Group #14545976 — subgroups

Subgroups — #14545976

Order 2176.

160 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 71 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×#10587666	C₂	1	1	yes	maximal, characteristic
2	C₂×#10587665	C₂	1	1	yes	maximal, characteristic
3	#11763912	C₂	1	2	yes	maximal, complemented
4	C₄⋊#63241	C₂	1	2	yes	maximal, complemented
5	#11763774	C₂	1	1	yes	maximal, characteristic
6	C₂×#63243	C₄	1	1	yes	characteristic
7	C₂×#63242	C₂²	1	1	yes	characteristic
8	C₂×#63241	C₂²	1	1	yes	characteristic
9	C₂×#63241	C₂²	1	2	yes	—
10	C₂×#63241	C₂²	1	1	yes	characteristic
11	#10587666	—	2	2	no	—
12	#10587665	—	2	2	no	—
13	#10587578	C₄	1	1	yes	characteristic
14	#10587542	C₄	1	2	yes	—
15	C₄×#3624	C₂²	1	2	yes	—
16	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂×#3624	C₂×C₄	1	2	yes	—
18	C₂×#3624	C₂×C₄	1	1	yes	characteristic
19	C₂×#3624	C₂³	1	1	yes	characteristic
20	#63243	—	2	2	no	—
21	#63242	D₄	1	2	yes	—
22	#63242	Q₈	1	2	yes	—
23	#63241	—	2	1	no	—
24	#63241	—	2	1	no	—
25	#63241	—	2	2	no	—
26	C₄⋊C₄×C₁₇	C₈	1	1	yes	characteristic
27	#63227	C₈	1	2	yes	—
28	C₄⋊Dic₁₇	C₈	1	1	yes	characteristic
29	C₄×Dic₁₇	C₂×C₄	1	2	yes	—
30	C₂²×D₁₇	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₁₇	C₂×C₈	1	2	yes	—
32	C₂×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
33	C₂×Dic₁₇	C₂×C₈	1	2	yes	—
34	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
35	C₄×D₁₇	C₄⋊C₄	1	2	yes	—
36	C₄×D₁₇	M₁₆	1	2	yes	—
37	#3624	—	2	2	no	—
38	#3624	C₄⋊C₄	1	2	yes	—
39	#3624	C₂×D₄	1	1	yes	characteristic
40	#3624	C₂×Q₈	1	1	yes	characteristic
41	#2151	—	17	1	no	maximal, Sylow
42	C₂²×C₁₇	C₂²×C₈	1	1	yes	characteristic
43	C₂×D₁₇	—	2	1	no	—
44	C₂×D₁₇	—	2	1	no	—
45	C₂×D₁₇	C₄⋊C₈	1	2	yes	—
46	C₄×C₁₇	—	2	2	no	—
47	C₄×C₁₇	C₄⋊C₈	1	2	yes	—
48	Dic₁₇	—	2	2	no	—
49	Dic₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
50	Dic₁₇	C₂×M₁₆	1	1	yes	characteristic
51	C₂×M₃₂	—	17	1	no	—
52	C₂²×C₁₆	—	17	1	no	—
53	#405	—	17	1	no	—
54	C₄⋊C₁₆	—	17	2	no	—
55	C₄⋊C₁₆	—	17	2	no	—
56	C₂×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₁₇	#504	1	1	yes	characteristic
58	C₂×C₁₇	#504	1	1	yes	characteristic
59	D₁₇	—	2	2	no	—
60	C₂×M₁₆	—	17	1	no	—
61	C₂²×C₈	—	17	1	no	—
62	#117	—	17	1	no	—
63	M₃₂	—	34	2	no	—
64	C₂×C₁₆	—	17	1	no	—
65	C₂×C₁₆	—	17	1	no	—
66	C₂×C₁₆	—	17	2	no	—
67	C₂×C₁₆	—	34	2	no	—
68	#97	—	17	2	no	—
69	C₄×C₈	—	17	2	no	—
70	C₁₇	#2151	1	1	yes	characteristic, complemented, Sylow
71	C₂²×C₄	—	17	1	no	—
72	M₁₆	—	34	2	no	—
73	C₂×C₈	—	17	2	no	—
74	C₂×C₈	—	17	2	no	—
75	C₂×C₈	—	17	1	no	—
76	C₂×C₈	—	17	1	no	—
77	C₂×C₈	—	17	2	no	—
78	C₄⋊C₄	—	17	1	no	—
79	C₄⋊C₄	#3633	1	1	yes	characteristic
80	C₂²⋊C₄	—	17	2	no	—
81	C₄²	—	17	2	no	—
82	C₁₆	—	34	1	no	—
83	C₁₆	—	34	1	no	—
84	C₁₆	—	34	2	no	—
85	C₂³	—	17	1	no	—
86	C₂×C₄	—	17	1	no	—
87	C₂×C₄	—	17	2	no	—
88	C₂×C₄	—	17	2	no	—
89	C₂×C₄	—	17	2	no	—
90	C₂×C₄	C₂×#3633	1	2	yes	—
91	C₂×C₄	C₂×#3633	1	1	yes	characteristic
92	C₈	—	17	1	no	—
93	C₈	—	17	2	no	—
94	C₈	—	17	1	no	—
95	C₈	—	34	2	no	—
96	C₂²	—	17	2	no	—
97	C₂²	—	34	1	no	—
98	C₂²	—	34	1	no	—
99	C₂²	C₂²×#3633	1	1	yes	characteristic
100	C₄	—	2	2	no	—
101	C₄	—	17	1	no	—
102	C₄	—	17	1	no	—
103	C₄	—	34	2	no	—
104	C₄	C₄⋊#3633	1	2	yes	—
105	C₂	—	34	2	no	—
106	C₂	C₂×C₄⋊#3633	1	1	yes	characteristic
107	C₂	#11763930	1	1	yes	characteristic
108	C₂	#11763931	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587666

