Subgroups — C₄×#16840806 #30614518

Order 15616.

1046 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840806	C₂	1	2	yes	maximal
2	C₄×#14519865	C₂	1	2	yes	maximal, complemented
3	C₄×#14519848	C₂	1	1	yes	maximal, characteristic, complemented
4	#21461623	C₂	1	2	yes	maximal, complemented
5	#21461596	C₂	1	4	yes	maximal, complemented
6	C₄×#14519842	C₂	1	2	yes	maximal, complemented
7	C₄×C₈×#6923	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#14519800	C₂	1	1	yes	maximal, characteristic
9	C₂×#14519865	C₂²	1	4	yes	—
10	#16840806	C₄	1	16	yes	direct product factor
11	C₂×#14519848	C₂²	1	2	yes	—
12	C₂×#14519842	C₂²	1	4	yes	—
13	#16840727	—	2	4	no	—
14	#16840727	C₂²	1	4	yes	—
15	C₄×#11758213	—	2	2	no	—
16	C₄×#11758213	C₂²	1	2	yes	—
17	C₄×D₆₁⋊C₈	—	2	2	no	—
18	C₄×D₆₁⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₈×#6923	C₂²	1	2	yes	—
20	C₂×C₈×#6923	C₂²	1	2	yes	—
21	C₄²×#6923	C₂²	1	1	yes	characteristic
22	#16840638	C₂²	1	2	yes	—
23	#16840632	C₂²	1	4	yes	—
24	C₂×#14519800	C₂²	1	1	yes	characteristic
25	C₄×#11758226	C₂²	1	2	yes	—
26	Q₁₆×Dic₆₁	C₄	1	1	yes	characteristic, complemented
27	C₄×#11758201	C₄	1	2	yes	complemented
28	#16839847	C₄	1	2	yes	complemented
29	C₄×D₄₈₈	C₄	1	1	yes	characteristic, complemented
30	#16839844	C₄	1	1	yes	characteristic, complemented
31	#16839799	C₂²	1	1	yes	characteristic
32	#16839757	C₂²	1	2	yes	—
33	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
34	C₄×Q₁₆×C₆₁	C₄	1	1	yes	characteristic, complemented
35	#14519865	—	2	8	no	—
36	#14519865	C₂×C₄	1	16	yes	complemented
37	C₂×#11758213	—	2	4	no	—
38	C₂×#11758213	C₂³	1	4	yes	—
39	C₂×D₆₁⋊C₈	—	2	4	no	—
40	C₂×D₆₁⋊C₈	C₂³	1	2	yes	—
41	C₂×C₄×#6923	—	2	1	no	—
42	C₂×C₄×#6923	—	2	1	no	—
43	C₂×C₄×#6923	C₂³	1	2	yes	—
44	#14519848	C₂×C₄	1	8	yes	complemented
45	#14519842	C₂×C₄	1	16	yes	complemented
46	#14519832	—	2	4	no	—
47	#14519832	—	2	2	no	—
48	C₄×#92420	—	2	4	no	—
49	C₄×#92420	—	2	2	no	—
50	C₄×#92420	D₄	1	1	yes	characteristic
51	C₄×#92420	D₄	1	1	yes	characteristic
52	C₄×#92420	D₄	1	2	yes	—
53	C₈×#6923	—	2	4	no	—
54	C₈×#6923	C₂×C₄	1	8	yes	complemented
55	C₂×#11758226	C₂³	1	2	yes	—
56	#14519800	C₂×C₄	1	8	yes	—
57	C₂×#11758201	C₂×C₄	1	2	yes	—
58	C₂×D₄₈₈	C₂×C₄	1	1	yes	characteristic
59	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
60	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
61	Q₈×Dic₆₁	C₂×C₄	1	2	yes	—
62	C₄×D₂₄₄	—	2	2	no	—
63	C₄×D₂₄₄	C₂×C₄	1	2	yes	—
64	#14519739	—	2	2	no	—
