Subgroups — C₂×C₄₁×#2318 #23810254

Order 10496.

3262 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₄×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2318	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄₁×#388	C₂	1	2	yes	maximal, complemented
4	C₂×C₄₁×#387	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₁×#386	C₂	1	2	yes	maximal
6	C₂×C₄₁×#385	C₂	1	2	yes	maximal, complemented
7	C₂×C₄₁×#383	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₁×#382	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄₁×#381	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄₁×#380	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×C₂²⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#95×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×D₄×C₄₁	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₄₁	C₂²	1	2	yes	—
15	C₂⁴×C₄×C₄₁	C₂²	1	1	yes	characteristic
16	C₂⁴×C₄×C₄₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₄₁	C₄	1	16	yes	complemented
18	C₂×C₄×D₄×C₄₁	C₂²	1	8	yes	—
19	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
20	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
21	C₂²×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
22	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
23	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
24	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
25	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
26	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
27	C₂²×C₄²×C₄₁	C₂²	1	2	yes	—
28	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
29	C₄₁×#388	C₂²	1	16	yes	complemented
30	C₄₁×#387	C₂²	1	16	yes	complemented
31	C₄₁×#386	C₂²	1	16	yes	—
32	C₄₁×#385	C₂²	1	16	yes	complemented
33	C₄₁×#383	C₂²	1	8	yes	complemented
34	C₄₁×#382	C₂²	1	8	yes	complemented
35	C₄₁×#381	C₂²	1	8	yes	complemented
36	C₄₁×#380	C₂²	1	8	yes	complemented
37	C₄×C₂²⋊C₄×C₄₁	C₂²	1	8	yes	complemented
38	C₂×#95×C₄₁	C₂²	1	1	yes	characteristic
39	C₂×#95×C₄₁	C₂²	1	1	yes	characteristic
40	C₂×#95×C₄₁	C₂²	1	1	yes	characteristic
41	C₂×#95×C₄₁	C₂²	1	1	yes	characteristic
42	C₂×#95×C₄₁	C₂²	1	2	yes	—
43	C₂×#95×C₄₁	C₂²	1	2	yes	—
44	C₂×#95×C₄₁	C₂²	1	2	yes	—
45	C₂×#95×C₄₁	C₂²	1	2	yes	—
46	C₂×#95×C₄₁	C₂²	1	2	yes	—
47	C₂×#95×C₄₁	C₂²	1	8	yes	complemented
48	C₂⁵×C₄₁	C₂³	1	1	yes	characteristic
49	C₂⁵×C₄₁	C₂³	1	1	yes	characteristic
50	C₂²×D₄×C₄₁	—	2	8	no	—
51	C₂²×D₄×C₄₁	C₂×C₄	1	8	yes	—
52	C₂²×D₄×C₄₁	C₂³	1	4	yes	—
53	C₂³×C₄×C₄₁	—	2	8	no	—
54	C₂³×C₄×C₄₁	—	4	4	no	—
55	C₂³×C₄×C₄₁	—	4	4	no	—
56	C₂³×C₄×C₄₁	—	4	4	no	—
57	C₂³×C₄×C₄₁	—	4	4	no	—
