Subgroups — C₄₁×#10696 #23786897

Order 10496.

1078 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#97×C₄₁	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₁×#336	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₁×#336	C₂	1	2	yes	maximal
5	C₄×#95×C₄₁	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₄₁	C₂²	1	2	yes	—
8	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₄₁	C₂²	1	2	yes	—
11	C₂×#97×C₄₁	—	2	4	no	—
12	C₂×#97×C₄₁	—	2	4	no	—
13	C₂×#97×C₄₁	C₄	1	8	yes	complemented
14	C₂×C₄×C₈×C₄₁	—	2	4	no	—
15	C₂×C₄×C₈×C₄₁	C₄	1	4	yes	complemented
16	C₂×#95×C₄₁	C₄	1	4	yes	—
17	C₄₁×#336	C₄	1	8	yes	complemented
18	C₄₁×#336	C₄	1	4	yes	complemented
19	C₂³×C₄×C₄₁	C₂×C₄	1	4	yes	—
20	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
22	C₂²×C₈×C₄₁	—	2	2	no	—
23	C₂²×C₈×C₄₁	—	2	2	no	—
24	C₂²×C₈×C₄₁	—	2	2	no	—
25	C₂²×C₈×C₄₁	—	2	1	no	—
26	C₂²×C₈×C₄₁	—	2	2	no	—
27	C₂²×C₈×C₄₁	—	2	1	no	—
28	C₂²×C₈×C₄₁	—	2	2	no	—
29	C₂²×C₈×C₄₁	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₄₁	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₄₁	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₄₁	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₄₁	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₄₁	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₄₁	C₂×C₄	1	4	yes	—
36	C₂²×C₈×C₄₁	D₄	1	4	yes	—
37	C₂²×C₈×C₄₁	D₄	1	2	yes	—
38	C₂²×C₈×C₄₁	Q₈	1	2	yes	—
39	C₂×C₄²×C₄₁	—	2	2	no	—
40	C₂×C₄²×C₄₁	—	2	2	no	—
41	C₂×C₄²×C₄₁	—	2	2	no	—
42	C₂×C₄²×C₄₁	—	2	2	no	—
43	C₂×C₄²×C₄₁	—	2	2	no	—
44	C₂×C₄²×C₄₁	—	2	2	no	—
45	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
49	#97×C₄₁	—	2	16	no	—
50	C₄×C₈×C₄₁	—	2	8	no	—
51	#95×C₄₁	—	2	8	no	—
52	C₂⁴×C₄₁	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₁	—	2	4	no	—
54	C₂²×C₄×C₄₁	—	2	4	no	—
55	C₂²×C₄×C₄₁	—	2	4	no	—
56	C₂²×C₄×C₄₁	—	2	4	no	—
57	C₂²×C₄×C₄₁	—	2	8	no	—
58	C₂²×C₄×C₄₁	—	2	8	no	—
59	C₂²×C₄×C₄₁	C₄²	1	8	yes	—
60	C₂²×C₄×C₄₁	C₄²	1	4	yes	—
61	C₂²×C₄×C₄₁	M₁₆	1	8	yes	—
62	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₁	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
70	C₂²×C₄×C₄₁	#41	1	2	yes	—
71	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
72	C₂×C₈×C₄₁	—	2	2	no	—
73	C₂×C₈×C₄₁	—	2	4	no	—
74	C₂×C₈×C₄₁	—	2	2	no	—
75	C₂×C₈×C₄₁	—	2	4	no	—
76	C₂×C₈×C₄₁	—	2	4	no	—
77	C₂×C₈×C₄₁	—	2	4	no	—
78	C₂×C₈×C₄₁	—	2	4	no	—
79	C₂×C₈×C₄₁	—	2	4	no	—
80	C₂×C₈×C₄₁	—	2	4	no	—
81	C₂×C₈×C₄₁	—	2	4	no	—
82	C₂×C₈×C₄₁	C₄²	1	4	yes	—
83	C₂×C₈×C₄₁	C₄²	1	8	yes	—
84	C₂×C₈×C₄₁	C₄²	1	4	yes	—
85	C₂×C₈×C₄₁	C₂²⋊C₄	1	4	yes	—
86	C₂×C₈×C₄₁	C₂²⋊C₄	1	8	yes	—
87	C₂×C₈×C₄₁	C₄⋊C₄	1	4	yes	—
88	C₂×C₈×C₄₁	C₄⋊C₄	1	8	yes	—
89	C₄²×C₄₁	—	2	2	no	—
