Subgroups — #44560535

Order 32128.

386 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632216	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#30632021	C₂	1	1	yes	maximal, characteristic, complemented
4	#30632004	C₂	1	1	yes	maximal, characteristic
5	#30631995	C₂	1	1	yes	maximal, characteristic, complemented
6	#30631994	C₂	1	1	yes	maximal, characteristic, complemented
7	#30631906	C₂	1	1	yes	maximal, characteristic, complemented
8	#30631841	C₂	1	1	yes	maximal, characteristic, complemented
9	#30631824	C₂	1	1	yes	maximal, characteristic
10	#30631794	C₂	1	1	yes	maximal, characteristic, complemented
11	#30631791	C₂	1	1	yes	maximal, characteristic, complemented
12	#30631788	C₂	1	1	yes	maximal, characteristic, complemented
13	#30631784	C₂	1	1	yes	maximal, characteristic, complemented
14	#30631743	C₂	1	1	yes	maximal, characteristic, complemented
15	#464×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₂₅₁	C₂²	1	1	yes	characteristic
17	C₂×#16860800	C₂²	1	1	yes	characteristic
18	C₂×#16860798	C₂²	1	1	yes	characteristic
19	#22548708	—	2	1	no	—
20	#22548707	C₂²	1	1	yes	characteristic
21	#22548697	—	2	1	no	—
22	#22548691	C₂²	1	1	yes	characteristic
23	#22548686	C₂²	1	1	yes	characteristic
24	Q₈×Dic₂₅₁	C₂²	1	1	yes	characteristic
25	Q₈×Dic₂₅₁	C₂²	1	1	yes	characteristic
26	C₄×Dic₅₀₂	—	2	1	no	—
27	C₄×Dic₅₀₂	C₂²	1	1	yes	characteristic
28	#22548680	—	2	1	no	—
29	#22548680	C₂²	1	1	yes	characteristic
30	D₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
31	(C₂²⋊D₂₅₁)⋊C₄	—	2	1	no	—
32	#22548671	—	2	1	no	—
33	C₄⋊C₄×D₂₅₁	—	2	1	no	—
34	C₄⋊C₄×D₂₅₁	C₂²	1	1	yes	characteristic
35	C₄²×D₂₅₁	—	2	1	no	—
36	C₄²×D₂₅₁	C₂²	1	1	yes	characteristic
37	#22548649	C₂²	1	1	yes	characteristic
38	#22548648	C₂²	1	1	yes	characteristic
39	#22548647	C₂²	1	1	yes	characteristic
40	#22548646	C₂²	1	1	yes	characteristic
41	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
42	#22548640	C₂²	1	1	yes	characteristic
43	#22548640	C₂²	1	1	yes	characteristic
44	#22548639	C₂²	1	1	yes	characteristic
45	#22548639	C₂²	1	1	yes	characteristic
46	#22548639	C₂²	1	1	yes	characteristic
47	#22548636	C₂²	1	1	yes	characteristic
48	(C₂×D₂₅₁)⋊C₈	C₂²	1	1	yes	characteristic
49	(C₂×D₂₅₁)⋊C₈	C₂²	1	1	yes	characteristic
50	#22548625	C₂²	1	1	yes	characteristic
51	#22548625	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
54	#124×C₂₅₁	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₂₅₁	C₂²	1	1	yes	characteristic
57	#103×C₂₅₁	C₂²	1	1	yes	characteristic
58	#102×C₂₅₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₅₀₂	—	2	1	no	—
60	C₂×Dic₅₀₂	C₂³	1	1	yes	characteristic
61	Q₈×D₂₅₁	—	2	2	no	—
62	Q₈×D₂₅₁	—	2	2	no	—
63	C₂×C₂²⋊D₂₅₁	—	2	1	no	—
64	C₂²×Dic₂₅₁	—	2	1	no	—
65	C₂×C₄×D₂₅₁	—	2	1	no	—
