Subgroups — C₂×#21445706 #29678028

Order 15616.

1062 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839800	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839720	C₂	1	2	yes	maximal
3	C₂²×#14519687	C₂	1	1	yes	maximal, characteristic
4	#21445706	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16839523	C₂	1	2	yes	maximal
6	C₂²×#108×C₆₁	C₂	1	1	yes	maximal, characteristic
7	C₂³×#92418	C₂²	1	1	yes	characteristic
8	C₂×#14519736	C₂²	1	2	yes	—
9	C₂²×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
10	#16839800	C₄	1	4	yes	complemented
11	#16839800	C₂²	1	4	yes	—
12	C₂×#14519711	C₄	1	2	yes	complemented
13	C₂×#14519709	C₄	1	2	yes	complemented
14	#16839720	—	2	4	no	—
15	#16839720	C₂²	1	8	yes	—
16	C₂×#14519687	—	2	2	no	—
17	C₂×#14519687	C₄	1	8	yes	complemented
18	C₂×#14519687	C₂²	1	4	yes	—
19	C₂×C₈×Dic₆₁	—	2	2	no	—
20	#16839523	C₂²	1	8	yes	—
21	C₂²×M₁₆×C₆₁	C₂²	1	2	yes	—
22	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×#108×C₆₁	C₄	1	8	yes	complemented
24	C₂×#108×C₆₁	C₂²	1	4	yes	—
25	C₂³×Dic₆₁	—	2	1	no	—
26	C₂²×#92418	C₂×C₄	1	4	yes	—
27	C₂²×#92418	D₄	1	2	yes	—
28	C₂²×#92418	D₄	1	2	yes	—
29	C₂²×#92418	Q₈	1	4	yes	—
30	C₂²×#92418	C₂³	1	2	yes	—
31	#14519736	—	2	2	no	—
32	#14519736	—	2	2	no	—
33	#14519736	C₂×C₄	1	4	yes	—
34	#14519736	C₂³	1	4	yes	—
35	C₂×C₄⋊Dic₆₁	C₂×C₄	1	4	yes	—
36	C₂×C₂²⋊Dic₆₁	—	2	2	no	—
37	C₂×C₄×Dic₆₁	—	2	2	no	—
38	C₂×C₄×Dic₆₁	—	2	2	no	—
39	C₂×C₄×Dic₆₁	—	2	1	no	—
40	C₂×C₄×Dic₆₁	C₂×C₄	1	4	yes	—
41	C₂×C₄×Dic₆₁	C₂³	1	2	yes	—
42	#14519711	C₂×C₄	1	8	yes	complemented
43	#14519709	C₂×C₄	1	8	yes	complemented
44	#14519687	—	2	8	no	—
45	#14519687	C₂×C₄	1	16	yes	complemented
46	C₈×Dic₆₁	—	2	8	no	—
47	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
48	C₂×M₁₆×C₆₁	—	2	2	no	—
49	C₂×M₁₆×C₆₁	—	2	4	no	—
50	C₂×M₁₆×C₆₁	C₂×C₄	1	8	yes	—
51	C₂×M₁₆×C₆₁	C₂³	1	4	yes	—
52	C₂²×C₈×C₆₁	—	2	2	no	—
53	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
54	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
57	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
58	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
59	#108×C₆₁	C₂×C₄	1	16	yes	complemented
60	C₂⁴×C₆₁	#41	1	1	yes	characteristic
61	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
69	C₂²×C₄×C₆₁	#41	1	2	yes	—
70	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₆₁	—	2	4	no	—
72	C₂²×Dic₆₁	—	2	4	no	—
73	C₂²×Dic₆₁	—	2	2	no	—
74	C₂²×Dic₆₁	—	4	4	no	—
75	M₁₆×C₆₁	—	2	8	no	—
76	M₁₆×C₆₁	—	2	4	no	—
77	M₁₆×C₆₁	C₂²×C₄	1	8	yes	—
78	C₂×C₈×C₆₁	—	2	8	no	—
79	C₂×C₈×C₆₁	—	2	4	no	—
80	C₂×C₈×C₆₁	C₄²	1	4	yes	—
81	C₂×C₈×C₆₁	C₄²	1	4	yes	—
82	C₂×C₈×C₆₁	C₄²	1	8	yes	—
83	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