58	C₂³×C₄×C₄₁	C₂×C₄	1	8	yes	—
59	C₂³×C₄×C₄₁	C₂×C₄	1	8	yes	complemented
60	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
61	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
62	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
63	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
64	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₄₁	C₂³	1	8	yes	—
70	C₂³×C₄×C₄₁	C₂³	1	4	yes	—
71	C₂³×C₄×C₄₁	C₂³	1	4	yes	—
72	C₄×D₄×C₄₁	C₂×C₄	1	64	yes	complemented
73	C₂×C₄⋊C₄×C₄₁	—	4	4	no	—
74	C₂×C₄⋊C₄×C₄₁	—	4	4	no	—
75	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	8	yes	complemented
76	C₂×C₄⋊C₄×C₄₁	C₂³	1	8	yes	—
77	C₂×C₄⋊C₄×C₄₁	C₂³	1	8	yes	—
78	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
79	C₂×C₂²⋊C₄×C₄₁	—	2	8	no	—
80	C₂×C₂²⋊C₄×C₄₁	—	4	4	no	—
81	C₂×C₂²⋊C₄×C₄₁	—	4	4	no	—
82	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	8	yes	—
83	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	8	yes	complemented
84	C₂×C₂²⋊C₄×C₄₁	C₂³	1	8	yes	—
85	C₂×C₂²⋊C₄×C₄₁	C₂³	1	8	yes	—
86	C₂×C₂²⋊C₄×C₄₁	C₂³	1	8	yes	—
87	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
88	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
89	C₂×C₄²×C₄₁	—	4	4	no	—
90	C₂×C₄²×C₄₁	C₂×C₄	1	8	yes	complemented
91	C₂×C₄²×C₄₁	C₂³	1	8	yes	—
92	C₂×C₄²×C₄₁	C₂³	1	4	yes	—
93	#95×C₄₁	C₂³	1	4	yes	—
94	#95×C₄₁	C₂³	1	4	yes	—
95	#95×C₄₁	C₂³	1	8	yes	—
96	#95×C₄₁	C₂³	1	4	yes	—
97	#95×C₄₁	C₂³	1	4	yes	—
98	#95×C₄₁	C₂³	1	8	yes	—
99	#95×C₄₁	C₂³	1	8	yes	—
100	#95×C₄₁	C₂³	1	8	yes	—
101	#95×C₄₁	C₂³	1	8	yes	—
102	C₂⁴×C₄₁	—	2	4	no	—
103	C₂⁴×C₄₁	—	2	2	no	—
104	C₂⁴×C₄₁	—	2	2	no	—
105	C₂⁴×C₄₁	—	2	4	no	—
106	C₂⁴×C₄₁	—	2	2	no	—
107	C₂⁴×C₄₁	—	2	4	no	—
108	C₂⁴×C₄₁	—	2	2	no	—
109	C₂⁴×C₄₁	C₂²×C₄	1	4	yes	—
110	C₂⁴×C₄₁	C₂²×C₄	1	4	yes	—
111	C₂⁴×C₄₁	#41	1	8	yes	—
112	C₂⁴×C₄₁	C₂⁴	1	2	yes	—
113	C₂⁴×C₄₁	C₂⁴	1	2	yes	—
114	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
115	C₂×D₄×C₄₁	—	2	32	no	—
116	C₂×D₄×C₄₁	—	2	16	no	—
117	C₂×D₄×C₄₁	C₂²×C₄	1	16	yes	—
118	C₂²×C₄×C₄₁	—	2	4	no	—
119	C₂²×C₄×C₄₁	—	2	4	no	—
120	C₂²×C₄×C₄₁	—	2	2	no	—
121	C₂²×C₄×C₄₁	—	2	4	no	—
122	C₂²×C₄×C₄₁	—	2	4	no	—
123	C₂²×C₄×C₄₁	—	2	2	no	—
124	C₂²×C₄×C₄₁	—	2	4	no	—
125	C₂²×C₄×C₄₁	—	2	4	no	—
126	C₂²×C₄×C₄₁	—	2	16	no	—
127	C₂²×C₄×C₄₁	—	2	4	no	—
128	C₂²×C₄×C₄₁	—	2	4	no	—
129	C₂²×C₄×C₄₁	—	2	4	no	—
130	C₂²×C₄×C₄₁	—	2	4	no	—
131	C₂²×C₄×C₄₁	—	2	4	no	—
132	C₂²×C₄×C₄₁	—	2	4	no	—
133	C₂²×C₄×C₄₁	—	2	4	no	—
134	C₂²×C₄×C₄₁	—	2	4	no	—
135	C₂²×C₄×C₄₁	—	2	4	no	—
136	C₂²×C₄×C₄₁	—	2	4	no	—
137	C₂²×C₄×C₄₁	—	4	16	no	—
138	C₂²×C₄×C₄₁	—	4	16	no	—
139	C₂²×C₄×C₄₁	—	4	8	no	—
