Subgroups — #29699712

Order 15616.

722 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14519799	C₂	1	1	yes	maximal, characteristic
2	#21454487	C₂	1	2	yes	maximal, complemented
3	#21446705	C₂	1	2	yes	maximal
4	#21446466	C₂	1	4	yes	maximal, complemented
5	#21445836	C₂	1	2	yes	maximal, complemented
6	#21445701	C₂	1	2	yes	maximal, complemented
7	C₈×C₄⋊Dic₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1848	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14519799	C₂²	1	1	yes	characteristic
10	C₄×#11758222	C₂²	1	2	yes	—
11	C₄×Dic₂₄₄	C₄	1	1	yes	characteristic, complemented
12	C₄×Dic₂₄₄	C₄	1	1	yes	characteristic, complemented
13	C₄×#11758190	C₄	1	2	yes	complemented
14	C₄×#11758190	C₄	1	2	yes	complemented
15	C₄×D₄₈₈	C₄	1	1	yes	characteristic, complemented
16	C₄×D₄₈₈	C₄	1	1	yes	characteristic, complemented
17	#16839822	—	2	4	no	—
18	#16839818	—	2	2	no	—
19	#16839814	—	2	2	no	—
20	#16839800	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊#92418	C₂²	1	1	yes	characteristic
22	C₂×#14519703	C₂²	1	4	yes	—
23	#16839759	C₂²	1	2	yes	—
24	#16839730	C₂²	1	4	yes	—
25	#16839720	C₂²	1	4	yes	—
26	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
27	C₄×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
28	#16839523	C₂²	1	4	yes	—
29	#16839516	C₂²	1	2	yes	—
30	C₂×#108×C₆₁	C₂²	1	2	yes	—
31	C₂×#108×C₆₁	C₂²	1	2	yes	—
32	C₆₁×#423	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
34	C₂×#11758222	C₂³	1	2	yes	—
35	#14519799	—	2	2	no	—
36	#14519799	—	2	2	no	—
37	C₂×Dic₂₄₄	C₂×C₄	1	1	yes	characteristic
38	C₂×#11758190	C₂×C₄	1	2	yes	—
39	C₂×D₄₈₈	C₂×C₄	1	1	yes	characteristic
40	C₂²×#92418	C₂³	1	2	yes	—
41	C₄×Dic₁₂₂	C₂×C₄	1	2	yes	—
42	C₄×Dic₁₂₂	C₂×C₄	1	2	yes	—
43	C₄×D₂₄₄	C₂×C₄	1	2	yes	—
44	C₄×D₂₄₄	C₂×C₄	1	2	yes	—
45	C₄×C₂²⋊D₆₁	—	2	2	no	—
46	C₄×C₂²⋊D₆₁	—	2	2	no	—
47	#14519736	C₂³	1	2	yes	—
48	#14519734	—	2	2	no	—
49	C₂×C₄⋊Dic₆₁	—	2	1	no	—
50	C₂×C₄×Dic₆₁	C₂³	1	2	yes	—
51	C₄²×D₆₁	—	2	2	no	—
52	#14519711	C₂×C₄	1	1	yes	characteristic
53	#14519711	C₂×C₄	1	1	yes	characteristic
54	#14519709	C₂×C₄	1	2	yes	—
55	C₄⋊#92418	D₄	1	1	yes	characteristic
56	C₄⋊#92418	D₄	1	1	yes	characteristic
57	C₄⋊#92418	D₄	1	2	yes	—
58	#14519703	—	4	4	no	—
59	#14519702	C₂×C₄	1	2	yes	—
60	#14519702	C₂×C₄	1	2	yes	—
61	#14519699	C₂×C₄	1	2	yes	—
62	#14519699	C₂×C₄	1	2	yes	—
63	#14519689	—	2	4	no	—
64	#14519689	—	2	4	no	—
65	#14519687	—	2	4	no	—
