Subgroups — C₂²×#379 #34003

Order 256.

16607 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁷	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₂²⋊C₄	C₂	1	6	yes	maximal, complemented
3	C₂×#379	C₂	1	24	yes	maximal, direct product factor
4	C₂⁶	—	2	48	no	—
5	C₂⁶	C₄	1	16	yes	complemented
6	C₂⁶	C₂²	1	12	yes	—
7	C₂⁶	C₂²	1	3	yes	—
8	C₂⁴×C₄	C₂²	1	4	yes	complemented
9	C₂²×C₂²⋊C₄	—	2	48	no	—
10	C₂²×C₂²⋊C₄	C₂²	1	72	yes	complemented
11	#379	C₂²	1	64	yes	direct product factor
12	C₂⁵	—	2	512	no	—
13	C₂⁵	—	2	384	no	—
14	C₂⁵	—	2	288	no	—
15	C₂⁵	—	2	48	no	—
16	C₂⁵	C₂×C₄	1	24	yes	—
17	C₂⁵	C₂×C₄	1	96	yes	complemented
18	C₂⁵	D₄	1	48	yes	—
19	C₂⁵	C₂³	1	16	yes	—
20	C₂⁵	C₂³	1	18	yes	—
21	C₂⁵	C₂³	1	1	yes	characteristic
22	C₂³×C₄	—	2	48	no	—
23	C₂³×C₄	C₂³	1	24	yes	complemented
24	C₂×C₂²⋊C₄	—	2	288	no	—
25	C₂×C₂²⋊C₄	C₂³	1	96	yes	complemented
26	C₂⁴	—	2	1536	no	—
27	C₂⁴	—	2	2048	no	—
28	C₂⁴	—	2	192	no	—
29	C₂⁴	—	2	1152	no	—
30	C₂⁴	—	2	384	no	—
31	C₂⁴	—	2	192	no	—
32	C₂⁴	—	2	144	no	—
33	C₂⁴	C₂²⋊C₄	1	192	yes	complemented
34	C₂⁴	C₂²×C₄	1	72	yes	—
35	C₂⁴	C₂²×C₄	1	4	yes	—
36	C₂⁴	C₂²×C₄	1	64	yes	complemented
37	C₂⁴	C₂×D₄	1	144	yes	—
38	C₂⁴	C₂×D₄	1	24	yes	—
39	C₂⁴	C₂⁴	1	12	yes	—
40	C₂⁴	C₂⁴	1	3	yes	—
41	C₂²×C₄	—	2	144	no	—
42	C₂²×C₄	—	4	64	no	—
43	C₂²×C₄	C₂⁴	1	16	yes	complemented
44	C₂²⋊C₄	—	2	192	no	—
45	C₂³	—	2	512	no	—
46	C₂³	—	2	3072	no	—
47	C₂³	—	2	288	no	—
48	C₂³	—	2	384	no	—
49	C₂³	—	2	576	no	—
50	C₂³	—	2	48	no	—
51	C₂³	—	2	768	no	—
52	C₂³	C₂×C₂²⋊C₄	1	48	yes	—
53	C₂³	C₂×C₂²⋊C₄	1	288	yes	complemented
54	C₂³	#120	1	64	yes	—
55	C₂³	C₂³×C₄	1	24	yes	—
56	C₂³	C₂³×C₄	1	6	yes	—
57	C₂³	C₂²×D₄	1	48	yes	—
58	C₂³	C₂²×D₄	1	36	yes	—
59	C₂³	C₂⁵	1	1	yes	characteristic
60	C₂×C₄	—	2	48	no	—
61	C₂×C₄	—	4	96	no	—
62	C₂²	—	2	512	no	—
63	C₂²	—	2	48	no	—
64	C₂²	—	2	96	no	—
65	C₂²	—	2	576	no	—
66	C₂²	#379	1	64	yes	direct product factor
67	C₂²	C₂²×C₂²⋊C₄	1	36	yes	—
68	C₂²	C₂²×C₂²⋊C₄	1	48	yes	complemented
69	C₂²	C₂×#120	1	48	yes	—
70	C₂²	C₂⁴×C₄	1	1	yes	characteristic
71	C₂²	C₂³×D₄	1	6	yes	—
72	C₄	—	4	16	no	—
73	C₂	—	2	48	no	—
74	C₂	C₂×#379	1	24	yes	direct product factor
75	C₂	C₂³×C₂²⋊C₄	1	3	yes	—
76	C₂	C₂²×#120	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁵	C₂³
commutator	C₂²	C₂⁴×C₄
Frattini, cosocle	C₂³	C₂⁵
prime radical	C₂⁷	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂³×C₂²⋊C₄	C₂	semidirect
3. C₂×#379	C₂	direct (N × H)
5. C₂⁶	C₄	semidirect
8. C₂⁴×C₄	C₂²	semidirect
10. C₂²×C₂²⋊C₄	C₂²	semidirect
11. #379	C₂²	direct (N × H)
17. C₂⁵	C₂×C₄	semidirect
23. C₂³×C₄	C₂³	semidirect
25. C₂×C₂²⋊C₄	C₂³	semidirect
33. C₂⁴	C₂²⋊C₄	semidirect
36. C₂⁴	C₂²×C₄	semidirect
43. C₂²×C₄	C₂⁴	semidirect
53. C₂³	C₂×C₂²⋊C₄	semidirect
66. C₂²	#379	direct (N × H)
68. C₂²	C₂²×C₂²⋊C₄	semidirect
74. C₂	C₂×#379	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁷	2
C₂³×C₂²⋊C₄	2
C₂×#379	2

Subgroups — C22×#379 #34003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#379 #34003