Subgroups — C₄₁×#21824 #23798025

Order 10496.

1006 conjugacy classes of subgroups.

The subgroup lattice has 280 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	C₂×C₄₁×#433	C₂	1	2	yes	maximal
3	C₄₁×#1985	C₂	1	4	yes	maximal, complemented
4	C₄₁×#1984	C₂	1	4	yes	maximal, complemented
5	C₄₁×#1818	C₂	1	2	yes	maximal
6	C₄₁×#1817	C₂	1	2	yes	maximal
7	C₄₁×#533	—	2	8	no	—
8	C₂×C₄⋊Q₈×C₄₁	C₄	1	2	yes	—
9	C₂×#125×C₄₁	C₄	1	2	yes	—
10	C₂×C₂²⋊Q₈×C₄₁	C₄	1	4	yes	—
11	C₂×C₄×Q₈×C₄₁	C₂²	1	4	yes	—
12	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
13	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
15	C₄₁×#433	—	2	8	no	—
16	C₂×C₄⋊C₈×C₄₁	C₂²	1	4	yes	—
17	C₂×C₂²⋊C₈×C₄₁	C₂²	1	4	yes	—
18	C₂×#97×C₄₁	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	—
20	C₄₁×#336	C₂²	1	2	yes	—
21	C₄₁×#336	C₂²	1	4	yes	—
22	C₄₁×#336	C₂²	1	2	yes	—
23	C₄₁×#336	C₂²	1	2	yes	—
24	C₄₁×#336	C₂²	1	4	yes	—
25	C₄₁×#336	C₂²	1	2	yes	—
26	C₂²×Q₈×C₄₁	C₂×C₄	1	4	yes	—
27	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
30	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
31	C₄⋊Q₈×C₄₁	—	2	8	no	—
32	#125×C₄₁	—	2	8	no	—
33	C₂²⋊Q₈×C₄₁	—	2	16	no	—
34	C₄×Q₈×C₄₁	—	2	8	no	—
35	C₄×Q₈×C₄₁	—	2	8	no	—
36	#117×C₄₁	—	2	4	no	—
37	#117×C₄₁	D₄	1	8	yes	—
38	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	8	yes	—
40	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
41	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
42	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
43	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₁	—	2	4	no	—
45	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
48	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
49	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
50	C₄⋊C₈×C₄₁	—	2	8	no	—
51	C₂²⋊C₈×C₄₁	—	2	8	no	—
52	#97×C₄₁	—	2	4	no	—
53	C₄×C₈×C₄₁	—	2	4	no	—
54	C₂⁴×C₄₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×Q₈×C₄₁	—	2	8	no	—
56	C₂×Q₈×C₄₁	—	2	8	no	—
57	C₂×Q₈×C₄₁	—	4	4	no	—
58	C₂²×C₄×C₄₁	—	2	2	no	—
59	C₂²×C₄×C₄₁	—	2	2	no	—
60	C₂²×C₄×C₄₁	—	2	4	no	—
61	C₂²×C₄×C₄₁	—	2	4	no	—
62	C₂²×C₄×C₄₁	—	4	2	no	—
63	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
69	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
70	C₂×C₈×C₄₁	—	2	4	no	—
71	C₂×C₈×C₄₁	—	2	4	no	—
72	C₂×C₈×C₄₁	—	2	4	no	—
73	C₂×C₈×C₄₁	—	2	4	no	—
74	C₂×C₈×C₄₁	—	2	4	no	—
75	C₂×C₈×C₄₁	—	2	4	no	—
76	C₄⋊C₄×C₄₁	—	2	4	no	—
77	C₄⋊C₄×C₄₁	—	2	16	no	—
78	C₄⋊C₄×C₄₁	—	2	8	no	—
79	C₄⋊C₄×C₄₁	C₂²⋊C₄	1	8	yes	—
80	C₂²⋊C₄×C₄₁	—	2	4	no	—
81	C₂²⋊C₄×C₄₁	C₂²⋊C₄	1	8	yes	—
82	C₄²×C₄₁	—	2	2	no	—
83	C₄²×C₄₁	—	2	2	no	—
84	C₄²×C₄₁	—	2	4	no	—
85	C₄²×C₄₁	—	2	4	no	—
86	C₄²×C₄₁	C₂×D₄	1	4	yes	—
87	C₄²×C₄₁	C₂×D₄	1	4	yes	—
88	C₂³×C₄₁	—	2	1	no	—
89	C₂³×C₄₁	—	2	1	no	—
90	C₂³×C₄₁	—	2	2	no	—
91	C₂³×C₄₁	—	2	2	no	—
92	C₂³×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
93	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
94	Q₈×C₄₁	—	4	16	no	—
95	C₂×C₄×C₄₁	—	2	2	no	—
96	C₂×C₄×C₄₁	—	2	1	no	—
