Subgroups — #30612116

Order 15616.

602 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462740	C₂	1	1	yes	maximal, characteristic
2	#21462736	C₂	1	1	yes	maximal, characteristic
3	#21461397	C₂	1	2	yes	maximal, complemented
4	#21461393	C₂	1	2	yes	maximal, complemented
5	#21461391	C₂	1	2	yes	maximal, complemented
6	#21461387	C₂	1	2	yes	maximal, complemented
7	#21461372	C₂	1	1	yes	maximal, characteristic, complemented
8	#21461370	C₂	1	1	yes	maximal, characteristic, complemented
9	#21461366	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16840624	C₂	1	1	yes	maximal, characteristic, complemented
11	#21454462	C₂	1	1	yes	maximal, characteristic
12	#16840812	—	2	1	no	—
13	#16840811	—	2	1	no	—
14	#16840796	—	2	1	no	—
15	#16840791	—	2	2	no	—
16	#16840790	—	2	1	no	—
17	#16840789	—	2	2	no	—
18	#16840784	C₂²	1	2	yes	—
19	#16840781	C₂²	1	2	yes	—
20	#16840778	C₂²	1	1	yes	characteristic
21	#16840778	C₂²	1	1	yes	characteristic
22	#16840766	C₂²	1	2	yes	—
23	D₆₁⋊(C₄⋊C₈)	C₂²	1	2	yes	—
24	C₂×C₄⋊#92420	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊#92420	C₂²	1	1	yes	characteristic
26	#16840626	C₂²	1	2	yes	—
27	#16840625	C₂²	1	1	yes	characteristic
28	#16840625	C₂²	1	1	yes	characteristic
29	#16840625	C₂²	1	2	yes	—
30	#16840624	C₂²	1	2	yes	—
31	#16840624	C₂²	1	2	yes	—
32	#16840624	C₂²	1	2	yes	—
33	#16840623	C₂²	1	1	yes	characteristic
34	#16840623	C₂²	1	1	yes	characteristic
35	#16840623	C₂²	1	2	yes	—
36	#16840230	C₂²	1	1	yes	characteristic
37	#16839822	C₄	1	2	yes	—
38	#16839822	C₄	1	2	yes	—
39	#16839814	C₂²	1	2	yes	—
40	#16839814	C₂²	1	2	yes	—
41	#16839789	C₄	1	1	yes	characteristic
42	#16839730	C₄	1	1	yes	characteristic
43	#16839728	C₄	1	1	yes	characteristic
44	M₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
45	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
46	C₆₁×#424	C₄	1	1	yes	characteristic
47	C₂×#11758213	—	2	2	no	—
48	C₂×D₆₁⋊C₈	—	2	2	no	—
49	C₂²×#92420	—	2	1	no	—
50	C₂²×#92420	—	2	1	no	—
51	#14519844	—	2	2	no	—
52	C₄⋊#92420	—	2	1	no	—
53	C₄⋊#92420	—	2	1	no	—
54	C₄⋊#92420	—	2	2	no	—
55	C₄⋊#92420	C₂³	1	2	yes	—
56	C₄⋊#92420	C₂³	1	2	yes	—
57	C₄⋊#92420	C₂³	1	1	yes	characteristic
58	C₄⋊#92420	C₂³	1	1	yes	characteristic
59	C₄⋊#92420	C₂³	1	1	yes	characteristic
60	C₄⋊#92420	C₂³	1	1	yes	characteristic
61	#14519834	—	2	2	no	—
62	#14519834	—	2	2	no	—
63	C₂²⋊#92420	—	2	1	no	—
64	C₂²⋊#92420	—	2	1	no	—
65	#14519832	—	2	2	no	—
66	C₄×#92420	—	2	1	no	—
67	C₄×#92420	—	2	1	no	—
68	C₄×#92420	—	2	2	no	—
69	C₂×#11758222	C₂×C₄	1	1	yes	characteristic
70	C₂×#11758194	C₂×C₄	1	1	yes	characteristic
71	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
