Subgroups — C₂×C₄₁×#1536 #23793861

Order 10496.

1234 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#432	C₂	1	2	yes	maximal, complemented
3	C₄₁×#1536	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄₁×#329	C₂	1	4	yes	maximal, complemented
5	C₂²×#41×C₄₁	C₄	1	2	yes	—
6	C₄₁×#532	—	2	4	no	—
7	C₄₁×#532	C₄	1	4	yes	—
8	C₄₁×#532	C₂²	1	4	yes	—
9	C₂×C₄⋊Q₈×C₄₁	C₄	1	1	yes	characteristic
10	C₂×#127×C₄₁	C₄	1	1	yes	characteristic
11	C₂×#124×C₄₁	C₂²	1	2	yes	—
12	C₂×#121×C₄₁	—	2	2	no	—
13	C₂×C₄×D₄×C₄₁	—	2	2	no	—
14	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
15	C₄₁×#432	—	2	4	no	—
16	C₄₁×#432	C₂²	1	8	yes	complemented
17	C₂×C₄⋊C₈×C₄₁	—	2	2	no	—
18	C₂×C₂²⋊C₈×C₄₁	—	2	2	no	—
19	C₂×#97×C₄₁	C₂²	1	4	yes	—
20	C₄₁×#329	C₂²	1	16	yes	complemented
21	C₂×#41×C₄₁	—	2	4	no	—
22	C₂×#41×C₄₁	—	4	8	no	—
23	C₂×#41×C₄₁	C₂×C₄	1	8	yes	—
24	C₂²×Q₈×C₄₁	C₂×C₄	1	2	yes	—
25	C₂²×D₄×C₄₁	—	2	2	no	—
26	C₂²×D₄×C₄₁	—	2	2	no	—
27	C₂²×D₄×C₄₁	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₄₁	—	2	2	no	—
29	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₄₁	D₄	1	2	yes	—
31	C₂²×C₈×C₄₁	—	2	4	no	—
32	C₄⋊Q₈×C₄₁	—	2	2	no	—
33	C₄⋊Q₈×C₄₁	C₂×C₄	1	4	yes	—
34	#127×C₄₁	—	2	2	no	—
35	#127×C₄₁	C₂×C₄	1	4	yes	—
36	#124×C₄₁	—	2	4	no	—
37	#124×C₄₁	C₂×C₄	1	4	yes	—
38	#124×C₄₁	C₂³	1	4	yes	—
39	#121×C₄₁	—	2	8	no	—
40	#121×C₄₁	—	2	8	no	—
41	C₄×D₄×C₄₁	—	2	8	no	—
42	C₄×D₄×C₄₁	—	2	8	no	—
43	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
44	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
45	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₄₁	D₄	1	2	yes	—
48	C₂×C₄²×C₄₁	D₄	1	4	yes	—
49	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
51	C₄⋊C₈×C₄₁	—	2	8	no	—
52	C₂²⋊C₈×C₄₁	—	2	8	no	—
53	#97×C₄₁	—	2	4	no	—
54	#97×C₄₁	C₂³	1	8	yes	—
55	C₂⁴×C₄₁	—	2	2	no	—
56	C₂⁴×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
57	#41×C₄₁	—	4	16	no	—
58	#41×C₄₁	—	4	16	no	—
59	C₂×Q₈×C₄₁	—	2	2	no	—
60	C₂×Q₈×C₄₁	—	4	4	no	—
61	C₂×Q₈×C₄₁	C₂²×C₄	1	4	yes	—
62	C₂×D₄×C₄₁	—	2	8	no	—
63	C₂×D₄×C₄₁	—	2	8	no	—
64	C₂×D₄×C₄₁	—	2	2	no	—
65	C₂×D₄×C₄₁	—	4	4	no	—
66	C₂×D₄×C₄₁	—	4	8	no	—
67	C₂×D₄×C₄₁	—	4	8	no	—
68	C₂×D₄×C₄₁	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₄₁	—	2	4	no	—
70	C₂²×C₄×C₄₁	—	2	2	no	—
71	C₂²×C₄×C₄₁	—	2	8	no	—
72	C₂²×C₄×C₄₁	—	4	4	no	—
73	C₂²×C₄×C₄₁	—	4	8	no	—
74	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
75	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
76	C₂²×C₄×C₄₁	C₂²⋊C₄	1	4	yes	—
77	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
82	C₂×C₈×C₄₁	—	2	8	no	—
83	C₂×C₈×C₄₁	—	4	8	no	—
84	C₄⋊C₄×C₄₁	—	2	8	no	—
85	C₂²⋊C₄×C₄₁	—	2	8	no	—
86	C₂²⋊C₄×C₄₁	—	2	8	no	—
87	C₄²×C₄₁	C₂²⋊C₄	1	4	yes	—
88	C₄²×C₄₁	C₂²⋊C₄	1	2	yes	—
89	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
91	C₄²×C₄₁	C₂×D₄	1	2	yes	—
92	C₄²×C₄₁	C₂×D₄	1	4	yes	—
93	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
94	C₂³×C₄₁	—	2	2	no	—
95	C₂³×C₄₁	—	2	4	no	—
96	C₂³×C₄₁	—	2	4	no	—
97	C₂³×C₄₁	—	4	4	no	—
98	C₂³×C₄₁	—	4	8	no	—
99	C₂³×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
100	C₂³×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	Q₈×C₄₁	—	4	4	no	—
102	Q₈×C₄₁	—	4	4	no	—
103	D₄×C₄₁	—	4	4	no	—
104	D₄×C₄₁	—	4	16	no	—
105	D₄×C₄₁	—	4	4	no	—
