Subgroups — C₄₁×#34190 #23810391

Order 10496.

2914 conjugacy classes of subgroups.

The subgroup lattice has 572 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×#2357	C₂	1	2	yes	maximal, complemented
3	C₄₁×#2353	C₂	1	4	yes	maximal, complemented
4	C₄₁×#2349	C₂	1	2	yes	maximal, complemented
5	C₄₁×#2344	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2342	C₂	1	4	yes	maximal, complemented
7	D₄×C₂²⋊C₄×C₄₁	C₂	1	2	yes	maximal
8	C₄₁×#2319	C₂	1	2	yes	maximal, complemented
9	C₄₁×#2318	C₂	1	4	yes	maximal, complemented
10	C₄₁×#2283	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₁×#2283	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄₁×#2280	C₂	1	2	yes	maximal, complemented
13	C₂×C₄₁×#381	C₂	1	2	yes	maximal, complemented
14	C₄₁×#2275	C₂	1	2	yes	maximal, complemented
15	C₂³×D₄×C₄₁	C₂²	1	1	yes	characteristic
16	C₂⁴×C₄×C₄₁	C₂²	1	2	yes	—
17	C₂×C₄×D₄×C₄₁	C₄	1	16	yes	complemented
18	C₂×C₄×D₄×C₄₁	C₂²	1	4	yes	—
19	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₄₁	C₂²	1	2	yes	—
22	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₄₁	C₂²	1	2	yes	—
24	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
25	C₂×#117×C₄₁	C₂²	1	4	yes	—
26	C₂²×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
27	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	4	yes	—
28	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
29	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
30	C₄₁×#390	C₂²	1	4	yes	—
31	C₄₁×#389	C₂²	1	4	yes	complemented
32	C₄₁×#388	C₂²	1	8	yes	complemented
33	C₄₁×#388	C₂²	1	8	yes	complemented
34	C₄₁×#387	C₂²	1	8	yes	complemented
35	C₄₁×#386	C₂²	1	8	yes	—
36	C₄₁×#385	C₂²	1	8	yes	complemented
37	C₄₁×#383	C₂²	1	2	yes	complemented
38	C₄₁×#383	C₂²	1	2	yes	complemented
39	C₄₁×#382	C₂²	1	2	yes	complemented
40	C₄₁×#382	C₂²	1	2	yes	complemented
41	C₄₁×#381	C₂²	1	1	yes	characteristic
42	C₄₁×#381	C₂²	1	2	yes	—
43	C₄₁×#381	C₂²	1	4	yes	—
44	C₄₁×#381	C₂²	1	1	yes	characteristic
45	C₄₁×#381	C₂²	1	4	yes	complemented
46	C₄₁×#381	C₂²	1	4	yes	complemented
47	C₄₁×#380	C₂²	1	2	yes	—
48	C₄₁×#380	C₂²	1	2	yes	—
49	C₄₁×#380	C₂²	1	4	yes	complemented
50	C₄₁×#380	C₂²	1	4	yes	complemented
51	C₄₁×#379	C₂²	1	4	yes	—
52	C₄×C₄⋊C₄×C₄₁	C₂²	1	2	yes	complemented
53	C₄×C₄⋊C₄×C₄₁	C₂²	1	4	yes	complemented
54	C₄×C₄⋊C₄×C₄₁	C₂²	1	2	yes	complemented
55	C₄×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
56	C₄×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
57	C₄×C₂²⋊C₄×C₄₁	C₂²	1	4	yes	complemented
58	C₄×C₂²⋊C₄×C₄₁	C₂²	1	8	yes	complemented
59	C₄×C₂²⋊C₄×C₄₁	C₂²	1	4	yes	complemented
60	C₄₁×#376	C₂²	1	2	yes	complemented
61	C₄₁×#376	C₂²	1	2	yes	complemented
62	C₂×#95×C₄₁	C₂²	1	4	yes	complemented
63	C₂×#95×C₄₁	C₂²	1	4	yes	complemented
64	C₂⁵×C₄₁	C₂³	1	2	yes	—
65	C₂²×D₄×C₄₁	—	2	8	no	—
66	C₂²×D₄×C₄₁	C₂×C₄	1	8	yes	—
67	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
