Subgroups — #28392334

Order 14464.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14268622	C₂	1	1	yes	maximal, characteristic
2	#20298553	C₂	1	1	yes	maximal, characteristic
3	#20298416	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×C₁₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14268631	C₄	1	1	yes	characteristic
6	#16813581	C₄	1	1	yes	characteristic
7	C₂×#14268622	—	2	2	no	—
8	C₂×#14268622	C₂²	1	2	yes	—
9	#16813532	C₄	1	2	yes	—
10	C₈×Dic₁₁₃	C₂²	1	2	yes	—
11	C₂²×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
12	M₃₂×C₁₁₃	—	2	2	no	—
13	C₂×C₁₆×C₁₁₃	C₂²	1	2	yes	—
14	#14268631	—	2	2	no	—
15	C₂×#11744751	C₂×C₄	1	2	yes	—
16	C₄⋊Dic₁₁₃	C₈	1	2	yes	—
17	C₂²⋊Dic₁₁₃	C₈	1	2	yes	—
18	C₄×Dic₁₁₃	C₂×C₄	1	2	yes	—
19	#14268622	—	2	2	no	—
20	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₁₃	D₄	1	2	yes	—
23	C₂×C₈×C₁₁₃	Q₈	1	2	yes	—
24	C₂×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
25	C₁₆×C₁₁₃	—	2	2	no	—
26	C₂³×C₁₁₃	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₁₁₃	C₄⋊C₄	1	2	yes	—
28	C₂×C₄×C₁₁₃	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₁₁₃	M₁₆	1	2	yes	—
30	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×Dic₁₁₃	C₂×C₈	1	4	yes	—
32	C₈×C₁₁₃	C₄⋊C₄	1	2	yes	—
33	C₈×C₁₁₃	C₂×D₄	1	1	yes	characteristic
34	C₈×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
35	#11744751	—	2	2	no	—
36	C₂²×C₁₁₃	—	2	2	no	—
37	C₂²×C₁₁₃	C₄⋊C₈	1	2	yes	—
38	C₂²×C₁₁₃	C₂²×C₈	1	1	yes	characteristic
39	C₄×C₁₁₃	C₄⋊C₈	1	2	yes	—
40	C₄×C₁₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₄×C₁₁₃	C₂×M₁₆	1	1	yes	characteristic
42	Dic₁₁₃	—	2	4	no	—
43	C₂×C₁₁₃	—	2	2	no	—
44	C₂×C₁₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
45	C₂×C₁₁₃	#504	1	2	yes	—
46	#2151	—	113	1	no	maximal, Sylow
47	C₁₁₃	#2151	1	1	yes	characteristic, complemented, Sylow
48	C₂×M₃₂	D₁₁₃	1	1	yes	characteristic
49	C₂²×C₁₆	—	113	1	no	—
50	#405	—	113	1	no	—
51	C₄⋊C₁₆	—	113	4	no	—
52	C₂×M₁₆	—	113	1	no	—
53	C₂²×C₈	C₂×D₁₁₃	1	1	yes	characteristic
54	#117	—	113	1	no	—
55	M₃₂	—	2	2	no	—
56	C₂×C₁₆	—	113	2	no	—
57	C₂×C₁₆	—	226	2	no	—
58	C₂×C₁₆	C₂×D₁₁₃	1	2	yes	—
59	#97	—	113	2	no	—
60	C₄×C₈	—	113	2	no	—
61	C₂²×C₄	C₄×D₁₁₃	1	1	yes	characteristic
62	M₁₆	—	226	2	no	—
63	C₂×C₈	—	113	2	no	—
64	C₂×C₈	Dic₂₂₆	1	2	yes	—
65	C₂×C₈	C₄×D₁₁₃	1	1	yes	characteristic
66	C₂×C₈	C₂²⋊D₁₁₃	1	2	yes	—
67	C₂×C₈	C₂²×D₁₁₃	1	1	yes	characteristic
68	C₄⋊C₄	—	113	2	no	—
69	C₂²⋊C₄	—	113	2	no	—
70	C₄²	—	113	2	no	—
71	C₁₆	—	2	2	no	—
72	C₁₆	—	226	2	no	—
73	C₂³	C₈×D₁₁₃	1	1	yes	characteristic
74	C₂×C₄	—	113	4	no	—
75	C₂×C₄	#14268626	1	2	yes	—
76	C₂×C₄	C₈×D₁₁₃	1	1	yes	characteristic
77	C₂×C₄	#14268630	1	2	yes	—
78	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
79	C₈	—	226	2	no	—
80	C₈	#14268626	1	2	yes	—
81	C₈	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
82	C₈	C₂×Dic₂₂₆	1	1	yes	characteristic
83	C₂²	—	2	2	no	—
84	C₂²	#16813551	1	2	yes	—
85	C₂²	C₂×C₈×D₁₁₃	1	1	yes	characteristic
86	C₄	—	226	4	no	—
87	C₄	#16813551	1	2	yes	—
88	C₄	C₂×#14268626	1	1	yes	characteristic
89	C₄	C₂×#14268630	1	1	yes	characteristic
90	C₂	—	2	2	no	—
91	C₂	C₂×#16813551	1	1	yes	characteristic
92	C₂	#20299016	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₁₃
commutator, socle	C₂²×C₁₁₃	C₂²×C₈
Fitting	C₂×M₃₂×C₁₁₃	C₂
prime radical	C₂³×C₁₁₃	C₂×C₈
cosocle	C₂×C₈×C₁₁₃	C₂³
hypercenter	C₂×M₃₂	D₁₁₃
hypocenter	C₁₁₃	#2151
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20298416	C₂	semidirect
47. C₁₁₃	#2151	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14268622	2
#20298553	2
#20298416	2
C₂×M₃₂×C₁₁₃	2
#2151	113