Subgroups — C₂×#21461383 #30612039

Order 15616.

758 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840783	C₂	1	1	yes	maximal, characteristic
2	C₂×#16840765	C₂	1	1	yes	maximal, characteristic
3	#21461383	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16840626	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16840625	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16840624	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16840623	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×Dic₆₁	C₂	1	1	yes	maximal, characteristic
9	C₂²×#11758217	—	2	1	no	—
10	#16840783	—	2	2	no	—
11	#16840783	C₂²	1	4	yes	—
12	#16840765	—	2	2	no	—
13	#16840765	C₂²	1	4	yes	—
14	C₂×#14519844	—	2	1	no	—
15	C₂×C₄⋊#92420	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊#92420	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊#92420	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊#92420	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊#92420	—	2	1	no	—
20	C₂×#14519832	—	2	1	no	—
21	#16840626	C₂²	1	4	yes	complemented
22	#16840625	C₂²	1	4	yes	complemented
23	#16840624	C₂²	1	4	yes	complemented
24	#16840623	C₂²	1	4	yes	complemented
25	C₂²×#11758194	C₄	1	1	yes	characteristic
26	C₂²×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
27	M₁₆×Dic₆₁	C₄	1	4	yes	—
28	M₁₆×Dic₆₁	C₂²	1	4	yes	—
29	C₂×#14519687	C₄	1	1	yes	characteristic
30	C₂×#14519687	C₄	1	1	yes	characteristic
31	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
32	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
33	C₂²×M₁₆×C₆₁	C₄	1	1	yes	characteristic
34	C₂×#11758217	—	2	4	no	—
35	C₂×#11758217	—	4	4	no	—
36	C₂²×#92420	—	2	1	no	—
37	C₂²×#92420	—	2	1	no	—
38	C₂²×#92420	—	2	1	no	—
39	C₂²×#92420	—	2	1	no	—
40	#14519844	—	2	4	no	—
41	C₄⋊#92420	—	2	1	no	—
42	C₄⋊#92420	—	2	1	no	—
43	C₄⋊#92420	—	2	1	no	—
44	C₄⋊#92420	—	2	1	no	—
45	C₄⋊#92420	C₂³	1	2	yes	—
46	C₄⋊#92420	C₂³	1	2	yes	—
47	C₄⋊#92420	C₂³	1	2	yes	—
48	C₄⋊#92420	C₂³	1	2	yes	—
49	C₂²⋊#92420	—	2	4	no	—
50	#14519832	—	2	4	no	—
51	C₂³×Dic₆₁	D₄	1	1	yes	characteristic
52	C₂³×Dic₆₁	Q₈	1	1	yes	characteristic
53	C₂×#11758194	—	2	4	no	—
54	C₂×#11758194	C₂×C₄	1	4	yes	—
55	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
56	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
59	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
60	C₂×C₄×Dic₆₁	D₄	1	2	yes	—
61	C₂×C₄×Dic₆₁	Q₈	1	1	yes	characteristic
62	C₂×C₄×Dic₆₁	Q₈	1	2	yes	—
63	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
64	C₂×C₄×Dic₆₁	C₂³	1	2	yes	—
65	#14519687	C₂×C₄	1	4	yes	—
