Subgroups — C₄₁×#21323 #23797524

Order 10496.

1182 conjugacy classes of subgroups.

The subgroup lattice has 278 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×C₄×M₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₁×#382	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1993	C₂	1	4	yes	maximal, complemented
5	C₄₁×#1991	C₂	1	4	yes	maximal, complemented
6	C₄₁×#1736	C₂	1	4	yes	maximal
7	C₂²×M₁₆×C₄₁	C₂²	1	2	yes	—
8	C₄₁×#533	—	2	4	no	—
9	C₄₁×#533	C₄	1	8	yes	complemented
10	C₂×C₄⋊Q₈×C₄₁	C₂²	1	2	yes	—
11	C₂×#125×C₄₁	C₂²	1	2	yes	—
12	C₂×C₂²⋊Q₈×C₄₁	—	2	2	no	—
13	C₂×C₄×Q₈×C₄₁	—	2	2	no	—
14	C₂×#117×C₄₁	C₂²	1	2	yes	—
15	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
16	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
17	C₄×M₁₆×C₄₁	—	2	4	no	—
18	C₂×#97×C₄₁	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	—
20	C₄₁×#382	—	2	2	no	—
21	C₄₁×#382	C₂²	1	2	yes	—
22	C₄₁×#382	C₂²	1	2	yes	—
23	C₂×#95×C₄₁	—	2	2	no	—
24	C₄₁×#340	C₂²	1	16	yes	—
25	C₂²×Q₈×C₄₁	—	2	2	no	—
26	C₂³×C₄×C₄₁	—	2	2	no	—
27	C₂³×C₄×C₄₁	D₄	1	2	yes	—
28	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
29	C₂×M₁₆×C₄₁	—	2	4	no	—
30	C₂×M₁₆×C₄₁	—	2	4	no	—
31	C₂×M₁₆×C₄₁	Q₈	1	8	yes	complemented
32	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
33	C₄⋊Q₈×C₄₁	—	2	4	no	—
34	C₄⋊Q₈×C₄₁	C₂×C₄	1	8	yes	complemented
35	#125×C₄₁	—	2	4	no	—
36	#125×C₄₁	C₂×C₄	1	8	yes	complemented
37	C₂²⋊Q₈×C₄₁	—	2	16	no	—
38	C₄×Q₈×C₄₁	—	2	16	no	—
39	#117×C₄₁	—	2	4	no	—
40	#117×C₄₁	C₂×C₄	1	8	yes	—
41	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
42	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
44	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
45	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
47	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
48	C₂×C₄²×C₄₁	—	2	2	no	—
49	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₄₁	D₄	1	2	yes	—
51	C₂×C₄²×C₄₁	D₄	1	4	yes	—
52	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
54	#97×C₄₁	—	2	4	no	—
55	C₄×C₈×C₄₁	—	2	4	no	—
56	#95×C₄₁	—	2	4	no	—
57	#95×C₄₁	—	2	4	no	—
58	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
59	C₂×Q₈×C₄₁	—	2	8	no	—
60	C₂×Q₈×C₄₁	—	4	8	no	—
61	C₂²×C₄×C₄₁	—	2	4	no	—
62	C₂²×C₄×C₄₁	—	2	4	no	—
63	C₂²×C₄×C₄₁	—	4	4	no	—
64	C₂²×C₄×C₄₁	—	4	8	no	—
65	C₂²×C₄×C₄₁	—	4	4	no	—
66	C₂²×C₄×C₄₁	C₂²⋊C₄	1	8	yes	—
67	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₄₁	C₂×Q₈	1	4	yes	—
73	C₂²×C₄×C₄₁	#41	1	4	yes	—
74	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
75	M₁₆×C₄₁	—	2	16	no	—
76	C₂×C₈×C₄₁	—	2	4	no	—
77	C₂×C₈×C₄₁	—	2	4	no	—
78	C₂×C₈×C₄₁	C₂×Q₈	1	8	yes	—
79	C₄⋊C₄×C₄₁	—	2	16	no	—
80	C₄⋊C₄×C₄₁	—	2	8	no	—
81	C₄⋊C₄×C₄₁	—	2	4	no	—
82	C₄⋊C₄×C₄₁	—	2	8	no	—
83	C₄⋊C₄×C₄₁	—	4	4	no	—
84	C₄⋊C₄×C₄₁	C₂²×C₄	1	8	yes	—
85	C₂²⋊C₄×C₄₁	—	2	8	no	—
86	C₄²×C₄₁	—	2	8	no	—
87	C₄²×C₄₁	C₂²⋊C₄	1	8	yes	—
88	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
89	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
90	C₄²×C₄₁	C₂×D₄	1	2	yes	—
91	C₄²×C₄₁	C₂×D₄	1	2	yes	—
92	C₂³×C₄₁	—	2	4	no	—
93	C₂³×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
94	C₂³×C₄₁	#124	1	2	yes	—
95	C₂³×C₄₁	C₄⋊Q₈	1	2	yes	—
96	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
97	Q₈×C₄₁	—	4	16	no	—
98	C₂×C₄×C₄₁	—	2	4	no	—
99	C₂×C₄×C₄₁	—	2	4	no	—
