Subgroups — C₄₁×#12682 #23788883

Order 10496.

930 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#131×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#349	C₂	1	4	yes	maximal, complemented
3	C₄₁×#2112	C₂	1	2	yes	maximal, complemented
4	C₂²×#41×C₄₁	C₄	1	1	yes	characteristic
5	C₂×#131×C₄₁	—	2	8	no	—
6	C₂×#131×C₄₁	—	2	4	no	—
7	C₂²×M₁₆×C₄₁	C₄	1	1	yes	characteristic
8	C₂²×M₁₆×C₄₁	C₂²	1	2	yes	—
9	C₂³×C₈×C₄₁	C₄	1	2	yes	—
10	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
11	C₂×M₃₂×C₄₁	C₂²	1	4	yes	—
12	C₄₁×#349	—	2	8	no	—
13	C₄₁×C₂²⋊C₁₆	—	2	4	no	—
14	C₂×#41×C₄₁	—	2	4	no	—
15	C₂×#41×C₄₁	—	2	2	no	—
16	C₂×#41×C₄₁	—	2	2	no	—
17	C₂×#41×C₄₁	—	2	4	no	—
18	C₂²×Q₈×C₄₁	C₈	1	1	yes	characteristic
19	C₂²×D₄×C₄₁	C₈	1	2	yes	—
20	C₂²×D₄×C₄₁	C₈	1	1	yes	characteristic
21	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
23	#131×C₄₁	—	2	32	no	—
24	C₂×M₁₆×C₄₁	—	2	2	no	—
25	C₂×M₁₆×C₄₁	—	2	4	no	—
26	C₂×M₁₆×C₄₁	—	2	4	no	—
27	C₂×M₁₆×C₄₁	—	2	4	no	—
28	C₂×M₁₆×C₄₁	D₄	1	8	yes	—
29	C₂²×C₈×C₄₁	—	2	4	no	—
30	C₂²×C₈×C₄₁	—	2	4	no	—
31	C₂²×C₈×C₄₁	—	2	4	no	—
32	C₂²×C₈×C₄₁	—	2	4	no	—
33	C₂²×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₄₁	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₄₁	D₄	1	4	yes	—
36	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
37	M₃₂×C₄₁	—	2	8	no	—
38	C₂×C₁₆×C₄₁	—	2	4	no	—
39	C₂⁴×C₄₁	C₂×C₈	1	1	yes	characteristic
40	C₂⁴×C₄₁	C₂×C₈	1	2	yes	—
41	#41×C₄₁	—	2	8	no	—
42	#41×C₄₁	—	2	16	no	—
43	#41×C₄₁	—	2	8	no	—
44	C₂×Q₈×C₄₁	—	2	4	no	—
45	C₂×Q₈×C₄₁	—	2	2	no	—
46	C₂×D₄×C₄₁	—	2	4	no	—
47	C₂×D₄×C₄₁	—	2	4	no	—
48	C₂×D₄×C₄₁	—	2	4	no	—
49	C₂×D₄×C₄₁	—	2	4	no	—
50	C₂×D₄×C₄₁	—	2	2	no	—
51	C₂²×C₄×C₄₁	—	2	2	no	—
52	C₂²×C₄×C₄₁	—	2	2	no	—
53	C₂²×C₄×C₄₁	—	2	2	no	—
54	C₂²×C₄×C₄₁	—	2	2	no	—
55	C₂²×C₄×C₄₁	—	2	2	no	—
56	C₂²×C₄×C₄₁	—	2	2	no	—
57	C₂²×C₄×C₄₁	C₂²⋊C₄	1	4	yes	—
58	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₄₁	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₄₁	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₄₁	M₁₆	1	2	yes	—
62	C₂²×C₄×C₄₁	M₁₆	1	4	yes	—
63	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
64	M₁₆×C₄₁	—	2	8	no	—
65	M₁₆×C₄₁	—	2	16	no	—
66	C₂×C₈×C₄₁	—	2	16	no	—
67	C₂×C₈×C₄₁	—	2	2	no	—
68	C₂×C₈×C₄₁	—	2	2	no	—
69	C₂×C₈×C₄₁	—	2	2	no	—
70	C₂×C₈×C₄₁	—	2	8	no	—
71	C₂×C₈×C₄₁	C₂²⋊C₄	1	4	yes	—
72	C₂×C₈×C₄₁	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₄₁	C₂×D₄	1	4	yes	—
74	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
75	C₁₆×C₄₁	—	4	4	no	—
76	C₂³×C₄₁	—	2	2	no	—
77	C₂³×C₄₁	—	2	2	no	—
78	C₂³×C₄₁	—	2	2	no	—
79	C₂³×C₄₁	—	4	2	no	—
80	C₂³×C₄₁	—	4	4	no	—
81	C₂³×C₄₁	C₂²⋊C₈	1	2	yes	—
82	C₂³×C₄₁	C₂²⋊C₈	1	4	yes	—
