Group #17960065 — subgroups

Subgroups — #17960065

Order 4352.

234 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 120 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	#14544331	C₂	1	2	yes	maximal
2	#14543809	C₂	1	2	yes	maximal, complemented
3	#14543808	C₂	1	2	yes	maximal, complemented
4	C₂×#11762584	C₂	1	1	yes	maximal, characteristic
5	#11763825	—	2	2	no	—
6	#11763672	—	2	2	no	—
7	#11763670	C₂²	1	4	yes	—
8	C₂²×#63214	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
10	#11762584	C₄	1	1	yes	characteristic
11	#11762584	C₄	1	1	yes	characteristic
12	#11762584	C₂²	1	1	yes	characteristic
13	#11762584	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₁₇	C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×Dic₁₇	D₄	1	1	yes	characteristic
17	C₂²×#3622	Q₈	1	1	yes	characteristic
18	C₂²⋊#3624	—	4	2	no	—
19	C₄⋊#3624	—	2	4	no	—
20	C₄×#3624	—	4	2	no	—
21	C₂×C₁₆×C₁₇	C₈	1	4	yes	complemented
22	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
25	C₈×Dic₁₇	D₄	1	1	yes	characteristic
26	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
27	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
28	C₂×#63214	—	2	2	no	—
29	C₂×#63214	C₂×C₄	1	1	yes	characteristic
30	C₂×#63214	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
32	C₂²×Dic₁₇	SD₁₆	1	2	yes	—
33	C₂×#3624	—	4	4	no	—
34	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
36	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
37	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
38	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
39	C₄×Dic₁₇	SD₁₆	1	2	yes	—
40	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
41	C₂×#3622	—	2	2	no	—
42	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
43	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
44	C₁₆×C₁₇	C₂×C₈	1	2	yes	—
45	C₁₆×C₁₇	C₂×C₈	1	2	yes	—
46	#63214	—	2	1	no	—
47	#63214	—	2	1	no	—
48	#11999	—	17	1	no	maximal, Sylow
49	C₂³×C₁₇	#106	1	1	yes	characteristic
50	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
51	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
52	C₂×C₄×C₁₇	#108	1	2	yes	—
53	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×Dic₁₇	—	4	2	no	—
55	C₂×Dic₁₇	#106	1	2	yes	—
56	C₂×Dic₁₇	C₂×SD₁₆	1	2	yes	—
57	#3624	—	8	4	no	—
58	C₈×C₁₇	C₄⋊C₈	1	2	yes	—
59	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
60	C₈×C₁₇	C₂×M₁₆	1	1	yes	characteristic
61	#3622	—	2	1	no	—
62	#3622	—	2	1	no	—
63	C₂×#346	—	17	1	no	—
64	#1564	—	17	2	no	—
