Group #16819186 — subgroups

Subgroups — #16819186

Order 3712.

480 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 221 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×#90777	C₂	1	2	yes	maximal
2	#14270918	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄²×C₂₉	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄×C₂₉	C₄	1	2	yes	—
5	C₂³×C₄×C₂₉	C₂²	1	1	yes	characteristic
6	C₂×C₄²×C₂₉	—	2	4	no	—
7	C₂×C₄²×C₂₉	C₄	1	2	yes	—
8	C₂×C₄²×C₂₉	C₂²	1	2	yes	—
9	C₂×#90777	—	2	4	no	—
10	C₂×#90777	—	2	4	no	—
11	C₂×#90777	C₄	1	8	yes	complemented
12	C₂²×#6597	C₂²	1	4	yes	—
13	C₂⁴×C₂₉	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₂₉	—	2	4	no	—
15	C₂²×C₄×C₂₉	—	2	8	no	—
16	C₂²×C₄×C₂₉	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₂₉	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₉	D₄	1	2	yes	—
20	C₂²×C₄×C₂₉	D₄	1	4	yes	—
21	C₂²×C₄×C₂₉	Q₈	1	2	yes	—
22	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
23	C₄²×C₂₉	—	2	4	no	—
24	C₄²×C₂₉	—	2	4	no	—
25	#90777	—	2	16	no	—
26	C₂×#6597	—	2	4	no	—
27	C₂×#6597	—	2	4	no	—
28	C₂×#6597	C₂×C₄	1	8	yes	complemented
29	C₂³×C₂₉	—	2	4	no	—
30	C₂³×C₂₉	C₄²	1	2	yes	—
31	C₂³×C₂₉	C₂²⋊C₄	1	2	yes	—
32	C₂³×C₂₉	C₄⋊C₄	1	2	yes	—
33	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₂₉	—	2	16	no	—
35	C₂×C₄×C₂₉	—	2	4	no	—
36	C₂×C₄×C₂₉	—	2	4	no	—
37	C₂×C₄×C₂₉	C₄²	1	2	yes	—
38	C₂×C₄×C₂₉	C₂²⋊C₄	1	8	yes	—
39	C₂×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₂₉	C₄⋊C₄	1	8	yes	—
41	C₂×C₄×C₂₉	C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₂₉	M₁₆	1	8	yes	complemented
43	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₂₉	C₂×Q₈	1	1	yes	characteristic
47	#6597	—	2	8	no	—
48	#1728	—	29	1	no	maximal, Sylow
49	C₂²×C₂₉	—	2	4	no	—
50	C₂²×C₂₉	—	2	4	no	—
51	C₂²×C₂₉	—	2	4	no	—
52	C₂²×C₂₉	#95	1	4	yes	—
53	C₂²×C₂₉	C₂×C₄²	1	1	yes	characteristic
54	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₂₉	C₂×M₁₆	1	4	yes	—
57	C₄×C₂₉	—	2	8	no	—
58	C₄×C₂₉	#95	1	4	yes	—
59	C₄×C₂₉	C₂×C₂²⋊C₄	1	2	yes	—
60	C₄×C₂₉	C₂×C₄⋊C₄	1	2	yes	—
61	C₂²×M₁₆	—	29	2	no	—
