Subgroups — C₄₁×#33849 #23810050

Order 10496.

676 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#508	C₂	1	2	yes	maximal, complemented
2	C₄₁×#3283	C₂	1	2	yes	maximal, complemented
3	C₄₁×#2247	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#2216	C₂	1	2	yes	maximal, complemented
5	C₄₁×#2208	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2189	C₂	1	4	yes	maximal, complemented
7	C₄×D₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
8	D₄×C₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₄₁	C₂²	1	4	yes	—
10	C₄₁×#547	—	2	2	no	—
11	C₄₁×#508	—	2	8	no	—
12	C₂×Q₃₂×C₄₁	C₂²	1	2	yes	—
13	C₂×Q₃₂×C₄₁	C₂²	1	2	yes	—
14	C₂×SD₃₂×C₄₁	C₂²	1	4	yes	—
15	C₂×D₁₆×C₄₁	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₄₁	C₂²	1	2	yes	—
17	C₄₁×#494	C₂²	1	2	yes	—
18	C₄₁×#467	C₂²	1	4	yes	—
19	C₄₁×#461	—	2	2	no	—
20	C₄₁×#452	—	2	4	no	—
21	C₄×D₈×C₄₁	C₂²	1	2	yes	—
22	C₈×D₄×C₄₁	C₂²	1	1	yes	characteristic
23	C₄₁×#366	C₂²	1	1	yes	characteristic
24	C₄₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
25	C₄₁×#358	C₂²	1	4	yes	—
26	C₄₁×#357	C₂²	1	2	yes	—
27	C₄₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₄×C₁₆×C₄₁	C₂²	1	1	yes	characteristic
29	C₂×#41×C₄₁	—	2	4	no	—
30	#135×C₄₁	—	2	8	no	—
31	#135×C₄₁	—	4	4	no	—
32	C₂×Q₁₆×C₄₁	—	2	2	no	—
33	C₂×Q₁₆×C₄₁	C₂³	1	4	yes	—
34	C₂×SD₁₆×C₄₁	—	2	4	no	—
35	C₂×D₈×C₄₁	C₂³	1	2	yes	—
36	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
37	C₄⋊Q₈×C₄₁	—	2	2	no	—
38	#123×C₄₁	—	4	4	no	—
39	C₂²⋊Q₈×C₄₁	—	2	4	no	—
40	#121×C₄₁	—	4	2	no	—
41	C₄×D₄×C₄₁	—	2	2	no	—
42	C₄×D₄×C₄₁	D₄	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₄₁	—	4	2	no	—
44	Q₃₂×C₄₁	—	2	4	no	—
45	Q₃₂×C₄₁	—	4	2	no	—
46	SD₃₂×C₄₁	—	2	8	no	—
47	D₁₆×C₄₁	D₄	1	4	yes	complemented
48	C₂×C₁₆×C₄₁	—	2	4	no	—
49	C₂×C₁₆×C₄₁	C₂³	1	2	yes	—
50	C₂×C₁₆×C₄₁	C₂³	1	1	yes	characteristic
51	C₂×C₁₆×C₄₁	C₂³	1	1	yes	characteristic
52	#107×C₄₁	C₂³	1	2	yes	—
53	C₄⋊C₈×C₄₁	D₄	1	1	yes	characteristic
54	#103×C₄₁	—	2	4	no	—
55	#102×C₄₁	—	2	2	no	—
56	C₂²⋊C₈×C₄₁	D₄	1	2	yes	—
57	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
58	#41×C₄₁	—	4	8	no	—
59	#41×C₄₁	—	8	4	no	—
60	C₂×Q₈×C₄₁	—	2	4	no	—
61	C₂×D₄×C₄₁	—	2	2	no	—
62	C₂×D₄×C₄₁	—	4	4	no	—
63	C₂×D₄×C₄₁	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₄₁	—	4	2	no	—
65	C₂²×C₄×C₄₁	—	4	4	no	—
66	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
67	Q₁₆×C₄₁	—	2	4	no	—
68	Q₁₆×C₄₁	—	2	4	no	—
69	Q₁₆×C₄₁	—	8	2	no	—
70	SD₁₆×C₄₁	—	4	8	no	—
71	D₈×C₄₁	—	2	2	no	—
72	D₈×C₄₁	C₂×D₄	1	4	yes	—
73	C₂×C₈×C₄₁	—	2	2	no	—
74	C₂×C₈×C₄₁	—	2	2	no	—
75	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₄₁	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₄₁	—	2	2	no	—
79	C₄⋊C₄×C₄₁	—	4	2	no	—
80	C₄⋊C₄×C₄₁	—	4	4	no	—
81	C₄⋊C₄×C₄₁	—	4	2	no	—
82	C₄⋊C₄×C₄₁	D₈	1	1	yes	characteristic
83	C₂²⋊C₄×C₄₁	—	4	4	no	—
84	C₂²⋊C₄×C₄₁	—	4	2	no	—
85	C₂²⋊C₄×C₄₁	D₈	1	2	yes	—
86	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
87	C₁₆×C₄₁	—	2	2	no	—
88	C₁₆×C₄₁	—	2	1	no	—
89	C₁₆×C₄₁	C₂×D₄	1	2	yes	—
90	C₂³×C₄₁	—	4	2	no	—
