Group #16949058 — subgroups

Subgroups — #16949058

Order 4352.

316 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 87 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	#14531039	C₂	1	2	yes	maximal
2	#14531036	C₂	1	2	yes	maximal, complemented
3	#14528402	C₂	1	1	yes	maximal, characteristic, complemented
4	#14526701	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#1709	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587500	—	2	4	no	—
7	C₂×#10587498	—	2	4	no	—
8	#11763214	—	2	2	no	—
9	#11763189	C₂²	1	2	yes	—
10	C₄×#63214	C₂²	1	2	yes	—
11	C₁₇×#494	C₂²	1	1	yes	characteristic
12	C₁₇×#493	C₂²	1	1	yes	characteristic
13	C₁₇×#486	—	2	2	no	—
14	C₈²×C₁₇	C₂²	1	1	yes	characteristic
15	C₂×Q₁₆×C₁₇	—	2	2	no	—
16	C₂×D₈×C₁₇	—	2	2	no	—
17	C₁₇×C₄⋊Q₈	—	2	1	no	—
18	C₁₇×#127	—	2	1	no	—
19	C₂×Dic₆₈	—	2	2	no	—
20	C₂×Dic₆₈	—	2	2	no	—
21	#10587555	—	2	2	no	—
22	C₁₇×#102	—	4	4	no	—
23	C₄×C₈×C₁₇	D₄	1	2	yes	—
24	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
25	#10587500	—	4	8	no	—
26	#10587498	—	4	8	no	—
27	#10587489	—	4	4	no	—
28	C₂×#63214	D₄	1	4	yes	—
29	C₂×Q₈×C₁₇	—	4	1	no	—
30	C₂×D₄×C₁₇	—	4	1	no	—
31	C₂×D₄×C₁₇	—	4	2	no	—
32	C₂×Dic₃₄	—	4	2	no	—
33	Q₁₆×C₁₇	—	4	4	no	—
34	D₈×C₁₇	—	4	4	no	—
35	C₂×C₈×C₁₇	D₈	1	4	yes	—
36	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
37	C₄⋊C₄×C₁₇	—	4	2	no	—
38	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
39	C₄⋊Dic₁₇	—	4	4	no	—
40	Dic₆₈	—	4	4	no	—
41	Dic₆₈	—	4	4	no	—
42	#63214	D₈	1	8	yes	—
43	#12520	—	17	1	no	maximal, Sylow
44	C₂³×C₁₇	—	8	1	no	—
45	Q₈×C₁₇	—	8	2	no	—
46	D₄×C₁₇	—	8	2	no	—
47	D₄×C₁₇	—	8	4	no	—
48	C₂×C₄×C₁₇	—	8	1	no	—
49	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
50	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
51	C₂×Dic₁₇	—	8	2	no	—
52	Dic₃₄	—	8	2	no	—
53	Dic₃₄	—	8	2	no	—
54	C₈×C₁₇	—	2	4	no	—
55	C₈×C₁₇	C₂×D₈	1	4	yes	—
56	#2248	—	17	2	no	—
57	#2247	—	17	2	no	—
58	#1713	—	17	1	no	—
59	#1709	D₁₇	1	1	yes	characteristic
60	#1311	—	17	1	no	—
61	C₂²×C₁₇	—	16	1	no	—
62	C₂²×C₁₇	—	16	2	no	—
63	C₂²×C₁₇	#493	1	1	yes	characteristic
64	C₄×C₁₇	—	16	1	no	—
65	C₄×C₁₇	#493	1	2	yes	—
66	C₄×C₁₇	#510	1	4	yes	—
67	Dic₁₇	—	16	2	no	—
68	C₂×Q₃₂	—	34	4	no	—
69	C₂×SD₃₂	—	34	4	no	—
70	#500	—	34	2	no	—
71	#494	—	17	2	no	—
72	#494	C₂×D₁₇	1	1	yes	characteristic
73	#493	C₂×D₁₇	1	1	yes	characteristic
74	#486	—	2	2	no	—
75	C₄×C₁₆	—	17	2	no	—
76	C₈²	C₂×D₁₇	1	1	yes	characteristic
77	C₂×C₁₇	—	32	1	no	—
78	C₂×C₁₇	#1712	1	1	yes	characteristic
79	C₂×C₁₇	#2251	1	2	yes	—
80	C₂×Q₁₆	—	2	2	no	—
81	C₂×Q₁₆	—	34	2	no	—
82	C₂×Q₁₆	—	34	2	no	—
83	C₂×D₈	—	2	2	no	—
84	C₄⋊Q₈	—	2	1	no	—
85	C₄⋊Q₈	—	34	2	no	—
86	#127	—	2	1	no	—
87	Q₃₂	—	68	8	no	—
88	SD₃₂	—	68	8	no	—
89	C₂×C₁₆	—	17	4	no	—
90	#107	—	68	4	no	—
91	#102	—	4	4	no	—
92	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
93	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
94	C₁₇	#12520	1	1	yes	characteristic, complemented, Sylow
95	C₂×Q₈	—	4	1	no	—
96	C₂×Q₈	—	68	2	no	—
97	C₂×D₄	—	4	2	no	—
98	C₂×D₄	—	4	1	no	—
99	Q₁₆	—	4	4	no	—
100	Q₁₆	—	68	4	no	—
101	Q₁₆	—	68	4	no	—
102	D₈	—	4	4	no	—
103	C₂×C₈	#63228	1	4	yes	—
104	C₂×C₈	D₄×D₁₇	1	2	yes	—
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	68	4	no	—
107	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
108	C₁₆	—	17	8	no	—
109	C₂³	—	8	1	no	—
110	Q₈	—	8	2	no	—
111	Q₈	—	136	2	no	—
112	Q₈	—	136	2	no	—
113	D₄	—	8	4	no	—
114	D₄	—	8	2	no	—
115	C₂×C₄	—	8	1	no	—
116	C₂×C₄	—	136	2	no	—
117	C₂×C₄	C₂×#63228	1	2	yes	—
118	C₂×C₄	#10587626	1	1	yes	characteristic
119	C₈	—	2	4	no	—
120	C₈	D₈×D₁₇	1	4	yes	—
121	C₂²	—	16	1	no	—
122	C₂²	—	16	2	no	—
123	C₂²	#11763185	1	1	yes	characteristic
124	C₄	—	16	1	no	—
125	C₄	—	272	2	no	—
126	C₄	#11763184	1	2	yes	—
127	C₄	#11763249	1	4	yes	—
128	C₂	—	32	1	no	—
129	C₂	#14528398	1	1	yes	characteristic
130	C₂	#14531056	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14531039

