Group #17982369 — subgroups

Subgroups — #17982369

Order 4352.

204 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 59 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	#14546006	C₂	1	2	yes	maximal, complemented
2	#14545847	C₂	1	1	yes	maximal, characteristic, complemented
3	#14545836	C₂	1	2	yes	maximal, complemented
4	#14545804	C₂	1	1	yes	maximal, characteristic, complemented
5	#14545188	C₂	1	1	yes	maximal, characteristic
6	#11763930	—	2	2	no	—
7	#11763913	C₂²	1	2	yes	—
8	#11763909	—	2	2	no	—
9	#11763904	C₂²	1	1	yes	characteristic
10	#11763893	C₂²	1	1	yes	characteristic
11	#11763851	C₂²	1	2	yes	—
12	#11763738	C₄	1	1	yes	characteristic
13	#11763723	C₄	1	2	yes	—
14	#11763696	C₂²	1	1	yes	characteristic
15	#11763238	C₄	1	1	yes	characteristic
16	#10587678	—	2	2	no	—
17	#10587670	—	4	2	no	—
18	C₂×#63241	—	4	2	no	—
19	C₄×#3633	—	2	1	no	—
20	#10587666	—	2	2	no	—
21	#10587665	D₄	1	2	yes	—
22	#10587598	C₂×C₄	1	2	yes	—
23	#10587548	—	2	2	no	—
24	#10587536	C₂×C₄	1	2	yes	—
25	C₈×#595	C₂³	1	1	yes	characteristic
26	#10587529	C₂×C₄	1	1	yes	characteristic
27	C₁₇×D₁₆	C₈	1	1	yes	characteristic, complemented
28	#10587498	C₈	1	2	yes	complemented
29	Dic₁₃₆	C₈	1	1	yes	characteristic, complemented
30	C₁₆×D₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂×#3633	—	4	1	no	—
32	#63254	—	2	2	no	—
33	#63247	—	4	2	no	—
34	C₂×#3624	—	4	2	no	—
35	C₄×#595	#41	1	1	yes	characteristic
36	#63243	—	2	2	no	—
37	#63242	C₂×D₄	1	1	yes	characteristic
38	#63241	—	2	2	no	—
39	#63241	—	4	2	no	—
40	D₈×C₁₇	C₂×C₈	1	2	yes	—
41	#63229	—	2	2	no	—
42	Dic₆₈	C₂×C₈	1	2	yes	—
43	C₈×D₁₇	C₂²×C₄	1	1	yes	characteristic
44	#63216	—	2	1	no	—
45	C₁₆×C₁₇	C₂×C₈	1	1	yes	characteristic
46	#63214	C₂×C₈	1	1	yes	characteristic
47	#12068	—	17	1	no	maximal, Sylow
48	C₂×#595	—	2	1	no	—
49	#3633	—	8	1	no	—
50	D₄×C₁₇	—	2	2	no	—
51	C₂²⋊D₁₇	—	4	2	no	—
52	C₂×Dic₁₇	—	4	2	no	—
53	C₄×D₁₇	C₄×D₄	1	1	yes	characteristic
54	Dic₃₄	—	2	2	no	—
55	#3624	—	4	2	no	—
