Group #16818551 — subgroups

Subgroups — #16818551

Order 3712.

240 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 57 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	#14271539	C₂	1	1	yes	maximal, characteristic
2	#14271363	C₂	1	2	yes	maximal, complemented
3	#14270907	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊#6597	C₂	1	2	yes	maximal, complemented
5	C₂₉×#559	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₉×#127	C₂²	1	1	yes	characteristic
7	C₂₉×#124	—	2	2	no	—
8	C₂₉×#123	—	2	2	no	—
9	C₄×Q₈×C₂₉	C₂²	1	2	yes	—
10	C₄×D₄×C₂₉	—	2	2	no	—
11	C₂×#90784	—	2	2	no	—
12	C₂×#90782	—	2	2	no	—
13	#11745236	C₂²	1	1	yes	characteristic
14	#11745157	—	2	2	no	—
15	C₄⋊#6597	C₂²	1	2	yes	—
16	C₄×#6597	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₂₉	D₄	1	2	yes	—
18	C₂×D₄×C₂₉	—	2	2	no	—
19	C₂×D₄×C₂₉	D₄	1	2	yes	—
20	C₂²×C₄×C₂₉	—	2	2	no	—
21	C₂×D₁₁₆	—	2	1	no	—
22	C₂×D₁₁₆	—	2	2	no	—
23	C₄⋊C₄×C₂₉	—	2	2	no	—
24	C₄⋊C₄×C₂₉	D₄	1	2	yes	—
25	C₂²⋊C₄×C₂₉	—	2	4	no	—
26	C₂²⋊C₄×C₂₉	—	2	2	no	—
27	C₄²×C₂₉	—	2	2	no	—
28	C₄²×C₂₉	C₂³	1	1	yes	characteristic
29	#90784	—	4	4	no	—
30	#90782	—	4	4	no	—
31	C₂×#6597	—	2	1	no	—
32	C₂×#6597	—	2	2	no	—
33	C₂³×C₂₉	—	2	2	no	—
34	C₂²×D₂₉	—	4	1	no	—
35	Q₈×C₂₉	—	2	2	no	—
36	Q₈×C₂₉	—	2	2	no	—
37	D₄×C₂₉	—	4	2	no	—
38	D₄×C₂₉	—	4	2	no	—
39	D₄×C₂₉	—	4	2	no	—
40	C₂×C₄×C₂₉	—	2	2	no	—
41	C₂×C₄×C₂₉	—	2	2	no	—
42	C₂×C₄×C₂₉	—	8	2	no	—
43	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
45	D₁₁₆	—	4	2	no	—
46	D₁₁₆	—	4	2	no	—
47	D₁₁₆	—	4	2	no	—
48	#6597	—	2	2	no	—
49	#6597	—	4	2	no	—
50	#1620	—	29	1	no	maximal, Sylow
51	C₂²×C₂₉	—	4	4	no	—
52	C₂²×C₂₉	—	4	2	no	—
53	C₂²×C₂₉	#120	1	1	yes	characteristic
54	C₂×D₂₉	—	8	2	no	—
55	C₂×D₂₉	—	8	1	no	—
56	C₄×C₂₉	—	2	1	no	—
57	C₄×C₂₉	—	4	2	no	—
58	C₄×C₂₉	—	4	2	no	—
59	C₄×C₂₉	#135	1	2	yes	—
60	C₄×C₂₉	AGL(1,ℤ/8)	1	2	yes	—
61	#559	D₂₉	1	1	yes	characteristic, complemented
62	#493	—	29	1	no	—
63	#460	—	29	2	no	—
64	#331	—	29	1	no	—
65	Q₈⋊C₈	—	29	2	no	—
66	C₂×C₂₉	—	8	2	no	—
67	C₂×C₂₉	#449	1	2	yes	—
68	C₂×C₂₉	#453	1	1	yes	characteristic
69	D₂₉	—	16	1	no	—
70	C₂×SD₁₆	—	58	2	no	—
71	C₂×D₈	—	58	2	no	—
72	#127	—	29	1	no	—
73	#127	C₂×D₂₉	1	1	yes	characteristic
74	#124	—	2	2	no	—
75	#123	—	2	2	no	—
76	C₄×Q₈	C₂×D₂₉	1	2	yes	—
77	C₄×D₄	—	2	2	no	—
78	C₄⋊C₈	—	29	2	no	—
79	#102	—	58	2	no	—
80	C₄×C₈	—	29	1	no	—
81	C₂₉	#1620	1	1	yes	characteristic, complemented, Sylow
82	C₂×Q₈	C₂²⋊D₂₉	1	2	yes	—
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	58	2	no	—
85	C₂×D₄	—	58	1	no	—
86	C₂×D₄	C₂²⋊D₂₉	1	2	yes	—
87	C₂²×C₄	—	2	2	no	—
88	SD₁₆	—	116	4	no	—
89	D₈	—	116	4	no	—
90	C₂×C₈	—	58	1	no	—
91	C₂×C₈	—	58	2	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	C₂²⋊D₂₉	1	2	yes	—
94	C₂²⋊C₄	—	2	4	no	—
95	C₂²⋊C₄	—	2	2	no	—
96	C₄²	—	2	2	no	—
97	C₄²	C₂²×D₂₉	1	1	yes	characteristic
98	C₂³	—	2	2	no	—
99	C₂³	—	116	1	no	—
100	Q₈	—	2	2	no	—
101	Q₈	—	2	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	116	2	no	—
106	D₄	—	116	2	no	—
107	D₄	—	116	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₂₉	1	2	yes	—
113	C₈	—	58	2	no	—
114	C₈	—	116	2	no	—
115	C₂²	—	4	4	no	—
116	C₂²	—	4	2	no	—
117	C₂²	—	232	2	no	—
118	C₂²	—	232	1	no	—
119	C₂²	#11745315	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	4	2	no	—
122	C₄	—	4	2	no	—
123	C₄	#11745311	1	2	yes	—
124	C₄	#11745312	1	2	yes	—
125	C₂	—	8	2	no	—
126	C₂	—	464	1	no	—
127	C₂	#14271308	1	2	yes	—
128	C₂	#14271331	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14271539

