Subgroups — C₄×#16840195 #29833295

Order 15616.

690 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840195	C₂	1	1	yes	maximal, characteristic
2	C₄×#14519800	C₂	1	2	yes	maximal, complemented
3	Q₃₂×Dic₆₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#14519720	C₂	1	2	yes	maximal, complemented
5	#21447668	C₂	1	2	yes	maximal, complemented
6	C₄×D₉₇₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#21447665	C₂	1	1	yes	maximal, characteristic, complemented
8	#21447600	C₂	1	1	yes	maximal, characteristic, complemented
9	#21447525	C₂	1	2	yes	maximal, complemented
10	C₄×C₁₆×D₆₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Q₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14519800	C₂²	1	2	yes	—
13	C₄×#11758226	—	2	2	no	—
14	#16840195	C₄	1	8	yes	direct product factor
15	C₂×#14519720	C₂²	1	2	yes	—
16	C₂×D₉₇₆	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×D₆₁	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₆₁	C₂²	1	1	yes	characteristic
19	Q₁₆×Dic₆₁	C₂²	1	2	yes	—
20	C₄×#11758201	—	2	2	no	—
21	#16839847	—	2	2	no	—
22	C₄×D₄₈₈	C₂²	1	2	yes	—
23	#16839844	C₂²	1	2	yes	—
24	#16839799	C₂²	1	2	yes	—
25	#16839757	—	2	2	no	—
26	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
27	#16839589	C₂²	1	2	yes	—
28	#16839588	C₂²	1	1	yes	characteristic
29	#16839566	C₂²	1	2	yes	—
30	#16839564	C₂²	1	2	yes	—
31	#16839562	C₂²	1	2	yes	—
32	C₄×#11758185	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
35	C₂×Q₃₂×C₆₁	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	—
37	C₆₁×#366	C₂²	1	1	yes	characteristic
38	C₆₁×#358	C₂²	1	2	yes	—
39	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
40	C₂×#11758226	—	2	2	no	—
41	#14519800	—	2	4	no	—
42	#14519800	C₂×C₄	1	8	yes	complemented
43	C₂×#11758201	—	2	2	no	—
44	C₂×D₄₈₈	C₂³	1	2	yes	—
45	C₂×C₈×D₆₁	D₄	1	1	yes	characteristic
46	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
47	Q₈×Dic₆₁	—	2	2	no	—
48	C₄×D₂₄₄	—	2	2	no	—
49	C₄×D₂₄₄	—	4	2	no	—
50	#14519739	—	2	2	no	—
51	#14519739	—	4	2	no	—
52	#14519735	—	2	2	no	—
53	#14519735	—	4	2	no	—
54	C₄²×D₆₁	—	4	2	no	—
55	C₄²×D₆₁	D₄	1	1	yes	characteristic
56	#14519720	C₂×C₄	1	8	yes	complemented
57	D₉₇₆	C₂×C₄	1	4	yes	complemented
58	C₂×#11758185	C₂³	1	1	yes	characteristic
59	C₂×#11758185	C₂³	1	1	yes	characteristic
60	C₁₆×D₆₁	—	2	2	no	—
61	C₁₆×D₆₁	C₂×C₄	1	4	yes	complemented
62	#14519711	C₂³	1	2	yes	—
63	#14519708	—	2	2	no	—
64	#14519701	—	2	2	no	—
65	#14519699	—	2	2	no	—
66	#14519697	—	2	2	no	—
67	C₄×#92418	D₄	1	1	yes	characteristic
68	C₈×Dic₆₁	D₄	1	1	yes	characteristic
69	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
70	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
71	C₄×Q₈×C₆₁	—	2	2	no	—
72	Q₃₂×C₆₁	C₂×C₄	1	4	yes	complemented
73	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
