Group #17982288 — subgroups

Subgroups — #17982288

Order 4352.

222 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 65 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	Q₁₆×#3633	C₂	1	2	yes	maximal
2	#14545851	C₂	1	1	yes	maximal, characteristic
3	#14545831	C₂	1	2	yes	maximal
4	#14545799	C₂	1	1	yes	maximal, characteristic
5	Q₃₂×#595	C₂	1	1	yes	maximal, characteristic
6	Q₈×#3633	—	2	2	no	—
7	#11763902	C₂²	1	1	yes	characteristic
8	#11763899	C₂²	1	2	yes	—
9	Q₈⋊#3633	—	2	2	no	—
10	C₈×#3633	C₂²	1	1	yes	characteristic
11	Q₁₆×#595	C₂²	1	2	yes	—
12	#11763735	C₄	1	1	yes	characteristic
13	#11763722	C₄	1	2	yes	—
14	C₁₆×#595	C₂²	1	1	yes	characteristic
15	Q₃₂×D₁₇	C₄	1	1	yes	characteristic
16	Q₈×#595	—	2	2	no	—
17	C₄⋊#3633	—	2	2	no	—
18	C₄⋊#3633	—	4	2	no	—
19	C₄×#3633	—	4	2	no	—
20	C₄×#3633	D₄	1	1	yes	characteristic
21	C₄×#3633	D₄	1	1	yes	characteristic
22	#10587665	—	2	1	no	—
23	Q₁₆×D₁₇	C₂×C₄	1	2	yes	—
24	#10587549	—	2	2	no	—
25	#10587534	C₂×C₄	1	2	yes	—
26	C₈×#595	C₂³	1	1	yes	characteristic
27	#10587529	C₂×C₄	1	1	yes	characteristic
28	C₁₇×Q₃₂	C₈	1	1	yes	characteristic, complemented
29	#10587500	C₈	1	2	yes	complemented
30	Dic₁₃₆	C₈	1	1	yes	characteristic, complemented
31	C₁₆×D₁₇	C₂×C₄	1	1	yes	characteristic
32	C₂×#3633	—	2	1	no	—
33	C₂×#3633	—	4	2	no	—
34	C₂×#3633	D₈	1	2	yes	—
35	Q₈×D₁₇	—	2	2	no	—
36	C₄⋊#595	—	2	2	no	—
37	C₄⋊#595	—	4	2	no	—
38	C₄×#595	—	4	2	no	—
39	C₄×#595	C₂×D₄	1	1	yes	characteristic
40	#63242	#41	1	1	yes	characteristic
41	#63241	—	2	2	no	—
42	Q₁₆×C₁₇	C₂×C₈	1	2	yes	—
43	#63231	—	2	2	no	—
44	Dic₆₈	C₂×C₈	1	2	yes	—
45	C₈×D₁₇	C₂²×C₄	1	1	yes	characteristic
46	#63216	—	2	1	no	—
47	C₁₆×C₁₇	C₂×C₈	1	1	yes	characteristic
48	#63214	C₂×C₈	1	1	yes	characteristic
49	#12044	—	17	1	no	maximal, Sylow
50	C₂×#595	—	4	2	no	—
51	C₂×#595	C₂×D₈	1	1	yes	characteristic
52	C₂×#595	#135	1	1	yes	characteristic
53	#3633	—	2	2	no	—
54	#3633	—	4	1	no	—
55	#3633	—	8	2	no	—
56	Q₈×C₁₇	—	2	2	no	—
57	C₄×D₁₇	—	4	2	no	—
58	C₄×D₁₇	C₄×D₄	1	1	yes	characteristic
59	Dic₃₄	—	2	2	no	—
60	Dic₃₄	—	4	2	no	—
61	#3624	—	2	1	no	—
62	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
