Subgroups — C₄₁×#21818 #23798019

Order 10496.

1306 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#431	C₂	1	2	yes	maximal
3	C₄₁×#1984	C₂	1	8	yes	maximal, complemented
4	C₄₁×#1817	C₂	1	4	yes	maximal
5	C₂³×Q₈×C₄₁	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₄₁	—	2	8	no	—
7	C₂×C₄×Q₈×C₄₁	C₄	1	4	yes	—
8	C₂×C₄×Q₈×C₄₁	C₂²	1	4	yes	—
9	C₂²×C₄⋊C₄×C₄₁	C₄	1	2	yes	—
10	C₂²×C₄⋊C₄×C₄₁	C₄	1	1	yes	characteristic
11	C₂²×C₄²×C₄₁	C₂²	1	2	yes	—
12	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
13	C₄₁×#431	—	2	8	no	—
14	C₂×C₄⋊C₈×C₄₁	C₂²	1	4	yes	—
15	C₂×C₂²⋊C₈×C₄₁	C₂²	1	4	yes	—
16	C₂×C₄×C₈×C₄₁	C₂²	1	4	yes	—
17	C₄₁×#336	C₂²	1	16	yes	—
18	C₂²×Q₈×C₄₁	—	2	4	no	—
19	C₂²×Q₈×C₄₁	—	2	4	no	—
20	C₂²×Q₈×C₄₁	C₈	1	8	yes	complemented
21	C₂²×Q₈×C₄₁	C₂×C₄	1	4	yes	—
22	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₄₁	C₂³	1	8	yes	—
28	C₄×Q₈×C₄₁	—	2	16	no	—
29	C₄×Q₈×C₄₁	—	2	8	no	—
30	C₄×Q₈×C₄₁	—	2	8	no	—
31	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
32	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
33	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
34	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	8	yes	—
35	C₂×C₄²×C₄₁	—	2	8	no	—
36	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₄₁	D₄	1	8	yes	—
39	C₂×C₄²×C₄₁	C₂³	1	4	yes	—
40	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
41	C₄⋊C₈×C₄₁	—	2	8	no	—
42	C₂²⋊C₈×C₄₁	—	2	8	no	—
43	C₄×C₈×C₄₁	—	2	8	no	—
44	C₂⁴×C₄₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×Q₈×C₄₁	—	2	32	no	—
46	C₂×Q₈×C₄₁	—	2	8	no	—
47	C₂×Q₈×C₄₁	—	2	8	no	—
48	C₂×Q₈×C₄₁	C₂×C₈	1	16	yes	—
49	C₂²×C₄×C₄₁	—	2	4	no	—
50	C₂²×C₄×C₄₁	—	2	4	no	—
51	C₂²×C₄×C₄₁	—	2	4	no	—
52	C₂²×C₄×C₄₁	—	2	4	no	—
53	C₂²×C₄×C₄₁	—	4	2	no	—
54	C₂²×C₄×C₄₁	—	4	4	no	—
55	C₂²×C₄×C₄₁	C₂²⋊C₄	1	4	yes	—
56	C₂²×C₄×C₄₁	C₂²⋊C₄	1	4	yes	—
57	C₂²×C₄×C₄₁	C₂×C₈	1	8	yes	—
58	C₂²×C₄×C₄₁	C₂×C₈	1	4	yes	—
59	C₂²×C₄×C₄₁	M₁₆	1	4	yes	—
60	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
66	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
67	C₂×C₈×C₄₁	—	2	8	no	—
68	C₂×C₈×C₄₁	—	2	16	no	—
69	C₄⋊C₄×C₄₁	—	2	16	no	—
70	C₄⋊C₄×C₄₁	—	2	8	no	—
71	C₄²×C₄₁	—	2	8	no	—
72	C₄²×C₄₁	—	2	8	no	—
73	C₄²×C₄₁	C₂²⋊C₄	1	8	yes	—
74	C₄²×C₄₁	C₂×D₄	1	4	yes	—
75	C₄²×C₄₁	C₂×D₄	1	4	yes	—
76	C₂³×C₄₁	—	2	2	no	—
77	C₂³×C₄₁	—	2	2	no	—
78	C₂³×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
79	C₂³×C₄₁	C₂²×C₈	1	2	yes	—
80	C₂³×C₄₁	C₂×M₁₆	1	2	yes	—
81	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
82	Q₈×C₄₁	—	2	32	no	—
83	C₂×C₄×C₄₁	—	2	4	no	—
84	C₂×C₄×C₄₁	—	2	2	no	—
85	C₂×C₄×C₄₁	—	2	4	no	—
86	C₂×C₄×C₄₁	—	2	4	no	—
87	C₂×C₄×C₄₁	—	2	16	no	—
88	C₂×C₄×C₄₁	—	2	8	no	—
89	C₂×C₄×C₄₁	—	2	4	no	—
90	C₂×C₄×C₄₁	—	2	8	no	—
91	C₂×C₄×C₄₁	—	4	4	no	—
92	C₂×C₄×C₄₁	—	4	8	no	—
93	C₂×C₄×C₄₁	C₂²⋊C₈	1	16	yes	—
94	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
95	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
