Subgroups — C₂×#21447720 #29833457

Order 15616.

710 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840198	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839883	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16839874	C₂	1	1	yes	maximal, characteristic, complemented
4	#21447720	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16839586	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16839573	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16839539	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₆₁×#361	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14519799	C₂²	1	1	yes	characteristic
10	C₂×#14519785	—	2	1	no	—
11	C₂×D₄×Dic₆₁	—	2	1	no	—
12	#16840198	C₄	1	4	yes	complemented
13	#16840198	C₂²	1	4	yes	—
14	C₂×#14519721	C₄	1	2	yes	complemented
15	C₂×#14519718	C₄	1	2	yes	complemented
16	C₂×#14519717	C₂²	1	1	yes	characteristic
17	#16839883	—	2	2	no	—
18	#16839883	C₂²	1	4	yes	complemented
19	#16839874	—	2	2	no	—
20	#16839874	C₂²	1	4	yes	complemented
21	C₂×#14519709	C₂²	1	1	yes	characteristic
22	C₂×#14519703	—	2	1	no	—
23	C₂×#14519698	—	2	1	no	—
24	C₂×#14519687	C₂²	1	1	yes	characteristic
25	#16839586	C₂²	1	4	yes	complemented
26	#16839573	C₂²	1	4	yes	complemented
27	#16839539	C₂²	1	4	yes	complemented
28	C₂²×D₈×C₆₁	C₂²	1	1	yes	characteristic
29	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
30	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
31	C₆₁×#361	C₂²	1	4	yes	complemented
32	C₂×#11758222	—	2	1	no	—
33	C₂³×Dic₆₁	—	4	1	no	—
34	#14519799	—	2	1	no	—
35	#14519799	—	2	1	no	—
36	#14519799	C₂×C₄	1	2	yes	complemented
37	#14519799	C₂³	1	2	yes	—
38	C₂×Dic₂₄₄	C₂×C₄	1	1	yes	characteristic, complemented
39	C₂×#11758190	—	2	1	no	—
40	C₂×D₄₈₈	C₂×C₄	1	1	yes	characteristic, complemented
41	#14519785	—	2	4	no	—
42	#14519785	—	4	2	no	—
43	C₂×#11758189	—	4	1	no	—
44	C₂²×#92418	D₄	1	1	yes	characteristic
45	C₂×C₈×D₆₁	—	4	1	no	—
46	D₄×Dic₆₁	—	2	4	no	—
47	D₄×Dic₆₁	—	4	2	no	—
48	C₂×C₄⋊Dic₆₁	—	2	1	no	—
49	C₂×C₂²⋊Dic₆₁	—	4	1	no	—
50	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
51	#14519721	C₂×C₄	1	8	yes	complemented
52	#14519718	C₂×C₄	1	8	yes	complemented
53	#14519717	C₂×C₄	1	2	yes	complemented
54	#14519717	C₂³	1	2	yes	—
55	C₂×#11758185	C₂×C₄	1	2	yes	complemented
56	#14519709	—	2	1	no	—
57	#14519709	C₂³	1	2	yes	—
58	#14519703	—	2	4	no	—
59	#14519698	—	2	4	no	—
60	#14519687	D₄	1	2	yes	—
61	#14519687	C₂³	1	2	yes	—
62	C₂²×D₄×C₆₁	—	2	1	no	—
63	C₂×D₈×C₆₁	—	2	1	no	—
64	C₂×D₈×C₆₁	—	2	1	no	—
65	C₂×D₈×C₆₁	—	2	1	no	—
