Group #11763528 — subgroups

Subgroups — #11763528

Order 1088.

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

normal subgroups only

This group has 212 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×#124	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#10587630	C₂	1	1	yes	maximal, characteristic, complemented
4	#10587622	C₂	1	1	yes	maximal, characteristic
5	D₄×Dic₁₇	C₂	1	1	yes	maximal, characteristic
6	#10587565	C₂	1	2	yes	maximal, complemented
7	#10587565	C₂	1	2	yes	maximal, complemented
8	#10587564	C₂	1	2	yes	maximal, complemented
9	#10587563	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₃₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Dic₃₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×C₁₇	C₂²	1	1	yes	characteristic
13	C₂×D₄×C₁₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₇	C₂²	1	2	yes	—
15	C₂×Dic₃₄	C₂²	1	1	yes	characteristic
16	C₂×Dic₃₄	C₂²	1	1	yes	characteristic
17	C₂²⋊C₄×C₁₇	C₂²	1	2	yes	—
18	C₂²⋊C₄×C₁₇	C₂²	1	2	yes	—
19	C₄²×C₁₇	C₂²	1	1	yes	characteristic
20	C₂²⋊Dic₁₇	C₂²	1	2	yes	—
21	C₂²⋊Dic₁₇	C₂²	1	2	yes	—
22	C₂²⋊Dic₁₇	C₂²	1	2	yes	—
23	C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
24	C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
25	C₄⋊Dic₁₇	C₂²	1	2	yes	—
26	#63225	C₂²	1	1	yes	characteristic
27	#63225	C₂²	1	2	yes	—
28	#63225	C₂²	1	1	yes	characteristic
29	#63225	C₂²	1	2	yes	—
30	#63225	C₂²	1	2	yes	—
31	C₄×Dic₁₇	C₂²	1	2	yes	—
32	C₄×Dic₁₇	C₂²	1	1	yes	characteristic
33	C₄×Dic₁₇	C₂²	1	1	yes	characteristic
34	C₄×Dic₁₇	C₂²	1	2	yes	—
35	C₂³×C₁₇	C₂³	1	2	yes	—
36	Q₈×C₁₇	—	2	2	no	—
37	D₄×C₁₇	—	2	2	no	—
38	C₂×C₄×C₁₇	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₁₇	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₁₇	C₂³	1	2	yes	—
41	C₂×C₄×C₁₇	C₂³	1	1	yes	characteristic
42	C₂×Dic₁₇	—	2	4	no	—
43	C₂×Dic₁₇	C₂³	1	2	yes	—
44	C₂×Dic₁₇	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
49	Dic₃₄	—	2	2	no	—
50	Dic₃₄	—	2	2	no	—
51	C₂²×C₁₇	—	2	2	no	—
52	C₂²×C₁₇	—	2	2	no	—
53	C₂²×C₁₇	—	2	2	no	—
54	C₂²×C₁₇	C₂⁴	1	1	yes	characteristic
55	C₄×C₁₇	—	2	1	no	—
56	C₄×C₁₇	—	2	2	no	—
