Subgroups — #47981352

Order 35456.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31800434	C₂	1	1	yes	maximal, characteristic
2	#31799832	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₂×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#22579902	—	2	1	no	—
5	Dic₂₂₁₆	C₄	1	1	yes	characteristic
6	#22579825	—	2	1	no	—
7	D₄₄₃₂	C₄	1	1	yes	characteristic
8	#22579816	—	2	1	no	—
9	C₂×C₁₆×C₂₇₇	C₂²	1	1	yes	characteristic
10	C₃₂×C₂₇₇	—	2	1	no	—
11	#17984563	—	4	1	no	—
12	Dic₁₁₀₈	—	2	1	no	—
13	#17984552	—	4	1	no	—
14	D₂₂₁₆	—	2	1	no	—
15	#17984547	—	4	1	no	—
16	C₂×C₈×C₂₇₇	D₄	1	1	yes	characteristic
17	C₁₆×C₂₇₇	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₂₇₇	D₄	1	1	yes	characteristic
19	Dic₅₅₄	—	4	1	no	—
20	D₁₁₀₈	—	4	1	no	—
21	C₂²⋊D₂₇₇	—	8	1	no	—
22	C₄×D₂₇₇	—	8	1	no	—
23	#14555660	—	8	1	no	—
24	C₂×C₄×C₂₇₇	D₈	1	1	yes	characteristic
25	C₈×C₂₇₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₂₇₇	SD₁₆	1	1	yes	characteristic
27	C₂²×C₂₇₇	SD₃₂	1	1	yes	characteristic
28	C₂×D₂₇₇	—	8	1	no	—
29	C₄×C₂₇₇	#102	1	1	yes	characteristic
30	C₄×C₂₇₇	D₁₆	1	1	yes	characteristic
31	Dic₂₇₇	—	8	1	no	—
32	C₂×C₂₇₇	—	2	1	no	—
33	C₂×C₂₇₇	#357	1	1	yes	characteristic
34	D₂₇₇	—	16	1	no	—
35	C₂₇₇	#1417	1	1	yes	characteristic, complemented, Sylow
36	#1417	—	277	1	no	maximal, Sylow
37	#508	—	277	1	no	—
38	C₂×C₃₂	D₂₇₇	1	1	yes	characteristic, complemented
39	#368	—	277	1	no	—
40	#135	—	554	1	no	—
41	Q₃₂	—	277	1	no	—
42	SD₃₂	—	554	1	no	—
43	D₁₆	—	277	1	no	—
44	C₂×C₁₆	C₂×D₂₇₇	1	1	yes	characteristic
45	#108	—	554	1	no	—
46	C₃₂	—	2	1	no	—
47	#41	—	1108	1	no	—
48	Q₁₆	—	554	1	no	—
49	SD₁₆	—	1108	1	no	—
50	D₈	—	554	1	no	—
51	M₁₆	—	1108	1	no	—
52	C₂×C₈	D₁₁₀₈	1	1	yes	characteristic
53	C₁₆	C₄×D₂₇₇	1	1	yes	characteristic
54	C₁₆	C₂²⋊D₂₇₇	1	1	yes	characteristic
55	Q₈	—	1108	1	no	—
56	D₄	—	1108	1	no	—
57	D₄	—	2216	1	no	—
58	C₂×C₄	—	2216	1	no	—
59	C₂×C₄	D₂₂₁₆	1	1	yes	characteristic
60	C₈	—	2216	1	no	—
61	C₈	#17984540	1	1	yes	characteristic
62	C₈	#17984552	1	1	yes	characteristic
63	C₂²	—	2216	1	no	—
64	C₂²	#22579825	1	1	yes	characteristic
65	C₄	—	2216	1	no	—
66	C₄	#22579802	1	1	yes	characteristic
67	C₄	D₄₄₃₂	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	4432	1	no	—
70	C₂	#31799804	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄₄₃₂
commutator	C₁₆×C₂₇₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₇₇
Fitting	C₂×C₃₂×C₂₇₇	C₂
socle	C₂×C₂₇₇	#357
prime radical	#31800434	C₂
cosocle	C₂×C₁₆×C₂₇₇	C₂²
hypercenter	C₂×C₃₂	D₂₇₇
hypocenter	C₂₇₇	#1417
LCS	G ▷ C₁₆×C₂₇₇ ▷ C₈×C₂₇₇ ▷ C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
2. #31799832	C₂	semidirect
3. C₂×C₃₂×C₂₇₇	C₂	semidirect
35. C₂₇₇	#1417	semidirect
38. C₂×C₃₂	D₂₇₇	semidirect

Maximal subgroups

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

subgroup	index
#31800434	2
#31799832	2
C₂×C₃₂×C₂₇₇	2
#1417	277