Subgroups — C₅×#33772 #11956492

Order 1280.

626 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3169	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2248	C₂	1	2	yes	maximal, complemented
4	C₅×#2241	C₂	1	2	yes	maximal, complemented
5	C₅×#2208	C₂	1	4	yes	maximal, complemented
6	C₅×#2185	C₂	1	4	yes	maximal, complemented
7	C₅×#2162	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₅×Q₁₆	—	2	1	no	—
9	C₂²×C₅×Q₁₆	C₂²	1	2	yes	—
10	C₅×#532	—	2	1	no	—
11	C₂×C₅×C₄⋊Q₈	—	2	1	no	—
12	C₂×C₅×SD₃₂	C₂²	1	8	yes	—
13	C₅×#500	C₂²	1	1	yes	characteristic
14	C₅×#494	—	2	1	no	—
15	C₅×#494	—	2	1	no	—
16	C₅×#494	C₂²	1	2	yes	—
17	C₅×#494	C₂²	1	2	yes	—
18	C₅×#493	C₂²	1	1	yes	characteristic
19	C₅×#486	—	2	1	no	—
20	C₅×#466	—	2	1	no	—
21	C₅×#466	C₂²	1	2	yes	—
22	C₄×C₅×D₈	—	2	1	no	—
23	C₄×C₅×D₈	C₂²	1	2	yes	—
24	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
25	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
26	C₅×#358	C₂²	1	8	yes	—
27	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₄×C₅×C₁₆	C₂²	1	2	yes	—
29	#33772	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂×C₅×#41	—	4	1	no	—
31	C₂²×C₅×Q₈	—	2	2	no	—
32	C₂×C₅×Q₁₆	—	2	2	no	—
33	C₂×C₅×Q₁₆	—	2	2	no	—
34	C₂×C₅×Q₁₆	—	2	2	no	—
35	C₂×C₅×Q₁₆	—	4	4	no	—
36	C₂×C₅×Q₁₆	—	4	4	no	—
37	C₂×C₅×Q₁₆	C₂³	1	4	yes	—
38	C₂×C₅×D₈	—	2	1	no	—
39	C₂×C₅×D₈	C₂³	1	2	yes	—
40	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
41	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
42	C₅×C₄⋊Q₈	—	2	1	no	—
43	C₅×C₄⋊Q₈	—	2	2	no	—
44	C₅×C₄⋊Q₈	—	2	2	no	—
45	C₅×C₄⋊Q₈	—	4	1	no	—
46	C₅×C₄⋊Q₈	—	4	1	no	—
47	C₅×#127	—	2	1	no	—
48	C₅×#124	—	4	1	no	—
49	C₅×#121	—	2	2	no	—
50	C₅×#121	—	4	1	no	—
51	C₄×C₅×D₄	—	2	2	no	—
52	C₄×C₅×D₄	—	4	1	no	—
53	C₂×C₅×C₄⋊C₄	—	4	1	no	—
54	C₂×C₅×C₄⋊C₄	—	4	1	no	—
55	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
56	C₅×SD₃₂	—	4	8	no	—
57	C₂×C₅×C₁₆	C₂³	1	4	yes	—
58	C₅×#107	—	2	1	no	—
59	C₅×#107	C₂³	1	2	yes	—
60	C₅×#102	—	2	2	no	—
61	C₅×#102	—	2	2	no	—
62	C₄×C₅×C₈	D₄	1	1	yes	characteristic
63	C₄×C₅×C₈	D₄	1	1	yes	characteristic
64	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
65	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
66	#3169	C₂×C₅	1	1	yes	characteristic
67	C₂×#494	C₂×C₅	1	1	yes	characteristic, complemented
68	#2248	C₂×C₅	1	2	yes	complemented
69	#2241	C₂×C₅	1	2	yes	complemented
70	#2208	C₂×C₅	1	4	yes	complemented
71	#2185	C₂×C₅	1	4	yes	complemented
72	#2162	C₂×C₅	1	1	yes	characteristic, complemented
73	C₅×#41	—	8	4	no	—
74	C₂×C₅×Q₈	—	2	4	no	—
75	C₂×C₅×Q₈	—	4	4	no	—
76	C₂×C₅×Q₈	—	4	4	no	—
77	C₂×C₅×Q₈	—	4	1	no	—
78	C₂×C₅×Q₈	—	4	2	no	—
79	C₂×C₅×D₄	—	2	2	no	—
80	C₂×C₅×D₄	—	4	2	no	—
81	C₂×C₅×D₄	—	4	1	no	—
82	C₂×C₅×D₄	—	4	1	no	—
83	C₂²×C₄×C₅	—	4	2	no	—
84	C₂²×C₄×C₅	—	4	2	no	—
85	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₅×Q₁₆	—	2	8	no	—
89	C₅×Q₁₆	—	4	4	no	—
90	C₅×Q₁₆	—	4	4	no	—
91	C₅×Q₁₆	—	4	4	no	—
92	C₅×D₈	—	2	4	no	—
93	C₅×D₈	—	4	2	no	—
94	C₂×C₅×C₈	—	2	2	no	—
95	C₂×C₅×C₈	—	2	2	no	—
96	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
98	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
100	C₅×C₄⋊C₄	—	2	2	no	—
101	C₅×C₄⋊C₄	—	4	2	no	—
102	C₅×C₄⋊C₄	—	4	2	no	—
103	C₅×C₄⋊C₄	—	4	2	no	—
104	C₅×C₄⋊C₄	—	4	2	no	—
