Subgroups — C₂×C₄₁×#1668 #23795313

Order 10496.

870 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#460	C₂	1	2	yes	maximal, complemented
3	C₂×C₈×Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#1668	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄₁×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄₁×#331	C₂	1	2	yes	maximal, complemented
7	C₂²×SD₁₆×C₄₁	—	2	2	no	—
8	C₂²×D₈×C₄₁	—	2	1	no	—
9	C₂²×D₈×C₄₁	—	2	1	no	—
10	C₂×#127×C₄₁	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
12	C₄₁×#493	—	2	2	no	—
13	C₄₁×#493	C₂²	1	4	yes	—
14	C₄₁×#460	—	2	4	no	—
15	C₄₁×#460	C₂²	1	8	yes	complemented
16	C₈×Q₈×C₄₁	—	2	2	no	—
17	C₈×Q₈×C₄₁	C₂²	1	4	yes	complemented
18	C₂×C₄⋊C₈×C₄₁	—	2	1	no	—
19	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
21	C₂×#102×C₄₁	—	2	2	no	—
22	C₂×C₄×C₈×C₄₁	—	2	1	no	—
23	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
24	C₄₁×#335	C₂²	1	4	yes	complemented
25	C₄₁×#331	C₂²	1	8	yes	complemented
26	C₂²×Q₈×C₄₁	D₄	1	1	yes	characteristic
27	C₂²×D₄×C₄₁	—	2	2	no	—
28	C₂²×D₄×C₄₁	—	2	2	no	—
29	C₂²×D₄×C₄₁	—	2	2	no	—
30	C₂×SD₁₆×C₄₁	—	2	8	no	—
31	C₂×SD₁₆×C₄₁	—	4	8	no	—
32	C₂×D₈×C₄₁	—	2	4	no	—
33	C₂×D₈×C₄₁	—	2	4	no	—
34	C₂×D₈×C₄₁	—	4	4	no	—
35	C₂×D₈×C₄₁	—	4	4	no	—
36	C₂²×C₈×C₄₁	—	2	2	no	—
37	C₂²×C₈×C₄₁	—	2	1	no	—
38	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
40	#127×C₄₁	—	2	2	no	—
41	#127×C₄₁	—	2	2	no	—
42	#127×C₄₁	—	2	2	no	—
43	#127×C₄₁	C₂³	1	4	yes	—
44	C₄×Q₈×C₄₁	—	2	1	no	—
45	C₄×Q₈×C₄₁	—	2	1	no	—
46	C₄×Q₈×C₄₁	D₄	1	2	yes	—
47	C₄×Q₈×C₄₁	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
49	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₁	—	2	1	no	—
51	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
52	C₄⋊C₈×C₄₁	—	2	4	no	—
53	C₄⋊C₈×C₄₁	—	2	1	no	—
54	C₄⋊C₈×C₄₁	—	2	2	no	—
55	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
56	C₄⋊C₈×C₄₁	C₂³	1	4	yes	—
57	#102×C₄₁	—	2	8	no	—
58	C₄×C₈×C₄₁	—	2	4	no	—
59	C₄×C₈×C₄₁	D₄	1	2	yes	—
60	C₄×C₈×C₄₁	C₂³	1	2	yes	—
61	C₂⁴×C₄₁	—	4	2	no	—
62	C₂×Q₈×C₄₁	—	2	1	no	—
63	C₂×Q₈×C₄₁	—	2	2	no	—
64	C₂×Q₈×C₄₁	D₈	1	4	yes	complemented
65	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
66	C₂×D₄×C₄₁	—	2	4	no	—
67	C₂×D₄×C₄₁	—	2	4	no	—
68	C₂×D₄×C₄₁	—	2	4	no	—
69	C₂×D₄×C₄₁	—	4	2	no	—
70	C₂×D₄×C₄₁	—	4	2	no	—
71	C₂×D₄×C₄₁	—	4	2	no	—
72	C₂×D₄×C₄₁	—	4	4	no	—
73	C₂×D₄×C₄₁	—	4	4	no	—
74	C₂×D₄×C₄₁	—	4	4	no	—
75	C₂×D₄×C₄₁	—	4	4	no	—
76	C₂×D₄×C₄₁	—	8	4	no	—
77	C₂²×C₄×C₄₁	—	2	1	no	—
78	C₂²×C₄×C₄₁	—	2	1	no	—
79	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
82	SD₁₆×C₄₁	—	4	16	no	—
83	D₈×C₄₁	—	4	8	no	—
84	D₈×C₄₁	—	4	8	no	—
85	C₂×C₈×C₄₁	—	2	2	no	—
86	C₂×C₈×C₄₁	—	2	2	no	—
87	C₂×C₈×C₄₁	—	2	4	no	—
88	C₂×C₈×C₄₁	—	4	4	no	—
89	C₂×C₈×C₄₁	—	4	2	no	—
90	C₂×C₈×C₄₁	SD₁₆	1	4	yes	complemented
