Group #16928012 — subgroups

Subgroups — #16928012

Order 4352.

262 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 107 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	#14528922	C₂	1	1	yes	maximal, characteristic
2	#14527067	C₂	1	1	yes	maximal, characteristic
3	#14526642	C₂	1	4	yes	maximal
4	C₁₇×#1835	C₂	1	1	yes	maximal, characteristic
5	#11762862	C₂²	1	2	yes	—
6	#11762833	C₂²	1	2	yes	—
7	#11762574	—	2	2	no	—
8	#11762530	C₄	1	2	yes	complemented
9	C₈×#3622	C₄	1	2	yes	complemented
10	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
11	C₁₇×#382	C₂²	1	2	yes	—
12	C₁₇×#340	—	2	2	no	—
13	C₂²×C₈×C₁₇	D₄	1	2	yes	—
14	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
15	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
16	C₂×#63223	—	2	2	no	—
17	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₄×C₈×C₁₇	Q₈	1	2	yes	—
20	C₁₇×#95	—	2	4	no	—
21	C₂²⋊#3622	—	2	2	no	—
22	C₄⋊#3622	—	2	2	no	—
23	C₄⋊#3622	—	2	4	no	—
24	C₄×#3622	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₁₇	—	4	2	no	—
26	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₇	#41	1	2	yes	—
28	C₂×C₈×C₁₇	—	2	4	no	—
29	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
30	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
31	C₄⋊C₄×C₁₇	—	4	4	no	—
32	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
33	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
34	C₄²×C₁₇	#41	1	2	yes	—
35	#63223	—	4	4	no	—
36	C₂×#3622	—	2	4	no	—
37	C₂×#3622	—	4	2	no	—
38	#8792	—	17	1	no	maximal, Sylow
39	C₂³×C₁₇	#124	1	1	yes	characteristic
40	C₂×C₄×C₁₇	—	4	2	no	—
41	C₂×C₄×C₁₇	—	8	4	no	—
42	C₂×C₄×C₁₇	—	8	2	no	—
43	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₇	#117	1	2	yes	—
45	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
46	C₂×C₄×C₁₇	#125	1	2	yes	—
47	C₂×C₄×C₁₇	#135	1	4	yes	—
48	C₈×C₁₇	—	2	2	no	—
49	C₈×C₁₇	—	2	2	no	—
50	#3622	—	4	4	no	—
51	#3622	—	8	2	no	—
52	#1837	—	17	1	no	—
53	#1835	D₁₇	1	1	yes	characteristic
54	#1450	—	17	1	no	—
55	#1286	—	17	4	no	—
56	C₂²×C₁₇	—	2	2	no	—
57	C₂²×C₁₇	#382	1	1	yes	characteristic
58	C₂²×C₁₇	#488	1	2	yes	—
59	C₄×C₁₇	—	2	2	no	—
60	C₄×C₁₇	—	2	2	no	—
61	C₄×C₁₇	—	16	2	no	—
62	C₄×C₁₇	#427	1	2	yes	—
63	C₄×C₁₇	#488	1	2	yes	—
64	#431	—	17	2	no	—
65	#423	—	17	2	no	—
66	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
67	#382	C₂×D₁₇	1	2	yes	—
68	#341	—	34	2	no	—
69	#340	—	2	2	no	—
70	#322	—	17	2	no	—
71	C₈²	—	17	2	no	—
72	C₂×C₁₇	—	2	2	no	—
73	C₂×C₁₇	#1833	1	2	yes	—
74	C₂×C₁₇	#1835	1	1	yes	characteristic
75	C₂×M₁₆	—	34	2	no	—
76	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
77	C₂×C₄⋊C₄	—	2	2	no	—
78	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
79	C₄⋊C₈	—	34	2	no	—
80	C₄⋊C₈	—	34	4	no	—
81	C₂²⋊C₈	—	34	2	no	—
82	C₄×C₈	—	17	4	no	—
83	C₄×C₈	Dic₃₄	1	2	yes	—
84	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
85	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
86	#95	—	2	4	no	—
87	C₁₇	#8792	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₄	—	4	2	no	—
89	C₂²×C₄	#63254	1	2	yes	—
90	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
91	M₁₆	—	68	4	no	—
92	C₂×C₈	—	2	4	no	—
93	C₂×C₈	—	34	4	no	—
94	C₂×C₈	—	68	2	no	—
95	C₂×C₈	#63225	1	2	yes	—
96	C₂×C₈	#63225	1	2	yes	—
97	C₄⋊C₄	—	4	4	no	—
98	C₄²	C₂×Dic₃₄	1	1	yes	characteristic
99	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
100	C₄²	#63256	1	2	yes	—
101	C₂³	#10587622	1	1	yes	characteristic
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	8	4	no	—
104	C₂×C₄	—	8	2	no	—
105	C₂×C₄	#10587576	1	2	yes	—
106	C₂×C₄	#10587578	1	2	yes	—
107	C₂×C₄	C₂×#63225	1	1	yes	characteristic
108	C₂×C₄	#10587633	1	1	yes	characteristic
109	C₂×C₄	#10587636	1	4	yes	—
110	C₈	—	2	2	no	—
111	C₈	—	2	2	no	—
112	C₈	—	68	4	no	—
113	C₈	—	136	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	#11762674	1	1	yes	characteristic
116	C₂²	#11763163	1	2	yes	—
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	16	2	no	—
120	C₄	#11762845	1	2	yes	—
121	C₄	#11763166	1	2	yes	—
122	C₂	—	2	2	no	—
123	C₂	#14528907	1	2	yes	—
124	C₂	#14528914	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14528922