65	#14519739	C₂×C₄	1	2	yes	—
66	#14519735	—	2	2	no	—
67	#14519735	C₂³	1	2	yes	—
68	C₄²×D₆₁	—	2	2	no	—
69	C₄²×D₆₁	C₂³	1	1	yes	characteristic
70	#14519711	C₂×C₄	1	1	yes	characteristic
71	#14519708	C₂×C₄	1	2	yes	—
72	#14519701	C₂×C₄	1	2	yes	—
73	#14519699	C₂×C₄	1	2	yes	—
74	#14519697	C₂×C₄	1	2	yes	—
75	C₄×#92418	C₂×C₄	1	1	yes	characteristic
76	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
77	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
78	C₂×Q₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
79	C₄×Q₈×C₆₁	C₂×C₄	1	2	yes	—
80	#107×C₆₁	C₂×C₄	1	1	yes	characteristic
81	#103×C₆₁	C₂×C₄	1	2	yes	—
82	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
83	C₂²×#6923	—	2	2	no	—
84	C₂×Q₈×C₆₁	C₂²×C₄	1	2	yes	—
85	#11758226	—	2	4	no	—
86	#11758226	C₂²×C₄	1	8	yes	—
87	C₂×D₂₄₄	—	2	2	no	—
88	C₂×D₂₄₄	C₂²×C₄	1	2	yes	—
89	C₂×C₄×D₆₁	—	2	2	no	—
90	C₂×C₄×D₆₁	—	2	2	no	—
91	C₂×C₄×D₆₁	—	2	2	no	—
92	C₂×C₄×D₆₁	#41	1	1	yes	characteristic
93	C₂×C₄×D₆₁	#41	1	1	yes	characteristic
94	C₂×C₄×D₆₁	C₂⁴	1	1	yes	characteristic
95	C₂×#92420	—	2	4	no	—
96	C₂×#92420	C₂×D₄	1	2	yes	—
97	C₂×#92420	C₂×D₄	1	2	yes	—
98	C₄×#6923	—	2	4	no	—
99	C₄×#6923	—	2	4	no	—
100	C₄×#6923	—	2	2	no	—
101	C₄×#6923	C₂²×C₄	1	4	yes	—
102	#11758213	—	2	4	no	—
103	#11758213	—	2	16	no	—
104	#11758213	C₂²×C₄	1	8	yes	—
105	D₆₁⋊C₈	—	2	16	no	—
106	D₆₁⋊C₈	C₂²×C₄	1	4	yes	—
107	D₆₁⋊C₈	#41	1	4	yes	—
108	Q₁₆×C₆₁	C₄²	1	4	yes	complemented
109	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
111	C₄⋊C₄×C₆₁	—	2	2	no	—
112	C₄⋊C₄×C₆₁	C₂²×C₄	1	2	yes	—
113	C₄²×C₆₁	—	2	2	no	—
114	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
115	#11758201	C₄²	1	8	yes	complemented
116	#11758198	—	2	2	no	—
117	#11758198	—	2	2	no	—
118	#11758198	—	2	2	no	—
119	C₄⋊Dic₆₁	—	2	2	no	—
120	C₄⋊Dic₆₁	C₂²×C₄	1	2	yes	—
121	C₄×Dic₆₁	—	2	2	no	—
122	C₄×Dic₆₁	—	2	2	no	—
123	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
124	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
125	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
126	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
127	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
128	D₄₈₈	C₄²	1	4	yes	complemented
129	C₈×D₆₁	—	2	2	no	—
130	C₈×D₆₁	C₂²×C₄	1	4	yes	—
131	C₂²×D₆₁	—	2	2	no	—
132	C₂²×D₆₁	C₂×#41	1	1	yes	characteristic
133	C₂×#6923	—	2	4	no	—
134	C₂×#6923	—	2	4	no	—
135	C₂×#6923	—	2	4	no	—
136	Q₈×C₆₁	—	2	2	no	—
137	Q₈×C₆₁	C₂×C₄²	1	4	yes	—
138	C₂×C₄×C₆₁	—	2	2	no	—
139	C₂×C₄×C₆₁	—	2	2	no	—
