Subgroups — #29725134

Order 15616.

656 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21454671	C₂	1	1	yes	maximal, characteristic, complemented
2	#21447279	C₂	1	2	yes	maximal, complemented
3	#21447089	C₂	1	2	yes	maximal, complemented
4	#21446730	C₂	1	1	yes	maximal, characteristic, complemented
5	#21446607	C₂	1	1	yes	maximal, characteristic, complemented
6	#21446308	C₂	1	2	yes	maximal, complemented
7	#21445944	C₂	1	1	yes	maximal, characteristic, complemented
8	#21445928	C₂	1	2	yes	maximal, complemented
9	#21445873	C₂	1	1	yes	maximal, characteristic, complemented
10	#21445619	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₆₁×#472	C₂	1	1	yes	maximal, characteristic
12	#16840237	C₂²	1	2	yes	—
13	#16839874	—	2	2	no	—
14	D₈×Dic₆₁	—	2	2	no	—
15	#16839810	C₂²	1	2	yes	—
16	C₂×#14519713	C₂²	1	1	yes	characteristic
17	#16839800	C₂²	1	1	yes	characteristic
18	#16839779	C₂²	1	2	yes	—
19	#16839767	C₂²	1	2	yes	—
20	#16839767	C₂²	1	2	yes	—
21	#16839746	C₂²	1	1	yes	characteristic
22	C₂×#14519694	C₂²	1	2	yes	—
23	#16839736	C₂²	1	1	yes	characteristic
24	#16839717	C₂²	1	1	yes	characteristic
25	#16839717	C₂²	1	1	yes	characteristic
26	#16839534	C₂²	1	2	yes	—
27	#16839534	C₂²	1	2	yes	—
28	#16839533	C₂²	1	1	yes	characteristic
29	#16839533	C₂²	1	2	yes	—
30	#16839533	C₂²	1	1	yes	characteristic
31	#16839519	C₂²	1	2	yes	—
32	C₂×AGL(1,ℤ/8)×C₆₁	C₂²	1	2	yes	—
33	C₂²×D₈×C₆₁	C₂²	1	1	yes	characteristic
34	C₂×#131×C₆₁	C₂²	1	1	yes	characteristic
35	C₆₁×#472	—	2	2	no	—
36	C₆₁×#472	—	2	2	no	—
37	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
38	C₂×#100×C₆₁	C₂²	1	2	yes	—
39	C₂×#11758194	C₂³	1	1	yes	characteristic
40	C₂×#11758194	C₂³	1	1	yes	characteristic
41	C₂×#11758194	C₂³	1	2	yes	—
42	D₄×Dic₆₁	—	2	4	no	—
43	D₄×Dic₆₁	—	4	2	no	—
44	#14519736	C₂³	1	1	yes	characteristic
45	#14519736	C₂³	1	2	yes	—
46	#14519736	C₂³	1	1	yes	characteristic
47	C₂×C₂²⋊Dic₆₁	—	2	4	no	—
48	#14519713	—	2	1	no	—
49	#14519713	—	2	1	no	—
50	#14519712	—	4	2	no	—
51	#14519711	—	2	1	no	—
52	#14519709	—	2	1	no	—
53	#14519704	—	4	2	no	—
54	#14519698	—	2	4	no	—
55	#14519698	—	4	2	no	—
56	#14519696	—	2	1	no	—
57	#14519694	—	2	1	no	—
58	#14519694	—	4	2	no	—
59	#14519692	—	2	2	no	—
60	#14519687	—	2	1	no	—
61	#14519687	—	2	1	no	—
62	C₈×Dic₆₁	—	2	1	no	—
63	C₈×Dic₆₁	—	2	1	no	—
64	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
65	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
66	AGL(1,ℤ/8)×C₆₁	—	2	4	no	—
67	AGL(1,ℤ/8)×C₆₁	—	4	2	no	—
68	C₂×SD₁₆×C₆₁	—	2	2	no	—
69	C₂×D₈×C₆₁	—	2	2	no	—
70	C₂×D₈×C₆₁	—	2	1	no	—