66	C₈×Dic₆₁	—	2	4	no	—
67	C₈×Dic₆₁	—	2	4	no	—
68	#14519684	—	2	1	no	—
69	C₂×M₁₆×C₆₁	C₂³	1	4	yes	—
70	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
73	#108×C₆₁	—	2	4	no	—
74	#108×C₆₁	—	4	2	no	—
75	C₄⋊C₈×C₆₁	—	2	2	no	—
76	C₄⋊C₈×C₆₁	—	2	2	no	—
77	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
78	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
79	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
80	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
82	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
83	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
84	C₂×C₂²⋊D₆₁	—	2	2	no	—
85	C₂²×Dic₆₁	—	2	2	no	—
86	#11758222	—	2	2	no	—
87	#11758222	—	2	2	no	—
88	#11758222	—	2	2	no	—
89	#11758222	—	2	2	no	—
90	C₂×D₂₄₄	C₂²×C₄	1	2	yes	—
91	C₂×C₄×D₆₁	—	2	2	no	—
92	C₂×C₄×D₆₁	—	2	2	no	—
93	C₂×C₄×D₆₁	—	2	2	no	—
94	C₂×Dic₁₂₂	C₂²×C₄	1	2	yes	—
95	M₁₆×C₆₁	—	2	4	no	—
96	M₁₆×C₆₁	—	2	4	no	—
97	C₂×C₈×C₆₁	—	2	1	no	—
98	C₂×C₈×C₆₁	—	2	1	no	—
99	C₂×C₈×C₆₁	—	2	4	no	—
100	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₆₁	#41	1	2	yes	—
105	C₂×C₈×C₆₁	#41	1	2	yes	—
106	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
110	C₂²⋊Dic₆₁	—	2	2	no	—
111	#11758198	—	2	2	no	—
112	#11758198	—	2	2	no	—
113	C₄⋊Dic₆₁	—	2	4	no	—
114	C₄⋊Dic₆₁	C₂²×C₄	1	2	yes	—
115	C₄⋊Dic₆₁	C₂²×C₄	1	2	yes	—
116	#11758196	—	2	2	no	—
117	#11758196	—	2	2	no	—
118	C₄×Dic₆₁	—	2	2	no	—
119	C₄×Dic₆₁	—	2	2	no	—
120	C₄×Dic₆₁	—	2	2	no	—
121	C₄×Dic₆₁	—	2	2	no	—
122	C₄×Dic₆₁	—	2	2	no	—
123	C₂×#92418	—	2	2	no	—
124	C₂×#92418	—	2	2	no	—
125	C₂×#92418	C₂×D₄	1	2	yes	—
126	C₂×#92418	C₂×D₄	1	2	yes	—
127	Dic₂₄₄	—	2	2	no	—
128	D₄₈₈	—	2	2	no	—
129	#11758190	—	2	4	no	—
130	C₂³×C₆₁	C₂×#41	1	1	yes	characteristic
131	C₂²×D₆₁	—	2	2	no	—
132	C₂×C₄×C₆₁	—	2	2	no	—
133	C₂×C₄×C₆₁	—	4	1	no	—
134	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
143	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
144	C₂²⋊D₆₁	—	4	2	no	—
145	C₂²⋊D₆₁	—	4	2	no	—
146	C₂×Dic₆₁	—	2	4	no	—
147	C₂×Dic₆₁	—	2	2	no	—
148	C₂×Dic₆₁	—	2	2	no	—
149	C₂×Dic₆₁	—	2	2	no	—
150	C₂×Dic₆₁	—	4	2	no	—
151	C₂×Dic₆₁	—	4	2	no	—
152	D₂₄₄	—	2	2	no	—
153	D₂₄₄	—	2	2	no	—
154	C₄×D₆₁	—	4	2	no	—
155	C₄×D₆₁	—	4	2	no	—
156	C₄×D₆₁	—	4	4	no	—
157	C₄×D₆₁	—	8	2	no	—
158	Dic₁₂₂	—	2	2	no	—
159	Dic₁₂₂	—	2	2	no	—
160	C₈×C₆₁	—	2	1	no	—
161	C₈×C₆₁	—	2	1	no	—
162	C₈×C₆₁	—	4	4	no	—
163	C₈×C₆₁	#117	1	2	yes	—
164	C₈×C₆₁	#117	1	2	yes	—