68	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
69	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₄₁	—	4	8	no	—
71	C₂³×C₄×C₄₁	C₂×C₄	1	16	yes	complemented
72	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
75	C₂³×C₄×C₄₁	C₂³	1	8	yes	—
76	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
77	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
78	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
79	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
80	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
81	C₄×D₄×C₄₁	—	2	16	no	—
82	C₄×D₄×C₄₁	C₂×C₄	1	8	yes	—
83	C₄×D₄×C₄₁	C₂×C₄	1	8	yes	—
84	C₄×D₄×C₄₁	C₂×C₄	1	8	yes	complemented
85	C₄×D₄×C₄₁	C₂×C₄	1	8	yes	complemented
86	C₄×D₄×C₄₁	C₂×C₄	1	16	yes	complemented
87	C₄×D₄×C₄₁	C₂×C₄	1	16	yes	complemented
88	#117×C₄₁	—	4	8	no	—
89	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	8	yes	complemented
90	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
91	C₂×C₄⋊C₄×C₄₁	C₂³	1	8	yes	—
92	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
93	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
95	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
96	C₂×C₂²⋊C₄×C₄₁	—	4	8	no	—
97	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	16	yes	complemented
98	C₂×C₂²⋊C₄×C₄₁	C₂³	1	2	yes	—
99	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
100	C₂×C₂²⋊C₄×C₄₁	C₂³	1	2	yes	—
101	C₂×C₂²⋊C₄×C₄₁	C₂³	1	8	yes	—
102	C₂×C₂²⋊C₄×C₄₁	C₂³	1	8	yes	—
103	C₂×C₂²⋊C₄×C₄₁	C₂³	1	8	yes	—
104	C₂×C₂²⋊C₄×C₄₁	C₂³	1	2	yes	—
105	C₂×C₂²⋊C₄×C₄₁	C₂³	1	2	yes	—
106	C₂×C₂²⋊C₄×C₄₁	C₂³	1	2	yes	—
107	C₂×C₂²⋊C₄×C₄₁	C₂³	1	2	yes	—
108	C₂×C₄²×C₄₁	C₂×C₄	1	8	yes	complemented
109	C₂×C₄²×C₄₁	C₂³	1	4	yes	—
110	C₂×C₄²×C₄₁	C₂³	1	8	yes	—
111	C₂×C₄²×C₄₁	C₂³	1	4	yes	—
112	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
113	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
114	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
116	#95×C₄₁	C₂³	1	2	yes	—
117	#95×C₄₁	C₂³	1	2	yes	—
118	#95×C₄₁	C₂³	1	4	yes	—
119	#95×C₄₁	C₂³	1	4	yes	—
120	#95×C₄₁	C₂³	1	8	yes	—
121	#95×C₄₁	C₂³	1	8	yes	—
122	#95×C₄₁	C₂³	1	4	yes	—
123	#95×C₄₁	C₂³	1	2	yes	—
124	#95×C₄₁	C₂³	1	4	yes	—
125	#95×C₄₁	C₂³	1	4	yes	—
126	#95×C₄₁	C₂³	1	4	yes	—
127	#95×C₄₁	C₂³	1	2	yes	—
128	C₂⁴×C₄₁	—	2	8	no	—
129	C₂⁴×C₄₁	—	2	8	no	—
130	C₂⁴×C₄₁	—	4	4	no	—
131	C₂⁴×C₄₁	C₂²×C₄	1	8	yes	—
132	C₂⁴×C₄₁	C₂⁴	1	2	yes	—
133	C₂⁴×C₄₁	C₂⁴	1	2	yes	—
134	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
135	C₂×D₄×C₄₁	—	2	8	no	—
136	C₂×D₄×C₄₁	—	2	32	no	—
137	C₂×D₄×C₄₁	—	2	4	no	—
138	C₂×D₄×C₄₁	C₂²×C₄	1	8	yes	—
139	C₂×D₄×C₄₁	C₂²×C₄	1	4	yes	—
140	C₂×D₄×C₄₁	C₂²×C₄	1	4	yes	—
141	C₂×D₄×C₄₁	#41	1	8	yes	—
142	C₂²×C₄×C₄₁	—	2	8	no	—
143	C₂²×C₄×C₄₁	—	2	4	no	—
144	C₂²×C₄×C₄₁	—	2	4	no	—
145	C₂²×C₄×C₄₁	—	2	16	no	—
146	C₂²×C₄×C₄₁	—	4	32	no	—
147	C₂²×C₄×C₄₁	—	4	4	no	—
148	C₂²×C₄×C₄₁	—	4	4	no	—
149	C₂²×C₄×C₄₁	—	4	4	no	—