100	C₂×C₄×C₄₁	—	2	8	no	—
101	C₂×C₄×C₄₁	—	4	8	no	—
102	C₂×C₄×C₄₁	—	4	8	no	—
103	C₂×C₄×C₄₁	—	4	4	no	—
104	C₂×C₄×C₄₁	—	4	8	no	—
105	C₂×C₄×C₄₁	—	8	8	no	—
106	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
107	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
108	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
109	C₂×C₄×C₄₁	C₄×Q₈	1	8	yes	—
110	C₂×C₄×C₄₁	C₂²⋊Q₈	1	8	yes	—
111	C₂×C₄×C₄₁	#124	1	2	yes	—
112	C₂×C₄×C₄₁	C₄⋊Q₈	1	2	yes	—
113	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₁	C₂²×Q₈	1	2	yes	—
116	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
117	C₈×C₄₁	—	2	8	no	—
118	#21323	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂²×C₄₁	—	2	4	no	—
120	C₂²×C₄₁	—	2	4	no	—
121	C₂²×C₄₁	—	2	4	no	—
122	C₂²×C₄₁	#391	1	4	yes	—
123	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
124	C₂²×C₄₁	C₂×#124	1	1	yes	characteristic
125	C₂²×C₄₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
126	C₂²×C₄₁	#575	1	4	yes	—
127	C₄×C₄₁	—	2	4	no	—
128	C₄×C₄₁	—	2	4	no	—
129	C₄×C₄₁	—	4	8	no	—
130	C₄×C₄₁	—	8	4	no	—
131	C₄×C₄₁	#391	1	4	yes	—
132	C₄×C₄₁	C₂×C₄×Q₈	1	2	yes	—
133	C₄×C₄₁	C₂×C₂²⋊Q₈	1	2	yes	—
134	C₂×#533	C₂×C₄₁	1	1	yes	characteristic
135	C₂×C₄×M₁₆	C₂×C₄₁	1	1	yes	characteristic
136	C₂×#382	C₂×C₄₁	1	1	yes	characteristic
137	#1993	C₂×C₄₁	1	4	yes	complemented
138	#1991	C₂×C₄₁	1	4	yes	complemented
139	#1736	C₂×C₄₁	1	4	yes	—
140	C₂×C₄₁	—	2	4	no	—
141	C₂×C₄₁	C₂×#391	1	1	yes	characteristic
142	C₂×C₄₁	#2896	1	2	yes	—
143	C₂×C₄₁	#3140	1	2	yes	—
144	C₂×C₄₁	#3166	1	2	yes	—
145	C₂²×M₁₆	C₂²×C₄₁	1	2	yes	—
146	#533	—	2	4	no	—
147	#533	C₄×C₄₁	1	8	yes	complemented
148	C₂×C₄⋊Q₈	C₂²×C₄₁	1	2	yes	—
149	C₂×#125	C₂²×C₄₁	1	2	yes	—
150	C₂×C₂²⋊Q₈	—	2	2	no	—
151	C₂×C₄×Q₈	—	2	2	no	—
152	C₂×#117	C₂²×C₄₁	1	2	yes	—
153	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
154	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
155	C₄×M₁₆	—	2	4	no	—
156	C₂×#97	C₂²×C₄₁	1	2	yes	—
157	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
158	#382	—	2	2	no	—
159	#382	C₂²×C₄₁	1	2	yes	—
160	#382	C₂²×C₄₁	1	2	yes	—
161	C₂×#95	—	2	2	no	—
162	#340	C₂²×C₄₁	1	16	yes	—
163	C₄₁	#21323	1	1	yes	characteristic, direct product factor, Sylow
164	C₂²×Q₈	—	2	2	no	—
165	C₂³×C₄	—	2	2	no	—
166	C₂³×C₄	D₄×C₄₁	1	2	yes	—
167	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
168	C₂×M₁₆	—	2	4	no	—
169	C₂×M₁₆	—	2	4	no	—
170	C₂×M₁₆	Q₈×C₄₁	1	8	yes	complemented
171	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
172	C₄⋊Q₈	—	2	4	no	—
173	C₄⋊Q₈	C₂×C₄×C₄₁	1	8	yes	complemented
174	#125	—	2	4	no	—
175	#125	C₂×C₄×C₄₁	1	8	yes	complemented
176	C₂²⋊Q₈	—	2	16	no	—
177	C₄×Q₈	—	2	16	no	—
178	#117	—	2	4	no	—
179	#117	C₂×C₄×C₄₁	1	8	yes	—
180	C₂×C₄⋊C₄	—	2	4	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	8	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
185	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
186	C₂×C₂²⋊C₄	—	2	2	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
189	C₂×C₄²	D₄×C₄₁	1	2	yes	—
190	C₂×C₄²	D₄×C₄₁	1	4	yes	—
191	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
192	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
193	#97	—	2	4	no	—
194	C₄×C₈	—	2	4	no	—
195	#95	—	2	4	no	—
196	#95	—	2	4	no	—
197	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
