Subgroups — #31823699

Order 17984.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₂₄₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#22582853	C₂	1	2	yes	maximal, complemented
3	#22582813	C₂	1	1	yes	maximal, characteristic, complemented
4	#22582801	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×C₂₈₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₁₂₄	C₂²	1	2	yes	—
7	C₂×C₂²⋊D₂₈₁	—	2	2	no	—
8	D₂₂₄₈	—	2	2	no	—
9	C₄⋊Dic₂₈₁	C₂²	1	2	yes	—
10	#17985115	—	2	2	no	—
11	C₂²×C₄×C₂₈₁	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₂₈₁	—	2	2	no	—
13	C₂×C₈×C₂₈₁	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₂₈₁	C₂²	1	1	yes	characteristic
15	C₂²×D₂₈₁	—	2	2	no	—
16	D₁₁₂₄	—	2	2	no	—
17	D₁₁₂₄	—	2	2	no	—
18	C₂²⋊D₂₈₁	—	4	4	no	—
19	C₂×Dic₂₈₁	—	2	2	no	—
20	C₂³×C₂₈₁	D₄	1	1	yes	characteristic
21	C₂×C₄×C₂₈₁	—	2	1	no	—
22	C₂×C₄×C₂₈₁	—	2	1	no	—
23	C₂×C₄×C₂₈₁	D₄	1	1	yes	characteristic
24	C₂×C₄×C₂₈₁	C₂³	1	1	yes	characteristic
25	C₈×C₂₈₁	—	2	1	no	—
26	C₈×C₂₈₁	D₄	1	2	yes	complemented
27	C₂²×C₂₈₁	—	2	1	no	—
28	C₂²×C₂₈₁	—	2	1	no	—
29	C₂²×C₂₈₁	D₈	1	2	yes	complemented
30	C₂²×C₂₈₁	C₂×D₄	1	1	yes	characteristic
31	C₂×D₂₈₁	—	4	2	no	—
32	C₂×D₂₈₁	—	4	2	no	—
33	C₂×D₂₈₁	—	4	2	no	—
34	C₄×C₂₈₁	—	2	1	no	—
35	C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
36	C₄×C₂₈₁	#41	1	1	yes	characteristic
37	Dic₂₈₁	—	4	2	no	—
38	C₂×C₂₈₁	—	2	2	no	—
39	C₂×C₂₈₁	#121	1	1	yes	characteristic
40	C₂×C₂₈₁	C₂×D₈	1	1	yes	characteristic
41	C₂×C₂₈₁	#135	1	1	yes	characteristic
42	D₂₈₁	—	8	2	no	—
43	C₂₈₁	#466	1	1	yes	characteristic, complemented, Sylow
44	#466	—	281	1	no	maximal, Sylow
45	C₂×D₈	—	281	1	no	—
46	C₂²×C₈	D₂₈₁	1	1	yes	characteristic, complemented
47	#121	—	281	2	no	—
48	#107	—	281	1	no	—
49	#102	—	281	2	no	—
50	C₂×D₄	—	281	2	no	—
51	C₂×D₄	—	562	2	no	—
52	C₂²×C₄	C₂×D₂₈₁	1	1	yes	characteristic
53	D₈	—	562	2	no	—
54	C₂×C₈	—	2	2	no	—
55	C₂×C₈	C₂×D₂₈₁	1	1	yes	characteristic
56	C₂×C₈	C₂×D₂₈₁	1	1	yes	characteristic
57	C₄⋊C₄	—	281	2	no	—
58	C₂²⋊C₄	—	562	2	no	—
59	C₂³	—	562	2	no	—
60	C₂³	D₁₁₂₄	1	1	yes	characteristic
61	D₄	—	562	2	no	—
62	D₄	—	562	2	no	—
63	D₄	—	1124	4	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	562	2	no	—
67	C₂×C₄	D₁₁₂₄	1	1	yes	characteristic
68	C₂×C₄	C₂²×D₂₈₁	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	C₂²⋊D₂₈₁	1	2	yes	complemented
71	C₂²	—	2	1	no	—
72	C₂²	—	2	1	no	—
73	C₂²	—	1124	2	no	—
74	C₂²	—	1124	2	no	—
75	C₂²	—	1124	2	no	—
76	C₂²	D₂₂₄₈	1	2	yes	complemented
77	C₂²	C₂×D₁₁₂₄	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	1124	2	no	—
80	C₄	C₂×C₂²⋊D₂₈₁	1	1	yes	characteristic
81	C₄	#17985138	1	1	yes	characteristic
82	C₂	—	2	2	no	—
83	C₂	—	2248	2	no	—
84	C₂	#22582853	1	1	yes	characteristic
85	C₂	C₂×D₂₂₄₈	1	1	yes	characteristic
86	C₂	#22582901	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₁₂₄
commutator, cosocle	C₂×C₄×C₂₈₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₈₁
Fitting	C₂²×C₈×C₂₈₁	C₂
socle	C₂²×C₂₈₁	C₂×D₄
hypercenter	C₂²×C₈	D₂₈₁
hypocenter	C₂₈₁	#466
LCS	G ▷ C₂×C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×D₂₂₄₈	C₂	semidirect
2. #22582853	C₂	semidirect
3. #22582813	C₂	semidirect
4. #22582801	C₂	semidirect
5. C₂²×C₈×C₂₈₁	C₂	semidirect
26. C₈×C₂₈₁	D₄	semidirect
29. C₂²×C₂₈₁	D₈	semidirect
43. C₂₈₁	#466	semidirect
46. C₂²×C₈	D₂₈₁	semidirect
70. C₈	C₂²⋊D₂₈₁	semidirect
76. C₂²	D₂₂₄₈	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×D₂₂₄₈	2
#22582853	2
#22582813	2
#22582801	2
C₂²×C₈×C₂₈₁	2
#466	281