66	C₂×D₈×C₆₁	C₂×C₄	1	2	yes	—
67	C₂×D₈×C₆₁	C₂×C₄	1	2	yes	—
68	C₂×D₈×C₆₁	C₂³	1	2	yes	—
69	C₂×M₁₆×C₆₁	—	2	1	no	—
70	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
71	M₃₂×C₆₁	—	2	1	no	—
72	M₃₂×C₆₁	C₂³	1	2	yes	—
73	C₂×C₁₆×C₆₁	—	2	1	no	—
74	#108×C₆₁	—	2	1	no	—
75	#108×C₆₁	C₂³	1	2	yes	—
76	C₂⁴×C₆₁	—	4	1	no	—
77	C₂×D₄×C₆₁	—	2	2	no	—
78	C₂×D₄×C₆₁	—	2	2	no	—
79	C₂×D₄×C₆₁	—	2	2	no	—
80	C₂×D₄×C₆₁	—	4	1	no	—
81	C₂×D₄×C₆₁	—	4	1	no	—
82	C₂×D₄×C₆₁	—	4	1	no	—
83	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
84	C₂×C₂²⋊D₆₁	—	4	1	no	—
85	C₂²×Dic₆₁	—	2	1	no	—
86	C₂²×Dic₆₁	—	4	4	no	—
87	C₂²×Dic₆₁	—	4	1	no	—
88	C₂²×Dic₆₁	—	8	4	no	—
89	#11758222	—	2	2	no	—
90	#11758222	—	2	2	no	—
91	#11758222	—	4	1	no	—
92	#11758222	—	4	1	no	—
93	C₂×D₂₄₄	—	2	1	no	—
94	C₂×C₄×D₆₁	—	4	1	no	—
95	C₂×Dic₁₂₂	—	2	1	no	—
96	D₈×C₆₁	—	2	2	no	—
97	D₈×C₆₁	—	2	2	no	—
98	D₈×C₆₁	—	2	2	no	—
99	D₈×C₆₁	C₂²×C₄	1	4	yes	—
100	M₁₆×C₆₁	—	2	2	no	—
101	M₁₆×C₆₁	—	4	1	no	—
102	C₂×C₈×C₆₁	—	4	1	no	—
103	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
107	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
108	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
109	C₂²⋊Dic₆₁	—	4	4	no	—
110	C₄⋊Dic₆₁	—	2	2	no	—
111	C₄⋊Dic₆₁	—	4	1	no	—
112	C₄×Dic₆₁	—	2	1	no	—
113	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
114	C₂×#92418	—	2	1	no	—
115	C₂×#92418	—	2	1	no	—
116	C₂×#92418	C₂×D₄	1	2	yes	—
117	Dic₂₄₄	—	2	1	no	—
118	Dic₂₄₄	C₂²×C₄	1	2	yes	complemented
119	D₄₈₈	—	2	1	no	—
120	D₄₈₈	C₂²×C₄	1	2	yes	complemented
121	#11758190	—	2	2	no	—
122	#11758190	—	2	2	no	—
123	#11758189	—	4	4	no	—
124	C₈×D₆₁	—	4	4	no	—
125	C₁₆×C₆₁	—	2	2	no	—
126	#11758185	C₂²×C₄	1	4	yes	complemented
127	C₂³×C₆₁	—	4	2	no	—
128	C₂³×C₆₁	—	4	2	no	—
129	C₂³×C₆₁	—	4	2	no	—
130	C₂³×C₆₁	—	8	2	no	—
131	C₂³×C₆₁	—	8	2	no	—
132	C₂³×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂²×D₆₁	—	4	1	no	—
134	D₄×C₆₁	—	2	4	no	—
135	D₄×C₆₁	—	4	2	no	—
136	D₄×C₆₁	—	4	2	no	—
137	D₄×C₆₁	—	4	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	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
144	C₂²⋊D₆₁	—	4	2	no	—
145	C₂²⋊D₆₁	—	4	2	no	—
146	C₂×Dic₆₁	—	2	2	no	—
147	C₂×Dic₆₁	—	4	2	no	—
148	C₂×Dic₆₁	—	4	1	no	—
149	C₂×Dic₆₁	—	4	2	no	—
150	C₂×Dic₆₁	—	8	8	no	—
151	C₂×Dic₆₁	—	8	2	no	—
152	D₂₄₄	—	2	2	no	—
153	D₂₄₄	—	4	1	no	—
154	C₄×D₆₁	—	4	2	no	—
155	C₄×D₆₁	—	4	2	no	—
156	Dic₁₂₂	—	2	2	no	—
157	Dic₁₂₂	—	4	1	no	—
158	C₈×C₆₁	—	4	2	no	—
159	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
160	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
