Group #17962823 — subgroups

Subgroups — #17962823

Order 4352.

198 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 124 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	C₂×#11763741	C₂	1	2	yes	maximal
2	#14543943	C₂	1	4	yes	maximal, complemented
3	#14530381	C₂	1	1	yes	maximal, characteristic
4	#11763741	—	2	2	no	—
5	#11763741	C₂²	1	4	yes	—
6	C₂×#10587467	—	2	2	no	—
7	C₂×#10587483	C₄	1	1	yes	characteristic
8	C₂²×#63214	C₂²	1	1	yes	characteristic
9	#11762585	C₄	1	2	yes	—
10	C₄×#63214	C₂²	1	2	yes	—
11	C₁₇×#405	C₄	1	1	yes	characteristic
12	C₂×M₁₆×C₁₇	C₈	1	1	yes	characteristic
13	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
14	C₁₇×#117	C₈	1	1	yes	characteristic
15	C₁₇×#97	C₈	1	2	yes	—
16	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
17	#10587483	—	2	2	no	—
18	C₂×#63214	C₂×C₄	1	2	yes	—
19	C₂×#63214	C₂×C₄	1	1	yes	characteristic
20	C₂×#63214	D₄	1	2	yes	—
21	C₂×#63214	Q₈	1	2	yes	—
22	C₂×#63214	C₂³	1	1	yes	characteristic
23	#10587467	—	2	4	no	—
24	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
25	M₁₆×C₁₇	C₁₆	1	4	yes	—
26	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
28	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
29	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
30	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
31	C₄⋊C₄×C₁₇	—	2	1	no	—
32	C₂²⋊C₄×C₁₇	—	2	1	no	—
33	C₄²×C₁₇	C₂×C₈	1	2	yes	—
34	#63214	—	2	2	no	—
35	#63214	C₄⋊C₄	1	2	yes	—
36	#63214	C₂×D₄	1	1	yes	characteristic
37	#63214	C₂×Q₈	1	1	yes	characteristic
38	#13630	—	17	1	no	maximal, Sylow
39	C₂³×C₁₇	M₃₂	1	1	yes	characteristic
40	C₂×C₄×C₁₇	—	2	2	no	—
41	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
42	C₂×C₄×C₁₇	C₂×C₁₆	1	2	yes	—
43	C₂×C₄×C₁₇	M₃₂	1	1	yes	characteristic
44	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
45	C₈×C₁₇	C₄⋊C₈	1	2	yes	—
46	C₈×C₁₇	C₂×C₁₆	1	4	yes	—
47	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₈×C₁₇	C₂×M₁₆	1	1	yes	characteristic
49	C₂×M₆₄	—	17	2	no	—
50	#2108	—	17	1	no	—
51	#1422	—	17	4	no	—
