Subgroups — C₂×C₄₁×#1657 #23795183

Order 10496.

1010 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#563	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₁×#496	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₁×#490	C₂	1	2	yes	maximal, complemented
4	C₄₁×#1657	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄₁×#330	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₁×#329	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×SD₁₆×C₄₁	—	2	1	no	—
8	C₂²×D₈×C₄₁	—	2	1	no	—
9	C₄₁×#563	—	2	4	no	—
10	C₄₁×#563	—	2	2	no	—
11	C₄₁×#563	C₂²	1	4	yes	complemented
12	C₂×C₄⋊Q₈×C₄₁	—	2	1	no	—
13	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
14	C₂×#126×C₄₁	—	2	2	no	—
15	C₂×#125×C₄₁	C₂²	1	2	yes	—
16	C₂×#124×C₄₁	C₂²	1	1	yes	characteristic
17	C₂×#123×C₄₁	—	2	1	no	—
18	C₂×C₂²⋊Q₈×C₄₁	—	2	2	no	—
19	C₄₁×#496	—	2	2	no	—
20	C₄₁×#496	C₂²	1	4	yes	—
21	C₄₁×#490	—	2	4	no	—
22	C₄₁×#490	C₂²	1	8	yes	complemented
23	C₂×#102×C₄₁	—	2	2	no	—
24	C₂×#102×C₄₁	—	2	2	no	—
25	C₂×#97×C₄₁	C₂²	1	2	yes	—
26	C₂×#97×C₄₁	C₂²	1	1	yes	characteristic
27	C₄₁×#330	C₂²	1	8	yes	complemented
28	C₄₁×#329	C₂²	1	4	yes	complemented
29	C₂²×Q₈×C₄₁	—	2	1	no	—
30	C₂²×Q₈×C₄₁	D₄	1	1	yes	characteristic
31	C₂²×D₄×C₄₁	—	2	1	no	—
32	C₂²×D₄×C₄₁	—	2	2	no	—
33	C₂²×D₄×C₄₁	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₄₁	—	2	1	no	—
35	C₂×SD₁₆×C₄₁	—	2	4	no	—
36	C₂×SD₁₆×C₄₁	—	4	4	no	—
37	C₂×D₈×C₄₁	—	2	4	no	—
38	C₂×D₈×C₄₁	—	4	4	no	—
39	C₂²×C₈×C₄₁	—	2	1	no	—
40	C₂²×C₈×C₄₁	—	2	2	no	—
41	C₄⋊Q₈×C₄₁	—	2	4	no	—
42	C₄⋊Q₈×C₄₁	—	2	4	no	—
43	#127×C₄₁	—	2	2	no	—
44	#127×C₄₁	—	2	1	no	—
45	#127×C₄₁	C₂³	1	2	yes	—
46	#126×C₄₁	—	2	8	no	—
47	#126×C₄₁	—	2	8	no	—
48	#125×C₄₁	—	2	2	no	—
49	#125×C₄₁	—	2	2	no	—
50	#125×C₄₁	D₄	1	4	yes	—
51	#125×C₄₁	C₂³	1	4	yes	—
52	#124×C₄₁	—	2	2	no	—
53	#124×C₄₁	D₄	1	2	yes	—
54	#124×C₄₁	C₂³	1	2	yes	—
55	#123×C₄₁	—	2	4	no	—
56	#123×C₄₁	—	2	4	no	—
57	C₂²⋊Q₈×C₄₁	—	2	8	no	—
58	C₂²⋊Q₈×C₄₁	—	2	8	no	—
59	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
60	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
61	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
62	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
63	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
64	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
65	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
66	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
67	C₂×C₄²×C₄₁	—	2	1	no	—
68	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
69	#102×C₄₁	—	2	8	no	—
70	#102×C₄₁	—	2	8	no	—
71	#97×C₄₁	—	2	2	no	—
72	#97×C₄₁	—	2	1	no	—
73	#97×C₄₁	C₂³	1	4	yes	—
74	#97×C₄₁	C₂³	1	2	yes	—
75	C₂⁴×C₄₁	—	2	1	no	—
76	C₂⁴×C₄₁	—	4	1	no	—
77	C₂×Q₈×C₄₁	—	2	4	no	—
78	C₂×Q₈×C₄₁	—	2	1	no	—
79	C₂×Q₈×C₄₁	—	4	2	no	—
80	C₂×Q₈×C₄₁	—	8	2	no	—
