Subgroups — C₅×#33826 #11956546

Order 1280.

488 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	C₅×#3177	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3176	C₂	1	1	yes	maximal, characteristic
3	C₅×#2229	C₂	1	2	yes	maximal, complemented
4	C₅×#2228	C₂	1	2	yes	maximal, complemented
5	C₅×#2220	C₂	1	2	yes	maximal, complemented
6	C₅×#2217	C₂	1	2	yes	maximal, complemented
7	C₅×#2208	C₂	1	2	yes	maximal, complemented
8	C₅×#2207	C₂	1	2	yes	maximal, complemented
9	C₅×#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#541	—	2	1	no	—
11	C₅×#540	—	2	1	no	—
12	C₂×C₅×SD₃₂	C₂²	1	2	yes	—
13	C₂×C₅×SD₃₂	C₂²	1	2	yes	—
14	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
15	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
16	C₅×#499	C₂²	1	1	yes	characteristic
17	C₅×#499	C₂²	1	1	yes	characteristic
18	C₅×#494	C₂²	1	1	yes	characteristic
19	C₅×#493	C₂²	1	1	yes	characteristic
20	C₅×#491	—	2	1	no	—
21	C₅×#490	—	2	1	no	—
22	C₅×#470	—	2	1	no	—
23	C₅×#469	—	2	1	no	—
24	C₅×#467	C₂²	1	2	yes	—
25	C₅×#466	C₂²	1	2	yes	—
26	C₅×#442	—	2	1	no	—
27	C₅×#441	—	2	1	no	—
28	C₄×C₅×Q₁₆	C₂²	1	2	yes	—
29	C₄×C₅×D₈	C₂²	1	2	yes	—
30	C₅×#405	C₂²	1	1	yes	characteristic
31	C₅×#367	C₂²	1	2	yes	—
32	C₅×#366	C₂²	1	2	yes	—
33	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
34	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
35	C₅×#358	C₂²	1	2	yes	—
36	C₅×#358	C₂²	1	2	yes	—
37	C₅×#357	C₂²	1	2	yes	—
38	C₅×#357	C₂²	1	2	yes	—
39	#33826	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂×C₅×Q₁₆	—	2	1	no	—
41	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
42	C₂×C₅×D₈	—	2	1	no	—
43	C₂×C₅×D₈	C₂³	1	2	yes	—
44	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
45	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
46	C₅×C₄⋊Q₈	—	2	1	no	—
47	C₅×#127	—	2	1	no	—
48	C₅×#126	—	4	2	no	—
49	C₅×#125	—	2	1	no	—
50	C₅×#125	—	2	1	no	—
51	C₅×#125	—	4	2	no	—
52	C₅×#123	—	4	2	no	—
53	C₅×C₂²⋊Q₈	—	2	2	no	—
54	C₅×#121	—	2	2	no	—
55	C₅×#121	—	4	2	no	—
56	C₄×C₅×Q₈	—	2	2	no	—
57	C₄×C₅×D₄	—	2	2	no	—
58	C₅×#117	D₄	1	1	yes	characteristic
59	C₅×SD₃₂	—	4	2	no	—
60	C₅×SD₃₂	—	4	2	no	—
61	C₅×M₃₂	—	4	1	no	—
62	C₂×C₅×C₁₆	—	4	1	no	—
63	C₂×C₅×C₁₆	C₂³	1	2	yes	—
64	C₂×C₅×C₁₆	C₂³	1	2	yes	—
65	C₅×#107	C₂³	1	2	yes	—
66	C₅×#107	C₂³	1	2	yes	—
67	C₅×#106	—	2	1	no	—
68	C₅×#106	—	2	1	no	—
69	C₅×#103	—	2	2	no	—
70	C₅×#103	—	2	2	no	—
71	C₅×#102	—	2	2	no	—
72	C₅×#102	—	2	2	no	—
73	C₅×#97	D₄	1	1	yes	characteristic
74	C₅×#97	D₄	1	1	yes	characteristic
75	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
76	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
77	#3177	C₂×C₅	1	1	yes	characteristic, complemented
78	#3176	C₂×C₅	1	1	yes	characteristic
79	#2229	C₂×C₅	1	2	yes	complemented
80	#2228	C₂×C₅	1	2	yes	complemented
81	#2220	C₂×C₅	1	2	yes	complemented
82	#2217	C₂×C₅	1	2	yes	complemented
83	#2208	C₂×C₅	1	2	yes	complemented
84	#2207	C₂×C₅	1	2	yes	complemented
85	#2151	C₂×C₅	1	1	yes	characteristic, complemented
86	C₂×C₅×Q₈	—	2	2	no	—
87	C₂×C₅×D₄	—	2	2	no	—
88	C₂×C₅×D₄	—	4	2	no	—
89	C₂×C₅×D₄	—	4	2	no	—
90	C₂²×C₄×C₅	—	4	2	no	—
91	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
92	C₅×Q₁₆	—	2	2	no	—
93	C₅×Q₁₆	—	2	2	no	—
94	C₅×Q₁₆	—	4	2	no	—
95	C₅×D₈	—	2	2	no	—
96	C₅×D₈	—	2	2	no	—
97	C₅×D₈	—	4	2	no	—
