Subgroups — #16845299

Order 3968.

214 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521445	C₂	1	2	yes	maximal, complemented
2	#14521423	C₂	1	2	yes	maximal
3	#14521422	C₂	1	1	yes	maximal, characteristic
4	C₈×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#498	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂×#92481	—	2	4	no	—
8	C₂×#92470	—	2	2	no	—
9	#11758451	C₂²	1	1	yes	characteristic
10	#11758446	C₂²	1	1	yes	characteristic
11	C₃₁×#106	—	2	2	no	—
12	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	#11758396	—	2	4	no	—
14	C₄×C₃₁⋊C₈	C₂²	1	2	yes	—
15	C₂×Q₈×C₃₁	—	2	2	no	—
16	C₂×D₁₂₄	—	2	1	no	—
17	C₂×D₁₂₄	—	2	2	no	—
18	C₂×Dic₆₂	—	2	1	no	—
19	C₂×C₈×C₃₁	D₄	1	2	yes	—
20	C₄⋊C₄×C₃₁	—	2	2	no	—
21	C₄⋊C₄×C₃₁	—	2	2	no	—
22	C₄²×C₃₁	C₂³	1	1	yes	characteristic
23	#92481	—	4	8	no	—
24	C₄⋊Dic₃₁	—	2	2	no	—
25	C₂×C₃₁⋊C₈	D₄	1	4	yes	—
26	#92470	—	4	4	no	—
27	C₂²×D₃₁	—	4	1	no	—
28	Q₈×C₃₁	—	4	4	no	—
29	C₂×C₄×C₃₁	—	4	2	no	—
30	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
31	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
32	C₂×Dic₃₁	—	4	1	no	—
33	D₁₂₄	—	4	2	no	—
34	D₁₂₄	—	4	4	no	—
35	Dic₆₂	—	4	2	no	—
36	C₈×C₃₁	SD₁₆	1	4	yes	complemented
37	C₃₁⋊C₈	SD₁₆	1	8	yes	complemented
38	#1705	—	31	1	no	maximal, Sylow
39	C₂²×C₃₁	#127	1	1	yes	characteristic
40	C₂×D₃₁	—	8	2	no	—
41	C₂×D₃₁	—	8	1	no	—
42	C₄×C₃₁	—	8	2	no	—
43	C₄×C₃₁	C₂×SD₁₆	1	4	yes	—
44	C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
45	Dic₃₁	—	8	1	no	—
46	#498	—	31	2	no	—
47	#498	D₃₁	1	1	yes	characteristic, complemented
48	#492	—	31	2	no	—
49	#492	—	31	1	no	—
50	C₈²	—	31	1	no	—
51	C₂×C₃₁	#492	1	2	yes	—
52	C₂×C₃₁	#492	1	1	yes	characteristic
53	D₃₁	—	16	1	no	—
54	C₂×SD₁₆	—	62	4	no	—
55	C₂×SD₁₆	—	62	2	no	—
56	C₄⋊Q₈	—	31	1	no	—
57	C₄⋊Q₈	C₂×D₃₁	1	2	yes	—
58	#127	—	31	1	no	—
59	#106	—	2	2	no	—
60	#106	—	62	4	no	—
61	C₄×C₈	—	31	2	no	—
62	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
63	C₃₁	#1705	1	1	yes	characteristic, complemented, Sylow
64	C₂×Q₈	—	2	2	no	—
65	C₂×Q₈	—	62	1	no	—
66	C₂×D₄	—	62	2	no	—
67	C₂×D₄	—	62	1	no	—
68	SD₁₆	—	124	8	no	—
69	SD₁₆	—	124	4	no	—
70	C₂×C₈	—	31	4	no	—
71	C₂×C₈	C₂²⋊D₃₁	1	2	yes	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	—	62	2	no	—
75	C₄²	C₂²×D₃₁	1	1	yes	characteristic
76	C₂³	—	124	1	no	—
77	Q₈	—	4	4	no	—
78	Q₈	—	124	2	no	—
79	D₄	—	124	4	no	—
80	D₄	—	124	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	124	1	no	—
83	C₂×C₄	D₄×D₃₁	1	2	yes	—
84	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
85	C₈	—	31	8	no	—
86	C₈	#92481	1	4	yes	complemented
87	C₂²	—	248	2	no	—
88	C₂²	—	248	1	no	—
89	C₂²	#11758517	1	1	yes	characteristic
90	C₄	—	8	2	no	—
91	C₄	—	248	1	no	—
92	C₄	SD₁₆×D₃₁	1	4	yes	—
93	C₄	C₂×#92481	1	2	yes	—
94	C₂	—	496	1	no	—
95	C₂	#14521420	1	2	yes	—
96	C₂	#14521423	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11758517
commutator, cosocle	C₄²×C₃₁	C₂³
Frattini	C₄²	C₂²×D₃₁
Fitting	C₃₁×#498	C₂
socle	C₂²×C₃₁	#127
prime radical	#11758451	C₂²
hypercenter	#498	D₃₁
hypocenter	C₃₁	#1705
LCS	G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

Complements

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

N	H ≅ G/N	type
1. #14521445	C₂	semidirect
4. C₈×C₃₁⋊C₈	C₂	semidirect
5. C₃₁×#498	C₂	semidirect
36. C₈×C₃₁	SD₁₆	semidirect
37. C₃₁⋊C₈	SD₁₆	semidirect
47. #498	D₃₁	semidirect
63. C₃₁	#1705	semidirect
86. C₈	#92481	semidirect

Maximal subgroups

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

subgroup	index
#14521445	2
#14521423	2
#14521422	2
C₈×C₃₁⋊C₈	2
C₃₁×#498	2
#1705	31