Subgroups — C₂×#22579947 #31801004

Order 17728.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14555667	C₂	1	2	yes	maximal
2	#22579947	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×Dic₂₇₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14555667	—	2	4	no	—
5	C₂×#14555667	C₂²	1	4	yes	—
6	C₂²×Dic₂₇₇	C₄	1	1	yes	characteristic
7	C₂²×Dic₂₇₇	C₂²	1	1	yes	characteristic
8	C₄×Dic₂₇₇	C₄	1	2	yes	—
9	C₄×Dic₂₇₇	C₂²	1	2	yes	—
10	C₂²×C₄×C₂₇₇	C₄	1	1	yes	characteristic
11	#14555667	—	2	4	no	—
12	C₂×Dic₂₇₇	C₈	1	4	yes	complemented
13	C₂×Dic₂₇₇	C₂×C₄	1	1	yes	characteristic
14	C₂×Dic₂₇₇	C₂×C₄	1	2	yes	—
15	C₂×Dic₂₇₇	D₄	1	2	yes	—
16	C₂×Dic₂₇₇	Q₈	1	2	yes	—
17	C₂×Dic₂₇₇	C₂³	1	1	yes	characteristic
18	C₂³×C₂₇₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₂₇₇	—	2	2	no	—
20	C₂×C₄×C₂₇₇	C₂×C₄	1	2	yes	—
21	C₂²×C₂₇₇	C₄⋊C₄	1	2	yes	—
22	C₂²×C₂₇₇	C₂×C₈	1	2	yes	—
23	C₂²×C₂₇₇	M₁₆	1	2	yes	—
24	C₂²×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
25	C₄×C₂₇₇	—	2	2	no	—
26	Dic₂₇₇	C₄⋊C₄	1	2	yes	—
27	Dic₂₇₇	C₂×C₈	1	4	yes	complemented
28	Dic₂₇₇	C₂×D₄	1	1	yes	characteristic
29	Dic₂₇₇	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₂₇₇	C₄⋊C₈	1	4	yes	complemented
31	C₂×C₂₇₇	C₂×C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₂₇₇	C₂²×C₈	1	1	yes	characteristic
33	C₂×C₂₇₇	C₂×M₁₆	1	1	yes	characteristic
34	C₂₇₇	C₂×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
35	C₂×C₄⋊C₈	—	277	1	no	maximal, Sylow
36	C₂²×C₈	—	277	2	no	—
37	C₂×C₄²	—	277	1	no	—
38	C₄⋊C₈	—	277	4	no	—
39	C₂²×C₄	—	277	1	no	—
40	C₂²×C₄	—	277	1	no	—
41	C₂²×C₄	#11764308	1	1	yes	characteristic
42	C₂×C₈	—	277	4	no	—
43	C₂×C₈	—	554	4	no	—
44	C₄²	—	277	2	no	—
45	C₄²	—	277	2	no	—
46	C₂³	C₂×#11764308	1	1	yes	characteristic
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	—	277	2	no	—
49	C₂×C₄	—	277	2	no	—
50	C₂×C₄	—	277	4	no	—
51	C₂×C₄	—	277	1	no	—
52	C₂×C₄	—	277	2	no	—
53	C₂×C₄	—	277	1	no	—
54	C₂×C₄	C₂×#11764308	1	2	yes	—
55	C₈	—	554	4	no	—
56	C₂²	C₄⋊#11764308	1	2	yes	—
57	C₂²	D₂₇₇⋊C₈	1	2	yes	—
58	C₂²	#17984572	1	2	yes	—
59	C₂²	C₂²×#11764308	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	—	277	1	no	—
62	C₄	—	277	2	no	—
63	C₄	—	277	1	no	—
64	C₄	—	277	4	no	—
65	C₂	#22579947	1	4	yes	direct product factor
66	C₂	C₂×C₄⋊#11764308	1	1	yes	characteristic
67	C₂	C₂×D₂₇₇⋊C₈	1	1	yes	characteristic
68	C₂	C₂×#17984572	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11764308
commutator	C₂×C₂₇₇	C₂²×C₈
Frattini	C₂²	C₂²×#11764308
Fitting	C₂²×C₄×C₂₇₇	C₄
socle, prime radical	C₂³×C₂₇₇	C₂×C₄
cosocle	C₂×Dic₂₇₇	C₂³
hypercenter	C₂²×C₄	#11764308
hypocenter	C₂₇₇	C₂×C₄⋊C₈
LCS	G ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. #22579947	C₂	direct (N × H)
12. C₂×Dic₂₇₇	C₈	semidirect
27. Dic₂₇₇	C₂×C₈	semidirect
30. C₂×C₂₇₇	C₄⋊C₈	semidirect
34. C₂₇₇	C₂×C₄⋊C₈	semidirect
65. C₂	#22579947	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14555667	2
#22579947	2
C₂×C₄×Dic₂₇₇	2
C₂×C₄⋊C₈	277

Subgroups — C2×#22579947 #31801004

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22579947 #31801004