56	#3624	AGL(1,ℤ/8)	1	2	yes	—
57	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
58	#3622	#131	1	1	yes	characteristic
59	#2179	—	17	1	no	—
60	#2171	—	17	2	no	—
61	#1369	—	17	1	no	—
62	#1336	—	17	2	no	—
63	#1313	—	17	1	no	—
64	C₂²×C₁₇	—	4	2	no	—
65	C₂×D₁₇	#444	1	1	yes	characteristic
66	#595	—	4	1	no	—
67	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
68	Dic₁₇	—	4	2	no	—
69	Dic₁₇	#442	1	1	yes	characteristic
70	#508	—	17	1	no	—
71	#504	—	34	2	no	—
72	#443	—	17	2	no	—
73	#368	—	17	1	no	—
74	#364	—	17	2	no	—
75	#359	—	17	2	no	—
76	#350	—	34	2	no	—
77	#346	—	17	1	no	—
78	#346	—	17	1	no	—
79	#322	—	17	1	no	—
80	C₂×C₁₇	—	8	2	no	—
81	C₂×C₁₇	#1580	1	1	yes	characteristic
82	D₁₇	—	2	1	no	—
83	#135	—	17	2	no	—
84	#131	—	34	2	no	—
85	Q₃₂	—	17	1	no	—
86	SD₃₂	—	17	2	no	—
87	D₁₆	#3633	1	1	yes	characteristic, complemented
88	M₃₂	—	34	2	no	—
89	M₃₂	—	68	2	no	—
90	C₂×C₁₆	—	17	1	no	—
91	C₂×C₁₆	—	17	2	no	—
92	C₂×C₁₆	—	17	1	no	—
93	C₂×C₁₆	—	68	2	no	—
94	#108	—	17	2	no	—
95	#104	—	34	2	no	—
96	C₄×C₈	—	17	1	no	—
97	C₄×C₈	—	34	1	no	—
98	C₁₇	#12068	1	1	yes	characteristic, complemented, Sylow
99	#41	—	34	2	no	—
100	Q₁₆	—	17	2	no	—
101	SD₁₆	—	34	2	no	—
102	D₈	C₂×#3633	1	2	yes	—
103	M₁₆	—	34	2	no	—
104	M₁₆	—	68	2	no	—
105	C₂×C₈	—	17	1	no	—
106	C₂×C₈	—	17	1	no	—
107	C₂×C₈	—	68	2	no	—
108	C₂×C₈	—	68	1	no	—
109	C₄²	—	17	1	no	—
110	C₁₆	—	17	1	no	—
111	C₁₆	—	34	2	no	—
112	C₁₆	—	34	1	no	—
113	C₁₆	—	68	2	no	—
114	C₁₆	C₂×#3633	1	1	yes	characteristic
115	Q₈	—	34	2	no	—
116	D₄	—	2	2	no	—
117	D₄	—	68	2	no	—
118	C₂×C₄	—	17	1	no	—
119	C₂×C₄	—	34	1	no	—
120	C₂×C₄	—	68	2	no	—
121	C₈	—	17	2	no	—
122	C₈	—	17	1	no	—
123	C₈	—	68	2	no	—
124	C₈	—	136	1	no	—
125	C₈	C₂²×#3633	1	1	yes	characteristic
126	C₂²	—	4	2	no	—
127	C₂²	—	17	1	no	—
128	C₄	—	17	1	no	—
129	C₄	—	68	1	no	—
130	C₄	—	68	2	no	—
131	C₄	D₄×#3633	1	1	yes	characteristic
132	C₂	—	8	2	no	—
133	C₂	—	34	1	no	—
134	C₂	#14545915	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14546006

