Subgroups — C₂×C₄₁×#3223 #23836996

Order 10496.

2022 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#541	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₁×#536	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#3223	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₄₁×#497	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄₁×#491	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₁×#488	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₁×#464	C₂	1	2	yes	maximal, complemented
9	C₂×C₄₁×#462	C₂	1	2	yes	maximal, complemented
10	C₂×C₄₁×#452	C₂	1	2	yes	maximal
11	C₂×C₄₁×#451	C₂	1	2	yes	maximal, complemented
12	C₂×C₄₁×#433	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₄₁	C₂²	1	1	yes	characteristic
14	C₂³×Q₈×C₄₁	C₂²	1	1	yes	characteristic
15	C₂²×Q₁₆×C₄₁	C₂²	1	2	yes	—
16	C₂²×Q₁₆×C₄₁	C₂²	1	2	yes	—
17	C₂²×Q₁₆×C₄₁	C₂²	1	2	yes	—
18	C₂²×SD₁₆×C₄₁	C₂²	1	2	yes	—
19	C₄₁×#541	—	2	4	no	—
20	C₄₁×#541	C₂²	1	8	yes	complemented
21	C₄₁×#536	—	2	4	no	—
22	C₄₁×#536	C₂²	1	8	yes	—
23	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
26	C₂×#126×C₄₁	—	2	2	no	—
27	C₂×#125×C₄₁	—	2	2	no	—
28	C₂×#125×C₄₁	C₂²	1	1	yes	characteristic
29	C₂×#124×C₄₁	C₂²	1	1	yes	characteristic
30	C₂×#124×C₄₁	C₂²	1	1	yes	characteristic
31	C₂×#123×C₄₁	—	2	2	no	—
32	C₂×C₂²⋊Q₈×C₄₁	—	2	2	no	—
33	C₂×C₂²⋊Q₈×C₄₁	C₂²	1	2	yes	—
34	C₂×C₄×Q₈×C₄₁	C₂²	1	2	yes	—
35	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
36	C₄₁×#497	C₂²	1	8	yes	complemented
37	C₄₁×#494	C₂²	1	8	yes	complemented
38	C₄₁×#491	C₂²	1	8	yes	complemented
39	C₄₁×#488	C₂²	1	8	yes	complemented
40	C₄₁×#464	C₂²	1	16	yes	complemented
41	C₄₁×#462	C₂²	1	16	yes	complemented
42	C₄₁×#452	C₂²	1	16	yes	—
43	C₄₁×#451	C₂²	1	16	yes	complemented
44	C₄₁×#433	C₂²	1	8	yes	complemented
45	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
46	C₂×#103×C₄₁	C₂²	1	2	yes	—
47	C₂×#103×C₄₁	C₂²	1	2	yes	—
48	C₂×#103×C₄₁	C₂²	1	2	yes	—
49	C₂×#102×C₄₁	C₂²	1	2	yes	—
50	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
51	C₂×#97×C₄₁	C₂²	1	1	yes	characteristic
52	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
53	C₂×#41×C₄₁	—	2	2	no	—
54	C₂×#41×C₄₁	—	4	4	no	—
55	C₂×#41×C₄₁	C₂³	1	4	yes	—
56	C₂²×Q₈×C₄₁	—	2	1	no	—
57	C₂²×Q₈×C₄₁	—	2	2	no	—
