Subgroups — C₄₁×#13756 #23789957

Order 10496.

3070 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#117×C₄₁	C₂	1	2	yes	maximal
2	C₂³×C₄²×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₄₁×C₄⋊#95	C₂	1	4	yes	maximal, complemented
4	C₄₁×C₂²⋊#95	C₂	1	4	yes	maximal
5	C₄×#95×C₄₁	C₂	1	4	yes	maximal, complemented
6	C₂⁴×C₄×C₄₁	C₂²	1	2	yes	—
7	C₂⁴×C₄×C₄₁	C₂²	1	1	yes	characteristic
8	C₂×#117×C₄₁	—	2	8	no	—
9	C₂×#117×C₄₁	—	2	8	no	—
10	C₂×#117×C₄₁	C₄	1	16	yes	complemented
11	C₂²×C₄⋊C₄×C₄₁	C₂²	1	4	yes	—
12	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	4	yes	—
13	C₂²×C₄²×C₄₁	—	2	8	no	—
14	C₂²×C₄²×C₄₁	C₄	1	8	yes	complemented
15	C₂²×C₄²×C₄₁	C₂²	1	2	yes	—
16	C₂²×C₄²×C₄₁	C₂²	1	2	yes	—
17	C₂²×C₄²×C₄₁	C₂²	1	4	yes	—
18	C₂×#95×C₄₁	C₂²	1	16	yes	—
19	C₂⁵×C₄₁	C₂³	1	1	yes	characteristic
20	C₂³×C₄×C₄₁	—	2	8	no	—
21	C₂³×C₄×C₄₁	—	2	16	no	—
22	C₂³×C₄×C₄₁	C₂×C₄	1	8	yes	—
23	C₂³×C₄×C₄₁	C₂×C₄	1	4	yes	—
24	C₂³×C₄×C₄₁	C₂×C₄	1	8	yes	—
25	C₂³×C₄×C₄₁	D₄	1	8	yes	—
26	C₂³×C₄×C₄₁	D₄	1	4	yes	—
27	C₂³×C₄×C₄₁	Q₈	1	4	yes	—
28	C₂³×C₄×C₄₁	C₂³	1	8	yes	—
29	C₂³×C₄×C₄₁	C₂³	1	4	yes	—
30	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
31	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
32	#117×C₄₁	—	2	64	no	—
33	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
34	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
35	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	16	yes	complemented
36	C₂×C₂²⋊C₄×C₄₁	—	2	8	no	—
37	C₂×C₂²⋊C₄×C₄₁	—	2	8	no	—
38	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	16	yes	—
39	C₂×C₄²×C₄₁	—	2	32	no	—
40	C₂×C₄²×C₄₁	—	2	8	no	—
41	C₂×C₄²×C₄₁	—	2	8	no	—
42	C₂×C₄²×C₄₁	—	2	8	no	—
43	C₂×C₄²×C₄₁	—	2	8	no	—
44	C₂×C₄²×C₄₁	C₂×C₄	1	8	yes	—
45	C₂×C₄²×C₄₁	C₂×C₄	1	8	yes	—
46	C₂×C₄²×C₄₁	C₂×C₄	1	16	yes	complemented
47	#95×C₄₁	—	2	32	no	—
48	C₂⁴×C₄₁	—	2	8	no	—
49	C₂⁴×C₄₁	C₂²×C₄	1	4	yes	—
50	C₂⁴×C₄₁	C₂²×C₄	1	2	yes	—
51	C₂⁴×C₄₁	C₂×D₄	1	2	yes	—
52	C₂⁴×C₄₁	C₂×D₄	1	4	yes	—
53	C₂⁴×C₄₁	C₂×Q₈	1	2	yes	—
54	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
55	C₂²×C₄×C₄₁	—	2	64	no	—
56	C₂²×C₄×C₄₁	—	2	16	no	—
57	C₂²×C₄×C₄₁	—	2	8	no	—
58	C₂²×C₄×C₄₁	—	2	8	no	—
59	C₂²×C₄×C₄₁	—	2	32	no	—
60	C₂²×C₄×C₄₁	—	2	8	no	—
61	C₂²×C₄×C₄₁	—	2	8	no	—
62	C₂²×C₄×C₄₁	—	2	32	no	—
63	C₂²×C₄×C₄₁	C₄²	1	16	yes	—
64	C₂²×C₄×C₄₁	C₂²⋊C₄	1	16	yes	—
65	C₂²×C₄×C₄₁	C₂²⋊C₄	1	32	yes	—
66	C₂²×C₄×C₄₁	C₄⋊C₄	1	16	yes	—
67	C₂²×C₄×C₄₁	C₄⋊C₄	1	32	yes	—
68	C₂²×C₄×C₄₁	C₂²×C₄	1	16	yes	—
69	C₂²×C₄×C₄₁	C₂²×C₄	1	8	yes	—
70	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
79	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
80	C₂²×C₄×C₄₁	#41	1	16	yes	—
81	C₄⋊C₄×C₄₁	—	2	32	no	—
82	C₂²⋊C₄×C₄₁	—	2	32	no	—
83	C₄²×C₄₁	—	2	16	no	—
84	C₄²×C₄₁	—	2	16	no	—
85	C₄²×C₄₁	—	2	32	no	—
86	C₂³×C₄₁	—	2	8	no	—
87	C₂³×C₄₁	—	2	16	no	—
88	C₂³×C₄₁	—	2	16	no	—
89	C₂³×C₄₁	—	2	8	no	—