84	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
87	C₂²⋊Dic₆₁	—	2	4	no	—
88	C₂²⋊Dic₆₁	—	2	4	no	—
89	C₄⋊Dic₆₁	—	2	4	no	—
90	C₄⋊Dic₆₁	C₂²×C₄	1	8	yes	—
91	C₄×Dic₆₁	—	2	4	no	—
92	C₄×Dic₆₁	—	2	4	no	—
93	C₄×Dic₆₁	—	2	4	no	—
94	C₄×Dic₆₁	—	2	2	no	—
95	C₄×Dic₆₁	C₂²×C₄	1	4	yes	—
96	C₂×#92418	C₄⋊C₄	1	8	yes	—
97	C₂×#92418	C₄⋊C₄	1	8	yes	—
98	C₂×#92418	C₂²×C₄	1	4	yes	—
99	C₂×#92418	C₂×D₄	1	4	yes	—
100	C₂×#92418	C₂×Q₈	1	4	yes	—
101	C₂³×C₆₁	—	2	4	no	—
102	C₂³×C₆₁	#117	1	2	yes	—
103	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
104	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
105	C₂³×C₆₁	C₄×Q₈	1	2	yes	—
106	C₂³×C₆₁	C₂×#41	1	1	yes	characteristic
107	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
108	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
109	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₆₁	#117	1	2	yes	—
112	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
117	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
118	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
119	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
125	C₂×Dic₆₁	—	2	4	no	—
126	C₂×Dic₆₁	—	2	8	no	—
127	C₂×Dic₆₁	—	4	4	no	—
128	C₂×Dic₆₁	—	4	8	no	—
129	C₂×Dic₆₁	—	4	8	no	—
130	C₈×C₆₁	—	2	8	no	—
131	C₈×C₆₁	C₂×C₄²	1	4	yes	—
132	C₈×C₆₁	C₂×C₄²	1	4	yes	—
133	#92418	C₂×C₄⋊C₄	1	8	yes	—
134	C₂×#1773	—	61	1	no	maximal, Sylow
135	C₂²×C₆₁	—	2	2	no	—
136	C₂²×C₆₁	—	2	2	no	—
137	C₂²×C₆₁	—	2	4	no	—
138	C₂²×C₆₁	—	2	4	no	—
139	C₂²×C₆₁	C₄×C₄⋊C₄	1	2	yes	—
140	C₂²×C₆₁	C₄×C₄⋊C₄	1	2	yes	—
141	C₂²×C₆₁	#444	1	4	yes	—
142	C₂²×C₆₁	C₂×#117	1	1	yes	characteristic
143	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
144	C₂²×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
145	C₄×C₆₁	C₄×C₄⋊C₄	1	2	yes	—
146	C₄×C₆₁	C₄×C₄⋊C₄	1	2	yes	—
147	C₄×C₆₁	C₂²×C₄²	1	1	yes	characteristic
148	C₄×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
149	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
150	C₄×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
151	Dic₆₁	—	4	4	no	—
152	Dic₆₁	—	4	8	no	—
153	C₂²×#108	D₆₁	1	1	yes	characteristic
154	C₂×#427	—	61	1	no	—
155	C₂×#405	—	61	2	no	—
156	C₂²×#97	—	61	1	no	—
157	#1773	—	61	8	no	—
158	C₂×#339	—	61	2	no	—
159	C₂×C₆₁	—	2	1	no	—
160	C₂×C₆₁	—	2	1	no	—
161	C₂×C₆₁	—	2	2	no	—
162	C₂×C₆₁	#1773	1	4	yes	complemented
163	C₂×C₆₁	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₆₁	C₂×#444	1	2	yes	—
165	C₂²×M₁₆	C₂×D₆₁	1	2	yes	—
166	C₂³×C₈	—	61	1	no	—
167	C₂³×C₈	C₂×D₆₁	1	1	yes	characteristic
168	C₂×#117	—	61	2	no	—
169	C₂²×C₄²	—	61	1	no	—
170	C₂×#108	Dic₆₁	1	8	yes	complemented
171	C₂×#108	C₂×D₆₁	1	4	yes	—
172	#427	—	61	4	no	—
173	#427	—	61	4	no	—
174	C₂×#107	—	61	2	no	—
175	C₂×#106	—	61	2	no	—
176	#405	—	61	8	no	—
