Subgroups — C₂×C₅×#3139 #11983334

Order 1280.

2118 conjugacy classes of subgroups.

The subgroup lattice has 310 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
2	C₂²×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₅×#3139	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅×#491	C₂	1	4	yes	maximal, complemented
5	C₂×C₅×#487	C₂	1	4	yes	maximal, complemented
6	C₂×C₅×#419	C₂	1	4	yes	maximal
7	C₂×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₅×Q₈	C₂²	1	2	yes	—
9	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
10	C₅×#533	—	2	4	no	—
11	C₅×#533	C₂²	1	8	yes	—
12	C₂×C₅×C₄⋊Q₈	—	2	2	no	—
13	C₂×C₅×C₄⋊Q₈	—	2	4	no	—
14	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
15	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
16	C₂×C₅×C₄⋊Q₈	C₂²	1	8	yes	—
17	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
18	C₂×C₅×#125	C₂²	1	2	yes	—
19	C₂×C₅×C₂²⋊Q₈	—	2	2	no	—
20	C₂×C₄×C₅×Q₈	—	2	2	no	—
21	C₂×C₅×#117	C₂²	1	2	yes	—
22	C₂²×C₅×C₄⋊C₄	—	2	2	no	—
23	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
24	C₅×#491	C₂²	1	32	yes	complemented
25	C₅×#487	C₂²	1	32	yes	complemented
26	C₅×#419	C₂²	1	32	yes	—
27	C₂×C₅×#103	C₂²	1	16	yes	—
28	C₄×C₅×M₁₆	C₂²	1	8	yes	—
29	C₂×C₅×#97	C₂²	1	2	yes	—
30	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
31	C₂×#3139	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂²×C₅×Q₈	—	2	2	no	—
33	C₂²×C₅×Q₈	—	2	4	no	—
34	C₂²×C₅×Q₈	—	2	2	no	—
35	C₂²×C₅×Q₈	—	2	16	no	—
36	C₂²×C₅×Q₈	C₂³	1	4	yes	—
37	C₂²×C₅×Q₈	C₂³	1	8	yes	—
38	C₂³×C₄×C₅	—	2	2	no	—
39	C₂³×C₄×C₅	D₄	1	2	yes	—
40	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₂×C₅×M₁₆	—	2	8	no	—
42	C₂×C₅×M₁₆	C₂³	1	8	yes	—
43	C₂²×C₅×C₈	C₂³	1	4	yes	—
44	C₅×C₄⋊Q₈	—	2	16	no	—
45	C₅×C₄⋊Q₈	—	2	4	no	—
46	C₅×C₄⋊Q₈	—	2	4	no	—
47	C₅×C₄⋊Q₈	—	2	4	no	—
48	C₅×C₄⋊Q₈	C₂³	1	8	yes	—
49	C₅×C₄⋊Q₈	C₂³	1	8	yes	—
50	C₅×C₄⋊Q₈	C₂³	1	8	yes	—
51	C₅×#125	—	2	4	no	—
52	C₅×#125	C₂³	1	8	yes	—
53	C₅×C₂²⋊Q₈	—	2	16	no	—
54	C₄×C₅×Q₈	—	2	16	no	—
55	C₅×#117	—	2	4	no	—
56	C₅×#117	C₂³	1	8	yes	—
57	C₂×C₅×C₄⋊C₄	—	2	16	no	—
58	C₂×C₅×C₄⋊C₄	—	2	4	no	—
59	C₂×C₅×C₄⋊C₄	—	2	4	no	—
60	C₂×C₅×C₄⋊C₄	—	2	4	no	—
61	C₂×C₅×C₄⋊C₄	—	2	2	no	—
62	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
63	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
64	C₂×C₄²×C₅	—	2	2	no	—
65	C₂×C₄²×C₅	D₄	1	2	yes	—
66	C₂×C₄²×C₅	D₄	1	4	yes	—
67	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₅	C₂³	1	4	yes	—
70	C₅×#103	C₂³	1	64	yes	—
71	C₅×#97	C₂³	1	8	yes	—
72	C₄×C₅×C₈	C₂³	1	8	yes	—
73	C₂×#533	C₂×C₅	1	1	yes	characteristic
74	C₂²×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
75	#3139	C₂×C₅	1	16	yes	direct product factor
