Subgroups — #47982765

Order 35456.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31800398	C₂	1	2	yes	maximal
2	#31800382	C₂	1	1	yes	maximal, characteristic
3	#31800352	C₂	1	2	yes	maximal
4	C₈×#14555660	C₂	1	1	yes	maximal, characteristic
5	#500×C₂₇₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₁₀₈	—	2	2	no	—
7	#22579880	C₂²	1	2	yes	—
8	#22579811	—	2	4	no	—
9	#22579804	—	2	4	no	—
10	C₄×#14555660	C₂²	1	1	yes	characteristic
11	C₄×#14555660	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₂₇₇	C₂²	1	2	yes	—
13	#107×C₂₇₇	—	2	2	no	—
14	C₄×C₈×C₂₇₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₅₅₄	—	2	2	no	—
16	Dic₁₁₀₈	—	4	4	no	—
17	C₂×#14555660	D₄	1	2	yes	—
18	C₂×#14555660	D₄	1	2	yes	—
19	C₄⋊Dic₂₇₇	—	2	2	no	—
20	C₄⋊Dic₂₇₇	—	2	2	no	—
21	C₂×Q₈×C₂₇₇	—	2	2	no	—
22	C₂×C₈×C₂₇₇	D₄	1	2	yes	—
23	C₄⋊C₄×C₂₇₇	—	2	2	no	—
24	C₄⋊C₄×C₂₇₇	—	2	2	no	—
25	C₄²×C₂₇₇	C₂³	1	1	yes	characteristic
26	Dic₅₅₄	—	4	4	no	—
27	C₂×Dic₂₇₇	—	4	2	no	—
28	#14555660	—	2	2	no	—
29	#14555660	Q₁₆	1	4	yes	complemented
30	Q₈×C₂₇₇	—	4	4	no	—
31	C₂×C₄×C₂₇₇	—	4	2	no	—
32	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
35	C₈×C₂₇₇	SD₁₆	1	4	yes	—
36	C₂²×C₂₇₇	#127	1	1	yes	characteristic
37	C₄×C₂₇₇	—	8	2	no	—
38	C₄×C₂₇₇	C₂×SD₁₆	1	2	yes	—
39	C₄×C₂₇₇	C₂×Q₁₆	1	2	yes	—
40	C₄×C₂₇₇	#135	1	2	yes	—
41	Dic₂₇₇	—	8	2	no	—
42	C₂×C₂₇₇	#492	1	1	yes	characteristic
43	C₂×C₂₇₇	#494	1	1	yes	characteristic
44	C₂×C₂₇₇	#495	1	1	yes	characteristic
45	C₂₇₇	#1707	1	1	yes	characteristic, complemented, Sylow
46	#1707	—	277	1	no	maximal, Sylow
47	#500	D₂₇₇	1	1	yes	characteristic
48	#498	—	277	2	no	—
49	#494	—	277	1	no	—
50	#487	—	277	2	no	—
51	C₈²	—	277	1	no	—
52	C₂×Q₁₆	—	554	2	no	—
53	C₄⋊Q₈	—	277	2	no	—
54	C₄⋊Q₈	C₂×D₂₇₇	1	2	yes	—
55	#107	—	2	2	no	—
56	#106	—	554	4	no	—
57	#103	—	554	4	no	—
58	C₄×C₈	—	277	1	no	—
59	C₄×C₈	—	277	1	no	—
60	C₄×C₈	C₂×D₂₇₇	1	1	yes	characteristic
61	C₂×Q₈	—	2	2	no	—
62	C₂×Q₈	—	554	2	no	—
63	Q₁₆	—	1108	4	no	—
64	C₂×C₈	—	277	2	no	—
65	C₂×C₈	—	277	2	no	—
66	C₂×C₈	C₂²⋊D₂₇₇	1	2	yes	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	2	2	no	—
69	C₄⋊C₄	—	554	2	no	—
70	C₄⋊C₄	—	554	2	no	—
71	C₄²	C₂²×D₂₇₇	1	1	yes	characteristic
72	Q₈	—	4	4	no	—
73	Q₈	—	1108	4	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	1108	2	no	—
76	C₂×C₄	D₄×D₂₇₇	1	1	yes	characteristic
77	C₂×C₄	D₄×D₂₇₇	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₂₇₇	1	1	yes	characteristic
79	C₈	—	277	4	no	—
80	C₈	—	554	2	no	—
81	C₈	#17984550	1	4	yes	—
82	C₂²	#22579876	1	1	yes	characteristic
83	C₄	—	8	2	no	—
84	C₄	—	2216	2	no	—
85	C₄	C₂×#17984550	1	2	yes	—
86	C₄	Q₁₆×D₂₇₇	1	2	yes	—
87	C₄	#22579899	1	2	yes	—
88	C₂	#31800376	1	1	yes	characteristic
89	C₂	#31800380	1	1	yes	characteristic
90	C₂	#31800385	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#22579876
commutator, cosocle	C₄²×C₂₇₇	C₂³
Frattini	C₄²	C₂²×D₂₇₇
Fitting	#500×C₂₇₇	C₂
socle, prime radical	C₂²×C₂₇₇	#127
hypercenter	#500	D₂₇₇
hypocenter	C₂₇₇	#1707
LCS	G ▷ C₄²×C₂₇₇ ▷ C₂²×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

Complements

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

N	H ≅ G/N	type
29. #14555660	Q₁₆	semidirect
45. C₂₇₇	#1707	semidirect

Maximal subgroups

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

subgroup	index
#31800398	2
#31800382	2
#31800352	2
C₈×#14555660	2
#500×C₂₇₇	2
#1707	277