Subgroups — C₂×C₄₁×#1798 #23796826

Order 10496.

1638 conjugacy classes of subgroups.

The subgroup lattice has 304 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	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₈×C₄₁	C₂	1	2	yes	maximal, complemented
3	C₂²×C₂²⋊C₈×C₄₁	C₂	1	2	yes	maximal
4	C₄₁×#1798	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄₁×#336	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₄×C₄₁	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₄₁	C₂²	1	4	yes	—
8	C₂×#117×C₄₁	—	2	8	no	—
9	C₂×#117×C₄₁	C₄	1	4	yes	—
10	C₂×#117×C₄₁	C₂²	1	4	yes	—
11	C₂²×C₄⋊C₄×C₄₁	C₄	1	2	yes	—
12	C₂²×C₂²⋊C₄×C₄₁	C₄	1	2	yes	—
13	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₄₁	—	2	8	no	—
16	C₂×C₄⋊C₈×C₄₁	C₂²	1	8	yes	complemented
17	C₂×C₂²⋊C₈×C₄₁	—	2	8	no	—
18	C₂×C₂²⋊C₈×C₄₁	C₂²	1	8	yes	—
19	C₄₁×#336	C₂²	1	8	yes	complemented
20	C₂⁵×C₄₁	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₄₁	—	2	4	no	—
22	C₂³×C₄×C₄₁	—	2	4	no	—
23	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₄₁	D₄	1	4	yes	—
28	C₂³×C₄×C₄₁	Q₈	1	4	yes	—
29	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
30	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₄₁	—	2	8	no	—
32	C₂²×C₈×C₄₁	—	2	8	no	—
33	C₂²×C₈×C₄₁	—	2	8	no	—
34	C₂²×C₈×C₄₁	C₂³	1	8	yes	—
35	#117×C₄₁	—	2	16	no	—
36	#117×C₄₁	—	2	16	no	—
37	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
38	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	8	yes	—
39	C₂×C₂²⋊C₄×C₄₁	—	2	8	no	—
40	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	8	yes	—
41	C₂×C₄²×C₄₁	—	2	4	no	—
42	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₁	D₄	1	8	yes	—
45	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
46	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
47	C₄⋊C₈×C₄₁	—	2	16	no	—
48	C₂²⋊C₈×C₄₁	—	2	16	no	—
49	C₂⁴×C₄₁	—	2	2	no	—
50	C₂⁴×C₄₁	—	2	2	no	—
51	C₂⁴×C₄₁	—	2	8	no	—
52	C₂⁴×C₄₁	C₄⋊C₄	1	4	yes	—
53	C₂⁴×C₄₁	C₂²×C₄	1	2	yes	—
54	C₂⁴×C₄₁	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₁	—	2	2	no	—
56	C₂²×C₄×C₄₁	—	2	2	no	—
57	C₂²×C₄×C₄₁	—	2	8	no	—
58	C₂²×C₄×C₄₁	—	2	4	no	—
59	C₂²×C₄×C₄₁	—	2	4	no	—
60	C₂²×C₄×C₄₁	—	2	16	no	—
61	C₂²×C₄×C₄₁	—	2	16	no	—
62	C₂²×C₄×C₄₁	—	2	4	no	—
63	C₂²×C₄×C₄₁	—	2	4	no	—
64	C₂²×C₄×C₄₁	—	2	4	no	—
65	C₂²×C₄×C₄₁	—	2	8	no	—
66	C₂²×C₄×C₄₁	C₂²⋊C₄	1	8	yes	—
67	C₂²×C₄×C₄₁	C₄⋊C₄	1	4	yes	—
68	C₂²×C₄×C₄₁	C₄⋊C₄	1	8	yes	—
69	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₁	C₂×D₄	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×Q₈	1	4	yes	—
77	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₄₁	#41	1	4	yes	—
79	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
80	C₂×C₈×C₄₁	—	2	8	no	—
81	C₂×C₈×C₄₁	—	2	8	no	—