76	C₂×#491	C₂×C₅	1	4	yes	complemented
77	C₂×#487	C₂×C₅	1	4	yes	complemented
78	C₂×#419	C₂×C₅	1	4	yes	—
79	C₂×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic
80	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂×C₅×Q₈	—	2	8	no	—
82	C₂×C₅×Q₈	—	2	8	no	—
83	C₂×C₅×Q₈	—	2	4	no	—
84	C₂×C₅×Q₈	—	2	16	no	—
85	C₂×C₅×Q₈	—	2	32	no	—
86	C₂×C₅×Q₈	—	4	16	no	—
87	C₂×C₅×Q₈	—	4	8	no	—
88	C₂×C₅×Q₈	—	4	8	no	—
89	C₂×C₅×Q₈	C₂⁴	1	8	yes	—
90	C₂²×C₄×C₅	—	2	8	no	—
91	C₂²×C₄×C₅	—	2	8	no	—
92	C₂²×C₄×C₅	—	2	4	no	—
93	C₂²×C₄×C₅	—	2	4	no	—
94	C₂²×C₄×C₅	—	4	8	no	—
95	C₂²×C₄×C₅	C₂×D₄	1	8	yes	—
96	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
97	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
98	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
100	C₂²×C₄×C₅	#41	1	8	yes	—
101	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
102	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
103	C₅×M₁₆	—	2	16	no	—
104	C₂×C₅×C₈	—	2	8	no	—
105	C₂×C₅×C₈	C₂⁴	1	8	yes	—
106	C₅×C₄⋊C₄	—	2	16	no	—
107	C₅×C₄⋊C₄	—	2	16	no	—
108	C₅×C₄⋊C₄	—	2	16	no	—
109	C₅×C₄⋊C₄	—	2	8	no	—
110	C₅×C₄⋊C₄	—	2	4	no	—
111	C₅×C₄⋊C₄	C₂⁴	1	8	yes	—
112	C₅×C₂²⋊C₄	—	2	8	no	—
113	C₄²×C₅	—	2	8	no	—
114	C₄²×C₅	C₂×D₄	1	8	yes	—
115	C₄²×C₅	C₂×D₄	1	2	yes	—
116	C₄²×C₅	C₂×D₄	1	2	yes	—
117	C₄²×C₅	C₂⁴	1	2	yes	—
118	C₄²×C₅	C₂⁴	1	2	yes	—
119	C₂³×Q₈	C₂²×C₅	1	2	yes	—
120	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
121	#533	—	2	4	no	—
122	#533	C₂²×C₅	1	8	yes	—
123	C₂×C₄⋊Q₈	—	2	2	no	—
124	C₂×C₄⋊Q₈	—	2	4	no	—
125	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
126	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
127	C₂×C₄⋊Q₈	C₂²×C₅	1	8	yes	—
128	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
129	C₂×#125	C₂²×C₅	1	2	yes	—
130	C₂×C₂²⋊Q₈	—	2	2	no	—
131	C₂×C₄×Q₈	—	2	2	no	—
132	C₂×#117	C₂²×C₅	1	2	yes	—
133	C₂²×C₄⋊C₄	—	2	2	no	—
134	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
135	#491	C₂²×C₅	1	32	yes	complemented
136	#487	C₂²×C₅	1	32	yes	complemented
137	#419	C₂²×C₅	1	32	yes	—
138	C₂×#103	C₂²×C₅	1	16	yes	—
139	C₄×M₁₆	C₂²×C₅	1	8	yes	—
140	C₂×#97	C₂²×C₅	1	2	yes	—
141	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
142	C₂³×C₅	—	2	4	no	—
143	C₂³×C₅	#124	1	4	yes	—
144	C₂³×C₅	C₂²×D₄	1	2	yes	—
145	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
146	C₅×Q₈	—	2	16	no	—
147	C₅×Q₈	—	4	16	no	—
148	C₅×Q₈	—	4	8	no	—
149	C₅×Q₈	—	4	16	no	—
150	C₂×C₄×C₅	—	2	4	no	—
151	C₂×C₄×C₅	—	2	16	no	—
152	C₂×C₄×C₅	—	2	8	no	—
153	C₂×C₄×C₅	—	2	8	no	—
154	C₂×C₄×C₅	—	4	16	no	—
155	C₂×C₄×C₅	—	4	8	no	—
156	C₂×C₄×C₅	—	4	8	no	—
157	C₂×C₄×C₅	#124	1	8	yes	—
158	C₂×C₄×C₅	#124	1	4	yes	—
159	C₂×C₄×C₅	#137	1	8	yes	—
