Subgroups — C₄₁×#22452 #23798653

Order 10496.

1368 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#135×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#444	C₂	1	2	yes	maximal, complemented
3	C₄₁×#2880	C₂	1	2	yes	maximal, complemented
4	C₄₁×#1975	C₂	1	2	yes	maximal, complemented
5	C₄₁×#1876	C₂	1	4	yes	maximal, complemented
6	C₄₁×#1810	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#1789	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1752	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₄₁	C₂²	1	2	yes	—
10	C₂×#135×C₄₁	—	2	8	no	—
11	C₂×#135×C₄₁	C₄	1	1	yes	characteristic, complemented
12	C₂×#135×C₄₁	C₄	1	2	yes	complemented
13	C₂×#135×C₄₁	C₄	1	1	yes	characteristic, complemented
14	C₂×#135×C₄₁	C₂²	1	2	yes	—
15	C₂×#135×C₄₁	C₂²	1	1	yes	characteristic
16	C₂×#135×C₄₁	C₂²	1	1	yes	characteristic
17	C₂²×Q₁₆×C₄₁	C₄	1	1	yes	characteristic, complemented
18	C₂²×SD₁₆×C₄₁	C₄	1	2	yes	complemented
19	C₂²×D₈×C₄₁	C₄	1	1	yes	characteristic, complemented
20	C₂×#131×C₄₁	C₂²	1	4	yes	complemented
21	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
22	C₄₁×#444	—	2	8	no	—
23	C₂×#108×C₄₁	C₂²	1	2	yes	complemented
24	C₂×#104×C₄₁	C₂²	1	4	yes	complemented
25	C₂×#104×C₄₁	C₂²	1	4	yes	complemented
26	C₄₁×#408	—	2	4	no	—
27	C₄₁×#408	C₂²	1	4	yes	complemented
28	C₄₁×#407	—	2	2	no	—
29	C₄₁×#407	C₂²	1	2	yes	—
30	C₂×C₂²⋊C₈×C₄₁	—	2	2	no	—
31	C₂×C₂²⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
32	C₂×#97×C₄₁	C₂²	1	2	yes	complemented
33	C₄×C₂²⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
34	C₄₁×#341	C₂²	1	2	yes	complemented
35	C₄₁×#336	C₂²	1	2	yes	complemented
36	C₂×#41×C₄₁	—	2	2	no	—
37	C₂×#41×C₄₁	—	2	2	no	—
38	C₂×#41×C₄₁	—	2	8	no	—
39	C₂×#41×C₄₁	—	4	4	no	—
40	C₂×#41×C₄₁	C₂×C₄	1	2	yes	—
41	C₂×#41×C₄₁	C₂×C₄	1	2	yes	—
42	C₂×#41×C₄₁	C₂³	1	2	yes	—
43	C₂×#41×C₄₁	C₂³	1	2	yes	—
44	C₂²×Q₈×C₄₁	C₂×C₄	1	2	yes	—
45	C₂²×D₄×C₄₁	—	2	2	no	—
46	C₂²×D₄×C₄₁	C₂×C₄	1	2	yes	—
47	C₂³×C₄×C₄₁	—	2	2	no	—
48	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
49	#135×C₄₁	—	2	8	no	—
50	#135×C₄₁	—	2	4	no	—
51	#135×C₄₁	—	2	8	no	—
52	#135×C₄₁	—	2	4	no	—
53	#135×C₄₁	—	2	4	no	—
54	#135×C₄₁	D₄	1	8	yes	—
55	C₂×Q₁₆×C₄₁	—	2	4	no	—
56	C₂×Q₁₆×C₄₁	C₂×C₄	1	2	yes	—
57	C₂×Q₁₆×C₄₁	C₂×C₄	1	1	yes	characteristic
58	C₂×Q₁₆×C₄₁	C₂×C₄	1	1	yes	characteristic
59	C₂×SD₁₆×C₄₁	—	2	8	no	—
60	C₂×SD₁₆×C₄₁	C₂×C₄	1	2	yes	—
61	C₂×SD₁₆×C₄₁	C₂×C₄	1	2	yes	—
62	C₂×SD₁₆×C₄₁	C₂×C₄	1	2	yes	—
63	C₂×SD₁₆×C₄₁	C₂×C₄	1	2	yes	—
64	C₂×D₈×C₄₁	—	2	4	no	—
65	C₂×D₈×C₄₁	C₂×C₄	1	2	yes	—
66	C₂×D₈×C₄₁	C₂×C₄	1	1	yes	characteristic
