Subgroups — C₄₁×#23667 #23799868

Order 10496.

874 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	C₂²×#107×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#424	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#2317	C₂	1	2	yes	maximal
4	C₄₁×#2087	C₂	1	4	yes	maximal, complemented
5	C₄₁×#2086	C₂	1	4	yes	maximal, complemented
6	C₄₁×#1736	C₂	1	2	yes	maximal
7	C₄₁×#1735	C₂	1	1	yes	maximal, characteristic
8	C₂²×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
10	C₂×#117×C₄₁	C₂²	1	2	yes	—
11	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
13	C₂×#107×C₄₁	—	2	4	no	—
14	C₂×#107×C₄₁	C₂²	1	2	yes	—
15	C₂×#107×C₄₁	C₂²	1	2	yes	—
16	C₄₁×#424	—	2	4	no	—
17	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
19	C₄₁×#389	C₂²	1	8	yes	—
20	C₄₁×#385	—	2	4	no	—
21	C₄₁×#384	—	2	2	no	—
22	C₄₁×#382	—	2	2	no	—
23	C₄₁×#340	C₂²	1	2	yes	—
24	C₄₁×#340	C₂²	1	8	yes	—
25	C₄₁×#340	C₂²	1	2	yes	—
26	C₂³×C₄×C₄₁	—	2	2	no	—
27	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₄₁	—	2	2	no	—
29	C₂×M₁₆×C₄₁	—	2	4	no	—
30	C₂²×C₈×C₄₁	—	2	4	no	—
31	C₂²×C₈×C₄₁	D₄	1	4	yes	—
32	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
33	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
35	#117×C₄₁	—	2	2	no	—
36	#117×C₄₁	—	2	4	no	—
37	#117×C₄₁	—	2	2	no	—
38	#117×C₄₁	Q₈	1	8	yes	—
39	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
40	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
41	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
42	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
43	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
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×C₂²⋊C₄×C₄₁	D₄	1	2	yes	—
49	C₂×C₄²×C₄₁	—	2	2	no	—
50	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
51	#107×C₄₁	—	2	4	no	—
52	#107×C₄₁	—	2	4	no	—
53	C₄⋊C₈×C₄₁	—	2	4	no	—
54	C₄⋊C₈×C₄₁	—	2	4	no	—
55	#95×C₄₁	—	2	8	no	—
56	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₁	—	2	4	no	—
58	C₂²×C₄×C₄₁	—	2	4	no	—
59	C₂²×C₄×C₄₁	—	4	4	no	—
60	C₂²×C₄×C₄₁	—	8	4	no	—
61	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₁	C₂×Q₈	1	4	yes	—
66	C₂²×C₄×C₄₁	#41	1	2	yes	—
67	C₂²×C₄×C₄₁	#41	1	4	yes	—
68	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
69	M₁₆×C₄₁	—	4	4	no	—
70	C₂×C₈×C₄₁	—	2	8	no	—
71	C₂×C₈×C₄₁	—	2	2	no	—
72	C₂×C₈×C₄₁	—	2	2	no	—
73	C₂×C₈×C₄₁	—	2	2	no	—
74	C₂×C₈×C₄₁	—	2	4	no	—
75	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
77	C₄⋊C₄×C₄₁	—	2	2	no	—
78	C₄⋊C₄×C₄₁	—	2	2	no	—
79	C₄⋊C₄×C₄₁	—	2	4	no	—
80	C₄⋊C₄×C₄₁	—	2	4	no	—
81	C₄⋊C₄×C₄₁	—	2	4	no	—
82	C₄⋊C₄×C₄₁	—	4	4	no	—
83	C₄⋊C₄×C₄₁	—	8	8	no	—
84	C₄⋊C₄×C₄₁	C₂×Q₈	1	8	yes	—
85	C₂²⋊C₄×C₄₁	—	2	4	no	—
86	C₂²⋊C₄×C₄₁	—	2	8	no	—
87	C₄²×C₄₁	—	2	8	no	—
88	C₄²×C₄₁	—	2	2	no	—
89	C₄²×C₄₁	—	2	2	no	—
90	C₂³×C₄₁	—	2	2	no	—
91	C₂³×C₄₁	—	2	2	no	—
92	C₂³×C₄₁	#121	1	2	yes	—
93	C₂³×C₄₁	C₂²⋊Q₈	1	4	yes	—
94	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₄₁	—	2	2	no	—
