Subgroups — C₄₁×#9730 #23785931

Order 10496.

454 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#500	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#1987	C₂	1	2	yes	maximal
3	C₈×C₄⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₁×#335	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₁×#333	C₂	1	2	yes	maximal
6	C₂×C₄⋊Q₈×C₄₁	C₂²	1	2	yes	—
7	C₄₁×#500	—	2	2	no	—
8	C₄₁×#500	C₄	1	4	yes	complemented
9	C₂×#107×C₄₁	—	2	1	no	—
10	C₂×#107×C₄₁	—	2	1	no	—
11	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₄₁	—	2	1	no	—
14	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
16	C₄₁×#340	—	2	4	no	—
17	C₄₁×#336	—	2	1	no	—
18	C₄₁×#335	—	2	2	no	—
19	C₄₁×#333	—	2	4	no	—
20	C₂²×Q₈×C₄₁	—	2	2	no	—
21	C₂²×C₈×C₄₁	—	2	1	no	—
22	C₂²×C₈×C₄₁	—	2	2	no	—
23	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
25	C₄⋊Q₈×C₄₁	—	2	2	no	—
26	C₄⋊Q₈×C₄₁	—	2	2	no	—
27	C₄⋊Q₈×C₄₁	—	2	2	no	—
28	C₄⋊Q₈×C₄₁	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
30	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
31	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
32	C₂×C₄²×C₄₁	—	2	1	no	—
33	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
34	#107×C₄₁	—	2	4	no	—
35	#107×C₄₁	—	2	4	no	—
36	C₄⋊C₈×C₄₁	—	2	2	no	—
37	C₄⋊C₈×C₄₁	—	2	2	no	—
38	C₄⋊C₈×C₄₁	—	2	1	no	—
39	C₄⋊C₈×C₄₁	D₄	1	2	yes	—
40	C₄×C₈×C₄₁	—	4	2	no	—
41	C₄×C₈×C₄₁	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₄₁	D₄	1	2	yes	—
43	#95×C₄₁	—	2	1	no	—
44	C₂×Q₈×C₄₁	—	2	4	no	—
45	C₂×Q₈×C₄₁	—	4	2	no	—
46	C₂×Q₈×C₄₁	—	8	4	no	—
47	C₂²×C₄×C₄₁	—	2	2	no	—
48	C₂²×C₄×C₄₁	—	4	2	no	—
49	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
52	C₂×C₈×C₄₁	—	2	1	no	—
53	C₂×C₈×C₄₁	—	2	1	no	—
54	C₂×C₈×C₄₁	—	2	2	no	—
55	C₂×C₈×C₄₁	—	4	1	no	—
56	C₂×C₈×C₄₁	—	4	1	no	—
57	C₂×C₈×C₄₁	—	4	2	no	—
58	C₂×C₈×C₄₁	—	4	4	no	—
59	C₂×C₈×C₄₁	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₄₁	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₄₁	SD₁₆	1	2	yes	—
62	C₂×C₈×C₄₁	Q₁₆	1	2	yes	—
63	C₄⋊C₄×C₄₁	—	2	2	no	—
64	C₄⋊C₄×C₄₁	—	2	4	no	—
65	C₄⋊C₄×C₄₁	—	2	4	no	—
66	C₄⋊C₄×C₄₁	—	4	2	no	—
67	C₄⋊C₄×C₄₁	—	4	2	no	—
68	C₄⋊C₄×C₄₁	Q₁₆	1	4	yes	—
69	C₄²×C₄₁	—	4	1	no	—
70	C₄²×C₄₁	—	4	1	no	—
71	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₄₁	#41	1	1	yes	characteristic
75	C₂³×C₄₁	#121	1	1	yes	characteristic
76	Q₈×C₄₁	—	8	4	no	—
77	Q₈×C₄₁	—	8	4	no	—
78	C₂×C₄×C₄₁	—	2	2	no	—
79	C₂×C₄×C₄₁	—	2	4	no	—
80	C₂×C₄×C₄₁	—	4	2	no	—
81	C₂×C₄×C₄₁	—	4	2	no	—
82	C₂×C₄×C₄₁	—	4	4	no	—
83	C₂×C₄×C₄₁	—	8	4	no	—
84	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
88	C₂×C₄×C₄₁	#124	1	1	yes	characteristic
89	C₂×C₄×C₄₁	#127	1	1	yes	characteristic
