Subgroups — C₄₁×#33469 #23809670

Order 10496.

646 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3414	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2957	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#2952	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×AGL(1,ℤ/16)	C₂	1	4	yes	maximal, complemented
5	C₄₁×#2180	C₂	1	4	yes	maximal, complemented
6	C₂×C₄₁×#365	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#2148	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₁×#360	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#2110	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#135×C₄₁	C₂²	1	1	yes	characteristic
11	C₂×#135×C₄₁	C₂²	1	1	yes	characteristic
12	C₄₁×#568	—	2	1	no	—
13	C₄₁×#520	—	2	1	no	—
14	C₄₁×#510	C₄	1	2	yes	complemented
15	C₄₁×#510	C₂²	1	2	yes	—
16	C₄₁×#509	C₄	1	2	yes	complemented
17	C₄₁×#509	C₂²	1	2	yes	—
18	C₄₁×#508	C₄	1	4	yes	complemented
19	C₂×M₃₂×C₄₁	C₂²	1	1	yes	characteristic
20	C₂×M₃₂×C₄₁	C₂²	1	1	yes	characteristic
21	C₄₁×#444	C₂²	1	4	yes	—
22	C₄₁×#443	—	2	2	no	—
23	C₄₁×#442	C₂²	1	2	yes	complemented
24	C₄₁×#441	C₂²	1	2	yes	complemented
25	C₄×SD₁₆×C₄₁	—	2	2	no	—
26	C₂×#108×C₄₁	C₂²	1	1	yes	characteristic
27	C₂×#106×C₄₁	C₂²	1	1	yes	characteristic
28	C₄₁×#421	—	2	1	no	—
29	C₄₁×#417	—	2	1	no	—
30	C₄₁×#405	C₂²	1	1	yes	characteristic
31	C₄₁×#365	C₂²	1	4	yes	complemented
32	C₄₁×#362	C₂²	1	2	yes	complemented
33	C₄₁×#361	C₂²	1	2	yes	complemented
34	C₄₁×#360	C₂²	1	4	yes	—
35	C₄₁×#347	C₂²	1	4	yes	complemented
36	C₂×#41×C₄₁	—	2	1	no	—
37	C₂×#41×C₄₁	—	2	1	no	—
38	#135×C₄₁	—	2	1	no	—
39	#135×C₄₁	—	2	1	no	—
40	#135×C₄₁	—	2	1	no	—
41	#135×C₄₁	—	2	1	no	—
42	#135×C₄₁	C₂×C₄	1	2	yes	—
43	#135×C₄₁	C₂×C₄	1	2	yes	—
44	#135×C₄₁	C₂×C₄	1	2	yes	complemented
45	#135×C₄₁	C₂³	1	2	yes	—
46	C₂×Q₁₆×C₄₁	C₂×C₄	1	1	yes	characteristic, complemented
47	C₂×Q₁₆×C₄₁	C₂³	1	1	yes	characteristic
48	C₂×SD₁₆×C₄₁	—	2	1	no	—
49	C₂×SD₁₆×C₄₁	—	2	1	no	—
50	C₂×D₈×C₄₁	C₂×C₄	1	1	yes	characteristic, complemented
51	C₂×D₈×C₄₁	C₂³	1	1	yes	characteristic
52	#131×C₄₁	—	2	4	no	—
53	#131×C₄₁	—	4	2	no	—
54	C₂×M₁₆×C₄₁	—	2	1	no	—
55	C₂×M₁₆×C₄₁	—	4	2	no	—
56	C₂×M₁₆×C₄₁	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
58	C₄×Q₈×C₄₁	—	2	2	no	—
59	C₄×D₄×C₄₁	—	2	2	no	—
60	C₄×D₄×C₄₁	—	4	2	no	—
61	#117×C₄₁	—	4	1	no	—
62	#117×C₄₁	D₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
64	C₂×C₄⋊C₄×C₄₁	—	4	1	no	—
65	Q₃₂×C₄₁	C₂×C₄	1	4	yes	complemented