67	C₂×D₈×C₄₁	C₂×C₄	1	1	yes	characteristic
68	#131×C₄₁	—	2	8	no	—
69	#131×C₄₁	—	4	4	no	—
70	C₂×M₁₆×C₄₁	—	2	4	no	—
71	C₂×M₁₆×C₄₁	C₂³	1	4	yes	—
72	C₂²×C₈×C₄₁	—	2	4	no	—
73	C₂²×C₈×C₄₁	—	2	4	no	—
74	C₂²×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
75	C₂²×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
76	C₂²×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
77	C₂²×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
78	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
80	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
81	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
82	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
83	#108×C₄₁	—	2	4	no	—
84	#104×C₄₁	—	2	8	no	—
85	#104×C₄₁	—	4	4	no	—
86	C₂²⋊C₈×C₄₁	—	2	2	no	—
87	C₂²⋊C₈×C₄₁	—	2	2	no	—
88	C₂²⋊C₈×C₄₁	—	2	4	no	—
89	C₂²⋊C₈×C₄₁	—	2	4	no	—
90	C₂²⋊C₈×C₄₁	D₄	1	2	yes	—
91	C₂²⋊C₈×C₄₁	D₄	1	2	yes	—
92	#97×C₄₁	—	2	4	no	—
93	#95×C₄₁	—	2	1	no	—
94	C₂⁴×C₄₁	—	2	2	no	—
95	C₂⁴×C₄₁	#41	1	1	yes	characteristic
96	#41×C₄₁	—	2	4	no	—
97	#41×C₄₁	—	2	4	no	—
98	#41×C₄₁	—	2	4	no	—
99	#41×C₄₁	—	2	4	no	—
100	#41×C₄₁	—	2	4	no	—
101	#41×C₄₁	—	4	8	no	—
102	#41×C₄₁	—	4	2	no	—
103	#41×C₄₁	—	4	2	no	—
104	#41×C₄₁	—	4	8	no	—
105	#41×C₄₁	C₂×D₄	1	8	yes	—
106	C₂×Q₈×C₄₁	—	2	2	no	—
107	C₂×Q₈×C₄₁	—	2	4	no	—
108	C₂×Q₈×C₄₁	—	4	2	no	—
109	C₂×Q₈×C₄₁	C₂²×C₄	1	2	yes	—
110	C₂×Q₈×C₄₁	C₂²×C₄	1	2	yes	—
111	C₂×D₄×C₄₁	—	2	2	no	—
112	C₂×D₄×C₄₁	—	2	4	no	—
113	C₂×D₄×C₄₁	—	2	2	no	—
114	C₂×D₄×C₄₁	—	2	2	no	—
115	C₂×D₄×C₄₁	—	2	2	no	—
116	C₂×D₄×C₄₁	—	2	2	no	—
117	C₂×D₄×C₄₁	—	4	2	no	—
118	C₂×D₄×C₄₁	—	4	8	no	—
119	C₂×D₄×C₄₁	C₂²×C₄	1	2	yes	—
120	C₂×D₄×C₄₁	C₂²×C₄	1	2	yes	—
121	C₂²×C₄×C₄₁	—	2	2	no	—
122	C₂²×C₄×C₄₁	—	2	2	no	—
123	C₂²×C₄×C₄₁	—	2	2	no	—
124	C₂²×C₄×C₄₁	—	2	2	no	—
125	C₂²×C₄×C₄₁	—	2	2	no	—
126	C₂²×C₄×C₄₁	—	2	2	no	—
127	C₂²×C₄×C₄₁	—	2	2	no	—
128	C₂²×C₄×C₄₁	—	2	2	no	—
129	C₂²×C₄×C₄₁	—	4	8	no	—
130	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
132	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
135	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
136	Q₁₆×C₄₁	—	2	4	no	—
137	Q₁₆×C₄₁	—	2	2	no	—
138	Q₁₆×C₄₁	C₂²⋊C₄	1	4	yes	complemented
139	SD₁₆×C₄₁	—	2	4	no	—
140	SD₁₆×C₄₁	—	2	4	no	—
141	SD₁₆×C₄₁	—	2	4	no	—
142	SD₁₆×C₄₁	C₂²⋊C₄	1	8	yes	complemented
143	D₈×C₄₁	—	2	4	no	—
144	D₈×C₄₁	—	2	2	no	—
145	D₈×C₄₁	C₂²⋊C₄	1	4	yes	complemented
146	M₁₆×C₄₁	—	2	4	no	—
147	M₁₆×C₄₁	—	2	4	no	—
148	M₁₆×C₄₁	—	4	8	no	—
149	C₂×C₈×C₄₁	—	2	2	no	—
150	C₂×C₈×C₄₁	—	2	2	no	—
151	C₂×C₈×C₄₁	—	2	2	no	—
152	C₂×C₈×C₄₁	—	2	2	no	—
153	C₂×C₈×C₄₁	—	2	2	no	—
154	C₂×C₈×C₄₁	—	2	2	no	—
