Subgroups — #46813205

Order 34432.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31771037	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₂₁₅₂	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₄₃₀₄	C₂	1	2	yes	maximal, complemented
4	#31770444	C₂	1	2	yes	maximal, complemented
5	M₆₄×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#22575475	—	2	1	no	—
7	C₂×Dic₁₀₇₆	—	2	1	no	—
8	Dic₂₁₅₂	—	2	1	no	—
9	Dic₂₁₅₂	C₂²	1	2	yes	—
10	Dic₂₁₅₂	C₂²	1	1	yes	characteristic
11	#22575398	—	2	1	no	—
12	D₄₃₀₄	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₂₆₉	C₂²	1	1	yes	characteristic
14	C₃₂×C₂₆₉	C₂²	1	2	yes	—
15	#16864402	—	4	1	no	—
16	C₂×Dic₅₃₈	—	4	1	no	—
17	Dic₁₀₇₆	—	2	1	no	—
18	Dic₁₀₇₆	—	2	2	no	—
19	Dic₁₀₇₆	—	4	1	no	—
20	#16864391	—	4	1	no	—
21	D₂₁₅₂	—	2	1	no	—
22	C₂×C₈×C₂₆₉	D₄	1	1	yes	characteristic
23	C₁₆×C₂₆₉	D₄	1	1	yes	characteristic
24	C₁₆×C₂₆₉	C₂³	1	1	yes	characteristic
25	Dic₅₃₈	—	4	1	no	—
26	Dic₅₃₈	—	4	2	no	—
27	Dic₅₃₈	—	8	1	no	—
28	D₁₀₇₆	—	4	1	no	—
29	C₂²⋊D₂₆₉	—	8	1	no	—
30	C₂×Dic₂₆₉	—	8	1	no	—
31	C₄×D₂₆₉	—	8	1	no	—
32	C₂×C₄×C₂₆₉	D₈	1	1	yes	characteristic
33	C₈×C₂₆₉	D₈	1	1	yes	characteristic
34	C₈×C₂₆₉	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₂₆₉	D₁₆	1	1	yes	characteristic
36	C₂×D₂₆₉	—	8	1	no	—
37	C₄×C₂₆₉	D₁₆	1	1	yes	characteristic
38	C₄×C₂₆₉	C₂×D₈	1	1	yes	characteristic
39	Dic₂₆₉	—	8	1	no	—
40	Dic₂₆₉	—	8	2	no	—
41	C₂×C₂₆₉	—	2	1	no	—
42	C₂×C₂₆₉	C₂×D₁₆	1	1	yes	characteristic
43	D₂₆₉	—	16	1	no	—
44	C₂₆₉	#2264	1	1	yes	characteristic, complemented, Sylow
45	#2264	—	269	1	no	maximal, Sylow
46	#508	—	269	1	no	—
47	C₂×Q₃₂	—	269	1	no	—
48	Q₆₄	—	269	2	no	—
49	SD₆₄	—	269	2	no	—
50	M₆₄	D₂₆₉	1	1	yes	characteristic, complemented
51	#135	—	538	1	no	—
52	C₂×Q₁₆	—	538	1	no	—
53	Q₃₂	—	269	2	no	—
54	Q₃₂	—	269	1	no	—
55	Q₃₂	—	538	1	no	—
56	SD₃₂	—	538	1	no	—
57	D₁₆	—	269	1	no	—
58	C₂×C₁₆	C₂×D₂₆₉	1	1	yes	characteristic
59	C₃₂	C₂×D₂₆₉	1	2	yes	—
60	#41	—	1076	1	no	—
61	C₂×Q₈	—	1076	1	no	—
62	Q₁₆	—	538	1	no	—
63	Q₁₆	—	538	2	no	—
64	Q₁₆	—	1076	1	no	—
65	SD₁₆	—	1076	1	no	—
66	D₈	—	538	1	no	—
67	C₂×C₈	D₁₀₇₆	1	1	yes	characteristic
68	C₁₆	D₁₀₇₆	1	1	yes	characteristic
69	C₁₆	C₂²×D₂₆₉	1	1	yes	characteristic
70	Q₈	—	1076	1	no	—
71	Q₈	—	1076	2	no	—
72	Q₈	—	2152	1	no	—
73	D₄	—	1076	1	no	—
74	D₄	—	2152	1	no	—
75	C₂×C₄	—	2152	1	no	—
76	C₂×C₄	—	2152	1	no	—
77	C₂×C₄	D₂₁₅₂	1	1	yes	characteristic
78	C₈	D₂₁₅₂	1	1	yes	characteristic
79	C₈	C₂×D₁₀₇₆	1	1	yes	characteristic
80	C₂²	—	2152	1	no	—
81	C₂²	D₄₃₀₄	1	1	yes	characteristic
82	C₄	—	2152	1	no	—
83	C₄	—	2152	2	no	—
84	C₄	D₄₃₀₄	1	1	yes	characteristic
85	C₄	C₂×D₂₁₅₂	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	4304	1	no	—
88	C₂	C₂×D₄₃₀₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄₃₀₄
commutator, cosocle	C₁₆×C₂₆₉	C₂³
Frattini	C₁₆	C₂²×D₂₆₉
Fitting	M₆₄×C₂₆₉	C₂
socle	C₂×C₂₆₉	C₂×D₁₆
prime radical	#31771037	C₂
hypercenter	M₆₄	D₂₆₉
hypocenter	C₂₆₉	#2264
LCS	G ▷ C₁₆×C₂₆₉ ▷ C₈×C₂₆₉ ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₂₁₅₂	C₂	semidirect
3. Dic₄₃₀₄	C₂	semidirect
4. #31770444	C₂	semidirect
5. M₆₄×C₂₆₉	C₂	semidirect
44. C₂₆₉	#2264	semidirect
50. M₆₄	D₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
#31771037	2
C₂×Dic₂₁₅₂	2
Dic₄₃₀₄	2
#31770444	2
M₆₄×C₂₆₉	2
#2264	269