Subgroups — C₄×#16840808 #30614526

Order 15616.

1022 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840808	C₂	1	2	yes	maximal
2	C₄×#14519865	C₂	1	2	yes	maximal, complemented
3	C₄×#14519847	C₂	1	1	yes	maximal, characteristic, complemented
4	#21461626	C₂	1	2	yes	maximal, complemented
5	#21461600	C₂	1	4	yes	maximal, complemented
6	C₄×#14519842	C₂	1	2	yes	maximal, complemented
7	C₄×#14519831	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#14519800	C₂	1	1	yes	maximal, characteristic
9	C₂×#14519865	C₂²	1	4	yes	—
10	#16840808	C₄	1	16	yes	direct product factor
11	C₂×#14519847	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	C₂×#14519831	C₂²	1	2	yes	—
18	C₂×#14519831	C₂²	1	2	yes	—
19	C₄×D₆₁⋊C₈	—	2	2	no	—
20	C₄×D₆₁⋊C₈	C₂²	1	1	yes	characteristic
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	#14519847	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	#14519831	—	2	4	no	—
49	#14519831	C₂×C₄	1	8	yes	complemented
50	C₄×#92420	—	2	4	no	—
51	C₄×#92420	—	2	2	no	—
52	C₄×#92420	D₄	1	2	yes	—
53	C₄×#92420	D₄	1	2	yes	—
54	C₂×#11758226	C₂³	1	2	yes	—
55	#14519800	C₂×C₄	1	8	yes	—
56	C₂×#11758201	C₂×C₄	1	2	yes	—
57	C₂×D₄₈₈	C₂×C₄	1	1	yes	characteristic
58	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
59	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
60	Q₈×Dic₆₁	C₂×C₄	1	2	yes	—
61	C₄×D₂₄₄	—	2	2	no	—
62	C₄×D₂₄₄	C₂×C₄	1	2	yes	—
63	#14519739	—	2	2	no	—
64	#14519739	C₂×C₄	1	2	yes	—
65	#14519735	—	2	2	no	—
66	#14519735	C₂³	1	2	yes	—
67	C₄²×D₆₁	—	2	2	no	—
68	C₄²×D₆₁	C₂³	1	1	yes	characteristic
69	#14519711	C₂×C₄	1	1	yes	characteristic
70	#14519708	C₂×C₄	1	2	yes	—
71	#14519701	C₂×C₄	1	2	yes	—
72	#14519699	C₂×C₄	1	2	yes	—
73	#14519697	C₂×C₄	1	2	yes	—
74	C₄×#92418	C₂×C₄	1	1	yes	characteristic
75	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
76	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
77	C₂×Q₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
78	C₄×Q₈×C₆₁	C₂×C₄	1	2	yes	—
79	#107×C₆₁	C₂×C₄	1	1	yes	characteristic
80	#103×C₆₁	C₂×C₄	1	2	yes	—
81	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
82	C₂²×#6923	—	2	2	no	—
83	C₂×Q₈×C₆₁	C₂²×C₄	1	2	yes	—
84	#11758226	—	2	4	no	—
85	#11758226	C₂²×C₄	1	8	yes	—
86	C₂×D₂₄₄	—	2	2	no	—
87	C₂×D₂₄₄	C₂²×C₄	1	2	yes	—
88	C₂×C₄×D₆₁	—	2	2	no	—
89	C₂×C₄×D₆₁	—	2	2	no	—
90	C₂×C₄×D₆₁	—	2	2	no	—
91	C₂×C₄×D₆₁	#41	1	1	yes	characteristic
92	C₂×C₄×D₆₁	#41	1	1	yes	characteristic
93	C₂×C₄×D₆₁	C₂⁴	1	1	yes	characteristic
94	C₂×#92420	—	2	4	no	—
95	C₂×#92420	C₂×D₄	1	4	yes	—
96	C₄×#6923	—	2	2	no	—
97	C₄×#6923	—	4	2	no	—
98	C₄×#6923	—	4	2	no	—
99	C₄×#6923	C₂²×C₄	1	4	yes	—
100	#11758213	—	2	4	no	—
101	#11758213	—	2	16	no	—
102	#11758213	C₂²×C₄	1	8	yes	—