91	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
93	C₄⋊C₄×C₄₁	—	2	2	no	—
94	C₄⋊C₄×C₄₁	—	2	1	no	—
95	C₄⋊C₄×C₄₁	—	2	2	no	—
96	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
97	C₄²×C₄₁	—	2	2	no	—
98	C₄²×C₄₁	—	2	2	no	—
99	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₄₁	#41	1	1	yes	characteristic
102	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
103	C₂³×C₄₁	—	4	4	no	—
104	C₂³×C₄₁	—	8	4	no	—
105	C₂³×C₄₁	—	8	4	no	—
106	C₂³×C₄₁	—	8	4	no	—
107	C₂³×C₄₁	#121	1	1	yes	characteristic
108	Q₈×C₄₁	—	2	2	no	—
109	Q₈×C₄₁	—	2	4	no	—
110	Q₈×C₄₁	C₂×D₈	1	4	yes	complemented
111	D₄×C₄₁	—	4	4	no	—
112	D₄×C₄₁	—	4	4	no	—
113	D₄×C₄₁	—	4	4	no	—
114	D₄×C₄₁	—	4	4	no	—
115	D₄×C₄₁	—	8	4	no	—
116	D₄×C₄₁	—	8	4	no	—
117	D₄×C₄₁	—	8	4	no	—
118	D₄×C₄₁	—	8	4	no	—
119	C₂×C₄×C₄₁	—	2	2	no	—
120	C₂×C₄×C₄₁	—	2	4	no	—
121	C₂×C₄×C₄₁	—	2	2	no	—
122	C₂×C₄×C₄₁	—	2	2	no	—
123	C₂×C₄×C₄₁	—	4	2	no	—
124	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
125	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
126	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
127	C₂×C₄×C₄₁	C₂×D₈	1	2	yes	—
128	C₂×C₄×C₄₁	C₂×SD₁₆	1	2	yes	—
129	C₂×C₄×C₄₁	#135	1	2	yes	—
130	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
131	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
134	C₈×C₄₁	—	2	2	no	—
135	C₈×C₄₁	—	4	4	no	—
136	C₈×C₄₁	—	4	2	no	—
137	C₈×C₄₁	C₂×SD₁₆	1	4	yes	complemented
138	C₂×#1668	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂²×C₄₁	—	8	4	no	—
140	C₂²×C₄₁	—	8	4	no	—
141	C₂²×C₄₁	—	8	4	no	—
142	C₂²×C₄₁	—	16	8	no	—
143	C₂²×C₄₁	#460	1	2	yes	—
144	C₂²×C₄₁	#466	1	2	yes	—
145	C₂²×C₄₁	#472	1	2	yes	—
146	C₂²×C₄₁	C₂×#121	1	1	yes	characteristic
147	C₄×C₄₁	—	2	4	no	—
148	C₄×C₄₁	—	2	1	no	—
149	C₄×C₄₁	—	2	1	no	—
150	C₄×C₄₁	—	4	2	no	—
151	C₄×C₄₁	#460	1	2	yes	—
152	C₄×C₄₁	#466	1	2	yes	—
153	C₄×C₄₁	C₂²×D₈	1	1	yes	characteristic
154	C₄×C₄₁	C₂²×SD₁₆	1	1	yes	characteristic
155	C₄×C₄₁	C₂×#135	1	1	yes	characteristic
156	C₄×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
157	C₂×#493	C₂×C₄₁	1	1	yes	characteristic
158	C₂×#460	C₂×C₄₁	1	2	yes	complemented
159	C₂×C₈×Q₈	C₂×C₄₁	1	1	yes	characteristic, complemented
160	#1668	C₂×C₄₁	1	8	yes	direct product factor
161	C₂×#335	C₂×C₄₁	1	1	yes	characteristic, complemented
162	C₂×#331	C₂×C₄₁	1	2	yes	complemented
163	C₂×C₄₁	—	16	4	no	—
164	C₂×C₄₁	#1668	1	4	yes	direct product factor
165	C₂×C₄₁	C₂×#460	1	1	yes	characteristic
166	C₂×C₄₁	C₂×#466	1	1	yes	characteristic
167	C₂×C₄₁	C₂×#472	1	1	yes	characteristic
168	C₂²×SD₁₆	—	2	2	no	—
169	C₂²×D₈	—	2	1	no	—
170	C₂²×D₈	—	2	1	no	—
171	C₂×#127	C₂²×C₄₁	1	2	yes	—
172	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
173	#493	—	2	2	no	—
174	#493	C₂²×C₄₁	1	4	yes	—