57	C₄×C₁₇	—	2	1	no	—
58	C₄×C₁₇	#41	1	2	yes	—
59	Dic₁₇	—	2	2	no	—
60	Dic₁₇	—	2	1	no	—
61	Dic₁₇	—	2	2	no	—
62	Dic₁₇	—	2	1	no	—
63	Dic₁₇	—	2	1	no	—
64	Dic₁₇	#41	1	2	yes	—
65	#556	—	17	1	no	maximal, Sylow
66	C₂×C₁₇	—	4	2	no	—
67	C₂×C₁₇	C₂×#41	1	1	yes	characteristic
68	C₂×C₁₇	C₂×#41	1	1	yes	characteristic
69	C₂×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
70	C₄⋊Q₈	—	17	1	no	—
71	#126	—	17	2	no	—
72	#126	—	17	2	no	—
73	#124	—	17	1	no	—
74	#124	D₁₇	1	1	yes	characteristic
75	C₂²⋊Q₈	—	17	2	no	—
76	C₄×Q₈	—	17	1	no	—
77	C₄×Q₈	—	17	1	no	—
78	C₄×Q₈	—	17	1	no	—
79	C₄×D₄	—	17	1	no	—
80	#117	—	17	2	no	—
81	C₁₇	#556	1	1	yes	characteristic, complemented, Sylow
82	C₂×Q₈	—	17	1	no	—
83	C₂×Q₈	—	17	1	no	—
84	C₂×Q₈	C₂×D₁₇	1	1	yes	characteristic
85	C₂×D₄	C₂×D₁₇	1	1	yes	characteristic
86	C₂²×C₄	—	17	2	no	—
87	C₄⋊C₄	—	17	1	no	—
88	C₄⋊C₄	—	17	2	no	—
89	C₄⋊C₄	—	17	1	no	—
90	C₄⋊C₄	—	17	2	no	—
91	C₄⋊C₄	—	17	2	no	—
92	C₄⋊C₄	—	17	1	no	—
93	C₄⋊C₄	—	17	1	no	—
94	C₄⋊C₄	—	17	2	no	—
95	C₂²⋊C₄	—	17	2	no	—
96	C₂²⋊C₄	—	17	2	no	—
97	C₂²⋊C₄	—	17	2	no	—
98	C₂²⋊C₄	C₂×D₁₇	1	2	yes	—
99	C₂²⋊C₄	C₂×D₁₇	1	2	yes	—
100	C₄²	—	17	2	no	—
101	C₄²	—	17	2	no	—
102	C₄²	—	17	1	no	—
103	C₄²	—	17	1	no	—
104	C₄²	C₂×D₁₇	1	1	yes	characteristic
105	C₂³	C₂²×D₁₇	1	2	yes	—
106	Q₈	—	2	2	no	—
107	Q₈	—	34	2	no	—
108	Q₈	—	34	2	no	—
109	D₄	—	2	2	no	—
110	C₂×C₄	—	17	1	no	—
111	C₂×C₄	—	17	1	no	—
112	C₂×C₄	—	17	2	no	—
113	C₂×C₄	—	17	1	no	—
114	C₂×C₄	—	17	2	no	—
115	C₂×C₄	—	17	1	no	—
116	C₂×C₄	—	34	4	no	—
117	C₂×C₄	C₂²×D₁₇	1	1	yes	characteristic
118	C₂×C₄	C₂²×D₁₇	1	2	yes	—
119	C₂×C₄	C₂²×D₁₇	1	1	yes	characteristic
120	C₂×C₄	C₂²×D₁₇	1	1	yes	characteristic
121	C₂²	—	2	2	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	C₂³×D₁₇	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	2	1	no	—
127	C₄	—	2	1	no	—
128	C₄	—	17	2	no	—
129	C₄	—	34	2	no	—
130	C₄	—	34	2	no	—
131	C₄	—	34	1	no	—
132	C₄	—	34	1	no	—
133	C₄	—	34	1	no	—
134	C₄	#63254	1	2	yes	—
135	C₂	—	4	2	no	—
136	C₂	C₂×#63254	1	1	yes	characteristic
137	C₂	#41×D₁₇	1	1	yes	characteristic
138	C₂	#10587697	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₇×#124

