Subgroups — C₄₁×#22456 #23798657

Order 10496.

1336 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#577	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2949	C₂	1	2	yes	maximal, complemented
3	C₄₁×#2884	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#2883	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1976	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1905	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1904	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1812	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1796	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#1795	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₁×#1757	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×2₋¹⁺⁴×C₄₁	C₂²	1	1	yes	characteristic
13	C₂×H(5,𝔽₂)×C₄₁	C₂²	1	1	yes	characteristic
14	C₄₁×#577	—	2	4	no	—
15	C₄₁×#577	—	2	4	no	—
16	C₂×#137×C₄₁	C₄	1	2	yes	complemented
17	C₂×#137×C₄₁	C₂²	1	1	yes	characteristic
18	C₂×AGL(1,ℤ/8)×C₄₁	C₄	1	2	yes	complemented
19	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	1	yes	characteristic
20	C₂×#135×C₄₁	C₄	1	2	yes	complemented
21	C₂×#135×C₄₁	C₂²	1	1	yes	characteristic
22	C₂²×SD₁₆×C₄₁	C₄	1	2	yes	complemented
23	C₂²×SD₁₆×C₄₁	C₂²	1	1	yes	characteristic
24	C₂×#131×C₄₁	C₂²	1	1	yes	characteristic
25	C₄₁×#519	C₂²	1	2	yes	complemented
26	C₄₁×#518	C₂²	1	2	yes	—
27	C₄₁×#440	—	2	4	no	—
28	C₄×SD₁₆×C₄₁	—	2	4	no	—
29	C₄₁×#427	C₂²	1	2	yes	complemented
30	C₄₁×#421	C₂²	1	2	yes	—
31	C₄₁×#421	C₂²	1	2	yes	complemented
32	C₄₁×#419	C₂²	1	2	yes	complemented
33	C₄₁×#418	C₂²	1	2	yes	—
34	C₄₁×#413	C₂²	1	1	yes	characteristic
35	C₄₁×#410	—	2	4	no	—
36	C₄₁×#410	C₂²	1	1	yes	characteristic
37	C₄₁×#410	C₂²	1	1	yes	characteristic
38	C₄₁×#410	C₂²	1	2	yes	—
39	C₄₁×#410	C₂²	1	1	yes	characteristic
40	C₄₁×#410	C₂²	1	2	yes	complemented
41	C₂×#99×C₄₁	—	2	4	no	—
42	C₄₁×#405	C₂²	1	2	yes	complemented
43	C₄₁×#344	C₂²	1	2	yes	complemented
44	C₄₁×H(3,ℤ/4)	C₂²	1	2	yes	complemented
45	2₋¹⁺⁴×C₄₁	—	2	4	no	—
46	2₋¹⁺⁴×C₄₁	—	4	2	no	—
47	H(5,𝔽₂)×C₄₁	—	2	4	no	—
48	H(5,𝔽₂)×C₄₁	—	4	2	no	—
49	C₂×#41×C₄₁	—	2	1	no	—
50	C₂×#41×C₄₁	—	2	2	no	—
51	C₂×#41×C₄₁	—	2	1	no	—
52	C₂×#41×C₄₁	C₂×C₄	1	2	yes	—
53	C₂×#41×C₄₁	C₂×C₄	1	2	yes	complemented
54	C₂×#41×C₄₁	C₂³	1	1	yes	characteristic
55	C₂×#41×C₄₁	C₂³	1	1	yes	characteristic
56	C₂×#41×C₄₁	C₂³	1	1	yes	characteristic
57	C₂²×Q₈×C₄₁	—	2	1	no	—
58	C₂²×Q₈×C₄₁	C₂×C₄	1	2	yes	complemented
59	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
60	C₂²×D₄×C₄₁	—	2	2	no	—
61	C₂²×D₄×C₄₁	—	2	1	no	—
62	C₂²×D₄×C₄₁	C₂×C₄	1	2	yes	—
63	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