90	C₄²×C₄₁	—	2	4	no	—
91	C₄²×C₄₁	—	2	2	no	—
92	C₄²×C₄₁	—	2	4	no	—
93	C₄²×C₄₁	—	2	4	no	—
94	C₄²×C₄₁	—	2	4	no	—
95	C₄²×C₄₁	—	2	4	no	—
96	C₂³×C₄₁	C₂×C₄²	1	1	yes	characteristic
97	C₂³×C₄₁	C₂×C₄²	1	2	yes	—
98	C₂³×C₄₁	C₄×D₄	1	2	yes	—
99	C₂³×C₄₁	C₄×D₄	1	1	yes	characteristic
100	C₂³×C₄₁	C₄×Q₈	1	1	yes	characteristic
101	C₂³×C₄₁	C₂×M₁₆	1	4	yes	—
102	C₂³×C₄₁	#131	1	4	yes	—
103	C₂×C₄×C₄₁	—	2	4	no	—
104	C₂×C₄×C₄₁	—	2	4	no	—
105	C₂×C₄×C₄₁	—	2	16	no	—
106	C₂×C₄×C₄₁	—	2	8	no	—
107	C₂×C₄×C₄₁	—	2	8	no	—
108	C₂×C₄×C₄₁	—	2	8	no	—
109	C₂×C₄×C₄₁	—	4	8	no	—
110	C₂×C₄×C₄₁	—	4	8	no	—
111	C₂×C₄×C₄₁	#97	1	8	yes	complemented
112	C₂×C₄×C₄₁	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₄₁	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₄₁	C₂×C₄²	1	2	yes	—
115	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
117	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₄₁	#117	1	1	yes	characteristic
120	C₂×C₄×C₄₁	#117	1	2	yes	—
121	C₂×C₄×C₄₁	#117	1	1	yes	characteristic
122	C₂×C₄×C₄₁	#117	1	2	yes	—
123	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₄₁	C₄×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₄₁	C₄×Q₈	1	2	yes	—
130	C₈×C₄₁	—	2	4	no	—
131	C₈×C₄₁	—	2	8	no	—
132	C₈×C₄₁	#95	1	8	yes	complemented
133	#10696	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂²×C₄₁	C₄³	1	1	yes	characteristic
135	C₂²×C₄₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
136	C₂²×C₄₁	C₄×C₂²⋊C₄	1	2	yes	—
137	C₂²×C₄₁	C₄×C₄⋊C₄	1	1	yes	characteristic
138	C₂²×C₄₁	C₄×C₄⋊C₄	1	2	yes	—
139	C₂²×C₄₁	C₂×#97	1	2	yes	—
140	C₂²×C₄₁	C₄×M₁₆	1	4	yes	—
141	C₂²×C₄₁	#405	1	2	yes	—
142	C₂²×C₄₁	#405	1	4	yes	—
143	C₂²×C₄₁	#435	1	8	yes	—
144	C₂²×C₄₁	#436	1	4	yes	—
145	C₂²×C₄₁	#446	1	4	yes	—
146	C₄×C₄₁	—	2	8	no	—
147	C₄×C₄₁	—	4	8	no	—
148	C₄×C₄₁	C₂×#95	1	1	yes	characteristic
149	C₄×C₄₁	#376	1	1	yes	characteristic
150	C₄×C₄₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
151	C₄×C₄₁	C₄×C₂²⋊C₄	1	2	yes	—
152	C₄×C₄₁	C₄×C₄⋊C₄	1	1	yes	characteristic
153	C₄×C₄₁	C₄×C₄⋊C₄	1	2	yes	—
154	C₂²×#97	C₂×C₄₁	1	2	yes	—
155	C₂²×C₄×C₈	C₂×C₄₁	1	1	yes	characteristic
156	C₂×#336	C₂×C₄₁	1	2	yes	—
157	C₂×#336	C₂×C₄₁	1	1	yes	characteristic
158	C₄×#95	C₂×C₄₁	1	1	yes	characteristic
159	C₂×C₄₁	C₄×#95	1	1	yes	characteristic
160	C₂×C₄₁	C₄×#97	1	1	yes	characteristic
161	C₂×C₄₁	#1726	1	1	yes	characteristic
162	C₂×C₄₁	#1752	1	2	yes	—
163	C₂×C₄₁	#1753	1	4	yes	—
164	C₂×C₄₁	#1770	1	2	yes	—
165	C₂×C₄₁	#1771	1	4	yes	—
166	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
167	C₂³×C₈	C₂²×C₄₁	1	2	yes	—
168	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
169	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