140	C₂²×C₄×C₄₁	—	4	8	no	—
141	C₂²×C₄×C₄₁	—	4	8	no	—
142	C₂²×C₄×C₄₁	—	4	8	no	—
143	C₂²×C₄×C₄₁	—	4	32	no	—
144	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
145	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
146	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
147	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
148	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
149	C₂²×C₄×C₄₁	C₂²×C₄	1	16	yes	—
150	C₂²×C₄×C₄₁	C₂²×C₄	1	16	yes	complemented
151	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
152	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
153	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
154	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
155	C₂²×C₄×C₄₁	C₂⁴	1	2	yes	—
156	C₂²×C₄×C₄₁	C₂⁴	1	2	yes	—
157	C₂²×C₄×C₄₁	C₂⁴	1	2	yes	—
158	C₂²×C₄×C₄₁	C₂⁴	1	2	yes	—
159	C₂²×C₄×C₄₁	C₂⁴	1	2	yes	—
160	C₄⋊C₄×C₄₁	—	4	8	no	—
161	C₄⋊C₄×C₄₁	—	4	8	no	—
162	C₄⋊C₄×C₄₁	C₂²×C₄	1	16	yes	complemented
163	C₂²⋊C₄×C₄₁	—	2	16	no	—
164	C₂²⋊C₄×C₄₁	—	4	8	no	—
165	C₂²⋊C₄×C₄₁	—	4	8	no	—
166	C₂²⋊C₄×C₄₁	C₂²×C₄	1	16	yes	—
167	C₂²⋊C₄×C₄₁	C₂²×C₄	1	16	yes	complemented
168	C₄²×C₄₁	—	4	8	no	—
169	C₄²×C₄₁	C₂²×C₄	1	16	yes	complemented
170	C₂³×C₄₁	—	2	8	no	—
171	C₂³×C₄₁	—	2	8	no	—
172	C₂³×C₄₁	—	2	16	no	—
173	C₂³×C₄₁	—	2	16	no	—
174	C₂³×C₄₁	—	2	4	no	—
175	C₂³×C₄₁	—	2	2	no	—
176	C₂³×C₄₁	—	2	2	no	—
177	C₂³×C₄₁	—	2	8	no	—
178	C₂³×C₄₁	—	2	8	no	—
179	C₂³×C₄₁	—	2	8	no	—
180	C₂³×C₄₁	—	2	4	no	—
181	C₂³×C₄₁	—	2	2	no	—
182	C₂³×C₄₁	—	2	4	no	—
183	C₂³×C₄₁	—	2	2	no	—
184	C₂³×C₄₁	—	4	8	no	—
185	C₂³×C₄₁	—	4	8	no	—
186	C₂³×C₄₁	#117	1	16	yes	—
187	C₂³×C₄₁	C₂³×C₄	1	4	yes	—
188	C₂³×C₄₁	C₂³×C₄	1	4	yes	—
189	C₂³×C₄₁	C₂³×C₄	1	2	yes	—
190	C₂³×C₄₁	C₂×#41	1	8	yes	—
191	C₂³×C₄₁	C₂×#41	1	4	yes	—
192	C₂³×C₄₁	H(5,𝔽₂)	1	4	yes	—
193	C₂³×C₄₁	H(5,𝔽₂)	1	2	yes	—
194	C₂³×C₄₁	2₋¹⁺⁴	1	2	yes	—
195	C₂³×C₄₁	C₂⁵	1	1	yes	characteristic
196	D₄×C₄₁	—	2	32	no	—
197	C₂×C₄×C₄₁	—	2	8	no	—
198	C₂×C₄×C₄₁	—	2	8	no	—
199	C₂×C₄×C₄₁	—	2	8	no	—
200	C₂×C₄×C₄₁	—	2	8	no	—
201	C₂×C₄×C₄₁	—	2	8	no	—
202	C₂×C₄×C₄₁	—	2	4	no	—
203	C₂×C₄×C₄₁	—	2	4	no	—
204	C₂×C₄×C₄₁	—	2	2	no	—
205	C₂×C₄×C₄₁	—	2	4	no	—
206	C₂×C₄×C₄₁	—	2	4	no	—
207	C₂×C₄×C₄₁	—	2	2	no	—
208	C₂×C₄×C₄₁	—	2	4	no	—
209	C₂×C₄×C₄₁	—	2	4	no	—
210	C₂×C₄×C₄₁	—	2	4	no	—
211	C₂×C₄×C₄₁	—	2	4	no	—
212	C₂×C₄×C₄₁	—	2	4	no	—
213	C₂×C₄×C₄₁	—	2	4	no	—
214	C₂×C₄×C₄₁	—	2	4	no	—
215	C₂×C₄×C₄₁	—	2	4	no	—
216	C₂×C₄×C₄₁	—	2	4	no	—
217	C₂×C₄×C₄₁	—	2	4	no	—
218	C₂×C₄×C₄₁	—	2	4	no	—
219	C₂×C₄×C₄₁	—	2	4	no	—
220	C₂×C₄×C₄₁	—	4	16	no	—
221	C₂×C₄×C₄₁	—	4	16	no	—
222	C₂×C₄×C₄₁	—	4	32	no	—
