Group #16945985 — subgroups

Subgroups — #16945985

Order 4352.

180 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 75 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	#14530599	C₂	1	2	yes	maximal
2	#14530379	C₂	1	1	yes	maximal, characteristic
3	#14526860	C₂	1	1	yes	maximal, characteristic, complemented
4	#14526851	C₂	1	2	yes	maximal, complemented
5	C₁₇×#1373	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10587469	C₄	1	1	yes	characteristic
7	C₂×C₁₆×D₁₇	C₄	1	1	yes	characteristic
8	#11762763	C₂²	1	1	yes	characteristic
9	#11762630	C₂²	1	2	yes	—
10	#11762590	—	2	2	no	—
11	#11762586	—	2	2	no	—
12	#11762585	C₂²	1	1	yes	characteristic
13	#11762584	C₄	1	1	yes	characteristic
14	C₁₆×Dic₁₇	C₄	1	1	yes	characteristic
15	C₁₇×#364	C₂²	1	2	yes	—
16	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×#63218	D₄	1	1	yes	characteristic
18	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
19	#10587558	Q₈	1	1	yes	characteristic
20	C₁₇×M₃₂	—	2	2	no	—
21	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
24	#10587486	D₄	1	1	yes	characteristic
25	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
26	#10587483	—	2	2	no	—
27	C₂×#63214	C₂×C₄	1	1	yes	characteristic
28	C₂×#63214	C₂×C₄	1	1	yes	characteristic
29	#10587474	Q₈	1	1	yes	characteristic
30	#10587469	—	2	2	no	—
31	C₁₆×D₁₇	—	2	2	no	—
32	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
35	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
36	#63227	—	2	1	no	—
37	#63225	—	2	1	no	—
38	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
40	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
41	#63218	—	2	1	no	—
42	#63218	—	2	1	no	—
43	C₈×D₁₇	—	2	1	no	—
44	C₈×D₁₇	—	2	1	no	—
45	C₁₆×C₁₇	—	2	1	no	—
46	C₁₆×C₁₇	—	4	2	no	—
47	C₁₆×C₁₇	M₁₆	1	2	yes	—
48	#63214	—	2	1	no	—
49	#63214	—	4	2	no	—
50	#63214	M₁₆	1	2	yes	—
51	#12027	—	17	1	no	maximal, Sylow
52	C₂²×D₁₇	#108	1	1	yes	characteristic
53	C₂×C₄×C₁₇	—	2	1	no	—
54	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×Dic₁₇	—	2	1	no	—
56	C₂×Dic₁₇	#108	1	1	yes	characteristic
57	C₄×D₁₇	—	2	1	no	—
58	C₄×D₁₇	—	2	1	no	—
59	C₈×C₁₇	C₂×M₁₆	1	1	yes	characteristic
60	C₈×C₁₇	C₂×M₁₆	1	1	yes	characteristic
61	C₈×C₁₇	#135	1	1	yes	characteristic
62	C₈×C₁₇	#135	1	1	yes	characteristic
63	#3622	—	2	1	no	—
64	#3622	—	2	1	no	—
65	#2166	—	17	2	no	—
66	#2108	—	17	1	no	—
67	#1373	—	17	1	no	—