C₂

yes

maximal, characteristic

C₂×#10587665

C₂

yes

maximal, characteristic

#11763912

C₂

yes

maximal, complemented

C₄⋊#63241

C₂

yes

maximal, complemented

#11763774

C₂

yes

maximal, characteristic

C₂×#63243

C₄

yes

characteristic

C₂×#63242

C₂²

yes

characteristic

C₂×#63241

C₂²

yes

characteristic

C₂×#63241

C₂²

yes

—

C₂×#63241

C₂²

yes

—

—

C₄

yes

characteristic

#10587542

C₄

yes

—

C₄×#3624

C₂²

yes

—

C₂×C₄×D₁₇

C₂×C₄

yes

characteristic

C₂×#3624

C₂×C₄

yes

—

C₂×#3624

C₂×C₄

yes

characteristic

C₂×#3624

C₂³

yes

characteristic

#63243

—

#63242

D₄

yes

—

#63242

Q₈

yes

—

#63241

—

#63241

—

#63241

—

C₄⋊C₄×C₁₇

C₈

yes

characteristic

#63227

C₈

yes

—

C₄⋊Dic₁₇

C₈

yes

characteristic

C₄×Dic₁₇

C₂×C₄

yes

—

C₂²×D₁₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₇

C₂×C₈

yes

—

C₂×C₄×C₁₇

C₂×C₈

yes

characteristic

C₂×Dic₁₇

C₂×C₈

yes

—

C₂×Dic₁₇

C₂²×C₄

yes

characteristic

C₄×D₁₇

C₄⋊C₄

yes

—

C₄×D₁₇

M₁₆

yes

—

#3624

—

#3624

C₄⋊C₄

yes

—

#3624

C₂×D₄

yes

characteristic

#3624

C₂×Q₈

yes

characteristic

#2151

—

maximal, Sylow

C₂²×C₁₇

C₂²×C₈

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

C₄⋊C₈

yes

—

C₄×C₁₇

—

C₄×C₁₇

C₄⋊C₈

yes

—

Dic₁₇

—

Dic₁₇

C₂×C₄⋊C₄

yes

characteristic

Dic₁₇

C₂×M₁₆

yes

characteristic

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂×C₁₇

C₂×C₄⋊C₈

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

D₁₇

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

C₄×C₈

—

C₁₇

#2151

yes

characteristic, complemented, Sylow

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

#3633

yes

characteristic

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×#3633

yes

—

C₂×C₄

C₂×#3633

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²×#3633

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

C₄⋊#3633

yes

—

105

C₂

—

106

C₂

C₂×C₄⋊#3633

yes

characteristic

107

C₂

#11763930

yes

characteristic

108

C₂

#11763931

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#3633
commutator, socle	C₂²×C₁₇	C₂²×C₈
Fitting	C₄⋊C₄×C₁₇	C₈
prime radical	C₂²×D₁₇	C₂×C₈
cosocle	C₂×#3624	C₂³
hypercenter	C₄⋊C₄	#3633
hypocenter	C₁₇	#2151
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

subgroup

G/H

center, Frattini

C₂²

C₂²×#3633

commutator, socle

C₂²×C₁₇

C₂²×C₈

Fitting

C₄⋊C₄×C₁₇

C₈

prime radical

C₂²×D₁₇

C₂×C₈

C₂³

C₄⋊C₄

C₁₇

G ▷ C₂²×C₁₇ ▷ C₁₇

C₁ ◁ C₂² ◁ C₄⋊C₄

N	H ≅ G/N	type
3. #11763912	C₂	semidirect
4. C₄⋊#63241	C₂	semidirect
70. C₁₇	#2151	semidirect

H ≅ G/N

type

3. #11763912

C₂

semidirect

4. C₄⋊#63241

C₂

semidirect

70. C₁₇

#2151

semidirect

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

subgroup	index
C₂×#10587666	2
C₂×#10587665	2
#11763912	2
C₄⋊#63241	2
#11763774	2
#2151	17

subgroup

index

Subgroups — #14545976

Conjugacy classes of subgroups

Complements

Maximal subgroups