223	C₂×C₄×C₄₁	—	4	8	no	—
224	C₂×C₄×C₄₁	—	4	8	no	—
225	C₂×C₄×C₄₁	—	4	8	no	—
226	C₂×C₄×C₄₁	—	4	8	no	—
227	C₂×C₄×C₄₁	C₂³×C₄	1	4	yes	—
228	C₂×C₄×C₄₁	C₂³×C₄	1	4	yes	—
229	C₂×C₄×C₄₁	C₂³×C₄	1	4	yes	—
230	C₂×C₄×C₄₁	C₂³×C₄	1	4	yes	—
231	C₂×C₄×C₄₁	C₂³×C₄	1	4	yes	—
232	C₂×#2318	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
233	C₂²×C₄₁	—	2	32	no	—
234	C₂²×C₄₁	—	2	4	no	—
235	C₂²×C₄₁	—	2	4	no	—
236	C₂²×C₄₁	—	2	8	no	—
237	C₂²×C₄₁	—	2	8	no	—
238	C₂²×C₄₁	—	2	4	no	—
239	C₂²×C₄₁	—	2	4	no	—
240	C₂²×C₄₁	—	2	4	no	—
241	C₂²×C₄₁	—	4	16	no	—
242	C₂²×C₄₁	—	4	4	no	—
243	C₂²×C₄₁	—	4	4	no	—
244	C₂²×C₄₁	C₂×#117	1	8	yes	—
245	C₂²×C₄₁	C₂×#117	1	4	yes	—
246	C₂²×C₄₁	#518	1	4	yes	—
247	C₂²×C₄₁	#520	1	4	yes	—
248	C₂²×C₄₁	#538	1	8	yes	—
249	C₂²×C₄₁	#539	1	8	yes	—
250	C₂²×C₄₁	C₂⁴×C₄	1	1	yes	characteristic
251	C₂²×C₄₁	C₂²×#41	1	2	yes	—
252	C₂²×C₄₁	C₂×H(5,𝔽₂)	1	2	yes	—
253	C₂²×C₄₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
254	C₂²×C₄₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
255	C₄×C₄₁	—	2	4	no	—
256	C₄×C₄₁	—	2	4	no	—
257	C₄×C₄₁	—	2	4	no	—
258	C₄×C₄₁	—	2	4	no	—
259	C₄×C₄₁	—	2	4	no	—
260	C₄×C₄₁	—	4	4	no	—
261	C₄×C₄₁	—	4	4	no	—
262	C₄×C₄₁	—	4	4	no	—
263	C₄×C₄₁	—	4	4	no	—
264	C₂²×C₄×D₄	C₂×C₄₁	1	1	yes	characteristic
265	#2318	C₂×C₄₁	1	16	yes	direct product factor
266	C₂×#388	C₂×C₄₁	1	2	yes	complemented
267	C₂×#387	C₂×C₄₁	1	2	yes	complemented
268	C₂×#386	C₂×C₄₁	1	2	yes	—
269	C₂×#385	C₂×C₄₁	1	2	yes	complemented
270	C₂×#383	C₂×C₄₁	1	1	yes	characteristic, complemented
271	C₂×#382	C₂×C₄₁	1	1	yes	characteristic, complemented
272	C₂×#381	C₂×C₄₁	1	1	yes	characteristic, complemented
273	C₂×#380	C₂×C₄₁	1	1	yes	characteristic, complemented
274	C₂×C₄×C₂²⋊C₄	C₂×C₄₁	1	1	yes	characteristic, complemented
275	C₂²×#95	C₂×C₄₁	1	1	yes	characteristic, complemented
276	C₂×C₄₁	—	2	8	no	—
277	C₂×C₄₁	—	4	4	no	—
278	C₂×C₄₁	#2318	1	8	yes	direct product factor
279	C₂×C₄₁	C₂²×#117	1	1	yes	characteristic
280	C₂×C₄₁	C₂×#518	1	1	yes	characteristic
281	C₂×C₄₁	C₂×#520	1	1	yes	characteristic
282	C₂×C₄₁	C₂×#538	1	2	yes	—
283	C₂×C₄₁	C₂×#539	1	2	yes	—
284	C₂³×D₄	C₂²×C₄₁	1	1	yes	characteristic
285	C₂⁴×C₄	C₂²×C₄₁	1	2	yes	—
286	C₂⁴×C₄	C₂²×C₄₁	1	1	yes	characteristic
287	C₂⁴×C₄	C₂²×C₄₁	1	1	yes	characteristic
288	C₂×C₄×D₄	C₄×C₄₁	1	16	yes	complemented
289	C₂×C₄×D₄	C₂²×C₄₁	1	8	yes	—
290	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
291	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
292	C₂²×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
293	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
