Subgroups — C₅×#10538 #11933258

Order 1280.

486 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#494	C₂	1	1	yes	maximal, characteristic
2	C₅×#1988	C₂	1	2	yes	maximal
3	C₅×#1939	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#332	C₂	1	2	yes	maximal
5	C₅×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅×Q₁₆	—	2	1	no	—
7	C₂²×C₅×Q₁₆	—	2	1	no	—
8	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₅×#494	—	2	4	no	—
10	C₂×C₅×C₄⋊C₈	—	2	1	no	—
11	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₅×#340	—	2	2	no	—
16	C₅×#340	—	2	2	no	—
17	C₅×#336	—	2	1	no	—
18	C₅×#332	—	2	4	no	—
19	#10538	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂²×C₅×Q₈	—	2	2	no	—
21	C₂×C₅×Q₁₆	—	2	4	no	—
22	C₂×C₅×Q₁₆	—	2	4	no	—
23	C₂×C₅×Q₁₆	—	4	2	no	—
24	C₂×C₅×Q₁₆	—	4	2	no	—
25	C₂×C₅×Q₁₆	—	4	2	no	—
26	C₂×C₅×Q₁₆	—	4	2	no	—
27	C₂²×C₅×C₈	—	2	1	no	—
28	C₂²×C₅×C₈	—	2	2	no	—
29	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
31	C₅×C₄⋊Q₈	—	2	2	no	—
32	C₅×C₄⋊Q₈	—	2	2	no	—
33	C₅×C₄⋊Q₈	—	2	2	no	—
34	C₅×C₄⋊Q₈	D₄	1	4	yes	—
35	C₂×C₅×C₄⋊C₄	—	2	2	no	—
36	C₂×C₅×C₄⋊C₄	—	2	2	no	—
37	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
38	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅	—	2	1	no	—
40	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
41	C₅×C₄⋊C₈	—	2	2	no	—
42	C₅×C₄⋊C₈	—	2	2	no	—
43	C₅×C₄⋊C₈	—	2	2	no	—
44	C₅×C₄⋊C₈	—	4	2	no	—
45	C₄×C₅×C₈	—	2	2	no	—
46	C₅×#95	—	2	1	no	—
47	C₂×#494	C₂×C₅	1	1	yes	characteristic
48	#1988	C₂×C₅	1	2	yes	—
49	#1939	C₂×C₅	1	1	yes	characteristic
50	C₂×#332	C₂×C₅	1	2	yes	—
51	#1277	C₂×C₅	1	1	yes	characteristic
52	C₂×C₅×Q₈	—	2	4	no	—
53	C₂×C₅×Q₈	—	4	2	no	—
54	C₂×C₅×Q₈	—	4	2	no	—
55	C₂×C₅×Q₈	—	4	2	no	—
56	C₂×C₅×Q₈	—	4	2	no	—
57	C₂×C₅×Q₈	—	4	2	no	—
58	C₂²×C₄×C₅	—	2	1	no	—
59	C₂²×C₄×C₅	—	2	1	no	—
60	C₂²×C₄×C₅	—	4	2	no	—
61	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	#41	1	1	yes	characteristic
64	C₅×Q₁₆	—	4	8	no	—
65	C₅×Q₁₆	—	4	8	no	—
66	C₂×C₅×C₈	—	2	1	no	—
67	C₂×C₅×C₈	—	2	1	no	—
68	C₂×C₅×C₈	—	2	1	no	—
69	C₂×C₅×C₈	—	2	1	no	—
70	C₂×C₅×C₈	—	4	1	no	—
71	C₂×C₅×C₈	—	4	2	no	—
72	C₂×C₅×C₈	—	4	1	no	—
73	C₂×C₅×C₈	—	4	2	no	—
74	C₂×C₅×C₈	—	4	1	no	—
75	C₂×C₅×C₈	—	4	2	no	—
76	C₂×C₅×C₈	D₈	1	2	yes	—
77	C₂×C₅×C₈	SD₁₆	1	2	yes	—
78	C₅×C₄⋊C₄	—	2	1	no	—
79	C₅×C₄⋊C₄	—	2	4	no	—
80	C₅×C₄⋊C₄	—	2	4	no	—
81	C₅×C₄⋊C₄	—	2	2	no	—
82	C₅×C₄⋊C₄	—	4	2	no	—
83	C₅×C₄⋊C₄	—	4	2	no	—
84	C₅×C₄⋊C₄	Q₁₆	1	2	yes	complemented
85	C₄²×C₅	—	4	1	no	—
86	C₄²×C₅	—	4	1	no	—
87	C₄²×C₅	C₂×D₄	1	2	yes	—
88	C₄²×C₅	#41	1	2	yes	—
89	C₂²×Q₁₆	—	2	1	no	—
90	C₂²×Q₁₆	—	2	1	no	—
91	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
92	#494	—	2	4	no	—
93	C₂×C₄⋊C₈	—	2	1	no	—
94	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
