Group #14583570 — subgroups

Subgroups — #14583570

Order 2752.

312 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 107 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×#10621898	C₂	1	2	yes	maximal, complemented
2	#13063322	C₂	1	1	yes	maximal, characteristic, complemented
3	#13063321	C₂	1	2	yes	maximal, complemented
4	#13063299	C₂	1	2	yes	maximal, complemented
5	#13063298	C₂	1	4	yes	maximal, complemented
6	C₄×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#13063287	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×D₄×C₄₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₄₃	C₂²	1	2	yes	—
11	C₂×C₂²⋊D₄₃	C₂²	1	4	yes	—
12	C₂×C₂²⋊D₄₃	C₂²	1	4	yes	—
13	#10621898	—	2	8	no	—
14	C₂×D₁₇₂	C₂²	1	2	yes	—
15	C₂×D₁₇₂	C₂²	1	2	yes	—
16	C₂×D₁₇₂	C₂²	1	2	yes	—
17	C₂×C₄×D₄₃	C₂²	1	4	yes	—
18	C₂×Dic₈₆	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
20	C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
21	C₄²×C₄₃	C₂²	1	1	yes	characteristic
22	(C₂×D₄₃)⋊C₄	C₂²	1	4	yes	—
23	C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
24	Dic₄₃⋊C₄	C₂²	1	2	yes	—
25	C₄×Dic₄₃	C₂²	1	2	yes	—
26	C₂³×C₄₃	C₂³	1	2	yes	—
27	C₂²×D₄₃	C₂³	1	4	yes	—
28	D₄×C₄₃	—	2	2	no	—
29	C₂×C₄×C₄₃	—	2	4	no	—
30	C₂×C₄×C₄₃	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₄₃	C₂³	1	2	yes	—
32	C₂×C₄×C₄₃	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₄₃	C₂³	1	1	yes	characteristic
34	C₂²⋊D₄₃	—	2	8	no	—
35	C₂²⋊D₄₃	—	4	4	no	—
36	C₂×Dic₄₃	C₂³	1	2	yes	—
37	C₂×Dic₄₃	C₂³	1	2	yes	—
38	D₁₇₂	—	2	4	no	—
39	D₁₇₂	—	2	4	no	—
40	D₁₇₂	—	4	2	no	—
41	C₄×D₄₃	—	2	8	no	—
42	Dic₈₆	D₄	1	4	yes	complemented
43	C₂²×C₄₃	—	2	2	no	—
44	C₂²×C₄₃	—	2	2	no	—
45	C₂²×C₄₃	—	2	2	no	—
46	C₂²×C₄₃	C₂⁴	1	1	yes	characteristic
47	C₂×D₄₃	—	2	4	no	—
48	C₂×D₄₃	—	2	4	no	—
49	C₂×D₄₃	—	2	4	no	—
50	C₄×C₄₃	—	2	2	no	—
51	C₄×C₄₃	—	2	1	no	—
52	C₄×C₄₃	C₂×D₄	1	2	yes	—
53	C₄×C₄₃	#41	1	2	yes	complemented
54	Dic₄₃	—	2	2	no	—
