Subgroups — C₂×C₅×#2187 #11956231

Order 1280.

1382 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×SD₃₂	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×#467	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2187	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×#357	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₅×D₄	—	2	1	no	—
8	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic, complemented
9	C₂²×C₅×D₈	—	2	8	no	—
10	C₂²×C₅×D₈	—	2	1	no	—
11	C₂²×C₅×D₈	—	2	2	no	—
12	C₂²×C₅×D₈	C₂²	1	4	yes	—
13	C₂²×C₅×D₈	C₂²	1	2	yes	—
14	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₅×C₂²⋊Q₈	—	2	1	no	—
16	C₂×C₅×SD₃₂	—	2	8	no	—
17	C₂×C₅×SD₃₂	C₂²	1	8	yes	complemented
18	C₂²×C₅×C₁₆	C₂²	1	2	yes	complemented
19	C₅×#467	—	2	2	no	—
20	C₅×#467	C₂²	1	4	yes	complemented
21	C₂×C₅×#107	C₂²	1	1	yes	characteristic, complemented
22	C₂×C₅×#103	—	2	1	no	—
23	C₅×#357	C₂²	1	8	yes	complemented
24	C₅×C₂²⋊C₁₆	C₂²	1	4	yes	complemented
25	C₂×#2187	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂⁵×C₅	—	4	1	no	—
27	C₂²×C₅×Q₈	—	2	1	no	—
28	C₂²×C₅×D₄	—	2	8	no	—
29	C₂²×C₅×D₄	—	2	4	no	—
30	C₂²×C₅×D₄	—	2	2	no	—
31	C₂²×C₅×D₄	—	4	4	no	—
32	C₂²×C₅×D₄	—	4	2	no	—
33	C₂²×C₅×D₄	—	4	1	no	—
34	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
35	C₂×C₅×Q₁₆	—	2	4	no	—
36	C₂×C₅×Q₁₆	—	2	1	no	—
37	C₂×C₅×Q₁₆	C₂³	1	2	yes	complemented
38	C₂×C₅×D₈	—	2	8	no	—
39	C₂×C₅×D₈	—	2	16	no	—
40	C₂×C₅×D₈	—	2	16	no	—
41	C₂×C₅×D₈	—	2	4	no	—
42	C₂×C₅×D₈	—	2	2	no	—
43	C₂×C₅×D₈	—	2	4	no	—
44	C₂×C₅×D₈	D₄	1	8	yes	—
45	C₂×C₅×D₈	C₂³	1	4	yes	—
46	C₂²×C₅×C₈	—	2	4	no	—
47	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
48	C₂²×C₅×C₈	D₄	1	2	yes	—
49	C₂²×C₅×C₈	C₂³	1	2	yes	—
50	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
51	C₅×C₂²⋊Q₈	—	2	4	no	—
52	C₅×C₂²⋊Q₈	—	4	2	no	—
53	C₂×C₅×C₄⋊C₄	—	2	1	no	—
54	C₂×C₅×C₄⋊C₄	—	4	1	no	—
55	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
56	C₅×SD₃₂	—	2	16	no	—
57	C₂×C₅×C₁₆	—	2	4	no	—
58	C₂×C₅×C₁₆	C₂³	1	4	yes	complemented
59	C₅×#107	—	2	1	no	—
60	C₅×#107	C₂³	1	2	yes	complemented
61	C₅×#103	—	2	4	no	—
62	C₂³×D₈	C₂×C₅	1	1	yes	characteristic
63	C₂²×SD₃₂	C₂×C₅	1	2	yes	complemented
64	C₂×#467	C₂×C₅	1	1	yes	characteristic, complemented
65	#2187	C₂×C₅	1	8	yes	direct product factor
66	C₂×#357	C₂×C₅	1	2	yes	complemented
67	C₂×C₂²⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
68	C₂⁴×C₅	—	4	8	no	—
69	C₂⁴×C₅	—	4	4	no	—
70	C₂⁴×C₅	—	4	2	no	—
71	C₂⁴×C₅	—	8	4	no	—
72	C₂⁴×C₅	—	8	4	no	—
73	C₂⁴×C₅	D₈	1	1	yes	characteristic
74	C₂×C₅×Q₈	—	2	2	no	—
75	C₂×C₅×Q₈	—	4	1	no	—
76	C₂×C₅×Q₈	—	4	2	no	—
77	C₂×C₅×Q₈	—	4	2	no	—
78	C₂×C₅×D₄	—	2	16	no	—
79	C₂×C₅×D₄	—	2	8	no	—
80	C₂×C₅×D₄	—	2	4	no	—
81	C₂×C₅×D₄	—	4	8	no	—
82	C₂×C₅×D₄	—	4	8	no	—
83	C₂×C₅×D₄	—	4	16	no	—
84	C₂×C₅×D₄	—	4	4	no	—
85	C₂×C₅×D₄	—	4	2	no	—
86	C₂×C₅×D₄	—	4	4	no	—
