Subgroups — #29836811

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	#21458957	C₂	1	1	yes	maximal, characteristic, complemented
2	#21457605	C₂	1	2	yes	maximal, complemented
3	#21448081	C₂	1	1	yes	maximal, characteristic, complemented
4	#21447941	C₂	1	1	yes	maximal, characteristic, complemented
5	#21447939	C₂	1	1	yes	maximal, characteristic, complemented
6	#21447890	C₂	1	2	yes	maximal, complemented
7	#21447787	C₂	1	2	yes	maximal, complemented
8	#21447783	C₂	1	2	yes	maximal, complemented
9	#21447700	C₂	1	1	yes	maximal, characteristic, complemented
10	#21447654	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#3417	C₂	1	1	yes	maximal, characteristic, complemented
12	#16840570	—	2	2	no	—
13	#16840564	C₂²	1	2	yes	—
14	#16840199	—	2	2	no	—
15	#16840197	C₂²	1	2	yes	—
16	#16840195	—	2	1	no	—
17	C₂×#14519720	C₂²	1	2	yes	complemented
18	C₂×D₉₇₆	C₂²	1	1	yes	characteristic
19	D₁₆×D₆₁	—	2	1	no	—
20	#16840178	C₂²	1	1	yes	characteristic
21	#16840147	C₂²	1	2	yes	complemented
22	#16840034	C₂²	1	2	yes	—
23	#16840032	C₂²	1	2	yes	complemented
24	#16840027	—	2	2	no	—
25	#16839938	—	2	2	no	—
26	#16839935	—	2	2	no	—
27	#16839879	C₂²	1	2	yes	complemented
28	#16839877	C₂²	1	1	yes	characteristic
29	#16839592	C₂²	1	1	yes	characteristic, complemented
30	#16839582	C₂²	1	1	yes	characteristic, complemented
31	#16839578	C₂²	1	2	yes	complemented
32	#16839576	C₂²	1	2	yes	complemented
33	#16839569	C₂²	1	2	yes	complemented
34	#16839547	C₂²	1	1	yes	characteristic
35	#16839546	C₂²	1	1	yes	characteristic, complemented
36	C₁₆×Dic₆₁	C₂²	1	1	yes	characteristic, complemented
37	C₆₁×#578	C₂²	1	2	yes	complemented
38	C₆₁×#510	—	2	2	no	—
39	C₆₁×#510	C₂²	1	2	yes	complemented
40	C₆₁×#508	—	2	1	no	—
41	C₆₁×#508	C₂²	1	1	yes	characteristic
42	C₂×Q₃₂×C₆₁	—	2	1	no	—
43	C₂×Q₃₂×C₆₁	C₂²	1	1	yes	characteristic, complemented
44	C₆₁×#504	C₂²	1	1	yes	characteristic, complemented
45	#14519823	—	2	2	no	—
46	#14519808	—	2	2	no	—
47	#14519808	—	4	2	no	—
48	#14519804	—	4	2	no	—
49	#14519802	—	2	2	no	—
50	#14519802	—	2	2	no	—
51	#14519802	—	4	2	no	—
52	#14519800	—	2	2	no	—
53	#14519797	—	4	2	no	—
54	#14519797	D₄	1	1	yes	characteristic
55	C₂×#11758201	—	2	2	no	—
56	C₂×#11758201	—	4	2	no	—
57	C₂×D₄₈₈	C₂³	1	2	yes	—
58	D₈×D₆₁	—	2	2	no	—
59	#14519784	—	2	2	no	—
60	#14519783	C₂³	1	1	yes	characteristic
61	#14519766	—	2	2	no	—
62	#14519720	—	2	2	no	—
63	#14519720	—	4	2	no	—
64	D₉₇₆	—	2	1	no	—
65	D₉₇₆	—	2	1	no	—
66	#14519718	—	2	2	no	—
67	#14519717	C₂³	1	1	yes	characteristic
68	#14519716	—	2	1	no	—
69	C₂×#11758185	C₂³	1	1	yes	characteristic
70	C₁₆×D₆₁	—	2	1	no	—
71	#14519713	C₂³	1	2	yes	—
72	#14519712	D₄	1	1	yes	characteristic
73	#14519704	—	2	2	no	—
74	#14519704	D₄	1	1	yes	characteristic
75	#14519703	D₄	1	1	yes	characteristic
