Subgroups — #29836814

Order 15616.

646 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21458968	C₂	1	1	yes	maximal, characteristic, complemented
2	#21458960	C₂	1	1	yes	maximal, characteristic, complemented
3	#21457608	C₂	1	2	yes	maximal, complemented
4	#21448081	C₂	1	1	yes	maximal, characteristic, complemented
5	#21447940	C₂	1	1	yes	maximal, characteristic, complemented
6	#21447891	C₂	1	2	yes	maximal, complemented
7	#21447788	C₂	1	2	yes	maximal, complemented
8	#21447786	C₂	1	2	yes	maximal, complemented
9	#21447699	C₂	1	1	yes	maximal, characteristic, complemented
10	#21447654	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#2222	C₂	1	1	yes	maximal, characteristic, complemented
12	#16840575	C₂²	1	2	yes	complemented
13	#16840571	—	2	2	no	—
14	#16840567	C₂²	1	2	yes	—
15	#16840204	—	2	2	no	—
16	#16840203	C₂²	1	2	yes	complemented
17	#16840200	—	2	2	no	—
18	#16840200	C₂²	1	2	yes	—
19	#16840195	—	2	1	no	—
20	#16840194	—	2	1	no	—
21	#16840194	C₂²	1	1	yes	characteristic
22	Q₃₂×D₆₁	—	2	1	no	—
23	C₂×#14519720	C₂²	1	2	yes	complemented
24	C₂×D₉₇₆	—	2	1	no	—
25	C₂×D₉₇₆	C₂²	1	1	yes	characteristic
26	#16840178	C₂²	1	1	yes	characteristic, complemented
27	#16840147	C₂²	1	2	yes	complemented
28	#16840033	C₂²	1	2	yes	complemented
29	#16840028	—	2	2	no	—
30	#16839938	—	2	2	no	—
31	#16839937	—	2	2	no	—
32	#16839878	C₂²	1	2	yes	complemented
33	#16839877	C₂²	1	1	yes	characteristic
34	#16839582	C₂²	1	1	yes	characteristic, complemented
35	#16839574	C₂²	1	2	yes	complemented
36	#16839568	C₂²	1	2	yes	complemented
37	#16839547	C₂²	1	1	yes	characteristic, complemented
38	#16839546	C₂²	1	1	yes	characteristic
39	C₁₆×Dic₆₁	C₂²	1	1	yes	characteristic, complemented
40	C₂×Q₃₂×C₆₁	C₂²	1	1	yes	characteristic, complemented
41	C₆₁×#504	C₂²	1	1	yes	characteristic
42	C₆₁×#473	C₂²	1	2	yes	—
43	C₆₁×#368	C₂²	1	1	yes	characteristic, complemented
44	C₆₁×#362	C₂²	1	2	yes	complemented
45	#14519825	—	2	2	no	—
46	#14519823	—	2	2	no	—
47	#14519809	—	4	2	no	—
48	#14519809	—	4	2	no	—
49	#14519808	—	2	2	no	—
50	#14519805	—	2	2	no	—
51	#14519805	—	2	2	no	—
52	#14519805	—	4	2	no	—
53	#14519804	—	4	2	no	—
54	#14519800	—	2	2	no	—
55	#14519800	—	4	2	no	—
56	#14519799	—	2	2	no	—
57	#14519799	C₂³	1	2	yes	—
58	#14519797	D₄	1	1	yes	characteristic
59	Q₁₆×D₆₁	—	2	2	no	—
60	C₂×#11758201	—	2	2	no	—
61	SD₁₆×D₆₁	—	4	2	no	—
62	C₂×D₄₈₈	—	2	2	no	—
63	C₂×D₄₈₈	C₂³	1	2	yes	—
64	#14519785	—	2	2	no	—
65	#14519783	C₂³	1	1	yes	characteristic
66	#14519766	—	2	2	no	—
67	Dic₄₈₈	D₄	1	1	yes	characteristic
68	#14519723	—	2	2	no	—
69	#14519722	—	2	2	no	—
70	#14519722	D₄	1	2	yes	—
71	#14519720	—	2	2	no	—
