Subgroups — C₂×C₅×#3138 #11983333

Order 1280.

2806 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₅×#127	C₂	1	1	yes	maximal, characteristic
3	C₅×#3138	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅×#490	C₂	1	4	yes	maximal, complemented
5	C₂×C₅×#486	C₂	1	4	yes	maximal, complemented
6	C₂×C₅×#418	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₅×D₄	—	2	2	no	—
9	C₂³×C₅×D₄	C₂²	1	2	yes	—
10	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
11	C₅×#533	—	2	4	no	—
12	C₅×#533	C₂²	1	8	yes	complemented
13	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	complemented
14	C₂×C₅×#127	—	2	2	no	—
15	C₂×C₅×#127	—	2	4	no	—
16	C₂×C₅×#127	C₂²	1	2	yes	—
17	C₂×C₅×#127	C₂²	1	2	yes	—
18	C₂×C₅×#127	C₂²	1	8	yes	—
19	C₂×C₅×#125	C₂²	1	2	yes	complemented
20	C₂×C₅×C₂²⋊Q₈	—	2	2	no	—
21	C₂×C₄×C₅×Q₈	—	2	2	no	—
22	C₂×C₅×#117	C₂²	1	2	yes	—
23	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
24	C₅×#490	C₂²	1	32	yes	complemented
25	C₅×#486	C₂²	1	32	yes	complemented
26	C₅×#418	C₂²	1	32	yes	complemented
27	C₂×C₅×#102	C₂²	1	16	yes	complemented
28	C₄×C₅×M₁₆	C₂²	1	8	yes	complemented
29	C₂×C₅×#97	C₂²	1	2	yes	complemented
30	C₂×C₄×C₅×C₈	C₂²	1	2	yes	complemented
31	C₂×#3138	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂⁵×C₅	—	2	2	no	—
33	C₂²×C₅×Q₈	—	2	2	no	—
34	C₂²×C₅×D₄	—	2	4	no	—
35	C₂²×C₅×D₄	—	2	2	no	—
36	C₂²×C₅×D₄	—	2	16	no	—
37	C₂²×C₅×D₄	—	2	4	no	—
38	C₂²×C₅×D₄	—	2	4	no	—
39	C₂²×C₅×D₄	—	2	16	no	—
40	C₂²×C₅×D₄	—	4	16	no	—
41	C₂²×C₅×D₄	C₂³	1	4	yes	—
42	C₂²×C₅×D₄	C₂³	1	8	yes	—
43	C₂³×C₄×C₅	D₄	1	2	yes	—
44	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
45	C₂×C₅×M₁₆	—	2	8	no	—
46	C₂×C₅×M₁₆	C₂³	1	8	yes	—
47	C₂²×C₅×C₈	C₂³	1	4	yes	—
48	C₅×C₄⋊Q₈	—	2	4	no	—
49	C₅×C₄⋊Q₈	C₂³	1	8	yes	complemented
50	C₅×#127	—	2	16	no	—
51	C₅×#127	—	2	4	no	—
52	C₅×#127	—	2	4	no	—
53	C₅×#127	C₂³	1	8	yes	—
54	C₅×#127	C₂³	1	8	yes	—
55	C₅×#125	—	2	4	no	—
56	C₅×#125	C₂³	1	8	yes	complemented
57	C₅×C₂²⋊Q₈	—	2	16	no	—
58	C₄×C₅×Q₈	—	2	16	no	—
59	C₅×#117	—	2	4	no	—
60	C₅×#117	C₂³	1	8	yes	—
61	C₂×C₅×C₄⋊C₄	—	2	4	no	—
62	C₂×C₅×C₄⋊C₄	—	2	2	no	—
63	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
64	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
65	C₂×C₄²×C₅	—	2	2	no	—
66	C₂×C₄²×C₅	D₄	1	2	yes	—
67	C₂×C₄²×C₅	D₄	1	4	yes	—
68	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₅	C₂³	1	4	yes	—
71	C₅×#102	C₂³	1	64	yes	complemented
72	C₅×#97	C₂³	1	8	yes	complemented