90	C₂×C₄×C₄₁	C₂×SD₁₆	1	1	yes	characteristic
91	C₂×C₄×C₄₁	C₂×Q₁₆	1	1	yes	characteristic
92	C₂×C₄×C₄₁	C₂×Q₁₆	1	2	yes	—
93	C₂×C₄×C₄₁	#135	1	2	yes	—
94	C₂×C₄×C₄₁	#137	1	1	yes	characteristic
95	C₂×C₄×C₄₁	#137	1	1	yes	characteristic
96	C₈×C₄₁	—	2	2	no	—
97	C₈×C₄₁	—	4	2	no	—
98	C₈×C₄₁	—	8	2	no	—
99	C₈×C₄₁	#103	1	4	yes	—
100	#9730	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂²×C₄₁	#390	1	1	yes	characteristic
102	C₂²×C₄₁	#461	1	1	yes	characteristic
103	C₂²×C₄₁	#462	1	1	yes	characteristic
104	C₂²×C₄₁	#467	1	2	yes	—
105	C₂²×C₄₁	#472	1	2	yes	—
106	C₄×C₄₁	—	2	1	no	—
107	C₄×C₄₁	—	2	1	no	—
108	C₄×C₄₁	—	4	4	no	—
109	C₄×C₄₁	—	8	4	no	—
110	C₄×C₄₁	C₂×#103	1	1	yes	characteristic
111	C₄×C₄₁	#419	1	1	yes	characteristic
112	C₄×C₄₁	C₄×Q₁₆	1	2	yes	—
113	C₄×C₄₁	#461	1	1	yes	characteristic
114	C₄×C₄₁	#462	1	1	yes	characteristic
115	C₄×C₄₁	#494	1	1	yes	characteristic
116	C₄×C₄₁	#495	1	1	yes	characteristic
117	C₂×#500	C₂×C₄₁	1	1	yes	characteristic
118	#1987	C₂×C₄₁	1	2	yes	—
119	C₈×C₄⋊C₄	C₂×C₄₁	1	1	yes	characteristic
120	C₂×#335	C₂×C₄₁	1	1	yes	characteristic
121	C₂×#333	C₂×C₄₁	1	2	yes	—
122	C₂×C₄₁	#1675	1	2	yes	—
123	C₂×C₄₁	#1676	1	2	yes	—
124	C₂×C₄₁	#1969	1	1	yes	characteristic
125	C₂×C₄₁	#1978	1	1	yes	characteristic
126	C₂×C₄₁	#1981	1	1	yes	characteristic
127	C₂×C₄⋊Q₈	C₂²×C₄₁	1	2	yes	—
128	#500	—	2	2	no	—
129	#500	C₄×C₄₁	1	4	yes	complemented
130	C₂×#107	—	2	1	no	—
131	C₂×#107	—	2	1	no	—
132	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
133	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
134	C₂×C₄×C₈	—	2	1	no	—
135	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
136	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
137	#340	—	2	4	no	—
138	#336	—	2	1	no	—
139	#335	—	2	2	no	—
140	#333	—	2	4	no	—
141	C₄₁	#9730	1	1	yes	characteristic, direct product factor, Sylow
142	C₂²×Q₈	—	2	2	no	—
143	C₂²×C₈	—	2	1	no	—
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
146	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
147	C₄⋊Q₈	—	2	2	no	—
148	C₄⋊Q₈	—	2	2	no	—
149	C₄⋊Q₈	—	2	2	no	—
150	C₄⋊Q₈	C₂×C₄×C₄₁	1	4	yes	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
154	C₂×C₄²	—	2	1	no	—
155	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
156	#107	—	2	4	no	—
157	#107	—	2	4	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	—	2	1	no	—
161	C₄⋊C₈	D₄×C₄₁	1	2	yes	—
162	C₄×C₈	—	4	2	no	—
163	C₄×C₈	C₂×C₄×C₄₁	1	2	yes	—
164	C₄×C₈	D₄×C₄₁	1	2	yes	—
165	#95	—	2	1	no	—
166	C₂×Q₈	—	2	4	no	—
167	C₂×Q₈	—	4	2	no	—
168	C₂×Q₈	—	8	4	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	4	2	no	—
171	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
