Group #14581051 — subgroups

Subgroups — #14581051

Order 2624.

236 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 99 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	#13050164	C₂	1	1	yes	maximal, characteristic, complemented
2	#13050150	C₂	1	4	yes	maximal, complemented
3	#13050138	C₂	1	2	yes	maximal, complemented
4	#13050135	C₂	1	2	yes	maximal, complemented
5	C₄×D₁₆₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#13050123	C₂	1	2	yes	maximal, complemented
7	#13050122	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×D₄×C₄₁	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₄₁	C₂²	1	2	yes	—
11	C₂×C₂²⋊D₄₁	C₂²	1	4	yes	—
12	C₂×D₁₆₄	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄₁	C₂²	1	2	yes	—
14	C₂×Dic₈₂	C₂²	1	2	yes	—
15	C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
16	C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
17	C₄²×C₄₁	C₂²	1	1	yes	characteristic
18	C₂²⋊Dic₄₁	C₂²	1	4	yes	—
19	#10620281	C₂²	1	4	yes	—
20	#10620281	C₂²	1	2	yes	—
21	C₄⋊Dic₄₁	C₂²	1	2	yes	—
22	C₄⋊Dic₄₁	C₂²	1	1	yes	characteristic
23	C₄⋊Dic₄₁	C₂²	1	2	yes	—
24	C₄×Dic₄₁	C₂²	1	4	yes	—
25	C₂³×C₄₁	C₂³	1	2	yes	—
26	C₂²×D₄₁	C₂³	1	2	yes	—
27	D₄×C₄₁	—	2	2	no	—
28	C₂×C₄×C₄₁	—	2	4	no	—
29	C₂×C₄×C₄₁	C₂³	1	2	yes	—
30	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
33	C₂²⋊D₄₁	—	4	4	no	—
34	C₂×Dic₄₁	C₂³	1	2	yes	—
35	C₂×Dic₄₁	C₂³	1	4	yes	—
36	D₁₆₄	—	2	2	no	—
37	C₄×D₄₁	—	2	4	no	—
38	Dic₈₂	—	4	2	no	—
39	C₂²×C₄₁	—	2	2	no	—
40	C₂²×C₄₁	—	2	2	no	—
41	C₂²×C₄₁	—	2	2	no	—
42	C₂²×C₄₁	C₂⁴	1	1	yes	characteristic
43	C₂×D₄₁	—	2	2	no	—
44	C₂×D₄₁	—	2	2	no	—
45	C₂×D₄₁	—	2	2	no	—
46	C₄×C₄₁	—	2	2	no	—
47	C₄×C₄₁	—	2	1	no	—
48	C₄×C₄₁	#41	1	2	yes	—
49	C₄×C₄₁	#41	1	2	yes	—
50	Dic₄₁	—	2	2	no	—
51	Dic₄₁	—	2	4	no	—
52	C₂×C₄₁	—	4	2	no	—
53	C₂×C₄₁	C₂×#41	1	1	yes	characteristic
54	C₂×C₄₁	C₂×#41	1	1	yes	characteristic
55	C₂×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
56	D₄₁	—	4	2	no	—
57	#555	—	41	1	no	maximal, Sylow
58	C₄₁	#555	1	1	yes	characteristic, complemented, Sylow
59	C₄⋊Q₈	—	41	1	no	—
60	#124	—	41	4	no	—
61	#121	—	41	2	no	—
62	#121	—	41	2	no	—
63	C₄×D₄	—	41	1	no	—
64	C₄×D₄	D₄₁	1	1	yes	characteristic, complemented
65	#117	—	41	2	no	—
66	#117	—	41	2	no	—
67	C₂×Q₈	—	41	2	no	—
68	C₂×D₄	—	41	4	no	—
69	C₂×D₄	—	41	1	no	—
70	C₂×D₄	C₂×D₄₁	1	1	yes	characteristic
71	C₂²×C₄	—	41	2	no	—
72	C₂²×C₄	C₂×D₄₁	1	2	yes	—
73	C₄⋊C₄	—	41	1	no	—
74	C₄⋊C₄	—	41	2	no	—
75	C₄⋊C₄	—	41	2	no	—
76	C₄⋊C₄	C₂×D₄₁	1	1	yes	characteristic
77	C₂²⋊C₄	—	41	4	no	—
78	C₂²⋊C₄	—	41	4	no	—
79	C₂²⋊C₄	—	41	2	no	—
80	C₂²⋊C₄	C₂×D₄₁	1	2	yes	—
81	C₄²	—	41	4	no	—
82	C₄²	C₂×D₄₁	1	1	yes	characteristic
83	C₂³	—	41	2	no	—
84	C₂³	C₂²×D₄₁	1	2	yes	—
85	Q₈	—	164	2	no	—
86	D₄	—	2	2	no	—
87	D₄	—	82	2	no	—
88	D₄	—	164	4	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	41	2	no	—
91	C₂×C₄	—	41	4	no	—
92	C₂×C₄	—	82	4	no	—
93	C₂×C₄	C₂²×D₄₁	1	2	yes	—
94	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
95	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
96	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
97	C₂²	—	2	2	no	—
98	C₂²	—	2	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	—	82	2	no	—
101	C₂²	—	82	2	no	—
102	C₂²	—	82	2	no	—
103	C₂²	C₂³×D₄₁	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	2	1	no	—
106	C₄	—	82	2	no	—
107	C₄	—	82	4	no	—
108	C₄	#10620306	1	2	yes	—
109	C₄	#10620308	1	2	yes	—
110	C₂	—	4	2	no	—
111	C₂	—	164	2	no	—
112	C₂	C₂×#10620308	1	1	yes	characteristic
113	C₂	C₂×#10620306	1	1	yes	characteristic
114	C₂	#13050210	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13050164

