Subgroups — C₂×#21446600 #29683554

Order 15616.

918 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839804	C₂	1	2	yes	maximal
2	C₂²×#14519709	C₂	1	1	yes	maximal, characteristic
3	C₂×#16839717	C₂	1	1	yes	maximal, characteristic
4	#21446600	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16839530	C₂	1	2	yes	maximal
6	C₂²×#108×C₆₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11758194	C₂²	1	1	yes	characteristic
8	C₂×#14519736	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊Dic₆₁	C₂²	1	2	yes	—
10	#16839804	—	2	4	no	—
11	#16839804	C₂²	1	8	yes	—
12	C₂×#14519711	—	2	2	no	—
13	C₂×#14519709	—	2	2	no	—
14	C₂×#14519709	—	2	1	no	—
15	C₂×#14519709	—	2	1	no	—
16	C₂×#14519709	—	2	2	no	—
17	C₂×#14519709	C₂²	1	4	yes	—
18	#16839717	—	2	2	no	—
19	#16839717	C₂²	1	4	yes	—
20	C₂×#14519687	—	2	1	no	—
21	C₂×C₈×Dic₆₁	—	2	1	no	—
22	#16839530	C₂²	1	8	yes	—
23	C₂²×M₁₆×C₆₁	C₂²	1	2	yes	—
24	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
25	C₂×#108×C₆₁	C₄	1	8	yes	complemented
26	C₂×#108×C₆₁	C₂²	1	4	yes	—
27	C₂³×Dic₆₁	—	2	2	no	—
28	C₂×#11758194	—	2	1	no	—
29	C₂×#11758194	—	2	2	no	—
30	C₂×#11758194	Q₈	1	4	yes	—
31	C₂×#11758194	C₂³	1	2	yes	—
32	C₂²×#92418	—	2	1	no	—
33	#14519736	—	2	1	no	—
34	#14519736	—	2	1	no	—
35	#14519736	D₄	1	2	yes	—
36	#14519736	C₂³	1	2	yes	—
37	C₂×C₄⋊Dic₆₁	—	2	2	no	—
38	C₂×C₄⋊Dic₆₁	—	2	2	no	—
39	C₂×C₄⋊Dic₆₁	—	2	2	no	—
40	C₂×C₄⋊Dic₆₁	—	2	2	no	—
41	C₂×C₄⋊Dic₆₁	—	2	2	no	—
42	C₂×C₄⋊Dic₆₁	D₄	1	2	yes	—
43	C₂×C₄⋊Dic₆₁	C₂³	1	4	yes	—
44	C₂×C₂²⋊Dic₆₁	—	2	1	no	—
45	C₂×C₄×Dic₆₁	—	2	1	no	—
46	#14519711	—	2	8	no	—
47	#14519709	—	2	4	no	—
48	#14519709	—	2	4	no	—
49	#14519709	—	2	8	no	—
50	#14519687	—	2	4	no	—
51	C₈×Dic₆₁	—	2	4	no	—
52	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
53	C₂×M₁₆×C₆₁	—	2	2	no	—
54	C₂×M₁₆×C₆₁	—	2	4	no	—
55	C₂×M₁₆×C₆₁	C₂×C₄	1	8	yes	—
56	C₂×M₁₆×C₆₁	C₂³	1	4	yes	—
57	C₂²×C₈×C₆₁	—	2	2	no	—
58	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
59	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₆₁	D₄	1	2	yes	—
62	C₂²×C₈×C₆₁	Q₈	1	4	yes	—
63	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
64	#108×C₆₁	C₂×C₄	1	16	yes	complemented
65	C₂⁴×C₆₁	#41	1	1	yes	characteristic
66	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₆₁	#41	1	2	yes	—
74	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
75	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
76	C₂²×Dic₆₁	—	2	2	no	—
77	C₂²×Dic₆₁	—	2	4	no	—
78	C₂²×Dic₆₁	—	4	2	no	—
79	C₂²×Dic₆₁	—	4	2	no	—
80	C₂²×Dic₆₁	—	8	4	no	—
81	M₁₆×C₆₁	—	2	8	no	—
82	M₁₆×C₆₁	—	2	4	no	—
83	M₁₆×C₆₁	C₂²×C₄	1	8	yes	—
84	C₂×C₈×C₆₁	—	2	8	no	—
85	C₂×C₈×C₆₁	—	2	4	no	—
86	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
87	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
