Group #14524949 — subgroups

Subgroups — C₁₇×#1943 #14524949

Order 2176.

240 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 128 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	C₂×C₁₇×#107	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₇×#106	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₇×#97	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#389	C₂	1	2	yes	maximal
5	C₁₇×#340	C₂	1	2	yes	maximal
6	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
9	C₂×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
10	C₂×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
12	C₁₇×#107	C₄	1	4	yes	complemented
13	C₁₇×#106	—	2	2	no	—
14	C₁₇×#97	—	2	2	no	—
15	C₁₇×#95	—	2	2	no	—
16	C₂²×C₄×C₁₇	—	2	2	no	—
17	C₂²×C₄×C₁₇	—	2	2	no	—
18	C₂²×C₄×C₁₇	D₄	1	2	yes	—
19	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
20	C₂×C₈×C₁₇	—	2	1	no	—
21	C₂×C₈×C₁₇	—	2	1	no	—
22	C₂×C₈×C₁₇	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₁₇	D₄	1	2	yes	—
24	C₂×C₈×C₁₇	Q₈	1	2	yes	—
25	C₂×C₈×C₁₇	Q₈	1	2	yes	—
26	C₄⋊C₄×C₁₇	—	2	2	no	—
27	C₄⋊C₄×C₁₇	—	2	2	no	—
28	C₄⋊C₄×C₁₇	—	2	2	no	—
29	C₄⋊C₄×C₁₇	—	4	4	no	—
30	C₄⋊C₄×C₁₇	C₂×C₄	1	4	yes	—
31	C₄²×C₁₇	—	2	1	no	—
32	C₄²×C₁₇	—	2	1	no	—
33	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₇	—	2	4	no	—
35	C₂×C₄×C₁₇	—	2	2	no	—
36	C₂×C₄×C₁₇	—	2	2	no	—
37	C₂×C₄×C₁₇	—	2	2	no	—
38	C₂×C₄×C₁₇	—	4	2	no	—
39	C₂×C₄×C₁₇	—	4	2	no	—
40	C₂×C₄×C₁₇	—	4	2	no	—
41	C₂×C₄×C₁₇	—	4	2	no	—
42	C₂×C₄×C₁₇	—	4	2	no	—
43	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
48	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
49	C₈×C₁₇	—	2	2	no	—
50	C₈×C₁₇	C₄⋊C₄	1	4	yes	complemented
51	#1943	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₁₇	#121	1	1	yes	characteristic
54	C₂²×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
55	C₂²×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
56	C₂²×C₁₇	#137	1	2	yes	—
57	C₄×C₁₇	—	4	4	no	—
58	C₄×C₁₇	—	4	2	no	—
59	C₄×C₁₇	—	8	2	no	—
60	C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
62	C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
63	C₄×C₁₇	#125	1	1	yes	characteristic
