Group #16939006 — subgroups

Subgroups — #16939006

Order 4352.

328 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 79 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	#14527575	C₂	1	3	yes	maximal, complemented
2	#14527453	C₂	1	3	yes	maximal
3	C₁₇×#1309	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#10587560	C₂²	1	1	yes	characteristic
5	#11762862	C₂²	1	3	yes	—
6	#11762561	—	2	3	no	—
7	#11762559	—	2	3	no	—
8	#11762552	—	2	3	no	—
9	#11762552	—	2	3	no	—
10	C₁₇×#423	C₂²	1	3	yes	—
11	C₂²×D₆₈	—	2	3	no	—
12	C₂×M₁₆×C₁₇	—	2	3	no	—
13	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
14	#10587560	—	2	3	no	—
15	#10587560	Q₈	1	2	yes	—
16	C₁₇×C₄⋊C₈	—	2	3	no	—
17	C₁₇×C₄⋊C₈	—	2	3	no	—
18	C₂²⋊#3622	—	2	3	no	—
19	C₄⋊#3622	D₄	1	3	yes	—
20	C₄⋊#3622	D₄	1	3	yes	—
21	C₄×#3622	—	2	3	no	—
22	C₂³×D₁₇	—	4	1	no	—
23	C₂²×C₄×C₁₇	#41	1	3	yes	—
24	C₂×D₆₈	—	2	6	no	—
25	C₂×D₆₈	—	4	3	no	—
26	C₂×D₆₈	—	8	3	no	—
27	C₂×D₆₈	—	8	3	no	—
28	M₁₆×C₁₇	—	8	3	no	—
29	C₂×C₈×C₁₇	—	4	3	no	—
30	C₄²×C₁₇	C₂×D₄	1	3	yes	—
31	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
32	C₂×#3622	—	2	3	no	—
33	C₂×#3622	—	2	3	no	—
34	#10611	—	17	1	no	maximal, Sylow
35	C₂³×C₁₇	#126	1	1	yes	characteristic
36	C₂²×D₁₇	—	4	2	no	—
37	C₂²×D₁₇	—	8	6	no	—
38	C₂×C₄×C₁₇	—	2	3	no	—
39	C₂×C₄×C₁₇	—	2	3	no	—
40	C₂×C₄×C₁₇	#121	1	3	yes	—
41	C₂×C₄×C₁₇	C₂²⋊Q₈	1	3	yes	—
42	D₆₈	—	8	6	no	—
43	D₆₈	—	8	3	no	—
44	D₆₈	—	8	3	no	—
45	C₈×C₁₇	—	8	3	no	—
46	#3622	—	4	3	no	—
47	#3622	—	4	3	no	—
48	#1574	—	17	3	no	—
49	#1545	—	17	3	no	—
50	#1309	D₁₇	1	1	yes	characteristic, complemented
51	C₂²×C₁₇	—	2	3	no	—
52	C₂²×C₁₇	#396	1	1	yes	characteristic
53	C₂×D₁₇	—	8	6	no	—
54	C₂×D₁₇	—	16	4	no	—
55	C₄×C₁₇	—	2	3	no	—
56	C₄×C₁₇	#471	1	3	yes	—
57	C₄×C₁₇	#472	1	3	yes	—
58	C₂×#127	—	17	1	no	—
59	#431	—	17	3	no	—
60	#423	C₂×D₁₇	1	3	yes	—
61	#335	—	34	3	no	—
62	#334	—	34	3	no	—
63	#331	—	34	3	no	—
64	#331	—	34	3	no	—
65	C₂×C₁₇	—	4	1	no	—
66	C₂×C₁₇	#2063	1	3	yes	—
67	D₁₇	—	16	2	no	—
68	C₂²×D₄	—	34	3	no	—
69	C₂×M₁₆	—	2	3	no	—
70	#127	—	17	2	no	—
71	#127	—	34	3	no	—
72	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
73	C₄⋊C₈	—	2	3	no	—
74	C₄⋊C₈	—	2	3	no	—
75	C₄⋊C₈	—	17	3	no	—
76	C₄⋊C₈	—	17	3	no	—
77	C₂²⋊C₈	—	34	3	no	—
78	C₄×C₈	—	34	3	no	—
79	C₁₇	#10611	1	1	yes	characteristic, complemented, Sylow
80	C₂⁴	—	68	1	no	—
81	C₂×D₄	—	34	6	no	—
82	C₂×D₄	—	68	3	no	—
83	C₂×D₄	—	136	3	no	—
84	C₂×D₄	—	136	3	no	—
85	C₂²×C₄	#63252	1	3	yes	—
86	M₁₆	—	8	3	no	—
87	C₂×C₈	—	4	3	no	—
88	C₂×C₈	—	34	3	no	—
89	C₂×C₈	—	34	3	no	—
90	C₄²	D₄×D₁₇	1	3	yes	—
91	C₄²	Q₈×D₁₇	1	1	yes	characteristic
92	C₂³	—	68	2	no	—
93	C₂³	—	136	6	no	—
94	C₂³	#10587562	1	1	yes	characteristic
95	D₄	—	136	6	no	—
96	D₄	—	136	3	no	—
97	D₄	—	136	3	no	—
98	C₂×C₄	—	2	3	no	—
99	C₂×C₄	—	2	3	no	—
100	C₂×C₄	#10587570	1	3	yes	—
101	C₂×C₄	#10587582	1	3	yes	—
102	C₈	—	8	3	no	—
103	C₈	—	68	3	no	—
104	C₈	—	68	3	no	—
105	C₂²	—	2	3	no	—
106	C₂²	—	272	4	no	—
107	C₂²	—	136	6	no	—
108	C₂²	#11762733	1	1	yes	characteristic
109	C₄	—	2	3	no	—
110	C₄	#11763075	1	3	yes	—
111	C₄	#11763078	1	3	yes	—
112	C₂	—	4	1	no	—
113	C₂	—	272	2	no	—
114	C₂	#14530145	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14527575