52	C₂²×C₁₇	—	4	1	no	—
53	C₂²×C₁₇	C₄⋊C₁₆	1	2	yes	—
54	C₂²×C₁₇	C₂×M₃₂	1	1	yes	characteristic
55	C₄×C₁₇	—	4	2	no	—
56	C₄×C₁₇	C₄⋊C₁₆	1	2	yes	—
57	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
58	C₄×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
59	C₂×M₃₂	—	17	1	no	—
60	C₂²×C₁₆	—	17	1	no	—
61	#405	#595	1	1	yes	characteristic
62	M₆₄	—	17	4	no	—
63	M₆₄	—	34	2	no	—
64	C₂×C₃₂	—	34	2	no	—
65	#346	—	17	2	no	—
66	C₄×C₁₆	—	17	2	no	—
67	C₂×C₁₇	—	2	1	no	—
68	C₂×C₁₇	—	2	2	no	—
69	C₂×C₁₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
70	C₂×M₁₆	#3624	1	1	yes	characteristic
71	C₂²×C₈	C₂×#595	1	1	yes	characteristic
72	#117	#3624	1	1	yes	characteristic
73	M₃₂	—	34	2	no	—
74	C₂×C₁₆	—	17	2	no	—
75	C₂×C₁₆	—	17	2	no	—
76	C₂×C₁₆	—	17	1	no	—
77	C₂×C₁₆	—	17	1	no	—
78	C₂×C₁₆	—	17	2	no	—
79	#97	#3624	1	2	yes	—
80	C₄×C₈	C₂×#595	1	2	yes	—
81	C₃₂	—	34	4	no	—
82	C₁₇	#13630	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
84	M₁₆	#63216	1	4	yes	—
85	C₂×C₈	#63243	1	2	yes	—
86	C₂×C₈	C₄⋊#595	1	2	yes	—
87	C₂×C₈	C₂×#3624	1	2	yes	—
88	C₂×C₈	C₂×#3624	1	1	yes	characteristic
89	C₂×C₈	C₂²×#595	1	1	yes	characteristic
90	C₄⋊C₄	—	2	1	no	—
91	C₂²⋊C₄	—	2	1	no	—
92	C₄²	C₂×#3624	1	2	yes	—
93	C₁₆	—	17	1	no	—
94	C₁₆	—	17	2	no	—
95	C₁₆	—	17	1	no	—
96	C₁₆	—	34	2	no	—
97	C₂³	#10587538	1	1	yes	characteristic
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	C₂×#63216	1	2	yes	—
100	C₂×C₄	#10587538	1	1	yes	characteristic
101	C₂×C₄	C₄⋊#3624	1	2	yes	—
102	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
103	C₈	C₂×#63216	1	4	yes	—
104	C₈	C₄⋊#3624	1	2	yes	—
105	C₈	C₂×#63243	1	1	yes	characteristic
106	C₈	C₂×C₄⋊#595	1	1	yes	characteristic
107	C₂²	—	4	1	no	—
108	C₂²	C₄⋊#63216	1	2	yes	—
109	C₂²	C₂×#10587538	1	1	yes	characteristic
110	C₄	—	4	2	no	—
111	C₄	C₄⋊#63216	1	2	yes	—
112	C₄	C₂×C₄⋊#3624	1	1	yes	characteristic
113	C₄	C₂²×#63216	1	1	yes	characteristic
114	C₂	—	2	1	no	—
115	C₂	—	2	2	no	—
116	C₂	C₂×C₄⋊#63216	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11763741