96	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
97	C₂×C₄×C₄₁	C₂²×C₈	1	8	yes	—
98	C₂×C₄×C₄₁	C₂²×C₈	1	4	yes	—
99	C₂×C₄×C₄₁	C₂×M₁₆	1	4	yes	—
100	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
101	C₂×C₄×C₄₁	2₋¹⁺⁴	1	4	yes	—
102	C₈×C₄₁	—	4	8	no	—
103	#21818	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂²×C₄₁	—	2	2	no	—
105	C₂²×C₄₁	—	2	2	no	—
106	C₂²×C₄₁	—	2	4	no	—
107	C₂²×C₄₁	—	2	4	no	—
108	C₂²×C₄₁	C₂×C₂²⋊C₈	1	4	yes	—
109	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
110	C₂²×C₄₁	#519	1	2	yes	—
111	C₂²×C₄₁	C₂³×C₈	1	1	yes	characteristic
112	C₂²×C₄₁	C₂²×M₁₆	1	1	yes	characteristic
113	C₂²×C₄₁	#568	1	2	yes	—
114	C₄×C₄₁	—	2	2	no	—
115	C₄×C₄₁	—	2	8	no	—
116	C₄×C₄₁	—	2	4	no	—
117	C₄×C₄₁	—	4	4	no	—
118	C₄×C₄₁	C₂×C₂²⋊C₈	1	8	yes	—
119	C₄×C₄₁	#536	1	4	yes	—
120	C₂²×C₄×Q₈	C₂×C₄₁	1	1	yes	characteristic
121	C₂×#431	C₂×C₄₁	1	2	yes	—
122	#1984	C₂×C₄₁	1	8	yes	complemented
123	#1817	C₂×C₄₁	1	4	yes	—
124	C₂×C₄₁	—	2	4	no	—
125	C₂×C₄₁	#2310	1	1	yes	characteristic
126	C₂×C₄₁	C₂²×C₂²⋊C₈	1	1	yes	characteristic
127	C₂×C₄₁	#2879	1	1	yes	characteristic
128	C₂×C₄₁	#2982	1	2	yes	—
129	C₂×C₄₁	#2983	1	2	yes	—
130	C₂³×Q₈	C₄×C₄₁	1	1	yes	characteristic
131	C₂×C₄×Q₈	—	2	8	no	—
132	C₂×C₄×Q₈	C₄×C₄₁	1	4	yes	—
133	C₂×C₄×Q₈	C₂²×C₄₁	1	4	yes	—
134	C₂²×C₄⋊C₄	C₄×C₄₁	1	2	yes	—
135	C₂²×C₄⋊C₄	C₄×C₄₁	1	1	yes	characteristic
136	C₂²×C₄²	C₂²×C₄₁	1	2	yes	—
137	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
138	#431	—	2	8	no	—
139	C₂×C₄⋊C₈	C₂²×C₄₁	1	4	yes	—
140	C₂×C₂²⋊C₈	C₂²×C₄₁	1	4	yes	—
141	C₂×C₄×C₈	C₂²×C₄₁	1	4	yes	—
142	#336	C₂²×C₄₁	1	16	yes	—
143	C₄₁	#21818	1	1	yes	characteristic, direct product factor, Sylow
144	C₂²×Q₈	—	2	4	no	—
145	C₂²×Q₈	—	2	4	no	—
146	C₂²×Q₈	C₈×C₄₁	1	8	yes	complemented
147	C₂²×Q₈	C₂×C₄×C₄₁	1	4	yes	—
148	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
149	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
150	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
151	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
152	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
153	C₂²×C₈	C₂³×C₄₁	1	8	yes	—
154	C₄×Q₈	—	2	16	no	—
155	C₄×Q₈	—	2	8	no	—
156	C₄×Q₈	—	2	8	no	—
157	C₂×C₄⋊C₄	—	2	8	no	—
158	C₂×C₄⋊C₄	—	2	4	no	—
159	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
160	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	8	yes	—
161	C₂×C₄²	—	2	8	no	—
162	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
163	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
164	C₂×C₄²	D₄×C₄₁	1	8	yes	—
165	C₂×C₄²	C₂³×C₄₁	1	4	yes	—
166	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
167	C₄⋊C₈	—	2	8	no	—
168	C₂²⋊C₈	—	2	8	no	—
169	C₄×C₈	—	2	8	no	—
170	C₂⁴	C₂²×C₄×C₄₁	1	1	yes	characteristic
171	C₂×Q₈	—	2	32	no	—
172	C₂×Q₈	—	2	8	no	—
173	C₂×Q₈	—	2	8	no	—
174	C₂×Q₈	C₂×C₈×C₄₁	1	16	yes	—
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	—	4	4	no	—
