Subgroups — #3069×C₂₁₁ #41010689

Order 27008.

446 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#575×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#137×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#137×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#568×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#473×C₂₁₁	C₂	1	4	yes	maximal, complemented
6	#471×C₂₁₁	C₂	1	4	yes	maximal, complemented
7	#430×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#413×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#101×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₂₁₁	C₂²	1	1	yes	characteristic
11	C₂×#41×C₂₁₁	C₂²	1	1	yes	characteristic
12	C₂²×Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
13	#137×C₂₁₁	—	2	2	no	—
14	#137×C₂₁₁	—	2	1	no	—
15	#137×C₂₁₁	—	2	2	no	—
16	#137×C₂₁₁	—	2	2	no	—
17	#137×C₂₁₁	C₂²	1	4	yes	—
18	#137×C₂₁₁	C₂²	1	2	yes	—
19	AGL(1,ℤ/8)×C₂₁₁	—	2	1	no	—
20	AGL(1,ℤ/8)×C₂₁₁	C₂²	1	2	yes	—
21	#135×C₂₁₁	—	2	2	no	—
22	C₂×Q₁₆×C₂₁₁	—	2	1	no	—
23	C₂×Q₁₆×C₂₁₁	C₂²	1	2	yes	complemented
24	C₂×SD₁₆×C₂₁₁	—	2	1	no	—
25	C₂×SD₁₆×C₂₁₁	C₂²	1	2	yes	complemented
26	#131×C₂₁₁	—	2	2	no	—
27	#131×C₂₁₁	C₂²	1	4	yes	—
28	C₂×M₁₆×C₂₁₁	—	2	2	no	—
29	C₂×M₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
30	C₂×M₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
31	C₂×M₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
32	C₂×M₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
33	#108×C₂₁₁	C₂²	1	4	yes	complemented
34	#101×C₂₁₁	C₂²	1	4	yes	complemented
35	#101×C₂₁₁	C₂²	1	2	yes	complemented
36	#100×C₂₁₁	C₂²	1	2	yes	complemented
37	#41×C₂₁₁	—	2	1	no	—
38	#41×C₂₁₁	—	2	1	no	—
39	#41×C₂₁₁	—	2	1	no	—
40	#41×C₂₁₁	—	2	1	no	—
41	#41×C₂₁₁	—	2	1	no	—
42	#41×C₂₁₁	—	2	1	no	—
43	#41×C₂₁₁	D₄	1	2	yes	—
44	#41×C₂₁₁	C₂³	1	2	yes	—
45	C₂×Q₈×C₂₁₁	—	2	1	no	—
46	C₂×Q₈×C₂₁₁	—	2	1	no	—
47	C₂×Q₈×C₂₁₁	—	2	1	no	—
48	C₂×Q₈×C₂₁₁	—	2	1	no	—
49	C₂×Q₈×C₂₁₁	—	2	1	no	—
50	C₂×Q₈×C₂₁₁	D₄	1	1	yes	characteristic
51	C₂×Q₈×C₂₁₁	C₂³	1	2	yes	—
52	C₂×Q₈×C₂₁₁	C₂³	1	1	yes	characteristic
53	C₂×D₄×C₂₁₁	—	2	1	no	—
54	C₂×D₄×C₂₁₁	—	2	1	no	—
55	C₂×D₄×C₂₁₁	D₄	1	1	yes	characteristic
56	C₂×D₄×C₂₁₁	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₂₁₁	—	2	1	no	—
58	C₂²×C₄×C₂₁₁	—	2	1	no	—
59	C₂²×C₄×C₂₁₁	—	2	1	no	—
60	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
61	Q₁₆×C₂₁₁	—	2	4	no	—
62	Q₁₆×C₂₁₁	—	2	2	no	—
63	Q₁₆×C₂₁₁	—	2	4	no	—
64	SD₁₆×C₂₁₁	—	2	2	no	—