64	#137×C₄₁	—	2	4	no	—
65	#137×C₄₁	—	2	8	no	—
66	AGL(1,ℤ/8)×C₄₁	—	2	4	no	—
67	AGL(1,ℤ/8)×C₄₁	—	2	8	no	—
68	#135×C₄₁	—	2	4	no	—
69	#135×C₄₁	—	2	4	no	—
70	#135×C₄₁	—	2	2	no	—
71	#135×C₄₁	—	2	2	no	—
72	C₂×Q₁₆×C₄₁	C₂×C₄	1	2	yes	complemented
73	C₂×Q₁₆×C₄₁	C₂×C₄	1	1	yes	characteristic, complemented
74	C₂×SD₁₆×C₄₁	—	2	4	no	—
75	C₂×SD₁₆×C₄₁	—	2	4	no	—
76	C₂×SD₁₆×C₄₁	C₂×C₄	1	2	yes	—
77	C₂×SD₁₆×C₄₁	C₂×C₄	1	2	yes	—
78	C₂×SD₁₆×C₄₁	C₂×C₄	1	2	yes	—
79	C₂×SD₁₆×C₄₁	C₂×C₄	1	1	yes	characteristic, complemented
80	C₂×SD₁₆×C₄₁	C₂×C₄	1	2	yes	complemented
81	C₂×SD₁₆×C₄₁	C₂×C₄	1	1	yes	characteristic, complemented
82	C₂×SD₁₆×C₄₁	D₄	1	4	yes	—
83	C₂×SD₁₆×C₄₁	D₄	1	4	yes	—
84	C₂×D₈×C₄₁	C₂×C₄	1	2	yes	—
85	C₂×D₈×C₄₁	C₂×C₄	1	1	yes	characteristic, complemented
86	#131×C₄₁	—	2	2	no	—
87	#131×C₄₁	—	2	2	no	—
88	C₂×M₁₆×C₄₁	C₂×C₄	1	2	yes	complemented
89	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
90	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
91	C₂²×C₈×C₄₁	C₂×C₄	1	2	yes	complemented
92	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
93	C₄×Q₈×C₄₁	—	2	4	no	—
94	C₄×Q₈×C₄₁	—	4	2	no	—
95	C₄×D₄×C₄₁	—	2	4	no	—
96	C₄×D₄×C₄₁	—	4	2	no	—
97	#117×C₄₁	—	2	4	no	—
98	#117×C₄₁	C₂³	1	2	yes	—
99	#117×C₄₁	C₂³	1	2	yes	—
100	#117×C₄₁	C₂³	1	2	yes	—
101	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
102	#107×C₄₁	—	2	2	no	—
103	#106×C₄₁	—	2	2	no	—
104	#104×C₄₁	—	4	2	no	—
105	#104×C₄₁	—	4	2	no	—
106	#103×C₄₁	—	2	4	no	—
107	#102×C₄₁	—	2	4	no	—
108	#101×C₄₁	—	2	1	no	—
109	#100×C₄₁	—	2	1	no	—
110	#99×C₄₁	—	2	2	no	—
111	#99×C₄₁	—	2	2	no	—
112	#99×C₄₁	—	2	4	no	—
113	#99×C₄₁	—	2	4	no	—
114	#99×C₄₁	D₄	1	4	yes	—
115	#97×C₄₁	—	2	2	no	—
116	C₄×C₈×C₄₁	—	2	2	no	—
117	C₂⁴×C₄₁	—	2	1	no	—
118	C₂⁴×C₄₁	—	2	2	no	—
119	#41×C₄₁	—	2	2	no	—
120	#41×C₄₁	—	2	2	no	—
121	#41×C₄₁	—	2	4	no	—
122	#41×C₄₁	—	2	4	no	—
123	#41×C₄₁	—	2	2	no	—
124	#41×C₄₁	—	2	2	no	—
125	#41×C₄₁	—	2	2	no	—
126	#41×C₄₁	—	2	2	no	—
127	#41×C₄₁	—	4	2	no	—
128	#41×C₄₁	—	4	2	no	—
129	#41×C₄₁	—	4	8	no	—
130	#41×C₄₁	—	4	4	no	—
131	#41×C₄₁	—	4	4	no	—
132	C₂×Q₈×C₄₁	—	2	2	no	—
133	C₂×Q₈×C₄₁	—	2	1	no	—
134	C₂×Q₈×C₄₁	—	2	2	no	—
135	C₂×Q₈×C₄₁	—	2	4	no	—
136	C₂×Q₈×C₄₁	—	4	2	no	—
137	C₂×Q₈×C₄₁	—	4	4	no	—
138	C₂×Q₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
139	C₂×Q₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
140	C₂×Q₈×C₄₁	C₂²×C₄	1	2	yes	—
141	C₂×Q₈×C₄₁	C₂×D₄	1	4	yes	—
142	C₂×Q₈×C₄₁	#41	1	1	yes	characteristic
143	C₂×D₄×C₄₁	—	2	4	no	—
144	C₂×D₄×C₄₁	—	2	2	no	—
145	C₂×D₄×C₄₁	—	2	2	no	—
146	C₂×D₄×C₄₁	—	2	1	no	—
147	C₂×D₄×C₄₁	—	2	1	no	—
148	C₂×D₄×C₄₁	—	2	2	no	—
149	C₂×D₄×C₄₁	—	2	1	no	—
