Group #16928142 — subgroups

Subgroups — #16928142

Order 4352.

306 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 155 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	#14528922	C₂	1	2	yes	maximal
2	#14527543	C₂	1	2	yes	maximal
3	#14527540	C₂	1	2	yes	maximal
4	C₂×C₁₇×#322	C₂	1	1	yes	maximal, characteristic
5	#11762833	C₂²	1	4	yes	—
6	#11762574	—	2	4	no	—
7	#11762562	—	2	4	no	—
8	#11762560	—	2	4	no	—
9	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
12	C₁₇×#322	C₄	1	4	yes	—
13	C₂²×C₈×C₁₇	D₄	1	2	yes	—
14	C₂²×C₈×C₁₇	Q₈	1	2	yes	—
15	C₂²×C₈×C₁₇	Q₈	1	2	yes	—
16	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
17	C₂×#63223	—	2	4	no	—
18	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
20	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₁₇	D₄	1	2	yes	—
22	C₄×C₈×C₁₇	D₄	1	2	yes	—
23	C₄×C₈×C₁₇	Q₈	1	2	yes	—
24	C₄⋊#3622	—	2	4	no	—
25	C₄⋊#3622	—	2	4	no	—
26	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
28	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
29	C₂×C₈×C₁₇	—	2	4	no	—
30	C₂×C₈×C₁₇	—	4	2	no	—
31	C₂×C₈×C₁₇	—	4	2	no	—
32	C₂×C₈×C₁₇	C₄⋊C₄	1	4	yes	—
33	C₂×C₈×C₁₇	C₄⋊C₄	1	4	yes	—
34	C₂×C₈×C₁₇	C₄⋊C₄	1	4	yes	—
35	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
37	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
38	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
39	#63223	—	4	8	no	—
40	C₂×#3622	—	4	4	no	—
41	#8827	—	17	1	no	maximal, Sylow
42	C₂³×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
43	C₂×C₄×C₁₇	—	2	2	no	—
44	C₂×C₄×C₁₇	—	2	2	no	—
45	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
49	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
50	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
51	C₂×C₄×C₁₇	#135	1	4	yes	—
52	C₈×C₁₇	—	2	4	no	—
53	C₈×C₁₇	—	2	4	no	—
54	C₈×C₁₇	—	2	4	no	—
55	#3622	—	8	4	no	—
56	#1837	—	17	2	no	—
57	#1567	—	17	2	no	—
58	#1566	—	17	2	no	—
59	C₂×#322	D₁₇	1	1	yes	characteristic
60	C₂²×C₁₇	—	2	1	no	—
61	C₂²×C₁₇	—	2	1	no	—
62	C₂²×C₁₇	#384	1	1	yes	characteristic
63	C₂²×C₁₇	#499	1	2	yes	—
64	C₄×C₁₇	—	2	1	no	—
65	C₄×C₁₇	—	2	1	no	—
66	C₄×C₁₇	#427	1	2	yes	—
67	C₄×C₁₇	#430	1	2	yes	—
68	C₄×C₁₇	#430	1	2	yes	—
69	C₄×C₁₇	#495	1	2	yes	—
70	#423	—	17	4	no	—
71	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
72	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
73	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
74	#341	—	34	4	no	—
75	#335	—	34	4	no	—
76	#334	—	34	4	no	—
77	#322	Dic₁₇	1	4	yes	—
78	C₂×C₁₇	—	2	2	no	—
79	C₂×C₁₇	#1846	1	1	yes	characteristic
80	C₂×C₁₇	#1849	1	1	yes	characteristic
81	C₂×C₁₇	#1849	1	1	yes	characteristic
82	C₂×M₁₆	—	34	4	no	—
83	C₂²×C₈	Dic₃₄	1	2	yes	—
84	C₂²×C₈	Dic₃₄	1	2	yes	—
85	C₂²×C₈	D₆₈	1	2	yes	—
86	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
87	C₄⋊C₈	—	34	4	no	—
88	C₄⋊C₈	—	34	4	no	—
89	C₄×C₈	Dic₃₄	1	2	yes	—
90	C₄×C₈	D₆₈	1	2	yes	—
91	C₄×C₈	D₆₈	1	2	yes	—
92	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
93	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
94	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
95	C₁₇	#8827	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
97	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
98	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
99	M₁₆	—	68	8	no	—
100	C₂×C₈	—	2	4	no	—
101	C₂×C₈	—	4	2	no	—
102	C₂×C₈	—	4	2	no	—
103	C₂×C₈	—	68	4	no	—
104	C₂×C₈	C₄⋊Dic₁₇	1	4	yes	—
105	C₂×C₈	C₄⋊Dic₁₇	1	4	yes	—
106	C₂×C₈	C₄⋊Dic₁₇	1	4	yes	—
107	C₄²	C₂×Dic₃₄	1	1	yes	characteristic
108	C₄²	C₂×D₆₈	1	1	yes	characteristic
109	C₄²	C₂×D₆₈	1	1	yes	characteristic
110	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
111	C₂³	#10587555	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	#10587555	1	1	yes	characteristic
115	C₂×C₄	#10587555	1	1	yes	characteristic
116	C₂×C₄	#10587560	1	1	yes	characteristic
117	C₂×C₄	#10587590	1	4	yes	—
118	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
119	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
120	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
121	C₈	—	2	4	no	—
122	C₈	—	2	4	no	—
123	C₈	—	2	4	no	—
124	C₈	—	136	4	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	#11762679	1	1	yes	characteristic
128	C₂²	#11763211	1	2	yes	—
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	#11762847	1	2	yes	—
132	C₄	#11762858	1	2	yes	—
133	C₄	#11762858	1	2	yes	—
134	C₄	#11763193	1	2	yes	—
135	C₂	—	2	2	no	—
136	C₂	#14528948	1	1	yes	characteristic
137	C₂	#14528959	1	1	yes	characteristic
138	C₂	#14528959	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14528922

