Group #11763501 — subgroups

Subgroups — #11763501

Order 1088.

214 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 99 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×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#10587632	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₂²⋊D₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#10587584	C₂	1	2	yes	maximal, complemented
7	#10587578	C₂	1	1	yes	maximal, characteristic, complemented
8	#10587576	C₂	1	1	yes	maximal, characteristic, complemented
9	#10587576	C₂	1	2	yes	maximal, complemented
10	#10587572	C₂	1	2	yes	maximal
11	#10587563	C₂	1	2	yes	maximal, complemented
12	C₂×Q₈×C₁₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₁₇	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₁₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₇	C₂²	1	2	yes	—
16	C₂×C₄×D₁₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
19	C₂²⋊C₄×C₁₇	C₂²	1	2	yes	—
20	C₂²⋊Dic₁₇	C₂²	1	1	yes	characteristic
21	#63227	C₂²	1	2	yes	—
22	#63227	C₂²	1	2	yes	—
23	#63227	C₂²	1	1	yes	characteristic
24	C₄⋊Dic₁₇	C₂²	1	2	yes	—
25	C₄⋊Dic₁₇	C₂²	1	2	yes	—
26	C₄⋊Dic₁₇	C₂²	1	2	yes	—
27	C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
28	C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
29	#63225	C₂²	1	2	yes	—
30	#63225	C₂²	1	2	yes	—
31	#63225	C₂²	1	1	yes	characteristic
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	C₂²×D₁₇	C₂³	1	1	yes	characteristic
37	Q₈×C₁₇	—	2	2	no	—
38	C₂×C₄×C₁₇	—	2	2	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	C₂×C₄×C₁₇	C₂³	1	1	yes	characteristic
43	C₂²⋊D₁₇	—	2	2	no	—
44	C₂×Dic₁₇	—	2	4	no	—
45	C₂×Dic₁₇	C₂³	1	1	yes	characteristic
46	C₂×Dic₁₇	C₂³	1	1	yes	characteristic
47	C₂×Dic₁₇	C₂³	1	2	yes	—
48	C₂×Dic₁₇	C₂³	1	2	yes	—
49	C₂×Dic₁₇	C₂³	1	1	yes	characteristic
50	C₄×D₁₇	—	2	2	no	—
51	C₂²×C₁₇	—	2	1	no	—
52	C₂²×C₁₇	—	2	1	no	—
53	C₂²×C₁₇	#41	1	2	yes	—
54	C₂²×C₁₇	C₂⁴	1	1	yes	characteristic
55	C₂×D₁₇	—	2	1	no	—
56	C₂×D₁₇	—	2	1	no	—
57	C₂×D₁₇	—	2	1	no	—
58	C₄×C₁₇	—	2	1	no	—
59	C₄×C₁₇	—	2	2	no	—
60	C₄×C₁₇	—	2	2	no	—
61	C₄×C₁₇	#41	1	2	yes	—
62	Dic₁₇	—	2	1	no	—
63	Dic₁₇	—	2	2	no	—
64	Dic₁₇	—	2	2	no	—
