Subgroups — #47982776

Order 35456.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31800407	C₂	1	1	yes	maximal, characteristic
2	#31800354	C₂	1	2	yes	maximal
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	#22579880	C₂²	1	2	yes	—
7	#22579814	—	2	2	no	—
8	#22579804	—	2	4	no	—
9	#22579803	—	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	C₂×#14555660	D₄	1	2	yes	—
17	C₂×#14555660	D₄	1	2	yes	—
18	C₄⋊Dic₂₇₇	—	2	2	no	—
19	C₄⋊Dic₂₇₇	—	2	2	no	—
20	C₂×Q₈×C₂₇₇	—	2	2	no	—
21	C₂×C₈×C₂₇₇	D₄	1	2	yes	—
22	C₄⋊C₄×C₂₇₇	—	2	2	no	—
23	C₄⋊C₄×C₂₇₇	—	2	2	no	—
24	C₄²×C₂₇₇	C₂³	1	1	yes	characteristic
25	Dic₅₅₄	—	4	4	no	—
26	C₂×Dic₂₇₇	—	4	2	no	—
27	#14555660	—	2	2	no	—
28	#14555660	—	2	2	no	—
29	Q₈×C₂₇₇	—	4	4	no	—
30	C₂×C₄×C₂₇₇	—	4	2	no	—
31	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
34	C₈×C₂₇₇	Q₁₆	1	4	yes	—
35	C₂²×C₂₇₇	#127	1	1	yes	characteristic
36	C₄×C₂₇₇	—	8	2	no	—
37	C₄×C₂₇₇	C₂×Q₁₆	1	2	yes	—
38	C₄×C₂₇₇	#135	1	2	yes	—
39	C₄×C₂₇₇	#135	1	2	yes	—
40	Dic₂₇₇	—	8	2	no	—
41	C₂×C₂₇₇	#494	1	1	yes	characteristic
42	C₂×C₂₇₇	#495	1	1	yes	characteristic
43	C₂×C₂₇₇	#495	1	1	yes	characteristic
44	C₂₇₇	#1710	1	1	yes	characteristic, complemented, Sylow
45	#1710	—	277	1	no	maximal, Sylow
46	#500	—	277	1	no	—
47	#500	D₂₇₇	1	1	yes	characteristic
48	#487	—	277	2	no	—
49	#487	—	277	2	no	—
50	C₈²	—	277	1	no	—
51	C₄⋊Q₈	—	277	2	no	—
52	C₄⋊Q₈	C₂×D₂₇₇	1	2	yes	—
53	#107	—	2	2	no	—
54	#107	—	554	2	no	—
55	#103	—	554	4	no	—
56	#103	—	554	4	no	—
57	C₄×C₈	—	277	1	no	—
58	C₄×C₈	—	277	1	no	—
59	C₄×C₈	C₂×D₂₇₇	1	1	yes	characteristic
60	C₂×Q₈	—	2	2	no	—
61	C₂×Q₈	—	554	2	no	—
62	C₂×C₈	—	277	2	no	—
63	C₂×C₈	—	277	2	no	—
64	C₂×C₈	C₂²⋊D₂₇₇	1	2	yes	—
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	—	554	2	no	—
68	C₄⋊C₄	—	554	2	no	—
69	C₄²	C₂²×D₂₇₇	1	1	yes	characteristic
70	Q₈	—	4	4	no	—
71	Q₈	—	1108	4	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	—	1108	2	no	—
74	C₂×C₄	D₄×D₂₇₇	1	1	yes	characteristic
75	C₂×C₄	D₄×D₂₇₇	1	1	yes	characteristic
76	C₂×C₄	C₂×C₂²⋊D₂₇₇	1	1	yes	characteristic
77	C₈	—	554	2	no	—
78	C₈	—	554	2	no	—
79	C₈	#17984553	1	4	yes	—
80	C₂²	#22579876	1	1	yes	characteristic
81	C₄	—	8	2	no	—
82	C₄	—	2216	2	no	—
83	C₄	C₂×#17984553	1	2	yes	—
84	C₄	#22579899	1	2	yes	—
85	C₄	#22579903	1	2	yes	—
86	C₂	#31800381	1	1	yes	characteristic
87	C₂	#31800385	1	1	yes	characteristic
88	C₂	#31800387	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	#1710
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
44. C₂₇₇	#1710	semidirect

Maximal subgroups

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

subgroup	index
#31800407	2
#31800354	2
#31800352	2
C₈×#14555660	2
#500×C₂₇₇	2
#1710	277