81	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
82	C₂×D₄×C₄₁	—	2	1	no	—
83	C₂×D₄×C₄₁	—	2	4	no	—
84	C₂×D₄×C₄₁	—	2	2	no	—
85	C₂×D₄×C₄₁	—	4	2	no	—
86	C₂×D₄×C₄₁	—	4	2	no	—
87	C₂×D₄×C₄₁	—	4	2	no	—
88	C₂×D₄×C₄₁	—	4	8	no	—
89	C₂×D₄×C₄₁	—	4	2	no	—
90	C₂×D₄×C₄₁	—	4	1	no	—
91	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₄₁	—	2	2	no	—
93	C₂²×C₄×C₄₁	—	2	1	no	—
94	C₂²×C₄×C₄₁	—	2	2	no	—
95	C₂²×C₄×C₄₁	—	2	4	no	—
96	C₂²×C₄×C₄₁	—	8	2	no	—
97	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
98	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
99	SD₁₆×C₄₁	—	4	8	no	—
100	D₈×C₄₁	—	4	8	no	—
101	C₂×C₈×C₄₁	—	2	4	no	—
102	C₂×C₈×C₄₁	—	2	2	no	—
103	C₂×C₈×C₄₁	—	2	4	no	—
104	C₂×C₈×C₄₁	—	4	4	no	—
105	C₄⋊C₄×C₄₁	—	2	8	no	—
106	C₄⋊C₄×C₄₁	—	2	4	no	—
107	C₄⋊C₄×C₄₁	—	2	4	no	—
108	C₄⋊C₄×C₄₁	—	2	2	no	—
109	C₄⋊C₄×C₄₁	—	2	2	no	—
110	C₄⋊C₄×C₄₁	—	2	4	no	—
111	C₄⋊C₄×C₄₁	—	2	4	no	—
112	C₄⋊C₄×C₄₁	C₂×D₄	1	4	yes	—
113	C₄⋊C₄×C₄₁	C₂×D₄	1	4	yes	—
114	C₂²⋊C₄×C₄₁	—	2	8	no	—
115	C₂²⋊C₄×C₄₁	—	2	4	no	—
116	C₂²⋊C₄×C₄₁	—	2	4	no	—
117	C₂²⋊C₄×C₄₁	—	2	4	no	—
118	C₄²×C₄₁	—	2	4	no	—
119	C₄²×C₄₁	C₂×D₄	1	2	yes	—
120	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
122	C₂³×C₄₁	—	2	2	no	—
123	C₂³×C₄₁	—	4	2	no	—
124	C₂³×C₄₁	—	4	4	no	—
125	C₂³×C₄₁	—	4	2	no	—
126	C₂³×C₄₁	—	8	2	no	—
127	C₂³×C₄₁	—	8	4	no	—
128	C₂³×C₄₁	#120	1	1	yes	characteristic
129	Q₈×C₄₁	—	4	2	no	—
130	Q₈×C₄₁	—	8	1	no	—
131	Q₈×C₄₁	—	8	4	no	—
132	D₄×C₄₁	—	4	2	no	—
133	D₄×C₄₁	—	4	2	no	—
134	D₄×C₄₁	—	4	2	no	—
135	D₄×C₄₁	—	4	8	no	—
136	D₄×C₄₁	—	8	1	no	—
137	D₄×C₄₁	—	8	2	no	—
138	D₄×C₄₁	—	8	4	no	—
139	C₂×C₄×C₄₁	—	2	2	no	—
140	C₂×C₄×C₄₁	—	2	4	no	—
141	C₂×C₄×C₄₁	—	2	4	no	—
142	C₂×C₄×C₄₁	—	2	2	no	—
143	C₂×C₄×C₄₁	—	2	4	no	—
144	C₂×C₄×C₄₁	—	4	4	no	—
145	C₂×C₄×C₄₁	—	4	2	no	—
146	C₂×C₄×C₄₁	—	4	4	no	—
147	C₂×C₄×C₄₁	—	8	4	no	—
148	C₂×C₄×C₄₁	#120	1	2	yes	—
149	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
150	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
151	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
152	C₈×C₄₁	—	2	4	no	—
153	C₈×C₄₁	—	4	4	no	—
154	C₂×#1657	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂²×C₄₁	—	4	2	no	—
156	C₂²×C₄₁	—	4	4	no	—
157	C₂²×C₄₁	—	8	4	no	—
158	C₂²×C₄₁	—	8	2	no	—
159	C₂²×C₄₁	—	8	4	no	—
160	C₂²×C₄₁	—	16	4	no	—
161	C₂²×C₄₁	#453	1	2	yes	—
162	C₂²×C₄₁	#454	1	4	yes	—
163	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
164	C₄×C₄₁	—	2	1	no	—
165	C₄×C₄₁	—	2	1	no	—
166	C₄×C₄₁	—	2	2	no	—
167	C₄×C₄₁	—	2	2	no	—
168	C₄×C₄₁	—	4	4	no	—
169	C₄×C₄₁	—	4	2	no	—
170	C₄×C₄₁	—	4	4	no	—
171	C₂×#563	C₂×C₄₁	1	1	yes	characteristic, complemented
172	C₂×#496	C₂×C₄₁	1	1	yes	characteristic
173	C₂×#490	C₂×C₄₁	1	2	yes	complemented
174	#1657	C₂×C₄₁	1	8	yes	direct product factor
175	C₂×#330	C₂×C₄₁	1	2	yes	complemented
176	C₂×#329	C₂×C₄₁	1	1	yes	characteristic, complemented