C₂

yes

maximal, characteristic

#14271363

C₂

yes

maximal, complemented

#14270907

C₂

yes

maximal, characteristic, complemented

Q₈⋊#6597

C₂

yes

maximal, complemented

C₂₉×#559

C₂

yes

maximal, characteristic, complemented

C₂₉×#127

C₂²

yes

characteristic

C₂₉×#124

—

C₂₉×#123

—

C₄×Q₈×C₂₉

C₂²

yes

—

C₄×D₄×C₂₉

—

C₂×#90784

—

C₂×#90782

—

#11745236

C₂²

yes

characteristic

#11745157

—

C₄⋊#6597

C₂²

yes

—

C₄×#6597

C₂²

yes

characteristic

C₂×Q₈×C₂₉

D₄

yes

—

C₂×D₄×C₂₉

—

C₂×D₄×C₂₉

D₄

yes

—

C₂²×C₄×C₂₉

—

C₂×D₁₁₆

—

C₂×D₁₁₆

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

D₄

yes

—

C₂²⋊C₄×C₂₉

—

C₂²⋊C₄×C₂₉

—

C₄²×C₂₉

—

C₄²×C₂₉

C₂³

yes

—

—

—

—

C₂³×C₂₉

—

C₂²×D₂₉

—

Q₈×C₂₉

—

Q₈×C₂₉

—

D₄×C₂₉

—

D₄×C₂₉

—

D₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

—

D₁₁₆

—

D₁₁₆

—

D₁₁₆

—

—

—

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

#120

yes

characteristic

C₂×D₂₉

—

C₂×D₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

#135

yes

—

C₄×C₂₉

AGL(1,ℤ/8)

yes

—

#559

D₂₉

yes

characteristic, complemented

#493

—

#460

—

#331

—

Q₈⋊C₈

—

C₂×C₂₉

—

C₂×C₂₉

#449

yes

—

C₂×C₂₉

#453

yes

characteristic

D₂₉

—

C₂×SD₁₆

—

C₂×D₈

—

#127

—

#127

C₂×D₂₉

yes

characteristic

#124

—

#123

—

C₄×Q₈

C₂×D₂₉

yes

—

C₄×D₄

—

C₄⋊C₈

—

#102

—

C₄×C₈

—

C₂₉

#1620

yes

characteristic, complemented, Sylow

C₂×Q₈

C₂²⋊D₂₉

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₂₉

yes

—

C₂²×C₄

—

SD₁₆

—

116

—

D₈

—

116

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₂₉

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×D₂₉

yes

characteristic

C₂³

—

C₂³

—

116

—

100

Q₈

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

116

—

106

D₄

—

116

—

107

D₄

—

116

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂×C₂²⋊D₂₉

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₂₉

yes

—

113

C₈

—

114

C₈

—

116

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

232

—

118

C₂²

—

232

—

119

C₂²

#11745315

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

#11745311

yes

—

124

C₄

#11745312

yes

—

125

C₂

—

126

C₂

—

464

—

127

C₂

#14271308

yes

—

128

C₂

#14271331

yes

characteristic

	subgroup	G/H
center	C₂²	#11745315
commutator, cosocle	C₄²×C₂₉	C₂³
Frattini	C₄²	C₂²×D₂₉
Fitting	C₂₉×#559	C₂
socle	C₂²×C₂₉	#120
prime radical	#14271539	C₂
hypercenter	#559	D₂₉
hypocenter	C₂₉	#1620
LCS	G ▷ C₄²×C₂₉ ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #559

subgroup

G/H

center

C₂²

#11745315

commutator, cosocle

C₄²×C₂₉

C₂³

Frattini

C₄²

C₂²×D₂₉

Fitting

C₂₉×#559

C₂

C₂²×C₂₉

C₂

D₂₉

C₂₉

G ▷ C₄²×C₂₉ ▷ C₂²×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #559

N	H ≅ G/N	type
2. #14271363	C₂	semidirect
3. #14270907	C₂	semidirect
4. Q₈⋊#6597	C₂	semidirect
5. C₂₉×#559	C₂	semidirect
61. #559	D₂₉	semidirect
81. C₂₉	#1620	semidirect

H ≅ G/N

type

2. #14271363

C₂

semidirect

3. #14270907

C₂

semidirect

4. Q₈⋊#6597

C₂

semidirect

5. C₂₉×#559

C₂

semidirect

61. #559

D₂₉

semidirect

81. C₂₉

#1620

semidirect

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

subgroup	index
#14271539	2
#14271363	2
#14270907	2
Q₈⋊#6597	2
C₂₉×#559	2
#1620	29

subgroup

index

C₂₉×#559

Subgroups — #16818551

Conjugacy classes of subgroups

Complements

Maximal subgroups