Subgroups — #28391679

Order 14464.

350 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299398	C₂	1	3	yes	maximal, complemented
2	#20298566	C₂	1	3	yes	maximal, complemented
3	#338×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14268647	C₂²	1	3	yes	—
5	#16813655	—	2	6	no	—
6	#16813650	—	2	6	no	—
7	#16813644	—	2	6	no	—
8	C₂²×#11744751	C₂²	1	1	yes	characteristic
9	#16813534	—	2	6	no	—
10	C₂×M₁₆×C₁₁₃	C₂²	1	3	yes	—
11	#14268647	—	2	6	no	—
12	#14268647	—	2	6	no	—
13	C₂×Dic₂₂₆	D₄	1	3	yes	—
14	C₂×D₄₅₂	D₄	1	3	yes	—
15	C₂×C₂²⋊D₁₁₃	—	2	3	no	—
16	C₂×C₄×D₁₁₃	—	2	3	no	—
17	Dic₄₅₂	—	2	6	no	—
18	#14268636	—	2	12	no	—
19	D₉₀₄	—	2	6	no	—
20	C₂×#11744751	—	2	3	no	—
21	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
22	M₁₆×C₁₁₃	—	2	6	no	—
23	C₂×C₈×C₁₁₃	D₄	1	6	yes	—
24	C₂³×C₁₁₃	#41	1	1	yes	characteristic
25	C₂²×D₁₁₃	—	2	3	no	—
26	C₂×C₄×C₁₁₃	C₂×D₄	1	3	yes	—
27	C₂×C₄×C₁₁₃	C₂×D₄	1	3	yes	—
28	C₂²⋊D₁₁₃	—	4	6	no	—
29	C₂×Dic₁₁₃	—	2	3	no	—
30	D₄₅₂	—	2	6	no	—
31	C₄×D₁₁₃	—	4	3	no	—
32	C₄×D₁₁₃	—	4	3	no	—
33	Dic₂₂₆	—	2	6	no	—
34	C₈×C₁₁₃	—	2	6	no	—
35	#11744751	—	4	1	no	—
36	C₂²×C₁₁₃	—	4	1	no	—
37	C₂²×C₁₁₃	#121	1	3	yes	—
38	C₂×D₁₁₃	—	4	6	no	—
39	C₂×D₁₁₃	—	4	3	no	—
40	C₄×C₁₁₃	#120	1	3	yes	—
41	C₄×C₁₁₃	#127	1	1	yes	characteristic
42	Dic₁₁₃	—	4	3	no	—
43	C₂×C₁₁₃	—	2	3	no	—
44	C₂×C₁₁₃	#392	1	1	yes	characteristic
45	D₁₁₃	—	8	3	no	—
46	#2017	—	113	1	no	maximal, Sylow
47	C₁₁₃	#2017	1	1	yes	characteristic, complemented, Sylow
48	#575	—	113	3	no	—
49	#408	—	113	3	no	—
50	#338	D₁₁₃	1	1	yes	characteristic, complemented
51	C₂×#41	—	113	3	no	—
52	#137	—	226	6	no	—
53	AGL(1,ℤ/8)	—	226	6	no	—
54	#135	—	226	6	no	—
55	C₂×M₁₆	C₂×D₁₁₃	1	3	yes	—
56	C₂²×C₈	—	113	1	no	—
57	C₂²⋊C₈	—	226	6	no	—
58	#41	—	226	6	no	—
59	#41	—	226	6	no	—
60	C₂×Q₈	—	113	3	no	—
61	C₂×D₄	—	113	3	no	—
62	C₂×D₄	—	226	3	no	—
63	C₂²×C₄	—	226	3	no	—
64	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
65	Q₁₆	—	226	6	no	—
66	SD₁₆	—	226	12	no	—
67	D₈	—	226	6	no	—
68	M₁₆	—	2	6	no	—
69	C₂×C₈	—	226	3	no	—
70	C₂×C₈	D₄₅₂	1	6	yes	—
71	C₂³	—	226	3	no	—
72	C₂³	#14268648	1	1	yes	characteristic
73	Q₈	—	226	6	no	—
74	D₄	—	452	6	no	—
75	D₄	—	226	6	no	—
76	C₂×C₄	—	452	3	no	—
77	C₂×C₄	—	452	3	no	—
78	C₂×C₄	—	226	3	no	—
79	C₂×C₄	D₄×D₁₁₃	1	3	yes	—
80	C₂×C₄	C₂×D₄₅₂	1	3	yes	—
81	C₈	—	2	6	no	—
82	C₈	—	452	1	no	—
83	C₂²	—	4	1	no	—
84	C₂²	—	452	6	no	—
85	C₂²	—	452	3	no	—
86	C₂²	#16813597	1	3	yes	—
87	C₄	—	452	3	no	—
88	C₄	#16813592	1	3	yes	—
89	C₄	#16813619	1	1	yes	characteristic
90	C₂	—	2	3	no	—
91	C₂	—	904	3	no	—
92	C₂	#20298508	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16813619
commutator, cosocle	C₂²×C₄×C₁₁₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₁₃
Fitting	#338×C₁₁₃	C₂
socle	C₂×C₁₁₃	#392
hypercenter	#338	D₁₁₃
hypocenter	C₁₁₃	#2017
LCS	G ▷ C₂²×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
1. #20299398	C₂	semidirect
2. #20298566	C₂	semidirect
3. #338×C₁₁₃	C₂	semidirect
47. C₁₁₃	#2017	semidirect
50. #338	D₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
#20299398	2
#20298566	2
#338×C₁₁₃	2
#2017	113