Group #17948786 — subgroups

Subgroups — #17948786

Order 4352.

180 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 46 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	#14545154	C₂	1	1	yes	maximal, characteristic
2	#14544003	C₂	1	1	yes	maximal, characteristic
3	#14528931	C₂	1	1	yes	maximal, characteristic
4	#11763826	—	2	1	no	—
5	#11763731	—	2	1	no	—
6	#11763705	—	2	1	no	—
7	#11763702	C₄	1	2	yes	complemented
8	#11763651	—	2	1	no	—
9	#11763650	C₄	1	1	yes	characteristic, complemented
10	C₈×#3624	C₄	1	1	yes	characteristic, complemented
11	M₁₆×Dic₁₇	C₂²	1	1	yes	characteristic
12	#11762579	—	2	1	no	—
13	#11762568	—	2	1	no	—
14	#11762567	C₄	1	1	yes	characteristic
15	C₁₇×H(3,ℤ/4)	C₄	1	1	yes	characteristic
16	C₂×M₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₁₇×#117	—	2	1	no	—
18	C₂×C₄×Dic₁₇	D₄	1	1	yes	characteristic
19	C₂×#63223	—	2	1	no	—
20	C₂×#63223	D₄	1	1	yes	characteristic
21	C₂²⋊#3624	—	2	2	no	—
22	#10587544	—	2	2	no	—
23	C₄×#3624	—	2	1	no	—
24	C₄×#3624	—	2	1	no	—
25	#10587538	—	2	2	no	—
26	#10587486	C₂×C₄	1	1	yes	characteristic
27	#10587486	Q₈	1	1	yes	characteristic
28	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₁₇	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₂²×Dic₁₇	—	2	1	no	—
32	C₂×#3624	—	2	2	no	—
33	C₂×#3624	—	2	2	no	—
34	M₁₆×C₁₇	—	4	1	no	—
35	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
37	C₄⋊C₄×C₁₇	—	4	1	no	—
38	C₂²⋊C₄×C₁₇	—	4	1	no	—
39	C₄²×C₁₇	—	4	1	no	—
40	C₄×Dic₁₇	—	2	1	no	—
41	C₄×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
43	#63223	—	4	1	no	—
44	#63223	—	4	1	no	—
45	#63223	—	4	1	no	—
46	C₂×#3622	—	4	1	no	—
47	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
49	#63216	—	4	2	no	—
50	#7226	—	17	1	no	maximal, Sylow
51	C₂³×C₁₇	#104	1	1	yes	characteristic
52	C₂×C₄×C₁₇	—	2	1	no	—
53	C₂×C₄×C₁₇	—	2	1	no	—
54	C₂×C₄×C₁₇	—	8	1	no	—
55	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
56	C₂×C₄×C₁₇	#104	1	1	yes	characteristic
57	C₂×Dic₁₇	—	2	1	no	—
58	C₂×Dic₁₇	—	2	1	no	—
59	C₂×Dic₁₇	—	4	2	no	—
60	#3624	—	4	2	no	—
61	#3624	—	4	2	no	—
62	C₈×C₁₇	—	2	1	no	—
63	C₈×C₁₇	—	2	1	no	—
64	#3622	—	2	1	no	—
65	#3622	—	2	1	no	—
66	#3622	—	8	1	no	—
67	#2165	—	17	1	no	—
68	#1841	—	17	1	no	—
69	#1453	—	17	1	no	—
70	C₂²×C₁₇	—	2	1	no	—
71	C₂²×C₁₇	—	4	1	no	—
72	C₂²×C₁₇	#339	1	1	yes	characteristic
73	C₄×C₁₇	—	2	1	no	—
74	C₄×C₁₇	—	16	1	no	—
75	C₄×C₁₇	H(3,ℤ/4)	1	1	yes	characteristic
76	C₄×C₁₇	#344	1	1	yes	characteristic
77	Dic₁₇	—	2	2	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)	#595	1	1	yes	characteristic
88	#322	—	17	1	no	—
89	#322	—	34	1	no	—
90	C₈²	—	17	1	no	—
91	C₂×C₁₇	—	2	1	no	—
92	C₂×C₁₇	—	4	1	no	—
93	C₂×C₁₇	#1288	1	1	yes	characteristic
94	C₂×M₁₆	—	17	1	no	—
95	C₂×M₁₆	—	34	1	no	—
96	C₂×M₁₆	C₂×#595	1	1	yes	characteristic
97	#117	—	2	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	—	17	1	no	—
105	C₄×C₈	—	17	1	no	—
106	C₄×C₈	—	34	1	no	—
107	C₄×C₈	—	34	1	no	—
108	C₁₇	#7226	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₄	—	34	1	no	—
110	C₂²×C₄	C₂²⋊#595	1	1	yes	characteristic
111	M₁₆	—	4	1	no	—
112	M₁₆	—	68	1	no	—
113	M₁₆	—	68	1	no	—
114	M₁₆	—	68	1	no	—
115	C₂×C₈	—	17	1	no	—
116	C₂×C₈	—	17	1	no	—
117	C₂×C₈	—	34	2	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	—	4	1	no	—
123	C₂²⋊C₄	—	4	1	no	—
124	C₄²	—	4	1	no	—
125	C₄²	—	17	1	no	—
126	C₄²	—	17	1	no	—
127	C₄²	—	34	1	no	—
128	C₁₆	—	68	2	no	—
129	C₂³	#10587548	1	1	yes	characteristic
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	8	1	no	—
133	C₂×C₄	—	34	1	no	—
134	C₂×C₄	—	34	1	no	—
135	C₂×C₄	—	68	2	no	—
136	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
137	C₂×C₄	#10587550	1	1	yes	characteristic
138	C₈	—	2	1	no	—
139	C₈	—	2	1	no	—
140	C₈	—	34	1	no	—
141	C₈	—	34	1	no	—
142	C₈	—	68	2	no	—
143	C₈	—	68	2	no	—
144	C₈	—	136	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	4	1	no	—
147	C₂²	#11763679	1	1	yes	characteristic
148	C₄	—	2	1	no	—
149	C₄	—	16	1	no	—
150	C₄	—	34	2	no	—
151	C₄	—	68	1	no	—
152	C₄	#11763676	1	1	yes	characteristic
153	C₄	#11763690	1	1	yes	characteristic
154	C₂	—	2	1	no	—
155	C₂	—	4	1	no	—
156	C₂	#14543643	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545154