66	C₂×C₄×D₂₅₁	—	2	1	no	—
67	C₂×C₄×D₂₅₁	D₄	1	1	yes	characteristic
68	C₂×C₄×D₂₅₁	D₄	1	1	yes	characteristic
69	C₂×C₄×D₂₅₁	C₂³	1	1	yes	characteristic
70	#16860800	—	2	2	no	—
71	#16860798	—	2	2	no	—
72	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₂₅₁	—	2	1	no	—
75	C₄⋊Dic₂₅₁	—	2	1	no	—
76	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
79	Dic₂₅₁⋊C₄	—	2	1	no	—
80	Dic₂₅₁⋊C₄	—	2	1	no	—
81	Dic₂₅₁⋊C₄	—	2	1	no	—
82	Dic₂₅₁⋊C₄	—	2	1	no	—
83	Dic₂₅₁⋊C₄	—	2	1	no	—
84	Dic₂₅₁⋊C₄	—	2	1	no	—
85	C₂²⋊Dic₂₅₁	—	2	1	no	—
86	C₂²⋊Dic₂₅₁	—	2	1	no	—
87	(C₂×D₂₅₁)⋊C₄	—	2	1	no	—
88	C₄×Dic₂₅₁	—	2	1	no	—
89	C₄×Dic₂₅₁	—	2	1	no	—
90	C₄×Dic₂₅₁	—	2	1	no	—
91	C₄×Dic₂₅₁	—	2	1	no	—
92	C₄×Dic₂₅₁	D₄	1	1	yes	characteristic
93	C₄×Dic₂₅₁	D₄	1	1	yes	characteristic
94	C₄×Dic₂₅₁	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₂₅₁	C₂³	1	1	yes	characteristic
98	Q₁₆×C₂₅₁	—	2	2	no	—
99	SD₁₆×C₂₅₁	—	2	2	no	—
100	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₂₅₁	—	2	1	no	—
103	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
104	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
105	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
106	C₄²×C₂₅₁	—	2	1	no	—
107	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
108	C₂²×D₂₅₁	C₂×D₄	1	1	yes	characteristic
109	Dic₅₀₂	—	2	4	no	—
110	Dic₅₀₂	—	2	1	no	—
111	Dic₅₀₂	#41	1	2	yes	—
112	C₂²⋊D₂₅₁	—	4	2	no	—
113	C₂×Dic₂₅₁	—	2	1	no	—
114	C₂×Dic₂₅₁	—	2	1	no	—
115	C₂×Dic₂₅₁	—	2	1	no	—
116	C₂×Dic₂₅₁	—	2	1	no	—
117	C₂×Dic₂₅₁	—	4	2	no	—
118	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
121	C₄×D₂₅₁	—	2	4	no	—
122	C₄×D₂₅₁	—	2	2	no	—
123	C₄×D₂₅₁	—	2	1	no	—
124	C₄×D₂₅₁	—	2	1	no	—
125	C₄×D₂₅₁	—	4	1	no	—
126	C₄×D₂₅₁	—	4	2	no	—
127	C₂₅₁⋊C₈	—	2	1	no	—
128	C₂₅₁⋊C₈	—	2	1	no	—
129	C₂³×C₂₅₁	—	2	1	no	—
130	Q₈×C₂₅₁	—	2	1	no	—
131	Q₈×C₂₅₁	—	2	1	no	—
132	Q₈×C₂₅₁	—	2	1	no	—
133	Q₈×C₂₅₁	#41	1	2	yes	—
134	D₄×C₂₅₁	—	2	1	no	—
135	D₄×C₂₅₁	—	2	1	no	—
136	C₂×C₄×C₂₅₁	—	2	1	no	—
137	C₂×C₄×C₂₅₁	—	2	1	no	—
138	C₂×C₄×C₂₅₁	—	2	1	no	—
139	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
140	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
141	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
142	C₈×C₂₅₁	—	2	1	no	—
143	C₈×C₂₅₁	—	2	1	no	—
144	C₂²×C₂₅₁	—	4	1	no	—
145	C₂²×C₂₅₁	—	4	1	no	—
146	C₂²×C₂₅₁	—	4	1	no	—
147	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
148	C₂×D₂₅₁	—	2	1	no	—
149	C₂×D₂₅₁	—	2	1	no	—
150	C₂×D₂₅₁	#137	1	2	yes	—
151	C₄×C₂₅₁	—	2	2	no	—
152	C₄×C₂₅₁	—	2	1	no	—
153	C₄×C₂₅₁	—	2	1	no	—
154	C₄×C₂₅₁	—	4	1	no	—
155	C₄×C₂₅₁	—	4	1	no	—
156	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
157	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
158	Dic₂₅₁	—	2	2	no	—
159	Dic₂₅₁	—	2	1	no	—
160	Dic₂₅₁	—	2	1	no	—
161	Dic₂₅₁	—	2	1	no	—