150	C₂²×C₄×C₄₁	—	4	4	no	—
151	C₂²×C₄×C₄₁	—	4	2	no	—
152	C₂²×C₄×C₄₁	—	4	8	no	—
153	C₂²×C₄×C₄₁	—	4	8	no	—
154	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
155	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
156	C₂²×C₄×C₄₁	C₂²×C₄	1	8	yes	—
157	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
158	C₂²×C₄×C₄₁	C₂²×C₄	1	8	yes	—
159	C₂²×C₄×C₄₁	C₂²×C₄	1	8	yes	—
160	C₂²×C₄×C₄₁	C₂²×C₄	1	8	yes	—
161	C₂²×C₄×C₄₁	C₂²×C₄	1	8	yes	—
162	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
163	C₂²×C₄×C₄₁	C₂⁴	1	8	yes	—
164	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
165	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
166	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
167	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
168	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
169	C₂²×C₄×C₄₁	C₂⁴	1	2	yes	—
170	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
171	C₂²×C₄×C₄₁	C₂⁴	1	2	yes	—
172	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
173	C₄⋊C₄×C₄₁	—	2	8	no	—
174	C₄⋊C₄×C₄₁	—	4	4	no	—
175	C₄⋊C₄×C₄₁	—	4	8	no	—
176	C₄⋊C₄×C₄₁	C₂²×C₄	1	4	yes	—
177	C₄⋊C₄×C₄₁	C₂²×C₄	1	4	yes	—
178	C₄⋊C₄×C₄₁	C₂²×C₄	1	8	yes	—
179	C₂²⋊C₄×C₄₁	—	2	16	no	—
180	C₂²⋊C₄×C₄₁	—	2	4	no	—
181	C₂²⋊C₄×C₄₁	—	2	8	no	—
182	C₂²⋊C₄×C₄₁	—	2	4	no	—
183	C₂²⋊C₄×C₄₁	—	4	8	no	—
184	C₂²⋊C₄×C₄₁	—	4	8	no	—
185	C₂²⋊C₄×C₄₁	C₂²×C₄	1	8	yes	—
186	C₂²⋊C₄×C₄₁	C₂²×C₄	1	8	yes	—
187	C₂²⋊C₄×C₄₁	C₂²×C₄	1	8	yes	—
188	C₂²⋊C₄×C₄₁	C₂²×C₄	1	8	yes	—
189	C₄²×C₄₁	—	2	8	no	—
190	C₄²×C₄₁	—	4	4	no	—
191	C₄²×C₄₁	—	4	8	no	—
192	C₄²×C₄₁	C₂²×C₄	1	4	yes	—
193	C₄²×C₄₁	C₂²×C₄	1	4	yes	—
194	C₄²×C₄₁	C₂²×C₄	1	4	yes	—
195	C₄²×C₄₁	C₂²×C₄	1	4	yes	—
196	C₂³×C₄₁	—	2	2	no	—
197	C₂³×C₄₁	—	2	4	no	—
198	C₂³×C₄₁	—	2	4	no	—
199	C₂³×C₄₁	—	2	2	no	—
200	C₂³×C₄₁	—	2	16	no	—
201	C₂³×C₄₁	—	2	16	no	—
202	C₂³×C₄₁	—	2	4	no	—
203	C₂³×C₄₁	—	2	4	no	—
204	C₂³×C₄₁	—	2	4	no	—
205	C₂³×C₄₁	—	4	8	no	—
206	C₂³×C₄₁	—	4	32	no	—
207	C₂³×C₄₁	C₂³×C₄	1	4	yes	—
208	C₂³×C₄₁	C₂³×C₄	1	4	yes	—
209	C₂³×C₄₁	C₂³×C₄	1	2	yes	—
210	C₂³×C₄₁	C₂×#41	1	8	yes	—
211	C₂³×C₄₁	H(5,𝔽₂)	1	4	yes	—
212	C₂³×C₄₁	C₂⁵	1	1	yes	characteristic
213	D₄×C₄₁	—	2	16	no	—
214	D₄×C₄₁	—	2	8	no	—
215	D₄×C₄₁	#117	1	16	yes	complemented
216	C₂×C₄×C₄₁	—	2	8	no	—
217	C₂×C₄×C₄₁	—	2	16	no	—
218	C₂×C₄×C₄₁	—	2	2	no	—
219	C₂×C₄×C₄₁	—	2	4	no	—
220	C₂×C₄×C₄₁	—	2	4	no	—
221	C₂×C₄×C₄₁	—	2	2	no	—
222	C₂×C₄×C₄₁	—	2	2	no	—
223	C₂×C₄×C₄₁	—	2	2	no	—
224	C₂×C₄×C₄₁	—	2	2	no	—
225	C₂×C₄×C₄₁	—	2	8	no	—
226	C₂×C₄×C₄₁	—	2	4	no	—
227	C₂×C₄×C₄₁	—	2	4	no	—
228	C₂×C₄×C₄₁	—	2	8	no	—
229	C₂×C₄×C₄₁	—	2	4	no	—
230	C₂×C₄×C₄₁	—	2	8	no	—
231	C₂×C₄×C₄₁	—	2	4	no	—
232	C₂×C₄×C₄₁	—	2	8	no	—
233	C₂×C₄×C₄₁	—	2	4	no	—
234	C₂×C₄×C₄₁	—	2	4	no	—
235	C₂×C₄×C₄₁	—	4	8	no	—
236	C₂×C₄×C₄₁	—	4	8	no	—
237	C₂×C₄×C₄₁	—	4	8	no	—
238	C₂×C₄×C₄₁	—	4	8	no	—
239	C₂×C₄×C₄₁	—	4	4	no	—
240	C₂×C₄×C₄₁	—	8	16	no	—
241	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