83	C₂³×C₄₁	C₂²×C₈	1	1	yes	characteristic
84	Q₈×C₄₁	—	2	8	no	—
85	D₄×C₄₁	—	2	16	no	—
86	D₄×C₄₁	—	2	8	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	4	no	—
91	C₂×C₄×C₄₁	—	2	2	no	—
92	C₂×C₄×C₄₁	—	2	4	no	—
93	C₂×C₄×C₄₁	—	2	2	no	—
94	C₂×C₄×C₄₁	—	2	4	no	—
95	C₂×C₄×C₄₁	—	2	4	no	—
96	C₂×C₄×C₄₁	—	2	4	no	—
97	C₂×C₄×C₄₁	C₂²⋊C₈	1	4	yes	—
98	C₂×C₄×C₄₁	C₂²⋊C₈	1	2	yes	—
99	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
100	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₄₁	C₂×M₁₆	1	1	yes	characteristic
102	C₂×C₄×C₄₁	C₂×M₁₆	1	2	yes	—
103	C₈×C₄₁	—	2	2	no	—
104	C₈×C₄₁	—	2	4	no	—
105	C₈×C₄₁	#120	1	4	yes	—
106	#12682	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂²×C₄₁	—	2	2	no	—
108	C₂²×C₄₁	—	2	4	no	—
109	C₂²×C₄₁	—	2	2	no	—
110	C₂²×C₄₁	—	2	4	no	—
111	C₂²×C₄₁	—	4	4	no	—
112	C₂²×C₄₁	—	4	4	no	—
113	C₂²×C₄₁	—	4	4	no	—
114	C₂²×C₄₁	—	4	4	no	—
115	C₂²×C₄₁	C₂×C₂²⋊C₈	1	2	yes	—
116	C₂²×C₄₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
117	C₄×C₄₁	—	2	2	no	—
118	C₄×C₄₁	—	2	4	no	—
119	C₄×C₄₁	#379	1	1	yes	characteristic
120	C₄×C₄₁	#407	1	2	yes	—
121	C₄×C₄₁	#407	1	1	yes	characteristic
122	C₂²×#131	C₂×C₄₁	1	1	yes	characteristic
123	C₂×#349	C₂×C₄₁	1	4	yes	complemented
124	#2112	C₂×C₄₁	1	2	yes	complemented
125	C₂×C₄₁	—	2	2	no	—
126	C₂×C₄₁	—	4	2	no	—
127	C₂×C₄₁	—	4	4	no	—
128	C₂×C₄₁	#1779	1	1	yes	characteristic
129	C₂×C₄₁	#2115	1	2	yes	—
130	C₂²×#41	C₄×C₄₁	1	1	yes	characteristic
131	C₂×#131	—	2	8	no	—
132	C₂×#131	—	2	4	no	—
133	C₂²×M₁₆	C₄×C₄₁	1	1	yes	characteristic
134	C₂²×M₁₆	C₂²×C₄₁	1	2	yes	—
135	C₂³×C₈	C₄×C₄₁	1	2	yes	—
136	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
137	C₂×M₃₂	C₂²×C₄₁	1	4	yes	—
138	#349	—	2	8	no	—
139	C₂²⋊C₁₆	—	2	4	no	—
140	C₄₁	#12682	1	1	yes	characteristic, direct product factor, Sylow
141	C₂×#41	—	2	4	no	—
142	C₂×#41	—	2	2	no	—
143	C₂×#41	—	2	2	no	—
144	C₂×#41	—	2	4	no	—
145	C₂²×Q₈	C₈×C₄₁	1	1	yes	characteristic
146	C₂²×D₄	C₈×C₄₁	1	2	yes	—
147	C₂²×D₄	C₈×C₄₁	1	1	yes	characteristic
148	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
149	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
150	#131	—	2	32	no	—
151	C₂×M₁₆	—	2	2	no	—
152	C₂×M₁₆	—	2	4	no	—
153	C₂×M₁₆	—	2	4	no	—
154	C₂×M₁₆	—	2	4	no	—
155	C₂×M₁₆	D₄×C₄₁	1	8	yes	—
156	C₂²×C₈	—	2	4	no	—
157	C₂²×C₈	—	2	4	no	—
158	C₂²×C₈	—	2	4	no	—
159	C₂²×C₈	—	2	4	no	—
160	C₂²×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
161	C₂²×C₈	C₂×C₄×C₄₁	1	2	yes	—
162	C₂²×C₈	D₄×C₄₁	1	4	yes	—
163	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
164	M₃₂	—	2	8	no	—
165	C₂×C₁₆	—	2	4	no	—
166	C₂⁴	C₂×C₈×C₄₁	1	1	yes	characteristic
167	C₂⁴	C₂×C₈×C₄₁	1	2	yes	—
168	#41	—	2	8	no	—
169	#41	—	2	16	no	—