160	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
161	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
162	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
164	C₂×C₄×C₅	C₂×#41	1	4	yes	—
165	C₂×C₄×C₅	C₂×#41	1	8	yes	—
166	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
167	C₅×C₈	—	2	8	no	—
168	C₂²×Q₈	—	2	2	no	—
169	C₂²×Q₈	—	2	2	no	—
170	C₂²×Q₈	—	2	4	no	—
171	C₂²×Q₈	—	2	16	no	—
172	C₂²×Q₈	C₂³×C₅	1	4	yes	—
173	C₂²×Q₈	C₂³×C₅	1	8	yes	—
174	C₂³×C₄	—	2	2	no	—
175	C₂³×C₄	C₅×D₄	1	2	yes	—
176	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
177	C₂×M₁₆	—	2	8	no	—
178	C₂×M₁₆	C₂³×C₅	1	8	yes	—
179	C₂²×C₈	C₂³×C₅	1	4	yes	—
180	C₄⋊Q₈	—	2	16	no	—
181	C₄⋊Q₈	—	2	4	no	—
182	C₄⋊Q₈	—	2	4	no	—
183	C₄⋊Q₈	—	2	4	no	—
184	C₄⋊Q₈	C₂³×C₅	1	8	yes	—
185	C₄⋊Q₈	C₂³×C₅	1	8	yes	—
186	C₄⋊Q₈	C₂³×C₅	1	8	yes	—
187	#125	—	2	4	no	—
188	#125	C₂³×C₅	1	8	yes	—
189	C₂²⋊Q₈	—	2	16	no	—
190	C₄×Q₈	—	2	16	no	—
191	#117	—	2	4	no	—
192	#117	C₂³×C₅	1	8	yes	—
193	C₂×C₄⋊C₄	—	2	4	no	—
194	C₂×C₄⋊C₄	—	2	16	no	—
195	C₂×C₄⋊C₄	—	2	4	no	—
196	C₂×C₄⋊C₄	—	2	4	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
199	C₂×C₂²⋊C₄	—	2	2	no	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	C₅×D₄	1	2	yes	—
202	C₂×C₄²	C₅×D₄	1	4	yes	—
203	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
204	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
205	C₂×C₄²	C₂³×C₅	1	4	yes	—
206	#103	C₂³×C₅	1	64	yes	—
207	#97	C₂³×C₅	1	8	yes	—
208	C₄×C₈	C₂³×C₅	1	8	yes	—
209	C₂²×C₅	—	2	8	no	—
210	C₂²×C₅	—	2	4	no	—
211	C₂²×C₅	C₂×#124	1	4	yes	—
212	C₂²×C₅	C₂×#124	1	2	yes	—
213	C₂²×C₅	C₂×#137	1	4	yes	—
214	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
215	C₄×C₅	—	2	4	no	—
216	C₄×C₅	—	4	8	no	—
217	C₄×C₅	—	4	8	no	—
218	C₄×C₅	C₂×#124	1	4	yes	—
219	C₄×C₅	C₂×#124	1	2	yes	—
220	C₄×C₅	C₂×#137	1	8	yes	—
221	C₄×C₅	C₂²×#41	1	2	yes	—
222	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
223	C₂×Q₈	—	2	8	no	—
224	C₂×Q₈	—	2	8	no	—
225	C₂×Q₈	—	2	4	no	—
226	C₂×Q₈	—	2	16	no	—
227	C₂×Q₈	—	2	32	no	—
228	C₂×Q₈	—	4	16	no	—
229	C₂×Q₈	—	4	8	no	—
230	C₂×Q₈	—	4	8	no	—
231	C₂×Q₈	C₂⁴×C₅	1	8	yes	—
232	C₂²×C₄	—	2	8	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	8	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	4	8	no	—
237	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
238	C₂²×C₄	C₂×C₅×D₄	1	8	yes	—
239	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
241	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
242	C₂²×C₄	C₅×#41	1	8	yes	—
243	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
244	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
245	M₁₆	—	2	16	no	—
246	C₂×C₈	—	2	8	no	—
