Group #16830044 — subgroups

Subgroups — #16830044

Order 3712.

348 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 165 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	#14271720	C₂	1	1	yes	maximal, characteristic
2	C₂×#11745232	C₂	1	1	yes	maximal, characteristic
3	#14271572	C₂	1	4	yes	maximal, complemented
4	#14271565	C₂	1	4	yes	maximal, complemented
5	#14271205	C₂	1	4	yes	maximal
6	C₄×M₁₆×C₂₉	C₂	1	1	yes	maximal, characteristic
7	C₂×M₁₆×C₂₉	C₂²	1	2	yes	—
8	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
9	#11745288	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₂₉	C₂²	1	2	yes	—
11	#11745286	—	2	2	no	—
12	C₂×#90779	—	2	2	no	—
13	#11745232	—	2	2	no	—
14	#11745232	C₂²	1	2	yes	—
15	#11745232	C₂²	1	2	yes	—
16	#11745232	C₂²	1	2	yes	—
17	#11745231	C₂²	1	2	yes	—
18	C₄×Dic₅₈	—	2	2	no	—
19	C₂₉×#97	C₂²	1	2	yes	—
20	C₄×C₈×C₂₉	C₂²	1	2	yes	—
21	#11745166	C₂²	1	8	yes	—
22	#11745165	C₂²	1	8	yes	—
23	C₂²×C₄×C₂₉	D₄	1	2	yes	—
24	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
25	C₂²×Dic₂₉	—	2	2	no	—
26	C₂×Dic₅₈	—	2	2	no	—
27	M₁₆×C₂₉	Q₈	1	8	yes	complemented
28	C₂×C₈×C₂₉	C₂³	1	4	yes	—
29	C₄²×C₂₉	D₄	1	2	yes	—
30	C₄²×C₂₉	C₂³	1	1	yes	characteristic
31	C₄²×C₂₉	C₂³	1	1	yes	characteristic
32	C₂²⋊Dic₂₉	—	2	2	no	—
33	C₄⋊Dic₂₉	—	2	2	no	—
34	C₄⋊Dic₂₉	—	2	2	no	—
35	C₄⋊Dic₂₉	C₂³	1	4	yes	—
36	C₄⋊Dic₂₉	C₂³	1	4	yes	—
37	#90779	—	2	4	no	—
38	#90779	—	2	4	no	—
39	#90779	—	2	4	no	—
40	C₄×Dic₂₉	—	2	2	no	—
41	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₉	—	4	2	no	—
43	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₂₉	C₂×Q₈	1	4	yes	—
47	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
48	C₂×Dic₂₉	—	2	4	no	—
49	C₂×Dic₂₉	—	2	4	no	—
50	C₂×Dic₂₉	—	4	4	no	—
51	Dic₅₈	—	4	4	no	—
52	C₈×C₂₉	C₂×Q₈	1	8	yes	—
53	#3166	—	29	1	no	maximal, Sylow
54	C₂²×C₂₉	—	2	2	no	—
55	C₂²×C₂₉	C₄⋊Q₈	1	2	yes	—
56	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
57	C₄×C₂₉	—	2	2	no	—
58	C₄×C₂₉	—	2	2	no	—
59	C₄×C₂₉	C₄⋊Q₈	1	2	yes	—
60	C₄×C₂₉	C₂²×Q₈	1	2	yes	—