150	C₂×D₄×C₄₁	—	2	2	no	—
151	C₂×D₄×C₄₁	—	4	2	no	—
152	C₂×D₄×C₄₁	—	4	4	no	—
153	C₂×D₄×C₄₁	—	4	8	no	—
154	C₂×D₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
155	C₂×D₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
156	C₂×D₄×C₄₁	C₂²×C₄	1	2	yes	—
157	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
158	C₂×D₄×C₄₁	C₂×D₄	1	4	yes	—
159	C₂×D₄×C₄₁	#41	1	1	yes	characteristic
160	C₂²×C₄×C₄₁	—	2	1	no	—
161	C₂²×C₄×C₄₁	—	2	2	no	—
162	C₂²×C₄×C₄₁	—	2	2	no	—
163	C₂²×C₄×C₄₁	—	2	2	no	—
164	C₂²×C₄×C₄₁	—	2	1	no	—
165	C₂²×C₄×C₄₁	—	2	1	no	—
166	C₂²×C₄×C₄₁	—	4	4	no	—
167	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
168	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
169	Q₁₆×C₄₁	—	2	4	no	—
170	Q₁₆×C₄₁	—	2	2	no	—
171	SD₁₆×C₄₁	—	2	4	no	—
172	SD₁₆×C₄₁	—	2	4	no	—
173	SD₁₆×C₄₁	—	2	1	no	—
174	SD₁₆×C₄₁	—	2	1	no	—
175	SD₁₆×C₄₁	—	2	1	no	—
176	SD₁₆×C₄₁	—	2	1	no	—
177	SD₁₆×C₄₁	C₂²⋊C₄	1	4	yes	—
178	SD₁₆×C₄₁	C₂²⋊C₄	1	4	yes	—
179	SD₁₆×C₄₁	C₂²⋊C₄	1	8	yes	—
180	D₈×C₄₁	—	2	4	no	—
181	D₈×C₄₁	—	2	2	no	—
182	M₁₆×C₄₁	—	2	2	no	—
183	M₁₆×C₄₁	—	2	4	no	—
184	M₁₆×C₄₁	—	2	2	no	—
185	M₁₆×C₄₁	—	2	2	no	—
186	C₂×C₈×C₄₁	—	2	2	no	—
187	C₂×C₈×C₄₁	—	2	2	no	—
188	C₂×C₈×C₄₁	—	2	2	no	—
189	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
190	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
191	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
192	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
193	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
194	C₄⋊C₄×C₄₁	—	2	2	no	—
195	C₄⋊C₄×C₄₁	—	2	2	no	—
196	C₄⋊C₄×C₄₁	—	2	2	no	—
197	C₄⋊C₄×C₄₁	—	4	4	no	—
198	C₂²⋊C₄×C₄₁	—	2	2	no	—
199	C₂²⋊C₄×C₄₁	—	2	2	no	—
200	C₂²⋊C₄×C₄₁	—	4	4	no	—
201	C₂²⋊C₄×C₄₁	C₂×D₄	1	4	yes	—
202	C₄²×C₄₁	—	2	2	no	—
203	C₄²×C₄₁	—	2	2	no	—
204	C₄²×C₄₁	—	2	2	no	—
205	C₄²×C₄₁	—	4	4	no	—
206	C₂³×C₄₁	—	2	2	no	—
207	C₂³×C₄₁	—	2	4	no	—
208	C₂³×C₄₁	—	2	1	no	—
209	C₂³×C₄₁	—	2	1	no	—
210	C₂³×C₄₁	—	4	4	no	—
211	C₂³×C₄₁	—	16	1	no	—
212	C₂³×C₄₁	—	16	2	no	—
213	C₂³×C₄₁	C₄×D₄	1	2	yes	—
214	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
215	Q₈×C₄₁	—	2	2	no	—
216	Q₈×C₄₁	—	2	4	no	—
217	Q₈×C₄₁	—	2	2	no	—
218	Q₈×C₄₁	—	2	1	no	—
219	Q₈×C₄₁	—	2	1	no	—
220	Q₈×C₄₁	—	2	1	no	—
221	Q₈×C₄₁	—	2	1	no	—
222	Q₈×C₄₁	—	4	4	no	—
223	Q₈×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
224	D₄×C₄₁	—	2	4	no	—
225	D₄×C₄₁	—	2	1	no	—
226	D₄×C₄₁	—	2	1	no	—
227	D₄×C₄₁	—	2	2	no	—
228	D₄×C₄₁	—	2	1	no	—
229	D₄×C₄₁	—	2	1	no	—
230	D₄×C₄₁	—	2	4	no	—
231	D₄×C₄₁	—	2	2	no	—
232	D₄×C₄₁	—	4	4	no	—
233	D₄×C₄₁	—	4	4	no	—
234	D₄×C₄₁	—	4	4	no	—
235	D₄×C₄₁	—	4	2	no	—
236	D₄×C₄₁	—	4	2	no	—
