Group #17953778 — subgroups

Subgroups — #17953778

Order 4352.

262 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 80 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	#14544709	C₂	1	1	yes	maximal, characteristic
2	#14544706	C₂	1	1	yes	maximal, characteristic
3	#14543638	C₂	1	4	yes	maximal
4	#14527069	C₂	1	1	yes	maximal, characteristic
5	#11763814	C₂²	1	2	yes	—
6	#11763753	C₂²	1	2	yes	—
7	#11763685	—	2	2	no	—
8	#11763683	—	2	2	no	—
9	#11762860	C₂²	1	2	yes	—
10	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
11	#11762530	C₄	1	1	yes	characteristic, complemented
12	C₈×#3622	C₄	1	2	yes	complemented
13	C₁₇×#322	C₄	1	1	yes	characteristic, complemented
14	C₂×C₄⋊#595	—	2	2	no	—
15	C₂×#63243	—	2	2	no	—
16	C₂×C₈×D₁₇	D₄	1	2	yes	—
17	C₄²×D₁₇	C₂³	1	1	yes	characteristic
18	(C₂×#595)⋊C₄	—	2	4	no	—
19	#10587544	—	2	4	no	—
20	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
22	#10587491	—	2	2	no	—
23	#10587485	—	2	2	no	—
24	C₈×Dic₁₇	Q₈	1	2	yes	—
25	C₄×#3622	C₂×C₄	1	2	yes	—
26	C₄×#3622	C₂×C₄	1	1	yes	characteristic
27	C₂²×#595	—	4	2	no	—
28	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×D₁₇	#41	1	2	yes	—
30	C₂×#3624	—	4	2	no	—
31	C₄⋊#595	—	4	4	no	—
32	#63243	—	4	4	no	—
33	C₂×C₈×C₁₇	—	2	2	no	—
34	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
35	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
36	C₄×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
37	C₄×Dic₁₇	#41	1	2	yes	—
38	C₂×#3622	—	2	2	no	—
39	C₂×#3622	C₄⋊C₄	1	2	yes	—
40	C₈×D₁₇	—	2	4	no	—
41	#8792	—	17	1	no	maximal, Sylow
42	C₂²×D₁₇	#124	1	1	yes	characteristic
43	C₂×#595	—	8	4	no	—
44	C₂×#595	—	8	2	no	—
45	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₇	#117	1	2	yes	—
47	C₂×Dic₁₇	#124	1	1	yes	characteristic
48	C₂×Dic₁₇	#125	1	2	yes	—
49	C₄×D₁₇	—	4	2	no	—
50	C₄×D₁₇	#135	1	4	yes	—
51	#3624	—	8	2	no	—
52	C₈×C₁₇	—	2	2	no	—
53	C₈×C₁₇	—	4	2	no	—
54	#3622	—	2	2	no	—
55	#3622	—	4	2	no	—
56	#1837	—	17	1	no	—
57	#1835	—	17	1	no	—
58	#1450	—	17	1	no	—
59	#1286	—	17	4	no	—
60	C₂²×C₁₇	#382	1	1	yes	characteristic
61	C₂×D₁₇	—	2	2	no	—
62	C₂×D₁₇	#488	1	2	yes	—
63	#595	—	16	2	no	—
64	C₄×C₁₇	—	2	2	no	—
65	C₄×C₁₇	#427	1	2	yes	—
66	Dic₁₇	—	2	2	no	—
67	Dic₁₇	#488	1	2	yes	—
68	#431	—	17	2	no	—
69	#423	—	17	2	no	—
70	C₂×C₄×C₈	—	17	1	no	—
71	#382	—	17	2	no	—
72	#341	—	34	2	no	—
73	#340	—	34	2	no	—
74	#322	—	17	1	no	—
75	#322	#595	1	1	yes	characteristic, complemented
76	C₈²	—	17	2	no	—
77	C₂×C₁₇	#1833	1	2	yes	—
78	C₂×C₁₇	#1835	1	1	yes	characteristic
79	D₁₇	—	2	2	no	—
80	C₂×M₁₆	—	34	2	no	—
81	C₂²×C₈	—	17	2	no	—
82	C₂×C₄⋊C₄	—	34	2	no	—
83	C₂×C₄²	—	17	1	no	—
84	C₄⋊C₈	—	34	2	no	—
85	C₄⋊C₈	—	34	4	no	—
86	C₂²⋊C₈	—	34	2	no	—
87	C₄×C₈	—	17	2	no	—
88	C₄×C₈	—	17	1	no	—
89	C₄×C₈	—	17	2	no	—
90	C₄×C₈	C₂×#595	1	2	yes	—
91	C₄×C₈	C₂×#595	1	1	yes	characteristic
92	#95	—	34	4	no	—
93	C₁₇	#8792	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₄	—	17	1	no	—
95	C₂²×C₄	—	17	2	no	—
96	C₂²×C₄	—	68	2	no	—
97	M₁₆	—	68	4	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	17	2	no	—
100	C₂×C₈	—	34	4	no	—
101	C₂×C₈	—	34	2	no	—
102	C₂×C₈	—	68	2	no	—
103	C₂×C₈	C₄⋊#595	1	2	yes	—
104	C₄⋊C₄	—	68	4	no	—
105	C₄²	—	17	1	no	—
106	C₄²	—	17	2	no	—
107	C₄²	C₂²×#595	1	1	yes	characteristic
108	C₂³	—	17	1	no	—
109	C₂×C₄	—	17	1	no	—
110	C₂×C₄	—	17	4	no	—
111	C₂×C₄	—	17	2	no	—
112	C₂×C₄	—	68	2	no	—
113	C₂×C₄	—	136	2	no	—
114	C₂×C₄	—	136	4	no	—
115	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
116	C₂×C₄	#10587677	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	4	2	no	—
119	C₈	—	34	2	no	—
120	C₈	—	68	2	no	—
121	C₈	—	136	2	no	—
122	C₂²	—	17	2	no	—
123	C₂²	—	34	2	no	—
124	C₂²	#11763753	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	272	2	no	—
127	C₄	—	17	2	no	—
128	C₄	—	34	2	no	—
129	C₄	#11763819	1	2	yes	—
130	C₂	—	34	2	no	—
131	C₂	#14544704	1	2	yes	—
132	C₂	#14544706	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14544709