C₂

yes

maximal, characteristic

Q₈×Dic₁₇

C₂

yes

maximal, characteristic, complemented

#10587630

C₂

yes

maximal, characteristic, complemented

#10587622

C₂

yes

maximal, characteristic

D₄×Dic₁₇

C₂

yes

maximal, characteristic

#10587565

C₂

yes

maximal, complemented

#10587565

C₂

yes

maximal, complemented

#10587564

C₂

yes

maximal, complemented

#10587563

C₂

yes

maximal, complemented

C₄×Dic₃₄

C₂

yes

maximal, characteristic, complemented

C₄×Dic₃₄

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₁₇

C₂²

yes

characteristic

C₂×D₄×C₁₇

C₂²

yes

characteristic

C₂²×Dic₁₇

C₂²

yes

—

C₂×Dic₃₄

C₂²

yes

characteristic

C₂×Dic₃₄

C₂²

yes

characteristic

C₂²⋊C₄×C₁₇

C₂²

yes

—

C₂²⋊C₄×C₁₇

C₂²

yes

—

C₄²×C₁₇

C₂²

yes

characteristic

C₂²⋊Dic₁₇

C₂²

yes

—

C₂²⋊Dic₁₇

C₂²

yes

—

C₂²⋊Dic₁₇

C₂²

yes

—

C₄⋊Dic₁₇

C₂²

yes

characteristic

C₄⋊Dic₁₇

C₂²

yes

characteristic

C₄⋊Dic₁₇

C₂²

yes

—

#63225

C₂²

yes

characteristic

#63225

C₂²

yes

—

#63225

C₂²

yes

characteristic

#63225

C₂²

yes

—

#63225

C₂²

yes

—

C₄×Dic₁₇

C₂²

yes

—

C₄×Dic₁₇

C₂²

yes

characteristic

C₄×Dic₁₇

C₂²

yes

characteristic

C₄×Dic₁₇

C₂²

yes

—

C₂³×C₁₇

C₂³

yes

—

Q₈×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

C₂³

yes

characteristic

C₂×C₄×C₁₇

C₂³

yes

characteristic

C₂×C₄×C₁₇

C₂³

yes

—

C₂×C₄×C₁₇

C₂³

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

C₂³

yes

—

C₂×Dic₁₇

C₂³

yes

characteristic

C₂×Dic₁₇

C₂³

yes

characteristic

C₂×Dic₁₇

C₂³

yes

characteristic

C₂×Dic₁₇

C₂³

yes

—

C₂×Dic₁₇

C₂³

yes

characteristic

Dic₃₄

—

Dic₃₄

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂⁴

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#41

yes

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

#41

yes

—

#556

—

maximal, Sylow

C₂×C₁₇

—

C₂×C₁₇

C₂×#41

yes

characteristic

C₂×C₁₇

C₂×#41

yes

characteristic

C₂×C₁₇

2₋¹⁺⁴

yes

characteristic

C₄⋊Q₈

—

—

—

—

D₁₇

yes

characteristic

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×D₄

—

#117

—

C₁₇

#556

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₁₇

yes

characteristic

C₂×D₄

C₂×D₁₇

yes

characteristic

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₁₇

yes

—

C₂²⋊C₄

C₂×D₁₇

yes

—

100

C₄²

—

101

C₄²

—

102

C₄²

—

103

C₄²

—

104

C₄²

C₂×D₁₇

yes

characteristic

105

C₂³

C₂²×D₁₇

yes

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

109

D₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂²×D₁₇

yes

characteristic

118

C₂×C₄

C₂²×D₁₇

yes

—

119

C₂×C₄

C₂²×D₁₇

yes

characteristic

120

C₂×C₄

C₂²×D₁₇

yes

characteristic

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂³×D₁₇

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

#63254

yes

—

135

C₂

—

136

C₂

C₂×#63254

yes

characteristic

137

C₂

#41×D₁₇

yes

characteristic

138

C₂

#10587697

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₇
commutator, socle, cosocle	C₂²×C₁₇	C₂⁴
Fitting	C₁₇×#124	C₂
prime radical	C₂×D₄×C₁₇	C₂²
hypercenter	#124	D₁₇
hypocenter	C₁₇	#556
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₇

commutator, socle, cosocle

C₂²×C₁₇

C₂⁴

Fitting

C₁₇×#124

C₂

prime radical

C₂×D₄×C₁₇

C₂²

D₁₇

C₁₇

G ▷ C₂²×C₁₇ ▷ C₁₇

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
2. Q₈×Dic₁₇	C₂	semidirect
3. #10587630	C₂	semidirect
6. #10587565	C₂	semidirect
7. #10587565	C₂	semidirect
8. #10587564	C₂	semidirect
9. #10587563	C₂	semidirect
10. C₄×Dic₃₄	C₂	semidirect
11. C₄×Dic₃₄	C₂	semidirect
81. C₁₇	#556	semidirect

H ≅ G/N

type

2. Q₈×Dic₁₇

C₂

semidirect

3. #10587630

C₂

semidirect

6. #10587565

C₂

semidirect

7. #10587565

C₂

semidirect

8. #10587564

C₂

semidirect

9. #10587563

C₂

semidirect

10. C₄×Dic₃₄

C₂

semidirect

11. C₄×Dic₃₄

C₂

semidirect

81. C₁₇

#556

semidirect

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

subgroup	index
C₁₇×#124	2
Q₈×Dic₁₇	2
#10587630	2
#10587622	2
D₄×Dic₁₇	2
#10587565	2
#10587564	2
#10587563	2
C₄×Dic₃₄	2
#556	17

subgroup

index

C₁₇×#124

Q₈×Dic₁₇

D₄×Dic₁₇

C₄×Dic₃₄

Subgroups — #11763528

Conjugacy classes of subgroups

Complements

Maximal subgroups