Group #11763523 — subgroups

Subgroups — #11763523

Order 1088.

236 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₁₇×#121	C₂	1	1	yes	maximal, characteristic, complemented
2	#10587625	C₂	1	2	yes	maximal, complemented
3	#10587624	C₂	1	1	yes	maximal, characteristic, complemented
4	#10587622	C₂	1	2	yes	maximal, complemented
5	D₄×Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₂²⋊D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#10587612	C₂	1	1	yes	maximal, characteristic, complemented
8	#10587578	C₂	1	1	yes	maximal, characteristic, complemented
9	#10587574	C₂	1	1	yes	maximal, characteristic, complemented
10	#10587571	C₂	1	2	yes	maximal, complemented
11	#10587563	C₂	1	2	yes	maximal, complemented
12	C₂×D₄×C₁₇	C₂²	1	1	yes	characteristic
13	C₂×D₄×C₁₇	C₂²	1	2	yes	—
14	C₂²×C₄×C₁₇	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊D₁₇	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊D₁₇	C₂²	1	2	yes	—
17	C₂²×Dic₁₇	C₂²	1	2	yes	—
18	C₂×C₄×D₁₇	C₂²	1	1	yes	characteristic
19	C₂×Dic₃₄	C₂²	1	2	yes	—
20	C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
21	C₂²⋊C₄×C₁₇	C₂²	1	2	yes	—
22	C₂²⋊Dic₁₇	C₂²	1	2	yes	—
23	C₂²⋊Dic₁₇	C₂²	1	2	yes	—
24	C₂²⋊Dic₁₇	C₂²	1	2	yes	—
25	C₂²⋊Dic₁₇	C₂²	1	1	yes	characteristic
26	#63227	C₂²	1	1	yes	characteristic
27	#63227	C₂²	1	2	yes	—
28	C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
29	C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
30	#63225	C₂²	1	1	yes	characteristic
31	#63225	C₂²	1	2	yes	—
32	C₄×Dic₁₇	C₂²	1	2	yes	—
33	C₄×Dic₁₇	C₂²	1	2	yes	—
34	C₄×Dic₁₇	C₂²	1	1	yes	characteristic
35	C₂³×C₁₇	C₂³	1	1	yes	characteristic
36	C₂³×C₁₇	C₂³	1	2	yes	—
37	C₂²×D₁₇	C₂³	1	1	yes	characteristic
38	D₄×C₁₇	—	2	2	no	—
39	D₄×C₁₇	—	4	2	no	—
40	C₂×C₄×C₁₇	—	2	2	no	—
41	C₂×C₄×C₁₇	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₁₇	C₂³	1	2	yes	—
43	C₂×C₄×C₁₇	C₂³	1	1	yes	characteristic
44	C₂²⋊D₁₇	—	2	2	no	—
45	C₂²⋊D₁₇	—	4	2	no	—
46	C₂×Dic₁₇	—	2	4	no	—
47	C₂×Dic₁₇	C₂³	1	1	yes	characteristic
48	C₂×Dic₁₇	C₂³	1	1	yes	characteristic
49	C₂×Dic₁₇	C₂³	1	2	yes	—
50	C₂×Dic₁₇	C₂³	1	2	yes	—
51	C₂×Dic₁₇	C₂³	1	1	yes	characteristic
52	C₄×D₁₇	—	2	2	no	—
53	Dic₃₄	—	4	2	no	—
54	C₂²×C₁₇	—	2	2	no	—
55	C₂²×C₁₇	—	2	1	no	—
56	C₂²×C₁₇	—	2	1	no	—
57	C₂²×C₁₇	—	2	2	no	—
58	C₂²×C₁₇	—	2	2	no	—
59	C₂²×C₁₇	—	2	1	no	—
60	C₂²×C₁₇	C₂⁴	1	1	yes	characteristic
61	C₂×D₁₇	—	2	1	no	—
62	C₂×D₁₇	—	2	1	no	—
63	C₂×D₁₇	—	2	1	no	—
64	C₄×C₁₇	—	2	1	no	—
