Subgroups — C₅×#5546 #11755931

Order 960.

588 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#237	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×D₄×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₅×#67	C₂	1	2	yes	maximal, complemented
5	C₂×C₅×Dic₃⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₅×D₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#887	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#67⋊C₄	C₂	1	2	yes	maximal, complemented
9	C₅×#877	C₂	1	2	yes	maximal, complemented
10	C₅×#872	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#520	—	3	1	no	maximal, Hall
13	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₅	C₂²	1	2	yes	—
15	C₂×C₅×#67	C₂²	1	2	yes	—
16	C₂²×C₅×Dic₃	C₂²	1	2	yes	—
17	C₂²×C₅×Dic₃	C₂²	1	2	yes	—
18	C₅×#237	C₄	1	8	yes	complemented
19	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
20	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
21	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
22	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
25	C₅×#217	C₂²	1	2	yes	—
26	C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
27	C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
28	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
29	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
30	C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
31	C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
32	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
33	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
34	C₄×C₅×Dic₃	C₂²	1	2	yes	—
35	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
36	C₄×C₅×Dic₃	C₂²	1	2	yes	—
37	#5546	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
38	C₂×C₅×#41	—	3	1	no	—
39	C₄×C₅×Q₈	—	3	1	no	—
40	C₄×C₅×Q₈	—	3	1	no	—
41	C₄×C₅×D₄	—	3	1	no	—
42	C₄×C₅×D₄	—	3	2	no	—
43	C₄×C₅×D₄	—	3	2	no	—
44	C₄×C₅×D₄	D₃	1	1	yes	characteristic, complemented
45	C₅×#117	—	3	2	no	—
46	C₅×#117	—	3	1	no	—
47	C₂×C₅×C₄⋊C₄	—	3	2	no	—
48	C₂×C₅×C₄⋊C₄	—	3	1	no	—
49	C₂³×C₃×C₅	C₂³	1	2	yes	—
50	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
51	C₃×C₅×D₄	C₂×C₄	1	4	yes	complemented
52	C₂×C₃×C₄×C₅	—	4	2	no	—
53	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
55	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
57	C₅×#67	C₂×C₄	1	8	yes	complemented
58	C₂×C₅×Dic₃	—	4	2	no	—
59	C₂×C₅×Dic₃	C₂×C₄	1	8	yes	complemented
60	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
61	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
62	C₂×C₅×Dic₃	C₂³	1	2	yes	—
63	C₂×C₅×Dic₃	C₂³	1	2	yes	—
64	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
65	C₄×C₅×D₃	—	4	2	no	—
66	C₄×C₅×D₃	C₂×C₄	1	4	yes	complemented
67	C₅×Dic₆	C₂×C₄	1	4	yes	complemented
68	C₂×#237	C₂×C₅	1	1	yes	characteristic
69	C₃×C₄×D₄	C₂×C₅	1	1	yes	characteristic, complemented
70	D₄×Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
71	C₄×#67	C₂×C₅	1	2	yes	complemented
72	C₂×Dic₃⋊C₄	C₂×C₅	1	2	yes	complemented
73	D₃×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
74	#887	C₂×C₅	1	1	yes	characteristic, complemented