95	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
96	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
97	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
98	#340	—	2	2	no	—
99	#340	—	2	2	no	—
100	#336	—	2	1	no	—
101	#332	—	2	4	no	—
102	C₂³×C₅	#121	1	1	yes	characteristic
103	C₅×Q₈	—	4	4	no	—
104	C₅×Q₈	—	4	4	no	—
105	C₅×Q₈	—	8	2	no	—
106	C₅×Q₈	—	8	2	no	—
107	C₂×C₄×C₅	—	2	1	no	—
108	C₂×C₄×C₅	—	2	1	no	—
109	C₂×C₄×C₅	—	2	2	no	—
110	C₂×C₄×C₅	—	4	1	no	—
111	C₂×C₄×C₅	—	4	1	no	—
112	C₂×C₄×C₅	—	4	4	no	—
113	C₂×C₄×C₅	—	4	1	no	—
114	C₂×C₄×C₅	—	4	1	no	—
115	C₂×C₄×C₅	—	4	1	no	—
116	C₂×C₄×C₅	—	8	2	no	—
117	C₂×C₄×C₅	—	8	2	no	—
118	C₂×C₄×C₅	#120	1	1	yes	characteristic
119	C₂×C₄×C₅	#121	1	2	yes	—
120	C₂×C₄×C₅	#123	1	2	yes	—
121	C₂×C₄×C₅	#124	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
125	C₂×C₄×C₅	#135	1	1	yes	characteristic
126	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×C₄×C₅	#137	1	2	yes	—
129	C₅×C₈	—	2	2	no	—
130	C₅×C₈	—	2	2	no	—
131	C₅×C₈	—	8	1	no	—
132	C₅×C₈	—	8	2	no	—
133	C₂²×Q₈	—	2	2	no	—
134	C₂×Q₁₆	—	2	4	no	—
135	C₂×Q₁₆	—	2	4	no	—
136	C₂×Q₁₆	—	4	2	no	—
137	C₂×Q₁₆	—	4	2	no	—
138	C₂×Q₁₆	—	4	2	no	—
139	C₂×Q₁₆	—	4	2	no	—
140	C₂²×C₈	—	2	1	no	—
141	C₂²×C₈	—	2	2	no	—
142	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
143	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
144	C₄⋊Q₈	—	2	2	no	—
145	C₄⋊Q₈	—	2	2	no	—
146	C₄⋊Q₈	—	2	2	no	—
147	C₄⋊Q₈	C₅×D₄	1	4	yes	—
148	C₂×C₄⋊C₄	—	2	2	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
151	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
152	C₂×C₄²	—	2	1	no	—
153	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
154	C₄⋊C₈	—	2	2	no	—
155	C₄⋊C₈	—	2	2	no	—
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	4	2	no	—
158	C₄×C₈	—	2	2	no	—
159	#95	—	2	1	no	—
160	C₂²×C₅	#394	1	1	yes	characteristic
161	C₂²×C₅	#459	1	1	yes	characteristic
162	C₂²×C₅	#460	1	1	yes	characteristic
163	C₂²×C₅	#467	1	1	yes	characteristic
164	C₂²×C₅	#470	1	1	yes	characteristic
165	C₂²×C₅	#473	1	1	yes	characteristic
166	C₂²×C₅	#473	1	1	yes	characteristic
167	C₄×C₅	—	2	2	no	—
168	C₄×C₅	—	4	2	no	—
169	C₄×C₅	—	8	1	no	—
170	C₄×C₅	—	8	4	no	—
171	C₄×C₅	#447	1	1	yes	characteristic
172	C₄×C₅	#450	1	1	yes	characteristic
173	C₄×C₅	#462	1	2	yes	—
174	C₄×C₅	#464	1	2	yes	—
175	C₄×C₅	#486	1	1	yes	characteristic
176	C₄×C₅	#490	1	1	yes	characteristic
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	—	4	2	no	—
179	C₂×Q₈	—	4	2	no	—
180	C₂×Q₈	—	4	2	no	—
181	C₂×Q₈	—	4	2	no	—
182	C₂×Q₈	—	4	2	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
187	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
188	C₂²×C₄	C₅×#41	1	1	yes	characteristic
189	Q₁₆	—	4	8	no	—
