Group #17948787 — subgroups

Subgroups — #17948787

Order 4352.

180 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 44 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	#14545156	C₂	1	1	yes	maximal, characteristic
2	#14544714	C₂	1	1	yes	maximal, characteristic
3	#14527078	C₂	1	1	yes	maximal, characteristic
4	#11763732	—	2	1	no	—
5	#11763706	—	2	1	no	—
6	#11763702	C₄	1	2	yes	complemented
7	#11763691	—	2	1	no	—
8	#11763680	—	2	1	no	—
9	#11763678	C₄	1	1	yes	characteristic
10	#11763677	C₄	1	1	yes	characteristic
11	#11762860	—	2	1	no	—
12	C₄×#63218	C₂²	1	1	yes	characteristic
13	#11762531	—	2	1	no	—
14	#11762530	C₄	1	1	yes	characteristic, complemented
15	C₈²×C₁₇	C₄	1	1	yes	characteristic, complemented
16	#10587677	—	2	1	no	—
17	D₁₇⋊M₁₆	—	2	1	no	—
18	C₂×#63218	C₂×C₄	1	1	yes	characteristic
19	C₂×#63218	D₄	1	1	yes	characteristic
20	C₄²×D₁₇	D₄	1	1	yes	characteristic
21	#10587538	—	2	2	no	—
22	C₄×C₈×C₁₇	—	2	1	no	—
23	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
24	#10587491	—	2	2	no	—
25	#10587486	C₂×C₄	1	1	yes	characteristic
26	#10587486	Q₈	1	1	yes	characteristic
27	#10587485	—	2	2	no	—
28	C₄×#3622	—	2	1	no	—
29	C₄×#3622	D₄	1	1	yes	characteristic
30	C₂×C₄×D₁₇	—	2	1	no	—
31	C₂×C₄×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²⋊#595	—	4	1	no	—
33	#63247	—	4	1	no	—
34	C₄⋊#595	—	4	1	no	—
35	C₄×#595	—	4	1	no	—
36	#63243	—	4	1	no	—
37	#63242	—	4	1	no	—
38	C₂×C₈×C₁₇	—	2	2	no	—
39	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
41	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₄×Dic₁₇	—	2	1	no	—
43	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂×#3622	—	2	2	no	—
45	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
47	#63218	—	4	1	no	—
48	#63218	—	4	1	no	—
49	#63216	—	4	2	no	—
50	#7226	—	17	1	no	maximal, Sylow
51	C₂²×D₁₇	#104	1	1	yes	characteristic
52	C₂×#595	—	8	1	no	—
53	C₂×C₄×C₁₇	—	2	1	no	—
54	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
55	C₂×Dic₁₇	—	2	1	no	—
56	C₂×Dic₁₇	#104	1	1	yes	characteristic
57	C₄×D₁₇	—	2	1	no	—
58	C₄×D₁₇	—	2	1	no	—
59	C₄×D₁₇	—	4	2	no	—
60	#3624	—	8	1	no	—
61	C₈×C₁₇	—	2	1	no	—
62	C₈×C₁₇	—	2	1	no	—
63	C₈×C₁₇	—	4	2	no	—
64	#3622	—	2	1	no	—
65	#3622	—	2	1	no	—
66	#3622	—	4	2	no	—
67	#2165	—	17	1	no	—
68	#1841	—	17	1	no	—
69	#1453	—	17	1	no	—
70	C₂²×C₁₇	#339	1	1	yes	characteristic
71	C₂×D₁₇	—	2	1	no	—
72	C₂×D₁₇	—	4	1	no	—
73	#595	—	16	1	no	—
74	C₄×C₁₇	—	2	2	no	—
75	C₄×C₁₇	H(3,ℤ/4)	1	1	yes	characteristic
76	C₄×C₁₇	#344	1	1	yes	characteristic
77	Dic₁₇	—	2	1	no	—
78	Dic₁₇	—	4	1	no	—
79	#431	—	34	1	no	—
80	C₄×M₁₆	—	17	1	no	—
81	#364	—	34	1	no	—
82	#349	—	34	1	no	—
83	#347	—	17	2	no	—
84	#344	—	34	1	no	—
85	#339	—	34	1	no	—
86	#338	—	17	1	no	—
87	H(3,ℤ/4)	—	17	1	no	—
88	#322	—	17	1	no	—
89	#322	—	34	1	no	—
90	C₈²	#595	1	1	yes	characteristic, complemented
91	C₂×C₁₇	—	2	1	no	—
92	C₂×C₁₇	#1288	1	1	yes	characteristic
93	D₁₇	—	4	1	no	—
94	C₂×M₁₆	—	17	1	no	—
95	C₂×M₁₆	—	17	1	no	—
96	C₂×M₁₆	—	34	1	no	—
97	#117	—	34	1	no	—
98	C₂×C₄²	—	17	1	no	—
99	M₃₂	—	34	2	no	—
100	C₄⋊C₈	—	34	2	no	—
101	C₂²⋊C₈	—	34	2	no	—
102	#97	—	17	1	no	—
103	#97	—	17	1	no	—
104	C₄×C₈	—	2	1	no	—
105	C₄×C₈	—	17	1	no	—
106	C₄×C₈	—	34	1	no	—
107	C₄×C₈	C₂×#595	1	1	yes	characteristic
108	C₁₇	#7226	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₄	—	17	1	no	—
110	C₂²×C₄	—	34	1	no	—
111	M₁₆	—	68	1	no	—
112	M₁₆	—	68	1	no	—
113	M₁₆	—	68	1	no	—
114	M₁₆	—	68	1	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	17	1	no	—
117	C₂×C₈	—	17	1	no	—
118	C₂×C₈	—	34	2	no	—
119	C₂×C₈	—	68	1	no	—
120	C₂×C₈	C₄×#595	1	1	yes	characteristic
121	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
122	C₄⋊C₄	—	68	1	no	—
123	C₂²⋊C₄	—	68	1	no	—
124	C₄²	—	17	1	no	—
125	C₄²	—	34	1	no	—
126	C₄²	—	68	1	no	—
127	C₄²	C₂²⋊#595	1	1	yes	characteristic
128	C₁₆	—	68	2	no	—
129	C₂³	—	17	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	17	1	no	—
132	C₂×C₄	—	34	1	no	—
133	C₂×C₄	—	34	1	no	—
134	C₂×C₄	—	34	1	no	—
135	C₂×C₄	—	68	2	no	—
136	C₂×C₄	—	136	1	no	—
137	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
138	C₈	—	2	1	no	—
139	C₈	—	2	1	no	—
140	C₈	—	4	2	no	—
141	C₈	—	34	1	no	—
142	C₈	—	34	1	no	—
143	C₈	—	68	2	no	—
144	C₈	—	136	1	no	—
145	C₂²	—	34	1	no	—
146	C₂²	—	68	1	no	—
147	C₂²	#11763680	1	1	yes	characteristic
148	C₄	—	2	2	no	—
149	C₄	—	272	1	no	—
150	C₄	—	34	1	no	—
151	C₄	—	68	1	no	—
152	C₄	#11763677	1	1	yes	characteristic
153	C₄	#11763691	1	1	yes	characteristic
154	C₂	—	2	1	no	—
155	C₂	—	68	1	no	—
156	C₂	#14543644	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545156

