Subgroups — C₂×C₄₁×#1533 #23793857

Order 10496.

1310 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#124×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#432	C₂	1	2	yes	maximal, complemented
3	C₄₁×#1533	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄₁×#329	C₂	1	4	yes	maximal, complemented
5	C₂³×Q₈×C₄₁	C₄	1	1	yes	characteristic
6	C₂³×D₄×C₄₁	C₄	1	1	yes	characteristic
7	C₂×#124×C₄₁	—	2	2	no	—
8	C₂×#124×C₄₁	—	2	4	no	—
9	C₂×#124×C₄₁	C₄	1	2	yes	—
10	C₂×#124×C₄₁	C₄	1	4	yes	—
11	C₂×#124×C₄₁	C₂²	1	2	yes	—
12	C₂×#124×C₄₁	C₂²	1	4	yes	—
13	C₂²×C₂²⋊C₄×C₄₁	—	2	2	no	—
14	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
15	C₄₁×#432	—	2	4	no	—
16	C₄₁×#432	C₂²	1	8	yes	complemented
17	C₂×C₄⋊C₈×C₄₁	—	2	2	no	—
18	C₂×C₂²⋊C₈×C₄₁	—	2	2	no	—
19	C₂×#97×C₄₁	C₂²	1	4	yes	complemented
20	C₄₁×#329	C₂²	1	16	yes	complemented
21	C₂⁵×C₄₁	—	2	1	no	—
22	C₂²×Q₈×C₄₁	—	2	2	no	—
23	C₂²×Q₈×C₄₁	—	2	1	no	—
24	C₂²×Q₈×C₄₁	—	4	4	no	—
25	C₂²×Q₈×C₄₁	C₂×C₄	1	4	yes	—
26	C₂²×Q₈×C₄₁	C₂×C₄	1	2	yes	—
27	C₂²×D₄×C₄₁	—	2	2	no	—
28	C₂²×D₄×C₄₁	—	2	1	no	—
29	C₂²×D₄×C₄₁	—	4	4	no	—
30	C₂²×D₄×C₄₁	C₂×C₄	1	4	yes	—
31	C₂²×D₄×C₄₁	C₂×C₄	1	2	yes	—
32	C₂³×C₄×C₄₁	—	2	1	no	—
33	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₄₁	D₄	1	2	yes	—
35	C₂²×C₈×C₄₁	—	2	4	no	—
36	#124×C₄₁	—	2	8	no	—
37	#124×C₄₁	—	2	4	no	—
38	#124×C₄₁	—	2	4	no	—
39	#124×C₄₁	—	2	8	no	—
40	#124×C₄₁	C₂×C₄	1	4	yes	—
41	#124×C₄₁	C₂×C₄	1	8	yes	—
42	#124×C₄₁	C₂³	1	4	yes	—
43	C₂×C₂²⋊C₄×C₄₁	—	2	8	no	—
44	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
45	C₂×C₂²⋊C₄×C₄₁	—	2	8	no	—
46	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
47	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₄₁	D₄	1	2	yes	—
50	C₂×C₄²×C₄₁	D₄	1	4	yes	—
51	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
53	C₄⋊C₈×C₄₁	—	2	8	no	—
54	C₂²⋊C₈×C₄₁	—	2	8	no	—
55	#97×C₄₁	—	2	4	no	—
56	#97×C₄₁	C₂³	1	8	yes	complemented
57	C₂⁴×C₄₁	—	2	4	no	—
58	C₂⁴×C₄₁	—	2	2	no	—
59	C₂⁴×C₄₁	—	4	8	no	—
60	C₂⁴×C₄₁	—	4	4	no	—
61	C₂⁴×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×Q₈×C₄₁	—	2	2	no	—
63	C₂×Q₈×C₄₁	—	2	4	no	—
64	C₂×Q₈×C₄₁	—	4	8	no	—
65	C₂×Q₈×C₄₁	—	4	4	no	—
66	C₂×Q₈×C₄₁	—	4	8	no	—
67	C₂×Q₈×C₄₁	—	4	4	no	—
68	C₂×Q₈×C₄₁	C₂²×C₄	1	4	yes	—
69	C₂×D₄×C₄₁	—	2	2	no	—
70	C₂×D₄×C₄₁	—	2	4	no	—
71	C₂×D₄×C₄₁	—	4	8	no	—
72	C₂×D₄×C₄₁	—	4	4	no	—
73	C₂×D₄×C₄₁	—	4	8	no	—
74	C₂×D₄×C₄₁	—	4	4	no	—
75	C₂×D₄×C₄₁	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₄₁	—	2	4	no	—
77	C₂²×C₄×C₄₁	—	2	4	no	—
78	C₂²×C₄×C₄₁	—	2	2	no	—
79	C₂²×C₄×C₄₁	—	4	4	no	—
80	C₂²×C₄×C₄₁	—	4	4	no	—
81	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
