Subgroups — C₄₁×#21790 #23797991

Order 10496.

1754 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×C₄×D₄×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#432	C₂	1	2	yes	maximal, complemented
3	C₄₁×#1955	C₂	1	8	yes	maximal, complemented
4	C₄₁×#1779	C₂	1	4	yes	maximal, complemented
5	C₂³×D₄×C₄₁	C₄	1	1	yes	characteristic
6	C₂⁴×C₄×C₄₁	C₂²	1	2	yes	—
7	C₂×C₄×D₄×C₄₁	—	2	8	no	—
8	C₂×C₄×D₄×C₄₁	C₄	1	4	yes	—
9	C₂×C₄×D₄×C₄₁	C₂²	1	4	yes	—
10	C₂²×C₄⋊C₄×C₄₁	C₄	1	1	yes	characteristic
11	C₂²×C₂²⋊C₄×C₄₁	C₄	1	2	yes	—
12	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
13	C₄₁×#432	—	2	8	no	—
14	C₂×C₄⋊C₈×C₄₁	C₂²	1	4	yes	complemented
15	C₂×C₂²⋊C₈×C₄₁	C₂²	1	4	yes	—
16	C₂×C₂²⋊C₈×C₄₁	C₂²	1	16	yes	complemented
17	C₂×#97×C₄₁	C₂²	1	4	yes	complemented
18	C₂⁵×C₄₁	C₂×C₄	1	2	yes	—
19	C₂²×D₄×C₄₁	—	2	8	no	—
20	C₂²×D₄×C₄₁	—	2	4	no	—
21	C₂²×D₄×C₄₁	C₂×C₄	1	4	yes	—
22	C₂³×C₄×C₄₁	—	2	8	no	—
23	C₂³×C₄×C₄₁	—	4	8	no	—
24	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₄₁	C₂×C₄	1	4	yes	—
27	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₄₁	C₂³	1	4	yes	—
29	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₄₁	C₂³	1	8	yes	—
31	C₄×D₄×C₄₁	—	2	16	no	—
32	C₄×D₄×C₄₁	—	2	8	no	—
33	C₄×D₄×C₄₁	—	2	8	no	—
34	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
35	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
36	C₂×C₂²⋊C₄×C₄₁	—	2	8	no	—
37	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	8	yes	—
38	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₄₁	D₄	1	8	yes	—
40	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
41	C₄⋊C₈×C₄₁	—	2	8	no	—
42	C₂²⋊C₈×C₄₁	—	2	8	no	—
43	C₂²⋊C₈×C₄₁	—	4	16	no	—
44	#97×C₄₁	—	2	8	no	—
45	C₂⁴×C₄₁	—	2	4	no	—
46	C₂⁴×C₄₁	—	2	4	no	—
47	C₂⁴×C₄₁	—	4	4	no	—
48	C₂⁴×C₄₁	—	4	4	no	—
49	C₂⁴×C₄₁	M₁₆	1	8	yes	—
50	C₂⁴×C₄₁	C₂²×C₄	1	4	yes	—
51	C₂⁴×C₄₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×D₄×C₄₁	—	2	32	no	—
53	C₂×D₄×C₄₁	—	2	16	no	—
54	C₂×D₄×C₄₁	—	2	8	no	—
55	C₂²×C₄×C₄₁	—	2	4	no	—
56	C₂²×C₄×C₄₁	—	2	4	no	—
57	C₂²×C₄×C₄₁	—	2	4	no	—
58	C₂²×C₄×C₄₁	—	2	8	no	—
59	C₂²×C₄×C₄₁	—	2	8	no	—
60	C₂²×C₄×C₄₁	—	2	16	no	—
61	C₂²×C₄×C₄₁	—	4	8	no	—
62	C₂²×C₄×C₄₁	—	4	4	no	—
63	C₂²×C₄×C₄₁	—	4	2	no	—
64	C₂²×C₄×C₄₁	—	4	32	no	—
65	C₂²×C₄×C₄₁	C₂²⋊C₄	1	4	yes	—
66	C₂²×C₄×C₄₁	C₂²⋊C₄	1	4	yes	—
67	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
72	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
73	C₂×C₈×C₄₁	—	2	8	no	—
74	C₂×C₈×C₄₁	—	2	16	no	—
75	C₄⋊C₄×C₄₁	—	2	8	no	—
76	C₂²⋊C₄×C₄₁	—	2	16	no	—
77	C₄²×C₄₁	C₂²⋊C₄	1	8	yes	—
78	C₄²×C₄₁	C₂×D₄	1	4	yes	—
79	C₄²×C₄₁	C₂×D₄	1	4	yes	—
80	C₂³×C₄₁	—	2	2	no	—
81	C₂³×C₄₁	—	2	2	no	—
82	C₂³×C₄₁	—	2	4	no	—
83	C₂³×C₄₁	—	2	8	no	—
84	C₂³×C₄₁	—	2	4	no	—
85	C₂³×C₄₁	—	2	8	no	—
86	C₂³×C₄₁	—	2	16	no	—
87	C₂³×C₄₁	—	4	16	no	—
88	C₂³×C₄₁	—	4	8	no	—
89	C₂³×C₄₁	—	4	8	no	—
90	C₂³×C₄₁	—	4	16	no	—
91	C₂³×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
92	C₂³×C₄₁	C₂×M₁₆	1	8	yes	—
93	C₂³×C₄₁	C₂×M₁₆	1	4	yes	—
94	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
95	D₄×C₄₁	—	2	32	no	—
96	C₂×C₄×C₄₁	—	2	2	no	—
97	C₂×C₄×C₄₁	—	2	4	no	—
98	C₂×C₄×C₄₁	—	2	4	no	—