294	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
295	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
296	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
297	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
298	C₂²×C₄²	C₂²×C₄₁	1	2	yes	—
299	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
300	#388	C₂²×C₄₁	1	16	yes	complemented
301	#387	C₂²×C₄₁	1	16	yes	complemented
302	#386	C₂²×C₄₁	1	16	yes	—
303	#385	C₂²×C₄₁	1	16	yes	complemented
304	#383	C₂²×C₄₁	1	8	yes	complemented
305	#382	C₂²×C₄₁	1	8	yes	complemented
306	#381	C₂²×C₄₁	1	8	yes	complemented
307	#380	C₂²×C₄₁	1	8	yes	complemented
308	C₄×C₂²⋊C₄	C₂²×C₄₁	1	8	yes	complemented
309	C₂×#95	C₂²×C₄₁	1	1	yes	characteristic
310	C₂×#95	C₂²×C₄₁	1	1	yes	characteristic
311	C₂×#95	C₂²×C₄₁	1	2	yes	—
312	C₂×#95	C₂²×C₄₁	1	1	yes	characteristic
313	C₂×#95	C₂²×C₄₁	1	1	yes	characteristic
314	C₂×#95	C₂²×C₄₁	1	2	yes	—
315	C₂×#95	C₂²×C₄₁	1	2	yes	—
316	C₂×#95	C₂²×C₄₁	1	2	yes	—
317	C₂×#95	C₂²×C₄₁	1	2	yes	—
318	C₂×#95	C₂²×C₄₁	1	8	yes	complemented
319	C₄₁	C₂×#2318	1	1	yes	characteristic, direct product factor, Sylow
320	C₂⁵	C₂³×C₄₁	1	1	yes	characteristic
321	C₂⁵	C₂³×C₄₁	1	1	yes	characteristic
322	C₂²×D₄	—	2	8	no	—
323	C₂²×D₄	C₂×C₄×C₄₁	1	8	yes	—
324	C₂²×D₄	C₂³×C₄₁	1	4	yes	—
325	C₂³×C₄	—	2	8	no	—
326	C₂³×C₄	—	4	4	no	—
327	C₂³×C₄	—	4	4	no	—
328	C₂³×C₄	—	4	4	no	—
329	C₂³×C₄	—	4	4	no	—
330	C₂³×C₄	C₂×C₄×C₄₁	1	8	yes	—
331	C₂³×C₄	C₂×C₄×C₄₁	1	8	yes	complemented
332	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
333	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
334	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
335	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
336	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
337	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
338	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
339	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
340	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
341	C₂³×C₄	C₂³×C₄₁	1	8	yes	—
342	C₂³×C₄	C₂³×C₄₁	1	4	yes	—
343	C₂³×C₄	C₂³×C₄₁	1	4	yes	—
344	C₄×D₄	C₂×C₄×C₄₁	1	64	yes	complemented
345	C₂×C₄⋊C₄	—	4	4	no	—
346	C₂×C₄⋊C₄	—	4	4	no	—
347	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	8	yes	complemented
348	C₂×C₄⋊C₄	C₂³×C₄₁	1	8	yes	—
349	C₂×C₄⋊C₄	C₂³×C₄₁	1	8	yes	—
350	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
351	C₂×C₂²⋊C₄	—	2	8	no	—
352	C₂×C₂²⋊C₄	—	4	4	no	—
353	C₂×C₂²⋊C₄	—	4	4	no	—
354	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	8	yes	—
355	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	8	yes	complemented