87	C₂²×C₄×C₅	—	2	2	no	—
88	C₂²×C₄×C₅	—	2	2	no	—
89	C₂²×C₄×C₅	—	4	1	no	—
90	C₂²×C₄×C₅	—	4	1	no	—
91	C₂²×C₄×C₅	D₈	1	2	yes	—
92	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
93	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
95	C₅×Q₁₆	—	2	4	no	—
96	C₅×Q₁₆	—	4	2	no	—
97	C₅×D₈	—	2	16	no	—
98	C₅×D₈	—	2	8	no	—
99	C₅×D₈	—	2	4	no	—
100	C₅×D₈	C₂×D₄	1	8	yes	—
101	C₂×C₅×C₈	—	2	2	no	—
102	C₂×C₅×C₈	—	2	4	no	—
103	C₂×C₅×C₈	—	2	4	no	—
104	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
105	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
106	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
107	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
108	C₅×C₄⋊C₄	—	2	2	no	—
109	C₅×C₄⋊C₄	—	4	1	no	—
110	C₅×C₄⋊C₄	—	4	4	no	—
111	C₅×C₂²⋊C₄	—	4	4	no	—
112	C₅×C₁₆	—	2	4	no	—
113	C₂³×D₄	—	2	1	no	—
114	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
115	C₂²×D₈	—	2	8	no	—
116	C₂²×D₈	—	2	1	no	—
117	C₂²×D₈	—	2	2	no	—
118	C₂²×D₈	C₂²×C₅	1	4	yes	—
119	C₂²×D₈	C₂²×C₅	1	2	yes	—
120	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
121	C₂×C₂²⋊Q₈	—	2	1	no	—
122	C₂×SD₃₂	—	2	8	no	—
123	C₂×SD₃₂	C₂²×C₅	1	8	yes	complemented
124	C₂²×C₁₆	C₂²×C₅	1	2	yes	complemented
125	#467	—	2	2	no	—
126	#467	C₂²×C₅	1	4	yes	complemented
127	C₂×#107	C₂²×C₅	1	1	yes	characteristic, complemented
128	C₂×#103	—	2	1	no	—
129	#357	C₂²×C₅	1	8	yes	complemented
130	C₂²⋊C₁₆	C₂²×C₅	1	4	yes	complemented
131	C₂³×C₅	—	2	2	no	—
132	C₂³×C₅	—	2	2	no	—
133	C₂³×C₅	—	4	16	no	—
134	C₂³×C₅	—	4	4	no	—
135	C₂³×C₅	—	4	8	no	—
136	C₂³×C₅	—	8	8	no	—
137	C₂³×C₅	—	8	32	no	—
138	C₂³×C₅	—	8	4	no	—
139	C₂³×C₅	—	8	8	no	—
140	C₂³×C₅	—	8	4	no	—
141	C₂³×C₅	SD₃₂	1	4	yes	complemented
142	C₂³×C₅	C₂×D₈	1	2	yes	—
143	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
144	C₅×Q₈	—	4	2	no	—
145	C₅×Q₈	—	4	2	no	—
146	C₅×Q₈	—	8	2	no	—
147	C₅×D₄	—	2	8	no	—
148	C₅×D₄	—	4	16	no	—
149	C₅×D₄	—	4	4	no	—
150	C₅×D₄	—	4	8	no	—
151	C₂×C₄×C₅	—	2	2	no	—
152	C₂×C₄×C₅	—	2	2	no	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	2	4	no	—
155	C₂×C₄×C₅	—	4	2	no	—
156	C₂×C₄×C₅	—	4	2	no	—
157	C₂×C₄×C₅	—	8	2	no	—
158	C₂×C₄×C₅	—	8	2	no	—
159	C₂×C₄×C₅	#120	1	2	yes	—
160	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
161	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
162	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
163	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
164	C₅×C₈	—	2	2	no	—
165	C₅×C₈	#120	1	2	yes	—
166	C₅×C₈	C₂²×D₄	1	2	yes	—
167	C₂⁵	—	4	1	no	—
168	C₂²×Q₈	—	2	1	no	—
169	C₂²×D₄	—	2	8	no	—
170	C₂²×D₄	—	2	4	no	—
171	C₂²×D₄	—	2	2	no	—
172	C₂²×D₄	—	4	4	no	—
173	C₂²×D₄	—	4	1	no	—
174	C₂²×D₄	—	4	2	no	—
175	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
176	C₂×Q₁₆	—	2	4	no	—
177	C₂×Q₁₆	—	2	1	no	—
178	C₂×Q₁₆	C₂³×C₅	1	2	yes	complemented
179	C₂×D₈	—	2	8	no	—
