Group #11763549 — subgroups

Subgroups — #11763549

Order 1088.

252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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×#123	C₂	1	1	yes	maximal, characteristic, complemented
2	#10587626	C₂	1	1	yes	maximal, characteristic, complemented
3	#10587623	C₂	1	1	yes	maximal, characteristic, complemented
4	#10587616	C₂	1	2	yes	maximal, complemented
5	#10587584	C₂	1	2	yes	maximal, complemented
6	#10587581	C₂	1	2	yes	maximal, complemented
7	#10587570	C₂	1	2	yes	maximal, complemented
8	#10587569	C₂	1	2	yes	maximal, complemented
9	#10587568	C₂	1	2	yes	maximal, complemented
10	C₂³×D₁₇	C₂²	1	1	yes	characteristic
11	C₂×D₄×C₁₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₁₇	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₂²⋊D₁₇	C₂²	1	2	yes	—
14	C₂×C₂²⋊D₁₇	C₂²	1	2	yes	—
15	C₂×C₂²⋊D₁₇	C₂²	1	2	yes	complemented
16	C₂×D₆₈	C₂²	1	2	yes	—
17	C₂×D₆₈	C₂²	1	2	yes	—
18	C₂×D₆₈	C₂²	1	1	yes	characteristic, complemented
19	C₂×C₄×D₁₇	C₂²	1	2	yes	complemented
20	C₄⋊C₄×C₁₇	C₂²	1	2	yes	complemented
21	C₂²⋊C₄×C₁₇	C₂²	1	2	yes	—
22	C₂²⋊C₄×C₁₇	C₂²	1	1	yes	characteristic, complemented
23	C₂²⋊Dic₁₇	C₂²	1	1	yes	characteristic
24	#63227	C₂²	1	2	yes	—
25	#63227	C₂²	1	2	yes	—
26	#63227	C₂²	1	2	yes	—
27	#63227	C₂²	1	2	yes	complemented
28	C₄⋊Dic₁₇	C₂²	1	2	yes	complemented
29	#63225	C₂²	1	2	yes	complemented
30	C₄×Dic₁₇	C₂²	1	1	yes	characteristic, complemented
31	C₂³×C₁₇	C₂³	1	1	yes	characteristic
32	C₂³×C₁₇	C₂³	1	1	yes	characteristic
33	C₂²×D₁₇	—	4	1	no	—
34	C₂²×D₁₇	—	4	2	no	—
35	C₂²×D₁₇	C₂³	1	2	yes	—
36	C₂²×D₁₇	C₂³	1	2	yes	—
37	D₄×C₁₇	—	4	1	no	—
38	C₂×C₄×C₁₇	—	4	1	no	—
39	C₂×C₄×C₁₇	C₂³	1	2	yes	—
40	C₂×C₄×C₁₇	C₂³	1	2	yes	—
41	C₂×C₄×C₁₇	C₂³	1	1	yes	characteristic
42	C₂²⋊D₁₇	—	4	2	no	—
43	C₂²⋊D₁₇	—	4	2	no	—
44	C₂²⋊D₁₇	—	4	2	no	—
45	C₂×Dic₁₇	C₂³	1	2	yes	—
46	C₂×Dic₁₇	C₂³	1	2	yes	—
47	D₆₈	—	4	2	no	—
48	D₆₈	—	4	2	no	—
49	D₆₈	—	4	1	no	—
50	C₄×D₁₇	—	4	2	no	—
51	C₂²×C₁₇	—	2	2	no	—
52	C₂²×C₁₇	—	2	2	no	—
53	C₂²×C₁₇	—	2	1	no	—
54	C₂²×C₁₇	—	2	1	no	—
55	C₂²×C₁₇	C₂⁴	1	1	yes	characteristic
56	C₂×D₁₇	—	2	2	no	—
57	C₂×D₁₇	—	2	2	no	—
58	C₂×D₁₇	—	2	2	no	—
59	C₂×D₁₇	—	2	2	no	—
60	C₂×D₁₇	—	2	2	no	—
61	C₂×D₁₇	—	2	2	no	—
