Group #17982292 — subgroups

Subgroups — #17982292

Order 4352.

216 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 61 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	#14546003	C₂	1	2	yes	maximal, complemented
2	#14545852	C₂	1	1	yes	maximal, characteristic, complemented
3	#14545835	C₂	1	2	yes	maximal, complemented
4	C₁₆×#3633	C₂	1	1	yes	maximal, characteristic, complemented
5	#14545185	C₂	1	1	yes	maximal, characteristic
6	#11763931	—	2	2	no	—
7	#11763914	C₂²	1	2	yes	—
8	#11763910	—	2	2	no	—
9	#11763902	C₂²	1	1	yes	characteristic
10	C₈×#3633	C₂²	1	1	yes	characteristic
11	#11763853	C₂²	1	2	yes	—
12	#11763737	C₄	1	1	yes	characteristic
13	#11763724	C₄	1	2	yes	—
14	C₁₆×#595	C₂²	1	1	yes	characteristic
15	#11763242	C₄	1	1	yes	characteristic
16	#10587680	—	2	2	no	—
17	C₄×#3633	—	2	1	no	—
18	#10587666	—	2	2	no	—
19	#10587666	—	4	2	no	—
20	#10587665	—	4	2	no	—
21	#10587665	D₄	1	1	yes	characteristic
22	#10587665	D₄	1	1	yes	characteristic
23	#10587605	C₂×C₄	1	2	yes	—
24	#10587550	—	2	2	no	—
25	#10587536	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	#10587499	C₈	1	2	yes	complemented
30	D₂₇₂	C₈	1	1	yes	characteristic, complemented
31	C₁₆×D₁₇	C₂×C₄	1	1	yes	characteristic
32	C₂×#3633	—	2	2	no	—
33	#63256	—	2	2	no	—
34	C₄×#595	#41	1	1	yes	characteristic
35	#63243	—	2	2	no	—
36	#63243	—	4	2	no	—
37	#63242	—	4	2	no	—
38	#63242	C₂×D₄	1	1	yes	characteristic
39	#63241	—	2	1	no	—
40	#63241	—	4	2	no	—
41	#63241	D₈	1	1	yes	characteristic
42	#63241	D₈	1	1	yes	characteristic
43	Q₁₆×C₁₇	C₂×C₈	1	2	yes	—
44	#63230	—	2	2	no	—
45	D₁₃₆	C₂×C₈	1	2	yes	—
46	C₈×D₁₇	C₂²×C₄	1	1	yes	characteristic
47	#63216	—	2	1	no	—
48	C₁₆×C₁₇	C₂×C₈	1	1	yes	characteristic
49	#63214	C₂×C₈	1	1	yes	characteristic
50	#12046	—	17	1	no	maximal, Sylow
51	C₂×#595	—	2	1	no	—
52	#3633	—	2	4	no	—
53	Q₈×C₁₇	—	2	2	no	—
54	D₆₈	—	2	2	no	—
55	D₆₈	—	4	2	no	—
56	C₄×D₁₇	—	4	2	no	—
57	C₄×D₁₇	C₄×D₄	1	1	yes	characteristic
58	#3624	—	4	2	no	—
59	#3624	C₂×D₈	1	1	yes	characteristic
60	#3624	#135	1	1	yes	characteristic
