Subgroups — C₂²×#88106 #13131797

Order 1664.

400 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#1055	C₂	1	1	yes	maximal, characteristic
2	C₂×#88106	C₂	1	12	yes	maximal, direct product factor
3	C₂²×#6092	C₂	1	2	yes	maximal, complemented
4	C₂³×C₄×C₁₃	C₄	1	1	yes	characteristic
5	C₂²×#1055	C₄	1	6	yes	—
6	C₂²×#1055	C₄	1	1	yes	characteristic
7	C₂²×#1055	C₂²	1	1	yes	characteristic
8	C₂²×#1055	C₂²	1	6	yes	—
9	#88106	C₂²	1	16	yes	direct product factor
10	C₂×#6092	C₂²	1	12	yes	complemented
11	C₂⁴×C₁₃	C₈	1	1	yes	characteristic
12	C₂²×C₄×C₁₃	C₈	1	6	yes	—
13	C₂²×C₄×C₁₃	C₈	1	1	yes	characteristic
14	C₂²×C₄×C₁₃	C₂×C₄	1	6	yes	—
15	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂×#1055	C₂×C₄	1	3	yes	—
17	C₂×#1055	C₂×C₄	1	12	yes	—
18	C₂×#1055	C₂×C₄	1	6	yes	—
19	C₂×#1055	C₂³	1	3	yes	—
20	C₂×#1055	C₂³	1	4	yes	—
21	#6092	C₂³	1	8	yes	complemented
22	C₂²×M₃₂	—	13	1	no	maximal, Sylow
23	C₂³×C₁₃	—	2	4	no	—
24	C₂³×C₁₃	C₂×C₈	1	6	yes	—
25	C₂³×C₁₃	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄×C₁₃	C₂×C₈	1	12	yes	—
27	C₂×C₄×C₁₃	C₂×C₈	1	6	yes	—
28	C₂×C₄×C₁₃	C₂×C₈	1	3	yes	—
29	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
30	C₂×C₄×C₁₃	C₂²×C₄	1	3	yes	—
31	#1055	C₂²×C₄	1	3	yes	—
32	#1055	C₂²×C₄	1	4	yes	—
33	#1055	C₂⁴	1	1	yes	characteristic
34	C₂³×C₈	—	13	1	no	—
35	C₂×M₃₂	—	13	12	no	—
36	C₂²×C₁₆	—	13	2	no	—
37	C₂²×C₁₃	—	2	12	no	—
38	C₂²×C₁₃	M₃₂	1	4	yes	complemented
39	C₂²×C₁₃	C₂²×C₈	1	4	yes	—
40	C₂²×C₁₃	C₂²×C₈	1	3	yes	—
41	C₄×C₁₃	C₂²×C₈	1	4	yes	—
42	C₄×C₁₃	C₂²×C₈	1	3	yes	—
43	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
44	C₂³×C₄	#261	1	1	yes	characteristic
45	C₂²×C₈	—	13	6	no	—
46	C₂²×C₈	—	13	6	no	—
47	C₂²×C₈	—	13	1	no	—
48	C₂²×C₈	—	13	1	no	—
49	M₃₂	—	13	16	no	—
50	C₂×C₁₆	—	13	12	no	—
51	C₂×C₁₃	—	2	4	no	—
52	C₂×C₁₃	C₂×M₃₂	1	6	yes	complemented
53	C₂×C₁₃	C₂³×C₈	1	1	yes	characteristic
54	C₂⁴	#1057	1	1	yes	characteristic
55	C₂²×C₄	#1057	1	1	yes	characteristic
56	C₂²×C₄	#1057	1	6	yes	—
57	C₂²×C₄	C₂×#261	1	6	yes	—
58	C₂²×C₄	C₂×#261	1	1	yes	characteristic
59	C₂×C₈	—	13	3	no	—
60	C₂×C₈	—	13	4	no	—
61	C₂×C₈	—	13	12	no	—
62	C₂×C₈	—	13	3	no	—
63	C₂×C₈	—	13	6	no	—
64	C₁₆	—	13	8	no	—
65	C₁₃	C₂²×M₃₂	1	1	yes	characteristic, complemented, Sylow
66	C₂³	—	2	4	no	—
67	C₂³	C₂×#1057	1	6	yes	—
68	C₂³	C₂×#1057	1	1	yes	characteristic
69	C₂×C₄	C₂×#1057	1	12	yes	—
70	C₂×C₄	C₂×#1057	1	6	yes	—
71	C₂×C₄	C₂×#1057	1	3	yes	—
72	C₂×C₄	C₂²×#261	1	4	yes	—
73	C₂×C₄	C₂²×#261	1	3	yes	—
74	C₈	—	13	1	no	—
75	C₈	—	13	3	no	—
76	C₈	—	13	4	no	—
77	C₂²	—	2	12	no	—
78	C₂²	#88106	1	4	yes	direct product factor
79	C₂²	C₂²×#1057	1	4	yes	—
80	C₂²	C₂²×#1057	1	3	yes	—
81	C₄	C₂²×#1057	1	4	yes	—
82	C₄	C₂²×#1057	1	3	yes	—
83	C₄	C₂³×#261	1	1	yes	characteristic
84	C₂	—	2	4	no	—
85	C₂	C₂×#88106	1	6	yes	direct product factor
86	C₂	C₂³×#1057	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×#261
commutator	C₂×C₁₃	C₂³×C₈
Frattini	C₄	C₂³×#261
Fitting	C₂³×C₄×C₁₃	C₄
socle	C₂³×C₁₃	C₂×C₈
prime radical	C₂⁴×C₁₃	C₈
cosocle	#1055	C₂⁴
hypercenter	C₂³×C₄	#261
hypocenter	C₁₃	C₂²×M₃₂
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#88106	C₂	direct (N × H)
3. C₂²×#6092	C₂	semidirect
9. #88106	C₂²	direct (N × H)
10. C₂×#6092	C₂²	semidirect
21. #6092	C₂³	semidirect
38. C₂²×C₁₃	M₃₂	semidirect
52. C₂×C₁₃	C₂×M₃₂	semidirect
65. C₁₃	C₂²×M₃₂	semidirect
78. C₂²	#88106	direct (N × H)
85. C₂	C₂×#88106	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#1055	2
C₂×#88106	2
C₂²×#6092	2
C₂²×M₃₂	13

Subgroups — C22×#88106 #13131797

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#88106 #13131797