—
29	D₄×C₁₈₁	C₁₆	1	4	yes	complemented
30	C₂×C₄×C₁₈₁	—	2	2	no	—
31	C₂×C₄×C₁₈₁	—	2	2	no	—
32	C₂×C₄×C₁₈₁	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₁₈₁	C₂×C₈	1	2	yes	—
34	C₂×C₄×C₁₈₁	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
36	C₈×C₁₈₁	—	2	2	no	—
37	C₈×C₁₈₁	—	2	1	no	—
38	C₈×C₁₈₁	C₂×D₄	1	1	yes	characteristic
39	C₈×C₁₈₁	#41	1	1	yes	characteristic
40	C₂²×C₁₈₁	—	2	2	no	—
41	C₂²×C₁₈₁	—	2	2	no	—
42	C₂²×C₁₈₁	C₂×C₁₆	1	4	yes	complemented
43	C₂²×C₁₈₁	C₂²×C₈	1	1	yes	characteristic
44	C₄×C₁₈₁	—	2	2	no	—
45	C₄×C₁₈₁	—	2	1	no	—
46	C₄×C₁₈₁	C₂×C₁₆	1	2	yes	complemented
47	C₄×C₁₈₁	C₄×D₄	1	1	yes	characteristic
48	C₄×C₁₈₁	#131	1	1	yes	characteristic
49	C₂×C₁₈₁	—	2	4	no	—
50	C₂×C₁₈₁	C₈×D₄	1	1	yes	characteristic
51	C₂×C₁₈₁	C₂²×C₁₆	1	1	yes	characteristic
52	C₂×C₁₈₁	#504	1	1	yes	characteristic
53	C₁₈₁	D₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
54	D₄×C₁₆	—	181	1	no	maximal, Sylow
55	C₂²×C₁₆	—	181	2	no	—
56	C₈×D₄	D₁₈₁	1	1	yes	characteristic
57	C₄⋊C₁₆	—	181	1	no	—
58	C₂²⋊C₁₆	—	181	2	no	—
59	C₄×C₁₆	—	181	1	no	—
60	C₂²×C₈	C₂×D₁₈₁	1	2	yes	—
61	C₄×D₄	Dic₁₈₁	1	1	yes	characteristic
62	C₂×C₁₆	—	362	4	no	—
63	C₂×C₁₆	—	181	1	no	—
64	C₂×C₁₆	—	181	2	no	—
65	C₂×C₁₆	—	181	1	no	—
66	C₄⋊C₈	Dic₁₈₁	1	1	yes	characteristic
67	C₂²⋊C₈	Dic₁₈₁	1	2	yes	—
68	C₄×C₈	C₂×D₁₈₁	1	1	yes	characteristic
69	C₂×D₄	#13089678	1	1	yes	characteristic
70	C₂²×C₄	C₂×Dic₁₈₁	1	2	yes	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	C₂×Dic₁₈₁	1	1	yes	characteristic
74	C₂×C₈	C₂×Dic₁₈₁	1	2	yes	—
75	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
76	C₄⋊C₄	#13089678	1	1	yes	characteristic
77	C₂²⋊C₄	#13089678	1	2	yes	—
78	C₄²	C₂×Dic₁₈₁	1	1	yes	characteristic
79	C₁₆	—	362	2	no	—
80	C₁₆	—	362	1	no	—
81	C₁₆	—	181	2	no	—
82	C₂³	C₂×#13089678	1	2	yes	—
83	D₄	#15668424	1	4	yes	direct product factor
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	C₂×#13089678	1	1	yes	characteristic
87	C₂×C₄	C₂×#13089678	1	2	yes	—
88	C₂×C₄	C₂×#13089678	1	1	yes	characteristic
89	C₂×C₄	C₂²×Dic₁₈₁	1	1	yes	characteristic
90	C₈	—	2	2	no	—
91	C₈	—	2	1	no	—
92	C₈	D₄×D₁₈₁	1	1	yes	characteristic
93	C₈	#15668448	1	1	yes	characteristic
94	C₂²	—	2	2	no	—
95	C₂²	—	2	2	no	—
96	C₂²	C₂×#15668424	1	4	yes	complemented
97	C₂²	C₂²×#13089678	1	1	yes	characteristic
98	C₄	—	2	2	no	—
99	C₄	—	2	1	no	—
100	C₄	C₂×#15668424	1	2	yes	complemented
101	C₄	D₄×Dic₁₈₁	1	1	yes	characteristic
102	C₄	#19153823	1	1	yes	characteristic
103	C₂	—	2	4	no	—
104	C₂	D₄×#13089678	1	1	yes	characteristic
105	C₂	C₂²×#15668424	1	1	yes	characteristic
106	C₂	#26041696	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#15668424

