Group #18010175 — subgroups

Subgroups — D₈×#64438 #18010175

Order 4736.

196 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 74 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	D₄×#64438	C₂	1	2	yes	maximal, complemented
2	#14558396	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊#64438	C₂	1	2	yes	maximal, complemented
4	C₈×#64438	C₂	1	1	yes	maximal, characteristic, complemented
5	D₈×Dic₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64438	—	2	2	no	—
7	C₄⋊#64438	C₂²	1	2	yes	—
8	C₂²⋊#64438	—	2	2	no	—
9	C₄×#64438	C₂²	1	1	yes	characteristic
10	C₄×#64438	C₂²	1	1	yes	characteristic
11	D₄×Dic₃₇	C₂²	1	2	yes	—
12	#11924470	C₄	1	1	yes	characteristic
13	#11924457	C₄	1	2	yes	—
14	C₈×Dic₃₇	C₂²	1	1	yes	characteristic
15	C₂×D₈×C₃₇	C₄	1	1	yes	characteristic
16	C₂×D₄×C₃₇	C₂×C₄	1	2	yes	—
17	C₂²×Dic₃₇	—	2	2	no	—
18	C₂×#64438	—	2	1	no	—
19	C₂×#64438	—	2	2	no	—
20	C₂×#64438	—	4	4	no	—
21	C₂×#64438	D₄	1	1	yes	characteristic
22	C₂×#64438	D₄	1	1	yes	characteristic
23	D₈×C₃₇	C₈	1	4	yes	complemented
24	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
25	C₂²⋊Dic₃₇	—	2	2	no	—
26	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
27	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
28	C₂×#64436	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₃₇	—	2	2	no	—
30	D₄×C₃₇	—	2	2	no	—
31	D₄×C₃₇	C₂×C₈	1	4	yes	complemented
32	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×Dic₃₇	—	2	2	no	—
34	C₂×Dic₃₇	—	4	2	no	—
35	C₂×Dic₃₇	—	4	2	no	—
36	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
37	C₂×Dic₃₇	#41	1	1	yes	characteristic
38	#64438	—	2	2	no	—
39	#64438	—	2	1	no	—
40	#64438	—	4	2	no	—
41	#64438	D₈	1	2	yes	direct product factor
42	C₈×C₃₇	C₂×C₈	1	2	yes	complemented
43	#64436	—	2	1	no	—
44	C₂²×C₃₇	—	2	4	no	—
45	C₂²×C₃₇	—	4	2	no	—
46	C₂²×C₃₇	—	4	2	no	—
47	C₂²×C₃₇	C₄×D₄	1	1	yes	characteristic