237	D₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
238	C₂×C₄×C₄₁	—	2	2	no	—
239	C₂×C₄×C₄₁	—	2	4	no	—
240	C₂×C₄×C₄₁	—	2	2	no	—
241	C₂×C₄×C₄₁	—	2	2	no	—
242	C₂×C₄×C₄₁	—	2	1	no	—
243	C₂×C₄×C₄₁	—	2	1	no	—
244	C₂×C₄×C₄₁	—	2	4	no	—
245	C₂×C₄×C₄₁	—	4	4	no	—
246	C₂×C₄×C₄₁	—	4	4	no	—
247	C₂×C₄×C₄₁	—	4	4	no	—
248	C₂×C₄×C₄₁	—	4	2	no	—
249	C₂×C₄×C₄₁	—	4	2	no	—
250	C₂×C₄×C₄₁	—	4	2	no	—
251	C₂×C₄×C₄₁	—	4	2	no	—
252	C₂×C₄×C₄₁	—	16	4	no	—
253	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
254	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
255	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
256	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
257	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
258	C₈×C₄₁	—	2	2	no	—
259	C₈×C₄₁	—	4	2	no	—
260	C₈×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
261	C₈×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
262	C₈×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
263	#22456	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
264	C₂²×C₄₁	—	2	2	no	—
265	C₂²×C₄₁	—	2	4	no	—
266	C₂²×C₄₁	—	4	1	no	—
267	C₂²×C₄₁	—	4	2	no	—
268	C₂²×C₄₁	—	4	1	no	—
269	C₂²×C₄₁	—	4	1	no	—
270	C₂²×C₄₁	—	8	4	no	—
271	C₂²×C₄₁	—	8	2	no	—
272	C₂²×C₄₁	—	8	1	no	—
273	C₂²×C₄₁	—	8	1	no	—
274	C₂²×C₄₁	—	16	4	no	—
275	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
276	C₂²×C₄₁	D₄²	1	2	yes	—
277	C₄×C₄₁	—	2	4	no	—
278	C₄×C₄₁	—	2	2	no	—
279	C₄×C₄₁	—	4	4	no	—
280	C₄×C₄₁	—	4	1	no	—
281	C₄×C₄₁	—	4	1	no	—
282	C₄×C₄₁	—	8	2	no	—
283	C₄×C₄₁	—	8	2	no	—
284	C₄×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
285	C₄×C₄₁	#518	1	1	yes	characteristic
286	C₄×C₄₁	#546	1	2	yes	—
287	C₂×#577	C₂×C₄₁	1	1	yes	characteristic
288	#2949	C₂×C₄₁	1	2	yes	complemented
289	#2884	C₂×C₄₁	1	1	yes	characteristic, complemented
290	#2883	C₂×C₄₁	1	1	yes	characteristic, complemented
291	#1976	C₂×C₄₁	1	2	yes	complemented
292	#1905	C₂×C₄₁	1	2	yes	complemented
293	#1904	C₂×C₄₁	1	2	yes	complemented
294	#1812	C₂×C₄₁	1	1	yes	characteristic, complemented
295	#1796	C₂×C₄₁	1	1	yes	characteristic, complemented
296	#1795	C₂×C₄₁	1	1	yes	characteristic, complemented
297	#1757	C₂×C₄₁	1	1	yes	characteristic, complemented
298	C₂×C₄₁	—	2	2	no	—
299	C₂×C₄₁	—	2	1	no	—
300	C₂×C₄₁	—	4	2	no	—
301	C₂×C₄₁	—	4	4	no	—
302	C₂×C₄₁	—	8	1	no	—
303	C₂×C₄₁	—	8	1	no	—
304	C₂×C₄₁	D₄×C₂²⋊C₄	1	1	yes	characteristic
305	C₂×2₋¹⁺⁴	C₂²×C₄₁	1	1	yes	characteristic
306	C₂×H(5,𝔽₂)	C₂²×C₄₁	1	1	yes	characteristic
307	#577	—	2	4	no	—
308	#577	—	2	4	no	—
309	C₂×#137	C₄×C₄₁	1	2	yes	complemented
310	C₂×#137	C₂²×C₄₁	1	1	yes	characteristic
311	C₂×AGL(1,ℤ/8)	C₄×C₄₁	1	2	yes	complemented
