Group #37559349 — subgroups

Subgroups — #37559349

Order 23872.

170 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 57 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×C₄⋊Dic₃₇₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#20238587	C₂	1	1	yes	maximal, characteristic
3	C₂×#20238587	C₂	1	2	yes	maximal
4	#26045699	C₂	1	2	yes	maximal, complemented
5	#95×C₃₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₃₇₃	C₂²	1	1	yes	characteristic
7	C₂²×Dic₃₇₃	C₂²	1	1	yes	characteristic
8	C₂²×Dic₃₇₃	C₂²	1	2	yes	—
9	C₄⋊Dic₃₇₃	—	2	2	no	—
10	#20238587	—	2	4	no	—
11	#20238587	—	2	2	no	—
12	C₂²×C₄×C₃₇₃	C₂²	1	2	yes	—
13	C₂²×C₄×C₃₇₃	C₂²	1	1	yes	characteristic
14	C₂²×D₃₇₃	—	2	1	no	—
15	C₂²×D₃₇₃	—	2	1	no	—
16	C₂²×D₃₇₃	—	2	2	no	—
17	C₂²×D₃₇₃	—	2	2	no	—
18	C₂²×D₃₇₃	D₄	1	2	yes	—
19	C₂×Dic₃₇₃	—	2	2	no	—
20	C₂×Dic₃₇₃	—	2	1	no	—
21	C₂×Dic₃₇₃	—	2	2	no	—
22	C₂×Dic₃₇₃	—	2	2	no	—
23	C₂×Dic₃₇₃	—	2	2	no	—
24	C₂³×C₃₇₃	C₂³	1	1	yes	characteristic
25	C₂×C₄×C₃₇₃	—	2	2	no	—
26	C₂×C₄×C₃₇₃	—	2	2	no	—
27	C₂×C₄×C₃₇₃	—	2	2	no	—
28	C₂×C₄×C₃₇₃	—	2	2	no	—
29	C₂×C₄×C₃₇₃	D₄	1	2	yes	—
30	C₂²×C₃₇₃	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₃₇₃	C₂×D₄	1	1	yes	characteristic
32	C₂²×C₃₇₃	#41	1	2	yes	—
33	C₂²×C₃₇₃	#41	1	2	yes	—
34	C₂²×C₃₇₃	#41	1	1	yes	characteristic
35	C₂×D₃₇₃	—	2	2	no	—
36	C₂×D₃₇₃	—	2	4	no	—
37	C₂×D₃₇₃	—	4	1	no	—
38	C₂×D₃₇₃	—	4	1	no	—
39	C₂×D₃₇₃	—	4	2	no	—
40	C₄×C₃₇₃	—	2	2	no	—
41	C₄×C₃₇₃	—	4	2	no	—
42	Dic₃₇₃	—	4	2	no	—
43	Dic₃₇₃	—	4	1	no	—
44	C₂×C₃₇₃	#121	1	1	yes	characteristic
45	C₂×C₃₇₃	#123	1	2	yes	—
46	C₂×C₃₇₃	#123	1	1	yes	characteristic
47	C₂×C₃₇₃	#124	1	1	yes	characteristic
48	C₂×C₃₇₃	#126	1	2	yes	—
49	D₃₇₃	—	4	2	no	—
50	C₃₇₃	#397	1	1	yes	characteristic, complemented, Sylow
51	#397	—	373	1	no	maximal, Sylow
52	C₂×C₄⋊C₄	—	373	1	no	—
53	C₂×C₂²⋊C₄	—	373	2	no	—
54	C₂×C₂²⋊C₄	—	373	1	no	—
55	#95	—	373	2	no	—
56	#95	D₃₇₃	1	1	yes	characteristic, complemented
57	C₂⁴	—	373	1	no	—
58	C₂²×C₄	—	373	1	no	—
59	C₂²×C₄	—	373	2	no	—
60	C₂²×C₄	C₂×D₃₇₃	1	2	yes	—
61	C₂²×C₄	C₂×D₃₇₃	1	1	yes	characteristic
62	C₄⋊C₄	—	746	2	no	—
63	C₂²⋊C₄	—	746	4	no	—
64	C₂²⋊C₄	—	746	2	no	—
65	C₂³	—	373	2	no	—
66	C₂³	—	746	1	no	—
67	C₂³	—	746	2	no	—
68	C₂³	—	746	2	no	—
69	C₂³	—	746	1	no	—
70	C₂³	C₂²×D₃₇₃	1	1	yes	characteristic
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	746	2	no	—
76	C₂×C₄	—	746	1	no	—
77	C₂×C₄	—	746	2	no	—
78	C₂×C₄	—	746	2	no	—
79	C₂×C₄	—	746	2	no	—
80	C₂×C₄	D₁₄₉₂	1	2	yes	—
81	C₂²	—	1492	1	no	—
82	C₂²	—	1492	1	no	—
83	C₂²	—	1492	2	no	—
84	C₂²	—	746	2	no	—
85	C₂²	—	746	4	no	—
86	C₂²	D₄×D₃₇₃	1	1	yes	characteristic
87	C₂²	C₂×D₁₄₉₂	1	1	yes	characteristic
88	C₂²	#20238609	1	2	yes	—
89	C₂²	#20238610	1	2	yes	—
90	C₂²	#20238611	1	1	yes	characteristic
91	C₄	—	2	2	no	—
92	C₄	—	4	2	no	—
93	C₄	—	1492	2	no	—
94	C₄	—	1492	1	no	—
95	C₂	—	1492	2	no	—
96	C₂	#26045767	1	1	yes	characteristic
97	C₂	#26045774	1	2	yes	—
98	C₂	#26045776	1	1	yes	characteristic
99	C₂	#26045780	1	1	yes	characteristic
100	C₂	#26045787	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊Dic₃₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#20238587