181	C₂²×C₄	C₂²⋊C₄×C₄₁	1	4	yes	—
182	C₂²×C₄	C₂²⋊C₄×C₄₁	1	4	yes	—
183	C₂²×C₄	C₂×C₈×C₄₁	1	8	yes	—
184	C₂²×C₄	C₂×C₈×C₄₁	1	4	yes	—
185	C₂²×C₄	M₁₆×C₄₁	1	4	yes	—
186	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
187	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
188	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
189	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
190	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
191	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
192	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
193	C₂×C₈	—	2	8	no	—
194	C₂×C₈	—	2	16	no	—
195	C₄⋊C₄	—	2	16	no	—
196	C₄⋊C₄	—	2	8	no	—
197	C₄²	—	2	8	no	—
198	C₄²	—	2	8	no	—
199	C₄²	C₂²⋊C₄×C₄₁	1	8	yes	—
200	C₄²	C₂×D₄×C₄₁	1	4	yes	—
201	C₄²	C₂×D₄×C₄₁	1	4	yes	—
202	C₂³	—	2	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
205	C₂³	C₂²×C₈×C₄₁	1	2	yes	—
206	C₂³	C₂×M₁₆×C₄₁	1	2	yes	—
207	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
208	Q₈	—	2	32	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	16	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	8	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	4	8	no	—
219	C₂×C₄	C₂²⋊C₈×C₄₁	1	16	yes	—
220	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
221	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
222	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
223	C₂×C₄	C₂²×C₈×C₄₁	1	8	yes	—
224	C₂×C₄	C₂²×C₈×C₄₁	1	4	yes	—
225	C₂×C₄	C₂×M₁₆×C₄₁	1	4	yes	—
226	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
227	C₂×C₄	2₋¹⁺⁴×C₄₁	1	4	yes	—
228	C₈	—	4	8	no	—
229	C₂²	—	2	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	4	no	—
232	C₂²	—	2	4	no	—
233	C₂²	C₂×C₂²⋊C₈×C₄₁	1	4	yes	—
234	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
235	C₂²	C₄₁×#519	1	2	yes	—
236	C₂²	C₂³×C₈×C₄₁	1	1	yes	characteristic
237	C₂²	C₂²×M₁₆×C₄₁	1	1	yes	characteristic
238	C₂²	C₄₁×#568	1	2	yes	—
239	C₄	—	2	2	no	—
240	C₄	—	2	8	no	—
241	C₄	—	2	4	no	—
242	C₄	—	4	4	no	—
243	C₄	C₂×C₂²⋊C₈×C₄₁	1	8	yes	—
244	C₄	C₄₁×#536	1	4	yes	—
245	C₂	—	2	4	no	—
246	C₂	C₄₁×#2310	1	1	yes	characteristic
247	C₂	C₂²×C₂²⋊C₈×C₄₁	1	1	yes	characteristic
248	C₂	C₄₁×#2879	1	1	yes	characteristic
249	C₂	C₄₁×#2982	1	2	yes	—
250	C₂	C₄₁×#2983	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₄₁	C₂⁴
commutator	C₂²	C₂³×C₈×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
socle	C₂³×C₄₁	C₂³×C₄
prime radical	C₂⁴×C₄₁	C₂²×C₄
LCS	G ▷ C₂² ▷ 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×#1984	C₂	semidirect
20. C₂²×Q₈×C₄₁	C₈	semidirect
103. #21818	C₄₁	direct (N × H)
122. #1984	C₂×C₄₁	semidirect
143. C₄₁	#21818	direct (N × H)
146. C₂²×Q₈	C₈×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×Q₈×C₄₁	2
C₂×C₄₁×#431	2
C₄₁×#1984	2
C₄₁×#1817	2
#21818	41

Subgroups — C41×#21818 #23798019

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#21818 #23798019