Group #21444563 — subgroups

Subgroups — C₂×#16839515 #21444563

Order 7808.

200 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 141 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×C₄⋊#92418	C₂	1	2	yes	maximal
2	#16839515	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×#92418	—	2	2	no	—
5	C₄⋊#92418	—	2	2	no	—
6	C₄⋊#92418	C₂²	1	4	yes	—
7	C₂²×C₈×C₆₁	C₄	1	1	yes	characteristic
8	C₂²×C₈×C₆₁	C₄	1	1	yes	characteristic
9	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
10	C₄×C₈×C₆₁	C₄	1	2	yes	—
11	C₄×C₈×C₆₁	C₂²	1	2	yes	—
12	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
14	C₂²×C₄×C₆₁	Q₈	1	1	yes	characteristic
15	C₂×C₈×C₆₁	—	2	2	no	—
16	C₂×C₈×C₆₁	C₈	1	4	yes	complemented
17	C₂×C₈×C₆₁	C₂×C₄	1	2	yes	—
18	C₂×C₈×C₆₁	C₂×C₄	1	2	yes	—
19	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₆₁	D₄	1	1	yes	characteristic
21	C₄²×C₆₁	Q₈	1	1	yes	characteristic
22	C₄²×C₆₁	C₂³	1	1	yes	characteristic
23	C₂×#92418	—	2	4	no	—
24	C₂×#92418	—	4	4	no	—
25	C₂³×C₆₁	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₆₁	—	2	2	no	—
27	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₆₁	C₂×C₈	1	2	yes	—
31	C₂×C₄×C₆₁	M₁₆	1	2	yes	—
32	C₂×C₄×C₆₁	D₈	1	2	yes	—
33	C₂×C₄×C₆₁	Q₁₆	1	2	yes	—
34	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
37	C₈×C₆₁	—	2	2	no	—
38	C₈×C₆₁	C₂×C₈	1	4	yes	complemented
39	#92418	—	4	4	no	—
40	C₂²×C₆₁	C₄⋊C₈	1	2	yes	—
41	C₂²×C₆₁	#107	1	2	yes	—
42	C₂²×C₆₁	#108	1	2	yes	—
43	C₂²×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₄×C₆₁	—	2	1	no	—
45	C₄×C₆₁	—	2	1	no	—
46	C₄×C₆₁	C₄⋊C₈	1	2	yes	—
47	C₄×C₆₁	#107	1	2	yes	—
48	C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
49	C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
50	C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
51	C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
52	C₂×#335	—	61	1	no	maximal, Sylow
53	C₂×C₆₁	#335	1	4	yes	complemented
54	C₂×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
55	C₂×C₆₁	C₂×#107	1	1	yes	characteristic
56	C₂×C₆₁	C₂×#108	1	1	yes	characteristic
57	C₂×C₄⋊C₈	—	61	2	no	—
58	C₂×C₄×C₈	D₆₁	1	1	yes	characteristic
59	#335	—	61	4	no	—
60	C₆₁	C₂×#335	1	1	yes	characteristic, complemented, Sylow
61	C₂²×C₈	—	122	2	no	—
62	C₂²×C₈	Dic₆₁	1	1	yes	characteristic
63	C₂²×C₈	Dic₆₁	1	1	yes	characteristic
64	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
65	C₄⋊C₈	—	61	4	no	—
66	C₄⋊C₈	—	122	2	no	—
67	C₄×C₈	Dic₆₁	1	2	yes	—
68	C₄×C₈	C₂×D₆₁	1	2	yes	—
69	C₂²×C₄	Dic₁₂₂	1	1	yes	characteristic
70	C₂²×C₄	D₂₄₄	1	1	yes	characteristic
71	C₂²×C₄	C₂×Dic₆₁	1	1	yes	characteristic
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	122	4	no	—
74	C₂×C₈	—	244	4	no	—
75	C₂×C₈	#92418	1	4	yes	complemented
76	C₂×C₈	C₂×Dic₆₁	1	2	yes	—
77	C₂×C₈	C₂×Dic₆₁	1	2	yes	—
78	C₄²	Dic₁₂₂	1	1	yes	characteristic
79	C₄²	D₂₄₄	1	1	yes	characteristic
80	C₄²	C₂×Dic₆₁	1	1	yes	characteristic
81	C₄²	C₂²×D₆₁	1	1	yes	characteristic
82	C₂³	C₄⋊Dic₆₁	1	1	yes	characteristic
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	D₄₈₈	1	2	yes	—
85	C₂×C₄	Dic₂₄₄	1	2	yes	—
86	C₂×C₄	C₂×#92418	1	2	yes	—
87	C₂×C₄	#11758194	1	2	yes	—
88	C₂×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
89	C₂×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
90	C₂×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
91	C₂×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
92	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
93	C₂×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
94	C₈	—	2	2	no	—
95	C₈	—	244	4	no	—
96	C₈	C₂×#92418	1	4	yes	complemented
97	C₂²	C₄⋊#92418	1	2	yes	—
98	C₂²	#14519711	1	2	yes	—
99	C₂²	#14519713	1	2	yes	—
100	C₂²	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
101	C₄	—	2	1	no	—
102	C₄	—	2	1	no	—
103	C₄	C₄⋊#92418	1	2	yes	—
104	C₄	#14519711	1	2	yes	—
105	C₄	C₂²×#92418	1	1	yes	characteristic
106	C₄	C₂×#11758194	1	1	yes	characteristic
107	C₄	C₂×D₄₈₈	1	1	yes	characteristic
108	C₄	C₂×Dic₂₄₄	1	1	yes	characteristic
109	C₂	#16839515	1	4	yes	direct product factor
110	C₂	C₂×C₄⋊#92418	1	1	yes	characteristic
111	C₂	C₂×#14519711	1	1	yes	characteristic
112	C₂	C₂×#14519713	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#92418

