Subgroups — #34209264

Order 20224.

270 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₅₂₈	C₂	1	4	yes	maximal, complemented
2	#23765650	C₂	1	2	yes	maximal, complemented
3	C₄×C₃₂×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₂₆₄	—	2	2	no	—
5	C₂×D₁₂₆₄	C₂²	1	4	yes	—
6	#19098940	—	2	2	no	—
7	D₂₅₂₈	—	2	8	no	—
8	C₂×C₃₂×C₇₉	C₂²	1	2	yes	—
9	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
10	C₂×D₆₃₂	—	2	4	no	—
11	C₂×D₆₃₂	—	4	2	no	—
12	#14579158	—	4	2	no	—
13	D₁₂₆₄	—	2	8	no	—
14	D₁₂₆₄	—	4	4	no	—
15	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
17	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
18	C₃₂×C₇₉	D₄	1	4	yes	complemented
19	C₂×D₃₁₆	—	4	4	no	—
20	C₂×D₃₁₆	—	8	4	no	—
21	D₆₃₂	—	4	4	no	—
22	D₆₃₂	—	4	4	no	—
23	D₆₃₂	—	8	4	no	—
24	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
25	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
26	C₄²×C₇₉	D₈	1	1	yes	characteristic
27	C₁₆×C₇₉	D₈	1	2	yes	—
28	C₁₆×C₇₉	C₂×D₄	1	2	yes	—
29	C₂²×D₇₉	—	8	4	no	—
30	C₂×C₄×C₇₉	D₁₆	1	2	yes	—
31	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
32	D₃₁₆	—	8	4	no	—
33	D₃₁₆	—	8	4	no	—
34	D₃₁₆	—	16	8	no	—
35	C₈×C₇₉	D₁₆	1	2	yes	—
36	C₈×C₇₉	#127	1	1	yes	characteristic
37	C₈×C₇₉	C₂×D₈	1	1	yes	characteristic
38	C₂²×C₇₉	C₂×D₁₆	1	1	yes	characteristic
39	C₂×D₇₉	—	16	4	no	—
40	C₂×D₇₉	—	16	8	no	—
41	C₄×C₇₉	D₃₂	1	4	yes	complemented
42	C₄×C₇₉	#493	1	1	yes	characteristic
43	C₄×C₇₉	C₂×D₁₆	1	1	yes	characteristic
44	#13725	—	79	1	no	maximal, Sylow
45	C₂×C₇₉	#2246	1	1	yes	characteristic
46	C₂×C₇₉	C₂×D₃₂	1	2	yes	—
47	D₇₉	—	32	4	no	—
48	C₂×D₃₂	—	79	4	no	—
49	#2246	—	79	2	no	—
50	C₄×C₃₂	D₇₉	1	1	yes	characteristic, complemented
51	C₇₉	#13725	1	1	yes	characteristic, complemented, Sylow
52	C₂×D₁₆	—	79	4	no	—
53	C₂×D₁₆	—	158	2	no	—
54	#493	—	158	2	no	—
55	D₃₂	—	158	8	no	—
56	C₂×C₃₂	C₂×D₇₉	1	2	yes	—
57	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
58	C₂×D₈	—	316	2	no	—
59	C₂×D₈	—	158	4	no	—
60	#127	—	316	2	no	—
61	D₁₆	—	316	4	no	—
62	D₁₆	—	158	8	no	—
63	C₂×C₁₆	D₃₁₆	1	1	yes	characteristic
64	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
65	C₄×C₈	D₃₁₆	1	1	yes	characteristic
66	C₃₂	D₃₁₆	1	4	yes	complemented
67	C₂×D₄	—	316	4	no	—
68	C₂×D₄	—	632	4	no	—
69	D₈	—	316	4	no	—
70	D₈	—	316	4	no	—
71	D₈	—	632	4	no	—
72	C₂×C₈	D₆₃₂	1	1	yes	characteristic
73	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
74	C₄²	D₆₃₂	1	1	yes	characteristic
75	C₁₆	D₆₃₂	1	2	yes	—
76	C₁₆	C₂×D₃₁₆	1	2	yes	—
77	C₂³	—	632	4	no	—
78	D₄	—	632	4	no	—
79	D₄	—	632	4	no	—
80	D₄	—	1264	8	no	—
81	C₂×C₄	D₁₂₆₄	1	2	yes	—
82	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
83	C₈	D₁₂₆₄	1	2	yes	—
84	C₈	#14579158	1	1	yes	characteristic
85	C₈	C₂×D₆₃₂	1	1	yes	characteristic
86	C₂²	—	1264	4	no	—
87	C₂²	—	1264	8	no	—
88	C₂²	C₂×D₁₂₆₄	1	1	yes	characteristic
89	C₄	D₂₅₂₈	1	4	yes	complemented
90	C₄	#19098940	1	1	yes	characteristic
91	C₄	C₂×D₁₂₆₄	1	1	yes	characteristic
92	C₂	—	2528	4	no	—
93	C₂	#23765650	1	1	yes	characteristic
94	C₂	C₂×D₂₅₂₈	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₂₆₄
commutator, cosocle	C₂×C₁₆×C₇₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₉
Fitting	C₄×C₃₂×C₇₉	C₂
socle	C₂²×C₇₉	C₂×D₁₆
hypercenter	C₄×C₃₂	D₇₉
hypocenter	C₇₉	#13725
LCS	G ▷ C₂×C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×D₂₅₂₈	C₂	semidirect
2. #23765650	C₂	semidirect
3. C₄×C₃₂×C₇₉	C₂	semidirect
18. C₃₂×C₇₉	D₄	semidirect
41. C₄×C₇₉	D₃₂	semidirect
50. C₄×C₃₂	D₇₉	semidirect
51. C₇₉	#13725	semidirect
66. C₃₂	D₃₁₆	semidirect
89. C₄	D₂₅₂₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₅₂₈	2
#23765650	2
C₄×C₃₂×C₇₉	2
#13725	79