Subgroups — C₂×#21444576 #29649013

Order 15616.

854 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839784	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×D₆₁	C₂	1	1	yes	maximal, characteristic
3	#21444576	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16839515	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16839514	C₂	1	2	yes	maximal, complemented
6	C₂×C₆₁×#335	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11758189	—	2	2	no	—
8	C₂²×C₈×D₆₁	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₆₁	C₄	1	1	yes	characteristic
10	C₂×C₄²×D₆₁	C₂²	1	1	yes	characteristic
11	#16839784	—	2	4	no	—
12	#16839784	C₂²	1	8	yes	—
13	C₂×C₄⋊#92418	C₂²	1	2	yes	—
14	C₂×#14519706	—	2	2	no	—
15	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₆₁	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₆₁	C₄	1	1	yes	characteristic
18	C₄×C₈×D₆₁	C₄	1	4	yes	—
19	C₄×C₈×D₆₁	C₂²	1	4	yes	—
20	#16839515	C₂²	1	4	yes	complemented
21	#16839514	C₂²	1	8	yes	complemented
22	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₆₁×#335	C₂²	1	4	yes	complemented
25	C₂²×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×D₆₁	D₄	1	1	yes	characteristic
27	C₂²×C₄×D₆₁	Q₈	1	1	yes	characteristic
28	C₂×#11758189	—	2	8	no	—
29	C₂×#11758189	—	4	8	no	—
30	C₂²×#92418	—	2	2	no	—
31	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
32	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₆₁	—	2	4	no	—
34	C₂×C₈×D₆₁	—	2	4	no	—
35	C₂×C₈×D₆₁	C₂×C₄	1	4	yes	—
36	C₂×C₈×D₆₁	C₂×C₄	1	4	yes	—
37	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
39	C₂×C₄×Dic₆₁	Q₈	1	1	yes	characteristic
40	C₄²×D₆₁	C₂×C₄	1	2	yes	—
41	C₄²×D₆₁	D₄	1	2	yes	—
42	C₄²×D₆₁	Q₈	1	2	yes	—
43	C₄²×D₆₁	C₂³	1	2	yes	—
44	C₄⋊#92418	—	2	2	no	—
45	C₄⋊#92418	C₂³	1	4	yes	—
46	#14519706	—	2	8	no	—
47	C₄×#92418	C₂×C₄	1	2	yes	—
48	C₄×#92418	C₂³	1	2	yes	—
49	C₈×Dic₆₁	C₂×C₄	1	4	yes	—
50	C₈×Dic₆₁	C₂×C₄	1	4	yes	—
51	C₂²×C₈×C₆₁	—	2	2	no	—
52	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₆₁	—	2	2	no	—
56	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
57	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₆₁	C₂³	1	2	yes	—
59	C₂³×D₆₁	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
63	C₂²×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×D₆₁	—	2	2	no	—
67	C₂×C₄×D₆₁	—	2	2	no	—
68	C₂×C₄×D₆₁	—	2	4	no	—
69	C₂×C₄×D₆₁	C₄⋊C₄	1	2	yes	—
70	C₂×C₄×D₆₁	C₄⋊C₄	1	2	yes	—
71	C₂×C₄×D₆₁	C₄⋊C₄	1	2	yes	—
72	C₂×C₄×D₆₁	D₈	1	4	yes	—
73	C₂×C₄×D₆₁	Q₁₆	1	4	yes	—
74	C₂×C₄×D₆₁	C₂²×C₄	1	2	yes	—
75	C₂×C₄×D₆₁	C₂×D₄	1	2	yes	—
76	C₂×C₄×D₆₁	C₂×Q₈	1	2	yes	—
77	C₂×C₈×C₆₁	—	2	2	no	—
78	C₂×C₈×C₆₁	—	2	4	no	—
79	C₂×C₈×C₆₁	—	4	4	no	—
80	C₂×C₈×C₆₁	M₁₆	1	4	yes	complemented
81	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
83	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