72	#14519742	D₄	1	1	yes	characteristic
73	#14519742	Q₈	1	1	yes	characteristic
74	#14519739	D₄	1	1	yes	characteristic
75	#14519739	Q₈	1	1	yes	characteristic
76	#14519737	D₄	1	2	yes	—
77	#14519737	Q₈	1	2	yes	—
78	#14519734	C₂×C₄	1	2	yes	—
79	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
80	C₄²×D₆₁	C₂³	1	2	yes	—
81	C₄⋊#92418	C₂×C₄	1	2	yes	—
82	C₄⋊#92418	C₂×C₄	1	2	yes	—
83	#14519689	C₂×C₄	1	2	yes	—
84	#14519689	C₂×C₄	1	2	yes	—
85	#14519689	C₂×C₄	1	2	yes	—
86	#14519689	C₂×C₄	1	2	yes	—
87	#14519687	C₂×C₄	1	2	yes	—
88	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
89	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
90	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
91	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
92	C₈×Dic₆₁	C₂³	1	2	yes	—
93	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
94	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
95	#117×C₆₁	C₂×C₄	1	1	yes	characteristic
96	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
97	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
98	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₂²⋊D₆₁	C₄⋊C₄	1	2	yes	—
100	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
101	C₂²×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
102	#11758222	—	2	2	no	—
103	#11758222	—	2	2	no	—
104	C₂×D₂₄₄	C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×D₆₁	C₂²×C₄	1	2	yes	—
106	C₂×C₄×D₆₁	C₂×D₄	1	2	yes	—
107	C₂×C₄×D₆₁	C₂×Q₈	1	2	yes	—
108	C₂×Dic₁₂₂	C₄⋊C₄	1	1	yes	characteristic
109	C₂×#92420	—	2	1	no	—
110	C₂×#92420	—	2	1	no	—
111	C₂×#92420	—	2	1	no	—
112	C₂×#92420	—	2	1	no	—
113	C₂×#92420	—	2	2	no	—
114	C₂×#92420	—	2	2	no	—
115	C₂×#92420	—	8	1	no	—
116	C₂×#92420	—	8	1	no	—
117	#11758213	—	8	2	no	—
118	D₆₁⋊C₈	—	8	2	no	—
119	M₁₆×C₆₁	—	2	2	no	—
120	C₂×C₈×C₆₁	—	2	2	no	—
121	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
124	C₄⋊C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
125	C₄⋊C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
126	C₂²⋊C₄×C₆₁	C₄⋊C₄	1	2	yes	—
127	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
128	#11758198	C₄⋊C₄	1	2	yes	—
129	#11758198	C₄⋊C₄	1	2	yes	—
130	#11758196	C₄⋊C₄	1	2	yes	—
131	#11758196	C₄⋊C₄	1	2	yes	—
132	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
133	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
134	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
135	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
136	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
137	C₄×Dic₆₁	C₂⁴	1	1	yes	characteristic
138	#11758194	—	2	2	no	—
139	C₂×#92418	—	2	2	no	—
140	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