58	C₂²×Q₈×C₄₁	—	4	4	no	—
59	C₂²×Q₈×C₄₁	C₂³	1	2	yes	—
60	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
61	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
62	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
63	C₂²×Q₈×C₄₁	C₂³	1	4	yes	—
64	C₂²×D₄×C₄₁	—	2	1	no	—
65	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₄₁	—	2	1	no	—
67	C₂³×C₄×C₄₁	—	2	1	no	—
68	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
69	C₂×Q₁₆×C₄₁	—	4	4	no	—
70	C₂×Q₁₆×C₄₁	—	4	4	no	—
71	C₂×Q₁₆×C₄₁	—	4	4	no	—
72	C₂×Q₁₆×C₄₁	C₂³	1	8	yes	—
73	C₂×Q₁₆×C₄₁	C₂³	1	8	yes	—
74	C₂×Q₁₆×C₄₁	C₂³	1	8	yes	—
75	C₂×SD₁₆×C₄₁	—	4	4	no	—
76	C₂×SD₁₆×C₄₁	C₂³	1	8	yes	—
77	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
78	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
79	C₄⋊Q₈×C₄₁	—	2	2	no	—
80	C₄⋊Q₈×C₄₁	—	2	2	no	—
81	C₄⋊Q₈×C₄₁	—	2	2	no	—
82	C₄⋊Q₈×C₄₁	C₂³	1	4	yes	—
83	C₄⋊Q₈×C₄₁	C₂³	1	4	yes	—
84	C₄⋊Q₈×C₄₁	C₂³	1	4	yes	—
85	#126×C₄₁	—	2	16	no	—
86	#125×C₄₁	—	2	16	no	—
87	#125×C₄₁	—	2	2	no	—
88	#125×C₄₁	C₂³	1	4	yes	—
89	#124×C₄₁	—	2	2	no	—
90	#124×C₄₁	—	2	2	no	—
91	#124×C₄₁	C₂³	1	4	yes	—
92	#124×C₄₁	C₂³	1	4	yes	—
93	#123×C₄₁	—	2	16	no	—
94	C₂²⋊Q₈×C₄₁	—	2	4	no	—
95	C₂²⋊Q₈×C₄₁	—	2	16	no	—
96	C₂²⋊Q₈×C₄₁	C₂³	1	8	yes	—
97	C₄×Q₈×C₄₁	—	2	4	no	—
98	C₄×Q₈×C₄₁	C₂³	1	8	yes	—
99	#117×C₄₁	D₄	1	4	yes	—
100	#117×C₄₁	C₂³	1	4	yes	—
101	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
102	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
103	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
104	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
105	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
106	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
107	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
108	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
109	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
110	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
111	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
112	C₂×C₂²⋊C₄×C₄₁	D₄	1	2	yes	—
113	C₂×C₄²×C₄₁	—	2	2	no	—
114	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
116	C₄⋊C₈×C₄₁	C₂³	1	8	yes	—
117	#103×C₄₁	C₂³	1	8	yes	—
118	#103×C₄₁	C₂³	1	8	yes	—
119	#103×C₄₁	C₂³	1	8	yes	—
120	#102×C₄₁	C₂³	1	8	yes	—