170	#41	—	2	8	no	—
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	—	2	2	no	—
173	C₂×D₄	—	2	4	no	—
174	C₂×D₄	—	2	4	no	—
175	C₂×D₄	—	2	4	no	—
176	C₂×D₄	—	2	4	no	—
177	C₂×D₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	C₂²⋊C₄×C₄₁	1	4	yes	—
185	C₂²×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
186	C₂²×C₄	C₂×C₈×C₄₁	1	2	yes	—
187	C₂²×C₄	C₂×C₈×C₄₁	1	1	yes	characteristic
188	C₂²×C₄	M₁₆×C₄₁	1	2	yes	—
189	C₂²×C₄	M₁₆×C₄₁	1	4	yes	—
190	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
191	M₁₆	—	2	8	no	—
192	M₁₆	—	2	16	no	—
193	C₂×C₈	—	2	16	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	8	no	—
198	C₂×C₈	C₂²⋊C₄×C₄₁	1	2	yes	—
199	C₂×C₈	C₂²⋊C₄×C₄₁	1	4	yes	—
200	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
201	C₂×C₈	C₂×D₄×C₄₁	1	4	yes	—
202	C₁₆	—	4	4	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	2	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	4	2	no	—
207	C₂³	—	4	4	no	—
208	C₂³	C₂²⋊C₈×C₄₁	1	2	yes	—
209	C₂³	C₂²⋊C₈×C₄₁	1	4	yes	—
210	C₂³	C₂²×C₈×C₄₁	1	1	yes	characteristic
211	Q₈	—	2	8	no	—
212	D₄	—	2	16	no	—
213	D₄	—	2	8	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	C₂²⋊C₈×C₄₁	1	4	yes	—
225	C₂×C₄	C₂²⋊C₈×C₄₁	1	2	yes	—
226	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
227	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
228	C₂×C₄	C₂×M₁₆×C₄₁	1	2	yes	—
229	C₂×C₄	C₂×M₁₆×C₄₁	1	1	yes	characteristic
230	C₈	—	2	2	no	—
231	C₈	—	2	4	no	—
232	C₈	#120×C₄₁	1	4	yes	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	2	4	no	—
236	C₂²	—	2	4	no	—
237	C₂²	—	4	4	no	—
238	C₂²	—	4	4	no	—
239	C₂²	—	4	4	no	—
240	C₂²	—	4	4	no	—
241	C₂²	C₂×C₂²⋊C₈×C₄₁	1	2	yes	—
242	C₂²	C₂×C₂²⋊C₈×C₄₁	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	2	4	no	—
245	C₄	C₄₁×#379	1	1	yes	characteristic
246	C₄	C₄₁×#407	1	2	yes	—
247	C₄	C₄₁×#407	1	1	yes	characteristic
248	C₂	—	2	2	no	—
249	C₂	—	4	2	no	—
250	C₂	—	4	4	no	—
251	C₂	C₄₁×#1779	1	1	yes	characteristic
252	C₂	C₄₁×#2115	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₄₁	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₈×C₄₁
Frattini, cosocle	C₂²×C₈	C₂³×C₄₁
socle	C₂²×C₄₁	C₂×C₂²⋊C₈
prime radical	C₂²×#41×C₄₁	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₁ ◁ C₂³×C₈×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₁×#349	C₂	semidirect
3. C₄₁×#2112	C₂	semidirect
106. #12682	C₄₁	direct (N × H)
123. C₂×#349	C₂×C₄₁	semidirect
124. #2112	C₂×C₄₁	semidirect
140. C₄₁	#12682	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×#131×C₄₁	2
C₂×C₄₁×#349	2
C₄₁×#2112	2
#12682	41

Subgroups — C41×#12682 #23788883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#12682 #23788883