Group #16869788 — subgroups

Subgroups — C₁₇×#12044 #16869788

Order 4352.

222 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 114 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	C₄×C₁₇×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	C₈×Q₁₆×C₁₇	C₂	1	2	yes	maximal
3	C₁₇×#1372	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#1332	C₂	1	2	yes	maximal
5	C₁₇×#1311	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₇×Q₃₂	C₄	1	1	yes	characteristic
7	C₈×Q₈×C₁₇	—	2	2	no	—
8	C₄×Q₁₆×C₁₇	C₂²	1	2	yes	—
9	C₁₇×#366	C₄	1	1	yes	characteristic
10	C₁₇×#358	C₄	1	2	yes	—
11	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
12	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
13	C₁₇×#335	C₂²	1	2	yes	—
14	C₁₇×Q₈⋊C₈	—	2	2	no	—
15	C₈²×C₁₇	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₁₇	C₂×C₄	1	2	yes	—
17	C₄×Q₈×C₁₇	—	2	2	no	—
18	C₁₇×Q₃₂	C₈	1	4	yes	complemented
19	C₂×C₁₆×C₁₇	—	2	1	no	—
20	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₁₇×#107	C₂×C₄	1	2	yes	—
23	C₁₇×C₄⋊C₈	—	2	2	no	—
24	C₁₇×C₄⋊C₈	—	4	2	no	—
25	C₁₇×#103	—	2	2	no	—
26	C₄×C₈×C₁₇	—	4	2	no	—
27	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
28	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
29	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
30	C₂×Q₈×C₁₇	—	2	2	no	—
31	Q₁₆×C₁₇	—	2	2	no	—
32	Q₁₆×C₁₇	C₂×C₈	1	4	yes	—
33	C₂×C₈×C₁₇	—	2	1	no	—
34	C₂×C₈×C₁₇	—	4	2	no	—
35	C₂×C₈×C₁₇	D₈	1	2	yes	—
36	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
38	C₄⋊C₄×C₁₇	—	2	2	no	—
39	C₄⋊C₄×C₁₇	—	4	2	no	—
40	C₄²×C₁₇	—	4	2	no	—
41	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₁₇	—	2	2	no	—
43	C₁₆×C₁₇	—	2	1	no	—
44	C₁₆×C₁₇	C₂×C₈	1	2	yes	—
45	#12044	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	Q₈×C₁₇	—	2	4	no	—
47	Q₈×C₁₇	—	4	2	no	—
48	C₂×C₄×C₁₇	—	4	2	no	—
49	C₂×C₄×C₁₇	—	4	2	no	—
50	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
53	C₈×C₁₇	—	2	1	no	—
54	C₈×C₁₇	—	2	2	no	—
55	C₈×C₁₇	—	4	1	no	—
56	C₈×C₁₇	—	8	2	no	—
57	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
58	C₈×C₁₇	#131	1	1	yes	characteristic
59	C₄×Q₃₂	C₂×C₁₇	1	1	yes	characteristic
60	C₈×Q₁₆	C₂×C₁₇	1	2	yes	—
61	#1372	C₂×C₁₇	1	1	yes	characteristic
62	#1332	C₂×C₁₇	1	2	yes	—
63	#1311	C₂×C₁₇	1	1	yes	characteristic
64	C₂²×C₁₇	C₄×D₈	1	1	yes	characteristic
65	C₄×C₁₇	—	2	1	no	—
66	C₄×C₁₇	—	4	4	no	—
67	C₄×C₁₇	—	8	2	no	—
68	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
69	C₄×C₁₇	#443	1	1	yes	characteristic
70	C₄×C₁₇	#510	1	2	yes	—
71	C₂×Q₃₂	C₄×C₁₇	1	1	yes	characteristic
72	C₈×Q₈	—	2	2	no	—
73	C₄×Q₁₆	C₂²×C₁₇	1	2	yes	—
74	#366	C₄×C₁₇	1	1	yes	characteristic
75	#358	C₄×C₁₇	1	2	yes	—
76	C₄×C₁₆	C₂²×C₁₇	1	1	yes	characteristic
77	C₄×C₁₆	C₂²×C₁₇	1	1	yes	characteristic
78	#335	C₂²×C₁₇	1	2	yes	—
79	Q₈⋊C₈	—	2	2	no	—
80	C₈²	C₂²×C₁₇	1	1	yes	characteristic
81	C₂×C₁₇	C₈×D₈	1	1	yes	characteristic
82	C₂×C₁₇	#2176	1	1	yes	characteristic
83	C₂×C₁₇	#2179	1	1	yes	characteristic
84	C₂×Q₁₆	C₂×C₄×C₁₇	1	2	yes	—
85	C₄×Q₈	—	2	2	no	—
86	Q₃₂	C₈×C₁₇	1	4	yes	complemented
87	C₂×C₁₆	—	2	1	no	—
88	C₂×C₁₆	C₂×C₄×C₁₇	1	1	yes	characteristic
89	C₂×C₁₆	C₂×C₄×C₁₇	1	1	yes	characteristic
90	#107	C₂×C₄×C₁₇	1	2	yes	—
91	C₄⋊C₈	—	2	2	no	—
92	C₄⋊C₈	—	4	2	no	—
93	#103	—	2	2	no	—
94	C₄×C₈	—	4	2	no	—
95	C₄×C₈	D₄×C₁₇	1	1	yes	characteristic
96	C₄×C₈	D₄×C₁₇	1	1	yes	characteristic
97	C₄×C₈	C₂³×C₁₇	1	1	yes	characteristic
98	C₁₇	#12044	1	1	yes	characteristic, direct product factor, Sylow
99	C₂×Q₈	—	2	2	no	—
100	Q₁₆	—	2	2	no	—
101	Q₁₆	C₂×C₈×C₁₇	1	4	yes	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	4	2	no	—
104	C₂×C₈	D₈×C₁₇	1	2	yes	—
105	C₂×C₈	C₂²×C₄×C₁₇	1	1	yes	characteristic
106	C₂×C₈	#41×C₁₇	1	1	yes	characteristic
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₄²	—	4	2	no	—
110	C₄²	C₂×D₄×C₁₇	1	1	yes	characteristic
111	C₁₆	—	2	2	no	—
112	C₁₆	—	2	1	no	—
113	C₁₆	C₂×C₈×C₁₇	1	2	yes	—
114	Q₈	—	2	4	no	—
115	Q₈	—	4	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	C₄×D₄×C₁₇	1	1	yes	characteristic
119	C₂×C₄	C₂×D₈×C₁₇	1	1	yes	characteristic
120	C₂×C₄	C₁₇×#135	1	1	yes	characteristic
121	C₈	—	2	2	no	—
122	C₈	—	2	1	no	—
123	C₈	—	4	1	no	—
124	C₈	—	8	2	no	—
125	C₈	C₂²×C₈×C₁₇	1	1	yes	characteristic
126	C₈	C₁₇×#131	1	1	yes	characteristic
127	C₂²	C₄×D₈×C₁₇	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	4	4	no	—
130	C₄	—	8	2	no	—
131	C₄	C₈×D₄×C₁₇	1	1	yes	characteristic
132	C₄	C₁₇×#443	1	1	yes	characteristic
133	C₄	C₁₇×#510	1	2	yes	—
134	C₂	C₈×D₈×C₁₇	1	1	yes	characteristic
135	C₂	C₁₇×#2176	1	1	yes	characteristic
136	C₂	C₁₇×#2179	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₁₇×Q₃₂

