Subgroups — C₂×C₄₁×#1360 #23788124

Order 10496.

434 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#495	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄₁×#334	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#1360	C₂	1	4	yes	maximal, direct product factor
5	C₂²×Q₁₆×C₄₁	C₄	1	1	yes	characteristic
6	C₂²×SD₁₆×C₄₁	—	2	1	no	—
7	C₂²×D₈×C₄₁	C₄	1	1	yes	characteristic
8	C₂×#124×C₄₁	—	2	1	no	—
9	C₂²×C₁₆×C₄₁	—	2	1	no	—
10	C₄₁×#495	—	2	1	no	—
11	C₄₁×#495	—	2	1	no	—
12	C₄₁×#495	C₄	1	2	yes	—
13	C₄₁×#495	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₄₁	—	2	1	no	—
15	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
16	C₄₁×C₄⋊C₁₆	—	2	1	no	—
17	C₄₁×C₄⋊C₁₆	C₂²	1	2	yes	complemented
18	C₄₁×#334	—	2	1	no	—
19	C₄₁×#334	C₂²	1	2	yes	complemented
20	C₂²×Q₈×C₄₁	—	2	1	no	—
21	C₂²×D₄×C₄₁	—	2	1	no	—
22	C₂×Q₁₆×C₄₁	—	2	1	no	—
23	C₂×Q₁₆×C₄₁	—	4	2	no	—
24	C₂×Q₁₆×C₄₁	C₂×C₄	1	2	yes	—
25	C₂×SD₁₆×C₄₁	—	2	2	no	—
26	C₂×SD₁₆×C₄₁	—	2	2	no	—
27	C₂×SD₁₆×C₄₁	—	4	2	no	—
28	C₂×SD₁₆×C₄₁	—	4	2	no	—
29	C₂×D₈×C₄₁	—	2	1	no	—
30	C₂×D₈×C₄₁	—	4	2	no	—
31	C₂×D₈×C₄₁	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₄₁	—	4	1	no	—
33	C₂²×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
35	#124×C₄₁	—	2	2	no	—
36	#124×C₄₁	—	2	2	no	—
37	#124×C₄₁	—	4	1	no	—
38	#124×C₄₁	—	4	1	no	—
39	C₂×C₂²⋊C₄×C₄₁	—	4	1	no	—
40	C₂×C₂²⋊C₄×C₄₁	—	4	1	no	—
41	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
42	C₂×C₁₆×C₄₁	—	2	2	no	—
43	C₂×C₁₆×C₄₁	—	4	2	no	—
44	C₄⋊C₈×C₄₁	—	2	2	no	—
45	C₄⋊C₈×C₄₁	—	4	1	no	—
46	C₄×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₄₁	D₄	1	1	yes	characteristic
48	C₄×C₈×C₄₁	D₄	1	1	yes	characteristic
49	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
50	C₂⁴×C₄₁	—	4	1	no	—
51	C₂×Q₈×C₄₁	—	2	2	no	—
52	C₂×Q₈×C₄₁	—	4	1	no	—
53	C₂×Q₈×C₄₁	—	4	2	no	—
54	C₂×Q₈×C₄₁	—	4	2	no	—
55	C₂×D₄×C₄₁	—	2	2	no	—
56	C₂×D₄×C₄₁	—	4	2	no	—
57	C₂×D₄×C₄₁	—	4	2	no	—
58	C₂×D₄×C₄₁	—	4	1	no	—
59	C₂²×C₄×C₄₁	—	4	1	no	—
60	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₁	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₄₁	SD₁₆	1	1	yes	characteristic
63	Q₁₆×C₄₁	—	4	2	no	—
64	Q₁₆×C₄₁	—	4	2	no	—
65	SD₁₆×C₄₁	—	4	4	no	—
66	SD₁₆×C₄₁	—	4	2	no	—
67	SD₁₆×C₄₁	—	4	2	no	—
68	D₈×C₄₁	—	4	2	no	—
69	D₈×C₄₁	—	4	2	no	—
70	C₂×C₈×C₄₁	—	2	2	no	—
71	C₂×C₈×C₄₁	—	4	2	no	—
72	C₂×C₈×C₄₁	—	8	2	no	—
73	C₂×C₈×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₄₁	D₈	1	2	yes	—