162	Dic₂₅₁	—	4	1	no	—
163	Dic₂₅₁	—	4	1	no	—
164	Dic₂₅₁	—	4	1	no	—
165	C₂×C₂₅₁	—	8	1	no	—
166	C₂×C₂₅₁	#525	1	1	yes	characteristic
167	C₂×C₂₅₁	C₂×#137	1	1	yes	characteristic
168	C₂×C₂₅₁	#575	1	1	yes	characteristic
169	D₂₅₁	—	2	2	no	—
170	C₂₅₁	#3117	1	1	yes	characteristic, complemented, Sylow
171	#3117	—	251	1	no	maximal, Sylow
172	#529	—	251	1	no	—
173	C₂×C₄×Q₈	—	251	1	no	—
174	#484	—	251	1	no	—
175	#481	—	251	1	no	—
176	#479	—	251	1	no	—
177	#479	—	251	1	no	—
178	#464	—	251	1	no	—
179	#464	D₂₅₁	1	1	yes	characteristic, complemented
180	#451	—	251	1	no	—
181	#448	—	251	1	no	—
182	#441	—	251	1	no	—
183	#441	—	251	1	no	—
184	#440	—	251	1	no	—
185	#440	—	251	1	no	—
186	#432	—	251	1	no	—
187	C₂²×Q₈	—	251	1	no	—
188	C₂×Q₁₆	—	251	1	no	—
189	C₂×Q₁₆	C₂×D₂₅₁	1	1	yes	characteristic
190	C₂×SD₁₆	—	251	1	no	—
191	C₂×SD₁₆	C₂×D₂₅₁	1	1	yes	characteristic
192	#126	—	502	1	no	—
193	#125	—	251	1	no	—
194	#124	C₂×D₂₅₁	1	1	yes	characteristic
195	#123	—	502	1	no	—
196	C₂²⋊Q₈	—	251	1	no	—
197	#121	—	251	1	no	—
198	C₄×Q₈	—	502	1	no	—
199	C₄×Q₈	—	502	1	no	—
200	C₄×Q₈	—	251	1	no	—
201	C₄×Q₈	—	251	1	no	—
202	C₄×Q₈	—	251	1	no	—
203	C₄×Q₈	—	251	1	no	—
204	C₄×Q₈	C₂×D₂₅₁	1	1	yes	characteristic
205	C₄×D₄	—	502	1	no	—
206	C₄×D₄	—	251	1	no	—
207	#117	—	502	1	no	—
208	C₂×C₄⋊C₄	—	502	1	no	—
209	C₂×C₄⋊C₄	—	251	1	no	—
210	C₂×C₄²	—	502	1	no	—
211	C₂×C₄²	—	251	1	no	—
212	#107	—	251	1	no	—
213	#107	—	251	1	no	—
214	#106	—	251	1	no	—
215	#106	—	251	1	no	—
216	C₄⋊C₈	—	251	1	no	—
217	C₄⋊C₈	C₂×D₂₅₁	1	1	yes	characteristic
218	#103	—	251	1	no	—
219	#103	—	251	1	no	—
220	#103	—	251	1	no	—
221	#103	—	251	1	no	—
222	#103	—	251	1	no	—
223	#103	C₂×D₂₅₁	1	1	yes	characteristic
224	#102	—	251	1	no	—
225	#102	C₂×D₂₅₁	1	1	yes	characteristic
226	C₂²⋊C₈	—	251	1	no	—
227	C₂²⋊C₈	—	251	1	no	—
228	#97	—	251	1	no	—
229	#97	—	251	1	no	—
230	C₂×Q₈	—	502	1	no	—
231	C₂×Q₈	—	502	2	no	—
232	C₂×Q₈	—	502	2	no	—
233	C₂×Q₈	—	251	1	no	—
234	C₂×Q₈	C₂²×D₂₅₁	1	1	yes	characteristic
235	C₂×Q₈	C₂²×D₂₅₁	1	1	yes	characteristic
236	C₂×D₄	—	502	1	no	—
237	C₂×D₄	C₂²×D₂₅₁	1	1	yes	characteristic
238	C₂²×C₄	—	502	1	no	—
239	C₂²×C₄	—	502	1	no	—
240	C₂²×C₄	—	502	1	no	—
241	C₂²×C₄	—	251	1	no	—
242	C₂²×C₄	—	251	1	no	—
243	C₂²×C₄	—	251	1	no	—
244	Q₁₆	—	2	2	no	—
245	Q₁₆	—	502	2	no	—
246	SD₁₆	—	2	2	no	—
247	SD₁₆	—	502	2	no	—
248	C₂×C₈	—	251	1	no	—
249	C₂×C₈	—	251	1	no	—
250	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
251	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	502	1	no	—
254	C₄⋊C₄	—	502	1	no	—
255	C₄⋊C₄	—	502	1	no	—
256	C₄⋊C₄	—	502	1	no	—
257	C₄⋊C₄	—	502	1	no	—
258	C₄⋊C₄	—	502	1	no	—
259	C₄⋊C₄	—	502	1	no	—
260	C₄⋊C₄	—	502	1	no	—
261	C₄⋊C₄	—	251	1	no	—
262	C₄⋊C₄	—	251	1	no	—
263	C₄⋊C₄	—	251	1	no	—
