Subgroups — #44559581

Order 32128.

764 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#16860810	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#22548735	C₂	1	2	yes	maximal, complemented
3	#30631847	C₂	1	2	yes	maximal, complemented
4	#30631842	C₂	1	1	yes	maximal, characteristic, complemented
5	#30631837	C₂	1	1	yes	maximal, characteristic, complemented
6	#30631830	C₂	1	2	yes	maximal, complemented
7	#30631825	C₂	1	1	yes	maximal, characteristic, complemented
8	#30631820	C₂	1	1	yes	maximal, characteristic, complemented
9	#30631681	C₂	1	2	yes	maximal, complemented
10	C₂×C₂²⋊(C₂₅₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
11	#525×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#16860810	—	2	8	no	—
13	C₂×#16860810	—	2	1	no	—
14	C₂×#16860810	—	2	1	no	—
15	C₂×#16860810	C₂²	1	2	yes	—
16	C₂×#16860810	C₂²	1	2	yes	—
17	C₂²×Dic₅₀₂	C₂²	1	1	yes	characteristic
18	C₂²×D₁₀₀₄	C₂²	1	1	yes	characteristic
19	C₂²×C₂²⋊D₂₅₁	—	2	1	no	—
20	C₂²×C₄×D₂₅₁	—	2	1	no	—
21	#22548735	—	2	8	no	—
22	C₂×#16860800	C₂²	1	2	yes	complemented
23	C₂×#16860798	C₂²	1	2	yes	complemented
24	C₂×#16860797	C₂²	1	2	yes	complemented
25	C₂×#16860795	C₂²	1	2	yes	complemented
26	C₂²×C₂₅₁⋊C₈	C₂²	1	2	yes	complemented
27	#22548638	C₂²	1	2	yes	complemented
28	#22548638	C₂²	1	2	yes	complemented
29	#22548635	C₂²	1	2	yes	complemented
30	#22548635	C₂²	1	2	yes	complemented
31	C₂²⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
32	C₂²⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
33	C₂²⋊(C₂₅₁⋊C₈)	C₂²	1	2	yes	complemented
34	C₂×#41×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
35	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
36	#123×C₂₅₁	—	2	1	no	—
37	C₂²⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
38	C₂²⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
39	#121×C₂₅₁	C₂²	1	1	yes	characteristic
40	#121×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
41	#120×C₂₅₁	—	2	1	no	—
42	C₄×D₄×C₂₅₁	—	2	1	no	—
43	C₄×D₄×C₂₅₁	—	2	1	no	—
44	#117×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
45	C₂³×D₂₅₁	—	2	1	no	—
46	#16860810	—	2	8	no	—
47	#16860810	—	2	8	no	—
48	#16860810	—	2	4	no	—
49	#16860810	—	2	4	no	—
50	#16860810	—	2	4	no	—
51	#16860810	D₄	1	8	yes	complemented
52	C₂×Dic₅₀₂	—	2	4	no	—
53	C₂×Dic₅₀₂	—	2	1	no	—
54	C₂×Dic₅₀₂	C₂³	1	2	yes	—
55	C₂×D₁₀₀₄	—	2	4	no	—
56	C₂×D₁₀₀₄	—	2	1	no	—
57	C₂×D₁₀₀₄	C₂³	1	2	yes	—
58	C₂×C₂²⋊D₂₅₁	—	2	8	no	—
59	C₂×C₂²⋊D₂₅₁	—	2	2	no	—
60	C₂×C₂²⋊D₂₅₁	—	2	2	no	—
61	C₂²×Dic₂₅₁	—	2	1	no	—
62	C₂×C₄×D₂₅₁	—	2	8	no	—
63	C₂×C₄×D₂₅₁	—	2	2	no	—
64	C₂×C₄×D₂₅₁	—	2	2	no	—
65	#16860800	—	2	4	no	—
66	#16860798	—	2	4	no	—
67	#16860797	—	2	4	no	—
68	#16860795	—	2	4	no	—
69	C₂×C₂₅₁⋊C₈	—	2	4	no	—
70	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
71	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
72	C₂⁴×C₂₅₁	D₄	1	1	yes	characteristic
73	#41×C₂₅₁	—	2	4	no	—
74	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
75	C₂×D₄×C₂₅₁	—	2	1	no	—
76	C₂×D₄×C₂₅₁	—	2	1	no	—
77	C₂×D₄×C₂₅₁	—	2	1	no	—
78	C₂×D₄×C₂₅₁	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₂₅₁	—	2	4	no	—