75	#107×C₆₁	C₂³	1	2	yes	—
76	#103×C₆₁	—	2	2	no	—
77	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
78	C₂×Q₈×C₆₁	—	2	2	no	—
79	#11758226	—	2	8	no	—
80	#11758226	—	4	4	no	—
81	C₂×D₂₄₄	—	2	2	no	—
82	C₂×D₂₄₄	—	4	2	no	—
83	C₂×C₄×D₆₁	—	4	2	no	—
84	C₂×C₄×D₆₁	—	4	2	no	—
85	C₂×C₄×D₆₁	—	4	2	no	—
86	C₂×C₄×D₆₁	D₈	1	1	yes	characteristic
87	C₂×C₄×D₆₁	D₈	1	1	yes	characteristic
88	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
89	Q₁₆×C₆₁	—	2	2	no	—
90	Q₁₆×C₆₁	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₆₁	—	2	2	no	—
94	C₄⋊C₄×C₆₁	—	4	2	no	—
95	C₄²×C₆₁	—	4	2	no	—
96	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
97	#11758201	—	2	8	no	—
98	#11758198	—	4	2	no	—
99	#11758198	—	4	2	no	—
100	#11758198	—	4	2	no	—
101	C₄⋊Dic₆₁	—	2	2	no	—
102	C₄⋊Dic₆₁	—	4	2	no	—
103	C₄×Dic₆₁	—	4	2	no	—
104	C₄×Dic₆₁	—	4	2	no	—
105	C₄×Dic₆₁	D₈	1	1	yes	characteristic
106	C₄×Dic₆₁	D₈	1	1	yes	characteristic
107	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
108	C₂×#92418	C₂×D₄	1	1	yes	characteristic
109	C₂×#92418	C₂×D₄	1	1	yes	characteristic
110	D₄₈₈	—	2	2	no	—
111	D₄₈₈	C₂²×C₄	1	4	yes	—
112	C₈×D₆₁	—	2	2	no	—
113	C₈×D₆₁	C₂²×C₄	1	2	yes	—
114	C₈×D₆₁	#41	1	2	yes	—
115	C₁₆×C₆₁	—	2	1	no	—
116	C₁₆×C₆₁	C₂²×C₄	1	2	yes	—
117	#11758185	—	2	1	no	—
118	#11758185	C₂²×C₄	1	2	yes	—
119	C₂²×D₆₁	—	4	2	no	—
120	C₂²×D₆₁	C₂×D₈	1	1	yes	characteristic
121	Q₈×C₆₁	—	2	4	no	—
122	Q₈×C₆₁	—	4	2	no	—
123	C₂×C₄×C₆₁	—	4	2	no	—
124	C₂×C₄×C₆₁	—	4	2	no	—
125	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
128	C₂×Dic₆₁	—	4	2	no	—
129	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
132	D₂₄₄	—	2	4	no	—
133	D₂₄₄	—	4	8	no	—
134	D₂₄₄	—	4	2	no	—
135	C₄×D₆₁	—	2	2	no	—
136	C₄×D₆₁	—	2	2	no	—
137	C₄×D₆₁	—	4	8	no	—
138	C₄×D₆₁	—	8	4	no	—
139	C₄×D₆₁	—	8	4	no	—
140	C₄×D₆₁	C₄×D₄	1	2	yes	—
141	C₄×D₆₁	#135	1	2	yes	—
142	C₈×C₆₁	—	2	1	no	—
143	C₈×C₆₁	C₂³×C₄	1	1	yes	characteristic
144	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
145	#92418	—	2	1	no	—
146	#92418	C₄×D₄	1	2	yes	—
147	C₄×#508	—	61	1	no	maximal, Sylow
148	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
149	C₂×D₆₁	—	2	1	no	—
150	C₂×D₆₁	—	2	1	no	—
151	C₂×D₆₁	—	4	4	no	—
152	C₂×D₆₁	—	8	2	no	—
153	C₂×D₆₁	—	8	2	no	—
154	C₂×D₆₁	C₄×D₈	1	2	yes	—
155	C₄×C₆₁	—	2	1	no	—
156	C₄×C₆₁	—	4	4	no	—
157	C₄×C₆₁	—	8	2	no	—
158	C₄×C₆₁	#508	1	2	yes	complemented
159	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
160	C₄×C₆₁	C₂×#135	1	1	yes	characteristic
161	Dic₆₁	—	2	1	no	—
162	Dic₆₁	—	8	2	no	—
163	Dic₆₁	C₄×D₈	1	2	yes	—
164	Dic₆₁	#508	1	2	yes	complemented
165	C₂×#508	—	61	1	no	—
166	C₄×#135	—	61	2	no	—
167	C₄×Q₃₂	—	61	1	no	—
168	C₄×Q₃₂	D₆₁	1	1	yes	characteristic, complemented
169	C₄×SD₃₂	—	61	2	no	—
170	C₄×SD₃₂	—	61	2	no	—
171	C₄×D₁₆	—	61	1	no	—
172	C₄×D₁₆	—	61	1	no	—
173	#2158	—	61	1	no	—
