Subgroups — C₂²×#17984547 #31800762

Order 17728.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984547	C₂	1	12	yes	maximal, direct product factor
2	C₂²×#14555660	C₂	1	2	yes	maximal, complemented
3	C₂³×C₄×C₂₇₇	C₂	1	1	yes	maximal, characteristic
4	#17984547	C₂²	1	16	yes	direct product factor
5	C₂×#14555660	C₂²	1	12	yes	complemented
6	C₂⁴×C₂₇₇	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₂₇₇	C₄	1	6	yes	—
8	C₂²×C₄×C₂₇₇	C₄	1	1	yes	characteristic
9	C₂²×C₄×C₂₇₇	C₂²	1	6	yes	—
10	C₂²×C₄×C₂₇₇	C₂²	1	1	yes	characteristic
11	#14555660	C₂³	1	8	yes	complemented
12	C₂³×C₂₇₇	—	2	4	no	—
13	C₂³×C₂₇₇	C₂×C₄	1	6	yes	—
14	C₂³×C₂₇₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₂₇₇	C₂×C₄	1	12	yes	—
16	C₂×C₄×C₂₇₇	C₂×C₄	1	6	yes	—
17	C₂×C₄×C₂₇₇	C₂×C₄	1	3	yes	—
18	C₂×C₄×C₂₇₇	C₂³	1	3	yes	—
19	C₂×C₄×C₂₇₇	C₂³	1	4	yes	—
20	C₂²×C₂₇₇	—	2	12	no	—
21	C₂²×C₂₇₇	M₁₆	1	4	yes	complemented
22	C₂²×C₂₇₇	C₂²×C₄	1	4	yes	—
23	C₂²×C₂₇₇	C₂²×C₄	1	3	yes	—
24	C₄×C₂₇₇	C₂²×C₄	1	3	yes	—
25	C₄×C₂₇₇	C₂²×C₄	1	4	yes	—
26	C₄×C₂₇₇	C₂⁴	1	1	yes	characteristic
27	C₂×C₂₇₇	—	2	4	no	—
28	C₂×C₂₇₇	C₂×M₁₆	1	6	yes	complemented
29	C₂×C₂₇₇	C₂³×C₄	1	1	yes	characteristic
30	C₂₇₇	C₂²×M₁₆	1	1	yes	characteristic, complemented, Sylow
31	C₂²×M₁₆	—	277	1	no	maximal, Sylow
32	C₂³×C₄	D₂₇₇	1	1	yes	characteristic
33	C₂×M₁₆	—	277	12	no	—
34	C₂²×C₈	—	277	2	no	—
35	C₂⁴	Dic₂₇₇	1	1	yes	characteristic
36	C₂²×C₄	Dic₂₇₇	1	1	yes	characteristic
37	C₂²×C₄	Dic₂₇₇	1	6	yes	—
38	C₂²×C₄	C₂×D₂₇₇	1	6	yes	—
39	C₂²×C₄	C₂×D₂₇₇	1	1	yes	characteristic
40	M₁₆	—	277	16	no	—
41	C₂×C₈	—	277	12	no	—
42	C₂³	—	2	4	no	—
43	C₂³	C₂×Dic₂₇₇	1	6	yes	—
44	C₂³	C₂×Dic₂₇₇	1	1	yes	characteristic
45	C₂×C₄	C₂×Dic₂₇₇	1	12	yes	—
46	C₂×C₄	C₂×Dic₂₇₇	1	6	yes	—
47	C₂×C₄	C₂×Dic₂₇₇	1	3	yes	—
48	C₂×C₄	C₂²×D₂₇₇	1	3	yes	—
49	C₂×C₄	C₂²×D₂₇₇	1	4	yes	—
50	C₈	—	277	8	no	—
51	C₂²	—	2	12	no	—
52	C₂²	#17984547	1	4	yes	direct product factor
53	C₂²	C₂²×Dic₂₇₇	1	4	yes	—
54	C₂²	C₂²×Dic₂₇₇	1	3	yes	—
55	C₄	C₂²×Dic₂₇₇	1	4	yes	—
56	C₄	C₂²×Dic₂₇₇	1	3	yes	—
57	C₄	C₂³×D₂₇₇	1	1	yes	characteristic
58	C₂	—	2	4	no	—
59	C₂	C₂×#17984547	1	6	yes	direct product factor
60	C₂	C₂³×Dic₂₇₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₇₇
commutator	C₂×C₂₇₇	C₂³×C₄
Frattini	C₄	C₂³×D₂₇₇
Fitting	C₂³×C₄×C₂₇₇	C₂
socle	C₂³×C₂₇₇	C₂×C₄
prime radical	C₂⁴×C₂₇₇	C₄
cosocle	C₄×C₂₇₇	C₂⁴
hypercenter	C₂³×C₄	D₂₇₇
hypocenter	C₂₇₇	C₂²×M₁₆
LCS	G ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#17984547	C₂	direct (N × H)
2. C₂²×#14555660	C₂	semidirect
4. #17984547	C₂²	direct (N × H)
5. C₂×#14555660	C₂²	semidirect
11. #14555660	C₂³	semidirect
21. C₂²×C₂₇₇	M₁₆	semidirect
28. C₂×C₂₇₇	C₂×M₁₆	semidirect
30. C₂₇₇	C₂²×M₁₆	semidirect
52. C₂²	#17984547	direct (N × H)
59. C₂	C₂×#17984547	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#17984547	2
C₂²×#14555660	2
C₂³×C₄×C₂₇₇	2
C₂²×M₁₆	277

Subgroups — C22×#17984547 #31800762

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17984547 #31800762