C₂

yes

maximal, characteristic

C₈×Q₁₆×C₁₇

C₂

yes

maximal

C₁₇×#1372

C₂

yes

maximal, characteristic

C₁₇×#1332

C₂

yes

maximal

C₁₇×#1311

C₂

yes

maximal, characteristic

C₂×C₁₇×Q₃₂

C₄

yes

characteristic

C₈×Q₈×C₁₇

—

C₄×Q₁₆×C₁₇

C₂²

yes

—

C₁₇×#366

C₄

yes

characteristic

C₁₇×#358

C₄

yes

—

C₄×C₁₆×C₁₇

C₂²

yes

characteristic

C₄×C₁₆×C₁₇

C₂²

yes

characteristic

C₁₇×#335

C₂²

yes

—

C₁₇×Q₈⋊C₈

—

C₈²×C₁₇

C₂²

yes

characteristic

C₂×Q₁₆×C₁₇

C₂×C₄

yes

—

C₄×Q₈×C₁₇

—

C₁₇×Q₃₂

C₈

yes

complemented

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₁₇×#107

C₂×C₄

yes

—

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₁₇×#103

—

C₄×C₈×C₁₇

—

C₄×C₈×C₁₇

D₄

yes

characteristic

C₄×C₈×C₁₇

D₄

yes

characteristic

C₄×C₈×C₁₇

C₂³

yes

characteristic

C₂×Q₈×C₁₇

—

Q₁₆×C₁₇

—

Q₁₆×C₁₇

C₂×C₈

yes

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

D₈

yes

—

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₇

#41

yes

characteristic

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₁₆×C₁₇

—

C₁₆×C₁₇

—

C₁₆×C₁₇

C₂×C₈

yes

—

#12044

C₁₇

yes

maximal, characteristic, direct product factor, Sylow

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₄×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