66	#14519687	C₂×C₄	1	4	yes	—
67	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
68	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
69	C₈×Dic₆₁	C₂³	1	2	yes	—
70	C₈×Dic₆₁	C₂³	1	2	yes	—
71	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
72	C₂×M₁₆×C₆₁	—	2	4	no	—
73	C₂×M₁₆×C₆₁	C₂×C₄	1	4	yes	—
74	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
75	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
76	C₂⁴×C₆₁	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₁	—	2	2	no	—
78	C₂²×C₄×C₆₁	—	2	2	no	—
79	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
81	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
83	C₂²×Dic₆₁	—	2	2	no	—
84	C₂²×Dic₆₁	—	2	2	no	—
85	C₂²×Dic₆₁	—	4	2	no	—
86	C₂²×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×Dic₆₁	C₄⋊C₄	1	2	yes	—
88	C₂²×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
89	C₂²×Dic₆₁	C₄⋊C₄	1	2	yes	—
90	C₂²×Dic₆₁	C₂×D₄	1	2	yes	—
91	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
92	C₂²×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
93	C₂²×Dic₆₁	C₂×Q₈	1	2	yes	—
94	#11758217	—	4	8	no	—
95	C₂×#92420	—	2	2	no	—
96	C₂×#92420	—	2	2	no	—
97	C₂×#92420	—	2	2	no	—
98	C₂×#92420	—	2	2	no	—
99	C₂×#92420	—	4	2	no	—
100	C₂×#92420	—	4	2	no	—
101	C₂×#92420	—	4	2	no	—
102	C₂×#92420	—	4	2	no	—
103	M₁₆×C₆₁	—	2	8	no	—
104	C₂×C₈×C₆₁	—	2	2	no	—
105	C₂×C₈×C₆₁	M₁₆	1	4	yes	complemented
106	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
107	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
108	C₄×Dic₆₁	C₄⋊C₄	1	2	yes	—
109	C₄×Dic₆₁	C₄⋊C₄	1	2	yes	—
110	C₄×Dic₆₁	C₄⋊C₄	1	2	yes	—
111	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
112	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
113	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
114	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
115	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
116	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
117	C₄×Dic₆₁	C₂⁴	1	1	yes	characteristic
118	#11758194	—	2	8	no	—
119	C₂×#92418	—	2	2	no	—
120	C₂×#92418	M₁₆	1	4	yes	complemented
121	C₂×#92418	C₂²×C₄	1	2	yes	—
122	C₂×#92418	C₂²×C₄	1	2	yes	—
123	C₂³×C₆₁	—	2	2	no	—
124	C₂³×C₆₁	—	2	2	no	—
125	C₂³×C₆₁	—	2	2	no	—
126	C₂³×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
127	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₆₁	—	2	2	no	—
129	C₂×C₄×C₆₁	—	2	2	no	—
130	C₂×C₄×C₆₁	—	2	2	no	—
131	C₂×C₄×C₆₁	—	2	4	no	—
132	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
133	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