68	#1373	D₁₇	1	1	yes	characteristic, complemented
69	#1369	—	17	2	no	—
70	C₂²×C₁₇	C₂×#108	1	1	yes	characteristic
71	C₂×D₁₇	—	2	1	no	—
72	C₂×D₁₇	—	4	1	no	—
73	C₄×C₁₇	—	4	1	no	—
74	C₄×C₁₇	#423	1	1	yes	characteristic
75	C₄×C₁₇	#427	1	1	yes	characteristic
76	Dic₁₇	—	2	1	no	—
77	Dic₁₇	—	4	1	no	—
78	C₂×M₃₂	—	17	1	no	—
79	C₂²×C₁₆	—	17	1	no	—
80	#405	—	17	1	no	—
81	#364	—	17	2	no	—
82	#364	C₂×D₁₇	1	2	yes	—
83	#349	—	34	2	no	—
84	#347	—	34	2	no	—
85	#346	—	17	1	no	—
86	#346	—	17	1	no	—
87	C₄×C₁₆	—	17	1	no	—
88	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
89	C₂×C₁₇	—	2	1	no	—
90	C₂×C₁₇	#1568	1	1	yes	characteristic
91	D₁₇	—	4	1	no	—
92	C₂×M₁₆	—	17	1	no	—
93	C₂²×C₈	—	17	1	no	—
94	#117	—	17	1	no	—
95	M₃₂	—	2	2	no	—
96	M₃₂	—	34	2	no	—
97	M₃₂	—	34	2	no	—
98	C₂×C₁₆	—	17	1	no	—
99	C₂×C₁₆	—	17	1	no	—
100	C₂×C₁₆	—	34	2	no	—
101	C₂×C₁₆	C₄×D₁₇	1	1	yes	characteristic
102	C₂×C₁₆	C₄×D₁₇	1	1	yes	characteristic
103	#97	—	17	1	no	—
104	#97	—	17	1	no	—
105	C₄×C₈	—	17	1	no	—
106	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
107	C₁₇	#12027	1	1	yes	characteristic, complemented, Sylow
108	C₂²×C₄	—	17	1	no	—
109	M₁₆	—	34	1	no	—
110	M₁₆	—	34	1	no	—
111	C₂×C₈	—	17	1	no	—
112	C₂×C₈	—	17	1	no	—
113	C₂×C₈	—	34	1	no	—
114	C₂×C₈	—	34	1	no	—
115	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
116	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
117	C₄⋊C₄	—	34	1	no	—
118	C₂²⋊C₄	—	34	1	no	—
119	C₄²	—	17	1	no	—
120	C₄²	Q₈×D₁₇	1	1	yes	characteristic
121	C₁₆	—	2	1	no	—
122	C₁₆	—	4	2	no	—
123	C₁₆	—	17	2	no	—
124	C₁₆	—	34	1	no	—
125	C₁₆	—	68	2	no	—
126	C₁₆	#63218	1	2	yes	—
127	C₂³	—	17	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	17	1	no	—
130	C₂×C₄	—	34	1	no	—
131	C₂×C₄	—	34	1	no	—
132	C₂×C₄	—	34	1	no	—
133	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
134	C₈	—	34	1	no	—
135	C₈	—	34	1	no	—
136	C₈	C₂×#63218	1	1	yes	characteristic
137	C₈	M₁₆×D₁₇	1	1	yes	characteristic
138	C₈	#10587598	1	1	yes	characteristic
139	C₈	#10587605	1	1	yes	characteristic
140	C₂²	—	34	1	no	—
141	C₂²	—	68	1	no	—
142	C₂²	#108×D₁₇	1	1	yes	characteristic
143	C₄	—	4	1	no	—
144	C₄	—	34	1	no	—
145	C₄	—	68	1	no	—
146	C₄	#11762831	1	1	yes	characteristic
147	C₄	#11762846	1	1	yes	characteristic
148	C₂	—	2	1	no	—
149	C₂	—	68	1	no	—
150	C₂	#14527546	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14530599

