Subgroups — C₄₁×#23553 #23799754

Order 10496.

1142 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#125×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#428	C₂	1	2	yes	maximal
3	C₄₁×#2296	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#2059	C₂	1	8	yes	maximal, complemented
5	C₄₁×#1736	C₂	1	2	yes	maximal
6	C₄₁×#1728	C₂	1	1	yes	maximal, characteristic
7	C₂²×M₁₆×C₄₁	C₂²	1	2	yes	—
8	C₂×#125×C₄₁	—	2	8	no	—
9	C₂×#125×C₄₁	—	2	2	no	—
10	C₂×#125×C₄₁	C₂²	1	4	yes	—
11	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
12	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
13	C₂²×C₄⋊C₄×C₄₁	—	2	2	no	—
14	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
15	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
16	C₄₁×#428	—	2	8	no	—
17	C₂×#107×C₄₁	C₂²	1	4	yes	—
18	C₂×#106×C₄₁	C₂²	1	4	yes	—
19	C₄₁×#389	C₂²	1	4	yes	—
20	C₄₁×#385	—	2	2	no	—
21	C₄×C₄⋊C₄×C₄₁	—	2	2	no	—
22	C₄₁×#340	C₂²	1	8	yes	—
23	C₄₁×#336	C₂²	1	4	yes	—
24	C₂³×C₄×C₄₁	—	2	2	no	—
25	C₂³×C₄×C₄₁	D₄	1	2	yes	—
26	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
27	C₂×M₁₆×C₄₁	—	2	4	no	—
28	C₂×M₁₆×C₄₁	—	2	4	no	—
29	C₂×M₁₆×C₄₁	Q₈	1	8	yes	—
30	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
31	#125×C₄₁	—	2	16	no	—
32	#125×C₄₁	—	2	16	no	—
33	#117×C₄₁	—	2	2	no	—
34	#117×C₄₁	—	2	4	no	—
35	#117×C₄₁	Q₈	1	4	yes	—
36	C₂×C₄⋊C₄×C₄₁	—	2	16	no	—
37	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
38	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
40	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
41	C₂×C₄⋊C₄×C₄₁	D₄	1	8	yes	—
42	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
44	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
45	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
46	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
47	C₂×C₄²×C₄₁	—	2	4	no	—
48	C₂×C₄²×C₄₁	—	2	1	no	—
49	C₂×C₄²×C₄₁	—	2	1	no	—
50	C₂×C₄²×C₄₁	D₄	1	2	yes	—
51	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
52	#107×C₄₁	—	2	8	no	—
53	#106×C₄₁	—	2	8	no	—
54	#95×C₄₁	—	2	4	no	—
55	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₁	—	2	4	no	—
57	C₂²×C₄×C₄₁	—	2	8	no	—
58	C₂²×C₄×C₄₁	—	2	2	no	—
59	C₂²×C₄×C₄₁	—	2	4	no	—
60	C₂²×C₄×C₄₁	—	2	2	no	—
61	C₂²×C₄×C₄₁	—	4	4	no	—
62	C₂²×C₄×C₄₁	—	4	8	no	—
63	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
64	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
66	C₂²×C₄×C₄₁	C₂×Q₈	1	4	yes	—
67	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₄₁	#41	1	2	yes	—
69	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
70	M₁₆×C₄₁	—	2	16	no	—
71	C₂×C₈×C₄₁	—	2	4	no	—
72	C₂×C₈×C₄₁	—	2	4	no	—
73	C₂×C₈×C₄₁	C₂×Q₈	1	8	yes	—
74	C₄⋊C₄×C₄₁	—	2	32	no	—
75	C₄⋊C₄×C₄₁	—	2	8	no	—
76	C₄⋊C₄×C₄₁	—	2	4	no	—
77	C₄⋊C₄×C₄₁	—	2	2	no	—
78	C₄⋊C₄×C₄₁	—	2	2	no	—
79	C₄⋊C₄×C₄₁	—	2	2	no	—
80	C₄⋊C₄×C₄₁	—	4	8	no	—
81	C₄⋊C₄×C₄₁	C₂×D₄	1	8	yes	—
82	C₄⋊C₄×C₄₁	C₂×Q₈	1	4	yes	—
83	C₂²⋊C₄×C₄₁	—	2	4	no	—
84	C₂²⋊C₄×C₄₁	—	4	2	no	—
