Group #37560086 — subgroups

Subgroups — Q₈×C₂²⋊D₃₇₃ #37560086

Order 23872.

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	C₂×Q₈×D₃₇₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#26045790	C₂	1	3	yes	maximal, complemented
3	#26045771	C₂	1	3	yes	maximal, complemented
4	#26045768	C₂	1	3	yes	maximal, complemented
5	Q₈×Dic₃₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₂²⋊D₃₇₃	C₂	1	3	yes	maximal
7	C₂²×Q₈×C₃₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₇₄₆	C₂²	1	3	yes	—
9	Q₈×D₃₇₃	—	2	4	no	—
10	C₂×C₂²⋊D₃₇₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₃₇₃	C₂²	1	3	yes	—
12	C₄⋊Dic₃₇₃	C₂²	1	3	yes	—
13	#20238589	C₂²	1	3	yes	—
14	#20238589	C₂²	1	6	yes	—
15	C₂²⋊Dic₃₇₃	C₂²	1	3	yes	—
16	#20238587	C₂²	1	3	yes	—
17	C₄×Dic₃₇₃	C₂²	1	3	yes	—
18	C₂×Q₈×C₃₇₃	—	2	4	no	—
19	C₂×Q₈×C₃₇₃	C₂²	1	3	yes	—
20	C₂×Q₈×C₃₇₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₃₇₃	C₂²	1	3	yes	—
22	C₂²×D₃₇₃	C₂³	1	1	yes	characteristic
23	Dic₇₄₆	—	2	6	no	—
24	C₂²⋊D₃₇₃	Q₈	1	4	yes	direct product factor
25	C₂×Dic₃₇₃	C₂³	1	3	yes	—
26	C₂×Dic₃₇₃	C₂³	1	3	yes	—
27	C₂×Dic₃₇₃	C₂³	1	1	yes	characteristic
28	C₄×D₃₇₃	—	2	6	no	—
29	C₂³×C₃₇₃	C₂³	1	1	yes	characteristic
30	Q₈×C₃₇₃	—	2	6	no	—
31	Q₈×C₃₇₃	D₄	1	4	yes	complemented
32	C₂×C₄×C₃₇₃	—	2	6	no	—
33	C₂×C₄×C₃₇₃	C₂³	1	3	yes	—
34	C₂×C₄×C₃₇₃	C₂³	1	3	yes	—
35	C₂²×C₃₇₃	—	2	1	no	—
36	C₂²×C₃₇₃	—	2	1	no	—
37	C₂²×C₃₇₃	C₂×Q₈	1	2	yes	complemented
38	C₂²×C₃₇₃	C₂⁴	1	1	yes	characteristic
39	C₂×D₃₇₃	—	2	1	no	—
40	C₂×D₃₇₃	—	2	1	no	—
41	C₂×D₃₇₃	C₂×Q₈	1	2	yes	complemented
42	C₄×C₃₇₃	—	2	3	no	—
43	C₄×C₃₇₃	C₂×D₄	1	6	yes	—
44	Dic₃₇₃	—	2	3	no	—
45	Dic₃₇₃	—	2	3	no	—
46	Dic₃₇₃	C₂×Q₈	1	2	yes	complemented
47	C₂×C₃₇₃	—	2	2	no	—
48	C₂×C₃₇₃	C₂²×D₄	1	1	yes	characteristic
49	C₂×C₃₇₃	C₂²×Q₈	1	1	yes	characteristic
50	C₂×C₃₇₃	2₋¹⁺⁴	1	1	yes	characteristic
51	D₃₇₃	—	2	2	no	—
52	C₃₇₃	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
53	D₄×Q₈	—	373	1	no	maximal, Sylow
54	C₂²×Q₈	—	373	1	no	—
55	C₂²×Q₈	D₃₇₃	1	1	yes	characteristic, complemented
56	C₄⋊Q₈	—	373	3	no	—
57	C₂²⋊Q₈	—	373	3	no	—
58	C₂²⋊Q₈	—	373	3	no	—
59	C₄×Q₈	—	373	1	no	—
60	C₄×D₄	—	373	3	no	—
61	C₂×Q₈	—	2	4	no	—
62	C₂×Q₈	—	373	3	no	—
63	C₂×Q₈	—	746	4	no	—
64	C₂×Q₈	C₂×D₃₇₃	1	3	yes	—
65	C₂×Q₈	C₂×D₃₇₃	1	1	yes	characteristic
66	C₂×D₄	—	373	1	no	—
67	C₂²×C₄	—	373	3	no	—
68	C₂²×C₄	C₂×D₃₇₃	1	3	yes	—
69	C₄⋊C₄	—	373	3	no	—
70	C₄⋊C₄	—	373	3	no	—
71	C₄⋊C₄	—	373	6	no	—
72	C₂²⋊C₄	—	373	3	no	—
73	C₂²⋊C₄	—	373	3	no	—
74	C₄²	—	373	3	no	—
75	C₂³	—	373	1	no	—
76	C₂³	C₂²×D₃₇₃	1	1	yes	characteristic
77	Q₈	—	2	6	no	—
78	Q₈	—	746	6	no	—
79	Q₈	C₂²⋊D₃₇₃	1	4	yes	direct product factor
80	D₄	—	373	4	no	—
81	C₂×C₄	—	2	6	no	—
82	C₂×C₄	—	373	1	no	—
83	C₂×C₄	—	373	3	no	—
84	C₂×C₄	—	373	3	no	—
85	C₂×C₄	—	746	6	no	—
86	C₂×C₄	C₂²×D₃₇₃	1	3	yes	—
87	C₂×C₄	C₂²×D₃₇₃	1	3	yes	—
88	C₂²	—	2	1	no	—
89	C₂²	—	2	1	no	—
90	C₂²	—	373	2	no	—
91	C₂²	—	746	1	no	—
92	C₂²	—	746	1	no	—
93	C₂²	Q₈×D₃₇₃	1	2	yes	complemented
94	C₂²	C₂³×D₃₇₃	1	1	yes	characteristic
95	C₄	—	2	3	no	—
96	C₄	—	373	2	no	—
97	C₄	—	746	3	no	—
98	C₄	—	746	3	no	—
99	C₄	C₂×C₂²⋊D₃₇₃	1	6	yes	—
100	C₂	—	2	2	no	—
101	C₂	—	746	2	no	—
102	C₂	C₂²×C₂²⋊D₃₇₃	1	1	yes	characteristic
103	C₂	C₂×Q₈×D₃₇₃	1	1	yes	characteristic
104	C₂	#26045840	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×D₃₇₃