86	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
87	C₄×Dic₆₁	C₄⋊C₄	1	2	yes	—
88	C₄×Dic₆₁	C₄⋊C₄	1	2	yes	—
89	C₄×Dic₆₁	C₄⋊C₄	1	2	yes	—
90	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
91	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
92	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
93	C₂×#92418	—	2	2	no	—
94	C₂×#92418	—	2	4	no	—
95	C₂×#92418	—	4	4	no	—
96	C₂×#92418	M₁₆	1	4	yes	complemented
97	C₂×#92418	C₂²×C₄	1	2	yes	—
98	C₂×#92418	C₂²×C₄	1	2	yes	—
99	#11758189	—	4	16	no	—
100	C₈×D₆₁	—	2	8	no	—
101	C₈×D₆₁	—	2	8	no	—
102	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂²×D₆₁	—	2	2	no	—
104	C₂²×D₆₁	—	2	2	no	—
105	C₂²×D₆₁	#107	1	4	yes	—
106	C₂²×D₆₁	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₆₁	—	2	2	no	—
108	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
112	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
113	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
114	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
115	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
118	C₂×Dic₆₁	—	2	2	no	—
119	C₂×Dic₆₁	#107	1	4	yes	—
120	C₂×Dic₆₁	#108	1	4	yes	—
121	C₂×Dic₆₁	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×Dic₆₁	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×Dic₆₁	C₂×C₄⋊C₄	1	2	yes	—
124	C₄×D₆₁	—	2	2	no	—
125	C₄×D₆₁	—	2	2	no	—
126	C₄×D₆₁	—	2	4	no	—
127	C₄×D₆₁	—	2	4	no	—
128	C₄×D₆₁	—	2	4	no	—
129	C₄×D₆₁	#107	1	8	yes	—
130	C₄×D₆₁	C₂×D₈	1	4	yes	—
131	C₄×D₆₁	C₂×Q₁₆	1	4	yes	—
132	C₈×C₆₁	—	2	2	no	—
133	C₈×C₆₁	—	4	4	no	—
134	C₈×C₆₁	C₂×M₁₆	1	4	yes	complemented
135	#92418	—	2	2	no	—
136	#92418	—	4	4	no	—
137	#92418	C₂×M₁₆	1	4	yes	complemented
138	C₂×#1567	—	61	1	no	maximal, Sylow
139	C₂²×C₆₁	#423	1	2	yes	—
140	C₂²×C₆₁	C₂×#107	1	2	yes	—
141	C₂²×C₆₁	C₂×#108	1	2	yes	—
142	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
143	C₂×D₆₁	—	2	8	no	—
144	C₂×D₆₁	—	2	2	no	—
145	C₂×D₆₁	—	2	2	no	—
146	C₂×D₆₁	C₂×#107	1	4	yes	—
147	C₄×C₆₁	—	2	1	no	—
148	C₄×C₆₁	—	2	1	no	—
149	C₄×C₆₁	#423	1	2	yes	—
150	C₄×C₆₁	C₂×#107	1	2	yes	—
151	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
152	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
153	C₄×C₆₁	C₂²×D₈	1	1	yes	characteristic
154	C₄×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
155	Dic₆₁	—	2	2	no	—
156	Dic₆₁	C₂×#107	1	4	yes	—
157	Dic₆₁	C₂×#108	1	4	yes	—
158	C₂×#423	—	61	2	no	—
159	C₂²×C₄×C₈	—	61	1	no	—
160	#1567	—	61	8	no	—
161	C₂×#335	—	61	2	no	—
162	C₂×#335	—	61	1	no	—
163	C₂×#335	D₆₁	1	1	yes	characteristic, complemented
164	C₂×C₆₁	#1567	1	4	yes	complemented
165	C₂×C₆₁	C₂×#423	1	1	yes	characteristic
166	C₂×C₆₁	C₂²×#107	1	1	yes	characteristic
167	C₂×C₆₁	C₂²×#108	1	1	yes	characteristic
168	D₆₁	—	2	4	no	—
169	C₂²×M₁₆	—	122	2	no	—
