Group #37559991 — subgroups

Subgroups — #37559991

Order 23872.

334 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 111 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×#20238610	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₁₄₉₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#26045789	C₂	1	2	yes	maximal, complemented
4	#26045780	C₂	1	2	yes	maximal, complemented
5	#26045767	C₂	1	4	yes	maximal, complemented
6	#26045762	C₂	1	4	yes	maximal, complemented
7	#117×C₃₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₃₇₃	C₂²	1	2	yes	—
9	#20238610	—	2	4	no	—
10	C₂×Dic₇₄₆	C₂²	1	1	yes	characteristic, complemented
11	C₂×D₁₄₉₂	—	2	4	no	—
12	C₂×D₁₄₉₂	C₂²	1	2	yes	—
13	C₂×D₁₄₉₂	C₂²	1	2	yes	—
14	C₂×D₁₄₉₂	C₂²	1	1	yes	characteristic, complemented
15	C₂×D₁₄₉₂	C₂²	1	8	yes	complemented
16	C₂×C₂²⋊D₃₇₃	C₂²	1	2	yes	—
17	C₂×C₄×D₃₇₃	C₂²	1	2	yes	complemented
18	#20238587	C₂²	1	8	yes	complemented
19	C₂²×C₄×C₃₇₃	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₃₇₃	C₂²	1	2	yes	complemented
21	C₂²⋊C₄×C₃₇₃	C₂²	1	2	yes	—
22	C₄²×C₃₇₃	C₂²	1	2	yes	complemented
23	C₂²×D₃₇₃	—	2	4	no	—
24	C₂²×D₃₇₃	—	4	4	no	—
25	C₂²×D₃₇₃	C₂³	1	2	yes	—
26	C₂²×D₃₇₃	C₂³	1	4	yes	—
27	Dic₇₄₆	—	2	2	no	—
28	D₁₄₉₂	—	2	4	no	—
29	D₁₄₉₂	—	2	4	no	—
30	D₁₄₉₂	—	2	2	no	—
31	D₁₄₉₂	—	4	8	no	—
32	C₂²⋊D₃₇₃	—	2	4	no	—
33	C₂×Dic₃₇₃	C₂³	1	2	yes	—
34	C₄×D₃₇₃	—	2	4	no	—
35	C₂³×C₃₇₃	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₃₇₃	D₄	1	4	yes	complemented
37	C₂×C₄×C₃₇₃	C₂³	1	4	yes	—
38	C₂×C₄×C₃₇₃	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₃₇₃	C₂³	1	1	yes	characteristic
40	C₂²×C₃₇₃	—	2	2	no	—
41	C₂²×C₃₇₃	C₂×D₄	1	2	yes	—
42	C₂²×C₃₇₃	C₂⁴	1	1	yes	characteristic
43	C₂×D₃₇₃	—	2	2	no	—
44	C₂×D₃₇₃	—	2	4	no	—
45	C₂×D₃₇₃	—	2	4	no	—
46	C₂×D₃₇₃	—	2	8	no	—
47	C₂×D₃₇₃	—	4	8	no	—
48	C₄×C₃₇₃	—	2	4	no	—
49	C₄×C₃₇₃	C₂×D₄	1	2	yes	—
50	C₄×C₃₇₃	C₂×D₄	1	2	yes	—
51	Dic₃₇₃	—	2	2	no	—
52	C₂×C₃₇₃	—	2	2	no	—
53	C₂×C₃₇₃	C₂²×D₄	1	1	yes	characteristic
54	C₂×C₃₇₃	H(5,𝔽₂)	1	2	yes	—
55	D₃₇₃	—	4	2	no	—
56	D₃₇₃	—	4	4	no	—
57	C₃₇₃	#535	1	1	yes	characteristic, complemented, Sylow
58	#535	—	373	1	no	maximal, Sylow
59	C₂×#41	—	373	1	no	—
60	C₂²×D₄	—	373	1	no	—
61	#127	—	373	2	no	—
62	#124	—	373	2	no	—
63	#121	—	373	4	no	—
64	#120	—	373	4	no	—
65	#117	D₃₇₃	1	1	yes	characteristic, complemented
66	C₂⁴	—	373	2	no	—
67	#41	—	746	4	no	—
68	C₂×Q₈	—	373	1	no	—
69	C₂×D₄	—	373	2	no	—
70	C₂×D₄	—	373	8	no	—
71	C₂×D₄	—	373	2	no	—
72	C₂×D₄	—	373	2	no	—
73	C₂×D₄	—	373	1	no	—
74	C₂×D₄	—	746	4	no	—
75	C₂²×C₄	—	373	2	no	—
76	C₂²×C₄	C₂×D₃₇₃	1	1	yes	characteristic
77	C₄⋊C₄	C₂×D₃₇₃	1	2	yes	complemented
78	C₂²⋊C₄	—	373	8	no	—
79	C₂²⋊C₄	C₂×D₃₇₃	1	2	yes	—
80	C₄²	C₂×D₃₇₃	1	2	yes	complemented
81	C₂³	—	373	4	no	—
82	C₂³	—	373	2	no	—
83	C₂³	—	1492	4	no	—
84	C₂³	—	746	4	no	—
85	C₂³	C₂²×D₃₇₃	1	1	yes	characteristic
86	Q₈	—	746	2	no	—
87	D₄	—	1492	8	no	—
88	D₄	—	746	4	no	—
89	D₄	—	746	4	no	—
90	D₄	—	746	4	no	—
91	D₄	—	746	2	no	—
92	C₂×C₄	—	373	2	no	—
93	C₂×C₄	—	746	4	no	—
94	C₂×C₄	D₁₄₉₂	1	4	yes	complemented
95	C₂×C₄	C₂²×D₃₇₃	1	4	yes	—
96	C₂×C₄	C₂²×D₃₇₃	1	1	yes	characteristic
97	C₂×C₄	C₂²×D₃₇₃	1	1	yes	characteristic
98	C₂²	—	2	2	no	—
99	C₂²	—	1492	8	no	—
100	C₂²	—	746	2	no	—
101	C₂²	—	746	4	no	—
102	C₂²	—	746	4	no	—
103	C₂²	—	746	8	no	—
104	C₂²	C₂×D₁₄₉₂	1	2	yes	—
105	C₂²	C₂³×D₃₇₃	1	1	yes	characteristic
106	C₄	—	2	4	no	—
107	C₄	—	746	2	no	—
108	C₄	C₂×D₁₄₉₂	1	2	yes	—
109	C₄	C₂×D₁₄₉₂	1	2	yes	—
110	C₂	—	2	2	no	—
111	C₂	—	1492	4	no	—
112	C₂	—	1492	2	no	—
113	C₂	C₂²×D₁₄₉₂	1	1	yes	characteristic
114	C₂	#26045838	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20238610