90	C₂³×C₄₁	—	2	8	no	—
91	C₂³×C₄₁	C₂×C₄²	1	4	yes	—
92	C₂³×C₄₁	C₂×C₂²⋊C₄	1	8	yes	—
93	C₂³×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
94	C₂³×C₄₁	C₂×C₄⋊C₄	1	4	yes	—
95	C₂³×C₄₁	C₂×C₄⋊C₄	1	8	yes	—
96	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
97	C₂³×C₄₁	C₂³×C₄	1	2	yes	—
98	C₂³×C₄₁	C₂²×D₄	1	2	yes	—
99	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
100	C₂³×C₄₁	C₂²×Q₈	1	1	yes	characteristic
101	C₂³×C₄₁	C₂×#41	1	8	yes	—
102	C₂×C₄×C₄₁	—	2	64	no	—
103	C₂×C₄×C₄₁	—	2	8	no	—
104	C₂×C₄×C₄₁	—	2	8	no	—
105	C₂×C₄×C₄₁	—	2	32	no	—
106	C₂×C₄×C₄₁	—	2	16	no	—
107	C₂×C₄×C₄₁	—	2	32	no	—
108	C₂×C₄×C₄₁	—	4	16	no	—
109	C₂×C₄×C₄₁	—	4	16	no	—
110	C₂×C₄×C₄₁	#95	1	64	yes	complemented
111	C₂×C₄×C₄₁	C₂×C₄²	1	4	yes	—
112	C₂×C₄×C₄₁	C₂×C₄²	1	4	yes	—
113	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
114	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
115	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	8	yes	—
116	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	8	yes	—
117	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	4	yes	—
118	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	4	yes	—
119	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	8	yes	—
120	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	8	yes	—
121	C₂×C₄×C₄₁	#117	1	16	yes	complemented
122	#13756	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂²×C₄₁	—	2	8	no	—
124	C₂²×C₄₁	—	2	8	no	—
125	C₂²×C₄₁	—	2	8	no	—
126	C₂²×C₄₁	—	2	16	no	—
127	C₂²×C₄₁	—	2	16	no	—
128	C₂²×C₄₁	C₂×#95	1	8	yes	—
129	C₂²×C₄₁	C₂²×C₄²	1	1	yes	characteristic
130	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
131	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	2	yes	—
132	C₂²×C₄₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
133	C₂²×C₄₁	C₂²×C₄⋊C₄	1	2	yes	—
134	C₂²×C₄₁	C₂×#117	1	4	yes	—
135	C₂²×C₄₁	C₄×#41	1	8	yes	—
136	C₂²×C₄₁	#525	1	8	yes	—
137	C₂²×C₄₁	#532	1	4	yes	—
138	C₂²×C₄₁	#533	1	4	yes	—
139	C₄×C₄₁	—	2	16	no	—
140	C₄×C₄₁	—	4	16	no	—
141	C₄×C₄₁	C₂×#95	1	8	yes	—
142	C₄×C₄₁	C₂×#95	1	8	yes	—
143	C₂²×#117	C₂×C₄₁	1	2	yes	—
144	C₂³×C₄²	C₂×C₄₁	1	1	yes	characteristic
145	C₄⋊#95	C₂×C₄₁	1	4	yes	complemented
146	C₂²⋊#95	C₂×C₄₁	1	4	yes	—
147	C₄×#95	C₂×C₄₁	1	4	yes	complemented
148	C₂×C₄₁	—	2	8	no	—
149	C₂×C₄₁	C₂²×#95	1	1	yes	characteristic
150	C₂×C₄₁	C₄×#117	1	2	yes	—
151	C₂×C₄₁	#2280	1	2	yes	—
152	C₂×C₄₁	#2285	1	2	yes	—
153	C₂×C₄₁	#2296	1	4	yes	—
154	C₂×C₄₁	#2297	1	4	yes	—
155	C₂⁴×C₄	C₂²×C₄₁	1	2	yes	—
156	C₂⁴×C₄	C₂²×C₄₁	1	1	yes	characteristic
157	C₂×#117	—	2	8	no	—
158	C₂×#117	—	2	8	no	—
159	C₂×#117	C₄×C₄₁	1	16	yes	complemented
160	C₂²×C₄⋊C₄	C₂²×C₄₁	1	4	yes	—
161	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	4	yes	—
162	C₂²×C₄²	—	2	8	no	—
163	C₂²×C₄²	C₄×C₄₁	1	8	yes	complemented
164	C₂²×C₄²	C₂²×C₄₁	1	2	yes	—
165	C₂²×C₄²	C₂²×C₄₁	1	2	yes	—