161	C₈×C₆₁	C₂³×C₄	1	1	yes	characteristic
162	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
163	#92418	—	2	2	no	—
164	C₂×AGL(1,ℤ/16)	—	61	1	no	maximal, Sylow
165	C₂²×C₆₁	—	2	2	no	—
166	C₂²×C₆₁	—	4	4	no	—
167	C₂²×C₆₁	—	8	2	no	—
168	C₂²×C₆₁	—	8	2	no	—
169	C₂²×C₆₁	—	8	4	no	—
170	C₂²×C₆₁	—	8	4	no	—
171	C₂²×C₆₁	#442	1	1	yes	characteristic
172	C₂²×C₆₁	#442	1	1	yes	characteristic
173	C₂²×C₆₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
174	C₂×D₆₁	—	4	2	no	—
175	C₂×D₆₁	—	8	2	no	—
176	C₄×C₆₁	#442	1	1	yes	characteristic
177	C₄×C₆₁	#442	1	1	yes	characteristic
178	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
179	C₄×C₆₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
180	Dic₆₁	—	4	2	no	—
181	Dic₆₁	—	4	2	no	—
182	Dic₆₁	—	8	2	no	—
183	C₂×#509	—	61	1	no	—
184	C₂×#444	—	61	1	no	—
185	C₂×#442	—	61	1	no	—
186	AGL(1,ℤ/16)	—	61	8	no	—
187	C₂×#365	—	61	1	no	—
188	C₂×#361	D₆₁	1	1	yes	characteristic, complemented
189	C₂×#360	—	61	1	no	—
190	C₂×#347	—	61	1	no	—
191	C₂×C₆₁	—	2	1	no	—
192	C₂×C₆₁	—	2	1	no	—
193	C₂×C₆₁	—	8	4	no	—
194	C₂×C₆₁	AGL(1,ℤ/16)	1	2	yes	complemented
195	C₂×C₆₁	C₂×#442	1	1	yes	characteristic
196	D₆₁	—	8	2	no	—
197	C₂×#135	—	61	1	no	—
198	C₂²×D₈	C₂×D₆₁	1	1	yes	characteristic
199	C₂×#131	—	122	1	no	—
200	C₂×C₄×D₄	—	122	1	no	—
201	#509	—	61	4	no	—
202	#509	—	61	4	no	—
203	C₂×SD₃₂	—	61	2	no	—
204	C₂×D₁₆	—	61	2	no	—
205	C₂×M₃₂	—	61	1	no	—
206	C₂×M₃₂	C₂×D₆₁	1	1	yes	characteristic
207	#444	—	61	4	no	—
208	#444	—	122	2	no	—
209	#442	—	61	4	no	—
210	#442	—	122	2	no	—
211	C₂×#108	C₂×D₆₁	1	1	yes	characteristic
212	C₂×#106	—	61	1	no	—
213	C₂×#104	—	122	1	no	—
214	C₂×#102	—	122	1	no	—
215	C₂×#97	—	61	1	no	—
216	#365	—	61	4	no	—
217	#361	C₂×D₆₁	1	4	yes	complemented
218	#360	—	61	4	no	—
219	#347	—	61	4	no	—
220	C₆₁	C₂×AGL(1,ℤ/16)	1	1	yes	characteristic, complemented, Sylow
221	C₂×#41	—	122	1	no	—
222	C₂²×D₄	—	2	1	no	—
223	C₂³×C₄	—	244	1	no	—
224	#135	—	61	2	no	—
225	#135	—	61	2	no	—
226	#135	—	122	1	no	—
227	#135	—	122	1	no	—
228	C₂×Q₁₆	—	61	1	no	—
229	C₂×SD₁₆	—	122	1	no	—
230	C₂×D₈	—	2	1	no	—
231	C₂×D₈	—	2	1	no	—
232	C₂×D₈	—	2	1	no	—
233	C₂×D₈	—	61	1	no	—
234	C₂×D₈	C₄×D₆₁	1	2	yes	—
235	C₂×D₈	C₄×D₆₁	1	2	yes	—
236	C₂×D₈	C₂²×D₆₁	1	2	yes	—
237	#131	—	122	4	no	—
238	#131	—	244	2	no	—
239	C₂×M₁₆	—	2	1	no	—
240	C₂×M₁₆	—	244	1	no	—
241	C₂²×C₈	—	61	1	no	—
242	C₂²×C₈	—	244	1	no	—
243	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
244	C₄×D₄	—	122	4	no	—
245	C₄×D₄	—	244	2	no	—
246	C₂×C₄⋊C₄	—	122	1	no	—
247	C₂×C₂²⋊C₄	—	244	1	no	—
248	C₂×C₄²	—	61	1	no	—