312	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	1	yes	characteristic
313	C₂×#135	C₄×C₄₁	1	2	yes	complemented
314	C₂×#135	C₂²×C₄₁	1	1	yes	characteristic
315	C₂²×SD₁₆	C₄×C₄₁	1	2	yes	complemented
316	C₂²×SD₁₆	C₂²×C₄₁	1	1	yes	characteristic
317	C₂×#131	C₂²×C₄₁	1	1	yes	characteristic
318	#519	C₂²×C₄₁	1	2	yes	complemented
319	#518	C₂²×C₄₁	1	2	yes	—
320	#440	—	2	4	no	—
321	C₄×SD₁₆	—	2	4	no	—
322	#427	C₂²×C₄₁	1	2	yes	complemented
323	#421	C₂²×C₄₁	1	2	yes	—
324	#421	C₂²×C₄₁	1	2	yes	complemented
325	#419	C₂²×C₄₁	1	2	yes	complemented
326	#418	C₂²×C₄₁	1	2	yes	—
327	#413	C₂²×C₄₁	1	1	yes	characteristic
328	#410	—	2	4	no	—
329	#410	C₂²×C₄₁	1	1	yes	characteristic
330	#410	C₂²×C₄₁	1	1	yes	characteristic
331	#410	C₂²×C₄₁	1	2	yes	—
332	#410	C₂²×C₄₁	1	1	yes	characteristic
333	#410	C₂²×C₄₁	1	2	yes	complemented
334	C₂×#99	—	2	4	no	—
335	#405	C₂²×C₄₁	1	2	yes	complemented
336	#344	C₂²×C₄₁	1	2	yes	complemented
337	H(3,ℤ/4)	C₂²×C₄₁	1	2	yes	complemented
338	C₄₁	#22456	1	1	yes	characteristic, direct product factor, Sylow
339	2₋¹⁺⁴	—	2	4	no	—
340	2₋¹⁺⁴	—	4	2	no	—
341	H(5,𝔽₂)	—	2	4	no	—
342	H(5,𝔽₂)	—	4	2	no	—
343	C₂×#41	—	2	1	no	—
344	C₂×#41	—	2	2	no	—
345	C₂×#41	—	2	1	no	—
346	C₂×#41	C₂×C₄×C₄₁	1	2	yes	—
347	C₂×#41	C₂×C₄×C₄₁	1	2	yes	complemented
348	C₂×#41	C₂³×C₄₁	1	1	yes	characteristic
349	C₂×#41	C₂³×C₄₁	1	1	yes	characteristic
350	C₂×#41	C₂³×C₄₁	1	1	yes	characteristic
351	C₂²×Q₈	—	2	1	no	—
352	C₂²×Q₈	C₂×C₄×C₄₁	1	2	yes	complemented
353	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
354	C₂²×D₄	—	2	2	no	—
355	C₂²×D₄	—	2	1	no	—
356	C₂²×D₄	C₂×C₄×C₄₁	1	2	yes	—
357	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
358	#137	—	2	4	no	—
359	#137	—	2	8	no	—
360	AGL(1,ℤ/8)	—	2	4	no	—
361	AGL(1,ℤ/8)	—	2	8	no	—
362	#135	—	2	4	no	—
363	#135	—	2	4	no	—
364	#135	—	2	2	no	—
365	#135	—	2	2	no	—
366	C₂×Q₁₆	C₂×C₄×C₄₁	1	2	yes	complemented
367	C₂×Q₁₆	C₂×C₄×C₄₁	1	1	yes	characteristic, complemented
368	C₂×SD₁₆	—	2	4	no	—
369	C₂×SD₁₆	—	2	4	no	—
370	C₂×SD₁₆	C₂×C₄×C₄₁	1	2	yes	—
371	C₂×SD₁₆	C₂×C₄×C₄₁	1	2	yes	—
372	C₂×SD₁₆	C₂×C₄×C₄₁	1	2	yes	—
373	C₂×SD₁₆	C₂×C₄×C₄₁	1	1	yes	characteristic, complemented
374	C₂×SD₁₆	C₂×C₄×C₄₁	1	2	yes	complemented
375	C₂×SD₁₆	C₂×C₄×C₄₁	1	1	yes	characteristic, complemented
376	C₂×SD₁₆	D₄×C₄₁	1	4	yes	—
377	C₂×SD₁₆	D₄×C₄₁	1	4	yes	—
378	C₂×D₈	C₂×C₄×C₄₁	1	2	yes	—
379	C₂×D₈	C₂×C₄×C₄₁	1	1	yes	characteristic, complemented
380	#131	—	2	2	no	—
381	#131	—	2	2	no	—
382	C₂×M₁₆	C₂×C₄×C₄₁	1	2	yes	complemented
383	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
384	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
