Subgroups — #28406971

Order 14464.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₁₆×#90673	C₂	1	2	yes	maximal
2	#20299525	C₂	1	1	yes	maximal, characteristic
3	#20299511	C₂	1	2	yes	maximal
4	#20299485	C₂	1	1	yes	maximal, characteristic
5	Q₃₂×D₁₁₃	C₂	1	1	yes	maximal, characteristic
6	Q₈×#90673	—	2	2	no	—
7	#16813695	C₂²	1	1	yes	characteristic
8	#16813691	C₂²	1	2	yes	—
9	#16813685	—	2	2	no	—
10	C₈×#90673	C₂²	1	1	yes	characteristic
11	Q₁₆×D₁₁₃	C₂²	1	2	yes	—
12	Dic₉₀₄	C₄	1	1	yes	characteristic, complemented
13	#16813568	C₄	1	2	yes	complemented
14	C₁₆×D₁₁₃	C₂²	1	1	yes	characteristic
15	Q₃₂×C₁₁₃	C₄	1	1	yes	characteristic, complemented
16	#14268655	D₄	1	1	yes	characteristic
17	C₄⋊#90673	—	2	2	no	—
18	C₄⋊#90673	—	4	2	no	—
19	C₄×#90673	—	4	2	no	—
20	C₄×#90673	D₄	1	1	yes	characteristic
21	#14268650	—	2	1	no	—
22	Q₈×D₁₁₃	—	2	2	no	—
23	Dic₄₅₂	C₂×C₄	1	2	yes	—
24	#14268637	—	2	2	no	—
25	C₈×D₁₁₃	C₂³	1	1	yes	characteristic
26	#14268622	C₂×C₄	1	1	yes	characteristic
27	Q₁₆×C₁₁₃	C₂×C₄	1	2	yes	—
28	C₁₆×C₁₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂×#90673	—	4	2	no	—
30	C₂×#90673	D₈	1	2	yes	—
31	Q₈×C₁₁₃	—	2	2	no	—
32	C₄×D₁₁₃	—	4	2	no	—
33	C₄×D₁₁₃	C₂×D₄	1	1	yes	characteristic
34	Dic₂₂₆	—	2	2	no	—
35	Dic₂₂₆	—	4	2	no	—
36	#11744753	—	2	1	no	—
37	C₈×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
38	#11744751	#41	1	1	yes	characteristic
39	C₂×D₁₁₃	C₂×D₈	1	1	yes	characteristic
40	#90673	—	2	2	no	—
41	#90673	—	8	2	no	—
42	C₄×C₁₁₃	—	4	2	no	—
43	C₄×C₁₁₃	C₄×D₄	1	1	yes	characteristic
44	Dic₁₁₃	—	4	2	no	—
45	Dic₁₁₃	#135	1	1	yes	characteristic
46	C₂×C₁₁₃	C₄×D₈	1	1	yes	characteristic
47	D₁₁₃	#510	1	2	yes	—
48	#2176	—	113	1	no	maximal, Sylow
49	C₁₁₃	#2176	1	1	yes	characteristic, complemented, Sylow
50	C₂×Q₃₂	—	113	1	no	—
51	C₄×Q₁₆	—	113	2	no	—
52	#367	—	113	1	no	—
53	#358	—	113	2	no	—
54	#346	—	113	1	no	—
55	C₂×Q₁₆	—	113	2	no	—
56	C₄×Q₈	—	226	2	no	—
57	Q₃₂	—	113	2	no	—
58	Q₃₂	—	113	1	no	—
59	Q₃₂	#90673	1	1	yes	characteristic, complemented
60	C₂×C₁₆	—	113	1	no	—
61	C₂×C₁₆	—	113	1	no	—
62	#107	—	113	2	no	—
63	#103	—	226	2	no	—
64	C₄×C₈	—	113	1	no	—
65	C₂×Q₈	—	226	2	no	—
66	Q₁₆	—	113	2	no	—
67	Q₁₆	—	226	2	no	—
68	Q₁₆	C₂×#90673	1	2	yes	—
69	C₂×C₈	—	113	1	no	—
70	C₂×C₈	—	113	1	no	—
71	C₄⋊C₄	—	452	2	no	—
72	C₄⋊C₄	—	226	2	no	—
73	C₄²	—	113	1	no	—
74	C₄²	—	452	2	no	—
75	C₁₆	—	113	1	no	—
76	C₁₆	—	226	1	no	—
77	C₁₆	C₂×#90673	1	1	yes	characteristic
78	Q₈	—	2	2	no	—
79	Q₈	—	452	2	no	—
80	Q₈	—	226	2	no	—
81	C₂×C₄	—	113	1	no	—
82	C₂×C₄	—	113	2	no	—
83	C₂×C₄	—	452	2	no	—
84	C₂×C₄	—	452	2	no	—
85	C₈	—	113	1	no	—
86	C₈	—	226	1	no	—
87	C₈	C₂²×#90673	1	1	yes	characteristic
88	C₂²	—	113	1	no	—
89	C₄	—	4	2	no	—
90	C₄	—	113	1	no	—
91	C₄	—	904	2	no	—
92	C₄	—	452	2	no	—
93	C₄	—	226	2	no	—
94	C₄	D₄×#90673	1	1	yes	characteristic
95	C₂	—	113	2	no	—
96	C₂	D₈×#90673	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×#90673
commutator	C₈×C₁₁₃	C₂²×C₄
Frattini	C₈	C₂²×#90673
Fitting	Q₃₂×C₁₁₃	C₄
socle	C₂×C₁₁₃	C₄×D₈
prime radical	C₂×D₁₁₃	C₂×D₈
cosocle	C₈×D₁₁₃	C₂³
hypercenter	Q₃₂	#90673
hypocenter	C₁₁₃	#2176
LCS	G ▷ C₈×C₁₁₃ ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
12. Dic₉₀₄	C₄	semidirect
13. #16813568	C₄	semidirect
15. Q₃₂×C₁₁₃	C₄	semidirect
49. C₁₁₃	#2176	semidirect
59. Q₃₂	#90673	semidirect

Maximal subgroups

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

subgroup	index
Q₁₆×#90673	2
#20299525	2
#20299511	2
#20299485	2
Q₃₂×D₁₁₃	2
#2176	113