Subgroups — C₄₁×#3276 #19107056

Order 5248.

472 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#137×C₄₁	C₂	1	2	yes	maximal, complemented
2	D₄×Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#547	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#470	C₂	1	2	yes	maximal, complemented
6	C₄₁×#462	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#461	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#452	C₂	1	2	yes	maximal, complemented
9	C₄₁×#451	C₂	1	2	yes	maximal, complemented
10	C₄×SD₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₁×#436	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₄₁	C₂²	1	2	yes	—
13	C₂²×Q₈×C₄₁	C₂²	1	2	yes	—
14	#137×C₄₁	—	2	8	no	—
15	C₂×Q₁₆×C₄₁	C₂²	1	2	yes	—
16	C₂×Q₁₆×C₄₁	C₂²	1	2	yes	—
17	C₂×Q₁₆×C₄₁	C₂²	1	2	yes	—
18	C₂×SD₁₆×C₄₁	C₂²	1	2	yes	—
19	C₂×SD₁₆×C₄₁	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₄₁	C₂²	1	2	yes	—
21	C₄⋊Q₈×C₄₁	—	2	1	no	—
22	C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
23	C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
24	#123×C₄₁	—	2	2	no	—
25	C₂²⋊Q₈×C₄₁	—	2	2	no	—
26	C₂²⋊Q₈×C₄₁	C₂²	1	2	yes	—
27	C₂²⋊Q₈×C₄₁	C₂²	1	2	yes	—
28	#121×C₄₁	—	2	1	no	—
29	C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
30	C₄×D₄×C₄₁	—	2	1	no	—
31	C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
32	C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
34	#106×C₄₁	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
36	#103×C₄₁	C₂²	1	1	yes	characteristic
37	#103×C₄₁	C₂²	1	2	yes	—
38	#103×C₄₁	C₂²	1	2	yes	—
39	#102×C₄₁	C₂²	1	1	yes	characteristic
40	C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
41	C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
42	#41×C₄₁	—	2	4	no	—
43	#41×C₄₁	—	4	2	no	—
44	C₂×Q₈×C₄₁	—	2	2	no	—
45	C₂×Q₈×C₄₁	—	2	4	no	—
46	C₂×Q₈×C₄₁	—	4	2	no	—
47	C₂×Q₈×C₄₁	C₂³	1	2	yes	—
48	C₂×Q₈×C₄₁	C₂³	1	2	yes	—
49	C₂×Q₈×C₄₁	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₄₁	—	2	2	no	—
51	C₂×D₄×C₄₁	D₄	1	1	yes	characteristic
52	C₂×D₄×C₄₁	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₄₁	—	2	2	no	—
54	C₂²×C₄×C₄₁	—	2	1	no	—
55	C₂²×C₄×C₄₁	—	2	2	no	—
56	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
57	Q₁₆×C₄₁	—	2	4	no	—
58	Q₁₆×C₄₁	—	2	4	no	—
59	Q₁₆×C₄₁	—	4	2	no	—
60	SD₁₆×C₄₁	—	2	4	no	—
61	SD₁₆×C₄₁	D₄	1	4	yes	complemented
62	M₁₆×C₄₁	—	2	4	no	—
63	C₂×C₈×C₄₁	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₄₁	C₂³	1	2	yes	—
65	C₂×C₈×C₄₁	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₄₁	—	2	1	no	—
67	C₄⋊C₄×C₄₁	—	2	2	no	—
68	C₄⋊C₄×C₄₁	—	2	1	no	—
69	C₄⋊C₄×C₄₁	—	2	1	no	—
70	C₄⋊C₄×C₄₁	—	2	1	no	—
71	C₄⋊C₄×C₄₁	—	2	2	no	—
72	C₄⋊C₄×C₄₁	—	2	1	no	—
73	C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
74	C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
76	C₂²⋊C₄×C₄₁	—	2	2	no	—
77	C₂²⋊C₄×C₄₁	—	2	2	no	—
78	C₂²⋊C₄×C₄₁	—	2	1	no	—
79	C₂²⋊C₄×C₄₁	D₄	1	2	yes	—
80	C₄²×C₄₁	—	2	1	no	—
81	C₄²×C₄₁	C₂³	1	1	yes	characteristic
82	C₂³×C₄₁	—	2	1	no	—
83	C₂³×C₄₁	C₂×D₄	1	2	yes	—
84	Q₈×C₄₁	—	2	1	no	—
85	Q₈×C₄₁	—	2	2	no	—
86	Q₈×C₄₁	—	2	2	no	—
87	Q₈×C₄₁	—	2	2	no	—
88	Q₈×C₄₁	—	2	2	no	—
89	Q₈×C₄₁	—	4	4	no	—