135	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
136	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
137	C₂×Dic₆₁	—	2	2	no	—
138	C₂×Dic₆₁	—	2	2	no	—
139	C₂×Dic₆₁	—	2	2	no	—
140	C₂×Dic₆₁	—	2	2	no	—
141	C₂×Dic₆₁	—	2	2	no	—
142	C₂×Dic₆₁	—	2	4	no	—
143	C₂×Dic₆₁	—	4	2	no	—
144	C₂×Dic₆₁	—	4	2	no	—
145	C₂×Dic₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×Dic₆₁	C₂×C₄⋊C₄	1	2	yes	—
147	C₂×Dic₆₁	C₂×C₄⋊C₄	1	2	yes	—
148	C₂×Dic₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×Dic₆₁	AGL(1,ℤ/8)	1	2	yes	—
150	C₂×Dic₆₁	#137	1	2	yes	—
151	C₂×Dic₆₁	C₂²×D₄	1	1	yes	characteristic
152	C₂×Dic₆₁	C₂²×Q₈	1	1	yes	characteristic
153	#92420	—	4	2	no	—
154	#92420	—	4	2	no	—
155	#92420	—	4	2	no	—
156	#92420	—	4	2	no	—
157	C₈×C₆₁	—	2	2	no	—
158	C₈×C₆₁	C₂×M₁₆	1	4	yes	complemented
159	#92418	—	2	2	no	—
160	#92418	C₂×M₁₆	1	4	yes	complemented
161	C₂×#1569	—	61	1	no	maximal, Sylow
162	C₂²×C₆₁	—	2	2	no	—
163	C₂²×C₆₁	—	2	2	no	—
164	C₂²×C₆₁	—	2	2	no	—
165	C₂²×C₆₁	—	4	4	no	—
166	C₂²×C₆₁	#423	1	2	yes	—
167	C₂²×C₆₁	#428	1	2	yes	—
168	C₂²×C₆₁	#430	1	2	yes	—
169	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
170	C₄×C₆₁	—	2	2	no	—
171	C₄×C₆₁	#423	1	2	yes	—
172	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
173	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
174	Dic₆₁	—	2	2	no	—
175	Dic₆₁	—	2	2	no	—
176	Dic₆₁	—	2	1	no	—
177	Dic₆₁	—	2	1	no	—
178	Dic₆₁	#428	1	2	yes	—
179	Dic₆₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
180	Dic₆₁	C₂×#137	1	1	yes	characteristic
181	C₂×#432	—	61	1	no	—
182	C₂×#423	—	61	1	no	—
183	C₂×C₄×M₁₆	—	61	1	no	—
184	#1569	—	61	8	no	—
185	C₂×#335	—	61	1	no	—
186	C₂×#335	—	61	1	no	—
187	C₂×#334	—	61	1	no	—
188	C₂×#334	—	61	1	no	—
189	C₂×C₆₁	—	4	2	no	—
190	C₂×C₆₁	#1569	1	4	yes	complemented
191	C₂×C₆₁	C₂×#423	1	1	yes	characteristic
192	C₂×C₆₁	C₂×#428	1	1	yes	characteristic
193	C₂×C₆₁	C₂×#430	1	1	yes	characteristic
194	C₂²×M₁₆	—	61	1	no	—
195	C₂²×M₁₆	—	122	1	no	—
196	C₂²×M₁₆	#6923	1	1	yes	characteristic
197	C₂²×C₄²	—	61	1	no	—
198	#432	—	61	4	no	—
199	#432	—	122	2	no	—
200	#423	—	61	4	no	—
201	#423	—	122	2	no	—
202	C₂×C₄⋊C₈	—	61	1	no	—
203	C₂×C₄⋊C₈	—	61	1	no	—
204	C₂×C₄⋊C₈	—	61	1	no	—
205	C₂×C₄⋊C₈	—	61	1	no	—
206	C₂×C₄⋊C₈	—	122	1	no	—
207	C₂×C₂²⋊C₈	—	122	1	no	—
208	C₄×M₁₆	—	61	4	no	—
209	C₄×M₁₆	—	61	4	no	—
210	C₂×#97	—	61	1	no	—
211	C₂×#97	—	61	1	no	—
212	C₂×#97	—	122	1	no	—
213	C₂×C₄×C₈	—	61	1	no	—
214	C₂×C₄×C₈	—	61	1	no	—
215	#335	—	61	4	no	—
216	#335	—	61	4	no	—
217	#334	—	61	4	no	—
218	#334	—	61	4	no	—