103	D₆₁⋊C₈	—	2	16	no	—
104	D₆₁⋊C₈	C₂²×C₄	1	4	yes	—
105	D₆₁⋊C₈	#41	1	4	yes	—
106	Q₁₆×C₆₁	C₄²	1	4	yes	complemented
107	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
109	C₄⋊C₄×C₆₁	—	2	2	no	—
110	C₄⋊C₄×C₆₁	C₂²×C₄	1	2	yes	—
111	C₄²×C₆₁	—	2	2	no	—
112	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
113	#11758201	C₄²	1	8	yes	complemented
114	#11758198	—	2	2	no	—
115	#11758198	—	2	2	no	—
116	#11758198	—	2	2	no	—
117	C₄⋊Dic₆₁	—	2	2	no	—
118	C₄⋊Dic₆₁	C₂²×C₄	1	2	yes	—
119	C₄×Dic₆₁	—	2	2	no	—
120	C₄×Dic₆₁	—	2	2	no	—
121	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
122	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
123	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
124	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
125	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
126	D₄₈₈	C₄²	1	4	yes	complemented
127	C₈×D₆₁	—	2	2	no	—
128	C₈×D₆₁	C₂²×C₄	1	4	yes	—
129	C₂²×D₆₁	—	2	2	no	—
130	C₂²×D₆₁	C₂×#41	1	1	yes	characteristic
131	C₂×#6923	—	2	4	no	—
132	C₂×#6923	—	4	2	no	—
133	C₂×#6923	—	4	2	no	—
134	Q₈×C₆₁	—	2	2	no	—
135	Q₈×C₆₁	C₂×C₄²	1	4	yes	—
136	C₂×C₄×C₆₁	—	2	2	no	—
137	C₂×C₄×C₆₁	—	2	2	no	—
138	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
141	C₂×Dic₆₁	—	2	2	no	—
142	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
143	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
144	C₂×Dic₆₁	C₂²×D₄	1	1	yes	characteristic
145	D₂₄₄	—	2	8	no	—
146	D₂₄₄	—	2	2	no	—
147	D₂₄₄	C₂×C₄²	1	4	yes	—
148	C₄×D₆₁	—	2	2	no	—
149	C₄×D₆₁	—	2	8	no	—
150	C₄×D₆₁	—	2	2	no	—
151	C₄×D₆₁	—	4	4	no	—
152	C₄×D₆₁	—	4	4	no	—
153	C₄×D₆₁	C₂³×C₄	1	2	yes	—
154	C₄×D₆₁	C₂×#41	1	2	yes	—
155	#92420	—	2	8	no	—
156	#92420	C₄×D₄	1	8	yes	—
157	C₈×C₆₁	—	2	1	no	—
158	C₈×C₆₁	C₂×C₄²	1	2	yes	—
159	#92418	—	2	1	no	—
160	#92418	C₂×C₄²	1	2	yes	—
161	C₄×#444	—	61	1	no	maximal, Sylow
162	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
163	C₂×D₆₁	—	2	1	no	—
164	C₂×D₆₁	—	2	1	no	—
165	C₂×D₆₁	—	2	4	no	—
166	C₂×D₆₁	—	4	2	no	—
167	C₂×D₆₁	—	4	2	no	—
168	C₂×D₆₁	C₄×#41	1	2	yes	—
169	#6923	—	4	4	no	—
170	C₄×C₆₁	—	2	1	no	—
171	C₄×C₆₁	—	2	4	no	—
172	C₄×C₆₁	—	4	2	no	—
173	C₄×C₆₁	#444	1	2	yes	complemented
174	C₄×C₆₁	C₂²×C₄²	1	1	yes	characteristic
175	C₄×C₆₁	C₄×#41	1	1	yes	characteristic
176	Dic₆₁	—	2	1	no	—
177	Dic₆₁	—	4	2	no	—
178	Dic₆₁	#444	1	2	yes	complemented
179	Dic₆₁	C₂×C₄×D₄	1	2	yes	—
180	C₂×#444	—	61	2	no	—
181	C₄×#135	—	61	1	no	—
182	C₄×#131	—	61	2	no	—
183	C₄×#108	—	61	1	no	—
184	#1773	—	61	2	no	—
185	#1765	—	61	4	no	—
186	C₄×#104	—	61	2	no	—
187	C₄×#97	—	61	1	no	—
188	C₂×C₆₁	C₄²×D₄	1	1	yes	characteristic
189	C₂×C₆₁	C₂×#444	1	1	yes	characteristic
190	C₂×C₆₁	C₂×#444	1	1	yes	characteristic