61	Dic₂₉	—	4	4	no	—
62	Dic₂₉	—	4	4	no	—
63	#533	—	29	1	no	—
64	C₂×#125	—	29	1	no	—
65	#501	—	29	4	no	—
66	#499	—	29	4	no	—
67	#428	—	29	4	no	—
68	C₄×M₁₆	D₂₉	1	1	yes	characteristic
69	C₂×C₂₉	—	2	2	no	—
70	C₂×C₂₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
71	C₂×C₂₉	#575	1	2	yes	—
72	C₂×M₁₆	C₂×D₂₉	1	2	yes	—
73	C₄⋊Q₈	—	29	2	no	—
74	#125	—	29	2	no	—
75	#125	—	29	2	no	—
76	#125	—	29	2	no	—
77	#125	—	58	2	no	—
78	C₂²⋊Q₈	—	58	2	no	—
79	C₄×Q₈	—	58	2	no	—
80	#117	—	29	2	no	—
81	C₂×C₄⋊C₄	—	29	2	no	—
82	C₂×C₄⋊C₄	—	58	2	no	—
83	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
84	#107	—	29	8	no	—
85	#106	—	29	8	no	—
86	#97	C₂×D₂₉	1	2	yes	—
87	C₄×C₈	C₂×D₂₉	1	2	yes	—
88	C₂₉	#3166	1	1	yes	characteristic, complemented, Sylow
89	C₂×Q₈	—	58	2	no	—
90	C₂²×C₄	—	58	2	no	—
91	C₂²×C₄	D₁₁₆	1	2	yes	—
92	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
93	M₁₆	Dic₅₈	1	8	yes	complemented
94	C₂×C₈	C₂²×D₂₉	1	4	yes	—
95	C₄⋊C₄	—	29	4	no	—
96	C₄⋊C₄	—	29	4	no	—
97	C₄⋊C₄	—	58	2	no	—
98	C₄⋊C₄	—	58	4	no	—
99	C₄⋊C₄	—	58	4	no	—
100	C₄⋊C₄	—	58	4	no	—
101	C₄⋊C₄	—	58	2	no	—
102	C₂²⋊C₄	—	58	2	no	—
103	C₄²	—	58	2	no	—
104	C₄²	D₁₁₆	1	2	yes	—
105	C₄²	C₂²×D₂₉	1	1	yes	characteristic
106	C₄²	C₂²×D₂₉	1	1	yes	characteristic
107	C₂³	C₂×D₁₁₆	1	1	yes	characteristic
108	Q₈	—	116	4	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	58	4	no	—
111	C₂×C₄	—	58	4	no	—
112	C₂×C₄	—	116	4	no	—
113	C₂×C₄	C₂×Dic₅₈	1	4	yes	—
114	C₂×C₄	C₂×D₁₁₆	1	2	yes	—
115	C₂×C₄	C₂×D₁₁₆	1	2	yes	—
116	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
117	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
118	C₈	C₂×Dic₅₈	1	8	yes	—
119	C₂²	—	2	2	no	—
120	C₂²	#11745231	1	2	yes	—
121	C₂²	C₂²×D₁₁₆	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	116	4	no	—
125	C₄	—	116	4	no	—
126	C₄	#11745231	1	2	yes	—
127	C₄	C₂²×Dic₅₈	1	2	yes	—
128	C₂	—	2	2	no	—
129	C₂	C₂×#11745231	1	1	yes	characteristic
130	C₂	#14271963	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14271720