88	C₂×C₈×C₆₁	C₄⋊C₄	1	8	yes	—
89	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
90	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₆₁	C₂×Q₈	1	4	yes	—
93	C₂²⋊Dic₆₁	—	2	2	no	—
94	C₂²⋊Dic₆₁	—	2	2	no	—
95	C₄⋊Dic₆₁	—	2	2	no	—
96	C₄⋊Dic₆₁	—	2	4	no	—
97	C₄⋊Dic₆₁	—	2	4	no	—
98	C₄⋊Dic₆₁	—	4	2	no	—
99	C₄⋊Dic₆₁	—	4	2	no	—
100	C₄⋊Dic₆₁	C₂×D₄	1	4	yes	—
101	C₄×Dic₆₁	—	2	2	no	—
102	C₄×Dic₆₁	—	2	2	no	—
103	#11758194	—	2	4	no	—
104	#11758194	—	2	2	no	—
105	#11758194	C₂×Q₈	1	4	yes	—
106	C₂×#92418	—	2	4	no	—
107	C₂×#92418	—	2	2	no	—
108	C₂³×C₆₁	—	2	4	no	—
109	C₂³×C₆₁	C₄×Q₈	1	2	yes	—
110	C₂³×C₆₁	#121	1	1	yes	characteristic
111	C₂³×C₆₁	#121	1	1	yes	characteristic
112	C₂³×C₆₁	#125	1	2	yes	—
113	C₂³×C₆₁	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
119	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
120	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
121	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
122	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₆₁	#125	1	2	yes	—
124	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
125	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
126	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
131	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
132	C₂×Dic₆₁	—	2	4	no	—
133	C₂×Dic₆₁	—	4	4	no	—
134	C₂×Dic₆₁	—	4	4	no	—
135	C₂×Dic₆₁	—	8	4	no	—
136	C₂×Dic₆₁	—	8	4	no	—
137	C₂×Dic₆₁	—	8	4	no	—
138	C₈×C₆₁	—	2	8	no	—
139	C₈×C₆₁	C₂×C₄⋊C₄	1	4	yes	—
140	C₈×C₆₁	C₂×C₄⋊C₄	1	4	yes	—
141	#92418	—	2	4	no	—
142	C₂×#1951	—	61	1	no	maximal, Sylow
143	C₂²×C₆₁	—	2	4	no	—
144	C₂²×C₆₁	—	2	4	no	—
145	C₂²×C₆₁	—	2	2	no	—
146	C₂²×C₆₁	—	2	2	no	—
147	C₂²×C₆₁	#389	1	2	yes	—
148	C₂²×C₆₁	#389	1	2	yes	—
149	C₂²×C₆₁	#471	1	4	yes	—
150	C₂²×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
151	C₂²×C₆₁	C₂×#121	1	1	yes	characteristic
152	C₂²×C₆₁	C₂×#125	1	1	yes	characteristic
153	C₄×C₆₁	#389	1	2	yes	—
154	C₄×C₆₁	#389	1	2	yes	—
155	C₄×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
156	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
157	C₄×C₆₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
158	C₄×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
159	Dic₆₁	—	4	4	no	—
160	Dic₆₁	—	8	4	no	—
161	C₂²×#108	D₆₁	1	1	yes	characteristic
162	C₂×#428	—	61	2	no	—
163	C₂²×#106	—	61	1	no	—
164	C₂×#405	—	61	1	no	—
165	#1951	—	61	8	no	—
166	C₂×#343	—	61	2	no	—
167	C₂×C₆₁	—	2	1	no	—
168	C₂×C₆₁	—	2	1	no	—
169	C₂×C₆₁	—	2	2	no	—
170	C₂×C₆₁	#1951	1	4	yes	complemented
171	C₂×C₆₁	C₂×#389	1	1	yes	characteristic
172	C₂×C₆₁	C₂×#471	1	2	yes	—
173	C₂²×M₁₆	—	61	1	no	—
174	C₂²×M₁₆	C₂×D₆₁	1	2	yes	—
175	C₂³×C₈	C₂×D₆₁	1	1	yes	characteristic
176	C₂×#117	—	61	1	no	—
177	C₂²×C₄⋊C₄	—	61	2	no	—
178	C₂×#108	Dic₆₁	1	8	yes	complemented