C₂

yes

maximal, characteristic

#14544003

C₂

yes

maximal, characteristic

#14528931

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

complemented

#11763651

—

#11763650

C₄

yes

characteristic, complemented

C₈×#3624

C₄

yes

characteristic, complemented

M₁₆×Dic₁₇

C₂²

yes

—

—

C₄

yes

characteristic

C₁₇×H(3,ℤ/4)

C₄

yes

characteristic

C₂×M₁₆×C₁₇

C₂×C₄

yes

characteristic

C₁₇×#117

—

C₂×C₄×Dic₁₇

D₄

yes

characteristic

C₂×#63223

—

C₂×#63223

D₄

yes

characteristic

C₂²⋊#3624

—

—

—

—

—

C₂×C₄

yes

characteristic

#10587486

Q₈

yes

characteristic

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

D₄

yes

characteristic

C₂²×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂²×Dic₁₇

—

C₂×#3624

—

C₂×#3624

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₄⋊C₄×C₁₇

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₇

C₄⋊C₄

yes

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

#3624

—

#3624

—

C₈×C₁₇

—

C₈×C₁₇

—

—

—

—

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#339

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

#431

—

C₄×M₁₆

—

—

—

—

—

—

—

yes

—

—

C₈²

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

#1288

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×#595

yes

characteristic

#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

—

108

C₁₇

#7226

yes

characteristic, complemented, Sylow

109

C₂²×C₄

—

110

C₂²×C₄

C₂²⋊#595

yes

characteristic

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

—

128

C₁₆

—

129

C₂³

#10587548

yes

characteristic

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

137

C₂×C₄

#10587550

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

136

—

145

C₂²

—

146

C₂²

—

147

C₂²

#11763679

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

#11763676

yes

characteristic

153

C₄

#11763690

yes

characteristic

154

C₂

—

155

C₂

—

156

C₂

#14543643

yes

characteristic

	subgroup	G/H
center	C₄	#11763676
commutator	C₂×C₈×C₁₇	C₄²
Frattini	C₂×M₁₆	C₂×#595
Fitting	C₁₇×H(3,ℤ/4)	C₄
socle	C₂×C₁₇	#1288
prime radical	C₂³×C₁₇	#104
cosocle	M₁₆×Dic₁₇	C₂²
hypercenter	H(3,ℤ/4)	#595
hypocenter	C₁₇	#7226
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#11763676

commutator

C₂×C₈×C₁₇

C₄²

Frattini

C₂×M₁₆

C₂×#595

Fitting

C₁₇×H(3,ℤ/4)

C₄

C₂×C₁₇

C₂³×C₁₇

M₁₆×Dic₁₇

C₂²

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
7. #11763702	C₄	semidirect
9. #11763650	C₄	semidirect
10. C₈×#3624	C₄	semidirect
108. C₁₇	#7226	semidirect

H ≅ G/N

type

7. #11763702

C₄

semidirect

9. #11763650

C₄

semidirect

10. C₈×#3624

C₄

semidirect

108. C₁₇

#7226

semidirect

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

subgroup	index
#14545154	2
#14544003	2
#14528931	2
#7226	17

subgroup

index

Subgroups — #17948786

Conjugacy classes of subgroups

Complements

Maximal subgroups