85	C₄²×C₄₁	—	2	4	no	—
86	C₄²×C₄₁	—	2	4	no	—
87	C₄²×C₄₁	—	2	4	no	—
88	C₄²×C₄₁	—	4	2	no	—
89	C₂³×C₄₁	—	2	4	no	—
90	C₂³×C₄₁	#120	1	2	yes	—
91	C₂³×C₄₁	C₂²⋊Q₈	1	2	yes	—
92	C₂³×C₄₁	C₄⋊Q₈	1	2	yes	—
93	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₄₁	—	2	4	no	—
95	C₂×C₄×C₄₁	—	2	8	no	—
96	C₂×C₄×C₄₁	—	2	4	no	—
97	C₂×C₄×C₄₁	—	2	4	no	—
98	C₂×C₄×C₄₁	—	2	4	no	—
99	C₂×C₄×C₄₁	—	4	8	no	—
100	C₂×C₄×C₄₁	—	4	16	no	—
101	C₂×C₄×C₄₁	—	4	4	no	—
102	C₂×C₄×C₄₁	—	4	8	no	—
103	C₂×C₄×C₄₁	—	4	2	no	—
104	C₂×C₄×C₄₁	—	4	4	no	—
105	C₂×C₄×C₄₁	#120	1	2	yes	—
106	C₂×C₄×C₄₁	C₂²⋊Q₈	1	8	yes	—
107	C₂×C₄×C₄₁	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₄₁	C₄⋊Q₈	1	8	yes	—
109	C₂×C₄×C₄₁	C₄⋊Q₈	1	2	yes	—
110	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
111	C₂×C₄×C₄₁	C₂²×Q₈	1	2	yes	—
112	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
114	C₈×C₄₁	—	2	8	no	—
115	#23553	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂²×C₄₁	—	2	4	no	—
117	C₂²×C₄₁	—	2	4	no	—
118	C₂²×C₄₁	—	2	4	no	—
119	C₂²×C₄₁	#395	1	4	yes	—
120	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
121	C₂²×C₄₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
122	C₂²×C₄₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
123	C₂²×C₄₁	#575	1	4	yes	—
124	C₄×C₄₁	—	4	4	no	—
125	C₄×C₄₁	—	4	4	no	—
126	C₄×C₄₁	—	4	8	no	—
127	C₄×C₄₁	—	8	2	no	—
128	C₄×C₄₁	#395	1	4	yes	—
129	C₄×C₄₁	C₂×C₂²⋊Q₈	1	2	yes	—
130	C₄×C₄₁	C₂×C₄⋊Q₈	1	2	yes	—
131	C₂²×#125	C₂×C₄₁	1	1	yes	characteristic
132	C₂×#428	C₂×C₄₁	1	2	yes	—
133	#2296	C₂×C₄₁	1	1	yes	characteristic
134	#2059	C₂×C₄₁	1	8	yes	complemented
135	#1736	C₂×C₄₁	1	2	yes	—
136	#1728	C₂×C₄₁	1	1	yes	characteristic
137	C₂×C₄₁	—	2	4	no	—
138	C₂×C₄₁	C₂×#395	1	1	yes	characteristic
139	C₂×C₄₁	#3005	1	2	yes	—
140	C₂×C₄₁	#3080	1	2	yes	—
141	C₂×C₄₁	#3166	1	2	yes	—
142	C₂²×M₁₆	C₂²×C₄₁	1	2	yes	—
143	C₂×#125	—	2	8	no	—
144	C₂×#125	—	2	2	no	—
145	C₂×#125	C₂²×C₄₁	1	4	yes	—
146	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
147	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
148	C₂²×C₄⋊C₄	—	2	2	no	—
149	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
150	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
151	#428	—	2	8	no	—
152	C₂×#107	C₂²×C₄₁	1	4	yes	—
153	C₂×#106	C₂²×C₄₁	1	4	yes	—
154	#389	C₂²×C₄₁	1	4	yes	—
155	#385	—	2	2	no	—
156	C₄×C₄⋊C₄	—	2	2	no	—
157	#340	C₂²×C₄₁	1	8	yes	—
158	#336	C₂²×C₄₁	1	4	yes	—
159	C₄₁	#23553	1	1	yes	characteristic, direct product factor, Sylow
160	C₂³×C₄	—	2	2	no	—
161	C₂³×C₄	D₄×C₄₁	1	2	yes	—
162	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
163	C₂×M₁₆	—	2	4	no	—
164	C₂×M₁₆	—	2	4	no	—
165	C₂×M₁₆	Q₈×C₄₁	1	8	yes	—
166	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
167	#125	—	2	16	no	—
168	#125	—	2	16	no	—
169	#117	—	2	2	no	—
170	#117	—	2	4	no	—
171	#117	Q₈×C₄₁	1	4	yes	—
172	C₂×C₄⋊C₄	—	2	16	no	—
173	C₂×C₄⋊C₄	—	2	8	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	—	2	4	no	—
176	C₂×C₄⋊C₄	—	2	4	no	—