179	C₂×#108	C₂×D₆₁	1	4	yes	—
180	#428	—	61	8	no	—
181	#428	—	122	4	no	—
182	C₂×#107	—	122	2	no	—
183	C₂×#106	—	61	4	no	—
184	C₂×#106	—	122	2	no	—
185	C₂×#106	—	122	1	no	—
186	C₂×#106	—	122	1	no	—
187	C₂×#106	—	122	2	no	—
188	#405	—	61	4	no	—
189	#405	—	122	2	no	—
190	C₂×#97	—	122	1	no	—
191	C₂×C₄×C₈	—	122	1	no	—
192	#343	—	61	8	no	—
193	C₆₁	C₂×#1951	1	1	yes	characteristic, complemented, Sylow
194	C₂³×C₄	—	122	2	no	—
195	C₂³×C₄	C₂²×D₆₁	1	1	yes	characteristic
196	C₂×M₁₆	—	2	2	no	—
197	C₂×M₁₆	—	2	4	no	—
198	C₂×M₁₆	—	61	4	no	—
199	C₂×M₁₆	—	61	2	no	—
200	C₂×M₁₆	—	122	1	no	—
201	C₂×M₁₆	—	122	2	no	—
202	C₂×M₁₆	C₂×Dic₆₁	1	8	yes	—
203	C₂×M₁₆	C₂²×D₆₁	1	4	yes	—
204	C₂²×C₈	—	2	2	no	—
205	C₂²×C₈	—	122	1	no	—
206	C₂²×C₈	Dic₁₂₂	1	4	yes	—
207	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
208	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
209	C₂²×C₈	D₂₄₄	1	2	yes	—
210	C₂²×C₈	C₂×Dic₆₁	1	4	yes	—
211	C₂²×C₈	C₂²×D₆₁	1	2	yes	—
212	#117	—	61	2	no	—
213	#117	—	61	2	no	—
214	#117	—	122	1	no	—
215	#117	—	122	1	no	—
216	C₂×C₄⋊C₄	—	61	2	no	—
217	C₂×C₄⋊C₄	—	61	4	no	—
218	C₂×C₄⋊C₄	—	122	2	no	—
219	C₂×C₄⋊C₄	—	122	2	no	—
220	C₂×C₄⋊C₄	—	122	2	no	—
221	C₂×C₄⋊C₄	—	122	2	no	—
222	C₂×C₄⋊C₄	—	122	2	no	—
223	C₂×C₂²⋊C₄	—	122	1	no	—
224	C₂×C₄²	—	122	1	no	—
225	#108	C₂×Dic₆₁	1	16	yes	complemented
226	#107	—	122	8	no	—
227	#106	—	122	8	no	—
228	#106	—	122	4	no	—
229	#106	—	122	4	no	—
230	#97	—	122	4	no	—
231	C₄×C₈	—	122	4	no	—
232	C₂⁴	#11758226	1	1	yes	characteristic
233	C₂²×C₄	—	122	2	no	—
234	C₂²×C₄	—	122	4	no	—
235	C₂²×C₄	—	488	4	no	—
236	C₂²×C₄	—	244	2	no	—
237	C₂²×C₄	—	244	2	no	—
238	C₂²×C₄	C₂×Dic₁₂₂	1	2	yes	—
239	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
241	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
242	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
243	C₂²×C₄	#11758224	1	2	yes	—
244	C₂²×C₄	Q₈×D₆₁	1	2	yes	—
245	C₂²×C₄	#11758226	1	1	yes	characteristic
246	C₂²×C₄	C₂²×Dic₆₁	1	2	yes	—
247	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
248	M₁₆	—	2	4	no	—
249	M₁₆	—	2	8	no	—
250	M₁₆	—	61	4	no	—
251	M₁₆	—	122	2	no	—
252	M₁₆	—	122	4	no	—
253	M₁₆	C₂²×Dic₆₁	1	8	yes	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	2	8	no	—
256	C₂×C₈	—	122	2	no	—
257	C₂×C₈	—	122	4	no	—
258	C₂×C₈	C₄⋊Dic₆₁	1	4	yes	—
259	C₂×C₈	C₄⋊Dic₆₁	1	4	yes	—
260	C₂×C₈	C₄⋊Dic₆₁	1	8	yes	—
261	C₂×C₈	C₂×Dic₁₂₂	1	4	yes	—
262	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
263	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
264	C₂×C₈	C₂²×Dic₆₁	1	4	yes	—
265	C₄⋊C₄	—	61	4	no	—
266	C₄⋊C₄	—	122	2	no	—
267	C₄⋊C₄	—	122	4	no	—
268	C₄⋊C₄	—	122	4	no	—
269	C₄⋊C₄	—	244	2	no	—
270	C₄⋊C₄	—	244	2	no	—
271	C₂²⋊C₄	—	122	2	no	—
272	C₂²⋊C₄	—	122	2	no	—
273	C₄²	—	122	2	no	—
