Group #16822678 — subgroups

Subgroups — #16822678

Order 3712.

408 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 213 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	#14271062	C₂	1	4	yes	maximal
2	#14271049	C₂	1	4	yes	maximal
3	#14271049	C₂	1	4	yes	maximal
4	#14271029	C₂	1	2	yes	maximal
5	C₂×C₂₉×#125	C₂	1	1	yes	maximal, characteristic
6	C₂₉×#125	C₄	1	8	yes	complemented
7	C₂×C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
8	C₂×C₄⋊C₄×C₂₉	C₂²	1	4	yes	—
9	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₂₉	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₂₉	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₂₉	C₂²	1	4	yes	—
13	C₂×C₄×Dic₂₉	C₂²	1	4	yes	—
14	#11745179	C₂²	1	4	yes	—
15	#11745179	C₂²	1	4	yes	—
16	#11745179	C₂²	1	4	yes	—
17	#11745179	C₂²	1	4	yes	—
18	C₂²×C₄×C₂₉	C₂³	1	4	yes	—
19	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
20	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
21	C₂²×Dic₂₉	C₂³	1	4	yes	—
22	C₂²×Dic₂₉	C₂³	1	4	yes	—
23	C₄⋊C₄×C₂₉	C₂×C₄	1	8	yes	—
24	C₄⋊C₄×C₂₉	C₂×C₄	1	16	yes	—
25	C₄²×C₂₉	C₂×C₄	1	4	yes	—
26	C₄⋊Dic₂₉	—	4	2	no	—
27	C₄⋊Dic₂₉	—	4	4	no	—
28	C₄⋊Dic₂₉	—	4	2	no	—
29	C₄×Dic₂₉	—	4	4	no	—
30	C₂³×C₂₉	C₂⁴	1	1	yes	characteristic
31	C₂×C₄×C₂₉	—	2	4	no	—
32	C₂×C₄×C₂₉	—	2	2	no	—
33	C₂×C₄×C₂₉	—	2	1	no	—
34	C₂×C₄×C₂₉	—	2	1	no	—
35	C₂×C₄×C₂₉	—	2	4	no	—
36	C₂×C₄×C₂₉	—	2	2	no	—
37	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₂₉	C₂²×C₄	1	4	yes	—
39	C₂×C₄×C₂₉	C₂²×C₄	1	8	yes	—
40	C₂×Dic₂₉	—	2	4	no	—
41	C₂×Dic₂₉	—	2	4	no	—
42	C₂×Dic₂₉	—	2	4	no	—
43	C₂×Dic₂₉	—	2	4	no	—
44	C₂×Dic₂₉	—	2	4	no	—
45	C₂×Dic₂₉	—	2	4	no	—
46	#2382	—	29	1	no	maximal, Sylow
47	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
48	C₂²×C₂₉	H(5,𝔽₂)	1	2	yes	—
49	C₂²×C₂₉	2₋¹⁺⁴	1	1	yes	characteristic
50	C₂²×C₂₉	2₋¹⁺⁴	1	1	yes	characteristic
51	C₂²×C₂₉	2₋¹⁺⁴	1	2	yes	—
52	C₄×C₂₉	—	2	2	no	—
53	C₄×C₂₉	—	2	8	no	—
54	C₄×C₂₉	—	2	4	no	—
55	Dic₂₉	—	4	4	no	—
56	Dic₂₉	—	4	4	no	—
57	C₂×#125	D₂₉	1	1	yes	characteristic
58	#389	—	29	4	no	—
59	#387	—	29	4	no	—
60	#387	—	29	4	no	—
61	#382	—	29	2	no	—
62	C₂×C₂₉	#519	1	1	yes	characteristic
63	C₂×C₂₉	#520	1	2	yes	—
64	C₂×C₂₉	#562	1	1	yes	characteristic
65	C₂×C₂₉	#563	1	2	yes	—
66	C₂×C₂₉	#564	1	1	yes	characteristic
67	#125	Dic₂₉	1	8	yes	complemented
68	C₂×C₄⋊C₄	—	29	2	no	—
69	C₂×C₄⋊C₄	—	29	2	no	—
70	C₂×C₄⋊C₄	—	29	4	no	—
71	C₂×C₄⋊C₄	C₂×D₂₉	1	2	yes	—
72	C₂×C₄⋊C₄	C₂×D₂₉	1	4	yes	—
73	C₂×C₄²	—	29	4	no	—
74	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
75	#95	—	29	4	no	—
76	#95	—	29	4	no	—
77	#95	—	29	4	no	—
78	#95	—	29	4	no	—
79	C₂₉	#2382	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₄	—	29	4	no	—
81	C₂²×C₄	—	29	4	no	—
82	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
83	C₂²×C₄	C₂²×D₂₉	1	4	yes	—
84	C₂²×C₄	C₂²×D₂₉	1	2	yes	—
85	C₄⋊C₄	—	116	4	no	—
86	C₄⋊C₄	—	116	2	no	—
87	C₄⋊C₄	—	116	2	no	—
88	C₄⋊C₄	C₂×Dic₂₉	1	8	yes	—
89	C₄⋊C₄	C₂×Dic₂₉	1	16	yes	—
90	C₄²	—	116	4	no	—
91	C₄²	C₂×Dic₂₉	1	4	yes	—
92	C₂³	C₂³×D₂₉	1	1	yes	characteristic
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	58	4	no	—
100	C₂×C₄	—	58	4	no	—
101	C₂×C₄	—	58	4	no	—
102	C₂×C₄	—	58	4	no	—
103	C₂×C₄	—	58	4	no	—
104	C₂×C₄	—	58	4	no	—
105	C₂×C₄	C₂²×Dic₂₉	1	2	yes	—
106	C₂×C₄	C₂²×Dic₂₉	1	4	yes	—
107	C₂×C₄	C₂²×Dic₂₉	1	8	yes	—
108	C₂²	#11745362	1	1	yes	characteristic
109	C₂²	#11745362	1	1	yes	characteristic
110	C₂²	#11745364	1	2	yes	—
111	C₂²	#11745365	1	2	yes	—
112	C₂²	C₂³×Dic₂₉	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	—	2	8	no	—
115	C₄	—	2	4	no	—
116	C₄	—	116	4	no	—
117	C₄	—	116	4	no	—
118	C₂	#14271641	1	1	yes	characteristic
119	C₂	#14271646	1	2	yes	—
120	C₂	#14271911	1	1	yes	characteristic
121	C₂	#14271917	1	2	yes	—
122	C₂	C₂₉⋊#564	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14271062