121	C₂²⋊C₈×C₄₁	C₂³	1	8	yes	—
122	#97×C₄₁	C₂³	1	4	yes	—
123	C₄×C₈×C₄₁	C₂³	1	4	yes	—
124	C₂⁴×C₄₁	—	2	1	no	—
125	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
126	#41×C₄₁	—	4	8	no	—
127	#41×C₄₁	—	4	8	no	—
128	C₂×Q₈×C₄₁	—	2	2	no	—
129	C₂×Q₈×C₄₁	—	2	4	no	—
130	C₂×Q₈×C₄₁	—	2	1	no	—
131	C₂×Q₈×C₄₁	—	2	1	no	—
132	C₂×Q₈×C₄₁	—	2	1	no	—
133	C₂×Q₈×C₄₁	—	2	4	no	—
134	C₂×Q₈×C₄₁	—	2	4	no	—
135	C₂×Q₈×C₄₁	—	2	4	no	—
136	C₂×Q₈×C₄₁	—	2	4	no	—
137	C₂×Q₈×C₄₁	—	2	4	no	—
138	C₂×Q₈×C₄₁	—	4	8	no	—
139	C₂×Q₈×C₄₁	—	4	4	no	—
140	C₂×Q₈×C₄₁	—	4	8	no	—
141	C₂×Q₈×C₄₁	C₂⁴	1	2	yes	—
142	C₂×Q₈×C₄₁	C₂⁴	1	2	yes	—
143	C₂×Q₈×C₄₁	C₂⁴	1	2	yes	—
144	C₂×Q₈×C₄₁	C₂⁴	1	4	yes	—
145	C₂×D₄×C₄₁	—	2	4	no	—
146	C₂×D₄×C₄₁	—	2	1	no	—
147	C₂×D₄×C₄₁	—	2	4	no	—
148	C₂×D₄×C₄₁	—	4	4	no	—
149	C₂×D₄×C₄₁	C₂⁴	1	2	yes	—
150	C₂²×C₄×C₄₁	—	2	4	no	—
151	C₂²×C₄×C₄₁	—	2	4	no	—
152	C₂²×C₄×C₄₁	—	2	2	no	—
153	C₂²×C₄×C₄₁	—	2	2	no	—
154	C₂²×C₄×C₄₁	—	2	1	no	—
155	C₂²×C₄×C₄₁	—	2	1	no	—
156	C₂²×C₄×C₄₁	—	2	1	no	—
157	C₂²×C₄×C₄₁	—	2	4	no	—
158	C₂²×C₄×C₄₁	—	4	4	no	—
159	C₂²×C₄×C₄₁	—	4	4	no	—
160	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
161	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
162	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
163	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₄₁	C₂⁴	1	2	yes	—
165	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
166	Q₁₆×C₄₁	—	4	8	no	—
167	Q₁₆×C₄₁	—	4	8	no	—
168	Q₁₆×C₄₁	—	4	8	no	—
169	SD₁₆×C₄₁	—	4	8	no	—
170	C₂×C₈×C₄₁	—	2	4	no	—
171	C₂×C₈×C₄₁	—	2	4	no	—
172	C₂×C₈×C₄₁	C₂⁴	1	4	yes	—
173	C₂×C₈×C₄₁	C₂⁴	1	4	yes	—
174	C₄⋊C₄×C₄₁	—	2	8	no	—
175	C₄⋊C₄×C₄₁	—	2	8	no	—
176	C₄⋊C₄×C₄₁	—	2	8	no	—
177	C₄⋊C₄×C₄₁	—	2	8	no	—
178	C₄⋊C₄×C₄₁	—	2	8	no	—
179	C₄⋊C₄×C₄₁	—	2	8	no	—
180	C₄⋊C₄×C₄₁	—	2	2	no	—
181	C₄⋊C₄×C₄₁	C₂×D₄	1	8	yes	—
182	C₄⋊C₄×C₄₁	C₂⁴	1	4	yes	—
183	C₂²⋊C₄×C₄₁	—	2	8	no	—
184	C₂²⋊C₄×C₄₁	—	2	8	no	—
185	C₂²⋊C₄×C₄₁	—	2	8	no	—
186	C₂²⋊C₄×C₄₁	C₂×D₄	1	8	yes	—
187	C₄²×C₄₁	—	2	8	no	—
188	C₄²×C₄₁	C₂×D₄	1	2	yes	—
189	C₄²×C₄₁	C₂×D₄	1	2	yes	—
190	C₄²×C₄₁	C₂⁴	1	2	yes	—
191	C₄²×C₄₁	C₂⁴	1	2	yes	—
192	C₂³×C₄₁	—	2	2	no	—
193	C₂³×C₄₁	—	2	4	no	—
194	C₂³×C₄₁	—	2	2	no	—
195	C₂³×C₄₁	—	4	2	no	—
196	C₂³×C₄₁	—	4	4	no	—
197	C₂³×C₄₁	C₂²×D₄	1	2	yes	—
198	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