C₂

yes

maximal, complemented

C₄⋊#15668424

C₂

yes

maximal, characteristic, complemented

C₂²⋊#15668424

C₂

yes

maximal, complemented

C₄×#15668424

C₂

yes

maximal, characteristic, complemented

C₈×D₄×C₁₈₁

C₂

yes

maximal, characteristic

C₂×#15668424

—

C₂×#15668424

C₂²

yes

—

C₂×#15668424

C₂²

yes

characteristic

C₂×#15668424

C₂²

yes

characteristic

C₂²×C₈×C₁₈₁

C₂²

yes

—

C₄×D₄×C₁₈₁

C₄

yes

characteristic

C₄⋊C₈×C₁₈₁

C₄

yes

characteristic

C₂²⋊C₈×C₁₈₁

C₄

yes

—

C₄×C₈×C₁₈₁

C₂²

yes

—

—

D₄

yes

direct product factor

C₂×D₄×C₁₈₁

C₈

yes

characteristic

C₂²×C₄×C₁₈₁

C₂×C₄

yes

—

C₂×C₈×C₁₈₁

—

C₂×C₈×C₁₈₁

—

C₂×C₈×C₁₈₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₈₁

C₂×C₄

yes

—

C₂×C₈×C₁₈₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₈₁

C₈

yes

characteristic

C₂²⋊C₄×C₁₈₁

C₈

yes

—

C₄²×C₁₈₁

C₂×C₄

yes

characteristic

C₂³×C₁₈₁

C₂×C₈

yes

—

D₄×C₁₈₁

C₁₆

yes

complemented

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

C₂×C₈

yes

characteristic

C₂×C₄×C₁₈₁

C₂×C₈

yes

—

C₂×C₄×C₁₈₁

C₂×C₈

yes

characteristic

C₂×C₄×C₁₈₁

C₂²×C₄

yes

characteristic

C₈×C₁₈₁

—

C₈×C₁₈₁

—

C₈×C₁₈₁

C₂×D₄

yes

characteristic

C₈×C₁₈₁

#41

yes

characteristic

C₂²×C₁₈₁

—

C₂²×C₁₈₁

—

C₂²×C₁₈₁

C₂×C₁₆

yes

complemented

C₂²×C₁₈₁

C₂²×C₈

yes

characteristic

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

C₂×C₁₆

yes

complemented

C₄×C₁₈₁

C₄×D₄

yes

characteristic

C₄×C₁₈₁

#131

yes

characteristic

C₂×C₁₈₁

—

C₂×C₁₈₁

C₈×D₄

yes

characteristic

C₂×C₁₈₁

C₂²×C₁₆

yes

characteristic

C₂×C₁₈₁

#504

yes

characteristic

C₁₈₁

D₄×C₁₆

yes

characteristic, complemented, Sylow

D₄×C₁₆

—

181

maximal, Sylow

C₂²×C₁₆

—

181

—

C₈×D₄

D₁₈₁

yes

characteristic

C₄⋊C₁₆

—

181

—

C₂²⋊C₁₆

—

181

—

C₄×C₁₆

—

181

—

C₂²×C₈

C₂×D₁₈₁

yes

—

C₄×D₄

Dic₁₈₁

yes

characteristic

C₂×C₁₆

—

362

—

C₂×C₁₆

—

181

—

C₂×C₁₆

—

181

—

C₂×C₁₆

—

181

—

C₄⋊C₈

Dic₁₈₁

yes

characteristic

C₂²⋊C₈

Dic₁₈₁

yes

—

C₄×C₈

C₂×D₁₈₁

yes

characteristic

C₂×D₄

#13089678

yes

characteristic

C₂²×C₄

C₂×Dic₁₈₁

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₁₈₁

yes

characteristic

C₂×C₈

C₂×Dic₁₈₁

yes

—

C₂×C₈

C₂²×D₁₈₁

yes

C₄⋊C₄

yes

C₂²⋊C₄

yes

—

C₄²

C₂×Dic₁₈₁

yes

characteristic

C₁₆

—

362

—

C₁₆

—

362

—

C₁₆

—

181

—

C₂³

C₂×#13089678

yes

—

D₄

