Finite Group Databasebeta

Subgroups — #44552136

Order 32128.

144 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₃₂×D₂₅₁	C₂	1	1	yes	maximal, characteristic
2	#108×D₂₅₁	C₂	1	1	yes	maximal, characteristic
3	#30631722	C₂	1	1	yes	maximal, characteristic
4	#30631511	C₂	1	1	yes	maximal, characteristic, complemented
5	#30631510	C₂	1	1	yes	maximal, characteristic, complemented
6	#30631509	C₂	1	1	yes	maximal, characteristic, complemented
7	#365×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	M₁₆×D₂₅₁	—	2	1	no	—
9	C₂×C₈×D₂₅₁	C₂²	1	1	yes	characteristic
10	#22548673	—	2	1	no	—
11	#22548655	C₂²	1	1	yes	characteristic
12	#22548654	C₄	1	2	yes	complemented
13	C₁₆×D₂₅₁	C₄	1	2	yes	complemented
14	#22548651	C₂²	1	1	yes	characteristic
15	#22548650	—	2	1	no	—
16	#22548648	—	2	1	no	—
17	#22548647	C₂²	1	1	yes	characteristic
18	M₃₂×C₂₅₁	C₂²	1	1	yes	characteristic
19	#108×C₂₅₁	C₂²	1	1	yes	characteristic
20	#106×C₂₅₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₂₅₁	D₄	1	1	yes	characteristic
22	#16860794	—	2	1	no	—
23	#16860793	—	4	1	no	—
24	C₂×C₂₅₁⋊C₈	D₄	1	1	yes	characteristic
25	C₈×D₂₅₁	—	4	1	no	—
26	C₈×D₂₅₁	C₂×C₄	1	1	yes	characteristic
27	C₈×D₂₅₁	C₂×C₄	1	1	yes	characteristic
28	C₈×D₂₅₁	Q₈	1	1	yes	characteristic
29	C₈×D₂₅₁	Q₈	1	1	yes	characteristic
30	C₄⋊Dic₂₅₁	—	2	1	no	—
31	(C₂×D₂₅₁)⋊C₄	—	4	1	no	—
32	C₄×Dic₂₅₁	—	4	1	no	—
33	C₂₅₁⋊C₁₆	C₂×C₄	1	2	yes	—
34	M₁₆×C₂₅₁	—	2	1	no	—
35	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₂₅₁	—	2	1	no	—
37	C₁₆×C₂₅₁	C₂×C₄	1	2	yes	—
38	C₂²×D₂₅₁	SD₁₆	1	1	yes	characteristic
39	C₂×Dic₂₅₁	—	4	1	no	—
40	C₂×Dic₂₅₁	SD₁₆	1	1	yes	characteristic
41	C₄×D₂₅₁	C₄⋊C₄	1	1	yes	characteristic
42	C₄×D₂₅₁	C₄⋊C₄	1	1	yes	characteristic
43	C₄×D₂₅₁	SD₁₆	1	1	yes	characteristic
44	C₄×D₂₅₁	SD₁₆	1	1	yes	characteristic
45	C₂₅₁⋊C₈	—	4	1	no	—
46	C₂₅₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
47	C₂₅₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₂₅₁	—	4	1	no	—
49	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
50	C₈×C₂₅₁	—	4	1	no	—
51	C₈×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
52	C₈×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₂₅₁	C₂×SD₁₆	1	1	yes	characteristic
54	C₂×D₂₅₁	—	4	1	no	—
55	C₂×D₂₅₁	#106	1	1	yes	characteristic
56	C₂×D₂₅₁	#106	1	1	yes	characteristic
57	C₄×C₂₅₁	—	8	1	no	—
58	C₄×C₂₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₄×C₂₅₁	C₂×SD₁₆	1	1	yes	characteristic
60	Dic₂₅₁	—	8	1	no	—
61	Dic₂₅₁	#106	1	1	yes	characteristic
62	Dic₂₅₁	#106	1	1	yes	characteristic
63	C₂×C₂₅₁	—	2	1	no	—
64	C₂×C₂₅₁	C₂×#106	1	1	yes	characteristic