96	C₂×C₄×C₄₁	—	2	2	no	—
97	C₂×C₄×C₄₁	—	2	8	no	—
98	C₂×C₄×C₄₁	—	2	4	no	—
99	C₂×C₄×C₄₁	—	2	4	no	—
100	C₂×C₄×C₄₁	—	4	4	no	—
101	C₂×C₄×C₄₁	—	4	4	no	—
102	C₂×C₄×C₄₁	—	4	4	no	—
103	C₂×C₄×C₄₁	—	4	8	no	—
104	C₂×C₄×C₄₁	—	8	8	no	—
105	C₂×C₄×C₄₁	#121	1	2	yes	—
106	C₂×C₄×C₄₁	C₂²⋊Q₈	1	4	yes	—
107	C₂×C₄×C₄₁	#123	1	4	yes	—
108	C₂×C₄×C₄₁	#125	1	8	yes	—
109	C₂×C₄×C₄₁	C₄⋊Q₈	1	4	yes	—
110	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₄₁	C₂²×Q₈	1	2	yes	—
112	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
113	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
114	C₈×C₄₁	—	2	2	no	—
115	C₈×C₄₁	—	2	2	no	—
116	C₈×C₄₁	—	4	2	no	—
117	#23667	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂²×C₄₁	—	2	8	no	—
119	C₂²×C₄₁	—	2	2	no	—
120	C₂²×C₄₁	—	2	2	no	—
121	C₂²×C₄₁	#398	1	4	yes	—
122	C₂²×C₄₁	C₂×#121	1	1	yes	characteristic
123	C₂²×C₄₁	C₂×C₂²⋊Q₈	1	2	yes	—
124	C₂²×C₄₁	#576	1	2	yes	—
125	C₂²×C₄₁	#578	1	2	yes	—
126	C₄×C₄₁	—	4	8	no	—
127	C₄×C₄₁	—	4	2	no	—
128	C₄×C₄₁	—	4	2	no	—
129	C₄×C₄₁	—	8	4	no	—
130	C₄×C₄₁	#398	1	4	yes	—
131	C₄×C₄₁	C₂×#123	1	1	yes	characteristic
132	C₄×C₄₁	C₂×#125	1	2	yes	—
133	C₄×C₄₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
134	C₂²×#107	C₂×C₄₁	1	1	yes	characteristic
135	C₂×#424	C₂×C₄₁	1	1	yes	characteristic
136	#2317	C₂×C₄₁	1	2	yes	—
137	#2087	C₂×C₄₁	1	4	yes	complemented
138	#2086	C₂×C₄₁	1	4	yes	complemented
139	#1736	C₂×C₄₁	1	2	yes	—
140	#1735	C₂×C₄₁	1	1	yes	characteristic
141	C₂×C₄₁	—	2	4	no	—
142	C₂×C₄₁	C₂×#398	1	1	yes	characteristic
143	C₂×C₄₁	#3058	1	1	yes	characteristic
144	C₂×C₄₁	#3059	1	1	yes	characteristic
145	C₂×C₄₁	#3083	1	4	yes	—
146	C₂²×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
147	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
148	C₂×#117	C₂²×C₄₁	1	2	yes	—
149	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
150	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
151	C₂×#107	—	2	4	no	—
152	C₂×#107	C₂²×C₄₁	1	2	yes	—
153	C₂×#107	C₂²×C₄₁	1	2	yes	—
154	#424	—	2	4	no	—
155	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
156	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
157	#389	C₂²×C₄₁	1	8	yes	—
158	#385	—	2	4	no	—
159	#384	—	2	2	no	—
160	#382	—	2	2	no	—
161	#340	C₂²×C₄₁	1	2	yes	—
162	#340	C₂²×C₄₁	1	8	yes	—
163	#340	C₂²×C₄₁	1	2	yes	—
164	C₄₁	#23667	1	1	yes	characteristic, direct product factor, Sylow
165	C₂³×C₄	—	2	2	no	—
166	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	4	no	—
169	C₂²×C₈	—	2	4	no	—
170	C₂²×C₈	D₄×C₄₁	1	4	yes	—
171	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
172	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
173	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
174	#117	—	2	2	no	—
175	#117	—	2	4	no	—
176	#117	—	2	2	no	—
177	#117	Q₈×C₄₁	1	8	yes	—
178	C₂×C₄⋊C₄	—	2	4	no	—
179	C₂×C₄⋊C₄	—	2	4	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	8	no	—
