Subgroups — C₄₁×#7880 #23784081

Order 10496.

618 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#431	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₁×#423	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1277	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₄₁	—	2	1	no	—
6	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
7	C₄₁×#431	—	2	2	no	—
8	C₄₁×#431	—	2	2	no	—
9	C₄₁×#431	—	2	2	no	—
10	C₄₁×#431	C₄	1	4	yes	—
11	C₄₁×#423	—	2	2	no	—
12	C₄₁×#423	C₄	1	4	yes	—
13	C₂×C₄⋊C₈×C₄₁	—	2	1	no	—
14	C₂×C₄⋊C₈×C₄₁	C₄	1	2	yes	—
15	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₈×C₄₁	—	2	1	no	—
18	C₂×C₂²⋊C₈×C₄₁	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₄₁	—	2	1	no	—
20	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	—
21	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₄₁	Q₈	1	1	yes	characteristic
24	C₂×M₁₆×C₄₁	—	2	4	no	—
25	C₂×M₁₆×C₄₁	—	4	4	no	—
26	C₂²×C₈×C₄₁	—	2	2	no	—
27	C₂²×C₈×C₄₁	—	2	2	no	—
28	C₂²×C₈×C₄₁	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₄₁	D₄	1	2	yes	—
34	C₂×C₄²×C₄₁	D₄	1	2	yes	—
35	C₂×C₄²×C₄₁	Q₈	1	1	yes	characteristic
36	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₄₁	—	2	4	no	—
38	C₄⋊C₈×C₄₁	—	2	4	no	—
39	C₄⋊C₈×C₄₁	—	2	2	no	—
40	C₄⋊C₈×C₄₁	—	2	2	no	—
41	C₄⋊C₈×C₄₁	C₂×C₄	1	4	yes	—
42	C₄⋊C₈×C₄₁	C₂×C₄	1	4	yes	—
43	C₂²⋊C₈×C₄₁	—	2	4	no	—
44	C₂²⋊C₈×C₄₁	—	2	4	no	—
45	C₄×C₈×C₄₁	—	2	4	no	—
46	C₄×C₈×C₄₁	—	2	2	no	—
47	C₄×C₈×C₄₁	—	2	2	no	—
48	C₂⁴×C₄₁	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₁	—	2	4	no	—
50	C₂²×C₄×C₄₁	—	2	4	no	—
51	C₂²×C₄×C₄₁	—	2	2	no	—
52	C₂²×C₄×C₄₁	—	2	2	no	—
53	C₂²×C₄×C₄₁	C₄²	1	2	yes	—
54	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₄₁	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₁	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₁	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₄₁	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₄₁	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₁	C₂×Q₈	1	1	yes	characteristic
65	M₁₆×C₄₁	—	4	8	no	—
66	C₂×C₈×C₄₁	—	2	4	no	—
67	C₂×C₈×C₄₁	—	2	4	no	—
68	C₂×C₈×C₄₁	—	2	4	no	—
69	C₂×C₈×C₄₁	—	2	8	no	—
70	C₂×C₈×C₄₁	—	4	2	no	—
71	C₂×C₈×C₄₁	—	4	4	no	—
72	C₂×C₈×C₄₁	—	4	2	no	—
73	C₄²×C₄₁	C₄²	1	2	yes	—
74	C₄²×C₄₁	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₄₁	C₂²⋊C₄	1	2	yes	—
76	C₄²×C₄₁	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₄₁	C₄⋊C₄	1	2	yes	—
78	C₄²×C₄₁	C₄⋊C₄	1	2	yes	—
79	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
83	C₂³×C₄₁	—	2	2	no	—
84	C₂³×C₄₁	—	2	2	no	—
85	C₂³×C₄₁	—	2	2	no	—
86	C₂³×C₄₁	#95	1	2	yes	—
87	C₂³×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
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	4	no	—
92	C₂×C₄×C₄₁	—	2	4	no	—
93	C₂×C₄×C₄₁	—	2	4	no	—
94	C₂×C₄×C₄₁	—	2	4	no	—
95	C₂×C₄×C₄₁	—	2	4	no	—
96	C₂×C₄×C₄₁	—	2	2	no	—
97	C₂×C₄×C₄₁	—	2	2	no	—
98	C₂×C₄×C₄₁	#95	1	2	yes	—
99	C₂×C₄×C₄₁	#95	1	2	yes	—
100	C₂×C₄×C₄₁	#95	1	2	yes	—
