Group #16817769 — subgroups

Subgroups — C₈×#90772 #16817769

Order 3712.

224 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 125 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	#14271214	C₂	1	1	yes	maximal, characteristic
2	C₄×#90772	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×D₂₉	C₂	1	1	yes	maximal, characteristic
4	#14270885	C₂	1	1	yes	maximal, characteristic, complemented
5	#14270884	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#6597	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈²×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#90772	C₄	1	2	yes	—
9	C₂×C₈×D₂₉	C₄	1	2	yes	—
10	C₄²×D₂₉	C₂²	1	1	yes	characteristic
11	C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
14	#11745168	C₄	1	2	yes	—
15	#11745163	C₄	1	2	yes	—
16	#11745162	C₄	1	2	yes	—
17	C₈×Dic₂₉	C₄	1	2	yes	—
18	C₄×#6597	C₂²	1	1	yes	characteristic
19	C₄×#6597	C₂²	1	1	yes	characteristic
20	C₄×#6597	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₂₉	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₂₉	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₂₉	C₂×C₄	1	2	yes	—
27	C₄²×C₂₉	C₂³	1	1	yes	characteristic
28	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
31	C₂×#6597	C₂×C₄	1	2	yes	—
32	C₂×#6597	C₂×C₄	1	2	yes	—
33	C₂×#6597	C₂×C₄	1	2	yes	—
34	#90772	C₈	1	8	yes	direct product factor
35	C₈×D₂₉	—	2	4	no	—
36	C₂²×D₂₉	C₄²	1	1	yes	characteristic
37	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×Dic₂₉	C₄²	1	1	yes	characteristic
41	C₂×Dic₂₉	C₄²	1	1	yes	characteristic
42	C₂×Dic₂₉	C₄²	1	1	yes	characteristic
43	C₄×D₂₉	—	2	2	no	—
44	C₄×D₂₉	—	2	2	no	—
45	C₄×D₂₉	C₂×C₈	1	4	yes	—
46	C₈×C₂₉	—	2	2	no	—
47	C₈×C₂₉	C₂×C₈	1	4	yes	complemented
48	C₈×C₂₉	M₁₆	1	4	yes	complemented
49	#6597	—	2	2	no	—
50	#6597	—	2	2	no	—
51	#6597	C₂×C₈	1	4	yes	complemented
52	C₈×M₁₆	—	29	1	no	maximal, Sylow
53	C₂²×C₂₉	C₂×C₄²	1	1	yes	characteristic
54	C₂×D₂₉	—	2	1	no	—
55	C₂×D₂₉	—	2	1	no	—
56	C₂×D₂₉	C₄×C₈	1	2	yes	—
57	C₄×C₂₉	C₂²×C₈	1	2	yes	—
58	C₄×C₂₉	C₂×M₁₆	1	2	yes	—
59	C₄×C₂₉	#131	1	2	yes	—
60	Dic₂₉	—	2	1	no	—
61	Dic₂₉	—	2	1	no	—
62	Dic₂₉	C₄×C₈	1	2	yes	—
63	#431	—	29	1	no	—
64	C₄×M₁₆	—	29	1	no	—
65	C₂×C₄×C₈	—	29	1	no	—
66	#322	—	29	1	no	—
67	#322	—	29	1	no	—
68	C₈²	—	29	1	no	—
69	C₈²	D₂₉	1	1	yes	characteristic, complemented
70	C₂×C₂₉	C₂×C₄×C₈	1	1	yes	characteristic
71	C₂×C₂₉	C₄×M₁₆	1	1	yes	characteristic
72	C₂×C₂₉	#405	1	1	yes	characteristic
73	D₂₉	—	2	2	no	—
74	C₂×M₁₆	—	29	2	no	—
75	C₂²×C₈	—	29	2	no	—
76	C₂×C₄²	—	29	1	no	—
77	C₄⋊C₈	—	29	2	no	—
78	C₂²⋊C₈	—	29	2	no	—
79	#97	—	29	2	no	—
80	C₄×C₈	—	29	2	no	—
81	C₄×C₈	—	29	1	no	—
82	C₄×C₈	—	29	1	no	—
83	C₄×C₈	—	29	1	no	—
84	C₄×C₈	C₂×D₂₉	1	1	yes	characteristic
85	C₄×C₈	C₂×D₂₉	1	1	yes	characteristic
86	C₄×C₈	C₂×D₂₉	1	1	yes	characteristic
87	C₂₉	C₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₄	—	29	1	no	—
89	C₂²×C₄	—	29	1	no	—
90	C₂²×C₄	—	29	1	no	—
91	M₁₆	—	29	8	no	—
92	C₂×C₈	—	29	2	no	—
93	C₂×C₈	—	29	2	no	—
94	C₂×C₈	—	29	2	no	—
95	C₂×C₈	—	58	4	no	—
96	C₂×C₈	C₄×D₂₉	1	2	yes	—
97	C₂×C₈	C₄×D₂₉	1	2	yes	—
98	C₂×C₈	C₄×D₂₉	1	2	yes	—
99	C₄²	—	29	1	no	—
100	C₄²	—	29	1	no	—
101	C₄²	—	29	1	no	—
102	C₄²	C₂²×D₂₉	1	1	yes	characteristic
103	C₂³	—	29	1	no	—
104	C₂×C₄	—	29	1	no	—
105	C₂×C₄	—	29	4	no	—
106	C₂×C₄	—	29	1	no	—
107	C₂×C₄	—	29	1	no	—
108	C₂×C₄	—	58	2	no	—
109	C₂×C₄	—	58	2	no	—
110	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
111	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
112	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	29	4	no	—
115	C₈	—	58	2	no	—
116	C₈	—	58	2	no	—
117	C₈	C₈×D₂₉	1	4	yes	complemented
118	C₈	#90772	1	4	yes	direct product factor
119	C₂²	—	29	2	no	—
120	C₂²	—	58	1	no	—
121	C₂²	—	58	1	no	—
122	C₂²	C₄²×D₂₉	1	1	yes	characteristic
123	C₄	—	29	2	no	—
124	C₄	—	58	1	no	—
125	C₄	—	58	1	no	—
126	C₄	C₂×C₈×D₂₉	1	2	yes	—
127	C₄	C₂×#90772	1	2	yes	—
128	C₄	#11745263	1	2	yes	—
129	C₂	—	58	2	no	—
130	C₂	C₄×C₈×D₂₉	1	1	yes	characteristic
131	C₂	C₄×#90772	1	1	yes	characteristic
132	C₂	#14271117	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14271214

