Group #16827971 — subgroups

Subgroups — #16827971

Order 3712.

756 conjugacy classes of subgroups; only the 289 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 289 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	#14271930	C₂	1	2	yes	maximal, complemented
2	#14271147	C₂	1	6	yes	maximal, complemented
3	C₂×#11745181	C₂	1	6	yes	maximal, complemented
4	C₂×C₂₉×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
5	C₂₉×H(5,𝔽₂)	—	2	8	no	—
6	C₂×#41×C₂₉	C₄	1	2	yes	—
7	C₂×#41×C₂₉	C₂²	1	3	yes	—
8	C₂×#41×C₂₉	C₂²	1	1	yes	characteristic
9	C₂²×D₄×C₂₉	C₄	1	6	yes	—
10	C₂²×D₄×C₂₉	C₂²	1	3	yes	—
11	#11745310	—	2	8	no	—
12	C₂×#90777	C₂²	1	12	yes	complemented
13	#11745184	C₂²	1	4	yes	complemented
14	#11745181	C₂²	1	12	yes	complemented
15	C₂⁴×C₂₉	C₂×C₄	1	6	yes	—
16	#41×C₂₉	—	2	8	no	—
17	#41×C₂₉	—	2	12	no	—
18	#41×C₂₉	D₄	1	8	yes	—
19	C₂×Q₈×C₂₉	C₂×C₄	1	1	yes	characteristic
20	C₂×Q₈×C₂₉	C₂³	1	1	yes	characteristic
21	C₂×D₄×C₂₉	—	2	24	no	—
22	C₂×D₄×C₂₉	—	2	12	no	—
23	C₂×D₄×C₂₉	C₂×C₄	1	6	yes	—
24	C₂×D₄×C₂₉	C₂×C₄	1	6	yes	—
25	C₂×D₄×C₂₉	C₂×C₄	1	3	yes	—
26	C₂×D₄×C₂₉	C₂³	1	3	yes	—
27	C₂²×C₄×C₂₉	C₂×C₄	1	6	yes	—
28	C₂²×C₄×C₂₉	C₂³	1	3	yes	—
29	#90777	—	2	12	no	—
30	#90777	C₂³	1	8	yes	—
31	C₂×#6597	—	2	12	no	—
32	C₂³×C₂₉	—	2	12	no	—
33	C₂³×C₂₉	—	8	6	no	—
34	C₂³×C₂₉	C₂²×C₄	1	6	yes	—
35	C₂³×C₂₉	C₂²×C₄	1	3	yes	—
36	Q₈×C₂₉	—	2	2	no	—
37	Q₈×C₂₉	C₂²⋊C₄	1	4	yes	—
38	D₄×C₂₉	—	2	12	no	—
39	D₄×C₂₉	—	2	12	no	—
40	D₄×C₂₉	—	2	6	no	—
41	D₄×C₂₉	C₂²⋊C₄	1	12	yes	—
42	C₂×C₄×C₂₉	—	2	12	no	—
43	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₂₉	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₂₉	C₂×D₄	1	12	yes	—
46	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
47	#6597	—	2	8	no	—
48	#2888	—	29	1	no	maximal, Sylow
49	C₂²×C₂₉	—	2	6	no	—
50	C₂²×C₂₉	—	4	3	no	—
51	C₂²×C₂₉	—	4	6	no	—
52	C₂²×C₂₉	—	8	12	no	—
53	C₂²×C₂₉	C₂×C₂²⋊C₄	1	6	yes	—
54	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
55	C₄×C₂₉	—	2	2	no	—
56	C₄×C₂₉	C₂×C₂²⋊C₄	1	6	yes	—
57	C₄×C₂₉	C₂²×D₄	1	2	yes	—
58	C₂×H(5,𝔽₂)	D₂₉	1	1	yes	characteristic
59	#568	—	29	2	no	—
60	#413	—	29	6	no	—
61	C₂×#100	—	29	6	no	—
62	C₂×C₂₉	—	2	1	no	—
63	C₂×C₂₉	—	2	6	no	—
64	C₂×C₂₉	—	4	6	no	—
65	C₂×C₂₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
66	H(5,𝔽₂)	—	2	8	no	—
67	C₂×#41	Dic₂₉	1	2	yes	—
68	C₂×#41	C₂×D₂₉	1	3	yes	—
69	C₂×#41	C₂×D₂₉	1	1	yes	characteristic
70	C₂²×D₄	Dic₂₉	1	6	yes	—
71	C₂²×D₄	C₂×D₂₉	1	3	yes	—
72	#131	—	58	8	no	—
73	C₂×M₁₆	—	29	12	no	—
74	#101	—	29	4	no	—
75	#100	—	29	12	no	—
76	C₂₉	#2888	1	1	yes	characteristic, complemented, Sylow
77	C₂⁴	C₂×Dic₂₉	1	6	yes	—
78	#41	—	2	8	no	—
79	#41	—	2	12	no	—
80	#41	C₂²⋊D₂₉	1	8	yes	—
81	C₂×Q₈	C₂×Dic₂₉	1	1	yes	characteristic
82	C₂×Q₈	C₂²×D₂₉	1	1	yes	characteristic
83	C₂×D₄	—	2	24	no	—
84	C₂×D₄	—	2	12	no	—
85	C₂×D₄	C₂×Dic₂₉	1	6	yes	—
86	C₂×D₄	C₂×Dic₂₉	1	6	yes	—
87	C₂×D₄	C₂×Dic₂₉	1	3	yes	—
88	C₂×D₄	C₂²×D₂₉	1	3	yes	—
89	C₂²×C₄	C₂×Dic₂₉	1	6	yes	—
90	C₂²×C₄	C₂²×D₂₉	1	3	yes	—
91	M₁₆	—	29	8	no	—
92	M₁₆	—	58	12	no	—
93	C₂×C₈	—	58	12	no	—
94	C₂³	—	2	12	no	—
95	C₂³	—	8	6	no	—
96	C₂³	C₂²×Dic₂₉	1	6	yes	—
97	C₂³	C₂²×Dic₂₉	1	3	yes	—
98	Q₈	—	2	2	no	—
99	Q₈	C₂²⋊Dic₂₉	1	4	yes	—
100	D₄	—	2	12	no	—
101	D₄	—	2	12	no	—
102	D₄	—	2	6	no	—
103	D₄	C₂²⋊Dic₂₉	1	12	yes	—
104	C₂×C₄	—	2	12	no	—
105	C₂×C₄	C₂²×Dic₂₉	1	2	yes	—
106	C₂×C₄	C₂²×Dic₂₉	1	3	yes	—
107	C₂×C₄	C₂×C₂²⋊D₂₉	1	12	yes	—
108	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
109	C₈	—	58	8	no	—
110	C₂²	—	2	6	no	—
111	C₂²	—	4	3	no	—
112	C₂²	—	4	6	no	—
113	C₂²	—	8	12	no	—
114	C₂²	C₂×C₂²⋊Dic₂₉	1	6	yes	—
115	C₂²	C₂³×Dic₂₉	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	C₂×C₂²⋊Dic₂₉	1	6	yes	—
118	C₄	C₂²×C₂²⋊D₂₉	1	2	yes	—
119	C₂	—	2	1	no	—
120	C₂	—	2	6	no	—
121	C₂	—	4	6	no	—
122	C₂	C₂²×C₂²⋊Dic₂₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14271930