82	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
83	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₁	C₂²⋊C₄	1	4	yes	—
85	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
89	C₂×C₈×C₄₁	—	2	8	no	—
90	C₂×C₈×C₄₁	—	4	8	no	—
91	C₂²⋊C₄×C₄₁	—	2	8	no	—
92	C₂²⋊C₄×C₄₁	—	2	16	no	—
93	C₂²⋊C₄×C₄₁	—	2	8	no	—
94	C₄²×C₄₁	C₂²⋊C₄	1	4	yes	—
95	C₄²×C₄₁	C₂²⋊C₄	1	2	yes	—
96	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
98	C₄²×C₄₁	C₂×D₄	1	2	yes	—
99	C₄²×C₄₁	C₂×D₄	1	4	yes	—
100	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
101	C₂³×C₄₁	—	2	4	no	—
102	C₂³×C₄₁	—	2	4	no	—
103	C₂³×C₄₁	—	4	16	no	—
104	C₂³×C₄₁	—	4	8	no	—
105	C₂³×C₄₁	—	4	4	no	—
106	C₂³×C₄₁	—	4	8	no	—
107	C₂³×C₄₁	—	8	16	no	—
108	C₂³×C₄₁	#100	1	4	yes	—
109	C₂³×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
110	C₂³×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	Q₈×C₄₁	—	4	4	no	—
112	Q₈×C₄₁	—	4	4	no	—
113	Q₈×C₄₁	—	4	8	no	—
114	D₄×C₄₁	—	4	4	no	—
115	D₄×C₄₁	—	4	4	no	—
116	D₄×C₄₁	—	4	8	no	—
117	C₂×C₄×C₄₁	—	2	4	no	—
118	C₂×C₄×C₄₁	—	2	2	no	—
119	C₂×C₄×C₄₁	—	2	4	no	—
120	C₂×C₄×C₄₁	—	2	2	no	—
121	C₂×C₄×C₄₁	—	2	4	no	—
122	C₂×C₄×C₄₁	—	2	4	no	—
123	C₂×C₄×C₄₁	—	2	4	no	—
124	C₂×C₄×C₄₁	—	4	4	no	—
125	C₂×C₄×C₄₁	—	4	4	no	—
126	C₂×C₄×C₄₁	—	4	8	no	—
127	C₂×C₄×C₄₁	—	4	4	no	—
128	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
129	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
130	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
132	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
134	C₈×C₄₁	—	4	8	no	—
135	C₂×#1533	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂²×C₄₁	—	2	4	no	—
137	C₂²×C₄₁	—	2	8	no	—
138	C₂²×C₄₁	—	4	4	no	—
139	C₂²×C₄₁	—	4	8	no	—
140	C₂²×C₄₁	—	8	16	no	—
141	C₂²×C₄₁	—	8	8	no	—
142	C₂²×C₄₁	C₂×#100	1	4	yes	—
143	C₂²×C₄₁	C₂×#100	1	2	yes	—
144	C₂²×C₄₁	#421	1	4	yes	—
145	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
146	C₄×C₄₁	—	2	1	no	—
147	C₄×C₄₁	—	2	2	no	—
148	C₄×C₄₁	—	2	1	no	—
149	C₄×C₄₁	—	2	2	no	—
150	C₄×C₄₁	—	2	2	no	—
151	C₄×C₄₁	—	2	4	no	—
152	C₄×C₄₁	—	4	4	no	—
153	C₂²×#124	C₂×C₄₁	1	1	yes	characteristic
154	C₂×#432	C₂×C₄₁	1	2	yes	complemented
155	#1533	C₂×C₄₁	1	8	yes	direct product factor
156	C₂×#329	C₂×C₄₁	1	4	yes	complemented
157	C₂×C₄₁	—	2	4	no	—
158	C₂×C₄₁	—	8	4	no	—
159	C₂×C₄₁	#1533	1	4	yes	direct product factor
160	C₂×C₄₁	C₂²×#100	1	1	yes	characteristic
161	C₂×C₄₁	C₂×#421	1	2	yes	—
162	C₂³×Q₈	C₄×C₄₁	1	1	yes	characteristic
163	C₂³×D₄	C₄×C₄₁	1	1	yes	characteristic
164	C₂×#124	—	2	2	no	—
165	C₂×#124	—	2	4	no	—
166	C₂×#124	C₄×C₄₁	1	2	yes	—