101	C₂×C₄×C₄₁	#108	1	4	yes	—
102	C₂×C₄×C₄₁	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₁	#131	1	4	yes	—
109	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
110	C₂×C₄×C₄₁	#137	1	2	yes	—
111	C₈×C₄₁	—	4	4	no	—
112	C₈×C₄₁	—	4	4	no	—
113	C₈×C₄₁	—	4	4	no	—
114	#7880	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂²×C₄₁	—	2	2	no	—
116	C₂²×C₄₁	—	2	2	no	—
117	C₂²×C₄₁	—	2	2	no	—
118	C₂²×C₄₁	—	4	4	no	—
119	C₂²×C₄₁	C₂×#95	1	1	yes	characteristic
120	C₂²×C₄₁	#424	1	2	yes	—
121	C₂²×C₄₁	#428	1	2	yes	—
122	C₂²×C₄₁	C₂×#108	1	2	yes	—
123	C₄×C₄₁	—	2	2	no	—
124	C₄×C₄₁	—	2	2	no	—
125	C₄×C₄₁	—	2	1	no	—
126	C₄×C₄₁	—	2	1	no	—
127	C₄×C₄₁	#341	1	4	yes	—
128	C₄×C₄₁	#405	1	2	yes	—
129	C₄×C₄₁	#408	1	2	yes	—
130	C₄×C₄₁	#417	1	2	yes	—
131	C₄×C₄₁	#418	1	1	yes	characteristic
132	C₄×C₄₁	#419	1	1	yes	characteristic
133	C₂×#431	C₂×C₄₁	1	1	yes	characteristic
134	C₂×#431	C₂×C₄₁	1	1	yes	characteristic
135	C₂×#423	C₂×C₄₁	1	1	yes	characteristic
136	#1277	C₂×C₄₁	1	4	yes	complemented
137	C₂×C₄₁	—	4	2	no	—
138	C₂×C₄₁	#1574	1	4	yes	—
139	C₂×C₄₁	#1729	1	1	yes	characteristic
140	C₂×C₄₁	#1736	1	1	yes	characteristic
141	C₂×C₄₁	C₂×#341	1	1	yes	characteristic
142	C₂²×M₁₆	—	2	1	no	—
143	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
144	#431	—	2	2	no	—
145	#431	—	2	2	no	—
146	#431	—	2	2	no	—
147	#431	C₄×C₄₁	1	4	yes	—
148	#423	—	2	2	no	—
149	#423	C₄×C₄₁	1	4	yes	—
150	C₂×C₄⋊C₈	—	2	1	no	—
151	C₂×C₄⋊C₈	C₄×C₄₁	1	2	yes	—
152	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
153	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
154	C₂×C₂²⋊C₈	—	2	1	no	—
155	C₂×C₂²⋊C₈	C₄×C₄₁	1	2	yes	—
156	C₂×C₄×C₈	—	2	1	no	—
157	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
158	C₄₁	#7880	1	1	yes	characteristic, direct product factor, Sylow
159	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
160	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
161	C₂³×C₄	Q₈×C₄₁	1	1	yes	characteristic
162	C₂×M₁₆	—	2	4	no	—
163	C₂×M₁₆	—	4	4	no	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	C₂×C₄×C₄₁	1	4	yes	—
167	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
168	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
169	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
170	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
171	C₂×C₄²	D₄×C₄₁	1	2	yes	—
172	C₂×C₄²	D₄×C₄₁	1	2	yes	—
173	C₂×C₄²	Q₈×C₄₁	1	1	yes	characteristic
174	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
175	C₄⋊C₈	—	2	4	no	—
176	C₄⋊C₈	—	2	4	no	—
177	C₄⋊C₈	—	2	2	no	—
178	C₄⋊C₈	—	2	2	no	—
179	C₄⋊C₈	C₂×C₄×C₄₁	1	4	yes	—
180	C₄⋊C₈	C₂×C₄×C₄₁	1	4	yes	—
181	C₂²⋊C₈	—	2	4	no	—
182	C₂²⋊C₈	—	2	4	no	—
183	C₄×C₈	—	2	4	no	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	—	2	2	no	—
186	C₂⁴	C₄⋊C₄×C₄₁	1	1	yes	characteristic
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	C₄²×C₄₁	1	2	yes	—
192	C₂²×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
