Subgroups — #2325×C₂₁₁ #41009945

Order 27008.

696 conjugacy classes of subgroups.

The subgroup lattice has 300 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	C₂×C₄×D₄×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	#385×C₂₁₁	C₂	1	2	yes	maximal, complemented
8	#380×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#95×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₄×C₂₁₁	C₂²	1	2	yes	—
11	C₂²×D₄×C₂₁₁	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₂₁₁	C₂²	1	2	yes	—
14	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
16	#121×C₂₁₁	C₄	1	8	yes	complemented
17	C₄×D₄×C₂₁₁	—	2	8	no	—
18	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
27	C₂×C₄²×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	#95×C₂₁₁	C₂²	1	2	yes	—
33	C₂⁴×C₂₁₁	C₂³	1	2	yes	—
34	C₂⁴×C₂₁₁	C₂³	1	1	yes	characteristic
35	C₂×D₄×C₂₁₁	—	2	4	no	—
36	C₂×D₄×C₂₁₁	—	2	4	no	—
37	C₂×D₄×C₂₁₁	—	4	1	no	—
38	C₂×D₄×C₂₁₁	—	4	1	no	—
39	C₂×D₄×C₂₁₁	C₂×C₄	1	8	yes	complemented
40	C₂×D₄×C₂₁₁	C₂×C₄	1	4	yes	complemented
41	C₂²×C₄×C₂₁₁	—	2	2	no	—
42	C₂²×C₄×C₂₁₁	—	2	4	no	—
43	C₂²×C₄×C₂₁₁	—	4	1	no	—
44	C₂²×C₄×C₂₁₁	—	4	1	no	—
45	C₂²×C₄×C₂₁₁	—	4	1	no	—
46	C₂²×C₄×C₂₁₁	—	4	1	no	—
47	C₂²×C₄×C₂₁₁	—	4	2	no	—
48	C₂²×C₄×C₂₁₁	—	4	2	no	—
49	C₂²×C₄×C₂₁₁	—	4	1	no	—
50	C₂²×C₄×C₂₁₁	—	4	1	no	—
51	C₂²×C₄×C₂₁₁	C₂×C₄	1	4	yes	complemented
52	C₂²×C₄×C₂₁₁	D₄	1	4	yes	complemented
53	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
54	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
55	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
56	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
58	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
60	C₄⋊C₄×C₂₁₁	—	2	4	no	—
61	C₄⋊C₄×C₂₁₁	—	4	1	no	—
62	C₄⋊C₄×C₂₁₁	—	4	1	no	—
63	C₄⋊C₄×C₂₁₁	C₂×C₄	1	4	yes	complemented
64	C₂²⋊C₄×C₂₁₁	—	2	4	no	—
65	C₂²⋊C₄×C₂₁₁	—	2	4	no	—
66	C₂²⋊C₄×C₂₁₁	—	4	2	no	—
67	C₂²⋊C₄×C₂₁₁	—	4	2	no	—
68	C₂²⋊C₄×C₂₁₁	C₂×C₄	1	8	yes	complemented
69	C₄²×C₂₁₁	—	2	4	no	—
70	C₂³×C₂₁₁	—	2	1	no	—
71	C₂³×C₂₁₁	—	2	1	no	—
72	C₂³×C₂₁₁	—	2	1	no	—
73	C₂³×C₂₁₁	—	2	1	no	—
74	C₂³×C₂₁₁	—	2	2	no	—
75	C₂³×C₂₁₁	—	2	2	no	—
76	C₂³×C₂₁₁	—	2	2	no	—
77	C₂³×C₂₁₁	—	2	2	no	—
78	C₂³×C₂₁₁	—	2	2	no	—
79	C₂³×C₂₁₁	—	2	2	no	—
80	C₂³×C₂₁₁	C₂²×C₄	1	4	yes	—
81	C₂³×C₂₁₁	C₂²×C₄	1	2	yes	—
82	C₂³×C₂₁₁	C₂×D₄	1	2	yes	—
83	C₂³×C₂₁₁	#41	1	2	yes	—
84	C₂³×C₂₁₁	C₂⁴	1	1	yes	characteristic
85	D₄×C₂₁₁	—	2	16	no	—
86	D₄×C₂₁₁	—	4	4	no	—
87	C₂×C₄×C₂₁₁	—	2	2	no	—
88	C₂×C₄×C₂₁₁	—	2	2	no	—
