Group #21458794 — subgroups

Subgroups — #21458794

Order 7808.

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

normal subgroups only

This group has 460 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	#16840434	C₂	1	2	yes	maximal, complemented
2	#16840139	C₂	1	3	yes	maximal, complemented
3	#16839958	C₂	1	6	yes	maximal, complemented
4	C₄×D₄₈₈	C₂	1	3	yes	maximal, complemented
5	C₈×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11758226	—	2	2	no	—
7	C₂×D₄₈₈	C₂²	1	3	yes	—
8	C₂×D₄₈₈	C₂²	1	6	yes	—
9	#14519774	C₂²	1	6	yes	—
10	#14519752	—	2	6	no	—
11	C₄×D₂₄₄	C₂²	1	6	yes	—
12	#14519711	C₂²	1	1	yes	characteristic
13	#14519699	C₂²	1	6	yes	—
14	C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₆₁	C₂²	1	3	yes	—
16	C₄×C₈×C₆₁	C₂²	1	3	yes	—
17	C₂×Q₈×C₆₁	D₄	1	1	yes	characteristic
18	#11758226	—	4	8	no	—
19	C₂×D₂₄₄	—	2	6	no	—
20	C₂×D₂₄₄	—	2	6	no	—
21	C₂×D₂₄₄	C₂³	1	6	yes	—
22	C₂×C₄×D₆₁	—	2	6	no	—
23	C₂×C₈×C₆₁	C₂³	1	3	yes	—
24	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₆₁	D₄	1	3	yes	—
26	C₄²×C₆₁	C₂³	1	3	yes	—
27	#11758198	—	2	6	no	—
28	C₄⋊Dic₆₁	C₂³	1	2	yes	—
29	D₄₈₈	—	2	6	no	—
30	D₄₈₈	—	4	6	no	—
31	C₂²×D₆₁	—	2	6	no	—
32	Q₈×C₆₁	D₈	1	4	yes	complemented
33	C₂×C₄×C₆₁	C₂×D₄	1	3	yes	—
34	C₂×C₄×C₆₁	C₂×D₄	1	3	yes	—
35	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
36	C₂×Dic₆₁	—	2	2	no	—
37	D₂₄₄	—	2	6	no	—
38	D₂₄₄	—	2	6	no	—
39	D₂₄₄	—	4	12	no	—
40	D₂₄₄	—	8	6	no	—
41	C₄×D₆₁	—	4	12	no	—
42	C₈×C₆₁	—	2	3	no	—
43	C₈×C₆₁	#41	1	2	yes	complemented
44	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
45	C₂×D₆₁	—	4	6	no	—
46	C₂×D₆₁	—	4	6	no	—
47	C₂×D₆₁	—	4	6	no	—
48	C₄×C₆₁	—	2	3	no	—
49	C₄×C₆₁	C₂×D₈	1	6	yes	—
50	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
51	C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
52	Dic₆₁	—	4	2	no	—
53	#3391	—	61	1	no	maximal, Sylow
54	C₂×C₆₁	#550	1	1	yes	characteristic
55	C₂×C₆₁	C₂²×D₈	1	1	yes	characteristic
56	C₂×C₆₁	#576	1	1	yes	characteristic
57	D₆₁	—	8	6	no	—
58	#550	—	61	2	no	—
59	#493	—	61	3	no	—
60	#459	—	61	6	no	—
61	C₈×Q₈	D₆₁	1	1	yes	characteristic, complemented
62	C₄×D₈	—	61	3	no	—
63	C₆₁	#3391	1	1	yes	characteristic, complemented, Sylow
64	C₂×#41	—	122	2	no	—
65	C₂×D₈	—	61	3	no	—
66	C₂×D₈	—	61	6	no	—
67	#127	—	61	6	no	—
68	#121	—	122	6	no	—
69	C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
70	C₄×D₄	—	61	6	no	—
71	#107	—	61	1	no	—
72	C₄⋊C₈	C₂×D₆₁	1	3	yes	—
73	#102	—	61	6	no	—
74	C₄×C₈	C₂×D₆₁	1	3	yes	—
75	#41	—	244	8	no	—
76	C₂×Q₈	D₂₄₄	1	1	yes	characteristic
77	C₂×D₄	—	61	6	no	—
78	C₂×D₄	—	122	6	no	—
79	C₂×D₄	—	122	6	no	—
80	C₂²×C₄	—	122	6	no	—
81	D₈	—	122	6	no	—
82	D₈	—	244	6	no	—
83	C₂×C₈	C₂²×D₆₁	1	3	yes	—
84	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
85	C₄⋊C₄	—	61	2	no	—
86	C₄⋊C₄	D₂₄₄	1	3	yes	—
87	C₂²⋊C₄	—	122	6	no	—
88	C₄²	C₂²×D₆₁	1	3	yes	—
89	C₂³	—	122	6	no	—
90	Q₈	D₄₈₈	1	4	yes	complemented
91	D₄	—	122	6	no	—
92	D₄	—	122	6	no	—
93	D₄	—	488	6	no	—
94	D₄	—	244	12	no	—
95	C₂×C₄	—	122	2	no	—
96	C₂×C₄	—	244	12	no	—
97	C₂×C₄	C₂×D₂₄₄	1	3	yes	—
98	C₂×C₄	C₂×D₂₄₄	1	3	yes	—
99	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
100	C₈	—	2	3	no	—
101	C₈	#11758226	1	2	yes	complemented
102	C₂²	—	244	6	no	—
103	C₂²	—	244	6	no	—
104	C₂²	—	244	6	no	—
105	C₂²	C₂²×D₂₄₄	1	1	yes	characteristic
106	C₄	—	2	3	no	—
107	C₄	—	244	2	no	—
108	C₄	C₂×D₄₈₈	1	6	yes	—
109	C₄	C₂×#11758226	1	1	yes	characteristic
110	C₄	#14519823	1	1	yes	characteristic
111	C₂	—	488	6	no	—
112	C₂	#16840434	1	1	yes	characteristic
113	C₂	C₂²×D₄₈₈	1	1	yes	characteristic
114	C₂	#16840567	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840434