167	C₂×#124	C₄×C₄₁	1	4	yes	—
168	C₂×#124	C₂²×C₄₁	1	2	yes	—
169	C₂×#124	C₂²×C₄₁	1	4	yes	—
170	C₂²×C₂²⋊C₄	—	2	2	no	—
171	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
172	#432	—	2	4	no	—
173	#432	C₂²×C₄₁	1	8	yes	complemented
174	C₂×C₄⋊C₈	—	2	2	no	—
175	C₂×C₂²⋊C₈	—	2	2	no	—
176	C₂×#97	C₂²×C₄₁	1	4	yes	complemented
177	#329	C₂²×C₄₁	1	16	yes	complemented
178	C₄₁	C₂×#1533	1	1	yes	characteristic, direct product factor, Sylow
179	C₂⁵	—	2	1	no	—
180	C₂²×Q₈	—	2	2	no	—
181	C₂²×Q₈	—	2	1	no	—
182	C₂²×Q₈	—	4	4	no	—
183	C₂²×Q₈	C₂×C₄×C₄₁	1	2	yes	—
184	C₂²×Q₈	C₂×C₄×C₄₁	1	4	yes	—
185	C₂²×D₄	—	2	2	no	—
186	C₂²×D₄	—	2	1	no	—
187	C₂²×D₄	—	4	4	no	—
188	C₂²×D₄	C₂×C₄×C₄₁	1	2	yes	—
189	C₂²×D₄	C₂×C₄×C₄₁	1	4	yes	—
190	C₂³×C₄	—	2	1	no	—
191	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
192	C₂³×C₄	D₄×C₄₁	1	2	yes	—
193	C₂²×C₈	—	2	4	no	—
194	#124	—	2	8	no	—
195	#124	—	2	4	no	—
196	#124	—	2	4	no	—
197	#124	—	2	8	no	—
198	#124	C₂×C₄×C₄₁	1	4	yes	—
199	#124	C₂×C₄×C₄₁	1	8	yes	—
200	#124	C₂³×C₄₁	1	4	yes	—
201	C₂×C₂²⋊C₄	—	2	4	no	—
202	C₂×C₂²⋊C₄	—	2	8	no	—
203	C₂×C₂²⋊C₄	—	2	4	no	—
204	C₂×C₂²⋊C₄	—	2	8	no	—
205	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
206	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
207	C₂×C₄²	D₄×C₄₁	1	2	yes	—
208	C₂×C₄²	D₄×C₄₁	1	4	yes	—
209	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
210	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
211	C₄⋊C₈	—	2	8	no	—
212	C₂²⋊C₈	—	2	8	no	—
213	#97	—	2	4	no	—
214	#97	C₂³×C₄₁	1	8	yes	complemented
215	C₂⁴	—	2	4	no	—
216	C₂⁴	—	2	2	no	—
217	C₂⁴	—	4	8	no	—
218	C₂⁴	—	4	4	no	—
219	C₂⁴	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
220	C₂×Q₈	—	2	2	no	—
221	C₂×Q₈	—	2	4	no	—
222	C₂×Q₈	—	4	8	no	—
223	C₂×Q₈	—	4	4	no	—
224	C₂×Q₈	—	4	8	no	—
225	C₂×Q₈	—	4	4	no	—
226	C₂×Q₈	C₂²×C₄×C₄₁	1	4	yes	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	4	8	no	—
230	C₂×D₄	—	4	4	no	—
231	C₂×D₄	—	4	8	no	—
232	C₂×D₄	—	4	4	no	—
233	C₂×D₄	C₂²×C₄×C₄₁	1	4	yes	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	4	4	no	—
238	C₂²×C₄	—	4	4	no	—
239	C₂²×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
240	C₂²×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
241	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
242	C₂²×C₄	C₂²⋊C₄×C₄₁	1	4	yes	—
243	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
244	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
246	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
247	C₂×C₈	—	2	8	no	—
248	C₂×C₈	—	4	8	no	—
249	C₂²⋊C₄	—	2	8	no	—
250	C₂²⋊C₄	—	2	16	no	—
251	C₂²⋊C₄	—	2	8	no	—
