Subgroups — #20299080

Order 7232.

294 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14268648	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₄₅₂	C₂	1	3	yes	maximal, complemented
4	#16813584	C₂	1	3	yes	maximal, complemented
5	#16813579	C₂	1	3	yes	maximal, complemented
6	C₄⋊C₄×D₁₁₃	C₂	1	3	yes	maximal, complemented
7	C₄×Q₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#14268648	C₄	1	8	yes	complemented
9	C₂×D₄₅₂	C₂²	1	3	yes	—
10	C₂×C₄×D₁₁₃	C₂²	1	6	yes	—
11	C₂×C₄×D₁₁₃	C₂²	1	3	yes	—
12	C₄⋊Dic₁₁₃	C₂²	1	3	yes	—
13	#14268626	C₂²	1	3	yes	—
14	#14268626	C₂²	1	1	yes	characteristic
15	#14268624	C₂²	1	3	yes	—
16	#14268624	C₂²	1	3	yes	—
17	C₄×Dic₁₁₃	C₂²	1	3	yes	—
18	C₂×Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₁₁₃	C₂²	1	3	yes	—
20	C₄²×C₁₁₃	C₂²	1	3	yes	—
21	C₂²×D₁₁₃	C₂³	1	3	yes	—
22	Q₈×C₁₁₃	C₂×C₄	1	4	yes	complemented
23	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₁₁₃	C₂³	1	3	yes	—
25	C₂×C₄×C₁₁₃	C₂³	1	3	yes	—
26	C₂×Dic₁₁₃	C₂³	1	1	yes	characteristic
27	C₂×Dic₁₁₃	C₂³	1	3	yes	—
28	C₂×Dic₁₁₃	C₂³	1	1	yes	characteristic
29	D₄₅₂	C₂×C₄	1	12	yes	complemented
30	C₄×D₁₁₃	—	4	6	no	—
31	C₄×D₁₁₃	C₂×C₄	1	12	yes	complemented
32	C₂²×C₁₁₃	C₂⁴	1	1	yes	characteristic
33	C₂×D₁₁₃	—	2	3	no	—
34	C₂×D₁₁₃	—	2	3	no	—
35	C₂×D₁₁₃	C₂²×C₄	1	6	yes	—
36	C₄×C₁₁₃	—	2	3	no	—
37	C₄×C₁₁₃	—	2	1	no	—
38	C₄×C₁₁₃	C₂²×C₄	1	6	yes	—
39	Dic₁₁₃	—	2	3	no	—
40	Dic₁₁₃	—	2	1	no	—
41	Dic₁₁₃	C₂²×C₄	1	2	yes	—
42	C₂×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
43	C₂×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
44	C₂×C₁₁₃	2₋¹⁺⁴	1	1	yes	characteristic
45	D₁₁₃	—	2	6	no	—
46	C₁₁₃	#520	1	1	yes	characteristic, complemented, Sylow
47	#520	—	113	1	no	maximal, Sylow
48	C₂×#41	—	113	1	no	—
49	C₄×Q₈	—	113	1	no	—
50	C₄×Q₈	D₁₁₃	1	1	yes	characteristic, complemented
51	C₄×D₄	—	113	3	no	—
52	C₄×D₄	—	113	3	no	—
53	#117	—	113	3	no	—
54	C₂×C₄⋊C₄	—	113	3	no	—
55	#41	—	113	8	no	—
56	C₂×Q₈	C₂×D₁₁₃	1	1	yes	characteristic
57	C₂×D₄	—	113	3	no	—
58	C₂²×C₄	—	113	3	no	—
59	C₂²×C₄	—	113	6	no	—
60	C₄⋊C₄	—	113	3	no	—
61	C₄⋊C₄	—	113	3	no	—
62	C₄⋊C₄	—	113	1	no	—
63	C₄⋊C₄	C₂×D₁₁₃	1	3	yes	—
64	C₂²⋊C₄	—	113	3	no	—
65	C₂²⋊C₄	—	113	3	no	—
66	C₄²	—	113	3	no	—
67	C₄²	C₂×D₁₁₃	1	3	yes	—
68	C₂³	—	113	3	no	—
69	Q₈	C₄×D₁₁₃	1	4	yes	complemented
70	D₄	—	113	12	no	—
71	C₂×C₄	—	113	1	no	—
72	C₂×C₄	—	113	12	no	—
73	C₂×C₄	—	113	1	no	—
74	C₂×C₄	—	113	3	no	—
75	C₂×C₄	—	452	6	no	—
76	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
77	C₂×C₄	C₂²×D₁₁₃	1	3	yes	—
78	C₂×C₄	C₂²×D₁₁₃	1	3	yes	—
79	C₂²	—	113	6	no	—
80	C₂²	—	226	3	no	—
81	C₂²	—	226	3	no	—
82	C₂²	C₂³×D₁₁₃	1	1	yes	characteristic
83	C₄	—	2	3	no	—
84	C₄	—	2	1	no	—
85	C₄	—	113	2	no	—
86	C₄	—	226	3	no	—
87	C₄	—	226	1	no	—
88	C₄	C₂×C₄×D₁₁₃	1	6	yes	—
89	C₂	—	226	6	no	—
90	C₂	C₂²×C₄×D₁₁₃	1	1	yes	characteristic
91	C₂	#16813668	1	1	yes	characteristic
92	C₂	#16813670	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₁₃
commutator	C₂×C₁₁₃	C₂³×C₄
Fitting	C₄×Q₈×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂⁴
hypercenter	C₄×Q₈	D₁₁₃
hypocenter	C₁₁₃	#520
LCS	G ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×Dic₁₁₃	C₂	semidirect
3. C₄×D₄₅₂	C₂	semidirect
4. #16813584	C₂	semidirect
5. #16813579	C₂	semidirect
6. C₄⋊C₄×D₁₁₃	C₂	semidirect
7. C₄×Q₈×C₁₁₃	C₂	semidirect
8. #14268648	C₄	semidirect
22. Q₈×C₁₁₃	C₂×C₄	semidirect
29. D₄₅₂	C₂×C₄	semidirect
31. C₄×D₁₁₃	C₂×C₄	semidirect
46. C₁₁₃	#520	semidirect
50. C₄×Q₈	D₁₁₃	semidirect
69. Q₈	C₄×D₁₁₃	semidirect