Subgroups — C₂×C₅×#2185 #11956232

Order 1280.

1178 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₅×SD₃₂	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×#466	C₂	1	1	yes	maximal, characteristic
4	C₅×#2185	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×#358	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₅×Q₈	—	2	1	no	—
8	C₂²×C₅×Q₁₆	—	2	8	no	—
9	C₂²×C₅×Q₁₆	—	2	1	no	—
10	C₂²×C₅×Q₁₆	—	2	2	no	—
11	C₂²×C₅×Q₁₆	C₂²	1	2	yes	—
12	C₂²×C₅×Q₁₆	C₂²	1	4	yes	complemented
13	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
14	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₅×#121	—	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	—
19	C₅×#466	—	2	2	no	—
20	C₅×#466	C₂²	1	4	yes	—
21	C₂×C₅×#107	C₂²	1	1	yes	characteristic
22	C₂×C₅×#102	—	2	1	no	—
23	C₅×#358	C₂²	1	8	yes	complemented
24	C₅×C₂²⋊C₁₆	C₂²	1	4	yes	complemented
25	C₂×#2185	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂²×C₅×Q₈	—	2	8	no	—
27	C₂²×C₅×Q₈	—	2	2	no	—
28	C₂²×C₅×Q₈	—	2	4	no	—
29	C₂²×C₅×Q₈	—	4	4	no	—
30	C₂²×C₅×Q₈	—	4	2	no	—
31	C₂²×C₅×Q₈	—	4	1	no	—
32	C₂²×C₅×D₄	—	2	1	no	—
33	C₂²×C₅×D₄	—	4	1	no	—
34	C₂³×C₄×C₅	—	4	1	no	—
35	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
36	C₂×C₅×Q₁₆	—	2	8	no	—
37	C₂×C₅×Q₁₆	—	2	16	no	—
38	C₂×C₅×Q₁₆	—	2	16	no	—
39	C₂×C₅×Q₁₆	—	2	2	no	—
40	C₂×C₅×Q₁₆	—	2	4	no	—
41	C₂×C₅×Q₁₆	—	2	4	no	—
42	C₂×C₅×Q₁₆	D₄	1	8	yes	complemented
43	C₂×C₅×Q₁₆	C₂³	1	4	yes	—
44	C₂×C₅×D₈	—	2	4	no	—
45	C₂×C₅×D₈	—	2	1	no	—
46	C₂×C₅×D₈	C₂³	1	2	yes	—
47	C₂²×C₅×C₈	—	2	4	no	—
48	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
49	C₂²×C₅×C₈	D₄	1	2	yes	—
50	C₂²×C₅×C₈	C₂³	1	2	yes	—
51	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
52	C₅×#121	—	2	4	no	—
53	C₅×#121	—	4	2	no	—
54	C₂×C₅×C₄⋊C₄	—	2	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	—
59	C₅×#107	—	2	1	no	—
60	C₅×#107	C₂³	1	2	yes	—
61	C₅×#102	—	2	4	no	—
62	C₂³×Q₁₆	C₂×C₅	1	1	yes	characteristic, complemented
63	C₂²×SD₃₂	C₂×C₅	1	2	yes	complemented
64	C₂×#466	C₂×C₅	1	1	yes	characteristic
65	#2185	C₂×C₅	1	8	yes	direct product factor
66	C₂×#358	C₂×C₅	1	2	yes	complemented
67	C₂×C₂²⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
68	C₂⁴×C₅	—	4	1	no	—
69	C₂⁴×C₅	D₈	1	1	yes	characteristic
70	C₂×C₅×Q₈	—	2	16	no	—
71	C₂×C₅×Q₈	—	2	8	no	—
72	C₂×C₅×Q₈	—	2	4	no	—
73	C₂×C₅×Q₈	—	4	8	no	—
74	C₂×C₅×Q₈	—	4	8	no	—
75	C₂×C₅×Q₈	—	4	16	no	—
76	C₂×C₅×Q₈	—	4	4	no	—
77	C₂×C₅×Q₈	—	4	2	no	—
78	C₂×C₅×Q₈	—	4	4	no	—
79	C₂×C₅×D₄	—	2	2	no	—
80	C₂×C₅×D₄	—	4	4	no	—
81	C₂×C₅×D₄	—	4	2	no	—
82	C₂×C₅×D₄	—	4	1	no	—
83	C₂×C₅×D₄	—	4	2	no	—