172	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
173	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	4	1	no	—
178	C₂×C₈	—	4	1	no	—
179	C₂×C₈	—	4	2	no	—
180	C₂×C₈	—	4	4	no	—
181	C₂×C₈	C₂²⋊C₄×C₄₁	1	2	yes	—
182	C₂×C₈	C₂²⋊C₄×C₄₁	1	2	yes	—
183	C₂×C₈	SD₁₆×C₄₁	1	2	yes	—
184	C₂×C₈	Q₁₆×C₄₁	1	2	yes	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	Q₁₆×C₄₁	1	4	yes	—
191	C₄²	—	4	1	no	—
192	C₄²	—	4	1	no	—
193	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
194	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
195	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
196	C₄²	#41×C₄₁	1	1	yes	characteristic
197	C₂³	#121×C₄₁	1	1	yes	characteristic
198	Q₈	—	8	4	no	—
199	Q₈	—	8	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	—	8	4	no	—
206	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
207	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
208	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
209	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
210	C₂×C₄	#124×C₄₁	1	1	yes	characteristic
211	C₂×C₄	#127×C₄₁	1	1	yes	characteristic
212	C₂×C₄	C₂×SD₁₆×C₄₁	1	1	yes	characteristic
213	C₂×C₄	C₂×Q₁₆×C₄₁	1	1	yes	characteristic
214	C₂×C₄	C₂×Q₁₆×C₄₁	1	2	yes	—
215	C₂×C₄	#135×C₄₁	1	2	yes	—
216	C₂×C₄	#137×C₄₁	1	1	yes	characteristic
217	C₂×C₄	#137×C₄₁	1	1	yes	characteristic
218	C₈	—	2	2	no	—
219	C₈	—	4	2	no	—
220	C₈	—	8	2	no	—
221	C₈	#103×C₄₁	1	4	yes	—
222	C₂²	C₄₁×#390	1	1	yes	characteristic
223	C₂²	C₄₁×#461	1	1	yes	characteristic
224	C₂²	C₄₁×#462	1	1	yes	characteristic
225	C₂²	C₄₁×#467	1	2	yes	—
226	C₂²	C₄₁×#472	1	2	yes	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	4	4	no	—
230	C₄	—	8	4	no	—
231	C₄	C₂×#103×C₄₁	1	1	yes	characteristic
232	C₄	C₄₁×#419	1	1	yes	characteristic
233	C₄	C₄×Q₁₆×C₄₁	1	2	yes	—
234	C₄	C₄₁×#461	1	1	yes	characteristic
235	C₄	C₄₁×#462	1	1	yes	characteristic
236	C₄	C₄₁×#494	1	1	yes	characteristic
237	C₄	C₄₁×#495	1	1	yes	characteristic
238	C₂	C₄₁×#1675	1	2	yes	—
239	C₂	C₄₁×#1676	1	2	yes	—
240	C₂	C₄₁×#1969	1	1	yes	characteristic
241	C₂	C₄₁×#1978	1	1	yes	characteristic
242	C₂	C₄₁×#1981	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₄₁	#121
commutator	C₄²	C₂²×C₄×C₄₁
Frattini, cosocle	C₂×C₄²	C₂³×C₄₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₄₁×#500	C₄	semidirect
100. #9730	C₄₁	direct (N × H)
129. #500	C₄×C₄₁	semidirect
141. C₄₁	#9730	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×#500	2
C₄₁×#1987	2
C₈×C₄⋊C₄×C₄₁	2
C₂×C₄₁×#335	2
C₂×C₄₁×#333	2
#9730	41

Subgroups — C41×#9730 #23785931

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#9730 #23785931