76	#14519696	—	2	2	no	—
77	#14519693	—	2	2	no	—
78	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
79	2₋¹⁺⁴×C₆₁	—	2	2	no	—
80	#137×C₆₁	—	2	2	no	—
81	#137×C₆₁	—	4	2	no	—
82	#135×C₆₁	—	2	2	no	—
83	#135×C₆₁	C₂³	1	2	yes	—
84	C₂×Q₁₆×C₆₁	—	2	2	no	—
85	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
86	#131×C₆₁	C₂³	1	1	yes	characteristic
87	Q₃₂×C₆₁	—	2	2	no	—
88	Q₃₂×C₆₁	—	2	1	no	—
89	Q₃₂×C₆₁	—	2	1	no	—
90	Q₃₂×C₆₁	D₄	1	1	yes	characteristic
91	SD₃₂×C₆₁	—	2	2	no	—
92	SD₃₂×C₆₁	D₄	1	2	yes	—
93	D₁₆×C₆₁	D₄	1	1	yes	characteristic
94	M₃₂×C₆₁	—	2	1	no	—
95	M₃₂×C₆₁	C₂³	1	1	yes	characteristic
96	C₂×C₁₆×C₆₁	—	2	1	no	—
97	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
98	#41×C₆₁	—	2	2	no	—
99	#41×C₆₁	—	4	2	no	—
100	#41×C₆₁	—	4	2	no	—
101	#41×C₆₁	—	8	2	no	—
102	#41×C₆₁	C₂×D₄	1	1	yes	characteristic
103	C₂×Q₈×C₆₁	—	2	2	no	—
104	C₂×Q₈×C₆₁	—	4	2	no	—
105	C₂×Q₈×C₆₁	—	4	2	no	—
106	#11758226	—	4	2	no	—
107	D₄×D₆₁	—	4	2	no	—
108	D₄×D₆₁	—	8	2	no	—
109	#11758222	—	4	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	Q₁₆×C₆₁	C₂×D₄	1	2	yes	—
117	SD₁₆×C₆₁	—	2	2	no	—
118	SD₁₆×C₆₁	—	4	2	no	—
119	D₈×C₆₁	C₂×D₄	1	2	yes	—
120	M₁₆×C₆₁	—	2	1	no	—
121	M₁₆×C₆₁	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₆₁	—	2	1	no	—
123	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
124	#11758202	—	4	2	no	—
125	#11758202	D₈	1	1	yes	characteristic
126	#11758201	—	4	2	no	—
127	#11758201	—	4	2	no	—
128	#11758201	—	4	2	no	—
129	#11758201	—	4	2	no	—
130	#11758201	—	4	2	no	—
131	#11758201	D₈	1	1	yes	characteristic
132	#11758200	D₈	1	1	yes	characteristic
133	#11758199	—	4	2	no	—
134	#11758199	—	4	2	no	—
135	#11758199	D₈	1	1	yes	characteristic
136	#11758198	—	8	2	no	—
137	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
138	#11758194	—	2	2	no	—
139	#11758194	—	4	2	no	—
140	#11758194	C₂×D₄	1	1	yes	characteristic
141	C₂×#92418	—	4	2	no	—
142	C₂×#92418	C₂×D₄	1	1	yes	characteristic
143	D₄₈₈	—	2	2	no	—
144	D₄₈₈	—	2	2	no	—
145	#11758189	—	2	1	no	—
146	C₈×D₆₁	—	2	1	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	—	2	2	no	—
155	Q₈×C₆₁	—	4	2	no	—
156	Q₈×C₆₁	—	4	2	no	—
157	Q₈×C₆₁	—	8	2	no	—
158	Q₈×C₆₁	C₂×D₈	1	1	yes	characteristic
159	D₄×C₆₁	—	2	1	no	—
160	D₄×C₆₁	—	2	2	no	—
161	D₄×C₆₁	—	4	2	no	—
162	D₄×C₆₁	—	8	2	no	—
163	D₄×C₆₁	C₂×D₈	1	1	yes	characteristic
164	C₂×C₄×C₆₁	—	2	1	no	—
165	C₂×C₄×C₆₁	—	4	2	no	—
166	C₂×C₄×C₆₁	—	4	2	no	—
167	C₂×C₄×C₆₁	—	8	2	no	—
168	C₂×C₄×C₆₁	—	8	2	no	—
169	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
170	C₂²⋊D₆₁	—	2	1	no	—
171	C₂²⋊D₆₁	—	8	2	no	—
172	C₂×Dic₆₁	—	2	1	no	—
173	D₂₄₄	—	4	2	no	—
174	D₂₄₄	—	4	2	no	—
175	D₂₄₄	—	8	2	no	—
176	D₂₄₄	C₂×D₈	1	1	yes	characteristic
177	C₄×D₆₁	—	2	1	no	—
178	C₄×D₆₁	—	8	2	no	—
179	Dic₁₂₂	C₂×D₈	1	1	yes	characteristic
