Subgroups — #15777628

Order 3328.

246 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13117065	C₂	1	4	yes	maximal
2	#13114454	C₂	1	1	yes	maximal, characteristic
3	#13112753	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1710	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#500	C₂²	1	2	yes	—
6	C₁₃×#487	—	2	2	no	—
7	#11718360	C₂²	1	2	yes	—
8	#11718343	—	2	2	no	—
9	C₈²×C₁₃	C₂²	1	1	yes	characteristic
10	#11718084	—	2	8	no	—
11	C₄×#6090	C₂²	1	2	yes	—
12	C₁₃×C₄⋊Q₈	—	2	2	no	—
13	C₂×Dic₅₂	—	2	4	no	—
14	#88123	—	2	2	no	—
15	C₁₃×#107	—	2	4	no	—
16	C₁₃×#103	—	4	4	no	—
17	C₄×C₈×C₁₃	D₄	1	2	yes	—
18	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
19	#88058	—	4	4	no	—
20	C₂×#6090	D₄	1	4	yes	—
21	#12514	—	13	1	no	maximal, Sylow
22	C₂×Q₈×C₁₃	—	4	2	no	—
23	C₂×Dic₂₆	—	4	2	no	—
24	C₂×C₈×C₁₃	D₈	1	4	yes	—
25	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
26	C₁₃×C₄⋊C₄	—	4	2	no	—
27	C₁₃×C₄⋊C₄	—	4	2	no	—
28	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
29	C₄⋊Dic₁₃	—	4	4	no	—
30	Dic₅₂	—	4	8	no	—
31	#6090	—	2	4	no	—
32	#2241	—	13	4	no	—
33	#1713	—	13	1	no	—
34	#1710	D₁₃	1	1	yes	characteristic
35	#1311	—	13	1	no	—
36	Q₈×C₁₃	—	8	4	no	—
37	C₂×C₄×C₁₃	—	8	2	no	—
38	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
39	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
40	C₂×Dic₁₃	—	8	2	no	—
41	Dic₂₆	—	8	4	no	—
42	C₈×C₁₃	—	2	4	no	—
43	C₈×C₁₃	#135	1	4	yes	—
44	#500	—	26	2	no	—
45	#500	C₂×D₁₃	1	2	yes	—
46	#494	—	13	2	no	—
47	#487	—	2	2	no	—
48	#358	—	26	8	no	—
49	C₄×C₁₆	—	13	2	no	—
50	C₈²	C₂×D₁₃	1	1	yes	characteristic
51	C₂²×C₁₃	#493	1	1	yes	characteristic
52	C₄×C₁₃	—	16	2	no	—
53	C₄×C₁₃	#495	1	2	yes	—
54	C₄×C₁₃	#510	1	4	yes	—
55	Dic₁₃	—	16	2	no	—
56	C₂×Q₁₆	—	26	4	no	—
57	C₄⋊Q₈	—	2	2	no	—
58	C₄⋊Q₈	—	26	2	no	—
59	C₂×C₁₆	—	13	4	no	—
60	#107	—	2	4	no	—
61	#107	—	52	4	no	—
62	#103	—	4	4	no	—
63	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
64	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
65	C₂×C₁₃	#1709	1	1	yes	characteristic
66	C₂×C₁₃	#2251	1	2	yes	—
67	C₂×Q₈	—	4	2	no	—
68	C₂×Q₈	—	52	2	no	—
69	Q₁₆	—	52	8	no	—
70	C₂×C₈	#6104	1	4	yes	—
71	C₂×C₈	D₄×D₁₃	1	2	yes	—
72	C₄⋊C₄	—	4	2	no	—
73	C₄⋊C₄	—	4	2	no	—
74	C₄⋊C₄	—	52	4	no	—
75	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
76	C₁₆	—	26	4	no	—
77	C₁₃	#12514	1	1	yes	characteristic, complemented, Sylow
78	Q₈	—	8	4	no	—
79	Q₈	—	104	4	no	—
80	C₂×C₄	—	8	2	no	—
81	C₂×C₄	—	104	2	no	—
82	C₂×C₄	C₂×#6104	1	2	yes	—
83	C₂×C₄	#88194	1	1	yes	characteristic
84	C₈	—	2	4	no	—
85	C₈	#88173	1	4	yes	—
86	C₂²	#11718734	1	1	yes	characteristic
87	C₄	—	16	2	no	—
88	C₄	—	208	2	no	—
89	C₄	#11718852	1	2	yes	—
90	C₄	#11718856	1	4	yes	—
91	C₂	#13114443	1	1	yes	characteristic
92	C₂	#13117108	1	2	yes	—

	subgroup	G/H
center	C₂²	#11718734
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1710	C₂
socle, prime radical	C₂²×C₁₃	#493
hypercenter	#1710	D₁₃
hypocenter	C₁₃	#12514
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

Complements

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

N	H ≅ G/N	type
77. C₁₃	#12514	semidirect

Maximal subgroups

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

subgroup	index
#13117065	2
#13114454	2
#13112753	2
C₁₃×#1710	2
#12514	13