180	C₂×D₈	—	2	16	no	—
181	C₂×D₈	—	2	16	no	—
182	C₂×D₈	—	2	4	no	—
183	C₂×D₈	—	2	2	no	—
184	C₂×D₈	—	2	4	no	—
185	C₂×D₈	C₅×D₄	1	8	yes	—
186	C₂×D₈	C₂³×C₅	1	4	yes	—
187	C₂²×C₈	—	2	4	no	—
188	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
189	C₂²×C₈	C₅×D₄	1	2	yes	—
190	C₂²×C₈	C₂³×C₅	1	2	yes	—
191	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
192	C₂²⋊Q₈	—	2	4	no	—
193	C₂²⋊Q₈	—	4	2	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	—	4	1	no	—
196	C₂×C₂²⋊C₄	—	4	1	no	—
197	SD₃₂	—	2	16	no	—
198	C₂×C₁₆	—	2	4	no	—
199	C₂×C₁₆	C₂³×C₅	1	4	yes	complemented
200	#107	—	2	1	no	—
201	#107	C₂³×C₅	1	2	yes	complemented
202	#103	—	2	4	no	—
203	C₂²×C₅	—	2	2	no	—
204	C₂²×C₅	—	2	8	no	—
205	C₂²×C₅	—	2	2	no	—
206	C₂²×C₅	—	4	8	no	—
207	C₂²×C₅	—	8	32	no	—
208	C₂²×C₅	—	8	4	no	—
209	C₂²×C₅	—	8	4	no	—
210	C₂²×C₅	—	8	16	no	—
211	C₂²×C₅	#447	1	2	yes	—
212	C₂²×C₅	C₂×SD₃₂	1	2	yes	—
213	C₂²×C₅	C₂×SD₃₂	1	4	yes	complemented
214	C₂²×C₅	#509	1	2	yes	—
215	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
216	C₄×C₅	—	2	2	no	—
217	C₄×C₅	—	8	2	no	—
218	C₄×C₅	—	8	2	no	—
219	C₄×C₅	#447	1	2	yes	—
220	C₄×C₅	C₂×#120	1	1	yes	characteristic
221	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
222	C₂⁴	—	4	8	no	—
223	C₂⁴	—	4	4	no	—
224	C₂⁴	—	4	2	no	—
225	C₂⁴	—	8	4	no	—
226	C₂⁴	—	8	4	no	—
227	C₂⁴	C₅×D₈	1	1	yes	characteristic
228	C₂×Q₈	—	2	2	no	—
229	C₂×Q₈	—	4	1	no	—
230	C₂×Q₈	—	4	2	no	—
231	C₂×Q₈	—	4	2	no	—
232	C₂×D₄	—	2	16	no	—
233	C₂×D₄	—	2	8	no	—
234	C₂×D₄	—	2	4	no	—
235	C₂×D₄	—	4	8	no	—
236	C₂×D₄	—	4	16	no	—
237	C₂×D₄	—	4	8	no	—
238	C₂×D₄	—	4	4	no	—
239	C₂×D₄	—	4	2	no	—
240	C₂×D₄	—	4	4	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	—	4	1	no	—
245	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
246	C₂²×C₄	C₅×D₈	1	2	yes	—
247	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
248	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
249	Q₁₆	—	2	4	no	—
250	Q₁₆	—	4	2	no	—
251	D₈	—	2	16	no	—
252	D₈	—	2	8	no	—
253	D₈	—	2	4	no	—
254	D₈	C₂×C₅×D₄	1	8	yes	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
259	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
260	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
261	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	4	no	—
265	C₂²⋊C₄	—	4	4	no	—
266	C₁₆	—	2	4	no	—
267	C₂×C₅	—	2	4	no	—
268	C₂×C₅	—	8	8	no	—
269	C₂×C₅	#2187	1	4	yes	direct product factor
270	C₂×C₅	C₂×#447	1	1	yes	characteristic
271	C₂×C₅	C₂²×SD₃₂	1	1	yes	characteristic
272	C₂×C₅	C₂×#509	1	1	yes	characteristic
273	C₂³	—	2	2	no	—
274	C₂³	—	2	2	no	—
275	C₂³	—	4	16	no	—
276	C₂³	—	4	8	no	—
277	C₂³	—	4	4	no	—
278	C₂³	—	8	8	no	—
279	C₂³	—	8	32	no	—
280	C₂³	—	8	4	no	—
281	C₂³	—	8	8	no	—
282	C₂³	—	8	4	no	—
283	C₂³	C₅×SD₃₂	1	4	yes	complemented
284	C₂³	C₂×C₅×D₈	1	2	yes	—
285	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