199	Q₈×C₄₁	—	2	4	no	—
200	Q₈×C₄₁	—	2	4	no	—
201	Q₈×C₄₁	—	2	4	no	—
202	Q₈×C₄₁	—	2	4	no	—
203	Q₈×C₄₁	—	2	4	no	—
204	Q₈×C₄₁	—	4	8	no	—
205	Q₈×C₄₁	—	4	4	no	—
206	Q₈×C₄₁	—	4	2	no	—
207	Q₈×C₄₁	—	4	2	no	—
208	Q₈×C₄₁	—	4	2	no	—
209	D₄×C₄₁	—	2	4	no	—
210	D₄×C₄₁	—	4	8	no	—
211	D₄×C₄₁	—	4	2	no	—
212	C₂×C₄×C₄₁	—	2	2	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	2	no	—
220	C₂×C₄×C₄₁	—	2	2	no	—
221	C₂×C₄×C₄₁	—	2	2	no	—
222	C₂×C₄×C₄₁	—	4	8	no	—
223	C₂×C₄×C₄₁	—	4	8	no	—
224	C₂×C₄×C₄₁	—	4	2	no	—
225	C₂×C₄×C₄₁	—	4	2	no	—
226	C₂×C₄×C₄₁	—	4	2	no	—
227	C₂×C₄×C₄₁	—	4	4	no	—
228	C₂×C₄×C₄₁	—	4	4	no	—
229	C₂×C₄×C₄₁	C₂²×D₄	1	4	yes	—
230	C₂×C₄×C₄₁	C₂²×D₄	1	4	yes	—
231	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
232	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
233	C₂×C₄×C₄₁	H(5,𝔽₂)	1	4	yes	—
234	C₂×C₄×C₄₁	C₂⁵	1	1	yes	characteristic
235	C₈×C₄₁	—	2	4	no	—
236	C₈×C₄₁	—	2	4	no	—
237	C₂×#3223	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
238	C₂²×C₄₁	—	2	2	no	—
239	C₂²×C₄₁	—	2	4	no	—
240	C₂²×C₄₁	—	4	4	no	—
241	C₂²×C₄₁	—	4	2	no	—
242	C₂²×C₄₁	—	4	4	no	—
243	C₂²×C₄₁	—	8	4	no	—
244	C₂²×C₄₁	#535	1	2	yes	—
245	C₂²×C₄₁	#578	1	4	yes	—
246	C₂²×C₄₁	C₂³×D₄	1	1	yes	characteristic
247	C₄×C₄₁	—	2	2	no	—
248	C₄×C₄₁	—	2	4	no	—
249	C₄×C₄₁	—	2	4	no	—
250	C₄×C₄₁	—	4	2	no	—
251	C₄×C₄₁	—	4	2	no	—
252	C₄×C₄₁	—	4	2	no	—
253	C₄×C₄₁	—	4	4	no	—
254	C₄×C₄₁	—	4	4	no	—
255	C₄×C₄₁	#535	1	2	yes	—
256	C₄×C₄₁	C₂×H(5,𝔽₂)	1	2	yes	—
257	C₂×#541	C₂×C₄₁	1	1	yes	characteristic, complemented
258	C₂×#536	C₂×C₄₁	1	1	yes	characteristic
259	#3223	C₂×C₄₁	1	16	yes	direct product factor
260	C₂×#497	C₂×C₄₁	1	1	yes	characteristic, complemented
261	C₂×#494	C₂×C₄₁	1	1	yes	characteristic, complemented
262	C₂×#491	C₂×C₄₁	1	1	yes	characteristic, complemented
263	C₂×#488	C₂×C₄₁	1	1	yes	characteristic, complemented
264	C₂×#464	C₂×C₄₁	1	2	yes	complemented
265	C₂×#462	C₂×C₄₁	1	2	yes	complemented
266	C₂×#452	C₂×C₄₁	1	2	yes	—
267	C₂×#451	C₂×C₄₁	1	2	yes	complemented
268	C₂×#433	C₂×C₄₁	1	1	yes	characteristic, complemented
269	C₂×C₄₁	—	4	2	no	—
270	C₂×C₄₁	—	8	2	no	—
271	C₂×C₄₁	#3223	1	4	yes	direct product factor
272	C₂×C₄₁	C₂×#535	1	1	yes	characteristic
273	C₂×C₄₁	C₂×#578	1	2	yes	—