C₂

yes

maximal, complemented

#14545847

C₂

yes

maximal, characteristic, complemented

#14545836

C₂

yes

maximal, complemented

#14545804

C₂

yes

maximal, characteristic, complemented

#14545188

C₂

yes

maximal, characteristic

#11763930

—

#11763913

C₂²

yes

—

#11763909

—

#11763904

C₂²

yes

characteristic

#11763893

C₂²

yes

characteristic

#11763851

C₂²

yes

—

#11763738

C₄

yes

characteristic

#11763723

C₄

yes

—

#11763696

C₂²

yes

characteristic

#11763238

C₄

yes

—

—

—

—

—

D₄

yes

—

#10587598

C₂×C₄

yes

—

#10587548

—

#10587536

C₂×C₄

yes

—

C₈×#595

C₂³

yes

characteristic

#10587529

C₂×C₄

yes

characteristic

C₁₇×D₁₆

C₈

yes

characteristic, complemented

#10587498

C₈

yes

complemented

Dic₁₃₆

C₈

yes

characteristic, complemented

C₁₆×D₁₇

C₂×C₄

yes

—

—

—

—

yes

—

C₂×D₄

yes

characteristic

#63241

—

#63241

—

D₈×C₁₇

C₂×C₈

yes

—

#63229

—

Dic₆₈

C₂×C₈

yes

—

C₈×D₁₇

C₂²×C₄

yes

characteristic

#63216

—

C₁₆×C₁₇

C₂×C₈

yes

characteristic

#63214

C₂×C₈

yes

—

—

—

D₄×C₁₇

—

C₂²⋊D₁₇

—

C₂×Dic₁₇

—

C₄×D₁₇

C₄×D₄

yes

characteristic

Dic₃₄

—

#3624

—

#3624

AGL(1,ℤ/8)

yes

—

C₈×C₁₇

C₂²×C₈

yes

yes

—

—

—

—

—

C₂²×C₁₇

—

C₂×D₁₇

#444

yes

characteristic

#595

—

C₄×C₁₇

C₈×D₄

yes

characteristic

Dic₁₇

—

Dic₁₇

yes

—

—

—

—

—

—

—

—

—

—

C₂×C₁₇

—

C₂×C₁₇

#1580

yes

characteristic

D₁₇

—

#135

—

#131

—

Q₃₂

—

SD₃₂

—

D₁₆

#3633

yes

characteristic, complemented

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#108

—

#104

—

C₄×C₈

—

C₄×C₈

—

C₁₇

#12068

yes

characteristic, complemented, Sylow

#41

—

100

Q₁₆

—

101

SD₁₆

—

102

D₈

C₂×#3633

yes

—

103

M₁₆

—

104

M₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₄²

—

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

113

C₁₆

—

114

C₁₆

C₂×#3633

yes

characteristic

115

Q₈

—

116

D₄

—

117

D₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

136

—

125

C₈

C₂²×#3633

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

D₄×#3633

yes

characteristic

132

C₂

—

133

C₂

—

134

C₂

#14545915

yes

characteristic

	subgroup	G/H
center	C₂	#14545915
commutator	C₈×C₁₇	C₂²×C₈
Frattini	C₈	C₂²×#3633
Fitting	C₁₇×D₁₆	C₈
socle	C₂×C₁₇	#1580
prime radical	#11763238	C₄
cosocle	C₈×#595	C₂³
hypercenter	D₁₆	#3633
hypocenter	C₁₇	#12068
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₁₆

subgroup

G/H

center

C₂

#14545915

commutator

C₈×C₁₇

C₂²×C₈

Frattini

C₈

C₂²×#3633

Fitting

C₁₇×D₁₆

C₈

C₂×C₁₇

C₄

C₂³

D₁₆

C₁₇

G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₁₆

N	H ≅ G/N	type
1. #14546006	C₂	semidirect
2. #14545847	C₂	semidirect
3. #14545836	C₂	semidirect
4. #14545804	C₂	semidirect
27. C₁₇×D₁₆	C₈	semidirect
28. #10587498	C₈	semidirect
29. Dic₁₃₆	C₈	semidirect
87. D₁₆	#3633	semidirect
98. C₁₇	#12068	semidirect

H ≅ G/N

type

1. #14546006

C₂

semidirect

2. #14545847

C₂

semidirect

3. #14545836

C₂

semidirect

4. #14545804

C₂

semidirect

27. C₁₇×D₁₆

C₈

semidirect

28. #10587498

C₈

semidirect

29. Dic₁₃₆

C₈

semidirect

87. D₁₆

#3633

semidirect

98. C₁₇

#12068

semidirect

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

subgroup	index
#14546006	2
#14545847	2
#14545836	2
#14545804	2
#14545188	2
#12068	17

subgroup

index

Subgroups — #17982369

Conjugacy classes of subgroups

Complements

Maximal subgroups