C₂

yes

maximal

#16839515

C₂

yes

maximal, direct product factor

C₂×C₄×C₈×C₆₁

C₂

yes

maximal, characteristic

C₂²×#92418

—

C₄⋊#92418

—

C₄⋊#92418

C₂²

yes

—

C₂²×C₈×C₆₁

C₄

yes

characteristic

C₂²×C₈×C₆₁

C₄

yes

characteristic

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₄×C₈×C₆₁

C₄

yes

—

C₄×C₈×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

Q₈

yes

characteristic

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₈

yes

complemented

C₂×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₈×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

Q₈

yes

characteristic

C₄²×C₆₁

C₂³

yes

characteristic

C₂×#92418

—

C₂×#92418

—

C₂³×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×C₈

yes

—

C₂×C₄×C₆₁

M₁₆

yes

—

C₂×C₄×C₆₁

D₈

yes

—

C₂×C₄×C₆₁

Q₁₆

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₈×C₆₁

—

C₈×C₆₁

C₂×C₈

yes

complemented

#92418

—

C₂²×C₆₁

C₄⋊C₈

yes

—

C₂²×C₆₁

#107

yes

—

C₂²×C₆₁

#108

yes

—

C₂²×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₄⋊C₈

yes

—

C₄×C₆₁

#107

yes

—

C₄×C₆₁

C₂²×C₈

yes

characteristic

C₄×C₆₁

C₂×M₁₆

yes

characteristic

C₄×C₆₁

C₂×D₈

yes

characteristic

C₄×C₆₁

C₂×Q₁₆

yes

—

C₂×C₆₁

yes

C₂×C₆₁

C₂×C₄⋊C₈

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₄⋊C₈

—

C₂×C₄×C₈

D₆₁

yes

characteristic

#335

—

C₆₁

C₂×#335

yes

characteristic, complemented, Sylow

C₂²×C₈

—

122

—

C₂²×C₈

Dic₆₁

yes

characteristic

C₂²×C₈

Dic₆₁

yes

characteristic

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

122

—

C₄×C₈

Dic₆₁

yes

—

C₄×C₈

C₂×D₆₁

yes

—

C₂²×C₄

Dic₁₂₂

yes

characteristic

C₂²×C₄

D₂₄₄

yes

characteristic

C₂²×C₄

C₂×Dic₆₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

244

—

C₂×C₈

#92418

yes

complemented

C₂×C₈

C₂×Dic₆₁

yes

—

C₂×C₈

C₂×Dic₆₁

yes

—

C₄²

Dic₁₂₂

yes

characteristic

C₄²

D₂₄₄

yes

characteristic

C₄²

C₂×Dic₆₁

yes

characteristic

C₄²

C₂²×D₆₁

yes

characteristic

C₂³

C₄⋊Dic₆₁

yes

characteristic

C₂×C₄

—

C₂×C₄

D₄₈₈