66	SD₃₂×C₄₁	C₂×C₄	1	4	yes	complemented
67	SD₃₂×C₄₁	C₂×C₄	1	4	yes	complemented
68	D₁₆×C₄₁	C₂×C₄	1	4	yes	complemented
69	M₃₂×C₄₁	—	2	1	no	—
70	M₃₂×C₄₁	C₂×C₄	1	2	yes	complemented
71	M₃₂×C₄₁	C₂³	1	2	yes	—
72	M₃₂×C₄₁	C₂³	1	2	yes	—
73	C₂×C₁₆×C₄₁	—	2	1	no	—
74	C₂×C₁₆×C₄₁	C₂×C₄	1	2	yes	complemented
75	#108×C₄₁	—	2	1	no	—
76	#108×C₄₁	C₂³	1	2	yes	—
77	#106×C₄₁	—	2	1	no	—
78	#106×C₄₁	C₂³	1	2	yes	—
79	#104×C₄₁	—	2	4	no	—
80	#103×C₄₁	—	2	2	no	—
81	#102×C₄₁	—	2	2	no	—
82	#97×C₄₁	C₂³	1	2	yes	—
83	C₄×C₈×C₄₁	D₄	1	2	yes	—
84	#41×C₄₁	—	2	2	no	—
85	#41×C₄₁	—	2	2	no	—
86	#41×C₄₁	—	2	2	no	—
87	#41×C₄₁	—	2	2	no	—
88	#41×C₄₁	—	4	1	no	—
89	#41×C₄₁	—	4	1	no	—
90	#41×C₄₁	—	4	1	no	—
91	#41×C₄₁	—	4	1	no	—
92	C₂×Q₈×C₄₁	—	2	1	no	—
93	C₂×Q₈×C₄₁	—	2	1	no	—
94	C₂×D₄×C₄₁	—	2	1	no	—
95	C₂×D₄×C₄₁	—	2	1	no	—
96	C₂×D₄×C₄₁	—	4	1	no	—
97	C₂×D₄×C₄₁	—	4	1	no	—
98	C₂²×C₄×C₄₁	—	4	2	no	—
99	C₂²×C₄×C₄₁	—	4	1	no	—
100	C₂²×C₄×C₄₁	—	4	1	no	—
101	C₂²×C₄×C₄₁	—	4	1	no	—
102	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
103	Q₁₆×C₄₁	—	2	1	no	—
104	Q₁₆×C₄₁	—	2	1	no	—
105	Q₁₆×C₄₁	C₂²×C₄	1	2	yes	—
106	Q₁₆×C₄₁	C₂²×C₄	1	2	yes	—
107	SD₁₆×C₄₁	—	2	2	no	—
108	SD₁₆×C₄₁	—	2	2	no	—
109	SD₁₆×C₄₁	—	2	2	no	—
110	SD₁₆×C₄₁	—	2	2	no	—
111	D₈×C₄₁	—	2	1	no	—
112	D₈×C₄₁	—	2	1	no	—
113	D₈×C₄₁	C₂²×C₄	1	2	yes	—
114	D₈×C₄₁	C₂²×C₄	1	2	yes	—
115	M₁₆×C₄₁	—	2	1	no	—
116	M₁₆×C₄₁	—	2	1	no	—
117	M₁₆×C₄₁	—	2	2	no	—
118	M₁₆×C₄₁	—	4	4	no	—
119	M₁₆×C₄₁	—	4	1	no	—
120	C₂×C₈×C₄₁	—	4	4	no	—
121	C₂×C₈×C₄₁	—	4	1	no	—
122	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
125	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₄₁	#41	1	1	yes	characteristic
127	C₂×C₈×C₄₁	#41	1	1	yes	characteristic
128	C₂×C₈×C₄₁	C₂⁴	1	1	yes	characteristic
129	C₄⋊C₄×C₄₁	—	2	1	no	—
130	C₄⋊C₄×C₄₁	—	2	2	no	—
131	C₄⋊C₄×C₄₁	—	4	2	no	—
132	C₄⋊C₄×C₄₁	—	4	1	no	—
133	C₂²⋊C₄×C₄₁	—	2	1	no	—
134	C₂²⋊C₄×C₄₁	—	4	2	no	—
135	C₄²×C₄₁	—	4	2	no	—
136	C₄²×C₄₁	C₂×D₄	1	2	yes	—
137	C₁₆×C₄₁	—	2	2	no	—
138	C₁₆×C₄₁	C₂²×C₄	1	4	yes	—
139	C₂³×C₄₁	—	4	1	no	—
140	C₂³×C₄₁	—	4	1	no	—
141	C₂³×C₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
142	Q₈×C₄₁	—	2	2	no	—
143	Q₈×C₄₁	—	2	2	no	—
144	Q₈×C₄₁	—	4	1	no	—
145	Q₈×C₄₁	—	4	1	no	—
146	D₄×C₄₁	—	2	2	no	—
147	D₄×C₄₁	—	2	2	no	—
148	D₄×C₄₁	—	4	2	no	—
149	D₄×C₄₁	—	4	2	no	—
150	D₄×C₄₁	—	4	2	no	—
151	D₄×C₄₁	—	4	2	no	—
152	D₄×C₄₁	—	4	1	no	—
153	D₄×C₄₁	—	4	1	no	—
154	C₂×C₄×C₄₁	—	2	2	no	—
155	C₂×C₄×C₄₁	—	4	1	no	—
156	C₂×C₄×C₄₁	—	4	1	no	—
