Group #21444705 — subgroups

Subgroups — #21444705

Order 7808.

180 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₄×D₄₈₈	C₂	1	1	yes	maximal, characteristic
2	#16839841	C₂	1	2	yes	maximal, complemented
3	#16839515	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839491	C₂	1	2	yes	maximal, complemented
5	C₈²×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄₈₈	C₄	1	1	yes	characteristic
7	C₂×#11758189	—	2	2	no	—
8	C₄×D₂₄₄	C₂²	1	2	yes	—
9	#14519711	C₄	1	1	yes	characteristic
10	C₄⋊#92418	C₂²	1	2	yes	—
11	#14519699	C₄	1	2	yes	—
12	#14519689	—	2	2	no	—
13	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×D₂₄₄	C₂×C₄	1	2	yes	—
17	C₂×C₄×D₆₁	—	2	2	no	—
18	C₂×C₈×C₆₁	—	2	1	no	—
19	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
22	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
23	C₄²×C₆₁	C₂³	1	1	yes	characteristic
24	#11758198	—	2	2	no	—
25	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
26	C₂×#92418	—	2	2	no	—
27	D₄₈₈	—	2	2	no	—
28	#11758189	—	4	4	no	—
29	C₂²×D₆₁	—	2	2	no	—
30	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
33	C₂×Dic₆₁	—	2	2	no	—
34	D₂₄₄	—	2	2	no	—
35	D₂₄₄	—	2	2	no	—
36	C₄×D₆₁	—	4	2	no	—
37	C₄×D₆₁	—	4	2	no	—
38	C₈×C₆₁	—	2	2	no	—
39	C₈×C₆₁	—	2	1	no	—
40	C₈×C₆₁	—	2	1	no	—
41	C₈×C₆₁	M₁₆	1	2	yes	complemented
42	C₈×C₆₁	D₈	1	2	yes	complemented
43	#92418	—	4	2	no	—
44	C₂²×C₆₁	C₄×D₄	1	1	yes	characteristic
45	C₂×D₆₁	—	4	2	no	—
46	C₂×D₆₁	—	4	2	no	—
47	C₂×D₆₁	—	4	2	no	—
48	C₄×C₆₁	—	2	1	no	—
49	C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
50	C₄×C₆₁	#131	1	1	yes	characteristic
51	C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
52	C₄×C₆₁	#135	1	1	yes	characteristic
53	Dic₆₁	—	4	2	no	—
54	#1589	—	61	1	no	maximal, Sylow
55	C₂×C₆₁	#436	1	1	yes	characteristic
56	C₂×C₆₁	C₄×D₈	1	1	yes	characteristic
57	C₂×C₆₁	#443	1	1	yes	characteristic
58	D₆₁	—	8	2	no	—
59	C₄×D₈	—	61	1	no	—
60	#436	—	61	2	no	—
61	#335	—	61	1	no	—
62	D₄⋊C₈	—	61	2	no	—
63	C₈²	D₆₁	1	1	yes	characteristic, complemented
64	C₆₁	#1589	1	1	yes	characteristic, complemented, Sylow
65	C₂×D₈	—	61	1	no	—
66	C₂×M₁₆	—	122	2	no	—
67	C₄×D₄	—	61	2	no	—
68	#107	—	61	1	no	—
69	C₄⋊C₈	—	61	2	no	—
70	#102	—	61	2	no	—
71	C₂²⋊C₈	—	122	2	no	—
72	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
73	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
74	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
75	C₂×D₄	—	61	2	no	—
76	C₂²×C₄	—	122	2	no	—
77	D₈	—	122	2	no	—
78	M₁₆	—	244	4	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	122	2	no	—
81	C₂×C₈	C₄×D₆₁	1	1	yes	characteristic
82	C₂×C₈	C₄×D₆₁	1	1	yes	characteristic
83	C₂×C₈	D₂₄₄	1	1	yes	characteristic
84	C₂×C₈	D₂₄₄	1	1	yes	characteristic
85	C₄⋊C₄	—	61	2	no	—
86	C₂²⋊C₄	—	122	2	no	—
87	C₄²	C₂²×D₆₁	1	1	yes	characteristic
88	C₂³	—	122	2	no	—
89	D₄	—	122	2	no	—
90	D₄	—	122	2	no	—
91	C₂×C₄	—	122	2	no	—
92	C₂×C₄	—	244	2	no	—
93	C₂×C₄	—	244	2	no	—
94	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
95	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
96	C₂×C₄	#11758222	1	1	yes	characteristic
97	C₈	—	2	2	no	—
98	C₈	—	2	1	no	—
99	C₈	—	2	1	no	—
100	C₈	—	244	2	no	—
101	C₈	#11758189	1	2	yes	complemented
102	C₈	D₄₈₈	1	2	yes	complemented
103	C₂²	—	244	2	no	—
104	C₂²	—	244	2	no	—
105	C₂²	—	244	2	no	—
106	C₂²	C₄×D₂₄₄	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	244	2	no	—
109	C₄	C₂×#11758189	1	1	yes	characteristic
110	C₄	#14519783	1	1	yes	characteristic
111	C₄	C₂×D₄₈₈	1	1	yes	characteristic
112	C₄	#14519799	1	1	yes	characteristic
113	C₂	—	488	2	no	—
114	C₂	#16839841	1	1	yes	characteristic
115	C₂	C₄×D₄₈₈	1	1	yes	characteristic
116	C₂	#16839880	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₄₈₈