C₂

yes

maximal, characteristic

C₄×#90772

C₂

yes

maximal, characteristic

C₄×C₈×D₂₉

C₂

yes

maximal, characteristic

#14270885

C₂

yes

maximal, characteristic, complemented

#14270884

C₂

yes

maximal, characteristic, complemented

C₈×#6597

C₂

yes

maximal, characteristic, complemented

C₈²×C₂₉

C₂

yes

maximal, characteristic, complemented

C₂×#90772

C₄

yes

—

C₂×C₈×D₂₉

C₄

yes

—

C₄²×D₂₉

C₂²

yes

characteristic

C₄×C₈×C₂₉

C₂²

yes

characteristic

C₄×C₈×C₂₉

C₂²

yes

characteristic

C₄×C₈×C₂₉

C₂²

yes

characteristic

#11745168

C₄

yes

—

#11745163

C₄

yes

—

#11745162

C₄

yes

—

C₈×Dic₂₉

C₄

yes

—

C₄×#6597

C₂²

yes

characteristic

C₄×#6597

C₂²

yes

characteristic

C₄×#6597

C₂²

yes

characteristic

C₂×C₄×D₂₉

C₂×C₄

yes

characteristic

C₂×C₄×D₂₉

C₂×C₄

yes

characteristic

C₂×C₄×D₂₉

C₂×C₄

yes

characteristic

C₂×C₈×C₂₉

C₂×C₄

yes

—

C₂×C₈×C₂₉

C₂×C₄

yes

—

C₂×C₈×C₂₉

C₂×C₄

yes

—

C₄²×C₂₉

C₂³

yes

characteristic

C₄×Dic₂₉

C₂×C₄

yes

characteristic

C₄×Dic₂₉

C₂×C₄

yes

characteristic

C₄×Dic₂₉

C₂×C₄

yes

characteristic

C₂×#6597

C₂×C₄

yes

—

C₂×#6597

C₂×C₄

yes

—

C₂×#6597

C₂×C₄

yes

—

#90772

C₈

yes

direct product factor

C₈×D₂₉

—

C₂²×D₂₉

C₄²

yes

characteristic

C₂×C₄×C₂₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉

C₂²×C₄

yes

characteristic

C₂×Dic₂₉

C₄²

yes

characteristic

C₂×Dic₂₉

C₄²

yes

characteristic

C₂×Dic₂₉

C₄²

yes

characteristic

C₄×D₂₉

—

C₄×D₂₉

—

C₄×D₂₉

C₂×C₈

yes

—

C₈×C₂₉

—

C₈×C₂₉

C₂×C₈

yes

complemented

C₈×C₂₉

M₁₆

yes

—

—

C₂×C₈

yes

complemented

C₈×M₁₆

—

maximal, Sylow

C₂²×C₂₉

C₂×C₄²

yes

characteristic

C₂×D₂₉

—

C₂×D₂₉

—

C₂×D₂₉

C₄×C₈

yes

—

C₄×C₂₉

C₂²×C₈

yes

—

C₄×C₂₉

C₂×M₁₆

yes

—

C₄×C₂₉

#131

yes

—

Dic₂₉

—

Dic₂₉

—

Dic₂₉

C₄×C₈

yes

—

#431

—

C₄×M₁₆

—

C₂×C₄×C₈

—

#322

—

#322

—

C₈²

—

C₈²

D₂₉

yes

characteristic, complemented

C₂×C₂₉

C₂×C₄×C₈

yes

characteristic

C₂×C₂₉

C₄×M₁₆

yes

characteristic

C₂×C₂₉

#405

yes

characteristic

D₂₉

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₂₉

yes

characteristic

C₄×C₈

C₂×D₂₉

yes

characteristic

C₄×C₈

C₂×D₂₉

yes

characteristic

C₂₉

C₈×M₁₆

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₂₉

yes

—

C₂×C₈

C₄×D₂₉

yes

—

C₂×C₈

C₄×D₂₉

yes

—

C₄²

—

100

C₄²

—

101

C₄²

