Group #14530071 — subgroups

Subgroups — #14530071

Order 2176.

266 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 65 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	#11763609	C₂	1	1	yes	maximal, characteristic
2	C₂×#10587476	C₂	1	2	yes	maximal, complemented
3	#11762791	C₂	1	2	yes	maximal, complemented
4	#11762567	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#423	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#63252	C₂²	1	2	yes	—
7	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
8	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
9	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
10	C₂×#63223	C₂²	1	2	yes	—
11	#10587595	—	2	2	no	—
12	#10587594	—	2	2	no	—
13	#10587590	—	2	2	no	—
14	C₁₇×C₄⋊C₈	—	2	2	no	—
15	#10587495	—	2	2	no	—
16	#10587494	—	2	2	no	—
17	#10587476	—	2	2	no	—
18	#10587476	—	2	2	no	—
19	C₂²×C₄×C₁₇	—	2	1	no	—
20	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
21	C₂×C₂²⋊D₁₇	—	2	2	no	—
22	#63252	—	2	2	no	—
23	#63252	—	2	2	no	—
24	#63252	—	2	2	no	—
25	#63252	—	2	2	no	—
26	C₂×D₆₈	D₄	1	2	yes	—
27	C₂×C₄×D₁₇	—	2	2	no	—
28	C₂×Dic₃₄	D₄	1	2	yes	—
29	M₁₆×C₁₇	—	2	2	no	—
30	M₁₆×C₁₇	—	2	2	no	—
31	C₂×C₈×C₁₇	—	2	1	no	—
32	C₂×C₈×C₁₇	D₄	1	2	yes	—
33	C₄²×C₁₇	—	2	1	no	—
34	C₄²×C₁₇	D₄	1	1	yes	characteristic
35	C₄²×C₁₇	D₄	1	1	yes	characteristic
36	#63223	—	2	2	no	—
37	#63223	—	2	2	no	—
38	C₂×#3622	—	2	2	no	—
39	Dic₆₈	—	2	2	no	—
40	D₁₃₆	—	2	2	no	—
41	#63219	—	2	4	no	—
42	C₂³×C₁₇	#41	1	1	yes	characteristic
43	C₂²×D₁₇	—	2	2	no	—
44	C₂×C₄×C₁₇	—	2	1	no	—
45	C₂×C₄×C₁₇	—	2	1	no	—
46	C₂×C₄×C₁₇	—	4	2	no	—
47	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₇	#41	1	2	yes	—
51	C₂²⋊D₁₇	—	4	2	no	—
52	C₂²⋊D₁₇	—	4	2	no	—
53	C₂×Dic₁₇	—	2	2	no	—
54	D₆₈	—	2	2	no	—
55	D₆₈	—	2	2	no	—
56	C₄×D₁₇	—	4	2	no	—
57	C₄×D₁₇	—	4	2	no	—
58	Dic₃₄	—	2	2	no	—
59	Dic₃₄	—	2	2	no	—
60	C₈×C₁₇	—	2	2	no	—
61	C₈×C₁₇	—	4	1	no	—
62	#3622	—	4	2	no	—
63	#2050	—	17	1	no	maximal, Sylow
64	C₂²×C₁₇	—	4	1	no	—
65	C₂²×C₁₇	#121	1	1	yes	characteristic
66	C₂²×C₁₇	#123	1	2	yes	—
67	C₂×D₁₇	—	4	2	no	—
68	C₂×D₁₇	—	8	2	no	—
69	C₄×C₁₇	—	2	2	no	—
70	C₄×C₁₇	—	4	1	no	—
71	C₄×C₁₇	#120	1	1	yes	characteristic
72	C₄×C₁₇	#121	1	2	yes	—
73	C₄×C₁₇	#124	1	1	yes	characteristic
