Subgroups — C₄₁×#33508 #23809709

Order 10496.

502 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#2966	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2183	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×C₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#2169	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#2168	C₂	1	2	yes	maximal, complemented
6	D₄×C₁₆×C₄₁	C₂	1	2	yes	maximal, complemented
7	D₄×C₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#2164	C₂	1	2	yes	maximal, complemented
9	C₄₁×#2162	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#2151	C₂	1	2	yes	maximal, complemented
11	C₂×C₄₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄₁×#529	C₄	1	1	yes	characteristic
13	C₂×M₃₂×C₄₁	C₂²	1	2	yes	—
14	C₂²×C₁₆×C₄₁	C₂²	1	2	yes	—
15	C₂²×C₁₆×C₄₁	C₂²	1	2	yes	—
16	C₄₁×#446	C₄	1	1	yes	characteristic
17	C₈×Q₈×C₄₁	C₂²	1	1	yes	characteristic
18	C₄₁×#436	C₄	1	1	yes	characteristic
19	C₄₁×#435	C₄	1	2	yes	—
20	C₈×D₄×C₄₁	C₂²	1	2	yes	—
21	C₈×D₄×C₄₁	C₂²	1	1	yes	characteristic
22	C₄₁×#433	C₄	1	2	yes	—
23	C₄₁×#432	C₄	1	1	yes	characteristic
24	C₄₁×#405	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
26	C₄₁×C₄⋊C₁₆	C₂²	1	2	yes	—
27	C₄₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
28	C₄₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
29	C₄₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
30	C₄₁×C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₄₁×C₄⋊C₁₆	C₂²	1	2	yes	—
32	C₄₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₄₁×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₄₁×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
35	C₄₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
36	C₄₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
37	C₄₁×#346	C₂²	1	2	yes	—
38	C₄×C₁₆×C₄₁	C₂²	1	1	yes	characteristic
39	C₄×C₁₆×C₄₁	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₄₁	C₂²	1	2	yes	—
41	C₂×M₁₆×C₄₁	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
45	#126×C₄₁	C₈	1	2	yes	—
46	#125×C₄₁	C₈	1	1	yes	characteristic
47	#124×C₄₁	C₈	1	1	yes	characteristic
48	#123×C₄₁	C₈	1	2	yes	—
49	C₂²⋊Q₈×C₄₁	C₈	1	1	yes	characteristic
50	#121×C₄₁	C₈	1	1	yes	characteristic
51	C₄×Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
52	C₄×D₄×C₄₁	C₂×C₄	1	2	yes	—
53	C₄×D₄×C₄₁	C₂×C₄	1	1	yes	characteristic
54	#117×C₄₁	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
56	M₃₂×C₄₁	—	4	2	no	—
57	C₂×C₁₆×C₄₁	—	4	2	no	—
58	C₂×C₁₆×C₄₁	—	4	2	no	—
59	C₂×C₁₆×C₄₁	C₂³	1	2	yes	—
60	C₂×C₁₆×C₄₁	C₂³	1	1	yes	characteristic
61	C₂×C₁₆×C₄₁	C₂³	1	2	yes	—
62	C₂×C₁₆×C₄₁	C₂³	1	1	yes	characteristic