84	C₂×C₅×D₄	—	8	4	no	—
85	C₂²×C₄×C₅	—	2	2	no	—
86	C₂²×C₄×C₅	—	2	2	no	—
87	C₂²×C₄×C₅	—	4	8	no	—
88	C₂²×C₄×C₅	—	4	4	no	—
89	C₂²×C₄×C₅	—	4	2	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	16	no	—
96	C₅×Q₁₆	—	2	8	no	—
97	C₅×Q₁₆	—	2	4	no	—
98	C₅×Q₁₆	C₂×D₄	1	8	yes	complemented
99	C₅×D₈	—	2	4	no	—
100	C₅×D₈	—	4	2	no	—
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×C₂²⋊C₄	—	4	4	no	—
111	C₅×C₁₆	—	2	4	no	—
112	C₂³×Q₈	—	2	1	no	—
113	C₂²×Q₁₆	—	2	8	no	—
114	C₂²×Q₁₆	—	2	1	no	—
115	C₂²×Q₁₆	—	2	2	no	—
116	C₂²×Q₁₆	C₂²×C₅	1	2	yes	—
117	C₂²×Q₁₆	C₂²×C₅	1	4	yes	complemented
118	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
119	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
120	C₂×#121	—	2	1	no	—
121	C₂×SD₃₂	—	2	8	no	—
122	C₂×SD₃₂	C₂²×C₅	1	8	yes	complemented
123	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
124	#466	—	2	2	no	—
125	#466	C₂²×C₅	1	4	yes	—
126	C₂×#107	C₂²×C₅	1	1	yes	characteristic
127	C₂×#102	—	2	1	no	—
128	#358	C₂²×C₅	1	8	yes	complemented
129	C₂²⋊C₁₆	C₂²×C₅	1	4	yes	complemented
130	C₂³×C₅	—	2	2	no	—
131	C₂³×C₅	—	2	2	no	—
132	C₂³×C₅	—	4	2	no	—
133	C₂³×C₅	—	8	2	no	—
134	C₂³×C₅	—	8	2	no	—
135	C₂³×C₅	—	8	2	no	—
136	C₂³×C₅	SD₃₂	1	4	yes	complemented
137	C₂³×C₅	C₂×D₈	1	2	yes	—
138	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
139	C₅×Q₈	—	2	8	no	—
140	C₅×Q₈	—	4	16	no	—
141	C₅×Q₈	—	4	4	no	—
142	C₅×Q₈	—	4	8	no	—
143	C₅×D₄	—	4	2	no	—
144	C₅×D₄	—	4	2	no	—
145	C₅×D₄	—	8	8	no	—
146	C₅×D₄	—	8	2	no	—
147	C₂×C₄×C₅	—	2	2	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	—	2	4	no	—
151	C₂×C₄×C₅	—	4	16	no	—
152	C₂×C₄×C₅	—	4	4	no	—
153	C₂×C₄×C₅	—	4	8	no	—
154	C₂×C₄×C₅	—	4	2	no	—
155	C₂×C₄×C₅	—	8	2	no	—
156	C₂×C₄×C₅	#120	1	2	yes	—
157	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
158	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
160	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
161	C₅×C₈	—	2	2	no	—
162	C₅×C₈	#120	1	2	yes	—
163	C₅×C₈	C₂²×D₄	1	2	yes	—
164	C₂²×Q₈	—	2	8	no	—
165	C₂²×Q₈	—	2	2	no	—
166	C₂²×Q₈	—	2	4	no	—
167	C₂²×Q₈	—	4	4	no	—
168	C₂²×Q₈	—	4	1	no	—
169	C₂²×Q₈	—	4	2	no	—
170	C₂²×D₄	—	2	1	no	—
171	C₂²×D₄	—	4	1	no	—
172	C₂³×C₄	—	4	1	no	—
173	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
174	C₂×Q₁₆	—	2	8	no	—
175	C₂×Q₁₆	—	2	16	no	—
176	C₂×Q₁₆	—	2	16	no	—
177	C₂×Q₁₆	—	2	2	no	—
178	C₂×Q₁₆	—	2	4	no	—
179	C₂×Q₁₆	—	2	4	no	—
180	C₂×Q₁₆	C₅×D₄	1	8	yes	complemented
181	C₂×Q₁₆	C₂³×C₅	1	4	yes	—
182	C₂×D₈	—	2	4	no	—
183	C₂×D₈	—	2	1	no	—
184	C₂×D₈	C₂³×C₅	1	2	yes	—
185	C₂²×C₈	—	2	4	no	—