91	C₂³×C₄₁	C₂×D₈	1	2	yes	—
92	Q₈×C₄₁	—	4	4	no	—
93	Q₈×C₄₁	—	4	4	no	—
94	D₄×C₄₁	—	2	4	no	—
95	D₄×C₄₁	—	2	2	no	—
96	D₄×C₄₁	—	4	2	no	—
97	D₄×C₄₁	—	8	8	no	—
98	C₂×C₄×C₄₁	—	2	2	no	—
99	C₂×C₄×C₄₁	—	2	2	no	—
100	C₂×C₄×C₄₁	—	4	2	no	—
101	C₂×C₄×C₄₁	—	4	4	no	—
102	C₂×C₄×C₄₁	—	8	8	no	—
103	C₂×C₄×C₄₁	—	8	4	no	—
104	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₄₁	C₂×D₈	1	2	yes	—
106	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
108	C₈×C₄₁	—	2	2	no	—
109	C₈×C₄₁	—	2	1	no	—
110	C₈×C₄₁	C₂²×D₄	1	1	yes	characteristic
111	C₈×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
112	#33849	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂²×C₄₁	—	2	2	no	—
114	C₂²×C₄₁	—	2	2	no	—
115	C₂²×C₄₁	—	2	2	no	—
116	C₂²×C₄₁	—	4	4	no	—
117	C₂²×C₄₁	—	8	2	no	—
118	C₂²×C₄₁	—	8	2	no	—
119	C₂²×C₄₁	C₂²×D₈	1	1	yes	characteristic
120	C₄×C₄₁	—	2	1	no	—
121	C₄×C₄₁	—	2	2	no	—
122	C₄×C₄₁	—	8	2	no	—
123	C₄×C₄₁	—	8	4	no	—
124	C₄×C₄₁	#508	1	2	yes	complemented
125	C₄×C₄₁	D₄²	1	1	yes	characteristic
126	C₄×C₄₁	#576	1	1	yes	characteristic
127	C₂×#508	C₂×C₄₁	1	2	yes	complemented
128	#3283	C₂×C₄₁	1	2	yes	complemented
129	#2247	C₂×C₄₁	1	1	yes	characteristic, complemented
130	#2216	C₂×C₄₁	1	2	yes	complemented
131	#2208	C₂×C₄₁	1	2	yes	complemented
132	#2189	C₂×C₄₁	1	4	yes	complemented
133	C₄×D₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
134	D₄×C₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
135	C₂×C₄₁	—	4	2	no	—
136	C₂×C₄₁	—	8	4	no	—
137	C₂×C₄₁	D₄×D₈	1	1	yes	characteristic
138	C₂×C₄₁	C₂×#508	1	1	yes	characteristic
139	C₂×C₄₁	#3417	1	1	yes	characteristic
140	C₂×#135	C₂²×C₄₁	1	4	yes	—
141	#547	—	2	2	no	—
142	#508	—	2	8	no	—
143	C₂×Q₃₂	C₂²×C₄₁	1	2	yes	—
144	C₂×Q₃₂	C₂²×C₄₁	1	2	yes	—
145	C₂×SD₃₂	C₂²×C₄₁	1	4	yes	—
146	C₂×D₁₆	C₂²×C₄₁	1	1	yes	characteristic
147	C₂²×C₁₆	C₂²×C₄₁	1	2	yes	—
148	#494	C₂²×C₄₁	1	2	yes	—
149	#467	C₂²×C₄₁	1	4	yes	—
150	#461	—	2	2	no	—
151	#452	—	2	4	no	—
152	C₄×D₈	C₂²×C₄₁	1	2	yes	—
153	C₈×D₄	C₂²×C₄₁	1	1	yes	characteristic
154	#366	C₂²×C₄₁	1	1	yes	characteristic
155	C₄⋊C₁₆	C₂²×C₄₁	1	1	yes	characteristic
156	#358	C₂²×C₄₁	1	4	yes	—
157	#357	C₂²×C₄₁	1	2	yes	—
158	C₂²⋊C₁₆	C₂²×C₄₁	1	2	yes	—
159	C₄×C₁₆	C₂²×C₄₁	1	1	yes	characteristic
160	C₄₁	#33849	1	1	yes	characteristic, direct product factor, Sylow
161	C₂×#41	—	2	4	no	—
162	#135	—	2	8	no	—
163	#135	—	4	4	no	—
164	C₂×Q₁₆	—	2	2	no	—
165	C₂×Q₁₆	C₂³×C₄₁	1	4	yes	—
166	C₂×SD₁₆	—	2	4	no	—
167	C₂×D₈	C₂³×C₄₁	1	2	yes	—
168	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
169	C₄⋊Q₈	—	2	2	no	—
170	#123	—	4	4	no	—
171	C₂²⋊Q₈	—	2	4	no	—
172	#121	—	4	2	no	—
173	C₄×D₄	—	2	2	no	—
174	C₄×D₄	D₄×C₄₁	1	1	yes	characteristic
175	C₂×C₄⋊C₄	—	4	2	no	—
176	Q₃₂	—	2	4	no	—
177	Q₃₂	—	4	2	no	—
178	SD₃₂	—	2	8	no	—
179	D₁₆	D₄×C₄₁	1	4	yes	complemented
180	C₂×C₁₆	—	2	4	no	—
181	C₂×C₁₆	C₂³×C₄₁	1	1	yes	characteristic
182	C₂×C₁₆	C₂³×C₄₁	1	2	yes	—
183	C₂×C₁₆	C₂³×C₄₁	1	1	yes	characteristic
184	#107	C₂³×C₄₁	1	2	yes	—
185	C₄⋊C₈	D₄×C₄₁	1	1	yes	characteristic
186	#103	—	2	4	no	—
187	#102	—	2	2	no	—
188	C₂²⋊C₈	D₄×C₄₁	1	2	yes	—