264	C₄⋊C₄	C₂²×D₂₅₁	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	502	1	no	—
268	C₂²⋊C₄	—	502	1	no	—
269	C₂²⋊C₄	—	502	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	502	1	no	—
272	C₄²	—	502	1	no	—
273	C₄²	—	502	1	no	—
274	C₄²	—	502	1	no	—
275	C₄²	—	251	1	no	—
276	C₄²	—	251	1	no	—
277	C₄²	—	251	1	no	—
278	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
279	C₂³	—	2	1	no	—
280	C₂³	—	251	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	2	1	no	—
284	Q₈	—	502	4	no	—
285	Q₈	—	502	1	no	—
286	Q₈	—	251	2	no	—
287	Q₈	#16860809	1	2	yes	—
288	D₄	—	2	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	1004	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	1004	1	no	—
295	C₂×C₄	—	1004	2	no	—
296	C₂×C₄	—	1004	2	no	—
297	C₂×C₄	—	502	4	no	—
298	C₂×C₄	—	502	2	no	—
299	C₂×C₄	—	502	1	no	—
300	C₂×C₄	—	502	1	no	—
301	C₂×C₄	—	502	1	no	—
302	C₂×C₄	—	502	1	no	—
303	C₂×C₄	—	502	1	no	—
304	C₂×C₄	—	502	1	no	—
305	C₂×C₄	—	251	1	no	—
306	C₂×C₄	—	251	1	no	—
307	C₂×C₄	—	251	1	no	—
308	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
309	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	502	1	no	—
314	C₈	—	502	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	502	1	no	—
319	C₂²	—	502	1	no	—
320	C₂²	—	251	2	no	—
321	C₂²	C₂×D₄×D₂₅₁	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	1004	1	no	—
328	C₄	—	1004	1	no	—
329	C₄	—	1004	1	no	—
330	C₄	—	502	2	no	—
331	C₄	—	502	1	no	—
332	C₄	—	502	1	no	—
333	C₄	—	502	1	no	—
334	C₄	#41×D₂₅₁	1	1	yes	characteristic
335	C₄	C₂×#16860809	1	1	yes	characteristic
336	C₂	—	8	1	no	—
337	C₂	—	502	2	no	—
338	C₂	#30632197	1	1	yes	characteristic
339	C₂	#137×D₂₅₁	1	1	yes	characteristic
340	C₂	#30632484	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₅₁
commutator, cosocle	C₂×C₄×C₂₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	#464×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
prime radical	#22548686	C₂²
hypercenter	#464	D₂₅₁
hypocenter	C₂₅₁	#3117
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #464

Complements

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

N	H ≅ G/N	type
2. C₄×Q₈×D₂₅₁	C₂	semidirect
3. #30632021	C₂	semidirect
5. #30631995	C₂	semidirect
6. #30631994	C₂	semidirect
7. #30631906	C₂	semidirect
8. #30631841	C₂	semidirect
10. #30631794	C₂	semidirect
11. #30631791	C₂	semidirect
12. #30631788	C₂	semidirect
13. #30631784	C₂	semidirect
14. #30631743	C₂	semidirect
15. #464×C₂₅₁	C₂	semidirect
170. C₂₅₁	#3117	semidirect
179. #464	D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#30632216	2
C₄×Q₈×D₂₅₁	2
#30632021	2
#30632004	2
#30631995	2
#30631994	2
#30631906	2
#30631841	2
#30631824	2
#30631794	2
#30631791	2
#30631788	2
#30631784	2
#30631743	2
#464×C₂₅₁	2
#3117	251