191	D₆₁	—	2	2	no	—
192	D₆₁	—	4	4	no	—
193	C₂×#135	—	61	1	no	—
194	C₂×#131	—	61	4	no	—
195	C₄×#41	—	61	2	no	—
196	#444	—	61	16	no	—
197	C₄×Q₁₆	—	61	1	no	—
198	C₄×Q₁₆	#6923	1	1	yes	characteristic, complemented
199	C₄×SD₁₆	—	61	2	no	—
200	C₄×SD₁₆	—	61	2	no	—
201	C₄×D₈	—	61	1	no	—
202	C₄×D₈	—	61	1	no	—
203	C₂×#108	—	61	2	no	—
204	#427	—	61	1	no	—
205	C₂×#104	—	61	4	no	—
206	#416	—	61	2	no	—
207	#405	—	61	4	no	—
208	#405	—	122	4	no	—
209	C₄×M₁₆	—	61	2	no	—
210	C₄×M₁₆	—	122	2	no	—
211	C₂×#97	—	61	2	no	—
212	C₂×#97	—	61	2	no	—
213	C₂×C₄×C₈	—	61	1	no	—
214	C₂×C₄×C₈	—	61	1	no	—
215	C₂×C₄×C₈	—	122	2	no	—
216	C₄³	—	61	1	no	—
217	#341	—	61	2	no	—
218	#339	—	61	4	no	—
219	C₆₁	C₄×#444	1	1	yes	characteristic, complemented, Sylow
220	C₂×#41	—	61	2	no	—
221	#135	—	61	8	no	—
222	C₂×Q₁₆	C₂×#6923	1	1	yes	characteristic
223	C₂×SD₁₆	—	61	2	no	—
224	C₂×D₈	—	61	1	no	—
225	#131	—	61	16	no	—
226	#131	—	122	8	no	—
227	C₂×M₁₆	—	61	4	no	—
228	C₂×M₁₆	—	122	4	no	—
229	C₂²×C₈	—	61	2	no	—
230	C₂²×C₈	—	61	1	no	—
231	C₂²×C₈	—	61	1	no	—
232	C₂²×C₈	—	122	4	no	—
233	C₄×Q₈	—	61	2	no	—
234	C₄×Q₈	C₂×#6923	1	2	yes	—
235	C₄×D₄	—	61	2	no	—
236	C₄×D₄	—	61	2	no	—
237	C₄×D₄	—	122	2	no	—
238	C₄×D₄	—	122	2	no	—
239	#117	—	61	2	no	—
240	#117	—	122	2	no	—
241	C₂×C₄²	—	61	1	no	—
242	C₂×C₄²	—	61	2	no	—
243	C₂×C₄²	—	122	2	no	—
244	C₂×C₄²	—	122	1	no	—
245	C₂×C₄²	—	122	1	no	—
246	#108	—	61	8	no	—
247	#107	—	61	1	no	—
248	#107	C₂×#6923	1	1	yes	characteristic
249	#106	—	61	2	no	—
250	#104	—	61	16	no	—
251	#103	—	61	2	no	—
252	#103	C₂×#6923	1	2	yes	—
253	#102	—	61	2	no	—
254	#102	—	61	2	no	—
255	#97	—	61	8	no	—
256	#97	—	122	2	no	—
257	#97	—	122	4	no	—
258	#97	—	122	4	no	—
259	C₄×C₈	—	61	2	no	—
260	C₄×C₈	—	61	2	no	—
261	C₄×C₈	—	61	1	no	—
262	C₄×C₈	—	61	1	no	—
263	C₄×C₈	—	61	1	no	—
264	C₄×C₈	—	122	2	no	—
265	C₄×C₈	—	122	4	no	—
266	C₄×C₈	C₂×#6923	1	1	yes	characteristic
267	#41	—	61	8	no	—
268	#41	—	122	4	no	—
269	C₂×Q₈	C₂²×#6923	1	2	yes	—
270	C₂×D₄	—	61	2	no	—
271	C₂×D₄	—	122	2	no	—
272	C₂²×C₄	—	61	1	no	—
273	C₂²×C₄	—	61	1	no	—
274	C₂²×C₄	—	61	1	no	—
275	C₂²×C₄	—	122	2	no	—
276	C₂²×C₄	—	122	2	no	—
277	C₂²×C₄	—	122	2	no	—
278	C₂²×C₄	—	122	2	no	—
279	Q₁₆	C₄×#6923	1	4	yes	complemented
280	SD₁₆	—	61	8	no	—
281	D₈	—	61	4	no	—
282	M₁₆	—	61	8	no	—
283	M₁₆	—	122	16	no	—
284	M₁₆	—	122	4	no	—
285	C₂×C₈	—	61	4	no	—
286	C₂×C₈	—	61	4	no	—
287	C₂×C₈	—	61	4	no	—
288	C₂×C₈	—	61	1	no	—
289	C₂×C₈	—	61	4	no	—
290	C₂×C₈	—	61	1	no	—
291	C₂×C₈	—	122	2	no	—
292	C₂×C₈	—	122	16	no	—
293	C₂×C₈	—	122	4	no	—
294	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
295	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	61	2	no	—
298	C₄⋊C₄	—	122	2	no	—
