Subgroups — C₂×C₄₁×#1985 #23797500

Order 10496.

1438 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₈×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×#97×C₄₁	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1985	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄₁×#336	C₂	1	4	yes	maximal
6	C₂³×Q₈×C₄₁	C₄	1	1	yes	characteristic
7	C₂³×C₈×C₄₁	C₂²	1	2	yes	—
8	C₂³×C₈×C₄₁	C₂²	1	2	yes	—
9	C₂×C₄×Q₈×C₄₁	—	2	4	no	—
10	C₂×C₄×Q₈×C₄₁	—	2	4	no	—
11	C₂×C₄×Q₈×C₄₁	C₄	1	4	yes	—
12	C₂×C₄×Q₈×C₄₁	C₂²	1	4	yes	—
13	C₂²×C₄⋊C₄×C₄₁	C₄	1	1	yes	characteristic
14	C₂²×C₄⋊C₄×C₄₁	C₄	1	2	yes	—
15	C₂²×C₄²×C₄₁	C₂²	1	2	yes	—
16	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₄₁	—	2	4	no	—
18	C₂×C₄⋊C₈×C₄₁	C₂²	1	4	yes	—
19	C₂×#97×C₄₁	—	2	4	no	—
20	C₂×#97×C₄₁	C₂²	1	4	yes	—
21	C₄₁×#336	C₂²	1	16	yes	—
22	C₂²×Q₈×C₄₁	—	2	4	no	—
23	C₂²×Q₈×C₄₁	—	2	8	no	—
24	C₂²×Q₈×C₄₁	C₂×C₄	1	4	yes	—
25	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₄₁	—	2	4	no	—
31	C₂²×C₈×C₄₁	—	2	4	no	—
32	C₂²×C₈×C₄₁	—	2	4	no	—
33	C₂²×C₈×C₄₁	—	2	8	no	—
34	C₂²×C₈×C₄₁	Q₈	1	8	yes	complemented
35	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
36	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
37	C₄×Q₈×C₄₁	—	2	16	no	—
38	C₄×Q₈×C₄₁	—	2	8	no	—
39	C₄×Q₈×C₄₁	—	2	8	no	—
40	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
41	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
42	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
43	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	8	yes	—
44	C₂×C₄²×C₄₁	—	2	4	no	—
45	C₂×C₄²×C₄₁	—	2	4	no	—
46	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₄₁	D₄	1	4	yes	—
49	C₂×C₄²×C₄₁	D₄	1	4	yes	—
50	C₂×C₄²×C₄₁	C₂³	1	4	yes	—
51	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
52	C₄⋊C₈×C₄₁	—	2	8	no	—
53	#97×C₄₁	—	2	8	no	—
54	C₂⁴×C₄₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×Q₈×C₄₁	—	2	32	no	—
56	C₂×Q₈×C₄₁	—	2	8	no	—
57	C₂×Q₈×C₄₁	—	2	16	no	—
58	C₂²×C₄×C₄₁	—	2	4	no	—
59	C₂²×C₄×C₄₁	—	2	4	no	—
60	C₂²×C₄×C₄₁	—	2	4	no	—
61	C₂²×C₄×C₄₁	—	2	4	no	—
62	C₂²×C₄×C₄₁	—	2	4	no	—
63	C₂²×C₄×C₄₁	—	2	8	no	—
64	C₂²×C₄×C₄₁	C₂²⋊C₄	1	4	yes	—
65	C₂²×C₄×C₄₁	C₂²⋊C₄	1	4	yes	—
66	C₂²×C₄×C₄₁	M₁₆	1	8	yes	—
67	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₄₁	C₂×Q₈	1	4	yes	—
75	C₂²×C₄×C₄₁	#41	1	4	yes	—
76	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₄₁	—	2	16	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	8	no	—
82	C₂×C₈×C₄₁	—	4	8	no	—
83	C₂×C₈×C₄₁	C₂×Q₈	1	8	yes	complemented
84	C₄⋊C₄×C₄₁	—	2	16	no	—
85	C₄⋊C₄×C₄₁	—	2	8	no	—
86	C₄²×C₄₁	—	2	8	no	—
87	C₄²×C₄₁	—	2	4	no	—
88	C₄²×C₄₁	—	2	4	no	—
89	C₄²×C₄₁	C₂²⋊C₄	1	8	yes	—
90	C₄²×C₄₁	C₂×D₄	1	4	yes	—
91	C₄²×C₄₁	C₂×D₄	1	4	yes	—
92	C₂³×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
93	C₂³×C₄₁	C₄×Q₈	1	4	yes	—
94	C₂³×C₄₁	C₂×M₁₆	1	4	yes	—
95	C₂³×C₄₁	#131	1	4	yes	—
96	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
97	Q₈×C₄₁	—	2	32	no	—
98	C₂×C₄×C₄₁	—	2	4	no	—
99	C₂×C₄×C₄₁	—	2	4	no	—
100	C₂×C₄×C₄₁	—	2	4	no	—
101	C₂×C₄×C₄₁	—	2	4	no	—
102	C₂×C₄×C₄₁	—	2	8	no	—
103	C₂×C₄×C₄₁	—	2	16	no	—
104	C₂×C₄×C₄₁	—	2	8	no	—
105	C₂×C₄×C₄₁	—	2	4	no	—
106	C₂×C₄×C₄₁	—	2	8	no	—
107	C₂×C₄×C₄₁	—	4	8	no	—
108	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
109	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
110	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