63	#3622	#131	1	1	yes	characteristic
64	C₄×Q₃₂	—	17	1	no	—
65	C₈×Q₁₆	—	17	2	no	—
66	#1372	—	17	1	no	—
67	#1332	—	17	2	no	—
68	#1311	—	17	1	no	—
69	C₂×D₁₇	C₄×D₈	1	1	yes	characteristic
70	#595	—	2	1	no	—
71	#595	—	8	2	no	—
72	#595	#510	1	2	yes	—
73	C₄×C₁₇	—	4	2	no	—
74	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
75	Dic₁₇	—	4	2	no	—
76	Dic₁₇	#443	1	1	yes	characteristic
77	C₂×Q₃₂	—	17	1	no	—
78	C₈×Q₈	—	34	2	no	—
79	C₄×Q₁₆	—	17	2	no	—
80	#366	—	17	1	no	—
81	#358	—	17	2	no	—
82	C₄×C₁₆	—	17	1	no	—
83	C₄×C₁₆	—	17	1	no	—
84	#335	—	17	2	no	—
85	Q₈⋊C₈	—	34	2	no	—
86	C₈²	—	17	1	no	—
87	C₂×C₁₇	C₈×D₈	1	1	yes	characteristic
88	D₁₇	#2176	1	1	yes	characteristic
89	D₁₇	#2179	1	1	yes	characteristic
90	C₂×Q₁₆	—	17	2	no	—
91	C₄×Q₈	—	34	2	no	—
92	Q₃₂	—	17	2	no	—
93	Q₃₂	—	17	1	no	—
94	Q₃₂	#3633	1	1	yes	characteristic, complemented
95	C₂×C₁₆	—	17	1	no	—
96	C₂×C₁₆	—	17	1	no	—
97	C₂×C₁₆	—	34	1	no	—
98	#107	—	17	2	no	—
99	C₄⋊C₈	—	34	2	no	—
100	C₄⋊C₈	—	68	2	no	—
101	#103	—	34	2	no	—
102	C₄×C₈	—	17	1	no	—
103	C₄×C₈	—	17	1	no	—
104	C₄×C₈	—	17	1	no	—
105	C₄×C₈	—	68	2	no	—
106	C₁₇	#12044	1	1	yes	characteristic, complemented, Sylow
107	C₂×Q₈	—	34	2	no	—
108	Q₁₆	—	17	2	no	—
109	Q₁₆	—	34	2	no	—
110	Q₁₆	C₂×#3633	1	2	yes	—
111	C₂×C₈	—	17	1	no	—
112	C₂×C₈	—	17	1	no	—
113	C₂×C₈	—	17	2	no	—
114	C₂×C₈	—	34	1	no	—
115	C₂×C₈	—	68	2	no	—
116	C₄⋊C₄	—	34	2	no	—
117	C₄⋊C₄	—	68	2	no	—
118	C₄²	—	17	1	no	—
119	C₄²	—	68	2	no	—
120	C₁₆	—	17	1	no	—
121	C₁₆	—	34	1	no	—
122	C₁₆	—	34	2	no	—
123	C₁₆	C₂×#3633	1	1	yes	characteristic
124	Q₈	—	2	2	no	—
125	Q₈	—	34	2	no	—
126	Q₈	—	68	2	no	—
127	C₂×C₄	—	17	1	no	—
128	C₂×C₄	—	17	1	no	—
129	C₂×C₄	—	17	1	no	—
130	C₂×C₄	—	68	2	no	—
131	C₂×C₄	—	68	2	no	—
132	C₈	—	17	1	no	—
133	C₈	—	34	1	no	—
134	C₈	—	34	2	no	—
135	C₈	—	68	1	no	—
136	C₈	—	136	2	no	—
137	C₈	C₂²×#3633	1	1	yes	characteristic
138	C₂²	—	17	1	no	—
139	C₄	—	4	2	no	—
140	C₄	—	17	2	no	—
141	C₄	—	17	1	no	—
142	C₄	—	34	1	no	—
143	C₄	—	68	2	no	—
144	C₄	—	136	2	no	—
145	C₄	D₄×#3633	1	1	yes	characteristic
146	C₂	—	17	1	no	—
147	C₂	—	17	1	no	—
148	C₂	D₈×#3633	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₁₆×#3633