75	#67⋊C₄	C₂×C₅	1	2	yes	complemented
76	#877	C₂×C₅	1	2	yes	complemented
77	#872	C₂×C₅	1	1	yes	characteristic, complemented
78	C₄×Dic₆	C₂×C₅	1	1	yes	characteristic, complemented
79	C₅×#41	—	3	8	no	—
80	C₂×C₅×Q₈	—	3	1	no	—
81	C₂×C₅×D₄	—	3	2	no	—
82	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic
83	C₂²×C₄×C₅	—	3	2	no	—
84	C₂²×C₄×C₅	—	3	1	no	—
85	C₂²×C₄×C₅	—	3	2	no	—
86	C₂²×C₄×C₅	—	3	2	no	—
87	C₂²×C₄×C₅	C₂×D₃	1	2	yes	—
88	C₅×C₄⋊C₄	—	3	2	no	—
89	C₅×C₄⋊C₄	—	3	1	no	—
90	C₅×C₄⋊C₄	—	3	2	no	—
91	C₅×C₄⋊C₄	—	3	2	no	—
92	C₅×C₄⋊C₄	—	3	1	no	—
93	C₅×C₄⋊C₄	—	3	1	no	—
94	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
95	C₅×C₂²⋊C₄	—	3	2	no	—
96	C₅×C₂²⋊C₄	—	3	1	no	—
97	C₅×C₂²⋊C₄	—	3	1	no	—
98	C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
99	C₄²×C₅	—	3	2	no	—
100	C₄²×C₅	—	3	2	no	—
101	C₄²×C₅	—	3	1	no	—
102	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
103	#520	—	3	1	no	Sylow
104	C₂²×C₃×C₅	—	2	2	no	—
105	C₂²×C₃×C₅	—	2	2	no	—
106	C₂²×C₃×C₅	C₂²×C₄	1	4	yes	—
107	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
108	C₂×C₅×D₃	—	2	1	no	—
109	C₂×C₅×D₃	—	2	1	no	—
110	C₂×C₅×D₃	C₂²×C₄	1	2	yes	—
111	C₃×C₄×C₅	—	2	2	no	—
112	C₃×C₄×C₅	—	2	1	no	—
113	C₃×C₄×C₅	—	2	1	no	—
114	C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
115	C₅×Dic₃	—	2	2	no	—
116	C₅×Dic₃	—	2	1	no	—
117	C₅×Dic₃	—	2	1	no	—
118	C₅×Dic₃	C₂²×C₄	1	4	yes	—
119	C₅×Dic₃	C₂²×C₄	1	2	yes	—
120	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
121	C₂²×C₃×C₄	C₂²×C₅	1	2	yes	—
122	C₂×#67	C₂²×C₅	1	2	yes	—
123	C₂²×Dic₃	C₂²×C₅	1	2	yes	—
124	C₂²×Dic₃	C₂²×C₅	1	2	yes	—
125	#237	C₄×C₅	1	8	yes	complemented
126	C₂×C₄×D₃	C₂²×C₅	1	2	yes	—
127	C₂×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
128	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
129	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
130	C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
131	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
132	#217	C₂²×C₅	1	2	yes	—
133	(C₂×D₃)⋊C₄	C₂²×C₅	1	1	yes	characteristic
134	(C₂×D₃)⋊C₄	C₂²×C₅	1	1	yes	characteristic
135	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
136	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
137	Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
138	Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
139	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
140	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
141	C₄×Dic₃	C₂²×C₅	1	2	yes	—
142	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
143	C₄×Dic₃	C₂²×C₅	1	2	yes	—
144	C₂³×C₅	—	3	1	no	—
145	C₂³×C₅	C₂²×D₃	1	2	yes	—
146	C₅×Q₈	—	3	4	no	—
147	C₅×D₄	—	3	8	no	—
148	C₅×D₄	C₄×D₃	1	4	yes	complemented
149	C₂×C₄×C₅	—	3	8	no	—
150	C₂×C₄×C₅	—	3	2	no	—
151	C₂×C₄×C₅	—	3	1	no	—
152	C₂×C₄×C₅	—	3	4	no	—
153	C₂×C₄×C₅	—	3	1	no	—
154	C₂×C₄×C₅	—	3	2	no	—
155	C₂×C₄×C₅	—	3	1	no	—
156	C₂×C₄×C₅	—	4	2	no	—
157	C₂×C₄×C₅	—	12	2	no	—
158	C₂×C₄×C₅	—	12	2	no	—
159	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
161	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
163	C₂×#41	—	3	1	no	—
164	C₄×Q₈	—	3	1	no	—