48	C₄×C₃₇	C₂²×C₈	1	1	yes	characteristic
49	C₄×C₃₇	#131	1	1	yes	characteristic
50	Dic₃₇	—	2	1	no	—
51	Dic₃₇	—	4	2	no	—
52	Dic₃₇	C₂×D₈	1	1	yes	characteristic
53	Dic₃₇	#135	1	1	yes	characteristic
54	C₈×D₈	—	37	1	no	maximal, Sylow
55	C₂×C₃₇	—	4	4	no	—
56	C₂×C₃₇	C₈×D₄	1	1	yes	characteristic
57	C₂×C₃₇	C₄×D₈	1	1	yes	characteristic
58	C₂×C₃₇	#443	1	1	yes	characteristic
59	C₄×D₈	—	37	1	no	—
60	C₈×D₄	—	37	2	no	—
61	#335	—	37	1	no	—
62	D₄⋊C₈	—	37	2	no	—
63	C₈²	—	37	1	no	—
64	C₃₇	C₈×D₈	1	1	yes	characteristic, complemented, Sylow
65	C₂×D₈	#3866	1	1	yes	characteristic
66	C₂²×C₈	—	74	2	no	—
67	C₄×D₄	—	37	2	no	—
68	#107	—	37	1	no	—
69	C₄⋊C₈	—	37	2	no	—
70	#102	—	37	2	no	—
71	C₂²⋊C₈	—	74	2	no	—
72	C₄×C₈	—	37	1	no	—
73	C₄×C₈	—	37	1	no	—
74	C₄×C₈	—	37	1	no	—
75	C₂×D₄	C₂×#3866	1	2	yes	—
76	C₂²×C₄	—	74	2	no	—
77	D₈	#64438	1	4	yes	direct product factor
78	C₂×C₈	—	37	1	no	—
79	C₂×C₈	—	37	1	no	—
80	C₂×C₈	—	37	1	no	—
81	C₂×C₈	—	74	1	no	—
82	C₂×C₈	—	74	2	no	—
83	C₂×C₈	—	148	4	no	—
84	C₂×C₈	C₂×#3866	1	1	yes	characteristic
85	C₄⋊C₄	—	37	2	no	—
86	C₂²⋊C₄	—	74	2	no	—
87	C₄²	—	37	1	no	—
88	C₂³	—	2	2	no	—
89	D₄	—	2	2	no	—
90	D₄	C₂×#64438	1	4	yes	complemented
91	C₂×C₄	—	37	1	no	—
92	C₂×C₄	—	37	1	no	—
93	C₂×C₄	—	74	2	no	—
94	C₂×C₄	—	148	2	no	—
95	C₂×C₄	—	148	2	no	—
96	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
97	C₈	—	37	2	no	—
98	C₈	—	74	1	no	—
99	C₈	—	74	2	no	—
100	C₈	—	74	1	no	—
101	C₈	—	148	2	no	—
102	C₈	C₂×#64438	1	2	yes	complemented
103	C₂²	—	2	4	no	—
104	C₂²	—	4	2	no	—
105	C₂²	—	4	2	no	—
106	C₂²	D₄×#3866	1	1	yes	characteristic
107	C₄	—	37	1	no	—
108	C₄	—	37	1	no	—
109	C₄	—	74	1	no	—
110	C₄	—	148	2	no	—
111	C₄	C₂²×#64438	1	1	yes	characteristic
112	C₄	#11924623	1	1	yes	characteristic
113	C₂	—	4	4	no	—
114	C₂	D₄×#64438	1	1	yes	characteristic
115	C₂	D₈×#3866	1	1	yes	characteristic
116	C₂	#14558575	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₄×#64438