C₂

yes

maximal, characteristic

C₂×#11745232

C₂

yes

maximal, characteristic

#14271572

C₂

yes

maximal, complemented

#14271565

C₂

yes

maximal, complemented

#14271205

C₂

yes

maximal

C₄×M₁₆×C₂₉

C₂

yes

maximal, characteristic

C₂×M₁₆×C₂₉

C₂²

yes

—

C₂×C₄²×C₂₉

C₂²

yes

characteristic

#11745288

C₂²

yes

—

C₂×C₄⋊Dic₂₉

C₂²

yes

—

—

—

—

C₂²

yes

—

#11745232

C₂²

yes

—

#11745232

C₂²

yes

—

#11745231

C₂²

yes

—

C₄×Dic₅₈

—

C₂₉×#97

C₂²

yes

—

C₄×C₈×C₂₉

C₂²

yes

—

#11745166

C₂²

yes

—

#11745165

C₂²

yes

—

C₂²×C₄×C₂₉

D₄

yes

—

C₂²×C₄×C₂₉

C₂³

yes

characteristic

C₂²×Dic₂₉

—

C₂×Dic₅₈

—

M₁₆×C₂₉

Q₈

yes

complemented

C₂×C₈×C₂₉

C₂³

yes

—

C₄²×C₂₉

D₄

yes

—

C₄²×C₂₉

C₂³

yes

characteristic

C₄²×C₂₉

C₂³

yes

characteristic

C₂²⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

C₂³

yes

—

C₄⋊Dic₂₉

C₂³

yes

—

#90779

—

#90779

—

#90779

—

C₄×Dic₂₉

—

C₂³×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂×D₄

yes

—

C₂×C₄×C₂₉

C₂×D₄

yes

—

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×Q₈

yes

—

C₂×C₄×C₂₉

C₂⁴

yes

characteristic

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

Dic₅₈

—

C₈×C₂₉

C₂×Q₈

yes

—

#3166

—

maximal, Sylow

C₂²×C₂₉

—

C₂²×C₂₉

C₄⋊Q₈

yes

—

C₂²×C₂₉

C₂²×D₄

yes

characteristic

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

C₄⋊Q₈

yes

—

C₄×C₂₉

C₂²×Q₈

yes

—

Dic₂₉

—

Dic₂₉

—

—

—

—

—

—

C₄×M₁₆

D₂₉

yes

characteristic

C₂×C₂₉

—

C₂×C₂₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₂₉

#575

yes

—

C₂×M₁₆

C₂×D₂₉

yes

—

C₄⋊Q₈

—

—

—

—

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₂₉

yes

—

—

C₂×D₂₉

yes

—

C₄×C₈

C₂×D₂₉

yes

—

C₂₉

#3166

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

D₁₁₆

yes

—

C₂²×C₄

C₂²×D₂₉

yes

characteristic

M₁₆

Dic₅₈

yes

complemented

C₂×C₈

C₂²×D₂₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

D₁₁₆

yes

—

105

C₄²

C₂²×D₂₉

yes

characteristic

106

C₄²

C₂²×D₂₉

yes

characteristic

107

C₂³

C₂×D₁₁₆

yes

characteristic

108

Q₈

—

116

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

116

—

113

C₂×C₄

C₂×Dic₅₈

yes

—

114

C₂×C₄

C₂×D₁₁₆

yes

—

115

C₂×C₄

C₂×D₁₁₆

yes

—

116

C₂×C₄

C₂×D₁₁₆

yes

characteristic

117

C₂×C₄

C₂³×D₂₉

yes

characteristic

118

C₈

C₂×Dic₅₈

yes

—

119

C₂²

—

120

C₂²

#11745231

yes

—

121

C₂²

C₂²×D₁₁₆

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

116

—

125

C₄

—

116

—

126

C₄

#11745231

yes

—

127

C₄

C₂²×Dic₅₈

yes

—

128

C₂

—

129

C₂

C₂×#11745231

yes

characteristic

130

C₂

#14271963

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₁₁₆
commutator, cosocle	C₂×C₄×C₂₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₉
Fitting	C₄×M₁₆×C₂₉	C₂
socle	C₂²×C₂₉	C₂²×D₄
prime radical	C₂³×C₂₉	C₂×D₄
hypercenter	C₄×M₁₆	D₂₉
hypocenter	C₂₉	#3166
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂²×D₁₁₆

commutator, cosocle

C₂×C₄×C₂₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₉

Fitting

C₄×M₁₆×C₂₉

C₂

socle

C₂²×C₂₉

C₂²×D₄

prime radical

C₂³×C₂₉

C₂×D₄

hypercenter

C₄×M₁₆

D₂₉

hypocenter

C₂₉

#3166

LCS

G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
3. #14271572	C₂	semidirect
4. #14271565	C₂	semidirect
27. M₁₆×C₂₉	Q₈	semidirect
88. C₂₉	#3166	semidirect
93. M₁₆	Dic₅₈	semidirect

H ≅ G/N

type

3. #14271572

C₂

semidirect

4. #14271565

C₂

semidirect

27. M₁₆×C₂₉

Q₈

semidirect

88. C₂₉

#3166

semidirect

93. M₁₆

Dic₅₈

semidirect

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

subgroup	index
#14271720	2
C₂×#11745232	2
#14271572	2
#14271565	2
#14271205	2
C₄×M₁₆×C₂₉	2
#3166	29

subgroup

index

C₄×M₁₆×C₂₉

Subgroups — #16830044

Conjugacy classes of subgroups

Complements

Maximal subgroups