Group #14528421 — subgroups

Subgroups — #14528421

Order 2176.

208 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 57 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	#11763185	C₂	1	1	yes	maximal, characteristic
2	#11763161	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763157	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763156	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763155	C₂	1	1	yes	maximal, characteristic, complemented
6	#11762531	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#492	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×SD₁₆×C₁₇	—	2	2	no	—
9	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₁₇×#127	C₂²	1	1	yes	characteristic
11	C₂×#63228	—	2	2	no	—
12	#10587560	C₂²	1	1	yes	characteristic
13	#10587555	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
15	#10587503	—	2	2	no	—
16	#10587492	—	2	2	no	—
17	#10587481	—	2	2	no	—
18	#10587480	—	2	2	no	—
19	C₄×#3622	C₂²	1	1	yes	characteristic
20	C₄×#3622	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₁₇	—	2	1	no	—
22	C₂×D₄×C₁₇	—	2	1	no	—
23	C₂×D₄×C₁₇	—	2	1	no	—
24	C₂×D₄×C₁₇	—	2	1	no	—
25	C₂×D₆₈	—	2	1	no	—
26	C₂×D₆₈	—	2	1	no	—
27	C₂×D₆₈	—	2	1	no	—
28	C₂×Dic₃₄	—	2	1	no	—
29	SD₁₆×C₁₇	—	4	4	no	—
30	C₂×C₈×C₁₇	D₄	1	2	yes	—
31	C₄⋊C₄×C₁₇	—	2	1	no	—
32	C₄⋊C₄×C₁₇	—	2	1	no	—
33	C₄²×C₁₇	C₂³	1	1	yes	characteristic
34	#63228	—	4	4	no	—
35	C₄⋊Dic₁₇	—	2	1	no	—
36	C₄⋊Dic₁₇	—	2	1	no	—
37	C₂×#3622	D₄	1	2	yes	—
38	C₂×#3622	D₄	1	2	yes	—
39	C₂³×C₁₇	—	4	1	no	—
40	C₂²×D₁₇	—	4	1	no	—
41	Q₈×C₁₇	—	4	2	no	—
42	D₄×C₁₇	—	4	2	no	—
43	D₄×C₁₇	—	4	2	no	—
44	D₄×C₁₇	—	4	2	no	—
45	C₂×C₄×C₁₇	—	4	1	no	—
46	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₁₇	—	4	1	no	—
50	D₆₈	—	4	2	no	—
51	D₆₈	—	4	2	no	—
52	D₆₈	—	4	2	no	—
53	Dic₃₄	—	4	2	no	—
54	C₈×C₁₇	—	2	2	no	—
55	#3622	—	2	2	no	—
56	#3622	—	2	2	no	—
57	#1717	—	17	1	no	maximal, Sylow
58	C₂²×C₁₇	—	8	1	no	—
59	C₂²×C₁₇	—	8	1	no	—
60	C₂²×C₁₇	—	8	1	no	—
61	C₂²×C₁₇	#127	1	1	yes	characteristic
62	C₂×D₁₇	—	8	1	no	—
63	C₂×D₁₇	—	8	1	no	—
64	C₂×D₁₇	—	8	1	no	—
65	C₄×C₁₇	—	8	1	no	—
66	C₄×C₁₇	#135	1	2	yes	—
67	C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
68	C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
69	Dic₁₇	—	8	1	no	—
70	#493	—	17	1	no	—
71	#492	D₁₇	1	1	yes	characteristic, complemented
72	#487	—	17	1	no	—
73	#486	—	17	1	no	—
74	#486	—	17	1	no	—
75	#486	—	17	1	no	—
76	#322	—	17	1	no	—
77	C₂×C₁₇	—	16	1	no	—
78	C₂×C₁₇	#495	1	1	yes	characteristic
79	C₂×C₁₇	#496	1	1	yes	characteristic
80	C₂×C₁₇	#496	1	1	yes	characteristic
81	D₁₇	—	16	1	no	—
82	C₂×SD₁₆	—	2	2	no	—
83	C₂×D₈	—	34	2	no	—
84	C₄⋊Q₈	—	17	1	no	—
85	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
86	#127	—	17	1	no	—
87	#127	C₂×D₁₇	1	1	yes	characteristic
88	#103	—	34	2	no	—
89	#102	—	34	2	no	—
90	#102	—	34	2	no	—
91	#102	—	34	2	no	—
92	C₄×C₈	—	17	1	no	—
93	C₄×C₈	—	17	1	no	—
94	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
95	C₁₇	#1717	1	1	yes	characteristic, complemented, Sylow
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	34	1	no	—
98	C₂×D₄	—	2	1	no	—
99	C₂×D₄	—	2	1	no	—
100	C₂×D₄	—	2	1	no	—
101	C₂×D₄	—	34	1	no	—
102	C₂×D₄	—	34	1	no	—
103	C₂×D₄	—	34	1	no	—
104	SD₁₆	—	4	4	no	—
105	D₈	—	68	4	no	—
106	C₂×C₈	—	17	2	no	—
107	C₂×C₈	—	17	2	no	—
108	C₂×C₈	C₂²⋊D₁₇	1	2	yes	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	34	1	no	—
112	C₄⋊C₄	—	34	1	no	—
113	C₄²	C₂²×D₁₇	1	1	yes	characteristic
114	C₂³	—	4	1	no	—
115	C₂³	—	68	1	no	—
116	Q₈	—	4	2	no	—
117	Q₈	—	68	2	no	—
118	D₄	—	4	2	no	—
119	D₄	—	4	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	68	2	no	—
122	D₄	—	68	2	no	—
123	D₄	—	68	2	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	68	1	no	—
126	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
127	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
128	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
129	C₈	—	2	2	no	—
130	C₈	—	34	2	no	—
131	C₈	—	34	2	no	—
132	C₂²	—	8	1	no	—
133	C₂²	—	8	1	no	—
134	C₂²	—	8	1	no	—
135	C₂²	—	136	1	no	—
136	C₂²	—	136	1	no	—
137	C₂²	—	136	1	no	—
138	C₂²	#10587626	1	1	yes	characteristic
139	C₄	—	8	1	no	—
140	C₄	—	136	1	no	—
141	C₄	#10587600	1	2	yes	—
142	C₄	#10587602	1	2	yes	—
143	C₄	#10587618	1	2	yes	—
144	C₂	—	16	1	no	—
145	C₂	—	272	1	no	—
146	C₂	#11763190	1	1	yes	characteristic
147	C₂	#11763195	1	1	yes	characteristic
148	C₂	#11763196	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763185