249	SD₃₂	—	61	8	no	—
250	D₁₆	—	61	8	no	—
251	M₃₂	—	2	1	no	—
252	M₃₂	—	61	2	no	—
253	M₃₂	—	61	2	no	—
254	M₃₂	C₂²×D₆₁	1	2	yes	—
255	C₂×C₁₆	—	2	1	no	—
256	C₂×C₁₆	—	61	2	no	—
257	#108	—	2	1	no	—
258	#108	C₂²×D₆₁	1	2	yes	—
259	#106	—	61	2	no	—
260	#106	—	122	1	no	—
261	#104	—	122	4	no	—
262	#102	—	122	4	no	—
263	#97	—	61	2	no	—
264	#97	—	61	2	no	—
265	C₂⁴	—	4	1	no	—
266	#41	—	122	2	no	—
267	#41	—	122	2	no	—
268	#41	—	244	1	no	—
269	#41	—	244	1	no	—
270	C₂×Q₈	—	122	1	no	—
271	C₂×D₄	—	2	2	no	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	—	2	2	no	—
274	C₂×D₄	—	4	1	no	—
275	C₂×D₄	—	4	1	no	—
276	C₂×D₄	—	4	1	no	—
277	C₂×D₄	—	122	1	no	—
278	C₂×D₄	—	244	1	no	—
279	C₂²×C₄	—	122	1	no	—
280	C₂²×C₄	—	488	4	no	—
281	C₂²×C₄	—	244	4	no	—
282	C₂²×C₄	—	244	1	no	—
283	C₂²×C₄	—	244	1	no	—
284	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
285	Q₁₆	—	61	2	no	—
286	Q₁₆	—	122	1	no	—
287	SD₁₆	—	122	2	no	—
288	SD₁₆	—	122	2	no	—
289	D₈	—	2	2	no	—
290	D₈	—	2	2	no	—
291	D₈	—	2	2	no	—
292	D₈	—	61	2	no	—
293	D₈	—	122	1	no	—
294	D₈	C₂×C₄×D₆₁	1	4	yes	—
295	M₁₆	—	2	2	no	—
296	M₁₆	—	4	1	no	—
297	M₁₆	—	244	4	no	—
298	C₂×C₈	—	4	1	no	—
299	C₂×C₈	—	61	2	no	—
300	C₂×C₈	—	122	1	no	—
301	C₂×C₈	—	122	1	no	—
302	C₂×C₈	—	244	4	no	—
303	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
304	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
305	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
306	C₂×C₈	#11758224	1	1	yes	characteristic
307	C₂×C₈	#11758224	1	1	yes	characteristic
308	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
309	C₄⋊C₄	—	122	2	no	—
310	C₄⋊C₄	—	244	1	no	—
311	C₂²⋊C₄	—	244	4	no	—
312	C₄²	—	61	2	no	—
313	C₄²	—	122	1	no	—
314	C₁₆	—	2	2	no	—
315	C₁₆	—	61	4	no	—
316	C₂³	—	4	2	no	—
317	C₂³	—	4	2	no	—
318	C₂³	—	4	2	no	—
319	C₂³	—	8	2	no	—
320	C₂³	—	8	2	no	—
321	C₂³	—	244	1	no	—
322	C₂³	#14519804	1	1	yes	characteristic
323	Q₈	—	122	2	no	—
324	Q₈	—	244	1	no	—
325	D₄	—	2	4	no	—
326	D₄	—	4	2	no	—
327	D₄	—	4	2	no	—
328	D₄	—	4	2	no	—
329	D₄	—	122	2	no	—
330	D₄	—	244	2	no	—
331	D₄	—	244	2	no	—
332	D₄	—	244	1	no	—
333	C₂×C₄	—	122	2	no	—
334	C₂×C₄	—	488	2	no	—
335	C₂×C₄	—	488	8	no	—
336	C₂×C₄	—	244	2	no	—
337	C₂×C₄	—	244	2	no	—
338	C₂×C₄	—	244	1	no	—
339	C₂×C₄	—	244	2	no	—
340	C₂×C₄	—	244	2	no	—
341	C₂×C₄	#14519737	1	1	yes	characteristic
342	C₂×C₄	#14519737	1	1	yes	characteristic
343	C₂×C₄	#14519801	1	1	yes	characteristic
344	C₂×C₄	#14519801	1	1	yes	characteristic
345	C₂×C₄	#14519804	1	1	yes	characteristic
346	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
347	C₈	—	4	2	no	—
