Subgroups — C₅×#22307 #11945027

Order 1280.

714 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#1984	C₂	1	4	yes	maximal
3	C₅×#1939	C₂	1	2	yes	maximal
4	C₅×#1916	C₂	1	4	yes	maximal
5	C₅×#1831	C₂	1	1	yes	maximal, characteristic
6	C₅×C₈×C₄⋊C₄	C₂	1	2	yes	maximal
7	C₅×#1744	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
9	C₂×C₄×C₅×Q₈	C₂²	1	2	yes	—
10	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×C₈	C₂²	1	4	yes	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₅×#391	C₄	1	2	yes	—
16	C₅×#389	C₄	1	2	yes	—
17	C₅×#389	C₄	1	2	yes	—
18	C₅×#384	C₄	1	1	yes	characteristic
19	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
20	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
21	C₄³×C₅	C₂²	1	1	yes	characteristic
22	C₅×#336	C₂²	1	4	yes	—
23	C₅×#336	C₂²	1	2	yes	—
24	C₅×#336	C₂²	1	2	yes	—
25	C₅×#336	C₂²	1	2	yes	—
26	C₅×#336	C₂²	1	4	yes	—
27	C₅×#336	C₂²	1	2	yes	—
28	#22307	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₂²×C₅×Q₈	C₂×C₄	1	2	yes	—
30	C₂²×C₅×C₈	C₂³	1	2	yes	—
31	C₂²×C₅×C₈	C₂³	1	4	yes	—
32	C₂²×C₅×C₈	C₂³	1	2	yes	—
33	C₅×C₄⋊Q₈	C₈	1	8	yes	complemented
34	C₄×C₅×Q₈	—	4	4	no	—
35	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
36	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
37	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
39	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
40	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₅	C₂³	1	4	yes	—
48	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
49	C₅×C₄⋊C₈	—	2	4	no	—
50	C₅×C₄⋊C₈	—	4	4	no	—
51	C₄×C₅×C₈	—	2	2	no	—
52	C₄×C₅×C₈	—	4	4	no	—
53	C₄×C₅×C₈	Q₈	1	4	yes	complemented
54	C₅×#95	C₂×C₄	1	4	yes	—
55	C₅×#95	C₂×C₄	1	4	yes	—
56	C₄×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
57	#1984	C₂×C₅	1	4	yes	—
58	#1939	C₂×C₅	1	2	yes	—
59	#1916	C₂×C₅	1	4	yes	—
60	#1831	C₂×C₅	1	1	yes	characteristic
61	C₈×C₄⋊C₄	C₂×C₅	1	2	yes	—
62	#1744	C₂×C₅	1	1	yes	characteristic
63	C₂×C₅×Q₈	—	4	2	no	—
64	C₂×C₅×Q₈	—	4	2	no	—
65	C₂×C₅×Q₈	C₂×C₈	1	8	yes	—
66	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
75	C₂×C₅×C₈	—	2	2	no	—
76	C₂×C₅×C₈	—	2	2	no	—
77	C₂×C₅×C₈	—	2	2	no	—
78	C₂×C₅×C₈	—	2	4	no	—
79	C₂×C₅×C₈	—	2	4	no	—
80	C₂×C₅×C₈	—	2	2	no	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	—	2	4	no	—
83	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
84	C₅×C₄⋊C₄	—	2	4	no	—
85	C₅×C₄⋊C₄	—	4	2	no	—
86	C₅×C₄⋊C₄	—	4	2	no	—
87	C₅×C₄⋊C₄	—	4	4	no	—
88	C₅×C₄⋊C₄	C₂×C₈	1	8	yes	—
89	C₅×C₄⋊C₄	C₂×C₈	1	8	yes	—
90	C₄²×C₅	—	2	2	no	—
91	C₄²×C₅	—	2	2	no	—
92	C₄²×C₅	—	2	4	no	—
93	C₄²×C₅	—	2	4	no	—
94	C₄²×C₅	—	2	2	no	—
95	C₄²×C₅	—	4	1	no	—
96	C₄²×C₅	—	4	1	no	—
97	C₄²×C₅	C₂×C₈	1	4	yes	—
98	C₄²×C₅	C₂×Q₈	1	2	yes	—
99	C₄²×C₅	#41	1	2	yes	—
100	C₂×C₄⋊Q₈	C₄×C₅	1	1	yes	characteristic
101	C₂×C₄×Q₈	C₂²×C₅	1	2	yes	—
102	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
103	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
104	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂²×C₅	1	4	yes	—
106	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
107	#391	C₄×C₅	1	2	yes	—
108	#389	C₄×C₅	1	2	yes	—
109	#389	C₄×C₅	1	2	yes	—
110	#384	C₄×C₅	1	1	yes	characteristic
111	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