274	C₄²	—	122	2	no	—
275	C₂³	—	2	4	no	—
276	C₂³	Q₈×Dic₆₁	1	2	yes	—
277	C₂³	#14519752	1	1	yes	characteristic
278	C₂³	#14519752	1	1	yes	characteristic
279	C₂³	#14519768	1	2	yes	—
280	C₂³	C₂×#11758226	1	1	yes	characteristic
281	C₂×C₄	—	122	4	no	—
282	C₂×C₄	—	488	4	no	—
283	C₂×C₄	—	488	4	no	—
284	C₂×C₄	—	488	4	no	—
285	C₂×C₄	—	244	4	no	—
286	C₂×C₄	—	244	4	no	—
287	C₂×C₄	C₂×C₄⋊Dic₆₁	1	2	yes	—
288	C₂×C₄	C₂×C₄⋊Dic₆₁	1	2	yes	—
289	C₂×C₄	D₄×Dic₆₁	1	2	yes	—
290	C₂×C₄	D₄×Dic₆₁	1	2	yes	—
291	C₂×C₄	Q₈×Dic₆₁	1	2	yes	—
292	C₂×C₄	#14519752	1	1	yes	characteristic
293	C₂×C₄	#14519752	1	1	yes	characteristic
294	C₂×C₄	#14519757	1	2	yes	—
295	C₂×C₄	#14519757	1	2	yes	—
296	C₂×C₄	#14519768	1	2	yes	—
297	C₂×C₄	#14519776	1	2	yes	—
298	C₂×C₄	#14519776	1	2	yes	—
299	C₂×C₄	C₂³×Dic₆₁	1	1	yes	characteristic
300	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
301	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
302	C₂×C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
303	C₂×C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
304	C₂×C₄	C₂×#11758224	1	1	yes	characteristic
305	C₈	—	2	8	no	—
306	C₈	—	122	4	no	—
307	C₈	C₂×C₄⋊Dic₆₁	1	4	yes	—
308	C₈	C₂×C₄⋊Dic₆₁	1	4	yes	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	4	no	—
312	C₂²	—	2	4	no	—
313	C₂²	#16839661	1	2	yes	—
314	C₂²	#16839661	1	2	yes	—
315	C₂²	#16840026	1	4	yes	—
316	C₂²	C₂×Q₈×Dic₆₁	1	1	yes	characteristic
317	C₂²	C₂×#14519752	1	1	yes	characteristic
318	C₂²	C₂×#14519768	1	1	yes	characteristic
319	C₄	—	488	4	no	—
320	C₄	—	244	4	no	—
321	C₄	#16839661	1	2	yes	—
322	C₄	#16839661	1	2	yes	—
323	C₄	C₂²×C₄⋊Dic₆₁	1	1	yes	characteristic
324	C₄	C₂×D₄×Dic₆₁	1	1	yes	characteristic
325	C₄	C₂×#14519757	1	1	yes	characteristic
326	C₄	C₂×#14519776	1	1	yes	characteristic
327	C₂	—	2	1	no	—
328	C₂	—	2	1	no	—
329	C₂	—	2	2	no	—
330	C₂	#21446600	1	4	yes	direct product factor
331	C₂	C₂×#16839661	1	1	yes	characteristic
332	C₂	C₂×#16840026	1	2	yes	—

	subgroup	G/H
center	C₂³	#14519752
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₂²×#108×C₆₁	C₂
socle	C₂³×C₆₁	#121
prime radical	C₂⁴×C₆₁	#41
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	C₂²×#108	D₆₁
hypocenter	C₆₁	C₂×#1951
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
4. #21446600	C₂	direct (N × H)
25. C₂×#108×C₆₁	C₄	semidirect
64. #108×C₆₁	C₂×C₄	semidirect
170. C₂×C₆₁	#1951	semidirect
178. C₂×#108	Dic₆₁	semidirect
193. C₆₁	C₂×#1951	semidirect
225. #108	C₂×Dic₆₁	semidirect
330. C₂	#21446600	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×#16839804	2
C₂²×#14519709	2
C₂×#16839717	2
#21446600	2
C₂×#16839530	2
C₂²×#108×C₆₁	2
C₂×#1951	61

Subgroups — C2×#21446600 #29683554

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21446600 #29683554