157	C₂×C₄×C₄₁	—	4	2	no	—
158	C₂×C₄×C₄₁	—	4	2	no	—
159	C₂×C₄×C₄₁	—	4	2	no	—
160	C₂×C₄×C₄₁	—	4	2	no	—
161	C₂×C₄×C₄₁	—	4	2	no	—
162	C₂×C₄×C₄₁	—	16	2	no	—
163	C₂×C₄×C₄₁	—	16	1	no	—
164	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
165	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
166	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
167	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
168	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
169	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
170	C₈×C₄₁	—	2	2	no	—
171	C₈×C₄₁	—	4	2	no	—
172	C₈×C₄₁	C₄×D₄	1	1	yes	characteristic
173	C₈×C₄₁	C₄×D₄	1	1	yes	characteristic
174	C₈×C₄₁	C₂³×C₄	1	1	yes	characteristic
175	C₈×C₄₁	C₂×#41	1	1	yes	characteristic
176	#33469	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
177	C₂²×C₄₁	—	4	1	no	—
178	C₂²×C₄₁	—	4	2	no	—
179	C₂²×C₄₁	—	4	2	no	—
180	C₂²×C₄₁	—	8	1	no	—
181	C₂²×C₄₁	—	8	1	no	—
182	C₂²×C₄₁	—	16	1	no	—
183	C₂²×C₄₁	#442	1	1	yes	characteristic
184	C₂²×C₄₁	#442	1	1	yes	characteristic
185	C₂²×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
186	C₄×C₄₁	—	4	2	no	—
187	C₄×C₄₁	—	4	2	no	—
188	C₄×C₄₁	—	4	2	no	—
189	C₄×C₄₁	—	8	2	no	—
190	C₄×C₄₁	#442	1	1	yes	characteristic
191	C₄×C₄₁	#442	1	1	yes	characteristic
192	C₄×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
193	C₄×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
194	#3414	C₂×C₄₁	1	1	yes	characteristic
195	#2957	C₂×C₄₁	1	1	yes	characteristic, complemented
196	#2952	C₂×C₄₁	1	1	yes	characteristic, complemented
197	AGL(1,ℤ/16)	C₂×C₄₁	1	4	yes	complemented
198	#2180	C₂×C₄₁	1	4	yes	complemented
199	C₂×#365	C₂×C₄₁	1	1	yes	characteristic, complemented
200	#2148	C₂×C₄₁	1	1	yes	characteristic, complemented
201	C₂×#360	C₂×C₄₁	1	1	yes	characteristic, complemented
202	#2110	C₂×C₄₁	1	1	yes	characteristic, complemented
203	C₂×C₄₁	—	2	1	no	—
204	C₂×C₄₁	—	2	1	no	—
205	C₂×C₄₁	—	2	1	no	—
206	C₂×C₄₁	—	8	2	no	—
207	C₂×C₄₁	—	8	2	no	—
208	C₂×C₄₁	C₂×#442	1	1	yes	characteristic
209	C₂×#135	C₂²×C₄₁	1	1	yes	characteristic
210	C₂×#135	C₂²×C₄₁	1	1	yes	characteristic
211	#568	—	2	1	no	—
212	#520	—	2	1	no	—
213	#510	C₄×C₄₁	1	2	yes	complemented
214	#510	C₂²×C₄₁	1	2	yes	—
215	#509	C₄×C₄₁	1	2	yes	complemented
216	#509	C₂²×C₄₁	1	2	yes	—
217	#508	C₄×C₄₁	1	4	yes	complemented
218	C₂×M₃₂	C₂²×C₄₁	1	1	yes	characteristic
219	C₂×M₃₂	C₂²×C₄₁	1	1	yes	characteristic
220	#444	C₂²×C₄₁	1	4	yes	—