C₂

yes

maximal

#14543943

C₂

yes

maximal, complemented

#14530381

C₂

yes

maximal, characteristic

#11763741

—

#11763741

C₂²

yes

—

C₂×#10587467

—

C₂×#10587483

C₄

yes

characteristic

C₂²×#63214

C₂²

yes

characteristic

#11762585

C₄

yes

—

C₄×#63214

C₂²

yes

—

C₁₇×#405

C₄

yes

characteristic

C₂×M₁₆×C₁₇

C₈

yes

characteristic

C₂²×C₈×C₁₇

C₂×C₄

yes

characteristic

C₁₇×#117

C₈

yes

characteristic

C₁₇×#97

C₈

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

#10587483

—

C₂×#63214

C₂×C₄

yes

—

C₂×#63214

C₂×C₄

yes

characteristic

C₂×#63214

D₄

yes

—

C₂×#63214

Q₈

yes

—

C₂×#63214

C₂³

yes

characteristic

#10587467

—

C₂²×C₄×C₁₇

C₂×C₈

yes

characteristic

M₁₆×C₁₇

C₁₆

yes

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

C₂×C₈

yes

characteristic

C₂×C₈×C₁₇

C₂×C₈

yes

—

C₂×C₈×C₁₇

M₁₆

yes

—

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₁₇

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

C₂×C₈

yes

—

#63214

—

#63214

C₄⋊C₄

yes

—

#63214

C₂×D₄

yes

characteristic

#63214

C₂×Q₈

yes

characteristic

#13630

—

maximal, Sylow

C₂³×C₁₇

M₃₂

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₄⋊C₈

yes

—

C₂×C₄×C₁₇

C₂×C₁₆

yes

—

C₂×C₄×C₁₇

M₃₂

yes

characteristic

C₂×C₄×C₁₇

C₂²×C₈

yes

characteristic

C₈×C₁₇

C₄⋊C₈

yes

—

C₈×C₁₇

C₂×C₁₆

yes

—

C₈×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₈×C₁₇

C₂×M₁₆

yes

characteristic

C₂×M₆₄

—

#2108

—

#1422

—

C₂²×C₁₇

—

C₂²×C₁₇

C₄⋊C₁₆

yes

—

C₂²×C₁₇

C₂×M₃₂

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

C₄⋊C₁₆

yes

—

C₄×C₁₇

C₂×C₄⋊C₈

yes

characteristic

C₄×C₁₇

C₂²×C₁₆

yes

characteristic

C₂×M₃₂

—

C₂²×C₁₆

—

#405

#595

yes

characteristic

M₆₄

—

M₆₄

—

C₂×C₃₂

—

#346

—

C₄×C₁₆

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

C₂×C₄⋊C₁₆

yes

C₂×M₁₆

yes

C₂²×C₈

yes

yes

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

#3624

yes

—

C₄×C₈

C₂×#595

yes

—

C₃₂

—

C₁₇

#13630

yes

characteristic, complemented, Sylow

C₂²×C₄

C₂×#3624

yes

characteristic

M₁₆

#63216

yes

—

C₂×C₈

#63243

yes

—

C₂×C₈

C₄⋊#595

yes

—

C₂×C₈

C₂×#3624

yes

—

C₂×C₈

C₂×#3624

yes

characteristic

C₂×C₈

C₂²×#595

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×#3624

yes

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

#10587538

yes

characteristic

C₂×C₄

—

C₂×C₄

C₂×#63216

yes

—

100

C₂×C₄

#10587538

yes

characteristic

101

C₂×C₄

C₄⋊#3624

yes

—

102

C₂×C₄

C₂²×#3624

yes

characteristic

103

C₈

C₂×#63216

yes

—

104

C₈

C₄⋊#3624

yes

—

105

C₈

C₂×#63243

yes

characteristic

106

C₈

C₂×C₄⋊#595

yes

characteristic

107

C₂²

—

108

C₂²

C₄⋊#63216

yes

—

109

C₂²

C₂×#10587538

yes

characteristic

110

C₄

—

111

C₄

C₄⋊#63216

yes

—

112

C₄

C₂×C₄⋊#3624

yes

characteristic

113

C₄

C₂²×#63216

yes

characteristic

114

C₂

—

115

C₂

—

116

C₂

C₂×C₄⋊#63216

yes

characteristic

	subgroup	G/H
center	C₈	C₂×C₄⋊#595
commutator	C₄×C₁₇	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×#595
Fitting	C₁₇×#405	C₄
socle	C₂×C₁₇	C₂×C₄⋊C₁₆
prime radical	C₂³×C₁₇	M₃₂
cosocle	C₂×#63214	C₂³
hypercenter	#405	#595
hypocenter	C₁₇	#13630
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #405

subgroup

G/H

center

C₈

C₂×C₄⋊#595

commutator

C₄×C₁₇

C₂²×C₁₆

Frattini

C₂×C₈

C₂²×#595

Fitting

C₁₇×#405

C₄

socle

C₂×C₁₇

C₂×C₄⋊C₁₆

prime radical

C₂³×C₁₇

M₃₂

C₂³

C₁₇

G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #405

N	H ≅ G/N	type
2. #14543943	C₂	semidirect
82. C₁₇	#13630	semidirect

H ≅ G/N

type

2. #14543943

C₂

semidirect

82. C₁₇

#13630

semidirect

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

subgroup	index
C₂×#11763741	2
#14543943	2
#14530381	2
#13630	17

subgroup

index

Subgroups — #17962823

Conjugacy classes of subgroups

Complements

Maximal subgroups