63	C₂×C₁₆×C₄₁	C₂³	1	1	yes	characteristic
64	C₂×C₁₆×C₄₁	C₂³	1	1	yes	characteristic
65	C₄⋊C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
66	C₄⋊C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
67	C₄⋊C₈×C₄₁	C₂×C₄	1	2	yes	—
68	C₄⋊C₈×C₄₁	C₂×C₄	1	2	yes	—
69	C₂²⋊C₈×C₄₁	C₂×C₄	1	2	yes	—
70	C₂²⋊C₈×C₄₁	C₂×C₄	1	2	yes	—
71	C₂²⋊C₈×C₄₁	C₂×C₄	1	2	yes	—
72	#97×C₄₁	C₂×C₄	1	2	yes	—
73	#97×C₄₁	C₂×C₄	1	2	yes	—
74	C₄×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
75	C₄×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
76	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₄₁	C₂³	1	2	yes	—
78	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₄₁	C₂×C₈	1	1	yes	characteristic
80	C₂×D₄×C₄₁	C₂×C₈	1	2	yes	—
81	C₂×D₄×C₄₁	C₂×C₈	1	1	yes	characteristic
82	C₂²×C₄×C₄₁	C₂×C₈	1	1	yes	characteristic
83	C₂²×C₄×C₄₁	C₂×C₈	1	1	yes	characteristic
84	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
86	M₁₆×C₄₁	—	4	2	no	—
87	C₂×C₈×C₄₁	—	2	1	no	—
88	C₂×C₈×C₄₁	—	2	2	no	—
89	C₂×C₈×C₄₁	—	2	1	no	—
90	C₂×C₈×C₄₁	—	2	2	no	—
91	C₂×C₈×C₄₁	—	4	1	no	—
92	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₄₁	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₄₁	C₂×C₈	1	2	yes	—
99	C₄⋊C₄×C₄₁	C₂×C₈	1	2	yes	—
100	C₄⋊C₄×C₄₁	C₂×C₈	1	1	yes	characteristic
101	C₄⋊C₄×C₄₁	C₂×C₈	1	2	yes	—
102	C₄⋊C₄×C₄₁	C₂×C₈	1	1	yes	characteristic
103	C₄⋊C₄×C₄₁	C₂×C₈	1	2	yes	—
104	C₂²⋊C₄×C₄₁	C₂×C₈	1	2	yes	—
105	C₂²⋊C₄×C₄₁	C₂×C₈	1	2	yes	—
106	C₂²⋊C₄×C₄₁	C₂×C₈	1	2	yes	—
107	C₂²⋊C₄×C₄₁	C₂×C₈	1	2	yes	—
108	C₂²⋊C₄×C₄₁	C₂×C₈	1	2	yes	—
109	C₄²×C₄₁	C₂×C₈	1	1	yes	characteristic
110	C₄²×C₄₁	C₂×C₈	1	1	yes	characteristic
111	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
114	C₁₆×C₄₁	—	2	1	no	—
115	C₁₆×C₄₁	—	2	2	no	—
116	C₁₆×C₄₁	—	2	2	no	—
117	C₁₆×C₄₁	—	2	1	no	—
118	C₁₆×C₄₁	—	2	1	no	—
119	C₁₆×C₄₁	—	2	1	no	—
120	C₂³×C₄₁	C₂²×C₈	1	2	yes	—
121	C₂³×C₄₁	C₂²×C₈	1	1	yes	characteristic
122	Q₈×C₄₁	—	2	2	no	—
123	D₄×C₄₁	—	2	4	no	—
124	D₄×C₄₁	—	2	2	no	—
125	C₂×C₄×C₄₁	—	2	2	no	—
126	C₂×C₄×C₄₁	—	2	1	no	—
127	C₂×C₄×C₄₁	—	2	2	no	—
128	C₂×C₄×C₄₁	—	2	1	no	—
129	C₂×C₄×C₄₁	—	2	2	no	—
130	C₂×C₄×C₄₁	—	2	2	no	—
131	C₂×C₄×C₄₁	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₄₁	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₄×C₄₁	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄×C₄₁	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₄×C₄₁	C₂²×C₈	1	2	yes	—
136	C₂×C₄×C₄₁	C₂²×C₈	1	2	yes	—
137	C₂×C₄×C₄₁	C₂²×C₈	1	2	yes	—
138	C₂×C₄×C₄₁	C₂²×C₈	1	1	yes	characteristic
139	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
140	C₈×C₄₁	—	2	1	no	—
141	C₈×C₄₁	—	2	2	no	—
142	C₈×C₄₁	—	2	1	no	—