90	Q₈×C₄₁	C₂×D₄	1	2	yes	—
91	D₄×C₄₁	—	2	2	no	—
92	D₄×C₄₁	—	2	1	no	—
93	D₄×C₄₁	—	4	4	no	—
94	D₄×C₄₁	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₄₁	—	2	1	no	—
96	C₂×C₄×C₄₁	—	2	2	no	—
97	C₂×C₄×C₄₁	—	2	2	no	—
98	C₂×C₄×C₄₁	—	2	1	no	—
99	C₂×C₄×C₄₁	—	2	1	no	—
100	C₂×C₄×C₄₁	—	2	2	no	—
101	C₂×C₄×C₄₁	—	2	2	no	—
102	C₂×C₄×C₄₁	—	4	4	no	—
103	C₂×C₄×C₄₁	—	4	4	no	—
104	C₂×C₄×C₄₁	—	4	2	no	—
105	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
107	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
109	C₈×C₄₁	—	2	2	no	—
110	C₈×C₄₁	—	2	1	no	—
111	C₈×C₄₁	C₂×D₄	1	2	yes	—
112	C₂²×C₄₁	—	2	2	no	—
113	C₂²×C₄₁	—	2	2	no	—
114	C₂²×C₄₁	—	2	2	no	—
115	C₂²×C₄₁	—	2	2	no	—
116	C₂²×C₄₁	—	4	1	no	—
117	C₂²×C₄₁	—	4	1	no	—
118	C₂²×C₄₁	C₂²×D₄	1	1	yes	characteristic
119	C₄×C₄₁	—	2	2	no	—
120	C₄×C₄₁	—	2	2	no	—
121	C₄×C₄₁	—	2	1	no	—
122	C₄×C₄₁	—	4	2	no	—
123	C₄×C₄₁	—	4	2	no	—
124	C₄×C₄₁	—	4	1	no	—
125	C₄×C₄₁	—	4	1	no	—
126	C₄×C₄₁	#137	1	2	yes	complemented
127	C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
128	C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
129	#3276	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂×C₄₁	—	4	2	no	—
131	C₂×C₄₁	—	4	2	no	—
132	C₂×C₄₁	D₄²	1	1	yes	characteristic
133	C₂×C₄₁	C₂×#137	1	1	yes	characteristic
134	C₂×C₄₁	#578	1	1	yes	characteristic
135	C₂×#137	C₂×C₄₁	1	2	yes	complemented
136	D₄×Q₈	C₂×C₄₁	1	1	yes	characteristic, complemented
137	#547	C₂×C₄₁	1	1	yes	characteristic
138	#494	C₂×C₄₁	1	1	yes	characteristic, complemented
139	#470	C₂×C₄₁	1	2	yes	complemented
140	#462	C₂×C₄₁	1	1	yes	characteristic, complemented
141	#461	C₂×C₄₁	1	1	yes	characteristic, complemented
142	#452	C₂×C₄₁	1	2	yes	complemented
143	#451	C₂×C₄₁	1	2	yes	complemented
144	C₄×SD₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
145	#436	C₂×C₄₁	1	1	yes	characteristic, complemented
146	C₄₁	#3276	1	1	yes	characteristic, direct product factor, Sylow
147	C₂×#41	C₂²×C₄₁	1	2	yes	—
148	C₂²×Q₈	C₂²×C₄₁	1	2	yes	—
149	#137	—	2	8	no	—
150	C₂×Q₁₆	C₂²×C₄₁	1	2	yes	—
151	C₂×Q₁₆	C₂²×C₄₁	1	2	yes	—
152	C₂×Q₁₆	C₂²×C₄₁	1	2	yes	—
153	C₂×SD₁₆	C₂²×C₄₁	1	2	yes	—
154	C₂×SD₁₆	C₂²×C₄₁	1	1	yes	characteristic
155	C₂×M₁₆	C₂²×C₄₁	1	2	yes	—
156	C₄⋊Q₈	—	2	1	no	—
157	C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
158	C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
159	#123	—	2	2	no	—
160	C₂²⋊Q₈	—	2	2	no	—
161	C₂²⋊Q₈	C₂²×C₄₁	1	2	yes	—
162	C₂²⋊Q₈	C₂²×C₄₁	1	2	yes	—
163	#121	—	2	1	no	—
164	C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
165	C₄×D₄	—	2	1	no	—
166	C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
167	C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
168	C₂×C₄⋊C₄	—	2	1	no	—
169	#106	C₂²×C₄₁	1	1	yes	characteristic
170	C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
171	#103	C₂²×C₄₁	1	1	yes	characteristic
172	#103	C₂²×C₄₁	1	2	yes	—
173	#103	C₂²×C₄₁	1	2	yes	—
174	#102	C₂²×C₄₁	1	1	yes	characteristic
175	C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
176	C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
177	#41	—	2	4	no	—
178	#41	—	4	2	no	—
179	C₂×Q₈	—	2	2	no	—
