Group #16949059 — subgroups

Subgroups — #16949059

Order 4352.

262 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 103 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	#14531057	C₂	1	4	yes	maximal
2	#14528402	C₂	1	1	yes	maximal, characteristic
3	C₈×#63214	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#1713	C₂	1	1	yes	maximal, characteristic
5	#11763214	—	2	1	no	—
6	#11763214	C₂²	1	2	yes	—
7	#11763189	—	2	1	no	—
8	#11762634	—	2	8	no	—
9	C₄×#63214	C₂²	1	2	yes	—
10	C₁₇×#500	—	2	1	no	—
11	C₁₇×#500	C₂²	1	2	yes	—
12	C₁₇×#494	—	2	1	no	—
13	C₈²×C₁₇	C₂²	1	1	yes	characteristic
14	C₂×Q₁₆×C₁₇	—	4	2	no	—
15	C₁₇×C₄⋊Q₈	—	2	2	no	—
16	C₂×Dic₆₈	—	4	2	no	—
17	#10587555	—	2	2	no	—
18	C₁₇×#107	—	2	4	no	—
19	C₁₇×#107	—	4	2	no	—
20	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
21	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
22	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
23	#10587489	—	2	4	no	—
24	#10587489	—	4	2	no	—
25	C₂×#63214	D₄	1	4	yes	—
26	C₂×Q₈×C₁₇	—	4	2	no	—
27	C₂×Dic₃₄	—	4	2	no	—
28	Q₁₆×C₁₇	—	8	4	no	—
29	C₂×C₈×C₁₇	D₈	1	2	yes	—
30	C₂×C₈×C₁₇	D₈	1	2	yes	—
31	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
32	C₄⋊C₄×C₁₇	—	4	2	no	—
33	C₄⋊C₄×C₁₇	—	4	2	no	—
34	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
35	C₄⋊Dic₁₇	—	4	2	no	—
36	C₄⋊Dic₁₇	—	4	2	no	—
37	Dic₆₈	—	8	4	no	—
38	#63214	Q₁₆	1	8	yes	complemented
39	#12521	—	17	1	no	maximal, Sylow
40	Q₈×C₁₇	—	8	4	no	—
41	C₂×C₄×C₁₇	—	8	2	no	—
42	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
43	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
44	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
45	C₂×Dic₁₇	—	8	2	no	—
46	Dic₃₄	—	8	4	no	—
47	C₈×C₁₇	D₁₆	1	4	yes	—
48	C₈×C₁₇	Q₃₂	1	4	yes	—
49	C₈×C₁₇	C₂×Q₁₆	1	4	yes	—
50	#2252	—	17	4	no	—
51	#1713	—	17	1	no	—
52	#1713	D₁₇	1	1	yes	characteristic
53	C₈×C₁₆	—	17	1	no	—
54	C₂²×C₁₇	#493	1	1	yes	characteristic
55	C₄×C₁₇	—	16	2	no	—
56	C₄×C₁₇	#494	1	2	yes	—
57	C₄×C₁₇	C₂×D₁₆	1	2	yes	—
58	C₄×C₁₇	C₂×Q₃₂	1	2	yes	—
59	Dic₁₇	—	16	2	no	—
60	#500	—	2	1	no	—
61	#500	—	17	2	no	—
62	#500	—	34	1	no	—
63	#500	C₂×D₁₇	1	2	yes	—
64	#494	—	2	1	no	—
65	#494	—	34	1	no	—
66	#366	—	34	8	no	—
67	C₄×C₁₆	—	17	2	no	—
68	C₈²	C₂×D₁₇	1	1	yes	characteristic
69	C₂×C₁₇	#1713	1	1	yes	characteristic
70	C₂×C₁₇	#2246	1	1	yes	characteristic
71	C₂×C₁₇	#2247	1	1	yes	characteristic
72	C₂×Q₁₆	—	4	2	no	—
73	C₂×Q₁₆	—	68	2	no	—
74	C₄⋊Q₈	—	2	2	no	—
75	C₄⋊Q₈	—	34	2	no	—
76	C₂×C₁₆	—	17	4	no	—
77	#107	—	2	4	no	—
78	#107	—	4	2	no	—
79	#107	—	34	4	no	—
80	#107	—	68	2	no	—
81	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
82	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
83	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
84	C₁₇	#12521	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₈	—	4	2	no	—
86	C₂×Q₈	—	68	2	no	—
87	Q₁₆	—	8	4	no	—
88	Q₁₆	—	136	4	no	—
89	C₂×C₈	#63228	1	2	yes	—
90	C₂×C₈	#63228	1	2	yes	—
91	C₂×C₈	D₄×D₁₇	1	2	yes	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	68	2	no	—
95	C₄⋊C₄	—	68	2	no	—
96	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
97	C₁₆	—	17	8	no	—
98	Q₈	—	8	4	no	—
99	Q₈	—	136	4	no	—
100	C₂×C₄	—	8	2	no	—
101	C₂×C₄	—	136	2	no	—
102	C₂×C₄	C₂×#63228	1	1	yes	characteristic
103	C₂×C₄	C₂×#63228	1	1	yes	characteristic
104	C₂×C₄	#10587626	1	1	yes	characteristic
105	C₈	#10587497	1	4	yes	—
106	C₈	#10587500	1	4	yes	—
107	C₈	Q₁₆×D₁₇	1	4	yes	—
108	C₂²	#11763185	1	1	yes	characteristic
109	C₄	—	16	2	no	—
110	C₄	—	272	2	no	—
111	C₄	#11763187	1	2	yes	—
112	C₄	C₂×#10587497	1	2	yes	—
113	C₄	C₂×#10587500	1	2	yes	—
114	C₂	#14528400	1	1	yes	characteristic
115	C₂	#14531033	1	1	yes	characteristic
116	C₂	#14531036	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14531057

