Subgroups — C₂×C₄₁×#3139 #23836815

Order 10496.

2118 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#533	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#3139	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₄₁×#491	C₂	1	4	yes	maximal, complemented
5	C₂×C₄₁×#487	C₂	1	4	yes	maximal, complemented
6	C₂×C₄₁×#419	C₂	1	4	yes	maximal
7	C₂×C₄×M₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic
8	C₂³×Q₈×C₄₁	C₂²	1	2	yes	—
9	C₂²×M₁₆×C₄₁	C₂²	1	2	yes	—
10	C₄₁×#533	—	2	4	no	—
11	C₄₁×#533	C₂²	1	8	yes	—
12	C₂×C₄⋊Q₈×C₄₁	—	2	2	no	—
13	C₂×C₄⋊Q₈×C₄₁	—	2	4	no	—
14	C₂×C₄⋊Q₈×C₄₁	C₂²	1	2	yes	—
15	C₂×C₄⋊Q₈×C₄₁	C₂²	1	2	yes	—
16	C₂×C₄⋊Q₈×C₄₁	C₂²	1	8	yes	—
17	C₂×C₄⋊Q₈×C₄₁	C₂²	1	2	yes	—
18	C₂×#125×C₄₁	C₂²	1	2	yes	—
19	C₂×C₂²⋊Q₈×C₄₁	—	2	2	no	—
20	C₂×C₄×Q₈×C₄₁	—	2	2	no	—
21	C₂×#117×C₄₁	C₂²	1	2	yes	—
22	C₂²×C₄⋊C₄×C₄₁	—	2	2	no	—
23	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
24	C₄₁×#491	C₂²	1	32	yes	complemented
25	C₄₁×#487	C₂²	1	32	yes	complemented
26	C₄₁×#419	C₂²	1	32	yes	—
27	C₂×#103×C₄₁	C₂²	1	16	yes	—
28	C₄×M₁₆×C₄₁	C₂²	1	8	yes	—
29	C₂×#97×C₄₁	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	—
31	C₂²×Q₈×C₄₁	—	2	2	no	—
32	C₂²×Q₈×C₄₁	—	2	4	no	—
33	C₂²×Q₈×C₄₁	—	2	2	no	—
34	C₂²×Q₈×C₄₁	—	2	16	no	—
35	C₂²×Q₈×C₄₁	C₂³	1	4	yes	—
36	C₂²×Q₈×C₄₁	C₂³	1	8	yes	—
37	C₂³×C₄×C₄₁	—	2	2	no	—
38	C₂³×C₄×C₄₁	D₄	1	2	yes	—
39	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
40	C₂×M₁₆×C₄₁	—	2	8	no	—
41	C₂×M₁₆×C₄₁	C₂³	1	8	yes	—
42	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
43	C₄⋊Q₈×C₄₁	—	2	16	no	—
44	C₄⋊Q₈×C₄₁	—	2	4	no	—
45	C₄⋊Q₈×C₄₁	—	2	4	no	—
46	C₄⋊Q₈×C₄₁	—	2	4	no	—
47	C₄⋊Q₈×C₄₁	C₂³	1	8	yes	—
48	C₄⋊Q₈×C₄₁	C₂³	1	8	yes	—
49	C₄⋊Q₈×C₄₁	C₂³	1	8	yes	—
50	#125×C₄₁	—	2	4	no	—
51	#125×C₄₁	C₂³	1	8	yes	—
52	C₂²⋊Q₈×C₄₁	—	2	16	no	—
53	C₄×Q₈×C₄₁	—	2	16	no	—
54	#117×C₄₁	—	2	4	no	—
55	#117×C₄₁	C₂³	1	8	yes	—
56	C₂×C₄⋊C₄×C₄₁	—	2	16	no	—
57	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
58	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
59	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
60	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
61	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
62	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
63	C₂×C₄²×C₄₁	—	2	2	no	—
64	C₂×C₄²×C₄₁	D₄	1	2	yes	—
65	C₂×C₄²×C₄₁	D₄	1	4	yes	—
66	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₄₁	C₂³	1	4	yes	—
69	#103×C₄₁	C₂³	1	64	yes	—
70	#97×C₄₁	C₂³	1	8	yes	—
71	C₄×C₈×C₄₁	C₂³	1	8	yes	—
72	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
73	C₂×Q₈×C₄₁	—	2	8	no	—
74	C₂×Q₈×C₄₁	—	2	8	no	—
75	C₂×Q₈×C₄₁	—	2	4	no	—