170	C₂³×C₈	—	61	1	no	—
171	C₂³×C₈	—	61	1	no	—
172	C₂²×C₄²	—	61	1	no	—
173	#423	—	61	8	no	—
174	#423	—	122	4	no	—
175	C₂×C₄⋊C₈	—	61	2	no	—
176	C₂×C₄⋊C₈	—	122	2	no	—
177	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
178	C₂×C₄×C₈	—	61	1	no	—
179	C₂×C₄×C₈	—	61	1	no	—
180	C₂×C₄×C₈	—	61	1	no	—
181	C₂×C₄×C₈	—	61	4	no	—
182	C₂×C₄×C₈	—	61	4	no	—
183	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
184	#335	—	61	8	no	—
185	#335	—	61	4	no	—
186	#335	C₂×D₆₁	1	4	yes	complemented
187	C₆₁	C₂×#1567	1	1	yes	characteristic, complemented, Sylow
188	C₂³×C₄	—	61	1	no	—
189	C₂³×C₄	—	61	1	no	—
190	C₂³×C₄	—	61	1	no	—
191	C₂×M₁₆	—	122	8	no	—
192	C₂×M₁₆	—	244	8	no	—
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	—	61	1	no	—
195	C₂²×C₈	—	61	1	no	—
196	C₂²×C₈	—	61	4	no	—
197	C₂²×C₈	—	61	4	no	—
198	C₂²×C₈	—	122	4	no	—
199	C₂²×C₈	—	122	4	no	—
200	C₂²×C₈	—	122	2	no	—
201	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
202	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
203	C₂×C₄²	—	61	1	no	—
204	C₂×C₄²	—	61	1	no	—
205	C₂×C₄²	—	61	1	no	—
206	C₂×C₄²	—	61	2	no	—
207	C₂×C₄²	—	61	2	no	—
208	C₂×C₄²	—	61	2	no	—
209	C₂×C₄²	—	61	2	no	—
210	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
211	C₄⋊C₈	—	2	2	no	—
212	C₄⋊C₈	—	61	4	no	—
213	C₄⋊C₈	—	122	8	no	—
214	C₄⋊C₈	—	122	2	no	—
215	C₄⋊C₈	C₂²×D₆₁	1	4	yes	—
216	C₄×C₈	—	61	4	no	—
217	C₄×C₈	—	61	4	no	—
218	C₄×C₈	—	61	2	no	—
219	C₄×C₈	—	61	2	no	—
220	C₄×C₈	C₄×D₆₁	1	2	yes	—
221	C₄×C₈	C₂²×D₆₁	1	2	yes	—
222	C₂⁴	—	61	1	no	—
223	C₂²×C₄	—	61	1	no	—
224	C₂²×C₄	—	61	1	no	—
225	C₂²×C₄	—	61	2	no	—
226	C₂²×C₄	—	61	2	no	—
227	C₂²×C₄	—	61	4	no	—
228	C₂²×C₄	—	61	4	no	—
229	C₂²×C₄	—	61	1	no	—
230	C₂²×C₄	—	61	2	no	—
231	C₂²×C₄	—	61	2	no	—
232	C₂²×C₄	—	61	2	no	—
233	C₂²×C₄	—	61	2	no	—
234	C₂²×C₄	—	122	4	no	—
235	C₂²×C₄	—	122	2	no	—
236	C₂²×C₄	—	122	2	no	—
237	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
238	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
239	C₂²×C₄	Q₈×D₆₁	1	1	yes	characteristic
240	M₁₆	—	244	16	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	4	4	no	—
244	C₂×C₈	—	61	2	no	—
245	C₂×C₈	—	61	4	no	—
246	C₂×C₈	—	61	2	no	—
247	C₂×C₈	—	122	8	no	—
248	C₂×C₈	—	122	2	no	—
249	C₂×C₈	—	122	8	no	—
250	C₂×C₈	—	122	4	no	—
251	C₂×C₈	—	244	4	no	—
252	C₂×C₈	#11758189	1	4	yes	complemented
253	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
254	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
255	C₄²	—	61	2	no	—
256	C₄²	—	61	2	no	—
257	C₄²	—	61	2	no	—
258	C₄²	—	61	2	no	—
259	C₄²	—	61	2	no	—
260	C₄²	—	61	2	no	—
261	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
262	C₄²	D₄×D₆₁	1	1	yes	characteristic
263	C₄²	Q₈×D₆₁	1	1	yes	characteristic
264	C₄²	C₂³×D₆₁	1	1	yes	characteristic
265	C₂³	—	61	2	no	—
266	C₂³	—	61	4	no	—