C₂

yes

maximal, characteristic

C₂×#20238587

C₂

yes

maximal

#26045699

C₂

yes

maximal, complemented

#95×C₃₇₃

C₂

yes

maximal, characteristic, complemented

C₂³×D₃₇₃

C₂²

yes

characteristic

C₂²×Dic₃₇₃

C₂²

yes

characteristic

C₂²×Dic₃₇₃

C₂²

yes

—

C₄⋊Dic₃₇₃

—

#20238587

—

#20238587

—

C₂²×C₄×C₃₇₃

C₂²

yes

—

C₂²×C₄×C₃₇₃

C₂²

yes

characteristic

C₂²×D₃₇₃

—

C₂²×D₃₇₃

—

C₂²×D₃₇₃

—

C₂²×D₃₇₃

—

C₂²×D₃₇₃

D₄

yes

—

C₂×Dic₃₇₃

—

C₂×Dic₃₇₃

—

C₂×Dic₃₇₃

—

C₂×Dic₃₇₃

—

C₂×Dic₃₇₃

—

C₂³×C₃₇₃

C₂³

yes

characteristic

C₂×C₄×C₃₇₃

—

C₂×C₄×C₃₇₃

—

C₂×C₄×C₃₇₃

—

C₂×C₄×C₃₇₃

—

C₂×C₄×C₃₇₃

D₄

yes

—

C₂²×C₃₇₃

C₂×D₄

yes

characteristic

C₂²×C₃₇₃

C₂×D₄

yes

characteristic

C₂²×C₃₇₃

#41

yes

—

C₂²×C₃₇₃

#41

yes

—

C₂²×C₃₇₃

#41

yes

characteristic

C₂×D₃₇₃

—

C₂×D₃₇₃

—

C₂×D₃₇₃

—

C₂×D₃₇₃

—

C₂×D₃₇₃

—

C₄×C₃₇₃

—

C₄×C₃₇₃

—

Dic₃₇₃

—

Dic₃₇₃

—

C₂×C₃₇₃

#121

yes

characteristic

C₂×C₃₇₃

#123

yes

—

C₂×C₃₇₃

yes

C₂×C₃₇₃

yes

C₂×C₃₇₃

yes

—

D₃₇₃

—

C₃₇₃

#397

yes

characteristic, complemented, Sylow

#397

—

373

maximal, Sylow

C₂×C₄⋊C₄

—

373

—

C₂×C₂²⋊C₄

—

373

—

C₂×C₂²⋊C₄

—

373

—

#95

—

373

—

#95

D₃₇₃

yes

characteristic, complemented

C₂⁴

—

373

—

C₂²×C₄

—

373

—

C₂²×C₄

—

373

—

C₂²×C₄

C₂×D₃₇₃

yes

—

C₂²×C₄

C₂×D₃₇₃

yes

characteristic

C₄⋊C₄

—

746

—

C₂²⋊C₄

—

746

—

C₂²⋊C₄

—

746

—

C₂³

—

373

—

C₂³

—

746

—

C₂³

—

746

—

C₂³

—

746

—

C₂³

—

746

—

C₂³

C₂²×D₃₇₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