140	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
143	C₂×Dic₆₁	—	2	2	no	—
144	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
145	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
146	C₂×Dic₆₁	C₂²×D₄	1	1	yes	characteristic
147	D₂₄₄	—	2	8	no	—
148	D₂₄₄	—	2	2	no	—
149	D₂₄₄	C₂×C₄²	1	4	yes	—
150	C₄×D₆₁	—	2	2	no	—
151	C₄×D₆₁	—	2	8	no	—
152	C₄×D₆₁	—	2	2	no	—
153	C₄×D₆₁	—	4	4	no	—
154	C₄×D₆₁	—	4	4	no	—
155	C₄×D₆₁	C₂³×C₄	1	2	yes	—
156	C₄×D₆₁	C₂×#41	1	2	yes	—
157	#92420	—	2	8	no	—
158	#92420	C₄×D₄	1	4	yes	—
159	#92420	C₄×D₄	1	4	yes	—
160	C₈×C₆₁	—	2	1	no	—
161	C₈×C₆₁	C₂×C₄²	1	2	yes	—
162	#92418	—	2	1	no	—
163	#92418	C₂×C₄²	1	2	yes	—
164	C₄×#443	—	61	1	no	maximal, Sylow
165	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
166	C₂×D₆₁	—	2	1	no	—
167	C₂×D₆₁	—	2	1	no	—
168	C₂×D₆₁	—	2	4	no	—
169	C₂×D₆₁	—	4	2	no	—
170	C₂×D₆₁	—	4	2	no	—
171	C₂×D₆₁	C₄×#41	1	2	yes	—
172	#6923	—	2	8	no	—
173	C₄×C₆₁	—	2	1	no	—
174	C₄×C₆₁	—	2	4	no	—
175	C₄×C₆₁	—	4	2	no	—
176	C₄×C₆₁	#443	1	2	yes	complemented
177	C₄×C₆₁	C₂²×C₄²	1	1	yes	characteristic
178	C₄×C₆₁	C₄×#41	1	1	yes	characteristic
179	Dic₆₁	—	2	1	no	—
180	Dic₆₁	—	4	2	no	—
181	Dic₆₁	#443	1	2	yes	complemented
182	Dic₆₁	C₂×C₄×D₄	1	2	yes	—
183	C₂×#443	—	61	2	no	—
184	C₄×#135	—	61	1	no	—
185	C₄×#131	—	61	2	no	—
186	C₄×#108	—	61	1	no	—
187	#1772	—	61	2	no	—
188	#1764	—	61	4	no	—
189	C₄×#104	—	61	2	no	—
190	C₄²×C₈	—	61	1	no	—
191	C₂×C₆₁	C₄²×D₄	1	1	yes	characteristic
192	C₂×C₆₁	C₂×#443	1	1	yes	characteristic
193	C₂×C₆₁	C₂×#443	1	1	yes	characteristic
194	D₆₁	—	2	2	no	—
195	D₆₁	—	4	4	no	—
196	C₂×#135	—	61	1	no	—
197	C₂×#131	—	61	4	no	—
198	C₄×#41	—	61	2	no	—
199	#443	—	61	16	no	—
200	C₄×Q₁₆	—	61	1	no	—
201	C₄×Q₁₆	#6923	1	1	yes	characteristic, complemented
202	C₄×SD₁₆	—	61	2	no	—
203	C₄×SD₁₆	—	61	2	no	—
204	C₄×D₈	—	61	1	no	—
205	C₄×D₈	—	61	1	no	—
206	C₂×#108	—	61	2	no	—
207	#427	—	61	1	no	—
208	C₂×#104	—	61	4	no	—
209	#416	—	61	2	no	—
210	#405	—	61	4	no	—
211	#405	—	122	4	no	—
212	C₄×M₁₆	—	61	2	no	—
213	C₄×M₁₆	—	122	2	no	—
214	C₂×C₄×C₈	—	61	2	no	—
215	C₂×C₄×C₈	—	61	1	no	—
216	C₂×C₄×C₈	—	61	1	no	—
217	C₂×C₄×C₈	—	61	2	no	—
218	C₂×C₄×C₈	—	122	2	no	—
219	C₄³	—	61	1	no	—
220	#341	—	61	2	no	—
221	#339	—	61	4	no	—
222	C₆₁	C₄×#443	1	1	yes	characteristic, complemented, Sylow
223	C₂×#41	—	61	2	no	—
224	#135	—	61	8	no	—
225	C₂×Q₁₆	C₂×#6923	1	1	yes	characteristic
226	C₂×SD₁₆	—	61	2	no	—
227	C₂×D₈	—	61	1	no	—
228	#131	—	61	16	no	—
229	#131	—	122	8	no	—
