Subgroups — D₄×Q₃₂×C₄₁ #23810053

Order 10496.

664 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₃₂×C₄₁	C₂	1	2	yes	maximal, complemented
2	D₄×Q₁₆×C₄₁	C₂	1	2	yes	maximal
3	C₄₁×#2247	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#2216	C₂	1	2	yes	maximal, complemented
5	C₄₁×#2209	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2188	C₂	1	4	yes	maximal, complemented
7	C₄×Q₃₂×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
8	D₄×C₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₄₁	C₂²	1	4	yes	—
10	D₄×Q₈×C₄₁	—	2	2	no	—
11	C₂×Q₃₂×C₄₁	—	2	8	no	—
12	C₂×Q₃₂×C₄₁	C₂²	1	4	yes	—
13	C₂×Q₃₂×C₄₁	C₂²	1	2	yes	—
14	C₂×Q₃₂×C₄₁	C₂²	1	2	yes	—
15	C₂×Q₃₂×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×#462	—	2	2	no	—
20	C₄₁×#451	—	2	4	no	—
21	C₄×Q₁₆×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	2	yes	—
26	C₄₁×#358	C₂²	1	4	yes	—
27	C₄₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₄×C₁₆×C₄₁	C₂²	1	1	yes	characteristic
29	C₂²×Q₈×C₄₁	—	2	4	no	—
30	C₂×Q₁₆×C₄₁	—	2	8	no	—
31	C₂×Q₁₆×C₄₁	—	2	4	no	—
32	C₂×Q₁₆×C₄₁	—	2	2	no	—
33	C₂×Q₁₆×C₄₁	—	4	4	no	—
34	C₂×Q₁₆×C₄₁	C₂³	1	4	yes	—
35	C₂×Q₁₆×C₄₁	C₂³	1	2	yes	—
36	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
37	C₄⋊Q₈×C₄₁	—	2	2	no	—
38	C₄⋊Q₈×C₄₁	—	4	2	no	—
39	C₂²⋊Q₈×C₄₁	—	2	4	no	—
40	C₂²⋊Q₈×C₄₁	—	4	4	no	—
41	C₄×Q₈×C₄₁	—	2	2	no	—
42	C₄×D₄×C₄₁	—	4	2	no	—
43	C₄×D₄×C₄₁	D₄	1	1	yes	characteristic
44	Q₃₂×C₄₁	—	2	8	no	—
45	Q₃₂×C₄₁	—	2	4	no	—
46	Q₃₂×C₄₁	—	4	2	no	—
47	Q₃₂×C₄₁	D₄	1	4	yes	direct product factor
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	2	no	—
55	#103×C₄₁	—	2	4	no	—
56	C₂²⋊C₈×C₄₁	D₄	1	2	yes	—
57	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
58	C₂×Q₈×C₄₁	—	2	2	no	—
59	C₂×Q₈×C₄₁	—	2	4	no	—
60	C₂×Q₈×C₄₁	—	4	8	no	—
61	C₂×Q₈×C₄₁	—	4	4	no	—
62	C₂×Q₈×C₄₁	—	8	4	no	—
63	C₂×D₄×C₄₁	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₄₁	—	4	4	no	—
65	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
66	Q₁₆×C₄₁	—	2	4	no	—
67	Q₁₆×C₄₁	—	2	2	no	—
68	Q₁₆×C₄₁	—	2	4	no	—
69	Q₁₆×C₄₁	—	4	8	no	—
70	Q₁₆×C₄₁	—	8	2	no	—
71	Q₁₆×C₄₁	C₂×D₄	1	4	yes	—
72	C₂×C₈×C₄₁	—	2	2	no	—
73	C₂×C₈×C₄₁	—	2	2	no	—
74	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₄₁	C₂⁴	1	1	yes	characteristic
77	C₄⋊C₄×C₄₁	—	2	2	no	—
78	C₄⋊C₄×C₄₁	—	4	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	D₈	1	2	yes	—
85	C₄²×C₄₁	—	4	2	no	—
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	C₂×D₈	1	2	yes	—
91	Q₈×C₄₁	—	2	4	no	—
92	Q₈×C₄₁	—	4	2	no	—
93	Q₈×C₄₁	—	4	4	no	—
94	Q₈×C₄₁	—	4	4	no	—
95	Q₈×C₄₁	—	8	8	no	—
96	D₄×C₄₁	Q₃₂	1	4	yes	direct product factor
97	C₂×C₄×C₄₁	—	2	2	no	—
98	C₂×C₄×C₄₁	—	2	2	no	—
99	C₂×C₄×C₄₁	—	4	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	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₄₁	C₂×D₈	1	2	yes	—
