Group #16933825 — subgroups

Subgroups — #16933825

Order 4352.

454 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 127 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	1	yes	maximal, characteristic
2	#14527456	C₂	1	4	yes	maximal
3	#14527076	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₇×#494	C₂	1	1	yes	maximal, characteristic
5	#11762862	C₂²	1	2	yes	—
6	#11762833	C₂²	1	2	yes	—
7	#11762574	—	2	2	no	—
8	#11762555	—	2	8	no	—
9	#11762531	—	2	2	no	—
10	C₂²×Q₁₆×C₁₇	—	2	2	no	—
11	C₂×C₁₇×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₁₇×#494	—	2	2	no	—
13	C₁₇×#494	C₄	1	4	yes	—
14	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₂²×Q₈×C₁₇	—	2	2	no	—
16	C₂×Q₁₆×C₁₇	—	2	8	no	—
17	C₂×Q₁₆×C₁₇	—	4	8	no	—
18	C₂²×C₈×C₁₇	D₄	1	2	yes	—
19	C₁₇×C₄⋊Q₈	—	2	4	no	—
20	C₁₇×C₄⋊Q₈	—	2	2	no	—
21	C₁₇×C₄⋊Q₈	C₂×C₄	1	4	yes	—
22	C₂×C₄⋊C₄×C₁₇	—	2	4	no	—
23	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
24	C₂×#63223	—	2	2	no	—
25	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₁₇	D₄	1	2	yes	—
27	C₂²⋊#3622	—	2	2	no	—
28	C₄⋊#3622	—	2	4	no	—
29	C₄⋊#3622	D₄	1	4	yes	—
30	C₄×#3622	—	2	2	no	—
31	C₂×Q₈×C₁₇	—	2	4	no	—
32	C₂×Q₈×C₁₇	—	4	2	no	—
33	C₂×Q₈×C₁₇	—	4	8	no	—
34	C₂²×C₄×C₁₇	—	4	2	no	—
35	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₇	#41	1	2	yes	—
37	Q₁₆×C₁₇	—	4	16	no	—
38	C₂×C₈×C₁₇	—	2	4	no	—
39	C₂×C₈×C₁₇	C₂²⋊C₄	1	4	yes	—
40	C₄⋊C₄×C₁₇	—	2	8	no	—
41	C₄⋊C₄×C₁₇	—	4	4	no	—
42	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
43	C₄²×C₁₇	C₂×D₄	1	2	yes	—
44	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
45	#63223	—	4	4	no	—
46	C₂×#3622	—	2	4	no	—
47	C₂×#3622	—	4	2	no	—
48	#9779	—	17	1	no	maximal, Sylow
49	C₂³×C₁₇	#124	1	1	yes	characteristic
50	Q₈×C₁₇	—	4	8	no	—
51	Q₈×C₁₇	—	8	4	no	—
52	C₂×C₄×C₁₇	—	2	4	no	—
53	C₂×C₄×C₁₇	—	4	4	no	—
54	C₂×C₄×C₁₇	—	8	4	no	—
55	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
57	C₂×C₄×C₁₇	#121	1	2	yes	—
58	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
59	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	4	yes	—
60	C₈×C₁₇	—	2	4	no	—
61	#3622	—	4	4	no	—
62	#3622	—	8	2	no	—
63	C₂×#494	D₁₇	1	1	yes	characteristic
64	#1837	—	17	1	no	—
65	#1546	—	17	4	no	—
66	#1452	—	17	1	no	—
67	C₂²×C₁₇	—	2	2	no	—
68	C₂²×C₁₇	#390	1	1	yes	characteristic
69	C₂²×C₁₇	#490	1	2	yes	—
70	C₄×C₁₇	—	2	2	no	—
71	C₄×C₁₇	—	8	4	no	—
72	C₄×C₁₇	#418	1	2	yes	—
73	C₄×C₁₇	#473	1	4	yes	—
74	C₄×C₁₇	#496	1	2	yes	—
75	C₂²×Q₁₆	—	2	2	no	—
76	C₂×C₄⋊Q₈	C₂×D₁₇	1	2	yes	—
77	#494	—	2	2	no	—
78	#494	Dic₁₇	1	4	yes	—
79	#431	—	17	2	no	—
80	#423	—	17	2	no	—
81	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
82	#341	—	34	2	no	—
83	#332	—	34	8	no	—
84	#322	—	34	2	no	—
85	C₂×C₁₇	—	2	2	no	—
86	C₂×C₁₇	#1961	1	1	yes	characteristic
87	C₂×C₁₇	#1979	1	2	yes	—
88	C₂²×Q₈	—	2	2	no	—
89	C₂×Q₁₆	—	2	8	no	—
90	C₂×Q₁₆	—	4	8	no	—
91	C₂×M₁₆	—	34	2	no	—
92	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
93	C₄⋊Q₈	—	2	4	no	—
94	C₄⋊Q₈	—	2	2	no	—
95	C₄⋊Q₈	C₂×Dic₁₇	1	4	yes	—
96	C₂×C₄⋊C₄	—	2	4	no	—
97	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
98	C₄⋊C₈	—	17	4	no	—
99	C₄⋊C₈	—	34	4	no	—
100	C₂²⋊C₈	—	34	2	no	—
101	C₄×C₈	—	34	2	no	—
102	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
103	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
104	C₁₇	#9779	1	1	yes	characteristic, complemented, Sylow
105	C₂×Q₈	—	2	4	no	—
106	C₂×Q₈	—	4	2	no	—
107	C₂×Q₈	—	4	8	no	—
108	C₂²×C₄	—	4	2	no	—
109	C₂²×C₄	#63254	1	2	yes	—
110	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
111	Q₁₆	—	4	16	no	—
112	M₁₆	—	68	4	no	—
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	—	34	4	no	—
115	C₂×C₈	—	68	2	no	—
116	C₂×C₈	C₂²⋊Dic₁₇	1	4	yes	—
117	C₄⋊C₄	—	2	8	no	—
118	C₄⋊C₄	—	4	4	no	—
119	C₄²	D₄×D₁₇	1	2	yes	—
120	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
121	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
122	C₂³	#10587622	1	1	yes	characteristic
123	Q₈	—	4	8	no	—
124	Q₈	—	8	4	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	4	4	no	—
127	C₂×C₄	—	8	4	no	—
128	C₂×C₄	#10587618	1	4	yes	—
129	C₂×C₄	D₄×Dic₁₇	1	2	yes	—
130	C₂×C₄	#10587624	1	2	yes	—
131	C₂×C₄	#10587626	1	1	yes	characteristic
132	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
133	C₈	—	2	4	no	—
134	C₈	—	68	4	no	—
135	C₈	—	136	2	no	—
136	C₂²	—	2	2	no	—
137	C₂²	#11762712	1	1	yes	characteristic
138	C₂²	#11763173	1	2	yes	—
139	C₄	—	2	2	no	—
140	C₄	—	8	4	no	—
141	C₄	#11762814	1	2	yes	—
142	C₄	#11763081	1	4	yes	—
143	C₄	#11763196	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	#14529609	1	1	yes	characteristic
146	C₂	#14529700	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14528922

