Group #14534730 — subgroups

Subgroups — #14534730

Order 2176.

584 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 149 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×#10587560	C₂	1	1	yes	maximal, characteristic
2	#11762726	C₂	1	8	yes	maximal, complemented
3	#11762710	C₂	1	4	yes	maximal, complemented
4	#11762655	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₇×#125	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₆₈	C₂²	1	2	yes	—
7	C₂²×D₆₈	C₂²	1	4	yes	—
8	C₁₇×#125	—	2	4	no	—
9	C₂×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
10	C₂×C₄⋊C₄×C₁₇	C₂²	1	4	yes	—
11	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
12	C₂×#63227	C₂²	1	16	yes	—
13	C₂×C₄×Dic₁₇	C₂²	1	2	yes	—
14	#10587560	—	2	4	no	—
15	#10587502	C₂²	1	4	yes	—
16	C₂³×D₁₇	C₂³	1	4	yes	—
17	C₂²×C₄×C₁₇	C₂³	1	4	yes	—
18	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
19	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
20	C₂²×Dic₁₇	C₂³	1	4	yes	—
21	C₂×D₆₈	—	2	8	no	—
22	C₂×D₆₈	—	2	4	no	—
23	C₂×D₆₈	—	2	8	no	—
24	C₂×D₆₈	—	4	8	no	—
25	C₄⋊C₄×C₁₇	—	2	8	no	—
26	C₄⋊C₄×C₁₇	—	2	4	no	—
27	C₄²×C₁₇	D₄	1	4	yes	—
28	#63227	—	4	16	no	—
29	C₄×Dic₁₇	—	2	4	no	—
30	C₂³×C₁₇	C₂⁴	1	1	yes	characteristic
31	C₂²×D₁₇	—	2	4	no	—
32	C₂²×D₁₇	—	2	8	no	—
33	C₂²×D₁₇	—	2	16	no	—
34	C₂×C₄×C₁₇	—	2	4	no	—
35	C₂×C₄×C₁₇	—	2	4	no	—
36	C₂×C₄×C₁₇	—	2	8	no	—
37	C₂×C₄×C₁₇	C₂×D₄	1	4	yes	—
38	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₇	#41	1	4	yes	—
40	C₂×Dic₁₇	—	2	4	no	—
41	C₂×Dic₁₇	—	2	8	no	—
42	D₆₈	—	4	16	no	—
43	D₆₈	—	4	8	no	—
44	#2641	—	17	1	no	maximal, Sylow
45	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
46	C₂²×C₁₇	C₂×#41	1	2	yes	—
47	C₂²×C₁₇	H(5,𝔽₂)	1	4	yes	—
48	C₂×D₁₇	—	4	4	no	—
49	C₂×D₁₇	—	4	8	no	—
50	C₂×D₁₇	—	4	16	no	—
51	C₄×C₁₇	—	2	4	no	—
52	C₄×C₁₇	—	4	4	no	—
53	C₄×C₁₇	#124	1	4	yes	—
54	Dic₁₇	—	4	4	no	—
55	C₂×#127	—	17	1	no	—
56	C₂×#125	D₁₇	1	1	yes	characteristic, complemented
57	#394	—	17	8	no	—
58	#390	—	17	4	no	—
59	#376	—	17	1	no	—
60	C₂×C₁₇	C₂×#124	1	1	yes	characteristic
61	C₂×C₁₇	#535	1	2	yes	—
62	C₂×C₁₇	#540	1	4	yes	—
63	D₁₇	—	8	4	no	—
64	C₂²×D₄	—	17	2	no	—
65	C₂²×D₄	—	17	4	no	—
66	#127	—	34	4	no	—
67	#125	—	2	4	no	—
68	C₂×C₄⋊C₄	C₂×D₁₇	1	2	yes	—
69	C₂×C₄⋊C₄	C₂×D₁₇	1	4	yes	—
70	C₂×C₂²⋊C₄	—	17	16	no	—
71	C₂×C₄²	—	17	2	no	—
72	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
73	#95	—	17	4	no	—
74	C₁₇	#2641	1	1	yes	characteristic, complemented, Sylow
75	C₂⁴	—	17	4	no	—
76	C₂×D₄	—	34	8	no	—
77	C₂×D₄	—	34	8	no	—
78	C₂×D₄	—	34	4	no	—
79	C₂×D₄	—	68	8	no	—
80	C₂²×C₄	—	17	4	no	—
81	C₂²×C₄	C₂²×D₁₇	1	4	yes	—
82	C₂²×C₄	C₂²×D₁₇	1	2	yes	—
83	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
84	C₄⋊C₄	—	2	8	no	—
85	C₄⋊C₄	—	2	4	no	—
86	C₂²⋊C₄	—	68	16	no	—
87	C₄²	—	34	4	no	—
88	C₄²	C₂²⋊D₁₇	1	4	yes	—
89	C₂³	—	34	8	no	—
90	C₂³	—	34	16	no	—
91	C₂³	—	34	4	no	—
92	C₂³	C₂³×D₁₇	1	1	yes	characteristic
93	D₄	—	68	16	no	—
94	D₄	—	68	8	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	8	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	34	4	no	—
99	C₂×C₄	—	34	8	no	—
100	C₂×C₄	#63256	1	4	yes	—
101	C₂×C₄	C₂×C₂²⋊D₁₇	1	4	yes	—
102	C₂×C₄	C₂×C₂²⋊D₁₇	1	2	yes	—
103	C₂²	—	68	4	no	—
104	C₂²	—	68	8	no	—
105	C₂²	—	68	16	no	—
106	C₂²	C₂×#63256	1	2	yes	—
107	C₂²	#10587696	1	4	yes	—
108	C₂²	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
109	C₄	—	2	4	no	—
110	C₄	—	4	4	no	—
111	C₄	—	68	4	no	—
112	C₄	#10587633	1	4	yes	—
113	C₂	—	136	4	no	—
114	C₂	C₂×#10587633	1	1	yes	characteristic
115	C₂	#11763373	1	2	yes	—
116	C₂	#11763415	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587560