141	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
142	C₂×#92418	C₂²×C₄	1	2	yes	—
143	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂²×D₆₁	C₂×C₄⋊C₄	1	2	yes	—
145	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
146	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
147	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₆₁	#131	1	2	yes	—
149	C₂×C₄×C₆₁	#131	1	2	yes	—
150	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
151	C₂²⋊D₆₁	—	2	4	no	—
152	C₂×Dic₆₁	—	2	1	no	—
153	C₂×Dic₆₁	—	2	1	no	—
154	C₂×Dic₆₁	—	4	1	no	—
155	C₂×Dic₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₂×Dic₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂×Dic₆₁	C₂×C₄⋊C₄	1	2	yes	—
158	C₂×Dic₆₁	C₂²×D₄	1	1	yes	characteristic
159	C₂×Dic₆₁	C₂²×Q₈	1	1	yes	characteristic
160	D₂₄₄	—	2	2	no	—
161	C₄×D₆₁	—	2	2	no	—
162	C₄×D₆₁	—	2	2	no	—
163	C₄×D₆₁	—	2	2	no	—
164	C₄×D₆₁	—	2	2	no	—
165	C₄×D₆₁	—	4	2	no	—
166	Dic₁₂₂	—	2	2	no	—
167	#92420	—	4	1	no	—
168	#92420	—	4	2	no	—
169	#92420	—	4	1	no	—
170	#92420	—	4	2	no	—
171	#92420	—	4	1	no	—
172	#92420	—	4	1	no	—
173	C₈×C₆₁	—	2	1	no	—
174	C₈×C₆₁	—	2	1	no	—
175	C₈×C₆₁	—	2	2	no	—
176	#92418	—	2	1	no	—
177	#92418	—	2	1	no	—
178	#92418	—	2	2	no	—
179	#17811	—	61	1	no	maximal, Sylow
180	C₂²×C₆₁	—	2	1	no	—
181	C₂²×C₆₁	—	2	1	no	—
182	C₂²×C₆₁	#424	1	2	yes	—
183	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
184	C₂×D₆₁	—	2	2	no	—
185	C₂×D₆₁	—	2	2	no	—
186	C₂×D₆₁	—	2	2	no	—
187	C₄×C₆₁	—	2	2	no	—
188	C₄×C₆₁	—	2	2	no	—
189	C₄×C₆₁	#424	1	2	yes	—
190	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
191	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
192	Dic₆₁	—	2	2	no	—
193	Dic₆₁	—	2	1	no	—
194	Dic₆₁	—	2	1	no	—
195	Dic₆₁	—	2	1	no	—
196	Dic₆₁	#577	1	1	yes	characteristic
197	Dic₆₁	#577	1	1	yes	characteristic
198	#2989	—	61	1	no	—
199	#2988	—	61	1	no	—
200	#2922	—	61	1	no	—
201	#1574	—	61	2	no	—
202	#1573	—	61	2	no	—
203	#1572	—	61	2	no	—
204	#1570	—	61	2	no	—
205	#1565	—	61	1	no	—
206	#1565	—	61	1	no	—
207	#1564	—	61	1	no	—
208	C₂×#334	—	61	1	no	—
209	C₂×C₆₁	—	2	2	no	—
210	C₂×C₆₁	C₂×#424	1	1	yes	characteristic
211	C₂×C₆₁	#2912	1	1	yes	characteristic
212	C₂×C₆₁	#2916	1	1	yes	characteristic
213	D₆₁	—	4	2	no	—
214	C₄×#41	—	61	1	no	—
215	#446	—	122	1	no	—
216	C₈×Q₈	—	122	1	no	—
217	#436	—	122	1	no	—
218	#435	—	122	2	no	—
219	C₈×D₄	—	122	2	no	—
220	C₈×D₄	—	122	1	no	—
221	#433	—	61	2	no	—
222	#433	—	61	2	no	—
223	#432	—	61	2	no	—
224	#431	—	61	2	no	—
225	#431	—	61	2	no	—
226	#431	—	61	1	no	—
227	#431	—	61	1	no	—
228	#431	—	61	2	no	—
229	#424	—	61	1	no	—
230	#424	#6923	1	1	yes	characteristic