80	C₂²×C₄×C₂₅₁	—	2	1	no	—
81	C₂²×C₄×C₂₅₁	D₄	1	1	yes	characteristic
82	C₂²×C₄×C₂₅₁	D₄	1	1	yes	characteristic
83	C₂²×C₄×C₂₅₁	D₄	1	1	yes	characteristic
84	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
85	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
86	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
87	C₄⋊C₄×C₂₅₁	—	2	1	no	—
88	C₄⋊C₄×C₂₅₁	—	2	1	no	—
89	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
90	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
91	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
92	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
93	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
94	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
95	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
96	C₄²×C₂₅₁	—	2	1	no	—
97	C₂²×D₂₅₁	—	2	4	no	—
98	C₂²×D₂₅₁	—	2	2	no	—
99	C₂²×D₂₅₁	—	4	2	no	—
100	C₂²×D₂₅₁	—	4	2	no	—
101	Dic₅₀₂	—	2	4	no	—
102	Dic₅₀₂	—	2	2	no	—
103	Dic₅₀₂	C₂×D₄	1	4	yes	—
104	D₁₀₀₄	—	2	4	no	—
105	D₁₀₀₄	—	2	2	no	—
106	D₁₀₀₄	C₂×D₄	1	4	yes	—
107	C₂²⋊D₂₅₁	—	2	8	no	—
108	C₂²⋊D₂₅₁	—	2	8	no	—
109	C₂×Dic₂₅₁	—	2	4	no	—
110	C₂×Dic₂₅₁	—	2	2	no	—
111	C₄×D₂₅₁	—	2	8	no	—
112	C₄×D₂₅₁	—	2	8	no	—
113	C₂₅₁⋊C₈	—	2	2	no	—
114	C₂₅₁⋊C₈	—	2	2	no	—
115	C₂³×C₂₅₁	—	2	2	no	—
116	C₂³×C₂₅₁	—	2	2	no	—
117	C₂³×C₂₅₁	—	2	1	no	—
118	C₂³×C₂₅₁	—	4	1	no	—
119	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
120	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
121	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
122	Q₈×C₂₅₁	—	2	2	no	—
123	D₄×C₂₅₁	—	2	2	no	—
124	D₄×C₂₅₁	—	4	2	no	—
125	D₄×C₂₅₁	—	4	2	no	—
126	D₄×C₂₅₁	—	4	2	no	—
127	C₂×C₄×C₂₅₁	—	2	2	no	—
128	C₂×C₄×C₂₅₁	—	2	2	no	—
129	C₂×C₄×C₂₅₁	—	2	2	no	—
130	C₂×C₄×C₂₅₁	—	2	2	no	—
131	C₂×C₄×C₂₅₁	—	2	2	no	—
132	C₂×C₄×C₂₅₁	—	2	1	no	—
133	C₂×C₄×C₂₅₁	—	2	1	no	—
134	C₂×C₄×C₂₅₁	—	2	1	no	—
135	C₂×C₄×C₂₅₁	—	4	2	no	—
136	C₂×C₄×C₂₅₁	C₂×D₄	1	4	yes	—
137	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
138	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
139	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
140	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
141	C₂²×C₂₅₁	—	2	1	no	—
142	C₂²×C₂₅₁	—	2	1	no	—
143	C₂²×C₂₅₁	—	2	1	no	—
144	C₂²×C₂₅₁	—	2	1	no	—
145	C₂²×C₂₅₁	—	2	1	no	—
146	C₂²×C₂₅₁	—	2	1	no	—
147	C₂²×C₂₅₁	—	2	1	no	—
148	C₂²×C₂₅₁	—	4	4	no	—
149	C₂²×C₂₅₁	—	4	1	no	—
150	C₂²×C₂₅₁	—	4	1	no	—
151	C₂²×C₂₅₁	—	4	1	no	—
152	C₂²×C₂₅₁	#120	1	2	yes	—
153	C₂²×C₂₅₁	#135	1	2	yes	complemented
154	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
155	C₂×D₂₅₁	—	2	4	no	—
156	C₂×D₂₅₁	—	4	8	no	—
157	C₂×D₂₅₁	—	4	2	no	—
158	C₂×D₂₅₁	—	4	2	no	—
159	C₄×C₂₅₁	—	2	1	no	—
160	C₄×C₂₅₁	—	2	1	no	—
161	C₄×C₂₅₁	—	4	1	no	—
162	C₄×C₂₅₁	—	4	1	no	—
163	C₄×C₂₅₁	—	4	1	no	—
164	C₄×C₂₅₁	#120	1	2	yes	—
165	C₄×C₂₅₁	C₂²×D₄	1	2	yes	—
166	Dic₂₅₁	—	2	4	no	—
167	C₂×C₂₅₁	—	2	2	no	—
168	C₂×C₂₅₁	—	2	2	no	—
169	C₂×C₂₅₁	—	4	1	no	—
170	C₂×C₂₅₁	—	8	1	no	—
171	C₂×C₂₅₁	C₂×#120	1	1	yes	characteristic
172	C₂×C₂₅₁	C₂×#135	1	1	yes	characteristic
173	C₂×C₂₅₁	#575	1	1	yes	characteristic
