Subgroups — #44559359

Order 32128.

372 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632221	C₂	1	1	yes	maximal, characteristic
2	#30631815	C₂	1	2	yes	maximal, complemented
3	#30631760	C₂	1	2	yes	maximal, complemented
4	#30631758	C₂	1	2	yes	maximal, complemented
5	C₈×C₂²⋊D₂₅₁	C₂	1	2	yes	maximal, complemented
6	#30631749	C₂	1	2	yes	maximal, complemented
7	#30631740	C₂	1	1	yes	maximal, characteristic, complemented
8	#30631644	C₂	1	2	yes	maximal, complemented
9	C₂×#97×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16860793	C₂²	1	2	yes	—
11	C₂×C₈×D₂₅₁	C₂²	1	2	yes	—
12	#22548709	C₄	1	2	yes	—
13	#22548707	C₄	1	1	yes	characteristic
14	#22548703	C₄	1	1	yes	characteristic
15	#22548698	C₄	1	2	yes	—
16	#22548694	C₄	1	1	yes	characteristic
17	#22548689	C₄	1	1	yes	characteristic
18	C₄×Dic₅₀₂	C₂²	1	1	yes	characteristic
19	C₄×D₁₀₀₄	C₂²	1	1	yes	characteristic
20	C₄×C₂²⋊D₂₅₁	C₂²	1	2	yes	—
21	#22548672	C₂²	1	2	yes	—
22	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	2	yes	—
23	Dic₂₅₁⋊C₈	C₂²	1	2	yes	—
24	Dic₂₅₁⋊C₈	C₂²	1	2	yes	—
25	C₂²⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
26	C₂²⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
27	(C₂×D₂₅₁)⋊C₈	C₂²	1	2	yes	—
28	(C₂×D₂₅₁)⋊C₈	C₂²	1	2	yes	—
29	#22548625	C₂²	1	2	yes	—
30	C₄×C₂₅₁⋊C₈	C₂²	1	2	yes	—
31	C₈×Dic₂₅₁	C₂²	1	2	yes	—
32	C₂²×C₈×C₂₅₁	C₂²	1	2	yes	—
33	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
34	#97×C₂₅₁	C₂²	1	1	yes	characteristic
35	#97×C₂₅₁	C₂²	1	2	yes	—
36	#97×C₂₅₁	C₂²	1	1	yes	characteristic
37	C₂×Dic₅₀₂	C₂×C₄	1	1	yes	characteristic
38	C₂×D₁₀₀₄	C₂×C₄	1	1	yes	characteristic
39	C₂×C₂²⋊D₂₅₁	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₂₅₁	C₂³	1	2	yes	—
41	#16860793	—	2	4	no	—
42	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
43	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
44	C₈×D₂₅₁	—	2	4	no	—
45	C₄⋊Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
46	C₄⋊Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
47	Dic₂₅₁⋊C₄	C₂×C₄	1	2	yes	—
48	Dic₂₅₁⋊C₄	C₂×C₄	1	2	yes	—
49	Dic₂₅₁⋊C₄	C₂×C₄	1	2	yes	—
50	Dic₂₅₁⋊C₄	C₂×C₄	1	2	yes	—
51	C₂²⋊Dic₂₅₁	C₂×C₄	1	2	yes	—
52	(C₂×D₂₅₁)⋊C₄	C₂×C₄	1	2	yes	—
53	(C₂×D₂₅₁)⋊C₄	C₂×C₄	1	2	yes	—
54	(C₂×D₂₅₁)⋊C₄	C₂×C₄	1	2	yes	—
55	(C₂×D₂₅₁)⋊C₄	C₂×C₄	1	2	yes	—
56	C₄×Dic₂₅₁	C₂³	1	2	yes	—
57	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₂₅₁	—	2	4	no	—
61	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
62	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
63	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
