Subgroups — #47985477

Order 35456.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31800435	C₂	1	2	yes	maximal, complemented
2	#31799839	C₂	1	2	yes	maximal, complemented
3	D₈₈₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₂×D₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₆₄×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#22579903	—	2	2	no	—
7	#22579823	—	2	2	no	—
8	D₄₄₃₂	C₂²	1	2	yes	—
9	C₁₆×D₂₇₇	C₂²	1	1	yes	characteristic
10	#22579786	C₂²	1	1	yes	characteristic
11	Q₃₂×C₂₇₇	C₂²	1	2	yes	—
12	C₃₂×C₂₇₇	C₂²	1	1	yes	characteristic
13	#17984564	—	4	2	no	—
14	#17984550	—	4	2	no	—
15	D₂₂₁₆	—	2	2	no	—
16	C₈×D₂₇₇	D₄	1	1	yes	characteristic
17	#17984538	D₄	1	1	yes	characteristic
18	Q₁₆×C₂₇₇	—	2	2	no	—
19	C₁₆×C₂₇₇	C₂³	1	1	yes	characteristic
20	D₁₁₀₈	—	4	2	no	—
21	D₁₁₀₈	—	8	2	no	—
22	C₄×D₂₇₇	—	8	2	no	—
23	C₄×D₂₇₇	D₈	1	1	yes	characteristic
24	#14555660	D₈	1	1	yes	characteristic
25	Q₈×C₂₇₇	—	4	2	no	—
26	C₈×C₂₇₇	C₂×D₄	1	1	yes	characteristic
27	C₂×D₂₇₇	—	8	2	no	—
28	C₂×D₂₇₇	D₁₆	1	1	yes	characteristic
29	C₄×C₂₇₇	—	8	2	no	—
30	C₄×C₂₇₇	C₂×D₈	1	1	yes	characteristic
31	Dic₂₇₇	D₁₆	1	1	yes	characteristic
32	C₂×C₂₇₇	C₂×D₁₆	1	1	yes	characteristic
33	D₂₇₇	—	2	1	no	—
34	D₂₇₇	—	16	2	no	—
35	C₂₇₇	#2262	1	1	yes	characteristic, complemented, Sylow
36	#2262	—	277	1	no	maximal, Sylow
37	#508	—	277	2	no	—
38	Q₆₄	D₂₇₇	1	1	yes	characteristic, complemented
39	SD₆₄	—	277	2	no	—
40	D₃₂	—	277	1	no	—
41	C₂×C₃₂	—	277	1	no	—
42	#135	—	554	2	no	—
43	Q₃₂	C₂×D₂₇₇	1	2	yes	—
44	SD₃₂	—	554	2	no	—
45	D₁₆	—	277	2	no	—
46	C₂×C₁₆	—	277	1	no	—
47	C₃₂	—	277	1	no	—
48	C₃₂	C₂×D₂₇₇	1	1	yes	characteristic
49	#41	—	1108	2	no	—
50	Q₁₆	—	2	2	no	—
51	SD₁₆	—	1108	2	no	—
52	D₈	—	554	2	no	—
53	C₂×C₈	—	277	1	no	—
54	C₁₆	—	277	1	no	—
55	C₁₆	C₂²×D₂₇₇	1	1	yes	characteristic
56	Q₈	—	4	2	no	—
57	D₄	—	1108	2	no	—
58	D₄	—	2216	2	no	—
59	C₂×C₄	—	277	1	no	—
60	C₂×C₄	—	2216	2	no	—
61	C₈	—	277	1	no	—
62	C₈	D₄×D₂₇₇	1	1	yes	characteristic
63	C₂²	—	277	1	no	—
64	C₂²	—	2216	2	no	—
65	C₄	—	8	2	no	—
66	C₄	—	277	1	no	—
67	C₄	D₈×D₂₇₇	1	1	yes	characteristic
68	C₂	—	554	1	no	—
69	C₂	—	4432	2	no	—
70	C₂	D₁₆×D₂₇₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₁₆×D₂₇₇
commutator, cosocle	C₁₆×C₂₇₇	C₂³
Frattini	C₁₆	C₂²×D₂₇₇
Fitting	Q₆₄×C₂₇₇	C₂
socle	C₂×C₂₇₇	C₂×D₁₆
hypercenter	Q₆₄	D₂₇₇
hypocenter	C₂₇₇	#2262
LCS	G ▷ C₁₆×C₂₇₇ ▷ C₈×C₂₇₇ ▷ C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₆₄

Complements

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

N	H ≅ G/N	type
1. #31800435	C₂	semidirect
2. #31799839	C₂	semidirect
3. D₈₈₆₄	C₂	semidirect
4. C₃₂×D₂₇₇	C₂	semidirect
5. Q₆₄×C₂₇₇	C₂	semidirect
35. C₂₇₇	#2262	semidirect
38. Q₆₄	D₂₇₇	semidirect

Maximal subgroups

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

subgroup	index
#31800435	2
#31799839	2
D₈₈₆₄	2
C₃₂×D₂₇₇	2
Q₆₄×C₂₇₇	2
#2262	277