165	#92418	—	2	2	no	—
166	#92418	—	2	2	no	—
167	#22802	—	61	1	no	maximal, Sylow
168	C₂²×C₆₁	—	2	1	no	—
169	C₂²×C₆₁	—	2	1	no	—
170	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂²×C₆₁	#555	1	1	yes	characteristic
172	C₂²×C₆₁	#556	1	1	yes	characteristic
173	C₂×D₆₁	—	4	2	no	—
174	C₂×D₆₁	—	4	2	no	—
175	C₂×D₆₁	—	4	2	no	—
176	C₄×C₆₁	—	2	1	no	—
177	C₄×C₆₁	—	2	1	no	—
178	C₄×C₆₁	—	2	1	no	—
179	C₄×C₆₁	#443	1	2	yes	—
180	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
181	C₄×C₆₁	#520	1	1	yes	characteristic
182	C₄×C₆₁	#547	1	1	yes	characteristic
183	C₄×C₆₁	#550	1	1	yes	characteristic
184	Dic₆₁	—	4	4	no	—
185	Dic₆₁	—	4	2	no	—
186	Dic₆₁	—	4	2	no	—
187	Dic₆₁	—	4	2	no	—
188	C₄×#135	—	61	1	no	—
189	#2925	—	61	2	no	—
190	#1974	—	61	2	no	—
191	#1929	—	61	4	no	—
192	#1848	D₆₁	1	1	yes	characteristic, complemented
193	#1806	—	61	2	no	—
194	#1772	—	61	2	no	—
195	C₈×C₄⋊C₄	—	61	1	no	—
196	C₂×C₆₁	—	2	1	no	—
197	C₂×C₆₁	—	2	1	no	—
198	C₂×C₆₁	#2354	1	1	yes	characteristic
199	C₂×C₆₁	C₂×#443	1	1	yes	characteristic
200	C₂×C₆₁	#2957	1	1	yes	characteristic
201	D₆₁	—	8	2	no	—
202	C₂×#135	—	61	1	no	—
203	C₄×#41	—	61	2	no	—
204	C₄×Q₁₆	—	61	1	no	—
205	C₄×Q₁₆	—	61	1	no	—
206	C₄×SD₁₆	—	61	2	no	—
207	C₄×SD₁₆	—	61	2	no	—
208	C₄×D₈	—	61	1	no	—
209	C₄×D₈	—	61	1	no	—
210	#433	—	122	4	no	—
211	#432	—	122	2	no	—
212	#431	—	122	2	no	—
213	C₂×#108	C₂×D₆₁	1	2	yes	—
214	C₂×#108	C₂×D₆₁	1	2	yes	—
215	#427	—	61	1	no	—
216	#423	C₂×D₆₁	1	2	yes	—
217	C₂×C₄⋊C₈	—	61	1	no	—
218	C₂×#104	—	61	4	no	—
219	#416	—	61	2	no	—
220	#408	—	61	4	no	—
221	#405	—	61	4	no	—
222	C₂×C₄×C₈	—	61	2	no	—
223	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
224	C₄×C₄⋊C₄	—	61	1	no	—
225	#339	—	61	4	no	—
226	#336	—	61	2	no	—
227	C₆₁	#22802	1	1	yes	characteristic, complemented, Sylow
228	C₂×#41	—	61	2	no	—
229	#135	—	122	2	no	—
230	#135	—	122	2	no	—
231	C₂×Q₁₆	—	61	1	no	—
232	C₂×SD₁₆	—	61	2	no	—
233	C₂×D₈	—	61	1	no	—
234	C₂×M₁₆	C₂²×D₆₁	1	4	yes	—
235	C₂²×C₈	—	61	2	no	—
236	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
237	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
238	C₄×Q₈	—	61	2	no	—
239	C₄×Q₈	—	61	2	no	—
240	C₄×D₄	—	61	2	no	—
241	C₄×D₄	—	61	2	no	—
242	C₄×D₄	—	122	2	no	—
243	C₄×D₄	—	122	2	no	—
244	#117	—	61	2	no	—
245	#117	—	122	2	no	—
246	C₂×C₄⋊C₄	—	122	1	no	—
247	C₂×C₄²	—	61	2	no	—
248	C₂×C₄²	—	122	2	no	—
249	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
