Subgroups — #31800732

Order 17728.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22579879	C₂	1	4	yes	maximal
2	#22579878	C₂	1	4	yes	maximal
3	C₄×Dic₅₅₄	C₂	1	2	yes	maximal
4	#22579845	C₂	1	4	yes	maximal
5	C₄⋊Q₈×C₂₇₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₅₄	C₂²	1	4	yes	—
7	C₂×Dic₅₅₄	C₂²	1	2	yes	—
8	C₄⋊Dic₂₇₇	C₂²	1	2	yes	—
9	#17984542	C₂²	1	8	yes	—
10	#17984542	C₂²	1	4	yes	—
11	C₄×Dic₂₇₇	C₂²	1	4	yes	—
12	C₄×Dic₂₇₇	C₂²	1	4	yes	—
13	C₂×Q₈×C₂₇₇	C₂²	1	2	yes	—
14	C₄⋊C₄×C₂₇₇	C₂²	1	4	yes	—
15	C₄²×C₂₇₇	C₂²	1	1	yes	characteristic
16	Dic₅₅₄	—	4	4	no	—
17	Dic₅₅₄	Q₈	1	8	yes	complemented
18	C₂×Dic₂₇₇	C₂³	1	4	yes	—
19	C₂×Dic₂₇₇	C₂³	1	4	yes	—
20	Q₈×C₂₇₇	—	4	2	no	—
21	C₂×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
22	C₂×C₄×C₂₇₇	C₂³	1	4	yes	—
23	C₂×C₄×C₂₇₇	C₂³	1	2	yes	—
24	C₂²×C₂₇₇	C₂⁴	1	1	yes	characteristic
25	C₄×C₂₇₇	—	2	1	no	—
26	C₄×C₂₇₇	—	2	4	no	—
27	C₄×C₂₇₇	C₂×Q₈	1	4	yes	—
28	Dic₂₇₇	—	2	4	no	—
29	Dic₂₇₇	C₂×Q₈	1	8	yes	—
30	C₂×C₂₇₇	C₂²×Q₈	1	2	yes	—
31	C₂×C₂₇₇	H(5,𝔽₂)	1	1	yes	characteristic
32	C₂₇₇	Q₈²	1	1	yes	characteristic, complemented, Sylow
33	Q₈²	—	277	1	no	maximal, Sylow
34	C₄⋊Q₈	—	277	4	no	—
35	C₄⋊Q₈	—	277	4	no	—
36	C₄⋊Q₈	D₂₇₇	1	1	yes	characteristic
37	C₄×Q₈	—	277	4	no	—
38	C₄×Q₈	—	277	2	no	—
39	C₂×Q₈	—	277	2	no	—
40	C₂×Q₈	—	277	4	no	—
41	C₂×Q₈	C₂×D₂₇₇	1	2	yes	—
42	C₄⋊C₄	—	277	8	no	—
43	C₄⋊C₄	—	277	4	no	—
44	C₄⋊C₄	—	277	2	no	—
45	C₄⋊C₄	C₂×D₂₇₇	1	4	yes	—
46	C₄²	—	277	4	no	—
47	C₄²	—	277	4	no	—
48	C₄²	C₂×D₂₇₇	1	1	yes	characteristic
49	Q₈	—	4	2	no	—
50	Q₈	—	277	8	no	—
51	Q₈	—	1108	4	no	—
52	C₂×C₄	—	277	4	no	—
53	C₂×C₄	—	277	4	no	—
54	C₂×C₄	C₂²×D₂₇₇	1	4	yes	—
55	C₂×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
56	C₂×C₄	C₂²×D₂₇₇	1	2	yes	—
57	C₂²	C₂³×D₂₇₇	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	4	no	—
60	C₄	—	277	8	no	—
61	C₄	—	554	4	no	—
62	C₄	Q₈×D₂₇₇	1	4	yes	—
63	C₂	C₂×Q₈×D₂₇₇	1	2	yes	—
64	C₂	#22579925	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₇₇
commutator, socle, prime radical, cosocle	C₂²×C₂₇₇	C₂⁴
Fitting	C₄⋊Q₈×C₂₇₇	C₂
hypercenter	C₄⋊Q₈	D₂₇₇
hypocenter	C₂₇₇	Q₈²
LCS	G ▷ C₂²×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
17. Dic₅₅₄	Q₈	semidirect
32. C₂₇₇	Q₈²	semidirect

Maximal subgroups

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

subgroup	index
#22579879	2
#22579878	2
C₄×Dic₅₅₄	2
#22579845	2
C₄⋊Q₈×C₂₇₇	2
Q₈²	277