Subgroups — C₄₁×#2396 #19106176

Order 5248.

528 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#117×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#400	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#399	C₂	1	1	yes	maximal, characteristic
5	C₄₁×#398	C₂	1	2	yes	maximal, complemented
6	C₄₁×#396	C₂	1	2	yes	maximal
7	C₄₁×#395	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#393	C₂	1	1	yes	maximal, characteristic
9	C₄×C₄⋊C₄×C₄₁	C₂	1	2	yes	maximal, complemented
10	C₄×C₄⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂²⋊C₄×C₄₁	C₂	1	2	yes	maximal
12	C₂²×Q₈×C₄₁	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₄₁	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₄₁	—	2	4	no	—
15	#117×C₄₁	—	2	4	no	—
16	C₂×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
27	C₂×C₄²×C₄₁	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₄₁	C₂²	1	2	yes	—
29	C₂×C₄²×C₄₁	C₂²	1	2	yes	—
30	C₂×C₄²×C₄₁	C₂²	1	1	yes	characteristic
31	#95×C₄₁	C₂²	1	2	yes	—
32	#95×C₄₁	C₂²	1	2	yes	—
33	#95×C₄₁	C₂²	1	2	yes	—
34	#95×C₄₁	C₂²	1	1	yes	characteristic
35	#95×C₄₁	C₂²	1	1	yes	characteristic
36	#95×C₄₁	C₂²	1	2	yes	—
37	C₂⁴×C₄₁	C₂³	1	1	yes	characteristic
38	C₂×Q₈×C₄₁	—	2	2	no	—
39	C₂×Q₈×C₄₁	—	2	4	no	—
40	C₂²×C₄×C₄₁	—	2	2	no	—
41	C₂²×C₄×C₄₁	—	2	2	no	—
42	C₂²×C₄×C₄₁	Q₈	1	4	yes	complemented
43	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
44	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
47	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
51	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
52	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
53	C₄⋊C₄×C₄₁	—	2	2	no	—
54	C₄⋊C₄×C₄₁	—	2	2	no	—
55	C₄⋊C₄×C₄₁	—	2	4	no	—
56	C₄⋊C₄×C₄₁	—	2	2	no	—
57	C₄⋊C₄×C₄₁	—	2	4	no	—
58	C₄⋊C₄×C₄₁	—	2	4	no	—
59	C₄⋊C₄×C₄₁	—	2	4	no	—
60	C₂²⋊C₄×C₄₁	—	2	2	no	—
61	C₂²⋊C₄×C₄₁	—	2	2	no	—
62	C₂²⋊C₄×C₄₁	—	2	4	no	—
63	C₂²⋊C₄×C₄₁	—	2	4	no	—
64	C₄²×C₄₁	—	2	2	no	—
65	C₄²×C₄₁	—	2	4	no	—
66	C₄²×C₄₁	—	2	4	no	—
67	C₄²×C₄₁	D₄	1	4	yes	—
68	C₂³×C₄₁	—	2	1	no	—
69	C₂³×C₄₁	—	2	2	no	—
70	C₂³×C₄₁	—	2	2	no	—
71	C₂³×C₄₁	—	2	1	no	—
72	C₂³×C₄₁	C₂×Q₈	1	2	yes	—
73	C₂³×C₄₁	C₂⁴	1	1	yes	characteristic
74	Q₈×C₄₁	—	2	8	no	—
75	C₂×C₄×C₄₁	—	2	2	no	—
76	C₂×C₄×C₄₁	—	2	1	no	—
77	C₂×C₄×C₄₁	—	2	2	no	—
78	C₂×C₄×C₄₁	—	2	2	no	—
79	C₂×C₄×C₄₁	—	2	2	no	—
80	C₂×C₄×C₄₁	—	2	1	no	—
81	C₂×C₄×C₄₁	—	2	2	no	—
82	C₂×C₄×C₄₁	—	2	2	no	—
83	C₂×C₄×C₄₁	—	2	8	no	—
84	C₂×C₄×C₄₁	—	2	1	no	—
85	C₂×C₄×C₄₁	—	2	1	no	—
86	C₂×C₄×C₄₁	—	2	2	no	—
87	C₂×C₄×C₄₁	—	2	2	no	—
88	C₂×C₄×C₄₁	—	2	2	no	—
89	C₂×C₄×C₄₁	—	2	2	no	—
90	C₂×C₄×C₄₁	—	2	2	no	—
91	C₂×C₄×C₄₁	—	2	2	no	—
92	C₂×C₄×C₄₁	—	2	2	no	—
93	C₂×C₄×C₄₁	—	2	2	no	—
94	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₄₁	C₂×Q₈	1	2	yes	—
98	C₂×C₄×C₄₁	C₂×Q₈	1	2	yes	—
99	C₂×C₄×C₄₁	#41	1	4	yes	—
100	C₂×C₄×C₄₁	#41	1	4	yes	—
101	C₂×C₄×C₄₁	#41	1	2	yes	—