180	C₈×C₆₁	—	2	1	no	—
181	C₈×C₆₁	C₂²×D₄	1	1	yes	characteristic
182	C₈×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
183	#92418	—	2	1	no	—
184	#92418	—	4	2	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	—	4	2	no	—
190	C₂²×C₆₁	#576	1	1	yes	characteristic
191	C₂×D₆₁	—	2	1	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	—	4	2	no	—
197	C₄×C₆₁	—	8	2	no	—
198	C₄×C₆₁	D₄²	1	1	yes	characteristic
199	C₄×C₆₁	C₂²×D₈	1	1	yes	characteristic
200	Dic₆₁	—	2	1	no	—
201	Dic₆₁	—	4	1	no	—
202	#3417	D₆₁	1	1	yes	characteristic, complemented
203	#3415	—	61	1	no	—
204	#3290	—	61	2	no	—
205	#2249	—	61	1	no	—
206	#2222	—	61	1	no	—
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	—	8	2	no	—
216	C₂×C₆₁	D₄×D₈	1	1	yes	characteristic
217	D₆₁	—	4	1	no	—
218	D₆₁	—	16	2	no	—
219	#578	C₂×D₆₁	1	2	yes	complemented
220	#577	—	122	2	no	—
221	#576	—	61	2	no	—
222	#510	—	2	2	no	—
223	#510	C₂×D₆₁	1	2	yes	complemented
224	#509	—	61	2	no	—
225	#509	—	122	2	no	—
226	#508	—	2	1	no	—
227	#508	—	122	1	no	—
228	#508	C₂×D₆₁	1	1	yes	characteristic
229	C₂×Q₃₂	—	2	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, complemented
236	#495	—	61	2	no	—
237	#473	—	61	2	no	—
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	—	61	1	no	—
245	#364	—	61	1	no	—
246	#362	—	61	2	no	—
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	—	2	2	no	—
254	H(5,𝔽₂)	—	122	2	no	—
255	#137	—	2	2	no	—
256	#137	—	4	2	no	—
257	#137	—	122	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	—	2	2	no	—
264	#135	—	61	1	no	—
265	#135	—	122	2	no	—
266	#135	—	244	2	no	—
267	#135	C₂²×D₆₁	1	2	yes	—
268	C₂×Q₁₆	—	2	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	2	no	—
279	Q₃₂	—	2	1	no	—
280	Q₃₂	—	2	1	no	—
281	Q₃₂	C₂²⋊D₆₁	1	1	yes	characteristic
282	SD₃₂	—	2	2	no	—
283	SD₃₂	—	122	2	no	—
284	SD₃₂	—	244	2	no	—
285	SD₃₂	C₂²⋊D₆₁	1	2	yes	—
286	D₁₆	—	122	2	no	—
287	D₁₆	—	122	1	no	—
288	D₁₆	—	122	1	no	—
289	D₁₆	C₂²⋊D₆₁	1	1	yes	characteristic
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	—	61	2	no	—
300	#104	—	61	1	no	—
301	#104	—	61	1	no	—
302	#104	—	122	2	no	—
303	#101	—	122	2	no	—
304	#100	—	122	2	no	—
305	C₄×C₈	—	61	1	no	—
306	#41	—	2	2	no	—
307	#41	—	4	2	no	—
308	#41	—	4	2	no	—
309	#41	—	8	2	no	—
310	#41	—	61	1	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	—	4	2	no	—
316	C₂×Q₈	—	4	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	—	244	2	no	—
326	Q₁₆	C₂×C₂²⋊D₆₁	1	2	yes	—
327	SD₁₆	—	2	2	no	—
328	SD₁₆	—	4	2	no	—
329	SD₁₆	—	61	1	no	—
330	SD₁₆	—	61	1	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	—	122	2	no	—
338	D₈	—	122	2	no	—
339	D₈	—	244	2	no	—
340	D₈	—	244	2	no	—
341	D₈	C₂×C₂²⋊D₆₁	1	2	yes	—
342	M₁₆	—	2	1	no	—
343	M₁₆	—	61	1	no	—
344	M₁₆	—	122	1	no	—