C₂

yes

maximal, characteristic

#14527456

C₂

yes

maximal

#14527076

C₂

yes

maximal, characteristic

C₂×C₁₇×#494

C₂

yes

maximal, characteristic

#11762862

C₂²

yes

—

#11762833

C₂²

yes

—

#11762574

—

#11762555

—

#11762531

—

C₂²×Q₁₆×C₁₇

—

C₂×C₁₇×C₄⋊Q₈

C₂²

yes

—

C₁₇×#494

—

C₁₇×#494

C₄

yes

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₂²×Q₈×C₁₇

—

C₂×Q₁₆×C₁₇

—

C₂×Q₁₆×C₁₇

—

C₂²×C₈×C₁₇

D₄

yes

—

C₁₇×C₄⋊Q₈

—

C₁₇×C₄⋊Q₈

—

C₁₇×C₄⋊Q₈

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂×#63223

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

D₄

yes

—

C₂²⋊#3622

—

C₄⋊#3622

—

C₄⋊#3622

D₄

yes

—

C₄×#3622

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

#41

yes

—

Q₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

—

C₄²×C₁₇

C₂×D₄

yes

—

—

—

—

C₂³×C₁₇

yes

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₄×D₄

yes

—

C₂×C₄×C₁₇

#121

yes

—

C₂×C₄×C₁₇

#127

yes

characteristic

C₂×C₄×C₁₇

AGL(1,ℤ/8)

yes

—

C₈×C₁₇

—

#3622

—

#3622

—

C₂×#494

D₁₇

yes

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

#390

yes

characteristic

C₂²×C₁₇

#490

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#418

yes

—

C₄×C₁₇

#473

yes

—

C₄×C₁₇

#496

yes

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂×D₁₇

yes

—

#494

—

#494

Dic₁₇

yes

—

#431

—

#423

—

C₂×C₄×C₈

C₂×D₁₇

yes

—

—

—

C₂×C₁₇

—

C₂×C₁₇

#1961

yes

characteristic

C₂×C₁₇

#1979

yes

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₁₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×Dic₁₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₁₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

100

C₂²⋊C₈

—

101

C₄×C₈

—

102

C₄×C₈

C₂×Dic₁₇

yes

—

103

C₄×C₈

C₂²⋊D₁₇

yes

—

104

C₁₇

#9779

yes

characteristic, complemented, Sylow

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂²×C₄

—

109

C₂²×C₄

#63254

yes

—

110

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

111

Q₁₆

—

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

C₂²⋊Dic₁₇

yes

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄²

D₄×D₁₇

yes

—

120

C₄²

C₂²×Dic₁₇

yes

characteristic

121

C₄²

C₂×C₂²⋊D₁₇

yes

characteristic

122

C₂³

#10587622

yes

characteristic

123

Q₈

—

124

Q₈

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

#10587618

yes

—

129

C₂×C₄

D₄×Dic₁₇

yes

—

130

C₂×C₄

#10587624

yes

—

131

C₂×C₄

#10587626

yes

characteristic

132

C₂×C₄

C₂×C₂²⋊Dic₁₇

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

—

136

C₂²

—

137

C₂²

#11762712

yes

characteristic

138

C₂²

#11763173

yes

—

139

C₄

—

140

C₄

—

141

C₄

#11762814

yes

—

142

C₄

#11763081

yes

—

143

C₄

#11763196

yes

—

144

C₂

—

145

C₂

#14529609

yes

characteristic

146

C₂

#14529700

yes

—

	subgroup	G/H
center	C₂²	#11762712
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₂×C₁₇×#494	C₂
socle	C₂²×C₁₇	#390
prime radical	C₂³×C₁₇	#124
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₂×#494	D₁₇
hypocenter	C₁₇	#9779
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

subgroup

G/H

center

C₂²

#11762712

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₂×C₁₇×#494

C₂

C₂²×C₁₇

C₂³×C₁₇

C₂×C₄²×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

N	H ≅ G/N	type
104. C₁₇	#9779	semidirect

H ≅ G/N

type

104. C₁₇

#9779

semidirect

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

subgroup	index
#14528922	2
#14527456	2
#14527076	2
C₂×C₁₇×#494	2
#9779	17

subgroup

index

C₂×C₁₇×#494

Subgroups — #16933825

Conjugacy classes of subgroups

Complements

Maximal subgroups