Subgroups — #44559431

Order 32128.

410 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632237	C₂	1	1	yes	maximal, characteristic
2	#30631766	C₂	1	2	yes	maximal, complemented
3	#30631764	C₂	1	2	yes	maximal, complemented
4	C₈×D₁₀₀₄	C₂	1	2	yes	maximal, complemented
5	#30631752	C₂	1	2	yes	maximal, complemented
6	#30631740	C₂	1	1	yes	maximal, characteristic, complemented
7	#30631652	C₂	1	4	yes	maximal, complemented
8	#431×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#16860810	C₂²	1	2	yes	—
10	C₂×#16860793	C₂²	1	4	yes	—
11	C₂×C₈×D₂₅₁	C₂²	1	4	yes	—
12	#22548715	C₄	1	1	yes	characteristic
13	#22548712	C₄	1	1	yes	characteristic
14	#22548703	C₄	1	2	yes	—
15	#22548689	C₄	1	2	yes	—
16	#22548689	C₄	1	2	yes	—
17	C₄×D₁₀₀₄	C₂²	1	2	yes	—
18	C₄×D₁₀₀₄	C₂²	1	2	yes	—
19	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
20	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
21	C₂²⋊(C₂₅₁⋊C₈)	C₂²	1	2	yes	—
22	(C₂×D₂₅₁)⋊C₈	C₂²	1	4	yes	—
23	(C₂×D₂₅₁)⋊C₈	C₂²	1	4	yes	—
24	C₄×C₂₅₁⋊C₈	C₂²	1	2	yes	—
25	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
26	C₄⋊C₈×C₂₅₁	C₂²	1	1	yes	characteristic
27	C₄⋊C₈×C₂₅₁	C₂²	1	1	yes	characteristic
28	C₂²⋊C₈×C₂₅₁	C₂²	1	2	yes	—
29	C₄×C₈×C₂₅₁	C₂²	1	2	yes	—
30	#16860810	—	4	2	no	—
31	#16860810	—	4	2	no	—
32	C₂×Dic₅₀₂	C₂×C₄	1	2	yes	—
33	C₂×D₁₀₀₄	C₂×C₄	1	2	yes	—
34	C₂×D₁₀₀₄	C₂×C₄	1	2	yes	—
35	C₂×D₁₀₀₄	C₂×C₄	1	2	yes	—
36	C₂×C₂²⋊D₂₅₁	C₂×C₄	1	4	yes	—
37	C₂×C₄×D₂₅₁	C₂³	1	4	yes	—
38	#16860793	—	4	4	no	—
39	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
40	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
41	C₈×D₂₅₁	—	4	4	no	—
42	C₄⋊Dic₂₅₁	C₂×C₄	1	2	yes	—
43	C₄⋊Dic₂₅₁	C₂×C₄	1	2	yes	—
44	(C₂×D₂₅₁)⋊C₄	C₂×C₄	1	4	yes	—
45	(C₂×D₂₅₁)⋊C₄	C₂×C₄	1	4	yes	—
46	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
50	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
51	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
52	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
53	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
54	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
55	C₂²×D₂₅₁	C₂²×C₄	1	4	yes	—
56	Dic₅₀₂	—	4	2	no	—
57	D₁₀₀₄	—	2	4	no	—
58	D₁₀₀₄	—	2	4	no	—
59	D₁₀₀₄	—	4	2	no	—
60	C₂²⋊D₂₅₁	—	4	4	no	—
61	C₂×Dic₂₅₁	C₂²×C₄	1	4	yes	—
62	C₄×D₂₅₁	—	2	4	no	—
63	C₄×D₂₅₁	—	2	4	no	—
64	C₂₅₁⋊C₈	—	2	2	no	—
65	C₂₅₁⋊C₈	—	2	2	no	—
66	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₂₅₁	—	2	2	no	—
68	C₂×C₄×C₂₅₁	—	2	2	no	—
69	C₂×C₄×C₂₅₁	—	2	1	no	—
70	C₂×C₄×C₂₅₁	—	2	1	no	—
71	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
77	C₈×C₂₅₁	—	2	2	no	—
78	C₈×C₂₅₁	—	2	2	no	—
79	C₂²×C₂₅₁	—	2	1	no	—
80	C₂²×C₂₅₁	—	2	1	no	—
81	C₂²×C₂₅₁	—	2	1	no	—
82	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
83	C₂×D₂₅₁	—	2	4	no	—
84	C₂×D₂₅₁	—	2	4	no	—
85	C₂×D₂₅₁	—	2	4	no	—
86	C₄×C₂₅₁	—	2	1	no	—
87	C₄×C₂₅₁	—	2	1	no	—
88	C₄×C₂₅₁	—	2	1	no	—
89	C₄×C₂₅₁	#131	1	2	yes	—
90	C₄×C₂₅₁	#131	1	2	yes	—
91	C₄×C₂₅₁	H(5,𝔽₂)	1	1	yes	characteristic
92	C₄×C₂₅₁	H(5,𝔽₂)	1	1	yes	characteristic
93	Dic₂₅₁	—	2	4	no	—
94	C₂×C₂₅₁	—	4	1	no	—
95	C₂×C₂₅₁	#518	1	1	yes	characteristic
96	C₂×C₂₅₁	C₂×#131	1	1	yes	characteristic
97	C₂×C₂₅₁	C₂×#131	1	1	yes	characteristic
98	D₂₅₁	—	4	4	no	—
99	C₂₅₁	#2979	1	1	yes	characteristic, complemented, Sylow
100	#2979	—	251	1	no	maximal, Sylow
101	#532	—	251	1	no	—
102	#436	—	251	2	no	—
103	#436	—	251	2	no	—
104	C₈×D₄	—	251	2	no	—
105	C₈×D₄	—	251	2	no	—