242	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
243	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
244	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
245	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
246	C₂×C₄×C₄₁	C₂³×C₄	1	4	yes	—
247	C₂×C₄×C₄₁	C₂³×C₄	1	4	yes	—
248	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
249	C₂×C₄×C₄₁	C₂×#41	1	4	yes	—
250	#34190	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
251	C₂²×C₄₁	—	2	2	no	—
252	C₂²×C₄₁	—	2	4	no	—
253	C₂²×C₄₁	—	2	2	no	—
254	C₂²×C₄₁	—	2	4	no	—
255	C₂²×C₄₁	—	2	16	no	—
256	C₂²×C₄₁	—	2	8	no	—
257	C₂²×C₄₁	—	2	4	no	—
258	C₂²×C₄₁	—	2	4	no	—
259	C₂²×C₄₁	—	2	4	no	—
260	C₂²×C₄₁	—	4	32	no	—
261	C₂²×C₄₁	—	4	8	no	—
262	C₂²×C₄₁	C₂×#117	1	8	yes	—
263	C₂²×C₄₁	#518	1	4	yes	—
264	C₂²×C₄₁	C₂⁴×C₄	1	1	yes	characteristic
265	C₂²×C₄₁	C₂²×#41	1	2	yes	—
266	C₂²×C₄₁	C₂×H(5,𝔽₂)	1	2	yes	—
267	C₂²×C₄₁	#585	1	2	yes	—
268	C₄×C₄₁	—	2	2	no	—
269	C₄×C₄₁	—	2	2	no	—
270	C₄×C₄₁	—	2	2	no	—
271	C₄×C₄₁	—	2	4	no	—
272	C₄×C₄₁	—	2	4	no	—
273	C₄×C₄₁	—	2	2	no	—
274	C₄×C₄₁	—	2	2	no	—
275	C₄×C₄₁	—	4	4	no	—
276	C₄×C₄₁	—	4	8	no	—
277	C₄×C₄₁	—	4	4	no	—
278	C₄×C₄₁	C₂×#117	1	4	yes	—
279	C₂²×C₄×D₄	C₂×C₄₁	1	1	yes	characteristic
280	#2357	C₂×C₄₁	1	2	yes	complemented
281	#2353	C₂×C₄₁	1	4	yes	complemented
282	#2349	C₂×C₄₁	1	2	yes	complemented
283	#2344	C₂×C₄₁	1	2	yes	complemented
284	#2342	C₂×C₄₁	1	4	yes	complemented
285	D₄×C₂²⋊C₄	C₂×C₄₁	1	2	yes	—
286	#2319	C₂×C₄₁	1	2	yes	complemented
287	#2318	C₂×C₄₁	1	4	yes	complemented
288	#2283	C₂×C₄₁	1	1	yes	characteristic, complemented
289	#2283	C₂×C₄₁	1	1	yes	characteristic, complemented
290	#2280	C₂×C₄₁	1	2	yes	complemented
291	C₂×#381	C₂×C₄₁	1	2	yes	complemented
292	#2275	C₂×C₄₁	1	2	yes	complemented
293	C₂×C₄₁	—	2	4	no	—
294	C₂×C₄₁	—	2	8	no	—
295	C₂×C₄₁	—	4	4	no	—
296	C₂×C₄₁	C₂²×#117	1	1	yes	characteristic
297	C₂×C₄₁	C₂×#518	1	1	yes	characteristic
298	C₂×C₄₁	#3428	1	1	yes	characteristic
299	C₂×C₄₁	#3513	1	4	yes	—
300	C₂³×D₄	C₂²×C₄₁	1	1	yes	characteristic
301	C₂⁴×C₄	C₂²×C₄₁	1	2	yes	—
302	C₂×C₄×D₄	C₄×C₄₁	1	16	yes	complemented
303	C₂×C₄×D₄	C₂²×C₄₁	1	4	yes	—
304	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
305	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
306	C₂×C₄×D₄	C₂²×C₄₁	1	2	yes	—
307	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
308	C₂×C₄×D₄	C₂²×C₄₁	1	2	yes	—
309	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
310	C₂×#117	C₂²×C₄₁	1	4	yes	—
311	C₂²×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
312	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	4	yes	—
313	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
314	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
315	#390	C₂²×C₄₁	1	4	yes	—
316	#389	C₂²×C₄₁	1	4	yes	complemented
317	#388	C₂²×C₄₁	1	8	yes	complemented
318	#388	C₂²×C₄₁	1	8	yes	complemented