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₄₉₂

C₂

yes

maximal, characteristic, complemented

#26045789

C₂

yes

maximal, complemented

#26045780

C₂

yes

maximal, complemented

#26045767

C₂

yes

maximal, complemented

#26045762

C₂

yes

maximal, complemented

#117×C₃₇₃

C₂

yes

maximal, characteristic, complemented

C₂³×D₃₇₃

C₂²

yes

—

#20238610

—

C₂×Dic₇₄₆

C₂²

yes

characteristic, complemented

C₂×D₁₄₉₂

—

C₂×D₁₄₉₂

C₂²

yes

—

C₂×D₁₄₉₂

C₂²

yes

—

C₂×D₁₄₉₂

C₂²

yes

characteristic, complemented

C₂×D₁₄₉₂

C₂²

yes

complemented

C₂×C₂²⋊D₃₇₃

C₂²

yes

—

C₂×C₄×D₃₇₃

C₂²

yes

complemented

#20238587

C₂²

yes

complemented

C₂²×C₄×C₃₇₃

C₂²

yes

characteristic

C₄⋊C₄×C₃₇₃

C₂²

yes

complemented

C₂²⋊C₄×C₃₇₃

C₂²

yes

—

C₄²×C₃₇₃

C₂²

yes

complemented

C₂²×D₃₇₃

—

C₂²×D₃₇₃

—

C₂²×D₃₇₃

C₂³

yes

—

C₂²×D₃₇₃

C₂³

yes

—

Dic₇₄₆

—

D₁₄₉₂

—

D₁₄₉₂

—

D₁₄₉₂

—

D₁₄₉₂

—

C₂²⋊D₃₇₃

—

C₂×Dic₃₇₃

C₂³

yes

—

C₄×D₃₇₃

—

C₂³×C₃₇₃

C₂³

yes

characteristic

C₂×C₄×C₃₇₃

D₄

yes

complemented

C₂×C₄×C₃₇₃

C₂³

yes

—

C₂×C₄×C₃₇₃

C₂³

yes

characteristic

C₂×C₄×C₃₇₃

C₂³

yes

characteristic

C₂²×C₃₇₃

—

C₂²×C₃₇₃

C₂×D₄

yes

—

C₂²×C₃₇₃

C₂⁴

yes

characteristic

C₂×D₃₇₃

—

C₂×D₃₇₃

—

C₂×D₃₇₃

—

C₂×D₃₇₃

—

C₂×D₃₇₃

—

C₄×C₃₇₃

—

C₄×C₃₇₃

C₂×D₄

yes

—

C₄×C₃₇₃

C₂×D₄

yes

—

Dic₃₇₃

—

C₂×C₃₇₃

—

C₂×C₃₇₃

C₂²×D₄

yes

characteristic

C₂×C₃₇₃

H(5,𝔽₂)

