Subgroups — #16847588

Order 3968.

172 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄₉₆	C₂	1	1	yes	maximal, characteristic
2	C₄×D₂₄₈	C₂	1	2	yes	maximal, complemented
3	#14520877	C₂	1	1	yes	maximal, characteristic, complemented
4	#14520851	C₂	1	2	yes	maximal, complemented
5	C₃₁×#346	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₂₄₈	C₂²	1	2	yes	—
7	C₄×D₁₂₄	—	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	#11758409	—	2	2	no	—
12	#11758406	C₂²	1	2	yes	—
13	D₄₉₆	C₄	1	4	yes	complemented
14	C₂×D₁₂₄	—	2	2	no	—
15	C₂×C₄×D₃₁	—	4	2	no	—
16	C₂×C₈×C₃₁	D₄	1	1	yes	characteristic
17	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
18	C₄²×C₃₁	D₄	1	1	yes	characteristic
19	(C₂×D₃₁)⋊C₄	—	4	2	no	—
20	C₄⋊Dic₃₁	—	2	2	no	—
21	D₂₄₈	—	2	2	no	—
22	D₂₄₈	C₂×C₄	1	4	yes	complemented
23	C₁₆×C₃₁	—	2	1	no	—
24	C₁₆×C₃₁	C₂×C₄	1	2	yes	complemented
25	C₂²×D₃₁	—	4	2	no	—
26	C₂×C₄×C₃₁	D₈	1	2	yes	—
27	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
28	C₂×Dic₃₁	—	4	2	no	—
29	D₁₂₄	—	2	4	no	—
30	D₁₂₄	—	4	2	no	—
31	C₄×D₃₁	—	8	4	no	—
32	C₈×C₃₁	—	2	1	no	—
33	C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
34	C₈×C₃₁	#41	1	1	yes	characteristic
35	#2177	—	31	1	no	maximal, Sylow
36	C₂²×C₃₁	C₂×D₈	1	1	yes	characteristic
37	C₂×D₃₁	—	4	4	no	—
38	C₂×D₃₁	—	8	2	no	—
39	C₂×D₃₁	—	8	2	no	—
40	C₄×C₃₁	—	2	2	no	—
41	C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
42	C₄×C₃₁	#135	1	1	yes	characteristic
43	Dic₃₁	—	8	2	no	—
44	C₂×D₁₆	—	31	1	no	—
45	C₄×D₈	—	31	2	no	—
46	#367	—	31	1	no	—
47	#357	—	31	2	no	—
48	#346	D₃₁	1	1	yes	characteristic, complemented
49	C₂×C₃₁	C₄×D₈	1	1	yes	characteristic
50	C₂×C₃₁	#509	1	2	yes	—
51	D₃₁	—	8	4	no	—
52	C₂×D₈	—	31	2	no	—
53	C₄×D₄	—	62	2	no	—
54	D₁₆	—	31	4	no	—
55	C₂×C₁₆	C₂×D₃₁	1	1	yes	characteristic
56	C₂×C₁₆	C₂×D₃₁	1	1	yes	characteristic
57	#107	—	31	2	no	—
58	#102	—	62	2	no	—
59	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
60	C₃₁	#2177	1	1	yes	characteristic, complemented, Sylow
61	C₂×D₄	—	62	2	no	—
62	C₂²×C₄	—	124	2	no	—
63	D₈	—	31	4	no	—
64	D₈	—	62	2	no	—
65	C₂×C₈	D₁₂₄	1	1	yes	characteristic
66	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
67	C₄⋊C₄	—	62	2	no	—
68	C₂²⋊C₄	—	124	2	no	—
69	C₄²	D₁₂₄	1	1	yes	characteristic
70	C₁₆	—	2	1	no	—
71	C₁₆	C₄×D₃₁	1	2	yes	complemented
72	C₂³	—	124	2	no	—
73	D₄	—	62	4	no	—
74	D₄	—	124	2	no	—
75	C₂×C₄	—	124	2	no	—
76	C₂×C₄	—	248	4	no	—
77	C₂×C₄	D₂₄₈	1	2	yes	—
78	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
79	C₈	—	2	1	no	—
80	C₈	C₂×C₄×D₃₁	1	1	yes	characteristic
81	C₈	#92495	1	1	yes	characteristic
82	C₂²	—	124	4	no	—
83	C₂²	—	248	2	no	—
84	C₂²	—	248	2	no	—
85	C₂²	C₂×D₂₄₈	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	248	2	no	—
88	C₄	C₄×D₁₂₄	1	1	yes	characteristic
89	C₄	#11758481	1	1	yes	characteristic
90	C₂	—	248	4	no	—
91	C₂	C₄×D₂₄₈	1	1	yes	characteristic
92	C₂	#14521487	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	C₂²×C₃₁	C₂×D₈
prime radical	C₂×D₄₉₆	C₂
cosocle	C₂×C₈×C₃₁	C₂³
hypercenter	#346	D₃₁
hypocenter	C₃₁	#2177
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
2. C₄×D₂₄₈	C₂	semidirect
3. #14520877	C₂	semidirect
4. #14520851	C₂	semidirect
5. C₃₁×#346	C₂	semidirect
13. D₄₉₆	C₄	semidirect
22. D₂₄₈	C₂×C₄	semidirect
24. C₁₆×C₃₁	C₂×C₄	semidirect
48. #346	D₃₁	semidirect
60. C₃₁	#2177	semidirect
71. C₁₆	C₄×D₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄₉₆	2
C₄×D₂₄₈	2
#14520877	2
#14520851	2
C₃₁×#346	2
#2177	31