175	#460	—	2	4	no	—
176	#460	C₂²×C₄₁	1	8	yes	complemented
177	C₈×Q₈	—	2	2	no	—
178	C₈×Q₈	C₂²×C₄₁	1	4	yes	complemented
179	C₂×C₄⋊C₈	—	2	1	no	—
180	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
181	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
182	C₂×#102	—	2	2	no	—
183	C₂×C₄×C₈	—	2	1	no	—
184	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
185	#335	C₂²×C₄₁	1	4	yes	complemented
186	#331	C₂²×C₄₁	1	8	yes	complemented
187	C₄₁	C₂×#1668	1	1	yes	characteristic, direct product factor, Sylow
188	C₂²×Q₈	D₄×C₄₁	1	1	yes	characteristic
189	C₂²×D₄	—	2	2	no	—
190	C₂²×D₄	—	2	2	no	—
191	C₂²×D₄	—	2	2	no	—
192	C₂×SD₁₆	—	2	8	no	—
193	C₂×SD₁₆	—	4	8	no	—
194	C₂×D₈	—	2	4	no	—
195	C₂×D₈	—	2	4	no	—
196	C₂×D₈	—	4	4	no	—
197	C₂×D₈	—	4	4	no	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	—	2	1	no	—
200	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
201	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
202	#127	—	2	2	no	—
203	#127	—	2	2	no	—
204	#127	—	2	2	no	—
205	#127	C₂³×C₄₁	1	4	yes	—
206	C₄×Q₈	—	2	1	no	—
207	C₄×Q₈	—	2	1	no	—
208	C₄×Q₈	D₄×C₄₁	1	2	yes	—
209	C₄×Q₈	C₂³×C₄₁	1	2	yes	—
210	C₂×C₄⋊C₄	—	2	1	no	—
211	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
212	C₂×C₄²	—	2	1	no	—
213	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
214	C₄⋊C₈	—	2	4	no	—
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	—	2	2	no	—
217	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
218	C₄⋊C₈	C₂³×C₄₁	1	4	yes	—
219	#102	—	2	8	no	—
220	C₄×C₈	—	2	4	no	—
221	C₄×C₈	D₄×C₄₁	1	2	yes	—
222	C₄×C₈	C₂³×C₄₁	1	2	yes	—
223	C₂⁴	—	4	2	no	—
224	C₂×Q₈	—	2	1	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	D₈×C₄₁	1	4	yes	complemented
227	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	2	4	no	—
230	C₂×D₄	—	2	4	no	—
231	C₂×D₄	—	4	2	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂×D₄	—	4	2	no	—
234	C₂×D₄	—	4	4	no	—
235	C₂×D₄	—	4	4	no	—
236	C₂×D₄	—	4	4	no	—
237	C₂×D₄	—	4	4	no	—
238	C₂×D₄	—	8	4	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
243	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
244	SD₁₆	—	4	16	no	—
245	D₈	—	4	8	no	—
246	D₈	—	4	8	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	4	4	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	SD₁₆×C₄₁	1	4	yes	complemented
253	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
254	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
262	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
263	C₄²	#41×C₄₁	1	1	yes	characteristic
264	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
265	C₂³	—	4	4	no	—
266	C₂³	—	8	4	no	—
267	C₂³	—	8	4	no	—
268	C₂³	—	8	4	no	—
269	C₂³	#121×C₄₁	1	1	yes	characteristic
270	Q₈	—	2	2	no	—
271	Q₈	—	2	4	no	—
