Group #17962817 — subgroups

Subgroups — #17962817

Order 4352.

206 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 128 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	1	yes	maximal, characteristic
2	C₂²×#10587467	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#10587467	C₂	1	4	yes	maximal, complemented
4	#14530381	C₂	1	1	yes	maximal, characteristic
5	#11763741	—	2	2	no	—
6	C₂×#10587467	—	2	2	no	—
7	C₂×#10587467	C₂²	1	2	yes	—
8	C₂×#10587467	C₂²	1	2	yes	—
9	C₂×#10587483	C₄	1	1	yes	characteristic
10	C₂²×#63214	C₂²	1	1	yes	characteristic
11	#11762585	C₄	1	2	yes	—
12	C₄×#63214	C₂²	1	2	yes	—
13	C₁₇×#405	C₄	1	1	yes	characteristic
14	C₂×M₁₆×C₁₇	C₈	1	1	yes	characteristic
15	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
16	C₁₇×#117	C₈	1	1	yes	characteristic
17	C₁₇×#97	C₈	1	2	yes	—
18	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
19	#10587483	—	2	2	no	—
20	C₂×#63214	C₂×C₄	1	2	yes	—
21	C₂×#63214	C₂×C₄	1	1	yes	characteristic
22	C₂×#63214	D₄	1	2	yes	—
23	C₂×#63214	Q₈	1	2	yes	—
24	C₂×#63214	C₂³	1	1	yes	characteristic
25	#10587467	—	2	2	no	—
26	#10587467	—	2	2	no	—
27	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
28	M₁₆×C₁₇	—	2	2	no	—
29	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
30	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
31	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
33	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
34	C₄⋊C₄×C₁₇	C₁₆	1	2	yes	—
35	C₂²⋊C₄×C₁₇	C₁₆	1	2	yes	—
36	C₄²×C₁₇	C₂×C₈	1	2	yes	—
37	#63214	—	2	2	no	—
38	#63214	C₄⋊C₄	1	2	yes	—
39	#63214	C₂×D₄	1	1	yes	characteristic
40	#63214	C₂×Q₈	1	1	yes	characteristic
41	#13628	—	17	1	no	maximal, Sylow
42	C₂³×C₁₇	C₂×C₁₆	1	1	yes	characteristic
43	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
44	C₂×C₄×C₁₇	C₂×C₁₆	1	4	yes	—
45	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
46	C₂×C₄×C₁₇	M₃₂	1	2	yes	—
47	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
48	C₈×C₁₇	—	2	2	no	—
49	C₈×C₁₇	C₄⋊C₈	1	2	yes	—
50	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₈×C₁₇	C₂×M₁₆	1	1	yes	characteristic
52	C₂×M₆₄	—	17	1	no	—
53	C₂²×C₃₂	—	17	1	no	—
54	#2108	—	17	1	no	—
55	C₄⋊C₃₂	—	17	4	no	—
56	C₂²×C₁₇	—	2	2	no	—
57	C₂²×C₁₇	C₄⋊C₁₆	1	2	yes	—
58	C₂²×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
59	C₄×C₁₇	—	2	4	no	—
60	C₄×C₁₇	C₄⋊C₁₆	1	2	yes	—
61	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
62	C₄×C₁₇	C₂×M₃₂	1	1	yes	characteristic
63	C₂×M₃₂	—	17	1	no	—
64	C₂²×C₁₆	—	17	1	no	—
65	#405	#595	1	1	yes	characteristic
66	M₆₄	—	34	2	no	—
67	C₂×C₃₂	—	17	2	no	—
68	C₂×C₃₂	—	17	2	no	—
69	C₂×C₃₂	—	34	2	no	—
70	#346	—	17	2	no	—
71	C₄×C₁₆	—	17	2	no	—
72	C₂×C₁₇	—	2	2	no	—
73	C₂×C₁₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
74	C₂×C₁₇	#2258	1	2	yes	—
75	C₂×M₁₆	#3624	1	1	yes	characteristic
76	C₂²×C₈	C₂×#595	1	1	yes	characteristic
77	#117	#3624	1	1	yes	characteristic
78	M₃₂	—	34	2	no	—
79	C₂×C₁₆	—	17	2	no	—
80	C₂×C₁₆	—	17	2	no	—
81	C₂×C₁₆	—	17	1	no	—
82	C₂×C₁₆	—	17	1	no	—
83	C₂×C₁₆	—	17	2	no	—
84	#97	#3624	1	2	yes	—
85	C₄×C₈	C₂×#595	1	2	yes	—
86	C₃₂	—	34	2	no	—
87	C₃₂	—	34	2	no	—
88	C₁₇	#13628	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	C₂×C₈	#63243	1	2	yes	—
92	C₂×C₈	C₄⋊#595	1	2	yes	—
93	C₂×C₈	C₂×#3624	1	2	yes	—
94	C₂×C₈	C₂×#3624	1	1	yes	characteristic
95	C₂×C₈	C₂²×#595	1	1	yes	characteristic
96	C₄⋊C₄	#63216	1	2	yes	—
97	C₂²⋊C₄	#63216	1	2	yes	—
98	C₄²	C₂×#3624	1	2	yes	—
99	C₁₆	—	17	1	no	—
100	C₁₆	—	17	2	no	—
101	C₁₆	—	17	1	no	—
102	C₁₆	—	34	2	no	—
103	C₂³	C₂×#63216	1	1	yes	characteristic
104	C₂×C₄	C₂×#63216	1	4	yes	—
105	C₂×C₄	C₂×#63216	1	1	yes	characteristic
106	C₂×C₄	#10587538	1	2	yes	—
107	C₂×C₄	C₄⋊#3624	1	2	yes	—
108	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	C₄⋊#3624	1	2	yes	—
111	C₈	C₂×#63243	1	1	yes	characteristic
112	C₈	C₂×C₄⋊#595	1	1	yes	characteristic
113	C₂²	—	2	2	no	—
114	C₂²	C₄⋊#63216	1	2	yes	—
115	C₂²	C₂²×#63216	1	1	yes	characteristic
116	C₄	—	2	4	no	—
117	C₄	C₄⋊#63216	1	2	yes	—
118	C₄	C₂×C₄⋊#3624	1	1	yes	characteristic
119	C₄	C₂×#10587538	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	C₂×C₄⋊#63216	1	1	yes	characteristic
122	C₂	#14545207	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11763741