65	#1368	—	17	2	no	—
66	#1368	—	17	2	no	—
67	C₂²×C₁₇	—	2	2	no	—
68	C₂²×C₁₇	#334	1	2	yes	—
69	C₂²×C₁₇	C₂×#106	1	1	yes	characteristic
70	C₄×C₁₇	#334	1	2	yes	—
71	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
72	C₄×C₁₇	C₂×#108	1	1	yes	characteristic
73	Dic₁₇	—	2	2	no	—
74	Dic₁₇	—	2	2	no	—
75	C₂²×C₁₆	—	17	1	no	—
76	C₂²×C₁₆	#595	1	1	yes	characteristic
77	#431	—	34	2	no	—
78	C₂×C₄×C₈	—	17	1	no	—
79	#346	—	17	1	no	—
80	#346	—	17	1	no	—
81	#346	—	17	1	no	—
82	#346	—	17	1	no	—
83	#335	—	34	2	no	—
84	#334	—	17	4	no	—
85	C₂×C₁₇	—	2	2	no	—
86	C₂×C₁₇	C₂×#334	1	1	yes	characteristic
87	C₂×C₁₇	#2155	1	2	yes	—
88	C₂²×C₈	—	17	1	no	—
89	C₂²×C₈	C₂×#595	1	1	yes	characteristic
90	C₂×C₄²	—	17	1	no	—
91	C₂×C₁₆	—	17	1	no	—
92	C₂×C₁₆	—	17	1	no	—
93	C₂×C₁₆	—	34	2	no	—
94	C₂×C₁₆	#3624	1	4	yes	complemented
95	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
96	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
97	C₄⋊C₈	—	34	4	no	—
98	C₂²⋊C₈	—	68	2	no	—
99	C₄×C₈	—	17	1	no	—
100	C₄×C₈	—	17	1	no	—
101	C₄×C₈	—	17	1	no	—
102	C₄×C₈	—	17	1	no	—
103	C₄×C₈	—	68	2	no	—
104	C₁₇	#11999	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₄	—	17	2	no	—
106	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
107	C₂×C₈	—	17	1	no	—
108	C₂×C₈	—	17	1	no	—
109	C₂×C₈	—	34	2	no	—
110	C₂×C₈	—	68	4	no	—
111	C₂×C₈	#63243	1	2	yes	—
112	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
113	C₂×C₈	C₂×#3624	1	2	yes	—
114	C₂×C₈	C₂²×#595	1	1	yes	characteristic
115	C₄²	—	17	1	no	—
116	C₄²	—	17	1	no	—
117	C₄²	—	17	2	no	—
118	C₁₆	—	34	1	no	—
119	C₁₆	—	34	1	no	—
120	C₁₆	C₂×#3624	1	2	yes	—
121	C₁₆	C₂×#3624	1	2	yes	—
122	C₂³	#10587533	1	1	yes	characteristic
123	C₂×C₄	—	17	2	no	—
124	C₂×C₄	—	17	2	no	—
125	C₂×C₄	—	68	2	no	—
126	C₂×C₄	#10587533	1	1	yes	characteristic
127	C₂×C₄	#10587535	1	2	yes	—
128	C₂×C₄	C₄⋊#3624	1	2	yes	—
129	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
130	C₈	—	34	1	no	—
131	C₈	—	34	1	no	—
132	C₈	—	136	4	no	—
133	C₈	C₄⋊#3624	1	2	yes	—
134	C₈	C₂×#63243	1	1	yes	characteristic
135	C₈	C₂²×#3624	1	1	yes	characteristic
136	C₂²	—	2	2	no	—
137	C₂²	#11763670	1	2	yes	—
138	C₂²	C₂×#10587533	1	1	yes	characteristic
139	C₄	—	34	2	no	—
140	C₄	—	34	2	no	—
141	C₄	#11763670	1	2	yes	—
142	C₄	C₂×C₄⋊#3624	1	1	yes	characteristic
143	C₄	C₂×#10587535	1	1	yes	characteristic
144	C₂	—	2	2	no	—
145	C₂	C₂×#11763670	1	1	yes	characteristic
146	C₂	#14545131	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14544331