99	C₂×C₄×C₄₁	—	2	8	no	—
100	C₂×C₄×C₄₁	—	2	8	no	—
101	C₂×C₄×C₄₁	—	2	4	no	—
102	C₂×C₄×C₄₁	—	2	8	no	—
103	C₂×C₄×C₄₁	—	4	4	no	—
104	C₂×C₄×C₄₁	—	4	16	no	—
105	C₂×C₄×C₄₁	—	4	8	no	—
106	C₂×C₄×C₄₁	—	4	16	no	—
107	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
108	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
109	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
110	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
111	C₂×C₄×C₄₁	H(5,𝔽₂)	1	4	yes	—
112	C₈×C₄₁	—	4	8	no	—
113	#21790	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂²×C₄₁	—	2	2	no	—
115	C₂²×C₄₁	—	2	2	no	—
116	C₂²×C₄₁	—	2	8	no	—
117	C₂²×C₄₁	—	2	8	no	—
118	C₂²×C₄₁	—	2	16	no	—
119	C₂²×C₄₁	—	4	32	no	—
120	C₂²×C₄₁	—	4	4	no	—
121	C₂²×C₄₁	—	4	4	no	—
122	C₂²×C₄₁	—	4	8	no	—
123	C₂²×C₄₁	#407	1	4	yes	—
124	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
125	C₂²×C₄₁	#518	1	2	yes	—
126	C₂²×C₄₁	C₂²×M₁₆	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	4	no	—
131	C₄×C₄₁	—	4	4	no	—
132	C₄×C₄₁	#535	1	4	yes	—
133	C₂²×C₄×D₄	C₂×C₄₁	1	1	yes	characteristic
134	C₂×#432	C₂×C₄₁	1	2	yes	complemented
135	#1955	C₂×C₄₁	1	8	yes	complemented
136	#1779	C₂×C₄₁	1	4	yes	complemented
137	C₂×C₄₁	—	2	4	no	—
138	C₂×C₄₁	—	4	8	no	—
139	C₂×C₄₁	#2309	1	1	yes	characteristic
140	C₂×C₄₁	C₂×#407	1	1	yes	characteristic
141	C₂×C₄₁	#2879	1	1	yes	characteristic
142	C₂×C₄₁	#2973	1	4	yes	—
143	C₂³×D₄	C₄×C₄₁	1	1	yes	characteristic
144	C₂⁴×C₄	C₂²×C₄₁	1	2	yes	—
145	C₂×C₄×D₄	—	2	8	no	—
146	C₂×C₄×D₄	C₄×C₄₁	1	4	yes	—
147	C₂×C₄×D₄	C₂²×C₄₁	1	4	yes	—
148	C₂²×C₄⋊C₄	C₄×C₄₁	1	1	yes	characteristic
149	C₂²×C₂²⋊C₄	C₄×C₄₁	1	2	yes	—
150	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
151	#432	—	2	8	no	—
152	C₂×C₄⋊C₈	C₂²×C₄₁	1	4	yes	complemented
153	C₂×C₂²⋊C₈	C₂²×C₄₁	1	4	yes	—
154	C₂×C₂²⋊C₈	C₂²×C₄₁	1	16	yes	complemented
155	C₂×#97	C₂²×C₄₁	1	4	yes	complemented
156	C₄₁	#21790	1	1	yes	characteristic, direct product factor, Sylow
157	C₂⁵	C₂×C₄×C₄₁	1	2	yes	—
158	C₂²×D₄	—	2	8	no	—
159	C₂²×D₄	—	2	4	no	—
160	C₂²×D₄	C₂×C₄×C₄₁	1	4	yes	—
161	C₂³×C₄	—	2	8	no	—
162	C₂³×C₄	—	4	8	no	—
163	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
164	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
165	C₂³×C₄	C₂×C₄×C₄₁	1	4	yes	—
166	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
167	C₂³×C₄	C₂³×C₄₁	1	4	yes	—
168	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
169	C₂²×C₈	C₂³×C₄₁	1	8	yes	—
170	C₄×D₄	—	2	16	no	—
171	C₄×D₄	—	2	8	no	—
172	C₄×D₄	—	2	8	no	—
173	C₂×C₄⋊C₄	—	2	4	no	—
174	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
175	C₂×C₂²⋊C₄	—	2	8	no	—
176	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	8	yes	—
177	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
178	C₂×C₄²	D₄×C₄₁	1	8	yes	—
179	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
180	C₄⋊C₈	—	2	8	no	—
181	C₂²⋊C₈	—	2	8	no	—
182	C₂²⋊C₈	—	4	16	no	—
183	#97	—	2	8	no	—
184	C₂⁴	—	2	4	no	—
185	C₂⁴	—	2	4	no	—
186	C₂⁴	—	4	4	no	—
187	C₂⁴	—	4	4	no	—
188	C₂⁴	M₁₆×C₄₁	1	8	yes	—
189	C₂⁴	C₂²×C₄×C₄₁	1	4	yes	—
190	C₂⁴	C₂²×C₄×C₄₁	1	1	yes	characteristic
191	C₂×D₄	—	2	32	no	—
192	C₂×D₄	—	2	16	no	—