#15668424

yes

direct product factor

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×#13089678

yes

characteristic

C₂×C₄

C₂×#13089678

yes

—

C₂×C₄

C₂×#13089678

yes

characteristic

C₂×C₄

C₂²×Dic₁₈₁

yes

characteristic

C₈

—

C₈

—

C₈

D₄×D₁₈₁

yes

characteristic

C₈

#15668448

yes

characteristic

C₂²

—

C₂²

—

C₂²

yes

C₂²

yes

C₄

—

C₄

—

100

C₄

C₂×#15668424

yes

complemented

101

C₄

D₄×Dic₁₈₁

yes

characteristic

102

C₄

#19153823

yes

characteristic

103

C₂

—

104

C₂

D₄×#13089678

yes

characteristic

105

C₂

C₂²×#15668424

yes

characteristic

106

C₂

#26041696

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₈₁
commutator	C₂×C₁₈₁	C₂²×C₁₆
Fitting	C₈×D₄×C₁₈₁	C₂
socle	C₂²×C₁₈₁	C₂²×C₈
prime radical	C₂×D₄×C₁₈₁	C₈
cosocle	C₂×C₈×C₁₈₁	C₂³
hypercenter	C₈×D₄	D₁₈₁
hypocenter	C₁₈₁	D₄×C₁₆
LCS	G ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂×C₈ ◁ C₈×D₄

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₈₁

commutator

C₂×C₁₈₁

C₂²×C₁₆

Fitting

C₈×D₄×C₁₈₁

C₂

socle

C₂²×C₁₈₁

C₂²×C₈

prime radical

C₂×D₄×C₁₈₁

C₈

cosocle

C₂×C₈×C₁₈₁

C₂³

hypercenter

C₈×D₄

D₁₈₁

hypocenter

C₁₈₁

D₄×C₁₆

LCS

G ▷ C₂×C₁₈₁ ▷ C₁₈₁

UCS

C₁ ◁ C₂×C₈ ◁ C₈×D₄

N	H ≅ G/N	type
1. C₂²×#15668424	C₂	semidirect
2. C₄⋊#15668424	C₂	semidirect
3. C₂²⋊#15668424	C₂	semidirect
4. C₄×#15668424	C₂	semidirect
17. #15668424	D₄	direct (N × H)
29. D₄×C₁₈₁	C₁₆	semidirect
42. C₂²×C₁₈₁	C₂×C₁₆	semidirect
46. C₄×C₁₈₁	C₂×C₁₆	semidirect
53. C₁₈₁	D₄×C₁₆	semidirect
83. D₄	#15668424	direct (N × H)
96. C₂²	C₂×#15668424	semidirect
100. C₄	C₂×#15668424	semidirect

H ≅ G/N

type

1. C₂²×#15668424

C₂

semidirect

2. C₄⋊#15668424

C₂

semidirect

3. C₂²⋊#15668424

C₂

semidirect

4. C₄×#15668424

C₂

semidirect

17. #15668424

D₄

direct (N × H)

29. D₄×C₁₈₁

C₁₆

semidirect

42. C₂²×C₁₈₁

C₂×C₁₆

semidirect

46. C₄×C₁₈₁

C₂×C₁₆

semidirect

53. C₁₈₁

D₄×C₁₆

semidirect

83. D₄

#15668424

direct (N × H)

96. C₂²

C₂×#15668424

semidirect

100. C₄

C₂×#15668424

semidirect

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

subgroup	index
C₂²×#15668424	2
C₄⋊#15668424	2
C₂²⋊#15668424	2
C₄×#15668424	2
C₈×D₄×C₁₈₁	2
D₄×C₁₆	181

subgroup

index

C₈×D₄×C₁₈₁

D₄×C₁₆

181

Subgroups — D₄×#15668424 #37535541

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D4×#15668424 #37535541

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#15668424 #37535541