C₂

yes

maximal

#14527543

C₂

yes

maximal

#14527540

C₂

yes

maximal

C₂×C₁₇×#322

C₂

yes

maximal, characteristic

#11762833

C₂²

yes

—

#11762574

—

#11762562

—

#11762560

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₁₇×#322

C₄

yes

—

C₂²×C₈×C₁₇

D₄

yes

—

C₂²×C₈×C₁₇

Q₈

yes

—

C₂²×C₈×C₁₇

Q₈

yes

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂×#63223

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

D₄

yes

—

C₄×C₈×C₁₇

D₄

yes

—

C₄×C₈×C₁₇

Q₈

yes

—

C₄⋊#3622

—

C₄⋊#3622

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂²×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

C₂×Q₈

yes

—

—

—

C₂³×C₁₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

#127

yes

characteristic

C₂×C₄×C₁₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₇

#135

yes

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

—

—

—

—

D₁₇

yes

characteristic

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#384

yes

characteristic

C₂²×C₁₇

#499

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#427

yes

—

C₄×C₁₇

#430

yes

—

C₄×C₁₇

#430

yes

—

C₄×C₁₇

#495

yes

—

#423

—

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

C₂×C₄×C₈

C₂×D₁₇

yes

—

—

—

Dic₁₇

yes

—

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×M₁₆

—

C₂²×C₈

Dic₃₄

yes

—

C₂²×C₈

Dic₃₄

yes

—

C₂²×C₈

D₆₈

yes

—

C₂×C₄²

C₂²×D₁₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

Dic₃₄

yes

—

C₄×C₈

D₆₈

yes

—

C₄×C₈

D₆₈

yes

—

C₄×C₈

C₂×Dic₁₇

yes

—

C₄×C₈

C₂×Dic₁₇

yes

—

C₄×C₈

C₂×Dic₁₇

yes

—

C₁₇

#8827

yes

characteristic, complemented, Sylow

C₂²×C₄

C₂×Dic₃₄

yes

characteristic

C₂²×C₄

C₂×Dic₃₄

yes

characteristic

C₂²×C₄

C₂×D₆₈

yes

characteristic

M₁₆

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

C₄⋊Dic₁₇

yes

—

105

C₂×C₈

C₄⋊Dic₁₇

yes

—

106

C₂×C₈

C₄⋊Dic₁₇

yes

—

107

C₄²

C₂×Dic₃₄

yes

characteristic

108

C₄²

C₂×D₆₈

yes

characteristic

109

C₄²

C₂×D₆₈

yes

characteristic

110

C₄²

C₂²×Dic₁₇

yes

characteristic

111

C₂³

#10587555

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

#10587555

yes

characteristic

115

C₂×C₄

#10587555

yes

characteristic

116

C₂×C₄

#10587560

yes

characteristic

117

C₂×C₄

#10587590

yes

—

118

C₂×C₄

C₂×C₄⋊Dic₁₇

yes

characteristic

119

C₂×C₄

C₂×C₄⋊Dic₁₇

yes

characteristic

120

C₂×C₄

C₂×C₄⋊Dic₁₇

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

136

—

125

C₂²

—

126

C₂²

—

127

C₂²

#11762679

yes

characteristic

128

C₂²

#11763211

yes

—

129

C₄

—

130

C₄

—

131

C₄

#11762847

yes

—

132

C₄

#11762858

yes

—

133

C₄

#11762858

yes

—

134

C₄

#11763193

yes

—

135

C₂

—

136

C₂

#14528948

yes

characteristic

137

C₂

#14528959

yes

characteristic

138

C₂

#14528959

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10587560
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₂×C₁₇×#322	C₂
socle	C₂²×C₁₇	#384
prime radical	C₂³×C₁₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₂×#322	D₁₇
hypocenter	C₁₇	#8827
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

#10587560

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₂×C₁₇×#322

C₂

socle

C₂²×C₁₇

#384

prime radical

C₂³×C₁₇

C₄⋊Q₈

cosocle

C₂×C₄²×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
95. C₁₇	#8827	semidirect

H ≅ G/N

type

95. C₁₇

#8827

semidirect

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

subgroup	index
#14528922	2
#14527543	2
#14527540	2
C₂×C₁₇×#322	2
#8827	17

subgroup

index

C₂×C₁₇×#322

Subgroups — #16928142

Conjugacy classes of subgroups

Complements

Maximal subgroups