221	#443	—	2	2	no	—
222	#442	C₂²×C₄₁	1	2	yes	complemented
223	#441	C₂²×C₄₁	1	2	yes	complemented
224	C₄×SD₁₆	—	2	2	no	—
225	C₂×#108	C₂²×C₄₁	1	1	yes	characteristic
226	C₂×#106	C₂²×C₄₁	1	1	yes	characteristic
227	#421	—	2	1	no	—
228	#417	—	2	1	no	—
229	#405	C₂²×C₄₁	1	1	yes	characteristic
230	#365	C₂²×C₄₁	1	4	yes	complemented
231	#362	C₂²×C₄₁	1	2	yes	complemented
232	#361	C₂²×C₄₁	1	2	yes	complemented
233	#360	C₂²×C₄₁	1	4	yes	—
234	#347	C₂²×C₄₁	1	4	yes	complemented
235	C₄₁	#33469	1	1	yes	characteristic, direct product factor, Sylow
236	C₂×#41	—	2	1	no	—
237	C₂×#41	—	2	1	no	—
238	#135	—	2	1	no	—
239	#135	—	2	1	no	—
240	#135	—	2	1	no	—
241	#135	—	2	1	no	—
242	#135	C₂×C₄×C₄₁	1	2	yes	—
243	#135	C₂×C₄×C₄₁	1	2	yes	—
244	#135	C₂×C₄×C₄₁	1	2	yes	complemented
245	#135	C₂³×C₄₁	1	2	yes	—
246	C₂×Q₁₆	C₂×C₄×C₄₁	1	1	yes	characteristic, complemented
247	C₂×Q₁₆	C₂³×C₄₁	1	1	yes	characteristic
248	C₂×SD₁₆	—	2	1	no	—
249	C₂×SD₁₆	—	2	1	no	—
250	C₂×D₈	C₂×C₄×C₄₁	1	1	yes	characteristic, complemented
251	C₂×D₈	C₂³×C₄₁	1	1	yes	characteristic
252	#131	—	2	4	no	—
253	#131	—	4	2	no	—
254	C₂×M₁₆	—	2	1	no	—
255	C₂×M₁₆	—	4	2	no	—
256	C₂×M₁₆	D₄×C₄₁	1	1	yes	characteristic
257	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
258	C₄×Q₈	—	2	2	no	—
259	C₄×D₄	—	2	2	no	—
260	C₄×D₄	—	4	2	no	—
261	#117	—	4	1	no	—
262	#117	D₄×C₄₁	1	1	yes	characteristic
263	C₂×C₄⋊C₄	—	2	1	no	—
264	C₂×C₄⋊C₄	—	4	1	no	—
265	Q₃₂	C₂×C₄×C₄₁	1	4	yes	complemented
266	SD₃₂	C₂×C₄×C₄₁	1	4	yes	complemented
267	SD₃₂	C₂×C₄×C₄₁	1	4	yes	complemented
268	D₁₆	C₂×C₄×C₄₁	1	4	yes	complemented
269	M₃₂	—	2	1	no	—
270	M₃₂	C₂×C₄×C₄₁	1	2	yes	complemented
271	M₃₂	C₂³×C₄₁	1	2	yes	—
272	M₃₂	C₂³×C₄₁	1	2	yes	—
273	C₂×C₁₆	—	2	1	no	—
274	C₂×C₁₆	C₂×C₄×C₄₁	1	2	yes	complemented
275	#108	—	2	1	no	—
276	#108	C₂³×C₄₁	1	2	yes	—
277	#106	—	2	1	no	—
278	#106	C₂³×C₄₁	1	2	yes	—
279	#104	—	2	4	no	—
280	#103	—	2	2	no	—
281	#102	—	2	2	no	—
282	#97	C₂³×C₄₁	1	2	yes	—
283	C₄×C₈	D₄×C₄₁	1	2	yes	—
284	#41	—	2	2	no	—
285	#41	—	2	2	no	—
286	#41	—	2	2	no	—
287	#41	—	2	2	no	—
288	#41	—	4	1	no	—
289	#41	—	4	1	no	—
290	#41	—	4	1	no	—
291	#41	—	4	1	no	—
292	C₂×Q₈	—	2	1	no	—
293	C₂×Q₈	—	2	1	no	—
294	C₂×D₄	—	2	1	no	—
295	C₂×D₄	—	2	1	no	—
296	C₂×D₄	—	4	1	no	—
297	C₂×D₄	—	4	1	no	—
298	C₂²×C₄	—	4	2	no	—
299	C₂²×C₄	—	4	1	no	—
300	C₂²×C₄	—	4	1	no	—
301	C₂²×C₄	—	4	1	no	—
302	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
303	Q₁₆	—	2	1	no	—
304	Q₁₆	—	2	1	no	—
