Subgroups — C₂×C₄₁×#2209 #23809778

Order 10496.

702 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₃₂×C₄₁	C₂	1	2	yes	maximal
2	C₂×C₄₁×#494	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Q₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#2209	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄₁×#358	C₂	1	2	yes	maximal
7	C₂²×Q₁₆×C₄₁	—	2	1	no	—
8	C₂²×Q₁₆×C₄₁	C₂²	1	2	yes	—
9	C₂²×Q₁₆×C₄₁	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₄₁	—	2	1	no	—
11	C₂×C₄×Q₈×C₄₁	—	2	1	no	—
12	C₂×Q₃₂×C₄₁	—	2	8	no	—
13	C₂×Q₃₂×C₄₁	C₂²	1	8	yes	—
14	C₂²×C₁₆×C₄₁	C₂²	1	2	yes	—
15	C₄₁×#494	—	2	2	no	—
16	C₄₁×#494	C₂²	1	4	yes	—
17	C₄×Q₁₆×C₄₁	—	2	2	no	—
18	C₄×Q₁₆×C₄₁	C₂²	1	4	yes	—
19	C₂×#107×C₄₁	C₂²	1	1	yes	characteristic
20	C₂×#103×C₄₁	—	2	1	no	—
21	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
22	C₄₁×C₄⋊C₁₆	C₂²	1	4	yes	—
23	C₄₁×#358	C₂²	1	8	yes	—
24	C₂²×Q₈×C₄₁	—	2	1	no	—
25	C₂²×Q₈×C₄₁	—	2	2	no	—
26	C₂×Q₁₆×C₄₁	—	2	1	no	—
27	C₂×Q₁₆×C₄₁	—	2	4	no	—
28	C₂×Q₁₆×C₄₁	—	2	4	no	—
29	C₂×Q₁₆×C₄₁	—	2	2	no	—
30	C₂×Q₁₆×C₄₁	—	2	2	no	—
31	C₂×Q₁₆×C₄₁	—	2	4	no	—
32	C₂×Q₁₆×C₄₁	—	4	4	no	—
33	C₂×Q₁₆×C₄₁	D₄	1	4	yes	—
34	C₂×Q₁₆×C₄₁	C₂³	1	4	yes	—
35	C₂×Q₁₆×C₄₁	C₂³	1	2	yes	—
36	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
38	C₄⋊Q₈×C₄₁	—	2	4	no	—
39	C₄⋊Q₈×C₄₁	—	4	2	no	—
40	C₄×Q₈×C₄₁	—	2	4	no	—
41	C₄×Q₈×C₄₁	—	4	2	no	—
42	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
43	C₂×C₄⋊C₄×C₄₁	—	4	1	no	—
44	C₂×C₄⋊C₄×C₄₁	—	4	1	no	—
45	C₂×C₄⋊C₄×C₄₁	—	4	1	no	—
46	C₂×C₄²×C₄₁	—	4	1	no	—
47	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
48	Q₃₂×C₄₁	—	2	16	no	—
49	C₂×C₁₆×C₄₁	—	2	4	no	—
50	C₂×C₁₆×C₄₁	C₂³	1	4	yes	—
51	#107×C₄₁	—	2	1	no	—
52	#107×C₄₁	C₂³	1	2	yes	—
53	#103×C₄₁	—	2	4	no	—
54	C₄×C₈×C₄₁	D₄	1	2	yes	—
55	C₄×C₈×C₄₁	C₂³	1	2	yes	—
56	C₂×Q₈×C₄₁	—	2	4	no	—
57	C₂×Q₈×C₄₁	—	2	2	no	—
58	C₂×Q₈×C₄₁	—	2	4	no	—
59	C₂×Q₈×C₄₁	—	4	1	no	—
60	C₂×Q₈×C₄₁	—	4	2	no	—
61	C₂×Q₈×C₄₁	—	4	2	no	—
62	C₂×Q₈×C₄₁	—	4	4	no	—
63	C₂×Q₈×C₄₁	—	4	4	no	—
64	C₂²×C₄×C₄₁	—	4	1	no	—
65	C₂²×C₄×C₄₁	—	4	1	no	—
66	C₂²×C₄×C₄₁	—	4	2	no	—
67	C₂²×C₄×C₄₁	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₄₁	D₈	1	1	yes	characteristic