C₂

yes

maximal, complemented

#16840139

C₂

yes

maximal, complemented

#16839958

C₂

yes

maximal, complemented

C₄×D₄₈₈

C₂

yes

maximal, complemented

C₈×Q₈×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂×#11758226

—

C₂×D₄₈₈

C₂²

yes

—

C₂×D₄₈₈

C₂²

yes

—

#14519774

C₂²

yes

—

#14519752

—

C₄×D₂₄₄

C₂²

yes

—

#14519711

C₂²

yes

characteristic

#14519699

C₂²

yes

—

C₄×Q₈×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

—

C₂×Q₈×C₆₁

D₄

yes

characteristic

#11758226

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

C₂³

yes

—

C₂×C₄×D₆₁

—

C₂×C₈×C₆₁

C₂³

yes

—

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄⋊C₄×C₆₁

D₄

yes

—

C₄²×C₆₁

C₂³

yes

—

#11758198

—

C₄⋊Dic₆₁

C₂³

yes

—

D₄₈₈

—

D₄₈₈

—

C₂²×D₆₁

—

Q₈×C₆₁

D₈

yes

complemented

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

C₄×D₆₁

—

C₈×C₆₁

—

C₈×C₆₁

#41

yes

complemented

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×D₈

yes

—

C₄×C₆₁

C₂×#41

yes

characteristic

C₄×C₆₁

H(5,𝔽₂)

yes

characteristic

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

C₂²×D₈

yes

characteristic

C₂×C₆₁

#576

yes

characteristic

D₆₁

—

#550

—

#493

—

#459

—

C₈×Q₈

D₆₁

yes

characteristic, complemented

C₄×D₈

—

C₆₁

#3391

yes

characteristic, complemented, Sylow

C₂×#41

—

122

—

C₂×D₈

—

C₂×D₈

—

#127

—

#121

—

122

—

C₄×Q₈

C₂×D₆₁

yes

characteristic

C₄×D₄

—

#107

—

C₄⋊C₈

C₂×D₆₁

yes

—

#102

—

C₄×C₈

C₂×D₆₁

yes

—

#41

—

244

—

C₂×Q₈

D₂₄₄

yes

characteristic

C₂×D₄

—

C₂×D₄

—

122

—

C₂×D₄

—

122

—

C₂²×C₄

—

122

—

D₈

—

122

—

D₈

—

244

—

C₂×C₈

C₂²×D₆₁

yes

—

C₂×C₈

C₂²×D₆₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

D₂₄₄

yes

—

C₂²⋊C₄

—

122

—

C₄²

C₂²×D₆₁

yes

—

C₂³

—

122

—

Q₈

D₄₈₈

yes

complemented

D₄

—

122

—

D₄

—

122

—

D₄

—

488

—

D₄

—

244

—

C₂×C₄

—

122

—

C₂×C₄

—

244

—

C₂×C₄

C₂×D₂₄₄

yes

—

C₂×C₄

C₂×D₂₄₄

yes

—

C₂×C₄

C₂³×D₆₁

yes

characteristic

100

C₈

—

101

C₈

#11758226

yes

complemented

102

C₂²

—

244

—

103

C₂²

—

244

—

104

C₂²

—

244

—

105

C₂²

C₂²×D₂₄₄

yes

characteristic

106

C₄

—

107

C₄

—

244

—

108

C₄

C₂×D₄₈₈

yes

—

109

C₄

C₂×#11758226

yes

characteristic

110

C₄

#14519823

yes

characteristic

111

C₂

—

488

—

112

C₂

#16840434

yes

characteristic

113

C₂

C₂²×D₄₈₈

yes

characteristic

114

C₂

#16840567

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₈×Q₈×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	C₈×Q₈	D₆₁
hypocenter	C₆₁	#3391
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₂₄₄

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₈×Q₈×C₆₁

C₂

socle

C₂²×C₆₁

C₂²×D₄

hypercenter

C₈×Q₈

D₆₁

hypocenter

C₆₁

#3391

LCS

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
1. #16840434	C₂	semidirect
2. #16840139	C₂	semidirect
3. #16839958	C₂	semidirect
4. C₄×D₄₈₈	C₂	semidirect
5. C₈×Q₈×C₆₁	C₂	semidirect
32. Q₈×C₆₁	D₈	semidirect
43. C₈×C₆₁	#41	semidirect
61. C₈×Q₈	D₆₁	semidirect
63. C₆₁	#3391	semidirect
90. Q₈	D₄₈₈	semidirect
101. C₈	#11758226	semidirect

H ≅ G/N

type

1. #16840434

C₂

semidirect

2. #16840139

C₂

semidirect

3. #16839958

C₂

semidirect

4. C₄×D₄₈₈

C₂

semidirect

5. C₈×Q₈×C₆₁

C₂

semidirect

32. Q₈×C₆₁

D₈

semidirect

43. C₈×C₆₁

#41

semidirect

61. C₈×Q₈

D₆₁

semidirect

63. C₆₁

#3391

semidirect

90. Q₈

D₄₈₈

semidirect

101. C₈

#11758226

semidirect

Subgroups — #21458794

Conjugacy classes of subgroups

Complements