170	C₂²×C₄²	C₂²×C₄₁	1	2	yes	—
171	C₂×#97	—	2	4	no	—
172	C₂×#97	—	2	4	no	—
173	C₂×#97	C₄×C₄₁	1	8	yes	complemented
174	C₂×C₄×C₈	—	2	4	no	—
175	C₂×C₄×C₈	C₄×C₄₁	1	4	yes	complemented
176	C₂×#95	C₄×C₄₁	1	4	yes	—
177	#336	C₄×C₄₁	1	8	yes	complemented
178	#336	C₄×C₄₁	1	4	yes	complemented
179	C₄₁	#10696	1	1	yes	characteristic, direct product factor, Sylow
180	C₂³×C₄	C₂×C₄×C₄₁	1	4	yes	—
181	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
182	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	C₂×C₄×C₄₁	1	4	yes	—
191	C₂²×C₈	C₂×C₄×C₄₁	1	4	yes	—
192	C₂²×C₈	C₂×C₄×C₄₁	1	4	yes	—
193	C₂²×C₈	C₂×C₄×C₄₁	1	2	yes	—
194	C₂²×C₈	C₂×C₄×C₄₁	1	4	yes	—
195	C₂²×C₈	C₂×C₄×C₄₁	1	2	yes	—
196	C₂²×C₈	C₂×C₄×C₄₁	1	4	yes	—
197	C₂²×C₈	D₄×C₄₁	1	4	yes	—
198	C₂²×C₈	D₄×C₄₁	1	2	yes	—
199	C₂²×C₈	Q₈×C₄₁	1	2	yes	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
207	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
208	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
209	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
210	#97	—	2	16	no	—
211	C₄×C₈	—	2	8	no	—
212	#95	—	2	8	no	—
213	C₂⁴	C₂²×C₄×C₄₁	1	1	yes	characteristic
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	8	no	—
219	C₂²×C₄	—	2	8	no	—
220	C₂²×C₄	C₄²×C₄₁	1	8	yes	—
221	C₂²×C₄	C₄²×C₄₁	1	4	yes	—
222	C₂²×C₄	M₁₆×C₄₁	1	8	yes	—
223	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
224	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
225	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
226	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
227	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
228	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
229	C₂²×C₄	C₂×Q₈×C₄₁	1	1	yes	characteristic
230	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
231	C₂²×C₄	#41×C₄₁	1	2	yes	—
232	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	C₄²×C₄₁	1	4	yes	—
244	C₂×C₈	C₄²×C₄₁	1	8	yes	—
245	C₂×C₈	C₄²×C₄₁	1	4	yes	—
246	C₂×C₈	C₂²⋊C₄×C₄₁	1	4	yes	—
247	C₂×C₈	C₂²⋊C₄×C₄₁	1	8	yes	—
248	C₂×C₈	C₄⋊C₄×C₄₁	1	4	yes	—
249	C₂×C₈	C₄⋊C₄×C₄₁	1	8	yes	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	4	no	—
254	C₄²	—	2	4	no	—
255	C₄²	—	2	4	no	—
256	C₄²	—	2	4	no	—
257	C₂³	C₂×C₄²×C₄₁	1	1	yes	characteristic
258	C₂³	C₂×C₄²×C₄₁	1	2	yes	—
259	C₂³	C₄×D₄×C₄₁	1	2	yes	—
260	C₂³	C₄×D₄×C₄₁	1	1	yes	characteristic
261	C₂³	C₄×Q₈×C₄₁	1	1	yes	characteristic
262	C₂³	C₂×M₁₆×C₄₁	1	4	yes	—
263	C₂³	#131×C₄₁	1	4	yes	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	16	no	—
267	C₂×C₄	—	2	8	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	4	8	no	—
271	C₂×C₄	—	4	8	no	—
272	C₂×C₄	#97×C₄₁	1	8	yes	complemented