267	C₂³	—	122	2	no	—
268	C₂³	—	122	2	no	—
269	C₂³	C₄⋊C₄×D₆₁	1	1	yes	characteristic
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	61	2	no	—
272	C₂×C₄	—	61	2	no	—
273	C₂×C₄	—	61	4	no	—
274	C₂×C₄	—	61	4	no	—
275	C₂×C₄	—	61	4	no	—
276	C₂×C₄	—	61	4	no	—
277	C₂×C₄	—	61	8	no	—
278	C₂×C₄	—	61	2	no	—
279	C₂×C₄	—	122	4	no	—
280	C₂×C₄	—	122	4	no	—
281	C₂×C₄	—	122	2	no	—
282	C₂×C₄	—	122	2	no	—
283	C₂×C₄	—	122	2	no	—
284	C₂×C₄	—	122	4	no	—
285	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
286	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
287	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
288	C₂×C₄	M₁₆×D₆₁	1	2	yes	—
289	C₂×C₄	C₂×#11758189	1	2	yes	—
290	C₂×C₄	D₈×D₆₁	1	2	yes	—
291	C₂×C₄	Q₁₆×D₆₁	1	2	yes	—
292	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
293	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
294	C₂×C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	4	4	no	—
297	C₈	—	61	4	no	—
298	C₈	—	122	2	no	—
299	C₈	—	244	4	no	—
300	C₈	C₂×#11758189	1	4	yes	complemented
301	C₂²	—	61	4	no	—
302	C₂²	—	122	2	no	—
303	C₂²	—	122	2	no	—
304	C₂²	—	122	8	no	—
305	C₂²	#16839784	1	2	yes	—
306	C₂²	#107×D₆₁	1	2	yes	—
307	C₂²	#108×D₆₁	1	2	yes	—
308	C₂²	C₂×C₄⋊C₄×D₆₁	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	61	4	no	—
312	C₄	—	61	4	no	—
313	C₄	—	122	2	no	—
314	C₄	#16839784	1	2	yes	—
315	C₄	#107×D₆₁	1	2	yes	—
316	C₄	C₂×M₁₆×D₆₁	1	1	yes	characteristic
317	C₄	C₂²×#11758189	1	1	yes	characteristic
318	C₄	C₂×D₈×D₆₁	1	1	yes	characteristic
319	C₄	C₂×Q₁₆×D₆₁	1	1	yes	characteristic
320	C₂	—	122	4	no	—
321	C₂	#21444576	1	4	yes	direct product factor
322	C₂	C₂×#16839784	1	1	yes	characteristic
323	C₂	C₂×#107×D₆₁	1	1	yes	characteristic
324	C₂	C₂×#108×D₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₄²	C₂³×D₆₁
Fitting	C₂×C₆₁×#335	C₂
socle	C₂³×C₆₁	C₂×C₄⋊C₄
prime radical	C₂³×D₆₁	C₄⋊C₄
cosocle	C₄²×C₆₁	C₂⁴
hypercenter	C₂×#335	D₆₁
hypocenter	C₆₁	C₂×#1567
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
3. #21444576	C₂	direct (N × H)
4. C₂×#16839515	C₂	semidirect
5. C₂×#16839514	C₂	semidirect
6. C₂×C₆₁×#335	C₂	semidirect
20. #16839515	C₂²	semidirect
21. #16839514	C₂²	semidirect
24. C₆₁×#335	C₂²	semidirect
80. C₂×C₈×C₆₁	M₁₆	semidirect
96. C₂×#92418	M₁₆	semidirect
134. C₈×C₆₁	C₂×M₁₆	semidirect
137. #92418	C₂×M₁₆	semidirect
163. C₂×#335	D₆₁	semidirect
164. C₂×C₆₁	#1567	semidirect
186. #335	C₂×D₆₁	semidirect
187. C₆₁	C₂×#1567	semidirect
252. C₂×C₈	#11758189	semidirect
300. C₈	C₂×#11758189	semidirect
321. C₂	#21444576	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×#16839784	2
C₂×C₄×C₈×D₆₁	2
#21444576	2
C₂×#16839515	2
C₂×#16839514	2
C₂×C₆₁×#335	2
C₂×#1567	61

Subgroups — C2×#21444576 #29649013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21444576 #29649013