Group #19136298 — subgroups

Subgroups — C₂×#14582704 #19136298

Order 5504.

232 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 101 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	C₂²×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#13063259	C₂	1	1	yes	maximal, characteristic
3	#14582704	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₈₆	—	2	1	no	—
6	C₂×Dic₁₇₂	—	2	1	no	—
7	C₂×Dic₁₇₂	—	2	2	no	—
8	C₂×Dic₁₇₂	C₄	1	4	yes	complemented
9	C₂×Dic₁₇₂	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₄₃	—	2	1	no	—
11	#13063259	—	2	1	no	—
12	#13063259	C₂²	1	2	yes	—
13	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₄₃	—	2	2	no	—
15	C₂×C₁₆×C₄₃	C₂²	1	2	yes	—
16	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
17	C₂²×Dic₄₃	—	4	1	no	—
18	C₂²×Dic₄₃	—	4	1	no	—
19	C₂×Dic₈₆	—	2	4	no	—
20	C₂×Dic₈₆	—	2	2	no	—
21	C₂×Dic₈₆	—	4	1	no	—
22	C₂×Dic₈₆	—	4	2	no	—
23	C₂×C₈×C₄₃	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₄₃	D₄	1	1	yes	characteristic
25	C₂×C₈×C₄₃	D₄	1	2	yes	—
26	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
27	C₄⋊Dic₄₃	—	2	2	no	—
28	C₄⋊Dic₄₃	—	4	1	no	—
29	Dic₁₇₂	—	2	4	no	—
30	Dic₁₇₂	—	2	2	no	—
31	Dic₁₇₂	C₂×C₄	1	4	yes	complemented
32	C₁₆×C₄₃	—	2	2	no	—
33	C₂³×C₄₃	D₈	1	1	yes	characteristic
34	C₂×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
35	C₂×C₄×C₄₃	D₈	1	1	yes	characteristic
36	C₂×C₄×C₄₃	SD₁₆	1	2	yes	—
37	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
38	C₂×Dic₄₃	—	4	4	no	—
39	C₂×Dic₄₃	—	4	2	no	—
40	C₂×Dic₄₃	—	4	2	no	—
41	C₂×Dic₄₃	—	8	2	no	—
42	Dic₈₆	—	2	4	no	—
43	Dic₈₆	—	4	4	no	—
44	Dic₈₆	—	4	2	no	—
45	C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
46	C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
47	C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄₃	#102	1	2	yes	—
49	C₂²×C₄₃	SD₃₂	1	2	yes	—
50	C₂²×C₄₃	Q₃₂	1	2	yes	—
51	C₂²×C₄₃	C₂×D₈	1	1	yes	characteristic
52	C₄×C₄₃	#102	1	2	yes	—
53	C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
55	Dic₄₃	—	4	4	no	—
56	Dic₄₃	—	8	2	no	—
57	C₂×#358	—	43	1	no	maximal, Sylow
58	C₂×C₄₃	#358	1	4	yes	complemented
59	C₂×C₄₃	C₂×#102	1	1	yes	characteristic
60	C₂×C₄₃	C₂×SD₃₂	1	1	yes	characteristic
61	C₂×C₄₃	C₂×Q₃₂	1	1	yes	characteristic
62	C₂²×Q₁₆	—	43	1	no	—
63	C₂²×C₁₆	D₄₃	1	1	yes	characteristic
64	C₂×#107	—	43	1	no	—
65	#358	—	43	4	no	—
66	C₄₃	C₂×#358	1	1	yes	characteristic, complemented, Sylow
67	C₂²×Q₈	—	86	1	no	—
68	C₂×Q₁₆	—	43	2	no	—
69	C₂×Q₁₆	—	43	4	no	—
70	C₂×Q₁₆	—	86	1	no	—
71	C₂×Q₁₆	—	86	2	no	—
72	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
73	C₂×C₄⋊C₄	—	86	1	no	—
74	C₂×C₁₆	—	2	2	no	—
75	C₂×C₁₆	C₂×D₄₃	1	2	yes	—
76	#107	—	43	2	no	—
77	#107	—	86	1	no	—
78	C₂×Q₈	—	86	4	no	—
79	C₂×Q₈	—	86	2	no	—
80	C₂×Q₈	—	172	2	no	—
81	C₂×Q₈	—	172	1	no	—
82	C₂²×C₄	—	172	1	no	—
83	C₂²×C₄	—	172	1	no	—
84	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
85	Q₁₆	—	43	4	no	—
86	Q₁₆	—	86	4	no	—
87	Q₁₆	—	86	2	no	—
88	C₂×C₈	C₄×D₄₃	1	2	yes	—
89	C₂×C₈	D₁₇₂	1	1	yes	characteristic
90	C₂×C₈	C₂²⋊D₄₃	1	2	yes	—
91	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
92	C₄⋊C₄	—	86	2	no	—
93	C₄⋊C₄	—	172	1	no	—
94	C₁₆	—	2	2	no	—
95	C₂³	D₃₄₄	1	1	yes	characteristic
96	Q₈	—	86	4	no	—
97	Q₈	—	172	4	no	—
98	Q₈	—	172	2	no	—
99	C₂×C₄	—	344	2	no	—
100	C₂×C₄	—	172	2	no	—
101	C₂×C₄	—	172	4	no	—
102	C₂×C₄	—	172	2	no	—
103	C₂×C₄	#10621873	1	2	yes	—
104	C₂×C₄	D₃₄₄	1	1	yes	characteristic
105	C₂×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
106	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
107	C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
108	C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
109	C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
110	C₂²	#13063247	1	2	yes	—
111	C₂²	#13063270	1	2	yes	—
112	C₂²	Dic₃₄₄	1	2	yes	—
113	C₂²	C₂×D₃₄₄	1	1	yes	characteristic
114	C₄	—	344	2	no	—
115	C₄	—	172	4	no	—
116	C₄	#13063247	1	2	yes	—
117	C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
118	C₄	C₂×#10621873	1	1	yes	characteristic
119	C₂	#14582704	1	4	yes	direct product factor
120	C₂	C₂×#13063247	1	1	yes	characteristic
121	C₂	C₂×#13063270	1	1	yes	characteristic
122	C₂	C₂×Dic₃₄₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₁₇₂