65	Dic₁₇	—	2	1	no	—
66	Dic₁₇	—	2	1	no	—
67	#552	—	17	1	no	maximal, Sylow
68	C₂×C₁₇	—	2	2	no	—
69	C₂×C₁₇	C₂×#41	1	1	yes	characteristic
70	C₂×C₁₇	C₂×#41	1	1	yes	characteristic
71	C₂×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
72	D₁₇	—	4	1	no	—
73	#126	—	17	2	no	—
74	#125	—	17	2	no	—
75	#125	—	17	1	no	—
76	#123	—	17	2	no	—
77	C₂²⋊Q₈	—	17	1	no	—
78	C₂²⋊Q₈	D₁₇	1	1	yes	characteristic, complemented
79	C₄×Q₈	—	17	1	no	—
80	C₄×D₄	—	17	1	no	—
81	#117	—	17	1	no	—
82	#117	—	17	2	no	—
83	C₂×C₄⋊C₄	—	17	1	no	—
84	C₁₇	#552	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₈	C₂×D₁₇	1	1	yes	characteristic
86	C₂×D₄	—	17	1	no	—
87	C₂²×C₄	—	17	2	no	—
88	C₂²×C₄	—	17	1	no	—
89	C₂²×C₄	C₂×D₁₇	1	1	yes	characteristic
90	C₄⋊C₄	—	17	1	no	—
91	C₄⋊C₄	—	17	1	no	—
92	C₄⋊C₄	—	17	2	no	—
93	C₄⋊C₄	—	17	2	no	—
94	C₄⋊C₄	—	17	2	no	—
95	C₄⋊C₄	—	17	1	no	—
96	C₄⋊C₄	—	17	2	no	—
97	C₄⋊C₄	—	17	2	no	—
98	C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
99	C₄⋊C₄	C₂×D₁₇	1	2	yes	—
100	C₂²⋊C₄	—	17	1	no	—
101	C₂²⋊C₄	—	17	2	no	—
102	C₂²⋊C₄	—	17	1	no	—
103	C₂²⋊C₄	—	17	2	no	—
104	C₂²⋊C₄	C₂×D₁₇	1	2	yes	—
105	C₄²	—	17	2	no	—
106	C₄²	—	17	2	no	—
107	C₄²	—	17	1	no	—
108	C₂³	—	17	1	no	—
109	C₂³	C₂²×D₁₇	1	1	yes	characteristic
110	Q₈	—	2	2	no	—
111	D₄	—	34	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	17	1	no	—
114	C₂×C₄	—	17	1	no	—
115	C₂×C₄	—	17	2	no	—
116	C₂×C₄	—	17	1	no	—
117	C₂×C₄	—	17	2	no	—
118	C₂×C₄	—	34	4	no	—
119	C₂×C₄	—	34	2	no	—
120	C₂×C₄	C₂²×D₁₇	1	2	yes	—
121	C₂×C₄	C₂²×D₁₇	1	2	yes	—
122	C₂×C₄	C₂²×D₁₇	1	1	yes	characteristic
123	C₂×C₄	C₂²×D₁₇	1	1	yes	characteristic
124	C₂²	—	2	1	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	34	1	no	—
127	C₂²	—	34	1	no	—
128	C₂²	—	34	1	no	—
129	C₂²	#63256	1	2	yes	—
130	C₂²	C₂³×D₁₇	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	34	1	no	—
135	C₄	—	34	2	no	—
136	C₄	—	34	2	no	—
137	C₄	—	34	1	no	—
138	C₄	—	34	1	no	—
139	C₄	#63254	1	2	yes	—
140	C₂	—	2	2	no	—
141	C₂	—	68	1	no	—
142	C₂	C₂×#63254	1	1	yes	characteristic
143	C₂	C₂×#63256	1	1	yes	characteristic
144	C₂	#10587697	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₇×C₂²⋊Q₈

