Group #14580254 — subgroups

Subgroups — #14580254

Order 2624.

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×D₁₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#10620279	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#10620281	C₂	1	2	yes	maximal, complemented
4	C₂×#10620281	C₂	1	2	yes	maximal, complemented
5	#95×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₄₁	C₂²	1	2	yes	—
7	C₂²×C₄×C₄₁	C₂²	1	2	yes	—
8	C₂²×C₄×C₄₁	C₂²	1	1	yes	characteristic
9	C₂²×Dic₄₁	C₂²	1	1	yes	characteristic
10	C₂²×Dic₄₁	C₂²	1	1	yes	characteristic
11	C₂×D₁₆₄	—	2	4	no	—
12	C₂×D₁₆₄	—	2	2	no	—
13	#10620281	—	2	4	no	—
14	#10620281	—	2	4	no	—
15	#10620279	—	2	2	no	—
16	C₂³×C₄₁	C₂³	1	1	yes	characteristic
17	C₂²×D₄₁	—	2	2	no	—
18	C₂²×D₄₁	—	2	2	no	—
19	C₂²×D₄₁	—	2	2	no	—
20	C₂²×D₄₁	—	2	2	no	—
21	C₂²×D₄₁	—	2	2	no	—
22	C₂²×D₄₁	—	2	2	no	—
23	C₂²×D₄₁	D₄	1	4	yes	—
24	C₂×C₄×C₄₁	—	2	2	no	—
25	C₂×C₄×C₄₁	—	2	2	no	—
26	C₂×C₄×C₄₁	—	2	2	no	—
27	C₂×C₄×C₄₁	—	2	2	no	—
28	C₂×C₄×C₄₁	D₄	1	2	yes	—
29	C₂×Dic₄₁	—	2	1	no	—
30	C₂×Dic₄₁	—	2	1	no	—
31	C₂×Dic₄₁	—	2	1	no	—
32	C₂×Dic₄₁	—	2	2	no	—
33	C₂×Dic₄₁	D₄	1	2	yes	complemented
34	D₁₆₄	—	2	8	no	—
35	C₂²×C₄₁	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄₁	C₂×D₄	1	2	yes	—
37	C₂²×C₄₁	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄₁	#41	1	2	yes	—
39	C₂²×C₄₁	#41	1	1	yes	characteristic
40	C₂×D₄₁	—	2	4	no	—
41	C₂×D₄₁	—	2	4	no	—
42	C₂×D₄₁	—	2	4	no	—
43	C₂×D₄₁	—	4	2	no	—
44	C₂×D₄₁	—	4	2	no	—
45	C₂×D₄₁	—	4	2	no	—
46	C₂×D₄₁	—	4	2	no	—
47	C₄×C₄₁	—	2	2	no	—
48	C₄×C₄₁	—	4	2	no	—
49	Dic₄₁	—	2	2	no	—
50	Dic₄₁	—	4	1	no	—
51	C₂×C₄₁	#120	1	1	yes	characteristic
52	C₂×C₄₁	#121	1	2	yes	—
53	C₂×C₄₁	#121	1	1	yes	characteristic
54	C₂×C₄₁	#123	1	2	yes	—
55	C₂×C₄₁	#124	1	1	yes	characteristic
56	D₄₁	—	4	4	no	—
57	#394	—	41	1	no	maximal, Sylow
58	C₄₁	#394	1	1	yes	characteristic, complemented, Sylow
59	C₂²×D₄	—	41	1	no	—
60	C₂×C₄⋊C₄	—	41	1	no	—
61	C₂×C₂²⋊C₄	—	41	2	no	—
62	C₂×C₂²⋊C₄	—	41	2	no	—
63	#95	D₄₁	1	1	yes	characteristic, complemented
64	C₂⁴	—	41	2	no	—
65	C₂×D₄	—	82	4	no	—
66	C₂×D₄	—	82	2	no	—
67	C₂²×C₄	—	41	1	no	—
68	C₂²×C₄	—	41	1	no	—
69	C₂²×C₄	C₂×D₄₁	1	2	yes	—
70	C₂²×C₄	C₂×D₄₁	1	1	yes	characteristic
71	C₄⋊C₄	—	82	2	no	—
72	C₂²⋊C₄	—	82	4	no	—
73	C₂²⋊C₄	—	82	4	no	—
74	C₂³	—	41	4	no	—
75	C₂³	—	82	2	no	—
76	C₂³	—	82	2	no	—
77	C₂³	—	82	2	no	—
78	C₂³	—	82	2	no	—
79	C₂³	—	82	2	no	—
80	C₂³	—	82	2	no	—
81	C₂³	C₂²×D₄₁	1	1	yes	characteristic
82	D₄	—	82	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	—	41	2	no	—
88	C₂×C₄	—	82	1	no	—
89	C₂×C₄	—	82	1	no	—
90	C₂×C₄	—	82	1	no	—
91	C₂×C₄	—	82	2	no	—
92	C₂×C₄	D₁₆₄	1	2	yes	—
93	C₂²	—	82	4	no	—
94	C₂²	—	82	4	no	—
95	C₂²	—	82	4	no	—
96	C₂²	—	164	2	no	—
97	C₂²	—	164	2	no	—
98	C₂²	—	164	2	no	—
99	C₂²	—	164	2	no	—
100	C₂²	C₂×D₁₆₄	1	1	yes	characteristic
101	C₂²	#10620306	1	2	yes	—
102	C₂²	D₄×D₄₁	1	2	yes	—
103	C₂²	D₄×D₄₁	1	1	yes	characteristic
104	C₂²	#10620310	1	1	yes	characteristic
105	C₄	—	2	2	no	—
106	C₄	—	4	2	no	—
107	C₄	—	82	2	no	—
108	C₄	—	164	1	no	—
109	C₂	—	164	4	no	—
110	C₂	#13050134	1	1	yes	characteristic
111	C₂	#13050137	1	2	yes	—
112	C₂	#13050139	1	1	yes	characteristic
113	C₂	#13050149	1	2	yes	—
114	C₂	#13050152	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₁₆₄

