Subgroups — #16845382

Order 3968.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521454	C₂	1	2	yes	maximal, complemented
2	#14521429	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521422	C₂	1	1	yes	maximal, characteristic
4	#14521396	C₂	1	2	yes	maximal
5	C₃₁×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₁₂₄	—	2	2	no	—
7	C₂×#92470	—	2	2	no	—
8	#11758451	C₂²	1	1	yes	characteristic
9	#11758446	C₂²	1	2	yes	—
10	#11758446	C₂²	1	1	yes	characteristic
11	C₄×C₈×C₃₁	C₂²	1	2	yes	—
12	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	#11758409	—	2	4	no	—
14	#11758406	—	2	4	no	—
15	C₂×D₁₂₄	—	2	2	no	—
16	C₂×D₁₂₄	—	2	1	no	—
17	C₂×Dic₆₂	—	2	1	no	—
18	C₂×Dic₆₂	—	2	2	no	—
19	C₂×C₈×C₃₁	D₄	1	4	yes	—
20	C₂×C₈×C₃₁	D₄	1	2	yes	—
21	C₄²×C₃₁	C₂³	1	1	yes	characteristic
22	C₄⋊Dic₃₁	—	2	2	no	—
23	C₄⋊Dic₃₁	—	2	2	no	—
24	C₄⋊Dic₃₁	—	2	2	no	—
25	Dic₁₂₄	—	4	4	no	—
26	#92470	—	4	4	no	—
27	C₂²×D₃₁	—	4	1	no	—
28	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
30	C₂×Dic₃₁	—	4	1	no	—
31	C₂×Dic₃₁	—	4	2	no	—
32	D₁₂₄	—	4	2	no	—
33	D₁₂₄	—	4	4	no	—
34	Dic₆₂	—	4	2	no	—
35	Dic₆₂	—	4	4	no	—
36	C₈×C₃₁	—	2	4	no	—
37	C₈×C₃₁	D₈	1	4	yes	—
38	#1722	—	31	1	no	maximal, Sylow
39	C₂²×C₃₁	#127	1	1	yes	characteristic
40	C₂×D₃₁	—	8	1	no	—
41	C₂×D₃₁	—	8	2	no	—
42	C₄×C₃₁	C₂×D₈	1	2	yes	—
43	C₄×C₃₁	#137	1	4	yes	—
44	Dic₃₁	—	8	1	no	—
45	Dic₃₁	—	8	2	no	—
46	#500	—	31	2	no	—
47	#494	—	31	1	no	—
48	#492	—	31	1	no	—
49	#486	—	31	2	no	—
50	#322	D₃₁	1	1	yes	characteristic, complemented
51	C₂×C₃₁	#493	1	1	yes	characteristic
52	C₂×C₃₁	#497	1	2	yes	—
53	D₃₁	—	16	1	no	—
54	C₂×Q₁₆	—	62	2	no	—
55	C₂×SD₁₆	—	62	2	no	—
56	C₄⋊Q₈	—	31	2	no	—
57	C₄⋊Q₈	—	31	1	no	—
58	#127	—	31	1	no	—
59	#107	—	62	4	no	—
60	#102	—	62	4	no	—
61	C₄×C₈	C₂×D₃₁	1	2	yes	—
62	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
63	C₃₁	#1722	1	1	yes	characteristic, complemented, Sylow
64	C₂×Q₈	—	62	1	no	—
65	C₂×Q₈	—	62	2	no	—
66	C₂×D₄	—	62	2	no	—
67	C₂×D₄	—	62	1	no	—
68	Q₁₆	—	124	4	no	—
69	SD₁₆	—	124	4	no	—
70	C₂×C₈	D₁₂₄	1	4	yes	—
71	C₂×C₈	D₁₂₄	1	2	yes	—
72	C₄⋊C₄	—	62	2	no	—
73	C₄⋊C₄	—	62	2	no	—
74	C₄⋊C₄	—	62	2	no	—
75	C₄²	C₂²×D₃₁	1	1	yes	characteristic
76	C₂³	—	124	1	no	—
77	Q₈	—	124	2	no	—
78	Q₈	—	124	4	no	—
79	D₄	—	124	2	no	—
80	D₄	—	124	4	no	—
81	C₂×C₄	—	124	2	no	—
82	C₂×C₄	—	124	1	no	—
83	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
84	C₂×C₄	C₂×D₁₂₄	1	2	yes	—
85	C₈	—	2	4	no	—
86	C₈	D₂₄₈	1	4	yes	—
87	C₂²	—	248	1	no	—
88	C₂²	—	248	2	no	—
89	C₂²	#11758451	1	1	yes	characteristic
90	C₄	—	248	1	no	—
91	C₄	—	248	2	no	—
92	C₄	C₂×D₂₄₈	1	2	yes	—
93	C₄	#11758486	1	4	yes	—
94	C₂	—	496	1	no	—
95	C₂	#14521426	1	1	yes	characteristic
96	C₂	#14521443	1	2	yes	—

	subgroup	G/H
center	C₂²	#11758451
commutator, cosocle	C₄²×C₃₁	C₂³
Frattini	C₄²	C₂²×D₃₁
Fitting	C₃₁×#322	C₂
socle	C₂²×C₃₁	#127
prime radical	#11758451	C₂²
hypercenter	#322	D₃₁
hypocenter	C₃₁	#1722
LCS	G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
1. #14521454	C₂	semidirect
2. #14521429	C₂	semidirect
5. C₃₁×#322	C₂	semidirect
50. #322	D₃₁	semidirect
63. C₃₁	#1722	semidirect

Maximal subgroups

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

subgroup	index
#14521454	2
#14521429	2
#14521422	2
#14521396	2
C₃₁×#322	2
#1722	31