Group #14539293 — subgroups

Subgroups — #14539293

Order 2176.

356 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 145 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	#11763364	C₂	1	2	yes	maximal
2	#11763212	C₂	1	2	yes	maximal, complemented
3	#11763209	C₂	1	4	yes	maximal, complemented
4	#11763207	C₂	1	1	yes	maximal, characteristic, complemented
5	#11762844	C₂	1	4	yes	maximal
6	C₄×C₈×D₁₇	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#498	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₇×C₄⋊Q₈	C₂²	1	2	yes	—
9	#10587632	—	2	2	no	—
10	Q₈×Dic₁₇	—	2	2	no	—
11	C₂×C₈×D₁₇	C₂²	1	2	yes	—
12	#10587583	—	2	2	no	—
13	#10587578	C₂²	1	4	yes	—
14	#10587576	C₂²	1	4	yes	—
15	#10587573	—	2	2	no	—
16	C₄²×D₁₇	C₂²	1	1	yes	characteristic
17	#10587555	C₂²	1	2	yes	—
18	C₁₇×#106	C₂²	1	4	yes	—
19	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
20	#10587488	C₂²	1	4	yes	—
21	C₈×Dic₁₇	C₂²	1	2	yes	—
22	#10587478	C₂²	1	8	yes	—
23	C₄×#3622	C₂²	1	1	yes	characteristic
24	C₂×Q₈×C₁₇	—	2	2	no	—
25	C₂×C₄×D₁₇	D₄	1	2	yes	—
26	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
27	C₂×Dic₃₄	—	2	2	no	—
28	C₂×C₈×C₁₇	C₂³	1	2	yes	—
29	C₄⋊C₄×C₁₇	—	2	2	no	—
30	C₄⋊C₄×C₁₇	C₂³	1	4	yes	—
31	C₄²×C₁₇	C₂³	1	1	yes	characteristic
32	#63227	—	2	4	no	—
33	C₄⋊Dic₁₇	—	2	2	no	—
34	C₄⋊Dic₁₇	—	2	4	no	—
35	C₄⋊Dic₁₇	C₂³	1	4	yes	—
36	#63225	—	2	4	no	—
37	C₄×Dic₁₇	—	2	4	no	—
38	C₄×Dic₁₇	D₄	1	2	yes	—
39	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
40	C₂×#3622	C₂³	1	2	yes	—
41	C₈×D₁₇	Q₈	1	8	yes	complemented
42	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₁₇	—	4	4	no	—
44	C₂×C₄×C₁₇	—	2	4	no	—
45	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₁₇	—	2	4	no	—
48	C₂×Dic₁₇	C₂×D₄	1	2	yes	—
49	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
50	C₄×D₁₇	—	2	4	no	—
51	C₄×D₁₇	C₂×Q₈	1	4	yes	—
52	Dic₃₄	—	4	4	no	—
53	C₈×C₁₇	C₂×Q₈	1	4	yes	—
54	#3622	C₂×Q₈	1	4	yes	—
55	#3160	—	17	1	no	maximal, Sylow
56	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
57	C₂×D₁₇	—	2	2	no	—
58	C₂×D₁₇	C₄⋊Q₈	1	2	yes	—
