Group #19136681 — subgroups

Subgroups — #19136681

Order 5504.

174 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 57 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×D₆₈₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583135	C₂	1	2	yes	maximal, complemented
3	#14582725	C₂	1	1	yes	maximal, characteristic, complemented
4	#14582701	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₃₄₄	C₂²	1	2	yes	—
7	#13063299	—	2	2	no	—
8	D₆₈₈	—	2	2	no	—
9	#13063259	C₂²	1	2	yes	—
10	#13063247	—	2	2	no	—
11	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₄₃	—	2	2	no	—
13	C₂×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
16	C₂×C₂²⋊D₄₃	—	4	2	no	—
17	C₂×D₁₇₂	—	2	2	no	—
18	C₂×C₈×C₄₃	—	2	1	no	—
19	C₂×C₈×C₄₃	—	2	1	no	—
20	C₂×C₈×C₄₃	D₄	1	1	yes	characteristic
21	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
22	(C₂×D₄₃)⋊C₄	—	4	2	no	—
23	C₄⋊Dic₄₃	—	2	2	no	—
24	D₃₄₄	—	2	2	no	—
25	D₃₄₄	—	2	2	no	—
26	C₁₆×C₄₃	—	2	1	no	—
27	C₁₆×C₄₃	D₄	1	2	yes	complemented
28	C₂³×C₄₃	D₈	1	1	yes	characteristic
29	C₂²×D₄₃	—	4	2	no	—
30	C₂×C₄×C₄₃	—	2	1	no	—
31	C₂×C₄×C₄₃	—	2	1	no	—
32	C₂×C₄×C₄₃	D₈	1	1	yes	characteristic
33	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
34	C₂²⋊D₄₃	—	8	4	no	—
35	C₂×Dic₄₃	—	4	2	no	—
36	D₁₇₂	—	4	2	no	—
37	D₁₇₂	—	4	2	no	—
38	C₈×C₄₃	—	2	1	no	—
39	C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
40	C₈×C₄₃	#41	1	1	yes	characteristic
41	C₂²×C₄₃	—	2	1	no	—
42	C₂²×C₄₃	—	2	1	no	—
43	C₂²×C₄₃	D₁₆	1	2	yes	complemented
44	C₂²×C₄₃	C₂×D₈	1	1	yes	characteristic
45	C₂×D₄₃	—	8	2	no	—
46	C₂×D₄₃	—	8	2	no	—
47	C₂×D₄₃	—	8	2	no	—
48	C₄×C₄₃	—	2	1	no	—
49	C₄×C₄₃	#121	1	1	yes	characteristic
50	C₄×C₄₃	#135	1	1	yes	characteristic
51	Dic₄₃	—	8	2	no	—
52	#2215	—	43	1	no	maximal, Sylow
53	C₂×C₄₃	—	2	2	no	—
54	C₂×C₄₃	#466	1	1	yes	characteristic
55	C₂×C₄₃	C₂×D₁₆	1	1	yes	characteristic
56	C₂×C₄₃	#508	1	1	yes	characteristic
57	D₄₃	—	16	2	no	—
58	C₂×D₁₆	—	43	1	no	—
59	C₂²×C₁₆	D₄₃	1	1	yes	characteristic, complemented
60	#466	—	43	2	no	—
61	#366	—	43	1	no	—
62	#357	—	43	2	no	—
63	C₄₃	#2215	1	1	yes	characteristic, complemented, Sylow
64	C₂×D₈	—	43	2	no	—
65	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
66	#121	—	86	2	no	—
67	D₁₆	—	86	2	no	—
68	C₂×C₁₆	—	2	2	no	—
69	C₂×C₁₆	C₂×D₄₃	1	1	yes	characteristic
70	C₂×C₁₆	C₂×D₄₃	1	1	yes	characteristic
71	#107	—	43	2	no	—
72	#102	—	86	2	no	—
73	C₂×D₄	—	86	2	no	—
74	C₂×D₄	—	172	2	no	—
75	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
76	D₈	—	86	2	no	—
77	D₈	—	86	2	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	D₁₇₂	1	1	yes	characteristic
81	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
82	C₄⋊C₄	—	86	2	no	—
83	C₂²⋊C₄	—	172	2	no	—
84	C₁₆	—	2	1	no	—
85	C₁₆	C₂²⋊D₄₃	1	2	yes	complemented
86	C₂³	—	172	2	no	—
87	C₂³	D₃₄₄	1	1	yes	characteristic
88	D₄	—	344	4	no	—
89	D₄	—	172	2	no	—
90	D₄	—	172	2	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	172	2	no	—
94	C₂×C₄	D₃₄₄	1	1	yes	characteristic
95	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
96	C₈	—	2	1	no	—
97	C₈	#10621898	1	1	yes	characteristic
98	C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
99	C₂²	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	—	344	2	no	—
102	C₂²	—	344	2	no	—
103	C₂²	—	344	2	no	—
104	C₂²	D₆₈₈	1	2	yes	complemented
105	C₂²	C₂×D₃₄₄	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	344	2	no	—
108	C₄	#13063299	1	1	yes	characteristic
109	C₄	#13063347	1	1	yes	characteristic
110	C₂	—	2	2	no	—
111	C₂	—	688	2	no	—
112	C₂	#14583135	1	1	yes	characteristic
113	C₂	C₂×D₆₈₈	1	1	yes	characteristic
114	C₂	#14583331	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₆₈₈