72	#14519720	—	4	2	no	—
73	D₉₇₆	—	2	2	no	—
74	D₉₇₆	—	2	1	no	—
75	D₉₇₆	—	2	1	no	—
76	D₉₇₆	D₄	1	1	yes	characteristic
77	#14519717	C₂³	1	1	yes	characteristic
78	#14519716	—	2	1	no	—
79	C₂×#11758185	C₂³	1	1	yes	characteristic
80	C₁₆×D₆₁	—	2	1	no	—
81	#14519712	D₄	1	1	yes	characteristic
82	#14519704	D₄	1	1	yes	characteristic
83	#14519703	—	2	2	no	—
84	#14519703	D₄	1	1	yes	characteristic
85	#14519693	—	2	2	no	—
86	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
87	#137×C₆₁	—	2	2	no	—
88	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
89	#131×C₆₁	C₂³	1	1	yes	characteristic
90	Q₃₂×C₆₁	—	2	1	no	—
91	Q₃₂×C₆₁	—	2	1	no	—
92	M₃₂×C₆₁	—	2	1	no	—
93	M₃₂×C₆₁	C₂³	1	1	yes	characteristic
94	C₂×C₁₆×C₆₁	—	2	1	no	—
95	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
96	#108×C₆₁	C₂³	1	2	yes	—
97	#101×C₆₁	—	2	2	no	—
98	#41×C₆₁	C₂×D₄	1	1	yes	characteristic
99	C₂×Q₈×C₆₁	—	2	2	no	—
100	#11758226	—	4	2	no	—
101	#11758226	—	4	2	no	—
102	#11758226	—	4	2	no	—
103	Q₈×D₆₁	—	4	2	no	—
104	Q₈×D₆₁	—	4	2	no	—
105	D₄×D₆₁	—	4	2	no	—
106	D₄×D₆₁	—	8	2	no	—
107	#11758222	—	2	2	no	—
108	#11758222	—	4	2	no	—
109	#11758222	—	8	2	no	—
110	#11758222	C₂×D₄	1	1	yes	characteristic
111	C₂×D₂₄₄	—	2	2	no	—
112	C₂×D₂₄₄	—	4	2	no	—
113	Q₁₆×C₆₁	—	2	2	no	—
114	Q₁₆×C₆₁	—	2	2	no	—
115	Q₁₆×C₆₁	—	4	2	no	—
116	SD₁₆×C₆₁	—	4	2	no	—
117	M₁₆×C₆₁	—	2	1	no	—
118	M₁₆×C₆₁	—	2	2	no	—
119	M₁₆×C₆₁	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₆₁	—	2	1	no	—
121	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
122	#11758202	—	4	2	no	—
123	#11758202	D₈	1	1	yes	characteristic
124	#11758201	—	4	2	no	—
125	#11758201	—	4	2	no	—
126	#11758201	—	4	2	no	—
127	#11758201	D₈	1	1	yes	characteristic
128	#11758200	D₈	1	1	yes	characteristic
129	#11758199	D₈	1	1	yes	characteristic
130	#11758198	—	8	2	no	—
131	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
132	#11758194	C₂×D₄	1	1	yes	characteristic
133	C₂×#92418	C₂×D₄	1	1	yes	characteristic
134	Dic₂₄₄	C₂×D₄	1	2	yes	—
135	D₄₈₈	—	2	2	no	—
136	D₄₈₈	—	2	2	no	—
137	D₄₈₈	—	4	2	no	—
138	D₄₈₈	—	4	2	no	—
139	D₄₈₈	C₂×D₄	1	2	yes	—
140	#11758190	—	2	2	no	—
141	#11758190	—	4	2	no	—
142	#11758190	—	4	2	no	—
143	#11758189	—	2	1	no	—
144	#11758189	—	4	2	no	—
145	C₈×D₆₁	—	2	1	no	—
146	C₈×D₆₁	—	4	2	no	—
147	C₁₆×C₆₁	—	2	1	no	—
148	C₁₆×C₆₁	C₂×D₄	1	1	yes	characteristic
149	C₁₆×C₆₁	C₂×D₄	1	1	yes	characteristic
150	#11758185	—	2	1	no	—
151	#11758185	—	2	1	no	—
152	C₂²×D₆₁	—	4	2	no	—
153	C₂²×D₆₁	—	8	2	no	—
154	Q₈×C₆₁	—	4	2	no	—
155	Q₈×C₆₁	—	4	2	no	—
156	Q₈×C₆₁	C₂×D₈	1	1	yes	characteristic
157	D₄×C₆₁	—	2	1	no	—
158	D₄×C₆₁	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₆₁	—	2	1	no	—