C₂

yes

maximal, characteristic

C₂²×#10587467

C₂

yes

maximal, characteristic

C₄⋊#10587467

C₂

yes

maximal, complemented

#14530381

C₂

yes

maximal, characteristic

#11763741

—

C₂×#10587467

—

C₂×#10587467

C₂²

yes

—

C₂×#10587467

C₂²

yes

—

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

—

#10587467

—

C₂²×C₄×C₁₇

C₂×C₈

yes

characteristic

M₁₆×C₁₇

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

C₂×C₈

yes

—

C₂×C₈×C₁₇

C₂×C₈

yes

characteristic

C₂×C₈×C₁₇

M₁₆

yes

—

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₁₇

C₁₆

yes

—

C₂²⋊C₄×C₁₇

C₁₆

yes

—

C₄²×C₁₇

C₂×C₈

yes

—

#63214

—

#63214

C₄⋊C₄

yes

—

#63214

C₂×D₄

yes

characteristic

#63214

C₂×Q₈

yes

characteristic

#13628

—

maximal, Sylow

C₂³×C₁₇

C₂×C₁₆

yes

characteristic

C₂×C₄×C₁₇

C₄⋊C₈

yes

—

C₂×C₄×C₁₇

C₂×C₁₆

yes

—

C₂×C₄×C₁₇

C₂×C₁₆

yes

characteristic

C₂×C₄×C₁₇

M₃₂

yes

—

C₂×C₄×C₁₇

C₂²×C₈

yes

characteristic

C₈×C₁₇

—

C₈×C₁₇

C₄⋊C₈

yes

—

C₈×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₈×C₁₇

C₂×M₁₆

yes

characteristic

C₂×M₆₄

—

C₂²×C₃₂

—

#2108

—

C₄⋊C₃₂

—

C₂²×C₁₇

—

C₂²×C₁₇

C₄⋊C₁₆

yes

—

C₂²×C₁₇

C₂²×C₁₆

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

C₄⋊C₁₆

yes

—

C₄×C₁₇

C₂×C₄⋊C₈

yes

characteristic

C₄×C₁₇

C₂×M₃₂

yes

characteristic

C₂×M₃₂

—

C₂²×C₁₆

—

#405

#595

yes

characteristic

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

#346

—

C₄×C₁₆

—

C₂×C₁₇

—

C₂×C₁₇

C₂×C₄⋊C₁₆

yes

characteristic

C₂×C₁₇

#2258

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

—

C₃₂

—

C₁₇

#13628

yes

characteristic, complemented, Sylow

C₂²×C₄

C₂×#3624

yes

characteristic

M₁₆

—

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

#63216

yes

—

C₂²⋊C₄

#63216

yes

—

C₄²

C₂×#3624

yes

—

C₁₆

—

100

C₁₆

—

101

C₁₆

—

102

C₁₆

—

103

C₂³

C₂×#63216

yes

characteristic

104

C₂×C₄

C₂×#63216

yes

—

105

C₂×C₄

C₂×#63216

yes

characteristic

106

C₂×C₄

#10587538

yes

—

107

C₂×C₄

C₄⋊#3624

yes

—

108

C₂×C₄

C₂²×#3624

yes

characteristic

109

C₈

—

110

C₈

C₄⋊#3624

yes

—

111

C₈

C₂×#63243

yes

characteristic

112

C₈

C₂×C₄⋊#595

yes

characteristic

113

C₂²

—

114

C₂²

C₄⋊#63216

yes

—

115

C₂²

C₂²×#63216

yes

characteristic

116

C₄

—

117

C₄

C₄⋊#63216

yes

—

118

C₄

C₂×C₄⋊#3624

yes

characteristic

119

C₄

C₂×#10587538

yes

characteristic

120

C₂

—

121

C₂

C₂×C₄⋊#63216

yes

characteristic

122

C₂

#14545207

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#595
commutator, socle	C₂²×C₁₇	C₂²×C₁₆
Fitting	C₁₇×#405	C₄
prime radical	C₂³×C₁₇	C₂×C₁₆
cosocle	C₂×#63214	C₂³
hypercenter	#405	#595
hypocenter	C₁₇	#13628
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ #405

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×#595

commutator, socle

C₂²×C₁₇

C₂²×C₁₆

Fitting

C₁₇×#405

C₄

prime radical

C₂³×C₁₇

C₂×C₁₆

C₂³

C₁₇

G ▷ C₂²×C₁₇ ▷ C₁₇

C₁ ◁ C₂×C₈ ◁ #405

N	H ≅ G/N	type
3. C₄⋊#10587467	C₂	semidirect
88. C₁₇	#13628	semidirect

H ≅ G/N

type

3. C₄⋊#10587467

C₂

semidirect

88. C₁₇

#13628

semidirect

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

subgroup	index
C₂×#11763741	2
C₂²×#10587467	2
C₄⋊#10587467	2
#14530381	2
#13628	17

subgroup

index

Subgroups — #17962817

Conjugacy classes of subgroups

Complements

Maximal subgroups