182	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
183	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
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×C₂²⋊C₄	D₄×C₄₁	1	2	yes	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
190	#107	—	2	4	no	—
191	#107	—	2	4	no	—
192	C₄⋊C₈	—	2	4	no	—
193	C₄⋊C₈	—	2	4	no	—
194	#95	—	2	8	no	—
195	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	4	4	no	—
199	C₂²×C₄	—	8	4	no	—
200	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
201	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
202	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
204	C₂²×C₄	C₂×Q₈×C₄₁	1	4	yes	—
205	C₂²×C₄	#41×C₄₁	1	2	yes	—
206	C₂²×C₄	#41×C₄₁	1	4	yes	—
207	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
208	M₁₆	—	4	4	no	—
209	C₂×C₈	—	2	8	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
215	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	4	4	no	—
222	C₄⋊C₄	—	8	8	no	—
223	C₄⋊C₄	C₂×Q₈×C₄₁	1	8	yes	—
224	C₂²⋊C₄	—	2	4	no	—
225	C₂²⋊C₄	—	2	8	no	—
226	C₄²	—	2	8	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	—	2	2	no	—
231	C₂³	#121×C₄₁	1	2	yes	—
232	C₂³	C₂²⋊Q₈×C₄₁	1	4	yes	—
233	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	8	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	8	no	—
243	C₂×C₄	—	8	8	no	—
244	C₂×C₄	#121×C₄₁	1	2	yes	—
245	C₂×C₄	C₂²⋊Q₈×C₄₁	1	4	yes	—
246	C₂×C₄	#123×C₄₁	1	4	yes	—
247	C₂×C₄	#125×C₄₁	1	8	yes	—
248	C₂×C₄	C₄⋊Q₈×C₄₁	1	4	yes	—
249	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
250	C₂×C₄	C₂²×Q₈×C₄₁	1	2	yes	—
251	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
252	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
253	C₈	—	2	2	no	—
254	C₈	—	2	2	no	—
255	C₈	—	4	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	8	no	—
259	C₂²	C₄₁×#398	1	4	yes	—
260	C₂²	C₂×#121×C₄₁	1	1	yes	characteristic
261	C₂²	C₂×C₂²⋊Q₈×C₄₁	1	2	yes	—
262	C₂²	C₄₁×#576	1	2	yes	—
263	C₂²	C₄₁×#578	1	2	yes	—
264	C₄	—	4	2	no	—
265	C₄	—	4	2	no	—
266	C₄	—	4	8	no	—
267	C₄	—	8	4	no	—
268	C₄	C₄₁×#398	1	4	yes	—
269	C₄	C₂×#123×C₄₁	1	1	yes	characteristic
270	C₄	C₂×#125×C₄₁	1	2	yes	—
271	C₄	C₂×C₄⋊Q₈×C₄₁	1	1	yes	characteristic
272	C₂	—	2	4	no	—
273	C₂	C₂×C₄₁×#398	1	1	yes	characteristic
274	C₂	C₄₁×#3058	1	1	yes	characteristic
275	C₂	C₄₁×#3059	1	1	yes	characteristic
276	C₂	C₄₁×#3083	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂⁴×C₄₁	C₂×D₄
LCS	G ▷ C₂²×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
4. C₄₁×#2087	C₂	semidirect
5. C₄₁×#2086	C₂	semidirect
117. #23667	C₄₁	direct (N × H)
137. #2087	C₂×C₄₁	semidirect
138. #2086	C₂×C₄₁	semidirect
164. C₄₁	#23667	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#107×C₄₁	2
C₂×C₄₁×#424	2
C₄₁×#2317	2
C₄₁×#2087	2
C₄₁×#2086	2
C₄₁×#1736	2
C₄₁×#1735	2
#23667	41

Subgroups — C41×#23667 #23799868

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#23667 #23799868