165	C₄×Q₈	—	3	1	no	—
166	C₄×D₄	—	3	1	no	—
167	C₄×D₄	—	3	2	no	—
168	C₄×D₄	—	3	2	no	—
169	C₄×D₄	C₅×D₃	1	1	yes	characteristic, complemented
170	#117	—	3	2	no	—
171	#117	—	3	1	no	—
172	C₂×C₄⋊C₄	—	3	2	no	—
173	C₂×C₄⋊C₄	—	3	1	no	—
174	C₂×C₃×C₅	—	2	4	no	—
175	C₂×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
176	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
177	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
178	C₅×D₃	—	2	2	no	—
179	C₂³×C₃	C₂³×C₅	1	2	yes	—
180	C₂²×D₃	C₂³×C₅	1	1	yes	characteristic
181	C₃×D₄	C₂×C₄×C₅	1	4	yes	complemented
182	C₂×C₃×C₄	—	4	2	no	—
183	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
185	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
187	#67	C₂×C₄×C₅	1	8	yes	complemented
188	C₂×Dic₃	—	4	2	no	—
189	C₂×Dic₃	C₂×C₄×C₅	1	8	yes	complemented
190	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
191	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
192	C₂×Dic₃	C₂³×C₅	1	2	yes	—
193	C₂×Dic₃	C₂³×C₅	1	2	yes	—
194	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
195	C₄×D₃	—	4	2	no	—
196	C₄×D₃	C₂×C₄×C₅	1	4	yes	complemented
197	Dic₆	C₂×C₄×C₅	1	4	yes	complemented
198	C₂²×C₅	—	2	2	no	—
199	C₂²×C₅	—	2	2	no	—
200	C₂²×C₅	—	3	2	no	—
201	C₂²×C₅	—	6	1	no	—
202	C₂²×C₅	—	6	1	no	—
203	C₂²×C₅	C₂×C₄×D₃	1	4	yes	—
204	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
205	C₄×C₅	—	2	1	no	—
206	C₄×C₅	—	2	2	no	—
207	C₄×C₅	—	2	1	no	—
208	C₄×C₅	—	3	2	no	—
209	C₄×C₅	—	3	4	no	—
210	C₄×C₅	—	6	2	no	—
211	C₄×C₅	—	6	1	no	—
212	C₄×C₅	—	6	1	no	—
213	C₄×C₅	C₂×C₄×D₃	1	2	yes	—
214	#41	—	3	8	no	—
215	C₂×Q₈	—	3	1	no	—
216	C₂×D₄	—	3	2	no	—
217	C₂×D₄	C₂×C₅×D₃	1	1	yes	characteristic
218	C₂²×C₄	—	3	2	no	—
219	C₂²×C₄	—	3	1	no	—
220	C₂²×C₄	—	3	2	no	—
221	C₂²×C₄	—	3	2	no	—
222	C₂²×C₄	C₂×C₅×D₃	1	2	yes	—
223	C₄⋊C₄	—	3	2	no	—
224	C₄⋊C₄	—	3	1	no	—
225	C₄⋊C₄	—	3	2	no	—
226	C₄⋊C₄	—	3	2	no	—
227	C₄⋊C₄	—	3	1	no	—
228	C₄⋊C₄	—	3	1	no	—
229	C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
230	C₂²⋊C₄	—	3	2	no	—
231	C₂²⋊C₄	—	3	1	no	—
232	C₂²⋊C₄	—	3	1	no	—
233	C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
234	C₄²	—	3	2	no	—
235	C₄²	—	3	2	no	—
236	C₄²	—	3	1	no	—
237	C₄²	C₂×C₅×D₃	1	1	yes	characteristic
238	C₃×C₅	#520	1	1	yes	characteristic, complemented, Hall
239	C₂²×C₃	—	2	2	no	—
240	C₂²×C₃	—	2	2	no	—
241	C₂²×C₃	C₂²×C₄×C₅	1	4	yes	—
242	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
243	C₂×D₃	—	2	1	no	—
244	C₂×D₃	—	2	1	no	—
245	C₂×D₃	C₂²×C₄×C₅	1	2	yes	—
246	C₃×C₄	—	2	2	no	—
247	C₃×C₄	—	2	1	no	—
248	C₃×C₄	—	2	1	no	—
249	C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
250	Dic₃	—	2	2	no	—
251	Dic₃	—	2	1	no	—
252	Dic₃	—	2	1	no	—
253	Dic₃	C₂²×C₄×C₅	1	4	yes	—
254	Dic₃	C₂²×C₄×C₅	1	2	yes	—
255	C₂×C₅	—	2	4	no	—
256	C₂×C₅	—	6	2	no	—
257	C₂×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
258	C₂×C₅	#1004	1	1	yes	characteristic
259	C₂×C₅	#1010	1	1	yes	characteristic
260	C₂³	—	3	1	no	—
261	C₂³	C₂²×C₅×D₃	1	2	yes	—
262	Q₈	—	3	4	no	—
263	D₄	—	3	8	no	—
264	D₄	C₄×C₅×D₃	1	4	yes	complemented
265	C₂×C₄	—	3	8	no	—
266	C₂×C₄	—	3	2	no	—
267	C₂×C₄	—	3	1	no	—