155	C₂×C₈×C₄₁	—	2	2	no	—
156	C₂×C₈×C₄₁	—	2	4	no	—
157	C₂×C₈×C₄₁	—	4	8	no	—
158	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
159	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
160	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
161	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
162	C₂×C₈×C₄₁	C₂×D₄	1	4	yes	—
163	C₂×C₈×C₄₁	C₂×D₄	1	4	yes	—
164	C₂²⋊C₄×C₄₁	—	4	2	no	—
165	C₄²×C₄₁	—	2	2	no	—
166	C₄²×C₄₁	—	2	2	no	—
167	C₂³×C₄₁	—	2	2	no	—
168	C₂³×C₄₁	—	2	2	no	—
169	C₂³×C₄₁	—	2	2	no	—
170	C₂³×C₄₁	—	2	2	no	—
171	C₂³×C₄₁	—	4	1	no	—
172	C₂³×C₄₁	—	4	4	no	—
173	C₂³×C₄₁	—	4	4	no	—
174	C₂³×C₄₁	—	8	2	no	—
175	C₂³×C₄₁	C₄×D₄	1	1	yes	characteristic
176	C₂³×C₄₁	C₄×D₄	1	1	yes	characteristic
177	C₂³×C₄₁	C₂×#41	1	1	yes	characteristic
178	Q₈×C₄₁	—	2	2	no	—
179	Q₈×C₄₁	—	2	2	no	—
180	Q₈×C₄₁	—	2	2	no	—
181	Q₈×C₄₁	—	4	4	no	—
182	Q₈×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
183	D₄×C₄₁	—	2	8	no	—
184	D₄×C₄₁	—	2	2	no	—
185	D₄×C₄₁	—	2	2	no	—
186	D₄×C₄₁	—	2	2	no	—
187	D₄×C₄₁	—	4	4	no	—
188	D₄×C₄₁	—	4	4	no	—
189	D₄×C₄₁	—	4	4	no	—
190	D₄×C₄₁	—	4	4	no	—
191	D₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
192	C₂×C₄×C₄₁	—	2	2	no	—
193	C₂×C₄×C₄₁	—	2	2	no	—
194	C₂×C₄×C₄₁	—	2	1	no	—
195	C₂×C₄×C₄₁	—	2	1	no	—
196	C₂×C₄×C₄₁	—	2	1	no	—
197	C₂×C₄×C₄₁	—	2	1	no	—
198	C₂×C₄×C₄₁	—	2	8	no	—
199	C₂×C₄×C₄₁	—	2	2	no	—
200	C₂×C₄×C₄₁	—	2	2	no	—
201	C₂×C₄×C₄₁	—	2	2	no	—
202	C₂×C₄×C₄₁	—	4	4	no	—
203	C₂×C₄×C₄₁	—	4	4	no	—
204	C₂×C₄×C₄₁	—	4	4	no	—
205	C₂×C₄×C₄₁	—	4	4	no	—
206	C₂×C₄×C₄₁	—	4	2	no	—
207	C₂×C₄×C₄₁	—	4	2	no	—
208	C₂×C₄×C₄₁	—	4	2	no	—
209	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
210	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
211	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
212	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
213	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
215	C₈×C₄₁	—	2	1	no	—
216	C₈×C₄₁	—	2	1	no	—
217	C₈×C₄₁	—	2	4	no	—
218	C₈×C₄₁	—	4	4	no	—
219	C₈×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
220	C₈×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
221	#22452	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
222	C₂²×C₄₁	—	2	1	no	—
223	C₂²×C₄₁	—	2	1	no	—
224	C₂²×C₄₁	—	2	1	no	—
225	C₂²×C₄₁	—	2	1	no	—
226	C₂²×C₄₁	—	2	1	no	—
227	C₂²×C₄₁	—	2	4	no	—
228	C₂²×C₄₁	—	2	1	no	—
229	C₂²×C₄₁	—	2	1	no	—
230	C₂²×C₄₁	—	4	4	no	—
231	C₂²×C₄₁	—	4	2	no	—
232	C₂²×C₄₁	—	4	2	no	—
233	C₂²×C₄₁	—	4	2	no	—
234	C₂²×C₄₁	—	4	2	no	—
235	C₂²×C₄₁	—	4	2	no	—
236	C₂²×C₄₁	—	8	8	no	—
237	C₂²×C₄₁	#444	1	2	yes	complemented