274	C₂²×#41	C₂²×C₄₁	1	1	yes	characteristic
275	C₂³×Q₈	C₂²×C₄₁	1	1	yes	characteristic
276	C₂²×Q₁₆	C₂²×C₄₁	1	2	yes	—
277	C₂²×Q₁₆	C₂²×C₄₁	1	2	yes	—
278	C₂²×Q₁₆	C₂²×C₄₁	1	2	yes	—
279	C₂²×SD₁₆	C₂²×C₄₁	1	2	yes	—
280	#541	—	2	4	no	—
281	#541	C₂²×C₄₁	1	8	yes	complemented
282	#536	—	2	4	no	—
283	#536	C₂²×C₄₁	1	8	yes	—
284	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
285	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
286	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
287	C₂×#126	—	2	2	no	—
288	C₂×#125	—	2	2	no	—
289	C₂×#125	C₂²×C₄₁	1	1	yes	characteristic
290	C₂×#124	C₂²×C₄₁	1	1	yes	characteristic
291	C₂×#124	C₂²×C₄₁	1	1	yes	characteristic
292	C₂×#123	—	2	2	no	—
293	C₂×C₂²⋊Q₈	—	2	2	no	—
294	C₂×C₂²⋊Q₈	C₂²×C₄₁	1	2	yes	—
295	C₂×C₄×Q₈	C₂²×C₄₁	1	2	yes	—
296	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
297	#497	C₂²×C₄₁	1	8	yes	complemented
298	#494	C₂²×C₄₁	1	8	yes	complemented
299	#491	C₂²×C₄₁	1	8	yes	complemented
300	#488	C₂²×C₄₁	1	8	yes	complemented
301	#464	C₂²×C₄₁	1	16	yes	complemented
302	#462	C₂²×C₄₁	1	16	yes	complemented
303	#452	C₂²×C₄₁	1	16	yes	—
304	#451	C₂²×C₄₁	1	16	yes	complemented
305	#433	C₂²×C₄₁	1	8	yes	complemented
306	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
307	C₂×#103	C₂²×C₄₁	1	2	yes	—
308	C₂×#103	C₂²×C₄₁	1	2	yes	—
309	C₂×#103	C₂²×C₄₁	1	2	yes	—
310	C₂×#102	C₂²×C₄₁	1	2	yes	—
311	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
312	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic
313	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
314	C₄₁	C₂×#3223	1	1	yes	characteristic, direct product factor, Sylow
315	C₂×#41	—	2	2	no	—
316	C₂×#41	—	4	4	no	—
317	C₂×#41	C₂³×C₄₁	1	4	yes	—
318	C₂²×Q₈	—	2	1	no	—
319	C₂²×Q₈	—	2	2	no	—
320	C₂²×Q₈	—	4	4	no	—
321	C₂²×Q₈	C₂³×C₄₁	1	2	yes	—
322	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
323	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
324	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
325	C₂²×Q₈	C₂³×C₄₁	1	4	yes	—
326	C₂²×D₄	—	2	1	no	—
327	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
328	C₂³×C₄	—	2	1	no	—
329	C₂³×C₄	—	2	1	no	—
330	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
331	C₂×Q₁₆	—	4	4	no	—
332	C₂×Q₁₆	—	4	4	no	—
333	C₂×Q₁₆	—	4	4	no	—
