Group #14544352 — subgroups

Subgroups — #14544352

Order 2176.

168 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 70 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	#11763813	C₂	1	1	yes	maximal, characteristic
2	#11763810	C₂	1	1	yes	maximal, characteristic
3	#11763671	C₂	1	2	yes	maximal, complemented
4	#11763670	C₂	1	2	yes	maximal, complemented
5	#11762861	C₂	1	1	yes	maximal, characteristic
6	C₂×#63243	—	2	1	no	—
7	C₂×#63242	—	2	1	no	—
8	C₄²×D₁₇	C₂²	1	1	yes	characteristic
9	#10587544	—	2	1	no	—
10	C₄⋊#3624	—	2	1	no	—
11	C₄⋊#3624	C₂²	1	1	yes	characteristic
12	C₄⋊#3624	C₂²	1	1	yes	characteristic
13	C₄⋊#3624	C₂²	1	2	yes	—
14	C₁₇×C₄⋊C₈	C₄	1	1	yes	characteristic
15	#10587491	C₄	1	2	yes	—
16	C₈×Dic₁₇	C₂²	1	2	yes	—
17	C₄⋊#3622	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	1	no	—
22	C₂×#3624	—	2	1	no	—
23	C₂×#3624	—	2	2	no	—
24	#63243	—	4	2	no	—
25	#63242	—	4	2	no	—
26	C₂×C₈×C₁₇	C₂×C₄	1	2	yes	—
27	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₁₇	D₄	1	1	yes	characteristic
29	C₄×Dic₁₇	Q₈	1	1	yes	characteristic
30	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
31	C₂×#3622	C₂×C₄	1	2	yes	—
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	1	no	—
40	C₄×D₁₇	—	2	1	no	—
41	C₄×D₁₇	—	4	1	no	—
42	C₄×D₁₇	SD₁₆	1	2	yes	—
43	C₄×D₁₇	SD₁₆	1	2	yes	—
44	#3624	—	4	1	no	—
45	#3624	—	4	1	no	—
46	#3624	—	4	2	no	—
47	C₈×C₁₇	—	2	2	no	—
48	#3622	—	2	2	no	—
49	#1570	—	17	1	no	maximal, Sylow
50	C₂²×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×D₁₇	—	2	1	no	—
52	C₂×D₁₇	—	2	1	no	—
53	C₂×D₁₇	#106	1	2	yes	—
54	C₄×C₁₇	—	2	1	no	—
55	C₄×C₁₇	#106	1	2	yes	—
56	C₄×C₁₇	#131	1	1	yes	characteristic
57	C₄×C₁₇	#131	1	1	yes	characteristic
58	Dic₁₇	—	2	1	no	—
59	Dic₁₇	—	2	1	no	—
60	Dic₁₇	C₂×SD₁₆	1	1	yes	characteristic
61	Dic₁₇	C₂×SD₁₆	1	1	yes	characteristic
62	#431	—	17	1	no	—
63	#423	—	17	1	no	—
64	C₂×C₄⋊C₈	—	17	1	no	—
65	#334	—	17	2	no	—
66	#334	—	17	2	no	—
67	C₂×C₁₇	#424	1	1	yes	characteristic
68	C₂×C₁₇	C₂×#106	1	1	yes	characteristic
69	C₂×C₁₇	#430	1	1	yes	characteristic
70	D₁₇	—	2	2	no	—
71	C₂×M₁₆	—	34	1	no	—
72	C₂²×C₈	—	34	1	no	—
73	C₂×C₄²	—	17	1	no	—
74	C₄⋊C₈	—	17	1	no	—
75	C₄⋊C₈	—	17	1	no	—
76	C₄⋊C₈	—	17	2	no	—
77	C₄⋊C₈	—	17	1	no	—
78	C₄⋊C₈	—	34	1	no	—
79	C₄⋊C₈	—	34	1	no	—
80	C₄⋊C₈	#595	1	1	yes	characteristic
81	C₂²⋊C₈	—	17	2	no	—
82	C₄×C₈	—	17	2	no	—
83	C₁₇	#1570	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₄	—	17	1	no	—
85	C₂²×C₄	—	17	1	no	—
86	C₂²×C₄	—	17	1	no	—
87	M₁₆	—	68	2	no	—
88	C₂×C₈	—	17	2	no	—
89	C₂×C₈	—	34	1	no	—
90	C₂×C₈	—	34	1	no	—
91	C₂×C₈	—	34	2	no	—
92	C₂×C₈	—	68	2	no	—
93	C₂×C₈	C₂×#595	1	2	yes	—
94	C₄²	—	17	1	no	—
95	C₄²	—	17	1	no	—
96	C₄²	—	17	1	no	—
97	C₄²	C₂×#595	1	1	yes	characteristic
98	C₂³	—	17	1	no	—
99	C₂×C₄	—	17	1	no	—
100	C₂×C₄	—	17	2	no	—
101	C₂×C₄	—	17	2	no	—
102	C₂×C₄	—	17	1	no	—
103	C₂×C₄	—	17	1	no	—
104	C₂×C₄	—	34	1	no	—
105	C₂×C₄	—	34	1	no	—
106	C₂×C₄	—	68	1	no	—
107	C₂×C₄	C₄⋊#595	1	1	yes	characteristic
108	C₂×C₄	C₄⋊#595	1	1	yes	characteristic
109	C₂×C₄	C₂²×#595	1	1	yes	characteristic
110	C₈	—	2	2	no	—
111	C₈	—	34	2	no	—
112	C₈	—	68	1	no	—
113	C₈	—	68	1	no	—
114	C₈	—	68	2	no	—
115	C₂²	—	17	2	no	—
116	C₂²	—	34	1	no	—
117	C₂²	—	34	1	no	—
118	C₂²	C₂×C₄⋊#595	1	1	yes	characteristic
119	C₄	—	2	1	no	—
120	C₄	—	17	1	no	—
121	C₄	—	17	1	no	—
122	C₄	—	34	1	no	—
123	C₄	—	34	1	no	—
124	C₄	#10587533	1	2	yes	—
125	C₄	#10587678	1	1	yes	characteristic
126	C₄	#10587680	1	1	yes	characteristic
127	C₂	—	34	2	no	—
128	C₂	#11763816	1	1	yes	characteristic
129	C₂	C₂×#10587533	1	1	yes	characteristic
130	C₂	#11763824	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763813

