Subgroups — #2333×C₂₁₁ #41009953

Order 27008.

596 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#121×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	#390×C₂₁₁	C₂	1	2	yes	maximal, complemented
3	#388×C₂₁₁	C₂	1	2	yes	maximal, complemented
4	#387×C₂₁₁	C₂	1	2	yes	maximal, complemented
5	#386×C₂₁₁	C₂	1	2	yes	maximal, complemented
6	#386×C₂₁₁	C₂	1	2	yes	maximal, complemented
7	#381×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₂²⋊C₄×C₂₁₁	C₂	1	2	yes	maximal, complemented
9	C₄×C₂²⋊C₄×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₄×C₂₁₁	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₂₁₁	C₂²	1	2	yes	—
12	C₂³×C₄×C₂₁₁	C₂²	1	2	yes	—
13	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
14	#121×C₂₁₁	C₄	1	8	yes	complemented
15	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
24	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
26	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
27	#95×C₂₁₁	C₂²	1	2	yes	—
28	#95×C₂₁₁	C₂²	1	2	yes	—
29	#95×C₂₁₁	C₂²	1	2	yes	—
30	#95×C₂₁₁	C₂²	1	2	yes	—
31	#95×C₂₁₁	C₂²	1	2	yes	—
32	C₂⁴×C₂₁₁	C₂³	1	1	yes	characteristic
33	C₂⁴×C₂₁₁	C₂³	1	2	yes	—
34	C₂×D₄×C₂₁₁	—	4	2	no	—
35	C₂×D₄×C₂₁₁	—	4	2	no	—
36	C₂×D₄×C₂₁₁	—	4	1	no	—
37	C₂×D₄×C₂₁₁	—	4	1	no	—
38	C₂×D₄×C₂₁₁	C₂×C₄	1	4	yes	—
39	C₂×D₄×C₂₁₁	C₂×C₄	1	8	yes	complemented
40	C₂²×C₄×C₂₁₁	—	2	4	no	—
41	C₂²×C₄×C₂₁₁	—	4	1	no	—
42	C₂²×C₄×C₂₁₁	—	4	1	no	—
43	C₂²×C₄×C₂₁₁	—	4	2	no	—
44	C₂²×C₄×C₂₁₁	—	4	2	no	—
45	C₂²×C₄×C₂₁₁	C₂×C₄	1	4	yes	complemented
46	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
47	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
48	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
49	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
52	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
53	C₄⋊C₄×C₂₁₁	—	4	2	no	—
54	C₄⋊C₄×C₂₁₁	C₂×C₄	1	4	yes	complemented
55	C₂²⋊C₄×C₂₁₁	—	2	4	no	—
56	C₂²⋊C₄×C₂₁₁	—	2	4	no	—
57	C₂²⋊C₄×C₂₁₁	—	4	2	no	—
58	C₂²⋊C₄×C₂₁₁	—	4	1	no	—
59	C₂²⋊C₄×C₂₁₁	—	4	2	no	—
60	C₂²⋊C₄×C₂₁₁	—	4	1	no	—
61	C₂²⋊C₄×C₂₁₁	C₂×C₄	1	8	yes	complemented
62	C₄²×C₂₁₁	—	4	1	no	—
63	C₄²×C₂₁₁	—	4	2	no	—
64	C₄²×C₂₁₁	—	4	1	no	—
65	C₂³×C₂₁₁	—	2	2	no	—
66	C₂³×C₂₁₁	—	2	2	no	—
67	C₂³×C₂₁₁	—	2	2	no	—
68	C₂³×C₂₁₁	—	2	1	no	—
69	C₂³×C₂₁₁	—	2	1	no	—
70	C₂³×C₂₁₁	—	2	2	no	—
71	C₂³×C₂₁₁	—	2	2	no	—
72	C₂³×C₂₁₁	—	2	2	no	—
73	C₂³×C₂₁₁	—	2	2	no	—
74	C₂³×C₂₁₁	—	2	2	no	—
75	C₂³×C₂₁₁	C₂²×C₄	1	4	yes	—
76	C₂³×C₂₁₁	C₂²×C₄	1	2	yes	—
77	C₂³×C₂₁₁	C₂⁴	1	1	yes	characteristic
78	D₄×C₂₁₁	—	4	4	no	—
79	D₄×C₂₁₁	—	4	8	no	—
80	C₂×C₄×C₂₁₁	—	2	1	no	—
81	C₂×C₄×C₂₁₁	—	2	1	no	—
82	C₂×C₄×C₂₁₁	—	2	2	no	—
83	C₂×C₄×C₂₁₁	—	2	2	no	—
84	C₂×C₄×C₂₁₁	—	2	2	no	—
85	C₂×C₄×C₂₁₁	—	2	2	no	—
86	C₂×C₄×C₂₁₁	—	2	1	no	—
87	C₂×C₄×C₂₁₁	—	2	1	no	—
88	C₂×C₄×C₂₁₁	—	2	2	no	—
89	C₂×C₄×C₂₁₁	—	2	2	no	—
90	C₂×C₄×C₂₁₁	—	2	2	no	—
91	C₂×C₄×C₂₁₁	—	2	2	no	—
92	C₂×C₄×C₂₁₁	—	2	2	no	—
93	C₂×C₄×C₂₁₁	—	2	2	no	—
94	C₂×C₄×C₂₁₁	—	2	2	no	—
95	C₂×C₄×C₂₁₁	—	2	2	no	—
96	C₂×C₄×C₂₁₁	—	2	2	no	—
97	C₂×C₄×C₂₁₁	—	4	4	no	—
98	C₂×C₄×C₂₁₁	—	4	8	no	—
99	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
100	C₂×C₄×C₂₁₁	C₂²×C₄	1	4	yes	—
101	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