177	#405	—	122	4	no	—
178	C₂×#97	—	61	8	no	—
179	C₂×#97	—	61	4	no	—
180	C₂×#97	—	122	2	no	—
181	C₂×C₄×C₈	—	122	2	no	—
182	#339	—	61	8	no	—
183	C₆₁	C₂×#1773	1	1	yes	characteristic, complemented, Sylow
184	C₂³×C₄	—	122	1	no	—
185	C₂³×C₄	C₂²×D₆₁	1	1	yes	characteristic
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	4	no	—
188	C₂×M₁₆	C₂×Dic₆₁	1	8	yes	—
189	C₂×M₁₆	C₂²×D₆₁	1	4	yes	—
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	—	61	4	no	—
192	C₂²×C₈	—	61	2	no	—
193	C₂²×C₈	—	61	2	no	—
194	C₂²×C₈	—	61	4	no	—
195	C₂²×C₈	—	61	2	no	—
196	C₂²×C₈	C₄×D₆₁	1	4	yes	—
197	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
198	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
199	C₂²×C₈	C₄×D₆₁	1	2	yes	—
200	C₂²×C₈	C₂×Dic₆₁	1	4	yes	—
201	C₂²×C₈	C₂²×D₆₁	1	2	yes	—
202	#117	—	61	4	no	—
203	#117	—	61	4	no	—
204	#117	—	122	2	no	—
205	#117	—	122	2	no	—
206	C₂×C₄⋊C₄	—	61	4	no	—
207	C₂×C₂²⋊C₄	—	122	2	no	—
208	C₂×C₄²	—	61	4	no	—
209	C₂×C₄²	—	61	2	no	—
210	C₂×C₄²	—	122	2	no	—
211	C₂×C₄²	—	122	2	no	—
212	C₂×C₄²	—	122	1	no	—
213	#108	C₂×Dic₆₁	1	16	yes	complemented
214	#107	—	61	8	no	—
215	#106	—	61	8	no	—
216	#97	—	61	16	no	—
217	#97	—	122	8	no	—
218	C₄×C₈	—	122	8	no	—
219	C₂⁴	#11758226	1	1	yes	characteristic
220	C₂²×C₄	—	122	2	no	—
221	C₂²×C₄	—	122	4	no	—
222	C₂²×C₄	—	122	4	no	—
223	C₂²×C₄	—	244	4	no	—
224	C₂²×C₄	C₂×C₄×D₆₁	1	2	yes	—
225	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
227	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
228	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
229	C₂²×C₄	#11758224	1	2	yes	—
230	C₂²×C₄	Q₈×D₆₁	1	2	yes	—
231	C₂²×C₄	#11758226	1	1	yes	characteristic
232	C₂²×C₄	C₂²×Dic₆₁	1	2	yes	—
233	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
234	M₁₆	—	2	4	no	—
235	M₁₆	—	2	8	no	—
236	M₁₆	C₂²×Dic₆₁	1	8	yes	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	8	no	—
239	C₂×C₈	—	61	4	no	—
240	C₂×C₈	—	61	4	no	—
241	C₂×C₈	—	61	4	no	—
242	C₂×C₈	—	61	8	no	—
243	C₂×C₈	—	61	8	no	—
244	C₂×C₈	C₄×Dic₆₁	1	4	yes	—
245	C₂×C₈	C₄×Dic₆₁	1	4	yes	—
246	C₂×C₈	C₄×Dic₆₁	1	8	yes	—
247	C₂×C₈	C₂×C₄×D₆₁	1	4	yes	—
248	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
249	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
250	C₂×C₈	C₂²×Dic₆₁	1	4	yes	—
251	C₄⋊C₄	—	61	8	no	—
252	C₄⋊C₄	—	122	4	no	—
253	C₂²⋊C₄	—	122	4	no	—
254	C₂²⋊C₄	—	122	4	no	—
255	C₄²	—	61	4	no	—
256	C₄²	—	122	4	no	—
257	C₄²	—	122	2	no	—
258	C₄²	—	122	4	no	—
259	C₄²	—	122	4	no	—
260	C₂³	—	2	4	no	—
261	C₂³	#14519735	1	2	yes	—
262	C₂³	#14519739	1	1	yes	characteristic
263	C₂³	#14519739	1	1	yes	characteristic
264	C₂³	Q₈×Dic₆₁	1	2	yes	—
265	C₂³	C₂×#11758226	1	1	yes	characteristic
266	C₂×C₄	—	122	4	no	—
267	C₂×C₄	—	122	8	no	—