174	D₂₅₁	—	4	4	no	—
175	C₂₅₁	#3002	1	1	yes	characteristic, complemented, Sylow
176	#3002	—	251	1	no	maximal, Sylow
177	C₂²×#41	—	251	1	no	—
178	C₂×#135	—	251	2	no	—
179	#525	D₂₅₁	1	1	yes	characteristic, complemented
180	#452	—	251	2	no	—
181	#451	—	251	1	no	—
182	#450	—	251	1	no	—
183	#449	—	251	2	no	—
184	#448	—	251	1	no	—
185	#447	—	251	1	no	—
186	#416	—	251	2	no	—
187	C₂×C₂²⋊C₈	—	251	1	no	—
188	C₂×#41	—	502	8	no	—
189	C₂×#41	—	502	1	no	—
190	C₂×#41	—	502	1	no	—
191	C₂×#41	—	251	2	no	—
192	C₂×#41	—	251	2	no	—
193	C₂×#41	C₂×D₂₅₁	1	1	yes	characteristic, complemented
194	C₂²×Q₈	—	251	1	no	—
195	C₂²×D₄	—	502	1	no	—
196	C₂²×D₄	—	251	1	no	—
197	C₂³×C₄	—	502	1	no	—
198	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
199	#135	—	502	8	no	—
200	C₂×Q₁₆	—	251	2	no	—
201	C₂×SD₁₆	—	251	2	no	—
202	C₂×SD₁₆	—	251	2	no	—
203	C₂×D₈	—	251	2	no	—
204	C₂²×C₈	—	251	2	no	—
205	#123	—	2	1	no	—
206	C₂²⋊Q₈	C₂×D₂₅₁	1	1	yes	characteristic
207	C₂²⋊Q₈	C₂×D₂₅₁	1	1	yes	characteristic, complemented
208	#121	C₂×D₂₅₁	1	1	yes	characteristic
209	#121	C₂×D₂₅₁	1	1	yes	characteristic, complemented
210	#120	—	2	1	no	—
211	C₄×D₄	—	2	1	no	—
212	C₄×D₄	—	2	1	no	—
213	#117	C₂×D₂₅₁	1	1	yes	characteristic, complemented
214	#103	—	251	2	no	—
215	#103	—	251	2	no	—
216	#102	—	251	2	no	—
217	#102	—	251	2	no	—
218	C₂²⋊C₈	—	251	1	no	—
219	C₂²⋊C₈	—	251	1	no	—
220	C₂²⋊C₈	—	251	2	no	—
221	C₂⁴	—	502	1	no	—
222	C₂⁴	C₂²⋊D₂₅₁	1	1	yes	characteristic
223	#41	—	2	4	no	—
224	#41	—	502	8	no	—
225	#41	—	502	8	no	—
226	#41	—	502	4	no	—
227	#41	—	502	4	no	—
228	#41	—	502	4	no	—
229	#41	—	251	8	no	—
230	C₂×Q₈	—	502	4	no	—
231	C₂×Q₈	—	502	1	no	—
232	C₂×Q₈	—	251	2	no	—
233	C₂×Q₈	C₂²×D₂₅₁	1	1	yes	characteristic
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	—	2	1	no	—
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	—	502	4	no	—
238	C₂×D₄	—	502	8	no	—
239	C₂×D₄	—	502	2	no	—
240	C₂×D₄	—	502	2	no	—
241	C₂×D₄	—	502	1	no	—
242	C₂×D₄	—	251	2	no	—
243	C₂×D₄	C₂²×D₂₅₁	1	1	yes	characteristic
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	502	8	no	—
247	C₂²×C₄	—	502	1	no	—
248	C₂²×C₄	—	502	2	no	—
249	C₂²×C₄	—	502	2	no	—
250	C₂²×C₄	C₂²⋊D₂₅₁	1	1	yes	characteristic
251	C₂²×C₄	C₂²⋊D₂₅₁	1	1	yes	characteristic
252	C₂²×C₄	C₂²⋊D₂₅₁	1	1	yes	characteristic
253	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
254	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
255	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
256	Q₁₆	—	502	4	no	—
257	SD₁₆	—	502	4	no	—
258	SD₁₆	—	502	4	no	—
259	D₈	—	502	4	no	—
260	C₂×C₈	—	502	4	no	—
261	C₂×C₈	—	251	2	no	—
262	C₂×C₈	—	251	2	no	—
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	C₂²×D₂₅₁	1	1	yes	characteristic
266	C₄⋊C₄	C₂²×D₂₅₁	1	1	yes	characteristic
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	2	1	no	—
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	2	1	no	—
272	C₄²	—	2	1	no	—
273	C₂³	—	2	2	no	—
274	C₂³	—	2	2	no	—
275	C₂³	—	2	1	no	—
276	C₂³	—	4	1	no	—
277	C₂³	—	1004	2	no	—
278	C₂³	—	1004	2	no	—
279	C₂³	—	502	4	no	—
280	C₂³	—	502	2	no	—
281	C₂³	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