102	C₂×C₄×C₂₁₁	#41	1	4	yes	—
103	C₂²×C₂₁₁	—	2	2	no	—
104	C₂²×C₂₁₁	—	2	4	no	—
105	C₂²×C₂₁₁	—	2	2	no	—
106	C₂²×C₂₁₁	—	2	2	no	—
107	C₂²×C₂₁₁	—	2	4	no	—
108	C₂²×C₂₁₁	—	2	4	no	—
109	C₂²×C₂₁₁	—	4	2	no	—
110	C₂²×C₂₁₁	—	4	2	no	—
111	C₂²×C₂₁₁	—	4	2	no	—
112	C₂²×C₂₁₁	—	4	2	no	—
113	C₂²×C₂₁₁	—	4	2	no	—
114	C₂²×C₂₁₁	—	4	2	no	—
115	C₂²×C₂₁₁	C₂³×C₄	1	1	yes	characteristic
116	C₂²×C₂₁₁	C₂×#41	1	2	yes	—
117	C₂²×C₂₁₁	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂²×C₂₁₁	H(5,𝔽₂)	1	2	yes	—
119	C₂²×C₂₁₁	H(5,𝔽₂)	1	1	yes	characteristic
120	C₄×C₂₁₁	—	2	2	no	—
121	C₄×C₂₁₁	—	2	2	no	—
122	C₄×C₂₁₁	—	2	4	no	—
123	C₄×C₂₁₁	—	2	4	no	—
124	C₄×C₂₁₁	—	4	2	no	—
125	C₄×C₂₁₁	—	4	2	no	—
126	C₄×C₂₁₁	—	4	2	no	—
127	C₂×C₂₁₁	—	4	2	no	—
128	C₂×C₂₁₁	—	4	4	no	—
129	C₂×C₂₁₁	C₄×#41	1	1	yes	characteristic
130	C₂×C₂₁₁	#518	1	1	yes	characteristic
131	C₂×C₂₁₁	#518	1	1	yes	characteristic
132	C₂×C₂₁₁	#538	1	2	yes	—
133	C₂×C₂₁₁	#540	1	2	yes	—
134	C₂₁₁	#2333	1	1	yes	characteristic, direct product factor, Sylow
135	#2333	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂×#121	C₂×C₂₁₁	1	1	yes	characteristic
137	#390	C₂×C₂₁₁	1	2	yes	complemented
138	#388	C₂×C₂₁₁	1	2	yes	complemented
139	#387	C₂×C₂₁₁	1	2	yes	complemented
140	#386	C₂×C₂₁₁	1	2	yes	complemented
141	#386	C₂×C₂₁₁	1	2	yes	complemented
142	#381	C₂×C₂₁₁	1	1	yes	characteristic, complemented
143	C₄×C₂²⋊C₄	C₂×C₂₁₁	1	2	yes	complemented
144	C₄×C₂²⋊C₄	C₂×C₂₁₁	1	1	yes	characteristic, complemented
145	C₂²×D₄	C₂²×C₂₁₁	1	1	yes	characteristic
146	C₂²×D₄	C₂²×C₂₁₁	1	2	yes	—
147	C₂³×C₄	C₂²×C₂₁₁	1	2	yes	—
148	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
149	#121	C₄×C₂₁₁	1	8	yes	complemented
150	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
151	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
152	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
153	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
154	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
155	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
156	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
157	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
158	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
159	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
160	C₂×C₄²	C₂²×C₂₁₁	1	2	yes	—
161	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
162	#95	C₂²×C₂₁₁	1	2	yes	—
163	#95	C₂²×C₂₁₁	1	2	yes	—
164	#95	C₂²×C₂₁₁	1	2	yes	—
165	#95	C₂²×C₂₁₁	1	2	yes	—
166	#95	C₂²×C₂₁₁	1	2	yes	—
167	C₂⁴	C₂³×C₂₁₁	1	1	yes	characteristic
168	C₂⁴	C₂³×C₂₁₁	1	2	yes	—
169	C₂×D₄	—	4	2	no	—
170	C₂×D₄	—	4	2	no	—
171	C₂×D₄	—	4	1	no	—
172	C₂×D₄	—	4	1	no	—
173	C₂×D₄	C₂×C₄×C₂₁₁	1	4	yes	—
174	C₂×D₄	C₂×C₄×C₂₁₁	1	8	yes	complemented
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	—	4	1	no	—
177	C₂²×C₄	—	4	1	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	C₂×C₄×C₂₁₁	1	4	yes	complemented
181	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
182	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
183	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
184	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
