Group #16948974 — subgroups

Subgroups — #16948974

Order 4352.

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

normal subgroups only

This group has 306 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	#14531059	C₂	1	1	yes	maximal, characteristic, complemented
2	#14531007	C₂	1	2	yes	maximal, complemented
3	#14528373	C₂	1	2	yes	maximal, complemented
4	C₁₆×#3622	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#2246	C₂	1	1	yes	maximal, characteristic
6	#11763214	C₂²	1	2	yes	—
7	#11763181	—	2	2	no	—
8	#11763155	—	2	2	no	—
9	#11762635	—	2	2	no	—
10	#11762610	—	2	4	no	—
11	C₄×#63214	C₂²	1	1	yes	characteristic
12	C₈×#3622	C₂²	1	1	yes	characteristic
13	C₂×C₁₇×D₁₆	—	2	2	no	—
14	C₁₇×#493	C₂²	1	2	yes	—
15	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
16	C₂×D₈×C₁₇	—	2	2	no	—
17	C₂×D₈×C₁₇	—	2	2	no	—
18	C₁₇×#127	—	2	2	no	—
19	C₂×#63229	—	4	4	no	—
20	#10587555	—	2	2	no	—
21	C₁₇×D₁₆	—	4	4	no	—
22	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
23	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
24	#10587503	—	4	4	no	—
25	#10587489	—	2	2	no	—
26	#10587489	—	2	2	no	—
27	C₂×#63214	D₄	1	2	yes	—
28	C₄×#3622	D₄	1	1	yes	characteristic
29	C₄×#3622	D₄	1	1	yes	characteristic
30	C₂×D₄×C₁₇	—	4	2	no	—
31	C₂×D₄×C₁₇	—	4	2	no	—
32	C₂×D₄×C₁₇	—	4	2	no	—
33	C₂×Dic₃₄	—	4	2	no	—
34	D₈×C₁₇	—	4	4	no	—
35	D₈×C₁₇	—	4	4	no	—
36	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
38	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
39	#63229	—	8	8	no	—
40	C₄⋊Dic₁₇	—	4	2	no	—
41	C₄⋊Dic₁₇	—	4	2	no	—
42	C₂×#3622	D₈	1	2	yes	—
43	C₂×#3622	D₈	1	2	yes	—
44	C₁₆×C₁₇	SD₁₆	1	4	yes	complemented
45	#63214	—	2	2	no	—
46	#12503	—	17	1	no	maximal, Sylow
47	C₂³×C₁₇	—	8	2	no	—
48	D₄×C₁₇	—	8	2	no	—
49	D₄×C₁₇	—	8	2	no	—
50	D₄×C₁₇	—	8	4	no	—
51	D₄×C₁₇	—	8	4	no	—
52	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
53	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
55	C₂×Dic₁₇	—	8	2	no	—
56	Dic₃₄	—	8	4	no	—
57	C₈×C₁₇	C₂×SD₁₆	1	2	yes	—
58	C₈×C₁₇	#135	1	2	yes	—
59	#3622	—	2	2	no	—
60	#3622	D₁₆	1	4	yes	complemented
61	#2252	—	17	1	no	—
62	#2246	D₁₇	1	1	yes	characteristic
63	#2240	—	17	2	no	—
64	#1706	—	17	2	no	—
65	C₈×C₁₆	—	17	1	no	—
66	C₂²×C₁₇	—	16	2	no	—
67	C₂²×C₁₇	—	16	2	no	—
68	C₂²×C₁₇	—	16	2	no	—
