Group #37559340 — subgroups

Subgroups — #37559340

Order 23872.

238 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 61 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×D₁₄₉₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#20238589	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#20238587	C₂	1	2	yes	maximal, complemented
4	C₂×#20238587	C₂	1	2	yes	maximal, complemented
5	#95×C₃₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₃₇₃	C₂²	1	2	yes	—
7	C₂×D₁₄₉₂	—	2	4	no	—
8	C₂×D₁₄₉₂	—	2	2	no	—
9	C₂²×Dic₃₇₃	C₂²	1	1	yes	characteristic
10	C₂²×Dic₃₇₃	C₂²	1	1	yes	characteristic
11	#20238589	—	2	2	no	—
12	#20238587	—	2	4	no	—
13	#20238587	—	2	4	no	—
14	C₂²×C₄×C₃₇₃	C₂²	1	2	yes	—
15	C₂²×C₄×C₃₇₃	C₂²	1	1	yes	characteristic
16	C₂²×D₃₇₃	—	2	2	no	—
17	C₂²×D₃₇₃	—	2	2	no	—
18	C₂²×D₃₇₃	—	2	2	no	—
19	C₂²×D₃₇₃	—	2	2	no	—
20	C₂²×D₃₇₃	—	2	2	no	—
21	C₂²×D₃₇₃	—	2	2	no	—
22	C₂²×D₃₇₃	D₄	1	4	yes	—
23	D₁₄₉₂	—	2	8	no	—
24	C₂×Dic₃₇₃	—	2	1	no	—
25	C₂×Dic₃₇₃	—	2	1	no	—
26	C₂×Dic₃₇₃	—	2	2	no	—
27	C₂×Dic₃₇₃	—	2	1	no	—
28	C₂×Dic₃₇₃	D₄	1	2	yes	complemented
29	C₂³×C₃₇₃	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₃₇₃	—	2	2	no	—
31	C₂×C₄×C₃₇₃	—	2	2	no	—
32	C₂×C₄×C₃₇₃	—	2	2	no	—
33	C₂×C₄×C₃₇₃	—	2	2	no	—
34	C₂×C₄×C₃₇₃	D₄	1	2	yes	—
35	C₂²×C₃₇₃	C₂×D₄	1	2	yes	—
36	C₂²×C₃₇₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₃₇₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₃₇₃	#41	1	2	yes	—
39	C₂²×C₃₇₃	#41	1	1	yes	characteristic
40	C₂×D₃₇₃	—	2	4	no	—
41	C₂×D₃₇₃	—	2	4	no	—
42	C₂×D₃₇₃	—	2	4	no	—
43	C₂×D₃₇₃	—	4	2	no	—
44	C₂×D₃₇₃	—	4	2	no	—
45	C₂×D₃₇₃	—	4	2	no	—
46	C₂×D₃₇₃	—	4	2	no	—
47	C₄×C₃₇₃	—	2	2	no	—
48	C₄×C₃₇₃	—	4	2	no	—
49	Dic₃₇₃	—	2	2	no	—
50	Dic₃₇₃	—	4	1	no	—