166	C₂²×C₄²	C₂²×C₄₁	1	4	yes	—
167	C₂×#95	C₂²×C₄₁	1	16	yes	—
168	C₄₁	#13756	1	1	yes	characteristic, direct product factor, Sylow
169	C₂⁵	C₂³×C₄₁	1	1	yes	characteristic
170	C₂³×C₄	—	2	8	no	—
171	C₂³×C₄	—	2	16	no	—
172	C₂³×C₄	C₂×C₄×C₄₁	1	8	yes	—
173	C₂³×C₄	C₂×C₄×C₄₁	1	8	yes	—
174	C₂³×C₄	C₂×C₄×C₄₁	1	4	yes	—
175	C₂³×C₄	D₄×C₄₁	1	4	yes	—
176	C₂³×C₄	D₄×C₄₁	1	8	yes	—
177	C₂³×C₄	Q₈×C₄₁	1	4	yes	—
178	C₂³×C₄	C₂³×C₄₁	1	8	yes	—
179	C₂³×C₄	C₂³×C₄₁	1	4	yes	—
180	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
181	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
182	#117	—	2	64	no	—
183	C₂×C₄⋊C₄	—	2	8	no	—
184	C₂×C₄⋊C₄	—	2	8	no	—
185	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	16	yes	complemented
186	C₂×C₂²⋊C₄	—	2	8	no	—
187	C₂×C₂²⋊C₄	—	2	8	no	—
188	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	16	yes	—
189	C₂×C₄²	—	2	32	no	—
190	C₂×C₄²	—	2	8	no	—
191	C₂×C₄²	—	2	8	no	—
192	C₂×C₄²	—	2	8	no	—
193	C₂×C₄²	—	2	8	no	—
194	C₂×C₄²	C₂×C₄×C₄₁	1	8	yes	—
195	C₂×C₄²	C₂×C₄×C₄₁	1	8	yes	—
196	C₂×C₄²	C₂×C₄×C₄₁	1	16	yes	complemented
197	#95	—	2	32	no	—
198	C₂⁴	—	2	8	no	—
199	C₂⁴	C₂²×C₄×C₄₁	1	4	yes	—
200	C₂⁴	C₂²×C₄×C₄₁	1	2	yes	—
201	C₂⁴	C₂×D₄×C₄₁	1	2	yes	—
202	C₂⁴	C₂×D₄×C₄₁	1	4	yes	—
203	C₂⁴	C₂×Q₈×C₄₁	1	2	yes	—
204	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
205	C₂²×C₄	—	2	64	no	—
206	C₂²×C₄	—	2	16	no	—
207	C₂²×C₄	—	2	8	no	—
208	C₂²×C₄	—	2	8	no	—
209	C₂²×C₄	—	2	32	no	—
210	C₂²×C₄	—	2	8	no	—
211	C₂²×C₄	—	2	8	no	—
212	C₂²×C₄	—	2	32	no	—
213	C₂²×C₄	C₄²×C₄₁	1	16	yes	—
214	C₂²×C₄	C₂²⋊C₄×C₄₁	1	16	yes	—
215	C₂²×C₄	C₂²⋊C₄×C₄₁	1	32	yes	—
216	C₂²×C₄	C₄⋊C₄×C₄₁	1	32	yes	—
217	C₂²×C₄	C₄⋊C₄×C₄₁	1	16	yes	—
218	C₂²×C₄	C₂²×C₄×C₄₁	1	16	yes	—
219	C₂²×C₄	C₂²×C₄×C₄₁	1	8	yes	—
220	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
221	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
222	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
223	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
224	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
225	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
226	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
227	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
228	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
229	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
230	C₂²×C₄	#41×C₄₁	1	16	yes	—
231	C₄⋊C₄	—	2	32	no	—
232	C₂²⋊C₄	—	2	32	no	—
233	C₄²	—	2	16	no	—
234	C₄²	—	2	16	no	—
235	C₄²	—	2	32	no	—
236	C₂³	—	2	8	no	—
237	C₂³	—	2	16	no	—
238	C₂³	—	2	16	no	—
239	C₂³	—	2	8	no	—
240	C₂³	—	2	8	no	—
241	C₂³	C₂×C₄²×C₄₁	1	4	yes	—
242	C₂³	C₂×C₂²⋊C₄×C₄₁	1	8	yes	—
243	C₂³	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
244	C₂³	C₂×C₄⋊C₄×C₄₁	1	4	yes	—
245	C₂³	C₂×C₄⋊C₄×C₄₁	1	8	yes	—
246	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
247	C₂³	C₂³×C₄×C₄₁	1	2	yes	—