385	C₂²×C₈	C₂×C₄×C₄₁	1	2	yes	complemented
386	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
387	C₄×Q₈	—	2	4	no	—
388	C₄×Q₈	—	4	2	no	—
389	C₄×D₄	—	2	4	no	—
390	C₄×D₄	—	4	2	no	—
391	#117	—	2	4	no	—
392	#117	C₂³×C₄₁	1	2	yes	—
393	#117	C₂³×C₄₁	1	2	yes	—
394	#117	C₂³×C₄₁	1	2	yes	—
395	C₂×C₂²⋊C₄	—	2	4	no	—
396	#107	—	2	2	no	—
397	#106	—	2	2	no	—
398	#104	—	4	2	no	—
399	#104	—	4	2	no	—
400	#103	—	2	4	no	—
401	#102	—	2	4	no	—
402	#101	—	2	1	no	—
403	#100	—	2	1	no	—
404	#99	—	2	2	no	—
405	#99	—	2	2	no	—
406	#99	—	2	4	no	—
407	#99	—	2	4	no	—
408	#99	D₄×C₄₁	1	4	yes	—
409	#97	—	2	2	no	—
410	C₄×C₈	—	2	2	no	—
411	C₂⁴	—	2	1	no	—
412	C₂⁴	—	2	2	no	—
413	#41	—	2	2	no	—
414	#41	—	2	2	no	—
415	#41	—	2	4	no	—
416	#41	—	2	4	no	—
417	#41	—	2	2	no	—
418	#41	—	2	2	no	—
419	#41	—	2	2	no	—
420	#41	—	2	2	no	—
421	#41	—	4	2	no	—
422	#41	—	4	2	no	—
423	#41	—	4	8	no	—
424	#41	—	4	4	no	—
425	#41	—	4	4	no	—
426	C₂×Q₈	—	2	2	no	—
427	C₂×Q₈	—	2	1	no	—
428	C₂×Q₈	—	2	2	no	—
429	C₂×Q₈	—	2	4	no	—
430	C₂×Q₈	—	4	2	no	—
431	C₂×Q₈	—	4	4	no	—
432	C₂×Q₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
433	C₂×Q₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
434	C₂×Q₈	C₂²×C₄×C₄₁	1	2	yes	—
435	C₂×Q₈	C₂×D₄×C₄₁	1	4	yes	—
436	C₂×Q₈	#41×C₄₁	1	1	yes	characteristic
437	C₂×D₄	—	2	4	no	—
438	C₂×D₄	—	2	2	no	—
439	C₂×D₄	—	2	2	no	—
440	C₂×D₄	—	2	1	no	—
441	C₂×D₄	—	2	1	no	—
442	C₂×D₄	—	2	2	no	—
443	C₂×D₄	—	2	1	no	—
444	C₂×D₄	—	2	2	no	—
445	C₂×D₄	—	4	2	no	—
446	C₂×D₄	—	4	4	no	—
447	C₂×D₄	—	4	8	no	—
448	C₂×D₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
449	C₂×D₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
450	C₂×D₄	C₂²×C₄×C₄₁	1	2	yes	—
451	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
452	C₂×D₄	C₂×D₄×C₄₁	1	4	yes	—
453	C₂×D₄	#41×C₄₁	1	1	yes	characteristic
454	C₂²×C₄	—	2	1	no	—
455	C₂²×C₄	—	2	2	no	—
456	C₂²×C₄	—	2	2	no	—
457	C₂²×C₄	—	2	2	no	—
458	C₂²×C₄	—	2	1	no	—
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	4	4	no	—
461	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
462	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
463	Q₁₆	—	2	4	no	—
464	Q₁₆	—	2	2	no	—
465	SD₁₆	—	2	4	no	—
466	SD₁₆	—	2	4	no	—
467	SD₁₆	—	2	1	no	—
468	SD₁₆	—	2	1	no	—
469	SD₁₆	—	2	1	no	—
470	SD₁₆	—	2	1	no	—
471	SD₁₆	C₂²⋊C₄×C₄₁	1	4	yes	—
472	SD₁₆	C₂²⋊C₄×C₄₁	1	4	yes	—
473	SD₁₆	C₂²⋊C₄×C₄₁	1	8	yes	—
474	D₈	—	2	4	no	—
475	D₈	—	2	2	no	—
476	M₁₆	—	2	2	no	—
477	M₁₆	—	2	4	no	—
478	M₁₆	—	2	2	no	—
479	M₁₆	—	2	2	no	—
480	C₂×C₈	—	2	2	no	—
481	C₂×C₈	—	2	2	no	—
482	C₂×C₈	—	2	2	no	—
483	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