yes

—

D₃₇₃

—

D₃₇₃

—

C₃₇₃

#535

yes

characteristic, complemented, Sylow

#535

—

373

maximal, Sylow

C₂×#41

—

373

—

C₂²×D₄

—

373

—

#127

—

373

—

#124

—

373

—

#121

—

373

—

#120

—

373

—

#117

D₃₇₃

yes

characteristic, complemented

C₂⁴

—

373

—

#41

—

746

—

C₂×Q₈

—

373

—

C₂×D₄

—

373

—

C₂×D₄

—

373

—

C₂×D₄

—

373

—

C₂×D₄

—

373

—

C₂×D₄

—

373

—

C₂×D₄

—

746

—

C₂²×C₄

—

373

—

C₂²×C₄

C₂×D₃₇₃

yes

characteristic

C₄⋊C₄

C₂×D₃₇₃

yes

complemented

C₂²⋊C₄

—

373

—

C₂²⋊C₄

C₂×D₃₇₃

yes

—

C₄²

C₂×D₃₇₃

yes

complemented

C₂³

—

373

—

C₂³

—

373

—

C₂³

—

1492

—

C₂³

—

746

—

C₂³

C₂²×D₃₇₃

yes

characteristic

Q₈

—

746

—

D₄

—

1492

—

D₄

—

746

—

D₄

—

746

—

D₄

—

746

—

D₄

—

746

—

C₂×C₄

—

373

—

C₂×C₄

—

746

—

C₂×C₄

D₁₄₉₂

yes

complemented

C₂×C₄

C₂²×D₃₇₃

yes

—

C₂×C₄

C₂²×D₃₇₃

yes

characteristic

C₂×C₄

C₂²×D₃₇₃

yes

characteristic

C₂²

—

C₂²

—

1492

—

100

C₂²

—

746

—

101

C₂²

—

746

—

102

C₂²

—

746

—

103

C₂²

—

746

—

104

C₂²

C₂×D₁₄₉₂

yes

—

105

C₂²

C₂³×D₃₇₃

yes

characteristic

106

C₄

—

107

C₄

—

746

—

108

C₄

C₂×D₁₄₉₂

yes

—

109

C₄

C₂×D₁₄₉₂

yes

—

110

C₂

—

111

C₂

—

1492

—

112

C₂

—

1492

—

113

C₂

C₂²×D₁₄₉₂

yes

characteristic

114

C₂

#26045838

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₇₃
commutator, socle, cosocle	C₂²×C₃₇₃	C₂⁴
Fitting	#117×C₃₇₃	C₂
hypercenter	#117	D₃₇₃
hypocenter	C₃₇₃	#535
LCS	G ▷ C₂²×C₃₇₃ ▷ C₃₇₃
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₇₃

commutator, socle, cosocle

C₂²×C₃₇₃

C₂⁴

Fitting

#117×C₃₇₃

C₂

D₃₇₃

C₃₇₃

G ▷ C₂²×C₃₇₃ ▷ C₃₇₃

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
1. C₂×#20238610	C₂	semidirect
2. C₂²×D₁₄₉₂	C₂	semidirect
3. #26045789	C₂	semidirect
4. #26045780	C₂	semidirect
5. #26045767	C₂	semidirect
6. #26045762	C₂	semidirect
7. #117×C₃₇₃	C₂	semidirect
10. C₂×Dic₇₄₆	C₂²	semidirect
14. C₂×D₁₄₉₂	C₂²	semidirect
15. C₂×D₁₄₉₂	C₂²	semidirect
17. C₂×C₄×D₃₇₃	C₂²	semidirect
18. #20238587	C₂²	semidirect
20. C₄⋊C₄×C₃₇₃	C₂²	semidirect
22. C₄²×C₃₇₃	C₂²	semidirect
36. C₂×C₄×C₃₇₃	D₄	semidirect
57. C₃₇₃	#535	semidirect
65. #117	D₃₇₃	semidirect
77. C₄⋊C₄	C₂×D₃₇₃	semidirect
80. C₄²	C₂×D₃₇₃	semidirect
94. C₂×C₄	D₁₄₉₂	semidirect

H ≅ G/N

type

1. C₂×#20238610

C₂

semidirect

2. C₂²×D₁₄₉₂

C₂

semidirect

3. #26045789

C₂

semidirect

4. #26045780

C₂

semidirect

5. #26045767

C₂

semidirect

6. #26045762

C₂

semidirect

7. #117×C₃₇₃

C₂

semidirect

10. C₂×Dic₇₄₆

C₂²

semidirect

14. C₂×D₁₄₉₂

C₂²

semidirect

15. C₂×D₁₄₉₂

C₂²

semidirect

17. C₂×C₄×D₃₇₃

C₂²

semidirect

18. #20238587

C₂²

semidirect

20. C₄⋊C₄×C₃₇₃

C₂²

semidirect

22. C₄²×C₃₇₃

C₂²

semidirect

36. C₂×C₄×C₃₇₃

D₄

semidirect

57. C₃₇₃

#535

semidirect

65. #117

D₃₇₃

semidirect

77. C₄⋊C₄

C₂×D₃₇₃

semidirect

80. C₄²

C₂×D₃₇₃

semidirect

94. C₂×C₄

D₁₄₉₂

semidirect

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

subgroup	index
C₂×#20238610	2
C₂²×D₁₄₉₂	2
#26045789	2
#26045780	2
#26045767	2
#26045762	2
#117×C₃₇₃	2
#535	373

subgroup

index

C₂²×D₁₄₉₂

#117×C₃₇₃

373

Subgroups — #37559991

Conjugacy classes of subgroups

Complements

Maximal subgroups