189	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
190	#41	—	4	8	no	—
191	#41	—	8	4	no	—
192	C₂×Q₈	—	2	4	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	4	4	no	—
195	C₂×D₄	D₈×C₄₁	1	1	yes	characteristic
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	—	4	4	no	—
198	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
199	Q₁₆	—	2	4	no	—
200	Q₁₆	—	2	4	no	—
201	Q₁₆	—	8	2	no	—
202	SD₁₆	—	4	8	no	—
203	D₈	—	2	2	no	—
204	D₈	C₂×D₄×C₄₁	1	4	yes	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
208	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
209	C₂×C₈	C₂⁴×C₄₁	1	1	yes	characteristic
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	4	4	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄⋊C₄	D₈×C₄₁	1	1	yes	characteristic
215	C₂²⋊C₄	—	4	4	no	—
216	C₂²⋊C₄	—	4	2	no	—
217	C₂²⋊C₄	D₈×C₄₁	1	2	yes	—
218	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
219	C₁₆	—	2	2	no	—
220	C₁₆	—	2	1	no	—
221	C₁₆	C₂×D₄×C₄₁	1	2	yes	—
222	C₂³	—	4	2	no	—
223	C₂³	C₂×D₈×C₄₁	1	2	yes	—
224	Q₈	—	4	4	no	—
225	Q₈	—	4	4	no	—
226	D₄	—	2	2	no	—
227	D₄	—	2	4	no	—
228	D₄	—	4	2	no	—
229	D₄	—	8	8	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	8	8	no	—
235	C₂×C₄	—	8	4	no	—
236	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
237	C₂×C₄	C₂×D₈×C₄₁	1	2	yes	—
238	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
239	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	—	2	1	no	—
242	C₈	C₂²×D₄×C₄₁	1	1	yes	characteristic
243	C₈	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	4	4	no	—
248	C₂²	—	8	2	no	—
249	C₂²	—	8	2	no	—
250	C₂²	C₂²×D₈×C₄₁	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	2	2	no	—
253	C₄	—	8	2	no	—
254	C₄	—	8	4	no	—
255	C₄	C₄₁×#508	1	2	yes	complemented
256	C₄	D₄²×C₄₁	1	1	yes	characteristic
257	C₄	C₄₁×#576	1	1	yes	characteristic
258	C₂	—	4	2	no	—
259	C₂	—	8	4	no	—
260	C₂	D₄×D₈×C₄₁	1	1	yes	characteristic
261	C₂	C₂×C₄₁×#508	1	1	yes	characteristic
262	C₂	C₄₁×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₄₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₄×D₄×C₄₁ ◁ C₈×D₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₁×#508	C₂	semidirect
2. C₄₁×#3283	C₂	semidirect
3. C₄₁×#2247	C₂	semidirect
4. C₄₁×#2216	C₂	semidirect
5. C₄₁×#2208	C₂	semidirect
6. C₄₁×#2189	C₂	semidirect
7. C₄×D₁₆×C₄₁	C₂	semidirect
8. D₄×C₁₆×C₄₁	C₂	semidirect
47. D₁₆×C₄₁	D₄	semidirect
112. #33849	C₄₁	direct (N × H)
124. C₄×C₄₁	#508	semidirect
127. C₂×#508	C₂×C₄₁	semidirect
128. #3283	C₂×C₄₁	semidirect
129. #2247	C₂×C₄₁	semidirect
130. #2216	C₂×C₄₁	semidirect
131. #2208	C₂×C₄₁	semidirect
132. #2189	C₂×C₄₁	semidirect
133. C₄×D₁₆	C₂×C₄₁	semidirect
134. D₄×C₁₆	C₂×C₄₁	semidirect
160. C₄₁	#33849	direct (N × H)
179. D₁₆	D₄×C₄₁	semidirect
255. C₄	C₄₁×#508	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#508	2
C₄₁×#3283	2
C₄₁×#2247	2
C₄₁×#2216	2
C₄₁×#2208	2
C₄₁×#2189	2
C₄×D₁₆×C₄₁	2
D₄×C₁₆×C₄₁	2
#33849	41

Subgroups — C41×#33849 #23810050

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#33849 #23810050