484	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
485	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
486	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
487	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
488	C₄⋊C₄	—	2	2	no	—
489	C₄⋊C₄	—	2	2	no	—
490	C₄⋊C₄	—	2	2	no	—
491	C₄⋊C₄	—	4	4	no	—
492	C₂²⋊C₄	—	2	2	no	—
493	C₂²⋊C₄	—	2	2	no	—
494	C₂²⋊C₄	—	4	4	no	—
495	C₂²⋊C₄	C₂×D₄×C₄₁	1	4	yes	—
496	C₄²	—	2	2	no	—
497	C₄²	—	2	2	no	—
498	C₄²	—	2	2	no	—
499	C₄²	—	4	4	no	—
500	C₂³	—	2	2	no	—
501	C₂³	—	2	4	no	—
502	C₂³	—	2	1	no	—
503	C₂³	—	2	1	no	—
504	C₂³	—	4	4	no	—
505	C₂³	—	16	1	no	—
506	C₂³	—	16	2	no	—
507	C₂³	C₄×D₄×C₄₁	1	2	yes	—
508	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
509	Q₈	—	2	2	no	—
510	Q₈	—	2	4	no	—
511	Q₈	—	2	2	no	—
512	Q₈	—	2	1	no	—
513	Q₈	—	2	1	no	—
514	Q₈	—	2	1	no	—
515	Q₈	—	2	1	no	—
516	Q₈	—	4	4	no	—
517	Q₈	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
518	D₄	—	2	4	no	—
519	D₄	—	2	1	no	—
520	D₄	—	2	1	no	—
521	D₄	—	2	2	no	—
522	D₄	—	2	1	no	—
523	D₄	—	2	1	no	—
524	D₄	—	2	4	no	—
525	D₄	—	2	2	no	—
526	D₄	—	4	4	no	—
527	D₄	—	4	4	no	—
528	D₄	—	4	4	no	—
529	D₄	—	4	2	no	—
530	D₄	—	4	2	no	—
531	D₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	4	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	4	4	no	—
540	C₂×C₄	—	4	4	no	—
541	C₂×C₄	—	4	4	no	—
542	C₂×C₄	—	4	2	no	—
543	C₂×C₄	—	4	2	no	—
544	C₂×C₄	—	4	2	no	—
545	C₂×C₄	—	4	2	no	—
546	C₂×C₄	—	16	4	no	—
547	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
548	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
549	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
550	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
551	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
552	C₈	—	2	2	no	—
553	C₈	—	4	2	no	—
554	C₈	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
555	C₈	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
556	C₈	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
557	C₂²	—	2	2	no	—
558	C₂²	—	2	4	no	—
559	C₂²	—	4	1	no	—
560	C₂²	—	4	2	no	—
561	C₂²	—	4	1	no	—
562	C₂²	—	4	1	no	—
563	C₂²	—	8	4	no	—
564	C₂²	—	8	2	no	—
565	C₂²	—	8	1	no	—
566	C₂²	—	8	1	no	—
567	C₂²	—	16	4	no	—
568	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
569	C₂²	D₄²×C₄₁	1	2	yes	—
570	C₄	—	2	4	no	—
571	C₄	—	2	2	no	—
572	C₄	—	4	4	no	—
573	C₄	—	4	1	no	—
574	C₄	—	4	1	no	—
575	C₄	—	8	2	no	—
576	C₄	—	8	2	no	—
577	C₄	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
578	C₄	C₄₁×#518	1	1	yes	characteristic
579	C₄	C₄₁×#546	1	2	yes	—
580	C₂	—	2	2	no	—
581	C₂	—	2	1	no	—
582	C₂	—	4	2	no	—
583	C₂	—	4	4	no	—
584	C₂	—	8	1	no	—