186	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
187	C₂²×C₈	C₅×D₄	1	2	yes	—
188	C₂²×C₈	C₂³×C₅	1	2	yes	—
189	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
190	#121	—	2	4	no	—
191	#121	—	4	2	no	—
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₂²⋊C₄	—	4	1	no	—
194	SD₃₂	—	2	16	no	—
195	C₂×C₁₆	—	2	4	no	—
196	C₂×C₁₆	C₂³×C₅	1	4	yes	—
197	#107	—	2	1	no	—
198	#107	C₂³×C₅	1	2	yes	—
199	#102	—	2	4	no	—
200	C₂²×C₅	—	2	2	no	—
201	C₂²×C₅	—	2	8	no	—
202	C₂²×C₅	—	2	2	no	—
203	C₂²×C₅	—	8	2	no	—
204	C₂²×C₅	—	8	2	no	—
205	C₂²×C₅	—	8	2	no	—
206	C₂²×C₅	—	16	4	no	—
207	C₂²×C₅	#447	1	2	yes	—
208	C₂²×C₅	C₂×SD₃₂	1	2	yes	—
209	C₂²×C₅	C₂×SD₃₂	1	4	yes	complemented
210	C₂²×C₅	#510	1	2	yes	—
211	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
212	C₄×C₅	—	2	2	no	—
213	C₄×C₅	—	4	8	no	—
214	C₄×C₅	—	8	2	no	—
215	C₄×C₅	#447	1	2	yes	—
216	C₄×C₅	C₂×#120	1	1	yes	characteristic
217	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
218	C₂⁴	—	4	1	no	—
219	C₂⁴	C₅×D₈	1	1	yes	characteristic
220	C₂×Q₈	—	2	16	no	—
221	C₂×Q₈	—	2	8	no	—
222	C₂×Q₈	—	2	4	no	—
223	C₂×Q₈	—	4	8	no	—
224	C₂×Q₈	—	4	16	no	—
225	C₂×Q₈	—	4	8	no	—
226	C₂×Q₈	—	4	4	no	—
227	C₂×Q₈	—	4	2	no	—
228	C₂×Q₈	—	4	4	no	—
229	C₂×D₄	—	2	2	no	—
230	C₂×D₄	—	4	4	no	—
231	C₂×D₄	—	4	2	no	—
232	C₂×D₄	—	4	1	no	—
233	C₂×D₄	—	4	2	no	—
234	C₂×D₄	—	8	4	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	4	8	no	—
238	C₂²×C₄	—	4	4	no	—
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
242	C₂²×C₄	C₅×D₈	1	2	yes	—
243	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
244	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
245	Q₁₆	—	2	16	no	—
246	Q₁₆	—	2	8	no	—
247	Q₁₆	—	2	4	no	—
248	Q₁₆	C₂×C₅×D₄	1	8	yes	complemented
249	D₈	—	2	4	no	—
250	D₈	—	4	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
255	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
256	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
257	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₂²⋊C₄	—	4	4	no	—
261	C₁₆	—	2	4	no	—
262	C₂×C₅	—	2	4	no	—
263	C₂×C₅	—	16	2	no	—
264	C₂×C₅	#2185	1	4	yes	direct product factor
265	C₂×C₅	C₂×#447	1	1	yes	characteristic
266	C₂×C₅	C₂²×SD₃₂	1	1	yes	characteristic
267	C₂×C₅	C₂×#510	1	1	yes	characteristic
268	C₂³	—	2	2	no	—
269	C₂³	—	2	2	no	—
270	C₂³	—	4	2	no	—
271	C₂³	—	8	2	no	—
272	C₂³	—	8	2	no	—
273	C₂³	—	8	2	no	—
274	C₂³	C₅×SD₃₂	1	4	yes	complemented
275	C₂³	C₂×C₅×D₈	1	2	yes	—
276	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
277	Q₈	—	2	8	no	—
278	Q₈	—	4	16	no	—
279	Q₈	—	4	8	no	—