69	C₂²×C₁₇	#493	1	1	yes	characteristic
70	C₄×C₁₇	#492	1	1	yes	characteristic
71	C₄×C₁₇	#495	1	1	yes	characteristic
72	C₄×C₁₇	C₂×D₁₆	1	2	yes	—
73	C₄×C₁₇	#508	1	2	yes	—
74	Dic₁₇	—	16	2	no	—
75	C₂×D₁₆	—	2	2	no	—
76	#500	—	17	2	no	—
77	#493	C₂×D₁₇	1	2	yes	—
78	#492	—	34	2	no	—
79	#486	—	34	2	no	—
80	#366	—	34	2	no	—
81	#357	—	34	4	no	—
82	C₄×C₁₆	—	17	1	no	—
83	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
84	C₈²	—	17	1	no	—
85	C₂×C₁₇	—	32	2	no	—
86	C₂×C₁₇	#1706	1	1	yes	characteristic
87	C₂×C₁₇	#2246	1	1	yes	characteristic
88	C₂×C₁₇	#2249	1	1	yes	characteristic
89	C₂×SD₁₆	—	68	4	no	—
90	C₂×D₈	—	2	2	no	—
91	C₂×D₈	—	2	2	no	—
92	C₄⋊Q₈	—	34	2	no	—
93	#127	—	2	2	no	—
94	D₁₆	—	4	4	no	—
95	C₂×C₁₆	—	17	2	no	—
96	C₂×C₁₆	C₂²⋊D₁₇	1	2	yes	—
97	#107	—	34	2	no	—
98	#107	—	34	2	no	—
99	#102	—	68	4	no	—
100	C₄×C₈	—	17	1	no	—
101	C₄×C₈	—	17	1	no	—
102	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
103	C₁₇	#12503	1	1	yes	characteristic, complemented, Sylow
104	C₂×Q₈	—	68	2	no	—
105	C₂×D₄	—	4	2	no	—
106	C₂×D₄	—	4	2	no	—
107	C₂×D₄	—	4	2	no	—
108	SD₁₆	—	136	8	no	—
109	D₈	—	4	4	no	—
110	D₈	—	4	4	no	—
111	C₂×C₈	—	17	2	no	—
112	C₂×C₈	—	17	2	no	—
113	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
114	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
115	C₄⋊C₄	—	68	2	no	—
116	C₄⋊C₄	—	68	2	no	—
117	C₄²	D₄×D₁₇	1	1	yes	characteristic
118	C₁₆	—	34	2	no	—
119	C₁₆	#63229	1	4	yes	complemented
120	C₂³	—	8	2	no	—
121	Q₈	—	136	4	no	—
122	D₄	—	8	4	no	—
123	D₄	—	8	4	no	—
124	D₄	—	8	2	no	—
125	D₄	—	8	2	no	—
126	C₂×C₄	—	136	2	no	—
127	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
128	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
129	C₂×C₄	#10587626	1	1	yes	characteristic
130	C₈	—	17	4	no	—
131	C₈	—	34	2	no	—
132	C₈	#10587598	1	2	yes	—
133	C₈	C₂×#63229	1	2	yes	—
134	C₂²	—	16	2	no	—
135	C₂²	—	16	2	no	—
136	C₂²	—	16	2	no	—
137	C₂²	#11763184	1	1	yes	characteristic
138	C₄	—	272	2	no	—
139	C₄	#11763181	1	1	yes	characteristic
140	C₄	#11763192	1	1	yes	characteristic
141	C₄	D₁₆×D₁₇	1	2	yes	—
142	C₄	#11763238	1	2	yes	—
143	C₂	—	32	2	no	—
144	C₂	#14528373	1	1	yes	characteristic
145	C₂	#14531032	1	1	yes	characteristic
146	C₂	#14531042	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14531059