yes

—

C₂×C₄

Dic₂₄₄

yes

—

C₂×C₄

C₂×#92418

yes

—

C₂×C₄

#11758194

yes

—

C₂×C₄

C₄⋊Dic₆₁

yes

characteristic

C₂×C₄

C₄⋊Dic₆₁

yes

characteristic

C₂×C₄

C₄⋊Dic₆₁

yes

characteristic

C₂×C₄

C₂×Dic₁₂₂

yes

characteristic

C₂×C₄

C₂×D₂₄₄

yes

characteristic

C₂×C₄

C₂²×Dic₆₁

yes

characteristic

C₈

—

C₈

—

244

—

C₈

C₂×#92418

yes

complemented

C₂²

C₄⋊#92418

yes

—

C₂²

#14519711

yes

—

C₂²

#14519713

yes

—

100

C₂²

C₂×C₄⋊Dic₆₁

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

C₄⋊#92418

yes

—

104

C₄

#14519711

yes

—

105

C₄

C₂²×#92418

yes

characteristic

106

C₄

C₂×#11758194

yes

characteristic

107

C₄

C₂×D₄₈₈

yes

characteristic

108

C₄

C₂×Dic₂₄₄

yes

characteristic

109

C₂

#16839515

yes

direct product factor

110

C₂

C₂×C₄⋊#92418

yes

characteristic

111

C₂

C₂×#14519711

yes

characteristic

112

C₂

C₂×#14519713

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	D₂₄₄
commutator	C₄×C₆₁	C₂²×C₈
Frattini	C₄²	C₂²×D₆₁
Fitting	C₂×C₄×C₈×C₆₁	C₂
socle	C₂³×C₆₁	C₄⋊C₄
hypercenter	C₂×C₄×C₈	D₆₁
hypocenter	C₆₁	C₂×#335
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂²×C₄

D₂₄₄

commutator

C₄×C₆₁

C₂²×C₈

Frattini

C₄²

C₂²×D₆₁

Fitting

C₂×C₄×C₈×C₆₁

C₂

socle

C₂³×C₆₁

C₄⋊C₄

hypercenter

C₂×C₄×C₈

D₆₁

hypocenter

C₆₁

C₂×#335

LCS

G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #16839515	C₂	direct (N × H)
16. C₂×C₈×C₆₁	C₈	semidirect
38. C₈×C₆₁	C₂×C₈	semidirect
53. C₂×C₆₁	#335	semidirect
60. C₆₁	C₂×#335	semidirect
75. C₂×C₈	#92418	semidirect
96. C₈	C₂×#92418	semidirect
109. C₂	#16839515	direct (N × H)

H ≅ G/N

type

2. #16839515

C₂

direct (N × H)

16. C₂×C₈×C₆₁

C₈

semidirect

38. C₈×C₆₁

C₂×C₈

semidirect

53. C₂×C₆₁

#335

semidirect

60. C₆₁

C₂×#335

semidirect

75. C₂×C₈

#92418

semidirect

96. C₈

C₂×#92418

semidirect

109. C₂

#16839515

direct (N × H)

Subgroups — C₂×#16839515 #21444563

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16839515 #21444563

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16839515 #21444563