305	Q₁₆	C₂²×C₄×C₄₁	1	2	yes	—
306	Q₁₆	C₂²×C₄×C₄₁	1	2	yes	—
307	SD₁₆	—	2	2	no	—
308	SD₁₆	—	2	2	no	—
309	SD₁₆	—	2	2	no	—
310	SD₁₆	—	2	2	no	—
311	D₈	—	2	1	no	—
312	D₈	—	2	1	no	—
313	D₈	C₂²×C₄×C₄₁	1	2	yes	—
314	D₈	C₂²×C₄×C₄₁	1	2	yes	—
315	M₁₆	—	2	1	no	—
316	M₁₆	—	2	1	no	—
317	M₁₆	—	2	2	no	—
318	M₁₆	—	4	4	no	—
319	M₁₆	—	4	1	no	—
320	C₂×C₈	—	4	4	no	—
321	C₂×C₈	—	4	1	no	—
322	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
323	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
324	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
325	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
326	C₂×C₈	#41×C₄₁	1	1	yes	characteristic
327	C₂×C₈	#41×C₄₁	1	1	yes	characteristic
328	C₂×C₈	C₂⁴×C₄₁	1	1	yes	characteristic
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	4	2	no	—
332	C₄⋊C₄	—	4	1	no	—
333	C₂²⋊C₄	—	2	1	no	—
334	C₂²⋊C₄	—	4	2	no	—
335	C₄²	—	4	2	no	—
336	C₄²	C₂×D₄×C₄₁	1	2	yes	—
337	C₁₆	—	2	2	no	—
338	C₁₆	C₂²×C₄×C₄₁	1	4	yes	—
339	C₂³	—	4	1	no	—
340	C₂³	—	4	1	no	—
341	C₂³	AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
342	Q₈	—	2	2	no	—
343	Q₈	—	2	2	no	—
344	Q₈	—	4	1	no	—
345	Q₈	—	4	1	no	—
346	D₄	—	2	2	no	—
347	D₄	—	2	2	no	—
348	D₄	—	4	2	no	—
349	D₄	—	4	2	no	—
350	D₄	—	4	2	no	—
351	D₄	—	4	2	no	—
352	D₄	—	4	1	no	—
353	D₄	—	4	1	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	4	1	no	—
356	C₂×C₄	—	4	1	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	16	2	no	—
363	C₂×C₄	—	16	1	no	—
364	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
365	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
366	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
367	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
368	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
369	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
370	C₈	—	2	2	no	—
371	C₈	—	4	2	no	—
372	C₈	C₄×D₄×C₄₁	1	1	yes	characteristic
373	C₈	C₄×D₄×C₄₁	1	1	yes	characteristic
374	C₈	C₂³×C₄×C₄₁	1	1	yes	characteristic
375	C₈	C₂×#41×C₄₁	1	1	yes	characteristic
376	C₂²	—	4	1	no	—
377	C₂²	—	4	2	no	—
378	C₂²	—	4	2	no	—
379	C₂²	—	8	1	no	—
380	C₂²	—	8	1	no	—
381	C₂²	—	16	1	no	—
382	C₂²	C₄₁×#442	1	1	yes	characteristic
383	C₂²	C₄₁×#442	1	1	yes	characteristic
384	C₂²	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
385	C₄	—	4	2	no	—
386	C₄	—	4	2	no	—
387	C₄	—	4	2	no	—
388	C₄	—	8	2	no	—
389	C₄	C₄₁×#442	1	1	yes	characteristic
390	C₄	C₄₁×#442	1	1	yes	characteristic