248	C₂³	C₂²×D₄×C₄₁	1	2	yes	—
249	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
250	C₂³	C₂²×Q₈×C₄₁	1	1	yes	characteristic
251	C₂³	C₂×#41×C₄₁	1	8	yes	—
252	C₂×C₄	—	2	64	no	—
253	C₂×C₄	—	2	8	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	2	32	no	—
256	C₂×C₄	—	2	16	no	—
257	C₂×C₄	—	2	32	no	—
258	C₂×C₄	—	4	16	no	—
259	C₂×C₄	—	4	16	no	—
260	C₂×C₄	#95×C₄₁	1	64	yes	complemented
261	C₂×C₄	C₂×C₄²×C₄₁	1	4	yes	—
262	C₂×C₄	C₂×C₄²×C₄₁	1	4	yes	—
263	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
264	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
265	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	8	yes	—
266	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	8	yes	—
267	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	4	yes	—
268	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	4	yes	—
269	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	8	yes	—
270	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	8	yes	—
271	C₂×C₄	#117×C₄₁	1	16	yes	complemented
272	C₂²	—	2	8	no	—
273	C₂²	—	2	8	no	—
274	C₂²	—	2	8	no	—
275	C₂²	—	2	16	no	—
276	C₂²	—	2	16	no	—
277	C₂²	C₂×#95×C₄₁	1	8	yes	—
278	C₂²	C₂²×C₄²×C₄₁	1	1	yes	characteristic
279	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
280	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	2	yes	—
281	C₂²	C₂²×C₄⋊C₄×C₄₁	1	1	yes	characteristic
282	C₂²	C₂²×C₄⋊C₄×C₄₁	1	2	yes	—
283	C₂²	C₂×#117×C₄₁	1	4	yes	—
284	C₂²	C₄×#41×C₄₁	1	8	yes	—
285	C₂²	C₄₁×#525	1	8	yes	—
286	C₂²	C₄₁×#532	1	4	yes	—
287	C₂²	C₄₁×#533	1	4	yes	—
288	C₄	—	2	16	no	—
289	C₄	—	4	16	no	—
290	C₄	C₂×#95×C₄₁	1	8	yes	—
291	C₄	C₂×#95×C₄₁	1	8	yes	—
292	C₂	—	2	8	no	—
293	C₂	C₂²×#95×C₄₁	1	1	yes	characteristic
294	C₂	C₄×#117×C₄₁	1	2	yes	—
295	C₂	C₄₁×#2280	1	2	yes	—
296	C₂	C₄₁×#2285	1	2	yes	—
297	C₂	C₄₁×#2296	1	4	yes	—
298	C₂	C₄₁×#2297	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄×C₄₁	C₂³
commutator	C₂²	C₂²×C₄²×C₄₁
Frattini, cosocle	C₂⁴	C₂⁴×C₄₁
socle	C₂⁴×C₄₁	C₂⁴
prime radical	C₂⁵×C₄₁	C₂³
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
3. C₄₁×C₄⋊#95	C₂	semidirect
5. C₄×#95×C₄₁	C₂	semidirect
10. C₂×#117×C₄₁	C₄	semidirect
14. C₂²×C₄²×C₄₁	C₄	semidirect
35. C₂×C₄⋊C₄×C₄₁	C₂×C₄	semidirect
46. C₂×C₄²×C₄₁	C₂×C₄	semidirect
110. C₂×C₄×C₄₁	#95	semidirect
121. C₂×C₄×C₄₁	#117	semidirect
122. #13756	C₄₁	direct (N × H)
145. C₄⋊#95	C₂×C₄₁	semidirect
147. C₄×#95	C₂×C₄₁	semidirect
159. C₂×#117	C₄×C₄₁	semidirect
163. C₂²×C₄²	C₄×C₄₁	semidirect
168. C₄₁	#13756	direct (N × H)
185. C₂×C₄⋊C₄	C₂×C₄×C₄₁	semidirect
196. C₂×C₄²	C₂×C₄×C₄₁	semidirect
260. C₂×C₄	#95×C₄₁	semidirect
271. C₂×C₄	#117×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#117×C₄₁	2
C₂³×C₄²×C₄₁	2
C₄₁×C₄⋊#95	2
C₄₁×C₂²⋊#95	2
C₄×#95×C₄₁	2
#13756	41

Subgroups — C41×#13756 #23789957

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#13756 #23789957