143	C₈×C₄₁	—	2	1	no	—
144	C₈×C₄₁	—	2	2	no	—
145	C₈×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
146	C₈×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
147	#33508	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂²×C₄₁	—	2	1	no	—
149	C₂²×C₄₁	—	2	1	no	—
150	C₂²×C₄₁	—	2	2	no	—
151	C₂²×C₄₁	—	2	2	no	—
152	C₂²×C₄₁	—	2	2	no	—
153	C₂²×C₄₁	#504	1	2	yes	—
154	C₂²×C₄₁	C₂³×C₈	1	1	yes	characteristic
155	C₄×C₄₁	—	2	1	no	—
156	C₄×C₄₁	—	2	1	no	—
157	C₄×C₄₁	—	2	1	no	—
158	C₄×C₄₁	—	2	2	no	—
159	C₄×C₄₁	—	2	2	no	—
160	C₄×C₄₁	—	2	1	no	—
161	C₄×C₄₁	—	2	2	no	—
162	C₄×C₄₁	#504	1	2	yes	—
163	C₄×C₄₁	#520	1	1	yes	characteristic
164	C₄×C₄₁	#568	1	1	yes	characteristic
165	#2966	C₂×C₄₁	1	1	yes	characteristic
166	#2183	C₂×C₄₁	1	1	yes	characteristic, complemented
167	Q₈×C₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
168	#2169	C₂×C₄₁	1	1	yes	characteristic, complemented
169	#2168	C₂×C₄₁	1	2	yes	complemented
170	D₄×C₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
171	D₄×C₁₆	C₂×C₄₁	1	2	yes	complemented
172	#2164	C₂×C₄₁	1	2	yes	complemented
173	#2162	C₂×C₄₁	1	1	yes	characteristic, complemented
174	#2151	C₂×C₄₁	1	2	yes	complemented
175	C₂×C₄⋊C₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
176	C₂×C₄₁	—	2	2	no	—
177	C₂×C₄₁	—	4	2	no	—
178	C₂×C₄₁	#2988	1	1	yes	characteristic
179	C₂×C₄₁	C₂×#504	1	1	yes	characteristic
180	C₂×C₄₁	C₂×#504	1	1	yes	characteristic
181	#529	C₄×C₄₁	1	1	yes	characteristic
182	C₂×M₃₂	C₂²×C₄₁	1	2	yes	—
183	C₂²×C₁₆	C₂²×C₄₁	1	2	yes	—
184	C₂²×C₁₆	C₂²×C₄₁	1	2	yes	—
185	#446	C₄×C₄₁	1	1	yes	characteristic
186	C₈×Q₈	C₂²×C₄₁	1	1	yes	characteristic
187	#436	C₄×C₄₁	1	1	yes	characteristic
188	#435	C₄×C₄₁	1	2	yes	—
189	C₈×D₄	C₂²×C₄₁	1	2	yes	—
190	C₈×D₄	C₂²×C₄₁	1	1	yes	characteristic
191	#433	C₄×C₄₁	1	2	yes	—
192	#432	C₄×C₄₁	1	1	yes	characteristic
193	#405	C₂²×C₄₁	1	2	yes	—
194	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
195	C₄⋊C₁₆	C₂²×C₄₁	1	2	yes	—
196	C₄⋊C₁₆	C₂²×C₄₁	1	1	yes	characteristic
197	C₄⋊C₁₆	C₂²×C₄₁	1	1	yes	characteristic
198	C₄⋊C₁₆	C₂²×C₄₁	1	1	yes	characteristic
199	C₄⋊C₁₆	C₂²×C₄₁	1	2	yes	—
200	C₄⋊C₁₆	C₂²×C₄₁	1	2	yes	—
201	C₄⋊C₁₆	C₂²×C₄₁	1	1	yes	characteristic
202	C₂²⋊C₁₆	C₂²×C₄₁	1	1	yes	characteristic
203	C₂²⋊C₁₆	C₂²×C₄₁	1	1	yes	characteristic
204	C₂²⋊C₁₆	C₂²×C₄₁	1	2	yes	—
205	C₂²⋊C₁₆	C₂²×C₄₁	1	2	yes	—
206	#346	C₂²×C₄₁	1	2	yes	—
207	C₄×C₁₆	C₂²×C₄₁	1	1	yes	characteristic
208	C₄×C₁₆	C₂²×C₄₁	1	1	yes	characteristic
209	C₄×C₁₆	C₂²×C₄₁	1	2	yes	—
210	C₄₁	#33508	1	1	yes	characteristic, direct product factor, Sylow
211	C₂×M₁₆	C₂×C₄×C₄₁	1	2	yes	—
212	C₂²×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
