Subgroups — C₂²×#365 #33402

Order 256.

351 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×#108	C₂	1	1	yes	maximal, characteristic
3	C₂²×#106	C₂	1	1	yes	maximal, characteristic
4	C₂×#365	C₂	1	12	yes	maximal, direct product factor
5	C₂²×M₁₆	—	2	1	no	—
6	C₂³×C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₄⋊C₄	—	2	1	no	—
8	C₂×M₃₂	C₄	1	6	yes	complemented
9	C₂×M₃₂	C₂²	1	6	yes	—
10	C₂²×C₁₆	C₄	1	2	yes	complemented
11	C₂×#108	—	2	3	no	—
12	C₂×#108	C₂²	1	6	yes	—
13	C₂×#106	—	2	3	no	—
14	C₂×#106	C₂²	1	6	yes	—
15	#365	C₂²	1	16	yes	direct product factor
16	C₂³×C₄	—	4	1	no	—
17	C₂³×C₄	D₄	1	1	yes	characteristic
18	C₂×M₁₆	—	2	6	no	—
19	C₂×M₁₆	—	4	3	no	—
20	C₂²×C₈	—	4	1	no	—
21	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈	C₂×C₄	1	3	yes	—
23	C₂²×C₈	D₄	1	3	yes	—
24	C₂²×C₈	Q₈	1	3	yes	—
25	C₂²×C₈	Q₈	1	1	yes	characteristic
26	C₂²×C₈	C₂³	1	3	yes	—
27	C₂×C₄⋊C₄	—	2	6	no	—
28	C₂×C₄⋊C₄	—	4	3	no	—
29	M₃₂	C₂×C₄	1	12	yes	complemented
30	M₃₂	C₂³	1	4	yes	—
31	C₂×C₁₆	C₂×C₄	1	12	yes	complemented
32	#108	—	2	6	no	—
33	#108	C₂³	1	4	yes	—
34	#106	—	2	6	no	—
35	#106	C₂³	1	4	yes	—
36	C₂⁴	SD₁₆	1	1	yes	characteristic
37	C₂²×C₄	—	4	6	no	—
38	C₂²×C₄	—	8	4	no	—
39	C₂²×C₄	C₄⋊C₄	1	3	yes	—
40	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₄	SD₁₆	1	3	yes	—
42	C₂²×C₄	SD₁₆	1	3	yes	—
43	C₂²×C₄	SD₁₆	1	1	yes	characteristic
44	C₂²×C₄	C₂×D₄	1	3	yes	—
45	M₁₆	—	2	4	no	—
46	M₁₆	—	4	6	no	—
47	C₂×C₈	—	4	6	no	—
48	C₂×C₈	C₄⋊C₄	1	3	yes	—
49	C₂×C₈	C₄⋊C₄	1	6	yes	—
50	C₂×C₈	C₄⋊C₄	1	3	yes	—
51	C₂×C₈	C₂²×C₄	1	3	yes	—
52	C₂×C₈	C₂²×C₄	1	3	yes	—
53	C₂×C₈	C₂×D₄	1	3	yes	—
54	C₂×C₈	C₂×Q₈	1	3	yes	—
55	C₂×C₈	C₂×Q₈	1	3	yes	—
56	C₂×C₈	C₂⁴	1	1	yes	characteristic
57	C₄⋊C₄	—	2	4	no	—
58	C₄⋊C₄	—	4	6	no	—
59	C₁₆	C₂²×C₄	1	8	yes	complemented
60	C₂³	—	2	4	no	—
61	C₂³	#106	1	3	yes	—
62	C₂³	#106	1	1	yes	characteristic
63	C₂³	C₂×SD₁₆	1	3	yes	—
64	C₂×C₄	—	4	4	no	—
65	C₂×C₄	—	8	12	no	—
66	C₂×C₄	#106	1	6	yes	—
67	C₂×C₄	#106	1	3	yes	—
68	C₂×C₄	#106	1	3	yes	—
69	C₂×C₄	C₂×C₄⋊C₄	1	3	yes	—
70	C₂×C₄	C₂×C₄⋊C₄	1	3	yes	—
71	C₂×C₄	C₂×SD₁₆	1	3	yes	—
72	C₂×C₄	C₂×SD₁₆	1	3	yes	—
73	C₂×C₄	C₂×SD₁₆	1	3	yes	—
74	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
75	C₈	—	4	4	no	—
76	C₈	C₂×C₄⋊C₄	1	3	yes	—
77	C₈	C₂×C₄⋊C₄	1	3	yes	—
78	C₈	C₂³×C₄	1	1	yes	characteristic
79	C₈	C₂²×Q₈	1	1	yes	characteristic
80	C₂²	—	2	6	no	—
81	C₂²	—	2	6	no	—
82	C₂²	#365	1	4	yes	direct product factor
83	C₂²	C₂×#106	1	3	yes	—
84	C₂²	C₂×#106	1	3	yes	—
85	C₂²	C₂²×SD₁₆	1	1	yes	characteristic
86	C₄	—	8	4	no	—
87	C₄	C₂×#106	1	3	yes	—
88	C₄	C₂×#106	1	3	yes	—
89	C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
90	C₄	C₂²×SD₁₆	1	1	yes	characteristic
91	C₂	—	2	1	no	—
92	C₂	—	2	3	no	—
93	C₂	C₂×#365	1	6	yes	direct product factor
94	C₂	C₂²×#106	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	#106
commutator	C₈	C₂³×C₄
Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂⁴	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. C₂×#365	C₂	direct (N × H)
8. C₂×M₃₂	C₄	semidirect
10. C₂²×C₁₆	C₄	semidirect
15. #365	C₂²	direct (N × H)
29. M₃₂	C₂×C₄	semidirect
31. C₂×C₁₆	C₂×C₄	semidirect
59. C₁₆	C₂²×C₄	semidirect
82. C₂²	#365	direct (N × H)
93. C₂	C₂×#365	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×M₃₂	2
C₂²×#108	2
C₂²×#106	2
C₂×#365	2

Subgroups — C22×#365 #33402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#365 #33402