C₂

yes

maximal

#14271049

C₂

yes

maximal

#14271049

C₂

yes

maximal

#14271029

C₂

yes

maximal

C₂×C₂₉×#125

C₂

yes

maximal, characteristic

C₂₉×#125

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₉

C₂²

yes

—

C₂×C₄⋊C₄×C₂₉

C₂²

yes

—

C₂×C₄²×C₂₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₉

C₂²

yes

—

C₂×C₄⋊Dic₂₉

C₂²

yes

—

C₂×C₄⋊Dic₂₉

C₂²

yes

—

C₂×C₄×Dic₂₉

C₂²

yes

—

#11745179

C₂²

yes

—

#11745179

C₂²

yes

—

#11745179

C₂²

yes

—

#11745179

C₂²

yes

—

C₂²×C₄×C₂₉

C₂³

yes

—

C₂²×C₄×C₂₉

C₂³

yes

—

C₂²×C₄×C₂₉

C₂³

yes

characteristic

C₂²×Dic₂₉

C₂³

yes

—

C₂²×Dic₂₉

C₂³

yes

—

C₄⋊C₄×C₂₉

C₂×C₄

yes

—

C₄⋊C₄×C₂₉

C₂×C₄

yes

—

C₄²×C₂₉

C₂×C₄

yes

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄×Dic₂₉

—

C₂³×C₂₉

C₂⁴

yes

characteristic

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂²×C₄

yes

—

C₂×C₄×C₂₉

C₂²×C₄

yes

—

C₂×C₄×C₂₉

C₂²×C₄

yes

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

#2382

—

maximal, Sylow

C₂²×C₂₉

C₂³×C₄

yes

characteristic

C₂²×C₂₉

H(5,𝔽₂)

yes

—

C₂²×C₂₉

2₋¹⁺⁴

yes

characteristic

C₂²×C₂₉

2₋¹⁺⁴

yes

characteristic

C₂²×C₂₉

2₋¹⁺⁴

yes

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

Dic₂₉

—

Dic₂₉

—

C₂×#125

D₂₉

yes

—

—

—

—

C₂×C₂₉

#519

yes

characteristic

C₂×C₂₉

#520

yes

—

C₂×C₂₉

#562

yes

characteristic

C₂×C₂₉

#563

yes

—

C₂×C₂₉

#564

yes

characteristic

#125

Dic₂₉

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₉

yes

—

C₂×C₄⋊C₄

C₂×D₂₉

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₉

yes

—

—

—

—

C₂₉

#2382

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₂₉

yes

characteristic

C₂²×C₄

C₂²×D₂₉

yes

—

C₂²×C₄

C₂²×D₂₉

yes

—

C₄⋊C₄

—

116

—

C₄⋊C₄

—

116

—

C₄⋊C₄

—

116

—

C₄⋊C₄

C₂×Dic₂₉

yes

—

C₄⋊C₄

C₂×Dic₂₉

yes

—

C₄²

—

116

—

C₄²

C₂×Dic₂₉

yes

—

C₂³

C₂³×D₂₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂²×Dic₂₉

yes

—

106

C₂×C₄

C₂²×Dic₂₉

yes

—

107

C₂×C₄

C₂²×Dic₂₉

yes

—

108

C₂²

#11745362

yes

characteristic

109

C₂²

#11745362

yes

characteristic

110

C₂²

#11745364

yes

—

111

C₂²

#11745365

yes

—

112

C₂²

C₂³×Dic₂₉

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

116

—

117

C₄

—

116

—

118

C₂

#14271641

yes

characteristic

119

C₂

#14271646

yes

—

120

C₂

#14271911

yes

characteristic

121

C₂

#14271917

yes

—

122

C₂

C₂₉⋊#564

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₉
commutator	C₂²×C₂₉	C₂³×C₄
Fitting	C₂×C₂₉×#125	C₂
socle, prime radical, cosocle	C₂³×C₂₉	C₂⁴
hypercenter	C₂×#125	D₂₉
hypocenter	C₂₉	#2382
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₂₉

commutator

C₂²×C₂₉

C₂³×C₄

Fitting

C₂×C₂₉×#125

C₂

socle, prime radical, cosocle

C₂³×C₂₉

C₂⁴

D₂₉

C₂₉

G ▷ C₂²×C₂₉ ▷ C₂₉

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
6. C₂₉×#125	C₄	semidirect
67. #125	Dic₂₉	semidirect
79. C₂₉	#2382	semidirect

H ≅ G/N

type

6. C₂₉×#125

C₄

semidirect

67. #125

Dic₂₉

semidirect

79. C₂₉

#2382

semidirect

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

subgroup	index
#14271062	2
#14271049	2
#14271029	2
C₂×C₂₉×#125	2
#2382	29

subgroup

index

C₂×C₂₉×#125

Subgroups — #16822678

Conjugacy classes of subgroups

Complements

Maximal subgroups