Group #17960308 — subgroups

Subgroups — #17960308

Order 4352.

204 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 66 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	#14545176	C₂	1	2	yes	maximal, complemented
2	#14543811	C₂	1	1	yes	maximal, characteristic, complemented
3	#14543749	C₂	1	2	yes	maximal, complemented
4	#14543700	C₂	1	1	yes	maximal, characteristic, complemented
5	#14530669	C₂	1	1	yes	maximal, characteristic
6	#11763868	—	2	2	no	—
7	#11763731	C₂²	1	2	yes	—
8	#11763708	—	2	2	no	—
9	#11763698	C₂²	1	1	yes	characteristic
10	#11763650	C₂²	1	1	yes	characteristic
11	#11763241	C₄	1	1	yes	characteristic
12	#11762925	C₂²	1	2	yes	—
13	#11762615	C₄	1	2	yes	—
14	#11762584	C₂²	1	1	yes	characteristic
15	C₁₇×#368	C₄	1	1	yes	characteristic
16	#10587593	—	2	2	no	—
17	#10587590	C₂×C₄	1	2	yes	—
18	C₄×#3624	—	2	1	no	—
19	#10587538	—	2	2	no	—
20	C₂×#63216	D₄	1	2	yes	—
21	#10587530	—	4	2	no	—
22	#10587529	—	4	2	no	—
23	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₁₇×#108	C₂×C₄	1	2	yes	—
25	#10587496	—	2	2	no	—
26	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
27	C₂×#63214	C₂×C₄	1	1	yes	characteristic
28	Dic₁₃₆	C₈	1	1	yes	characteristic, complemented
29	#10587471	C₈	1	2	yes	complemented
30	D₂₇₂	C₈	1	1	yes	characteristic, complemented
31	#63252	—	2	2	no	—
32	C₂×#3624	—	4	1	no	—
33	M₁₆×C₁₇	—	2	2	no	—
34	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
35	C₄×Dic₁₇	#41	1	1	yes	characteristic
36	C₂×#3622	C₂×D₄	1	1	yes	characteristic
37	Dic₆₈	C₂×C₈	1	2	yes	—
38	D₁₃₆	C₂×C₈	1	2	yes	—
39	#63219	—	2	2	no	—
40	#63218	—	4	2	no	—
41	C₈×D₁₇	—	4	2	no	—
42	#63216	—	2	2	no	—
43	#63216	—	4	2	no	—
44	C₁₆×C₁₇	C₂×C₈	1	1	yes	characteristic
45	C₁₆×C₁₇	C₂×C₈	1	1	yes	characteristic
46	#63214	—	2	1	no	—
47	#12068	—	17	1	no	maximal, Sylow
48	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
49	C₂²⋊D₁₇	—	4	2	no	—
50	C₂×Dic₁₇	—	2	1	no	—
51	D₆₈	—	2	2	no	—
52	C₄×D₁₇	—	4	2	no	—
53	Dic₃₄	—	2	2	no	—
54	#3624	—	8	1	no	—
55	C₈×C₁₇	—	4	2	no	—
56	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
57	C₈×C₁₇	#131	1	1	yes	characteristic
58	#3622	AGL(1,ℤ/8)	1	2	yes	—
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	#444	1	1	yes	characteristic
65	C₂×D₁₇	—	4	2	no	—
66	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
67	C₄×C₁₇	#442	1	1	yes	characteristic
68	Dic₁₇	—	4	1	no	—
69	Dic₁₇	—	4	2	no	—
70	#508	—	17	1	no	—
71	#504	—	34	2	no	—
72	#443	—	17	2	no	—
73	#368	#595	1	1	yes	characteristic
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	—	2	1	no	—
81	C₂×C₁₇	#1580	1	1	yes	characteristic
82	D₁₇	—	8	2	no	—
83	#135	—	17	2	no	—
84	#131	—	34	2	no	—
85	Q₃₂	—	17	1	no	—
86	SD₃₂	—	17	2	no	—
87	D₁₆	—	17	1	no	—
88	M₃₂	—	34	2	no	—
89	M₃₂	—	68	2	no	—
90	C₂×C₁₆	—	17	2	no	—
91	C₂×C₁₆	—	17	1	no	—
92	C₂×C₁₆	—	68	2	no	—
93	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
94	#108	C₂×#595	1	2	yes	—
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	—	17	2	no	—
103	M₁₆	—	2	2	no	—
104	M₁₆	—	68	2	no	—
105	C₂×C₈	—	17	1	no	—
106	C₂×C₈	—	68	2	no	—
107	C₂×C₈	—	68	1	no	—
108	C₂×C₈	C₂²×#595	1	1	yes	characteristic
109	C₄²	—	17	1	no	—
110	C₁₆	—	34	2	no	—
111	C₁₆	—	34	1	no	—
112	C₁₆	—	68	2	no	—
113	C₁₆	#63242	1	1	yes	characteristic
114	C₁₆	#63242	1	1	yes	characteristic
115	Q₈	—	34	2	no	—
116	D₄	—	34	2	no	—
117	D₄	—	68	2	no	—
118	C₂×C₄	—	34	1	no	—
119	C₂×C₄	—	68	2	no	—
120	C₂×C₄	D₄×#595	1	1	yes	characteristic
121	C₈	—	4	2	no	—
122	C₈	—	17	2	no	—
123	C₈	—	136	1	no	—
124	C₈	C₂×#63242	1	1	yes	characteristic
125	C₈	#10587680	1	1	yes	characteristic
126	C₂²	—	68	2	no	—
127	C₂²	#11763855	1	1	yes	characteristic
128	C₄	—	68	1	no	—
129	C₄	—	68	2	no	—
130	C₄	#11763837	1	1	yes	characteristic
131	C₄	#11763850	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	136	2	no	—
134	C₂	#14544382	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545176