98	C₅×M₁₆	—	4	1	no	—
99	C₂×C₅×C₈	—	2	1	no	—
100	C₂×C₅×C₈	—	2	1	no	—
101	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
102	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
105	C₅×C₄⋊C₄	—	2	2	no	—
106	C₅×C₄⋊C₄	—	2	2	no	—
107	C₅×C₄⋊C₄	—	4	2	no	—
108	C₅×C₄⋊C₄	—	4	2	no	—
109	C₅×C₄⋊C₄	—	4	2	no	—
110	C₅×C₄⋊C₄	—	4	2	no	—
111	C₅×C₄⋊C₄	—	4	2	no	—
112	C₅×C₄⋊C₄	D₈	1	2	yes	—
113	C₅×C₂²⋊C₄	—	4	2	no	—
114	C₅×C₂²⋊C₄	—	4	2	no	—
115	C₅×C₂²⋊C₄	—	4	2	no	—
116	C₅×C₂²⋊C₄	D₈	1	2	yes	—
117	C₄²×C₅	—	4	2	no	—
118	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
120	C₅×C₁₆	—	2	2	no	—
121	C₅×C₁₆	—	2	2	no	—
122	#541	—	2	1	no	—
123	#540	—	2	1	no	—
124	C₂×SD₃₂	C₂²×C₅	1	2	yes	—
125	C₂×SD₃₂	C₂²×C₅	1	2	yes	—
126	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
127	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
128	#499	C₂²×C₅	1	1	yes	characteristic
129	#499	C₂²×C₅	1	1	yes	characteristic
130	#494	C₂²×C₅	1	1	yes	characteristic
131	#493	C₂²×C₅	1	1	yes	characteristic
132	#491	—	2	1	no	—
133	#490	—	2	1	no	—
134	#470	—	2	1	no	—
135	#469	—	2	1	no	—
136	#467	C₂²×C₅	1	2	yes	—
137	#466	C₂²×C₅	1	2	yes	—
138	#442	—	2	1	no	—
139	#441	—	2	1	no	—
140	C₄×Q₁₆	C₂²×C₅	1	2	yes	—
141	C₄×D₈	C₂²×C₅	1	2	yes	—
142	#405	C₂²×C₅	1	1	yes	characteristic
143	#367	C₂²×C₅	1	2	yes	—
144	#366	C₂²×C₅	1	2	yes	—
145	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
146	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
147	#358	C₂²×C₅	1	2	yes	—
148	#358	C₂²×C₅	1	2	yes	—
149	#357	C₂²×C₅	1	2	yes	—
150	#357	C₂²×C₅	1	2	yes	—
151	C₂³×C₅	—	4	2	no	—
152	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
153	C₅×Q₈	—	4	2	no	—
154	C₅×Q₈	—	4	2	no	—
155	C₅×D₄	—	4	2	no	—
156	C₅×D₄	—	4	2	no	—
157	C₅×D₄	—	16	2	no	—
158	C₅×D₄	—	16	2	no	—
159	C₂×C₄×C₅	—	2	1	no	—
160	C₂×C₄×C₅	—	2	1	no	—
161	C₂×C₄×C₅	—	4	2	no	—
162	C₂×C₄×C₅	—	4	2	no	—
163	C₂×C₄×C₅	—	4	2	no	—
164	C₂×C₄×C₅	—	16	2	no	—
165	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
166	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
167	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
169	C₅×C₈	—	2	1	no	—
170	C₅×C₈	—	2	1	no	—
171	C₅×C₈	—	2	1	no	—
172	C₅×C₈	H(5,𝔽₂)	1	1	yes	characteristic
173	C₅×C₈	2₋¹⁺⁴	1	1	yes	characteristic
174	C₂×Q₁₆	—	2	1	no	—
175	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
176	C₂×D₈	—	2	1	no	—
177	C₂×D₈	C₂³×C₅	1	2	yes	—
178	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
179	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
180	C₄⋊Q₈	—	2	1	no	—
181	#127	—	2	1	no	—
182	#126	—	4	2	no	—
183	#125	—	2	1	no	—
184	#125	—	2	1	no	—
185	#125	—	4	2	no	—
186	#123	—	4	2	no	—
187	C₂²⋊Q₈	—	2	2	no	—
188	#121	—	2	2	no	—
189	#121	—	4	2	no	—
190	C₄×Q₈	—	2	2	no	—
191	C₄×D₄	—	2	2	no	—
192	#117	C₅×D₄	1	1	yes	characteristic
193	SD₃₂	—	4	2	no	—
194	SD₃₂	—	4	2	no	—
195	M₃₂	—	4	1	no	—
196	C₂×C₁₆	—	4	1	no	—
197	C₂×C₁₆	C₂³×C₅	1	2	yes	—
198	C₂×C₁₆	C₂³×C₅	1	2	yes	—
199	#107	C₂³×C₅	1	2	yes	—
200	#107	C₂³×C₅	1	2	yes	—
201	#106	—	2	1	no	—
202	#106	—	2	1	no	—
203	#103	—	2	2	no	—
204	#103	—	2	2	no	—
205	#102	—	2	2	no	—
206	#102	—	2	2	no	—
207	#97	C₅×D₄	1	1	yes	characteristic
208	#97	C₅×D₄	1	1	yes	characteristic
209	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
210	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