51	C₂×C₃₇₃	#120	1	1	yes	characteristic
52	C₂×C₃₇₃	#121	1	2	yes	—
53	C₂×C₃₇₃	#121	1	1	yes	characteristic
54	C₂×C₃₇₃	#123	1	2	yes	—
55	C₂×C₃₇₃	#124	1	1	yes	characteristic
56	D₃₇₃	—	4	4	no	—
57	C₃₇₃	#394	1	1	yes	characteristic, complemented, Sylow
58	#394	—	373	1	no	maximal, Sylow
59	C₂²×D₄	—	373	1	no	—
60	C₂×C₄⋊C₄	—	373	1	no	—
61	C₂×C₂²⋊C₄	—	373	2	no	—
62	C₂×C₂²⋊C₄	—	373	2	no	—
63	#95	D₃₇₃	1	1	yes	characteristic, complemented
64	C₂⁴	—	373	2	no	—
65	C₂×D₄	—	746	4	no	—
66	C₂×D₄	—	746	2	no	—
67	C₂²×C₄	—	373	1	no	—
68	C₂²×C₄	—	373	1	no	—
69	C₂²×C₄	C₂×D₃₇₃	1	2	yes	—
70	C₂²×C₄	C₂×D₃₇₃	1	1	yes	characteristic
71	C₄⋊C₄	—	746	2	no	—
72	C₂²⋊C₄	—	746	4	no	—
73	C₂²⋊C₄	—	746	4	no	—
74	C₂³	—	373	4	no	—
75	C₂³	—	746	2	no	—
76	C₂³	—	746	2	no	—
77	C₂³	—	746	2	no	—
78	C₂³	—	746	2	no	—
79	C₂³	—	746	2	no	—
80	C₂³	—	746	2	no	—
81	C₂³	C₂²×D₃₇₃	1	1	yes	characteristic
82	D₄	—	746	8	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	373	2	no	—
88	C₂×C₄	—	746	2	no	—
89	C₂×C₄	—	746	1	no	—
90	C₂×C₄	—	746	1	no	—
91	C₂×C₄	—	746	1	no	—
92	C₂×C₄	D₁₄₉₂	1	2	yes	—
93	C₂²	—	1492	2	no	—
94	C₂²	—	1492	2	no	—
95	C₂²	—	1492	2	no	—
96	C₂²	—	1492	2	no	—
97	C₂²	—	746	4	no	—
98	C₂²	—	746	4	no	—
99	C₂²	—	746	4	no	—
100	C₂²	D₄×D₃₇₃	1	2	yes	—
101	C₂²	D₄×D₃₇₃	1	1	yes	characteristic
102	C₂²	C₂×D₁₄₉₂	1	1	yes	characteristic
103	C₂²	#20238610	1	2	yes	—
104	C₂²	#20238611	1	1	yes	characteristic
105	C₄	—	2	2	no	—
106	C₄	—	4	2	no	—
107	C₄	—	1492	1	no	—
108	C₄	—	746	2	no	—
109	C₂	—	1492	4	no	—
110	C₂	#26045762	1	1	yes	characteristic
111	C₂	#26045765	1	2	yes	—
112	C₂	#26045767	1	1	yes	characteristic
113	C₂	#26045777	1	2	yes	—
114	C₂	#26045780	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₁₄₉₂