190	Q₁₆	—	4	8	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	4	1	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	C₅×D₈	1	2	yes	—
202	C₂×C₈	C₅×SD₁₆	1	2	yes	—
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄⋊C₄	C₅×Q₁₆	1	2	yes	complemented
210	C₄²	—	4	1	no	—
211	C₄²	—	4	1	no	—
212	C₄²	C₂×C₅×D₄	1	2	yes	—
213	C₄²	C₅×#41	1	2	yes	—
214	C₂×C₅	#1669	1	1	yes	characteristic
215	C₂×C₅	#1670	1	1	yes	characteristic
216	C₂×C₅	#1689	1	1	yes	characteristic
217	C₂×C₅	#1690	1	1	yes	characteristic
218	C₂×C₅	#2041	1	1	yes	characteristic
219	C₂×C₅	#2047	1	1	yes	characteristic
220	C₂×C₅	#2056	1	1	yes	characteristic
221	C₂³	C₅×#121	1	1	yes	characteristic
222	Q₈	—	4	4	no	—
223	Q₈	—	4	4	no	—
224	Q₈	—	8	2	no	—
225	Q₈	—	8	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	4	1	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	C₅×#120	1	1	yes	characteristic
238	C₂×C₄	C₅×#121	1	2	yes	—
239	C₂×C₄	C₅×#123	1	2	yes	—
240	C₂×C₄	C₅×#124	1	1	yes	characteristic
241	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
242	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
243	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
244	C₂×C₄	C₅×#135	1	1	yes	characteristic
245	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
246	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
247	C₂×C₄	C₅×#137	1	2	yes	—
248	C₈	—	2	2	no	—
249	C₈	—	2	2	no	—
250	C₈	—	8	1	no	—
251	C₈	—	8	2	no	—
252	C₅	#10538	1	1	yes	characteristic, direct product factor, Sylow
253	C₂²	C₅×#394	1	1	yes	characteristic
254	C₂²	C₅×#459	1	1	yes	characteristic
255	C₂²	C₅×#460	1	1	yes	characteristic
256	C₂²	C₅×#467	1	1	yes	characteristic
257	C₂²	C₅×#470	1	1	yes	characteristic
258	C₂²	C₅×#473	1	1	yes	characteristic
259	C₂²	C₅×#473	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	4	2	no	—
262	C₄	—	8	4	no	—
263	C₄	—	8	1	no	—
264	C₄	C₅×#447	1	1	yes	characteristic
265	C₄	C₅×#450	1	1	yes	characteristic
266	C₄	C₅×#462	1	2	yes	—
267	C₄	C₅×#464	1	2	yes	—
268	C₄	C₅×#486	1	1	yes	characteristic
269	C₄	C₅×#490	1	1	yes	characteristic
270	C₂	C₅×#1669	1	1	yes	characteristic
271	C₂	C₅×#1670	1	1	yes	characteristic
272	C₂	C₅×#1689	1	1	yes	characteristic
273	C₂	C₅×#1690	1	1	yes	characteristic
274	C₂	C₅×#2041	1	1	yes	characteristic
275	C₂	C₅×#2047	1	1	yes	characteristic
276	C₂	C₅×#2056	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
19. #10538	C₅	direct (N × H)
84. C₅×C₄⋊C₄	Q₁₆	semidirect
209. C₄⋊C₄	C₅×Q₁₆	semidirect
252. C₅	#10538	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#494	2
C₅×#1988	2
C₅×#1939	2
C₂×C₅×#332	2
C₅×#1277	2
#10538	5

Subgroups — C5×#10538 #11933258

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10538 #11933258