Subgroups — #36378866

Order 22592.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26015080	C₂	1	2	yes	maximal, complemented
2	#26015079	C₂	1	6	yes	maximal, complemented
3	C₂×#19150747	C₂	1	3	yes	maximal, complemented
4	D₃₅₃⋊M₁₆	C₂	1	3	yes	maximal, complemented
5	#41×D₃₅₃	C₂	1	1	yes	maximal, characteristic
6	#19150747	C₂²	1	1	yes	characteristic
7	#19150747	C₂²	1	3	yes	—
8	#19150747	C₂²	1	6	yes	complemented
9	#19150746	C₂²	1	6	yes	complemented
10	#19150745	C₂²	1	6	yes	complemented
11	C₂×#15667732	C₂²	1	6	yes	complemented
12	#19150738	C₂²	1	1	yes	characteristic
13	#19150737	C₄	1	3	yes	—
14	#19150736	C₂²	1	3	yes	—
15	Q₈×D₃₅₃	C₄	1	1	yes	characteristic
16	D₄×D₃₅₃	C₄	1	3	yes	—
17	C₂×C₄×D₃₅₃	C₂²	1	3	yes	—
18	#41×C₃₅₃	C₄	1	1	yes	characteristic
19	#15667732	C₂³	1	6	yes	—
20	#15667732	C₂³	1	2	yes	—
21	C₂²×D₃₅₃	C₂×C₄	1	3	yes	—
22	Dic₇₀₆	C₂×C₄	1	3	yes	—
23	D₁₄₁₂	C₂×C₄	1	3	yes	—
24	C₂²⋊D₃₅₃	C₂×C₄	1	6	yes	—
25	C₂×Dic₃₅₃	C₂³	1	3	yes	—
26	C₄×D₃₅₃	C₂×C₄	1	3	yes	—
27	C₄×D₃₅₃	C₂×C₄	1	3	yes	—
28	C₄×D₃₅₃	C₂³	1	1	yes	characteristic
29	C₄×D₃₅₃	C₂³	1	3	yes	—
30	Q₈×C₃₅₃	C₂×C₄	1	1	yes	characteristic
31	D₄×C₃₅₃	C₂×C₄	1	3	yes	—
32	C₂×C₄×C₃₅₃	C₂×C₄	1	3	yes	—
33	C₂²×C₃₅₃	C₂²×C₄	1	3	yes	—
34	C₂×D₃₅₃	—	4	3	no	—
35	C₂×D₃₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×D₃₅₃	C₂²×C₄	1	3	yes	—
37	C₄×C₃₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₄×C₃₅₃	C₂²×C₄	1	3	yes	—
39	Dic₃₅₃	C₂²×C₄	1	3	yes	—
40	Dic₃₅₃	C₂⁴	1	1	yes	characteristic
41	C₂×C₃₅₃	—	2	3	no	—
42	C₂×C₃₅₃	C₂³×C₄	1	1	yes	characteristic
43	D₃₅₃	—	2	1	no	—
44	D₃₅₃	—	2	3	no	—
45	C₃₅₃	#568	1	1	yes	characteristic, complemented, Sylow
46	#568	—	353	1	no	maximal, Sylow
47	C₂×#41	—	353	1	no	—
48	#131	—	353	6	no	—
49	#131	—	353	2	no	—
50	C₂×M₁₆	—	353	3	no	—
51	C₂×M₁₆	—	353	3	no	—
52	#41	—	353	1	no	—
53	#41	—	353	3	no	—
54	#41	—	353	3	no	—
55	#41	#13083443	1	1	yes	characteristic
56	C₂×Q₈	—	353	1	no	—
57	C₂×D₄	—	353	3	no	—
58	C₂²×C₄	—	353	3	no	—
59	M₁₆	—	353	1	no	—
60	M₁₆	—	353	3	no	—
61	M₁₆	—	353	6	no	—
62	M₁₆	—	353	6	no	—
63	C₂×C₈	—	353	6	no	—
64	C₂×C₈	—	353	6	no	—
65	C₂³	—	353	3	no	—
66	Q₈	—	353	3	no	—
67	Q₈	C₂×#13083443	1	1	yes	characteristic
68	D₄	—	353	6	no	—
69	D₄	—	353	3	no	—
70	D₄	C₂×#13083443	1	3	yes	—
71	C₂×C₄	—	353	3	no	—
72	C₂×C₄	—	353	3	no	—
73	C₂×C₄	—	353	3	no	—
74	C₂×C₄	—	353	3	no	—
75	C₂×C₄	—	353	1	no	—
76	C₂×C₄	C₂×#13083443	1	3	yes	—
77	C₈	—	353	6	no	—
78	C₈	—	353	2	no	—
79	C₂²	—	353	1	no	—
80	C₂²	—	353	3	no	—
81	C₂²	—	1412	3	no	—
82	C₂²	C₂²×#13083443	1	3	yes	—
83	C₄	—	353	1	no	—
84	C₄	—	353	3	no	—
85	C₄	C₂²×#13083443	1	1	yes	characteristic
86	C₄	C₂²×#13083443	1	3	yes	—
87	C₂	—	2	3	no	—
88	C₂	—	706	1	no	—
89	C₂	—	706	3	no	—
90	C₂	C₂³×#13083443	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×#13083443
commutator, socle	C₂×C₃₅₃	C₂³×C₄
Fitting	#41×C₃₅₃	C₄
prime radical	#41×D₃₅₃	C₂
cosocle	Dic₃₅₃	C₂⁴
hypercenter	#41	#13083443
hypocenter	C₃₅₃	#568
LCS	G ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂ ◁ #41

Complements

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

N	H ≅ G/N	type
1. #26015080	C₂	semidirect
2. #26015079	C₂	semidirect
3. C₂×#19150747	C₂	semidirect
4. D₃₅₃⋊M₁₆	C₂	semidirect
8. #19150747	C₂²	semidirect
9. #19150746	C₂²	semidirect
10. #19150745	C₂²	semidirect
11. C₂×#15667732	C₂²	semidirect
45. C₃₅₃	#568	semidirect

Maximal subgroups

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

subgroup	index
#26015080	2
#26015079	2
C₂×#19150747	2
D₃₅₃⋊M₁₆	2
#41×D₃₅₃	2
#568	353