61	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
62	#3622	#131	1	1	yes	characteristic
63	#2178	—	17	1	no	—
64	#2170	—	17	2	no	—
65	#1373	—	17	1	no	—
66	#1335	—	17	2	no	—
67	C₈×C₁₆	—	17	1	no	—
68	C₂×D₁₇	—	4	2	no	—
69	C₂×D₁₇	#443	1	1	yes	characteristic
70	#595	—	2	2	no	—
71	C₄×C₁₇	—	4	2	no	—
72	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
73	Dic₁₇	C₄×D₈	1	1	yes	characteristic
74	#508	—	17	1	no	—
75	#504	—	34	2	no	—
76	#443	—	17	2	no	—
77	#368	—	17	1	no	—
78	#364	—	17	2	no	—
79	#359	—	17	2	no	—
80	#350	—	34	2	no	—
81	C₄×C₁₆	—	17	1	no	—
82	C₄×C₁₆	—	17	1	no	—
83	C₈²	—	17	1	no	—
84	C₂×C₁₇	C₈×D₈	1	1	yes	characteristic
85	D₁₇	—	2	1	no	—
86	D₁₇	—	8	2	no	—
87	#135	—	17	2	no	—
88	#131	—	34	2	no	—
89	Q₃₂	#3633	1	1	yes	characteristic, complemented
90	SD₃₂	—	17	2	no	—
91	D₁₆	—	17	1	no	—
92	M₃₂	—	34	2	no	—
93	M₃₂	—	68	2	no	—
94	C₂×C₁₆	—	17	1	no	—
95	C₂×C₁₆	—	17	1	no	—
96	C₂×C₁₆	—	17	1	no	—
97	C₂×C₁₆	—	17	1	no	—
98	C₂×C₁₆	—	68	2	no	—
99	#108	—	17	2	no	—
100	#104	—	34	2	no	—
101	C₄×C₈	—	17	1	no	—
102	C₄×C₈	—	34	1	no	—
103	C₁₇	#12046	1	1	yes	characteristic, complemented, Sylow
104	#41	—	34	2	no	—
105	Q₁₆	C₂×#3633	1	2	yes	—
106	SD₁₆	—	34	2	no	—
107	D₈	—	17	2	no	—
108	M₁₆	—	34	2	no	—
109	M₁₆	—	68	2	no	—
110	C₂×C₈	—	17	1	no	—
111	C₂×C₈	—	17	1	no	—
112	C₂×C₈	—	34	2	no	—
113	C₂×C₈	—	68	2	no	—
114	C₄²	—	17	1	no	—
115	C₁₆	—	17	1	no	—
116	C₁₆	—	17	1	no	—
117	C₁₆	—	17	1	no	—
118	C₁₆	—	34	1	no	—
119	C₁₆	—	34	1	no	—
120	C₁₆	—	68	2	no	—
121	C₁₆	C₂×#3633	1	1	yes	characteristic
122	Q₈	—	2	2	no	—
123	D₄	—	34	2	no	—
124	D₄	—	68	2	no	—
125	C₂×C₄	—	17	1	no	—
126	C₂×C₄	—	34	1	no	—
127	C₂×C₄	—	68	2	no	—
128	C₈	—	17	1	no	—
129	C₈	—	17	1	no	—
130	C₈	—	17	1	no	—
131	C₈	—	34	4	no	—
132	C₈	—	68	2	no	—
133	C₈	C₂²×#3633	1	1	yes	characteristic
134	C₂²	—	17	1	no	—
135	C₂²	—	68	2	no	—
136	C₄	—	4	2	no	—
137	C₄	—	17	1	no	—
138	C₄	—	34	2	no	—
139	C₄	D₄×#3633	1	1	yes	characteristic
140	C₂	—	34	1	no	—
141	C₂	—	136	2	no	—
142	C₂	D₈×#3633	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14546003