105	C₅×C₄⋊C₄	—	4	1	no	—
106	C₅×C₂²⋊C₄	—	4	2	no	—
107	C₅×C₂²⋊C₄	—	4	2	no	—
108	C₄²×C₅	D₈	1	1	yes	characteristic
109	C₄²×C₅	D₈	1	1	yes	characteristic
110	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
112	C₅×C₁₆	—	2	4	no	—
113	C₂²×Q₁₆	—	2	1	no	—
114	C₂²×Q₁₆	C₂²×C₅	1	2	yes	—
115	#532	—	2	1	no	—
116	C₂×C₄⋊Q₈	—	2	1	no	—
117	C₂×SD₃₂	C₂²×C₅	1	8	yes	—
118	#500	C₂²×C₅	1	1	yes	characteristic
119	#494	—	2	1	no	—
120	#494	—	2	1	no	—
121	#494	C₂²×C₅	1	2	yes	—
122	#494	C₂²×C₅	1	2	yes	—
123	#493	C₂²×C₅	1	1	yes	characteristic
124	#486	—	2	1	no	—
125	#466	—	2	1	no	—
126	#466	C₂²×C₅	1	2	yes	—
127	C₄×D₈	—	2	1	no	—
128	C₄×D₈	C₂²×C₅	1	2	yes	—
129	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
130	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
131	#358	C₂²×C₅	1	8	yes	—
132	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
133	C₄×C₁₆	C₂²×C₅	1	2	yes	—
134	C₂³×C₅	—	4	2	no	—
135	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
136	C₅×Q₈	—	4	4	no	—
137	C₅×Q₈	—	4	4	no	—
138	C₅×Q₈	—	8	4	no	—
139	C₅×Q₈	—	8	2	no	—
140	C₅×D₄	—	4	2	no	—
141	C₅×D₄	—	4	2	no	—
142	C₅×D₄	—	8	4	no	—
143	C₅×D₄	—	8	2	no	—
144	C₅×D₄	—	8	2	no	—
145	C₂×C₄×C₅	—	2	1	no	—
146	C₂×C₄×C₅	—	2	1	no	—
147	C₂×C₄×C₅	—	2	2	no	—
148	C₂×C₄×C₅	—	4	2	no	—
149	C₂×C₄×C₅	—	4	4	no	—
150	C₂×C₄×C₅	—	8	4	no	—
151	C₂×C₄×C₅	—	8	4	no	—
152	C₂×C₄×C₅	SD₃₂	1	4	yes	complemented
153	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
156	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
159	C₅×C₈	—	2	1	no	—
160	C₅×C₈	—	2	1	no	—
161	C₅×C₈	AGL(1,ℤ/8)	1	2	yes	—
162	C₅×C₈	H(5,𝔽₂)	1	2	yes	—
163	C₂×#41	—	4	1	no	—
164	C₂²×Q₈	—	2	2	no	—
165	C₂×Q₁₆	—	2	2	no	—
166	C₂×Q₁₆	—	2	2	no	—
167	C₂×Q₁₆	—	2	2	no	—
168	C₂×Q₁₆	—	4	4	no	—
169	C₂×Q₁₆	—	4	4	no	—
170	C₂×Q₁₆	C₂³×C₅	1	4	yes	—
171	C₂×D₈	—	2	1	no	—
172	C₂×D₈	C₂³×C₅	1	2	yes	—
173	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
174	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
175	C₄⋊Q₈	—	2	1	no	—
176	C₄⋊Q₈	—	2	2	no	—
177	C₄⋊Q₈	—	2	2	no	—
178	C₄⋊Q₈	—	4	1	no	—
179	C₄⋊Q₈	—	4	1	no	—
180	#127	—	2	1	no	—
181	#124	—	4	1	no	—
182	#121	—	2	2	no	—
183	#121	—	4	1	no	—
184	C₄×D₄	—	2	2	no	—
185	C₄×D₄	—	4	1	no	—
186	C₂×C₄⋊C₄	—	4	1	no	—
187	C₂×C₄⋊C₄	—	4	1	no	—
188	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
189	SD₃₂	—	4	8	no	—
190	C₂×C₁₆	C₂³×C₅	1	4	yes	—
191	#107	—	2	1	no	—
192	#107	C₂³×C₅	1	2	yes	—
193	#102	—	2	2	no	—
194	#102	—	2	2	no	—
195	C₄×C₈	C₅×D₄	1	1	yes	characteristic
196	C₄×C₈	C₅×D₄	1	1	yes	characteristic
197	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
198	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
199	C₂²×C₅	—	2	1	no	—
200	C₂²×C₅	—	2	1	no	—
201	C₂²×C₅	—	8	2	no	—
202	C₂²×C₅	—	8	2	no	—
203	C₂²×C₅	—	8	2	no	—
204	C₂²×C₅	C₂×SD₃₂	1	2	yes	—
205	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
206	C₄×C₅	—	2	1	no	—
207	C₄×C₅	—	2	1	no	—
208	C₄×C₅	—	8	2	no	—
209	C₄×C₅	—	8	4	no	—
210	C₄×C₅	C₂×SD₃₂	1	2	yes	—
211	C₄×C₅	C₂×SD₃₂	1	2	yes	—
212	C₄×C₅	#510	1	2	yes	—
213	C₄×C₅	#535	1	1	yes	characteristic
214	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
215	#41	—	8	4	no	—
216	C₂×Q₈	—	2	4	no	—
217	C₂×Q₈	—	4	4	no	—
218	C₂×Q₈	—	4	4	no	—
219	C₂×Q₈	—	4	1	no	—
220	C₂×Q₈	—	4	2	no	—