177	C₂×C₄₁	—	8	2	no	—
178	C₂×C₄₁	—	16	2	no	—
179	C₂×C₄₁	#1657	1	4	yes	direct product factor
180	C₂×C₄₁	C₂×#453	1	1	yes	characteristic
181	C₂×C₄₁	C₂×#454	1	2	yes	—
182	C₂²×SD₁₆	—	2	1	no	—
183	C₂²×D₈	—	2	1	no	—
184	#563	—	2	4	no	—
185	#563	—	2	2	no	—
186	#563	C₂²×C₄₁	1	4	yes	complemented
187	C₂×C₄⋊Q₈	—	2	1	no	—
188	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
189	C₂×#126	—	2	2	no	—
190	C₂×#125	C₂²×C₄₁	1	2	yes	—
191	C₂×#124	C₂²×C₄₁	1	1	yes	characteristic
192	C₂×#123	—	2	1	no	—
193	C₂×C₂²⋊Q₈	—	2	2	no	—
194	#496	—	2	2	no	—
195	#496	C₂²×C₄₁	1	4	yes	—
196	#490	—	2	4	no	—
197	#490	C₂²×C₄₁	1	8	yes	complemented
198	C₂×#102	—	2	2	no	—
199	C₂×#102	—	2	2	no	—
200	C₂×#97	C₂²×C₄₁	1	2	yes	—
201	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic
202	#330	C₂²×C₄₁	1	8	yes	complemented
203	#329	C₂²×C₄₁	1	4	yes	complemented
204	C₄₁	C₂×#1657	1	1	yes	characteristic, direct product factor, Sylow
205	C₂²×Q₈	—	2	1	no	—
206	C₂²×Q₈	D₄×C₄₁	1	1	yes	characteristic
207	C₂²×D₄	—	2	1	no	—
208	C₂²×D₄	—	2	2	no	—
209	C₂²×D₄	D₄×C₄₁	1	1	yes	characteristic
210	C₂³×C₄	—	2	1	no	—
211	C₂×SD₁₆	—	2	4	no	—
212	C₂×SD₁₆	—	4	4	no	—
213	C₂×D₈	—	2	4	no	—
214	C₂×D₈	—	4	4	no	—
215	C₂²×C₈	—	2	1	no	—
216	C₂²×C₈	—	2	2	no	—
217	C₄⋊Q₈	—	2	4	no	—
218	C₄⋊Q₈	—	2	4	no	—
219	#127	—	2	2	no	—
220	#127	—	2	1	no	—
221	#127	C₂³×C₄₁	1	2	yes	—
222	#126	—	2	8	no	—
223	#126	—	2	8	no	—
224	#125	—	2	2	no	—
225	#125	—	2	2	no	—
226	#125	D₄×C₄₁	1	4	yes	—
227	#125	C₂³×C₄₁	1	4	yes	—
228	#124	—	2	2	no	—
229	#124	D₄×C₄₁	1	2	yes	—
230	#124	C₂³×C₄₁	1	2	yes	—
231	#123	—	2	4	no	—
232	#123	—	2	4	no	—
233	C₂²⋊Q₈	—	2	8	no	—
234	C₂²⋊Q₈	—	2	8	no	—
235	C₂×C₄⋊C₄	—	2	2	no	—
236	C₂×C₄⋊C₄	—	2	2	no	—
237	C₂×C₄⋊C₄	—	2	2	no	—
238	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
239	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
240	C₂×C₂²⋊C₄	—	2	2	no	—
241	C₂×C₂²⋊C₄	—	2	1	no	—
242	C₂×C₂²⋊C₄	—	2	2	no	—
243	C₂×C₄²	—	2	1	no	—
244	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
245	#102	—	2	8	no	—
246	#102	—	2	8	no	—
247	#97	—	2	2	no	—
248	#97	—	2	1	no	—
249	#97	C₂³×C₄₁	1	4	yes	—
250	#97	C₂³×C₄₁	1	2	yes	—
251	C₂⁴	—	2	1	no	—
252	C₂⁴	—	4	1	no	—
253	C₂×Q₈	—	2	4	no	—
254	C₂×Q₈	—	2	1	no	—
255	C₂×Q₈	—	4	2	no	—
256	C₂×Q₈	—	8	2	no	—
257	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
258	C₂×D₄	—	2	1	no	—
259	C₂×D₄	—	2	4	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	4	2	no	—
262	C₂×D₄	—	4	2	no	—
263	C₂×D₄	—	4	2	no	—
264	C₂×D₄	—	4	8	no	—
265	C₂×D₄	—	4	2	no	—
266	C₂×D₄	—	4	1	no	—
267	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	—	2	4	no	—
272	C₂²×C₄	—	8	2	no	—
273	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
274	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
275	SD₁₆	—	4	8	no	—
276	D₈	—	4	8	no	—