62	C₂×D₁₇	—	4	4	no	—
63	C₄×C₁₇	—	2	2	no	—
64	C₄×C₁₇	—	2	2	no	—
65	C₄×C₁₇	—	2	1	no	—
66	Dic₁₇	—	2	2	no	—
67	Dic₁₇	—	2	2	no	—
68	#560	—	17	1	no	maximal, Sylow
69	C₂×C₁₇	—	4	1	no	—
70	C₂×C₁₇	—	4	1	no	—
71	C₂×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
72	C₂×C₁₇	H(5,𝔽₂)	1	2	yes	—
73	D₁₇	—	4	2	no	—
74	D₁₇	—	4	2	no	—
75	#127	—	17	1	no	—
76	#126	—	17	2	no	—
77	#123	—	17	2	no	—
78	#123	D₁₇	1	1	yes	characteristic, complemented
79	#121	—	17	2	no	—
80	#121	—	17	2	no	—
81	#121	—	17	2	no	—
82	#120	—	17	1	no	—
83	#120	—	17	2	no	—
84	C₁₇	#560	1	1	yes	characteristic, complemented, Sylow
85	C₂⁴	—	17	1	no	—
86	C₂×D₄	—	17	2	no	—
87	C₂×D₄	—	17	2	no	—
88	C₂×D₄	—	17	2	no	—
89	C₂×D₄	—	17	2	no	—
90	C₂×D₄	—	17	2	no	—
91	C₂×D₄	—	17	1	no	—
92	C₂×D₄	C₂×D₁₇	1	1	yes	characteristic
93	C₂²×C₄	—	17	2	no	—
94	C₂²×C₄	C₂×D₁₇	1	1	yes	characteristic, complemented
95	C₄⋊C₄	—	17	2	no	—
96	C₄⋊C₄	—	17	2	no	—
97	C₄⋊C₄	C₂×D₁₇	1	2	yes	complemented
98	C₂²⋊C₄	—	17	1	no	—
99	C₂²⋊C₄	—	17	2	no	—
100	C₂²⋊C₄	—	17	2	no	—
101	C₂²⋊C₄	—	17	2	no	—
102	C₂²⋊C₄	—	17	2	no	—
103	C₂²⋊C₄	C₂×D₁₇	1	2	yes	—
104	C₂²⋊C₄	C₂×D₁₇	1	1	yes	characteristic, complemented
105	C₄²	—	17	1	no	—
106	C₂³	—	17	2	no	—
107	C₂³	—	17	2	no	—
108	C₂³	—	68	1	no	—
109	C₂³	—	68	2	no	—
110	C₂³	C₂²×D₁₇	1	1	yes	characteristic
111	C₂³	C₂²×D₁₇	1	1	yes	characteristic
112	D₄	—	4	1	no	—
113	D₄	—	68	2	no	—
114	D₄	—	68	1	no	—
115	D₄	—	68	2	no	—
116	D₄	—	68	2	no	—
117	D₄	—	68	2	no	—
118	D₄	—	68	2	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	17	2	no	—
121	C₂×C₄	—	17	2	no	—
122	C₂×C₄	—	68	2	no	—
123	C₂×C₄	C₂²×D₁₇	1	1	yes	characteristic
124	C₂×C₄	C₂²×D₁₇	1	2	yes	—
125	C₂×C₄	C₂²×D₁₇	1	2	yes	—
126	C₂²	—	2	1	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	2	no	—
130	C₂²	—	34	2	no	—
131	C₂²	—	34	2	no	—
132	C₂²	—	34	2	no	—
133	C₂²	—	34	2	no	—
134	C₂²	—	34	2	no	—
135	C₂²	—	34	2	no	—
136	C₂²	—	68	4	no	—
137	C₂²	C₂³×D₁₇	1	1	yes	characteristic
138	C₄	—	2	2	no	—
139	C₄	—	2	1	no	—
140	C₄	—	2	2	no	—
141	C₄	—	34	2	no	—
142	C₄	—	34	2	no	—
143	C₂	—	4	1	no	—
144	C₂	—	4	1	no	—
145	C₂	—	68	2	no	—
146	C₂	—	68	2	no	—
147	C₂	#10587691	1	1	yes	characteristic
148	C₂	#10587696	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₇×#123