71	C₂×D₈×C₆₁	—	2	1	no	—
72	C₂×D₈×C₆₁	—	2	1	no	—
73	C₂×D₈×C₆₁	—	2	1	no	—
74	C₂×D₈×C₆₁	D₄	1	2	yes	—
75	C₂×D₈×C₆₁	D₄	1	2	yes	—
76	#131×C₆₁	—	2	1	no	—
77	#131×C₆₁	—	2	1	no	—
78	#131×C₆₁	—	2	1	no	—
79	#131×C₆₁	—	2	1	no	—
80	C₂×M₁₆×C₆₁	—	2	1	no	—
81	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
82	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
83	C₂²×C₈×C₆₁	—	2	1	no	—
84	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
85	#108×C₆₁	—	2	1	no	—
86	#108×C₆₁	—	2	1	no	—
87	#100×C₆₁	—	2	4	no	—
88	C₂⁴×C₆₁	—	2	2	no	—
89	#41×C₆₁	—	2	1	no	—
90	#41×C₆₁	—	2	1	no	—
91	#41×C₆₁	—	2	1	no	—
92	#41×C₆₁	—	2	1	no	—
93	C₂×Q₈×C₆₁	#41	1	1	yes	characteristic
94	C₂×D₄×C₆₁	—	2	1	no	—
95	C₂×D₄×C₆₁	—	2	2	no	—
96	C₂×D₄×C₆₁	—	2	2	no	—
97	C₂×D₄×C₆₁	—	2	2	no	—
98	C₂×D₄×C₆₁	—	2	2	no	—
99	C₂×D₄×C₆₁	—	2	2	no	—
100	C₂×D₄×C₆₁	C₂×D₄	1	4	yes	—
101	C₂×D₄×C₆₁	#41	1	1	yes	characteristic
102	C₂²×C₄×C₆₁	—	2	1	no	—
103	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
104	C₂²×Dic₆₁	—	4	4	no	—
105	SD₁₆×C₆₁	—	4	4	no	—
106	D₈×C₆₁	—	2	2	no	—
107	D₈×C₆₁	—	2	2	no	—
108	D₈×C₆₁	—	2	2	no	—
109	D₈×C₆₁	—	2	2	no	—
110	D₈×C₆₁	—	4	4	no	—
111	M₁₆×C₆₁	—	2	1	no	—
112	M₁₆×C₆₁	—	2	2	no	—
113	M₁₆×C₆₁	—	2	1	no	—
114	M₁₆×C₆₁	—	2	2	no	—
115	M₁₆×C₆₁	—	4	1	no	—
116	M₁₆×C₆₁	—	4	1	no	—
117	C₂×C₈×C₆₁	—	2	1	no	—
118	C₂×C₈×C₆₁	—	2	2	no	—
119	C₂×C₈×C₆₁	—	4	1	no	—
120	C₂×C₈×C₆₁	—	4	1	no	—
121	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
124	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
125	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
126	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
127	C₂²⋊Dic₆₁	—	2	1	no	—
128	C₂²⋊Dic₆₁	—	2	2	no	—
129	C₂²⋊Dic₆₁	—	4	4	no	—
130	C₂²⋊Dic₆₁	#41	1	2	yes	—
131	C₄⋊Dic₆₁	—	2	1	no	—
132	C₄⋊Dic₆₁	—	2	2	no	—
133	C₄⋊Dic₆₁	—	2	1	no	—
134	C₄×Dic₆₁	—	2	1	no	—
135	C₄×Dic₆₁	—	2	1	no	—
136	C₄×Dic₆₁	—	2	2	no	—
137	#11758194	—	2	1	no	—
138	#11758194	—	2	1	no	—
139	#11758194	—	2	2	no	—
140	#11758194	—	2	1	no	—
141	#11758194	—	2	2	no	—
142	#11758194	—	2	1	no	—
143	C₂×#92418	—	2	1	no	—
144	C₂×#92418	—	2	1	no	—
145	C₂×#92418	—	2	2	no	—
146	C₂³×C₆₁	—	2	1	no	—
147	C₂³×C₆₁	—	2	4	no	—
148	C₂³×C₆₁	—	4	2	no	—
149	C₂³×C₆₁	—	4	2	no	—
150	C₂³×C₆₁	—	16	2	no	—
151	C₂³×C₆₁	C₂×#41	1	1	yes	characteristic
152	Q₈×C₆₁	—	2	1	no	—
153	Q₈×C₆₁	—	2	1	no	—
154	D₄×C₆₁	—	2	4	no	—
155	D₄×C₆₁	—	2	1	no	—
156	D₄×C₆₁	—	2	4	no	—
157	D₄×C₆₁	—	2	1	no	—
158	D₄×C₆₁	—	4	1	no	—
159	D₄×C₆₁	—	4	1	no	—
160	D₄×C₆₁	—	4	2	no	—
161	D₄×C₆₁	—	4	2	no	—
162	C₂×C₄×C₆₁	—	2	1	no	—
163	C₂×C₄×C₆₁	—	4	1	no	—
164	C₂×C₄×C₆₁	—	4	1	no	—