334	C₂×Q₁₆	C₂³×C₄₁	1	8	yes	—
335	C₂×Q₁₆	C₂³×C₄₁	1	8	yes	—
336	C₂×Q₁₆	C₂³×C₄₁	1	8	yes	—
337	C₂×SD₁₆	—	4	4	no	—
338	C₂×SD₁₆	C₂³×C₄₁	1	8	yes	—
339	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
340	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
341	C₄⋊Q₈	—	2	2	no	—
342	C₄⋊Q₈	—	2	2	no	—
343	C₄⋊Q₈	—	2	2	no	—
344	C₄⋊Q₈	C₂³×C₄₁	1	4	yes	—
345	C₄⋊Q₈	C₂³×C₄₁	1	4	yes	—
346	C₄⋊Q₈	C₂³×C₄₁	1	4	yes	—
347	#126	—	2	16	no	—
348	#125	—	2	16	no	—
349	#125	—	2	2	no	—
350	#125	C₂³×C₄₁	1	4	yes	—
351	#124	—	2	2	no	—
352	#124	—	2	2	no	—
353	#124	C₂³×C₄₁	1	4	yes	—
354	#124	C₂³×C₄₁	1	4	yes	—
355	#123	—	2	16	no	—
356	C₂²⋊Q₈	—	2	4	no	—
357	C₂²⋊Q₈	—	2	16	no	—
358	C₂²⋊Q₈	C₂³×C₄₁	1	8	yes	—
359	C₄×Q₈	—	2	4	no	—
360	C₄×Q₈	C₂³×C₄₁	1	8	yes	—
361	#117	D₄×C₄₁	1	4	yes	—
362	#117	C₂³×C₄₁	1	4	yes	—
363	C₂×C₄⋊C₄	—	2	2	no	—
364	C₂×C₄⋊C₄	—	2	2	no	—
365	C₂×C₄⋊C₄	—	2	2	no	—
366	C₂×C₄⋊C₄	—	2	2	no	—
367	C₂×C₄⋊C₄	—	2	2	no	—
368	C₂×C₄⋊C₄	—	2	2	no	—
369	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
370	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
371	C₂×C₂²⋊C₄	—	2	2	no	—
372	C₂×C₂²⋊C₄	—	2	2	no	—
373	C₂×C₂²⋊C₄	—	2	2	no	—
374	C₂×C₂²⋊C₄	D₄×C₄₁	1	2	yes	—
375	C₂×C₄²	—	2	2	no	—
376	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
377	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
378	C₄⋊C₈	C₂³×C₄₁	1	8	yes	—
379	#103	C₂³×C₄₁	1	8	yes	—
380	#103	C₂³×C₄₁	1	8	yes	—
381	#103	C₂³×C₄₁	1	8	yes	—
382	#102	C₂³×C₄₁	1	8	yes	—
383	C₂²⋊C₈	C₂³×C₄₁	1	8	yes	—
384	#97	C₂³×C₄₁	1	4	yes	—
385	C₄×C₈	C₂³×C₄₁	1	4	yes	—
386	C₂⁴	—	2	1	no	—
387	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
388	#41	—	4	8	no	—
389	#41	—	4	8	no	—
390	C₂×Q₈	—	2	2	no	—
391	C₂×Q₈	—	2	4	no	—
392	C₂×Q₈	—	2	1	no	—
393	C₂×Q₈	—	2	1	no	—
394	C₂×Q₈	—	2	1	no	—
395	C₂×Q₈	—	2	4	no	—
396	C₂×Q₈	—	2	4	no	—
397	C₂×Q₈	—	2	4	no	—
398	C₂×Q₈	—	2	4	no	—
399	C₂×Q₈	—	2	4	no	—
400	C₂×Q₈	—	4	8	no	—
401	C₂×Q₈	—	4	4	no	—
402	C₂×Q₈	—	4	8	no	—
403	C₂×Q₈	C₂⁴×C₄₁	1	2	yes	—
404	C₂×Q₈	C₂⁴×C₄₁	1	2	yes	—
405	C₂×Q₈	C₂⁴×C₄₁	1	2	yes	—
406	C₂×Q₈	C₂⁴×C₄₁	1	4	yes	—
407	C₂×D₄	—	2	4	no	—
408	C₂×D₄	—	2	1	no	—
409	C₂×D₄	—	2	4	no	—
410	C₂×D₄	—	4	4	no	—
411	C₂×D₄	C₂⁴×C₄₁	1	2	yes	—
412	C₂²×C₄	—	2	4	no	—
413	C₂²×C₄	—	2	2	no	—
414	C₂²×C₄	—	2	2	no	—
415	C₂²×C₄	—	2	1	no	—