C₂

yes

maximal, characteristic, complemented

#10587626

C₂

yes

maximal, characteristic, complemented

#10587623

C₂

yes

maximal, characteristic, complemented

#10587616

C₂

yes

maximal, complemented

#10587584

C₂

yes

maximal, complemented

#10587581

C₂

yes

maximal, complemented

#10587570

C₂

yes

maximal, complemented

#10587569

C₂

yes

maximal, complemented

#10587568

C₂

yes

maximal, complemented

C₂³×D₁₇

C₂²

yes

characteristic

C₂×D₄×C₁₇

C₂²

yes

characteristic

C₂²×C₄×C₁₇

C₂²

yes

characteristic, complemented

C₂×C₂²⋊D₁₇

C₂²

yes

—

C₂×C₂²⋊D₁₇

C₂²

yes

—

C₂×C₂²⋊D₁₇

C₂²

yes

complemented

C₂×D₆₈

C₂²

yes

—

C₂×D₆₈

C₂²

yes

—

C₂×D₆₈

C₂²

yes

characteristic, complemented

C₂×C₄×D₁₇

C₂²

yes

complemented

C₄⋊C₄×C₁₇

C₂²

yes

complemented

C₂²⋊C₄×C₁₇

C₂²

yes

—

C₂²⋊C₄×C₁₇

C₂²

yes

characteristic, complemented

C₂²⋊Dic₁₇

C₂²

yes

characteristic

#63227

C₂²

yes

—

#63227

C₂²

yes

—

#63227

C₂²

yes

—

#63227

C₂²

yes

complemented

C₄⋊Dic₁₇

C₂²

yes

complemented

#63225

C₂²

yes

complemented

C₄×Dic₁₇

C₂²

yes

characteristic, complemented

C₂³×C₁₇

C₂³

yes

characteristic

C₂³×C₁₇

C₂³

yes

characteristic

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂²×D₁₇

C₂³

yes

—

C₂²×D₁₇

C₂³

yes

—

D₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂³

yes

—

C₂×C₄×C₁₇

C₂³

yes

—

C₂×C₄×C₁₇

C₂³

yes

characteristic

C₂²⋊D₁₇

—

C₂²⋊D₁₇

—

C₂²⋊D₁₇

—

C₂×Dic₁₇

C₂³

yes

—

C₂×Dic₁₇

C₂³

yes

—

D₆₈

—

D₆₈

—

D₆₈

—

C₄×D₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂⁴

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

Dic₁₇

—

Dic₁₇

—

#560

—

maximal, Sylow

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

H(5,𝔽₂)

yes

characteristic

C₂×C₁₇

H(5,𝔽₂)

yes

—

D₁₇

—

D₁₇

—

—

—

—

D₁₇

yes

characteristic, complemented

—

—

—

—

—

C₁₇

#560

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₁₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×D₁₇