165	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
167	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
168	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
169	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
170	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
171	C₂×Dic₆₁	—	2	2	no	—
172	C₂×Dic₆₁	—	4	2	no	—
173	C₂×Dic₆₁	—	4	1	no	—
174	C₂×Dic₆₁	—	16	4	no	—
175	C₈×C₆₁	—	4	1	no	—
176	C₈×C₆₁	—	4	2	no	—
177	C₈×C₆₁	#123	1	1	yes	characteristic
178	C₈×C₆₁	#123	1	1	yes	characteristic
179	C₈×C₆₁	#123	1	1	yes	characteristic
180	C₈×C₆₁	#123	1	1	yes	characteristic
181	#92418	—	4	2	no	—
182	#92418	—	4	1	no	—
183	#92418	—	4	1	no	—
184	#24988	—	61	1	no	maximal, Sylow
185	C₂²×C₆₁	—	4	1	no	—
186	C₂²×C₆₁	—	4	1	no	—
187	C₂²×C₆₁	—	4	4	no	—
188	C₂²×C₆₁	—	8	4	no	—
189	C₂²×C₆₁	—	8	1	no	—
190	C₂²×C₆₁	—	16	2	no	—
191	C₂²×C₆₁	—	16	2	no	—
192	C₂²×C₆₁	#529	1	1	yes	characteristic
193	C₂²×C₆₁	#546	1	1	yes	characteristic
194	C₂²×C₆₁	#555	1	1	yes	characteristic
195	C₄×C₆₁	—	4	1	no	—
196	C₄×C₆₁	C₂×#123	1	1	yes	characteristic
197	C₄×C₆₁	#540	1	1	yes	characteristic
198	C₄×C₆₁	#546	1	1	yes	characteristic
199	C₄×C₆₁	#553	1	1	yes	characteristic
200	Dic₆₁	—	4	2	no	—
201	Dic₆₁	—	8	1	no	—
202	Dic₆₁	—	8	2	no	—
203	C₂×#472	D₆₁	1	1	yes	characteristic
204	#2948	—	61	1	no	—
205	#2081	—	61	2	no	—
206	#2048	—	61	2	no	—
207	#1978	—	61	1	no	—
208	#1953	—	61	1	no	—
209	#1904	—	61	2	no	—
210	#1833	—	61	1	no	—
211	#1829	—	61	2	no	—
212	#1814	—	61	1	no	—
213	#1757	—	61	1	no	—
214	C₂×C₆₁	—	2	1	no	—
215	C₂×C₆₁	—	2	1	no	—
216	C₂×C₆₁	—	2	1	no	—
217	C₂×C₆₁	—	8	1	no	—
218	C₂×C₆₁	—	8	4	no	—
219	C₂×C₆₁	#2420	1	1	yes	characteristic
220	C₂×AGL(1,ℤ/8)	C₂×D₆₁	1	2	yes	—
221	C₂²×D₈	C₂×D₆₁	1	1	yes	characteristic
222	C₂×#131	C₂×D₆₁	1	1	yes	characteristic
223	#518	—	61	2	no	—
224	#472	—	2	2	no	—
225	#472	—	2	2	no	—
226	#442	—	122	2	no	—
227	C₄×D₈	—	122	2	no	—
228	#430	—	61	2	no	—
229	C₂×#108	—	61	1	no	—
230	C₂×#108	C₂×D₆₁	1	1	yes	characteristic
231	#427	—	61	1	no	—
232	#421	—	61	2	no	—
233	#418	—	61	2	no	—
234	#418	—	61	2	no	—
235	#413	—	61	1	no	—
236	C₂×#100	—	61	2	no	—
237	C₂×#100	C₂×D₆₁	1	2	yes	—
238	#410	—	61	1	no	—
239	#405	—	61	1	no	—
240	#405	—	61	1	no	—
241	#344	—	61	2	no	—
242	#344	—	61	2	no	—
243	#344	—	61	1	no	—
244	#344	—	61	1	no	—
245	#344	—	61	2	no	—
246	H(3,ℤ/4)	—	61	2	no	—
247	C₆₁	#24988	1	1	yes	characteristic, complemented, Sylow
248	C₂×#41	C₂²×D₆₁	1	1	yes	characteristic
249	C₂²×D₄	C₂²×D₆₁	1	2	yes	—
250	AGL(1,ℤ/8)	—	2	4	no	—
251	AGL(1,ℤ/8)	—	4	2	no	—
252	C₂×SD₁₆	—	2	2	no	—
253	C₂×D₈	—	2	2	no	—