69	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
70	Q₁₆×C₄₁	—	2	4	no	—
71	Q₁₆×C₄₁	—	2	4	no	—
72	Q₁₆×C₄₁	—	2	2	no	—
73	Q₁₆×C₄₁	—	2	4	no	—
74	Q₁₆×C₄₁	—	4	8	no	—
75	Q₁₆×C₄₁	—	4	4	no	—
76	Q₁₆×C₄₁	C₂×D₄	1	4	yes	—
77	C₂×C₈×C₄₁	—	2	2	no	—
78	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₄₁	#41	1	2	yes	—
82	C₂×C₈×C₄₁	C₂⁴	1	1	yes	characteristic
83	C₄⋊C₄×C₄₁	—	2	2	no	—
84	C₄⋊C₄×C₄₁	—	4	1	no	—
85	C₄⋊C₄×C₄₁	—	4	4	no	—
86	C₄⋊C₄×C₄₁	—	4	4	no	—
87	C₄⋊C₄×C₄₁	—	4	4	no	—
88	C₄²×C₄₁	—	4	4	no	—
89	C₄²×C₄₁	D₈	1	2	yes	—
90	C₄²×C₄₁	C₂×D₄	1	2	yes	—
91	C₁₆×C₄₁	—	2	4	no	—
92	C₂³×C₄₁	C₂×D₈	1	1	yes	characteristic
93	Q₈×C₄₁	—	2	4	no	—
94	Q₈×C₄₁	—	4	2	no	—
95	Q₈×C₄₁	—	4	4	no	—
96	Q₈×C₄₁	—	4	4	no	—
97	Q₈×C₄₁	—	4	4	no	—
98	Q₈×C₄₁	—	8	4	no	—
99	C₂×C₄×C₄₁	—	2	2	no	—
100	C₂×C₄×C₄₁	—	4	2	no	—
101	C₂×C₄×C₄₁	—	4	4	no	—
102	C₂×C₄×C₄₁	—	4	2	no	—
103	C₂×C₄×C₄₁	—	4	4	no	—
104	C₂×C₄×C₄₁	—	8	4	no	—
105	C₂×C₄×C₄₁	—	8	2	no	—
106	C₂×C₄×C₄₁	Q₃₂	1	4	yes	complemented
107	C₂×C₄×C₄₁	#121	1	2	yes	—
108	C₂×C₄×C₄₁	C₂×D₈	1	2	yes	—
109	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₄₁	C₂×D₈	1	2	yes	—
111	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
112	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
113	C₈×C₄₁	—	2	2	no	—
114	C₈×C₄₁	#121	1	2	yes	—
115	C₈×C₄₁	C₂²×D₄	1	1	yes	characteristic
116	C₈×C₄₁	C₂×#41	1	1	yes	characteristic
117	C₂×#2209	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂²×C₄₁	#459	1	2	yes	—
119	C₂²×C₄₁	C₂×Q₃₂	1	2	yes	—
120	C₂²×C₄₁	#510	1	2	yes	—
121	C₂²×C₄₁	C₂²×D₈	1	1	yes	characteristic
122	C₄×C₄₁	—	2	2	no	—
123	C₄×C₄₁	—	4	4	no	—
124	C₄×C₄₁	—	8	2	no	—
125	C₄×C₄₁	—	8	4	no	—
126	C₄×C₄₁	#459	1	2	yes	—
127	C₄×C₄₁	C₂×Q₃₂	1	4	yes	complemented
128	C₄×C₄₁	C₂×#121	1	1	yes	characteristic
129	C₄×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₂²×Q₃₂	C₂×C₄₁	1	2	yes	—
131	C₂×#494	C₂×C₄₁	1	1	yes	characteristic
132	C₂×C₄×Q₁₆	C₂×C₄₁	1	1	yes	characteristic
133	#2209	C₂×C₄₁	1	8	yes	direct product factor
134	C₂×C₄⋊C₁₆	C₂×C₄₁	1	1	yes	characteristic
135	C₂×#358	C₂×C₄₁	1	2	yes	—
136	C₂×C₄₁	#2209	1	4	yes	direct product factor
137	C₂×C₄₁	C₂×#459	1	1	yes	characteristic
138	C₂×C₄₁	C₂²×Q₃₂	1	1	yes	characteristic
139	C₂×C₄₁	C₂×#510	1	1	yes	characteristic