319	#387	C₂²×C₄₁	1	8	yes	complemented
320	#386	C₂²×C₄₁	1	8	yes	—
321	#385	C₂²×C₄₁	1	8	yes	complemented
322	#383	C₂²×C₄₁	1	2	yes	complemented
323	#383	C₂²×C₄₁	1	2	yes	complemented
324	#382	C₂²×C₄₁	1	2	yes	complemented
325	#382	C₂²×C₄₁	1	2	yes	complemented
326	#381	C₂²×C₄₁	1	4	yes	—
327	#381	C₂²×C₄₁	1	1	yes	characteristic
328	#381	C₂²×C₄₁	1	2	yes	—
329	#381	C₂²×C₄₁	1	1	yes	characteristic
330	#381	C₂²×C₄₁	1	4	yes	complemented
331	#381	C₂²×C₄₁	1	4	yes	complemented
332	#380	C₂²×C₄₁	1	2	yes	—
333	#380	C₂²×C₄₁	1	2	yes	—
334	#380	C₂²×C₄₁	1	4	yes	complemented
335	#380	C₂²×C₄₁	1	4	yes	complemented
336	#379	C₂²×C₄₁	1	4	yes	—
337	C₄×C₄⋊C₄	C₂²×C₄₁	1	2	yes	complemented
338	C₄×C₄⋊C₄	C₂²×C₄₁	1	4	yes	complemented
339	C₄×C₄⋊C₄	C₂²×C₄₁	1	2	yes	complemented
340	C₄×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
341	C₄×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
342	C₄×C₂²⋊C₄	C₂²×C₄₁	1	4	yes	complemented
343	C₄×C₂²⋊C₄	C₂²×C₄₁	1	8	yes	complemented
344	C₄×C₂²⋊C₄	C₂²×C₄₁	1	4	yes	complemented
345	#376	C₂²×C₄₁	1	2	yes	complemented
346	#376	C₂²×C₄₁	1	2	yes	complemented
347	C₂×#95	C₂²×C₄₁	1	4	yes	complemented
348	C₂×#95	C₂²×C₄₁	1	4	yes	complemented
349	C₄₁	#34190	1	1	yes	characteristic, direct product factor, Sylow
350	C₂⁵	C₂³×C₄₁	1	2	yes	—
351	C₂²×D₄	—	2	8	no	—
352	C₂²×D₄	C₂×C₄×C₄₁	1	8	yes	—
353	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
354	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
355	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
356	C₂³×C₄	—	4	8	no	—
357	C₂³×C₄	C₂×C₄×C₄₁	1	16	yes	complemented
358	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
359	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
360	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
361	C₂³×C₄	C₂³×C₄₁	1	8	yes	—
362	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
363	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
364	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
365	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
366	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
367	C₄×D₄	—	2	16	no	—
368	C₄×D₄	C₂×C₄×C₄₁	1	8	yes	—
369	C₄×D₄	C₂×C₄×C₄₁	1	8	yes	—
370	C₄×D₄	C₂×C₄×C₄₁	1	8	yes	complemented
371	C₄×D₄	C₂×C₄×C₄₁	1	8	yes	complemented
372	C₄×D₄	C₂×C₄×C₄₁	1	16	yes	complemented
373	C₄×D₄	C₂×C₄×C₄₁	1	16	yes	complemented
374	#117	—	4	8	no	—
375	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	8	yes	complemented
376	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
377	C₂×C₄⋊C₄	C₂³×C₄₁	1	8	yes	—
378	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
379	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
380	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
381	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
382	C₂×C₂²⋊C₄	—	4	8	no	—
383	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	16	yes	complemented
384	C₂×C₂²⋊C₄	C₂³×C₄₁	1	2	yes	—