105	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
107	C₈×C₄₁	—	2	2	no	—
108	C₈×C₄₁	—	2	1	no	—
109	C₈×C₄₁	C₂²×D₄	1	1	yes	characteristic
110	C₈×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
111	D₄×Q₃₂	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂²×C₄₁	—	2	2	no	—
113	C₂²×C₄₁	—	2	2	no	—
114	C₂²×C₄₁	C₂×Q₃₂	1	4	yes	complemented
115	C₂²×C₄₁	C₂²×D₈	1	1	yes	characteristic
116	C₄×C₄₁	—	2	1	no	—
117	C₄×C₄₁	—	2	2	no	—
118	C₄×C₄₁	—	4	4	no	—
119	C₄×C₄₁	—	8	2	no	—
120	C₄×C₄₁	—	8	4	no	—
121	C₄×C₄₁	C₂×Q₃₂	1	2	yes	complemented
122	C₄×C₄₁	D₄²	1	1	yes	characteristic
123	C₄×C₄₁	#576	1	1	yes	characteristic
124	C₂²×Q₃₂	C₂×C₄₁	1	2	yes	complemented
125	D₄×Q₁₆	C₂×C₄₁	1	2	yes	—
126	#2247	C₂×C₄₁	1	1	yes	characteristic, complemented
127	#2216	C₂×C₄₁	1	2	yes	complemented
128	#2209	C₂×C₄₁	1	2	yes	complemented
129	#2188	C₂×C₄₁	1	4	yes	complemented
130	C₄×Q₃₂	C₂×C₄₁	1	1	yes	characteristic, complemented
131	D₄×C₁₆	C₂×C₄₁	1	1	yes	characteristic
132	C₂×C₄₁	—	2	4	no	—
133	C₂×C₄₁	D₄×D₈	1	1	yes	characteristic
134	C₂×C₄₁	C₂²×Q₃₂	1	1	yes	characteristic
135	C₂×C₄₁	#3417	1	1	yes	characteristic
136	C₂²×Q₁₆	C₂²×C₄₁	1	4	yes	—
137	D₄×Q₈	—	2	2	no	—
138	C₂×Q₃₂	—	2	8	no	—
139	C₂×Q₃₂	C₂²×C₄₁	1	4	yes	—
140	C₂×Q₃₂	C₂²×C₄₁	1	2	yes	—
141	C₂×Q₃₂	C₂²×C₄₁	1	2	yes	—
142	C₂×Q₃₂	C₂²×C₄₁	1	1	yes	characteristic
143	C₂²×C₁₆	C₂²×C₄₁	1	2	yes	—
144	#494	C₂²×C₄₁	1	2	yes	—
145	#467	C₂²×C₄₁	1	4	yes	—
146	#462	—	2	2	no	—
147	#451	—	2	4	no	—
148	C₄×Q₁₆	C₂²×C₄₁	1	2	yes	—
149	C₈×D₄	C₂²×C₄₁	1	1	yes	characteristic
150	#366	C₂²×C₄₁	1	1	yes	characteristic
151	C₄⋊C₁₆	C₂²×C₄₁	1	1	yes	characteristic
152	#358	C₂²×C₄₁	1	2	yes	—
153	#358	C₂²×C₄₁	1	4	yes	—
154	C₂²⋊C₁₆	C₂²×C₄₁	1	2	yes	—
155	C₄×C₁₆	C₂²×C₄₁	1	1	yes	characteristic
156	C₄₁	D₄×Q₃₂	1	1	yes	characteristic, direct product factor, Sylow
157	C₂²×Q₈	—	2	4	no	—
158	C₂×Q₁₆	—	2	8	no	—
159	C₂×Q₁₆	—	2	4	no	—
160	C₂×Q₁₆	—	2	2	no	—
161	C₂×Q₁₆	—	4	4	no	—
162	C₂×Q₁₆	C₂³×C₄₁	1	4	yes	—
163	C₂×Q₁₆	C₂³×C₄₁	1	2	yes	—
164	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
165	C₄⋊Q₈	—	2	2	no	—
166	C₄⋊Q₈	—	4	2	no	—
167	C₂²⋊Q₈	—	2	4	no	—
168	C₂²⋊Q₈	—	4	4	no	—
169	C₄×Q₈	—	2	2	no	—
170	C₄×D₄	—	4	2	no	—
171	C₄×D₄	D₄×C₄₁	1	1	yes	characteristic
172	Q₃₂	—	2	8	no	—
173	Q₃₂	—	2	4	no	—
174	Q₃₂	—	4	2	no	—
175	Q₃₂	D₄×C₄₁	1	4	yes	direct product factor
176	C₂×C₁₆	—	2	4	no	—
177	C₂×C₁₆	C₂³×C₄₁	1	1	yes	characteristic
178	C₂×C₁₆	C₂³×C₄₁	1	2	yes	—
179	C₂×C₁₆	C₂³×C₄₁	1	1	yes	characteristic
180	#107	C₂³×C₄₁	1	2	yes	—
181	C₄⋊C₈	D₄×C₄₁	1	1	yes	characteristic
182	#103	—	2	2	no	—
183	#103	—	2	4	no	—
184	C₂²⋊C₈	D₄×C₄₁	1	2	yes	—
185	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
186	C₂×Q₈	—	2	2	no	—
187	C₂×Q₈	—	2	4	no	—
188	C₂×Q₈	—	4	8	no	—
189	C₂×Q₈	—	4	4	no	—
190	C₂×Q₈	—	8	4	no	—