C₂

yes

maximal

#14530379

C₂

yes

maximal, characteristic

#14526860

C₂

yes

maximal, characteristic, complemented

#14526851

C₂

yes

maximal, complemented

C₁₇×#1373

C₂

yes

maximal, characteristic, complemented

C₂×#10587469

C₄

yes

characteristic

C₂×C₁₆×D₁₇

C₄

yes

characteristic

#11762763

C₂²

yes

characteristic

#11762630

C₂²

yes

—

#11762590

—

#11762586

—

#11762585

C₂²

yes

characteristic

#11762584

C₄

yes

characteristic

C₁₆×Dic₁₇

C₄

yes

characteristic

C₁₇×#364

C₂²

yes

—

C₄×C₁₆×C₁₇

C₂²

yes

characteristic

C₂×#63218

D₄

yes

characteristic

C₂×C₈×D₁₇

C₂×C₄

yes

characteristic

#10587558

Q₈

yes

characteristic

C₁₇×M₃₂

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

C₂³

yes

characteristic

#10587486

D₄

yes

characteristic

C₈×Dic₁₇

C₂×C₄

yes

characteristic

#10587483

—

C₂×#63214

C₂×C₄

yes

characteristic

C₂×#63214

C₂×C₄

yes

characteristic

#10587474

Q₈

yes

characteristic

#10587469

—

C₁₆×D₁₇

—

C₂×C₄×D₁₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

C₂×Q₈

yes

characteristic

#63227

—

#63225

—

C₄×Dic₁₇

C₄⋊C₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

#63218

—

#63218

—

C₈×D₁₇

—

C₈×D₁₇

—

C₁₆×C₁₇

—

C₁₆×C₁₇

—

C₁₆×C₁₇

M₁₆

yes

—

#63214

—

#63214

—

#63214

M₁₆

yes

—

—

C₂²×D₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

#108

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

—

C₈×C₁₇

C₂×M₁₆

yes

characteristic

C₈×C₁₇

C₂×M₁₆

yes

C₈×C₁₇

yes

C₈×C₁₇

yes

—

—

—

—

—

D₁₇

yes

characteristic, complemented

#1369

—

C₂²×C₁₇

C₂×#108

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

#364

—

#364

C₂×D₁₇

yes

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₁₇

yes

characteristic

C₂×C₁₇

—

C₂×C₁₇

#1568

yes

characteristic

D₁₇

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

C₄×D₁₇

yes

characteristic

102

C₂×C₁₆

C₄×D₁₇

yes

characteristic

103

#97

—

104

#97

—

105

C₄×C₈

—

106

C₄×C₈

C₂²×D₁₇

yes

characteristic

107

C₁₇

#12027

yes

characteristic, complemented, Sylow

108

C₂²×C₄

—

109

M₁₆

—

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

C₂×C₄×D₁₇

yes

characteristic

116

C₂×C₈

D₄×D₁₇

yes

characteristic

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

119

C₄²

—

120

C₄²

Q₈×D₁₇

yes

characteristic

121

C₁₆

—

122

C₁₆

—

123

C₁₆

—

124

C₁₆

—

125

C₁₆

—

126

C₁₆

#63218

yes

—

127

C₂³

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₄⋊C₄×D₁₇

yes

characteristic

134

C₈

—

135

C₈

—

136

C₈

C₂×#63218

yes

characteristic

137

C₈

M₁₆×D₁₇

yes

characteristic

138

C₈

#10587598

yes

characteristic

139

C₈

#10587605

yes

characteristic

140

C₂²

—

141

C₂²

—

142

C₂²

#108×D₁₇

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

#11762831

yes

characteristic

147

C₄

#11762846

yes

characteristic

148

C₂

—

149

C₂

—

150

C₂

#14527546

yes

characteristic

	subgroup	G/H
center	C₄	#11762846
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#1373	C₂
socle	C₂×C₁₇	#1568
prime radical	C₂²×D₁₇	#108
cosocle	C₄×C₈×C₁₇	C₂³
hypercenter	#1373	D₁₇
hypocenter	C₁₇	#12027
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1373

subgroup

G/H

center

C₄

#11762846

commutator

C₂×C₈×C₁₇

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₁₇

Fitting

C₁₇×#1373

C₂

C₂×C₁₇

C₂²×D₁₇

C₄×C₈×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1373

N	H ≅ G/N	type
3. #14526860	C₂	semidirect
4. #14526851	C₂	semidirect
5. C₁₇×#1373	C₂	semidirect
68. #1373	D₁₇	semidirect
107. C₁₇	#12027	semidirect

H ≅ G/N

type

3. #14526860

C₂

semidirect

4. #14526851

C₂

semidirect

5. C₁₇×#1373

C₂

semidirect

68. #1373

D₁₇

semidirect

107. C₁₇

#12027

semidirect

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

subgroup	index
#14530599	2
#14530379	2
#14526860	2
#14526851	2
C₁₇×#1373	2
#12027	17

subgroup

index

C₁₇×#1373

Subgroups — #16945985

Conjugacy classes of subgroups

Complements

Maximal subgroups