82	C₂×C₈×C₄₁	—	2	8	no	—
83	C₂×C₈×C₄₁	—	4	16	no	—
84	C₄⋊C₄×C₄₁	—	2	16	no	—
85	C₂²⋊C₄×C₄₁	—	2	16	no	—
86	C₄²×C₄₁	—	2	4	no	—
87	C₄²×C₄₁	—	2	4	no	—
88	C₄²×C₄₁	C₂²⋊C₄	1	8	yes	—
89	C₄²×C₄₁	C₂×D₄	1	8	yes	—
90	C₂³×C₄₁	—	2	2	no	—
91	C₂³×C₄₁	—	2	2	no	—
92	C₂³×C₄₁	—	2	8	no	—
93	C₂³×C₄₁	—	2	8	no	—
94	C₂³×C₄₁	—	2	16	no	—
95	C₂³×C₄₁	—	4	8	no	—
96	C₂³×C₄₁	—	4	8	no	—
97	C₂³×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂³×C₄₁	C₂×C₄⋊C₄	1	4	yes	—
99	C₂³×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
100	C₂³×C₄₁	#117	1	2	yes	—
101	C₂³×C₄₁	#131	1	8	yes	—
102	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₄₁	—	2	2	no	—
104	C₂×C₄×C₄₁	—	2	2	no	—
105	C₂×C₄×C₄₁	—	2	8	no	—
106	C₂×C₄×C₄₁	—	2	8	no	—
107	C₂×C₄×C₄₁	—	2	16	no	—
108	C₂×C₄×C₄₁	—	2	8	no	—
109	C₂×C₄×C₄₁	—	2	8	no	—
110	C₂×C₄×C₄₁	—	2	16	no	—
111	C₂×C₄×C₄₁	—	2	4	no	—
112	C₂×C₄×C₄₁	—	2	8	no	—
113	C₂×C₄×C₄₁	—	4	8	no	—
114	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	8	yes	—
115	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
116	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	4	yes	—
117	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₄₁	#117	1	2	yes	—
119	C₂×C₄×C₄₁	#121	1	8	yes	—
120	C₂×C₄×C₄₁	C₂²⋊Q₈	1	8	yes	—
121	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
123	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
125	C₈×C₄₁	—	4	8	no	—
126	C₂×#1798	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂²×C₄₁	—	2	8	no	—
128	C₂²×C₄₁	—	2	4	no	—
129	C₂²×C₄₁	—	4	16	no	—
130	C₂²×C₄₁	—	4	4	no	—
131	C₂²×C₄₁	—	4	4	no	—
132	C₂²×C₄₁	#380	1	4	yes	—
133	C₂²×C₄₁	#408	1	8	yes	—
134	C₂²×C₄₁	#424	1	8	yes	—
135	C₂²×C₄₁	#433	1	8	yes	—
136	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₄₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
138	C₂²×C₄₁	C₂×#117	1	1	yes	characteristic
139	C₂²×C₄₁	C₂×#131	1	4	yes	—
140	C₄×C₄₁	—	2	8	no	—
141	C₄×C₄₁	—	2	4	no	—
142	C₄×C₄₁	—	4	4	no	—
143	C₄×C₄₁	#380	1	4	yes	—
144	C₄×C₄₁	C₂×#121	1	2	yes	—
145	C₄×C₄₁	C₂×C₂²⋊Q₈	1	2	yes	—
146	C₂²×#117	C₂×C₄₁	1	1	yes	characteristic
147	C₂²×C₄⋊C₈	C₂×C₄₁	1	2	yes	complemented
148	C₂²×C₂²⋊C₈	C₂×C₄₁	1	2	yes	—
149	#1798	C₂×C₄₁	1	8	yes	direct product factor
150	C₂×#336	C₂×C₄₁	1	2	yes	complemented
151	C₂×C₄₁	—	4	4	no	—
152	C₂×C₄₁	#1798	1	8	yes	direct product factor
153	C₂×C₄₁	C₂×#380	1	1	yes	characteristic
154	C₂×C₄₁	C₂×#408	1	2	yes	—
155	C₂×C₄₁	C₂×#424	1	2	yes	—
156	C₂×C₄₁	C₂×#433	1	2	yes	—
157	C₂⁴×C₄	C₂²×C₄₁	1	1	yes	characteristic
158	C₂³×C₈	C₂²×C₄₁	1	4	yes	—
159	C₂×#117	—	2	8	no	—
160	C₂×#117	C₄×C₄₁	1	4	yes	—
161	C₂×#117	C₂²×C₄₁	1	4	yes	—
162	C₂²×C₄⋊C₄	C₄×C₄₁	1	2	yes	—
163	C₂²×C₂²⋊C₄	C₄×C₄₁	1	2	yes	—
164	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