177	C₂×C₄⋊C₄	D₄×C₄₁	1	8	yes	—
178	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
179	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
180	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
181	C₂×C₂²⋊C₄	—	2	1	no	—
182	C₂×C₂²⋊C₄	—	2	1	no	—
183	C₂×C₄²	—	2	1	no	—
184	C₂×C₄²	—	2	4	no	—
185	C₂×C₄²	—	2	1	no	—
186	C₂×C₄²	D₄×C₄₁	1	2	yes	—
187	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
188	#107	—	2	8	no	—
189	#106	—	2	8	no	—
190	#95	—	2	4	no	—
191	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	8	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	4	4	no	—
198	C₂²×C₄	—	4	8	no	—
199	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
200	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
202	C₂²×C₄	C₂×Q₈×C₄₁	1	4	yes	—
203	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
204	C₂²×C₄	#41×C₄₁	1	2	yes	—
205	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
206	M₁₆	—	2	16	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	C₂×Q₈×C₄₁	1	8	yes	—
210	C₄⋊C₄	—	2	32	no	—
211	C₄⋊C₄	—	2	8	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	4	8	no	—
217	C₄⋊C₄	C₂×D₄×C₄₁	1	8	yes	—
218	C₄⋊C₄	C₂×Q₈×C₄₁	1	4	yes	—
219	C₂²⋊C₄	—	2	4	no	—
220	C₂²⋊C₄	—	4	2	no	—
221	C₄²	—	2	4	no	—
222	C₄²	—	2	4	no	—
223	C₄²	—	2	4	no	—
224	C₄²	—	4	2	no	—
225	C₂³	—	2	4	no	—
226	C₂³	#120×C₄₁	1	2	yes	—
227	C₂³	C₂²⋊Q₈×C₄₁	1	2	yes	—
228	C₂³	C₄⋊Q₈×C₄₁	1	2	yes	—
229	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	4	8	no	—
236	C₂×C₄	—	4	16	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	8	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	#120×C₄₁	1	2	yes	—
242	C₂×C₄	C₂²⋊Q₈×C₄₁	1	8	yes	—
243	C₂×C₄	C₂²⋊Q₈×C₄₁	1	2	yes	—
244	C₂×C₄	C₄⋊Q₈×C₄₁	1	8	yes	—
245	C₂×C₄	C₄⋊Q₈×C₄₁	1	2	yes	—
246	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
247	C₂×C₄	C₂²×Q₈×C₄₁	1	2	yes	—
248	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
249	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
250	C₈	—	2	8	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	4	no	—
254	C₂²	C₄₁×#395	1	4	yes	—
255	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
256	C₂²	C₂×C₂²⋊Q₈×C₄₁	1	1	yes	characteristic
257	C₂²	C₂×C₄⋊Q₈×C₄₁	1	1	yes	characteristic
258	C₂²	C₄₁×#575	1	4	yes	—
259	C₄	—	4	4	no	—
260	C₄	—	4	4	no	—
261	C₄	—	4	8	no	—
262	C₄	—	8	2	no	—
263	C₄	C₄₁×#395	1	4	yes	—
264	C₄	C₂×C₂²⋊Q₈×C₄₁	1	2	yes	—
265	C₄	C₂×C₄⋊Q₈×C₄₁	1	2	yes	—
266	C₂	—	2	4	no	—
267	C₂	C₂×C₄₁×#395	1	1	yes	characteristic
268	C₂	C₄₁×#3005	1	2	yes	—
269	C₂	C₄₁×#3080	1	2	yes	—
270	C₂	C₄₁×#3166	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₄₁×#2059	C₂	semidirect
115. #23553	C₄₁	direct (N × H)
134. #2059	C₂×C₄₁	semidirect
159. C₄₁	#23553	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#125×C₄₁	2
C₂×C₄₁×#428	2
C₄₁×#2296	2
C₄₁×#2059	2
C₄₁×#1736	2
C₄₁×#1728	2
#23553	41

Subgroups — C41×#23553 #23799754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#23553 #23799754