C₂

yes

maximal, characteristic

#11762726

C₂

yes

maximal, complemented

#11762710

C₂

yes

maximal, complemented

#11762655

C₂

yes

maximal, characteristic, complemented

C₂×C₁₇×#125

C₂

yes

maximal, characteristic, complemented

C₂²×D₆₈

C₂²

yes

—

C₂²×D₆₈

C₂²

yes

—

C₁₇×#125

—

C₂×C₄⋊C₄×C₁₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₇

C₂²

yes

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×#63227

C₂²

yes

—

C₂×C₄×Dic₁₇

C₂²

yes

—

#10587560

—

#10587502

C₂²

yes

—

C₂³×D₁₇

C₂³

yes

—

C₂²×C₄×C₁₇

C₂³

yes

—

C₂²×C₄×C₁₇

C₂³

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

C₂³

yes

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

D₄

yes

—

#63227

—

C₄×Dic₁₇

—

C₂³×C₁₇

C₂⁴

yes

characteristic

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

#41

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

D₆₈

—

D₆₈

—

#2641

—

maximal, Sylow

C₂²×C₁₇

C₂²×D₄

yes

characteristic

C₂²×C₁₇

C₂×#41

yes

—

C₂²×C₁₇

H(5,𝔽₂)

yes

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#124

yes

—

Dic₁₇

—

C₂×#127

—

C₂×#125

D₁₇

yes

characteristic, complemented

#394

—

#390

—

#376

—

C₂×C₁₇

C₂×#124

yes

characteristic

C₂×C₁₇

#535

yes

—

C₂×C₁₇

#540

yes

—

D₁₇

—

C₂²×D₄

—

C₂²×D₄

—

#127

—

#125

—

C₂×C₄⋊C₄

C₂×D₁₇

yes

—

C₂×C₄⋊C₄

C₂×D₁₇

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₇

yes

characteristic

#95

—

C₁₇

#2641

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₇

yes

—

C₂²×C₄

C₂²×D₁₇

yes

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²⋊D₁₇

yes

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂³×D₁₇

yes

characteristic

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

#63256

yes

—

101

C₂×C₄

C₂×C₂²⋊D₁₇

yes

—

102

C₂×C₄

C₂×C₂²⋊D₁₇

yes

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

C₂×#63256

yes

—

107

C₂²

#10587696

yes

—

108

C₂²

C₂²×C₂²⋊D₁₇

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

#10587633

yes

—

113

C₂

—

136

—

114

C₂

C₂×#10587633

yes

characteristic

115

C₂

#11763373

yes

—

116

C₂

#11763415

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₇
commutator, socle, cosocle	C₂³×C₁₇	C₂⁴
Fitting	C₂×C₁₇×#125	C₂
prime radical	C₂×#10587560	C₂
hypercenter	C₂×#125	D₁₇
hypocenter	C₁₇	#2641
LCS	G ▷ C₂³×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₇

commutator, socle, cosocle

C₂³×C₁₇

C₂⁴

Fitting

C₂×C₁₇×#125

C₂

C₂

D₁₇

C₁₇

G ▷ C₂³×C₁₇ ▷ C₁₇

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
2. #11762726	C₂	semidirect
3. #11762710	C₂	semidirect
4. #11762655	C₂	semidirect
5. C₂×C₁₇×#125	C₂	semidirect
56. C₂×#125	D₁₇	semidirect
74. C₁₇	#2641	semidirect

H ≅ G/N

type

2. #11762726

C₂

semidirect

3. #11762710

C₂

semidirect

4. #11762655

C₂

semidirect

5. C₂×C₁₇×#125

C₂

semidirect

56. C₂×#125

D₁₇

semidirect

74. C₁₇

#2641

semidirect

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

subgroup	index
C₂×#10587560	2
#11762726	2
#11762710	2
#11762655	2
C₂×C₁₇×#125	2
#2641	17

subgroup

index

C₂×C₁₇×#125

Subgroups — #14534730

Conjugacy classes of subgroups

Complements

Maximal subgroups