76	C₂×Q₈×C₄₁	—	2	16	no	—
77	C₂×Q₈×C₄₁	—	2	32	no	—
78	C₂×Q₈×C₄₁	—	4	16	no	—
79	C₂×Q₈×C₄₁	—	4	8	no	—
80	C₂×Q₈×C₄₁	—	4	8	no	—
81	C₂×Q₈×C₄₁	C₂⁴	1	8	yes	—
82	C₂²×C₄×C₄₁	—	2	8	no	—
83	C₂²×C₄×C₄₁	—	2	8	no	—
84	C₂²×C₄×C₄₁	—	2	4	no	—
85	C₂²×C₄×C₄₁	—	2	4	no	—
86	C₂²×C₄×C₄₁	—	4	8	no	—
87	C₂²×C₄×C₄₁	C₂×D₄	1	8	yes	—
88	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₄₁	#41	1	8	yes	—
93	C₂²×C₄×C₄₁	C₂⁴	1	2	yes	—
94	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
95	M₁₆×C₄₁	—	2	16	no	—
96	C₂×C₈×C₄₁	—	2	8	no	—
97	C₂×C₈×C₄₁	C₂⁴	1	8	yes	—
98	C₄⋊C₄×C₄₁	—	2	16	no	—
99	C₄⋊C₄×C₄₁	—	2	16	no	—
100	C₄⋊C₄×C₄₁	—	2	16	no	—
101	C₄⋊C₄×C₄₁	—	2	8	no	—
102	C₄⋊C₄×C₄₁	—	2	4	no	—
103	C₄⋊C₄×C₄₁	C₂⁴	1	8	yes	—
104	C₂²⋊C₄×C₄₁	—	2	8	no	—
105	C₄²×C₄₁	—	2	8	no	—
106	C₄²×C₄₁	C₂×D₄	1	8	yes	—
107	C₄²×C₄₁	C₂×D₄	1	2	yes	—
108	C₄²×C₄₁	C₂×D₄	1	2	yes	—
109	C₄²×C₄₁	C₂⁴	1	2	yes	—
110	C₄²×C₄₁	C₂⁴	1	2	yes	—
111	C₂³×C₄₁	—	2	4	no	—
112	C₂³×C₄₁	#124	1	4	yes	—
113	C₂³×C₄₁	C₂²×D₄	1	2	yes	—
114	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
115	Q₈×C₄₁	—	2	16	no	—
116	Q₈×C₄₁	—	4	16	no	—
117	Q₈×C₄₁	—	4	8	no	—
118	Q₈×C₄₁	—	4	16	no	—
119	C₂×C₄×C₄₁	—	2	4	no	—
120	C₂×C₄×C₄₁	—	2	16	no	—
121	C₂×C₄×C₄₁	—	2	8	no	—
122	C₂×C₄×C₄₁	—	2	8	no	—
123	C₂×C₄×C₄₁	—	4	16	no	—
124	C₂×C₄×C₄₁	—	4	8	no	—
125	C₂×C₄×C₄₁	—	4	8	no	—
126	C₂×C₄×C₄₁	#124	1	8	yes	—
127	C₂×C₄×C₄₁	#124	1	4	yes	—
128	C₂×C₄×C₄₁	#137	1	8	yes	—
129	C₂×C₄×C₄₁	C₂²×D₄	1	4	yes	—
130	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
131	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₄₁	C₂²×D₄	1	4	yes	—
133	C₂×C₄×C₄₁	C₂×#41	1	4	yes	—
134	C₂×C₄×C₄₁	C₂×#41	1	8	yes	—
135	C₂×C₄×C₄₁	C₂⁵	1	1	yes	characteristic
136	C₈×C₄₁	—	2	8	no	—
137	C₂×#3139	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₂²×C₄₁	—	2	8	no	—
139	C₂²×C₄₁	—	2	4	no	—
140	C₂²×C₄₁	C₂×#124	1	4	yes	—
141	C₂²×C₄₁	C₂×#124	1	2	yes	—
142	C₂²×C₄₁	C₂×#137	1	4	yes	—
143	C₂²×C₄₁	C₂³×D₄	1	1	yes	characteristic
144	C₄×C₄₁	—	2	4	no	—
145	C₄×C₄₁	—	4	8	no	—
146	C₄×C₄₁	—	4	8	no	—
147	C₄×C₄₁	C₂×#124	1	4	yes	—
148	C₄×C₄₁	C₂×#124	1	2	yes	—
149	C₄×C₄₁	C₂×#137	1	8	yes	—
150	C₄×C₄₁	C₂²×#41	1	2	yes	—
151	C₂×#533	C₂×C₄₁	1	1	yes	characteristic
152	C₂²×C₄⋊Q₈	C₂×C₄₁	1	1	yes	characteristic
153	#3139	C₂×C₄₁	1	16	yes	direct product factor
154	C₂×#491	C₂×C₄₁	1	4	yes	complemented
155	C₂×#487	C₂×C₄₁	1	4	yes	complemented
156	C₂×#419	C₂×C₄₁	1	4	yes	—
157	C₂×C₄×M₁₆	C₂×C₄₁	1	1	yes	characteristic
158	C₂×C₄₁	—	2	4	no	—
159	C₂×C₄₁	#3139	1	4	yes	direct product factor
160	C₂×C₄₁	C₂²×#124	1	1	yes	characteristic
161	C₂×C₄₁	C₂²×#137	1	2	yes	—
162	C₂³×Q₈	C₂²×C₄₁	1	2	yes	—