C₂

yes

maximal, characteristic

C₂×#10620279

C₂

yes

maximal, characteristic, complemented

C₂×#10620281

C₂

yes

maximal, complemented

C₂×#10620281

C₂

yes

maximal, complemented

#95×C₄₁

C₂

yes

maximal, characteristic, complemented

C₂³×D₄₁

C₂²

yes

—

C₂²×C₄×C₄₁

C₂²

yes

—

C₂²×C₄×C₄₁

C₂²

yes

characteristic

C₂²×Dic₄₁

C₂²

yes

characteristic

C₂²×Dic₄₁

C₂²

yes

characteristic

C₂×D₁₆₄

—

C₂×D₁₆₄

—

#10620281

—

#10620281

—

#10620279

—

C₂³×C₄₁

C₂³

yes

characteristic

C₂²×D₄₁

—

C₂²×D₄₁

—

C₂²×D₄₁

—

C₂²×D₄₁

—

C₂²×D₄₁

—

C₂²×D₄₁

—

C₂²×D₄₁

D₄

yes

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

D₄

yes

—

C₂×Dic₄₁

—

C₂×Dic₄₁

—

C₂×Dic₄₁

—

C₂×Dic₄₁

—

C₂×Dic₄₁

D₄

yes

complemented

D₁₆₄

—

C₂²×C₄₁

C₂×D₄

yes

characteristic

C₂²×C₄₁

C₂×D₄

yes

—

C₂²×C₄₁

C₂×D₄

yes

characteristic

C₂²×C₄₁

#41

yes

—

C₂²×C₄₁

#41

yes

characteristic

C₂×D₄₁

—

C₂×D₄₁

—

C₂×D₄₁

—

C₂×D₄₁

—

C₂×D₄₁

—

C₂×D₄₁

—

C₂×D₄₁

—

C₄×C₄₁

—

C₄×C₄₁

—

Dic₄₁

—

Dic₄₁

—

C₂×C₄₁

#120

yes

characteristic

C₂×C₄₁

#121

yes

—

C₂×C₄₁

#121

yes

characteristic

C₂×C₄₁

#123

yes

—

C₂×C₄₁

#124

yes

characteristic

D₄₁

—

#394

—

maximal, Sylow

C₄₁

#394

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

#95

D₄₁

yes

characteristic, complemented

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄₁

yes

—

C₂²×C₄

C₂×D₄₁

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²×D₄₁

yes

characteristic

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₁₆₄

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

164

—

C₂²

—

164

—

C₂²

—

164

—

C₂²

—

164

—

100

C₂²

C₂×D₁₆₄

yes

characteristic

101

C₂²

#10620306

yes

—

102

C₂²

D₄×D₄₁

yes

—

103

C₂²

D₄×D₄₁

yes

characteristic

104

C₂²

#10620310

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

164

—

109

C₂

—

164

—

110

C₂

#13050134

yes

characteristic

111

C₂

#13050137

yes

—

112

C₂

#13050139

yes

characteristic

113

C₂

#13050149

yes

—

114

C₂

#13050152

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₄₁
commutator, socle, cosocle	C₂³×C₄₁	C₂³
Fitting	#95×C₄₁	C₂
prime radical	C₂²×D₁₆₄	C₂
hypercenter	#95	D₄₁
hypocenter	C₄₁	#394
LCS	G ▷ C₂³×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂³ ◁ #95

subgroup

G/H

center, Frattini

C₂³

C₂²×D₄₁

commutator, socle, cosocle

C₂³×C₄₁

C₂³

Fitting

#95×C₄₁

C₂

prime radical

C₂²×D₁₆₄

C₂

D₄₁

C₄₁

G ▷ C₂³×C₄₁ ▷ C₄₁

C₁ ◁ C₂³ ◁ #95

N	H ≅ G/N	type
2. C₂×#10620279	C₂	semidirect
3. C₂×#10620281	C₂	semidirect
4. C₂×#10620281	C₂	semidirect
5. #95×C₄₁	C₂	semidirect
33. C₂×Dic₄₁	D₄	semidirect
58. C₄₁	#394	semidirect
63. #95	D₄₁	semidirect

H ≅ G/N

type

2. C₂×#10620279

C₂

semidirect

3. C₂×#10620281

C₂

semidirect

4. C₂×#10620281

C₂

semidirect

5. #95×C₄₁

C₂

semidirect

33. C₂×Dic₄₁

D₄

semidirect

58. C₄₁

#394

semidirect

63. #95

D₄₁

semidirect

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

subgroup	index
C₂²×D₁₆₄	2
C₂×#10620279	2
C₂×#10620281	2
#95×C₄₁	2
#394	41

subgroup

index

C₂²×D₁₆₄

C₂×#10620279

C₂×#10620281

#95×C₄₁

#394

Subgroups — #14580254

Conjugacy classes of subgroups

Complements

Maximal subgroups