106	#431	—	251	1	no	—
107	#431	D₂₅₁	1	1	yes	characteristic, complemented
108	#408	—	251	4	no	—
109	C₂×#41	—	251	2	no	—
110	C₂×M₁₆	—	251	4	no	—
111	C₂²×C₈	—	251	4	no	—
112	C₄⋊Q₈	—	251	1	no	—
113	#127	—	251	1	no	—
114	#124	—	251	2	no	—
115	#121	—	251	2	no	—
116	#121	—	251	2	no	—
117	C₄×D₄	—	251	2	no	—
118	C₄×D₄	—	251	2	no	—
119	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
120	C₄⋊C₈	—	251	1	no	—
121	C₄⋊C₈	—	251	1	no	—
122	C₄⋊C₈	C₂×D₂₅₁	1	1	yes	characteristic
123	C₄⋊C₈	C₂×D₂₅₁	1	1	yes	characteristic
124	C₂²⋊C₈	—	251	2	no	—
125	C₂²⋊C₈	—	251	4	no	—
126	C₂²⋊C₈	—	251	4	no	—
127	C₂²⋊C₈	C₂×D₂₅₁	1	2	yes	—
128	C₄×C₈	—	251	2	no	—
129	C₄×C₈	C₂×D₂₅₁	1	2	yes	—
130	#41	—	1004	2	no	—
131	#41	—	1004	2	no	—
132	C₂×Q₈	—	251	2	no	—
133	C₂×D₄	—	251	4	no	—
134	C₂×D₄	—	251	2	no	—
135	C₂×D₄	—	251	2	no	—
136	C₂×D₄	—	251	2	no	—
137	C₂²×C₄	—	251	4	no	—
138	C₂²×C₄	C₄×D₂₅₁	1	1	yes	characteristic
139	C₂²×C₄	C₄×D₂₅₁	1	1	yes	characteristic
140	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
141	M₁₆	—	1004	4	no	—
142	C₂×C₈	—	1004	4	no	—
143	C₂×C₈	—	251	2	no	—
144	C₂×C₈	—	251	2	no	—
145	C₂×C₈	C₂²×D₂₅₁	1	2	yes	—
146	C₂×C₈	C₂²×D₂₅₁	1	2	yes	—
147	C₄⋊C₄	—	251	2	no	—
148	C₄⋊C₄	—	251	2	no	—
149	C₂²⋊C₄	—	251	4	no	—
150	C₂²⋊C₄	—	251	4	no	—
151	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
152	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
153	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
154	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
155	C₂³	—	251	4	no	—
156	C₂³	C₂×C₄×D₂₅₁	1	1	yes	characteristic
157	Q₈	—	1004	2	no	—
158	D₄	—	1004	2	no	—
159	D₄	—	1004	4	no	—
160	D₄	—	502	4	no	—
161	D₄	—	502	4	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	502	4	no	—
167	C₂×C₄	—	502	4	no	—
168	C₂×C₄	—	251	4	no	—
169	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
170	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
171	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
172	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
173	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
174	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
175	C₈	—	2	2	no	—
176	C₈	—	2	2	no	—
177	C₈	—	502	2	no	—
178	C₈	—	502	2	no	—
179	C₂²	—	2	1	no	—
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	—	502	4	no	—
183	C₂²	—	502	4	no	—
184	C₂²	—	502	4	no	—
185	C₂²	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	—	2	1	no	—
189	C₄	—	502	4	no	—
190	C₄	#22548721	1	2	yes	—
191	C₄	#22548723	1	2	yes	—
192	C₄	#22548761	1	1	yes	characteristic
193	C₄	#22548761	1	1	yes	characteristic
194	C₂	—	4	1	no	—
195	C₂	—	1004	4	no	—
196	C₂	#30632161	1	1	yes	characteristic
197	C₂	C₂×#22548721	1	1	yes	characteristic
198	C₂	C₂×#22548723	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₅₁
commutator, socle	C₂²×C₂₅₁	C₂³×C₄
Fitting	#431×C₂₅₁	C₂
prime radical	#30632237	C₂
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	#431	D₂₅₁
hypocenter	C₂₅₁	#2979
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
2. #30631766	C₂	semidirect
3. #30631764	C₂	semidirect
4. C₈×D₁₀₀₄	C₂	semidirect
5. #30631752	C₂	semidirect
6. #30631740	C₂	semidirect
7. #30631652	C₂	semidirect
8. #431×C₂₅₁	C₂	semidirect
99. C₂₅₁	#2979	semidirect
107. #431	D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#30632237	2
#30631766	2
#30631764	2
C₈×D₁₀₀₄	2
#30631752	2
#30631740	2
#30631652	2
#431×C₂₅₁	2
#2979	251