193	C₂²×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
194	C₂²×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
195	C₂²×C₄	C₄⋊C₄×C₄₁	1	1	yes	characteristic
196	C₂²×C₄	C₄⋊C₄×C₄₁	1	1	yes	characteristic
197	C₂²×C₄	C₄⋊C₄×C₄₁	1	2	yes	—
198	C₂²×C₄	C₄⋊C₄×C₄₁	1	2	yes	—
199	C₂²×C₄	C₄⋊C₄×C₄₁	1	1	yes	characteristic
200	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
202	C₂²×C₄	C₂×Q₈×C₄₁	1	1	yes	characteristic
203	M₁₆	—	4	8	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	8	no	—
208	C₂×C₈	—	4	4	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	—	4	2	no	—
211	C₄²	C₄²×C₄₁	1	2	yes	—
212	C₄²	C₂²⋊C₄×C₄₁	1	2	yes	—
213	C₄²	C₂²⋊C₄×C₄₁	1	2	yes	—
214	C₄²	C₂²⋊C₄×C₄₁	1	2	yes	—
215	C₄²	C₄⋊C₄×C₄₁	1	2	yes	—
216	C₄²	C₄⋊C₄×C₄₁	1	2	yes	—
217	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
218	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
219	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
220	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	2	2	no	—
224	C₂³	#95×C₄₁	1	2	yes	—
225	C₂³	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	#95×C₄₁	1	2	yes	—
237	C₂×C₄	#95×C₄₁	1	2	yes	—
238	C₂×C₄	#95×C₄₁	1	2	yes	—
239	C₂×C₄	#108×C₄₁	1	4	yes	—
240	C₂×C₄	C₂×C₄²×C₄₁	1	1	yes	characteristic
241	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
242	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
243	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
244	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
245	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
246	C₂×C₄	#131×C₄₁	1	4	yes	—
247	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
248	C₂×C₄	#137×C₄₁	1	2	yes	—
249	C₈	—	4	4	no	—
250	C₈	—	4	4	no	—
251	C₈	—	4	4	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	4	4	no	—
256	C₂²	C₂×#95×C₄₁	1	1	yes	characteristic
257	C₂²	C₄₁×#424	1	2	yes	—
258	C₂²	C₄₁×#428	1	2	yes	—
259	C₂²	C₂×#108×C₄₁	1	2	yes	—
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	C₄₁×#341	1	4	yes	—
265	C₄	C₄₁×#405	1	2	yes	—
266	C₄	C₄₁×#408	1	2	yes	—
267	C₄	C₄₁×#417	1	2	yes	—
268	C₄	C₄₁×#418	1	1	yes	characteristic
269	C₄	C₄₁×#419	1	1	yes	characteristic
270	C₂	—	4	2	no	—
271	C₂	C₄₁×#1574	1	4	yes	—
272	C₂	C₄₁×#1729	1	1	yes	characteristic
273	C₂	C₄₁×#1736	1	1	yes	characteristic
274	C₂	C₂×C₄₁×#341	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₄₁	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₄₁
Frattini, cosocle	C₂×C₄²	C₂³×C₄₁
socle	C₂³×C₄₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₄₁	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₄₁ ◁ C₂²×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄₁×#1277	C₂	semidirect
114. #7880	C₄₁	direct (N × H)
136. #1277	C₂×C₄₁	semidirect
158. C₄₁	#7880	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#431	2
C₂×C₄₁×#423	2
C₄₁×#1277	2
#7880	41

Subgroups — C41×#7880 #23784081

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#7880 #23784081