73	C₄×C₅×C₈	C₂³	1	8	yes	complemented
74	C₂×#533	C₂×C₅	1	1	yes	characteristic, complemented
75	C₂²×#127	C₂×C₅	1	1	yes	characteristic
76	#3138	C₂×C₅	1	16	yes	direct product factor
77	C₂×#490	C₂×C₅	1	4	yes	complemented
78	C₂×#486	C₂×C₅	1	4	yes	complemented
79	C₂×#418	C₂×C₅	1	4	yes	complemented
80	C₂×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic, complemented
81	C₂⁴×C₅	—	2	8	no	—
82	C₂⁴×C₅	—	2	4	no	—
83	C₂⁴×C₅	—	4	16	no	—
84	C₂⁴×C₅	—	4	8	no	—
85	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂×C₅×Q₈	—	2	8	no	—
87	C₂×C₅×Q₈	—	4	8	no	—
88	C₂×C₅×D₄	—	2	8	no	—
89	C₂×C₅×D₄	—	2	4	no	—
90	C₂×C₅×D₄	—	2	16	no	—
91	C₂×C₅×D₄	—	2	16	no	—
92	C₂×C₅×D₄	—	2	16	no	—
93	C₂×C₅×D₄	—	2	32	no	—
94	C₂×C₅×D₄	—	4	16	no	—
95	C₂×C₅×D₄	—	4	64	no	—
96	C₂×C₅×D₄	—	4	16	no	—
97	C₂×C₅×D₄	—	4	8	no	—
98	C₂×C₅×D₄	—	4	16	no	—
99	C₂×C₅×D₄	C₂⁴	1	8	yes	—
100	C₂²×C₄×C₅	—	2	8	no	—
101	C₂²×C₄×C₅	—	2	4	no	—
102	C₂²×C₄×C₅	C₂×D₄	1	8	yes	—
103	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
107	C₂²×C₄×C₅	#41	1	8	yes	—
108	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
109	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
110	C₅×M₁₆	—	2	16	no	—
111	C₂×C₅×C₈	—	2	8	no	—
112	C₂×C₅×C₈	C₂⁴	1	8	yes	—
113	C₅×C₄⋊C₄	—	2	16	no	—
114	C₅×C₄⋊C₄	—	2	8	no	—
115	C₅×C₄⋊C₄	—	2	4	no	—
116	C₅×C₄⋊C₄	C₂⁴	1	8	yes	—
117	C₅×C₂²⋊C₄	—	2	8	no	—
118	C₄²×C₅	—	2	8	no	—
119	C₄²×C₅	C₂×D₄	1	8	yes	—
120	C₄²×C₅	C₂×D₄	1	2	yes	—
121	C₄²×C₅	C₂×D₄	1	2	yes	—
122	C₄²×C₅	C₂⁴	1	2	yes	—
123	C₄²×C₅	C₂⁴	1	2	yes	—
124	C₂³×D₄	—	2	2	no	—
125	C₂³×D₄	C₂²×C₅	1	2	yes	—
126	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
127	#533	—	2	4	no	—
128	#533	C₂²×C₅	1	8	yes	complemented
129	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	complemented
130	C₂×#127	—	2	2	no	—
131	C₂×#127	—	2	4	no	—
132	C₂×#127	C₂²×C₅	1	2	yes	—
133	C₂×#127	C₂²×C₅	1	2	yes	—
134	C₂×#127	C₂²×C₅	1	8	yes	—
135	C₂×#125	C₂²×C₅	1	2	yes	complemented
136	C₂×C₂²⋊Q₈	—	2	2	no	—
137	C₂×C₄×Q₈	—	2	2	no	—
138	C₂×#117	C₂²×C₅	1	2	yes	—
139	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
140	#490	C₂²×C₅	1	32	yes	complemented
141	#486	C₂²×C₅	1	32	yes	complemented
142	#418	C₂²×C₅	1	32	yes	complemented
143	C₂×#102	C₂²×C₅	1	16	yes	complemented
144	C₄×M₁₆	C₂²×C₅	1	8	yes	complemented
145	C₂×#97	C₂²×C₅	1	2	yes	complemented
146	C₂×C₄×C₈	C₂²×C₅	1	2	yes	complemented
147	C₂³×C₅	—	2	4	no	—
148	C₂³×C₅	—	2	8	no	—