C₂

yes

maximal

#14528402

C₂

yes

maximal, characteristic

C₈×#63214

C₂

yes

maximal, characteristic

C₁₇×#1713

C₂

yes

maximal, characteristic

#11763214

—

#11763214

C₂²

yes

—

#11763189

—

#11762634

—

C₄×#63214

C₂²

yes

—

C₁₇×#500

—

C₁₇×#500

C₂²

yes

—

C₁₇×#494

—

C₈²×C₁₇

C₂²

yes

characteristic

C₂×Q₁₆×C₁₇

—

C₁₇×C₄⋊Q₈

—

C₂×Dic₆₈

—

#10587555

—

C₁₇×#107

—

C₁₇×#107

—

C₄×C₈×C₁₇

D₄

yes

characteristic

C₄×C₈×C₁₇

D₄

yes

characteristic

C₄×C₈×C₁₇

C₂³

yes

—

—

D₄

yes

—

C₂×Q₈×C₁₇

—

C₂×Dic₃₄

—

Q₁₆×C₁₇

—

C₂×C₈×C₁₇

D₈

yes

—

C₂×C₈×C₁₇

D₈

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

Dic₆₈

—

#63214

Q₁₆

yes

complemented

#12521

—

maximal, Sylow

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#127

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×Dic₁₇

—

Dic₃₄

—

C₈×C₁₇

D₁₆

yes

—

C₈×C₁₇

Q₃₂

yes

—

C₈×C₁₇

C₂×Q₁₆

yes

—

#2252

—

#1713

—

#1713

D₁₇

yes

characteristic

C₈×C₁₆

—

C₂²×C₁₇

#493

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

#494

yes

—

C₄×C₁₇

C₂×D₁₆

yes

—

C₄×C₁₇

C₂×Q₃₂

yes

—

Dic₁₇

—

—

—

—

C₂×D₁₇

yes

—

#494

—

#494

—

#366

—

C₄×C₁₆

—

C₈²

C₂×D₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

—

—

—

—

—

C₄×C₈

C₂²⋊D₁₇

yes

characteristic

C₄×C₈

C₂²⋊D₁₇

yes

characteristic

C₄×C₈

C₂²×D₁₇

yes

characteristic

C₁₇

#12521

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

136

—

C₂×C₈

#63228

yes

—

C₂×C₈

#63228

yes

—

C₂×C₈

D₄×D₁₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₂²⋊D₁₇

yes

characteristic

C₁₆

—

Q₈

—

Q₈

—

136

—

100

C₂×C₄

—

101

C₂×C₄

—

136

—

102

C₂×C₄

C₂×#63228

yes

characteristic

103

C₂×C₄

C₂×#63228

yes

characteristic

104

C₂×C₄

#10587626

yes

characteristic

105

C₈

#10587497

yes

—

106

C₈

#10587500

yes

—

107

C₈

Q₁₆×D₁₇

yes

—

108

C₂²

#11763185

yes

characteristic

109

C₄

—

110

C₄

—

272

—

111

C₄

#11763187

yes

—

112

C₄

C₂×#10587497

yes

—

113

C₄

C₂×#10587500

yes

—

114

C₂

#14528400

yes

characteristic

115

C₂

#14531033

yes

characteristic

116

C₂

#14531036

yes

characteristic

	subgroup	G/H
center	C₂²	#11763185
commutator, cosocle	C₄×C₈×C₁₇	C₂³
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#1713	C₂
socle, prime radical	C₂²×C₁₇	#493
hypercenter	#1713	D₁₇
hypocenter	C₁₇	#12521
LCS	G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

subgroup

G/H

center

C₂²

#11763185

commutator, cosocle

C₄×C₈×C₁₇

C₂³

Frattini

C₄×C₈

C₂²×D₁₇

Fitting

C₁₇×#1713

C₂

C₂²×C₁₇

D₁₇

C₁₇

G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

N	H ≅ G/N	type
38. #63214	Q₁₆	semidirect
84. C₁₇	#12521	semidirect

H ≅ G/N

type

38. #63214

Q₁₆

semidirect

84. C₁₇

#12521

semidirect

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

subgroup	index
#14531057	2
#14528402	2
C₈×#63214	2
C₁₇×#1713	2
#12521	17

subgroup

index

C₁₇×#1713

Subgroups — #16949059

Conjugacy classes of subgroups

Complements

Maximal subgroups