273	C₂×C₄	C₂×C₄²×C₄₁	1	1	yes	characteristic
274	C₂×C₄	C₂×C₄²×C₄₁	1	1	yes	characteristic
275	C₂×C₄	C₂×C₄²×C₄₁	1	2	yes	—
276	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
277	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
278	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
280	C₂×C₄	#117×C₄₁	1	1	yes	characteristic
281	C₂×C₄	#117×C₄₁	1	2	yes	—
282	C₂×C₄	#117×C₄₁	1	1	yes	characteristic
283	C₂×C₄	#117×C₄₁	1	2	yes	—
284	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
285	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
286	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
287	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
288	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
289	C₂×C₄	C₄×Q₈×C₄₁	1	1	yes	characteristic
290	C₂×C₄	C₄×Q₈×C₄₁	1	2	yes	—
291	C₈	—	2	4	no	—
292	C₈	—	2	8	no	—
293	C₈	#95×C₄₁	1	8	yes	complemented
294	C₂²	C₄³×C₄₁	1	1	yes	characteristic
295	C₂²	C₄×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
296	C₂²	C₄×C₂²⋊C₄×C₄₁	1	2	yes	—
297	C₂²	C₄×C₄⋊C₄×C₄₁	1	1	yes	characteristic
298	C₂²	C₄×C₄⋊C₄×C₄₁	1	2	yes	—
299	C₂²	C₂×#97×C₄₁	1	2	yes	—
300	C₂²	C₄×M₁₆×C₄₁	1	4	yes	—
301	C₂²	C₄₁×#405	1	2	yes	—
302	C₂²	C₄₁×#405	1	4	yes	—
303	C₂²	C₄₁×#435	1	8	yes	—
304	C₂²	C₄₁×#436	1	4	yes	—
305	C₂²	C₄₁×#446	1	4	yes	—
306	C₄	—	2	8	no	—
307	C₄	—	4	8	no	—
308	C₄	C₂×#95×C₄₁	1	1	yes	characteristic
309	C₄	C₄₁×#376	1	1	yes	characteristic
310	C₄	C₄×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
311	C₄	C₄×C₂²⋊C₄×C₄₁	1	2	yes	—
312	C₄	C₄×C₄⋊C₄×C₄₁	1	1	yes	characteristic
313	C₄	C₄×C₄⋊C₄×C₄₁	1	2	yes	—
314	C₂	C₄×#95×C₄₁	1	1	yes	characteristic
315	C₂	C₄×#97×C₄₁	1	1	yes	characteristic
316	C₂	C₄₁×#1726	1	1	yes	characteristic
317	C₂	C₄₁×#1752	1	2	yes	—
318	C₂	C₄₁×#1753	1	4	yes	—
319	C₂	C₄₁×#1770	1	2	yes	—
320	C₂	C₄₁×#1771	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄×C₄₁	C₂³
commutator	C₂²	C₄³×C₄₁
Frattini, cosocle	C₂³×C₄	C₂³×C₄₁
socle, prime radical	C₂⁴×C₄₁	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₂×#97×C₄₁	C₄	semidirect
15. C₂×C₄×C₈×C₄₁	C₄	semidirect
17. C₄₁×#336	C₄	semidirect
18. C₄₁×#336	C₄	semidirect
111. C₂×C₄×C₄₁	#97	semidirect
132. C₈×C₄₁	#95	semidirect
133. #10696	C₄₁	direct (N × H)
173. C₂×#97	C₄×C₄₁	semidirect
175. C₂×C₄×C₈	C₄×C₄₁	semidirect
177. #336	C₄×C₄₁	semidirect
178. #336	C₄×C₄₁	semidirect
179. C₄₁	#10696	direct (N × H)
272. C₂×C₄	#97×C₄₁	semidirect
293. C₈	#95×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#97×C₄₁	2
C₂²×C₄×C₈×C₄₁	2
C₂×C₄₁×#336	2
C₄×#95×C₄₁	2
#10696	41

Subgroups — C41×#10696 #23786897

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#10696 #23786897