—

102

C₄²

C₂²×D₂₉

yes

characteristic

103

C₂³

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₂×C₄×D₂₉

yes

characteristic

111

C₂×C₄

C₂×C₄×D₂₉

yes

characteristic

112

C₂×C₄

C₂×C₄×D₂₉

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

C₈×D₂₉

yes

complemented

118

C₈

#90772

yes

direct product factor

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₄²×D₂₉

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

C₂×C₈×D₂₉

yes

—

127

C₄

C₂×#90772

yes

—

128

C₄

#11745263

yes

—

129

C₂

—

130

C₂

C₄×C₈×D₂₉

yes

characteristic

131

C₂

C₄×#90772

yes

characteristic

132

C₂

#14271117

yes

characteristic

	subgroup	G/H
center	C₄×C₈	C₂×D₂₉
commutator	C₂×C₂₉	C₂×C₄×C₈
Frattini	C₄²	C₂²×D₂₉
Fitting	C₈²×C₂₉	C₂
socle	C₂²×C₂₉	C₂×C₄²
prime radical	C₂²×D₂₉	C₄²
cosocle	C₄²×C₂₉	C₂³
hypercenter	C₈²	D₂₉
hypocenter	C₂₉	C₈×M₁₆
LCS	G ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₄×C₈ ◁ C₈²

subgroup

G/H

center

C₄×C₈

C₂×D₂₉

commutator

C₂×C₂₉

C₂×C₄×C₈

Frattini

C₄²

C₂²×D₂₉

Fitting

C₈²×C₂₉

C₂

socle

C₂²×C₂₉

C₂×C₄²

prime radical

C₂²×D₂₉

C₄²

cosocle

C₄²×C₂₉

C₂³

hypercenter

C₈²

D₂₉

hypocenter

C₂₉

C₈×M₁₆

LCS

G ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₄×C₈ ◁ C₈²

N	H ≅ G/N	type
4. #14270885	C₂	semidirect
5. #14270884	C₂	semidirect
6. C₈×#6597	C₂	semidirect
7. C₈²×C₂₉	C₂	semidirect
34. #90772	C₈	direct (N × H)
47. C₈×C₂₉	C₂×C₈	semidirect
48. C₈×C₂₉	M₁₆	semidirect
51. #6597	C₂×C₈	semidirect
69. C₈²	D₂₉	semidirect
87. C₂₉	C₈×M₁₆	semidirect
117. C₈	C₈×D₂₉	semidirect
118. C₈	#90772	direct (N × H)

H ≅ G/N

type

4. #14270885

C₂

semidirect

5. #14270884

C₂

semidirect

6. C₈×#6597

C₂

semidirect

7. C₈²×C₂₉

C₂

semidirect

34. #90772

C₈

direct (N × H)

47. C₈×C₂₉

C₂×C₈

semidirect

48. C₈×C₂₉

M₁₆

semidirect

51. #6597

C₂×C₈

semidirect

69. C₈²

D₂₉

semidirect

87. C₂₉

C₈×M₁₆

semidirect

117. C₈

C₈×D₂₉

semidirect

118. C₈

#90772

direct (N × H)

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

subgroup	index
#14271214	2
C₄×#90772	2
C₄×C₈×D₂₉	2
#14270885	2
#14270884	2
C₈×#6597	2
C₈²×C₂₉	2
C₈×M₁₆	29

subgroup

index

C₄×C₈×D₂₉

C₈²×C₂₉

C₈×M₁₆

Subgroups — C₈×#90772 #16817769

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C8×#90772 #16817769

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#90772 #16817769