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	—	4	2	no	—
223	C₂×D₄	—	4	1	no	—
224	C₂×D₄	—	4	1	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	—	4	2	no	—
227	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
228	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
230	Q₁₆	—	2	8	no	—
231	Q₁₆	—	4	4	no	—
232	Q₁₆	—	4	4	no	—
233	Q₁₆	—	4	4	no	—
234	D₈	—	2	4	no	—
235	D₈	—	4	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
239	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
240	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
241	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	4	1	no	—
248	C₂²⋊C₄	—	4	2	no	—
249	C₂²⋊C₄	—	4	2	no	—
250	C₄²	C₅×D₈	1	1	yes	characteristic
251	C₄²	C₅×D₈	1	1	yes	characteristic
252	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
253	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
254	C₁₆	—	2	4	no	—
255	C₂×C₅	—	2	2	no	—
256	C₂×C₅	—	16	2	no	—
257	C₂×C₅	#3205	1	1	yes	characteristic
258	C₂×C₅	C₂²×SD₃₂	1	1	yes	characteristic
259	C₂×C₅	C₂×#510	1	1	yes	characteristic
260	C₂³	—	4	2	no	—
261	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
262	Q₈	—	4	4	no	—
263	Q₈	—	4	4	no	—
264	Q₈	—	8	4	no	—
265	Q₈	—	8	2	no	—
266	D₄	—	4	2	no	—
267	D₄	—	4	2	no	—
268	D₄	—	8	4	no	—
269	D₄	—	8	2	no	—
270	D₄	—	8	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	8	4	no	—
277	C₂×C₄	—	8	4	no	—
278	C₂×C₄	C₅×SD₃₂	1	4	yes	complemented
279	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
280	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
281	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
282	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
283	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
284	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	C₅×AGL(1,ℤ/8)	1	2	yes	—
288	C₈	C₅×H(5,𝔽₂)	1	2	yes	—
289	C₅	#33772	1	1	yes	characteristic, direct product factor, Sylow
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	8	2	no	—
293	C₂²	—	8	2	no	—
294	C₂²	—	8	2	no	—
295	C₂²	C₂×C₅×SD₃₂	1	2	yes	—
296	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	8	2	no	—
300	C₄	—	8	4	no	—
301	C₄	C₂×C₅×SD₃₂	1	2	yes	—
302	C₄	C₂×C₅×SD₃₂	1	2	yes	—
303	C₄	C₅×#510	1	2	yes	—
304	C₄	C₅×#535	1	1	yes	characteristic
305	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
306	C₂	—	2	2	no	—
307	C₂	—	16	2	no	—
308	C₂	C₅×#3205	1	1	yes	characteristic
309	C₂	C₂²×C₅×SD₃₂	1	1	yes	characteristic
310	C₂	C₂×C₅×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₅×#3169	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#494	C₂	semidirect
3. C₅×#2248	C₂	semidirect
4. C₅×#2241	C₂	semidirect
5. C₅×#2208	C₂	semidirect
6. C₅×#2185	C₂	semidirect
7. C₅×#2162	C₂	semidirect
29. #33772	C₅	direct (N × H)
67. C₂×#494	C₂×C₅	semidirect
68. #2248	C₂×C₅	semidirect
69. #2241	C₂×C₅	semidirect
70. #2208	C₂×C₅	semidirect
71. #2185	C₂×C₅	semidirect
72. #2162	C₂×C₅	semidirect
152. C₂×C₄×C₅	SD₃₂	semidirect
278. C₂×C₄	C₅×SD₃₂	semidirect
289. C₅	#33772	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#3169	2
C₂×C₅×#494	2
C₅×#2248	2
C₅×#2241	2
C₅×#2208	2
C₅×#2185	2
C₅×#2162	2
#33772	5

Subgroups — C5×#33772 #11956492

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33772 #11956492