112	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
113	C₄³	C₂²×C₅	1	1	yes	characteristic
114	#336	C₂²×C₅	1	2	yes	—
115	#336	C₂²×C₅	1	2	yes	—
116	#336	C₂²×C₅	1	2	yes	—
117	#336	C₂²×C₅	1	4	yes	—
118	#336	C₂²×C₅	1	4	yes	—
119	#336	C₂²×C₅	1	2	yes	—
120	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
121	C₅×Q₈	—	4	8	no	—
122	C₂×C₄×C₅	—	2	1	no	—
123	C₂×C₄×C₅	—	2	4	no	—
124	C₂×C₄×C₅	—	2	1	no	—
125	C₂×C₄×C₅	—	2	1	no	—
126	C₂×C₄×C₅	—	2	1	no	—
127	C₂×C₄×C₅	—	2	1	no	—
128	C₂×C₄×C₅	—	2	1	no	—
129	C₂×C₄×C₅	—	2	4	no	—
130	C₂×C₄×C₅	—	2	1	no	—
131	C₂×C₄×C₅	—	2	4	no	—
132	C₂×C₄×C₅	—	2	1	no	—
133	C₂×C₄×C₅	—	2	1	no	—
134	C₂×C₄×C₅	—	2	1	no	—
135	C₂×C₄×C₅	—	2	4	no	—
136	C₂×C₄×C₅	—	2	1	no	—
137	C₂×C₄×C₅	—	2	1	no	—
138	C₂×C₄×C₅	—	2	4	no	—
139	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
140	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
141	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
142	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
143	C₂×C₄×C₅	C₂²×C₈	1	8	yes	—
144	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
145	C₂×C₄×C₅	#131	1	2	yes	—
146	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
148	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
149	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
150	C₅×C₈	—	2	4	no	—
151	C₅×C₈	—	4	2	no	—
152	C₅×C₈	—	4	4	no	—
153	C₂²×Q₈	C₂×C₄×C₅	1	2	yes	—
154	C₂²×C₈	C₂³×C₅	1	2	yes	—
155	C₂²×C₈	C₂³×C₅	1	4	yes	—
156	C₂²×C₈	C₂³×C₅	1	2	yes	—
157	C₄⋊Q₈	C₅×C₈	1	8	yes	complemented
158	C₄×Q₈	—	4	4	no	—
159	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
160	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
161	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
162	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
163	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
164	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
165	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
166	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
167	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
168	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
169	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
170	C₂×C₄²	C₂³×C₅	1	4	yes	—
171	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
172	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
173	C₄⋊C₈	—	2	4	no	—
174	C₄⋊C₈	—	4	4	no	—
175	C₄×C₈	—	2	2	no	—
176	C₄×C₈	—	4	4	no	—
177	C₄×C₈	C₅×Q₈	1	4	yes	complemented
178	#95	C₂×C₄×C₅	1	4	yes	—
179	#95	C₂×C₄×C₅	1	4	yes	—
180	C₂²×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
181	C₂²×C₅	#518	1	1	yes	characteristic
182	C₂²×C₅	#519	1	1	yes	characteristic
183	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
184	C₂²×C₅	C₂×#131	1	1	yes	characteristic
185	C₂²×C₅	#568	1	1	yes	characteristic
186	C₂²×C₅	#568	1	1	yes	characteristic
187	C₄×C₅	—	2	2	no	—
188	C₄×C₅	—	2	8	no	—
189	C₄×C₅	—	2	2	no	—
190	C₄×C₅	—	2	2	no	—
191	C₄×C₅	—	4	1	no	—
192	C₄×C₅	—	4	4	no	—
193	C₄×C₅	—	4	1	no	—
194	C₄×C₅	C₈×Q₈	1	4	yes	—
195	C₄×C₅	#542	1	2	yes	—
196	C₄×C₅	#544	1	2	yes	—
197	C₂×Q₈	—	4	2	no	—
198	C₂×Q₈	—	4	2	no	—
199	C₂×Q₈	C₂×C₅×C₈	1	8	yes	—
200	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
