Group #19137213 — subgroups

Subgroups — #19137213

Order 5504.

600 conjugacy classes of subgroups; only the 309 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 309 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×#13063284	C₂	1	2	yes	maximal
2	#14582798	C₂	1	8	yes	maximal, complemented
3	#14582759	C₂	1	4	yes	maximal
4	C₂²×C₄⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic
5	#13063284	—	2	8	no	—
6	C₂×C₄⋊Dic₄₃	C₂²	1	4	yes	—
7	C₂×C₄⋊Dic₄₃	C₂²	1	8	yes	—
8	C₂×C₂²⋊Dic₄₃	C₂²	1	4	yes	—
9	C₂×C₄×Dic₄₃	C₂²	1	4	yes	—
10	C₄₃⋊#95	C₂²	1	8	yes	—
11	C₂³×C₄×C₄₃	C₂²	1	2	yes	—
12	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₄₃	C₄	1	8	yes	complemented
14	C₂×C₄⋊C₄×C₄₃	C₂²	1	4	yes	—
15	C₂⁴×C₄₃	C₂³	1	1	yes	characteristic
16	C₂²×C₄×C₄₃	—	4	4	no	—
17	C₂²×C₄×C₄₃	C₂×C₄	1	8	yes	—
18	C₂²×C₄×C₄₃	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₄₃	D₄	1	4	yes	—
20	C₂²×C₄×C₄₃	Q₈	1	4	yes	—
21	C₂²×C₄×C₄₃	C₂³	1	4	yes	—
22	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
23	C₂²×Dic₄₃	C₂³	1	8	yes	—
24	C₄⋊C₄×C₄₃	C₂×C₄	1	16	yes	—
25	C₂²⋊Dic₄₃	—	2	8	no	—
26	C₄⋊Dic₄₃	—	2	8	no	—
27	C₄⋊Dic₄₃	—	4	8	no	—
28	C₄×Dic₄₃	—	2	8	no	—
29	C₂³×C₄₃	—	2	2	no	—
30	C₂³×C₄₃	—	2	2	no	—
31	C₂³×C₄₃	C₂²×C₄	1	2	yes	—
32	C₂³×C₄₃	C₂×D₄	1	2	yes	—
33	C₂³×C₄₃	C₂×Q₈	1	2	yes	—
34	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
35	C₂×C₄×C₄₃	—	2	8	no	—
36	C₂×C₄×C₄₃	—	4	8	no	—
37	C₂×C₄×C₄₃	C₄⋊C₄	1	16	yes	complemented
38	C₂×C₄×C₄₃	C₂²×C₄	1	8	yes	—
39	C₂×C₄×C₄₃	C₂²×C₄	1	4	yes	—
40	C₂×C₄×C₄₃	C₂×D₄	1	4	yes	—
41	C₂×C₄×C₄₃	C₂×Q₈	1	4	yes	—
42	C₂×Dic₄₃	—	2	8	no	—
43	C₂×Dic₄₃	—	2	8	no	—
44	C₂×Dic₄₃	—	2	8	no	—
45	C₂²×C₄₃	—	2	2	no	—
46	C₂²×C₄₃	—	2	2	no	—
47	C₂²×C₄₃	—	2	4	no	—
48	C₂²×C₄₃	—	2	4	no	—
49	C₂²×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
50	C₂²×C₄₃	C₂³×C₄	1	1	yes	characteristic
51	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
52	C₂²×C₄₃	C₂²×Q₈	1	1	yes	characteristic
53	C₂²×C₄₃	H(5,𝔽₂)	1	2	yes	—
54	C₂²×C₄₃	2₋¹⁺⁴	1	2	yes	—
55	C₄×C₄₃	—	2	8	no	—
56	C₄×C₄₃	C₂×C₄⋊C₄	1	8	yes	—
57	Dic₄₃	—	4	8	no	—
58	#2316	—	43	1	no	maximal, Sylow