C₂

yes

maximal

#14545851

C₂

yes

maximal, characteristic

#14545831

C₂

yes

maximal

#14545799

C₂

yes

maximal, characteristic

Q₃₂×#595

C₂

yes

maximal, characteristic

Q₈×#3633

—

#11763902

C₂²

yes

characteristic

#11763899

C₂²

yes

—

Q₈⋊#3633

—

C₈×#3633

C₂²

yes

characteristic

Q₁₆×#595

C₂²

yes

—

#11763735

C₄

yes

characteristic

#11763722

C₄

yes

—

C₁₆×#595

C₂²

yes

characteristic

Q₃₂×D₁₇

C₄

yes

—

—

—

—

D₄

yes

characteristic

C₄×#3633

D₄

yes

characteristic

#10587665

—

Q₁₆×D₁₇

C₂×C₄

yes

—

#10587549

—

#10587534

C₂×C₄

yes

—

C₈×#595

C₂³

yes

characteristic

#10587529

C₂×C₄

yes

characteristic

C₁₇×Q₃₂

C₈

yes

characteristic, complemented

#10587500

C₈

yes

complemented

Dic₁₃₆

C₈

yes

characteristic, complemented

C₁₆×D₁₇

C₂×C₄

yes

—

—

D₈

yes

—

Q₈×D₁₇

—

—

—

—

C₂×D₄

yes

yes

—

Q₁₆×C₁₇

C₂×C₈

yes

—

#63231

—

Dic₆₈

C₂×C₈

yes

—

C₈×D₁₇

C₂²×C₄

yes

characteristic

#63216

—

C₁₆×C₁₇

C₂×C₈

yes

characteristic

#63214

C₂×C₈

yes

—

—

C₂×D₈

yes

yes

—

—

—

Q₈×C₁₇

—

C₄×D₁₇

—

C₄×D₁₇

C₄×D₄

yes

characteristic

Dic₃₄

—

Dic₃₄

—

#3624

—

C₈×C₁₇

C₂²×C₈

yes

yes

C₄×Q₃₂

—

C₈×Q₁₆

—

#1372

—

#1332

—

#1311

—

C₂×D₁₇

C₄×D₈

yes

—

—

yes

—

C₄×C₁₇

—

C₄×C₁₇

C₈×D₄

yes

characteristic

Dic₁₇

—

Dic₁₇

#443

yes

characteristic

C₂×Q₃₂

—

C₈×Q₈

—

C₄×Q₁₆

—

#366

—

#358

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

C₈²

—

C₂×C₁₇

C₈×D₈

yes

D₁₇

yes

D₁₇

yes

C₂×Q₁₆

—

C₄×Q₈

—

Q₃₂

—

Q₃₂

—

Q₃₂

#3633

yes

characteristic, complemented

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

C₄⋊C₈

—

100

C₄⋊C₈

—

101

#103

—

102

C₄×C₈

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₁₇

#12044

yes

characteristic, complemented, Sylow

107

C₂×Q₈

—

108

Q₁₆

—

109

Q₁₆

—

110

Q₁₆

C₂×#3633

yes

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄²

—

119

C₄²

—

120

C₁₆

—

121

C₁₆

—

122

C₁₆

—

123

C₁₆

C₂×#3633

yes

characteristic

124

Q₈

—

125

Q₈

—

126

Q₈

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

136

—

137

C₈

C₂²×#3633

yes

characteristic

138

C₂²

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

136

—

145

C₄

D₄×#3633

yes

characteristic

146

C₂

—

147

C₂

—

148

C₂

D₈×#3633

yes

characteristic

	subgroup	G/H
center	C₂	D₈×#3633
commutator	C₈×C₁₇	C₂²×C₈
Frattini	C₈	C₂²×#3633
Fitting	C₁₇×Q₃₂	C₈
socle	C₂×C₁₇	C₈×D₈
prime radical	C₂×D₁₇	C₄×D₈
cosocle	C₈×#595	C₂³
hypercenter	Q₃₂	#3633
hypocenter	C₁₇	#12044
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

subgroup

G/H

center

C₂

D₈×#3633

commutator

C₈×C₁₇

C₂²×C₈

Frattini

C₈

C₂²×#3633

Fitting

C₁₇×Q₃₂

C₈

socle

C₂×C₁₇

C₈×D₈

prime radical

C₂×D₁₇

C₄×D₈

C₂³

Q₃₂

C₁₇

G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

N	H ≅ G/N	type
28. C₁₇×Q₃₂	C₈	semidirect
29. #10587500	C₈	semidirect
30. Dic₁₃₆	C₈	semidirect
94. Q₃₂	#3633	semidirect
106. C₁₇	#12044	semidirect

H ≅ G/N

type

28. C₁₇×Q₃₂

C₈

semidirect

29. #10587500

C₈

semidirect

30. Dic₁₃₆

C₈

semidirect

94. Q₃₂

#3633

semidirect

106. C₁₇

#12044

semidirect

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

subgroup	index
Q₁₆×#3633	2
#14545851	2
#14545831	2
#14545799	2
Q₃₂×#595	2
#12044	17

subgroup

index

Q₁₆×#3633

Q₃₂×#595

Subgroups — #17982288

Conjugacy classes of subgroups

Complements

Maximal subgroups