C₂

yes

maximal, complemented

#14558396

C₂

yes

maximal, characteristic, complemented

D₄⋊#64438

C₂

yes

maximal, complemented

C₈×#64438

C₂

yes

maximal, characteristic, complemented

D₈×Dic₃₇

C₂

yes

maximal, characteristic

C₂²×#64438

—

C₄⋊#64438

C₂²

yes

—

C₂²⋊#64438

—

C₄×#64438

C₂²

yes

characteristic

C₄×#64438

C₂²

yes

characteristic

D₄×Dic₃₇

C₂²

yes

—

#11924470

C₄

yes

characteristic

#11924457

C₄

yes

—

C₈×Dic₃₇

C₂²

yes

characteristic

C₂×D₈×C₃₇

C₄

yes

characteristic

C₂×D₄×C₃₇

C₂×C₄

yes

—

C₂²×Dic₃₇

—

—

—

—

D₄

yes

characteristic

C₂×#64438

D₄

yes

characteristic

D₈×C₃₇

C₈

yes

complemented

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₂²⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

C₄×Dic₃₇

C₂³

yes

characteristic

C₂×#64436

C₂×C₄

yes

characteristic

C₂³×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

C₂×C₈

yes

complemented

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂×D₄

yes

C₂×Dic₃₇

yes

—

—

—

D₈

yes

direct product factor

C₈×C₃₇

C₂×C₈

yes

complemented

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₄×D₄

yes

characteristic

C₄×C₃₇

C₂²×C₈

yes

characteristic

C₄×C₃₇

#131

yes

characteristic

Dic₃₇

—

Dic₃₇

—

Dic₃₇

C₂×D₈

yes

characteristic

Dic₃₇

#135

yes

characteristic

C₈×D₈

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₈×D₄

yes

characteristic

C₂×C₃₇

C₄×D₈

yes

characteristic

C₂×C₃₇

#443

yes

characteristic

C₄×D₈

—

C₈×D₄

—

#335

—

D₄⋊C₈

—

C₈²

—

C₃₇

C₈×D₈

yes

characteristic, complemented, Sylow

C₂×D₈

#3866

yes

characteristic

C₂²×C₈

—

C₄×D₄

—

#107

—

C₄⋊C₈

—

#102

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×D₄

C₂×#3866

yes

—

C₂²×C₄

—

D₈

#64438

yes

direct product factor

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

148

—

C₂×C₈

C₂×#3866

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

D₄

—

D₄

C₂×#64438

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

148

—

C₂×C₄

—

148

—

C₂×C₄

C₂²×#3866

yes

characteristic

C₈

—

C₈

—

C₈

—

100

C₈

—

101

C₈

—

148

—

102

C₈

C₂×#64438

yes

complemented

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

D₄×#3866

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

148

—

111

C₄

C₂²×#64438

yes

characteristic

112

C₄

#11924623

yes

characteristic

113

C₂

—

114

C₂

D₄×#64438

yes

characteristic

115

C₂

D₈×#3866

yes

characteristic

116

C₂

#14558575

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#3866
commutator	C₄×C₃₇	C₂²×C₈
Frattini	C₂×C₄	C₂²×#3866
Fitting, prime radical	C₂×D₈×C₃₇	C₄
socle	C₂²×C₃₇	C₄×D₄
cosocle	C₄×Dic₃₇	C₂³
hypercenter	C₂×D₈	#3866
hypocenter	C₃₇	C₈×D₈
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

subgroup

G/H

center

C₂²

D₄×#3866

commutator

C₄×C₃₇

C₂²×C₈

Frattini

C₂×C₄

C₂²×#3866

Fitting, prime radical

C₂×D₈×C₃₇

C₄

socle

C₂²×C₃₇

C₄×D₄

cosocle

C₄×Dic₃₇

C₂³

hypercenter

C₂×D₈

#3866

hypocenter

C₃₇

C₈×D₈

LCS

G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

N	H ≅ G/N	type
1. D₄×#64438	C₂	semidirect
2. #14558396	C₂	semidirect
3. D₄⋊#64438	C₂	semidirect
4. C₈×#64438	C₂	semidirect
23. D₈×C₃₇	C₈	semidirect
31. D₄×C₃₇	C₂×C₈	semidirect
41. #64438	D₈	direct (N × H)
42. C₈×C₃₇	C₂×C₈	semidirect
64. C₃₇	C₈×D₈	semidirect
77. D₈	#64438	direct (N × H)
90. D₄	C₂×#64438	semidirect
102. C₈	C₂×#64438	semidirect

H ≅ G/N

type

1. D₄×#64438

C₂

semidirect

2. #14558396

C₂

semidirect

3. D₄⋊#64438

C₂

semidirect

4. C₈×#64438

C₂

semidirect

23. D₈×C₃₇

C₈

semidirect

31. D₄×C₃₇

C₂×C₈

semidirect

41. #64438

D₈

direct (N × H)

42. C₈×C₃₇

C₂×C₈

semidirect

64. C₃₇

C₈×D₈

semidirect

77. D₈

#64438

direct (N × H)

90. D₄

C₂×#64438

semidirect

102. C₈

C₂×#64438

semidirect

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

subgroup	index
D₄×#64438	2
#14558396	2
D₄⋊#64438	2
C₈×#64438	2
D₈×Dic₃₇	2
C₈×D₈	37

subgroup

index

D₈×Dic₃₇

C₈×D₈

Subgroups — D₈×#64438 #18010175

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D8×#64438 #18010175

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₈×#64438 #18010175