C₂

yes

maximal, characteristic, complemented

#26045790

C₂

yes

maximal, complemented

#26045771

C₂

yes

maximal, complemented

#26045768

C₂

yes

maximal, complemented

Q₈×Dic₃₇₃

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₃₇₃

C₂

yes

maximal

C₂²×Q₈×C₃₇₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₇₄₆

C₂²

yes

—

Q₈×D₃₇₃

—

C₂×C₂²⋊D₃₇₃

C₂²

yes

characteristic

C₂×C₄×D₃₇₃

C₂²

yes

—

C₄⋊Dic₃₇₃

C₂²

yes

—

#20238589

C₂²

yes

—

#20238589

C₂²

yes

—

C₂²⋊Dic₃₇₃

C₂²

yes

—

#20238587

C₂²

yes

—

C₄×Dic₃₇₃

C₂²

yes

—

C₂×Q₈×C₃₇₃

—

C₂×Q₈×C₃₇₃

C₂²

yes

—

C₂×Q₈×C₃₇₃

C₂²

yes

characteristic

C₂²×C₄×C₃₇₃

C₂²

yes

—

C₂²×D₃₇₃

C₂³

yes

characteristic

Dic₇₄₆

—

C₂²⋊D₃₇₃

Q₈

yes

direct product factor

C₂×Dic₃₇₃

C₂³

yes

—

C₂×Dic₃₇₃

C₂³

yes

—

C₂×Dic₃₇₃

C₂³

yes

characteristic

C₄×D₃₇₃

—

C₂³×C₃₇₃

C₂³

yes

characteristic

Q₈×C₃₇₃

—

Q₈×C₃₇₃

D₄

yes

complemented

C₂×C₄×C₃₇₃

—

C₂×C₄×C₃₇₃

C₂³

yes

—

C₂×C₄×C₃₇₃

C₂³

yes

—

C₂²×C₃₇₃

—

C₂²×C₃₇₃

—

C₂²×C₃₇₃

C₂×Q₈

yes

complemented

C₂²×C₃₇₃

C₂⁴

yes

characteristic

C₂×D₃₇₃

—

C₂×D₃₇₃

—

C₂×D₃₇₃

C₂×Q₈

yes

complemented

C₄×C₃₇₃

—

C₄×C₃₇₃

C₂×D₄

yes

—

Dic₃₇₃

—

Dic₃₇₃

—

Dic₃₇₃

C₂×Q₈

yes

complemented

C₂×C₃₇₃

—

C₂×C₃₇₃

C₂²×D₄

yes

characteristic

C₂×C₃₇₃

C₂²×Q₈

yes

characteristic

C₂×C₃₇₃

2₋¹⁺⁴

yes

characteristic

D₃₇₃

—

C₃₇₃

D₄×Q₈

yes

characteristic, complemented, Sylow

D₄×Q₈

—

373

maximal, Sylow

C₂²×Q₈

—

373

—

C₂²×Q₈

D₃₇₃

yes

characteristic, complemented

C₄⋊Q₈

—

373

—

C₂²⋊Q₈

—

373

—

C₂²⋊Q₈

—

373

—

C₄×Q₈

—

373

—

C₄×D₄

—

373

—

C₂×Q₈

—

C₂×Q₈

—

373

—

C₂×Q₈

—

746

—

C₂×Q₈

C₂×D₃₇₃

yes

—

C₂×Q₈

C₂×D₃₇₃

yes

characteristic

C₂×D₄

—

373

—

C₂²×C₄

—

373

—

C₂²×C₄

C₂×D₃₇₃

yes

—

C₄⋊C₄

—

373

—

C₄⋊C₄

—

373

—

C₄⋊C₄

—

373

—

C₂²⋊C₄

—

373

—

C₂²⋊C₄

—

373

—

C₄²

—

373

—

C₂³

—

373

—

C₂³

C₂²×D₃₇₃

yes

characteristic

Q₈

—

Q₈

—

746

—

Q₈

C₂²⋊D₃₇₃

yes

direct product factor

D₄

—

373

—

C₂×C₄

—

C₂×C₄

—

373

—

C₂×C₄

—

373

—

C₂×C₄

—

373

—

C₂×C₄

—

746

—

C₂×C₄

C₂²×D₃₇₃

yes

—

C₂×C₄

C₂²×D₃₇₃

yes

—

C₂²

—

C₂²

—

C₂²

—

373

—