74	Dic₁₇	—	4	2	no	—
75	#575	—	17	1	no	—
76	#423	D₁₇	1	1	yes	characteristic, complemented
77	C₂×#104	—	17	2	no	—
78	#413	—	17	2	no	—
79	#338	—	17	1	no	—
80	C₂×C₁₇	—	2	1	no	—
81	C₂×C₁₇	—	2	2	no	—
82	C₂×C₁₇	#394	1	1	yes	characteristic
83	D₁₇	—	8	2	no	—
84	C₂×#41	—	17	2	no	—
85	#137	—	34	2	no	—
86	AGL(1,ℤ/8)	—	34	2	no	—
87	#135	—	34	2	no	—
88	C₂×M₁₆	—	17	2	no	—
89	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
90	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
91	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
92	C₄⋊C₈	—	2	2	no	—
93	#104	—	34	2	no	—
94	#104	—	34	2	no	—
95	#101	—	34	2	no	—
96	#100	—	34	2	no	—
97	C₁₇	#2050	1	1	yes	characteristic, complemented, Sylow
98	#41	—	34	2	no	—
99	#41	—	34	2	no	—
100	#41	—	34	2	no	—
101	#41	—	34	2	no	—
102	C₂×Q₈	—	17	2	no	—
103	C₂×D₄	—	17	2	no	—
104	C₂×D₄	—	34	2	no	—
105	C₂²×C₄	—	2	1	no	—
106	C₂²×C₄	—	34	2	no	—
107	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
108	Q₁₆	—	34	2	no	—
109	SD₁₆	—	34	4	no	—
110	D₈	—	34	2	no	—
111	M₁₆	—	2	2	no	—
112	M₁₆	—	2	2	no	—
113	M₁₆	—	34	2	no	—
114	M₁₆	—	34	2	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	34	2	no	—
117	C₂×C₈	C₂²⋊D₁₇	1	2	yes	—
118	C₄²	—	2	1	no	—
119	C₄²	D₆₈	1	1	yes	characteristic
120	C₄²	D₆₈	1	1	yes	characteristic
121	C₂³	—	34	2	no	—
122	C₂³	#63252	1	1	yes	characteristic
123	Q₈	—	34	2	no	—
124	Q₈	—	34	2	no	—
125	D₄	—	34	2	no	—
126	D₄	—	34	2	no	—
127	D₄	—	68	2	no	—
128	D₄	—	68	2	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	34	2	no	—
133	C₂×C₄	—	68	2	no	—
134	C₂×C₄	—	68	2	no	—
135	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
136	C₂×C₄	D₄×D₁₇	1	2	yes	—
137	C₂×C₄	#63256	1	2	yes	—
138	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	—	4	1	no	—
141	C₈	—	68	2	no	—
142	C₂²	—	4	1	no	—
143	C₂²	—	68	2	no	—
144	C₂²	—	136	2	no	—
145	C₂²	#10587580	1	2	yes	—
146	C₂²	#10587616	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	4	1	no	—
149	C₄	—	68	2	no	—
150	C₄	#10587581	1	2	yes	—
151	C₄	#10587623	1	1	yes	characteristic
152	C₄	#10587633	1	1	yes	characteristic
153	C₂	—	2	1	no	—
154	C₂	—	2	2	no	—
155	C₂	—	136	2	no	—
156	C₂	#11762726	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763609