C₂

yes

maximal, characteristic

#14527067

C₂

yes

maximal, characteristic

#14526642

C₂

yes

maximal

C₁₇×#1835

C₂

yes

maximal, characteristic

#11762862

C₂²

yes

—

#11762833

C₂²

yes

—

#11762574

—

#11762530

C₄

yes

complemented

C₈×#3622

C₄

yes

complemented

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₁₇×#382

C₂²

yes

—

C₁₇×#340

—

C₂²×C₈×C₁₇

D₄

yes

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂×#63223

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

Q₈

yes

—

C₁₇×#95

—

C₂²⋊#3622

—

C₄⋊#3622

—

C₄⋊#3622

—

C₄×#3622

C₂×C₄

yes

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

#41

yes

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₂×Q₈

yes

characteristic

C₄²×C₁₇

#41

yes

—

—

—

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

#117

yes

—

C₂×C₄×C₁₇

#124

yes

characteristic

C₂×C₄×C₁₇

#125

yes

—

C₂×C₄×C₁₇

#135

yes

—

C₈×C₁₇

—

C₈×C₁₇

—

—

—

—

D₁₇

yes

characteristic

#1450

—

#1286

—

C₂²×C₁₇

—

C₂²×C₁₇

#382

yes

characteristic

C₂²×C₁₇

#488

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#427

yes

—

C₄×C₁₇

#488

yes

—

#431

—

#423

—

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

#382

C₂×D₁₇

yes

—

#341

—

#340

—

#322

—

C₈²

—

C₂×C₁₇

—

C₂×C₁₇

#1833

yes

—

C₂×C₁₇

#1835

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₁₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₁₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

Dic₃₄

yes

—

C₄×C₈

C₄×D₁₇

yes

characteristic

C₄×C₈

C₄×D₁₇

yes

characteristic

#95

—

C₁₇

#8792

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

#63254

yes

—

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#63225

yes

—

C₂×C₈

#63225

yes

—

C₄⋊C₄

—

C₄²

C₂×Dic₃₄

yes

characteristic

C₄²

C₂×C₄×D₁₇

yes

characteristic

100

C₄²

#63256

yes

—

101

C₂³

#10587622

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

#10587576

yes

—

106

C₂×C₄

#10587578

yes

—

107

C₂×C₄

C₂×#63225

yes

characteristic

108

C₂×C₄

#10587633

yes

characteristic

109

C₂×C₄

#10587636

yes

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

136

—

114

C₂²

—

115

C₂²

#11762674

yes

characteristic

116

C₂²

#11763163

yes

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

#11762845

yes

—

121

C₄

#11763166

yes

—

122

C₂

—

123

C₂

#14528907

yes

—

124

C₂

#14528914

yes

characteristic

	subgroup	G/H
center	C₂²	#11762674
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1835	C₂
socle	C₂²×C₁₇	#382
prime radical	C₂³×C₁₇	#124
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1835	D₁₇
hypocenter	C₁₇	#8792
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1835

subgroup

G/H

center

C₂²

#11762674

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#1835

C₂

C₂²×C₁₇

C₂³×C₁₇

C₂×C₄²×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1835

N	H ≅ G/N	type
8. #11762530	C₄	semidirect
9. C₈×#3622	C₄	semidirect
87. C₁₇	#8792	semidirect

H ≅ G/N

type

8. #11762530

C₄

semidirect

9. C₈×#3622

C₄

semidirect

87. C₁₇

#8792

semidirect

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

subgroup	index
#14528922	2
#14527067	2
#14526642	2
C₁₇×#1835	2
#8792	17

subgroup

index

C₁₇×#1835

Subgroups — #16928012

Conjugacy classes of subgroups

Complements

Maximal subgroups