yes

characteristic, complemented

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₁₇

yes

complemented

C₂²⋊C₄

—

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

C₂×D₁₇

yes

—

104

C₂²⋊C₄

C₂×D₁₇

yes

characteristic, complemented

105

C₄²

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

—

110

C₂³

C₂²×D₁₇

yes

characteristic

111

C₂³

C₂²×D₁₇

yes

characteristic

112

D₄

—

113

D₄

—

114

D₄

—

115

D₄

—

116

D₄

—

117

D₄

—

118

D₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

C₂²×D₁₇

yes

characteristic

124

C₂×C₄

C₂²×D₁₇

yes

—

125

C₂×C₄

C₂²×D₁₇

yes

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂³×D₁₇

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₂

—

144

C₂

—

145

C₂

—

146

C₂

—

147

C₂

#10587691

yes

characteristic

148

C₂

#10587696

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₇
commutator, socle, cosocle	C₂²×C₁₇	C₂⁴
Fitting	C₁₇×#123	C₂
hypercenter	#123	D₁₇
hypocenter	C₁₇	#560
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ #123

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₇

commutator, socle, cosocle

C₂²×C₁₇

C₂⁴

Fitting

C₁₇×#123

C₂

D₁₇

C₁₇

G ▷ C₂²×C₁₇ ▷ C₁₇

C₁ ◁ C₂² ◁ #123

N	H ≅ G/N	type
1. C₁₇×#123	C₂	semidirect
2. #10587626	C₂	semidirect
3. #10587623	C₂	semidirect
4. #10587616	C₂	semidirect
5. #10587584	C₂	semidirect
6. #10587581	C₂	semidirect
7. #10587570	C₂	semidirect
8. #10587569	C₂	semidirect
9. #10587568	C₂	semidirect
12. C₂²×C₄×C₁₇	C₂²	semidirect
15. C₂×C₂²⋊D₁₇	C₂²	semidirect
18. C₂×D₆₈	C₂²	semidirect
19. C₂×C₄×D₁₇	C₂²	semidirect
20. C₄⋊C₄×C₁₇	C₂²	semidirect
22. C₂²⋊C₄×C₁₇	C₂²	semidirect
27. #63227	C₂²	semidirect
28. C₄⋊Dic₁₇	C₂²	semidirect
29. #63225	C₂²	semidirect
30. C₄×Dic₁₇	C₂²	semidirect
78. #123	D₁₇	semidirect
84. C₁₇	#560	semidirect
94. C₂²×C₄	C₂×D₁₇	semidirect
97. C₄⋊C₄	C₂×D₁₇	semidirect
104. C₂²⋊C₄	C₂×D₁₇	semidirect

H ≅ G/N

type

1. C₁₇×#123

C₂

semidirect

2. #10587626

C₂

semidirect

3. #10587623

C₂

semidirect

4. #10587616

C₂

semidirect

5. #10587584

C₂

semidirect

6. #10587581

C₂

semidirect

7. #10587570

C₂

semidirect

8. #10587569

C₂

semidirect

9. #10587568

C₂

semidirect

12. C₂²×C₄×C₁₇

C₂²

semidirect

15. C₂×C₂²⋊D₁₇

C₂²

semidirect

18. C₂×D₆₈

C₂²

semidirect

19. C₂×C₄×D₁₇

C₂²

semidirect

20. C₄⋊C₄×C₁₇

C₂²

semidirect

22. C₂²⋊C₄×C₁₇

C₂²

semidirect

27. #63227

C₂²

semidirect

28. C₄⋊Dic₁₇

C₂²

semidirect

29. #63225

C₂²

semidirect

30. C₄×Dic₁₇

C₂²

semidirect

78. #123

D₁₇

semidirect

84. C₁₇

#560

semidirect

94. C₂²×C₄

C₂×D₁₇

semidirect

97. C₄⋊C₄

C₂×D₁₇

semidirect

104. C₂²⋊C₄

C₂×D₁₇

semidirect

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

subgroup	index
C₁₇×#123	2
#10587626	2
#10587623	2
#10587616	2
#10587584	2
#10587581	2
#10587570	2
#10587569	2
#10587568	2
#560	17

subgroup

index

C₁₇×#123

Subgroups — #11763549

Conjugacy classes of subgroups

Complements

Maximal subgroups