Subgroups — #13116706

Order 1664.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₀₄	C₂	1	1	yes	maximal, characteristic
2	#11718534	C₂	1	2	yes	maximal
3	C₁₃×#367	C₂	1	1	yes	maximal, characteristic
4	#11718094	C₂	1	1	yes	maximal, characteristic
5	#11718084	C₂	1	2	yes	maximal
6	C₂×Dic₅₂	C₂²	1	2	yes	—
7	#88141	—	2	2	no	—
8	C₂×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
9	C₁₃×#107	C₂²	1	2	yes	—
10	C₈×Dic₁₃	C₂²	1	1	yes	characteristic
11	C₂×#6090	C₂²	1	1	yes	characteristic
12	#88050	—	2	2	no	—
13	Dic₁₀₄	C₄	1	4	yes	complemented
14	C₂×Dic₂₆	—	2	2	no	—
15	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
16	C₁₃×C₄⋊C₄	—	2	2	no	—
17	#6101	—	4	2	no	—
18	C₄×Dic₁₃	—	4	2	no	—
19	C₄×Dic₁₃	D₄	1	1	yes	characteristic
20	C₂×#1055	D₄	1	1	yes	characteristic
21	Dic₅₂	—	2	2	no	—
22	Dic₅₂	C₂×C₄	1	4	yes	—
23	C₁₃×C₁₆	C₂×C₄	1	2	yes	—
24	#6090	—	2	1	no	—
25	#2176	—	13	1	no	maximal, Sylow
26	C₂×C₄×C₁₃	—	4	2	no	—
27	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
28	C₂×Dic₁₃	—	4	2	no	—
29	C₂×Dic₁₃	D₈	1	2	yes	—
30	Dic₂₆	—	2	4	no	—
31	Dic₂₆	—	4	2	no	—
32	C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₁₃	#41	1	1	yes	characteristic
34	#1055	—	2	1	no	—
35	C₂×Q₃₂	—	13	1	no	—
36	C₄×Q₁₆	—	13	2	no	—
37	#367	D₁₃	1	1	yes	characteristic
38	#358	—	13	2	no	—
39	#346	—	13	1	no	—
40	C₂²×C₁₃	C₂×D₈	1	1	yes	characteristic
41	C₄×C₁₃	—	8	2	no	—
42	C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
43	C₄×C₁₃	#135	1	1	yes	characteristic
44	Dic₁₃	—	2	2	no	—
45	Dic₁₃	—	4	4	no	—
46	C₂×Q₁₆	—	13	2	no	—
47	C₄×Q₈	—	26	2	no	—
48	Q₃₂	—	13	4	no	—
49	C₂×C₁₆	—	13	1	no	—
50	C₂×C₁₆	C₂×D₁₃	1	1	yes	characteristic
51	#107	C₂×D₁₃	1	2	yes	—
52	#103	—	26	2	no	—
53	C₄×C₈	—	13	1	no	—
54	C₂×C₁₃	C₄×D₈	1	1	yes	characteristic
55	C₂×C₁₃	#510	1	2	yes	—
56	C₂×Q₈	—	26	2	no	—
57	Q₁₆	—	13	4	no	—
58	Q₁₆	—	26	2	no	—
59	C₂×C₈	—	13	1	no	—
60	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
61	C₄⋊C₄	—	2	2	no	—
62	C₄⋊C₄	—	52	2	no	—
63	C₄²	—	13	1	no	—
64	C₄²	—	52	2	no	—
65	C₁₆	—	26	1	no	—
66	C₁₆	C₄×D₁₃	1	2	yes	—
67	C₁₃	#2176	1	1	yes	characteristic, complemented, Sylow
68	Q₈	—	26	4	no	—
69	Q₈	—	52	2	no	—
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	—	13	2	no	—
72	C₂×C₄	—	52	2	no	—
73	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
74	C₈	—	26	1	no	—
75	C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
76	C₈	#6131	1	1	yes	characteristic
77	C₂²	D₈×D₁₃	1	1	yes	characteristic
78	C₄	—	8	2	no	—
79	C₄	—	26	2	no	—
80	C₄	—	52	4	no	—
81	C₄	#88147	1	1	yes	characteristic
82	C₄	#88173	1	1	yes	characteristic
83	C₂	#11718533	1	1	yes	characteristic
84	C₂	#11718576	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₁₃
commutator	C₈×C₁₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₃
Fitting	C₁₃×#367	C₂
socle, prime radical	C₂²×C₁₃	C₂×D₈
cosocle	C₂×C₈×C₁₃	C₂³
hypercenter	#367	D₁₃
hypocenter	C₁₃	#2176
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

Complements

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

N	H ≅ G/N	type
13. Dic₁₀₄	C₄	semidirect
67. C₁₃	#2176	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₀₄	2
#11718534	2
C₁₃×#367	2
#11718094	2
#11718084	2
#2176	13