185	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
186	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
187	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	C₂×C₄×C₂₁₁	1	4	yes	complemented
190	C₂²⋊C₄	—	2	4	no	—
191	C₂²⋊C₄	—	2	4	no	—
192	C₂²⋊C₄	—	4	2	no	—
193	C₂²⋊C₄	—	4	1	no	—
194	C₂²⋊C₄	—	4	2	no	—
195	C₂²⋊C₄	—	4	1	no	—
196	C₂²⋊C₄	C₂×C₄×C₂₁₁	1	8	yes	complemented
197	C₄²	—	4	1	no	—
198	C₄²	—	4	2	no	—
199	C₄²	—	4	1	no	—
200	C₂³	—	2	2	no	—
201	C₂³	—	2	2	no	—
202	C₂³	—	2	2	no	—
203	C₂³	—	2	1	no	—
204	C₂³	—	2	1	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	2	2	no	—
207	C₂³	—	2	2	no	—
208	C₂³	—	2	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	C₂²×C₄×C₂₁₁	1	4	yes	—
211	C₂³	C₂²×C₄×C₂₁₁	1	2	yes	—
212	C₂³	C₂⁴×C₂₁₁	1	1	yes	characteristic
213	D₄	—	4	4	no	—
214	D₄	—	4	8	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	8	no	—
234	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
235	C₂×C₄	C₂²×C₄×C₂₁₁	1	4	yes	—
236	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
237	C₂×C₄	#41×C₂₁₁	1	4	yes	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	4	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	4	no	—
243	C₂²	—	2	4	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	C₂³×C₄×C₂₁₁	1	1	yes	characteristic
251	C₂²	C₂×#41×C₂₁₁	1	2	yes	—
252	C₂²	H(5,𝔽₂)×C₂₁₁	1	1	yes	characteristic
253	C₂²	H(5,𝔽₂)×C₂₁₁	1	2	yes	—
254	C₂²	H(5,𝔽₂)×C₂₁₁	1	1	yes	characteristic
255	C₄	—	2	4	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	4	no	—
259	C₄	—	4	2	no	—
260	C₄	—	4	2	no	—
261	C₄	—	4	2	no	—
262	C₂	—	4	2	no	—
263	C₂	—	4	4	no	—
264	C₂	C₄×#41×C₂₁₁	1	1	yes	characteristic
265	C₂	#518×C₂₁₁	1	1	yes	characteristic
266	C₂	#518×C₂₁₁	1	1	yes	characteristic
267	C₂	#538×C₂₁₁	1	2	yes	—
268	C₂	#540×C₂₁₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₁₁	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₁₁
Frattini, cosocle	C₂³	C₂⁴×C₂₁₁
prime radical	C₂×#121×C₂₁₁	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #390×C₂₁₁	C₂	semidirect
3. #388×C₂₁₁	C₂	semidirect
4. #387×C₂₁₁	C₂	semidirect
5. #386×C₂₁₁	C₂	semidirect
6. #386×C₂₁₁	C₂	semidirect
7. #381×C₂₁₁	C₂	semidirect
8. C₄×C₂²⋊C₄×C₂₁₁	C₂	semidirect
9. C₄×C₂²⋊C₄×C₂₁₁	C₂	semidirect
14. #121×C₂₁₁	C₄	semidirect
39. C₂×D₄×C₂₁₁	C₂×C₄	semidirect
45. C₂²×C₄×C₂₁₁	C₂×C₄	semidirect
54. C₄⋊C₄×C₂₁₁	C₂×C₄	semidirect
61. C₂²⋊C₄×C₂₁₁	C₂×C₄	semidirect
134. C₂₁₁	#2333	direct (N × H)
135. #2333	C₂₁₁	direct (N × H)
137. #390	C₂×C₂₁₁	semidirect
138. #388	C₂×C₂₁₁	semidirect
139. #387	C₂×C₂₁₁	semidirect
140. #386	C₂×C₂₁₁	semidirect
141. #386	C₂×C₂₁₁	semidirect
142. #381	C₂×C₂₁₁	semidirect
143. C₄×C₂²⋊C₄	C₂×C₂₁₁	semidirect
144. C₄×C₂²⋊C₄	C₂×C₂₁₁	semidirect
149. #121	C₄×C₂₁₁	semidirect
174. C₂×D₄	C₂×C₄×C₂₁₁	semidirect
180. C₂²×C₄	C₂×C₄×C₂₁₁	semidirect
189. C₄⋊C₄	C₂×C₄×C₂₁₁	semidirect
196. C₂²⋊C₄	C₂×C₄×C₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#121×C₂₁₁	2
#390×C₂₁₁	2
#388×C₂₁₁	2
#387×C₂₁₁	2
#386×C₂₁₁	2
#381×C₂₁₁	2
C₄×C₂²⋊C₄×C₂₁₁	2
#2333	211

Subgroups — #2333×C211 #41009953

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2333×C₂₁₁ #41009953