250	#108	—	2	4	no	—
251	#108	—	4	2	no	—
252	#107	—	61	1	no	—
253	#107	—	61	1	no	—
254	#106	—	61	2	no	—
255	C₄⋊C₈	—	2	2	no	—
256	C₄⋊C₈	—	2	2	no	—
257	C₄⋊C₈	—	61	1	no	—
258	C₄⋊C₈	—	61	1	no	—
259	C₄⋊C₈	—	61	2	no	—
260	#104	—	244	4	no	—
261	#103	—	61	2	no	—
262	#103	—	61	2	no	—
263	#102	—	61	2	no	—
264	#102	—	61	2	no	—
265	C₂²⋊C₈	—	122	4	no	—
266	C₂²⋊C₈	—	122	4	no	—
267	#97	—	122	4	no	—
268	C₄×C₈	—	122	4	no	—
269	C₄×C₈	—	122	4	no	—
270	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
271	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
272	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
273	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
274	#95	—	122	1	no	—
275	#41	—	122	2	no	—
276	#41	—	122	2	no	—
277	#41	—	122	2	no	—
278	#41	—	122	2	no	—
279	C₂×Q₈	—	61	2	no	—
280	C₂×D₄	—	61	2	no	—
281	C₂×D₄	—	122	2	no	—
282	C₂²×C₄	—	122	2	no	—
283	C₂²×C₄	—	122	2	no	—
284	C₂²×C₄	—	122	2	no	—
285	C₂²×C₄	—	122	2	no	—
286	C₂²×C₄	#11758226	1	1	yes	characteristic
287	C₂²×C₄	#11758226	1	1	yes	characteristic
288	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
289	Q₁₆	—	122	2	no	—
290	SD₁₆	—	122	4	no	—
291	D₈	—	122	2	no	—
292	M₁₆	—	2	4	no	—
293	M₁₆	—	2	4	no	—
294	C₂×C₈	—	2	1	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	2	4	no	—
297	C₂×C₈	—	61	2	no	—
298	C₂×C₈	—	61	2	no	—
299	C₂×C₈	—	122	2	no	—
300	C₂×C₈	—	122	2	no	—
301	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
302	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
303	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
304	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
305	C₂×C₈	#11758224	1	2	yes	—
306	C₂×C₈	#11758226	1	2	yes	—
307	C₄⋊C₄	—	61	2	no	—
308	C₄⋊C₄	—	61	2	no	—
309	C₄⋊C₄	—	122	4	no	—
310	C₄⋊C₄	—	122	2	no	—
311	C₄⋊C₄	—	122	2	no	—
312	C₂²⋊C₄	—	122	2	no	—
313	C₂²⋊C₄	—	122	2	no	—
314	C₂²⋊C₄	—	122	2	no	—
315	C₄²	—	122	2	no	—
316	C₄²	—	122	2	no	—
317	C₄²	—	122	2	no	—
318	C₄²	—	122	2	no	—
319	C₄²	—	122	2	no	—
320	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
321	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
322	C₄²	D₄×D₆₁	1	1	yes	characteristic
323	C₄²	D₄×D₆₁	1	1	yes	characteristic
324	C₂³	—	122	2	no	—
325	C₂³	C₂×#11758226	1	1	yes	characteristic
326	Q₈	—	122	2	no	—
327	Q₈	—	122	2	no	—
328	D₄	—	122	2	no	—
329	D₄	—	122	2	no	—
330	D₄	—	244	2	no	—
331	D₄	—	244	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	122	4	no	—