385	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
386	C₂×C₂²⋊C₄	C₂³×C₄₁	1	2	yes	—
387	C₂×C₂²⋊C₄	C₂³×C₄₁	1	8	yes	—
388	C₂×C₂²⋊C₄	C₂³×C₄₁	1	8	yes	—
389	C₂×C₂²⋊C₄	C₂³×C₄₁	1	8	yes	—
390	C₂×C₂²⋊C₄	C₂³×C₄₁	1	2	yes	—
391	C₂×C₂²⋊C₄	C₂³×C₄₁	1	2	yes	—
392	C₂×C₂²⋊C₄	C₂³×C₄₁	1	2	yes	—
393	C₂×C₂²⋊C₄	C₂³×C₄₁	1	2	yes	—
394	C₂×C₄²	C₂×C₄×C₄₁	1	8	yes	complemented
395	C₂×C₄²	C₂³×C₄₁	1	4	yes	—
396	C₂×C₄²	C₂³×C₄₁	1	8	yes	—
397	C₂×C₄²	C₂³×C₄₁	1	4	yes	—
398	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
399	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
400	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
401	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
402	#95	C₂³×C₄₁	1	2	yes	—
403	#95	C₂³×C₄₁	1	2	yes	—
404	#95	C₂³×C₄₁	1	4	yes	—
405	#95	C₂³×C₄₁	1	4	yes	—
406	#95	C₂³×C₄₁	1	8	yes	—
407	#95	C₂³×C₄₁	1	8	yes	—
408	#95	C₂³×C₄₁	1	4	yes	—
409	#95	C₂³×C₄₁	1	2	yes	—
410	#95	C₂³×C₄₁	1	4	yes	—
411	#95	C₂³×C₄₁	1	4	yes	—
412	#95	C₂³×C₄₁	1	4	yes	—
413	#95	C₂³×C₄₁	1	2	yes	—
414	C₂⁴	—	2	8	no	—
415	C₂⁴	—	2	8	no	—
416	C₂⁴	—	4	4	no	—
417	C₂⁴	C₂²×C₄×C₄₁	1	8	yes	—
418	C₂⁴	C₂⁴×C₄₁	1	2	yes	—
419	C₂⁴	C₂⁴×C₄₁	1	2	yes	—
420	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
421	C₂×D₄	—	2	8	no	—
422	C₂×D₄	—	2	32	no	—
423	C₂×D₄	—	2	4	no	—
424	C₂×D₄	C₂²×C₄×C₄₁	1	8	yes	—
425	C₂×D₄	C₂²×C₄×C₄₁	1	4	yes	—
426	C₂×D₄	C₂²×C₄×C₄₁	1	4	yes	—
427	C₂×D₄	#41×C₄₁	1	8	yes	—
428	C₂²×C₄	—	2	8	no	—
429	C₂²×C₄	—	2	4	no	—
430	C₂²×C₄	—	2	4	no	—
431	C₂²×C₄	—	2	16	no	—
432	C₂²×C₄	—	4	32	no	—
433	C₂²×C₄	—	4	4	no	—
434	C₂²×C₄	—	4	4	no	—
435	C₂²×C₄	—	4	4	no	—
436	C₂²×C₄	—	4	4	no	—
437	C₂²×C₄	—	4	2	no	—
438	C₂²×C₄	—	4	8	no	—
439	C₂²×C₄	—	4	8	no	—
440	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
441	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
442	C₂²×C₄	C₂²×C₄×C₄₁	1	8	yes	—
443	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
444	C₂²×C₄	C₂²×C₄×C₄₁	1	8	yes	—
445	C₂²×C₄	C₂²×C₄×C₄₁	1	8	yes	—
446	C₂²×C₄	C₂²×C₄×C₄₁	1	8	yes	—
447	C₂²×C₄	C₂²×C₄×C₄₁	1	8	yes	—
448	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
449	C₂²×C₄	C₂⁴×C₄₁	1	8	yes	—
450	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
451	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
452	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
453	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
454	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
455	C₂²×C₄	C₂⁴×C₄₁	1	2	yes	—
456	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
457	C₂²×C₄	C₂⁴×C₄₁	1	2	yes	—
458	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
459	C₄⋊C₄	—	2	8	no	—
460	C₄⋊C₄	—	4	8	no	—
461	C₄⋊C₄	—	4	4	no	—
462	C₄⋊C₄	C₂²×C₄×C₄₁	1	4	yes	—
463	C₄⋊C₄	C₂²×C₄×C₄₁	1	4	yes	—
464	C₄⋊C₄	C₂²×C₄×C₄₁	1	8	yes	—
465	C₂²⋊C₄	—	2	16	no	—
466	C₂²⋊C₄	—	2	4	no	—
467	C₂²⋊C₄	—	2	8	no	—
468	C₂²⋊C₄	—	2	4	no	—
469	C₂²⋊C₄	—	4	8	no	—
470	C₂²⋊C₄	—	4	8	no	—