149	C₂³×C₅	—	4	32	no	—
150	C₂³×C₅	—	4	16	no	—
151	C₂³×C₅	—	4	8	no	—
152	C₂³×C₅	—	4	16	no	—
153	C₂³×C₅	—	8	32	no	—
154	C₂³×C₅	#124	1	4	yes	—
155	C₂³×C₅	C₂²×D₄	1	2	yes	—
156	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
157	C₅×Q₈	—	4	16	no	—
158	C₅×D₄	—	2	16	no	—
159	C₅×D₄	—	4	32	no	—
160	C₅×D₄	—	4	16	no	—
161	C₅×D₄	—	4	8	no	—
162	C₅×D₄	—	4	32	no	—
163	C₂×C₄×C₅	—	2	4	no	—
164	C₂×C₄×C₅	—	2	16	no	—
165	C₂×C₄×C₅	—	2	8	no	—
166	C₂×C₄×C₅	—	4	8	no	—
167	C₂×C₄×C₅	#124	1	8	yes	—
168	C₂×C₄×C₅	#124	1	4	yes	—
169	C₂×C₄×C₅	AGL(1,ℤ/8)	1	8	yes	—
170	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
171	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
172	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
174	C₂×C₄×C₅	C₂×#41	1	4	yes	—
175	C₂×C₄×C₅	C₂×#41	1	8	yes	—
176	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
177	C₅×C₈	—	2	8	no	—
178	C₂⁵	—	2	2	no	—
179	C₂²×Q₈	—	2	2	no	—
180	C₂²×D₄	—	2	2	no	—
181	C₂²×D₄	—	2	4	no	—
182	C₂²×D₄	—	2	4	no	—
183	C₂²×D₄	—	2	16	no	—
184	C₂²×D₄	—	2	4	no	—
185	C₂²×D₄	—	2	16	no	—
186	C₂²×D₄	—	4	16	no	—
187	C₂²×D₄	C₂³×C₅	1	4	yes	—
188	C₂²×D₄	C₂³×C₅	1	8	yes	—
189	C₂³×C₄	C₅×D₄	1	2	yes	—
190	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
191	C₂×M₁₆	—	2	8	no	—
192	C₂×M₁₆	C₂³×C₅	1	8	yes	—
193	C₂²×C₈	C₂³×C₅	1	4	yes	—
194	C₄⋊Q₈	—	2	4	no	—
195	C₄⋊Q₈	C₂³×C₅	1	8	yes	complemented
196	#127	—	2	16	no	—
197	#127	—	2	4	no	—
198	#127	—	2	4	no	—
199	#127	C₂³×C₅	1	8	yes	—
200	#127	C₂³×C₅	1	8	yes	—
201	#125	—	2	4	no	—
202	#125	C₂³×C₅	1	8	yes	complemented
203	C₂²⋊Q₈	—	2	16	no	—
204	C₄×Q₈	—	2	16	no	—
205	#117	—	2	4	no	—
206	#117	C₂³×C₅	1	8	yes	—
207	C₂×C₄⋊C₄	—	2	4	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
210	C₂×C₂²⋊C₄	—	2	2	no	—
211	C₂×C₄²	—	2	2	no	—
212	C₂×C₄²	C₅×D₄	1	2	yes	—
213	C₂×C₄²	C₅×D₄	1	4	yes	—
214	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
215	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
216	C₂×C₄²	C₂³×C₅	1	4	yes	—
217	#102	C₂³×C₅	1	64	yes	complemented
218	#97	C₂³×C₅	1	8	yes	complemented
219	C₄×C₈	C₂³×C₅	1	8	yes	complemented
220	C₂²×C₅	—	2	8	no	—
221	C₂²×C₅	—	2	4	no	—
222	C₂²×C₅	—	4	8	no	—
223	C₂²×C₅	—	4	16	no	—
224	C₂²×C₅	—	8	32	no	—
225	C₂²×C₅	—	8	16	no	—
226	C₂²×C₅	C₂×#124	1	4	yes	—
227	C₂²×C₅	C₂×#124	1	2	yes	—
228	C₂²×C₅	C₂×AGL(1,ℤ/8)	1	4	yes	—
229	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
230	C₄×C₅	—	2	4	no	—
231	C₄×C₅	—	4	8	no	—
232	C₄×C₅	C₂×#124	1	4	yes	—