174	#2138	—	61	2	no	—
175	C₂×C₄×C₁₆	—	61	1	no	—
176	C₂×C₆₁	C₂×C₄×D₈	1	1	yes	characteristic
177	C₂×C₆₁	C₂×#508	1	1	yes	characteristic
178	C₂×C₆₁	C₂×#508	1	1	yes	characteristic
179	D₆₁	—	2	2	no	—
180	D₆₁	—	8	4	no	—
181	C₂×#135	—	61	2	no	—
182	C₄×#41	—	122	2	no	—
183	#508	—	61	8	no	—
184	C₂×Q₃₂	C₂×D₆₁	1	1	yes	characteristic
185	C₂×SD₃₂	—	61	2	no	—
186	C₂×D₁₆	—	61	1	no	—
187	C₂²×C₁₆	—	61	1	no	—
188	C₂²×C₁₆	—	61	1	no	—
189	C₄×Q₁₆	—	61	2	no	—
190	C₄×Q₁₆	C₂×D₆₁	1	2	yes	—
191	C₄×SD₁₆	—	122	2	no	—
192	C₄×SD₁₆	—	122	2	no	—
193	C₄×D₈	—	61	2	no	—
194	C₄×D₈	—	61	2	no	—
195	#427	—	61	2	no	—
196	#416	—	122	2	no	—
197	C₂×C₄×C₈	—	61	1	no	—
198	#367	—	61	2	no	—
199	#366	—	61	1	no	—
200	#366	C₂×D₆₁	1	1	yes	characteristic
201	#358	—	61	2	no	—
202	#358	C₂×D₆₁	1	2	yes	—
203	#357	—	61	2	no	—
204	#357	—	61	2	no	—
205	C₄×C₁₆	—	61	1	no	—
206	C₄×C₁₆	—	61	1	no	—
207	C₄×C₁₆	—	61	1	no	—
208	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
209	C₆₁	C₄×#508	1	1	yes	characteristic, complemented, Sylow
210	C₂×#41	—	122	2	no	—
211	#135	—	61	8	no	—
212	#135	—	122	4	no	—
213	C₂×Q₁₆	C₂²×D₆₁	1	2	yes	—
214	C₂×SD₁₆	—	122	2	no	—
215	C₂×D₈	—	61	2	no	—
216	C₂²×C₈	—	61	1	no	—
217	C₂²×C₈	—	61	1	no	—
218	C₄×Q₈	—	2	2	no	—
219	C₄×Q₈	—	122	2	no	—
220	C₄×D₄	—	122	2	no	—
221	C₄×D₄	—	122	2	no	—
222	C₄×D₄	—	244	2	no	—
223	C₄×D₄	—	244	2	no	—
224	#117	—	122	2	no	—
225	#117	—	244	2	no	—
226	C₂×C₄²	—	61	1	no	—
227	C₂×C₄²	—	244	2	no	—
228	Q₃₂	C₄×D₆₁	1	4	yes	complemented
229	SD₃₂	—	61	8	no	—
230	D₁₆	—	61	4	no	—
231	C₂×C₁₆	—	61	1	no	—
232	C₂×C₁₆	—	61	4	no	—
233	C₂×C₁₆	—	61	1	no	—
234	C₂×C₁₆	—	122	2	no	—
235	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
236	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
237	#107	—	61	2	no	—
238	#107	C₂²×D₆₁	1	2	yes	—
239	#106	—	122	2	no	—
240	#103	—	2	2	no	—
241	#103	—	122	2	no	—
242	#102	—	122	2	no	—
243	#102	—	122	2	no	—
244	C₄×C₈	—	61	1	no	—
245	C₄×C₈	—	61	1	no	—
246	C₄×C₈	—	61	1	no	—
247	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
248	#41	—	122	8	no	—
249	#41	—	244	4	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×D₄	—	122	2	no	—
252	C₂×D₄	—	244	2	no	—
253	C₂²×C₄	—	61	1	no	—
254	C₂²×C₄	—	61	1	no	—
255	C₂²×C₄	—	61	1	no	—
256	C₂²×C₄	—	244	2	no	—
257	C₂²×C₄	—	244	2	no	—
258	C₂²×C₄	—	244	2	no	—
259	Q₁₆	—	2	2	no	—
260	Q₁₆	C₂×C₄×D₆₁	1	4	yes	—
261	SD₁₆	—	122	8	no	—
262	D₈	—	61	4	no	—
263	D₈	—	122	2	no	—
264	C₂×C₈	—	61	1	no	—
265	C₂×C₈	—	61	2	no	—
266	C₂×C₈	—	61	2	no	—
267	C₂×C₈	—	61	1	no	—
268	C₂×C₈	—	122	2	no	—
269	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
270	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	122	2	no	—
274	C₄⋊C₄	—	244	2	no	—
275	C₂²⋊C₄	—	244	2	no	—
276	C₂²⋊C₄	—	244	2	no	—
277	C₂²⋊C₄	—	244	2	no	—
278	C₄²	—	4	2	no	—
279	C₄²	—	61	1	no	—
280	C₄²	—	61	1	no	—