247	C₂×C₈	C₂⁴×C₅	1	8	yes	—
248	C₄⋊C₄	—	2	16	no	—
249	C₄⋊C₄	—	2	16	no	—
250	C₄⋊C₄	—	2	16	no	—
251	C₄⋊C₄	—	2	8	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	C₂⁴×C₅	1	8	yes	—
254	C₂²⋊C₄	—	2	8	no	—
255	C₄²	—	2	8	no	—
256	C₄²	C₂×C₅×D₄	1	8	yes	—
257	C₄²	C₂×C₅×D₄	1	2	yes	—
258	C₄²	C₂×C₅×D₄	1	2	yes	—
259	C₄²	C₂⁴×C₅	1	2	yes	—
260	C₄²	C₂⁴×C₅	1	2	yes	—
261	C₂×C₅	—	2	4	no	—
262	C₂×C₅	#3139	1	4	yes	direct product factor
263	C₂×C₅	C₂²×#124	1	1	yes	characteristic
264	C₂×C₅	C₂²×#137	1	2	yes	—
265	C₂³	—	2	4	no	—
266	C₂³	C₅×#124	1	4	yes	—
267	C₂³	C₂²×C₅×D₄	1	2	yes	—
268	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
269	Q₈	—	2	16	no	—
270	Q₈	—	4	16	no	—
271	Q₈	—	4	8	no	—
272	Q₈	—	4	16	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	16	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	4	16	no	—
278	C₂×C₄	—	4	8	no	—
279	C₂×C₄	—	4	8	no	—
280	C₂×C₄	C₅×#124	1	8	yes	—
281	C₂×C₄	C₅×#124	1	4	yes	—
282	C₂×C₄	C₅×#137	1	8	yes	—
283	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
284	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
285	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
286	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
287	C₂×C₄	C₂×C₅×#41	1	4	yes	—
288	C₂×C₄	C₂×C₅×#41	1	8	yes	—
289	C₂×C₄	C₂⁵×C₅	1	1	yes	characteristic
290	C₈	—	2	8	no	—
291	C₅	C₂×#3139	1	1	yes	characteristic, direct product factor, Sylow
292	C₂²	—	2	8	no	—
293	C₂²	—	2	4	no	—
294	C₂²	C₂×C₅×#124	1	4	yes	—
295	C₂²	C₂×C₅×#124	1	2	yes	—
296	C₂²	C₂×C₅×#137	1	4	yes	—
297	C₂²	C₂³×C₅×D₄	1	1	yes	characteristic
298	C₄	—	2	4	no	—
299	C₄	—	4	8	no	—
300	C₄	—	4	8	no	—
301	C₄	C₂×C₅×#124	1	4	yes	—
302	C₄	C₂×C₅×#124	1	2	yes	—
303	C₄	C₂×C₅×#137	1	8	yes	—
304	C₄	C₂²×C₅×#41	1	2	yes	—
305	C₂	—	2	4	no	—
306	C₂	C₅×#3139	1	4	yes	direct product factor
307	C₂	C₂²×C₅×#124	1	1	yes	characteristic
308	C₂	C₂²×C₅×#137	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅
prime radical	C₂⁴×C₅	C₂×D₄
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
3. C₅×#3139	C₂	direct (N × H)
4. C₂×C₅×#491	C₂	semidirect
5. C₂×C₅×#487	C₂	semidirect
24. C₅×#491	C₂²	semidirect
25. C₅×#487	C₂²	semidirect
31. C₂×#3139	C₅	direct (N × H)
75. #3139	C₂×C₅	direct (N × H)
76. C₂×#491	C₂×C₅	semidirect
77. C₂×#487	C₂×C₅	semidirect
135. #491	C₂²×C₅	semidirect
136. #487	C₂²×C₅	semidirect
262. C₂×C₅	#3139	direct (N × H)
291. C₅	C₂×#3139	direct (N × H)
306. C₂	C₅×#3139	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×C₄⋊Q₈	2
C₅×#3139	2
C₂×C₅×#491	2
C₂×C₅×#487	2
C₂×C₅×#419	2
C₂×C₄×C₅×M₁₆	2
C₂×#3139	5

Subgroups — C2×C5×#3139 #11983334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#3139 #11983334