C₂

yes

maximal, complemented

#14543811

C₂

yes

maximal, characteristic, complemented

#14543749

C₂

yes

maximal, complemented

#14543700

C₂

yes

maximal, characteristic, complemented

#14530669

C₂

yes

maximal, characteristic

#11763868

—

#11763731

C₂²

yes

—

#11763708

—

#11763698

C₂²

yes

characteristic

#11763650

C₂²

yes

characteristic

#11763241

C₄

yes

characteristic

#11762925

C₂²

yes

—

#11762615

C₄

yes

—

#11762584

C₂²

yes

characteristic

C₁₇×#368

C₄

yes

characteristic

#10587593

—

#10587590

C₂×C₄

yes

—

C₄×#3624

—

#10587538

—

C₂×#63216

D₄

yes

—

#10587530

—

#10587529

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₁₇×#108

C₂×C₄

yes

—

#10587496

—

C₈×Dic₁₇

C₂³

yes

characteristic

C₂×#63214

C₂×C₄

yes

characteristic

Dic₁₃₆

C₈

yes

characteristic, complemented

#10587471

C₈

yes

complemented

D₂₇₂

C₈

yes

characteristic, complemented

#63252

—

C₂×#3624

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

C₂²×C₄

yes

C₄×Dic₁₇

yes

C₂×D₄

yes

characteristic

Dic₆₈

C₂×C₈

yes

—

D₁₃₆

C₂×C₈

yes

—

#63219

—

#63218

—

C₈×D₁₇

—

#63216

—

#63216

—

C₁₆×C₁₇

C₂×C₈

yes

characteristic

C₁₆×C₁₇

C₂×C₈

yes

—

—

C₂×C₄×C₁₇

C₄×D₄

yes

characteristic

C₂²⋊D₁₇

—

C₂×Dic₁₇

—

D₆₈

—

C₄×D₁₇

—

Dic₃₄

—

#3624

—

C₈×C₁₇

—

C₈×C₁₇

C₂²×C₈

yes

C₈×C₁₇

yes

yes

—

—

—

—

—

—

C₂²×C₁₇

#444

yes

characteristic

C₂×D₁₇

—

C₄×C₁₇

C₈×D₄

yes

characteristic

C₄×C₁₇

#442

yes

characteristic

Dic₁₇

—

Dic₁₇

—

—

—

—

yes

—

—

—

—

—

—

C₂×C₁₇

—

C₂×C₁₇

#1580

yes

characteristic

D₁₇

—

#135

—

#131

—

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

yes

—

#104

—

C₄×C₈

—

C₄×C₈

—

C₁₇

#12068

yes

characteristic, complemented, Sylow

#41

—

100

Q₁₆

—

101

SD₁₆

—

102

D₈

—

103

M₁₆

—

104

M₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂²×#595

yes

characteristic

109

C₄²

—

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

113

C₁₆

#63242

yes

characteristic

114

C₁₆

#63242

yes

characteristic

115

Q₈

—

116

D₄

—

117

D₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

D₄×#595

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

136

—

124

C₈

C₂×#63242

yes

characteristic

125

C₈

#10587680

yes

characteristic

126

C₂²

—

127

C₂²

#11763855

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

#11763837

yes

characteristic

131

C₄

#11763850

yes

characteristic

132

C₂

—

133

C₂

—

136

—

134

C₂

#14544382

yes

characteristic

	subgroup	G/H
center	C₄	#11763850
commutator	C₈×C₁₇	C₂²×C₈
Frattini	C₂×C₈	C₂²×#595
Fitting	C₁₇×#368	C₄
socle	C₂×C₁₇	#1580
prime radical	#11763241	C₄
cosocle	C₈×Dic₁₇	C₂³
hypercenter	#368	#595
hypocenter	C₁₇	#12068
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

subgroup

G/H

center

C₄

#11763850

commutator

C₈×C₁₇

C₂²×C₈

Frattini

C₂×C₈

C₂²×#595

Fitting

C₁₇×#368

C₄

C₂×C₁₇

C₄

C₈×Dic₁₇

C₂³

C₁₇

G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

N	H ≅ G/N	type
1. #14545176	C₂	semidirect
2. #14543811	C₂	semidirect
3. #14543749	C₂	semidirect
4. #14543700	C₂	semidirect
28. Dic₁₃₆	C₈	semidirect
29. #10587471	C₈	semidirect
30. D₂₇₂	C₈	semidirect
98. C₁₇	#12068	semidirect

H ≅ G/N

type

1. #14545176

C₂

semidirect

2. #14543811

C₂

semidirect

3. #14543749

C₂

semidirect

4. #14543700

C₂

semidirect

28. Dic₁₃₆

C₈

semidirect

29. #10587471

C₈

semidirect

30. D₂₇₂

C₈

semidirect

98. C₁₇

#12068

semidirect

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

subgroup	index
#14545176	2
#14543811	2
#14543749	2
#14543700	2
#14530669	2
#12068	17

subgroup

index

Subgroups — #17960308

Conjugacy classes of subgroups

Complements

Maximal subgroups