160	C₂×C₄×C₆₁	—	4	2	no	—
161	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
162	C₂²⋊D₆₁	—	2	1	no	—
163	C₂²⋊D₆₁	—	4	2	no	—
164	C₂²⋊D₆₁	—	8	2	no	—
165	C₂×Dic₆₁	—	2	1	no	—
166	D₂₄₄	—	2	2	no	—
167	D₂₄₄	—	4	2	no	—
168	D₂₄₄	—	4	2	no	—
169	D₂₄₄	—	8	2	no	—
170	D₂₄₄	—	8	2	no	—
171	D₂₄₄	C₂×D₈	1	1	yes	characteristic
172	C₄×D₆₁	—	2	1	no	—
173	C₄×D₆₁	—	4	2	no	—
174	C₄×D₆₁	—	8	2	no	—
175	C₄×D₆₁	—	8	2	no	—
176	C₄×D₆₁	—	8	2	no	—
177	Dic₁₂₂	—	2	2	no	—
178	Dic₁₂₂	—	8	2	no	—
179	Dic₁₂₂	C₂×D₈	1	1	yes	characteristic
180	C₈×C₆₁	—	2	1	no	—
181	C₈×C₆₁	—	4	2	no	—
182	C₈×C₆₁	C₂²×D₄	1	1	yes	characteristic
183	C₈×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
184	#92418	—	2	1	no	—
185	#92418	C₂×D₈	1	1	yes	characteristic
186	#92418	C₂×D₈	1	1	yes	characteristic
187	#33859	—	61	1	no	maximal, Sylow
188	C₂²×C₆₁	—	2	1	no	—
189	C₂²×C₆₁	#576	1	1	yes	characteristic
190	C₂×D₆₁	—	2	1	no	—
191	C₂×D₆₁	—	4	2	no	—
192	C₂×D₆₁	—	8	2	no	—
193	C₂×D₆₁	—	16	2	no	—
194	C₂×D₆₁	—	32	2	no	—
195	C₄×C₆₁	—	2	1	no	—
196	C₄×C₆₁	—	8	2	no	—
197	C₄×C₆₁	D₄²	1	1	yes	characteristic
198	C₄×C₆₁	C₂²×D₈	1	1	yes	characteristic
199	Dic₆₁	—	2	1	no	—
200	Dic₆₁	—	4	1	no	—
201	Dic₆₁	—	4	2	no	—
202	#3417	—	61	1	no	—
203	#3415	—	61	1	no	—
204	#3290	—	61	2	no	—
205	#2249	—	61	1	no	—
206	#2222	D₆₁	1	1	yes	characteristic, complemented
207	#2221	—	61	1	no	—
208	#2212	—	61	2	no	—
209	#2194	—	61	2	no	—
210	#2193	—	61	2	no	—
211	#2178	—	61	1	no	—
212	#2170	—	61	1	no	—
213	C₂×C₆₁	—	2	1	no	—
214	C₂×C₆₁	—	4	1	no	—
215	C₂×C₆₁	D₄×D₈	1	1	yes	characteristic
216	D₆₁	—	4	1	no	—
217	D₆₁	—	8	2	no	—
218	D₆₁	—	16	2	no	—
219	#578	—	61	2	no	—
220	#577	—	122	2	no	—
221	#576	—	61	2	no	—
222	#510	—	61	2	no	—
223	#510	—	122	2	no	—
224	#509	—	61	2	no	—
225	#509	—	122	2	no	—
226	#508	—	61	1	no	—
227	#508	—	122	1	no	—
228	#508	—	122	1	no	—
229	C₂×Q₃₂	—	122	1	no	—
230	C₂×Q₃₂	C₂×D₆₁	1	1	yes	characteristic, complemented
231	C₂×SD₃₂	—	61	2	no	—
232	C₂×D₁₆	—	61	1	no	—
233	C₂×D₁₆	—	122	1	no	—
234	#504	—	61	1	no	—
235	#504	C₂×D₆₁	1	1	yes	characteristic
236	#495	—	61	2	no	—
237	#473	C₂×D₆₁	1	2	yes	—
238	#472	—	61	2	no	—
239	#471	—	122	2	no	—
240	#455	—	122	2	no	—
241	#454	—	122	2	no	—
242	#443	—	61	1	no	—
243	#443	—	61	2	no	—
244	#368	C₂×D₆₁	1	1	yes	characteristic, complemented
245	#364	—	61	1	no	—
246	#362	C₂×D₆₁	1	2	yes	complemented
247	#361	—	61	2	no	—
248	#359	—	61	2	no	—
249	#350	—	61	1	no	—
250	#350	—	61	1	no	—
251	C₄×C₁₆	—	61	1	no	—
252	C₆₁	#33859	1	1	yes	characteristic, complemented, Sylow
253	2₋¹⁺⁴	—	122	2	no	—
254	H(5,𝔽₂)	—	122	2	no	—
255	#137	—	2	2	no	—
256	#137	—	122	2	no	—
257	#137	—	244	2	no	—