C₂

yes

maximal, characteristic, complemented

#14583135

C₂

yes

maximal, complemented

#14582725

C₂

yes

maximal, characteristic, complemented

#14582701

C₂

yes

maximal, complemented

C₂²×C₁₆×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂×D₃₄₄

C₂²

yes

—

#13063299

—

D₆₈₈

—

#13063259

C₂²

yes

—

#13063247

—

C₂²×C₈×C₄₃

C₂²

yes

characteristic

C₂×C₁₆×C₄₃

—

C₂×C₁₆×C₄₃

C₂²

yes

characteristic

C₂×C₁₆×C₄₃

C₂²

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂×C₂²⋊D₄₃

—

C₂×D₁₇₂

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

D₄

yes

characteristic

C₂×C₈×C₄₃

C₂³

yes

characteristic

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

D₃₄₄

—

D₃₄₄

—

C₁₆×C₄₃

—

C₁₆×C₄₃

D₄

yes

complemented

C₂³×C₄₃

D₈

yes

characteristic

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

D₈

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

C₈×C₄₃

—

C₈×C₄₃

C₂×D₄

yes

characteristic

C₈×C₄₃

#41

yes

characteristic

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

D₁₆

yes

complemented

C₂²×C₄₃

C₂×D₈

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

#2215

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

#466

yes

characteristic

C₂×C₄₃

C₂×D₁₆

yes

characteristic

C₂×C₄₃

#508

yes

characteristic

D₄₃

—

C₂×D₁₆

—

C₂²×C₁₆

D₄₃

yes

characteristic, complemented

#466

—

#366

—

#357

—

C₄₃

#2215

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂²×C₈

C₂×D₄₃

yes

characteristic

#121

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄₃

yes

characteristic

C₂×C₁₆

C₂×D₄₃

yes

characteristic

#107

—

#102

—

C₂×D₄

—

C₂×D₄

—

172

—

C₂²×C₄

D₁₇₂

yes

characteristic

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₁₇₂

yes

characteristic

C₂×C₈

C₂²×D₄₃

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

172

—

C₁₆

—

C₁₆

C₂²⋊D₄₃

yes

complemented

C₂³

—

172

—

C₂³

D₃₄₄

yes

characteristic

D₄

—

344

—

D₄

—

172

—

D₄

—

172

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

172

—

C₂×C₄

D₃₄₄

yes

characteristic

C₂×C₄

C₂×D₁₇₂

yes

characteristic

C₈

—

C₈

#10621898

yes

characteristic

C₈

C₂×C₂²⋊D₄₃

yes

characteristic

C₂²

—

100

C₂²

—

101

C₂²

—

344

—

102

C₂²

—

344

—

103

C₂²

—

344

—

104

C₂²

D₆₈₈

yes

complemented

105

C₂²

C₂×D₃₄₄

yes

characteristic

106

C₄

—

107

C₄

—

344

—

108

C₄

#13063299

yes

characteristic

109

C₄

#13063347

yes

characteristic

110

C₂

—

111

C₂

—

688

—

112

C₂

#14583135

yes

characteristic

113

C₂

C₂×D₆₈₈

yes

characteristic

114

C₂

#14583331

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₃₄₄
commutator	C₂×C₈×C₄₃	C₂³
Frattini	C₂×C₈	C₂²×D₄₃
Fitting	C₂²×C₁₆×C₄₃	C₂
socle	C₂²×C₄₃	C₂×D₈
hypercenter	C₂²×C₁₆	D₄₃
hypocenter	C₄₃	#2215
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂²

C₂×D₃₄₄

commutator

C₂×C₈×C₄₃

C₂³

Frattini

C₂×C₈

C₂²×D₄₃

Fitting

C₂²×C₁₆×C₄₃

C₂

socle

C₂²×C₄₃

C₂×D₈

hypercenter

C₂²×C₁₆

D₄₃

hypocenter

C₄₃

#2215

LCS

G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
1. C₂×D₆₈₈	C₂	semidirect
2. #14583135	C₂	semidirect
3. #14582725	C₂	semidirect
4. #14582701	C₂	semidirect
5. C₂²×C₁₆×C₄₃	C₂	semidirect
27. C₁₆×C₄₃	D₄	semidirect
43. C₂²×C₄₃	D₁₆	semidirect
59. C₂²×C₁₆	D₄₃	semidirect
63. C₄₃	#2215	semidirect
85. C₁₆	C₂²⋊D₄₃	semidirect
104. C₂²	D₆₈₈	semidirect

H ≅ G/N

type

1. C₂×D₆₈₈

C₂

semidirect

2. #14583135

C₂

semidirect

3. #14582725

C₂

semidirect

4. #14582701

C₂

semidirect

5. C₂²×C₁₆×C₄₃

C₂

semidirect

27. C₁₆×C₄₃

D₄

semidirect

43. C₂²×C₄₃

D₁₆

semidirect

59. C₂²×C₁₆

D₄₃

semidirect

63. C₄₃

#2215

semidirect

85. C₁₆

C₂²⋊D₄₃

semidirect

104. C₂²

D₆₈₈

semidirect

Subgroups — #19136681

Conjugacy classes of subgroups

Complements