282	C₂³	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
283	C₂³	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
284	Q₈	—	2	2	no	—
285	Q₈	—	502	4	no	—
286	Q₈	—	502	2	no	—
287	Q₈	—	251	4	no	—
288	D₄	—	2	2	no	—
289	D₄	—	4	2	no	—
290	D₄	—	4	2	no	—
291	D₄	—	4	2	no	—
292	D₄	—	502	4	no	—
293	D₄	—	502	8	no	—
294	D₄	—	502	8	no	—
295	D₄	—	502	2	no	—
296	D₄	—	251	4	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	502	8	no	—
307	C₂×C₄	—	502	4	no	—
308	C₂×C₄	—	502	2	no	—
309	C₂×C₄	—	502	8	no	—
310	C₂×C₄	D₄×D₂₅₁	1	4	yes	—
311	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
312	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
313	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
314	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
315	C₈	—	502	2	no	—
316	C₈	—	502	2	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	4	4	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	1004	8	no	—
329	C₂²	—	1004	2	no	—
330	C₂²	—	1004	2	no	—
331	C₂²	—	502	4	no	—
332	C₂²	#22548683	1	2	yes	—
333	C₂²	#22548735	1	2	yes	complemented
334	C₂²	C₂²×C₂²⋊D₂₅₁	1	1	yes	characteristic
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	502	4	no	—
341	C₄	#22548683	1	2	yes	—
342	C₄	C₂×D₄×D₂₅₁	1	2	yes	—
343	C₂	—	2	2	no	—
344	C₂	—	2	2	no	—
345	C₂	—	4	1	no	—
346	C₂	—	8	1	no	—
347	C₂	—	1004	4	no	—
348	C₂	C₂×#22548683	1	1	yes	characteristic
349	C₂	C₂×#22548735	1	1	yes	characteristic
350	C₂	#30632485	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₅₁
commutator, cosocle	C₂×C₄×C₂₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	#525×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
hypercenter	#525	D₂₅₁
hypocenter	C₂₅₁	#3002
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #525

Complements

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

N	H ≅ G/N	type
1. C₂²×#16860810	C₂	semidirect
2. C₂×#22548735	C₂	semidirect
3. #30631847	C₂	semidirect
4. #30631842	C₂	semidirect
5. #30631837	C₂	semidirect
6. #30631830	C₂	semidirect
7. #30631825	C₂	semidirect
8. #30631820	C₂	semidirect
9. #30631681	C₂	semidirect
10. C₂×C₂²⋊(C₂₅₁⋊C₈)	C₂	semidirect
11. #525×C₂₅₁	C₂	semidirect
22. C₂×#16860800	C₂²	semidirect
23. C₂×#16860798	C₂²	semidirect
24. C₂×#16860797	C₂²	semidirect
25. C₂×#16860795	C₂²	semidirect
26. C₂²×C₂₅₁⋊C₈	C₂²	semidirect
27. #22548638	C₂²	semidirect
28. #22548638	C₂²	semidirect
29. #22548635	C₂²	semidirect
30. #22548635	C₂²	semidirect
33. C₂²⋊(C₂₅₁⋊C₈)	C₂²	semidirect
34. C₂×#41×C₂₅₁	C₂²	semidirect
38. C₂²⋊Q₈×C₂₅₁	C₂²	semidirect
40. #121×C₂₅₁	C₂²	semidirect
44. #117×C₂₅₁	C₂²	semidirect
51. #16860810	D₄	semidirect
153. C₂²×C₂₅₁	#135	semidirect
175. C₂₅₁	#3002	semidirect
179. #525	D₂₅₁	semidirect
193. C₂×#41	C₂×D₂₅₁	semidirect
207. C₂²⋊Q₈	C₂×D₂₅₁	semidirect
209. #121	C₂×D₂₅₁	semidirect
213. #117	C₂×D₂₅₁	semidirect
333. C₂²	#22548735	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#16860810	2
C₂×#22548735	2
#30631847	2
#30631842	2
#30631837	2
#30631830	2
#30631825	2
#30631820	2
#30631681	2
C₂×C₂²⋊(C₂₅₁⋊C₈)	2
#525×C₂₅₁	2
#3002	251