286	Q₈	—	4	2	no	—
287	Q₈	—	4	2	no	—
288	Q₈	—	8	2	no	—
289	D₄	—	2	8	no	—
290	D₄	—	4	16	no	—
291	D₄	—	4	8	no	—
292	D₄	—	4	4	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	8	2	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	C₅×#120	1	2	yes	—
302	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
303	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
304	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
305	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	C₅×#120	1	2	yes	—
308	C₈	C₂²×C₅×D₄	1	2	yes	—
309	C₅	C₂×#2187	1	1	yes	characteristic, direct product factor, Sylow
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	8	no	—
313	C₂²	—	4	8	no	—
314	C₂²	—	8	32	no	—
315	C₂²	—	8	4	no	—
316	C₂²	—	8	4	no	—
317	C₂²	—	8	16	no	—
318	C₂²	C₅×#447	1	2	yes	—
319	C₂²	C₂×C₅×SD₃₂	1	2	yes	—
320	C₂²	C₂×C₅×SD₃₂	1	4	yes	complemented
321	C₂²	C₅×#509	1	2	yes	—
322	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
323	C₄	—	2	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	C₅×#447	1	2	yes	—
327	C₄	C₂×C₅×#120	1	1	yes	characteristic
328	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
329	C₂	—	2	4	no	—
330	C₂	—	8	8	no	—
331	C₂	C₅×#2187	1	4	yes	direct product factor
332	C₂	C₂×C₅×#447	1	1	yes	characteristic
333	C₂	C₂²×C₅×SD₃₂	1	1	yes	characteristic
334	C₂	C₂×C₅×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂³×C₅×D₈	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×SD₃₂	C₂	semidirect
3. C₂×C₅×#467	C₂	semidirect
4. C₅×#2187	C₂	direct (N × H)
5. C₂×C₅×#357	C₂	semidirect
6. C₂×C₅×C₂²⋊C₁₆	C₂	semidirect
8. C₂²×C₅×Q₁₆	C₂²	semidirect
17. C₂×C₅×SD₃₂	C₂²	semidirect
18. C₂²×C₅×C₁₆	C₂²	semidirect
20. C₅×#467	C₂²	semidirect
21. C₂×C₅×#107	C₂²	semidirect
23. C₅×#357	C₂²	semidirect
24. C₅×C₂²⋊C₁₆	C₂²	semidirect
25. C₂×#2187	C₅	direct (N × H)
37. C₂×C₅×Q₁₆	C₂³	semidirect
58. C₂×C₅×C₁₆	C₂³	semidirect
60. C₅×#107	C₂³	semidirect
63. C₂²×SD₃₂	C₂×C₅	semidirect
64. C₂×#467	C₂×C₅	semidirect
65. #2187	C₂×C₅	direct (N × H)
66. C₂×#357	C₂×C₅	semidirect
67. C₂×C₂²⋊C₁₆	C₂×C₅	semidirect
114. C₂²×Q₁₆	C₂²×C₅	semidirect
123. C₂×SD₃₂	C₂²×C₅	semidirect
124. C₂²×C₁₆	C₂²×C₅	semidirect
126. #467	C₂²×C₅	semidirect
127. C₂×#107	C₂²×C₅	semidirect
129. #357	C₂²×C₅	semidirect
130. C₂²⋊C₁₆	C₂²×C₅	semidirect
141. C₂³×C₅	SD₃₂	semidirect
178. C₂×Q₁₆	C₂³×C₅	semidirect
199. C₂×C₁₆	C₂³×C₅	semidirect
201. #107	C₂³×C₅	semidirect
213. C₂²×C₅	C₂×SD₃₂	semidirect
269. C₂×C₅	#2187	direct (N × H)
283. C₂³	C₅×SD₃₂	semidirect
309. C₅	C₂×#2187	direct (N × H)
320. C₂²	C₂×C₅×SD₃₂	semidirect
331. C₂	C₅×#2187	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₅×D₈	2
C₂²×C₅×SD₃₂	2
C₂×C₅×#467	2
C₅×#2187	2
C₂×C₅×#357	2
C₂×C₅×C₂²⋊C₁₆	2
C₂×#2187	5

Subgroups — C2×C5×#2187 #11956231

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2187 #11956231