348	C₈	—	122	2	no	—
349	C₈	#14519737	1	1	yes	characteristic
350	C₈	#14519737	1	1	yes	characteristic
351	C₈	C₂²×C₄×D₆₁	1	1	yes	characteristic
352	C₈	C₂×#11758224	1	1	yes	characteristic
353	C₂²	—	2	2	no	—
354	C₂²	—	4	4	no	—
355	C₂²	—	8	2	no	—
356	C₂²	—	8	2	no	—
357	C₂²	—	8	4	no	—
358	C₂²	—	8	4	no	—
359	C₂²	—	488	2	no	—
360	C₂²	—	244	2	no	—
361	C₂²	#16839871	1	1	yes	characteristic
362	C₂²	#16839871	1	1	yes	characteristic
363	C₂²	C₂×#14519804	1	1	yes	characteristic
364	C₄	—	488	2	no	—
365	C₄	—	244	2	no	—
366	C₄	—	244	2	no	—
367	C₄	#16839871	1	1	yes	characteristic
368	C₄	#16839871	1	1	yes	characteristic
369	C₄	C₂×#14519737	1	1	yes	characteristic
370	C₄	C₂×#14519801	1	1	yes	characteristic
371	C₂	—	2	1	no	—
372	C₂	—	2	1	no	—
373	C₂	—	8	4	no	—
374	C₂	—	488	2	no	—
375	C₂	#21447720	1	2	yes	direct product factor
376	C₂	C₂×#16839871	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16839871
commutator	C₈×C₆₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₂×C₆₁×#361	C₂
socle	C₂²×C₆₁	#442
prime radical	C₂×#16840198	C₂
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	C₂×#361	D₆₁
hypocenter	C₆₁	C₂×AGL(1,ℤ/16)
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#361

Complements

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

N	H ≅ G/N	type
2. C₂×#16839883	C₂	semidirect
3. C₂×#16839874	C₂	semidirect
4. #21447720	C₂	direct (N × H)
5. C₂×#16839586	C₂	semidirect
6. C₂×#16839573	C₂	semidirect
7. C₂×#16839539	C₂	semidirect
8. C₂×C₆₁×#361	C₂	semidirect
12. #16840198	C₄	semidirect
14. C₂×#14519721	C₄	semidirect
15. C₂×#14519718	C₄	semidirect
18. #16839883	C₂²	semidirect
20. #16839874	C₂²	semidirect
25. #16839586	C₂²	semidirect
26. #16839573	C₂²	semidirect
27. #16839539	C₂²	semidirect
31. C₆₁×#361	C₂²	semidirect
36. #14519799	C₂×C₄	semidirect
38. C₂×Dic₂₄₄	C₂×C₄	semidirect
40. C₂×D₄₈₈	C₂×C₄	semidirect
51. #14519721	C₂×C₄	semidirect
52. #14519718	C₂×C₄	semidirect
53. #14519717	C₂×C₄	semidirect
55. C₂×#11758185	C₂×C₄	semidirect
118. Dic₂₄₄	C₂²×C₄	semidirect
120. D₄₈₈	C₂²×C₄	semidirect
126. #11758185	C₂²×C₄	semidirect
188. C₂×#361	D₆₁	semidirect
194. C₂×C₆₁	AGL(1,ℤ/16)	semidirect
217. #361	C₂×D₆₁	semidirect
220. C₆₁	C₂×AGL(1,ℤ/16)	semidirect
375. C₂	#21447720	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×#16840198	2
C₂×#16839883	2
C₂×#16839874	2
#21447720	2
C₂×#16839586	2
C₂×#16839573	2
C₂×#16839539	2
C₂×C₆₁×#361	2
C₂×AGL(1,ℤ/16)	61

Subgroups — C2×#21447720 #29833457

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21447720 #29833457