55	Dic₄₃	C₂×D₄	1	4	yes	—
56	C₂×C₄₃	—	4	2	no	—
57	C₂×C₄₃	C₂²×D₄	1	1	yes	characteristic
58	C₂×C₄₃	C₂×#41	1	1	yes	characteristic
59	C₂×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
60	D₄₃	—	4	4	no	—
61	#550	—	43	1	no	maximal, Sylow
62	C₄₃	#550	1	1	yes	characteristic, complemented, Sylow
63	C₂×#41	—	43	2	no	—
64	#127	—	43	2	no	—
65	#127	—	43	1	no	—
66	#121	—	43	2	no	—
67	#121	—	43	4	no	—
68	C₄×Q₈	—	43	1	no	—
69	C₄×D₄	—	43	2	no	—
70	C₄×D₄	D₄₃	1	1	yes	characteristic, complemented
71	#41	—	86	8	no	—
72	C₂×Q₈	—	43	1	no	—
73	C₂×D₄	—	43	4	no	—
74	C₂×D₄	—	43	4	no	—
75	C₂×D₄	—	43	2	no	—
76	C₂×D₄	—	43	2	no	—
77	C₂×D₄	—	43	2	no	—
78	C₂×D₄	C₂×D₄₃	1	1	yes	characteristic
79	C₂²×C₄	—	43	4	no	—
80	C₂²×C₄	C₂×D₄₃	1	2	yes	—
81	C₄⋊C₄	—	43	2	no	—
82	C₄⋊C₄	—	43	1	no	—
83	C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
84	C₂²⋊C₄	—	43	4	no	—
85	C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
86	C₄²	—	43	2	no	—
87	C₄²	C₂×D₄₃	1	1	yes	characteristic
88	C₂³	—	43	4	no	—
89	C₂³	C₂²×D₄₃	1	2	yes	—
90	Q₈	—	43	4	no	—
91	D₄	—	2	2	no	—
92	D₄	—	86	8	no	—
93	D₄	—	86	4	no	—
94	D₄	—	86	4	no	—
95	D₄	—	172	2	no	—
96	D₄	—	172	4	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	43	2	no	—
99	C₂×C₄	—	43	2	no	—
100	C₂×C₄	—	86	8	no	—
101	C₂×C₄	C₂²×D₄₃	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₄₃	1	2	yes	—
103	C₂×C₄	C₂²×D₄₃	1	1	yes	characteristic
104	C₂×C₄	C₂²×D₄₃	1	1	yes	characteristic
105	C₂²	—	2	2	no	—
106	C₂²	—	2	2	no	—
107	C₂²	—	2	2	no	—
108	C₂²	—	86	4	no	—
109	C₂²	—	86	4	no	—
110	C₂²	—	86	4	no	—
111	C₂²	C₂³×D₄₃	1	1	yes	characteristic
112	C₄	—	2	2	no	—
113	C₄	—	2	1	no	—
114	C₄	—	43	4	no	—
115	C₄	—	86	2	no	—
116	C₄	#10621898	1	2	yes	complemented
117	C₄	D₄×D₄₃	1	2	yes	—
118	C₂	—	4	2	no	—
119	C₂	—	172	4	no	—
120	C₂	C₂×D₄×D₄₃	1	1	yes	characteristic
121	C₂	C₂×#10621898	1	1	yes	characteristic
122	C₂	#13063371	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10621898