391	C₄	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
392	C₄	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
393	C₂	—	2	1	no	—
394	C₂	—	2	1	no	—
395	C₂	—	2	1	no	—
396	C₂	—	8	2	no	—
397	C₂	—	8	2	no	—
398	C₂	C₂×C₄₁×#442	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₄₁	C₂×#442
commutator	C₈	C₂³×C₄×C₄₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₄₁
prime radical	C₄₁×#3414	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄₁ ◁ C₂²×C₄×C₄₁ ◁ C₄₁×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#2957	C₂	semidirect
3. C₄₁×#2952	C₂	semidirect
4. C₄₁×AGL(1,ℤ/16)	C₂	semidirect
5. C₄₁×#2180	C₂	semidirect
6. C₂×C₄₁×#365	C₂	semidirect
7. C₄₁×#2148	C₂	semidirect
8. C₂×C₄₁×#360	C₂	semidirect
9. C₄₁×#2110	C₂	semidirect
14. C₄₁×#510	C₄	semidirect
16. C₄₁×#509	C₄	semidirect
18. C₄₁×#508	C₄	semidirect
23. C₄₁×#442	C₂²	semidirect
24. C₄₁×#441	C₂²	semidirect
31. C₄₁×#365	C₂²	semidirect
32. C₄₁×#362	C₂²	semidirect
33. C₄₁×#361	C₂²	semidirect
35. C₄₁×#347	C₂²	semidirect
44. #135×C₄₁	C₂×C₄	semidirect
46. C₂×Q₁₆×C₄₁	C₂×C₄	semidirect
50. C₂×D₈×C₄₁	C₂×C₄	semidirect
65. Q₃₂×C₄₁	C₂×C₄	semidirect
66. SD₃₂×C₄₁	C₂×C₄	semidirect
67. SD₃₂×C₄₁	C₂×C₄	semidirect
68. D₁₆×C₄₁	C₂×C₄	semidirect
70. M₃₂×C₄₁	C₂×C₄	semidirect
74. C₂×C₁₆×C₄₁	C₂×C₄	semidirect
176. #33469	C₄₁	direct (N × H)
195. #2957	C₂×C₄₁	semidirect
196. #2952	C₂×C₄₁	semidirect
197. AGL(1,ℤ/16)	C₂×C₄₁	semidirect
198. #2180	C₂×C₄₁	semidirect
199. C₂×#365	C₂×C₄₁	semidirect
200. #2148	C₂×C₄₁	semidirect
201. C₂×#360	C₂×C₄₁	semidirect
202. #2110	C₂×C₄₁	semidirect
213. #510	C₄×C₄₁	semidirect
215. #509	C₄×C₄₁	semidirect
217. #508	C₄×C₄₁	semidirect
222. #442	C₂²×C₄₁	semidirect
223. #441	C₂²×C₄₁	semidirect
230. #365	C₂²×C₄₁	semidirect
231. #362	C₂²×C₄₁	semidirect
232. #361	C₂²×C₄₁	semidirect
234. #347	C₂²×C₄₁	semidirect
235. C₄₁	#33469	direct (N × H)
244. #135	C₂×C₄×C₄₁	semidirect
246. C₂×Q₁₆	C₂×C₄×C₄₁	semidirect
250. C₂×D₈	C₂×C₄×C₄₁	semidirect
265. Q₃₂	C₂×C₄×C₄₁	semidirect
266. SD₃₂	C₂×C₄×C₄₁	semidirect
267. SD₃₂	C₂×C₄×C₄₁	semidirect
268. D₁₆	C₂×C₄×C₄₁	semidirect
270. M₃₂	C₂×C₄×C₄₁	semidirect
274. C₂×C₁₆	C₂×C₄×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×#3414	2
C₄₁×#2957	2
C₄₁×#2952	2
C₄₁×AGL(1,ℤ/16)	2
C₄₁×#2180	2
C₂×C₄₁×#365	2
C₄₁×#2148	2
C₂×C₄₁×#360	2
C₄₁×#2110	2
#33469	41

Subgroups — C41×#33469 #23809670

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#33469 #23809670