258	#137	—	244	2	no	—
259	AGL(1,ℤ/8)	—	122	2	no	—
260	AGL(1,ℤ/8)	—	122	2	no	—
261	AGL(1,ℤ/8)	—	244	2	no	—
262	AGL(1,ℤ/8)	—	244	2	no	—
263	#135	—	61	1	no	—
264	#135	—	61	2	no	—
265	#135	—	122	2	no	—
266	#135	—	122	2	no	—
267	#135	—	244	2	no	—
268	C₂×Q₁₆	—	122	2	no	—
269	C₂×Q₁₆	C₂²×D₆₁	1	2	yes	—
270	C₂×SD₁₆	—	122	2	no	—
271	C₂×SD₁₆	—	244	2	no	—
272	C₂×D₈	—	61	2	no	—
273	C₂×D₈	—	122	2	no	—
274	#131	—	61	1	no	—
275	#131	—	122	2	no	—
276	#131	C₂²×D₆₁	1	1	yes	characteristic
277	#124	—	122	2	no	—
278	Q₃₂	—	2	1	no	—
279	Q₃₂	—	2	1	no	—
280	Q₃₂	—	61	1	no	—
281	Q₃₂	—	122	2	no	—
282	SD₃₂	—	61	2	no	—
283	SD₃₂	—	122	2	no	—
284	SD₃₂	—	122	2	no	—
285	SD₃₂	—	244	2	no	—
286	D₁₆	—	61	1	no	—
287	D₁₆	—	122	2	no	—
288	D₁₆	—	122	1	no	—
289	D₁₆	—	122	1	no	—
290	M₃₂	—	2	1	no	—
291	M₃₂	—	61	1	no	—
292	M₃₂	—	122	1	no	—
293	M₃₂	C₂²×D₆₁	1	1	yes	characteristic
294	C₂×C₁₆	—	2	1	no	—
295	C₂×C₁₆	—	61	1	no	—
296	C₂×C₁₆	—	122	1	no	—
297	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
298	#108	—	61	1	no	—
299	#108	C₂²×D₆₁	1	2	yes	—
300	#104	—	61	1	no	—
301	#104	—	61	1	no	—
302	#104	—	122	2	no	—
303	#101	—	2	2	no	—
304	#100	—	122	2	no	—
305	C₄×C₈	—	61	1	no	—
306	#41	—	61	1	no	—
307	#41	—	122	2	no	—
308	#41	—	488	2	no	—
309	#41	—	244	2	no	—
310	#41	—	244	2	no	—
311	#41	—	244	2	no	—
312	#41	—	244	2	no	—
313	#41	D₄×D₆₁	1	1	yes	characteristic
314	C₂×Q₈	—	2	2	no	—
315	C₂×Q₈	—	244	2	no	—
316	C₂×Q₈	—	244	2	no	—
317	C₂×D₄	—	122	2	no	—
318	C₂×D₄	—	488	2	no	—
319	C₂×D₄	—	244	2	no	—
320	C₂×D₄	—	244	2	no	—
321	Q₁₆	—	2	2	no	—
322	Q₁₆	—	2	2	no	—
323	Q₁₆	—	4	2	no	—
324	Q₁₆	—	61	1	no	—
325	Q₁₆	—	61	2	no	—
326	Q₁₆	—	244	2	no	—
327	SD₁₆	—	4	2	no	—
328	SD₁₆	—	61	1	no	—
329	SD₁₆	—	61	1	no	—
330	SD₁₆	—	122	2	no	—
331	SD₁₆	—	244	2	no	—
332	SD₁₆	—	244	2	no	—
333	SD₁₆	—	244	2	no	—
334	SD₁₆	—	244	2	no	—
335	SD₁₆	—	244	2	no	—
336	D₈	—	61	1	no	—
337	D₈	—	61	2	no	—
338	D₈	—	122	2	no	—
339	D₈	—	122	2	no	—
340	D₈	—	244	2	no	—
341	D₈	—	244	2	no	—
342	M₁₆	—	2	1	no	—
343	M₁₆	—	2	2	no	—
344	M₁₆	—	61	1	no	—
345	M₁₆	—	122	1	no	—
346	M₁₆	—	244	2	no	—
347	M₁₆	D₄×D₆₁	1	1	yes	characteristic
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	61	1	no	—
350	C₂×C₈	—	122	1	no	—
351	C₂×C₈	—	244	2	no	—
352	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
353	C₂²⋊C₄	—	488	2	no	—
354	C₄²	—	61	1	no	—
355	C₁₆	—	2	1	no	—
356	C₁₆	—	122	1	no	—
357	C₁₆	—	122	1	no	—
358	C₁₆	D₄×D₆₁	1	1	yes	characteristic
359	C₁₆	D₄×D₆₁	1	1	yes	characteristic
360	C₂³	—	488	2	no	—
361	C₂³	—	244	2	no	—
362	Q₈	—	4	2	no	—
363	Q₈	—	4	2	no	—
364	Q₈	—	61	1	no	—
365	Q₈	—	122	2	no	—
366	Q₈	—	488	2	no	—
367	Q₈	D₈×D₆₁	1	1	yes	characteristic