C₂

yes

maximal, characteristic

C₂×#13063259

C₂

yes

maximal, characteristic

#14582704

C₂

yes

maximal, direct product factor

C₂²×C₁₆×C₄₃

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

C₄

yes

complemented

C₂×Dic₁₇₂

C₂²

yes

—

C₂×C₄⋊Dic₄₃

—

#13063259

—

#13063259

C₂²

yes

—

C₂²×C₈×C₄₃

C₂²

yes

characteristic

C₂×C₁₆×C₄₃

—

C₂×C₁₆×C₄₃

C₂²

yes

—

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

C₂×C₄

yes

—

C₂×C₈×C₄₃

D₄

yes

characteristic

C₂×C₈×C₄₃

D₄

yes

—

C₂×C₈×C₄₃

C₂³

yes

characteristic

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₁₇₂

—

Dic₁₇₂

—

Dic₁₇₂

C₂×C₄

yes

complemented

C₁₆×C₄₃

—

C₂³×C₄₃

D₈

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂×C₄×C₄₃

D₈

yes

characteristic

C₂×C₄×C₄₃

SD₁₆

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

C₂²⋊C₄

yes

—

C₈×C₄₃

C₂²×C₄

yes

characteristic

C₈×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄₃

#102

yes

—

C₂²×C₄₃

SD₃₂

yes

—

C₂²×C₄₃

Q₃₂

yes

—

C₂²×C₄₃

C₂×D₈

yes

characteristic

C₄×C₄₃

#102

yes

—

C₄×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₄₃

C₂×SD₁₆

yes

characteristic

Dic₄₃

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

C₂×SD₃₂

yes

characteristic

C₂×C₄₃

C₂×Q₃₂

yes

characteristic

C₂²×Q₁₆

—

C₂²×C₁₆

D₄₃

yes

characteristic

C₂×#107

—

#358

—

C₄₃

C₂×#358

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

C₂×D₄₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄₃

yes

—

#107

—

#107

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

172

—

C₂×Q₈

—

172

—

C₂²×C₄

—

172

—

C₂²×C₄

—

172

—

C₂²×C₄

D₁₇₂

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

C₄×D₄₃

yes

—

C₂×C₈

D₁₇₂

yes

characteristic

C₂×C₈

C₂²⋊D₄₃

yes

—

C₂×C₈

C₂²×D₄₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

172

—

C₁₆

—

C₂³

D₃₄₄

yes

characteristic

Q₈

—

Q₈

—

172

—

Q₈

—

172

—

C₂×C₄

—

344

—

100

C₂×C₄

—

172

—

101

C₂×C₄

—

172

—

102

C₂×C₄

—

172

—

103

C₂×C₄

#10621873

yes

—

104

C₂×C₄

D₃₄₄

yes

characteristic

105

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

—

106

C₂×C₄

C₂×D₁₇₂

yes

characteristic

107

C₈

(C₂×D₄₃)⋊C₄

yes

—

108

C₈

C₂×C₄×D₄₃

yes

characteristic

109

C₈

C₂×C₂²⋊D₄₃

yes

characteristic

110

C₂²

#13063247

yes

—

111

C₂²

#13063270

yes

—

112

C₂²

Dic₃₄₄

yes

—

113

C₂²

C₂×D₃₄₄

yes

characteristic

114

C₄

—

344

—

115

C₄

—

172

—

116

C₄

#13063247

yes

—

117

C₄

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

118

C₄

C₂×#10621873

yes

characteristic

119

C₂

#14582704

yes

direct product factor

120

C₂

C₂×#13063247

yes

characteristic

121

C₂

C₂×#13063270

yes

characteristic

122

C₂

C₂×Dic₃₄₄

yes

characteristic

	subgroup	G/H
center	C₂³	D₃₄₄
commutator	C₈×C₄₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₄₃
Fitting	C₂²×C₁₆×C₄₃	C₂
socle	C₂³×C₄₃	D₈
hypercenter	C₂²×C₁₆	D₄₃
hypocenter	C₄₃	C₂×#358
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂³

D₃₄₄

commutator

C₈×C₄₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₄₃

Fitting

C₂²×C₁₆×C₄₃

C₂

socle

C₂³×C₄₃

D₈

hypercenter

C₂²×C₁₆

D₄₃

hypocenter

C₄₃

C₂×#358

LCS

G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
3. #14582704	C₂	direct (N × H)
8. C₂×Dic₁₇₂	C₄	semidirect
31. Dic₁₇₂	C₂×C₄	semidirect
58. C₂×C₄₃	#358	semidirect
66. C₄₃	C₂×#358	semidirect
119. C₂	#14582704	direct (N × H)

H ≅ G/N

type

3. #14582704

C₂

direct (N × H)

8. C₂×Dic₁₇₂

C₄

semidirect

31. Dic₁₇₂

C₂×C₄

semidirect

58. C₂×C₄₃

#358

semidirect

66. C₄₃

C₂×#358

semidirect

119. C₂

#14582704

direct (N × H)

Subgroups — C₂×#14582704 #19136298

Conjugacy classes of subgroups

Complements

Subgroups — C2×#14582704 #19136298

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#14582704 #19136298