252	C₄²	C₂²⋊C₄×C₄₁	1	4	yes	—
253	C₄²	C₂²⋊C₄×C₄₁	1	2	yes	—
254	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
255	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
256	C₄²	C₂×D₄×C₄₁	1	2	yes	—
257	C₄²	C₂×D₄×C₄₁	1	4	yes	—
258	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
259	C₂³	—	2	4	no	—
260	C₂³	—	2	4	no	—
261	C₂³	—	4	16	no	—
262	C₂³	—	4	8	no	—
263	C₂³	—	4	4	no	—
264	C₂³	—	4	8	no	—
265	C₂³	—	8	16	no	—
266	C₂³	#100×C₄₁	1	4	yes	—
267	C₂³	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
268	C₂³	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
269	Q₈	—	4	4	no	—
270	Q₈	—	4	4	no	—
271	Q₈	—	4	8	no	—
272	D₄	—	4	4	no	—
273	D₄	—	4	4	no	—
274	D₄	—	4	8	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	8	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
287	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
288	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
289	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
290	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
291	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
292	C₈	—	4	8	no	—
293	C₂²	—	2	8	no	—
294	C₂²	—	2	4	no	—
295	C₂²	—	4	4	no	—
296	C₂²	—	4	8	no	—
297	C₂²	—	8	16	no	—
298	C₂²	—	8	8	no	—
299	C₂²	C₂×#100×C₄₁	1	4	yes	—
300	C₂²	C₂×#100×C₄₁	1	2	yes	—
301	C₂²	C₄₁×#421	1	4	yes	—
302	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
303	C₄	—	2	1	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	2	4	no	—
309	C₄	—	4	4	no	—
310	C₂	—	2	4	no	—
311	C₂	—	8	4	no	—
312	C₂	C₄₁×#1533	1	4	yes	direct product factor
313	C₂	C₂²×#100×C₄₁	1	1	yes	characteristic
314	C₂	C₂×C₄₁×#421	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₄²	C₂⁴×C₄₁
prime radical	C₂³×D₄×C₄₁	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂²×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₁×#432	C₂	semidirect
3. C₄₁×#1533	C₂	direct (N × H)
4. C₂×C₄₁×#329	C₂	semidirect
16. C₄₁×#432	C₂²	semidirect
19. C₂×#97×C₄₁	C₂²	semidirect
20. C₄₁×#329	C₂²	semidirect
56. #97×C₄₁	C₂³	semidirect
135. C₂×#1533	C₄₁	direct (N × H)
154. C₂×#432	C₂×C₄₁	semidirect
155. #1533	C₂×C₄₁	direct (N × H)
156. C₂×#329	C₂×C₄₁	semidirect
159. C₂×C₄₁	#1533	direct (N × H)
173. #432	C₂²×C₄₁	semidirect
176. C₂×#97	C₂²×C₄₁	semidirect
177. #329	C₂²×C₄₁	semidirect
178. C₄₁	C₂×#1533	direct (N × H)
214. #97	C₂³×C₄₁	semidirect
312. C₂	C₄₁×#1533	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#124×C₄₁	2
C₂×C₄₁×#432	2
C₄₁×#1533	2
C₂×C₄₁×#329	2
C₂×#1533	41

Subgroups — C2×C41×#1533 #23793857

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#1533 #23793857