59	C₂×C₄₃	—	2	4	no	—
60	C₂×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄₃	#518	1	1	yes	characteristic
62	C₂×C₄₃	#519	1	1	yes	characteristic
63	C₂×C₄₃	#537	1	2	yes	—
64	C₂×C₄₃	#544	1	2	yes	—
65	C₂×#117	—	43	2	no	—
66	C₂²×C₄⋊C₄	D₄₃	1	1	yes	characteristic
67	#389	—	43	8	no	—
68	#380	—	43	4	no	—
69	C₄₃	#2316	1	1	yes	characteristic, complemented, Sylow
70	C₂³×C₄	C₂×D₄₃	1	2	yes	—
71	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic
72	#117	—	86	8	no	—
73	C₂×C₄⋊C₄	—	43	4	no	—
74	C₂×C₄⋊C₄	—	43	8	no	—
75	C₂×C₄⋊C₄	Dic₄₃	1	8	yes	complemented
76	C₂×C₄⋊C₄	C₂×D₄₃	1	4	yes	—
77	C₂×C₂²⋊C₄	—	43	4	no	—
78	C₂×C₄²	—	43	4	no	—
79	#95	—	43	8	no	—
80	C₂⁴	C₂²×D₄₃	1	1	yes	characteristic
81	C₂²×C₄	—	4	4	no	—
82	C₂²×C₄	—	43	8	no	—
83	C₂²×C₄	Dic₈₆	1	4	yes	—
84	C₂²×C₄	D₁₇₂	1	4	yes	—
85	C₂²×C₄	C₂×Dic₄₃	1	8	yes	—
86	C₂²×C₄	C₂×Dic₄₃	1	4	yes	—
87	C₂²×C₄	C₂²×D₄₃	1	4	yes	—
88	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
89	C₄⋊C₄	—	86	8	no	—
90	C₄⋊C₄	—	172	8	no	—
91	C₄⋊C₄	C₂×Dic₄₃	1	16	yes	—
92	C₂²⋊C₄	—	86	8	no	—
93	C₄²	—	86	8	no	—
94	C₂³	—	2	2	no	—
95	C₂³	—	2	2	no	—
96	C₂³	C₂×Dic₈₆	1	2	yes	—
97	C₂³	C₂×D₁₇₂	1	2	yes	—
98	C₂³	C₂²×Dic₄₃	1	2	yes	—
99	C₂³	C₂³×D₄₃	1	1	yes	characteristic
100	C₂×C₄	—	2	8	no	—
101	C₂×C₄	—	4	8	no	—
102	C₂×C₄	—	86	8	no	—
103	C₂×C₄	—	86	8	no	—
104	C₂×C₄	—	86	8	no	—
105	C₂×C₄	C₄⋊Dic₄₃	1	16	yes	complemented
106	C₂×C₄	C₂×Dic₈₆	1	4	yes	—
107	C₂×C₄	C₂×D₁₇₂	1	4	yes	—
108	C₂×C₄	C₂²×Dic₄₃	1	8	yes	—
109	C₂×C₄	C₂²×Dic₄₃	1	4	yes	—
110	C₂²	—	2	2	no	—
111	C₂²	—	2	2	no	—
112	C₂²	—	2	4	no	—
113	C₂²	—	2	4	no	—
114	C₂²	C₂×C₄⋊Dic₄₃	1	4	yes	—
115	C₂²	C₂³×Dic₄₃	1	1	yes	characteristic
116	C₂²	C₂²×D₁₇₂	1	1	yes	characteristic
117	C₂²	C₂²×Dic₈₆	1	1	yes	characteristic
118	C₂²	#13063370	1	2	yes	—
119	C₂²	#13063373	1	2	yes	—
120	C₄	—	2	8	no	—
121	C₄	—	172	8	no	—
122	C₄	C₂×C₄⋊Dic₄₃	1	8	yes	—
123	C₂	—	2	4	no	—
124	C₂	C₂²×C₄⋊Dic₄₃	1	1	yes	characteristic
125	C₂	#14583374	1	1	yes	characteristic
126	C₂	#14583377	1	1	yes	characteristic
127	C₂	#14583480	1	2	yes	—
128	C₂	#14583530	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063284