111	C₂×C₄×C₄₁	C₄×Q₈	1	4	yes	—
112	C₂×C₄×C₄₁	C₂²⋊Q₈	1	8	yes	—
113	C₂×C₄×C₄₁	#124	1	4	yes	—
114	C₂×C₄×C₄₁	C₄⋊Q₈	1	4	yes	—
115	C₂×C₄×C₄₁	C₂×M₁₆	1	8	yes	—
116	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₄₁	C₂²×Q₈	1	2	yes	—
119	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
120	C₈×C₄₁	—	2	8	no	—
121	C₈×C₄₁	—	4	4	no	—
122	C₂×#1985	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂²×C₄₁	#391	1	4	yes	—
124	C₂²×C₄₁	#407	1	4	yes	—
125	C₂²×C₄₁	#408	1	4	yes	—
126	C₂²×C₄₁	#446	1	16	yes	—
127	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
128	C₂²×C₄₁	C₂×C₄×Q₈	1	2	yes	—
129	C₂²×C₄₁	C₂²×M₁₆	1	2	yes	—
130	C₂²×C₄₁	C₂×#131	1	2	yes	—
131	C₄×C₄₁	—	2	4	no	—
132	C₄×C₄₁	—	2	8	no	—
133	C₄×C₄₁	—	2	4	no	—
134	C₄×C₄₁	—	4	4	no	—
135	C₄×C₄₁	#391	1	4	yes	—
136	C₄×C₄₁	C₂×C₂²⋊Q₈	1	2	yes	—
137	C₄×C₄₁	C₂×#124	1	1	yes	characteristic
138	C₄×C₄₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
139	C₂²×C₄×Q₈	C₂×C₄₁	1	1	yes	characteristic
140	C₂²×C₄⋊C₈	C₂×C₄₁	1	1	yes	characteristic
141	C₂²×#97	C₂×C₄₁	1	1	yes	characteristic
142	#1985	C₂×C₄₁	1	8	yes	direct product factor
143	C₂×#336	C₂×C₄₁	1	4	yes	—
144	C₂×C₄₁	#1985	1	8	yes	direct product factor
145	C₂×C₄₁	C₂×#391	1	1	yes	characteristic
146	C₂×C₄₁	C₂×#407	1	1	yes	characteristic
147	C₂×C₄₁	C₂×#408	1	1	yes	characteristic
148	C₂×C₄₁	C₂×#446	1	4	yes	—
149	C₂³×Q₈	C₄×C₄₁	1	1	yes	characteristic
150	C₂³×C₈	C₂²×C₄₁	1	2	yes	—
151	C₂³×C₈	C₂²×C₄₁	1	2	yes	—
152	C₂×C₄×Q₈	—	2	4	no	—
153	C₂×C₄×Q₈	—	2	4	no	—
154	C₂×C₄×Q₈	C₄×C₄₁	1	4	yes	—
155	C₂×C₄×Q₈	C₂²×C₄₁	1	4	yes	—
156	C₂²×C₄⋊C₄	C₄×C₄₁	1	1	yes	characteristic
157	C₂²×C₄⋊C₄	C₄×C₄₁	1	2	yes	—
158	C₂²×C₄²	C₂²×C₄₁	1	2	yes	—
159	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
160	C₂×C₄⋊C₈	—	2	4	no	—
161	C₂×C₄⋊C₈	C₂²×C₄₁	1	4	yes	—
162	C₂×#97	—	2	4	no	—
163	C₂×#97	C₂²×C₄₁	1	4	yes	—
164	#336	C₂²×C₄₁	1	16	yes	—
165	C₄₁	C₂×#1985	1	1	yes	characteristic, direct product factor, Sylow
166	C₂²×Q₈	—	2	4	no	—
167	C₂²×Q₈	—	2	8	no	—
168	C₂²×Q₈	C₂×C₄×C₄₁	1	4	yes	—
169	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
170	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
171	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
172	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
173	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
174	C₂²×C₈	—	2	4	no	—
175	C₂²×C₈	—	2	4	no	—
176	C₂²×C₈	—	2	4	no	—
177	C₂²×C₈	—	2	8	no	—
178	C₂²×C₈	Q₈×C₄₁	1	8	yes	complemented
179	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
180	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
181	C₄×Q₈	—	2	16	no	—
182	C₄×Q₈	—	2	8	no	—
183	C₄×Q₈	—	2	8	no	—
184	C₂×C₄⋊C₄	—	2	8	no	—
185	C₂×C₄⋊C₄	—	2	4	no	—
186	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
187	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	8	yes	—
188	C₂×C₄²	—	2	4	no	—
189	C₂×C₄²	—	2	4	no	—
190	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
191	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
192	C₂×C₄²	D₄×C₄₁	1	4	yes	—
193	C₂×C₄²	D₄×C₄₁	1	4	yes	—
194	C₂×C₄²	C₂³×C₄₁	1	4	yes	—
195	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
196	C₄⋊C₈	—	2	8	no	—
197	#97	—	2	8	no	—
198	C₂⁴	C₂²×C₄×C₄₁	1	1	yes	characteristic
199	C₂×Q₈	—	2	32	no	—
200	C₂×Q₈	—	2	8	no	—
201	C₂×Q₈	—	2	16	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	8	no	—