#135

yes

characteristic

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

C₂²×C₈

yes

characteristic

C₈×C₁₇

#131

yes

characteristic

C₄×Q₃₂

C₂×C₁₇

yes

characteristic

C₈×Q₁₆

C₂×C₁₇

yes

—

#1372

C₂×C₁₇

yes

characteristic

#1332

C₂×C₁₇

yes

—

#1311

C₂×C₁₇

yes

characteristic

C₂²×C₁₇

C₄×D₈

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₈×D₄

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

C₂×Q₃₂

C₄×C₁₇

yes

characteristic

C₈×Q₈

—

C₄×Q₁₆

C₂²×C₁₇

yes

—

#366

C₄×C₁₇

yes

characteristic

#358

C₄×C₁₇

yes

—

C₄×C₁₆

C₂²×C₁₇

yes

characteristic

C₄×C₁₆

C₂²×C₁₇

yes

characteristic

#335

C₂²×C₁₇

yes

—

Q₈⋊C₈

—

C₈²

C₂²×C₁₇

yes

characteristic

C₂×C₁₇

C₈×D₈

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×Q₁₆

C₂×C₄×C₁₇

yes

—

C₄×Q₈

—

Q₃₂

C₈×C₁₇

yes

complemented

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄×C₁₇

yes

characteristic

C₂×C₁₆

C₂×C₄×C₁₇

yes

characteristic

#107

C₂×C₄×C₁₇

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

D₄×C₁₇

yes

characteristic

C₄×C₈

D₄×C₁₇

yes

characteristic

C₄×C₈

C₂³×C₁₇

yes

characteristic

C₁₇

#12044

yes

characteristic, direct product factor, Sylow

C₂×Q₈

—

100

Q₁₆

—

101

Q₁₆

C₂×C₈×C₁₇

yes

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

D₈×C₁₇

yes

—

105

C₂×C₈

C₂²×C₄×C₁₇

yes

characteristic

106

C₂×C₈

#41×C₁₇

yes

characteristic

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

—

110

C₄²

C₂×D₄×C₁₇

yes

characteristic

111

C₁₆

—

112

C₁₆

—

113

C₁₆

C₂×C₈×C₁₇

yes

—

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₄×D₄×C₁₇

yes

characteristic

119

C₂×C₄

C₂×D₈×C₁₇

yes

characteristic

120

C₂×C₄

C₁₇×#135

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

C₂²×C₈×C₁₇

yes

characteristic

126

C₈

C₁₇×#131

yes

characteristic

127

C₂²

C₄×D₈×C₁₇

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

C₈×D₄×C₁₇

yes

characteristic

132

C₄

C₁₇×#443

yes

characteristic

133

C₄

C₁₇×#510

yes

—

134

C₂

C₈×D₈×C₁₇

yes

characteristic

135

C₂

C₁₇×#2176

yes

characteristic

136

C₂

C₁₇×#2179

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₁₇	C₂×D₈
commutator	C₈	C₂²×C₈×C₁₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₇
socle, prime radical	C₂²×C₁₇	C₄×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₇ ◁ C₄×C₈×C₁₇ ◁ C₈²×C₁₇ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₇

C₂×D₈

commutator

C₈

C₂²×C₈×C₁₇

Frattini, cosocle

C₄×C₈

C₂³×C₁₇

socle, prime radical

C₂²×C₁₇

C₄×D₈

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₁₇ ◁ C₄×C₈×C₁₇ ◁ C₈²×C₁₇ ◁ G

N	H ≅ G/N	type
18. C₁₇×Q₃₂	C₈	semidirect
45. #12044	C₁₇	direct (N × H)
86. Q₃₂	C₈×C₁₇	semidirect
98. C₁₇	#12044	direct (N × H)

H ≅ G/N

type

18. C₁₇×Q₃₂

C₈

semidirect

45. #12044

C₁₇

direct (N × H)

86. Q₃₂

C₈×C₁₇

semidirect

98. C₁₇

#12044

direct (N × H)

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

subgroup	index
C₄×C₁₇×Q₃₂	2
C₈×Q₁₆×C₁₇	2
C₁₇×#1372	2
C₁₇×#1332	2
C₁₇×#1311	2
#12044	17

subgroup

index

C₄×C₁₇×Q₃₂

C₈×Q₁₆×C₁₇

C₁₇×#1372

C₁₇×#1332

C₁₇×#1311

#12044

Subgroups — C₁₇×#12044 #16869788

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C17×#12044 #16869788

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#12044 #16869788