64	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
65	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
66	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
67	C₂²×D₂₅₁	C₂²×C₄	1	2	yes	—
68	Dic₅₀₂	—	4	1	no	—
69	D₁₀₀₄	—	4	1	no	—
70	C₂²⋊D₂₅₁	—	2	4	no	—
71	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
72	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
73	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
74	C₄×D₂₅₁	—	2	2	no	—
75	C₄×D₂₅₁	—	2	2	no	—
76	C₂₅₁⋊C₈	—	2	2	no	—
77	C₂₅₁⋊C₈	—	2	2	no	—
78	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₂₅₁	—	2	2	no	—
80	C₂×C₄×C₂₅₁	—	2	2	no	—
81	C₂×C₄×C₂₅₁	—	4	1	no	—
82	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
88	C₈×C₂₅₁	—	2	2	no	—
89	C₈×C₂₅₁	#41	1	4	yes	—
90	C₂²×C₂₅₁	—	2	1	no	—
91	C₂²×C₂₅₁	—	2	1	no	—
92	C₂²×C₂₅₁	#131	1	2	yes	—
93	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
94	C₂×D₂₅₁	—	2	2	no	—
95	C₂×D₂₅₁	—	2	2	no	—
96	C₂×D₂₅₁	—	2	2	no	—
97	C₄×C₂₅₁	—	2	1	no	—
98	C₄×C₂₅₁	—	2	1	no	—
99	C₄×C₂₅₁	—	2	1	no	—
100	C₄×C₂₅₁	—	2	1	no	—
101	C₄×C₂₅₁	—	2	1	no	—
102	C₄×C₂₅₁	C₂×#41	1	2	yes	—
103	Dic₂₅₁	—	2	2	no	—
104	Dic₂₅₁	—	2	2	no	—
105	Dic₂₅₁	—	2	2	no	—
106	C₂×C₂₅₁	—	2	2	no	—
107	C₂×C₂₅₁	C₄×#41	1	1	yes	characteristic
108	C₂×C₂₅₁	C₂×#131	1	1	yes	characteristic
109	C₂×C₂₅₁	#568	1	1	yes	characteristic
110	D₂₅₁	—	4	2	no	—
111	C₂₅₁	#2968	1	1	yes	characteristic, complemented, Sylow
112	#2968	—	251	1	no	maximal, Sylow
113	#529	—	251	1	no	—
114	#446	—	251	2	no	—
115	#435	—	251	2	no	—
116	#435	—	251	2	no	—
117	C₈×D₄	—	251	2	no	—
118	#433	—	251	2	no	—
119	#431	—	251	1	no	—
120	#405	—	251	2	no	—
121	C₂×#97	D₂₅₁	1	1	yes	characteristic, complemented
122	C₂×M₁₆	—	251	2	no	—
123	C₂²×C₈	—	251	2	no	—
124	C₂²×C₈	C₂×D₂₅₁	1	2	yes	—
125	#126	—	251	2	no	—
126	#125	—	251	1	no	—
127	#124	—	251	1	no	—
128	#123	—	251	2	no	—
129	C₂²⋊Q₈	—	251	1	no	—
130	#121	—	251	1	no	—
131	C₄×Q₈	—	251	1	no	—
132	C₄×D₄	—	251	2	no	—
133	C₄×D₄	—	251	1	no	—
134	#117	—	251	2	no	—
135	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
136	C₄⋊C₈	—	251	2	no	—
137	C₄⋊C₈	—	251	2	no	—
138	C₄⋊C₈	—	251	2	no	—
139	C₂²⋊C₈	—	251	1	no	—
140	C₂²⋊C₈	—	251	1	no	—
141	C₂²⋊C₈	—	251	2	no	—
142	C₂²⋊C₈	—	251	2	no	—
143	#97	—	251	2	no	—
144	#97	C₂×D₂₅₁	1	1	yes	characteristic
145	#97	C₂×D₂₅₁	1	1	yes	characteristic
146	#97	C₂×D₂₅₁	1	2	yes	—
147	C₄×C₈	—	251	2	no	—
148	C₄×C₈	—	251	2	no	—
149	C₂×Q₈	—	251	1	no	—
150	C₂×D₄	—	251	2	no	—
151	C₂×D₄	—	251	1	no	—
152	C₂²×C₄	—	251	2	no	—
153	C₂²×C₄	C₄×D₂₅₁	1	1	yes	characteristic
154	C₂²×C₄	C₄×D₂₅₁	1	1	yes	characteristic