97	C₂×C₄×C₄₁	—	2	1	no	—
98	C₂×C₄×C₄₁	—	2	4	no	—
99	C₂×C₄×C₄₁	—	2	2	no	—
100	C₂×C₄×C₄₁	—	2	4	no	—
101	C₂×C₄×C₄₁	—	2	8	no	—
102	C₂×C₄×C₄₁	—	2	2	no	—
103	C₂×C₄×C₄₁	—	2	4	no	—
104	C₂×C₄×C₄₁	—	2	2	no	—
105	C₂×C₄×C₄₁	—	2	4	no	—
106	C₂×C₄×C₄₁	—	2	2	no	—
107	C₂×C₄×C₄₁	—	2	4	no	—
108	C₂×C₄×C₄₁	—	4	8	no	—
109	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
110	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	8	yes	—
111	C₂×C₄×C₄₁	#131	1	4	yes	—
112	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
113	C₂×C₄×C₄₁	2₋¹⁺⁴	1	2	yes	—
114	C₂×C₄×C₄₁	2₋¹⁺⁴	1	2	yes	—
115	C₈×C₄₁	—	4	4	no	—
116	C₈×C₄₁	—	4	4	no	—
117	#21824	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂²×C₄₁	—	2	2	no	—
119	C₂²×C₄₁	—	2	4	no	—
120	C₂²×C₄₁	—	4	2	no	—
121	C₂²×C₄₁	—	4	1	no	—
122	C₂²×C₄₁	—	4	1	no	—
123	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
124	C₂²×C₄₁	#519	1	1	yes	characteristic
125	C₂²×C₄₁	#519	1	1	yes	characteristic
126	C₂²×C₄₁	C₂×#131	1	2	yes	—
127	C₂²×C₄₁	#568	1	2	yes	—
128	C₄×C₄₁	—	2	2	no	—
129	C₄×C₄₁	—	2	4	no	—
130	C₄×C₄₁	—	2	8	no	—
131	C₄×C₄₁	—	4	2	no	—
132	C₄×C₄₁	—	4	4	no	—
133	C₄×C₄₁	#536	1	4	yes	—
134	C₂×#533	C₂×C₄₁	1	1	yes	characteristic
135	C₂×#433	C₂×C₄₁	1	2	yes	—
136	#1985	C₂×C₄₁	1	4	yes	complemented
137	#1984	C₂×C₄₁	1	4	yes	complemented
138	#1818	C₂×C₄₁	1	2	yes	—
139	#1817	C₂×C₄₁	1	2	yes	—
140	C₂×C₄₁	—	4	2	no	—
141	C₂×C₄₁	#2310	1	1	yes	characteristic
142	C₂×C₄₁	#2880	1	2	yes	—
143	C₂×C₄₁	#2985	1	2	yes	—
144	C₂×C₄₁	#2986	1	1	yes	characteristic
145	C₂×C₄₁	#2987	1	1	yes	characteristic
146	#533	—	2	8	no	—
147	C₂×C₄⋊Q₈	C₄×C₄₁	1	2	yes	—
148	C₂×#125	C₄×C₄₁	1	2	yes	—
149	C₂×C₂²⋊Q₈	C₄×C₄₁	1	4	yes	—
150	C₂×C₄×Q₈	C₂²×C₄₁	1	4	yes	—
151	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
152	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
153	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
154	#433	—	2	8	no	—
155	C₂×C₄⋊C₈	C₂²×C₄₁	1	4	yes	—
156	C₂×C₂²⋊C₈	C₂²×C₄₁	1	4	yes	—
157	C₂×#97	C₂²×C₄₁	1	2	yes	—
158	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
159	#336	C₂²×C₄₁	1	2	yes	—
160	#336	C₂²×C₄₁	1	4	yes	—
161	#336	C₂²×C₄₁	1	2	yes	—
162	#336	C₂²×C₄₁	1	2	yes	—
163	#336	C₂²×C₄₁	1	4	yes	—
164	#336	C₂²×C₄₁	1	2	yes	—
165	C₄₁	#21824	1	1	yes	characteristic, direct product factor, Sylow
166	C₂²×Q₈	C₂×C₄×C₄₁	1	4	yes	—
167	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
168	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
169	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
170	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
171	C₄⋊Q₈	—	2	8	no	—
172	#125	—	2	8	no	—
173	C₂²⋊Q₈	—	2	16	no	—
174	C₄×Q₈	—	2	8	no	—
175	C₄×Q₈	—	2	8	no	—
176	#117	—	2	4	no	—
177	#117	D₄×C₄₁	1	8	yes	—
178	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
179	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	8	yes	—
180	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
181	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