C₂

yes

maximal, characteristic

#11763161

C₂

yes

maximal, characteristic, complemented

#11763157

C₂

yes

maximal, characteristic, complemented

#11763156

C₂

yes

maximal, characteristic, complemented

#11763155

C₂

yes

maximal, characteristic, complemented

#11762531

C₂

yes

maximal, characteristic, complemented

C₁₇×#492

C₂

yes

maximal, characteristic, complemented

C₂×SD₁₆×C₁₇

—

C₁₇×C₄⋊Q₈

C₂²

yes

characteristic

C₁₇×#127

C₂²

yes

characteristic

C₂×#63228

—

#10587560

C₂²

yes

characteristic

#10587555

C₂²

yes

characteristic

C₄×C₈×C₁₇

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄×#3622

C₂²

yes

characteristic

C₂×Q₈×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×Dic₃₄

—

SD₁₆×C₁₇

—

C₂×C₈×C₁₇

D₄

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂³

yes

characteristic

#63228

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₂×#3622

D₄

yes

—

C₂×#3622

D₄

yes

—

C₂³×C₁₇

—

C₂²×D₁₇

—

Q₈×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×Dic₁₇

—

D₆₈

—

D₆₈

—

D₆₈

—

Dic₃₄

—

C₈×C₁₇

—

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#127

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#135

yes

—

C₄×C₁₇

AGL(1,ℤ/8)

yes

—

C₄×C₁₇

AGL(1,ℤ/8)

yes

—

Dic₁₇

—

#493

—

#492

D₁₇

yes

characteristic, complemented

—

—

—

—

—

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

D₁₇

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₇

yes

characteristic

#127

—

#127

C₂×D₁₇

yes

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₁₇

yes

characteristic

C₁₇

#1717

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

SD₁₆

—

105

D₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂²⋊D₁₇

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄²

C₂²×D₁₇

yes

characteristic

114

C₂³

—

115

C₂³

—

116

Q₈

—

117

Q₈

—

118

D₄

—

119

D₄

—

120

D₄

—

121

D₄

—

122

D₄

—

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

D₄×D₁₇

yes

characteristic

127

C₂×C₄

D₄×D₁₇

yes

characteristic

128

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

129

C₈

—

130

C₈

—

131

C₈

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

—

136

C₂²

—

136

—

137

C₂²

—

136

—

138

C₂²

#10587626

yes

characteristic

139

C₄

—

140

C₄

—

136

—

141

C₄

#10587600

yes

—

142

C₄

#10587602

yes

—

143

C₄

#10587618

yes

—

144

C₂

—

145

C₂

—

272

—

146

C₂

#11763190

yes

characteristic

147

C₂

#11763195

yes

characteristic

148

C₂

#11763196

yes

characteristic

	subgroup	G/H
center	C₂²	#10587626
commutator, cosocle	C₄²×C₁₇	C₂³
Frattini	C₄²	C₂²×D₁₇
Fitting	C₁₇×#492	C₂
socle	C₂²×C₁₇	#127
prime radical	#11763185	C₂
hypercenter	#492	D₁₇
hypocenter	C₁₇	#1717
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#10587626

commutator, cosocle

C₄²×C₁₇

C₂³

Frattini

C₄²

C₂²×D₁₇

Fitting

C₁₇×#492

C₂

C₂²×C₁₇

C₂

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
2. #11763161	C₂	semidirect
3. #11763157	C₂	semidirect
4. #11763156	C₂	semidirect
5. #11763155	C₂	semidirect
6. #11762531	C₂	semidirect
7. C₁₇×#492	C₂	semidirect
71. #492	D₁₇	semidirect
95. C₁₇	#1717	semidirect

H ≅ G/N

type

2. #11763161

C₂

semidirect

3. #11763157

C₂

semidirect

4. #11763156

C₂

semidirect

5. #11763155

C₂

semidirect

6. #11762531

C₂

semidirect

7. C₁₇×#492

C₂

semidirect

71. #492

D₁₇

semidirect

95. C₁₇

#1717

semidirect

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

subgroup	index
#11763185	2
#11763161	2
#11763157	2
#11763156	2
#11763155	2
#11762531	2
C₁₇×#492	2
#1717	17

subgroup

index

C₁₇×#492

Subgroups — #14528421

Conjugacy classes of subgroups

Complements

Maximal subgroups