Group #16948959 — subgroups

Subgroups — #16948959

Order 4352.

212 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 69 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	#14530937	C₂	1	2	yes	maximal
2	#14528402	C₂	1	1	yes	maximal, characteristic
3	#14527682	C₂	1	1	yes	maximal, characteristic
4	#14526841	C₂	1	2	yes	maximal
5	C₁₇×#1367	C₂	1	1	yes	maximal, characteristic
6	#11763214	—	2	1	no	—
7	#11763214	C₂²	1	2	yes	—
8	#11763189	—	2	1	no	—
9	#11762939	—	2	1	no	—
10	C₄×Dic₆₈	C₂²	1	1	yes	characteristic
11	#11762635	—	2	2	no	—
12	#11762614	—	2	2	no	—
13	#11762562	C₂²	1	1	yes	characteristic
14	#11762541	—	2	1	no	—
15	C₁₇×C₄⋊C₁₆	C₂²	1	2	yes	—
16	C₈²×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₆₈	—	4	2	no	—
18	C₂×Dic₆₈	D₄	1	1	yes	characteristic
19	#10587555	—	2	2	no	—
20	C₄×Dic₃₄	—	2	1	no	—
21	C₂×C₁₆×C₁₇	—	2	2	no	—
22	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
23	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
24	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
25	#10587489	—	2	2	no	—
26	#10587489	—	2	2	no	—
27	#10587489	—	4	1	no	—
28	#10587489	—	4	1	no	—
29	#10587489	D₄	1	1	yes	characteristic
30	#10587487	—	2	1	no	—
31	#10587486	—	4	1	no	—
32	#10587485	—	4	1	no	—
33	C₄⋊#3622	—	2	1	no	—
34	C₂×Dic₃₄	—	2	1	no	—
35	C₂×Dic₃₄	—	4	2	no	—
36	C₂×C₈×C₁₇	—	2	1	no	—
37	C₂×C₈×C₁₇	D₈	1	1	yes	characteristic
38	C₂×C₈×C₁₇	D₈	1	1	yes	characteristic
39	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
41	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
42	C₄⋊Dic₁₇	—	2	1	no	—
43	C₄⋊Dic₁₇	—	4	2	no	—
44	C₄⋊Dic₁₇	—	4	2	no	—
45	#63225	—	4	1	no	—
46	C₄×Dic₁₇	—	4	1	no	—
47	C₂×#3622	—	4	1	no	—
48	Dic₆₈	—	2	2	no	—
49	Dic₆₈	—	8	4	no	—
50	C₁₆×C₁₇	—	4	2	no	—
51	#12500	—	17	1	no	maximal, Sylow
52	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
53	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₂×Dic₁₇	—	4	1	no	—
56	C₂×Dic₁₇	—	4	1	no	—
57	C₂×Dic₁₇	—	8	2	no	—
58	Dic₃₄	—	4	1	no	—
59	Dic₃₄	—	4	1	no	—
60	Dic₃₄	—	8	4	no	—
61	C₈×C₁₇	—	2	2	no	—
62	C₈×C₁₇	—	2	1	no	—
63	C₈×C₁₇	—	2	1	no	—
64	C₈×C₁₇	Q₃₂	1	2	yes	—
65	C₈×C₁₇	#137	1	2	yes	—
66	#3622	—	8	1	no	—
67	#2227	—	17	2	no	—
68	#1713	—	17	1	no	—
69	#1591	—	17	1	no	—
70	#1367	D₁₇	1	1	yes	characteristic
71	#1365	—	17	2	no	—
72	C₂²×C₁₇	#459	1	1	yes	characteristic
73	C₄×C₁₇	—	2	1	no	—
74	C₄×C₁₇	#470	1	1	yes	characteristic