102	C₂²×C₄₁	—	2	2	no	—
103	C₂²×C₄₁	—	2	4	no	—
104	C₂²×C₄₁	—	4	1	no	—
105	C₂²×C₄₁	—	4	2	no	—
106	C₂²×C₄₁	—	4	1	no	—
107	C₂²×C₄₁	C₂²×D₄	1	1	yes	characteristic
108	C₂²×C₄₁	C₂²×Q₈	1	1	yes	characteristic
109	C₂²×C₄₁	C₂×#41	1	1	yes	characteristic
110	C₂²×C₄₁	C₂×#41	1	2	yes	—
111	C₂²×C₄₁	C₂×#41	1	2	yes	—
112	C₄×C₄₁	—	2	2	no	—
113	C₄×C₄₁	—	2	2	no	—
114	C₄×C₄₁	—	2	4	no	—
115	C₄×C₄₁	—	2	4	no	—
116	C₄×C₄₁	—	2	2	no	—
117	C₄×C₄₁	—	4	1	no	—
118	C₄×C₄₁	—	4	2	no	—
119	C₄×C₄₁	—	4	1	no	—
120	C₄×C₄₁	—	4	2	no	—
121	C₄×C₄₁	C₂²⋊Q₈	1	4	yes	—
122	#2396	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂×C₄₁	—	4	2	no	—
124	C₂×C₄₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄₁	#525	1	1	yes	characteristic
126	C₂×C₄₁	#529	1	2	yes	—
127	C₂×C₄₁	#532	1	1	yes	characteristic
128	C₂×C₄₁	#533	1	2	yes	—
129	C₂×C₄×Q₈	C₂×C₄₁	1	1	yes	characteristic, complemented
130	C₂×#117	C₂×C₄₁	1	1	yes	characteristic
131	#400	C₂×C₄₁	1	1	yes	characteristic, complemented
132	#399	C₂×C₄₁	1	1	yes	characteristic
133	#398	C₂×C₄₁	1	2	yes	complemented
134	#396	C₂×C₄₁	1	2	yes	—
135	#395	C₂×C₄₁	1	1	yes	characteristic, complemented
136	#393	C₂×C₄₁	1	1	yes	characteristic
137	C₄×C₄⋊C₄	C₂×C₄₁	1	2	yes	complemented
138	C₄×C₄⋊C₄	C₂×C₄₁	1	1	yes	characteristic, complemented
139	C₄×C₂²⋊C₄	C₂×C₄₁	1	2	yes	—
140	C₄₁	#2396	1	1	yes	characteristic, direct product factor, Sylow
141	C₂²×Q₈	C₂²×C₄₁	1	1	yes	characteristic
142	C₂³×C₄	C₂²×C₄₁	1	1	yes	characteristic
143	C₄×Q₈	—	2	4	no	—
144	#117	—	2	4	no	—
145	C₂×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
146	C₂×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
147	C₂×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
148	C₂×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
149	C₂×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
150	C₂×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
151	C₂×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
152	C₂×C₂²⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
153	C₂×C₂²⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
154	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
155	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
156	C₂×C₄²	C₂²×C₄₁	1	1	yes	characteristic
157	C₂×C₄²	C₂²×C₄₁	1	2	yes	—
158	C₂×C₄²	C₂²×C₄₁	1	2	yes	—
159	C₂×C₄²	C₂²×C₄₁	1	1	yes	characteristic
160	#95	C₂²×C₄₁	1	2	yes	—
161	#95	C₂²×C₄₁	1	2	yes	—
162	#95	C₂²×C₄₁	1	2	yes	—
163	#95	C₂²×C₄₁	1	1	yes	characteristic
164	#95	C₂²×C₄₁	1	1	yes	characteristic
165	#95	C₂²×C₄₁	1	2	yes	—
166	C₂⁴	C₂³×C₄₁	1	1	yes	characteristic
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	—	2	4	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	Q₈×C₄₁	1	4	yes	complemented
172	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
173	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