191	C₂×D₄	D₈×C₄₁	1	1	yes	characteristic
192	C₂²×C₄	—	4	4	no	—
193	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
194	Q₁₆	—	2	4	no	—
195	Q₁₆	—	2	2	no	—
196	Q₁₆	—	2	4	no	—
197	Q₁₆	—	4	8	no	—
198	Q₁₆	—	8	2	no	—
199	Q₁₆	C₂×D₄×C₄₁	1	4	yes	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
203	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
204	C₂×C₈	C₂⁴×C₄₁	1	1	yes	characteristic
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	—	4	4	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	D₈×C₄₁	1	1	yes	characteristic
211	C₂²⋊C₄	—	4	4	no	—
212	C₂²⋊C₄	D₈×C₄₁	1	2	yes	—
213	C₄²	—	4	2	no	—
214	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
215	C₁₆	—	2	2	no	—
216	C₁₆	—	2	1	no	—
217	C₁₆	C₂×D₄×C₄₁	1	2	yes	—
218	C₂³	C₂×D₈×C₄₁	1	2	yes	—
219	Q₈	—	2	4	no	—
220	Q₈	—	4	2	no	—
221	Q₈	—	4	4	no	—
222	Q₈	—	4	4	no	—
223	Q₈	—	8	8	no	—
224	D₄	Q₃₂×C₄₁	1	4	yes	direct product factor
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	8	8	no	—
231	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
232	C₂×C₄	C₂×D₈×C₄₁	1	2	yes	—
233	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
234	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
235	C₈	—	2	2	no	—
236	C₈	—	2	1	no	—
237	C₈	C₂²×D₄×C₄₁	1	1	yes	characteristic
238	C₈	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	C₂×Q₃₂×C₄₁	1	4	yes	complemented
242	C₂²	C₂²×D₈×C₄₁	1	1	yes	characteristic
243	C₄	—	2	1	no	—
244	C₄	—	2	2	no	—
245	C₄	—	4	4	no	—
246	C₄	—	8	2	no	—
247	C₄	—	8	4	no	—
248	C₄	C₂×Q₃₂×C₄₁	1	2	yes	complemented
249	C₄	D₄²×C₄₁	1	1	yes	characteristic
250	C₄	C₄₁×#576	1	1	yes	characteristic
251	C₂	—	2	4	no	—
252	C₂	D₄×D₈×C₄₁	1	1	yes	characteristic
253	C₂	C₂²×Q₃₂×C₄₁	1	1	yes	characteristic
254	C₂	C₄₁×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₄₁
prime radical	C₂×D₄×C₄₁	D₈
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. C₂²×Q₃₂×C₄₁	C₂	semidirect
3. C₄₁×#2247	C₂	semidirect
4. C₄₁×#2216	C₂	semidirect
5. C₄₁×#2209	C₂	semidirect
6. C₄₁×#2188	C₂	semidirect
7. C₄×Q₃₂×C₄₁	C₂	semidirect
47. Q₃₂×C₄₁	D₄	direct (N × H)
96. D₄×C₄₁	Q₃₂	direct (N × H)
111. D₄×Q₃₂	C₄₁	direct (N × H)
114. C₂²×C₄₁	C₂×Q₃₂	semidirect
121. C₄×C₄₁	C₂×Q₃₂	semidirect
124. C₂²×Q₃₂	C₂×C₄₁	semidirect
126. #2247	C₂×C₄₁	semidirect
127. #2216	C₂×C₄₁	semidirect
128. #2209	C₂×C₄₁	semidirect
129. #2188	C₂×C₄₁	semidirect
130. C₄×Q₃₂	C₂×C₄₁	semidirect
156. C₄₁	D₄×Q₃₂	direct (N × H)
175. Q₃₂	D₄×C₄₁	direct (N × H)
224. D₄	Q₃₂×C₄₁	direct (N × H)
241. C₂²	C₂×Q₃₂×C₄₁	semidirect
248. C₄	C₂×Q₃₂×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Q₃₂×C₄₁	2
D₄×Q₁₆×C₄₁	2
C₄₁×#2247	2
C₄₁×#2216	2
C₄₁×#2209	2
C₄₁×#2188	2
C₄×Q₃₂×C₄₁	2
D₄×C₁₆×C₄₁	2
D₄×Q₃₂	41

Subgroups — D4×Q32×C41 #23810053

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×Q₃₂×C₄₁ #23810053