65	D₂₅₁	—	2	1	no	—
66	D₂₅₁	—	2	1	no	—
67	C₂₅₁	#2155	1	1	yes	characteristic, complemented, Sylow
68	#2155	—	251	1	no	maximal, Sylow
69	C₂×M₃₂	—	251	1	no	—
70	C₂×#108	—	251	1	no	—
71	#427	—	251	1	no	—
72	#365	—	251	1	no	—
73	#365	—	251	1	no	—
74	#365	—	251	1	no	—
75	#365	D₂₅₁	1	1	yes	characteristic, complemented
76	C₂×M₁₆	—	502	1	no	—
77	C₂²×C₈	—	251	1	no	—
78	#117	—	502	1	no	—
79	M₃₂	—	251	1	no	—
80	M₃₂	—	251	2	no	—
81	M₃₂	C₂×D₂₅₁	1	1	yes	characteristic
82	C₂×C₁₆	—	251	2	no	—
83	#108	—	502	1	no	—
84	#108	—	251	1	no	—
85	#108	C₂×D₂₅₁	1	1	yes	characteristic
86	#107	—	502	1	no	—
87	#106	—	251	1	no	—
88	#106	C₂×D₂₅₁	1	1	yes	characteristic
89	C₂²×C₄	—	251	1	no	—
90	M₁₆	—	2	1	no	—
91	M₁₆	—	1004	1	no	—
92	M₁₆	—	502	1	no	—
93	C₂×C₈	—	1004	1	no	—
94	C₂×C₈	—	251	1	no	—
95	C₂×C₈	—	251	1	no	—
96	C₂×C₈	—	251	1	no	—
97	C₂×C₈	—	251	1	no	—
98	C₂×C₈	—	251	1	no	—
99	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	502	1	no	—
102	C₂²⋊C₄	—	1004	1	no	—
103	C₄²	—	1004	1	no	—
104	C₁₆	—	251	2	no	—
105	C₁₆	C₄×D₂₅₁	1	2	yes	—
106	C₂³	—	251	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	1004	1	no	—
109	C₂×C₄	—	251	1	no	—
110	C₂×C₄	—	251	1	no	—
111	C₂×C₄	—	251	1	no	—
112	C₂×C₄	—	251	1	no	—
113	C₂×C₄	—	251	1	no	—
114	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
115	C₈	—	4	1	no	—
116	C₈	—	1004	1	no	—
117	C₈	—	251	1	no	—
118	C₈	—	251	1	no	—
119	C₈	C₂×C₄×D₂₅₁	1	1	yes	characteristic
120	C₈	Q₈×D₂₅₁	1	1	yes	characteristic
121	C₂²	—	1004	1	no	—
122	C₂²	—	251	1	no	—
123	C₂²	—	251	1	no	—
124	C₂²	SD₁₆×D₂₅₁	1	1	yes	characteristic
125	C₄	—	8	1	no	—
126	C₄	—	2008	1	no	—
127	C₄	—	251	1	no	—
128	C₄	—	251	1	no	—
129	C₄	C₄⋊C₄×D₂₅₁	1	1	yes	characteristic
130	C₄	SD₁₆×D₂₅₁	1	1	yes	characteristic
131	C₂	—	2	1	no	—
132	C₂	—	502	1	no	—
133	C₂	—	502	1	no	—
134	C₂	#106×D₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#106×D₂₅₁
commutator	C₈×C₂₅₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₂₅₁
Fitting	#365×C₂₅₁	C₂
socle	C₂×C₂₅₁	C₂×#106
prime radical	C₂²×D₂₅₁	SD₁₆
cosocle	C₂×C₈×C₂₅₁	C₂³
hypercenter	#365	D₂₅₁
hypocenter	C₂₅₁	#2155
LCS	G ▷ C₈×C₂₅₁ ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

Complements

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

N	H ≅ G/N	type
4. #30631511	C₂	semidirect
5. #30631510	C₂	semidirect
6. #30631509	C₂	semidirect
7. #365×C₂₅₁	C₂	semidirect
12. #22548654	C₄	semidirect
13. C₁₆×D₂₅₁	C₄	semidirect
67. C₂₅₁	#2155	semidirect
75. #365	D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
M₃₂×D₂₅₁	2
#108×D₂₅₁	2
#30631722	2
#30631511	2
#30631510	2
#30631509	2
#365×C₂₅₁	2
#2155	251