C₂

yes

maximal, characteristic, complemented

#14531007

C₂

yes

maximal, complemented

#14528373

C₂

yes

maximal, complemented

C₁₆×#3622

C₂

yes

maximal, characteristic, complemented

C₁₇×#2246

C₂

yes

maximal, characteristic

#11763214

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₈×#3622

C₂²

yes

characteristic

C₂×C₁₇×D₁₆

—

C₁₇×#493

C₂²

yes

—

C₄×C₁₆×C₁₇

C₂²

yes

characteristic

C₂×D₈×C₁₇

—

C₂×D₈×C₁₇

—

C₁₇×#127

—

C₂×#63229

—

#10587555

—

C₁₇×D₁₆

—

C₂×C₁₆×C₁₇

D₄

yes

—

C₄×C₈×C₁₇

C₂³

yes

—

—

—

D₄

yes

—

C₄×#3622

D₄

yes

characteristic

C₄×#3622

D₄

yes

characteristic

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×Dic₃₄

—

D₈×C₁₇

—

D₈×C₁₇

—

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

characteristic

#63229

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₂×#3622

D₈

yes

—

C₂×#3622

D₈

yes

—

C₁₆×C₁₇

SD₁₆

yes

—

—

C₂³×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

#127

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×Dic₁₇

—

Dic₃₄

—

C₈×C₁₇

C₂×SD₁₆

yes

—

C₈×C₁₇

#135

yes

—

#3622

—

#3622

D₁₆

yes

complemented

#2252

—

#2246

D₁₇

yes

characteristic

#2240

—

#1706

—

C₈×C₁₆

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

C₂×D₁₆

yes

—

C₄×C₁₇

#508

yes

—

Dic₁₇

—

C₂×D₁₆

—

#500

—

#493

C₂×D₁₇

yes

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₁₇

yes

characteristic

C₈²

—

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₁₇

yes

—

#107

—

#107

—

#102

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

C₂²×D₁₇

yes

characteristic

103

C₁₇

#12503

yes

characteristic, complemented, Sylow

104

C₂×Q₈

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

SD₁₆

—

136

—

109

D₈

—

110

D₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

D₄×D₁₇

yes

characteristic

114

C₂×C₈

C₂×C₂²⋊D₁₇

yes

characteristic

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄²

D₄×D₁₇

yes

characteristic

118

C₁₆

—

119

C₁₆

#63229

yes

complemented

120

C₂³

—

121

Q₈

—

136

—

122

D₄

—

123

D₄

—

124

D₄

—

125

D₄

—

126

C₂×C₄

—

136

—

127

C₂×C₄

D₈×D₁₇

yes

characteristic

128

C₂×C₄

D₈×D₁₇

yes

characteristic

129

C₂×C₄

#10587626

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

#10587598

yes

—

133

C₈

C₂×#63229

yes

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

#11763184

yes

characteristic

138

C₄

—

272

—

139

C₄

#11763181

yes

characteristic

140

C₄

#11763192

yes

characteristic

141

C₄

D₁₆×D₁₇

yes

—

142

C₄

#11763238

yes

—

143

C₂

—

144

C₂

#14528373

yes

characteristic

145

C₂

#14531032

yes

characteristic

146

C₂

#14531042

yes

characteristic

	subgroup	G/H
center	C₂²	#11763184
commutator, cosocle	C₄×C₈×C₁₇	C₂³
Frattini	C₄×C₈	C₂²×D₁₇
Fitting, prime radical	C₁₇×#2246	C₂
socle	C₂²×C₁₇	#493
hypercenter	#2246	D₁₇
hypocenter	C₁₇	#12503
LCS	G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

subgroup

G/H

center

C₂²

#11763184

commutator, cosocle

C₄×C₈×C₁₇

C₂³

Frattini

C₄×C₈

C₂²×D₁₇

Fitting, prime radical

C₁₇×#2246

C₂

C₂²×C₁₇

D₁₇

C₁₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

N	H ≅ G/N	type
1. #14531059	C₂	semidirect
2. #14531007	C₂	semidirect
3. #14528373	C₂	semidirect
4. C₁₆×#3622	C₂	semidirect
44. C₁₆×C₁₇	SD₁₆	semidirect
60. #3622	D₁₆	semidirect
103. C₁₇	#12503	semidirect
119. C₁₆	#63229	semidirect

H ≅ G/N

type

1. #14531059

C₂

semidirect

2. #14531007

C₂

semidirect

3. #14528373

C₂

semidirect

4. C₁₆×#3622

C₂

semidirect

44. C₁₆×C₁₇

SD₁₆

semidirect

60. #3622

D₁₆

semidirect

103. C₁₇

#12503

semidirect

119. C₁₆

#63229

semidirect

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

subgroup	index
#14531059	2
#14531007	2
#14528373	2
C₁₆×#3622	2
C₁₇×#2246	2
#12503	17

subgroup

index

#14531059

#14531007

#14528373

C₁₆×#3622

C₁₇×#2246

#12503

Subgroups — #16948974

Conjugacy classes of subgroups

Complements

Maximal subgroups