230	C₂×M₁₆	—	61	4	no	—
231	C₂×M₁₆	—	122	4	no	—
232	C₂²×C₈	—	61	2	no	—
233	C₂²×C₈	—	61	1	no	—
234	C₂²×C₈	—	61	1	no	—
235	C₂²×C₈	—	122	4	no	—
236	C₄×Q₈	—	61	2	no	—
237	C₄×Q₈	C₂×#6923	1	2	yes	—
238	C₄×D₄	—	61	2	no	—
239	C₄×D₄	—	61	2	no	—
240	C₄×D₄	—	122	2	no	—
241	C₄×D₄	—	122	2	no	—
242	#117	—	61	2	no	—
243	#117	—	122	2	no	—
244	C₂×C₄²	—	61	1	no	—
245	C₂×C₄²	—	61	2	no	—
246	C₂×C₄²	—	122	2	no	—
247	C₂×C₄²	—	122	1	no	—
248	C₂×C₄²	—	122	1	no	—
249	#108	—	61	8	no	—
250	#107	—	61	1	no	—
251	#107	C₂×#6923	1	1	yes	characteristic
252	#106	—	61	2	no	—
253	#104	—	61	16	no	—
254	#103	—	61	2	no	—
255	#103	C₂×#6923	1	2	yes	—
256	#102	—	61	2	no	—
257	#102	—	61	2	no	—
258	#97	—	122	2	no	—
259	#97	—	122	4	no	—
260	C₄×C₈	—	61	8	no	—
261	C₄×C₈	—	61	1	no	—
262	C₄×C₈	—	61	1	no	—
263	C₄×C₈	—	61	2	no	—
264	C₄×C₈	—	61	1	no	—
265	C₄×C₈	—	61	1	no	—
266	C₄×C₈	—	61	1	no	—
267	C₄×C₈	—	122	2	no	—
268	C₄×C₈	—	122	4	no	—
269	C₄×C₈	—	122	4	no	—
270	C₄×C₈	C₂×#6923	1	1	yes	characteristic
271	#41	—	61	8	no	—
272	#41	—	122	4	no	—
273	C₂×Q₈	C₂²×#6923	1	2	yes	—
274	C₂×D₄	—	61	2	no	—
275	C₂×D₄	—	122	2	no	—
276	C₂²×C₄	—	61	1	no	—
277	C₂²×C₄	—	61	1	no	—
278	C₂²×C₄	—	61	1	no	—
279	C₂²×C₄	—	122	2	no	—
280	C₂²×C₄	—	122	2	no	—
281	C₂²×C₄	—	122	2	no	—
282	C₂²×C₄	—	122	2	no	—
283	Q₁₆	C₄×#6923	1	4	yes	complemented
284	SD₁₆	—	61	8	no	—
285	D₈	—	61	4	no	—
286	M₁₆	—	61	8	no	—
287	M₁₆	—	122	16	no	—
288	M₁₆	—	122	4	no	—
289	C₂×C₈	—	61	2	no	—
290	C₂×C₈	—	61	2	no	—
291	C₂×C₈	—	61	4	no	—
292	C₂×C₈	—	61	4	no	—
293	C₂×C₈	—	61	1	no	—
294	C₂×C₈	—	61	4	no	—
295	C₂×C₈	—	61	1	no	—
296	C₂×C₈	—	122	2	no	—
297	C₂×C₈	—	122	16	no	—
298	C₂×C₈	—	122	4	no	—
299	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
300	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	61	2	no	—
303	C₄⋊C₄	—	122	2	no	—
304	C₄⋊C₄	C₂²×#6923	1	2	yes	—
305	C₂²⋊C₄	—	122	2	no	—
306	C₂²⋊C₄	—	122	2	no	—
307	C₂²⋊C₄	—	122	2	no	—
308	C₄²	—	2	2	no	—
309	C₄²	—	61	1	no	—
310	C₄²	—	61	1	no	—
311	C₄²	—	61	1	no	—
312	C₄²	—	61	4	no	—
313	C₄²	—	122	2	no	—
314	C₄²	—	122	2	no	—
315	C₄²	—	122	2	no	—
316	C₄²	—	122	4	no	—
317	C₄²	—	122	4	no	—
318	C₄²	C₂²×#6923	1	1	yes	characteristic
319	C₂³	—	61	1	no	—
320	C₂³	—	122	2	no	—
321	Q₈	—	2	2	no	—
322	Q₈	C₂×C₄×#6923	1	4	yes	—
323	D₄	—	61	4	no	—
324	D₄	—	122	8	no	—
325	D₄	—	122	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	61	1	no	—
329	C₂×C₄	—	61	1	no	—
330	C₂×C₄	—	61	2	no	—
331	C₂×C₄	—	61	2	no	—
332	C₂×C₄	—	61	1	no	—
333	C₂×C₄	—	122	2	no	—