C₂

yes

maximal, characteristic

#11763810

C₂

yes

maximal, characteristic

#11763671

C₂

yes

maximal, complemented

#11763670

C₂

yes

maximal, complemented

#11762861

C₂

yes

maximal, characteristic

C₂×#63243

—

C₂×#63242

—

C₄²×D₁₇

C₂²

yes

—

—

C₂²

yes

characteristic

C₄⋊#3624

C₂²

yes

characteristic

C₄⋊#3624

C₂²

yes

—

C₁₇×C₄⋊C₈

C₄

yes

characteristic

#10587491

C₄

yes

—

C₈×Dic₁₇

C₂²

yes

—

C₄⋊#3622

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

—

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

SD₁₆

yes

—

C₄×D₁₇

SD₁₆

yes

—

#3624

—

#3624

—

#3624

—

C₈×C₁₇

—

#3622

—

#1570

—

maximal, Sylow

C₂²×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

#106

yes

—

C₄×C₁₇

—

C₄×C₁₇

#106

yes

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

Dic₁₇

C₂×SD₁₆

yes

characteristic

Dic₁₇

C₂×SD₁₆

yes

characteristic

#431

—

#423

—

C₂×C₄⋊C₈

—

#334

—

#334

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

D₁₇

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

#595

yes

characteristic

C₂²⋊C₈

—

C₄×C₈

—

C₁₇

#1570

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#595

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×#595

yes

characteristic

C₂³

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₄⋊#595

yes

characteristic

108

C₂×C₄

C₄⋊#595

yes

characteristic

109

C₂×C₄

C₂²×#595

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

C₂×C₄⋊#595

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

#10587533

yes

—

125

C₄

#10587678

yes

characteristic

126

C₄

#10587680

yes

characteristic

127

C₂

—

128

C₂

#11763816

yes

characteristic

129

C₂

C₂×#10587533

yes

characteristic

130

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×C₄⋊C₈	C₄
socle	C₂²×C₁₇	C₂×C₄⋊C₄
prime radical	C₂²×D₁₇	C₄⋊C₄
cosocle	C₄×Dic₁₇	C₂³
hypercenter	C₄⋊C₈	#595
hypocenter	C₁₇	#1570
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊#595

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#595

Fitting

C₁₇×C₄⋊C₈

C₄

socle

C₂²×C₁₇

C₂×C₄⋊C₄

prime radical

C₂²×D₁₇

C₄⋊C₄

cosocle

C₄×Dic₁₇

C₂³

C₄⋊C₈

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

N	H ≅ G/N	type
3. #11763671	C₂	semidirect
4. #11763670	C₂	semidirect
83. C₁₇	#1570	semidirect

H ≅ G/N

type

3. #11763671

C₂

semidirect

4. #11763670

C₂

semidirect

83. C₁₇

#1570

semidirect

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

subgroup	index
#11763813	2
#11763810	2
#11763671	2
#11763670	2
#11762861	2
#1570	17

subgroup

index

Subgroups — #14544352

Conjugacy classes of subgroups

Complements

Maximal subgroups