62	C₂²×C₄²	D₂₉	1	1	yes	characteristic
63	#336	—	29	4	no	—
64	C₂×C₂₉	—	2	4	no	—
65	C₂×C₂₉	C₂×#95	1	1	yes	characteristic
66	C₂×C₂₉	C₄×M₁₆	1	2	yes	—
67	C₂×C₂₉	#407	1	2	yes	—
68	C₂×C₂₉	#423	1	2	yes	—
69	C₂³×C₄	Dic₂₉	1	2	yes	—
70	C₂³×C₄	C₂×D₂₉	1	1	yes	characteristic
71	C₂×M₁₆	—	29	8	no	—
72	C₂×M₁₆	—	58	4	no	—
73	C₂×M₁₆	—	58	4	no	—
74	C₂²×C₈	—	29	4	no	—
75	C₂×C₄²	—	2	4	no	—
76	C₂×C₄²	Dic₂₉	1	2	yes	—
77	C₂×C₄²	C₂×D₂₉	1	2	yes	—
78	C₂₉	#1728	1	1	yes	characteristic, complemented, Sylow
79	C₂⁴	C₂×Dic₂₉	1	1	yes	characteristic
80	C₂²×C₄	—	2	4	no	—
81	C₂²×C₄	—	2	8	no	—
82	C₂²×C₄	Dic₅₈	1	2	yes	—
83	C₂²×C₄	C₄×D₂₉	1	4	yes	—
84	C₂²×C₄	D₁₁₆	1	2	yes	—
85	C₂²×C₄	C₂×Dic₂₉	1	4	yes	—
86	C₂²×C₄	C₂×Dic₂₉	1	1	yes	characteristic
87	C₂²×C₄	C₂²⋊D₂₉	1	4	yes	—
88	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
89	M₁₆	—	58	16	no	—
90	C₂×C₈	—	29	8	no	—
91	C₂×C₈	—	58	4	no	—
92	C₂×C₈	—	58	4	no	—
93	C₄²	—	2	4	no	—
94	C₄²	—	2	4	no	—
95	C₂³	—	2	4	no	—
96	C₂³	C₄×Dic₂₉	1	2	yes	—
97	C₂³	C₄⋊Dic₂₉	1	2	yes	—
98	C₂³	C₂²⋊Dic₂₉	1	2	yes	—
99	C₂³	C₂²×Dic₂₉	1	1	yes	characteristic
100	C₂×C₄	—	2	16	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	#90777	1	8	yes	complemented
104	C₂×C₄	C₄×Dic₂₉	1	2	yes	—
105	C₂×C₄	#90779	1	8	yes	—
106	C₂×C₄	C₄⋊Dic₂₉	1	2	yes	—
107	C₂×C₄	#90781	1	8	yes	—
108	C₂×C₄	C₂²⋊Dic₂₉	1	2	yes	—
109	C₂×C₄	C₂×Dic₅₈	1	1	yes	characteristic
110	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
111	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₂₉	1	2	yes	—
113	C₈	—	58	8	no	—
114	C₂²	—	2	4	no	—
115	C₂²	—	2	4	no	—
116	C₂²	—	2	4	no	—
117	C₂²	#11745179	1	4	yes	—
118	C₂²	C₂×#90777	1	4	yes	—
119	C₂²	C₂×C₄×Dic₂₉	1	1	yes	characteristic
120	C₂²	C₂×C₄⋊Dic₂₉	1	1	yes	characteristic
121	C₂²	C₂×C₂²⋊Dic₂₉	1	1	yes	characteristic
122	C₄	—	2	8	no	—
123	C₄	#11745179	1	4	yes	—
124	C₄	C₂×#90779	1	2	yes	—
125	C₄	C₂×#90781	1	2	yes	—
126	C₂	—	2	4	no	—
127	C₂	C₂×#11745179	1	1	yes	characteristic
128	C₂	C₄×#90777	1	2	yes	—
129	C₂	#14271127	1	2	yes	—
130	C₂	#14271187	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#90777