238	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
239	C₂²×C₄₁	#546	1	2	yes	—
240	C₄×C₄₁	—	2	1	no	—
241	C₄×C₄₁	—	2	1	no	—
242	C₄×C₄₁	—	2	4	no	—
243	C₄×C₄₁	—	4	2	no	—
244	C₄×C₄₁	—	8	2	no	—
245	C₄×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
246	C₄×C₄₁	#518	1	1	yes	characteristic
247	C₄×C₄₁	D₄²	1	2	yes	—
248	C₂²×#135	C₂×C₄₁	1	1	yes	characteristic
249	C₂×#444	C₂×C₄₁	1	2	yes	complemented
250	#2880	C₂×C₄₁	1	2	yes	complemented
251	#1975	C₂×C₄₁	1	2	yes	complemented
252	#1876	C₂×C₄₁	1	4	yes	complemented
253	#1810	C₂×C₄₁	1	1	yes	characteristic, complemented
254	#1789	C₂×C₄₁	1	2	yes	complemented
255	#1752	C₂×C₄₁	1	1	yes	characteristic, complemented
256	C₂×C₄₁	—	2	2	no	—
257	C₂×C₄₁	—	2	2	no	—
258	C₂×C₄₁	—	4	1	no	—
259	C₂×C₄₁	—	4	4	no	—
260	C₂×C₄₁	—	8	2	no	—
261	C₂×C₄₁	D₄×C₂²⋊C₄	1	1	yes	characteristic
262	C₂×C₄₁	C₂×#444	1	1	yes	characteristic
263	C₂×C₄₁	#2957	1	1	yes	characteristic
264	C₂²×#41	C₂²×C₄₁	1	2	yes	—
265	C₂×#135	—	2	8	no	—
266	C₂×#135	C₄×C₄₁	1	1	yes	characteristic, complemented
267	C₂×#135	C₄×C₄₁	1	2	yes	complemented
268	C₂×#135	C₄×C₄₁	1	1	yes	characteristic, complemented
269	C₂×#135	C₂²×C₄₁	1	2	yes	—
270	C₂×#135	C₂²×C₄₁	1	1	yes	characteristic
271	C₂×#135	C₂²×C₄₁	1	1	yes	characteristic
272	C₂²×Q₁₆	C₄×C₄₁	1	1	yes	characteristic, complemented
273	C₂²×SD₁₆	C₄×C₄₁	1	2	yes	complemented
274	C₂²×D₈	C₄×C₄₁	1	1	yes	characteristic, complemented
275	C₂×#131	C₂²×C₄₁	1	4	yes	complemented
276	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
277	#444	—	2	8	no	—
278	C₂×#108	C₂²×C₄₁	1	2	yes	complemented
279	C₂×#104	C₂²×C₄₁	1	4	yes	complemented
280	C₂×#104	C₂²×C₄₁	1	4	yes	complemented
281	#408	—	2	4	no	—
282	#408	C₂²×C₄₁	1	4	yes	complemented
283	#407	—	2	2	no	—
284	#407	C₂²×C₄₁	1	2	yes	—
285	C₂×C₂²⋊C₈	—	2	2	no	—
286	C₂×C₂²⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
287	C₂×#97	C₂²×C₄₁	1	2	yes	complemented
288	C₄×C₂²⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
289	#341	C₂²×C₄₁	1	2	yes	complemented
290	#336	C₂²×C₄₁	1	2	yes	complemented
291	C₄₁	#22452	1	1	yes	characteristic, direct product factor, Sylow
292	C₂×#41	—	2	2	no	—
293	C₂×#41	—	2	2	no	—
294	C₂×#41	—	2	8	no	—
295	C₂×#41	—	4	4	no	—
296	C₂×#41	C₂×C₄×C₄₁	1	2	yes	—
297	C₂×#41	C₂×C₄×C₄₁	1	2	yes	—
298	C₂×#41	C₂³×C₄₁	1	2	yes	—
299	C₂×#41	C₂³×C₄₁	1	2	yes	—
300	C₂²×Q₈	C₂×C₄×C₄₁	1	2	yes	—
301	C₂²×D₄	—	2	2	no	—
302	C₂²×D₄	C₂×C₄×C₄₁	1	2	yes	—
303	C₂³×C₄	—	2	2	no	—
304	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
305	#135	—	2	8	no	—
306	#135	—	2	4	no	—
307	#135	—	2	8	no	—
308	#135	—	2	4	no	—