299	C₄⋊C₄	C₂²×#6923	1	2	yes	—
300	C₂²⋊C₄	—	122	2	no	—
301	C₂²⋊C₄	—	122	2	no	—
302	C₂²⋊C₄	—	122	2	no	—
303	C₄²	—	2	2	no	—
304	C₄²	—	61	1	no	—
305	C₄²	—	61	1	no	—
306	C₄²	—	61	1	no	—
307	C₄²	—	61	4	no	—
308	C₄²	—	122	2	no	—
309	C₄²	—	122	2	no	—
310	C₄²	—	122	2	no	—
311	C₄²	—	244	2	no	—
312	C₄²	—	244	2	no	—
313	C₄²	C₂²×#6923	1	1	yes	characteristic
314	C₂³	—	61	1	no	—
315	C₂³	—	122	2	no	—
316	Q₈	—	2	2	no	—
317	Q₈	C₂×C₄×#6923	1	4	yes	—
318	D₄	—	61	4	no	—
319	D₄	—	122	8	no	—
320	D₄	—	122	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	61	1	no	—
324	C₂×C₄	—	61	1	no	—
325	C₂×C₄	—	61	2	no	—
326	C₂×C₄	—	61	2	no	—
327	C₂×C₄	—	61	1	no	—
328	C₂×C₄	—	122	2	no	—
329	C₂×C₄	—	122	8	no	—
330	C₂×C₄	—	122	2	no	—
331	C₂×C₄	—	122	4	no	—
332	C₂×C₄	—	122	2	no	—
333	C₂×C₄	—	244	4	no	—
334	C₂×C₄	—	244	2	no	—
335	C₂×C₄	—	244	2	no	—
336	C₂×C₄	—	244	4	no	—
337	C₂×C₄	D₄×#6923	1	1	yes	characteristic
338	C₂×C₄	D₄×#6923	1	1	yes	characteristic
339	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
340	C₈	—	2	1	no	—
341	C₈	—	61	8	no	—
342	C₈	—	61	2	no	—
343	C₈	—	122	8	no	—
344	C₈	—	122	1	no	—
345	C₈	C₂×C₄×#6923	1	2	yes	—
346	C₂²	—	61	2	no	—
347	C₂²	—	122	1	no	—
348	C₂²	—	122	1	no	—
349	C₂²	—	122	4	no	—
350	C₂²	—	244	2	no	—
351	C₂²	—	244	2	no	—
352	C₂²	C₂×D₄×#6923	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	4	no	—
355	C₄	—	4	2	no	—
356	C₄	—	61	2	no	—
357	C₄	—	61	2	no	—
358	C₄	—	122	1	no	—
359	C₄	—	244	4	no	—
360	C₄	—	244	2	no	—
361	C₄	#16840808	1	2	yes	direct product factor
362	C₄	C₂²×C₄×#6923	1	1	yes	characteristic
363	C₄	#41×#6923	1	1	yes	characteristic
364	C₂	—	122	2	no	—
365	C₂	—	244	4	no	—
366	C₂	C₄×D₄×#6923	1	1	yes	characteristic
367	C₂	C₂×#16840808	1	1	yes	characteristic
368	C₂	D₆₁⋊#444	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×#444
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×#14519847	C₂	semidirect
4. #21461626	C₂	semidirect
5. #21461600	C₂	semidirect
6. C₄×#14519842	C₂	semidirect
7. C₄×#14519831	C₂	semidirect
10. #16840808	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. #14519847	C₂×C₄	semidirect
45. #14519842	C₂×C₄	semidirect
49. #14519831	C₂×C₄	semidirect
106. Q₁₆×C₆₁	C₄²	semidirect
113. #11758201	C₄²	semidirect
126. D₄₈₈	C₄²	semidirect
173. C₄×C₆₁	#444	semidirect
178. Dic₆₁	#444	semidirect
198. C₄×Q₁₆	#6923	semidirect
219. C₆₁	C₄×#444	semidirect
279. Q₁₆	C₄×#6923	semidirect
361. C₄	#16840808	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×#16840808	2
C₄×#14519865	2
C₄×#14519847	2
#21461626	2
#21461600	2
C₄×#14519842	2
C₄×#14519831	2
C₄×#14519800	2
C₄×#444	61

Subgroups — C4×#16840808 #30614526

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16840808 #30614526