Group #21423688 — subgroups

Subgroups — #21423688

Order 7552.

224 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 77 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	#16837807	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837779	C₂	1	1	yes	maximal, characteristic
3	#16837775	C₂	1	2	yes	maximal, complemented
4	#16837749	C₂	1	2	yes	maximal, complemented
5	C₈²×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11745660	—	2	4	no	—
7	C₂×D₄₇₂	—	2	2	no	—
8	#14273727	C₂²	1	2	yes	—
9	#14273724	C₂²	1	2	yes	—
10	#14273661	—	2	2	no	—
11	#14273649	—	2	4	no	—
12	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×D₂₃₆	—	2	2	no	—
16	C₂×D₂₃₆	—	2	2	no	—
17	C₂×D₂₃₆	—	2	2	no	—
18	C₂×Dic₁₁₈	—	2	2	no	—
19	C₂×C₈×C₅₉	D₄	1	2	yes	—
20	C₂×C₈×C₅₉	D₄	1	2	yes	—
21	C₂×C₈×C₅₉	D₄	1	2	yes	—
22	C₄²×C₅₉	C₂³	1	1	yes	characteristic
23	C₄⋊Dic₅₉	—	2	2	no	—
24	C₄⋊Dic₅₉	—	2	2	no	—
25	D₄₇₂	—	4	4	no	—
26	#11745660	—	4	8	no	—
27	C₂²×D₅₉	—	4	2	no	—
28	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
31	C₂×Dic₅₉	—	4	2	no	—
32	D₂₃₆	—	4	4	no	—
33	D₂₃₆	—	4	4	no	—
34	D₂₃₆	—	4	4	no	—
35	Dic₁₁₈	—	4	4	no	—
36	C₈×C₅₉	—	2	2	no	—
37	C₈×C₅₉	D₈	1	4	yes	complemented
38	C₈×C₅₉	SD₁₆	1	4	yes	complemented
39	C₂²×C₅₉	#127	1	1	yes	characteristic
40	C₂×D₅₉	—	8	2	no	—
41	C₂×D₅₉	—	8	2	no	—
42	C₂×D₅₉	—	8	2	no	—
43	C₄×C₅₉	C₂×D₈	1	2	yes	—
44	C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
45	C₄×C₅₉	#135	1	2	yes	—
46	Dic₅₉	—	8	2	no	—
47	#1706	—	59	1	no	maximal, Sylow
48	C₂×C₅₉	#492	1	1	yes	characteristic
49	C₂×C₅₉	#493	1	1	yes	characteristic
50	C₂×C₅₉	#495	1	1	yes	characteristic
51	D₅₉	—	16	2	no	—
52	#500	—	59	1	no	—
53	#493	—	59	1	no	—
54	#492	—	59	2	no	—
55	#486	—	59	2	no	—
56	C₈²	D₅₉	1	1	yes	characteristic, complemented
57	C₅₉	#1706	1	1	yes	characteristic, complemented, Sylow
58	C₂×SD₁₆	—	118	4	no	—
59	C₂×D₈	—	118	2	no	—
60	C₄⋊Q₈	—	59	2	no	—
61	#127	—	59	2	no	—
62	#107	—	118	2	no	—
63	#102	—	118	4	no	—
64	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
65	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
66	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
67	C₂×Q₈	—	118	2	no	—
68	C₂×D₄	—	118	2	no	—
69	C₂×D₄	—	118	2	no	—
70	C₂×D₄	—	118	2	no	—
71	SD₁₆	—	236	8	no	—
72	D₈	—	236	4	no	—
73	C₂×C₈	D₂₃₆	1	2	yes	—
74	C₂×C₈	D₂₃₆	1	2	yes	—
75	C₂×C₈	D₂₃₆	1	2	yes	—
76	C₄⋊C₄	—	118	2	no	—
77	C₄⋊C₄	—	118	2	no	—
78	C₄²	C₂²×D₅₉	1	1	yes	characteristic
79	C₂³	—	236	2	no	—
80	Q₈	—	236	4	no	—
81	D₄	—	236	4	no	—
82	D₄	—	236	4	no	—
83	D₄	—	236	4	no	—
84	C₂×C₄	—	236	2	no	—
85	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
86	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
87	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
88	C₈	—	2	2	no	—
89	C₈	#11745660	1	4	yes	complemented
90	C₈	D₄₇₂	1	4	yes	complemented
91	C₂²	—	472	2	no	—
92	C₂²	—	472	2	no	—
93	C₂²	—	472	2	no	—
94	C₂²	#14273724	1	1	yes	characteristic
95	C₄	—	472	2	no	—
96	C₄	C₂×D₄₇₂	1	2	yes	—
97	C₄	C₂×#11745660	1	2	yes	—
98	C₄	#14273749	1	2	yes	—
99	C₂	—	944	2	no	—
100	C₂	#16837775	1	1	yes	characteristic
101	C₂	#16837779	1	1	yes	characteristic
102	C₂	#16837786	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837807