C₂

yes

maximal, characteristic, complemented

#10587632

C₂

yes

maximal, characteristic, complemented

Q₈×Dic₁₇

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₁₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₁₇

C₂

yes

maximal, characteristic, complemented

#10587584

C₂

yes

maximal, complemented

#10587578

C₂

yes

maximal, characteristic, complemented

#10587576

C₂

yes

maximal, characteristic, complemented

#10587576

C₂

yes

maximal, complemented

#10587572

C₂

yes

maximal

#10587563

C₂

yes

maximal, complemented

C₂×Q₈×C₁₇

C₂²

yes

characteristic

C₂²×C₄×C₁₇

C₂²

yes

characteristic

C₂×C₂²⋊D₁₇

C₂²

yes

characteristic

C₂²×Dic₁₇

C₂²

yes

—

C₂×C₄×D₁₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₇

C₂²

yes

—

C₂²⋊C₄×C₁₇

C₂²

yes

—

C₂²⋊Dic₁₇

C₂²

yes

characteristic

#63227

C₂²

yes

—

#63227

C₂²

yes

—

#63227

C₂²

yes

characteristic

C₄⋊Dic₁₇

C₂²

yes

—

C₄⋊Dic₁₇

C₂²

yes

—

C₄⋊Dic₁₇

C₂²

yes

—

C₄⋊Dic₁₇

C₂²

yes

characteristic

C₄⋊Dic₁₇

C₂²

yes

characteristic

#63225

C₂²

yes

—

#63225

C₂²

yes

—

#63225

C₂²

yes

characteristic

C₄×Dic₁₇

C₂²

yes

—

C₄×Dic₁₇

C₂²

yes

—

C₄×Dic₁₇

C₂²

yes

characteristic

C₂³×C₁₇

C₂³

yes

characteristic

C₂²×D₁₇

C₂³

yes

characteristic

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂³

yes

—

C₂×C₄×C₁₇

C₂³

yes

—

C₂×C₄×C₁₇

C₂³

yes

characteristic

C₂×C₄×C₁₇

C₂³

yes

characteristic

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

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#41

yes

—

C₂²×C₁₇

C₂⁴

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#41

yes

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

#552

—

maximal, Sylow

C₂×C₁₇

—

C₂×C₁₇

C₂×#41

yes

characteristic

C₂×C₁₇

C₂×#41

yes

characteristic

C₂×C₁₇

2₋¹⁺⁴

yes

characteristic

D₁₇

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

D₁₇

yes

characteristic, complemented

C₄×Q₈

—

C₄×D₄

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₁₇

#552

yes

characteristic, complemented, Sylow

C₂×Q₈

C₂×D₁₇

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₁₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₁₇

yes

characteristic

C₄⋊C₄

C₂×D₁₇

yes

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

104

C₂²⋊C₄

C₂×D₁₇

yes

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

108

C₂³

—

109

C₂³

C₂²×D₁₇

yes

characteristic

110

Q₈

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₂²×D₁₇

yes

—

121

C₂×C₄

C₂²×D₁₇

yes

—

122

C₂×C₄

C₂²×D₁₇

yes

characteristic

123

C₂×C₄

C₂²×D₁₇

yes

characteristic

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

#63256

yes

—

130

C₂²

C₂³×D₁₇

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

#63254

yes

—

140

C₂

—

141

C₂

—

142

C₂

C₂×#63254

yes

characteristic

143

C₂

C₂×#63256

yes

characteristic

144

C₂

#10587697

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₇
commutator, socle, cosocle	C₂²×C₁₇	C₂⁴
Fitting	C₁₇×C₂²⋊Q₈	C₂
prime radical	C₂×C₂²⋊D₁₇	C₂²
hypercenter	C₂²⋊Q₈	D₁₇
hypocenter	C₁₇	#552
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₇

commutator, socle, cosocle

C₂²×C₁₇

C₂⁴

Fitting

C₁₇×C₂²⋊Q₈

C₂

prime radical

C₂×C₂²⋊D₁₇

C₂²

hypercenter

C₂²⋊Q₈

D₁₇

C₁₇

G ▷ C₂²×C₁₇ ▷ C₁₇

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. C₁₇×C₂²⋊Q₈	C₂	semidirect
2. #10587632	C₂	semidirect
3. Q₈×Dic₁₇	C₂	semidirect
5. C₂×C₄⋊Dic₁₇	C₂	semidirect
6. #10587584	C₂	semidirect
7. #10587578	C₂	semidirect
8. #10587576	C₂	semidirect
9. #10587576	C₂	semidirect
11. #10587563	C₂	semidirect
78. C₂²⋊Q₈	D₁₇	semidirect
84. C₁₇	#552	semidirect

H ≅ G/N

type

1. C₁₇×C₂²⋊Q₈

C₂

semidirect

2. #10587632

C₂

semidirect

3. Q₈×Dic₁₇

C₂

semidirect

5. C₂×C₄⋊Dic₁₇

C₂

semidirect

6. #10587584

C₂

semidirect

7. #10587578

C₂

semidirect

8. #10587576

C₂

semidirect

9. #10587576

C₂

semidirect

11. #10587563

C₂

semidirect

78. C₂²⋊Q₈

D₁₇

semidirect

84. C₁₇

#552

semidirect

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

subgroup	index
C₁₇×C₂²⋊Q₈	2
#10587632	2
Q₈×Dic₁₇	2
C₄×C₂²⋊D₁₇	2
C₂×C₄⋊Dic₁₇	2
#10587584	2
#10587578	2
#10587576	2
#10587572	2
#10587563	2
#552	17

subgroup

index

C₁₇×C₂²⋊Q₈

#10587632

Q₈×Dic₁₇

C₄×C₂²⋊D₁₇

C₂×C₄⋊Dic₁₇

Subgroups — #11763501

Conjugacy classes of subgroups

Complements

Maximal subgroups