C₂

yes

maximal, complemented

#14271147

C₂

yes

maximal, complemented

C₂×#11745181

C₂

yes

maximal, complemented

C₂×C₂₉×H(5,𝔽₂)

C₂

yes

maximal, characteristic

C₂₉×H(5,𝔽₂)

—

C₂×#41×C₂₉

C₄

yes

—

C₂×#41×C₂₉

C₂²

yes

—

C₂×#41×C₂₉

C₂²

yes

characteristic

C₂²×D₄×C₂₉

C₄

yes

—

C₂²×D₄×C₂₉

C₂²

yes

—

#11745310

—

C₂×#90777

C₂²

yes

complemented

#11745184

C₂²

yes

complemented

#11745181

C₂²

yes

complemented

C₂⁴×C₂₉

C₂×C₄

yes

—

#41×C₂₉

—

#41×C₂₉

—

#41×C₂₉

D₄

yes

—

C₂×Q₈×C₂₉

C₂×C₄

yes

characteristic

C₂×Q₈×C₂₉

C₂³

yes

characteristic

C₂×D₄×C₂₉

—

C₂×D₄×C₂₉

—

C₂×D₄×C₂₉

C₂×C₄

yes

—

C₂×D₄×C₂₉

C₂×C₄

yes

—

C₂×D₄×C₂₉

C₂×C₄

yes

—

C₂×D₄×C₂₉

C₂³

yes

—

C₂²×C₄×C₂₉

C₂×C₄

yes

—

C₂²×C₄×C₂₉

C₂³

yes

—

#90777

—

#90777

C₂³

yes

—

C₂×#6597

—

C₂³×C₂₉

—

C₂³×C₂₉

—

C₂³×C₂₉

C₂²×C₄

yes

—

C₂³×C₂₉

C₂²×C₄

yes

—

Q₈×C₂₉

—

Q₈×C₂₉

C₂²⋊C₄

yes

—

D₄×C₂₉

—

D₄×C₂₉

—

D₄×C₂₉

—

D₄×C₂₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂²×C₄

yes

—

C₂×C₄×C₂₉

C₂²×C₄

yes

—

C₂×C₄×C₂₉

C₂×D₄

yes

—

C₂×C₄×C₂₉

C₂⁴

yes

—

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

C₂×C₂²⋊C₄

yes

—

C₂²×C₂₉

C₂³×C₄

yes

characteristic

C₄×C₂₉

—

C₄×C₂₉

C₂×C₂²⋊C₄

yes

—

C₄×C₂₉

C₂²×D₄

yes

—

C₂×H(5,𝔽₂)