219	C₆₁	C₂×#1569	1	1	yes	characteristic, complemented, Sylow
220	C₂³×C₄	—	61	1	no	—
221	C₂³×C₄	—	61	1	no	—
222	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
223	C₂×M₁₆	—	2	4	no	—
224	C₂×M₁₆	—	61	4	no	—
225	C₂×M₁₆	—	122	4	no	—
226	C₂×M₁₆	—	122	4	no	—
227	C₂×M₁₆	—	244	4	no	—
228	C₂×M₁₆	C₂×#6923	1	4	yes	—
229	C₂²×C₈	—	61	1	no	—
230	C₂²×C₈	—	61	1	no	—
231	C₂²×C₈	—	122	1	no	—
232	C₂²×C₈	—	122	1	no	—
233	C₂²×C₈	—	122	1	no	—
234	C₂²×C₈	—	122	1	no	—
235	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
236	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
237	C₂×C₄²	—	61	1	no	—
238	C₂×C₄²	—	61	2	no	—
239	C₂×C₄²	—	61	2	no	—
240	C₂×C₄²	—	61	1	no	—
241	C₂×C₄²	—	61	1	no	—
242	C₂×C₄²	—	61	1	no	—
243	C₂×C₄²	—	61	2	no	—
244	C₂×C₄²	—	61	2	no	—
245	C₄⋊C₈	—	61	2	no	—
246	C₄⋊C₈	—	61	2	no	—
247	C₄⋊C₈	—	61	2	no	—
248	C₄⋊C₈	—	61	2	no	—
249	C₄⋊C₈	—	122	1	no	—
250	C₄⋊C₈	—	122	1	no	—
251	C₄⋊C₈	—	122	1	no	—
252	C₄⋊C₈	—	122	1	no	—
253	C₄⋊C₈	—	122	4	no	—
254	C₂²⋊C₈	—	122	4	no	—
255	#97	—	61	4	no	—
256	#97	—	61	4	no	—
257	#97	—	122	4	no	—
258	C₄×C₈	—	61	2	no	—
259	C₄×C₈	—	61	2	no	—
260	C₄×C₈	—	61	2	no	—
261	C₄×C₈	—	61	2	no	—
262	C₂⁴	C₄⋊#6923	1	1	yes	characteristic
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	61	1	no	—
266	C₂²×C₄	—	61	2	no	—
267	C₂²×C₄	—	61	2	no	—
268	C₂²×C₄	—	61	1	no	—
269	C₂²×C₄	—	61	1	no	—
270	C₂²×C₄	—	61	1	no	—
271	C₂²×C₄	—	61	2	no	—
272	C₂²×C₄	—	61	2	no	—
273	C₂²×C₄	—	122	2	no	—
274	C₂²×C₄	—	122	2	no	—
275	C₂²×C₄	—	244	2	no	—
276	C₂²×C₄	C₄⋊#6923	1	2	yes	—
277	C₂²×C₄	C₄⋊#6923	1	1	yes	characteristic
278	C₂²×C₄	C₂²×#6923	1	2	yes	—
279	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
280	M₁₆	—	2	8	no	—
281	M₁₆	—	122	8	no	—
282	M₁₆	—	244	8	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	61	4	no	—
285	C₂×C₈	—	61	2	no	—
286	C₂×C₈	—	61	2	no	—
287	C₂×C₈	—	122	2	no	—
288	C₂×C₈	—	122	2	no	—
289	C₂×C₈	—	122	2	no	—
290	C₂×C₈	—	122	2	no	—
291	C₂×C₈	—	122	2	no	—
292	C₂×C₈	—	244	2	no	—
293	C₂×C₈	—	244	2	no	—
294	C₂×C₈	—	244	2	no	—
295	C₂×C₈	—	244	2	no	—
296	C₂×C₈	#11758217	1	4	yes	complemented
297	C₂×C₈	C₂²×#6923	1	2	yes	—
298	C₂×C₈	C₂²×#6923	1	2	yes	—
299	C₄²	—	61	2	no	—
300	C₄²	—	61	1	no	—
301	C₄²	—	61	2	no	—
302	C₄²	—	61	2	no	—
303	C₄²	—	61	2	no	—
304	C₄²	—	61	1	no	—
305	C₄²	—	61	2	no	—
306	C₄²	—	61	1	no	—
307	C₄²	—	61	1	no	—
308	C₄²	—	61	2	no	—
309	C₂³	—	2	2	no	—
310	C₂³	—	2	2	no	—
311	C₂³	—	2	2	no	—
312	C₂³	C₂×C₄⋊#6923	1	2	yes	—
313	C₂³	C₂×C₄⋊#6923	1	1	yes	characteristic
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	61	2	no	—