334	C₂×C₄	—	122	8	no	—
335	C₂×C₄	—	122	4	no	—
336	C₂×C₄	—	122	4	no	—
337	C₂×C₄	—	122	4	no	—
338	C₂×C₄	—	122	2	no	—
339	C₂×C₄	—	122	2	no	—
340	C₂×C₄	—	244	4	no	—
341	C₂×C₄	—	244	4	no	—
342	C₂×C₄	D₄×#6923	1	1	yes	characteristic
343	C₂×C₄	D₄×#6923	1	1	yes	characteristic
344	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
345	C₈	—	2	1	no	—
346	C₈	—	61	4	no	—
347	C₈	—	61	4	no	—
348	C₈	—	61	2	no	—
349	C₈	—	122	8	no	—
350	C₈	—	122	1	no	—
351	C₈	C₂×C₄×#6923	1	2	yes	—
352	C₂²	—	61	2	no	—
353	C₂²	—	122	1	no	—
354	C₂²	—	122	1	no	—
355	C₂²	—	122	4	no	—
356	C₂²	—	244	2	no	—
357	C₂²	—	244	2	no	—
358	C₂²	C₂×D₄×#6923	1	1	yes	characteristic
359	C₄	—	2	1	no	—
360	C₄	—	2	4	no	—
361	C₄	—	4	2	no	—
362	C₄	—	61	2	no	—
363	C₄	—	61	2	no	—
364	C₄	—	122	8	no	—
365	C₄	—	122	1	no	—
366	C₄	—	244	2	no	—
367	C₄	#16840806	1	2	yes	direct product factor
368	C₄	C₂²×C₄×#6923	1	1	yes	characteristic
369	C₄	#41×#6923	1	1	yes	characteristic
370	C₂	—	122	2	no	—
371	C₂	—	244	4	no	—
372	C₂	C₄×D₄×#6923	1	1	yes	characteristic
373	C₂	D₆₁⋊#443	1	1	yes	characteristic
374	C₂	C₂×#16840806	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6923
commutator	C₄×C₆₁	C₂²×C₄²
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₄×Q₁₆×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	C₂×#14519800	C₂²
cosocle	C₂×C₄×D₆₁	C₂⁴
hypercenter	C₄×Q₁₆	#6923
hypocenter	C₆₁	C₄×#443
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. C₄×#14519865	C₂	semidirect
3. C₄×#14519848	C₂	semidirect
4. #21461623	C₂	semidirect
5. #21461596	C₂	semidirect
6. C₄×#14519842	C₂	semidirect
7. C₄×C₈×#6923	C₂	semidirect
10. #16840806	C₄	direct (N × H)
26. Q₁₆×Dic₆₁	C₄	semidirect
27. C₄×#11758201	C₄	semidirect
28. #16839847	C₄	semidirect
29. C₄×D₄₈₈	C₄	semidirect
30. #16839844	C₄	semidirect
34. C₄×Q₁₆×C₆₁	C₄	semidirect
36. #14519865	C₂×C₄	semidirect
44. #14519848	C₂×C₄	semidirect
45. #14519842	C₂×C₄	semidirect
54. C₈×#6923	C₂×C₄	semidirect
108. Q₁₆×C₆₁	C₄²	semidirect
115. #11758201	C₄²	semidirect
128. D₄₈₈	C₄²	semidirect
176. C₄×C₆₁	#443	semidirect
181. Dic₆₁	#443	semidirect
201. C₄×Q₁₆	#6923	semidirect
222. C₆₁	C₄×#443	semidirect
283. Q₁₆	C₄×#6923	semidirect
367. C₄	#16840806	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16840806	2
C₄×#14519865	2
C₄×#14519848	2
#21461623	2
#21461596	2
C₄×#14519842	2
C₄×C₈×#6923	2
C₄×#14519800	2
C₄×#443	61

Subgroups — C4×#16840806 #30614518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16840806 #30614518