345	M₁₆	—	122	2	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	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
359	C₁₆	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
360	C₂³	—	488	2	no	—
361	C₂³	—	244	2	no	—
362	Q₈	—	2	2	no	—
363	Q₈	—	4	2	no	—
364	Q₈	—	4	2	no	—
365	Q₈	—	8	2	no	—
366	Q₈	—	61	1	no	—
367	Q₈	D₈×D₆₁	1	1	yes	characteristic
368	D₄	—	2	1	no	—
369	D₄	—	2	2	no	—
370	D₄	—	4	2	no	—
371	D₄	—	8	2	no	—
372	D₄	—	61	1	no	—
373	D₄	—	122	1	no	—
374	D₄	—	488	2	no	—
375	D₄	—	488	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	—	4	2	no	—
382	C₂×C₄	—	8	2	no	—
383	C₂×C₄	—	8	2	no	—
384	C₂×C₄	—	122	1	no	—
385	C₂×C₄	—	122	1	no	—
386	C₂×C₄	—	488	2	no	—
387	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
388	C₈	—	2	1	no	—
389	C₈	—	61	1	no	—
390	C₈	—	61	1	no	—
391	C₈	—	122	1	no	—
392	C₈	—	244	2	no	—
393	C₈	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
394	C₈	#14519822	1	1	yes	characteristic
395	C₂²	—	2	1	no	—
396	C₂²	—	4	2	no	—
397	C₂²	—	122	1	no	—
398	C₂²	—	1952	2	no	—
399	C₂²	—	976	2	no	—
400	C₂²	—	488	2	no	—
401	C₂²	#16840563	1	1	yes	characteristic
402	C₄	—	2	1	no	—
403	C₄	—	4	2	no	—
404	C₄	—	8	2	no	—
405	C₄	—	122	1	no	—
406	C₄	—	244	1	no	—
407	C₄	D₄×C₂²⋊D₆₁	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	—	8	2	no	—
412	C₂	—	976	2	no	—
413	C₂	—	244	1	no	—
414	C₂	D₈×C₂²⋊D₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×C₂²⋊D₆₁
commutator, cosocle	C₂×C₈×C₆₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₆₁×#3417	C₂
socle	C₂×C₆₁	D₄×D₈
hypercenter	#3417	D₆₁
hypocenter	C₆₁	#33859
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #3417

Complements

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

N	H ≅ G/N	type
1. #21458957	C₂	semidirect
2. #21457605	C₂	semidirect
3. #21448081	C₂	semidirect
4. #21447941	C₂	semidirect
5. #21447939	C₂	semidirect
6. #21447890	C₂	semidirect
7. #21447787	C₂	semidirect
8. #21447783	C₂	semidirect
9. #21447700	C₂	semidirect
10. #21447654	C₂	semidirect
11. C₆₁×#3417	C₂	semidirect
17. C₂×#14519720	C₂²	semidirect
21. #16840147	C₂²	semidirect
23. #16840032	C₂²	semidirect
27. #16839879	C₂²	semidirect
29. #16839592	C₂²	semidirect
30. #16839582	C₂²	semidirect
31. #16839578	C₂²	semidirect
32. #16839576	C₂²	semidirect
33. #16839569	C₂²	semidirect
35. #16839546	C₂²	semidirect
36. C₁₆×Dic₆₁	C₂²	semidirect
37. C₆₁×#578	C₂²	semidirect
39. C₆₁×#510	C₂²	semidirect
43. C₂×Q₃₂×C₆₁	C₂²	semidirect
44. C₆₁×#504	C₂²	semidirect
202. #3417	D₆₁	semidirect
219. #578	C₂×D₆₁	semidirect
223. #510	C₂×D₆₁	semidirect
230. C₂×Q₃₂	C₂×D₆₁	semidirect
235. #504	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
#21458957	2
#21457605	2
#21448081	2
#21447941	2
#21447939	2
#21447890	2
#21447787	2
#21447783	2
#21447700	2
#21447654	2
C₆₁×#3417	2
#33859	61