746

—

C₂×C₄

—

746

—

C₂×C₄

—

746

—

C₂×C₄

—

746

—

C₂×C₄

—

746

—

C₂×C₄

D₁₄₉₂

yes

—

C₂²

—

1492

—

C₂²

—

1492

—

C₂²

—

1492

—

C₂²

—

746

—

C₂²

—

746

—

C₂²

D₄×D₃₇₃

yes

characteristic

C₂²

C₂×D₁₄₉₂

yes

characteristic

C₂²

#20238609

yes

—

C₂²

#20238610

yes

—

C₂²

#20238611

yes

characteristic

C₄

—

C₄

—

C₄

—

1492

—

C₄

—

1492

—

C₂

—

1492

—

C₂

#26045767

yes

characteristic

C₂

#26045774

yes

—

C₂

yes

C₂

yes

100

C₂

#26045787

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₃₇₃
commutator, socle, cosocle	C₂³×C₃₇₃	C₂³
Fitting	#95×C₃₇₃	C₂
prime radical	C₂³×D₃₇₃	C₂²
hypercenter	#95	D₃₇₃
hypocenter	C₃₇₃	#397
LCS	G ▷ C₂³×C₃₇₃ ▷ C₃₇₃
UCS	C₁ ◁ C₂³ ◁ #95

subgroup

G/H

center, Frattini

C₂³

C₂²×D₃₇₃

commutator, socle, cosocle

C₂³×C₃₇₃

C₂³

Fitting

#95×C₃₇₃

C₂

prime radical

C₂³×D₃₇₃

C₂²

D₃₇₃

C₃₇₃

G ▷ C₂³×C₃₇₃ ▷ C₃₇₃

C₁ ◁ C₂³ ◁ #95

N	H ≅ G/N	type
1. C₂×C₄⋊Dic₃₇₃	C₂	semidirect
4. #26045699	C₂	semidirect
5. #95×C₃₇₃	C₂	semidirect
50. C₃₇₃	#397	semidirect
56. #95	D₃₇₃	semidirect

H ≅ G/N

type

1. C₂×C₄⋊Dic₃₇₃

C₂

semidirect

4. #26045699

C₂

semidirect

5. #95×C₃₇₃

C₂

semidirect

50. C₃₇₃

#397

semidirect

56. #95

D₃₇₃

semidirect

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

subgroup	index
C₂×C₄⋊Dic₃₇₃	2
C₂×#20238587	2
#26045699	2
#95×C₃₇₃	2
#397	373

subgroup

index

C₂×C₄⋊Dic₃₇₃

C₂×#20238587

#26045699

#95×C₃₇₃

#397

373

Subgroups — #37559349

Conjugacy classes of subgroups

Complements

Maximal subgroups