471	C₂²⋊C₄	C₂²×C₄×C₄₁	1	8	yes	—
472	C₂²⋊C₄	C₂²×C₄×C₄₁	1	8	yes	—
473	C₂²⋊C₄	C₂²×C₄×C₄₁	1	8	yes	—
474	C₂²⋊C₄	C₂²×C₄×C₄₁	1	8	yes	—
475	C₄²	—	2	8	no	—
476	C₄²	—	4	8	no	—
477	C₄²	—	4	4	no	—
478	C₄²	C₂²×C₄×C₄₁	1	4	yes	—
479	C₄²	C₂²×C₄×C₄₁	1	4	yes	—
480	C₄²	C₂²×C₄×C₄₁	1	4	yes	—
481	C₄²	C₂²×C₄×C₄₁	1	4	yes	—
482	C₂³	—	2	2	no	—
483	C₂³	—	2	4	no	—
484	C₂³	—	2	4	no	—
485	C₂³	—	2	2	no	—
486	C₂³	—	2	16	no	—
487	C₂³	—	2	16	no	—
488	C₂³	—	2	4	no	—
489	C₂³	—	2	4	no	—
490	C₂³	—	2	4	no	—
491	C₂³	—	4	8	no	—
492	C₂³	—	4	32	no	—
493	C₂³	C₂³×C₄×C₄₁	1	4	yes	—
494	C₂³	C₂³×C₄×C₄₁	1	4	yes	—
495	C₂³	C₂³×C₄×C₄₁	1	2	yes	—
496	C₂³	C₂×#41×C₄₁	1	8	yes	—
497	C₂³	H(5,𝔽₂)×C₄₁	1	4	yes	—
498	C₂³	C₂⁵×C₄₁	1	1	yes	characteristic
499	D₄	—	2	16	no	—
500	D₄	—	2	8	no	—
501	D₄	#117×C₄₁	1	16	yes	complemented
502	C₂×C₄	—	2	8	no	—
503	C₂×C₄	—	2	16	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	4	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	8	no	—
512	C₂×C₄	—	2	4	no	—
513	C₂×C₄	—	2	4	no	—
514	C₂×C₄	—	2	8	no	—
515	C₂×C₄	—	2	4	no	—
516	C₂×C₄	—	2	8	no	—
517	C₂×C₄	—	2	4	no	—
518	C₂×C₄	—	2	8	no	—
519	C₂×C₄	—	2	4	no	—
520	C₂×C₄	—	2	4	no	—
521	C₂×C₄	—	4	8	no	—
522	C₂×C₄	—	4	8	no	—
523	C₂×C₄	—	4	8	no	—
524	C₂×C₄	—	4	8	no	—
525	C₂×C₄	—	4	4	no	—
526	C₂×C₄	—	8	16	no	—
527	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
528	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
529	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
530	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
531	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
532	C₂×C₄	C₂³×C₄×C₄₁	1	4	yes	—
533	C₂×C₄	C₂³×C₄×C₄₁	1	4	yes	—
534	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
535	C₂×C₄	C₂×#41×C₄₁	1	4	yes	—
536	C₂²	—	2	2	no	—
537	C₂²	—	2	4	no	—
538	C₂²	—	2	2	no	—
539	C₂²	—	2	4	no	—
540	C₂²	—	2	16	no	—
541	C₂²	—	2	8	no	—
542	C₂²	—	2	4	no	—
543	C₂²	—	2	4	no	—
544	C₂²	—	2	4	no	—
545	C₂²	—	4	32	no	—
546	C₂²	—	4	8	no	—
547	C₂²	C₂×#117×C₄₁	1	8	yes	—
548	C₂²	C₄₁×#518	1	4	yes	—
549	C₂²	C₂⁴×C₄×C₄₁	1	1	yes	characteristic
550	C₂²	C₂²×#41×C₄₁	1	2	yes	—
551	C₂²	C₂×H(5,𝔽₂)×C₄₁	1	2	yes	—
552	C₂²	C₄₁×#585	1	2	yes	—
553	C₄	—	2	2	no	—
554	C₄	—	2	2	no	—
555	C₄	—	2	2	no	—
556	C₄	—	2	4	no	—
557	C₄	—	2	4	no	—
558	C₄	—	2	2	no	—
559	C₄	—	2	2	no	—
560	C₄	—	4	4	no	—
561	C₄	—	4	8	no	—
562	C₄	—	4	4	no	—
563	C₄	C₂×#117×C₄₁	1	4	yes	—
564	C₂	—	2	4	no	—
565	C₂	—	2	8	no	—
566	C₂	—	4	4	no	—
567	C₂	C₂²×#117×C₄₁	1	1	yes	characteristic
568	C₂	C₂×C₄₁×#518	1	1	yes	characteristic
569	C₂	C₄₁×#3428	1	1	yes	characteristic
570	C₂	C₄₁×#3513	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₄₁×#2357	C₂	semidirect
3. C₄₁×#2353	C₂	semidirect
4. C₄₁×#2349	C₂	semidirect
5. C₄₁×#2344	C₂	semidirect
6. C₄₁×#2342	C₂	semidirect
8. C₄₁×#2319	C₂	semidirect
9. C₄₁×#2318	C₂	semidirect
10. C₄₁×#2283	C₂	semidirect
11. C₄₁×#2283	C₂	semidirect
12. C₄₁×#2280	C₂	semidirect