65	SD₁₆×C₂₁₁	—	2	4	no	—
66	SD₁₆×C₂₁₁	—	2	2	no	—
67	SD₁₆×C₂₁₁	—	2	4	no	—
68	D₈×C₂₁₁	—	2	2	no	—
69	M₁₆×C₂₁₁	—	2	1	no	—
70	M₁₆×C₂₁₁	—	2	1	no	—
71	M₁₆×C₂₁₁	—	2	1	no	—
72	M₁₆×C₂₁₁	—	2	4	no	—
73	M₁₆×C₂₁₁	D₄	1	2	yes	—
74	M₁₆×C₂₁₁	D₄	1	2	yes	—
75	M₁₆×C₂₁₁	C₂³	1	2	yes	—
76	M₁₆×C₂₁₁	C₂³	1	2	yes	—
77	M₁₆×C₂₁₁	C₂³	1	2	yes	—
78	C₂×C₈×C₂₁₁	—	2	1	no	—
79	C₂×C₈×C₂₁₁	—	2	1	no	—
80	C₂×C₈×C₂₁₁	—	2	1	no	—
81	C₂×C₈×C₂₁₁	—	2	4	no	—
82	C₂×C₈×C₂₁₁	C₂³	1	2	yes	—
83	C₂³×C₂₁₁	—	2	1	no	—
84	C₂³×C₂₁₁	—	2	1	no	—
85	C₂³×C₂₁₁	#41	1	1	yes	characteristic
86	Q₈×C₂₁₁	—	2	1	no	—
87	Q₈×C₂₁₁	—	2	2	no	—
88	Q₈×C₂₁₁	—	2	2	no	—
89	Q₈×C₂₁₁	—	2	1	no	—
90	Q₈×C₂₁₁	—	2	1	no	—
91	Q₈×C₂₁₁	—	4	1	no	—
92	Q₈×C₂₁₁	—	4	1	no	—
93	Q₈×C₂₁₁	C₂×D₄	1	2	yes	—
94	D₄×C₂₁₁	—	2	2	no	—
95	D₄×C₂₁₁	—	2	2	no	—
96	D₄×C₂₁₁	—	2	1	no	—
97	D₄×C₂₁₁	—	2	1	no	—
98	D₄×C₂₁₁	—	2	1	no	—
99	D₄×C₂₁₁	—	4	1	no	—
100	D₄×C₂₁₁	—	4	1	no	—
101	D₄×C₂₁₁	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₂₁₁	—	2	1	no	—
103	C₂×C₄×C₂₁₁	—	2	2	no	—
104	C₂×C₄×C₂₁₁	—	2	2	no	—
105	C₂×C₄×C₂₁₁	—	2	1	no	—
106	C₂×C₄×C₂₁₁	—	2	2	no	—
107	C₂×C₄×C₂₁₁	—	4	1	no	—
108	C₂×C₄×C₂₁₁	—	4	1	no	—
109	C₂×C₄×C₂₁₁	—	4	1	no	—
110	C₂×C₄×C₂₁₁	—	4	1	no	—
111	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
116	C₂×C₄×C₂₁₁	C₂⁴	1	1	yes	characteristic
117	C₈×C₂₁₁	—	2	2	no	—
118	C₈×C₂₁₁	—	2	2	no	—
119	C₈×C₂₁₁	—	2	2	no	—
120	C₈×C₂₁₁	C₂×D₄	1	4	yes	—
121	C₂²×C₂₁₁	—	2	2	no	—
122	C₂²×C₂₁₁	—	4	1	no	—
123	C₂²×C₂₁₁	—	4	1	no	—
124	C₂²×C₂₁₁	—	8	1	no	—
125	C₂²×C₂₁₁	—	8	1	no	—
126	C₂²×C₂₁₁	#121	1	1	yes	characteristic
127	C₂²×C₂₁₁	#121	1	1	yes	characteristic
128	C₂²×C₂₁₁	C₂×#41	1	1	yes	characteristic
129	C₄×C₂₁₁	—	2	2	no	—
130	C₄×C₂₁₁	—	4	1	no	—
131	C₄×C₂₁₁	—	4	2	no	—
132	C₄×C₂₁₁	#121	1	1	yes	characteristic
133	C₄×C₂₁₁	#121	1	1	yes	characteristic
134	C₄×C₂₁₁	C₂²×D₄	1	1	yes	characteristic
135	C₄×C₂₁₁	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₂₁₁	—	2	1	no	—
137	C₂×C₂₁₁	—	2	1	no	—
138	C₂×C₂₁₁	—	2	1	no	—
139	C₂×C₂₁₁	—	4	2	no	—
140	C₂×C₂₁₁	—	8	1	no	—
141	C₂×C₂₁₁	C₂×#121	1	1	yes	characteristic
142	C₂₁₁	#3069	1	1	yes	characteristic, direct product factor, Sylow
143	#3069	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	#575	C₂×C₂₁₁	1	1	yes	characteristic
145	C₂×#137	C₂×C₂₁₁	1	1	yes	characteristic, complemented
146	C₂×#137	C₂×C₂₁₁	1	1	yes	characteristic, complemented
147	#568	C₂×C₂₁₁	1	1	yes	characteristic, complemented
148	#473	C₂×C₂₁₁	1	4	yes	complemented
149	#471	C₂×C₂₁₁	1	4	yes	complemented