76	C₂×C₈×C₄₁	SD₁₆	1	2	yes	—
77	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
79	C₂²⋊C₄×C₄₁	—	4	2	no	—
80	C₂²⋊C₄×C₄₁	—	4	2	no	—
81	C₂²⋊C₄×C₄₁	—	4	2	no	—
82	C₂²⋊C₄×C₄₁	—	4	2	no	—
83	C₄²×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
84	C₄²×C₄₁	D₈	1	1	yes	characteristic
85	C₄²×C₄₁	SD₁₆	1	1	yes	characteristic
86	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
87	C₁₆×C₄₁	—	4	2	no	—
88	C₂³×C₄₁	—	4	2	no	—
89	C₂³×C₄₁	—	8	2	no	—
90	C₂³×C₄₁	—	8	2	no	—
91	C₂³×C₄₁	—	8	2	no	—
92	C₂³×C₄₁	#102	1	1	yes	characteristic
93	Q₈×C₄₁	—	4	2	no	—
94	Q₈×C₄₁	—	4	2	no	—
95	Q₈×C₄₁	—	8	2	no	—
96	D₄×C₄₁	—	4	2	no	—
97	D₄×C₄₁	—	4	2	no	—
98	D₄×C₄₁	—	8	2	no	—
99	C₂×C₄×C₄₁	—	2	2	no	—
100	C₂×C₄×C₄₁	—	2	2	no	—
101	C₂×C₄×C₄₁	—	4	2	no	—
102	C₂×C₄×C₄₁	—	8	2	no	—
103	C₂×C₄×C₄₁	#100	1	2	yes	—
104	C₂×C₄×C₄₁	#102	1	1	yes	characteristic
105	C₂×C₄×C₄₁	#102	1	1	yes	characteristic
106	C₂×C₄×C₄₁	#102	1	1	yes	characteristic
107	C₂×C₄×C₄₁	#102	1	2	yes	—
108	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₄₁	C₂×SD₁₆	1	1	yes	characteristic
111	C₈×C₄₁	—	2	1	no	—
112	C₈×C₄₁	—	2	1	no	—
113	C₈×C₄₁	—	8	2	no	—
114	C₈×C₄₁	#102	1	2	yes	—
115	C₈×C₄₁	C₂×D₈	1	1	yes	characteristic
116	C₈×C₄₁	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×#1360	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂²×C₄₁	—	8	2	no	—
119	C₂²×C₄₁	—	8	2	no	—
120	C₂²×C₄₁	—	8	2	no	—
121	C₂²×C₄₁	—	16	4	no	—
122	C₂²×C₄₁	#331	1	2	yes	—
123	C₂²×C₄₁	#359	1	2	yes	—
124	C₂²×C₄₁	#360	1	2	yes	—
125	C₂²×C₄₁	C₂×#102	1	1	yes	characteristic
126	C₄×C₄₁	—	2	1	no	—
127	C₄×C₄₁	—	2	1	no	—
128	C₄×C₄₁	—	2	1	no	—
129	C₄×C₄₁	—	2	1	no	—
130	C₄×C₄₁	—	8	2	no	—
131	C₄×C₄₁	#331	1	2	yes	—
132	C₄×C₄₁	C₂×#100	1	1	yes	characteristic
133	C₄×C₄₁	C₂×#102	1	1	yes	characteristic
134	C₂×#495	C₂×C₄₁	1	1	yes	characteristic
135	C₂×C₄⋊C₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
136	C₂×#334	C₂×C₄₁	1	1	yes	characteristic, complemented
137	#1360	C₂×C₄₁	1	4	yes	direct product factor
138	C₂×C₄₁	—	16	2	no	—
139	C₂×C₄₁	#1360	1	4	yes	direct product factor
140	C₂×C₄₁	C₂×#331	1	1	yes	characteristic
141	C₂×C₄₁	C₂×#359	1	1	yes	characteristic
142	C₂×C₄₁	C₂×#360	1	1	yes	characteristic
143	C₂²×Q₁₆	C₄×C₄₁	1	1	yes	characteristic
144	C₂²×SD₁₆	—	2	1	no	—
145	C₂²×D₈	C₄×C₄₁	1	1	yes	characteristic
146	C₂×#124	—	2	1	no	—
147	C₂²×C₁₆	—	2	1	no	—
148	#495	—	2	1	no	—