155	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
156	M₁₆	—	502	4	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	502	4	no	—
159	C₂×C₈	—	251	2	no	—
160	C₂×C₈	—	251	2	no	—
161	C₂×C₈	C₂²×D₂₅₁	1	2	yes	—
162	C₂×C₈	C₂²×D₂₅₁	1	2	yes	—
163	C₄⋊C₄	—	251	2	no	—
164	C₄⋊C₄	—	251	2	no	—
165	C₄⋊C₄	—	251	2	no	—
166	C₄⋊C₄	—	251	2	no	—
167	C₄⋊C₄	—	251	1	no	—
168	C₄⋊C₄	—	251	1	no	—
169	C₂²⋊C₄	—	251	2	no	—
170	C₂²⋊C₄	—	251	2	no	—
171	C₂²⋊C₄	—	251	2	no	—
172	C₂²⋊C₄	—	251	2	no	—
173	C₂²⋊C₄	—	251	2	no	—
174	C₄²	—	251	2	no	—
175	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
176	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
177	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
178	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
179	C₂³	—	251	2	no	—
180	C₂³	C₂×C₄×D₂₅₁	1	1	yes	characteristic
181	Q₈	—	1004	1	no	—
182	D₄	—	1004	1	no	—
183	D₄	—	502	4	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	4	1	no	—
187	C₂×C₄	—	502	2	no	—
188	C₂×C₄	—	502	2	no	—
189	C₂×C₄	—	251	2	no	—
190	C₂×C₄	—	251	2	no	—
191	C₂×C₄	—	251	2	no	—
192	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
195	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
196	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
197	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	—	502	2	no	—
200	C₈	—	502	2	no	—
201	C₈	#16860810	1	4	yes	—
202	C₂²	—	2	1	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	502	2	no	—
205	C₂²	—	502	2	no	—
206	C₂²	—	502	2	no	—
207	C₂²	#22548723	1	2	yes	—
208	C₂²	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	502	2	no	—
215	C₄	—	502	2	no	—
216	C₄	—	502	2	no	—
217	C₄	C₂×#16860810	1	2	yes	—
218	C₂	—	2	2	no	—
219	C₂	—	1004	2	no	—
220	C₂	C₄×#16860810	1	1	yes	characteristic
221	C₂	C₂×#22548723	1	1	yes	characteristic
222	C₂	#30632453	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₅₁
commutator, socle	C₂²×C₂₅₁	C₂³×C₄
Fitting	C₂×#97×C₂₅₁	C₂
prime radical	#22548689	C₄
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	C₂×#97	D₂₅₁
hypocenter	C₂₅₁	#2968
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
2. #30631815	C₂	semidirect
3. #30631760	C₂	semidirect
4. #30631758	C₂	semidirect
5. C₈×C₂²⋊D₂₅₁	C₂	semidirect
6. #30631749	C₂	semidirect
7. #30631740	C₂	semidirect
8. #30631644	C₂	semidirect
9. C₂×#97×C₂₅₁	C₂	semidirect
111. C₂₅₁	#2968	semidirect
121. C₂×#97	D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#30632221	2
#30631815	2
#30631760	2
#30631758	2
C₈×C₂²⋊D₂₅₁	2
#30631749	2
#30631740	2
#30631644	2
C₂×#97×C₂₅₁	2
#2968	251