356	C₂×C₂²⋊C₄	C₂³×C₄₁	1	8	yes	—
357	C₂×C₂²⋊C₄	C₂³×C₄₁	1	8	yes	—
358	C₂×C₂²⋊C₄	C₂³×C₄₁	1	8	yes	—
359	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
360	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
361	C₂×C₄²	—	4	4	no	—
362	C₂×C₄²	C₂×C₄×C₄₁	1	8	yes	complemented
363	C₂×C₄²	C₂³×C₄₁	1	8	yes	—
364	C₂×C₄²	C₂³×C₄₁	1	4	yes	—
365	#95	C₂³×C₄₁	1	4	yes	—
366	#95	C₂³×C₄₁	1	4	yes	—
367	#95	C₂³×C₄₁	1	8	yes	—
368	#95	C₂³×C₄₁	1	4	yes	—
369	#95	C₂³×C₄₁	1	4	yes	—
370	#95	C₂³×C₄₁	1	8	yes	—
371	#95	C₂³×C₄₁	1	8	yes	—
372	#95	C₂³×C₄₁	1	8	yes	—
373	#95	C₂³×C₄₁	1	8	yes	—
374	C₂⁴	—	2	4	no	—
375	C₂⁴	—	2	2	no	—
376	C₂⁴	—	2	2	no	—
377	C₂⁴	—	2	4	no	—
378	C₂⁴	—	2	2	no	—
379	C₂⁴	—	2	4	no	—
380	C₂⁴	—	2	2	no	—
381	C₂⁴	C₂²×C₄×C₄₁	1	4	yes	—
382	C₂⁴	C₂²×C₄×C₄₁	1	4	yes	—
383	C₂⁴	#41×C₄₁	1	8	yes	—
384	C₂⁴	C₂⁴×C₄₁	1	2	yes	—
385	C₂⁴	C₂⁴×C₄₁	1	2	yes	—
386	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
387	C₂×D₄	—	2	32	no	—
388	C₂×D₄	—	2	16	no	—
389	C₂×D₄	C₂²×C₄×C₄₁	1	16	yes	—
390	C₂²×C₄	—	2	4	no	—
391	C₂²×C₄	—	2	4	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	2	4	no	—
394	C₂²×C₄	—	2	4	no	—
395	C₂²×C₄	—	2	2	no	—
396	C₂²×C₄	—	2	4	no	—
397	C₂²×C₄	—	2	4	no	—
398	C₂²×C₄	—	2	16	no	—
399	C₂²×C₄	—	2	4	no	—
400	C₂²×C₄	—	2	4	no	—
401	C₂²×C₄	—	2	4	no	—
402	C₂²×C₄	—	2	4	no	—
403	C₂²×C₄	—	2	4	no	—
404	C₂²×C₄	—	2	4	no	—
405	C₂²×C₄	—	2	4	no	—
406	C₂²×C₄	—	2	4	no	—
407	C₂²×C₄	—	2	4	no	—
408	C₂²×C₄	—	2	4	no	—
409	C₂²×C₄	—	4	16	no	—
410	C₂²×C₄	—	4	16	no	—
411	C₂²×C₄	—	4	8	no	—
412	C₂²×C₄	—	4	8	no	—
413	C₂²×C₄	—	4	8	no	—
414	C₂²×C₄	—	4	8	no	—
415	C₂²×C₄	—	4	32	no	—
416	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
417	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
418	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
419	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
420	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
421	C₂²×C₄	C₂²×C₄×C₄₁	1	16	yes	—
422	C₂²×C₄	C₂²×C₄×C₄₁	1	16	yes	complemented
423	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
424	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
425	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
426	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
427	C₂²×C₄	C₂⁴×C₄₁	1	2	yes	—
428	C₂²×C₄	C₂⁴×C₄₁	1	2	yes	—
429	C₂²×C₄	C₂⁴×C₄₁	1	2	yes	—
430	C₂²×C₄	C₂⁴×C₄₁	1	2	yes	—
431	C₂²×C₄	C₂⁴×C₄₁	1	2	yes	—
432	C₄⋊C₄	—	4	8	no	—
433	C₄⋊C₄	—	4	8	no	—
434	C₄⋊C₄	C₂²×C₄×C₄₁	1	16	yes	complemented
435	C₂²⋊C₄	—	2	16	no	—
436	C₂²⋊C₄	—	4	8	no	—
437	C₂²⋊C₄	—	4	8	no	—
438	C₂²⋊C₄	C₂²×C₄×C₄₁	1	16	yes	—