C₂

yes

maximal

#14582798

C₂

yes

maximal, complemented

#14582759

C₂

yes

maximal

C₂²×C₄⋊C₄×C₄₃

C₂

yes

maximal, characteristic

#13063284

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×C₂²⋊Dic₄₃

C₂²

yes

—

C₂×C₄×Dic₄₃

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₂³×C₄×C₄₃

C₂²

yes

—

C₂³×C₄×C₄₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₃

C₄

yes

complemented

C₂×C₄⋊C₄×C₄₃

C₂²

yes

—

C₂⁴×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

Q₈

yes

—

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

—

C₄⋊C₄×C₄₃

C₂×C₄

yes

—

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄×Dic₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

C₂²×C₄

yes

—

C₂³×C₄₃

C₂×D₄

yes

—

C₂³×C₄₃

C₂×Q₈

yes

—

C₂³×C₄₃

C₂⁴

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄⋊C₄

yes

complemented

C₂×C₄×C₄₃

C₂²×C₄

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×Q₈

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×C₄⋊C₄

yes

—

C₂²×C₄₃

C₂³×C₄

yes

characteristic

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂²×C₄₃

C₂²×Q₈

yes

characteristic

C₂²×C₄₃

H(5,𝔽₂)

yes

—

C₂²×C₄₃

2₋¹⁺⁴

yes

—

C₄×C₄₃

—

C₄×C₄₃

C₂×C₄⋊C₄

yes

—

Dic₄₃

—

#2316

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

C₂²×C₄⋊C₄

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄₃

#544

yes

—

C₂×#117

—

C₂²×C₄⋊C₄

D₄₃

yes

characteristic

#389

—

#380

—

C₄₃

#2316

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₄₃

yes

—

C₂³×C₄

C₂×D₄₃

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₄₃

yes

complemented

C₂×C₄⋊C₄

C₂×D₄₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#95

—

C₂⁴

C₂²×D₄₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₈₆

yes

—

C₂²×C₄

D₁₇₂

yes

—

C₂²×C₄

C₂×Dic₄₃

yes

—

C₂²×C₄

C₂×Dic₄₃

yes

—

C₂²×C₄

C₂²×D₄₃

yes

—

C₂²×C₄

C₂²×D₄₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

172

—

C₄⋊C₄

C₂×Dic₄₃

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

—

C₂³

C₂×Dic₈₆

yes

—

C₂³

C₂×D₁₇₂

yes

—

C₂³

C₂²×Dic₄₃

yes

—

C₂³

C₂³×D₄₃

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₄⋊Dic₄₃

yes

complemented

106

C₂×C₄

C₂×Dic₈₆

yes

—

107

C₂×C₄

C₂×D₁₇₂

yes

—

108

C₂×C₄

C₂²×Dic₄₃

yes

—

109

C₂×C₄

C₂²×Dic₄₃

yes

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂×C₄⋊Dic₄₃

yes

—

115

C₂²

C₂³×Dic₄₃

yes

characteristic

116

C₂²

C₂²×D₁₇₂

yes

characteristic

117

C₂²

C₂²×Dic₈₆

yes

characteristic

118

C₂²

#13063370

yes

—

119

C₂²

#13063373

yes

—

120

C₄

—

121

C₄

—

172

—

122

C₄

C₂×C₄⋊Dic₄₃

yes

—

123

C₂

—

124

C₂

C₂²×C₄⋊Dic₄₃

yes

characteristic

125

C₂

#14583374

yes

characteristic

126

C₂

#14583377

yes

characteristic

127

C₂

#14583480

yes

—

128

C₂

#14583530

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂³×C₄
Fitting	C₂²×C₄⋊C₄×C₄₃	C₂
socle	C₂³×C₄₃	C₂⁴
hypercenter	C₂²×C₄⋊C₄	D₄₃
hypocenter	C₄₃	#2316
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₃

commutator

C₂²×C₄₃

C₂³×C₄

Fitting

C₂²×C₄⋊C₄×C₄₃

C₂

socle

C₂³×C₄₃

C₂⁴

hypercenter

C₂²×C₄⋊C₄

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₄₃

C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
2. #14582798	C₂	semidirect
13. C₂×C₄⋊C₄×C₄₃	C₄	semidirect
37. C₂×C₄×C₄₃	C₄⋊C₄	semidirect
69. C₄₃	#2316	semidirect
75. C₂×C₄⋊C₄	Dic₄₃	semidirect
105. C₂×C₄	C₄⋊Dic₄₃	semidirect

H ≅ G/N

type

2. #14582798

C₂

semidirect

13. C₂×C₄⋊C₄×C₄₃

C₄

semidirect

37. C₂×C₄×C₄₃

C₄⋊C₄

semidirect

69. C₄₃

#2316

semidirect

75. C₂×C₄⋊C₄

Dic₄₃

semidirect

105. C₂×C₄

C₄⋊Dic₄₃

semidirect

Subgroups — #19137213

Conjugacy classes of subgroups

Complements