280	Q₈	—	4	4	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	8	8	no	—
284	D₄	—	8	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	4	16	no	—
290	C₂×C₄	—	4	8	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	8	2	no	—
294	C₂×C₄	C₅×#120	1	2	yes	—
295	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
296	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
297	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
298	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
299	C₈	—	2	2	no	—
300	C₈	C₅×#120	1	2	yes	—
301	C₈	C₂²×C₅×D₄	1	2	yes	—
302	C₅	C₂×#2185	1	1	yes	characteristic, direct product factor, Sylow
303	C₂²	—	2	2	no	—
304	C₂²	—	2	8	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	—	8	2	no	—
308	C₂²	—	8	2	no	—
309	C₂²	—	16	4	no	—
310	C₂²	C₅×#447	1	2	yes	—
311	C₂²	C₂×C₅×SD₃₂	1	2	yes	—
312	C₂²	C₂×C₅×SD₃₂	1	4	yes	complemented
313	C₂²	C₅×#510	1	2	yes	—
314	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
315	C₄	—	2	2	no	—
316	C₄	—	4	8	no	—
317	C₄	—	8	2	no	—
318	C₄	C₅×#447	1	2	yes	—
319	C₄	C₂×C₅×#120	1	1	yes	characteristic
320	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
321	C₂	—	2	4	no	—
322	C₂	—	16	2	no	—
323	C₂	C₅×#2185	1	4	yes	direct product factor
324	C₂	C₂×C₅×#447	1	1	yes	characteristic
325	C₂	C₂²×C₅×SD₃₂	1	1	yes	characteristic
326	C₂	C₂×C₅×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂×C₅×#466	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
1. C₂³×C₅×Q₁₆	C₂	semidirect
2. C₂²×C₅×SD₃₂	C₂	semidirect
4. C₅×#2185	C₂	direct (N × H)
5. C₂×C₅×#358	C₂	semidirect
6. C₂×C₅×C₂²⋊C₁₆	C₂	semidirect
12. C₂²×C₅×Q₁₆	C₂²	semidirect
17. C₂×C₅×SD₃₂	C₂²	semidirect
23. C₅×#358	C₂²	semidirect
24. C₅×C₂²⋊C₁₆	C₂²	semidirect
25. C₂×#2185	C₅	direct (N × H)
42. C₂×C₅×Q₁₆	D₄	semidirect
62. C₂³×Q₁₆	C₂×C₅	semidirect
63. C₂²×SD₃₂	C₂×C₅	semidirect
65. #2185	C₂×C₅	direct (N × H)
66. C₂×#358	C₂×C₅	semidirect
67. C₂×C₂²⋊C₁₆	C₂×C₅	semidirect
98. C₅×Q₁₆	C₂×D₄	semidirect
117. C₂²×Q₁₆	C₂²×C₅	semidirect
122. C₂×SD₃₂	C₂²×C₅	semidirect
128. #358	C₂²×C₅	semidirect
129. C₂²⋊C₁₆	C₂²×C₅	semidirect
136. C₂³×C₅	SD₃₂	semidirect
180. C₂×Q₁₆	C₅×D₄	semidirect
209. C₂²×C₅	C₂×SD₃₂	semidirect
248. Q₁₆	C₂×C₅×D₄	semidirect
264. C₂×C₅	#2185	direct (N × H)
274. C₂³	C₅×SD₃₂	semidirect
302. C₅	C₂×#2185	direct (N × H)
312. C₂²	C₂×C₅×SD₃₂	semidirect
323. C₂	C₅×#2185	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×Q₁₆	2
C₂²×C₅×SD₃₂	2
C₂×C₅×#466	2
C₅×#2185	2
C₂×C₅×#358	2
C₂×C₅×C₂²⋊C₁₆	2
C₂×#2185	5

Subgroups — C2×C5×#2185 #11956232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2185 #11956232