254	C₂×D₈	—	2	1	no	—
255	C₂×D₈	—	2	1	no	—
256	C₂×D₈	—	2	1	no	—
257	C₂×D₈	—	2	1	no	—
258	C₂×D₈	C₂²⋊D₆₁	1	2	yes	—
259	C₂×D₈	C₂²⋊D₆₁	1	2	yes	—
260	#131	—	2	1	no	—
261	#131	—	2	1	no	—
262	#131	—	2	1	no	—
263	#131	—	2	1	no	—
264	C₂×M₁₆	—	2	1	no	—
265	C₂×M₁₆	—	61	1	no	—
266	C₂×M₁₆	—	61	2	no	—
267	C₂×M₁₆	—	61	1	no	—
268	C₂×M₁₆	C₂²×D₆₁	1	1	yes	characteristic
269	C₂×M₁₆	C₂²×D₆₁	1	2	yes	—
270	C₂²×C₈	—	2	1	no	—
271	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
272	C₄×D₄	—	122	4	no	—
273	C₄×D₄	—	244	2	no	—
274	#117	—	61	1	no	—
275	#117	—	61	2	no	—
276	#117	—	61	1	no	—
277	C₂×C₂²⋊C₄	—	122	4	no	—
278	#108	—	2	1	no	—
279	#108	—	2	1	no	—
280	#108	—	122	1	no	—
281	#108	—	122	1	no	—
282	#108	—	244	2	no	—
283	#107	—	122	1	no	—
284	#106	—	122	1	no	—
285	#104	—	244	2	no	—
286	#102	—	122	4	no	—
287	#102	—	244	2	no	—
288	#101	—	122	1	no	—
289	#100	—	2	4	no	—
290	#100	—	122	1	no	—
291	#100	—	244	2	no	—
292	#99	—	122	2	no	—
293	#97	—	122	1	no	—
294	#97	—	122	1	no	—
295	C₄×C₈	—	122	1	no	—
296	C₄×C₈	—	122	1	no	—
297	C₂⁴	—	2	2	no	—
298	#41	—	2	1	no	—
299	#41	—	2	1	no	—
300	#41	—	2	1	no	—
301	#41	—	2	1	no	—
302	C₂×Q₈	#11758224	1	1	yes	characteristic
303	C₂×D₄	—	2	1	no	—
304	C₂×D₄	—	2	2	no	—
305	C₂×D₄	—	2	2	no	—
306	C₂×D₄	—	2	2	no	—
307	C₂×D₄	—	2	2	no	—
308	C₂×D₄	—	2	2	no	—
309	C₂×D₄	#11758224	1	1	yes	characteristic
310	C₂×D₄	C₂×C₂²⋊D₆₁	1	4	yes	—
311	C₂²×C₄	—	2	1	no	—
312	C₂²×C₄	—	244	4	no	—
313	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
314	SD₁₆	—	4	4	no	—
315	D₈	—	2	2	no	—
316	D₈	—	2	2	no	—
317	D₈	—	2	2	no	—
318	D₈	—	2	2	no	—
319	D₈	—	4	4	no	—
320	M₁₆	—	2	1	no	—
321	M₁₆	—	2	2	no	—
322	M₁₆	—	2	1	no	—
323	M₁₆	—	2	2	no	—
324	M₁₆	—	4	1	no	—
325	M₁₆	—	4	1	no	—
326	M₁₆	—	122	1	no	—
327	M₁₆	—	122	1	no	—
328	M₁₆	—	122	2	no	—
329	M₁₆	—	122	1	no	—
330	M₁₆	—	122	1	no	—
331	M₁₆	—	122	2	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	4	1	no	—
335	C₂×C₈	—	4	1	no	—
336	C₂×C₈	—	122	1	no	—
337	C₂×C₈	—	122	1	no	—
338	C₂×C₈	—	122	2	no	—
339	C₂×C₈	#11758224	1	1	yes	characteristic
340	C₂×C₈	#11758224	1	1	yes	characteristic
341	C₂×C₈	#11758224	1	1	yes	characteristic
342	C₂×C₈	#11758224	1	1	yes	characteristic
343	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
344	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
345	C₄⋊C₄	—	122	1	no	—
346	C₄⋊C₄	—	122	2	no	—
347	C₄⋊C₄	—	122	1	no	—
348	C₂²⋊C₄	—	61	2	no	—
349	C₂²⋊C₄	—	122	1	no	—
350	C₂²⋊C₄	—	122	2	no	—
351	C₂²⋊C₄	—	244	4	no	—
352	C₄²	—	122	1	no	—
353	C₄²	—	122	1	no	—
354	C₄²	—	122	2	no	—
355	C₂³	—	2	1	no	—
356	C₂³	—	2	4	no	—
357	C₂³	—	4	2	no	—