C₂

yes

maximal, characteristic

#16839841

C₂

yes

maximal, complemented

#16839515

C₂

yes

maximal, characteristic, complemented

#16839491

C₂

yes

maximal, complemented

C₈²×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂×D₄₈₈

C₄

yes

characteristic

C₂×#11758189

—

C₄×D₂₄₄

C₂²

yes

—

#14519711

C₄

yes

characteristic

C₄⋊#92418

C₂²

yes

—

#14519699

C₄

yes

—

#14519689

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×D₂₄₄

C₂×C₄

yes

—

C₂×C₄×D₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₈×C₆₁

D₄

yes

characteristic

C₂×C₈×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

C₂³

yes

characteristic

#11758198

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

C₂×#92418

—

D₄₈₈

—

#11758189

—

C₂²×D₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

characteristic

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

C₄×D₆₁

—

C₄×D₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

M₁₆

yes

complemented

C₈×C₆₁

D₈

yes

complemented

#92418

—

C₂²×C₆₁

C₄×D₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×M₁₆

yes

characteristic

C₄×C₆₁

#131

yes

characteristic

C₄×C₆₁

C₂×D₈

yes

characteristic

C₄×C₆₁

#135

yes

characteristic

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

C₄×D₈

yes

characteristic

C₂×C₆₁

#443

yes

characteristic

D₆₁

—

C₄×D₈

—

#436

—

#335

—

D₄⋊C₈

—

C₈²

D₆₁

yes

characteristic, complemented

C₆₁

#1589

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×M₁₆

—

122

—

C₄×D₄

—

#107

—

C₄⋊C₈

—

#102

—

C₂²⋊C₈

—

122

—

C₄×C₈

C₂×D₆₁

yes

characteristic

C₄×C₈

C₂×D₆₁

yes

characteristic

C₄×C₈

C₂×D₆₁

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

122

—

D₈

—

122

—

M₁₆

—

244

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

C₄×D₆₁

yes

characteristic

C₂×C₈

C₄×D₆₁

yes

characteristic

C₂×C₈

D₂₄₄

yes

characteristic

C₂×C₈

D₂₄₄

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

122

—

C₄²

C₂²×D₆₁

yes

characteristic

C₂³

—

122

—

D₄

—

122

—

D₄

—

122

—

C₂×C₄

—

122

—

C₂×C₄

—

244

—

C₂×C₄

—

244

—

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

C₂×C₄

C₂×D₂₄₄

yes

characteristic

C₂×C₄

#11758222

yes

characteristic

C₈

—

C₈

—

C₈

—

100

C₈

—

244

—

101

C₈

#11758189

yes

complemented

102

C₈

D₄₈₈

yes

complemented

103

C₂²

—

244

—

104

C₂²

—

244

—

105

C₂²

—

244

—

106

C₂²

C₄×D₂₄₄

yes

characteristic

107

C₄

—

108

C₄

—

244

—

109

C₄

C₂×#11758189

yes

characteristic

110

C₄

#14519783

yes

characteristic

111

C₄

C₂×D₄₈₈

yes

characteristic

112

C₄

#14519799

yes

characteristic

113

C₂

—

488

—

114

C₂

#16839841

yes

characteristic

115

C₂

C₄×D₄₈₈

yes

characteristic

116

C₂

#16839880

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₄²	C₂²×D₆₁
Fitting	C₈²×C₆₁	C₂
socle	C₂²×C₆₁	C₄×D₄
hypercenter	C₈²	D₆₁
hypocenter	C₆₁	#1589
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈²

subgroup

G/H

center

C₂×C₄

C₂×D₂₄₄

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₄²

C₂²×D₆₁

Fitting

C₈²×C₆₁

C₂

socle

C₂²×C₆₁

C₄×D₄

hypercenter

C₈²

D₆₁

hypocenter

C₆₁

#1589

LCS

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈²

N	H ≅ G/N	type
2. #16839841	C₂	semidirect
3. #16839515	C₂	semidirect
4. #16839491	C₂	semidirect
5. C₈²×C₆₁	C₂	semidirect
41. C₈×C₆₁	M₁₆	semidirect
42. C₈×C₆₁	D₈	semidirect
63. C₈²	D₆₁	semidirect
64. C₆₁	#1589	semidirect
101. C₈	#11758189	semidirect
102. C₈	D₄₈₈	semidirect

H ≅ G/N

type

2. #16839841

C₂

semidirect

3. #16839515

C₂

semidirect

4. #16839491

C₂

semidirect

5. C₈²×C₆₁

C₂

semidirect

41. C₈×C₆₁

M₁₆

semidirect

42. C₈×C₆₁

D₈

semidirect

63. C₈²

D₆₁

semidirect

64. C₆₁

#1589

semidirect

101. C₈

#11758189

semidirect

102. C₈

D₄₈₈

semidirect

Subgroups — #21444705

Conjugacy classes of subgroups

Complements