C₂

yes

maximal

#14531036

C₂

yes

maximal, complemented

#14528402

C₂

yes

maximal, characteristic, complemented

#14526701

C₂

yes

maximal, characteristic, complemented

C₁₇×#1709

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

C₄×#63214

C₂²

yes

—

C₁₇×#494

C₂²

yes

characteristic

C₁₇×#493

C₂²

yes

characteristic

C₁₇×#486

—

C₈²×C₁₇

C₂²

yes

characteristic

C₂×Q₁₆×C₁₇

—

C₂×D₈×C₁₇

—

C₁₇×C₄⋊Q₈

—

C₁₇×#127

—

C₂×Dic₆₈

—

C₂×Dic₆₈

—

#10587555

—

C₁₇×#102

—

C₄×C₈×C₁₇

D₄

yes

—

C₄×C₈×C₁₇

C₂³

yes

—

—

—

D₄

yes

—

C₂×Q₈×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×Dic₃₄

—

Q₁₆×C₁₇

—

D₈×C₁₇

—

C₂×C₈×C₁₇

D₈

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄⋊Dic₁₇

—

Dic₆₈

—

Dic₆₈

—

#63214

D₈

yes

—

#12520

—

maximal, Sylow

C₂³×C₁₇

—

Q₈×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#127

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

C₂×D₈

yes

—

—

—

—

D₁₇

yes

characteristic

#1311

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#493

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

#493

yes

—

C₄×C₁₇

#510

yes

—

Dic₁₇

—

C₂×Q₃₂

—

C₂×SD₃₂

—

#500

—

#494

—

#494

C₂×D₁₇

yes

characteristic

#493

C₂×D₁₇

yes

characteristic

#486

—

C₄×C₁₆

—

C₈²

C₂×D₁₇

yes

characteristic

C₂×C₁₇

—

C₂×C₁₇

#1712

yes

characteristic

C₂×C₁₇

#2251

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

Q₃₂

—

SD₃₂

—

C₂×C₁₆

—

#107

—

#102

—

C₄×C₈

C₂²⋊D₁₇

yes

—

C₄×C₈

C₂²×D₁₇

yes

characteristic

C₁₇

#12520

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

100

Q₁₆

—

101

Q₁₆

—

102

D₈

—

103

C₂×C₈

#63228

yes

—

104

C₂×C₈

D₄×D₁₇

yes

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄²

C₂×C₂²⋊D₁₇

yes

characteristic

108

C₁₆

—

109

C₂³

—

110

Q₈

—

111

Q₈

—

136

—

112

Q₈

—

136

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

136

—

117

C₂×C₄

C₂×#63228

yes

—

118

C₂×C₄

#10587626

yes

characteristic

119

C₈

—

120

C₈

D₈×D₁₇

yes

—

121

C₂²

—

122

C₂²

—

123

C₂²

#11763185

yes

characteristic

124

C₄

—

125

C₄

—

272

—

126

C₄

#11763184

yes

—

127

C₄

#11763249

yes

—

128

C₂

—

129

C₂

#14528398

yes

characteristic

130

C₂

#14531056

yes

—

	subgroup	G/H
center	C₂²	#11763185
commutator, cosocle	C₄×C₈×C₁₇	C₂³
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#1709	C₂
socle	C₂²×C₁₇	#493
prime radical	C₁₇×#493	C₂²
hypercenter	#1709	D₁₇
hypocenter	C₁₇	#12520
LCS	G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#11763185

commutator, cosocle

C₄×C₈×C₁₇

C₂³

Frattini

C₄×C₈

C₂²×D₁₇

Fitting

C₁₇×#1709

C₂

socle

C₂²×C₁₇

#493

prime radical

C₁₇×#493

C₂²

D₁₇

C₁₇

G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
2. #14531036	C₂	semidirect
3. #14528402	C₂	semidirect
4. #14526701	C₂	semidirect
94. C₁₇	#12520	semidirect

H ≅ G/N

type

2. #14531036

C₂

semidirect

3. #14528402

C₂

semidirect

4. #14526701

C₂

semidirect

94. C₁₇

#12520

semidirect

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

subgroup	index
#14531039	2
#14531036	2
#14528402	2
#14526701	2
C₁₇×#1709	2
#12520	17

subgroup

index

C₁₇×#1709

Subgroups — #16949058

Conjugacy classes of subgroups

Complements

Maximal subgroups