368	D₄	—	2	1	no	—
369	D₄	—	61	1	no	—
370	D₄	—	122	1	no	—
371	D₄	—	122	2	no	—
372	D₄	—	488	2	no	—
373	D₄	—	488	2	no	—
374	D₄	—	488	2	no	—
375	D₄	—	244	2	no	—
376	D₄	—	244	2	no	—
377	D₄	—	244	2	no	—
378	D₄	D₈×D₆₁	1	1	yes	characteristic
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	122	1	no	—
382	C₂×C₄	—	122	1	no	—
383	C₂×C₄	—	488	2	no	—
384	C₂×C₄	—	488	2	no	—
385	C₂×C₄	—	488	2	no	—
386	C₂×C₄	—	244	2	no	—
387	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
388	C₈	—	2	1	no	—
389	C₈	—	4	2	no	—
390	C₈	—	61	1	no	—
391	C₈	—	61	1	no	—
392	C₈	—	122	1	no	—
393	C₈	C₂×D₄×D₆₁	1	1	yes	characteristic
394	C₈	#14519823	1	1	yes	characteristic
395	C₂²	—	2	1	no	—
396	C₂²	—	122	1	no	—
397	C₂²	—	1952	2	no	—
398	C₂²	—	976	2	no	—
399	C₂²	—	488	2	no	—
400	C₂²	—	244	2	no	—
401	C₂²	#16840564	1	1	yes	characteristic
402	C₄	—	2	1	no	—
403	C₄	—	8	2	no	—
404	C₄	—	122	1	no	—
405	C₄	—	244	1	no	—
406	C₄	—	244	2	no	—
407	C₄	#16840396	1	1	yes	characteristic
408	C₄	C₂×D₈×D₆₁	1	1	yes	characteristic
409	C₂	—	2	1	no	—
410	C₂	—	4	1	no	—
411	C₂	—	976	2	no	—
412	C₂	—	488	2	no	—
413	C₂	—	244	1	no	—
414	C₂	#21457493	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#21457493
commutator, cosocle	C₂×C₈×C₆₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₆₁×#2222	C₂
socle	C₂×C₆₁	D₄×D₈
hypercenter	#2222	D₆₁
hypocenter	C₆₁	#33859
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #2222

Complements

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

N	H ≅ G/N	type
1. #21458968	C₂	semidirect
2. #21458960	C₂	semidirect
3. #21457608	C₂	semidirect
4. #21448081	C₂	semidirect
5. #21447940	C₂	semidirect
6. #21447891	C₂	semidirect
7. #21447788	C₂	semidirect
8. #21447786	C₂	semidirect
9. #21447699	C₂	semidirect
10. #21447654	C₂	semidirect
11. C₆₁×#2222	C₂	semidirect
12. #16840575	C₂²	semidirect
16. #16840203	C₂²	semidirect
23. C₂×#14519720	C₂²	semidirect
26. #16840178	C₂²	semidirect
27. #16840147	C₂²	semidirect
28. #16840033	C₂²	semidirect
32. #16839878	C₂²	semidirect
34. #16839582	C₂²	semidirect
35. #16839574	C₂²	semidirect
36. #16839568	C₂²	semidirect
37. #16839547	C₂²	semidirect
39. C₁₆×Dic₆₁	C₂²	semidirect
40. C₂×Q₃₂×C₆₁	C₂²	semidirect
43. C₆₁×#368	C₂²	semidirect
44. C₆₁×#362	C₂²	semidirect
206. #2222	D₆₁	semidirect
230. C₂×Q₃₂	C₂×D₆₁	semidirect
244. #368	C₂×D₆₁	semidirect
246. #362	C₂×D₆₁	semidirect
252. C₆₁	#33859	semidirect

Maximal subgroups

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

subgroup	index
#21458968	2
#21458960	2
#21457608	2
#21448081	2
#21447940	2
#21447891	2
#21447788	2
#21447786	2
#21447699	2
#21447654	2
C₆₁×#2222	2
#33859	61