C₂

yes

maximal, characteristic

#14544706

C₂

yes

maximal, characteristic

#14543638

C₂

yes

maximal

#14527069

C₂

yes

maximal, characteristic

#11763814

C₂²

yes

—

#11763753

C₂²

yes

—

#11763685

—

#11763683

—

#11762860

C₂²

yes

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762530

C₄

yes

characteristic, complemented

C₈×#3622

C₄

yes

complemented

C₁₇×#322

C₄

yes

characteristic, complemented

C₂×C₄⋊#595

—

C₂×#63243

—

C₂×C₈×D₁₇

D₄

yes

—

C₄²×D₁₇

C₂³

yes

characteristic

(C₂×#595)⋊C₄

—

#10587544

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

#10587491

—

#10587485

—

C₈×Dic₁₇

Q₈

yes

—

C₄×#3622

C₂×C₄

yes

—

C₄×#3622

C₂×C₄

yes

characteristic

C₂²×#595

—

C₂×C₄×D₁₇

C₂×D₄

yes

characteristic

C₂×C₄×D₁₇

#41

yes

—

C₂×#3624

—

C₄⋊#595

—

#63243

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄×Dic₁₇

C₂×Q₈

yes

characteristic

C₄×Dic₁₇

#41

yes

—

C₂×#3622

—

C₂×#3622

C₄⋊C₄

yes

—

C₈×D₁₇

—

—

C₂²×D₁₇

yes

—

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

#117

yes

—

C₂×Dic₁₇

#124

yes

characteristic

C₂×Dic₁₇

#125

yes

—

C₄×D₁₇

—

C₄×D₁₇

#135

yes

—

#3624

—

C₈×C₁₇

—

C₈×C₁₇

—

—

—

—

—

—

—

C₂²×C₁₇

#382

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

#488

yes

—

#595

—

C₄×C₁₇

—

C₄×C₁₇

#427

yes

—

Dic₁₇

—

Dic₁₇

#488

yes

—

#431

—

#423

—

C₂×C₄×C₈

—

—

—

—

—

yes

characteristic, complemented

C₈²

—

C₂×C₁₇

#1833

yes

—

C₂×C₁₇

#1835

yes

characteristic

D₁₇

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#595

yes

—

C₄×C₈

C₂×#595

yes

characteristic

#95

—

C₁₇

#8792

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

C₄⋊#595

yes

—

104

C₄⋊C₄

—

105

C₄²

—

106

C₄²

—

107

C₄²

C₂²×#595

yes

characteristic

108

C₂³

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

136

—

114

C₂×C₄

—

136

—

115

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

116

C₂×C₄

#10587677

yes

—

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

—

136

—

122

C₂²

—

123

C₂²

—

124

C₂²

#11763753

yes

characteristic

125

C₄

—

126

C₄

—

272

—

127

C₄

—

128

C₄

—

129

C₄

#11763819

yes

—

130

C₂

—

131

C₂

#14544704

yes

—

132

C₂

#14544706

yes

characteristic

	subgroup	G/H
center	C₂²	#11763753
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₄²	C₂²×#595
Fitting	C₁₇×#322	C₄
socle	C₂²×C₁₇	#382
prime radical	C₂²×D₁₇	#124
cosocle	C₄²×D₁₇	C₂³
hypercenter	#322	#595
hypocenter	C₁₇	#8792
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#11763753

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₄²

C₂²×#595

Fitting

C₁₇×#322

C₄

C₂²×C₁₇

C₂²×D₁₇

C₄²×D₁₇

C₂³

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
11. #11762530	C₄	semidirect
12. C₈×#3622	C₄	semidirect
13. C₁₇×#322	C₄	semidirect
75. #322	#595	semidirect
93. C₁₇	#8792	semidirect

H ≅ G/N

type

11. #11762530

C₄

semidirect

12. C₈×#3622

C₄

semidirect

13. C₁₇×#322

C₄

semidirect

75. #322

#595

semidirect

93. C₁₇

#8792

semidirect

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

subgroup	index
#14544709	2
#14544706	2
#14543638	2
#14527069	2
#8792	17

subgroup

index

Subgroups — #17953778

Conjugacy classes of subgroups

Complements

Maximal subgroups