106	D₄×C₄₁	—	4	16	no	—
107	C₂×C₄×C₄₁	—	2	2	no	—
108	C₂×C₄×C₄₁	—	2	4	no	—
109	C₂×C₄×C₄₁	—	2	4	no	—
110	C₂×C₄×C₄₁	—	2	4	no	—
111	C₂×C₄×C₄₁	—	2	4	no	—
112	C₂×C₄×C₄₁	—	2	4	no	—
113	C₂×C₄×C₄₁	—	4	4	no	—
114	C₂×C₄×C₄₁	—	4	4	no	—
115	C₂×C₄×C₄₁	—	4	4	no	—
116	C₂×C₄×C₄₁	—	4	16	no	—
117	C₂×C₄×C₄₁	#100	1	4	yes	—
118	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
119	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
120	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
122	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
124	C₈×C₄₁	—	4	8	no	—
125	C₂×#1536	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂²×C₄₁	—	2	2	no	—
127	C₂²×C₄₁	—	2	4	no	—
128	C₂²×C₄₁	—	4	4	no	—
129	C₂²×C₄₁	—	4	4	no	—
130	C₂²×C₄₁	—	4	8	no	—
131	C₂²×C₄₁	—	8	8	no	—
132	C₂²×C₄₁	C₂×#100	1	2	yes	—
133	C₂²×C₄₁	#421	1	4	yes	—
134	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
135	C₄×C₄₁	—	2	1	no	—
136	C₄×C₄₁	—	2	1	no	—
137	C₄×C₄₁	—	2	2	no	—
138	C₄×C₄₁	—	2	2	no	—
139	C₄×C₄₁	—	2	4	no	—
140	C₄×C₄₁	—	4	4	no	—
141	C₄×C₄₁	C₂×#100	1	4	yes	—
142	C₂×#532	C₂×C₄₁	1	1	yes	characteristic
143	C₂×#432	C₂×C₄₁	1	2	yes	complemented
144	#1536	C₂×C₄₁	1	8	yes	direct product factor
145	C₂×#329	C₂×C₄₁	1	4	yes	complemented
146	C₂×C₄₁	—	4	2	no	—
147	C₂×C₄₁	—	8	4	no	—
148	C₂×C₄₁	#1536	1	4	yes	direct product factor
149	C₂×C₄₁	C₂²×#100	1	1	yes	characteristic
150	C₂×C₄₁	C₂×#421	1	2	yes	—
151	C₂²×#41	C₄×C₄₁	1	2	yes	—
152	#532	—	2	4	no	—
153	#532	C₄×C₄₁	1	4	yes	—
154	#532	C₂²×C₄₁	1	4	yes	—
155	C₂×C₄⋊Q₈	C₄×C₄₁	1	1	yes	characteristic
156	C₂×#127	C₄×C₄₁	1	1	yes	characteristic
157	C₂×#124	C₂²×C₄₁	1	2	yes	—
158	C₂×#121	—	2	2	no	—
159	C₂×C₄×D₄	—	2	2	no	—
160	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
161	#432	—	2	4	no	—
162	#432	C₂²×C₄₁	1	8	yes	complemented
163	C₂×C₄⋊C₈	—	2	2	no	—
164	C₂×C₂²⋊C₈	—	2	2	no	—
165	C₂×#97	C₂²×C₄₁	1	4	yes	—
166	#329	C₂²×C₄₁	1	16	yes	complemented
167	C₄₁	C₂×#1536	1	1	yes	characteristic, direct product factor, Sylow
168	C₂×#41	—	2	4	no	—
169	C₂×#41	—	4	8	no	—
170	C₂×#41	C₂×C₄×C₄₁	1	8	yes	—
171	C₂²×Q₈	C₂×C₄×C₄₁	1	2	yes	—
172	C₂²×D₄	—	2	2	no	—
173	C₂²×D₄	—	2	2	no	—
174	C₂²×D₄	C₂×C₄×C₄₁	1	2	yes	—
175	C₂³×C₄	—	2	2	no	—
176	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
177	C₂³×C₄	D₄×C₄₁	1	2	yes	—
178	C₂²×C₈	—	2	4	no	—
179	C₄⋊Q₈	—	2	2	no	—
180	C₄⋊Q₈	C₂×C₄×C₄₁	1	4	yes	—
181	#127	—	2	2	no	—
182	#127	C₂×C₄×C₄₁	1	4	yes	—
183	#124	—	2	4	no	—
184	#124	C₂×C₄×C₄₁	1	4	yes	—
185	#124	C₂³×C₄₁	1	4	yes	—
186	#121	—	2	8	no	—
187	#121	—	2	8	no	—
188	C₄×D₄	—	2	8	no	—
189	C₄×D₄	—	2	8	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₂²⋊C₄	—	2	4	no	—
192	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
193	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
194	C₂×C₄²	D₄×C₄₁	1	2	yes	—
195	C₂×C₄²	D₄×C₄₁	1	4	yes	—
196	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
197	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
198	C₄⋊C₈	—	2	8	no	—
199	C₂²⋊C₈	—	2	8	no	—
200	#97	—	2	4	no	—
201	#97	C₂³×C₄₁	1	8	yes	—
202	C₂⁴	—	2	2	no	—
203	C₂⁴	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
204	#41	—	4	16	no	—
205	#41	—	4	16	no	—
206	C₂×Q₈	—	2	2	no	—
207	C₂×Q₈	—	4	4	no	—
208	C₂×Q₈	C₂²×C₄×C₄₁	1	4	yes	—
209	C₂×D₄	—	2	8	no	—