C₂

yes

maximal, complemented

#14527453

C₂

yes

maximal

C₁₇×#1309

C₂

yes

maximal, characteristic, complemented

C₂×#10587560

C₂²

yes

characteristic

#11762862

C₂²

yes

—

—

—

—

—

C₁₇×#423

C₂²

yes

—

C₂²×D₆₈

—

C₂×M₁₆×C₁₇

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

#10587560

—

#10587560

Q₈

yes

—

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₂²⋊#3622

—

C₄⋊#3622

D₄

yes

—

C₄⋊#3622

D₄

yes

—

C₄×#3622

—

C₂³×D₁₇

—

C₂²×C₄×C₁₇

#41

yes

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₄²×C₁₇

C₂×D₄

yes

—

C₄²×C₁₇

C₂×Q₈

yes

—

—

—

C₂³×C₁₇

yes

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#121

yes

—

C₂×C₄×C₁₇

C₂²⋊Q₈

yes

—

D₆₈

—

D₆₈

—

D₆₈

—

C₈×C₁₇

—

—

—

—

—

D₁₇

yes

characteristic, complemented

C₂²×C₁₇

—

C₂²×C₁₇

#396

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#471

yes

—

C₄×C₁₇

#472

yes

—

C₂×#127

—

#431

—

#423

C₂×D₁₇

yes

—

—

—

—

—

C₂×C₁₇

—

C₂×C₁₇

#2063

yes

—

D₁₇

—

C₂²×D₄

—

C₂×M₁₆

—

#127

—

#127

—

C₂×C₄²

C₂²×D₁₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₁₇

#10611

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

136

—

C₂×D₄

—

136

—

C₂²×C₄

#63252

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

D₄×D₁₇

yes

—

C₄²

Q₈×D₁₇

yes

characteristic

C₂³

—

C₂³

—

136

—

C₂³

#10587562

yes

characteristic

D₄

—

136

—

D₄

—

136

—

D₄

—

136

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

#10587570

yes

—

101

C₂×C₄

#10587582

yes

—

102

C₈

—

103

C₈

—

104

C₈

—

105

C₂²

—

106

C₂²

—

272

—

107

C₂²

—

136

—

108

C₂²

#11762733

yes

characteristic

109

C₄

—

110

C₄

#11763075

yes

—

111

C₄

#11763078

yes

—

112

C₂

—

113

C₂

—

272

—

114

C₂

#14530145

yes

—

	subgroup	G/H
center	C₂²	#11762733
commutator, cosocle	C₂×C₄²×C₁₇	C₂³
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1309	C₂
socle	C₂²×C₁₇	#396
prime radical	C₂×#10587560	C₂²
hypercenter	#1309	D₁₇
hypocenter	C₁₇	#10611
LCS	G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

subgroup

G/H

center

C₂²

#11762733

commutator, cosocle

C₂×C₄²×C₁₇

C₂³

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#1309

C₂

C₂²×C₁₇

C₂²

D₁₇

C₁₇

G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

N	H ≅ G/N	type
1. #14527575	C₂	semidirect
3. C₁₇×#1309	C₂	semidirect
50. #1309	D₁₇	semidirect
79. C₁₇	#10611	semidirect

H ≅ G/N

type

1. #14527575

C₂

semidirect

3. C₁₇×#1309

C₂

semidirect

50. #1309

D₁₇

semidirect

79. C₁₇

#10611

semidirect

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

subgroup	index
#14527575	2
#14527453	2
C₁₇×#1309	2
#10611	17

subgroup

index

#14527575

#14527453

C₁₇×#1309

#10611

Subgroups — #16939006

Conjugacy classes of subgroups

Complements

Maximal subgroups