59	C₄×C₁₇	—	4	4	no	—
60	C₄×C₁₇	#135	1	4	yes	—
61	C₄×C₁₇	C₂²×Q₈	1	2	yes	—
62	Dic₁₇	—	2	2	no	—
63	Dic₁₇	—	4	4	no	—
64	Dic₁₇	C₄⋊Q₈	1	2	yes	—
65	#533	—	17	2	no	—
66	#500	—	17	2	no	—
67	#499	—	17	4	no	—
68	#498	—	17	1	no	—
69	#498	D₁₇	1	1	yes	characteristic, complemented
70	#427	—	17	4	no	—
71	C₂×C₄×C₈	—	17	1	no	—
72	C₂×C₁₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₁₇	C₂×#135	1	2	yes	—
74	D₁₇	—	2	2	no	—
75	C₂²×C₈	—	17	2	no	—
76	C₄⋊Q₈	—	17	2	no	—
77	C₄⋊Q₈	C₂×D₁₇	1	2	yes	—
78	#125	—	17	4	no	—
79	C₂²⋊Q₈	—	34	2	no	—
80	C₂²⋊Q₈	—	34	2	no	—
81	C₄×Q₈	—	34	2	no	—
82	C₄×Q₈	—	34	2	no	—
83	#117	—	17	4	no	—
84	C₂×C₄²	—	17	1	no	—
85	#107	—	17	8	no	—
86	#106	—	17	4	no	—
87	#106	C₂×D₁₇	1	4	yes	—
88	C₄×C₈	—	17	2	no	—
89	C₄×C₈	—	17	1	no	—
90	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
91	C₁₇	#3160	1	1	yes	characteristic, complemented, Sylow
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	34	2	no	—
94	C₂²×C₄	—	17	1	no	—
95	C₂²×C₄	—	17	2	no	—
96	C₂×C₈	—	17	2	no	—
97	C₂×C₈	—	17	8	no	—
98	C₂×C₈	C₂²×D₁₇	1	2	yes	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	17	4	no	—
101	C₄⋊C₄	—	34	4	no	—
102	C₄⋊C₄	—	34	4	no	—
103	C₄⋊C₄	—	34	2	no	—
104	C₄⋊C₄	C₂²×D₁₇	1	4	yes	—
105	C₂²⋊C₄	—	34	4	no	—
106	C₄²	—	17	2	no	—
107	C₄²	—	17	1	no	—
108	C₄²	—	34	4	no	—
109	C₄²	C₂²×D₁₇	1	1	yes	characteristic
110	C₂³	—	17	1	no	—
111	Q₈	—	4	4	no	—
112	Q₈	—	68	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	17	1	no	—
115	C₂×C₄	—	17	4	no	—
116	C₂×C₄	—	17	2	no	—
117	C₂×C₄	—	34	4	no	—
118	C₂×C₄	—	34	4	no	—
119	C₂×C₄	D₄×D₁₇	1	2	yes	—
120	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
121	C₈	—	17	4	no	—
122	C₈	Q₈×D₁₇	1	4	yes	—
123	C₂²	—	17	2	no	—
124	C₂²	—	34	2	no	—
125	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
126	C₄	—	4	4	no	—
127	C₄	—	17	2	no	—
128	C₄	—	34	2	no	—
129	C₄	—	68	4	no	—
130	C₄	#10587602	1	4	yes	—
131	C₄	C₂×Q₈×D₁₇	1	2	yes	—
132	C₂	—	34	2	no	—
133	C₂	C₄⋊Q₈×D₁₇	1	1	yes	characteristic
134	C₂	C₂×#10587602	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11763364