211	C₂²×C₅	—	2	2	no	—
212	C₂²×C₅	—	2	1	no	—
213	C₂²×C₅	—	8	2	no	—
214	C₂²×C₅	—	8	2	no	—
215	C₂²×C₅	—	8	2	no	—
216	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
217	C₄×C₅	—	2	2	no	—
218	C₄×C₅	—	2	2	no	—
219	C₄×C₅	—	2	1	no	—
220	C₄×C₅	—	8	2	no	—
221	C₄×C₅	—	8	2	no	—
222	C₄×C₅	—	8	2	no	—
223	C₄×C₅	#537	1	1	yes	characteristic
224	C₄×C₅	#575	1	1	yes	characteristic
225	C₂×Q₈	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
231	Q₁₆	—	2	2	no	—
232	Q₁₆	—	2	2	no	—
233	Q₁₆	—	4	2	no	—
234	D₈	—	2	2	no	—
235	D₈	—	2	2	no	—
236	D₈	—	4	2	no	—
237	M₁₆	—	4	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
241	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
242	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
243	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	C₅×D₈	1	2	yes	—
252	C₂²⋊C₄	—	4	2	no	—
253	C₂²⋊C₄	—	4	2	no	—
254	C₂²⋊C₄	—	4	2	no	—
255	C₂²⋊C₄	C₅×D₈	1	2	yes	—
256	C₄²	—	4	2	no	—
257	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
258	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
259	C₁₆	—	2	2	no	—
260	C₁₆	—	2	2	no	—
261	C₂×C₅	—	4	1	no	—
262	C₂×C₅	—	16	2	no	—
263	C₂×C₅	#3245	1	1	yes	characteristic
264	C₂×C₅	#3416	1	2	yes	—
265	C₂³	—	4	2	no	—
266	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
267	Q₈	—	4	2	no	—
268	Q₈	—	4	2	no	—
269	D₄	—	4	2	no	—
270	D₄	—	4	2	no	—
271	D₄	—	16	2	no	—
272	D₄	—	16	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	16	2	no	—
279	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
280	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
281	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
282	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
283	C₈	—	2	1	no	—
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	C₅×H(5,𝔽₂)	1	1	yes	characteristic
287	C₈	C₅×2₋¹⁺⁴	1	1	yes	characteristic
288	C₅	#33826	1	1	yes	characteristic, direct product factor, Sylow
289	C₂²	—	2	1	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	8	2	no	—
292	C₂²	—	8	2	no	—
293	C₂²	—	8	2	no	—
294	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
295	C₄	—	2	1	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	8	2	no	—
299	C₄	—	8	2	no	—
300	C₄	—	8	2	no	—
301	C₄	C₅×#537	1	1	yes	characteristic
302	C₄	C₅×#575	1	1	yes	characteristic
303	C₂	—	4	1	no	—
304	C₂	—	16	2	no	—
305	C₂	C₅×#3245	1	1	yes	characteristic
306	C₂	C₅×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₅×#3176	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#117 ◁ C₅×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#3177	C₂	semidirect
3. C₅×#2229	C₂	semidirect
4. C₅×#2228	C₂	semidirect
5. C₅×#2220	C₂	semidirect
6. C₅×#2217	C₂	semidirect
7. C₅×#2208	C₂	semidirect
8. C₅×#2207	C₂	semidirect
9. C₅×#2151	C₂	semidirect
39. #33826	C₅	direct (N × H)
77. #3177	C₂×C₅	semidirect
79. #2229	C₂×C₅	semidirect
80. #2228	C₂×C₅	semidirect
81. #2220	C₂×C₅	semidirect
82. #2217	C₂×C₅	semidirect
83. #2208	C₂×C₅	semidirect
84. #2207	C₂×C₅	semidirect
85. #2151	C₂×C₅	semidirect
288. C₅	#33826	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×#3177	2
C₅×#3176	2
C₅×#2229	2
C₅×#2228	2
C₅×#2220	2
C₅×#2217	2
C₅×#2208	2
C₅×#2207	2
C₅×#2151	2
#33826	5

Subgroups — C5×#33826 #11956546

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33826 #11956546