193	C₂×D₄	—	2	8	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	8	no	—
198	C₂²×C₄	—	2	8	no	—
199	C₂²×C₄	—	2	16	no	—
200	C₂²×C₄	—	4	8	no	—
201	C₂²×C₄	—	4	4	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	—	4	32	no	—
204	C₂²×C₄	C₂²⋊C₄×C₄₁	1	4	yes	—
205	C₂²×C₄	C₂²⋊C₄×C₄₁	1	4	yes	—
206	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
207	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
208	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
209	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
210	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
211	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
212	C₂×C₈	—	2	8	no	—
213	C₂×C₈	—	2	16	no	—
214	C₄⋊C₄	—	2	8	no	—
215	C₂²⋊C₄	—	2	16	no	—
216	C₄²	C₂²⋊C₄×C₄₁	1	8	yes	—
217	C₄²	C₂×D₄×C₄₁	1	4	yes	—
218	C₄²	C₂×D₄×C₄₁	1	4	yes	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	4	no	—
222	C₂³	—	2	8	no	—
223	C₂³	—	2	4	no	—
224	C₂³	—	2	8	no	—
225	C₂³	—	2	16	no	—
226	C₂³	—	4	16	no	—
227	C₂³	—	4	8	no	—
228	C₂³	—	4	8	no	—
229	C₂³	—	4	16	no	—
230	C₂³	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
231	C₂³	C₂×M₁₆×C₄₁	1	8	yes	—
232	C₂³	C₂×M₁₆×C₄₁	1	4	yes	—
233	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
234	D₄	—	2	32	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	8	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	16	no	—
244	C₂×C₄	—	4	8	no	—
245	C₂×C₄	—	4	16	no	—
246	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
247	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
248	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
249	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
250	C₂×C₄	H(5,𝔽₂)×C₄₁	1	4	yes	—
251	C₈	—	4	8	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	8	no	—
255	C₂²	—	2	8	no	—
256	C₂²	—	2	16	no	—
257	C₂²	—	4	32	no	—
258	C₂²	—	4	4	no	—
259	C₂²	—	4	4	no	—
260	C₂²	—	4	8	no	—
261	C₂²	C₄₁×#407	1	4	yes	—
262	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
263	C₂²	C₄₁×#518	1	2	yes	—
264	C₂²	C₂²×M₁₆×C₄₁	1	2	yes	—
265	C₂²	C₄₁×#568	1	2	yes	—
266	C₄	—	2	2	no	—
267	C₄	—	2	4	no	—
268	C₄	—	2	4	no	—
269	C₄	—	4	4	no	—
270	C₄	C₄₁×#535	1	4	yes	—
271	C₂	—	2	4	no	—
272	C₂	—	4	8	no	—
273	C₂	C₄₁×#2309	1	1	yes	characteristic
274	C₂	C₂×C₄₁×#407	1	1	yes	characteristic
275	C₂	C₄₁×#2879	1	1	yes	characteristic
276	C₂	C₄₁×#2973	1	4	yes	—

	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	C₂³×D₄×C₄₁	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
2. C₂×C₄₁×#432	C₂	semidirect
3. C₄₁×#1955	C₂	semidirect
4. C₄₁×#1779	C₂	semidirect
14. C₂×C₄⋊C₈×C₄₁	C₂²	semidirect
16. C₂×C₂²⋊C₈×C₄₁	C₂²	semidirect
17. C₂×#97×C₄₁	C₂²	semidirect
113. #21790	C₄₁	direct (N × H)
134. C₂×#432	C₂×C₄₁	semidirect
135. #1955	C₂×C₄₁	semidirect
136. #1779	C₂×C₄₁	semidirect
152. C₂×C₄⋊C₈	C₂²×C₄₁	semidirect
154. C₂×C₂²⋊C₈	C₂²×C₄₁	semidirect
155. C₂×#97	C₂²×C₄₁	semidirect
156. C₄₁	#21790	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×C₄×D₄×C₄₁	2
C₂×C₄₁×#432	2
C₄₁×#1955	2
C₄₁×#1779	2
#21790	41

Subgroups — C41×#21790 #23797991

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#21790 #23797991