C₂

yes

maximal

#11763212

C₂

yes

maximal, complemented

#11763209

C₂

yes

maximal, complemented

#11763207

C₂

yes

maximal, characteristic, complemented

#11762844

C₂

yes

maximal

C₄×C₈×D₁₇

C₂

yes

maximal, characteristic

C₁₇×#498

C₂

yes

maximal, characteristic, complemented

C₁₇×C₄⋊Q₈

C₂²

yes

—

#10587632

—

Q₈×Dic₁₇

—

C₂×C₈×D₁₇

C₂²

yes

—

#10587583

—

#10587578

C₂²

yes

—

#10587576

C₂²

yes

—

#10587573

—

C₄²×D₁₇

C₂²

yes

characteristic

#10587555

C₂²

yes

—

C₁₇×#106

C₂²

yes

—

C₄×C₈×C₁₇

C₂²

yes

characteristic

#10587488

C₂²

yes

—

C₈×Dic₁₇

C₂²

yes

—

#10587478

C₂²

yes

—

C₄×#3622

C₂²

yes

characteristic

C₂×Q₈×C₁₇

—

C₂×C₄×D₁₇

D₄

yes

—

C₂×C₄×D₁₇

C₂³

yes

characteristic

C₂×Dic₃₄

—

C₂×C₈×C₁₇

C₂³

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

C₂³

yes

—

C₄²×C₁₇

C₂³

yes

characteristic

#63227

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

C₂³

yes

—

#63225

—

C₄×Dic₁₇

—

C₄×Dic₁₇

D₄

yes

—

C₄×Dic₁₇

C₂³

yes

characteristic

C₂×#3622

C₂³

yes

—

C₈×D₁₇

Q₈

yes

complemented

C₂²×D₁₇

C₂×D₄

yes

characteristic

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂⁴

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

C₂×D₄

yes

—

C₂×Dic₁₇

C₂×D₄

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

C₂×Q₈

yes

—

Dic₃₄

—

C₈×C₁₇

C₂×Q₈

yes

—

#3622

C₂×Q₈

yes

—

#3160

—

maximal, Sylow

C₂²×C₁₇

C₂²×D₄

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

C₄⋊Q₈

yes

—

C₄×C₁₇

—

C₄×C₁₇

#135

yes

—

C₄×C₁₇

C₂²×Q₈

yes

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

C₄⋊Q₈

yes

—

—

—

—

—

D₁₇

yes

characteristic, complemented

#427

—

C₂×C₄×C₈

—

C₂×C₁₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₇

C₂×#135

yes

—

D₁₇

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₇

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

#107

—

#106

—

#106

C₂×D₁₇

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₁₇

yes

characteristic

C₁₇

#3160

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×D₁₇

yes

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

C₂²×D₁₇

yes

—

105

C₂²⋊C₄

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₄²

C₂²×D₁₇

yes

characteristic

110

C₂³

—

111

Q₈

—

112

Q₈

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

D₄×D₁₇

yes

—

120

C₂×C₄

C₂³×D₁₇

yes

characteristic

121

C₈

—

122

C₈

Q₈×D₁₇

yes

—

123

C₂²

—

124

C₂²

—

125

C₂²

C₂×D₄×D₁₇

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#10587602

yes

—

131

C₄

C₂×Q₈×D₁₇

yes

—

132

C₂

—

133

C₂

C₄⋊Q₈×D₁₇

yes

characteristic

134

C₂

C₂×#10587602

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#498	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂²×D₁₇	C₂×D₄
hypercenter	#498	D₁₇
hypocenter	C₁₇	#3160
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

C₂×D₄×D₁₇

commutator, cosocle

C₂×C₄×C₁₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₇

Fitting

C₁₇×#498

C₂

socle

C₂²×C₁₇

C₂²×D₄

prime radical

C₂²×D₁₇

C₂×D₄

D₁₇

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
2. #11763212	C₂	semidirect
3. #11763209	C₂	semidirect
4. #11763207	C₂	semidirect
7. C₁₇×#498	C₂	semidirect
41. C₈×D₁₇	Q₈	semidirect
69. #498	D₁₇	semidirect
91. C₁₇	#3160	semidirect

H ≅ G/N

type

2. #11763212

C₂

semidirect

3. #11763209

C₂

semidirect

4. #11763207

C₂

semidirect

7. C₁₇×#498

C₂

semidirect

41. C₈×D₁₇

Q₈

semidirect

69. #498

D₁₇

semidirect

91. C₁₇

#3160

semidirect

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

subgroup	index
#11763364	2
#11763212	2
#11763209	2
#11763207	2
#11762844	2
C₄×C₈×D₁₇	2
C₁₇×#498	2
#3160	17

subgroup

index

C₄×C₈×D₁₇

C₁₇×#498

#3160

Subgroups — #14539293

Conjugacy classes of subgroups

Complements

Maximal subgroups