416	C₂²×C₄	—	2	1	no	—
417	C₂²×C₄	—	2	1	no	—
418	C₂²×C₄	—	2	4	no	—
419	C₂²×C₄	—	2	4	no	—
420	C₂²×C₄	—	4	4	no	—
421	C₂²×C₄	—	4	4	no	—
422	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
423	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
424	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
425	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
426	C₂²×C₄	C₂⁴×C₄₁	1	2	yes	—
427	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
428	Q₁₆	—	4	8	no	—
429	Q₁₆	—	4	8	no	—
430	Q₁₆	—	4	8	no	—
431	SD₁₆	—	4	8	no	—
432	C₂×C₈	—	2	4	no	—
433	C₂×C₈	—	2	4	no	—
434	C₂×C₈	C₂⁴×C₄₁	1	4	yes	—
435	C₂×C₈	C₂⁴×C₄₁	1	4	yes	—
436	C₄⋊C₄	—	2	8	no	—
437	C₄⋊C₄	—	2	8	no	—
438	C₄⋊C₄	—	2	8	no	—
439	C₄⋊C₄	—	2	8	no	—
440	C₄⋊C₄	—	2	8	no	—
441	C₄⋊C₄	—	2	8	no	—
442	C₄⋊C₄	—	2	2	no	—
443	C₄⋊C₄	C₂×D₄×C₄₁	1	8	yes	—
444	C₄⋊C₄	C₂⁴×C₄₁	1	4	yes	—
445	C₂²⋊C₄	—	2	8	no	—
446	C₂²⋊C₄	—	2	8	no	—
447	C₂²⋊C₄	—	2	8	no	—
448	C₂²⋊C₄	C₂×D₄×C₄₁	1	8	yes	—
449	C₄²	—	2	8	no	—
450	C₄²	C₂×D₄×C₄₁	1	2	yes	—
451	C₄²	C₂×D₄×C₄₁	1	2	yes	—
452	C₄²	C₂⁴×C₄₁	1	2	yes	—
453	C₄²	C₂⁴×C₄₁	1	2	yes	—
454	C₂³	—	2	2	no	—
455	C₂³	—	2	4	no	—
456	C₂³	—	2	2	no	—
457	C₂³	—	4	2	no	—
458	C₂³	—	4	4	no	—
459	C₂³	C₂²×D₄×C₄₁	1	2	yes	—
460	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
461	Q₈	—	2	4	no	—
462	Q₈	—	2	4	no	—
463	Q₈	—	2	4	no	—
464	Q₈	—	2	4	no	—
465	Q₈	—	2	4	no	—
466	Q₈	—	4	8	no	—
467	Q₈	—	4	2	no	—
468	Q₈	—	4	2	no	—
469	Q₈	—	4	2	no	—
470	Q₈	—	4	4	no	—
471	D₄	—	2	4	no	—
472	D₄	—	4	8	no	—
473	D₄	—	4	2	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	2	4	no	—
477	C₂×C₄	—	2	4	no	—
478	C₂×C₄	—	2	4	no	—
479	C₂×C₄	—	2	4	no	—
480	C₂×C₄	—	2	4	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	4	8	no	—
485	C₂×C₄	—	4	8	no	—
486	C₂×C₄	—	4	2	no	—
487	C₂×C₄	—	4	2	no	—
488	C₂×C₄	—	4	2	no	—
489	C₂×C₄	—	4	4	no	—
490	C₂×C₄	—	4	4	no	—
491	C₂×C₄	C₂²×D₄×C₄₁	1	4	yes	—
492	C₂×C₄	C₂²×D₄×C₄₁	1	4	yes	—
493	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
494	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
495	C₂×C₄	H(5,𝔽₂)×C₄₁	1	4	yes	—
496	C₂×C₄	C₂⁵×C₄₁	1	1	yes	characteristic
497	C₈	—	2	4	no	—
498	C₈	—	2	4	no	—
499	C₂²	—	2	4	no	—
500	C₂²	—	2	2	no	—
501	C₂²	—	4	4	no	—
502	C₂²	—	4	4	no	—
503	C₂²	—	4	2	no	—
504	C₂²	—	8	4	no	—