585	C₂	—	8	1	no	—
586	C₂	D₄×C₂²⋊C₄×C₄₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₄₁	D₄×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂×C₄₁×#577	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄₁ ◁ C₂²×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#2949	C₂	semidirect
3. C₄₁×#2884	C₂	semidirect
4. C₄₁×#2883	C₂	semidirect
5. C₄₁×#1976	C₂	semidirect
6. C₄₁×#1905	C₂	semidirect
7. C₄₁×#1904	C₂	semidirect
8. C₄₁×#1812	C₂	semidirect
9. C₄₁×#1796	C₂	semidirect
10. C₄₁×#1795	C₂	semidirect
11. C₄₁×#1757	C₂	semidirect
16. C₂×#137×C₄₁	C₄	semidirect
18. C₂×AGL(1,ℤ/8)×C₄₁	C₄	semidirect
20. C₂×#135×C₄₁	C₄	semidirect
22. C₂²×SD₁₆×C₄₁	C₄	semidirect
25. C₄₁×#519	C₂²	semidirect
29. C₄₁×#427	C₂²	semidirect
31. C₄₁×#421	C₂²	semidirect
32. C₄₁×#419	C₂²	semidirect
40. C₄₁×#410	C₂²	semidirect
42. C₄₁×#405	C₂²	semidirect
43. C₄₁×#344	C₂²	semidirect
44. C₄₁×H(3,ℤ/4)	C₂²	semidirect
53. C₂×#41×C₄₁	C₂×C₄	semidirect
58. C₂²×Q₈×C₄₁	C₂×C₄	semidirect
72. C₂×Q₁₆×C₄₁	C₂×C₄	semidirect
73. C₂×Q₁₆×C₄₁	C₂×C₄	semidirect
79. C₂×SD₁₆×C₄₁	C₂×C₄	semidirect
80. C₂×SD₁₆×C₄₁	C₂×C₄	semidirect
81. C₂×SD₁₆×C₄₁	C₂×C₄	semidirect
85. C₂×D₈×C₄₁	C₂×C₄	semidirect
88. C₂×M₁₆×C₄₁	C₂×C₄	semidirect
91. C₂²×C₈×C₄₁	C₂×C₄	semidirect
263. #22456	C₄₁	direct (N × H)
288. #2949	C₂×C₄₁	semidirect
289. #2884	C₂×C₄₁	semidirect
290. #2883	C₂×C₄₁	semidirect
291. #1976	C₂×C₄₁	semidirect
292. #1905	C₂×C₄₁	semidirect
293. #1904	C₂×C₄₁	semidirect
294. #1812	C₂×C₄₁	semidirect
295. #1796	C₂×C₄₁	semidirect
296. #1795	C₂×C₄₁	semidirect
297. #1757	C₂×C₄₁	semidirect
309. C₂×#137	C₄×C₄₁	semidirect
311. C₂×AGL(1,ℤ/8)	C₄×C₄₁	semidirect
313. C₂×#135	C₄×C₄₁	semidirect
315. C₂²×SD₁₆	C₄×C₄₁	semidirect
318. #519	C₂²×C₄₁	semidirect
322. #427	C₂²×C₄₁	semidirect
324. #421	C₂²×C₄₁	semidirect
325. #419	C₂²×C₄₁	semidirect
333. #410	C₂²×C₄₁	semidirect
335. #405	C₂²×C₄₁	semidirect
336. #344	C₂²×C₄₁	semidirect
337. H(3,ℤ/4)	C₂²×C₄₁	semidirect
338. C₄₁	#22456	direct (N × H)
347. C₂×#41	C₂×C₄×C₄₁	semidirect
352. C₂²×Q₈	C₂×C₄×C₄₁	semidirect
366. C₂×Q₁₆	C₂×C₄×C₄₁	semidirect
367. C₂×Q₁₆	C₂×C₄×C₄₁	semidirect
373. C₂×SD₁₆	C₂×C₄×C₄₁	semidirect
374. C₂×SD₁₆	C₂×C₄×C₄₁	semidirect
375. C₂×SD₁₆	C₂×C₄×C₄₁	semidirect
379. C₂×D₈	C₂×C₄×C₄₁	semidirect
382. C₂×M₁₆	C₂×C₄×C₄₁	semidirect
385. C₂²×C₈	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
C₂×C₄₁×#577	2
C₄₁×#2949	2
C₄₁×#2884	2
C₄₁×#2883	2
C₄₁×#1976	2
C₄₁×#1905	2
C₄₁×#1904	2
C₄₁×#1812	2
C₄₁×#1796	2
C₄₁×#1795	2
C₄₁×#1757	2
#22456	41

Subgroups — C41×#22456 #23798657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#22456 #23798657