C₂²

—

746

—

C₂²

—

746

—

C₂²

Q₈×D₃₇₃

yes

complemented

C₂²

C₂³×D₃₇₃

yes

characteristic

C₄

—

C₄

—

373

—

C₄

—

746

—

C₄

—

746

—

C₄

C₂×C₂²⋊D₃₇₃

yes

—

100

C₂

—

101

C₂

—

746

—

102

C₂

C₂²×C₂²⋊D₃₇₃

yes

characteristic

103

C₂

C₂×Q₈×D₃₇₃

yes

characteristic

104

C₂

#26045840

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₇₃
commutator, socle, cosocle	C₂²×C₃₇₃	C₂⁴
Fitting	C₂²×Q₈×C₃₇₃	C₂
prime radical	C₂×C₂²⋊D₃₇₃	C₂²
hypercenter	C₂²×Q₈	D₃₇₃
hypocenter	C₃₇₃	D₄×Q₈
LCS	G ▷ C₂²×C₃₇₃ ▷ C₃₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₇₃

commutator, socle, cosocle

C₂²×C₃₇₃

C₂⁴

Fitting

C₂²×Q₈×C₃₇₃

C₂

prime radical

C₂×C₂²⋊D₃₇₃

C₂²

hypercenter

C₂²×Q₈

D₃₇₃

hypocenter

C₃₇₃

D₄×Q₈

LCS

G ▷ C₂²×C₃₇₃ ▷ C₃₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×Q₈

N	H ≅ G/N	type
1. C₂×Q₈×D₃₇₃	C₂	semidirect
2. #26045790	C₂	semidirect
3. #26045771	C₂	semidirect
4. #26045768	C₂	semidirect
5. Q₈×Dic₃₇₃	C₂	semidirect
7. C₂²×Q₈×C₃₇₃	C₂	semidirect
24. C₂²⋊D₃₇₃	Q₈	direct (N × H)
31. Q₈×C₃₇₃	D₄	semidirect
37. C₂²×C₃₇₃	C₂×Q₈	semidirect
41. C₂×D₃₇₃	C₂×Q₈	semidirect
46. Dic₃₇₃	C₂×Q₈	semidirect
52. C₃₇₃	D₄×Q₈	semidirect
55. C₂²×Q₈	D₃₇₃	semidirect
79. Q₈	C₂²⋊D₃₇₃	direct (N × H)
93. C₂²	Q₈×D₃₇₃	semidirect

H ≅ G/N

type

1. C₂×Q₈×D₃₇₃

C₂

semidirect

2. #26045790

C₂

semidirect

3. #26045771

C₂

semidirect

4. #26045768

C₂

semidirect

5. Q₈×Dic₃₇₃

C₂

semidirect

7. C₂²×Q₈×C₃₇₃

C₂

semidirect

24. C₂²⋊D₃₇₃

Q₈

direct (N × H)

31. Q₈×C₃₇₃

D₄

semidirect

37. C₂²×C₃₇₃

C₂×Q₈

semidirect

41. C₂×D₃₇₃

C₂×Q₈

semidirect

46. Dic₃₇₃

C₂×Q₈

semidirect

52. C₃₇₃

D₄×Q₈

semidirect

55. C₂²×Q₈

D₃₇₃

semidirect

79. Q₈

C₂²⋊D₃₇₃

direct (N × H)

93. C₂²

Q₈×D₃₇₃

semidirect

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

subgroup	index
C₂×Q₈×D₃₇₃	2
#26045790	2
#26045771	2
#26045768	2
Q₈×Dic₃₇₃	2
C₄×C₂²⋊D₃₇₃	2
C₂²×Q₈×C₃₇₃	2
D₄×Q₈	373

subgroup

index

C₂×Q₈×D₃₇₃

#26045790

#26045771

#26045768

Q₈×Dic₃₇₃

C₄×C₂²⋊D₃₇₃

C₂²×Q₈×C₃₇₃

D₄×Q₈

373

Subgroups — Q₈×C₂²⋊D₃₇₃ #37560086

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q8×C22⋊D373 #37560086

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₂²⋊D₃₇₃ #37560086