C₂

yes

maximal

#14270918

C₂

yes

maximal, complemented

C₂²×C₄²×C₂₉

C₂

yes

maximal, characteristic

C₂³×C₄×C₂₉

C₄

yes

—

C₂³×C₄×C₂₉

C₂²

yes

characteristic

C₂×C₄²×C₂₉

—

C₂×C₄²×C₂₉

C₄

yes

—

C₂×C₄²×C₂₉

C₂²

yes

—

C₂×#90777

—

C₂×#90777

—

C₂×#90777

C₄

yes

complemented

C₂²×#6597

C₂²

yes

—

C₂⁴×C₂₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₉

—

C₂²×C₄×C₂₉

—

C₂²×C₄×C₂₉

C₂×C₄

yes

—

C₂²×C₄×C₂₉

C₂×C₄

yes

—

C₂²×C₄×C₂₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₉

D₄

yes

—

C₂²×C₄×C₂₉

D₄

yes

—

C₂²×C₄×C₂₉

Q₈

yes

—

C₂²×C₄×C₂₉

C₂³

yes

characteristic

C₄²×C₂₉

—

C₄²×C₂₉

—

—

—

—

C₂×C₄

yes

complemented

C₂³×C₂₉

—

C₂³×C₂₉

C₄²

yes

—

C₂³×C₂₉

C₂²⋊C₄

yes

—

C₂³×C₂₉

C₄⋊C₄

yes

—

C₂³×C₂₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₄²

yes

—

C₂×C₄×C₂₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₉

C₄⋊C₄

yes

—

C₂×C₄×C₂₉

C₄⋊C₄

yes

—

C₂×C₄×C₂₉

M₁₆

yes

complemented

C₂×C₄×C₂₉

C₂²×C₄

yes

—

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

—

C₂×C₄×C₂₉

C₂×Q₈

yes

—

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

#95

yes

—

C₂²×C₂₉

C₂×C₄²

yes

characteristic

C₂²×C₂₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₉

C₂×M₁₆

yes

—

C₄×C₂₉

—

C₄×C₂₉

#95

yes

—

C₄×C₂₉

C₂×C₂²⋊C₄

yes

—

C₄×C₂₉

C₂×C₄⋊C₄

yes

—

C₂²×M₁₆

—

C₂²×C₄²

D₂₉

yes

characteristic

#336

—

C₂×C₂₉

—

C₂×C₂₉

C₂×#95

yes

characteristic

C₂×C₂₉

C₄×M₁₆

yes

—

C₂×C₂₉

#407

yes

—

C₂×C₂₉

#423

yes

—

C₂³×C₄

Dic₂₉

yes

—

C₂³×C₄

C₂×D₂₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

Dic₂₉

yes

—

C₂×C₄²

C₂×D₂₉

yes

—

C₂₉

#1728

yes

characteristic, complemented, Sylow

C₂⁴

C₂×Dic₂₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₅₈

yes

—

C₂²×C₄

C₄×D₂₉

yes

—

C₂²×C₄

D₁₁₆

yes

—

C₂²×C₄

C₂×Dic₂₉

yes

—

C₂²×C₄

C₂×Dic₂₉

yes

characteristic

C₂²×C₄

C₂²⋊D₂₉

yes

—

C₂²×C₄

C₂²×D₂₉

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₂³

—

C₂³

C₄×Dic₂₉

yes

—

C₂³

C₄⋊Dic₂₉

yes

—

C₂³

C₂²⋊Dic₂₉

yes

—

C₂³

C₂²×Dic₂₉

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

#90777

yes

complemented

104

C₂×C₄

C₄×Dic₂₉

yes

—

105

C₂×C₄

#90779

yes

—

106

C₂×C₄

C₄⋊Dic₂₉

yes

—

107

C₂×C₄

#90781

yes

—

108

C₂×C₄

C₂²⋊Dic₂₉

yes

—

109

C₂×C₄

C₂×Dic₅₈

yes

characteristic

110

C₂×C₄

C₂×C₄×D₂₉

yes

—

111

C₂×C₄

C₂×D₁₁₆

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₂₉

yes

—

113

C₈

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

#11745179

yes

—

118

C₂²

C₂×#90777

yes

—

119

C₂²

C₂×C₄×Dic₂₉

yes

characteristic

120

C₂²

C₂×C₄⋊Dic₂₉

yes

characteristic

121

C₂²

C₂×C₂²⋊Dic₂₉

yes

characteristic

122

C₄

—

123

C₄

#11745179

yes

—

124

C₄

C₂×#90779

yes

—

125

C₄

C₂×#90781

yes

—

126

C₂

—

127

C₂

C₂×#11745179

yes

characteristic

128

C₂

C₄×#90777

yes

—

129

C₂

#14271127

yes

—

130

C₂

#14271187

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₉
commutator	C₂²×C₂₉	C₂×C₄²
Fitting	C₂²×C₄²×C₂₉	C₂
socle	C₂³×C₂₉	C₂²×C₄
prime radical	C₂⁴×C₂₉	C₂×C₄
cosocle	C₂²×C₄×C₂₉	C₂³
hypercenter	C₂²×C₄²	D₂₉
hypocenter	C₂₉	#1728
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₉

commutator

C₂²×C₂₉

C₂×C₄²

Fitting

C₂²×C₄²×C₂₉

C₂

socle

C₂³×C₂₉

C₂²×C₄

prime radical

C₂⁴×C₂₉

C₂×C₄

cosocle

C₂²×C₄×C₂₉

C₂³

hypercenter

C₂²×C₄²

D₂₉

C₂₉

G ▷ C₂²×C₂₉ ▷ C₂₉

C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

N	H ≅ G/N	type
2. #14270918	C₂	semidirect
11. C₂×#90777	C₄	semidirect
28. C₂×#6597	C₂×C₄	semidirect
42. C₂×C₄×C₂₉	M₁₆	semidirect
78. C₂₉	#1728	semidirect
103. C₂×C₄	#90777	semidirect

H ≅ G/N

type

2. #14270918

C₂

semidirect

11. C₂×#90777

C₄

semidirect

28. C₂×#6597

C₂×C₄

semidirect

42. C₂×C₄×C₂₉

M₁₆

semidirect

78. C₂₉

#1728

semidirect

103. C₂×C₄

#90777

semidirect

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

subgroup	index
C₂²×#90777	2
#14270918	2
C₂²×C₄²×C₂₉	2
#1728	29

subgroup

index

C₂²×#90777

#14270918

C₂²×C₄²×C₂₉

#1728

Subgroups — #16819186

Conjugacy classes of subgroups

Complements

Maximal subgroups