439	C₂²⋊C₄	C₂²×C₄×C₄₁	1	16	yes	complemented
440	C₄²	—	4	8	no	—
441	C₄²	C₂²×C₄×C₄₁	1	16	yes	complemented
442	C₂³	—	2	8	no	—
443	C₂³	—	2	8	no	—
444	C₂³	—	2	16	no	—
445	C₂³	—	2	16	no	—
446	C₂³	—	2	4	no	—
447	C₂³	—	2	2	no	—
448	C₂³	—	2	2	no	—
449	C₂³	—	2	8	no	—
450	C₂³	—	2	8	no	—
451	C₂³	—	2	8	no	—
452	C₂³	—	2	4	no	—
453	C₂³	—	2	2	no	—
454	C₂³	—	2	4	no	—
455	C₂³	—	2	2	no	—
456	C₂³	—	4	8	no	—
457	C₂³	—	4	8	no	—
458	C₂³	#117×C₄₁	1	16	yes	—
459	C₂³	C₂³×C₄×C₄₁	1	4	yes	—
460	C₂³	C₂³×C₄×C₄₁	1	4	yes	—
461	C₂³	C₂³×C₄×C₄₁	1	2	yes	—
462	C₂³	C₂×#41×C₄₁	1	8	yes	—
463	C₂³	C₂×#41×C₄₁	1	4	yes	—
464	C₂³	H(5,𝔽₂)×C₄₁	1	4	yes	—
465	C₂³	H(5,𝔽₂)×C₄₁	1	2	yes	—
466	C₂³	2₋¹⁺⁴×C₄₁	1	2	yes	—
467	C₂³	C₂⁵×C₄₁	1	1	yes	characteristic
468	D₄	—	2	32	no	—
469	C₂×C₄	—	2	8	no	—
470	C₂×C₄	—	2	8	no	—
471	C₂×C₄	—	2	8	no	—
472	C₂×C₄	—	2	8	no	—
473	C₂×C₄	—	2	8	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	4	no	—
478	C₂×C₄	—	2	4	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	4	no	—
481	C₂×C₄	—	2	4	no	—
482	C₂×C₄	—	2	4	no	—
483	C₂×C₄	—	2	4	no	—
484	C₂×C₄	—	2	4	no	—
485	C₂×C₄	—	2	4	no	—
486	C₂×C₄	—	2	4	no	—
487	C₂×C₄	—	2	4	no	—
488	C₂×C₄	—	2	4	no	—
489	C₂×C₄	—	2	4	no	—
490	C₂×C₄	—	2	4	no	—
491	C₂×C₄	—	2	4	no	—
492	C₂×C₄	—	4	16	no	—
493	C₂×C₄	—	4	16	no	—
494	C₂×C₄	—	4	32	no	—
495	C₂×C₄	—	4	8	no	—
496	C₂×C₄	—	4	8	no	—
497	C₂×C₄	—	4	8	no	—
498	C₂×C₄	—	4	8	no	—
499	C₂×C₄	C₂³×C₄×C₄₁	1	4	yes	—
500	C₂×C₄	C₂³×C₄×C₄₁	1	4	yes	—
501	C₂×C₄	C₂³×C₄×C₄₁	1	4	yes	—
502	C₂×C₄	C₂³×C₄×C₄₁	1	4	yes	—
503	C₂×C₄	C₂³×C₄×C₄₁	1	4	yes	—
504	C₂²	—	2	32	no	—
505	C₂²	—	2	4	no	—
506	C₂²	—	2	4	no	—
507	C₂²	—	2	8	no	—
508	C₂²	—	2	8	no	—
509	C₂²	—	2	4	no	—
510	C₂²	—	2	4	no	—
511	C₂²	—	2	4	no	—
512	C₂²	—	4	16	no	—
513	C₂²	—	4	4	no	—
514	C₂²	—	4	4	no	—
515	C₂²	C₂×#117×C₄₁	1	8	yes	—
516	C₂²	C₂×#117×C₄₁	1	4	yes	—
517	C₂²	C₄₁×#518	1	4	yes	—
518	C₂²	C₄₁×#520	1	4	yes	—
519	C₂²	C₄₁×#538	1	8	yes	—
520	C₂²	C₄₁×#539	1	8	yes	—
521	C₂²	C₂⁴×C₄×C₄₁	1	1	yes	characteristic
522	C₂²	C₂²×#41×C₄₁	1	2	yes	—
523	C₂²	C₂×H(5,𝔽₂)×C₄₁	1	2	yes	—
524	C₂²	C₂×H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
525	C₂²	C₂×2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
526	C₄	—	2	4	no	—
527	C₄	—	2	4	no	—
528	C₄	—	2	4	no	—
529	C₄	—	2	4	no	—
530	C₄	—	2	4	no	—
531	C₄	—	4	4	no	—
532	C₄	—	4	4	no	—
533	C₄	—	4	4	no	—
534	C₄	—	4	4	no	—
535	C₂	—	2	8	no	—
536	C₂	—	4	4	no	—
537	C₂	C₄₁×#2318	1	8	yes	direct product factor
538	C₂	C₂²×#117×C₄₁	1	1	yes	characteristic
539	C₂	C₂×C₄₁×#518	1	1	yes	characteristic
540	C₂	C₂×C₄₁×#520	1	1	yes	characteristic