182	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
183	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
184	C₂×C₄²	—	2	4	no	—
185	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
186	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
187	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
188	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
189	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
190	C₄⋊C₈	—	2	8	no	—
191	C₂²⋊C₈	—	2	8	no	—
192	#97	—	2	4	no	—
193	C₄×C₈	—	2	4	no	—
194	C₂⁴	C₂²×C₄×C₄₁	1	1	yes	characteristic
195	C₂×Q₈	—	2	8	no	—
196	C₂×Q₈	—	2	8	no	—
197	C₂×Q₈	—	4	4	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
204	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
205	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
206	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
207	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
208	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
209	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	4	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	2	16	no	—
218	C₄⋊C₄	—	2	8	no	—
219	C₄⋊C₄	C₂²⋊C₄×C₄₁	1	8	yes	—
220	C₂²⋊C₄	—	2	4	no	—
221	C₂²⋊C₄	C₂²⋊C₄×C₄₁	1	8	yes	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	4	no	—
225	C₄²	—	2	4	no	—
226	C₄²	C₂×D₄×C₄₁	1	4	yes	—
227	C₄²	C₂×D₄×C₄₁	1	4	yes	—
228	C₂³	—	2	1	no	—
229	C₂³	—	2	1	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
233	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
234	Q₈	—	4	16	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	4	8	no	—
249	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
250	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	8	yes	—
251	C₂×C₄	#131×C₄₁	1	4	yes	—
252	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
253	C₂×C₄	2₋¹⁺⁴×C₄₁	1	2	yes	—
254	C₂×C₄	2₋¹⁺⁴×C₄₁	1	2	yes	—
255	C₈	—	4	4	no	—
256	C₈	—	4	4	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	4	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	1	no	—
262	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
263	C₂²	C₄₁×#519	1	1	yes	characteristic
264	C₂²	C₄₁×#519	1	1	yes	characteristic
265	C₂²	C₂×#131×C₄₁	1	2	yes	—
266	C₂²	C₄₁×#568	1	2	yes	—
267	C₄	—	2	2	no	—
268	C₄	—	2	4	no	—
269	C₄	—	2	8	no	—
270	C₄	—	4	2	no	—
271	C₄	—	4	4	no	—
272	C₄	C₄₁×#536	1	4	yes	—
273	C₂	—	4	2	no	—
274	C₂	C₄₁×#2310	1	1	yes	characteristic
275	C₂	C₄₁×#2880	1	2	yes	—
276	C₂	C₄₁×#2985	1	2	yes	—
277	C₂	C₄₁×#2986	1	1	yes	characteristic
278	C₂	C₄₁×#2987	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₄₁×#1985	C₂	semidirect
4. C₄₁×#1984	C₂	semidirect
117. #21824	C₄₁	direct (N × H)
136. #1985	C₂×C₄₁	semidirect
137. #1984	C₂×C₄₁	semidirect
165. C₄₁	#21824	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
C₂×C₄₁×#433	2
C₄₁×#1985	2
C₄₁×#1984	2
C₄₁×#1818	2
C₄₁×#1817	2
#21824	41

Subgroups — C41×#21824 #23798025

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#21824 #23798025