150	#430	C₂×C₂₁₁	1	1	yes	characteristic, complemented
151	#413	C₂×C₂₁₁	1	1	yes	characteristic, complemented
152	C₂×#101	C₂×C₂₁₁	1	1	yes	characteristic, complemented
153	C₂×#41	C₂²×C₂₁₁	1	1	yes	characteristic
154	C₂×#41	C₂²×C₂₁₁	1	1	yes	characteristic
155	C₂²×Q₈	C₂²×C₂₁₁	1	1	yes	characteristic
156	#137	—	2	2	no	—
157	#137	—	2	1	no	—
158	#137	—	2	2	no	—
159	#137	—	2	2	no	—
160	#137	C₂²×C₂₁₁	1	4	yes	—
161	#137	C₂²×C₂₁₁	1	2	yes	—
162	AGL(1,ℤ/8)	—	2	1	no	—
163	AGL(1,ℤ/8)	C₂²×C₂₁₁	1	2	yes	—
164	#135	—	2	2	no	—
165	C₂×Q₁₆	—	2	1	no	—
166	C₂×Q₁₆	C₂²×C₂₁₁	1	2	yes	complemented
167	C₂×SD₁₆	—	2	1	no	—
168	C₂×SD₁₆	C₂²×C₂₁₁	1	2	yes	complemented
169	#131	—	2	2	no	—
170	#131	C₂²×C₂₁₁	1	4	yes	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	C₂²×C₂₁₁	1	1	yes	characteristic
173	C₂×M₁₆	C₂²×C₂₁₁	1	1	yes	characteristic
174	C₂×M₁₆	C₂²×C₂₁₁	1	1	yes	characteristic
175	C₂×M₁₆	C₂²×C₂₁₁	1	1	yes	characteristic
176	#108	C₂²×C₂₁₁	1	4	yes	complemented
177	#101	C₂²×C₂₁₁	1	4	yes	complemented
178	#101	C₂²×C₂₁₁	1	2	yes	complemented
179	#100	C₂²×C₂₁₁	1	2	yes	complemented
180	#41	—	2	1	no	—
181	#41	—	2	1	no	—
182	#41	—	2	1	no	—
183	#41	—	2	1	no	—
184	#41	—	2	1	no	—
185	#41	—	2	1	no	—
186	#41	D₄×C₂₁₁	1	2	yes	—
187	#41	C₂³×C₂₁₁	1	2	yes	—
188	C₂×Q₈	—	2	1	no	—
189	C₂×Q₈	—	2	1	no	—
190	C₂×Q₈	—	2	1	no	—
191	C₂×Q₈	—	2	1	no	—
192	C₂×Q₈	—	2	1	no	—
193	C₂×Q₈	D₄×C₂₁₁	1	1	yes	characteristic
194	C₂×Q₈	C₂³×C₂₁₁	1	2	yes	—
195	C₂×Q₈	C₂³×C₂₁₁	1	1	yes	characteristic
196	C₂×D₄	—	2	1	no	—
197	C₂×D₄	—	2	1	no	—
198	C₂×D₄	D₄×C₂₁₁	1	1	yes	characteristic
199	C₂×D₄	C₂³×C₂₁₁	1	1	yes	characteristic
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
204	Q₁₆	—	2	4	no	—
205	Q₁₆	—	2	2	no	—
206	Q₁₆	—	2	4	no	—
207	SD₁₆	—	2	2	no	—
208	SD₁₆	—	2	4	no	—
209	SD₁₆	—	2	2	no	—
210	SD₁₆	—	2	4	no	—
211	D₈	—	2	2	no	—
212	M₁₆	—	2	1	no	—
213	M₁₆	—	2	1	no	—
214	M₁₆	—	2	1	no	—
215	M₁₆	—	2	4	no	—
216	M₁₆	D₄×C₂₁₁	1	2	yes	—
217	M₁₆	D₄×C₂₁₁	1	2	yes	—
218	M₁₆	C₂³×C₂₁₁	1	2	yes	—
219	M₁₆	C₂³×C₂₁₁	1	2	yes	—
220	M₁₆	C₂³×C₂₁₁	1	2	yes	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	C₂³×C₂₁₁	1	2	yes	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	#41×C₂₁₁	1	1	yes	characteristic
229	Q₈	—	2	1	no	—
230	Q₈	—	2	2	no	—
231	Q₈	—	2	2	no	—
232	Q₈	—	2	1	no	—
233	Q₈	—	2	1	no	—
234	Q₈	—	4	1	no	—
235	Q₈	—	4	1	no	—
236	Q₈	C₂×D₄×C₂₁₁	1	2	yes	—
237	D₄	—	2	2	no	—
238	D₄	—	2	2	no	—
239	D₄	—	2	1	no	—
240	D₄	—	2	1	no	—
241	D₄	—	2	1	no	—