309	#135	—	2	4	no	—
310	#135	D₄×C₄₁	1	8	yes	—
311	C₂×Q₁₆	—	2	4	no	—
312	C₂×Q₁₆	C₂×C₄×C₄₁	1	2	yes	—
313	C₂×Q₁₆	C₂×C₄×C₄₁	1	1	yes	characteristic
314	C₂×Q₁₆	C₂×C₄×C₄₁	1	1	yes	characteristic
315	C₂×SD₁₆	—	2	8	no	—
316	C₂×SD₁₆	C₂×C₄×C₄₁	1	2	yes	—
317	C₂×SD₁₆	C₂×C₄×C₄₁	1	2	yes	—
318	C₂×SD₁₆	C₂×C₄×C₄₁	1	2	yes	—
319	C₂×SD₁₆	C₂×C₄×C₄₁	1	2	yes	—
320	C₂×D₈	—	2	4	no	—
321	C₂×D₈	C₂×C₄×C₄₁	1	2	yes	—
322	C₂×D₈	C₂×C₄×C₄₁	1	1	yes	characteristic
323	C₂×D₈	C₂×C₄×C₄₁	1	1	yes	characteristic
324	#131	—	2	8	no	—
325	#131	—	4	4	no	—
326	C₂×M₁₆	—	2	4	no	—
327	C₂×M₁₆	C₂³×C₄₁	1	4	yes	—
328	C₂²×C₈	—	2	4	no	—
329	C₂²×C₈	—	2	4	no	—
330	C₂²×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
331	C₂²×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
332	C₂²×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
333	C₂²×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
334	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
335	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
336	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
337	C₂×C₂²⋊C₄	—	2	1	no	—
338	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
339	#108	—	2	4	no	—
340	#104	—	2	8	no	—
341	#104	—	4	4	no	—
342	C₂²⋊C₈	—	2	2	no	—
343	C₂²⋊C₈	—	2	2	no	—
344	C₂²⋊C₈	—	2	4	no	—
345	C₂²⋊C₈	—	2	4	no	—
346	C₂²⋊C₈	D₄×C₄₁	1	2	yes	—
347	C₂²⋊C₈	D₄×C₄₁	1	2	yes	—
348	#97	—	2	4	no	—
349	#95	—	2	1	no	—
350	C₂⁴	—	2	2	no	—
351	C₂⁴	#41×C₄₁	1	1	yes	characteristic
352	#41	—	2	4	no	—
353	#41	—	2	4	no	—
354	#41	—	2	4	no	—
355	#41	—	2	4	no	—
356	#41	—	2	4	no	—
357	#41	—	4	8	no	—
358	#41	—	4	2	no	—
359	#41	—	4	2	no	—
360	#41	—	4	8	no	—
361	#41	C₂×D₄×C₄₁	1	8	yes	—
362	C₂×Q₈	—	2	2	no	—
363	C₂×Q₈	—	2	4	no	—
364	C₂×Q₈	—	4	2	no	—
365	C₂×Q₈	C₂²×C₄×C₄₁	1	2	yes	—
366	C₂×Q₈	C₂²×C₄×C₄₁	1	2	yes	—
367	C₂×D₄	—	2	2	no	—
368	C₂×D₄	—	2	4	no	—
369	C₂×D₄	—	2	2	no	—
370	C₂×D₄	—	2	2	no	—
371	C₂×D₄	—	2	2	no	—
372	C₂×D₄	—	2	2	no	—
373	C₂×D₄	—	4	2	no	—
374	C₂×D₄	—	4	8	no	—
375	C₂×D₄	C₂²×C₄×C₄₁	1	2	yes	—
376	C₂×D₄	C₂²×C₄×C₄₁	1	2	yes	—
377	C₂²×C₄	—	2	2	no	—
378	C₂²×C₄	—	2	2	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	4	8	no	—
386	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
387	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
388	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
389	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
390	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
391	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
392	Q₁₆	—	2	4	no	—
393	Q₁₆	—	2	2	no	—
394	Q₁₆	C₂²⋊C₄×C₄₁	1	4	yes	complemented
395	SD₁₆	—	2	4	no	—