140	C₂²×Q₁₆	—	2	1	no	—
141	C₂²×Q₁₆	C₂²×C₄₁	1	2	yes	—
142	C₂²×Q₁₆	C₂²×C₄₁	1	1	yes	characteristic
143	C₂×C₄⋊Q₈	—	2	1	no	—
144	C₂×C₄×Q₈	—	2	1	no	—
145	C₂×Q₃₂	—	2	8	no	—
146	C₂×Q₃₂	C₂²×C₄₁	1	8	yes	—
147	C₂²×C₁₆	C₂²×C₄₁	1	2	yes	—
148	#494	—	2	2	no	—
149	#494	C₂²×C₄₁	1	4	yes	—
150	C₄×Q₁₆	—	2	2	no	—
151	C₄×Q₁₆	C₂²×C₄₁	1	4	yes	—
152	C₂×#107	C₂²×C₄₁	1	1	yes	characteristic
153	C₂×#103	—	2	1	no	—
154	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
155	C₄⋊C₁₆	C₂²×C₄₁	1	4	yes	—
156	#358	C₂²×C₄₁	1	8	yes	—
157	C₄₁	C₂×#2209	1	1	yes	characteristic, direct product factor, Sylow
158	C₂²×Q₈	—	2	1	no	—
159	C₂²×Q₈	—	2	2	no	—
160	C₂×Q₁₆	—	2	1	no	—
161	C₂×Q₁₆	—	2	4	no	—
162	C₂×Q₁₆	—	2	4	no	—
163	C₂×Q₁₆	—	2	2	no	—
164	C₂×Q₁₆	—	2	2	no	—
165	C₂×Q₁₆	—	2	4	no	—
166	C₂×Q₁₆	—	4	4	no	—
167	C₂×Q₁₆	D₄×C₄₁	1	4	yes	—
168	C₂×Q₁₆	C₂³×C₄₁	1	4	yes	—
169	C₂×Q₁₆	C₂³×C₄₁	1	2	yes	—
170	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
171	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
172	C₄⋊Q₈	—	2	4	no	—
173	C₄⋊Q₈	—	4	2	no	—
174	C₄×Q₈	—	2	4	no	—
175	C₄×Q₈	—	4	2	no	—
176	C₂×C₄⋊C₄	—	2	1	no	—
177	C₂×C₄⋊C₄	—	4	1	no	—
178	C₂×C₄⋊C₄	—	4	1	no	—
179	C₂×C₄⋊C₄	—	4	1	no	—
180	C₂×C₄²	—	4	1	no	—
181	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
182	Q₃₂	—	2	16	no	—
183	C₂×C₁₆	—	2	4	no	—
184	C₂×C₁₆	C₂³×C₄₁	1	4	yes	—
185	#107	—	2	1	no	—
186	#107	C₂³×C₄₁	1	2	yes	—
187	#103	—	2	4	no	—
188	C₄×C₈	D₄×C₄₁	1	2	yes	—
189	C₄×C₈	C₂³×C₄₁	1	2	yes	—
190	C₂×Q₈	—	2	4	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×Q₈	—	2	4	no	—
193	C₂×Q₈	—	4	1	no	—
194	C₂×Q₈	—	4	2	no	—
195	C₂×Q₈	—	4	2	no	—
196	C₂×Q₈	—	4	4	no	—
197	C₂×Q₈	—	4	4	no	—
198	C₂²×C₄	—	4	1	no	—
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	D₈×C₄₁	1	1	yes	characteristic
202	C₂²×C₄	D₈×C₄₁	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
204	Q₁₆	—	2	4	no	—
205	Q₁₆	—	2	4	no	—
206	Q₁₆	—	2	2	no	—
207	Q₁₆	—	2	4	no	—
208	Q₁₆	—	4	8	no	—
209	Q₁₆	—	4	4	no	—
210	Q₁₆	C₂×D₄×C₄₁	1	4	yes	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
213	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
214	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
