Subgroups — #16847583

Order 3968.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₄₈	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₁₂₄	C₂	1	2	yes	maximal
3	#14520877	C₂	1	1	yes	maximal, characteristic
4	#14520854	C₂	1	2	yes	maximal
5	C₃₁×#346	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₂₄	C₂²	1	2	yes	—
7	C₄×Dic₆₂	—	2	2	no	—
8	C₂×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
10	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	#11758406	C₂²	1	2	yes	—
12	#11758404	—	2	2	no	—
13	Dic₂₄₈	C₄	1	4	yes	complemented
14	C₂×Dic₆₂	—	2	2	no	—
15	C₂×C₈×C₃₁	D₄	1	1	yes	characteristic
16	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
17	C₄²×C₃₁	D₄	1	1	yes	characteristic
18	C₄⋊Dic₃₁	—	2	2	no	—
19	Dic₃₁⋊C₄	—	4	2	no	—
20	C₄×Dic₃₁	—	4	2	no	—
21	Dic₁₂₄	—	2	2	no	—
22	Dic₁₂₄	C₂×C₄	1	4	yes	—
23	C₁₆×C₃₁	—	2	1	no	—
24	C₁₆×C₃₁	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₃₁	D₈	1	2	yes	—
26	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
27	C₂×Dic₃₁	—	4	2	no	—
28	C₂×Dic₃₁	—	4	2	no	—
29	Dic₆₂	—	2	4	no	—
30	Dic₆₂	—	4	2	no	—
31	C₈×C₃₁	—	2	1	no	—
32	C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₃₁	#41	1	1	yes	characteristic
34	#2176	—	31	1	no	maximal, Sylow
35	C₂²×C₃₁	C₂×D₈	1	1	yes	characteristic
36	C₄×C₃₁	—	2	2	no	—
37	C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
38	C₄×C₃₁	#135	1	1	yes	characteristic
39	Dic₃₁	—	4	4	no	—
40	Dic₃₁	—	8	2	no	—
41	C₂×Q₃₂	—	31	1	no	—
42	C₄×Q₁₆	—	31	2	no	—
43	#367	—	31	1	no	—
44	#358	—	31	2	no	—
45	#346	D₃₁	1	1	yes	characteristic
46	C₂×C₃₁	C₄×D₈	1	1	yes	characteristic
47	C₂×C₃₁	#510	1	2	yes	—
48	C₂×Q₁₆	—	31	2	no	—
49	C₄×Q₈	—	62	2	no	—
50	Q₃₂	—	31	4	no	—
51	C₂×C₁₆	C₂×D₃₁	1	1	yes	characteristic
52	C₂×C₁₆	C₂×D₃₁	1	1	yes	characteristic
53	#107	—	31	2	no	—
54	#103	—	62	2	no	—
55	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
56	C₃₁	#2176	1	1	yes	characteristic, complemented, Sylow
57	C₂×Q₈	—	62	2	no	—
58	Q₁₆	—	31	4	no	—
59	Q₁₆	—	62	2	no	—
60	C₂×C₈	D₁₂₄	1	1	yes	characteristic
61	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
62	C₄⋊C₄	—	62	2	no	—
63	C₄⋊C₄	—	124	2	no	—
64	C₄²	—	124	2	no	—
65	C₄²	D₁₂₄	1	1	yes	characteristic
66	C₁₆	—	2	1	no	—
67	C₁₆	C₄×D₃₁	1	2	yes	—
68	Q₈	—	62	4	no	—
69	Q₈	—	124	2	no	—
70	C₂×C₄	—	124	2	no	—
71	C₂×C₄	—	124	2	no	—
72	C₂×C₄	D₂₄₈	1	2	yes	—
73	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
74	C₈	—	2	1	no	—
75	C₈	C₂×C₄×D₃₁	1	1	yes	characteristic
76	C₈	#92495	1	1	yes	characteristic
77	C₂²	C₂×D₂₄₈	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	124	4	no	—
80	C₄	—	248	2	no	—
81	C₄	C₄×D₁₂₄	1	1	yes	characteristic
82	C₄	#11758481	1	1	yes	characteristic
83	C₂	C₄×D₂₄₈	1	1	yes	characteristic
84	C₂	#14521492	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₂₄₈
commutator	C₈×C₃₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₃₁
Fitting	C₃₁×#346	C₂
socle, prime radical	C₂²×C₃₁	C₂×D₈
cosocle	C₂×C₈×C₃₁	C₂³
hypercenter	#346	D₃₁
hypocenter	C₃₁	#2176
LCS	G ▷ C₈×C₃₁ ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
13. Dic₂₄₈	C₄	semidirect
56. C₃₁	#2176	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₄₈	2
C₄×Dic₁₂₄	2
#14520877	2
#14520854	2
C₃₁×#346	2
#2176	31