64	C₂×#107	C₂×C₁₇	1	1	yes	characteristic
65	C₂×#106	C₂×C₁₇	1	1	yes	characteristic
66	C₂×#97	C₂×C₁₇	1	1	yes	characteristic
67	#389	C₂×C₁₇	1	2	yes	—
68	#340	C₂×C₁₇	1	2	yes	—
69	C₂×C₁₇	#389	1	1	yes	characteristic
70	C₂×C₁₇	#441	1	1	yes	characteristic
71	C₂×C₁₇	#442	1	1	yes	characteristic
72	C₂×C₁₇	#468	1	2	yes	—
73	C₂×C₁₇	#501	1	2	yes	—
74	C₂²×C₈	C₂²×C₁₇	1	1	yes	characteristic
75	C₂²×C₈	C₂²×C₁₇	1	1	yes	characteristic
76	C₂×C₄⋊C₄	—	2	2	no	—
77	C₂×C₄⋊C₄	C₂²×C₁₇	1	2	yes	—
78	C₂×C₄⋊C₄	C₂²×C₁₇	1	2	yes	—
79	C₂×C₄²	C₂²×C₁₇	1	1	yes	characteristic
80	#107	C₄×C₁₇	1	4	yes	complemented
81	#106	—	2	2	no	—
82	#97	—	2	2	no	—
83	#95	—	2	2	no	—
84	C₁₇	#1943	1	1	yes	characteristic, direct product factor, Sylow
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	D₄×C₁₇	1	2	yes	—
88	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	C₂×C₄×C₁₇	1	2	yes	—
92	C₂×C₈	D₄×C₁₇	1	2	yes	—
93	C₂×C₈	Q₈×C₁₇	1	2	yes	—
94	C₂×C₈	Q₈×C₁₇	1	2	yes	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	4	4	no	—
99	C₄⋊C₄	C₂×C₄×C₁₇	1	4	yes	—
100	C₄²	—	2	1	no	—
101	C₄²	—	2	1	no	—
102	C₂³	C₂×D₄×C₁₇	1	1	yes	characteristic
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
113	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
114	C₂×C₄	C₂×Q₈×C₁₇	1	1	yes	characteristic
115	C₂×C₄	C₂×Q₈×C₁₇	1	1	yes	characteristic
116	C₂×C₄	#41×C₁₇	1	1	yes	characteristic
117	C₂×C₄	#41×C₁₇	1	1	yes	characteristic
118	C₈	—	2	2	no	—
119	C₈	C₄⋊C₄×C₁₇	1	4	yes	complemented
120	C₂²	C₄×D₄×C₁₇	1	1	yes	characteristic
121	C₂²	C₁₇×#121	1	1	yes	characteristic
122	C₂²	C₁₇×C₄⋊Q₈	1	1	yes	characteristic
123	C₂²	C₁₇×AGL(1,ℤ/8)	1	2	yes	—
124	C₂²	C₁₇×#137	1	2	yes	—
125	C₄	—	4	4	no	—
126	C₄	—	4	2	no	—
127	C₄	—	8	2	no	—
128	C₄	C₂×C₄⋊C₄×C₁₇	1	1	yes	characteristic
129	C₄	C₄×Q₈×C₁₇	1	1	yes	characteristic
130	C₄	C₁₇×C₂²⋊Q₈	1	1	yes	characteristic
131	C₄	C₁₇×#125	1	1	yes	characteristic
132	C₂	C₁₇×#389	1	1	yes	characteristic
133	C₂	C₁₇×#441	1	1	yes	characteristic
134	C₂	C₁₇×#442	1	1	yes	characteristic
135	C₂	C₁₇×#468	1	2	yes	—
136	C₂	C₁₇×#501	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₇×#107