D₂₉

yes

—

—

—

C₂×C₂₉

—

C₂×C₂₉

—

C₂×C₂₉

—

C₂×C₂₉

C₂²×C₂²⋊C₄

yes

characteristic

H(5,𝔽₂)

—

C₂×#41

Dic₂₉

yes

—

C₂×#41

C₂×D₂₉

yes

—

C₂×#41

C₂×D₂₉

yes

characteristic

C₂²×D₄

Dic₂₉

yes

—

C₂²×D₄

C₂×D₂₉

yes

—

#131

—

C₂×M₁₆

—

#101

—

#100

—

C₂₉

#2888

yes

characteristic, complemented, Sylow

C₂⁴

C₂×Dic₂₉

yes

—

#41

—

#41

—

#41

C₂²⋊D₂₉

yes

—

C₂×Q₈

C₂×Dic₂₉

yes

characteristic

C₂×Q₈

C₂²×D₂₉

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₂₉

yes

—

C₂×D₄

C₂×Dic₂₉

yes

—

C₂×D₄

C₂×Dic₂₉

yes

—

C₂×D₄

C₂²×D₂₉

yes

—

C₂²×C₄

C₂×Dic₂₉

yes

—

C₂²×C₄

C₂²×D₂₉

yes

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂³

—

C₂³

—

C₂³

C₂²×Dic₂₉

yes

—

C₂³

C₂²×Dic₂₉

yes

—

Q₈

—

Q₈

C₂²⋊Dic₂₉

yes

—

100

D₄

—

101

D₄

—

102

D₄

—

103

D₄

C₂²⋊Dic₂₉

yes

—

104

C₂×C₄

—

105

C₂×C₄

C₂²×Dic₂₉

yes

—

106

C₂×C₄

C₂²×Dic₂₉

yes

—

107

C₂×C₄

C₂×C₂²⋊D₂₉

yes

—

108

C₂×C₄

C₂³×D₂₉

yes

characteristic

109

C₈

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂×C₂²⋊Dic₂₉

yes

—

115

C₂²

C₂³×Dic₂₉

yes

characteristic

116

C₄

—

117

C₄

C₂×C₂²⋊Dic₂₉

yes

—

118

C₄

C₂²×C₂²⋊D₂₉

yes

—

119

C₂

—

120

C₂

—

121

C₂

—

122

C₂

C₂²×C₂²⋊Dic₂₉

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊Dic₂₉
commutator	C₂²×C₂₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₉
Fitting, prime radical	C₂×C₂₉×H(5,𝔽₂)	C₂
socle	C₂×C₂₉	C₂²×C₂²⋊C₄
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	C₂×H(5,𝔽₂)	D₂₉
hypocenter	C₂₉	#2888
LCS	G ▷ C₂²×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ C₂×H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊Dic₂₉

commutator

C₂²×C₂₉

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₂₉

Fitting, prime radical

C₂×C₂₉×H(5,𝔽₂)

C₂

socle

C₂×C₂₉

C₂²×C₂²⋊C₄

cosocle

C₂×C₄×C₂₉

C₂⁴

hypercenter

C₂×H(5,𝔽₂)

D₂₉

hypocenter

C₂₉

#2888

LCS

G ▷ C₂²×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂ ◁ C₂×#41 ◁ C₂×H(5,𝔽₂)

N	H ≅ G/N	type
1. #14271930	C₂	semidirect
2. #14271147	C₂	semidirect
3. C₂×#11745181	C₂	semidirect
12. C₂×#90777	C₂²	semidirect
13. #11745184	C₂²	semidirect
14. #11745181	C₂²	semidirect
76. C₂₉	#2888	semidirect

H ≅ G/N

type

1. #14271930

C₂

semidirect

2. #14271147

C₂

semidirect

3. C₂×#11745181

C₂

semidirect

12. C₂×#90777

C₂²

semidirect

13. #11745184

C₂²

semidirect

14. #11745181

C₂²

semidirect

76. C₂₉

#2888

semidirect

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

subgroup	index
#14271930	2
#14271147	2
C₂×#11745181	2
C₂×C₂₉×H(5,𝔽₂)	2
#2888	29

subgroup

index

C₂×C₂₉×H(5,𝔽₂)

Subgroups — #16827971

Conjugacy classes of subgroups

Complements

Maximal subgroups