233	C₄×C₅	C₂×#124	1	2	yes	—
234	C₄×C₅	C₂×AGL(1,ℤ/8)	1	8	yes	—
235	C₄×C₅	C₂²×#41	1	2	yes	—
236	C₂⁴	—	2	8	no	—
237	C₂⁴	—	2	4	no	—
238	C₂⁴	—	4	16	no	—
239	C₂⁴	—	4	8	no	—
240	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
241	C₂×Q₈	—	2	8	no	—
242	C₂×Q₈	—	4	8	no	—
243	C₂×D₄	—	2	8	no	—
244	C₂×D₄	—	2	4	no	—
245	C₂×D₄	—	2	16	no	—
246	C₂×D₄	—	2	16	no	—
247	C₂×D₄	—	2	16	no	—
248	C₂×D₄	—	2	32	no	—
249	C₂×D₄	—	4	16	no	—
250	C₂×D₄	—	4	64	no	—
251	C₂×D₄	—	4	16	no	—
252	C₂×D₄	—	4	8	no	—
253	C₂×D₄	—	4	16	no	—
254	C₂×D₄	C₂⁴×C₅	1	8	yes	—
255	C₂²×C₄	—	2	8	no	—
256	C₂²×C₄	—	2	4	no	—
257	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
258	C₂²×C₄	C₂×C₅×D₄	1	8	yes	—
259	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
260	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
261	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
262	C₂²×C₄	C₅×#41	1	8	yes	—
263	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
264	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
265	M₁₆	—	2	16	no	—
266	C₂×C₈	—	2	8	no	—
267	C₂×C₈	C₂⁴×C₅	1	8	yes	—
268	C₄⋊C₄	—	2	16	no	—
269	C₄⋊C₄	—	2	8	no	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	C₂⁴×C₅	1	8	yes	—
272	C₂²⋊C₄	—	2	8	no	—
273	C₄²	—	2	8	no	—
274	C₄²	C₂×C₅×D₄	1	8	yes	—
275	C₄²	C₂×C₅×D₄	1	2	yes	—
276	C₄²	C₂×C₅×D₄	1	2	yes	—
277	C₄²	C₂⁴×C₅	1	2	yes	—
278	C₄²	C₂⁴×C₅	1	2	yes	—
279	C₂×C₅	—	2	4	no	—
280	C₂×C₅	—	8	8	no	—
281	C₂×C₅	#3138	1	4	yes	direct product factor
282	C₂×C₅	C₂²×#124	1	1	yes	characteristic
283	C₂×C₅	C₂²×AGL(1,ℤ/8)	1	2	yes	—
284	C₂³	—	2	4	no	—
285	C₂³	—	2	8	no	—
286	C₂³	—	4	32	no	—
287	C₂³	—	4	16	no	—
288	C₂³	—	4	8	no	—
289	C₂³	—	4	16	no	—
290	C₂³	—	8	32	no	—
291	C₂³	C₅×#124	1	4	yes	—
292	C₂³	C₂²×C₅×D₄	1	2	yes	—
293	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
294	Q₈	—	4	16	no	—
295	D₄	—	2	16	no	—
296	D₄	—	4	32	no	—
297	D₄	—	4	16	no	—
298	D₄	—	4	8	no	—
299	D₄	—	4	32	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	16	no	—
302	C₂×C₄	—	2	8	no	—
303	C₂×C₄	—	4	8	no	—
304	C₂×C₄	C₅×#124	1	8	yes	—
305	C₂×C₄	C₅×#124	1	4	yes	—
306	C₂×C₄	C₅×AGL(1,ℤ/8)	1	8	yes	—
307	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
308	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
309	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
310	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
311	C₂×C₄	C₂×C₅×#41	1	4	yes	—
312	C₂×C₄	C₂×C₅×#41	1	8	yes	—