C₂

yes

maximal, characteristic

C₂×C₁₇×#106

C₂

yes

maximal, characteristic

C₂×C₁₇×#97

C₂

yes

maximal, characteristic

C₁₇×#389

C₂

yes

maximal

C₁₇×#340

C₂

yes

maximal

C₂²×C₈×C₁₇

C₂²

yes

characteristic

C₂²×C₈×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₇

C₂²

yes

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₁₇×#107

C₄

yes

complemented

C₁₇×#106

—

C₁₇×#97

—

C₁₇×#95

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

D₄

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂×C₄

yes

—

C₂×C₈×C₁₇

D₄

yes

—

C₂×C₈×C₁₇

Q₈

yes

—

C₂×C₈×C₁₇

Q₈

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

C₂×C₄

yes

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₈

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₈×C₁₇

—

C₈×C₁₇

C₄⋊C₄

yes

complemented

#1943

C₁₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₁₇

C₄×D₄

yes

characteristic

C₂²×C₁₇

#121

yes

characteristic

C₂²×C₁₇

C₄⋊Q₈

yes

characteristic

C₂²×C₁₇

AGL(1,ℤ/8)

yes

—

C₂²×C₁₇

#137

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₇

C₄×Q₈

yes

characteristic

C₄×C₁₇

C₂²⋊Q₈

yes

C₄×C₁₇

yes

C₂×C₁₇

yes

characteristic

C₂×#106

C₂×C₁₇

yes

characteristic

C₂×#97

C₂×C₁₇

yes

characteristic

#389

C₂×C₁₇

yes

—

#340

C₂×C₁₇

yes

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

—

C₂×C₁₇

#501

yes

—

C₂²×C₈

C₂²×C₁₇

yes

characteristic

C₂²×C₈

C₂²×C₁₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₁₇

yes

—

C₂×C₄⋊C₄

C₂²×C₁₇

yes

—

C₂×C₄²

C₂²×C₁₇

yes

characteristic

#107

C₄×C₁₇

yes

—

—

—

C₁₇

#1943

yes

characteristic, direct product factor, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₁₇

yes

—

C₂²×C₄

C₂³×C₁₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₁₇

yes

—

C₂×C₈

D₄×C₁₇

yes

—

C₂×C₈

Q₈×C₁₇

yes

—

C₂×C₈

Q₈×C₁₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₄×C₁₇

yes

—

100

C₄²

—

101

C₄²

—

102

C₂³

C₂×D₄×C₁₇

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂²×C₄×C₁₇

yes

characteristic

113

C₂×C₄

C₂×D₄×C₁₇

yes

characteristic

114

C₂×C₄

C₂×Q₈×C₁₇

yes

characteristic

115

C₂×C₄

C₂×Q₈×C₁₇

yes

characteristic

116

C₂×C₄

#41×C₁₇

yes

characteristic

117

C₂×C₄

#41×C₁₇

yes

characteristic

118

C₈

—

119

C₈

C₄⋊C₄×C₁₇

yes

complemented

120

C₂²

C₄×D₄×C₁₇

yes

characteristic

121

C₂²

C₁₇×#121

yes

characteristic

122

C₂²

C₁₇×C₄⋊Q₈

yes

characteristic

123

C₂²

C₁₇×AGL(1,ℤ/8)

yes

—

124

C₂²

C₁₇×#137

yes

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

C₂×C₄⋊C₄×C₁₇

yes

characteristic

129

C₄

C₄×Q₈×C₁₇

yes

characteristic

130

C₄

C₁₇×C₂²⋊Q₈

yes

characteristic

131

C₄

C₁₇×#125

yes

characteristic

132

C₂

C₁₇×#389

yes

characteristic

133

C₂

C₁₇×#441

yes

characteristic

134

C₂

C₁₇×#442

yes

characteristic

135

C₂

C₁₇×#468

yes

—

136

C₂

C₁₇×#501

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₁₇
Frattini, cosocle	C₂²×C₄	C₂³×C₁₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₁₇

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₁₇

Frattini, cosocle

C₂²×C₄

C₂³×C₁₇

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

N	H ≅ G/N	type
12. C₁₇×#107	C₄	semidirect
50. C₈×C₁₇	C₄⋊C₄	semidirect
51. #1943	C₁₇	direct (N × H)
80. #107	C₄×C₁₇	semidirect
84. C₁₇	#1943	direct (N × H)
119. C₈	C₄⋊C₄×C₁₇	semidirect

H ≅ G/N

type

12. C₁₇×#107

C₄

semidirect

50. C₈×C₁₇

C₄⋊C₄

semidirect

51. #1943

C₁₇

direct (N × H)

80. #107

C₄×C₁₇

semidirect

84. C₁₇

#1943

direct (N × H)

119. C₈

C₄⋊C₄×C₁₇

semidirect

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

subgroup	index
C₂×C₁₇×#107	2
C₂×C₁₇×#106	2
C₂×C₁₇×#97	2
C₁₇×#389	2
C₁₇×#340	2
#1943	17

subgroup

index

C₂×C₁₇×#107

C₂×C₁₇×#106

C₂×C₁₇×#97

C₁₇×#389

C₁₇×#340

#1943

Subgroups — C₁₇×#1943 #14524949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C17×#1943 #14524949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#1943 #14524949