268	C₂×C₄	—	244	8	no	—
269	C₂×C₄	—	244	4	no	—
270	C₂×C₄	—	244	8	no	—
271	C₂×C₄	C₂×C₄×Dic₆₁	1	2	yes	—
272	C₂×C₄	C₂×C₄×Dic₆₁	1	2	yes	—
273	C₂×C₄	C₄⋊C₄×D₆₁	1	2	yes	—
274	C₂×C₄	C₄⋊C₄×D₆₁	1	2	yes	—
275	C₂×C₄	#14519735	1	2	yes	—
276	C₂×C₄	#14519739	1	1	yes	characteristic
277	C₂×C₄	#14519739	1	1	yes	characteristic
278	C₂×C₄	D₄×Dic₆₁	1	2	yes	—
279	C₂×C₄	D₄×Dic₆₁	1	2	yes	—
280	C₂×C₄	#14519742	1	2	yes	—
281	C₂×C₄	#14519742	1	2	yes	—
282	C₂×C₄	Q₈×Dic₆₁	1	2	yes	—
283	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
284	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
285	C₂×C₄	C₂³×Dic₆₁	1	1	yes	characteristic
286	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
287	C₂×C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
288	C₂×C₄	C₂×#11758224	1	1	yes	characteristic
289	C₈	—	2	8	no	—
290	C₈	—	61	8	no	—
291	C₈	C₂×C₄×Dic₆₁	1	4	yes	—
292	C₈	C₂×C₄×Dic₆₁	1	4	yes	—
293	C₂²	—	2	4	no	—
294	C₂²	—	2	4	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	C₄⋊C₄×Dic₆₁	1	2	yes	—
298	C₂²	C₄⋊C₄×Dic₆₁	1	2	yes	—
299	C₂²	#16839883	1	4	yes	—
300	C₂²	C₂×#14519735	1	1	yes	characteristic
301	C₂²	C₂×#14519739	1	1	yes	characteristic
302	C₂²	C₂×Q₈×Dic₆₁	1	1	yes	characteristic
303	C₄	—	244	4	no	—
304	C₄	—	244	8	no	—
305	C₄	C₄⋊C₄×Dic₆₁	1	2	yes	—
306	C₄	C₄⋊C₄×Dic₆₁	1	2	yes	—
307	C₄	C₂²×C₄×Dic₆₁	1	1	yes	characteristic
308	C₄	C₂×C₄⋊C₄×D₆₁	1	1	yes	characteristic
309	C₄	C₂×D₄×Dic₆₁	1	1	yes	characteristic
310	C₄	C₂×#14519742	1	1	yes	characteristic
311	C₂	—	2	1	no	—
312	C₂	—	2	1	no	—
313	C₂	—	2	2	no	—
314	C₂	#21445706	1	4	yes	direct product factor
315	C₂	C₂×C₄⋊C₄×Dic₆₁	1	1	yes	characteristic
316	C₂	C₂×#16839883	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₁
commutator	C₄×C₆₁	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₂²×#108×C₆₁	C₂
socle	C₂³×C₆₁	C₄×D₄
prime radical	C₂⁴×C₆₁	#41
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	C₂²×#108	D₆₁
hypocenter	C₆₁	C₂×#1773
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
4. #21445706	C₂	direct (N × H)
10. #16839800	C₄	semidirect
12. C₂×#14519711	C₄	semidirect
13. C₂×#14519709	C₄	semidirect
17. C₂×#14519687	C₄	semidirect
23. C₂×#108×C₆₁	C₄	semidirect
42. #14519711	C₂×C₄	semidirect
43. #14519709	C₂×C₄	semidirect
45. #14519687	C₂×C₄	semidirect
59. #108×C₆₁	C₂×C₄	semidirect
162. C₂×C₆₁	#1773	semidirect
170. C₂×#108	Dic₆₁	semidirect
183. C₆₁	C₂×#1773	semidirect
213. #108	C₂×Dic₆₁	semidirect
314. C₂	#21445706	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×#16839800	2
C₂×#16839720	2
C₂²×#14519687	2
#21445706	2
C₂×#16839523	2
C₂²×#108×C₆₁	2
C₂×#1773	61

Subgroups — C2×#21445706 #29678028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21445706 #29678028