C₂

yes

maximal, complemented

#14545852

C₂

yes

maximal, characteristic, complemented

#14545835

C₂

yes

maximal, complemented

C₁₆×#3633

C₂

yes

maximal, characteristic, complemented

#14545185

C₂

yes

maximal, characteristic

#11763931

—

#11763914

C₂²

yes

—

#11763910

—

#11763902

C₂²

yes

characteristic

C₈×#3633

C₂²

yes

characteristic

#11763853

C₂²

yes

—

#11763737

C₄

yes

characteristic

#11763724

C₄

yes

—

C₁₆×#595

C₂²

yes

characteristic

#11763242

C₄

yes

—

—

—

—

—

D₄

yes

characteristic

#10587665

D₄

yes

characteristic

#10587605

C₂×C₄

yes

—

#10587550

—

#10587536

C₂×C₄

yes

—

C₈×#595

C₂³

yes

characteristic

#10587529

C₂×C₄

yes

characteristic

C₁₇×Q₃₂

C₈

yes

characteristic, complemented

#10587499

C₈

yes

complemented

D₂₇₂

C₈

yes

characteristic, complemented

C₁₆×D₁₇

C₂×C₄

yes

—

—

yes

—

—

—

C₂×D₄

yes

—

—

D₈

yes

characteristic

#63241

D₈

yes

characteristic

Q₁₆×C₁₇

C₂×C₈

yes

—

#63230

—

D₁₃₆

C₂×C₈

yes

—

C₈×D₁₇

C₂²×C₄

yes

characteristic

#63216

—

C₁₆×C₁₇

C₂×C₈

yes

characteristic

#63214

C₂×C₈

yes

—

—

—

Q₈×C₁₇

—

D₆₈

—

D₆₈

—

C₄×D₁₇

—

C₄×D₁₇

C₄×D₄

yes

characteristic

#3624

—

#3624

C₂×D₈

yes

yes

C₈×C₁₇

C₂²×C₈

yes

yes

—

—

—

—

C₈×C₁₆

—

C₂×D₁₇

—

C₂×D₁₇

#443

yes

characteristic

#595

—

C₄×C₁₇

—

C₄×C₁₇

C₈×D₄

yes

characteristic

Dic₁₇

C₄×D₈

yes

—

—

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

—

C₈²

—

C₂×C₁₇

C₈×D₈

yes

characteristic

D₁₇

—

D₁₇

—

#135

—

#131

—

Q₃₂

#3633

yes

characteristic, complemented

SD₃₂

—

D₁₆

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#108

—

100

#104

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₁₇

#12046

yes

characteristic, complemented, Sylow

104

#41

—

105

Q₁₆

C₂×#3633

yes

—

106

SD₁₆

—

107

D₈

—

108

M₁₆

—

109

M₁₆

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₄²

—

115

C₁₆

—

116

C₁₆

—

117

C₁₆

—

118

C₁₆

—

119

C₁₆

—

120

C₁₆

—

121

C₁₆

C₂×#3633

yes

characteristic

122

Q₈

—

123

D₄

—

124

D₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

C₂²×#3633

yes

characteristic

134

C₂²

—

135

C₂²

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

D₄×#3633

yes

characteristic

140

C₂

—

141

C₂

—

136

—

142

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	#11763242	C₄
cosocle	C₈×#595	C₂³
hypercenter	Q₃₂	#3633
hypocenter	C₁₇	#12046
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

C₄

C₂³

Q₃₂

C₁₇

G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

N	H ≅ G/N	type
1. #14546003	C₂	semidirect
2. #14545852	C₂	semidirect
3. #14545835	C₂	semidirect
4. C₁₆×#3633	C₂	semidirect
28. C₁₇×Q₃₂	C₈	semidirect
29. #10587499	C₈	semidirect
30. D₂₇₂	C₈	semidirect
89. Q₃₂	#3633	semidirect
103. C₁₇	#12046	semidirect

H ≅ G/N

type

1. #14546003

C₂

semidirect

2. #14545852

C₂

semidirect

3. #14545835

C₂

semidirect

4. C₁₆×#3633

C₂

semidirect

28. C₁₇×Q₃₂

C₈

semidirect

29. #10587499

C₈

semidirect

30. D₂₇₂

C₈

semidirect

89. Q₃₂

#3633

semidirect

103. C₁₇

#12046

semidirect

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

subgroup	index
#14546003	2
#14545852	2
#14545835	2
C₁₆×#3633	2
#14545185	2
#12046	17

subgroup

index

C₁₆×#3633

Subgroups — #17982292

Conjugacy classes of subgroups

Complements

Maximal subgroups