208	C₂²×C₄	C₂²⋊C₄×C₄₁	1	4	yes	—
209	C₂²×C₄	C₂²⋊C₄×C₄₁	1	4	yes	—
210	C₂²×C₄	M₁₆×C₄₁	1	8	yes	—
211	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
212	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
213	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
214	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
215	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
217	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
218	C₂²×C₄	C₂×Q₈×C₄₁	1	4	yes	—
219	C₂²×C₄	#41×C₄₁	1	4	yes	—
220	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
221	C₂×C₈	—	2	16	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	8	no	—
226	C₂×C₈	—	4	8	no	—
227	C₂×C₈	C₂×Q₈×C₄₁	1	8	yes	complemented
228	C₄⋊C₄	—	2	16	no	—
229	C₄⋊C₄	—	2	8	no	—
230	C₄²	—	2	8	no	—
231	C₄²	—	2	4	no	—
232	C₄²	—	2	4	no	—
233	C₄²	C₂²⋊C₄×C₄₁	1	8	yes	—
234	C₄²	C₂×D₄×C₄₁	1	4	yes	—
235	C₄²	C₂×D₄×C₄₁	1	4	yes	—
236	C₂³	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
237	C₂³	C₄×Q₈×C₄₁	1	4	yes	—
238	C₂³	C₂×M₁₆×C₄₁	1	4	yes	—
239	C₂³	#131×C₄₁	1	4	yes	—
240	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
241	Q₈	—	2	32	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	2	16	no	—
248	C₂×C₄	—	2	8	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	4	8	no	—
252	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
253	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
254	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
255	C₂×C₄	C₄×Q₈×C₄₁	1	4	yes	—
256	C₂×C₄	C₂²⋊Q₈×C₄₁	1	8	yes	—
257	C₂×C₄	#124×C₄₁	1	4	yes	—
258	C₂×C₄	C₄⋊Q₈×C₄₁	1	4	yes	—
259	C₂×C₄	C₂×M₁₆×C₄₁	1	8	yes	—
260	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
261	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
262	C₂×C₄	C₂²×Q₈×C₄₁	1	2	yes	—
263	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
264	C₈	—	2	8	no	—
265	C₈	—	4	4	no	—
266	C₂²	C₄₁×#391	1	4	yes	—
267	C₂²	C₄₁×#407	1	4	yes	—
268	C₂²	C₄₁×#408	1	4	yes	—
269	C₂²	C₄₁×#446	1	16	yes	—
270	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
271	C₂²	C₂×C₄×Q₈×C₄₁	1	2	yes	—
272	C₂²	C₂²×M₁₆×C₄₁	1	2	yes	—
273	C₂²	C₂×#131×C₄₁	1	2	yes	—
274	C₄	—	2	4	no	—
275	C₄	—	2	8	no	—
276	C₄	—	2	4	no	—
277	C₄	—	4	4	no	—
278	C₄	C₄₁×#391	1	4	yes	—
279	C₄	C₂×C₂²⋊Q₈×C₄₁	1	2	yes	—
280	C₄	C₂×#124×C₄₁	1	1	yes	characteristic
281	C₄	C₂×C₄⋊Q₈×C₄₁	1	1	yes	characteristic
282	C₂	C₄₁×#1985	1	8	yes	direct product factor
283	C₂	C₂×C₄₁×#391	1	1	yes	characteristic
284	C₂	C₂×C₄₁×#407	1	1	yes	characteristic
285	C₂	C₂×C₄₁×#408	1	1	yes	characteristic
286	C₂	C₂×C₄₁×#446	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄×C₄₁	C₂³
commutator	C₂³	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
socle, prime radical	C₂⁴×C₄₁	C₂²×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
4. C₄₁×#1985	C₂	direct (N × H)
34. C₂²×C₈×C₄₁	Q₈	semidirect
83. C₂×C₈×C₄₁	C₂×Q₈	semidirect
122. C₂×#1985	C₄₁	direct (N × H)
142. #1985	C₂×C₄₁	direct (N × H)
144. C₂×C₄₁	#1985	direct (N × H)
165. C₄₁	C₂×#1985	direct (N × H)
178. C₂²×C₈	Q₈×C₄₁	semidirect
227. C₂×C₈	C₂×Q₈×C₄₁	semidirect
282. C₂	C₄₁×#1985	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×C₄×Q₈×C₄₁	2
C₂²×C₄⋊C₈×C₄₁	2
C₂²×#97×C₄₁	2
C₄₁×#1985	2
C₂×C₄₁×#336	2
C₂×#1985	41

Subgroups — C2×C41×#1985 #23797500

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#1985 #23797500