149	#495	—	2	1	no	—
150	#495	C₄×C₄₁	1	2	yes	—
151	#495	C₂²×C₄₁	1	2	yes	—
152	C₂×C₄⋊C₈	—	2	1	no	—
153	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
154	C₄⋊C₁₆	—	2	1	no	—
155	C₄⋊C₁₆	C₂²×C₄₁	1	2	yes	complemented
156	#334	—	2	1	no	—
157	#334	C₂²×C₄₁	1	2	yes	complemented
158	C₄₁	C₂×#1360	1	1	yes	characteristic, direct product factor, Sylow
159	C₂²×Q₈	—	2	1	no	—
160	C₂²×D₄	—	2	1	no	—
161	C₂×Q₁₆	—	2	1	no	—
162	C₂×Q₁₆	—	4	2	no	—
163	C₂×Q₁₆	C₂×C₄×C₄₁	1	2	yes	—
164	C₂×SD₁₆	—	2	2	no	—
165	C₂×SD₁₆	—	2	2	no	—
166	C₂×SD₁₆	—	4	2	no	—
167	C₂×SD₁₆	—	4	2	no	—
168	C₂×D₈	—	2	1	no	—
169	C₂×D₈	—	4	2	no	—
170	C₂×D₈	C₂×C₄×C₄₁	1	2	yes	—
171	C₂²×C₈	—	4	1	no	—
172	C₂²×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
173	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
174	#124	—	2	2	no	—
175	#124	—	2	2	no	—
176	#124	—	4	1	no	—
177	#124	—	4	1	no	—
178	C₂×C₂²⋊C₄	—	4	1	no	—
179	C₂×C₂²⋊C₄	—	4	1	no	—
180	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
181	C₂×C₁₆	—	2	2	no	—
182	C₂×C₁₆	—	4	2	no	—
183	C₄⋊C₈	—	2	2	no	—
184	C₄⋊C₈	—	4	1	no	—
185	C₄×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
186	C₄×C₈	D₄×C₄₁	1	1	yes	characteristic
187	C₄×C₈	D₄×C₄₁	1	1	yes	characteristic
188	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
189	C₂⁴	—	4	1	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	—	4	1	no	—
192	C₂×Q₈	—	4	2	no	—
193	C₂×Q₈	—	4	2	no	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	—	4	2	no	—
196	C₂×D₄	—	4	2	no	—
197	C₂×D₄	—	4	1	no	—
198	C₂²×C₄	—	4	1	no	—
199	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
200	C₂²×C₄	D₈×C₄₁	1	1	yes	characteristic
201	C₂²×C₄	SD₁₆×C₄₁	1	1	yes	characteristic
202	Q₁₆	—	4	2	no	—
203	Q₁₆	—	4	2	no	—
204	SD₁₆	—	4	4	no	—
205	SD₁₆	—	4	2	no	—
206	SD₁₆	—	4	2	no	—
207	D₈	—	4	2	no	—
208	D₈	—	4	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	4	2	no	—
211	C₂×C₈	—	8	2	no	—
212	C₂×C₈	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
213	C₂×C₈	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
214	C₂×C₈	D₈×C₄₁	1	2	yes	—
215	C₂×C₈	SD₁₆×C₄₁	1	2	yes	—
216	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
217	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
218	C₂²⋊C₄	—	4	2	no	—
219	C₂²⋊C₄	—	4	2	no	—
220	C₂²⋊C₄	—	4	2	no	—
221	C₂²⋊C₄	—	4	2	no	—
222	C₄²	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
223	C₄²	D₈×C₄₁	1	1	yes	characteristic