272	Q₈	C₂×D₈×C₄₁	1	4	yes	complemented
273	D₄	—	4	4	no	—
274	D₄	—	4	4	no	—
275	D₄	—	4	4	no	—
276	D₄	—	4	4	no	—
277	D₄	—	8	4	no	—
278	D₄	—	8	4	no	—
279	D₄	—	8	4	no	—
280	D₄	—	8	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
287	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
288	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
289	C₂×C₄	C₂×D₈×C₄₁	1	2	yes	—
290	C₂×C₄	C₂×SD₁₆×C₄₁	1	2	yes	—
291	C₂×C₄	#135×C₄₁	1	2	yes	—
292	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
293	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
295	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
296	C₈	—	2	2	no	—
297	C₈	—	4	4	no	—
298	C₈	—	4	2	no	—
299	C₈	C₂×SD₁₆×C₄₁	1	4	yes	complemented
300	C₂²	—	8	4	no	—
301	C₂²	—	8	4	no	—
302	C₂²	—	8	4	no	—
303	C₂²	—	16	8	no	—
304	C₂²	C₄₁×#460	1	2	yes	—
305	C₂²	C₄₁×#466	1	2	yes	—
306	C₂²	C₄₁×#472	1	2	yes	—
307	C₂²	C₂×#121×C₄₁	1	1	yes	characteristic
308	C₄	—	2	4	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	4	2	no	—
312	C₄	C₄₁×#460	1	2	yes	—
313	C₄	C₄₁×#466	1	2	yes	—
314	C₄	C₂²×D₈×C₄₁	1	1	yes	characteristic
315	C₄	C₂²×SD₁₆×C₄₁	1	1	yes	characteristic
316	C₄	C₂×#135×C₄₁	1	1	yes	characteristic
317	C₄	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
318	C₂	—	16	4	no	—
319	C₂	C₄₁×#1668	1	4	yes	direct product factor
320	C₂	C₂×C₄₁×#460	1	1	yes	characteristic
321	C₂	C₂×C₄₁×#466	1	1	yes	characteristic
322	C₂	C₂×C₄₁×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₁	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₁
prime radical	C₂×C₄₁×#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₁×#460	C₂	semidirect
3. C₂×C₈×Q₈×C₄₁	C₂	semidirect
4. C₄₁×#1668	C₂	direct (N × H)
5. C₂×C₄₁×#335	C₂	semidirect
6. C₂×C₄₁×#331	C₂	semidirect
15. C₄₁×#460	C₂²	semidirect
17. C₈×Q₈×C₄₁	C₂²	semidirect
24. C₄₁×#335	C₂²	semidirect
25. C₄₁×#331	C₂²	semidirect
64. C₂×Q₈×C₄₁	D₈	semidirect
90. C₂×C₈×C₄₁	SD₁₆	semidirect
110. Q₈×C₄₁	C₂×D₈	semidirect
137. C₈×C₄₁	C₂×SD₁₆	semidirect
138. C₂×#1668	C₄₁	direct (N × H)
158. C₂×#460	C₂×C₄₁	semidirect
159. C₂×C₈×Q₈	C₂×C₄₁	semidirect
160. #1668	C₂×C₄₁	direct (N × H)
161. C₂×#335	C₂×C₄₁	semidirect
162. C₂×#331	C₂×C₄₁	semidirect
164. C₂×C₄₁	#1668	direct (N × H)
176. #460	C₂²×C₄₁	semidirect
178. C₈×Q₈	C₂²×C₄₁	semidirect
185. #335	C₂²×C₄₁	semidirect
186. #331	C₂²×C₄₁	semidirect
187. C₄₁	C₂×#1668	direct (N × H)
226. C₂×Q₈	D₈×C₄₁	semidirect
252. C₂×C₈	SD₁₆×C₄₁	semidirect
272. Q₈	C₂×D₈×C₄₁	semidirect
299. C₈	C₂×SD₁₆×C₄₁	semidirect
319. C₂	C₄₁×#1668	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#493	2
C₂×C₄₁×#460	2
C₂×C₈×Q₈×C₄₁	2
C₄₁×#1668	2
C₂×C₄₁×#335	2
C₂×C₄₁×#331	2
C₂×#1668	41

Subgroups — C2×C41×#1668 #23795313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#1668 #23795313