268	C₂×C₄	—	3	4	no	—
269	C₂×C₄	—	3	1	no	—
270	C₂×C₄	—	3	2	no	—
271	C₂×C₄	—	3	1	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	12	2	no	—
274	C₂×C₄	—	12	2	no	—
275	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
276	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
277	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
278	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
279	C₂×C₃	—	2	4	no	—
280	C₂×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
281	C₂×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
282	C₂×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
283	D₃	—	2	2	no	—
284	C₅	#5546	1	1	yes	characteristic, direct product factor, Sylow
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	3	2	no	—
288	C₂²	—	6	1	no	—
289	C₂²	—	6	1	no	—
290	C₂²	C₂×C₄×C₅×D₃	1	4	yes	—
291	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	1	no	—
295	C₄	—	3	2	no	—
296	C₄	—	3	4	no	—
297	C₄	—	6	2	no	—
298	C₄	—	6	1	no	—
299	C₄	—	6	1	no	—
300	C₄	C₂×C₄×C₅×D₃	1	2	yes	—
301	C₃	C₅×#520	1	1	yes	characteristic, complemented, Sylow
302	C₂	—	2	4	no	—
303	C₂	—	6	2	no	—
304	C₂	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
305	C₂	C₅×#1004	1	1	yes	characteristic
306	C₂	C₅×#1010	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂³×D₃
commutator	C₂×C₃	C₂³×C₄×C₅
Frattini	C₂²	C₂³×C₅×D₃
Fitting	C₃×C₄×C₅×D₄	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂×C₅×#237	C₂
cosocle	C₂²×C₃	C₂⁴×C₅
hypercenter	C₄×C₅×D₄	D₃
hypocenter	C₃	C₅×#520
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄×C₅×D₄

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₅×D₄	C₂	semidirect
3. C₅×D₄×Dic₃	C₂	semidirect
4. C₄×C₅×#67	C₂	semidirect
5. C₂×C₅×Dic₃⋊C₄	C₂	semidirect
6. C₅×D₃×C₄⋊C₄	C₂	semidirect
7. C₅×#887	C₂	semidirect
8. C₅×#67⋊C₄	C₂	semidirect
9. C₅×#877	C₂	semidirect
10. C₅×#872	C₂	semidirect
11. C₄×C₅×Dic₆	C₂	semidirect
18. C₅×#237	C₄	semidirect
37. #5546	C₅	direct (N × H)
44. C₄×C₅×D₄	D₃	semidirect
51. C₃×C₅×D₄	C₂×C₄	semidirect
57. C₅×#67	C₂×C₄	semidirect
59. C₂×C₅×Dic₃	C₂×C₄	semidirect
66. C₄×C₅×D₃	C₂×C₄	semidirect
67. C₅×Dic₆	C₂×C₄	semidirect
69. C₃×C₄×D₄	C₂×C₅	semidirect
70. D₄×Dic₃	C₂×C₅	semidirect
71. C₄×#67	C₂×C₅	semidirect
72. C₂×Dic₃⋊C₄	C₂×C₅	semidirect
73. D₃×C₄⋊C₄	C₂×C₅	semidirect
74. #887	C₂×C₅	semidirect
75. #67⋊C₄	C₂×C₅	semidirect
76. #877	C₂×C₅	semidirect
77. #872	C₂×C₅	semidirect
78. C₄×Dic₆	C₂×C₅	semidirect
125. #237	C₄×C₅	semidirect
148. C₅×D₄	C₄×D₃	semidirect
169. C₄×D₄	C₅×D₃	semidirect
181. C₃×D₄	C₂×C₄×C₅	semidirect
187. #67	C₂×C₄×C₅	semidirect
189. C₂×Dic₃	C₂×C₄×C₅	semidirect
196. C₄×D₃	C₂×C₄×C₅	semidirect
197. Dic₆	C₂×C₄×C₅	semidirect
238. C₃×C₅	#520	semidirect
264. D₄	C₄×C₅×D₃	semidirect
284. C₅	#5546	direct (N × H)
301. C₃	C₅×#520	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#237	2
C₃×C₄×C₅×D₄	2
C₅×D₄×Dic₃	2
C₄×C₅×#67	2
C₂×C₅×Dic₃⋊C₄	2
C₅×D₃×C₄⋊C₄	2
C₅×#887	2
C₅×#67⋊C₄	2
C₅×#877	2
C₅×#872	2
C₄×C₅×Dic₆	2
C₅×#520	3
#5546	5

Subgroups — C5×#5546 #11755931

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5546 #11755931