281	C₄²	—	61	1	no	—
282	C₄²	—	244	2	no	—
283	C₄²	—	244	2	no	—
284	C₄²	D₄×D₆₁	1	1	yes	characteristic
285	C₁₆	—	2	1	no	—
286	C₁₆	—	61	2	no	—
287	C₁₆	—	122	1	no	—
288	C₁₆	C₂×C₄×D₆₁	1	2	yes	—
289	C₂³	—	61	1	no	—
290	C₂³	—	244	2	no	—
291	Q₈	—	2	4	no	—
292	Q₈	—	4	2	no	—
293	D₄	—	122	4	no	—
294	D₄	—	244	8	no	—
295	D₄	—	244	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	61	1	no	—
299	C₂×C₄	—	61	1	no	—
300	C₂×C₄	—	61	2	no	—
301	C₂×C₄	—	61	2	no	—
302	C₂×C₄	—	61	1	no	—
303	C₂×C₄	—	122	2	no	—
304	C₂×C₄	—	122	2	no	—
305	C₂×C₄	—	488	4	no	—
306	C₂×C₄	—	488	4	no	—
307	C₂×C₄	—	244	8	no	—
308	C₂×C₄	—	244	2	no	—
309	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
310	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
311	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	61	2	no	—
314	C₈	—	122	1	no	—
315	C₈	C₂²×C₄×D₆₁	1	1	yes	characteristic
316	C₈	#41×D₆₁	1	1	yes	characteristic
317	C₂²	—	61	2	no	—
318	C₂²	—	122	1	no	—
319	C₂²	—	122	1	no	—
320	C₂²	—	488	2	no	—
321	C₂²	—	488	2	no	—
322	C₂²	—	244	4	no	—
323	C₂²	C₂×D₈×D₆₁	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	4	4	no	—
326	C₄	—	8	2	no	—
327	C₄	—	61	2	no	—
328	C₄	—	61	2	no	—
329	C₄	—	122	1	no	—
330	C₄	—	488	2	no	—
331	C₄	#16840195	1	2	yes	direct product factor
332	C₄	C₄×D₄×D₆₁	1	1	yes	characteristic
333	C₄	#135×D₆₁	1	1	yes	characteristic
334	C₂	—	122	2	no	—
335	C₂	—	488	4	no	—
336	C₂	C₄×D₈×D₆₁	1	1	yes	characteristic
337	C₂	#508×D₆₁	1	1	yes	characteristic
338	C₂	C₂×#16840195	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₆₁
commutator	C₈×C₆₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₄×Q₃₂×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×D₈
prime radical	C₂×#16840195	C₂
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	C₄×Q₃₂	D₆₁
hypocenter	C₆₁	C₄×#508
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
2. C₄×#14519800	C₂	semidirect
3. Q₃₂×Dic₆₁	C₂	semidirect
4. C₄×#14519720	C₂	semidirect
5. #21447668	C₂	semidirect
6. C₄×D₉₇₆	C₂	semidirect
7. #21447665	C₂	semidirect
8. #21447600	C₂	semidirect
9. #21447525	C₂	semidirect
10. C₄×C₁₆×D₆₁	C₂	semidirect
11. C₄×Q₃₂×C₆₁	C₂	semidirect
14. #16840195	C₄	direct (N × H)
42. #14519800	C₂×C₄	semidirect
56. #14519720	C₂×C₄	semidirect
57. D₉₇₆	C₂×C₄	semidirect
61. C₁₆×D₆₁	C₂×C₄	semidirect
72. Q₃₂×C₆₁	C₂×C₄	semidirect
158. C₄×C₆₁	#508	semidirect
164. Dic₆₁	#508	semidirect
168. C₄×Q₃₂	D₆₁	semidirect
209. C₆₁	C₄×#508	semidirect
228. Q₃₂	C₄×D₆₁	semidirect
331. C₄	#16840195	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×#16840195	2
C₄×#14519800	2
Q₃₂×Dic₆₁	2
C₄×#14519720	2
#21447668	2
C₄×D₉₇₆	2
#21447665	2
#21447600	2
#21447525	2
C₄×C₁₆×D₆₁	2
C₄×Q₃₂×C₆₁	2
C₄×#508	61

Subgroups — C4×#16840195 #29833295

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16840195 #29833295