C₂

yes

maximal, characteristic, complemented

#16837779

C₂

yes

maximal, characteristic

#16837775

C₂

yes

maximal, complemented

#16837749

C₂

yes

maximal, complemented

C₈²×C₅₉

C₂

yes

maximal, characteristic, complemented

C₂×#11745660

—

C₂×D₄₇₂

—

#14273727

C₂²

yes

—

#14273724

C₂²

yes

—

#14273661

—

#14273649

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

D₄

yes

—

C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

D₄₇₂

—

#11745660

—

C₂²×D₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

D₈

yes

complemented

C₈×C₅₉

SD₁₆

yes

complemented

C₂²×C₅₉

#127

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

C₂×D₈

yes

—

C₄×C₅₉

C₂×SD₁₆

yes

—

C₄×C₅₉

#135

yes

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

—

—

—

—

C₈²

D₅₉

yes

characteristic, complemented

C₅₉

#1706

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

118

—

C₂×D₈

—

118

—

C₄⋊Q₈

—

#127

—

#107

—

118

—

#102

—

118

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₄×C₈

C₂×D₅₉

yes

characteristic

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

118

—

C₂×D₄

—

118

—

C₂×D₄

—

118

—

C₂×D₄

—

118

—

SD₁₆

—

236

—

D₈

—

236

—

C₂×C₈

D₂₃₆

yes

—

C₂×C₈

D₂₃₆

yes

—

C₂×C₈

D₂₃₆

yes

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄²

C₂²×D₅₉

yes

characteristic

C₂³

—

236

—

Q₈

—

236

—

D₄

—

236

—

D₄

—

236

—

D₄

—

236

—

C₂×C₄

—

236

—

C₂×C₄

C₂×D₂₃₆

yes

characteristic

C₂×C₄

C₂×D₂₃₆

yes

characteristic

C₂×C₄

C₂×D₂₃₆

yes

characteristic

C₈

—

C₈

#11745660

yes

complemented

C₈

D₄₇₂

yes

complemented

C₂²

—

472

—

C₂²

—

472

—

C₂²

—

472

—

C₂²

#14273724

yes

characteristic

C₄

—

472

—

C₄

C₂×D₄₇₂

yes

—

C₄

C₂×#11745660

yes

—

C₄

#14273749

yes

—

C₂

—

944

—

100

C₂

#16837775

yes

characteristic

101

C₂

#16837779

yes

characteristic

102

C₂

#16837786

yes

characteristic

	subgroup	G/H
center	C₂²	#14273724
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₈²×C₅₉	C₂
socle	C₂²×C₅₉	#127
hypercenter	C₈²	D₅₉
hypocenter	C₅₉	#1706
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

subgroup

G/H

center

C₂²

#14273724

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₈²×C₅₉

C₂

socle

C₂²×C₅₉

#127

hypercenter

C₈²

D₅₉

hypocenter

C₅₉

#1706

LCS

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
1. #16837807	C₂	semidirect
3. #16837775	C₂	semidirect
4. #16837749	C₂	semidirect
5. C₈²×C₅₉	C₂	semidirect
37. C₈×C₅₉	D₈	semidirect
38. C₈×C₅₉	SD₁₆	semidirect
56. C₈²	D₅₉	semidirect
57. C₅₉	#1706	semidirect
89. C₈	#11745660	semidirect
90. C₈	D₄₇₂	semidirect

H ≅ G/N

type

1. #16837807

C₂

semidirect

3. #16837775

C₂

semidirect

4. #16837749

C₂

semidirect

5. C₈²×C₅₉

C₂

semidirect

37. C₈×C₅₉

D₈

semidirect

38. C₈×C₅₉

SD₁₆

semidirect

56. C₈²

D₅₉

semidirect

57. C₅₉

#1706

semidirect

89. C₈

#11745660

semidirect

90. C₈

D₄₇₂

semidirect

Subgroups — #21423688

Conjugacy classes of subgroups

Complements