C₂

yes

maximal, characteristic

C₂×#20238589

C₂

yes

maximal, characteristic, complemented

C₂×#20238587

C₂

yes

maximal, complemented

C₂×#20238587

C₂

yes

maximal, complemented

#95×C₃₇₃

C₂

yes

maximal, characteristic, complemented

C₂³×D₃₇₃

C₂²

yes

—

C₂×D₁₄₉₂

—

C₂×D₁₄₉₂

—

C₂²×Dic₃₇₃

C₂²

yes

characteristic

C₂²×Dic₃₇₃

C₂²

yes

—

—

—

C₂²×C₄×C₃₇₃

C₂²

yes

—

C₂²×C₄×C₃₇₃

C₂²

yes

characteristic

C₂²×D₃₇₃

—

C₂²×D₃₇₃

—

C₂²×D₃₇₃

—

C₂²×D₃₇₃

—

C₂²×D₃₇₃

—

C₂²×D₃₇₃

—

C₂²×D₃₇₃

D₄

yes

—

D₁₄₉₂

—

C₂×Dic₃₇₃

—

C₂×Dic₃₇₃

—

C₂×Dic₃₇₃

—

C₂×Dic₃₇₃

—

C₂×Dic₃₇₃

D₄

yes

complemented

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

—

C₂²×C₃₇₃

C₂×D₄

yes

characteristic

C₂²×C₃₇₃

C₂×D₄

yes

characteristic

C₂²×C₃₇₃

#41

yes

—

C₂²×C₃₇₃

#41

yes

characteristic

C₂×D₃₇₃

—

C₂×D₃₇₃

—

C₂×D₃₇₃

—

C₂×D₃₇₃

—

C₂×D₃₇₃

—

C₂×D₃₇₃

—

C₂×D₃₇₃

—

C₄×C₃₇₃

—

C₄×C₃₇₃

—

Dic₃₇₃

—

Dic₃₇₃

—

C₂×C₃₇₃

#120

yes

characteristic

C₂×C₃₇₃

#121

yes

—

C₂×C₃₇₃

#121

yes

characteristic

C₂×C₃₇₃

#123

yes

—

C₂×C₃₇₃

#124

yes

characteristic

D₃₇₃

—

C₃₇₃

#394

yes

characteristic, complemented, Sylow

#394

—

373

maximal, Sylow

C₂²×D₄

—

373

—

C₂×C₄⋊C₄

—

373

—

C₂×C₂²⋊C₄

—

373

—

C₂×C₂²⋊C₄

—

373

—

#95

D₃₇₃

yes

characteristic, complemented

C₂⁴

—

373

—

C₂×D₄

—

746

—

C₂×D₄

—

746

—

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

—

746

—

C₂³

—

746

—

C₂³

C₂²×D₃₇₃

yes

characteristic

D₄

—

746

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

373

—

C₂×C₄

—

746

—

C₂×C₄

—

746

—

C₂×C₄

—

746

—

C₂×C₄

—

746

—

C₂×C₄

D₁₄₉₂

yes

—

C₂²

—

1492

—

C₂²

—

1492

—

C₂²

—

1492

—

C₂²

—

1492

—

C₂²

—

746

—

C₂²

—

746

—

C₂²

—

746

—

100

C₂²

D₄×D₃₇₃

yes

—

101

C₂²

D₄×D₃₇₃

yes

characteristic

102

C₂²

C₂×D₁₄₉₂

yes

characteristic

103

C₂²

#20238610

yes

—

104

C₂²

#20238611

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

1492

—

108

C₄

—

746

—

109

C₂

—

1492

—

110

C₂

#26045762

yes

characteristic

111

C₂

#26045765

yes

—

112

C₂

#26045767

yes

characteristic

113

C₂

#26045777

yes

—

114

C₂

#26045780

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₃₇₃
commutator, socle, cosocle	C₂³×C₃₇₃	C₂³
Fitting	#95×C₃₇₃	C₂
prime radical	C₂²×D₁₄₉₂	C₂
hypercenter	#95	D₃₇₃
hypocenter	C₃₇₃	#394
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

C₂²×D₁₄₉₂

C₂

D₃₇₃

C₃₇₃

G ▷ C₂³×C₃₇₃ ▷ C₃₇₃

C₁ ◁ C₂³ ◁ #95

N	H ≅ G/N	type
2. C₂×#20238589	C₂	semidirect
3. C₂×#20238587	C₂	semidirect
4. C₂×#20238587	C₂	semidirect
5. #95×C₃₇₃	C₂	semidirect
28. C₂×Dic₃₇₃	D₄	semidirect
57. C₃₇₃	#394	semidirect
63. #95	D₃₇₃	semidirect

H ≅ G/N

type

2. C₂×#20238589

C₂

semidirect

3. C₂×#20238587

C₂

semidirect

4. C₂×#20238587

C₂

semidirect

5. #95×C₃₇₃

C₂

semidirect

28. C₂×Dic₃₇₃

D₄

semidirect

57. C₃₇₃

#394

semidirect

63. #95

D₃₇₃

semidirect

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

subgroup	index
C₂²×D₁₄₉₂	2
C₂×#20238589	2
C₂×#20238587	2
#95×C₃₇₃	2
#394	373

subgroup

index

C₂²×D₁₄₉₂

C₂×#20238589

C₂×#20238587

#95×C₃₇₃

#394

373

Subgroups — #37559340

Conjugacy classes of subgroups

Complements

Maximal subgroups