242	D₄	—	4	1	no	—
243	D₄	—	4	1	no	—
244	D₄	C₂×D₄×C₂₁₁	1	2	yes	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
255	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
256	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
257	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
258	C₂×C₄	#41×C₂₁₁	1	1	yes	characteristic
259	C₂×C₄	C₂⁴×C₂₁₁	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₈	—	2	2	no	—
262	C₈	—	2	2	no	—
263	C₈	C₂×D₄×C₂₁₁	1	4	yes	—
264	C₂²	—	2	2	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	8	1	no	—
268	C₂²	—	8	1	no	—
269	C₂²	#121×C₂₁₁	1	1	yes	characteristic
270	C₂²	#121×C₂₁₁	1	1	yes	characteristic
271	C₂²	C₂×#41×C₂₁₁	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	2	no	—
275	C₄	#121×C₂₁₁	1	1	yes	characteristic
276	C₄	#121×C₂₁₁	1	1	yes	characteristic
277	C₄	C₂²×D₄×C₂₁₁	1	1	yes	characteristic
278	C₄	C₂²×D₄×C₂₁₁	1	1	yes	characteristic
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	4	2	no	—
283	C₂	—	8	1	no	—
284	C₂	C₂×#121×C₂₁₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₁₁	C₂×#121
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₁₁
prime radical	#575×C₂₁₁	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₁₁ ◁ C₂²×C₄×C₂₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#137×C₂₁₁	C₂	semidirect
3. C₂×#137×C₂₁₁	C₂	semidirect
4. #568×C₂₁₁	C₂	semidirect
5. #473×C₂₁₁	C₂	semidirect
6. #471×C₂₁₁	C₂	semidirect
7. #430×C₂₁₁	C₂	semidirect
8. #413×C₂₁₁	C₂	semidirect
9. C₂×#101×C₂₁₁	C₂	semidirect
23. C₂×Q₁₆×C₂₁₁	C₂²	semidirect
25. C₂×SD₁₆×C₂₁₁	C₂²	semidirect
33. #108×C₂₁₁	C₂²	semidirect
34. #101×C₂₁₁	C₂²	semidirect
35. #101×C₂₁₁	C₂²	semidirect
36. #100×C₂₁₁	C₂²	semidirect
142. C₂₁₁	#3069	direct (N × H)
143. #3069	C₂₁₁	direct (N × H)
145. C₂×#137	C₂×C₂₁₁	semidirect
146. C₂×#137	C₂×C₂₁₁	semidirect
147. #568	C₂×C₂₁₁	semidirect
148. #473	C₂×C₂₁₁	semidirect
149. #471	C₂×C₂₁₁	semidirect
150. #430	C₂×C₂₁₁	semidirect
151. #413	C₂×C₂₁₁	semidirect
152. C₂×#101	C₂×C₂₁₁	semidirect
166. C₂×Q₁₆	C₂²×C₂₁₁	semidirect
168. C₂×SD₁₆	C₂²×C₂₁₁	semidirect
176. #108	C₂²×C₂₁₁	semidirect
177. #101	C₂²×C₂₁₁	semidirect
178. #101	C₂²×C₂₁₁	semidirect
179. #100	C₂²×C₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#575×C₂₁₁	2
C₂×#137×C₂₁₁	2
#568×C₂₁₁	2
#473×C₂₁₁	2
#471×C₂₁₁	2
#430×C₂₁₁	2
#413×C₂₁₁	2
C₂×#101×C₂₁₁	2
#3069	211

Subgroups — #3069×C211 #41010689

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3069×C₂₁₁ #41010689