215	C₂×C₈	#41×C₄₁	1	2	yes	—
216	C₂×C₈	C₂⁴×C₄₁	1	1	yes	characteristic
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	4	1	no	—
219	C₄⋊C₄	—	4	4	no	—
220	C₄⋊C₄	—	4	4	no	—
221	C₄⋊C₄	—	4	4	no	—
222	C₄²	—	4	4	no	—
223	C₄²	D₈×C₄₁	1	2	yes	—
224	C₄²	C₂×D₄×C₄₁	1	2	yes	—
225	C₁₆	—	2	4	no	—
226	C₂³	C₂×D₈×C₄₁	1	1	yes	characteristic
227	Q₈	—	2	4	no	—
228	Q₈	—	4	2	no	—
229	Q₈	—	4	4	no	—
230	Q₈	—	4	4	no	—
231	Q₈	—	4	4	no	—
232	Q₈	—	8	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	—	8	4	no	—
240	C₂×C₄	Q₃₂×C₄₁	1	4	yes	complemented
241	C₂×C₄	#121×C₄₁	1	2	yes	—
242	C₂×C₄	C₂×D₈×C₄₁	1	2	yes	—
243	C₂×C₄	C₂×D₈×C₄₁	1	2	yes	—
244	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
245	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
246	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	#121×C₄₁	1	2	yes	—
249	C₈	C₂²×D₄×C₄₁	1	1	yes	characteristic
250	C₈	C₂×#41×C₄₁	1	1	yes	characteristic
251	C₂²	C₄₁×#459	1	2	yes	—
252	C₂²	C₂×Q₃₂×C₄₁	1	2	yes	—
253	C₂²	C₄₁×#510	1	2	yes	—
254	C₂²	C₂²×D₈×C₄₁	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	4	4	no	—
257	C₄	—	8	2	no	—
258	C₄	—	8	4	no	—
259	C₄	C₄₁×#459	1	2	yes	—
260	C₄	C₂×Q₃₂×C₄₁	1	4	yes	complemented
261	C₄	C₂×#121×C₄₁	1	1	yes	characteristic
262	C₄	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
263	C₂	C₄₁×#2209	1	4	yes	direct product factor
264	C₂	C₂×C₄₁×#459	1	1	yes	characteristic
265	C₂	C₂²×Q₃₂×C₄₁	1	1	yes	characteristic
266	C₂	C₂×C₄₁×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₄₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₄₁
LCS	G ▷ C₂×C₈ ▷ 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
4. C₄₁×#2209	C₂	direct (N × H)
106. C₂×C₄×C₄₁	Q₃₂	semidirect
117. C₂×#2209	C₄₁	direct (N × H)
127. C₄×C₄₁	C₂×Q₃₂	semidirect
133. #2209	C₂×C₄₁	direct (N × H)
136. C₂×C₄₁	#2209	direct (N × H)
157. C₄₁	C₂×#2209	direct (N × H)
240. C₂×C₄	Q₃₂×C₄₁	semidirect
260. C₄	C₂×Q₃₂×C₄₁	semidirect
263. C₂	C₄₁×#2209	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×Q₃₂×C₄₁	2
C₂×C₄₁×#494	2
C₂×C₄×Q₁₆×C₄₁	2
C₄₁×#2209	2
C₂×C₄₁×C₄⋊C₁₆	2
C₂×C₄₁×#358	2
C₂×#2209	41

Subgroups — C2×C41×#2209 #23809778

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#2209 #23809778