13. C₂×C₄₁×#381	C₂	semidirect
14. C₄₁×#2275	C₂	semidirect
17. C₂×C₄×D₄×C₄₁	C₄	semidirect
31. C₄₁×#389	C₂²	semidirect
32. C₄₁×#388	C₂²	semidirect
33. C₄₁×#388	C₂²	semidirect
34. C₄₁×#387	C₂²	semidirect
36. C₄₁×#385	C₂²	semidirect
37. C₄₁×#383	C₂²	semidirect
38. C₄₁×#383	C₂²	semidirect
39. C₄₁×#382	C₂²	semidirect
40. C₄₁×#382	C₂²	semidirect
45. C₄₁×#381	C₂²	semidirect
46. C₄₁×#381	C₂²	semidirect
49. C₄₁×#380	C₂²	semidirect
50. C₄₁×#380	C₂²	semidirect
52. C₄×C₄⋊C₄×C₄₁	C₂²	semidirect
53. C₄×C₄⋊C₄×C₄₁	C₂²	semidirect
54. C₄×C₄⋊C₄×C₄₁	C₂²	semidirect
57. C₄×C₂²⋊C₄×C₄₁	C₂²	semidirect
58. C₄×C₂²⋊C₄×C₄₁	C₂²	semidirect
59. C₄×C₂²⋊C₄×C₄₁	C₂²	semidirect
60. C₄₁×#376	C₂²	semidirect
61. C₄₁×#376	C₂²	semidirect
62. C₂×#95×C₄₁	C₂²	semidirect
63. C₂×#95×C₄₁	C₂²	semidirect
71. C₂³×C₄×C₄₁	C₂×C₄	semidirect
84. C₄×D₄×C₄₁	C₂×C₄	semidirect
85. C₄×D₄×C₄₁	C₂×C₄	semidirect
86. C₄×D₄×C₄₁	C₂×C₄	semidirect
87. C₄×D₄×C₄₁	C₂×C₄	semidirect
89. C₂×C₄⋊C₄×C₄₁	C₂×C₄	semidirect
97. C₂×C₂²⋊C₄×C₄₁	C₂×C₄	semidirect
108. C₂×C₄²×C₄₁	C₂×C₄	semidirect
215. D₄×C₄₁	#117	semidirect
250. #34190	C₄₁	direct (N × H)
280. #2357	C₂×C₄₁	semidirect
281. #2353	C₂×C₄₁	semidirect
282. #2349	C₂×C₄₁	semidirect
283. #2344	C₂×C₄₁	semidirect
284. #2342	C₂×C₄₁	semidirect
286. #2319	C₂×C₄₁	semidirect
287. #2318	C₂×C₄₁	semidirect
288. #2283	C₂×C₄₁	semidirect
289. #2283	C₂×C₄₁	semidirect
290. #2280	C₂×C₄₁	semidirect
291. C₂×#381	C₂×C₄₁	semidirect
292. #2275	C₂×C₄₁	semidirect
302. C₂×C₄×D₄	C₄×C₄₁	semidirect
316. #389	C₂²×C₄₁	semidirect
317. #388	C₂²×C₄₁	semidirect
318. #388	C₂²×C₄₁	semidirect
319. #387	C₂²×C₄₁	semidirect
321. #385	C₂²×C₄₁	semidirect
322. #383	C₂²×C₄₁	semidirect
323. #383	C₂²×C₄₁	semidirect
324. #382	C₂²×C₄₁	semidirect
325. #382	C₂²×C₄₁	semidirect
330. #381	C₂²×C₄₁	semidirect
331. #381	C₂²×C₄₁	semidirect
334. #380	C₂²×C₄₁	semidirect
335. #380	C₂²×C₄₁	semidirect
337. C₄×C₄⋊C₄	C₂²×C₄₁	semidirect
338. C₄×C₄⋊C₄	C₂²×C₄₁	semidirect
339. C₄×C₄⋊C₄	C₂²×C₄₁	semidirect
342. C₄×C₂²⋊C₄	C₂²×C₄₁	semidirect
343. C₄×C₂²⋊C₄	C₂²×C₄₁	semidirect
344. C₄×C₂²⋊C₄	C₂²×C₄₁	semidirect
345. #376	C₂²×C₄₁	semidirect
346. #376	C₂²×C₄₁	semidirect
347. C₂×#95	C₂²×C₄₁	semidirect
348. C₂×#95	C₂²×C₄₁	semidirect
349. C₄₁	#34190	direct (N × H)
357. C₂³×C₄	C₂×C₄×C₄₁	semidirect
370. C₄×D₄	C₂×C₄×C₄₁	semidirect
371. C₄×D₄	C₂×C₄×C₄₁	semidirect
372. C₄×D₄	C₂×C₄×C₄₁	semidirect
373. C₄×D₄	C₂×C₄×C₄₁	semidirect
375. C₂×C₄⋊C₄	C₂×C₄×C₄₁	semidirect
383. C₂×C₂²⋊C₄	C₂×C₄×C₄₁	semidirect
394. C₂×C₄²	C₂×C₄×C₄₁	semidirect
501. D₄	#117×C₄₁	semidirect

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×#2357	2
C₄₁×#2353	2
C₄₁×#2349	2
C₄₁×#2344	2
C₄₁×#2342	2
D₄×C₂²⋊C₄×C₄₁	2
C₄₁×#2319	2
C₄₁×#2318	2
C₄₁×#2283	2
C₄₁×#2280	2
C₂×C₄₁×#381	2
C₄₁×#2275	2
#34190	41

Subgroups — C41×#34190 #23810391

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#34190 #23810391