C₂

yes

maximal, characteristic

#14544714

C₂

yes

maximal, characteristic

#14527078

C₂

yes

maximal, characteristic

#11763732

—

#11763706

—

#11763702

C₄

yes

—

—

C₄

yes

characteristic

#11763677

C₄

yes

characteristic

#11762860

—

C₄×#63218

C₂²

yes

characteristic

#11762531

—

#11762530

C₄

yes

characteristic, complemented

C₈²×C₁₇

C₄

yes

characteristic, complemented

#10587677

—

D₁₇⋊M₁₆

—

C₂×#63218

C₂×C₄

yes

characteristic

C₂×#63218

D₄

yes

characteristic

C₄²×D₁₇

D₄

yes

characteristic

#10587538

—

C₄×C₈×C₁₇

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

#10587491

—

#10587486

C₂×C₄

yes

characteristic

#10587486

Q₈

yes

—

—

D₄

yes

characteristic

C₂×C₄×D₁₇

—

C₂×C₄×D₁₇

C₂²⋊C₄

yes

characteristic

C₂²⋊#595

—

—

—

—

—

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₄²×C₁₇

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₇

—

C₄×Dic₁₇

C₄⋊C₄

yes

characteristic

C₂×#3622

—

C₂×#3622

C₂²⋊C₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

—

—

—

—

C₂²×D₁₇

yes

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#95

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

#104

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

#3624

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

—

—

—

—

—

—

C₂²×C₁₇

#339

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

#595

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

#431

—

C₄×M₁₆

—

—

—

—

—

—

—

—

—

—

C₈²

#595

yes

characteristic, complemented

C₂×C₁₇

—

C₂×C₁₇

#1288

yes

characteristic

D₁₇

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

#117

—

C₂×C₄²

—

M₃₂

—

100

C₄⋊C₈

—

101

C₂²⋊C₈

—

102

#97

—

103

#97

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

C₂×#595

yes

characteristic

108

C₁₇

#7226

yes

characteristic, complemented, Sylow

109

C₂²×C₄

—

110

C₂²×C₄

—

111

M₁₆

—

112

M₁₆

—

113

M₁₆

—

114

M₁₆

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

C₄×#595

yes

characteristic

121

C₂×C₈

C₄⋊#595

yes

characteristic

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

C₂²⋊#595

yes

characteristic

128

C₁₆

—

129

C₂³

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

136

—

137

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

136

—

145

C₂²

—

146

C₂²

—

147

C₂²

#11763680

yes

characteristic

148

C₄

—

149

C₄

—

272

—

150

C₄

—

151

C₄

—

152

C₄

#11763677

yes

characteristic

153

C₄

#11763691

yes

characteristic

154

C₂

—

155

C₂

—

156

C₂

#14543644

yes

characteristic

	subgroup	G/H
center	C₄	#11763677
commutator	C₂×C₈×C₁₇	C₄²
Frattini	C₄×C₈	C₂×#595
Fitting	C₈²×C₁₇	C₄
socle	C₂×C₁₇	#1288
prime radical	C₂²×D₁₇	#104
cosocle	C₄×#63218	C₂²
hypercenter	C₈²	#595
hypocenter	C₁₇	#7226
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₈²

subgroup

G/H

center

C₄

#11763677

commutator

C₂×C₈×C₁₇

C₄²

Frattini

C₄×C₈

C₂×#595

Fitting

C₈²×C₁₇

C₄

C₂×C₁₇

C₂²×D₁₇

C₂²

C₈²

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₈²

N	H ≅ G/N	type
6. #11763702	C₄	semidirect
14. #11762530	C₄	semidirect
15. C₈²×C₁₇	C₄	semidirect
90. C₈²	#595	semidirect
108. C₁₇	#7226	semidirect

H ≅ G/N

type

6. #11763702

C₄

semidirect

14. #11762530

C₄

semidirect

15. C₈²×C₁₇

C₄

semidirect

90. C₈²

#595

semidirect

108. C₁₇

#7226

semidirect

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

subgroup	index
#14545156	2
#14544714	2
#14527078	2
#7226	17

subgroup

index

Subgroups — #17948787

Conjugacy classes of subgroups

Complements

Maximal subgroups