65	C₄×C₁₇	—	2	2	no	—
66	C₄×C₁₇	#41	1	2	yes	—
67	Dic₁₇	—	2	1	no	—
68	Dic₁₇	—	2	2	no	—
69	Dic₁₇	—	2	1	no	—
70	Dic₁₇	—	2	2	no	—
71	Dic₁₇	#41	1	2	yes	—
72	#555	—	17	1	no	maximal, Sylow
73	C₂×C₁₇	—	4	2	no	—
74	C₂×C₁₇	—	4	1	no	—
75	C₂×C₁₇	C₂×#41	1	1	yes	characteristic
76	C₂×C₁₇	C₂×#41	1	1	yes	characteristic
77	C₂×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
78	D₁₇	—	4	1	no	—
79	C₄⋊Q₈	—	17	1	no	—
80	#124	—	17	2	no	—
81	#124	—	17	2	no	—
82	#121	—	17	1	no	—
83	#121	—	17	2	no	—
84	#121	D₁₇	1	1	yes	characteristic, complemented
85	C₄×D₄	—	17	1	no	—
86	C₄×D₄	—	17	1	no	—
87	#117	—	17	1	no	—
88	#117	—	17	1	no	—
89	#117	—	17	2	no	—
90	C₁₇	#555	1	1	yes	characteristic, complemented, Sylow
91	C₂×Q₈	—	17	2	no	—
92	C₂×D₄	—	17	1	no	—
93	C₂×D₄	—	17	2	no	—
94	C₂×D₄	C₂×D₁₇	1	1	yes	characteristic
95	C₂×D₄	C₂×D₁₇	1	2	yes	—
96	C₂²×C₄	—	17	2	no	—
97	C₂²×C₄	—	17	1	no	—
98	C₂²×C₄	C₂×D₁₇	1	1	yes	characteristic
99	C₄⋊C₄	—	17	1	no	—
100	C₄⋊C₄	—	17	1	no	—
101	C₄⋊C₄	—	17	2	no	—
102	C₄⋊C₄	—	17	1	no	—
103	C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
104	C₂²⋊C₄	—	17	2	no	—
105	C₂²⋊C₄	—	17	2	no	—
106	C₂²⋊C₄	—	17	1	no	—
107	C₂²⋊C₄	—	17	2	no	—
108	C₂²⋊C₄	—	17	2	no	—
109	C₂²⋊C₄	—	17	1	no	—
110	C₂²⋊C₄	C₂×D₁₇	1	2	yes	—
111	C₄²	—	17	2	no	—
112	C₄²	—	17	2	no	—
113	C₄²	—	17	1	no	—
114	C₂³	—	17	1	no	—
115	C₂³	C₂²×D₁₇	1	1	yes	characteristic
116	C₂³	C₂²×D₁₇	1	2	yes	—
117	Q₈	—	68	2	no	—
118	D₄	—	2	2	no	—
119	D₄	—	4	2	no	—
120	D₄	—	34	2	no	—
121	D₄	—	68	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	17	1	no	—
124	C₂×C₄	—	17	1	no	—
125	C₂×C₄	—	17	2	no	—
126	C₂×C₄	—	17	1	no	—
127	C₂×C₄	—	17	2	no	—
128	C₂×C₄	—	34	4	no	—
129	C₂×C₄	—	34	2	no	—
130	C₂×C₄	C₂²×D₁₇	1	1	yes	characteristic
131	C₂×C₄	C₂²×D₁₇	1	2	yes	—
132	C₂×C₄	C₂²×D₁₇	1	1	yes	characteristic
133	C₂²	—	2	1	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	2	1	no	—
136	C₂²	—	2	2	no	—
137	C₂²	—	2	2	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	34	1	no	—
140	C₂²	—	34	1	no	—
141	C₂²	—	34	1	no	—
142	C₂²	C₂³×D₁₇	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	2	no	—
145	C₄	—	17	2	no	—
146	C₄	—	34	1	no	—
147	C₄	—	34	2	no	—
148	C₄	—	34	1	no	—
149	C₄	—	34	2	no	—
150	C₄	#63254	1	2	yes	—
151	C₂	—	4	1	no	—
152	C₂	—	4	2	no	—
153	C₂	—	68	1	no	—
154	C₂	C₂×#63254	1	1	yes	characteristic
155	C₂	#10587691	1	1	yes	characteristic
156	C₂	#41×D₁₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₇×#121