231	#423	—	61	2	no	—
232	C₂×C₄⋊C₈	—	61	1	no	—
233	C₂×C₄⋊C₈	—	61	1	no	—
234	C₂×C₄⋊C₈	—	61	2	no	—
235	#408	—	61	1	no	—
236	#408	—	61	1	no	—
237	C₄×M₁₆	—	61	1	no	—
238	C₂×C₄×C₈	—	61	1	no	—
239	#335	—	61	1	no	—
240	#335	—	61	1	no	—
241	#335	—	61	2	no	—
242	#335	—	61	2	no	—
243	#334	—	61	2	no	—
244	#334	—	61	2	no	—
245	#334	—	61	1	no	—
246	#334	—	61	1	no	—
247	#334	—	61	2	no	—
248	#334	—	61	2	no	—
249	C₆₁	#17811	1	1	yes	characteristic, complemented, Sylow
250	C₂×#41	—	61	1	no	—
251	C₂×M₁₆	—	61	1	no	—
252	C₂×M₁₆	—	122	2	no	—
253	C₂×M₁₆	C₂×#6923	1	1	yes	characteristic
254	C₂²×C₈	—	61	1	no	—
255	C₂²×C₈	—	122	1	no	—
256	C₂²×C₈	—	122	1	no	—
257	C₂²×C₈	—	122	2	no	—
258	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
259	C₄×Q₈	—	61	1	no	—
260	C₄×Q₈	—	61	1	no	—
261	C₄×D₄	—	61	1	no	—
262	C₄×D₄	—	61	1	no	—
263	C₄×D₄	—	61	2	no	—
264	C₄×D₄	—	61	2	no	—
265	#117	—	61	2	no	—
266	#117	C₂×#6923	1	1	yes	characteristic
267	C₂×C₄²	—	61	1	no	—
268	C₂×C₄²	—	61	2	no	—
269	C₄⋊C₈	—	61	2	no	—
270	C₄⋊C₈	—	61	1	no	—
271	C₄⋊C₈	—	61	2	no	—
272	C₄⋊C₈	—	61	1	no	—
273	C₄⋊C₈	—	61	1	no	—
274	C₄⋊C₈	—	61	1	no	—
275	C₄⋊C₈	—	61	2	no	—
276	C₄⋊C₈	—	61	2	no	—
277	C₄⋊C₈	—	122	1	no	—
278	C₄⋊C₈	—	122	1	no	—
279	C₄⋊C₈	—	122	2	no	—
280	C₄⋊C₈	—	122	2	no	—
281	C₄⋊C₈	C₂×#6923	1	2	yes	—
282	C₄⋊C₈	C₂×#6923	1	2	yes	—
283	C₂²⋊C₈	—	61	2	no	—
284	C₂²⋊C₈	—	61	2	no	—
285	C₂²⋊C₈	—	61	2	no	—
286	C₂²⋊C₈	—	61	2	no	—
287	C₂²⋊C₈	—	122	1	no	—
288	C₂²⋊C₈	—	122	1	no	—
289	C₂²⋊C₈	—	122	2	no	—
290	C₂²⋊C₈	—	122	2	no	—
291	#97	—	61	2	no	—
292	#97	—	122	2	no	—
293	C₄×C₈	—	61	1	no	—
294	C₄×C₈	—	61	1	no	—
295	C₄×C₈	—	61	2	no	—
296	C₄×C₈	—	61	1	no	—
297	C₄×C₈	—	61	1	no	—
298	C₄×C₈	—	122	1	no	—
299	C₄×C₈	—	122	2	no	—
300	C₄×C₈	—	122	1	no	—
301	#41	—	122	2	no	—
302	#41	—	122	2	no	—
303	C₂×Q₈	—	61	1	no	—
304	C₂×D₄	—	61	2	no	—
305	C₂×D₄	—	61	1	no	—
306	C₂²×C₄	—	61	1	no	—
307	C₂²×C₄	—	61	1	no	—
308	C₂²×C₄	—	61	2	no	—
309	C₂²×C₄	—	61	2	no	—
310	C₂²×C₄	—	61	2	no	—
311	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
312	M₁₆	—	2	2	no	—
313	M₁₆	—	122	2	no	—
314	M₁₆	—	488	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	61	1	no	—
317	C₂×C₈	—	61	1	no	—
318	C₂×C₈	—	61	2	no	—
319	C₂×C₈	—	122	2	no	—
320	C₂×C₈	—	122	1	no	—
321	C₂×C₈	—	122	1	no	—
322	C₂×C₈	—	122	1	no	—
323	C₂×C₈	—	122	2	no	—
324	C₂×C₈	—	122	1	no	—
325	C₂×C₈	—	122	2	no	—
326	C₂×C₈	—	488	2	no	—
327	C₂×C₈	—	488	1	no	—
328	C₂×C₈	—	488	1	no	—
329	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
330	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
331	C₂×C₈	C₂²×#6923	1	2	yes	—
332	C₄⋊C₄	—	61	2	no	—
333	C₄⋊C₄	—	61	2	no	—