163	C₂²×M₁₆	C₂²×C₄₁	1	2	yes	—
164	#533	—	2	4	no	—
165	#533	C₂²×C₄₁	1	8	yes	—
166	C₂×C₄⋊Q₈	—	2	2	no	—
167	C₂×C₄⋊Q₈	—	2	4	no	—
168	C₂×C₄⋊Q₈	C₂²×C₄₁	1	2	yes	—
169	C₂×C₄⋊Q₈	C₂²×C₄₁	1	2	yes	—
170	C₂×C₄⋊Q₈	C₂²×C₄₁	1	8	yes	—
171	C₂×C₄⋊Q₈	C₂²×C₄₁	1	2	yes	—
172	C₂×#125	C₂²×C₄₁	1	2	yes	—
173	C₂×C₂²⋊Q₈	—	2	2	no	—
174	C₂×C₄×Q₈	—	2	2	no	—
175	C₂×#117	C₂²×C₄₁	1	2	yes	—
176	C₂²×C₄⋊C₄	—	2	2	no	—
177	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
178	#491	C₂²×C₄₁	1	32	yes	complemented
179	#487	C₂²×C₄₁	1	32	yes	complemented
180	#419	C₂²×C₄₁	1	32	yes	—
181	C₂×#103	C₂²×C₄₁	1	16	yes	—
182	C₄×M₁₆	C₂²×C₄₁	1	8	yes	—
183	C₂×#97	C₂²×C₄₁	1	2	yes	—
184	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
185	C₄₁	C₂×#3139	1	1	yes	characteristic, direct product factor, Sylow
186	C₂²×Q₈	—	2	2	no	—
187	C₂²×Q₈	—	2	2	no	—
188	C₂²×Q₈	—	2	4	no	—
189	C₂²×Q₈	—	2	16	no	—
190	C₂²×Q₈	C₂³×C₄₁	1	4	yes	—
191	C₂²×Q₈	C₂³×C₄₁	1	8	yes	—
192	C₂³×C₄	—	2	2	no	—
193	C₂³×C₄	D₄×C₄₁	1	2	yes	—
194	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
195	C₂×M₁₆	—	2	8	no	—
196	C₂×M₁₆	C₂³×C₄₁	1	8	yes	—
197	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
198	C₄⋊Q₈	—	2	16	no	—
199	C₄⋊Q₈	—	2	4	no	—
200	C₄⋊Q₈	—	2	4	no	—
201	C₄⋊Q₈	—	2	4	no	—
202	C₄⋊Q₈	C₂³×C₄₁	1	8	yes	—
203	C₄⋊Q₈	C₂³×C₄₁	1	8	yes	—
204	C₄⋊Q₈	C₂³×C₄₁	1	8	yes	—
205	#125	—	2	4	no	—
206	#125	C₂³×C₄₁	1	8	yes	—
207	C₂²⋊Q₈	—	2	16	no	—
208	C₄×Q₈	—	2	16	no	—
209	#117	—	2	4	no	—
210	#117	C₂³×C₄₁	1	8	yes	—
211	C₂×C₄⋊C₄	—	2	4	no	—
212	C₂×C₄⋊C₄	—	2	16	no	—
213	C₂×C₄⋊C₄	—	2	4	no	—
214	C₂×C₄⋊C₄	—	2	4	no	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
217	C₂×C₂²⋊C₄	—	2	2	no	—
218	C₂×C₄²	—	2	2	no	—
219	C₂×C₄²	D₄×C₄₁	1	2	yes	—
220	C₂×C₄²	D₄×C₄₁	1	4	yes	—
221	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
222	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
223	C₂×C₄²	C₂³×C₄₁	1	4	yes	—
224	#103	C₂³×C₄₁	1	64	yes	—
225	#97	C₂³×C₄₁	1	8	yes	—
226	C₄×C₈	C₂³×C₄₁	1	8	yes	—
227	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
228	C₂×Q₈	—	2	8	no	—
229	C₂×Q₈	—	2	8	no	—
230	C₂×Q₈	—	2	4	no	—
231	C₂×Q₈	—	2	16	no	—
232	C₂×Q₈	—	2	32	no	—
233	C₂×Q₈	—	4	16	no	—
234	C₂×Q₈	—	4	8	no	—
235	C₂×Q₈	—	4	8	no	—
236	C₂×Q₈	C₂⁴×C₄₁	1	8	yes	—
237	C₂²×C₄	—	2	8	no	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	2	8	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	4	8	no	—
242	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
243	C₂²×C₄	C₂×D₄×C₄₁	1	8	yes	—
244	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
246	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
247	C₂²×C₄	#41×C₄₁	1	8	yes	—