396	SD₁₆	—	2	4	no	—
397	SD₁₆	—	2	4	no	—
398	SD₁₆	C₂²⋊C₄×C₄₁	1	8	yes	complemented
399	D₈	—	2	4	no	—
400	D₈	—	2	2	no	—
401	D₈	C₂²⋊C₄×C₄₁	1	4	yes	complemented
402	M₁₆	—	2	4	no	—
403	M₁₆	—	2	4	no	—
404	M₁₆	—	4	8	no	—
405	C₂×C₈	—	2	2	no	—
406	C₂×C₈	—	2	2	no	—
407	C₂×C₈	—	2	2	no	—
408	C₂×C₈	—	2	2	no	—
409	C₂×C₈	—	2	2	no	—
410	C₂×C₈	—	2	2	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	2	4	no	—
413	C₂×C₈	—	4	8	no	—
414	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
415	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
416	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
417	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
418	C₂×C₈	C₂×D₄×C₄₁	1	4	yes	—
419	C₂×C₈	C₂×D₄×C₄₁	1	4	yes	—
420	C₂²⋊C₄	—	4	2	no	—
421	C₄²	—	2	2	no	—
422	C₄²	—	2	2	no	—
423	C₂³	—	2	2	no	—
424	C₂³	—	2	2	no	—
425	C₂³	—	2	2	no	—
426	C₂³	—	2	2	no	—
427	C₂³	—	4	1	no	—
428	C₂³	—	4	4	no	—
429	C₂³	—	4	4	no	—
430	C₂³	—	8	2	no	—
431	C₂³	C₄×D₄×C₄₁	1	1	yes	characteristic
432	C₂³	C₄×D₄×C₄₁	1	1	yes	characteristic
433	C₂³	C₂×#41×C₄₁	1	1	yes	characteristic
434	Q₈	—	2	2	no	—
435	Q₈	—	2	2	no	—
436	Q₈	—	2	2	no	—
437	Q₈	—	4	4	no	—
438	Q₈	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
439	D₄	—	2	8	no	—
440	D₄	—	2	2	no	—
441	D₄	—	2	2	no	—
442	D₄	—	2	2	no	—
443	D₄	—	4	4	no	—
444	D₄	—	4	4	no	—
445	D₄	—	4	4	no	—
446	D₄	—	4	4	no	—
447	D₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	8	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	4	4	no	—
459	C₂×C₄	—	4	4	no	—
460	C₂×C₄	—	4	4	no	—
461	C₂×C₄	—	4	4	no	—
462	C₂×C₄	—	4	2	no	—
463	C₂×C₄	—	4	2	no	—
464	C₂×C₄	—	4	2	no	—
465	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
466	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
467	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
468	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
469	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
470	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
471	C₈	—	2	1	no	—
472	C₈	—	2	1	no	—
473	C₈	—	2	4	no	—
474	C₈	—	4	4	no	—
475	C₈	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
476	C₈	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
477	C₂²	—	2	1	no	—
478	C₂²	—	2	1	no	—
479	C₂²	—	2	1	no	—
480	C₂²	—	2	1	no	—
481	C₂²	—	2	1	no	—
482	C₂²	—	2	4	no	—
483	C₂²	—	2	1	no	—
484	C₂²	—	2	1	no	—
485	C₂²	—	4	4	no	—
486	C₂²	—	4	2	no	—
487	C₂²	—	4	2	no	—
488	C₂²	—	4	2	no	—
489	C₂²	—	4	2	no	—
490	C₂²	—	4	2	no	—
491	C₂²	—	8	8	no	—
492	C₂²	C₄₁×#444	1	2	yes	complemented
493	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
