Subgroups — #396×C₂₅₇ #30656265

Order 16448.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₄×C₂₅₇	C₂	1	3	yes	maximal, complemented
2	C₂×C₂²⋊C₄×C₂₅₇	C₂	1	3	yes	maximal
3	#95×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×C₂₅₇	C₂²	1	1	yes	characteristic
5	C₂²×C₄×C₂₅₇	C₂²	1	3	yes	—
6	C₂²×C₄×C₂₅₇	C₂²	1	3	yes	—
7	C₄⋊C₄×C₂₅₇	—	2	6	no	—
8	C₂²⋊C₄×C₂₅₇	—	2	6	no	—
9	C₂³×C₂₅₇	—	2	3	no	—
10	C₂³×C₂₅₇	—	2	3	no	—
11	C₂³×C₂₅₇	Q₈	1	2	yes	—
12	C₂³×C₂₅₇	C₂³	1	1	yes	characteristic
13	C₂×C₄×C₂₅₇	—	2	3	no	—
14	C₂×C₄×C₂₅₇	—	2	3	no	—
15	C₂×C₄×C₂₅₇	—	2	3	no	—
16	C₂×C₄×C₂₅₇	—	2	3	no	—
17	C₂×C₄×C₂₅₇	—	2	3	no	—
18	C₂×C₄×C₂₅₇	D₄	1	6	yes	—
19	C₂²×C₂₅₇	—	2	6	no	—
20	C₂²×C₂₅₇	—	4	3	no	—
21	C₂²×C₂₅₇	—	4	1	no	—
22	C₂²×C₂₅₇	C₂×D₄	1	3	yes	—
23	C₂²×C₂₅₇	C₂×Q₈	1	1	yes	characteristic
24	C₂²×C₂₅₇	#41	1	3	yes	—
25	C₄×C₂₅₇	—	2	6	no	—
26	C₄×C₂₅₇	—	4	3	no	—
27	C₂×C₂₅₇	—	4	2	no	—
28	C₂×C₂₅₇	#121	1	3	yes	—
29	C₂×C₂₅₇	C₂²⋊Q₈	1	3	yes	—
30	C₂×C₂₅₇	#126	1	1	yes	characteristic
31	C₂₅₇	#396	1	1	yes	characteristic, direct product factor, Sylow
32	#396	C₂₅₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂×C₄⋊C₄	C₂×C₂₅₇	1	3	yes	complemented
34	C₂×C₂²⋊C₄	C₂×C₂₅₇	1	3	yes	—
35	#95	C₂×C₂₅₇	1	1	yes	characteristic, complemented
36	C₂⁴	C₂²×C₂₅₇	1	1	yes	characteristic
37	C₂²×C₄	C₂²×C₂₅₇	1	3	yes	—
38	C₂²×C₄	C₂²×C₂₅₇	1	3	yes	—
39	C₄⋊C₄	—	2	6	no	—
40	C₂²⋊C₄	—	2	6	no	—
41	C₂³	—	2	3	no	—
42	C₂³	—	2	3	no	—
43	C₂³	Q₈×C₂₅₇	1	2	yes	—
44	C₂³	C₂³×C₂₅₇	1	1	yes	characteristic
45	C₂×C₄	—	2	3	no	—
46	C₂×C₄	—	2	3	no	—
47	C₂×C₄	—	2	3	no	—
48	C₂×C₄	—	2	3	no	—
49	C₂×C₄	—	2	3	no	—
50	C₂×C₄	D₄×C₂₅₇	1	6	yes	—
51	C₂²	—	2	6	no	—
52	C₂²	—	4	3	no	—
53	C₂²	—	4	1	no	—
54	C₂²	C₂×D₄×C₂₅₇	1	3	yes	—
55	C₂²	C₂×Q₈×C₂₅₇	1	1	yes	characteristic
56	C₂²	#41×C₂₅₇	1	3	yes	—
57	C₄	—	2	6	no	—
58	C₄	—	4	3	no	—
59	C₂	—	4	2	no	—
60	C₂	#121×C₂₅₇	1	3	yes	—
61	C₂	C₂²⋊Q₈×C₂₅₇	1	3	yes	—
62	C₂	#126×C₂₅₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₅₇	C₂³
commutator, Frattini, cosocle	C₂³	C₂³×C₂₅₇
prime radical	C₂⁴×C₂₅₇	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₅₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊C₄×C₂₅₇	C₂	semidirect
3. #95×C₂₅₇	C₂	semidirect
31. C₂₅₇	#396	direct (N × H)
32. #396	C₂₅₇	direct (N × H)
33. C₂×C₄⋊C₄	C₂×C₂₅₇	semidirect
35. #95	C₂×C₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₄×C₂₅₇	2
C₂×C₂²⋊C₄×C₂₅₇	2
#95×C₂₅₇	2
#396	257

Subgroups — #396×C257 #30656265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #396×C₂₅₇ #30656265