C₂

yes

maximal, characteristic

C₂×#10587476

C₂

yes

maximal, complemented

#11762791

C₂

yes

maximal, complemented

#11762567

C₂

yes

maximal, characteristic, complemented

C₁₇×#423

C₂

yes

maximal, characteristic, complemented

C₂×#63252

C₂²

yes

—

C₂×M₁₆×C₁₇

C₂²

yes

characteristic

C₂×M₁₆×C₁₇

C₂²

yes

characteristic

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×#63223

C₂²

yes

—

#10587595

—

#10587594

—

#10587590

—

C₁₇×C₄⋊C₈

—

—

—

—

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂×C₂²⋊D₁₇

—

—

—

—

—

C₂×D₆₈

D₄

yes

—

C₂×C₄×D₁₇

—

C₂×Dic₃₄

D₄

yes

—

M₁₆×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

D₄

yes

—

C₄²×C₁₇

—

C₄²×C₁₇

D₄

yes

characteristic

C₄²×C₁₇

D₄

yes

—

—

—

Dic₆₈

—

D₁₃₆

—

#63219

—

C₂³×C₁₇

#41

yes

characteristic

C₂²×D₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

#41

yes

—

C₂²⋊D₁₇

—

C₂²⋊D₁₇

—

C₂×Dic₁₇

—

D₆₈

—

D₆₈

—

C₄×D₁₇

—

C₄×D₁₇

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

#3622

—

#2050

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

#121

yes

characteristic

C₂²×C₁₇

#123

yes

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#120

yes

characteristic

C₄×C₁₇

#121

yes

—

C₄×C₁₇

#124

yes

characteristic

Dic₁₇

—

#575

—

#423

D₁₇

yes

characteristic, complemented

C₂×#104

—

#413

—

#338

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

#394

yes

characteristic

D₁₇

—

C₂×#41

—

#137

—

AGL(1,ℤ/8)

—

#135

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₇

yes

characteristic

C₂×M₁₆

C₂×D₁₇

yes

characteristic

C₂×C₄²

C₂×D₁₇

yes

characteristic

C₄⋊C₈

—

—

—

—

—

C₁₇

#2050

yes

characteristic, complemented, Sylow

#41

—

#41

—

100

#41

—

101

#41

—

102

C₂×Q₈

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

C₂²×D₁₇

yes

characteristic

108

Q₁₆

—

109

SD₁₆

—

110

D₈

—

111

M₁₆

—

112

M₁₆

—

113

M₁₆

—

114

M₁₆

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

C₂²⋊D₁₇

yes

—

118

C₄²

—

119

C₄²

D₆₈

yes

characteristic

120

C₄²

D₆₈

yes

characteristic

121

C₂³

—

122

C₂³

#63252

yes

characteristic

123

Q₈

—

124

Q₈

—

125

D₄

—

126

D₄

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₂×D₆₈

yes

characteristic

136

C₂×C₄

D₄×D₁₇

yes

—

137

C₂×C₄

#63256

yes

—

138

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

136

—

145

C₂²

#10587580

yes

—

146

C₂²

#10587616

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#10587581

yes

—

151

C₄

#10587623

yes

characteristic

152

C₄

#10587633

yes

characteristic

153

C₂

—

154

C₂

—

155

C₂

—

136

—

156

C₂

#11762726

yes

characteristic

	subgroup	G/H
center	C₄	#10587633
commutator, cosocle	C₂²×C₄×C₁₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#423	C₂
socle	C₂×C₁₇	#394
prime radical	#11763609	C₂
hypercenter	#423	D₁₇
hypocenter	C₁₇	#2050
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #423

subgroup

G/H

center

C₄

#10587633

commutator, cosocle

C₂²×C₄×C₁₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₁₇×#423

C₂

C₂×C₁₇

C₂

D₁₇

C₁₇

G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #423

N	H ≅ G/N	type
2. C₂×#10587476	C₂	semidirect
3. #11762791	C₂	semidirect
4. #11762567	C₂	semidirect
5. C₁₇×#423	C₂	semidirect
76. #423	D₁₇	semidirect
97. C₁₇	#2050	semidirect

H ≅ G/N

type

2. C₂×#10587476

C₂

semidirect

3. #11762791

C₂

semidirect

4. #11762567

C₂

semidirect

5. C₁₇×#423

C₂

semidirect

76. #423

D₁₇

semidirect

97. C₁₇

#2050

semidirect

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

subgroup	index
#11763609	2
C₂×#10587476	2
#11762791	2
#11762567	2
C₁₇×#423	2
#2050	17

subgroup

index

C₁₇×#423

Subgroups — #14530071

Conjugacy classes of subgroups

Complements

Maximal subgroups