201	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
202	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
203	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
204	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
205	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
206	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
207	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
208	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₄⋊C₄	—	4	4	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₄⋊C₄	C₂×C₅×C₈	1	8	yes	—
223	C₄⋊C₄	C₂×C₅×C₈	1	8	yes	—
224	C₄²	—	2	2	no	—
225	C₄²	—	2	4	no	—
226	C₄²	—	2	4	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	4	1	no	—
230	C₄²	—	4	1	no	—
231	C₄²	C₂×C₅×C₈	1	4	yes	—
232	C₄²	C₂×C₅×Q₈	1	2	yes	—
233	C₄²	C₅×#41	1	2	yes	—
234	C₂×C₅	#2331	1	1	yes	characteristic
235	C₂×C₅	C₂×C₈×Q₈	1	1	yes	characteristic
236	C₂×C₅	#2961	1	1	yes	characteristic
237	C₂×C₅	#2972	1	1	yes	characteristic
238	C₂×C₅	#2981	1	1	yes	characteristic
239	C₂×C₅	#2982	1	1	yes	characteristic
240	C₂×C₅	#2985	1	1	yes	characteristic
241	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
242	Q₈	—	4	8	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
261	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
262	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
263	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
264	C₂×C₄	C₂²×C₅×C₈	1	8	yes	—
265	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
266	C₂×C₄	C₅×#131	1	2	yes	—
267	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
268	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
269	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
270	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
271	C₈	—	2	4	no	—
272	C₈	—	4	2	no	—
273	C₈	—	4	4	no	—
274	C₅	#22307	1	1	yes	characteristic, direct product factor, Sylow
275	C₂²	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
276	C₂²	C₅×#518	1	1	yes	characteristic
277	C₂²	C₅×#519	1	1	yes	characteristic
278	C₂²	C₂³×C₅×C₈	1	1	yes	characteristic
279	C₂²	C₂×C₅×#131	1	1	yes	characteristic
280	C₂²	C₅×#568	1	1	yes	characteristic
281	C₂²	C₅×#568	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	2	8	no	—
286	C₄	—	4	1	no	—
287	C₄	—	4	4	no	—
288	C₄	—	4	1	no	—
289	C₄	C₅×C₈×Q₈	1	4	yes	—
290	C₄	C₅×#542	1	2	yes	—
291	C₄	C₅×#544	1	2	yes	—
292	C₂	C₅×#2331	1	1	yes	characteristic
293	C₂	C₂×C₅×C₈×Q₈	1	1	yes	characteristic
294	C₂	C₅×#2961	1	1	yes	characteristic
295	C₂	C₅×#2972	1	1	yes	characteristic
296	C₂	C₅×#2981	1	1	yes	characteristic
297	C₂	C₅×#2982	1	1	yes	characteristic
298	C₂	C₅×#2985	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂⁴
commutator	C₂²	C₂³×C₅×C₈
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle, prime radical	C₂³×C₅	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
28. #22307	C₅	direct (N × H)
33. C₅×C₄⋊Q₈	C₈	semidirect
53. C₄×C₅×C₈	Q₈	semidirect
157. C₄⋊Q₈	C₅×C₈	semidirect
177. C₄×C₈	C₅×Q₈	semidirect
274. C₅	#22307	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₅×C₄⋊Q₈	2
C₅×#1984	2
C₅×#1939	2
C₅×#1916	2
C₅×#1831	2
C₅×C₈×C₄⋊C₄	2
C₅×#1744	2
#22307	5

Subgroups — C5×#22307 #11945027

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22307 #11945027