358	C₂³	—	4	2	no	—
359	C₂³	—	16	2	no	—
360	C₂³	#41×D₆₁	1	1	yes	characteristic
361	Q₈	—	2	1	no	—
362	Q₈	—	2	1	no	—
363	D₄	—	2	1	no	—
364	D₄	—	2	4	no	—
365	D₄	—	2	1	no	—
366	D₄	—	2	4	no	—
367	D₄	—	4	1	no	—
368	D₄	—	4	1	no	—
369	D₄	—	4	2	no	—
370	D₄	—	4	2	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	4	1	no	—
373	C₂×C₄	—	4	1	no	—
374	C₂×C₄	—	122	2	no	—
375	C₂×C₄	—	976	4	no	—
376	C₂×C₄	—	244	1	no	—
377	C₂×C₄	—	244	2	no	—
378	C₂×C₄	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
379	C₂×C₄	C₂×#11758224	1	1	yes	characteristic
380	C₂×C₄	C₂×#11758224	1	1	yes	characteristic
381	C₂×C₄	C₂×#11758224	1	1	yes	characteristic
382	C₂×C₄	#14519822	1	1	yes	characteristic
383	C₂×C₄	#14519823	1	1	yes	characteristic
384	C₈	—	4	1	no	—
385	C₈	—	4	2	no	—
386	C₈	—	244	1	no	—
387	C₈	—	244	2	no	—
388	C₈	—	244	1	no	—
389	C₈	#14519758	1	1	yes	characteristic
390	C₈	#14519758	1	1	yes	characteristic
391	C₈	#14519758	1	1	yes	characteristic
392	C₈	#14519758	1	1	yes	characteristic
393	C₂²	—	4	1	no	—
394	C₂²	—	4	1	no	—
395	C₂²	—	4	4	no	—
396	C₂²	—	8	4	no	—
397	C₂²	—	8	1	no	—
398	C₂²	—	16	2	no	—
399	C₂²	—	16	2	no	—
400	C₂²	#16840293	1	1	yes	characteristic
401	C₂²	#16840400	1	1	yes	characteristic
402	C₂²	#16840476	1	1	yes	characteristic
403	C₄	—	4	1	no	—
404	C₄	—	488	2	no	—
405	C₄	—	488	1	no	—
406	C₄	—	244	2	no	—
407	C₄	C₂×#14519758	1	1	yes	characteristic
408	C₄	#16840365	1	1	yes	characteristic
409	C₄	#16840406	1	1	yes	characteristic
410	C₄	#16840464	1	1	yes	characteristic
411	C₂	—	2	1	no	—
412	C₂	—	2	1	no	—
413	C₂	—	2	1	no	—
414	C₂	—	8	4	no	—
415	C₂	—	8	1	no	—
416	C₂	#21449307	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#21449307
commutator, cosocle	C₂²×C₄×C₆₁	C₂⁴
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting, prime radical	C₂×C₆₁×#472	C₂
socle	C₂×C₆₁	#2420
hypercenter	C₂×#472	D₆₁
hypocenter	C₆₁	#24988
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#472

Complements

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

N	H ≅ G/N	type
1. #21454671	C₂	semidirect
2. #21447279	C₂	semidirect
3. #21447089	C₂	semidirect
4. #21446730	C₂	semidirect
5. #21446607	C₂	semidirect
6. #21446308	C₂	semidirect
7. #21445944	C₂	semidirect
8. #21445928	C₂	semidirect
9. #21445873	C₂	semidirect
10. #21445619	C₂	semidirect
247. C₆₁	#24988	semidirect

Maximal subgroups

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

subgroup	index
#21454671	2
#21447279	2
#21447089	2
#21446730	2
#21446607	2
#21446308	2
#21445944	2
#21445928	2
#21445873	2
#21445619	2
C₂×C₆₁×#472	2
#24988	61