277	C₂×C₈	—	2	4	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	4	no	—
280	C₂×C₈	—	4	4	no	—
281	C₄⋊C₄	—	2	8	no	—
282	C₄⋊C₄	—	2	4	no	—
283	C₄⋊C₄	—	2	4	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	4	no	—
287	C₄⋊C₄	—	2	4	no	—
288	C₄⋊C₄	C₂×D₄×C₄₁	1	4	yes	—
289	C₄⋊C₄	C₂×D₄×C₄₁	1	4	yes	—
290	C₂²⋊C₄	—	2	8	no	—
291	C₂²⋊C₄	—	2	4	no	—
292	C₂²⋊C₄	—	2	4	no	—
293	C₂²⋊C₄	—	2	4	no	—
294	C₄²	—	2	4	no	—
295	C₄²	C₂×D₄×C₄₁	1	2	yes	—
296	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
297	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
298	C₂³	—	2	2	no	—
299	C₂³	—	4	2	no	—
300	C₂³	—	4	4	no	—
301	C₂³	—	4	2	no	—
302	C₂³	—	8	2	no	—
303	C₂³	—	8	4	no	—
304	C₂³	#120×C₄₁	1	1	yes	characteristic
305	Q₈	—	4	2	no	—
306	Q₈	—	8	1	no	—
307	Q₈	—	8	4	no	—
308	D₄	—	4	2	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	8	no	—
312	D₄	—	8	1	no	—
313	D₄	—	8	2	no	—
314	D₄	—	8	4	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	4	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	4	4	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	4	no	—
323	C₂×C₄	—	8	4	no	—
324	C₂×C₄	#120×C₄₁	1	2	yes	—
325	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
326	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
327	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
328	C₈	—	2	4	no	—
329	C₈	—	4	4	no	—
330	C₂²	—	4	2	no	—
331	C₂²	—	4	4	no	—
332	C₂²	—	8	4	no	—
333	C₂²	—	8	2	no	—
334	C₂²	—	8	4	no	—
335	C₂²	—	16	4	no	—
336	C₂²	C₄₁×#453	1	2	yes	—
337	C₂²	C₄₁×#454	1	4	yes	—
338	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	2	no	—
343	C₄	—	4	4	no	—
344	C₄	—	4	2	no	—
345	C₄	—	4	4	no	—
346	C₂	—	8	2	no	—
347	C₂	—	16	2	no	—
348	C₂	C₄₁×#1657	1	4	yes	direct product factor
349	C₂	C₂×C₄₁×#453	1	1	yes	characteristic
350	C₂	C₂×C₄₁×#454	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₁
prime radical	C₂×C₄₁×#496	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₁×#563	C₂	semidirect
3. C₂×C₄₁×#490	C₂	semidirect
4. C₄₁×#1657	C₂	direct (N × H)
5. C₂×C₄₁×#330	C₂	semidirect
6. C₂×C₄₁×#329	C₂	semidirect
11. C₄₁×#563	C₂²	semidirect
22. C₄₁×#490	C₂²	semidirect
27. C₄₁×#330	C₂²	semidirect
28. C₄₁×#329	C₂²	semidirect
154. C₂×#1657	C₄₁	direct (N × H)
171. C₂×#563	C₂×C₄₁	semidirect
173. C₂×#490	C₂×C₄₁	semidirect
174. #1657	C₂×C₄₁	direct (N × H)
175. C₂×#330	C₂×C₄₁	semidirect
176. C₂×#329	C₂×C₄₁	semidirect
179. C₂×C₄₁	#1657	direct (N × H)
186. #563	C₂²×C₄₁	semidirect
197. #490	C₂²×C₄₁	semidirect
202. #330	C₂²×C₄₁	semidirect
203. #329	C₂²×C₄₁	semidirect
204. C₄₁	C₂×#1657	direct (N × H)
348. C₂	C₄₁×#1657	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×C₄₁×#563	2
C₂×C₄₁×#496	2
C₂×C₄₁×#490	2
C₄₁×#1657	2
C₂×C₄₁×#330	2
C₂×C₄₁×#329	2
C₂×#1657	41

Subgroups — C2×C41×#1657 #23795183

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#1657 #23795183