165	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
166	C₂×C₄⋊C₈	—	2	8	no	—
167	C₂×C₄⋊C₈	C₂²×C₄₁	1	8	yes	complemented
168	C₂×C₂²⋊C₈	—	2	8	no	—
169	C₂×C₂²⋊C₈	C₂²×C₄₁	1	8	yes	—
170	#336	C₂²×C₄₁	1	8	yes	complemented
171	C₄₁	C₂×#1798	1	1	yes	characteristic, direct product factor, Sylow
172	C₂⁵	C₂×C₄×C₄₁	1	1	yes	characteristic
173	C₂³×C₄	—	2	4	no	—
174	C₂³×C₄	—	2	4	no	—
175	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
176	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
177	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
178	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
179	C₂³×C₄	D₄×C₄₁	1	4	yes	—
180	C₂³×C₄	Q₈×C₄₁	1	4	yes	—
181	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
182	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
183	C₂²×C₈	—	2	8	no	—
184	C₂²×C₈	—	2	8	no	—
185	C₂²×C₈	—	2	8	no	—
186	C₂²×C₈	C₂³×C₄₁	1	8	yes	—
187	#117	—	2	16	no	—
188	#117	—	2	16	no	—
189	C₂×C₄⋊C₄	—	2	8	no	—
190	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	8	yes	—
191	C₂×C₂²⋊C₄	—	2	8	no	—
192	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	8	yes	—
193	C₂×C₄²	—	2	4	no	—
194	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
195	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
196	C₂×C₄²	D₄×C₄₁	1	8	yes	—
197	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
198	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
199	C₄⋊C₈	—	2	16	no	—
200	C₂²⋊C₈	—	2	16	no	—
201	C₂⁴	—	2	2	no	—
202	C₂⁴	—	2	2	no	—
203	C₂⁴	—	2	8	no	—
204	C₂⁴	C₄⋊C₄×C₄₁	1	4	yes	—
205	C₂⁴	C₂²×C₄×C₄₁	1	2	yes	—
206	C₂⁴	C₂²×C₄×C₄₁	1	1	yes	characteristic
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	8	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	16	no	—
213	C₂²×C₄	—	2	16	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	8	no	—
218	C₂²×C₄	C₂²⋊C₄×C₄₁	1	8	yes	—
219	C₂²×C₄	C₄⋊C₄×C₄₁	1	4	yes	—
220	C₂²×C₄	C₄⋊C₄×C₄₁	1	8	yes	—
221	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
222	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
223	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
224	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
225	C₂²×C₄	C₂×D₄×C₄₁	1	4	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	4	yes	—
229	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
230	C₂²×C₄	#41×C₄₁	1	4	yes	—
231	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
232	C₂×C₈	—	2	8	no	—
233	C₂×C₈	—	2	8	no	—
234	C₂×C₈	—	2	8	no	—
235	C₂×C₈	—	4	16	no	—
236	C₄⋊C₄	—	2	16	no	—
237	C₂²⋊C₄	—	2	16	no	—
238	C₄²	—	2	4	no	—
239	C₄²	—	2	4	no	—
240	C₄²	C₂²⋊C₄×C₄₁	1	8	yes	—
241	C₄²	C₂×D₄×C₄₁	1	8	yes	—
242	C₂³	—	2	2	no	—
243	C₂³	—	2	2	no	—
244	C₂³	—	2	8	no	—
245	C₂³	—	2	8	no	—
246	C₂³	—	2	16	no	—