224	C₄²	SD₁₆×C₄₁	1	1	yes	characteristic
225	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
226	C₁₆	—	4	2	no	—
227	C₂³	—	4	2	no	—
228	C₂³	—	8	2	no	—
229	C₂³	—	8	2	no	—
230	C₂³	—	8	2	no	—
231	C₂³	#102×C₄₁	1	1	yes	characteristic
232	Q₈	—	4	2	no	—
233	Q₈	—	4	2	no	—
234	Q₈	—	8	2	no	—
235	D₄	—	4	2	no	—
236	D₄	—	4	2	no	—
237	D₄	—	8	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	8	2	no	—
242	C₂×C₄	#100×C₄₁	1	2	yes	—
243	C₂×C₄	#102×C₄₁	1	1	yes	characteristic
244	C₂×C₄	#102×C₄₁	1	1	yes	characteristic
245	C₂×C₄	#102×C₄₁	1	1	yes	characteristic
246	C₂×C₄	#102×C₄₁	1	2	yes	—
247	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
248	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
249	C₂×C₄	C₂×SD₁₆×C₄₁	1	1	yes	characteristic
250	C₈	—	2	1	no	—
251	C₈	—	2	1	no	—
252	C₈	—	8	2	no	—
253	C₈	#102×C₄₁	1	2	yes	—
254	C₈	C₂×D₈×C₄₁	1	1	yes	characteristic
255	C₈	C₂×SD₁₆×C₄₁	1	1	yes	characteristic
256	C₂²	—	8	2	no	—
257	C₂²	—	8	2	no	—
258	C₂²	—	8	2	no	—
259	C₂²	—	16	4	no	—
260	C₂²	C₄₁×#331	1	2	yes	—
261	C₂²	C₄₁×#359	1	2	yes	—
262	C₂²	C₄₁×#360	1	2	yes	—
263	C₂²	C₂×#102×C₄₁	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	8	2	no	—
269	C₄	C₄₁×#331	1	2	yes	—
270	C₄	C₂×#100×C₄₁	1	1	yes	characteristic
271	C₄	C₂×#102×C₄₁	1	1	yes	characteristic
272	C₂	—	16	2	no	—
273	C₂	C₄₁×#1360	1	4	yes	direct product factor
274	C₂	C₂×C₄₁×#331	1	1	yes	characteristic
275	C₂	C₂×C₄₁×#359	1	1	yes	characteristic
276	C₂	C₂×C₄₁×#360	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₁	#102
commutator	C₂×C₈	C₂²×C₄×C₄₁
Frattini, cosocle	C₄×C₈	C₂³×C₄₁
prime radical	C₂²×D₈×C₄₁	C₄
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ C₂×C₄×C₈×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₁×C₄⋊C₁₆	C₂	semidirect
3. C₂×C₄₁×#334	C₂	semidirect
4. C₄₁×#1360	C₂	direct (N × H)
17. C₄₁×C₄⋊C₁₆	C₂²	semidirect
19. C₄₁×#334	C₂²	semidirect
117. C₂×#1360	C₄₁	direct (N × H)
135. C₂×C₄⋊C₁₆	C₂×C₄₁	semidirect
136. C₂×#334	C₂×C₄₁	semidirect
137. #1360	C₂×C₄₁	direct (N × H)
139. C₂×C₄₁	#1360	direct (N × H)
155. C₄⋊C₁₆	C₂²×C₄₁	semidirect
157. #334	C₂²×C₄₁	semidirect
158. C₄₁	C₂×#1360	direct (N × H)
273. C₂	C₄₁×#1360	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×#495	2
C₂×C₄₁×C₄⋊C₁₆	2
C₂×C₄₁×#334	2
C₄₁×#1360	2
C₂×#1360	41

Subgroups — C2×C41×#1360 #23788124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#1360 #23788124