213	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
214	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
215	#126	C₈×C₄₁	1	2	yes	—
216	#125	C₈×C₄₁	1	1	yes	characteristic
217	#124	C₈×C₄₁	1	1	yes	characteristic
218	#123	C₈×C₄₁	1	2	yes	—
219	C₂²⋊Q₈	C₈×C₄₁	1	1	yes	characteristic
220	#121	C₈×C₄₁	1	1	yes	characteristic
221	C₄×Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
222	C₄×D₄	C₂×C₄×C₄₁	1	2	yes	—
223	C₄×D₄	C₂×C₄×C₄₁	1	1	yes	characteristic
224	#117	C₂×C₄×C₄₁	1	2	yes	—
225	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
226	M₃₂	—	4	2	no	—
227	C₂×C₁₆	—	4	2	no	—
228	C₂×C₁₆	—	4	2	no	—
229	C₂×C₁₆	C₂³×C₄₁	1	1	yes	characteristic
230	C₂×C₁₆	C₂³×C₄₁	1	1	yes	characteristic
231	C₂×C₁₆	C₂³×C₄₁	1	1	yes	characteristic
232	C₂×C₁₆	C₂³×C₄₁	1	2	yes	—
233	C₂×C₁₆	C₂³×C₄₁	1	1	yes	characteristic
234	C₂×C₁₆	C₂³×C₄₁	1	2	yes	—
235	C₄⋊C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
236	C₄⋊C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
237	C₄⋊C₈	C₂×C₄×C₄₁	1	2	yes	—
238	C₄⋊C₈	C₂×C₄×C₄₁	1	2	yes	—
239	C₂²⋊C₈	C₂×C₄×C₄₁	1	2	yes	—
240	C₂²⋊C₈	C₂×C₄×C₄₁	1	2	yes	—
241	C₂²⋊C₈	C₂×C₄×C₄₁	1	2	yes	—
242	#97	C₂×C₄×C₄₁	1	2	yes	—
243	#97	C₂×C₄×C₄₁	1	2	yes	—
244	C₄×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
245	C₄×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
246	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
247	C₄×C₈	C₂³×C₄₁	1	2	yes	—
248	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
249	C₂×Q₈	C₂×C₈×C₄₁	1	1	yes	characteristic
250	C₂×D₄	C₂×C₈×C₄₁	1	2	yes	—
251	C₂×D₄	C₂×C₈×C₄₁	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₈×C₄₁	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₈×C₄₁	1	1	yes	characteristic
254	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
255	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
256	M₁₆	—	4	2	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
263	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
264	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
265	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
266	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
267	C₂×C₈	C₂⁴×C₄₁	1	1	yes	characteristic
268	C₄⋊C₄	C₂×C₈×C₄₁	1	2	yes	—
269	C₄⋊C₄	C₂×C₈×C₄₁	1	2	yes	—
270	C₄⋊C₄	C₂×C₈×C₄₁	1	1	yes	characteristic
271	C₄⋊C₄	C₂×C₈×C₄₁	1	2	yes	—
272	C₄⋊C₄	C₂×C₈×C₄₁	1	1	yes	characteristic
273	C₄⋊C₄	C₂×C₈×C₄₁	1	2	yes	—
274	C₂²⋊C₄	C₂×C₈×C₄₁	1	2	yes	—
275	C₂²⋊C₄	C₂×C₈×C₄₁	1	2	yes	—
276	C₂²⋊C₄	C₂×C₈×C₄₁	1	2	yes	—
277	C₂²⋊C₄	C₂×C₈×C₄₁	1	2	yes	—
278	C₂²⋊C₄	C₂×C₈×C₄₁	1	2	yes	—
279	C₄²	C₂×C₈×C₄₁	1	1	yes	characteristic