89	C₂×C₄×C₂₁₁	—	2	1	no	—
90	C₂×C₄×C₂₁₁	—	2	1	no	—
91	C₂×C₄×C₂₁₁	—	2	2	no	—
92	C₂×C₄×C₂₁₁	—	2	2	no	—
93	C₂×C₄×C₂₁₁	—	2	1	no	—
94	C₂×C₄×C₂₁₁	—	2	1	no	—
95	C₂×C₄×C₂₁₁	—	2	2	no	—
96	C₂×C₄×C₂₁₁	—	2	2	no	—
97	C₂×C₄×C₂₁₁	—	2	2	no	—
98	C₂×C₄×C₂₁₁	—	2	2	no	—
99	C₂×C₄×C₂₁₁	—	2	2	no	—
100	C₂×C₄×C₂₁₁	—	2	2	no	—
101	C₂×C₄×C₂₁₁	—	2	2	no	—
102	C₂×C₄×C₂₁₁	—	2	2	no	—
103	C₂×C₄×C₂₁₁	—	2	2	no	—
104	C₂×C₄×C₂₁₁	—	4	4	no	—
105	C₂×C₄×C₂₁₁	—	4	4	no	—
106	C₂×C₄×C₂₁₁	—	4	8	no	—
107	C₂×C₄×C₂₁₁	—	4	4	no	—
108	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
109	C₂×C₄×C₂₁₁	C₂²×C₄	1	4	yes	—
110	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
111	C₂×C₄×C₂₁₁	C₂×D₄	1	4	yes	—
112	C₂²×C₂₁₁	—	2	2	no	—
113	C₂²×C₂₁₁	—	2	2	no	—
114	C₂²×C₂₁₁	—	2	2	no	—
115	C₂²×C₂₁₁	—	2	2	no	—
116	C₂²×C₂₁₁	—	2	2	no	—
117	C₂²×C₂₁₁	—	2	2	no	—
118	C₂²×C₂₁₁	—	2	4	no	—
119	C₂²×C₂₁₁	—	2	4	no	—
120	C₂²×C₂₁₁	—	2	4	no	—
121	C₂²×C₂₁₁	—	4	2	no	—
122	C₂²×C₂₁₁	—	4	2	no	—
123	C₂²×C₂₁₁	—	4	2	no	—
124	C₂²×C₂₁₁	—	4	2	no	—
125	C₂²×C₂₁₁	C₄×D₄	1	4	yes	—
126	C₂²×C₂₁₁	C₂³×C₄	1	1	yes	characteristic
127	C₂²×C₂₁₁	C₂²×D₄	1	1	yes	characteristic
128	C₂²×C₂₁₁	C₂×#41	1	1	yes	characteristic
129	C₂²×C₂₁₁	H(5,𝔽₂)	1	1	yes	characteristic
130	C₂²×C₂₁₁	H(5,𝔽₂)	1	1	yes	characteristic
131	C₂²×C₂₁₁	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂²×C₂₁₁	2₋¹⁺⁴	1	1	yes	characteristic
133	C₄×C₂₁₁	—	2	4	no	—
134	C₄×C₂₁₁	—	2	2	no	—
135	C₄×C₂₁₁	—	2	2	no	—
136	C₄×C₂₁₁	—	2	4	no	—
137	C₄×C₂₁₁	—	4	2	no	—
138	C₄×C₂₁₁	—	4	2	no	—
139	C₄×C₂₁₁	—	4	2	no	—
140	C₂×C₂₁₁	—	2	4	no	—
141	C₂×C₂₁₁	—	4	4	no	—
142	C₂×C₂₁₁	C₂×C₄×D₄	1	1	yes	characteristic
143	C₂×C₂₁₁	#518	1	1	yes	characteristic
144	C₂×C₂₁₁	#520	1	1	yes	characteristic
145	C₂×C₂₁₁	#534	1	1	yes	characteristic
146	C₂×C₂₁₁	#537	1	1	yes	characteristic
147	C₂×C₂₁₁	#538	1	1	yes	characteristic
148	C₂×C₂₁₁	#539	1	1	yes	characteristic
149	C₂₁₁	#2325	1	1	yes	characteristic, direct product factor, Sylow
150	#2325	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂×#121	C₂×C₂₁₁	1	1	yes	characteristic
152	C₂×C₄×D₄	C₂×C₂₁₁	1	2	yes	complemented
153	#388	C₂×C₂₁₁	1	2	yes	complemented
154	#387	C₂×C₂₁₁	1	2	yes	complemented
155	#386	C₂×C₂₁₁	1	2	yes	complemented
156	#386	C₂×C₂₁₁	1	2	yes	complemented
157	#385	C₂×C₂₁₁	1	2	yes	complemented
158	#380	C₂×C₂₁₁	1	1	yes	characteristic, complemented
159	C₂×#95	C₂×C₂₁₁	1	1	yes	characteristic, complemented
160	C₂²×D₄	C₂²×C₂₁₁	1	2	yes	—
161	C₂²×D₄	C₂²×C₂₁₁	1	1	yes	characteristic