541	C₂	C₂×C₄₁×#538	1	2	yes	—
542	C₂	C₂×C₄₁×#539	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₄₁	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₄₁
Frattini, cosocle	C₂³	C₂⁵×C₄₁
prime radical	C₂³×D₄×C₄₁	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#2318	C₂	direct (N × H)
3. C₂×C₄₁×#388	C₂	semidirect
4. C₂×C₄₁×#387	C₂	semidirect
6. C₂×C₄₁×#385	C₂	semidirect
7. C₂×C₄₁×#383	C₂	semidirect
8. C₂×C₄₁×#382	C₂	semidirect
9. C₂×C₄₁×#381	C₂	semidirect
10. C₂×C₄₁×#380	C₂	semidirect
11. C₂×C₄×C₂²⋊C₄×C₄₁	C₂	semidirect
12. C₂²×#95×C₄₁	C₂	semidirect
17. C₂×C₄×D₄×C₄₁	C₄	semidirect
29. C₄₁×#388	C₂²	semidirect
30. C₄₁×#387	C₂²	semidirect
32. C₄₁×#385	C₂²	semidirect
33. C₄₁×#383	C₂²	semidirect
34. C₄₁×#382	C₂²	semidirect
35. C₄₁×#381	C₂²	semidirect
36. C₄₁×#380	C₂²	semidirect
37. C₄×C₂²⋊C₄×C₄₁	C₂²	semidirect
47. C₂×#95×C₄₁	C₂²	semidirect
59. C₂³×C₄×C₄₁	C₂×C₄	semidirect
72. C₄×D₄×C₄₁	C₂×C₄	semidirect
75. C₂×C₄⋊C₄×C₄₁	C₂×C₄	semidirect
83. C₂×C₂²⋊C₄×C₄₁	C₂×C₄	semidirect
90. C₂×C₄²×C₄₁	C₂×C₄	semidirect
150. C₂²×C₄×C₄₁	C₂²×C₄	semidirect
162. C₄⋊C₄×C₄₁	C₂²×C₄	semidirect
167. C₂²⋊C₄×C₄₁	C₂²×C₄	semidirect
169. C₄²×C₄₁	C₂²×C₄	semidirect
232. C₂×#2318	C₄₁	direct (N × H)
265. #2318	C₂×C₄₁	direct (N × H)
266. C₂×#388	C₂×C₄₁	semidirect
267. C₂×#387	C₂×C₄₁	semidirect
269. C₂×#385	C₂×C₄₁	semidirect
270. C₂×#383	C₂×C₄₁	semidirect
271. C₂×#382	C₂×C₄₁	semidirect
272. C₂×#381	C₂×C₄₁	semidirect
273. C₂×#380	C₂×C₄₁	semidirect
274. C₂×C₄×C₂²⋊C₄	C₂×C₄₁	semidirect
275. C₂²×#95	C₂×C₄₁	semidirect
278. C₂×C₄₁	#2318	direct (N × H)
288. C₂×C₄×D₄	C₄×C₄₁	semidirect
300. #388	C₂²×C₄₁	semidirect
301. #387	C₂²×C₄₁	semidirect
303. #385	C₂²×C₄₁	semidirect
304. #383	C₂²×C₄₁	semidirect
305. #382	C₂²×C₄₁	semidirect
306. #381	C₂²×C₄₁	semidirect
307. #380	C₂²×C₄₁	semidirect
308. C₄×C₂²⋊C₄	C₂²×C₄₁	semidirect
318. C₂×#95	C₂²×C₄₁	semidirect
319. C₄₁	C₂×#2318	direct (N × H)
331. C₂³×C₄	C₂×C₄×C₄₁	semidirect
344. C₄×D₄	C₂×C₄×C₄₁	semidirect
347. C₂×C₄⋊C₄	C₂×C₄×C₄₁	semidirect
355. C₂×C₂²⋊C₄	C₂×C₄×C₄₁	semidirect
362. C₂×C₄²	C₂×C₄×C₄₁	semidirect
422. C₂²×C₄	C₂²×C₄×C₄₁	semidirect
434. C₄⋊C₄	C₂²×C₄×C₄₁	semidirect
439. C₂²⋊C₄	C₂²×C₄×C₄₁	semidirect
441. C₄²	C₂²×C₄×C₄₁	semidirect
537. C₂	C₄₁×#2318	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₄×C₄₁	2
C₄₁×#2318	2
C₂×C₄₁×#388	2
C₂×C₄₁×#387	2
C₂×C₄₁×#386	2
C₂×C₄₁×#385	2
C₂×C₄₁×#383	2
C₂×C₄₁×#382	2
C₂×C₄₁×#381	2
C₂×C₄₁×#380	2
C₂×C₄×C₂²⋊C₄×C₄₁	2
C₂²×#95×C₄₁	2
C₂×#2318	41

Subgroups — C2×C41×#2318 #23810254

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#2318 #23810254