247	C₂³	—	4	8	no	—
248	C₂³	—	4	8	no	—
249	C₂³	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
250	C₂³	C₂×C₄⋊C₄×C₄₁	1	4	yes	—
251	C₂³	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
252	C₂³	#117×C₄₁	1	2	yes	—
253	C₂³	#131×C₄₁	1	8	yes	—
254	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	2	16	no	—
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	2	16	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	4	8	no	—
266	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	8	yes	—
267	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
268	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	4	yes	—
269	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
270	C₂×C₄	#117×C₄₁	1	2	yes	—
271	C₂×C₄	#121×C₄₁	1	8	yes	—
272	C₂×C₄	C₂²⋊Q₈×C₄₁	1	8	yes	—
273	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
274	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
275	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
276	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
277	C₈	—	4	8	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	2	8	no	—
280	C₂²	—	4	16	no	—
281	C₂²	—	4	4	no	—
282	C₂²	—	4	4	no	—
283	C₂²	C₄₁×#380	1	4	yes	—
284	C₂²	C₄₁×#408	1	8	yes	—
285	C₂²	C₄₁×#424	1	8	yes	—
286	C₂²	C₄₁×#433	1	8	yes	—
287	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
288	C₂²	C₂²×C₄⋊C₄×C₄₁	1	1	yes	characteristic
289	C₂²	C₂×#117×C₄₁	1	1	yes	characteristic
290	C₂²	C₂×#131×C₄₁	1	4	yes	—
291	C₄	—	2	4	no	—
292	C₄	—	2	8	no	—
293	C₄	—	4	4	no	—
294	C₄	C₄₁×#380	1	4	yes	—
295	C₄	C₂×#121×C₄₁	1	2	yes	—
296	C₄	C₂×C₂²⋊Q₈×C₄₁	1	2	yes	—
297	C₂	—	4	4	no	—
298	C₂	C₄₁×#1798	1	8	yes	direct product factor
299	C₂	C₂×C₄₁×#380	1	1	yes	characteristic
300	C₂	C₂×C₄₁×#408	1	2	yes	—
301	C₂	C₂×C₄₁×#424	1	2	yes	—
302	C₂	C₂×C₄₁×#433	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₄₁	C₂³
commutator	C₂³	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
socle	C₂⁴×C₄₁	C₂²×C₄
prime radical	C₂⁵×C₄₁	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄⋊C₈×C₄₁	C₂	semidirect
4. C₄₁×#1798	C₂	direct (N × H)
5. C₂×C₄₁×#336	C₂	semidirect
16. C₂×C₄⋊C₈×C₄₁	C₂²	semidirect
19. C₄₁×#336	C₂²	semidirect
126. C₂×#1798	C₄₁	direct (N × H)
147. C₂²×C₄⋊C₈	C₂×C₄₁	semidirect
149. #1798	C₂×C₄₁	direct (N × H)
150. C₂×#336	C₂×C₄₁	semidirect
152. C₂×C₄₁	#1798	direct (N × H)
167. C₂×C₄⋊C₈	C₂²×C₄₁	semidirect
170. #336	C₂²×C₄₁	semidirect
171. C₄₁	C₂×#1798	direct (N × H)
298. C₂	C₄₁×#1798	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×#117×C₄₁	2
C₂²×C₄⋊C₈×C₄₁	2
C₂²×C₂²⋊C₈×C₄₁	2
C₄₁×#1798	2
C₂×C₄₁×#336	2
C₂×#1798	41

Subgroups — C2×C41×#1798 #23796826

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#1798 #23796826