162	C₂³×C₄	C₂²×C₂₁₁	1	2	yes	—
163	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
164	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
165	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
166	#121	C₄×C₂₁₁	1	8	yes	complemented
167	C₄×D₄	—	2	8	no	—
168	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
169	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
170	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
171	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
172	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
173	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
174	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
175	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
176	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
177	C₂×C₄²	C₂²×C₂₁₁	1	2	yes	—
178	#95	C₂²×C₂₁₁	1	2	yes	—
179	#95	C₂²×C₂₁₁	1	2	yes	—
180	#95	C₂²×C₂₁₁	1	2	yes	—
181	#95	C₂²×C₂₁₁	1	2	yes	—
182	#95	C₂²×C₂₁₁	1	2	yes	—
183	C₂⁴	C₂³×C₂₁₁	1	2	yes	—
184	C₂⁴	C₂³×C₂₁₁	1	1	yes	characteristic
185	C₂×D₄	—	2	4	no	—
186	C₂×D₄	—	2	4	no	—
187	C₂×D₄	—	4	1	no	—
188	C₂×D₄	—	4	1	no	—
189	C₂×D₄	C₂×C₄×C₂₁₁	1	8	yes	complemented
190	C₂×D₄	C₂×C₄×C₂₁₁	1	4	yes	complemented
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	—	4	1	no	—
196	C₂²×C₄	—	4	1	no	—
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	—	4	1	no	—
201	C₂²×C₄	C₂×C₄×C₂₁₁	1	4	yes	complemented
202	C₂²×C₄	D₄×C₂₁₁	1	4	yes	complemented
203	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
204	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
205	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
206	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
207	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
208	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
209	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	4	1	no	—
212	C₄⋊C₄	—	4	1	no	—
213	C₄⋊C₄	C₂×C₄×C₂₁₁	1	4	yes	complemented
214	C₂²⋊C₄	—	2	4	no	—
215	C₂²⋊C₄	—	2	4	no	—
216	C₂²⋊C₄	—	4	2	no	—
217	C₂²⋊C₄	—	4	2	no	—
218	C₂²⋊C₄	C₂×C₄×C₂₁₁	1	8	yes	complemented
219	C₄²	—	2	4	no	—
220	C₂³	—	2	1	no	—
221	C₂³	—	2	1	no	—
222	C₂³	—	2	1	no	—
223	C₂³	—	2	1	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	2	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	C₂²×C₄×C₂₁₁	1	4	yes	—
231	C₂³	C₂²×C₄×C₂₁₁	1	2	yes	—
232	C₂³	C₂×D₄×C₂₁₁	1	2	yes	—
233	C₂³	#41×C₂₁₁	1	2	yes	—
234	C₂³	C₂⁴×C₂₁₁	1	1	yes	characteristic
235	D₄	—	2	16	no	—
