Group #14544350 — subgroups

Subgroups — #14544350

Order 2176.

166 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 64 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	#11763831	C₂	1	1	yes	maximal, characteristic
2	#11763813	C₂	1	1	yes	maximal, characteristic
3	#11763673	C₂	1	2	yes	maximal, complemented
4	#11763671	C₂	1	2	yes	maximal, complemented
5	#11762761	C₂	1	1	yes	maximal, characteristic
6	C₂×#63243	—	2	1	no	—
7	C₂×#63218	C₄	1	2	yes	—
8	C₄²×D₁₇	C₂²	1	1	yes	characteristic
9	#10587543	—	2	1	no	—
10	#10587542	—	2	1	no	—
11	C₄⋊#3624	—	2	1	no	—
12	C₄⋊#3624	C₂²	1	2	yes	—
13	C₄⋊#3624	C₂²	1	1	yes	characteristic
14	C₄⋊#3624	C₂²	1	1	yes	characteristic
15	C₁₇×#97	C₄	1	1	yes	characteristic
16	C₈×Dic₁₇	C₂²	1	2	yes	—
17	#10587474	C₄	1	1	yes	characteristic
18	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
20	C₂×C₄×D₁₇	Q₈	1	1	yes	characteristic
21	C₂×#3624	—	2	2	no	—
22	C₂×#3624	—	2	1	no	—
23	C₂×#3624	—	2	1	no	—
24	#63243	—	4	2	no	—
25	C₂×C₈×C₁₇	C₂×C₄	1	2	yes	—
26	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₁₇	D₄	1	1	yes	characteristic
28	C₄×Dic₁₇	Q₈	1	1	yes	characteristic
29	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
30	C₂×#3622	C₂×C₄	1	2	yes	—
31	#63218	—	2	4	no	—
32	C₂²×D₁₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
38	C₂×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
39	C₄×D₁₇	—	2	2	no	—
40	C₄×D₁₇	—	2	1	no	—
41	C₄×D₁₇	—	2	1	no	—
42	C₄×D₁₇	—	4	1	no	—
43	#3624	—	4	2	no	—
44	#3624	—	4	1	no	—
45	#3624	—	4	1	no	—
46	C₈×C₁₇	—	2	2	no	—
47	#3622	M₁₆	1	4	yes	complemented
48	#1569	—	17	1	no	maximal, Sylow
49	C₂²×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×D₁₇	—	2	1	no	—
51	C₂×D₁₇	—	2	1	no	—
52	C₂×D₁₇	—	2	1	no	—
53	C₄×C₁₇	—	2	1	no	—
54	C₄×C₁₇	—	2	1	no	—
55	C₄×C₁₇	C₂×M₁₆	1	2	yes	—
56	Dic₁₇	—	2	1	no	—
57	Dic₁₇	—	2	1	no	—
58	Dic₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
59	Dic₁₇	#137	1	1	yes	characteristic
60	#432	—	17	1	no	—
61	#423	—	17	1	no	—
62	C₄×M₁₆	—	17	1	no	—
63	#335	—	17	2	no	—
64	#334	—	17	2	no	—
65	C₂×C₁₇	#423	1	1	yes	characteristic
66	C₂×C₁₇	#428	1	1	yes	characteristic
67	C₂×C₁₇	#430	1	1	yes	characteristic
68	D₁₇	—	4	1	no	—
69	C₂×M₁₆	—	17	2	no	—
70	C₂×M₁₆	—	34	1	no	—
71	C₂×C₄²	—	17	1	no	—
72	C₄⋊C₈	—	17	1	no	—
73	C₄⋊C₈	—	17	2	no	—
74	C₄⋊C₈	—	17	1	no	—
75	C₄⋊C₈	—	34	1	no	—
76	C₂²⋊C₈	—	34	1	no	—
77	#97	—	17	1	no	—
78	#97	—	34	1	no	—
79	#97	#595	1	1	yes	characteristic
80	C₄×C₈	—	17	2	no	—
81	C₁₇	#1569	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₄	—	17	1	no	—
83	C₂²×C₄	—	17	1	no	—
84	C₂²×C₄	—	17	1	no	—
85	M₁₆	—	34	4	no	—
86	M₁₆	—	68	2	no	—
87	C₂×C₈	—	17	2	no	—
88	C₂×C₈	—	34	1	no	—
89	C₂×C₈	—	34	1	no	—
90	C₂×C₈	—	34	2	no	—
91	C₂×C₈	C₂×#595	1	2	yes	—
92	C₄²	—	17	1	no	—
93	C₄²	—	17	1	no	—
94	C₄²	—	17	1	no	—
95	C₄²	C₂×#595	1	1	yes	characteristic
96	C₂³	—	17	1	no	—
97	C₂×C₄	—	17	1	no	—
98	C₂×C₄	—	17	1	no	—
99	C₂×C₄	—	17	1	no	—
100	C₂×C₄	—	34	1	no	—
101	C₂×C₄	—	34	1	no	—
102	C₂×C₄	—	34	2	no	—
103	C₂×C₄	—	68	1	no	—
104	C₂×C₄	C₄⋊#595	1	1	yes	characteristic
105	C₂×C₄	C₄⋊#595	1	1	yes	characteristic
106	C₂×C₄	C₂²×#595	1	1	yes	characteristic
107	C₈	—	2	2	no	—
108	C₈	—	17	4	no	—
109	C₈	—	68	2	no	—
110	C₈	—	68	1	no	—
111	C₈	—	68	1	no	—
112	C₂²	—	34	1	no	—
113	C₂²	—	34	1	no	—
114	C₂²	—	34	1	no	—
115	C₂²	C₂×C₄⋊#595	1	1	yes	characteristic
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	17	1	no	—
119	C₄	—	17	1	no	—
120	C₄	—	34	1	no	—
121	C₄	—	34	1	no	—
122	C₄	D₁₇⋊M₁₆	1	2	yes	—
123	C₂	—	68	1	no	—
124	C₂	#11763814	1	1	yes	characteristic
125	C₂	#11763820	1	1	yes	characteristic
126	C₂	#11763824	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763831