C₂

yes

maximal, complemented

#13063322

C₂

yes

maximal, characteristic, complemented

#13063321

C₂

yes

maximal, complemented

#13063299

C₂

yes

maximal, complemented

#13063298

C₂

yes

maximal, complemented

C₄×Dic₈₆

C₂

yes

maximal, characteristic, complemented

#13063287

C₂

yes

maximal, complemented

C₄×D₄×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂×D₄×C₄₃

C₂²

yes

characteristic

C₂²×C₄×C₄₃

C₂²

yes

—

C₂×C₂²⋊D₄₃

C₂²

yes

—

C₂×C₂²⋊D₄₃

C₂²

yes

—

#10621898

—

C₂×D₁₇₂

C₂²

yes

—

C₂×D₁₇₂

C₂²

yes

—

C₂×D₁₇₂

C₂²

yes

—

C₂×C₄×D₄₃

C₂²

yes

—

C₂×Dic₈₆

C₂²

yes

characteristic

C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₂²⋊C₄×C₄₃

C₂²

yes

—

C₄²×C₄₃

C₂²

yes

characteristic

(C₂×D₄₃)⋊C₄

C₂²

yes

—

C₄⋊Dic₄₃

C₂²

yes

characteristic

Dic₄₃⋊C₄

C₂²

yes

—

C₄×Dic₄₃

C₂²

yes

—

C₂³×C₄₃

C₂³

yes

—

C₂²×D₄₃

C₂³

yes

—

D₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂³

yes

characteristic

C₂×C₄×C₄₃

C₂³

yes

—

C₂×C₄×C₄₃

C₂³

yes

characteristic

C₂×C₄×C₄₃

C₂³

yes

characteristic

C₂²⋊D₄₃

—

C₂²⋊D₄₃

—

C₂×Dic₄₃

C₂³

yes

—

C₂×Dic₄₃

C₂³

yes

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

Dic₈₆

D₄

yes

complemented

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂⁴

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×D₄

yes

—

C₄×C₄₃

#41

yes

complemented

Dic₄₃

—

Dic₄₃

C₂×D₄

yes

—

C₂×C₄₃

—

C₂×C₄₃

C₂²×D₄

yes

characteristic

C₂×C₄₃

C₂×#41

yes

characteristic

C₂×C₄₃

H(5,𝔽₂)

yes

characteristic

D₄₃

—

#550

—

maximal, Sylow

C₄₃

#550

yes

characteristic, complemented, Sylow

C₂×#41

—

—

—

—

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

D₄₃

yes

characteristic, complemented

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₄₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×D₄₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₄₃

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₄₃

yes

—

C₄²

—

C₄²

C₂×D₄₃

yes

characteristic

C₂³

—

C₂³

C₂²×D₄₃

yes

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

172

—

D₄

—

172

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₂²×D₄₃

yes

characteristic

102

C₂×C₄

C₂²×D₄₃

yes

—

103

C₂×C₄

C₂²×D₄₃

yes

characteristic

104

C₂×C₄

C₂²×D₄₃

yes

characteristic

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

C₂³×D₄₃

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

#10621898

yes

complemented

117

C₄

D₄×D₄₃

yes

—

118

C₂

—

119

C₂

—

172

—

120

C₂

C₂×D₄×D₄₃

yes

characteristic

121

C₂

C₂×#10621898

yes

characteristic

122

C₂

#13063371

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₃
commutator, socle, cosocle	C₂²×C₄₃	C₂⁴
Fitting	C₄×D₄×C₄₃	C₂
hypercenter	C₄×D₄	D₄₃
hypocenter	C₄₃	#550
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₃

commutator, socle, cosocle

C₂²×C₄₃

C₂⁴

Fitting

C₄×D₄×C₄₃

C₂

hypercenter

C₄×D₄

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₄₃

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. C₂×#10621898	C₂	semidirect
2. #13063322	C₂	semidirect
3. #13063321	C₂	semidirect
4. #13063299	C₂	semidirect
5. #13063298	C₂	semidirect
6. C₄×Dic₈₆	C₂	semidirect
7. #13063287	C₂	semidirect
8. C₄×D₄×C₄₃	C₂	semidirect
42. Dic₈₆	D₄	semidirect
53. C₄×C₄₃	#41	semidirect
62. C₄₃	#550	semidirect
70. C₄×D₄	D₄₃	semidirect
116. C₄	#10621898	semidirect

H ≅ G/N

type

1. C₂×#10621898

C₂

semidirect

2. #13063322

C₂

semidirect

3. #13063321

C₂

semidirect

4. #13063299

C₂

semidirect

5. #13063298

C₂

semidirect

6. C₄×Dic₈₆

C₂

semidirect

7. #13063287

C₂

semidirect

8. C₄×D₄×C₄₃

C₂

semidirect

42. Dic₈₆

D₄

semidirect

53. C₄×C₄₃

#41

semidirect

62. C₄₃

#550

semidirect

70. C₄×D₄

D₄₃

semidirect

116. C₄

#10621898

semidirect

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

subgroup	index
C₂×#10621898	2
#13063322	2
#13063321	2
#13063299	2
#13063298	2
C₄×Dic₈₆	2
#13063287	2
C₄×D₄×C₄₃	2
#550	43

subgroup

index

C₄×Dic₈₆

C₄×D₄×C₄₃

Subgroups — #14583570

Conjugacy classes of subgroups

Complements

Maximal subgroups