Subgroups — #31800106

Order 17728.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984553	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₂₇₇	C₂	1	1	yes	maximal, characteristic
3	#22579845	C₂	1	1	yes	maximal, characteristic
4	#22579811	C₂	1	1	yes	maximal, characteristic
5	#22579805	C₂	1	1	yes	maximal, characteristic
6	#22579790	C₂	1	1	yes	maximal, characteristic
7	#103×C₂₇₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₅₅₄	C₂²	1	1	yes	characteristic
9	#17984553	C₄	1	4	yes	complemented
10	C₂×#14555660	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₇₇	—	2	1	no	—
12	C₄⋊Dic₂₇₇	C₂²	1	1	yes	characteristic
13	#17984542	—	2	1	no	—
14	C₄×Dic₂₇₇	—	2	1	no	—
15	C₄×Dic₂₇₇	—	2	1	no	—
16	C₄×Dic₂₇₇	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₂₇₇	C₂²	1	1	yes	characteristic
18	C₂×C₈×C₂₇₇	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₂₇₇	C₂²	1	1	yes	characteristic
20	Dic₅₅₄	—	2	1	no	—
21	Dic₅₅₄	C₂×C₄	1	2	yes	—
22	C₂×Dic₂₇₇	—	2	1	no	—
23	C₂×Dic₂₇₇	—	2	1	no	—
24	C₂×Dic₂₇₇	D₄	1	1	yes	characteristic
25	C₂×Dic₂₇₇	D₄	1	1	yes	characteristic
26	#14555660	C₂×C₄	1	2	yes	—
27	Q₈×C₂₇₇	—	2	1	no	—
28	Q₈×C₂₇₇	C₂×C₄	1	2	yes	—
29	C₂×C₄×C₂₇₇	—	2	1	no	—
30	C₂×C₄×C₂₇₇	—	2	1	no	—
31	C₂×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
32	C₈×C₂₇₇	—	2	1	no	—
33	C₂²×C₂₇₇	C₂×D₄	1	1	yes	characteristic
34	C₄×C₂₇₇	—	2	2	no	—
35	C₄×C₂₇₇	—	4	1	no	—
36	C₄×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
37	C₄×C₂₇₇	#41	1	1	yes	characteristic
38	Dic₂₇₇	—	2	2	no	—
39	Dic₂₇₇	—	2	1	no	—
40	Dic₂₇₇	—	2	1	no	—
41	Dic₂₇₇	—	4	1	no	—
42	C₂×C₂₇₇	C₄×D₄	1	1	yes	characteristic
43	C₂×C₂₇₇	#137	1	1	yes	characteristic
44	C₂×C₂₇₇	#137	1	1	yes	characteristic
45	C₂₇₇	#441	1	1	yes	characteristic, complemented, Sylow
46	#441	—	277	1	no	maximal, Sylow
47	C₂×Q₁₆	—	277	1	no	—
48	C₄×Q₈	—	277	1	no	—
49	C₄×Q₈	—	277	1	no	—
50	#106	—	277	1	no	—
51	#103	—	277	1	no	—
52	#103	D₂₇₇	1	1	yes	characteristic
53	#97	—	277	1	no	—
54	C₂×Q₈	—	277	1	no	—
55	C₂×Q₈	C₂×D₂₇₇	1	1	yes	characteristic
56	Q₁₆	—	277	4	no	—
57	C₂×C₈	—	277	1	no	—
58	C₂×C₈	C₂×D₂₇₇	1	1	yes	characteristic
59	C₄⋊C₄	—	277	1	no	—
60	C₄⋊C₄	—	554	1	no	—
61	C₄⋊C₄	—	554	1	no	—
62	C₄⋊C₄	C₂×D₂₇₇	1	1	yes	characteristic
63	C₄²	—	277	1	no	—
64	C₄²	—	554	1	no	—
65	C₄²	—	554	1	no	—
66	Q₈	—	2	1	no	—
67	Q₈	—	277	2	no	—
68	Q₈	—	554	1	no	—
69	Q₈	C₄×D₂₇₇	1	2	yes	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	277	1	no	—
73	C₂×C₄	—	277	1	no	—
74	C₂×C₄	—	554	1	no	—
75	C₂×C₄	—	554	1	no	—
76	C₂×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	277	2	no	—
79	C₂²	D₄×D₂₇₇	1	1	yes	characteristic
80	C₄	—	2	2	no	—
81	C₄	—	4	1	no	—
82	C₄	—	554	2	no	—
83	C₄	—	554	1	no	—
84	C₄	—	554	1	no	—
85	C₄	—	1108	1	no	—
86	C₄	C₂×C₄×D₂₇₇	1	1	yes	characteristic
87	C₄	#17984562	1	1	yes	characteristic
88	C₂	#22579840	1	1	yes	characteristic
89	C₂	#22579909	1	1	yes	characteristic
90	C₂	#22579912	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₇₇
commutator	C₄×C₂₇₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₇₇
Fitting	#103×C₂₇₇	C₂
socle, prime radical	C₂²×C₂₇₇	C₂×D₄
cosocle	C₂×C₄×C₂₇₇	C₂³
hypercenter	#103	D₂₇₇
hypocenter	C₂₇₇	#441
LCS	G ▷ C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
9. #17984553	C₄	semidirect
45. C₂₇₇	#441	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984553	2
Q₈×Dic₂₇₇	2
#22579845	2
#22579811	2
#22579805	2
#22579790	2
#103×C₂₇₇	2
#441	277