75	C₄×C₁₇	#473	1	1	yes	characteristic
76	C₄×C₁₇	C₂×Q₃₂	1	1	yes	characteristic
77	C₄×C₁₇	#510	1	1	yes	characteristic
78	Dic₁₇	—	8	1	no	—
79	Dic₁₇	—	8	1	no	—
80	Dic₁₇	—	16	2	no	—
81	#500	—	17	2	no	—
82	#500	—	34	1	no	—
83	#494	—	34	1	no	—
84	#446	—	34	1	no	—
85	C₄×Q₁₆	—	17	1	no	—
86	#366	—	34	2	no	—
87	C₄⋊C₁₆	C₂×D₁₇	1	2	yes	—
88	#358	—	34	2	no	—
89	#335	—	17	1	no	—
90	Q₈⋊C₈	—	34	1	no	—
91	C₈²	C₂×D₁₇	1	1	yes	characteristic
92	C₂×C₁₇	#1689	1	1	yes	characteristic
93	C₂×C₁₇	#2209	1	1	yes	characteristic
94	C₂×C₁₇	#2213	1	1	yes	characteristic
95	C₂×Q₁₆	—	17	1	no	—
96	C₂×Q₁₆	—	68	2	no	—
97	C₄⋊Q₈	—	34	2	no	—
98	C₄×Q₈	—	34	1	no	—
99	C₂×C₁₆	—	2	2	no	—
100	#107	—	17	1	no	—
101	#107	—	34	2	no	—
102	#107	—	34	2	no	—
103	#107	—	68	1	no	—
104	#107	—	68	1	no	—
105	C₄⋊C₈	—	34	1	no	—
106	C₄⋊C₈	—	68	1	no	—
107	#103	—	34	1	no	—
108	#97	—	68	1	no	—
109	C₄×C₈	D₆₈	1	1	yes	characteristic
110	C₄×C₈	D₆₈	1	1	yes	characteristic
111	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
112	C₁₇	#12500	1	1	yes	characteristic, complemented, Sylow
113	C₂×Q₈	—	34	1	no	—
114	C₂×Q₈	—	68	2	no	—
115	Q₁₆	—	34	2	no	—
116	Q₁₆	—	136	4	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	68	1	no	—
119	C₂×C₈	D₁₃₆	1	1	yes	characteristic
120	C₂×C₈	D₁₃₆	1	1	yes	characteristic
121	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
122	C₂×C₈	#63256	1	1	yes	characteristic
123	C₄⋊C₄	—	34	1	no	—
124	C₄⋊C₄	—	68	1	no	—
125	C₄⋊C₄	—	68	2	no	—
126	C₄⋊C₄	—	68	2	no	—
127	C₄²	—	68	1	no	—
128	C₄²	C₂×D₆₈	1	1	yes	characteristic
129	C₁₆	—	4	2	no	—
130	Q₈	—	68	1	no	—
131	Q₈	—	68	1	no	—
132	Q₈	—	136	4	no	—
133	C₂×C₄	—	68	1	no	—
134	C₂×C₄	—	68	1	no	—
135	C₂×C₄	—	136	2	no	—
136	C₂×C₄	#10587581	1	1	yes	characteristic
137	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
138	C₂×C₄	#10587594	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	136	1	no	—
143	C₈	Dic₁₃₆	1	2	yes	—
144	C₈	#10587601	1	2	yes	—
145	C₂²	#11763005	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	272	2	no	—
148	C₄	—	136	1	no	—
149	C₄	—	136	1	no	—
150	C₄	#11763068	1	1	yes	characteristic
151	C₄	#11763085	1	1	yes	characteristic
152	C₄	C₂×Dic₁₃₆	1	1	yes	characteristic
153	C₄	#11763252	1	1	yes	characteristic
154	C₂	#14528274	1	1	yes	characteristic
155	C₂	#14530835	1	1	yes	characteristic
156	C₂	#14530861	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14530937