C₂

yes

maximal, characteristic

#11763813

C₂

yes

maximal, characteristic

#11763673

C₂

yes

maximal, complemented

#11763671

C₂

yes

maximal, complemented

#11762761

C₂

yes

maximal, characteristic

C₂×#63243

—

C₂×#63218

C₄

yes

—

C₄²×D₁₇

C₂²

yes

—

—

—

C₂²

yes

—

C₄⋊#3624

C₂²

yes

characteristic

C₄⋊#3624

C₂²

yes

characteristic

C₁₇×#97

C₄

yes

characteristic

C₈×Dic₁₇

C₂²

yes

—

#10587474

C₄

yes

characteristic

C₂×C₄×D₁₇

C₂×C₄

yes

characteristic

C₂×C₄×D₁₇

D₄

yes

characteristic

C₂×C₄×D₁₇

Q₈

yes

—

—

—

—

C₂×C₈×C₁₇

C₂×C₄

yes

—

C₄²×C₁₇

C₂×C₄

yes

characteristic

C₄×Dic₁₇

D₄

yes

characteristic

C₄×Dic₁₇

Q₈

yes

characteristic

C₄×Dic₁₇

C₂³

yes

characteristic

C₂×#3622

C₂×C₄

yes

—

#63218

—

C₂²×D₁₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×Dic₁₇

C₄⋊C₄

yes

characteristic

C₂×Dic₁₇

C₂×D₄

yes

characteristic

C₂×Dic₁₇

C₂×Q₈

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

#3624

—

#3624

—

#3624

—

C₈×C₁₇

—

#3622

M₁₆

yes

complemented

#1569

—

maximal, Sylow

C₂²×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂×M₁₆

yes

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

yes

Dic₁₇

yes

—

—

C₄×M₁₆

—

#335

—

#334

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

D₁₇

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

—

—

yes

C₄×C₈

—

C₁₇

#1569

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#595

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×#595

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₄⋊#595

yes

characteristic

105

C₂×C₄

C₄⋊#595

yes

characteristic

106

C₂×C₄

C₂²×#595

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂×C₄⋊#595

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

D₁₇⋊M₁₆

yes

—

123

C₂

—

124

C₂

#11763814

yes

characteristic

125

C₂

#11763820

yes

characteristic

126

C₂

#11763824

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#595
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#595
Fitting	C₁₇×#97	C₄
socle	C₂²×C₁₇	C₂×C₄⋊C₄
prime radical	C₂²×D₁₇	C₄⋊C₄
cosocle	C₄×Dic₁₇	C₂³
hypercenter	#97	#595
hypocenter	C₁₇	#1569
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

C₂×C₄⋊#595

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#595

Fitting

C₁₇×#97

C₄

socle

C₂²×C₁₇

C₂×C₄⋊C₄

prime radical

C₂²×D₁₇

C₄⋊C₄

cosocle

C₄×Dic₁₇

C₂³

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
3. #11763673	C₂	semidirect
4. #11763671	C₂	semidirect
47. #3622	M₁₆	semidirect
81. C₁₇	#1569	semidirect

H ≅ G/N

type

3. #11763673

C₂

semidirect

4. #11763671

C₂

semidirect

47. #3622

M₁₆

semidirect

81. C₁₇

#1569

semidirect

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

subgroup	index
#11763831	2
#11763813	2
#11763673	2
#11763671	2
#11762761	2
#1569	17

subgroup

index

Subgroups — #14544350

Conjugacy classes of subgroups

Complements

Maximal subgroups