180	C₂×Q₈	—	2	4	no	—
181	C₂×Q₈	—	4	2	no	—
182	C₂×Q₈	C₂³×C₄₁	1	2	yes	—
183	C₂×Q₈	C₂³×C₄₁	1	2	yes	—
184	C₂×Q₈	C₂³×C₄₁	1	1	yes	characteristic
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	D₄×C₄₁	1	1	yes	characteristic
187	C₂×D₄	C₂³×C₄₁	1	1	yes	characteristic
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
192	Q₁₆	—	2	4	no	—
193	Q₁₆	—	2	4	no	—
194	Q₁₆	—	4	2	no	—
195	SD₁₆	—	2	4	no	—
196	SD₁₆	D₄×C₄₁	1	4	yes	complemented
197	M₁₆	—	2	4	no	—
198	C₂×C₈	C₂³×C₄₁	1	1	yes	characteristic
199	C₂×C₈	C₂³×C₄₁	1	2	yes	—
200	C₂×C₈	C₂³×C₄₁	1	1	yes	characteristic
201	C₄⋊C₄	—	2	1	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
209	C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
210	C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₂²⋊C₄	D₄×C₄₁	1	2	yes	—
215	C₄²	—	2	1	no	—
216	C₄²	C₂³×C₄₁	1	1	yes	characteristic
217	C₂³	—	2	1	no	—
218	C₂³	C₂×D₄×C₄₁	1	2	yes	—
219	Q₈	—	2	1	no	—
220	Q₈	—	2	2	no	—
221	Q₈	—	2	2	no	—
222	Q₈	—	2	2	no	—
223	Q₈	—	2	2	no	—
224	Q₈	—	4	4	no	—
225	Q₈	C₂×D₄×C₄₁	1	2	yes	—
226	D₄	—	2	2	no	—
227	D₄	—	2	1	no	—
228	D₄	—	4	4	no	—
229	D₄	C₂×D₄×C₄₁	1	2	yes	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
241	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
242	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
243	C₂×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	2	1	no	—
246	C₈	C₂×D₄×C₄₁	1	2	yes	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	4	1	no	—
252	C₂²	—	4	1	no	—
253	C₂²	C₂²×D₄×C₄₁	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	1	no	—
257	C₄	—	4	2	no	—
258	C₄	—	4	2	no	—
259	C₄	—	4	1	no	—
260	C₄	—	4	1	no	—
261	C₄	#137×C₄₁	1	2	yes	complemented
262	C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
263	C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
264	C₂	—	4	2	no	—
265	C₂	—	4	2	no	—
266	C₂	D₄²×C₄₁	1	1	yes	characteristic
267	C₂	C₂×#137×C₄₁	1	1	yes	characteristic
268	C₂	C₄₁×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₄₁
prime radical	C₄₁×#547	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₄×D₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#137×C₄₁	C₂	semidirect
2. D₄×Q₈×C₄₁	C₂	semidirect
4. C₄₁×#494	C₂	semidirect
5. C₄₁×#470	C₂	semidirect
6. C₄₁×#462	C₂	semidirect
7. C₄₁×#461	C₂	semidirect
8. C₄₁×#452	C₂	semidirect
9. C₄₁×#451	C₂	semidirect
10. C₄×SD₁₆×C₄₁	C₂	semidirect
11. C₄₁×#436	C₂	semidirect
61. SD₁₆×C₄₁	D₄	semidirect
126. C₄×C₄₁	#137	semidirect
129. #3276	C₄₁	direct (N × H)
135. C₂×#137	C₂×C₄₁	semidirect
136. D₄×Q₈	C₂×C₄₁	semidirect
138. #494	C₂×C₄₁	semidirect
139. #470	C₂×C₄₁	semidirect
140. #462	C₂×C₄₁	semidirect
141. #461	C₂×C₄₁	semidirect
142. #452	C₂×C₄₁	semidirect
143. #451	C₂×C₄₁	semidirect
144. C₄×SD₁₆	C₂×C₄₁	semidirect
145. #436	C₂×C₄₁	semidirect
146. C₄₁	#3276	direct (N × H)
196. SD₁₆	D₄×C₄₁	semidirect
261. C₄	#137×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#137×C₄₁	2
D₄×Q₈×C₄₁	2
C₄₁×#547	2
C₄₁×#494	2
C₄₁×#470	2
C₄₁×#462	2
C₄₁×#461	2
C₄₁×#452	2
C₄₁×#451	2
C₄×SD₁₆×C₄₁	2
C₄₁×#436	2
#3276	41

Subgroups — C41×#3276 #19107056

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#3276 #19107056