319	C₂×C₄	—	61	2	no	—
320	C₂×C₄	—	61	1	no	—
321	C₂×C₄	—	61	1	no	—
322	C₂×C₄	—	61	2	no	—
323	C₂×C₄	—	61	2	no	—
324	C₂×C₄	—	61	1	no	—
325	C₂×C₄	—	61	1	no	—
326	C₂×C₄	—	122	2	no	—
327	C₂×C₄	—	122	2	no	—
328	C₂×C₄	—	122	4	no	—
329	C₂×C₄	—	122	2	no	—
330	C₂×C₄	—	122	2	no	—
331	C₂×C₄	—	122	2	no	—
332	C₂×C₄	—	244	2	no	—
333	C₂×C₄	—	244	2	no	—
334	C₂×C₄	C₂×C₄⋊#6923	1	2	yes	—
335	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
336	C₂×C₄	C₂×#11758217	1	2	yes	—
337	C₂×C₄	D₆₁⋊M₁₆	1	2	yes	—
338	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
339	C₈	—	2	2	no	—
340	C₈	—	61	4	no	—
341	C₈	—	122	2	no	—
342	C₈	—	244	2	no	—
343	C₈	—	244	2	no	—
344	C₈	—	244	2	no	—
345	C₈	—	244	2	no	—
346	C₈	C₂×#11758217	1	4	yes	complemented
347	C₂²	—	2	2	no	—
348	C₂²	—	2	2	no	—
349	C₂²	—	2	2	no	—
350	C₂²	—	4	4	no	—
351	C₂²	#16840765	1	2	yes	—
352	C₂²	#16840773	1	2	yes	—
353	C₂²	#16840777	1	2	yes	—
354	C₂²	C₂²×C₄⋊#6923	1	1	yes	characteristic
355	C₄	—	2	2	no	—
356	C₄	—	61	1	no	—
357	C₄	—	61	1	no	—
358	C₄	—	61	2	no	—
359	C₄	—	122	2	no	—
360	C₄	—	122	2	no	—
361	C₄	—	122	1	no	—
362	C₄	—	122	1	no	—
363	C₄	#16840765	1	2	yes	—
364	C₄	C₂²×#11758217	1	1	yes	characteristic
365	C₄	C₂×D₆₁⋊M₁₆	1	1	yes	characteristic
366	C₂	—	4	2	no	—
367	C₂	#21461383	1	4	yes	direct product factor
368	C₂	C₂×#16840765	1	1	yes	characteristic
369	C₂	C₂×#16840773	1	1	yes	characteristic
370	C₂	C₂×#16840777	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#6923
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₂²×M₁₆×C₆₁	C₄
socle	C₂³×C₆₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₆₁	C₄⋊C₄
cosocle	C₄×Dic₆₁	C₂⁴
hypercenter	C₂²×M₁₆	#6923
hypocenter	C₆₁	C₂×#1569
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #21461383	C₂	direct (N × H)
4. C₂×#16840626	C₂	semidirect
5. C₂×#16840625	C₂	semidirect
6. C₂×#16840624	C₂	semidirect
7. C₂×#16840623	C₂	semidirect
21. #16840626	C₂²	semidirect
22. #16840625	C₂²	semidirect
23. #16840624	C₂²	semidirect
24. #16840623	C₂²	semidirect
105. C₂×C₈×C₆₁	M₁₆	semidirect
120. C₂×#92418	M₁₆	semidirect
158. C₈×C₆₁	C₂×M₁₆	semidirect
160. #92418	C₂×M₁₆	semidirect
190. C₂×C₆₁	#1569	semidirect
219. C₆₁	C₂×#1569	semidirect
296. C₂×C₈	#11758217	semidirect
346. C₈	C₂×#11758217	semidirect
367. C₂	#21461383	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×#16840783	2
C₂×#16840765	2
#21461383	2
C₂×#16840626	2
C₂×#16840625	2
C₂×#16840624	2
C₂×#16840623	2
C₂×M₁₆×Dic₆₁	2
C₂×#1569	61

Subgroups — C2×#21461383 #30612039

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21461383 #30612039