210	C₂×D₄	—	2	8	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	—	4	4	no	—
213	C₂×D₄	—	4	8	no	—
214	C₂×D₄	—	4	8	no	—
215	C₂×D₄	C₂²×C₄×C₄₁	1	4	yes	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	8	no	—
219	C₂²×C₄	—	4	4	no	—
220	C₂²×C₄	—	4	8	no	—
221	C₂²×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
222	C₂²×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
223	C₂²×C₄	C₂²⋊C₄×C₄₁	1	4	yes	—
224	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
225	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
226	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
227	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
228	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
229	C₂×C₈	—	2	8	no	—
230	C₂×C₈	—	4	8	no	—
231	C₄⋊C₄	—	2	8	no	—
232	C₂²⋊C₄	—	2	8	no	—
233	C₂²⋊C₄	—	2	8	no	—
234	C₄²	C₂²⋊C₄×C₄₁	1	4	yes	—
235	C₄²	C₂²⋊C₄×C₄₁	1	2	yes	—
236	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
237	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
238	C₄²	C₂×D₄×C₄₁	1	2	yes	—
239	C₄²	C₂×D₄×C₄₁	1	4	yes	—
240	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
241	C₂³	—	2	2	no	—
242	C₂³	—	2	4	no	—
243	C₂³	—	2	4	no	—
244	C₂³	—	4	4	no	—
245	C₂³	—	4	8	no	—
246	C₂³	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
247	C₂³	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
248	Q₈	—	4	4	no	—
249	Q₈	—	4	4	no	—
250	D₄	—	4	4	no	—
251	D₄	—	4	4	no	—
252	D₄	—	4	16	no	—
253	D₄	—	4	16	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	16	no	—
264	C₂×C₄	#100×C₄₁	1	4	yes	—
265	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
266	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
267	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
268	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
269	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
270	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
271	C₈	—	4	8	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	4	4	no	—
275	C₂²	—	4	4	no	—
276	C₂²	—	4	8	no	—
277	C₂²	—	8	8	no	—
278	C₂²	C₂×#100×C₄₁	1	2	yes	—
279	C₂²	C₄₁×#421	1	4	yes	—
280	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	4	no	—
285	C₄	—	2	2	no	—
286	C₄	—	4	4	no	—
287	C₄	C₂×#100×C₄₁	1	4	yes	—
288	C₂	—	4	2	no	—
289	C₂	—	8	4	no	—
290	C₂	C₄₁×#1536	1	4	yes	direct product factor
291	C₂	C₂²×#100×C₄₁	1	1	yes	characteristic
292	C₂	C₂×C₄₁×#421	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₄²	C₂⁴×C₄₁
prime radical	C₂×C₄₁×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ 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
2. C₂×C₄₁×#432	C₂	semidirect
3. C₄₁×#1536	C₂	direct (N × H)
4. C₂×C₄₁×#329	C₂	semidirect
16. C₄₁×#432	C₂²	semidirect
20. C₄₁×#329	C₂²	semidirect
125. C₂×#1536	C₄₁	direct (N × H)
143. C₂×#432	C₂×C₄₁	semidirect
144. #1536	C₂×C₄₁	direct (N × H)
145. C₂×#329	C₂×C₄₁	semidirect
148. C₂×C₄₁	#1536	direct (N × H)
162. #432	C₂²×C₄₁	semidirect
166. #329	C₂²×C₄₁	semidirect
167. C₄₁	C₂×#1536	direct (N × H)
290. C₂	C₄₁×#1536	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×#532	2
C₂×C₄₁×#432	2
C₄₁×#1536	2
C₂×C₄₁×#329	2
C₂×#1536	41

Subgroups — C2×C41×#1536 #23793861

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#1536 #23793861