C₂

yes

maximal, characteristic, complemented

#13050150

C₂

yes

maximal, complemented

#13050138

C₂

yes

maximal, complemented

#13050135

C₂

yes

maximal, complemented

C₄×D₁₆₄

C₂

yes

maximal, characteristic, complemented

#13050123

C₂

yes

maximal, complemented

#13050122

C₂

yes

maximal, complemented

C₄×D₄×C₄₁

C₂

yes

maximal, characteristic, complemented

C₂×D₄×C₄₁

C₂²

yes

characteristic

C₂²×C₄×C₄₁

C₂²

yes

—

C₂×C₂²⋊D₄₁

C₂²

yes

—

C₂×D₁₆₄

C₂²

yes

characteristic

C₂×C₄×D₄₁

C₂²

yes

—

C₂×Dic₈₂

C₂²

yes

—

C₄⋊C₄×C₄₁

C₂²

yes

characteristic

C₂²⋊C₄×C₄₁

C₂²

yes

—

C₄²×C₄₁

C₂²

yes

characteristic

C₂²⋊Dic₄₁

C₂²

yes

—

#10620281

C₂²

yes

—

#10620281

C₂²

yes

—

C₄⋊Dic₄₁

C₂²

yes

—

C₄⋊Dic₄₁

C₂²

yes

characteristic

C₄⋊Dic₄₁

C₂²

yes

—

C₄×Dic₄₁

C₂²

yes

—

C₂³×C₄₁

C₂³

yes

—

C₂²×D₄₁

C₂³

yes

—

D₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

C₂³

yes

—

C₂×C₄×C₄₁

C₂³

yes

characteristic

C₂×C₄×C₄₁

C₂³

yes

characteristic

C₂×C₄×C₄₁

C₂³

yes

characteristic

C₂²⋊D₄₁

—

C₂×Dic₄₁

C₂³

yes

—

C₂×Dic₄₁

C₂³

yes

—

D₁₆₄

—

C₄×D₄₁

—

Dic₈₂

—

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

C₂⁴

yes

characteristic

C₂×D₄₁

—

C₂×D₄₁

—

C₂×D₄₁

—

C₄×C₄₁

—

C₄×C₄₁

—

C₄×C₄₁

#41

yes

—

C₄×C₄₁

#41

yes

—

Dic₄₁

—

Dic₄₁

—

C₂×C₄₁

—

C₂×C₄₁

C₂×#41

yes

characteristic

C₂×C₄₁

C₂×#41

yes

characteristic

C₂×C₄₁

H(5,𝔽₂)

yes

characteristic

D₄₁

—

#555

—

maximal, Sylow

C₄₁

#555

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

#124

—

#121

—

#121

—

C₄×D₄

—

C₄×D₄

D₄₁

yes

characteristic, complemented

#117

—

#117

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₄₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×D₄₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₄₁

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₄₁

yes

—

C₄²

—

C₄²

C₂×D₄₁

yes

characteristic

C₂³

—

C₂³

C₂²×D₄₁

yes

—

Q₈

—

164

—

D₄

—

D₄

—

D₄

—

164

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×D₄₁

yes

—

C₂×C₄

C₂²×D₄₁

yes

characteristic

C₂×C₄

C₂²×D₄₁

yes

characteristic

C₂×C₄

C₂²×D₄₁

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

103

C₂²

C₂³×D₄₁

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

#10620306

yes

—

109

C₄

#10620308

yes

—

110

C₂

—

111

C₂

—

164

—

112

C₂

C₂×#10620308

yes

characteristic

113

C₂

C₂×#10620306

yes

characteristic

114

C₂

#13050210

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₁
commutator, socle, cosocle	C₂²×C₄₁	C₂⁴
Fitting	C₄×D₄×C₄₁	C₂
hypercenter	C₄×D₄	D₄₁
hypocenter	C₄₁	#555
LCS	G ▷ C₂²×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₁

commutator, socle, cosocle

C₂²×C₄₁

C₂⁴

Fitting

C₄×D₄×C₄₁

C₂

hypercenter

C₄×D₄

D₄₁

C₄₁

G ▷ C₂²×C₄₁ ▷ C₄₁

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. #13050164	C₂	semidirect
2. #13050150	C₂	semidirect
3. #13050138	C₂	semidirect
4. #13050135	C₂	semidirect
5. C₄×D₁₆₄	C₂	semidirect
6. #13050123	C₂	semidirect
7. #13050122	C₂	semidirect
8. C₄×D₄×C₄₁	C₂	semidirect
58. C₄₁	#555	semidirect
64. C₄×D₄	D₄₁	semidirect

H ≅ G/N

type

1. #13050164

C₂

semidirect

2. #13050150

C₂

semidirect

3. #13050138

C₂

semidirect

4. #13050135

C₂

semidirect

5. C₄×D₁₆₄

C₂

semidirect

6. #13050123

C₂

semidirect

7. #13050122

C₂

semidirect

8. C₄×D₄×C₄₁

C₂

semidirect

58. C₄₁

#555

semidirect

64. C₄×D₄

D₄₁

semidirect

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

subgroup	index
#13050164	2
#13050150	2
#13050138	2
#13050135	2
C₄×D₁₆₄	2
#13050123	2
#13050122	2
C₄×D₄×C₄₁	2
#555	41

subgroup

index

C₄×D₁₆₄

C₄×D₄×C₄₁

Subgroups — #14581051

Conjugacy classes of subgroups

Complements

Maximal subgroups