280	C₄²	C₂×C₈×C₄₁	1	1	yes	characteristic
281	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
282	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
283	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
284	C₁₆	—	2	1	no	—
285	C₁₆	—	2	2	no	—
286	C₁₆	—	2	2	no	—
287	C₁₆	—	2	1	no	—
288	C₁₆	—	2	1	no	—
289	C₁₆	—	2	1	no	—
290	C₂³	C₂²×C₈×C₄₁	1	2	yes	—
291	C₂³	C₂²×C₈×C₄₁	1	1	yes	characteristic
292	Q₈	—	2	2	no	—
293	D₄	—	2	4	no	—
294	D₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	C₂²×C₈×C₄₁	1	1	yes	characteristic
302	C₂×C₄	C₂²×C₈×C₄₁	1	1	yes	characteristic
303	C₂×C₄	C₂²×C₈×C₄₁	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₈×C₄₁	1	1	yes	characteristic
305	C₂×C₄	C₂²×C₈×C₄₁	1	2	yes	—
306	C₂×C₄	C₂²×C₈×C₄₁	1	2	yes	—
307	C₂×C₄	C₂²×C₈×C₄₁	1	2	yes	—
308	C₂×C₄	C₂²×C₈×C₄₁	1	1	yes	characteristic
309	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	2	no	—
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	2	2	no	—
315	C₈	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
316	C₈	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	C₄₁×#504	1	2	yes	—
323	C₂²	C₂³×C₈×C₄₁	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	2	no	—
331	C₄	C₄₁×#504	1	2	yes	—
332	C₄	C₄₁×#520	1	1	yes	characteristic
333	C₄	C₄₁×#568	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	4	2	no	—
336	C₂	C₄₁×#2988	1	1	yes	characteristic
337	C₂	C₂×C₄₁×#504	1	1	yes	characteristic
338	C₂	C₂×C₄₁×#504	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₄₁	C₂⁴
commutator	C₂²	C₂³×C₈×C₄₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₄₁
socle	C₂²×C₄₁	C₂³×C₈
prime radical	#121×C₄₁	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#2183	C₂	semidirect
3. Q₈×C₁₆×C₄₁	C₂	semidirect
4. C₄₁×#2169	C₂	semidirect
5. C₄₁×#2168	C₂	semidirect
6. D₄×C₁₆×C₄₁	C₂	semidirect
7. D₄×C₁₆×C₄₁	C₂	semidirect
8. C₄₁×#2164	C₂	semidirect
9. C₄₁×#2162	C₂	semidirect
10. C₄₁×#2151	C₂	semidirect
11. C₂×C₄₁×C₄⋊C₁₆	C₂	semidirect
147. #33508	C₄₁	direct (N × H)
166. #2183	C₂×C₄₁	semidirect
167. Q₈×C₁₆	C₂×C₄₁	semidirect
168. #2169	C₂×C₄₁	semidirect
169. #2168	C₂×C₄₁	semidirect
170. D₄×C₁₆	C₂×C₄₁	semidirect
171. D₄×C₁₆	C₂×C₄₁	semidirect
172. #2164	C₂×C₄₁	semidirect
173. #2162	C₂×C₄₁	semidirect
174. #2151	C₂×C₄₁	semidirect
175. C₂×C₄⋊C₁₆	C₂×C₄₁	semidirect
210. C₄₁	#33508	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₁×#2966	2
C₄₁×#2183	2
Q₈×C₁₆×C₄₁	2
C₄₁×#2169	2
C₄₁×#2168	2
D₄×C₁₆×C₄₁	2
C₄₁×#2164	2
C₄₁×#2162	2
C₄₁×#2151	2
C₂×C₄₁×C₄⋊C₁₆	2
#33508	41

Subgroups — C41×#33508 #23809709

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#33508 #23809709