248	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
249	C₂²×C₄	C₂⁴×C₄₁	1	2	yes	—
250	M₁₆	—	2	16	no	—
251	C₂×C₈	—	2	8	no	—
252	C₂×C₈	C₂⁴×C₄₁	1	8	yes	—
253	C₄⋊C₄	—	2	16	no	—
254	C₄⋊C₄	—	2	16	no	—
255	C₄⋊C₄	—	2	16	no	—
256	C₄⋊C₄	—	2	8	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	C₂⁴×C₄₁	1	8	yes	—
259	C₂²⋊C₄	—	2	8	no	—
260	C₄²	—	2	8	no	—
261	C₄²	C₂×D₄×C₄₁	1	8	yes	—
262	C₄²	C₂×D₄×C₄₁	1	2	yes	—
263	C₄²	C₂×D₄×C₄₁	1	2	yes	—
264	C₄²	C₂⁴×C₄₁	1	2	yes	—
265	C₄²	C₂⁴×C₄₁	1	2	yes	—
266	C₂³	—	2	4	no	—
267	C₂³	#124×C₄₁	1	4	yes	—
268	C₂³	C₂²×D₄×C₄₁	1	2	yes	—
269	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
270	Q₈	—	2	16	no	—
271	Q₈	—	4	16	no	—
272	Q₈	—	4	8	no	—
273	Q₈	—	4	16	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	16	no	—
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	4	16	no	—
279	C₂×C₄	—	4	8	no	—
280	C₂×C₄	—	4	8	no	—
281	C₂×C₄	#124×C₄₁	1	8	yes	—
282	C₂×C₄	#124×C₄₁	1	4	yes	—
283	C₂×C₄	#137×C₄₁	1	8	yes	—
284	C₂×C₄	C₂²×D₄×C₄₁	1	4	yes	—
285	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
286	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₄×C₄₁	1	4	yes	—
288	C₂×C₄	C₂×#41×C₄₁	1	4	yes	—
289	C₂×C₄	C₂×#41×C₄₁	1	8	yes	—
290	C₂×C₄	C₂⁵×C₄₁	1	1	yes	characteristic
291	C₈	—	2	8	no	—
292	C₂²	—	2	4	no	—
293	C₂²	—	2	8	no	—
294	C₂²	C₂×#124×C₄₁	1	4	yes	—
295	C₂²	C₂×#124×C₄₁	1	2	yes	—
296	C₂²	C₂×#137×C₄₁	1	4	yes	—
297	C₂²	C₂³×D₄×C₄₁	1	1	yes	characteristic
298	C₄	—	2	4	no	—
299	C₄	—	4	8	no	—
300	C₄	—	4	8	no	—
301	C₄	C₂×#124×C₄₁	1	4	yes	—
302	C₄	C₂×#124×C₄₁	1	2	yes	—
303	C₄	C₂×#137×C₄₁	1	8	yes	—
304	C₄	C₂²×#41×C₄₁	1	2	yes	—
305	C₂	—	2	4	no	—
306	C₂	C₄₁×#3139	1	4	yes	direct product factor
307	C₂	C₂²×#124×C₄₁	1	1	yes	characteristic
308	C₂	C₂²×#137×C₄₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₄₁
prime radical	C₂⁴×C₄₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂²×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₁×#3139	C₂	direct (N × H)
4. C₂×C₄₁×#491	C₂	semidirect
5. C₂×C₄₁×#487	C₂	semidirect
24. C₄₁×#491	C₂²	semidirect
25. C₄₁×#487	C₂²	semidirect
137. C₂×#3139	C₄₁	direct (N × H)
153. #3139	C₂×C₄₁	direct (N × H)
154. C₂×#491	C₂×C₄₁	semidirect
155. C₂×#487	C₂×C₄₁	semidirect
159. C₂×C₄₁	#3139	direct (N × H)
178. #491	C₂²×C₄₁	semidirect
179. #487	C₂²×C₄₁	semidirect
185. C₄₁	C₂×#3139	direct (N × H)
306. C₂	C₄₁×#3139	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×#533	2
C₂²×C₄⋊Q₈×C₄₁	2
C₄₁×#3139	2
C₂×C₄₁×#491	2
C₂×C₄₁×#487	2
C₂×C₄₁×#419	2
C₂×C₄×M₁₆×C₄₁	2
C₂×#3139	41

Subgroups — C2×C41×#3139 #23836815

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#3139 #23836815