313	C₂×C₄	C₂⁵×C₅	1	1	yes	characteristic
314	C₈	—	2	8	no	—
315	C₅	C₂×#3138	1	1	yes	characteristic, direct product factor, Sylow
316	C₂²	—	2	8	no	—
317	C₂²	—	2	4	no	—
318	C₂²	—	4	8	no	—
319	C₂²	—	4	16	no	—
320	C₂²	—	8	32	no	—
321	C₂²	—	8	16	no	—
322	C₂²	C₂×C₅×#124	1	4	yes	—
323	C₂²	C₂×C₅×#124	1	2	yes	—
324	C₂²	C₂×C₅×AGL(1,ℤ/8)	1	4	yes	—
325	C₂²	C₂³×C₅×D₄	1	1	yes	characteristic
326	C₄	—	2	4	no	—
327	C₄	—	4	8	no	—
328	C₄	C₂×C₅×#124	1	4	yes	—
329	C₄	C₂×C₅×#124	1	2	yes	—
330	C₄	C₂×C₅×AGL(1,ℤ/8)	1	8	yes	—
331	C₄	C₂²×C₅×#41	1	2	yes	—
332	C₂	—	2	4	no	—
333	C₂	—	8	8	no	—
334	C₂	C₅×#3138	1	4	yes	direct product factor
335	C₂	C₂²×C₅×#124	1	1	yes	characteristic
336	C₂	C₂²×C₅×AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅
prime radical	C₂²×C₅×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#533	C₂	semidirect
3. C₅×#3138	C₂	direct (N × H)
4. C₂×C₅×#490	C₂	semidirect
5. C₂×C₅×#486	C₂	semidirect
6. C₂×C₅×#418	C₂	semidirect
7. C₂×C₄×C₅×M₁₆	C₂	semidirect
12. C₅×#533	C₂²	semidirect
13. C₂×C₅×C₄⋊Q₈	C₂²	semidirect
19. C₂×C₅×#125	C₂²	semidirect
24. C₅×#490	C₂²	semidirect
25. C₅×#486	C₂²	semidirect
26. C₅×#418	C₂²	semidirect
27. C₂×C₅×#102	C₂²	semidirect
28. C₄×C₅×M₁₆	C₂²	semidirect
29. C₂×C₅×#97	C₂²	semidirect
30. C₂×C₄×C₅×C₈	C₂²	semidirect
31. C₂×#3138	C₅	direct (N × H)
49. C₅×C₄⋊Q₈	C₂³	semidirect
56. C₅×#125	C₂³	semidirect
71. C₅×#102	C₂³	semidirect
72. C₅×#97	C₂³	semidirect
73. C₄×C₅×C₈	C₂³	semidirect
74. C₂×#533	C₂×C₅	semidirect
76. #3138	C₂×C₅	direct (N × H)
77. C₂×#490	C₂×C₅	semidirect
78. C₂×#486	C₂×C₅	semidirect
79. C₂×#418	C₂×C₅	semidirect
80. C₂×C₄×M₁₆	C₂×C₅	semidirect
128. #533	C₂²×C₅	semidirect
129. C₂×C₄⋊Q₈	C₂²×C₅	semidirect
135. C₂×#125	C₂²×C₅	semidirect
140. #490	C₂²×C₅	semidirect
141. #486	C₂²×C₅	semidirect
142. #418	C₂²×C₅	semidirect
143. C₂×#102	C₂²×C₅	semidirect
144. C₄×M₁₆	C₂²×C₅	semidirect
145. C₂×#97	C₂²×C₅	semidirect
146. C₂×C₄×C₈	C₂²×C₅	semidirect
195. C₄⋊Q₈	C₂³×C₅	semidirect
202. #125	C₂³×C₅	semidirect
217. #102	C₂³×C₅	semidirect
218. #97	C₂³×C₅	semidirect
219. C₄×C₈	C₂³×C₅	semidirect
281. C₂×C₅	#3138	direct (N × H)
315. C₅	C₂×#3138	direct (N × H)
334. C₂	C₅×#3138	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#533	2
C₂²×C₅×#127	2
C₅×#3138	2
C₂×C₅×#490	2
C₂×C₅×#486	2
C₂×C₅×#418	2
C₂×C₄×C₅×M₁₆	2
C₂×#3138	5

Subgroups — C2×C5×#3138 #11983333

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#3138 #11983333