198	C₂×Q₈	—	2	8	no	—
199	C₂×Q₈	—	4	8	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	4	4	no	—
203	C₂²×C₄	—	4	8	no	—
204	C₂²×C₄	—	4	4	no	—
205	C₂²×C₄	C₂²⋊C₄×C₄₁	1	8	yes	—
206	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
207	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
208	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
209	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
211	C₂²×C₄	C₂×Q₈×C₄₁	1	4	yes	—
212	C₂²×C₄	#41×C₄₁	1	4	yes	—
213	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
214	M₁₆	—	2	16	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	C₂×Q₈×C₄₁	1	8	yes	—
218	C₄⋊C₄	—	2	16	no	—
219	C₄⋊C₄	—	2	8	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	2	8	no	—
222	C₄⋊C₄	—	4	4	no	—
223	C₄⋊C₄	C₂²×C₄×C₄₁	1	8	yes	—
224	C₂²⋊C₄	—	2	8	no	—
225	C₄²	—	2	8	no	—
226	C₄²	C₂²⋊C₄×C₄₁	1	8	yes	—
227	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
228	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
229	C₄²	C₂×D₄×C₄₁	1	2	yes	—
230	C₄²	C₂×D₄×C₄₁	1	2	yes	—
231	C₂³	—	2	4	no	—
232	C₂³	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
233	C₂³	#124×C₄₁	1	2	yes	—
234	C₂³	C₄⋊Q₈×C₄₁	1	2	yes	—
235	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
236	Q₈	—	4	16	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	4	8	no	—
241	C₂×C₄	—	4	8	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	8	no	—
244	C₂×C₄	—	8	8	no	—
245	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
246	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
247	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
248	C₂×C₄	C₄×Q₈×C₄₁	1	8	yes	—
249	C₂×C₄	C₂²⋊Q₈×C₄₁	1	8	yes	—
250	C₂×C₄	#124×C₄₁	1	2	yes	—
251	C₂×C₄	C₄⋊Q₈×C₄₁	1	2	yes	—
252	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
253	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
254	C₂×C₄	C₂²×Q₈×C₄₁	1	2	yes	—
255	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
256	C₈	—	2	8	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	2	4	no	—
259	C₂²	—	2	4	no	—
260	C₂²	C₄₁×#391	1	4	yes	—
261	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
262	C₂²	C₂×#124×C₄₁	1	1	yes	characteristic
263	C₂²	C₂×C₄⋊Q₈×C₄₁	1	1	yes	characteristic
264	C₂²	C₄₁×#575	1	4	yes	—
265	C₄	—	2	4	no	—
266	C₄	—	2	4	no	—
267	C₄	—	4	8	no	—
268	C₄	—	8	4	no	—
269	C₄	C₄₁×#391	1	4	yes	—
270	C₄	C₂×C₄×Q₈×C₄₁	1	2	yes	—
271	C₄	C₂×C₂²⋊Q₈×C₄₁	1	2	yes	—
272	C₂	—	2	4	no	—
273	C₂	C₂×C₄₁×#391	1	1	yes	characteristic
274	C₂	C₄₁×#2896	1	2	yes	—
275	C₂	C₄₁×#3140	1	2	yes	—
276	C₂	C₄₁×#3166	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×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
4. C₄₁×#1993	C₂	semidirect
5. C₄₁×#1991	C₂	semidirect
9. C₄₁×#533	C₄	semidirect
31. C₂×M₁₆×C₄₁	Q₈	semidirect
34. C₄⋊Q₈×C₄₁	C₂×C₄	semidirect
36. #125×C₄₁	C₂×C₄	semidirect
118. #21323	C₄₁	direct (N × H)
137. #1993	C₂×C₄₁	semidirect
138. #1991	C₂×C₄₁	semidirect
147. #533	C₄×C₄₁	semidirect
163. C₄₁	#21323	direct (N × H)
170. C₂×M₁₆	Q₈×C₄₁	semidirect
173. C₄⋊Q₈	C₂×C₄×C₄₁	semidirect
175. #125	C₂×C₄×C₄₁	semidirect

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×C₄×M₁₆×C₄₁	2
C₂×C₄₁×#382	2
C₄₁×#1993	2
C₄₁×#1991	2
C₄₁×#1736	2
#21323	41

Subgroups — C41×#21323 #23797524

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#21323 #23797524