C₂

yes

maximal, characteristic, complemented

#10587625

C₂

yes

maximal, complemented

#10587624

C₂

yes

maximal, characteristic, complemented

#10587622

C₂

yes

maximal, complemented

D₄×Dic₁₇

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₁₇

C₂

yes

maximal, characteristic, complemented

#10587612

C₂

yes

maximal, characteristic, complemented

#10587578

C₂

yes

maximal, characteristic, complemented

#10587574

C₂

yes

maximal, characteristic, complemented

#10587571

C₂

yes

maximal, complemented

#10587563

C₂

yes

maximal, complemented

C₂×D₄×C₁₇

C₂²

yes

characteristic

C₂×D₄×C₁₇

C₂²

yes

—

C₂²×C₄×C₁₇

C₂²

yes

characteristic

C₂×C₂²⋊D₁₇

C₂²

yes

characteristic

C₂×C₂²⋊D₁₇

C₂²

yes

—

C₂²×Dic₁₇

C₂²

yes

—

C₂×C₄×D₁₇

C₂²

yes

characteristic

C₂×Dic₃₄

C₂²

yes

—

C₄⋊C₄×C₁₇

C₂²

yes

characteristic

C₂²⋊C₄×C₁₇

C₂²

yes

—

C₂²⋊Dic₁₇

C₂²

yes

—

C₂²⋊Dic₁₇

C₂²

yes

—

C₂²⋊Dic₁₇

C₂²

yes

—

C₂²⋊Dic₁₇

C₂²

yes

characteristic

#63227

C₂²

yes

characteristic

#63227

C₂²

yes

—

C₄⋊Dic₁₇

C₂²

yes

characteristic

C₄⋊Dic₁₇

C₂²

yes

characteristic

#63225

C₂²

yes

characteristic

#63225

C₂²

yes

—

C₄×Dic₁₇

C₂²

yes

—

C₄×Dic₁₇

C₂²

yes

—

C₄×Dic₁₇

C₂²

yes

characteristic

C₂³×C₁₇

C₂³

yes

characteristic

C₂³×C₁₇

C₂³

yes

—

C₂²×D₁₇

C₂³

yes

characteristic

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂³

yes

characteristic

C₂×C₄×C₁₇

C₂³

yes

—

C₂×C₄×C₁₇

C₂³

yes

characteristic

C₂²⋊D₁₇

—

C₂²⋊D₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

C₂³

yes

characteristic

C₂×Dic₁₇

C₂³

yes

characteristic

C₂×Dic₁₇

C₂³

yes

—

C₂×Dic₁₇

C₂³

yes

—

C₂×Dic₁₇

C₂³

yes

characteristic

C₄×D₁₇

—

Dic₃₄

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂⁴

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#41

yes

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

#41

yes

—

#555

—

maximal, Sylow

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

C₂×#41

yes

characteristic

C₂×C₁₇

C₂×#41

yes

characteristic

C₂×C₁₇

H(5,𝔽₂)

yes

characteristic

D₁₇

—

C₄⋊Q₈

—

—

—

—

—

D₁₇

yes

characteristic, complemented

C₄×D₄

—

C₄×D₄

—

#117

—

#117

—

#117

—

C₁₇

#555

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₁₇

yes

characteristic

C₂×D₄

C₂×D₁₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₁₇

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

C₂×D₁₇

yes

characteristic

104

C₂²⋊C₄

—

105

C₂²⋊C₄

—

106

C₂²⋊C₄

—

107

C₂²⋊C₄

—

108

C₂²⋊C₄

—

109

C₂²⋊C₄

—

110

C₂²⋊C₄

C₂×D₁₇

yes

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₂³

—

115

C₂³

C₂²×D₁₇

yes

characteristic

116

C₂³

C₂²×D₁₇

yes

—

117

Q₈

—

118

D₄

—

119

D₄

—

120

D₄

—

121

D₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₂²×D₁₇

yes

characteristic

131

C₂×C₄

C₂²×D₁₇

yes

—

132

C₂×C₄

C₂²×D₁₇

yes

characteristic

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₂³×D₁₇

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#63254

yes

—

151

C₂

—

152

C₂

—

153

C₂

—

154

C₂

C₂×#63254

yes

characteristic

155

C₂

#10587691

yes

characteristic

156

C₂

#41×D₁₇

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₇
commutator, socle, cosocle	C₂²×C₁₇	C₂⁴
Fitting	C₁₇×#121	C₂
hypercenter	#121	D₁₇
hypocenter	C₁₇	#555
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ #121

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₇

commutator, socle, cosocle

C₂²×C₁₇

C₂⁴

Fitting

C₁₇×#121

C₂

D₁₇

C₁₇

G ▷ C₂²×C₁₇ ▷ C₁₇

C₁ ◁ C₂² ◁ #121

N	H ≅ G/N	type
1. C₁₇×#121	C₂	semidirect
2. #10587625	C₂	semidirect
3. #10587624	C₂	semidirect
4. #10587622	C₂	semidirect
5. D₄×Dic₁₇	C₂	semidirect
6. C₄×C₂²⋊D₁₇	C₂	semidirect
7. #10587612	C₂	semidirect
8. #10587578	C₂	semidirect
9. #10587574	C₂	semidirect
10. #10587571	C₂	semidirect
11. #10587563	C₂	semidirect
84. #121	D₁₇	semidirect
90. C₁₇	#555	semidirect

H ≅ G/N

type

1. C₁₇×#121

C₂

semidirect

2. #10587625

C₂

semidirect

3. #10587624

C₂

semidirect

4. #10587622

C₂

semidirect

5. D₄×Dic₁₇

C₂

semidirect

6. C₄×C₂²⋊D₁₇

C₂

semidirect

7. #10587612

C₂

semidirect

8. #10587578

C₂

semidirect

9. #10587574

C₂

semidirect

10. #10587571

C₂

semidirect

11. #10587563

C₂

semidirect

84. #121

D₁₇

semidirect

90. C₁₇

#555

semidirect

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

subgroup	index
C₁₇×#121	2
#10587625	2
#10587624	2
#10587622	2
D₄×Dic₁₇	2
C₄×C₂²⋊D₁₇	2
#10587612	2
#10587578	2
#10587574	2
#10587571	2
#10587563	2
#555	17

subgroup

index

C₁₇×#121

#10587625

#10587624

#10587622

D₄×Dic₁₇

C₄×C₂²⋊D₁₇

Subgroups — #11763523

Conjugacy classes of subgroups

Complements

Maximal subgroups