494	C₂²	C₄₁×#546	1	2	yes	—
495	C₄	—	2	1	no	—
496	C₄	—	2	1	no	—
497	C₄	—	2	4	no	—
498	C₄	—	4	2	no	—
499	C₄	—	8	2	no	—
500	C₄	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
501	C₄	C₄₁×#518	1	1	yes	characteristic
502	C₄	D₄²×C₄₁	1	2	yes	—
503	C₂	—	2	2	no	—
504	C₂	—	2	2	no	—
505	C₂	—	4	1	no	—
506	C₂	—	4	4	no	—
507	C₂	—	8	2	no	—
508	C₂	D₄×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
509	C₂	C₂×C₄₁×#444	1	1	yes	characteristic
510	C₂	C₄₁×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₄₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
socle	C₂²×C₄₁	C₂×C₄×D₄
prime radical	C₂²×#135×C₄₁	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₁ ◁ C₂×C₂²⋊C₈×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₁×#444	C₂	semidirect
3. C₄₁×#2880	C₂	semidirect
4. C₄₁×#1975	C₂	semidirect
5. C₄₁×#1876	C₂	semidirect
6. C₄₁×#1810	C₂	semidirect
7. C₄₁×#1789	C₂	semidirect
8. C₄₁×#1752	C₂	semidirect
11. C₂×#135×C₄₁	C₄	semidirect
12. C₂×#135×C₄₁	C₄	semidirect
13. C₂×#135×C₄₁	C₄	semidirect
17. C₂²×Q₁₆×C₄₁	C₄	semidirect
18. C₂²×SD₁₆×C₄₁	C₄	semidirect
19. C₂²×D₈×C₄₁	C₄	semidirect
20. C₂×#131×C₄₁	C₂²	semidirect
23. C₂×#108×C₄₁	C₂²	semidirect
24. C₂×#104×C₄₁	C₂²	semidirect
25. C₂×#104×C₄₁	C₂²	semidirect
27. C₄₁×#408	C₂²	semidirect
32. C₂×#97×C₄₁	C₂²	semidirect
34. C₄₁×#341	C₂²	semidirect
35. C₄₁×#336	C₂²	semidirect
138. Q₁₆×C₄₁	C₂²⋊C₄	semidirect
142. SD₁₆×C₄₁	C₂²⋊C₄	semidirect
145. D₈×C₄₁	C₂²⋊C₄	semidirect
221. #22452	C₄₁	direct (N × H)
237. C₂²×C₄₁	#444	semidirect
249. C₂×#444	C₂×C₄₁	semidirect
250. #2880	C₂×C₄₁	semidirect
251. #1975	C₂×C₄₁	semidirect
252. #1876	C₂×C₄₁	semidirect
253. #1810	C₂×C₄₁	semidirect
254. #1789	C₂×C₄₁	semidirect
255. #1752	C₂×C₄₁	semidirect
266. C₂×#135	C₄×C₄₁	semidirect
267. C₂×#135	C₄×C₄₁	semidirect
268. C₂×#135	C₄×C₄₁	semidirect
272. C₂²×Q₁₆	C₄×C₄₁	semidirect
273. C₂²×SD₁₆	C₄×C₄₁	semidirect
274. C₂²×D₈	C₄×C₄₁	semidirect
275. C₂×#131	C₂²×C₄₁	semidirect
278. C₂×#108	C₂²×C₄₁	semidirect
279. C₂×#104	C₂²×C₄₁	semidirect
280. C₂×#104	C₂²×C₄₁	semidirect
282. #408	C₂²×C₄₁	semidirect
287. C₂×#97	C₂²×C₄₁	semidirect
289. #341	C₂²×C₄₁	semidirect
290. #336	C₂²×C₄₁	semidirect
291. C₄₁	#22452	direct (N × H)
394. Q₁₆	C₂²⋊C₄×C₄₁	semidirect
398. SD₁₆	C₂²⋊C₄×C₄₁	semidirect
401. D₈	C₂²⋊C₄×C₄₁	semidirect
492. C₂²	C₄₁×#444	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#135×C₄₁	2
C₂×C₄₁×#444	2
C₄₁×#2880	2
C₄₁×#1975	2
C₄₁×#1876	2
C₄₁×#1810	2
C₄₁×#1789	2
C₄₁×#1752	2
#22452	41

Subgroups — C41×#22452 #23798653

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#22452 #23798653