334	C₄⋊C₄	C₄⋊#6923	1	1	yes	characteristic
335	C₄⋊C₄	C₄⋊#6923	1	1	yes	characteristic
336	C₂²⋊C₄	—	61	2	no	—
337	C₂²⋊C₄	—	61	2	no	—
338	C₂²⋊C₄	C₄⋊#6923	1	2	yes	—
339	C₄²	—	61	1	no	—
340	C₄²	—	61	2	no	—
341	C₄²	—	61	1	no	—
342	C₄²	—	61	2	no	—
343	C₄²	—	61	1	no	—
344	C₄²	—	61	1	no	—
345	C₄²	C₂²×#6923	1	2	yes	—
346	C₂³	—	61	2	no	—
347	C₂³	C₂×C₄⋊#6923	1	1	yes	characteristic
348	Q₈	—	122	2	no	—
349	D₄	—	122	4	no	—
350	D₄	—	122	2	no	—
351	C₂×C₄	—	61	1	no	—
352	C₂×C₄	—	61	1	no	—
353	C₂×C₄	—	61	1	no	—
354	C₂×C₄	—	61	2	no	—
355	C₂×C₄	—	61	1	no	—
356	C₂×C₄	—	122	2	no	—
357	C₂×C₄	—	122	2	no	—
358	C₂×C₄	—	122	1	no	—
359	C₂×C₄	—	122	1	no	—
360	C₂×C₄	—	122	2	no	—
361	C₂×C₄	—	122	2	no	—
362	C₂×C₄	—	244	2	no	—
363	C₂×C₄	—	244	1	no	—
364	C₂×C₄	C₂×C₄⋊#6923	1	2	yes	—
365	C₂×C₄	C₂×C₄⋊#6923	1	2	yes	—
366	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
367	C₂×C₄	#14519864	1	2	yes	—
368	C₂×C₄	#14519865	1	2	yes	—
369	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	2	2	no	—
373	C₈	—	122	1	no	—
374	C₈	—	122	1	no	—
375	C₈	—	122	2	no	—
376	C₈	—	244	1	no	—
377	C₈	—	244	1	no	—
378	C₈	—	244	1	no	—
379	C₈	—	244	2	no	—
380	C₈	—	244	2	no	—
381	C₈	—	244	1	no	—
382	C₂²	—	2	1	no	—
383	C₂²	—	2	1	no	—
384	C₂²	—	122	2	no	—
385	C₂²	—	122	2	no	—
386	C₂²	—	122	2	no	—
387	C₂²	#16840769	1	2	yes	—
388	C₂²	C₂²×C₄⋊#6923	1	1	yes	characteristic
389	C₄	—	2	2	no	—
390	C₄	—	2	2	no	—
391	C₄	—	61	1	no	—
392	C₄	—	61	1	no	—
393	C₄	—	122	2	no	—
394	C₄	—	122	1	no	—
395	C₄	—	122	1	no	—
396	C₄	—	122	1	no	—
397	C₄	#16840769	1	2	yes	—
398	C₄	C₂×#14519864	1	1	yes	characteristic
399	C₄	C₂×#14519865	1	1	yes	characteristic
400	C₂	—	2	2	no	—
401	C₂	—	244	2	no	—
402	C₂	C₂×#16840769	1	1	yes	characteristic
403	C₂	#21462473	1	1	yes	characteristic
404	C₂	#21462481	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#6923
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₆₁×#424	C₄
socle	C₂²×C₆₁	C₂²×C₄⋊C₄
prime radical	C₂×#11758222	C₂×C₄
cosocle	C₄×Dic₆₁	C₂⁴
hypercenter	#424	#6923
hypocenter	C₆₁	#17811
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #424

Complements

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

N	H ≅ G/N	type
3. #21461397	C₂	semidirect
4. #21461393	C₂	semidirect
5. #21461391	C₂	semidirect
6. #21461387	C₂	semidirect
7. #21461372	C₂	semidirect
8. #21461370	C₂	semidirect
9. #21461366	C₂	semidirect
10. C₂×#16840624	C₂	semidirect
249. C₆₁	#17811	semidirect

Maximal subgroups

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

subgroup	index
#21462740	2
#21462736	2
#21461397	2
#21461393	2
#21461391	2
#21461387	2
#21461372	2
#21461370	2
#21461366	2
C₂×#16840624	2
#21454462	2
#17811	61