236	D₄	—	4	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	8	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
259	C₂×C₄	C₂²×C₄×C₂₁₁	1	4	yes	—
260	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
261	C₂×C₄	C₂×D₄×C₂₁₁	1	4	yes	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	4	2	no	—
272	C₂²	—	4	2	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	4	2	no	—
275	C₂²	C₄×D₄×C₂₁₁	1	4	yes	—
276	C₂²	C₂³×C₄×C₂₁₁	1	1	yes	characteristic
277	C₂²	C₂²×D₄×C₂₁₁	1	1	yes	characteristic
278	C₂²	C₂×#41×C₂₁₁	1	1	yes	characteristic
279	C₂²	H(5,𝔽₂)×C₂₁₁	1	1	yes	characteristic
280	C₂²	H(5,𝔽₂)×C₂₁₁	1	1	yes	characteristic
281	C₂²	H(5,𝔽₂)×C₂₁₁	1	1	yes	characteristic
282	C₂²	2₋¹⁺⁴×C₂₁₁	1	1	yes	characteristic
283	C₄	—	2	4	no	—
284	C₄	—	2	4	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	2	no	—
289	C₄	—	4	2	no	—
290	C₂	—	2	4	no	—
291	C₂	—	4	4	no	—
292	C₂	C₂×C₄×D₄×C₂₁₁	1	1	yes	characteristic
293	C₂	#518×C₂₁₁	1	1	yes	characteristic
294	C₂	#520×C₂₁₁	1	1	yes	characteristic
295	C₂	#534×C₂₁₁	1	1	yes	characteristic
296	C₂	#537×C₂₁₁	1	1	yes	characteristic
297	C₂	#538×C₂₁₁	1	1	yes	characteristic
298	C₂	#539×C₂₁₁	1	1	yes	characteristic

	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. C₂×C₄×D₄×C₂₁₁	C₂	semidirect
3. #388×C₂₁₁	C₂	semidirect
4. #387×C₂₁₁	C₂	semidirect
5. #386×C₂₁₁	C₂	semidirect
6. #386×C₂₁₁	C₂	semidirect
7. #385×C₂₁₁	C₂	semidirect
8. #380×C₂₁₁	C₂	semidirect
9. C₂×#95×C₂₁₁	C₂	semidirect
16. #121×C₂₁₁	C₄	semidirect
39. C₂×D₄×C₂₁₁	C₂×C₄	semidirect
40. C₂×D₄×C₂₁₁	C₂×C₄	semidirect
51. C₂²×C₄×C₂₁₁	C₂×C₄	semidirect
52. C₂²×C₄×C₂₁₁	D₄	semidirect
63. C₄⋊C₄×C₂₁₁	C₂×C₄	semidirect
68. C₂²⋊C₄×C₂₁₁	C₂×C₄	semidirect
149. C₂₁₁	#2325	direct (N × H)
150. #2325	C₂₁₁	direct (N × H)
152. C₂×C₄×D₄	C₂×C₂₁₁	semidirect
153. #388	C₂×C₂₁₁	semidirect
154. #387	C₂×C₂₁₁	semidirect
155. #386	C₂×C₂₁₁	semidirect
156. #386	C₂×C₂₁₁	semidirect
157. #385	C₂×C₂₁₁	semidirect
158. #380	C₂×C₂₁₁	semidirect
159. C₂×#95	C₂×C₂₁₁	semidirect
166. #121	C₄×C₂₁₁	semidirect
189. C₂×D₄	C₂×C₄×C₂₁₁	semidirect
190. C₂×D₄	C₂×C₄×C₂₁₁	semidirect
201. C₂²×C₄	C₂×C₄×C₂₁₁	semidirect
202. C₂²×C₄	D₄×C₂₁₁	semidirect
213. C₄⋊C₄	C₂×C₄×C₂₁₁	semidirect
218. 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
C₂×C₄×D₄×C₂₁₁	2
#388×C₂₁₁	2
#387×C₂₁₁	2
#386×C₂₁₁	2
#385×C₂₁₁	2
#380×C₂₁₁	2
C₂×#95×C₂₁₁	2
#2325	211

Subgroups — #2325×C211 #41009945

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2325×C₂₁₁ #41009945