505	C₂²	C₄₁×#535	1	2	yes	—
506	C₂²	C₄₁×#578	1	4	yes	—
507	C₂²	C₂³×D₄×C₄₁	1	1	yes	characteristic
508	C₄	—	2	4	no	—
509	C₄	—	2	4	no	—
510	C₄	—	2	2	no	—
511	C₄	—	4	4	no	—
512	C₄	—	4	2	no	—
513	C₄	—	4	2	no	—
514	C₄	—	4	2	no	—
515	C₄	—	4	4	no	—
516	C₄	C₄₁×#535	1	2	yes	—
517	C₄	C₂×H(5,𝔽₂)×C₄₁	1	2	yes	—
518	C₂	—	4	2	no	—
519	C₂	—	8	2	no	—
520	C₂	C₄₁×#3223	1	4	yes	direct product factor
521	C₂	C₂×C₄₁×#535	1	1	yes	characteristic
522	C₂	C₂×C₄₁×#578	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₄₁
prime radical	C₂²×#41×C₄₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂×#117×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₁×#541	C₂	semidirect
3. C₄₁×#3223	C₂	direct (N × H)
4. C₂×C₄₁×#497	C₂	semidirect
5. C₂×C₄₁×#494	C₂	semidirect
6. C₂×C₄₁×#491	C₂	semidirect
7. C₂×C₄₁×#488	C₂	semidirect
8. C₂×C₄₁×#464	C₂	semidirect
9. C₂×C₄₁×#462	C₂	semidirect
11. C₂×C₄₁×#451	C₂	semidirect
12. C₂×C₄₁×#433	C₂	semidirect
20. C₄₁×#541	C₂²	semidirect
36. C₄₁×#497	C₂²	semidirect
37. C₄₁×#494	C₂²	semidirect
38. C₄₁×#491	C₂²	semidirect
39. C₄₁×#488	C₂²	semidirect
40. C₄₁×#464	C₂²	semidirect
41. C₄₁×#462	C₂²	semidirect
43. C₄₁×#451	C₂²	semidirect
44. C₄₁×#433	C₂²	semidirect
237. C₂×#3223	C₄₁	direct (N × H)
257. C₂×#541	C₂×C₄₁	semidirect
259. #3223	C₂×C₄₁	direct (N × H)
260. C₂×#497	C₂×C₄₁	semidirect
261. C₂×#494	C₂×C₄₁	semidirect
262. C₂×#491	C₂×C₄₁	semidirect
263. C₂×#488	C₂×C₄₁	semidirect
264. C₂×#464	C₂×C₄₁	semidirect
265. C₂×#462	C₂×C₄₁	semidirect
267. C₂×#451	C₂×C₄₁	semidirect
268. C₂×#433	C₂×C₄₁	semidirect
271. C₂×C₄₁	#3223	direct (N × H)
281. #541	C₂²×C₄₁	semidirect
297. #497	C₂²×C₄₁	semidirect
298. #494	C₂²×C₄₁	semidirect
299. #491	C₂²×C₄₁	semidirect
300. #488	C₂²×C₄₁	semidirect
301. #464	C₂²×C₄₁	semidirect
302. #462	C₂²×C₄₁	semidirect
304. #451	C₂²×C₄₁	semidirect
305. #433	C₂²×C₄₁	semidirect
314. C₄₁	C₂×#3223	direct (N × H)
520. C₂	C₄₁×#3223	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#541	2
C₂×C₄₁×#536	2
C₄₁×#3223	2
C₂×C₄₁×#497	2
C₂×C₄₁×#494	2
C₂×C₄₁×#491	2
C₂×C₄₁×#488	2
C₂×C₄₁×#464	2
C₂×C₄₁×#462	2
C₂×C₄₁×#452	2
C₂×C₄₁×#451	2
C₂×C₄₁×#433	2
C₂×#3223	41

Subgroups — C2×C41×#3223 #23836996

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#3223 #23836996