Subgroups — #20299326

Order 7232.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16813622	C₂	1	1	yes	maximal, characteristic
2	#16813620	C₂	1	2	yes	maximal
3	#16813614	C₂	1	2	yes	maximal
4	#16813612	C₂	1	4	yes	maximal
5	C₄×Dic₂₂₆	C₂	1	2	yes	maximal
6	C₄×Dic₂₂₆	C₂	1	1	yes	maximal, characteristic
7	#16813587	C₂	1	2	yes	maximal
8	C₄×Q₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₂₂₆	C₂²	1	2	yes	—
10	C₂×Dic₂₂₆	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
13	C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
14	#14268626	C₂²	1	4	yes	—
15	#14268626	C₂²	1	2	yes	—
16	#14268626	C₂²	1	4	yes	—
17	#14268626	C₂²	1	4	yes	—
18	C₄×Dic₁₁₃	C₂²	1	4	yes	—
19	C₄×Dic₁₁₃	C₂²	1	2	yes	—
20	C₂×Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
23	C₄²×C₁₁₃	C₂²	1	2	yes	—
24	C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
25	Q₈×C₁₁₃	—	2	2	no	—
26	C₂×C₄×C₁₁₃	C₂³	1	2	yes	—
27	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₁₁₃	C₂³	1	2	yes	—
29	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₁₃	C₂³	1	2	yes	—
32	C₂×Dic₁₁₃	C₂³	1	4	yes	—
33	C₂×Dic₁₁₃	C₂³	1	2	yes	—
34	Dic₂₂₆	—	2	4	no	—
35	Dic₂₂₆	Q₈	1	4	yes	complemented
36	C₂²×C₁₁₃	C₂⁴	1	1	yes	characteristic
37	C₄×C₁₁₃	—	2	2	no	—
38	C₄×C₁₁₃	—	2	2	no	—
39	C₄×C₁₁₃	—	2	1	no	—
40	C₄×C₁₁₃	C₂×Q₈	1	2	yes	—
41	C₄×C₁₁₃	#41	1	2	yes	—
42	Dic₁₁₃	—	2	4	no	—
43	Dic₁₁₃	—	2	2	no	—
44	Dic₁₁₃	C₂×Q₈	1	4	yes	—
45	C₂×C₁₁₃	C₂²×Q₈	1	1	yes	characteristic
46	C₂×C₁₁₃	C₂×#41	1	1	yes	characteristic
47	C₂×C₁₁₃	2₋¹⁺⁴	1	1	yes	characteristic
48	C₁₁₃	#557	1	1	yes	characteristic, complemented, Sylow
49	#557	—	113	1	no	maximal, Sylow
50	C₄⋊Q₈	—	113	2	no	—
51	C₄⋊Q₈	—	113	1	no	—
52	#125	—	113	4	no	—
53	#125	—	113	2	no	—
54	C₄×Q₈	—	113	2	no	—
55	C₄×Q₈	—	113	2	no	—
56	C₄×Q₈	—	113	1	no	—
57	C₄×Q₈	D₁₁₃	1	1	yes	characteristic
58	C₂×Q₈	—	113	1	no	—
59	C₂×Q₈	—	113	2	no	—
60	C₂×Q₈	C₂×D₁₁₃	1	1	yes	characteristic
61	C₄⋊C₄	—	113	2	no	—
62	C₄⋊C₄	—	113	4	no	—
63	C₄⋊C₄	—	113	1	no	—
64	C₄⋊C₄	—	113	4	no	—
65	C₄⋊C₄	—	113	2	no	—
66	C₄⋊C₄	—	113	4	no	—
67	C₄⋊C₄	—	113	2	no	—
68	C₄⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
69	C₄⋊C₄	C₂×D₁₁₃	1	2	yes	—
70	C₄²	—	113	4	no	—
71	C₄²	—	113	2	no	—
72	C₄²	C₂×D₁₁₃	1	2	yes	—
73	C₄²	C₂×D₁₁₃	1	1	yes	characteristic
74	Q₈	—	2	2	no	—
75	Q₈	—	113	4	no	—
76	Q₈	—	226	4	no	—
77	C₂×C₄	—	113	2	no	—
78	C₂×C₄	—	113	2	no	—
79	C₂×C₄	—	113	4	no	—
80	C₂×C₄	C₂²×D₁₁₃	1	2	yes	—
81	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
82	C₂×C₄	C₂²×D₁₁₃	1	2	yes	—
83	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
84	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
85	C₂²	C₂³×D₁₁₃	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	2	2	no	—
88	C₄	—	2	1	no	—
89	C₄	—	113	4	no	—
90	C₄	—	226	4	no	—
91	C₄	—	226	2	no	—
92	C₄	Q₈×D₁₁₃	1	2	yes	—
93	C₄	#14268647	1	2	yes	—
94	C₂	C₂×Q₈×D₁₁₃	1	1	yes	characteristic
95	C₂	C₂×#14268647	1	1	yes	characteristic
96	C₂	#16813669	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₁₃
commutator, socle	C₂²×C₁₁₃	C₂⁴
Fitting	C₄×Q₈×C₁₁₃	C₂
hypercenter	C₄×Q₈	D₁₁₃
hypocenter	C₁₁₃	#557
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
35. Dic₂₂₆	Q₈	semidirect
48. C₁₁₃	#557	semidirect