Subgroups — #34094481

Order 20032.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23756473	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×D₃₁₃	C₂	1	1	yes	maximal, characteristic
3	#23756429	C₂	1	1	yes	maximal, characteristic, complemented
4	#23756428	C₂	1	1	yes	maximal, characteristic, complemented
5	#23756421	C₂	1	1	yes	maximal, characteristic, complemented
6	#23756409	C₂	1	1	yes	maximal, characteristic
7	#103×C₃₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₃₁₃	—	2	1	no	—
9	C₂×C₄×D₃₁₃	C₂²	1	1	yes	characteristic
10	C₂×#14578809	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₃₁₃	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₃₁₃	C₂²	1	1	yes	characteristic
13	#19097540	—	2	1	no	—
14	#19097540	—	2	1	no	—
15	#19097538	—	2	1	no	—
16	C₂×Q₈×C₃₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₃₁₃	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₃₁₃	C₂²	1	1	yes	characteristic
19	C₂²×D₃₁₃	D₄	1	1	yes	characteristic
20	C₂×Dic₃₁₃	—	2	1	no	—
21	C₂×Dic₃₁₃	—	2	1	no	—
22	C₂×Dic₃₁₃	D₄	1	1	yes	characteristic
23	C₄×D₃₁₃	—	2	1	no	—
24	C₄×D₃₁₃	—	2	1	no	—
25	C₄×D₃₁₃	—	4	2	no	—
26	#14578809	—	2	1	no	—
27	Q₈×C₃₁₃	—	2	1	no	—
28	Q₈×C₃₁₃	—	2	1	no	—
29	C₂×C₄×C₃₁₃	—	2	1	no	—
30	C₂×C₄×C₃₁₃	—	2	1	no	—
31	C₂×C₄×C₃₁₃	C₂³	1	1	yes	characteristic
32	C₈×C₃₁₃	—	2	1	no	—
33	C₂²×C₃₁₃	C₂×D₄	1	1	yes	characteristic
34	C₂×D₃₁₃	—	2	1	no	—
35	C₂×D₃₁₃	—	2	1	no	—
36	C₂×D₃₁₃	SD₁₆	1	2	yes	—
37	C₄×C₃₁₃	—	4	1	no	—
38	C₄×C₃₁₃	—	4	1	no	—
39	C₄×C₃₁₃	#41	1	1	yes	characteristic
40	C₄×C₃₁₃	#41	1	1	yes	characteristic
41	Dic₃₁₃	—	2	1	no	—
42	Dic₃₁₃	—	4	1	no	—
43	Dic₃₁₃	—	4	1	no	—
44	C₂×C₃₁₃	#123	1	1	yes	characteristic
45	C₂×C₃₁₃	C₂×SD₁₆	1	1	yes	characteristic
46	C₂×C₃₁₃	#137	1	1	yes	characteristic
47	D₃₁₃	—	2	2	no	—
48	C₃₁₃	#483	1	1	yes	characteristic, complemented, Sylow
49	#483	—	313	1	no	maximal, Sylow
50	C₂²⋊Q₈	—	313	1	no	—
51	C₂×C₄⋊C₄	—	313	1	no	—
52	#106	—	313	1	no	—
53	#106	—	313	1	no	—
54	#103	—	313	1	no	—
55	#103	D₃₁₃	1	1	yes	characteristic, complemented
56	C₂²⋊C₈	—	313	1	no	—
57	C₂×Q₈	C₂×D₃₁₃	1	1	yes	characteristic
58	C₂²×C₄	—	313	1	no	—
59	C₂²×C₄	—	626	1	no	—
60	C₂×C₈	—	313	1	no	—
61	C₂×C₈	C₂×D₃₁₃	1	1	yes	characteristic
62	C₄⋊C₄	—	313	1	no	—
63	C₄⋊C₄	—	313	1	no	—
64	C₄⋊C₄	—	626	1	no	—
65	C₄⋊C₄	—	626	1	no	—
66	C₄⋊C₄	C₂×D₃₁₃	1	1	yes	characteristic
67	C₂²⋊C₄	—	626	1	no	—
68	C₂³	—	313	1	no	—
69	Q₈	—	2	1	no	—
70	Q₈	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	313	1	no	—
74	C₂×C₄	—	626	1	no	—
75	C₂×C₄	—	626	1	no	—
76	C₂×C₄	—	626	1	no	—
77	C₂×C₄	—	626	1	no	—
78	C₂×C₄	—	1252	2	no	—
79	C₂×C₄	C₂²×D₃₁₃	1	1	yes	characteristic
80	C₈	—	2	1	no	—
81	C₈	—	626	1	no	—
82	C₂²	—	313	2	no	—
83	C₂²	—	626	1	no	—
84	C₂²	—	626	1	no	—
85	C₂²	D₄×D₃₁₃	1	1	yes	characteristic
86	C₄	—	4	1	no	—
87	C₄	—	4	1	no	—
88	C₄	—	626	1	no	—
89	C₄	—	1252	1	no	—
90	C₄	—	1252	1	no	—
91	C₄	#19097560	1	1	yes	characteristic
92	C₄	#19097561	1	1	yes	characteristic
93	C₂	—	626	2	no	—
94	C₂	#23756479	1	1	yes	characteristic
95	C₂	SD₁₆×D₃₁₃	1	1	yes	characteristic
96	C₂	#23756529	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₃₁₃
commutator, cosocle	C₂×C₄×C₃₁₃	C₂³
Frattini	C₂×C₄	C₂²×D₃₁₃
Fitting	#103×C₃₁₃	C₂
socle	C₂²×C₃₁₃	C₂×D₄
prime radical	C₂²×D₃₁₃	D₄
hypercenter	#103	D₃₁₃
hypocenter	C₃₁₃	#483
LCS	G ▷ C₂×C₄×C₃₁₃ ▷ C₂×C₃₁₃ ▷ C₃₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
3. #23756429	C₂	semidirect
4. #23756428	C₂	semidirect
5. #23756421	C₂	semidirect
7. #103×C₃₁₃	C₂	semidirect
48. C₃₁₃	#483	semidirect
55. #103	D₃₁₃	semidirect

Maximal subgroups

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

subgroup	index
#23756473	2
C₄⋊C₄×D₃₁₃	2
#23756429	2
#23756428	2
#23756421	2
#23756409	2
#103×C₃₁₃	2
#483	313