Subgroups — C₄×#22548651 #44550267

Order 32128.

228 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22548651	C₂	1	2	yes	maximal
2	C₄×#16860794	C₂	1	2	yes	maximal
3	#30631452	C₂	1	2	yes	maximal
4	C₂×C₄×C₈×C₂₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#16860794	C₂²	1	4	yes	—
6	#22548651	C₄	1	8	yes	direct product factor
7	#22548626	—	2	2	no	—
8	C₄×C₂₅₁⋊C₈	—	2	2	no	—
9	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₂₅₁	C₄	1	4	yes	—
13	#16860794	—	2	4	no	—
14	#16860794	C₂×C₄	1	8	yes	—
15	C₂×C₂₅₁⋊C₈	—	2	4	no	—
16	C₂²×C₄×C₂₅₁	Q₈	1	2	yes	—
17	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
18	C₂×C₈×C₂₅₁	—	2	2	no	—
19	C₂×C₈×C₂₅₁	C₂×C₄	1	4	yes	—
20	C₂×C₈×C₂₅₁	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₂₅₁	C₂×C₄	1	2	yes	—
22	C₄²×C₂₅₁	C₂×C₄	1	2	yes	—
23	C₄²×C₂₅₁	D₄	1	1	yes	characteristic
24	C₄²×C₂₅₁	D₄	1	1	yes	characteristic
25	C₂₅₁⋊C₈	—	2	8	no	—
26	C₂³×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
27	C₂×C₄×C₂₅₁	—	2	4	no	—
28	C₂×C₄×C₂₅₁	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₂₅₁	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
31	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₂₅₁	#41	1	2	yes	—
34	C₈×C₂₅₁	—	2	2	no	—
35	C₈×C₂₅₁	C₄²	1	4	yes	—
36	C₂²×C₂₅₁	—	2	2	no	—
37	C₂²×C₂₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₂₅₁	C₄×Q₈	1	2	yes	—
39	C₄×C₂₅₁	—	2	2	no	—
40	C₄×C₂₅₁	#108	1	4	yes	complemented
41	C₄×C₂₅₁	C₂×C₄²	1	1	yes	characteristic
42	C₄×C₂₅₁	#117	1	1	yes	characteristic
43	C₄×C₂₅₁	C₄×D₄	1	2	yes	—
44	C₂×C₂₅₁	—	2	2	no	—
45	C₂×C₂₅₁	C₄×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₂₅₁	C₂×#108	1	2	yes	—
47	C₂₅₁	C₄×#108	1	1	yes	characteristic, complemented, Sylow
48	C₄×#108	—	251	1	no	maximal, Sylow
49	C₂×#108	—	251	2	no	—
50	C₄×M₁₆	—	251	2	no	—
51	C₂×C₄×C₈	D₂₅₁	1	1	yes	characteristic
52	#341	—	251	2	no	—
53	C₂×M₁₆	—	251	4	no	—
54	C₂²×C₈	C₂×D₂₅₁	1	1	yes	characteristic
55	C₂²×C₈	C₂×D₂₅₁	1	1	yes	characteristic
56	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
57	#108	—	251	8	no	—
58	#97	—	502	2	no	—
59	C₄×C₈	—	502	2	no	—
60	C₄×C₈	Dic₂₅₁	1	4	yes	—
61	C₂²×C₄	Dic₅₀₂	1	2	yes	—
62	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
63	M₁₆	—	502	4	no	—
64	M₁₆	—	251	8	no	—
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	—	502	4	no	—
67	C₂×C₈	C₄×D₂₅₁	1	4	yes	—
68	C₂×C₈	C₂×Dic₂₅₁	1	2	yes	—
69	C₂×C₈	C₂×Dic₂₅₁	1	2	yes	—
70	C₄²	C₂×Dic₂₅₁	1	2	yes	—
71	C₄²	D₁₀₀₄	1	1	yes	characteristic
72	C₄²	D₁₀₀₄	1	1	yes	characteristic
73	C₂³	C₂×Dic₅₀₂	1	1	yes	characteristic
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	C₄⋊Dic₂₅₁	1	2	yes	—
76	C₂×C₄	C₄⋊Dic₂₅₁	1	2	yes	—
77	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
78	C₂×C₄	C₂²×Dic₂₅₁	1	1	yes	characteristic
79	C₂×C₄	C₂×D₁₀₀₄	1	1	yes	characteristic
80	C₂×C₄	#16860810	1	2	yes	—
81	C₈	—	2	2	no	—
82	C₈	—	502	8	no	—
83	C₈	C₄×Dic₂₅₁	1	4	yes	—
84	C₂²	—	2	2	no	—
85	C₂²	C₂×C₄⋊Dic₂₅₁	1	1	yes	characteristic
86	C₂²	C₄×Dic₅₀₂	1	2	yes	—
87	C₄	—	2	2	no	—
88	C₄	#22548651	1	4	yes	direct product factor
89	C₄	C₂×C₄×Dic₂₅₁	1	1	yes	characteristic
90	C₄	#22548674	1	1	yes	characteristic
91	C₄	C₄×D₁₀₀₄	1	2	yes	—
92	C₂	—	2	2	no	—
93	C₂	C₄×C₄⋊Dic₂₅₁	1	1	yes	characteristic
94	C₂	C₂×#22548651	1	2	yes	—

	subgroup	G/H
center	C₄²	D₁₀₀₄
commutator	C₄×C₂₅₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₅₁
Fitting	C₂×C₄×C₈×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂×C₄⋊C₄
prime radical	C₂³×C₂₅₁	C₂×Q₈
cosocle	C₂²×C₄×C₂₅₁	C₂³
hypercenter	C₂×C₄×C₈	D₂₅₁
hypocenter	C₂₅₁	C₄×#108
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. #22548651	C₄	direct (N × H)
40. C₄×C₂₅₁	#108	semidirect
47. C₂₅₁	C₄×#108	semidirect
88. C₄	#22548651	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#22548651	2
C₄×#16860794	2
#30631452	2
C₂×C₄×C₈×C₂₅₁	2
C₄×#108	251

Subgroups — C4×#22548651 #44550267

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#22548651 #44550267