C₂

yes

maximal

#14543809

C₂

yes

maximal, complemented

#14543808

C₂

yes

maximal, complemented

C₂×#11762584

C₂

yes

maximal, characteristic

#11763825

—

#11763672

—

#11763670

C₂²

yes

—

C₂²×#63214

C₄

yes

characteristic

C₂×C₈×Dic₁₇

C₂²

yes

characteristic

#11762584

C₄

yes

characteristic

#11762584

C₄

yes

characteristic

#11762584

C₂²

yes

characteristic

#11762584

C₂²

yes

characteristic

C₂²×C₁₆×C₁₇

C₄

yes

characteristic

C₂²×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₇

D₄

yes

characteristic

C₂²×#3622

Q₈

yes

characteristic

C₂²⋊#3624

—

C₄⋊#3624

—

C₄×#3624

—

C₂×C₁₆×C₁₇

C₈

yes

complemented

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

D₄

yes

characteristic

C₈×Dic₁₇

Q₈

yes

characteristic

C₈×Dic₁₇

C₂³

yes

characteristic

C₂×#63214

—

C₂×#63214

C₂×C₄

yes

characteristic

C₂×#63214

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₇

C₄⋊C₄

yes

characteristic

C₂²×Dic₁₇

SD₁₆

yes

—

C₂×#3624

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×C₈

yes

—

C₂×C₈×C₁₇

M₁₆

yes

—

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₄×Dic₁₇

C₄⋊C₄

yes

characteristic

C₄×Dic₁₇

SD₁₆

yes

—

C₄×Dic₁₇

C₂×D₄

yes

characteristic

C₂×#3622

—

C₂×#3622

C₄⋊C₄

yes

characteristic

C₂×#3622

C₂×Q₈

yes

characteristic

C₁₆×C₁₇

C₂×C₈

yes

—

C₁₆×C₁₇

C₂×C₈

yes

—

—

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

C₄⋊C₈

yes

—

C₂×C₄×C₁₇

#106

yes

characteristic

C₂×C₄×C₁₇

#108

yes

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

#106

yes

—

C₂×Dic₁₇

C₂×SD₁₆

yes

—

#3624

—

C₈×C₁₇

C₄⋊C₈

yes

—

C₈×C₁₇

C₂²×C₈

yes

characteristic

C₈×C₁₇

C₂×M₁₆

yes

—

—

—

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

#334

yes

—

C₂²×C₁₇

C₂×#106

yes

characteristic

C₄×C₁₇

#334

yes

—

C₄×C₁₇

C₂×C₄⋊C₈

yes

characteristic

C₄×C₁₇

C₂×#108

yes

characteristic

Dic₁₇

—

Dic₁₇

—

C₂²×C₁₆

—

C₂²×C₁₆

#595

yes

characteristic

#431

—

C₂×C₄×C₈

—

—

—

—

—

—

—

C₂×C₁₇

—

C₂×C₁₇

C₂×#334

yes

characteristic

C₂×C₁₇

#2155

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×#595

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

C₂×C₁₆

yes

C₂×C₁₆

yes

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

—

104

C₁₇

#11999

yes

characteristic, complemented, Sylow

105

C₂²×C₄

—

106

C₂²×C₄

C₄⋊#595

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

#63243

yes

—

112

C₂×C₈

C₄⋊#595

yes

characteristic

113

C₂×C₈

C₂×#3624

yes

—

114

C₂×C₈

C₂²×#595

yes

characteristic

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₁₆

—

119

C₁₆

—

120

C₁₆

C₂×#3624

yes

—

121

C₁₆

C₂×#3624

yes

—

122

C₂³

#10587533

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

#10587533

yes

characteristic

127

C₂×C₄

#10587535

yes

—

128

C₂×C₄

C₄⋊#3624

yes

—

129

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

136

—

133

C₈

C₄⋊#3624

yes

—

134

C₈

C₂×#63243

yes

characteristic

135

C₈

C₂²×#3624

yes

characteristic

136

C₂²

—

137

C₂²

#11763670

yes

—

138

C₂²

C₂×#10587533

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

#11763670

yes

—

142

C₄

C₂×C₄⋊#3624

yes

characteristic

143

C₄

C₂×#10587535

yes

characteristic

144

C₂

—

145

C₂

C₂×#11763670

yes

characteristic

146

C₂

#14545131

yes

—

	subgroup	G/H
center	C₂×C₄	#10587533
commutator	C₈×C₁₇	C₂²×C₈
Frattini	C₂×C₈	C₂²×#595
Fitting	C₂²×C₁₆×C₁₇	C₄
socle	C₂²×C₁₇	C₂×#106
prime radical	C₂³×C₁₇	#106
cosocle	C₈×Dic₁₇	C₂³
hypercenter	C₂²×C₁₆	#595
hypocenter	C₁₇	#11999
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂×C₄

#10587533

commutator

C₈×C₁₇

C₂²×C₈

Frattini

C₂×C₈

C₂²×#595

Fitting

C₂²×C₁₆×C₁₇

C₄

C₂²×C₁₇

C₂³×C₁₇

C₈×Dic₁₇

C₂³

C₂²×C₁₆

C₁₇

G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
2. #14543809	C₂	semidirect
3. #14543808	C₂	semidirect
21. C₂×C₁₆×C₁₇	C₈	semidirect
94. C₂×C₁₆	#3624	semidirect
104. C₁₇	#11999	semidirect

H ≅ G/N

type

2. #14543809

C₂

semidirect

3. #14543808

C₂

semidirect

21. C₂×C₁₆×C₁₇

C₈

semidirect

94. C₂×C₁₆

#3624

semidirect

104. C₁₇

#11999

semidirect

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

subgroup	index
#14544331	2
#14543809	2
#14543808	2
C₂×#11762584	2
#11999	17

subgroup

index

Subgroups — #17960065

Conjugacy classes of subgroups

Complements

Maximal subgroups