C₂

yes

maximal

#14528402

C₂

yes

maximal, characteristic

#14527682

C₂

yes

maximal, characteristic

#14526841

C₂

yes

maximal

C₁₇×#1367

C₂

yes

maximal, characteristic

#11763214

—

#11763214

C₂²

yes

—

#11763189

—

#11762939

—

C₄×Dic₆₈

C₂²

yes

—

—

C₂²

yes

characteristic

#11762541

—

C₁₇×C₄⋊C₁₆

C₂²

yes

—

C₈²×C₁₇

C₂²

yes

characteristic

C₂×Dic₆₈

—

C₂×Dic₆₈

D₄

yes

characteristic

#10587555

—

C₄×Dic₃₄

—

C₂×C₁₆×C₁₇

—

C₄×C₈×C₁₇

D₄

yes

characteristic

C₄×C₈×C₁₇

D₄

yes

characteristic

C₄×C₈×C₁₇

C₂³

yes

—

—

—

—

D₄

yes

—

—

—

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

D₈

yes

characteristic

C₂×C₈×C₁₇

D₈

yes

characteristic

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

#41

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63225

—

C₄×Dic₁₇

—

C₂×#3622

—

Dic₆₈

—

Dic₆₈

—

C₁₆×C₁₇

—

—

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

Q₃₂

yes

—

C₈×C₁₇

#137

yes

—

—

—

—

—

D₁₇

yes

characteristic

#1365

—

C₂²×C₁₇

#459

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

C₂×Q₃₂

yes

characteristic

C₄×C₁₇

#510

yes

characteristic

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

—

—

—

—

C₄×Q₁₆

—

#366

—

C₄⋊C₁₆

C₂×D₁₇

yes

—

#358

—

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₁₆

—

100

#107

—

101

#107

—

102

#107

—

103

#107

—

104

#107

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

#103

—

108

#97

—

109

C₄×C₈

D₆₈

yes

characteristic

110

C₄×C₈

D₆₈

yes

characteristic

111

C₄×C₈

C₂²×D₁₇

yes

characteristic

112

C₁₇

#12500

yes

characteristic, complemented, Sylow

113

C₂×Q₈

—

114

C₂×Q₈

—

115

Q₁₆

—

116

Q₁₆

—

136

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

D₁₃₆

yes

characteristic

120

C₂×C₈

D₁₃₆

yes

characteristic

121

C₂×C₈

D₄×D₁₇

yes

characteristic

122

C₂×C₈

#63256

yes

characteristic

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄²

—

128

C₄²

C₂×D₆₈

yes

characteristic

129

C₁₆

—

130

Q₈

—

131

Q₈

—

132

Q₈

—

136

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

—

136

C₂×C₄

#10587581

yes

characteristic

137

C₂×C₄

C₂×D₁₃₆

yes

characteristic

138

C₂×C₄

#10587594

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

136

—

143

C₈

Dic₁₃₆

yes

—

144

C₈

#10587601

yes

—

145

C₂²

#11763005

yes

characteristic

146

C₄

—

147

C₄

—

272

—

148

C₄

—

136

—

149

C₄

—

136

—

150

C₄

#11763068

yes

characteristic

151

C₄

#11763085

yes

characteristic

152

C₄

C₂×Dic₁₃₆

yes

characteristic

153

C₄

#11763252

yes

characteristic

154

C₂

#14528274

yes

characteristic

155

C₂

#14530835

yes

characteristic

156

C₂

#14530861

yes

characteristic

	subgroup	G/H
center	C₂²	#11763005
commutator, cosocle	C₄×C₈×C₁₇	C₂³
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#1367	C₂
socle, prime radical	C₂²×C₁₇	#459
hypercenter	#1367	D₁₇
hypocenter	C₁₇	#12500
LCS	G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1367

subgroup

G/H

center

C₂²

#11763005

commutator, cosocle

C₄×C₈×C₁₇

C₂³

Frattini

C₄×C₈

C₂²×D₁₇

Fitting

C₁₇×#1367

C₂

C₂²×C₁₇

D₁₇

C₁₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1367

N	H ≅ G/N	type
112. C₁₇	#12500	semidirect

H ≅ G/N

type

112. C₁₇

#12500

semidirect

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

subgroup	index
#14530937	2
#14528402	2
#14527682	2
#14526841	2
C₁₇×#1367	2
#12500	17

subgroup

index

C₁₇×#1367

Subgroups — #16948959

Conjugacy classes of subgroups

Complements

Maximal subgroups