174	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
175	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
176	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
177	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
178	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
179	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
180	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
181	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	4	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	2	4	no	—
192	C₂²⋊C₄	—	2	4	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	2	4	no	—
195	C₄²	—	2	4	no	—
196	C₄²	D₄×C₄₁	1	4	yes	—
197	C₂³	—	2	1	no	—
198	C₂³	—	2	2	no	—
199	C₂³	—	2	2	no	—
200	C₂³	—	2	1	no	—
201	C₂³	C₂×Q₈×C₄₁	1	2	yes	—
202	C₂³	C₂⁴×C₄₁	1	1	yes	characteristic
203	Q₈	—	2	8	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	8	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
224	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
225	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
226	C₂×C₄	C₂×Q₈×C₄₁	1	2	yes	—
227	C₂×C₄	C₂×Q₈×C₄₁	1	2	yes	—
228	C₂×C₄	#41×C₄₁	1	4	yes	—
229	C₂×C₄	#41×C₄₁	1	4	yes	—
230	C₂×C₄	#41×C₄₁	1	2	yes	—
231	C₂²	—	2	2	no	—
232	C₂²	—	2	4	no	—
233	C₂²	—	4	1	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	1	no	—
236	C₂²	C₂²×D₄×C₄₁	1	1	yes	characteristic
237	C₂²	C₂²×Q₈×C₄₁	1	1	yes	characteristic
238	C₂²	C₂×#41×C₄₁	1	1	yes	characteristic
239	C₂²	C₂×#41×C₄₁	1	2	yes	—
240	C₂²	C₂×#41×C₄₁	1	2	yes	—
241	C₄	—	2	2	no	—
242	C₄	—	2	2	no	—
243	C₄	—	2	4	no	—
244	C₄	—	2	4	no	—
245	C₄	—	2	2	no	—
246	C₄	—	4	1	no	—
247	C₄	—	4	2	no	—
248	C₄	—	4	1	no	—
249	C₄	—	4	2	no	—
250	C₄	C₂²⋊Q₈×C₄₁	1	4	yes	—
251	C₂	—	4	2	no	—
252	C₂	C₂×C₂²⋊Q₈×C₄₁	1	1	yes	characteristic
253	C₂	C₄₁×#525	1	1	yes	characteristic
254	C₂	C₄₁×#529	1	2	yes	—
255	C₂	C₄₁×#532	1	1	yes	characteristic
256	C₂	C₄₁×#533	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₄₁	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁴×C₄₁
socle	C₂³×C₄₁	C₂⁴
prime radical	C₂⁴×C₄₁	C₂³
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
1. C₂×C₄×Q₈×C₄₁	C₂	semidirect
3. C₄₁×#400	C₂	semidirect
5. C₄₁×#398	C₂	semidirect
7. C₄₁×#395	C₂	semidirect
9. C₄×C₄⋊C₄×C₄₁	C₂	semidirect
10. C₄×C₄⋊C₄×C₄₁	C₂	semidirect
42. C₂²×C₄×C₄₁	Q₈	semidirect
122. #2396	C₄₁	direct (N × H)
129. C₂×C₄×Q₈	C₂×C₄₁	semidirect
131. #400	C₂×C₄₁	semidirect
133. #398	C₂×C₄₁	semidirect
135. #395	C₂×C₄₁	semidirect
137. C₄×C₄⋊C₄	C₂×C₄₁	semidirect
138. C₄×C₄⋊C₄	C₂×C₄₁	semidirect
140. C₄₁	#2396	direct (N × H)
171. C₂²×C₄	Q₈×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₈×C₄₁	2
C₂×#117×C₄₁	2
C₄₁×#400	2
C₄₁×#399	2
C₄₁×#398	2
C₄₁×#396	2
C₄₁×#395	2
C₄₁×#393	2
C₄×C₄⋊C₄×C₄₁	2
C₄×C₂²⋊C₄×C₄₁	2
#2396	41

Subgroups — C41×#2396 #19106176

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#2396 #19106176