335	C₂×C₄	—	122	2	no	—
336	C₂×C₄	—	122	2	no	—
337	C₂×C₄	—	122	2	no	—
338	C₂×C₄	—	488	2	no	—
339	C₂×C₄	—	244	2	no	—
340	C₂×C₄	—	244	2	no	—
341	C₂×C₄	—	244	4	no	—
342	C₂×C₄	—	244	2	no	—
343	C₂×C₄	—	244	2	no	—
344	C₂×C₄	#14519739	1	1	yes	characteristic
345	C₂×C₄	#14519739	1	1	yes	characteristic
346	C₂×C₄	#14519739	1	1	yes	characteristic
347	C₂×C₄	#14519739	1	1	yes	characteristic
348	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
349	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
350	C₂×C₄	C₂×#11758224	1	1	yes	characteristic
351	C₂×C₄	C₂×#11758226	1	1	yes	characteristic
352	C₂×C₄	#14519822	1	1	yes	characteristic
353	C₂×C₄	#14519825	1	1	yes	characteristic
354	C₈	—	2	1	no	—
355	C₈	—	2	1	no	—
356	C₈	—	4	4	no	—
357	C₈	—	122	2	no	—
358	C₈	—	122	2	no	—
359	C₈	#14519735	1	2	yes	—
360	C₈	#14519735	1	2	yes	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	244	2	no	—
364	C₂²	—	244	2	no	—
365	C₂²	—	244	2	no	—
366	C₂²	C₂×#14519739	1	1	yes	characteristic
367	C₂²	#16840478	1	1	yes	characteristic
368	C₂²	#16840486	1	1	yes	characteristic
369	C₄	—	2	1	no	—
370	C₄	—	2	1	no	—
371	C₄	—	2	1	no	—
372	C₄	—	244	4	no	—
373	C₄	—	244	2	no	—
374	C₄	—	244	2	no	—
375	C₄	—	244	2	no	—
376	C₄	#16839878	1	2	yes	—
377	C₄	C₂×#14519735	1	1	yes	characteristic
378	C₄	#16840243	1	1	yes	characteristic
379	C₄	#16840415	1	1	yes	characteristic
380	C₄	#16840432	1	1	yes	characteristic
381	C₂	—	2	1	no	—
382	C₂	—	2	1	no	—
383	C₂	—	488	2	no	—
384	C₂	#21448770	1	1	yes	characteristic
385	C₂	C₂×#16839878	1	1	yes	characteristic
386	C₂	#21454735	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₆₁×#1848	C₂
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	C₂×#14519799	C₂²
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	#1848	D₆₁
hypocenter	C₆₁	#22802
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
2. #21454487	C₂	semidirect
4. #21446466	C₂	semidirect
5. #21445836	C₂	semidirect
6. #21445701	C₂	semidirect
7. C₈×C₄⋊Dic₆₁	C₂	semidirect
8. C₆₁×#1848	C₂	semidirect
11. C₄×Dic₂₄₄	C₄	semidirect
12. C₄×Dic₂₄₄	C₄	semidirect
13. C₄×#11758190	C₄	semidirect
14. C₄×#11758190	C₄	semidirect
15. C₄×D₄₈₈	C₄	semidirect
16. C₄×D₄₈₈	C₄	semidirect
192. #1848	D₆₁	semidirect
227. C₆₁	#22802	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14519799	2
#21454487	2
#21446705	2
#21446466	2
#21445836	2
#21445701	2
C₈×C₄⋊Dic₆₁	2
C₆₁×#1848	2
#22802	61