Subgroups — C₅×#10618 #11933338

Order 1280.

366 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1984	C₂	1	1	yes	maximal, characteristic
2	C₅×#1845	C₂	1	1	yes	maximal, characteristic
3	C₅×#1803	C₂	1	2	yes	maximal
4	C₅×#1296	C₂	1	2	yes	maximal
5	C₅×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₅×#107	—	2	1	no	—
8	C₂×C₅×#107	—	2	1	no	—
9	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₅×#103	—	2	2	no	—
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₅×#384	C₂²	1	2	yes	—
14	C₅×#340	—	2	2	no	—
15	C₅×#336	—	2	1	no	—
16	C₅×#336	—	2	1	no	—
17	#10618	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂²×C₅×Q₈	D₄	1	1	yes	characteristic
19	C₂²×C₅×C₈	—	2	1	no	—
20	C₂²×C₅×C₈	—	2	2	no	—
21	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
23	C₄×C₅×Q₈	—	2	2	no	—
24	C₄×C₅×Q₈	—	2	2	no	—
25	C₂×C₅×C₄⋊C₄	—	2	1	no	—
26	C₂×C₅×C₄⋊C₄	—	2	2	no	—
27	C₂×C₅×C₄⋊C₄	—	2	2	no	—
28	C₂×C₅×C₄⋊C₄	—	2	2	no	—
29	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₅	—	2	1	no	—
31	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
32	C₅×#107	—	4	2	no	—
33	C₅×#107	—	4	2	no	—
34	C₅×C₄⋊C₈	—	2	2	no	—
35	C₅×C₄⋊C₈	—	2	2	no	—
36	C₅×C₄⋊C₈	—	2	2	no	—
37	C₅×#103	—	4	4	no	—
38	C₄×C₅×C₈	—	2	2	no	—
39	#1984	C₂×C₅	1	1	yes	characteristic
40	#1845	C₂×C₅	1	1	yes	characteristic
41	#1803	C₂×C₅	1	2	yes	—
42	#1296	C₂×C₅	1	2	yes	—
43	#1277	C₂×C₅	1	1	yes	characteristic
44	C₂×C₅×Q₈	—	2	1	no	—
45	C₂×C₅×Q₈	—	2	2	no	—
46	C₂×C₅×Q₈	—	2	2	no	—
47	C₂×C₅×Q₈	Q₁₆	1	2	yes	—
48	C₂²×C₄×C₅	—	2	1	no	—
49	C₂²×C₄×C₅	—	2	1	no	—
50	C₂²×C₄×C₅	—	4	2	no	—
51	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅	#41	1	1	yes	characteristic
54	C₂×C₅×C₈	—	2	1	no	—
55	C₂×C₅×C₈	—	2	1	no	—
56	C₂×C₅×C₈	—	2	1	no	—
57	C₂×C₅×C₈	—	2	1	no	—
58	C₂×C₅×C₈	—	4	2	no	—
59	C₂×C₅×C₈	—	4	1	no	—
60	C₂×C₅×C₈	—	4	2	no	—
61	C₂×C₅×C₈	—	4	1	no	—
62	C₂×C₅×C₈	—	4	2	no	—
63	C₂×C₅×C₈	—	4	1	no	—
64	C₂×C₅×C₈	SD₁₆	1	2	yes	—
65	C₂×C₅×C₈	Q₁₆	1	2	yes	—
66	C₅×C₄⋊C₄	—	2	2	no	—
67	C₅×C₄⋊C₄	—	4	2	no	—
68	C₅×C₄⋊C₄	—	4	2	no	—
69	C₅×C₄⋊C₄	—	4	2	no	—
70	C₅×C₄⋊C₄	—	4	2	no	—
71	C₅×C₄⋊C₄	—	4	2	no	—
72	C₅×C₄⋊C₄	—	8	2	no	—
73	C₄²×C₅	—	4	1	no	—
74	C₄²×C₅	—	4	1	no	—
75	C₄²×C₅	#41	1	2	yes	—
76	C₄²×C₅	#41	1	2	yes	—
77	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
78	C₂×#107	—	2	1	no	—
79	C₂×#107	—	2	1	no	—
80	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
81	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
82	C₂×#103	—	2	2	no	—
83	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
84	#384	C₂²×C₅	1	2	yes	—
85	#340	—	2	2	no	—
86	#336	—	2	1	no	—
87	#336	—	2	1	no	—
88	C₂³×C₅	#121	1	1	yes	characteristic
89	C₅×Q₈	—	2	4	no	—
90	C₅×Q₈	—	4	2	no	—
91	C₂×C₄×C₅	—	2	1	no	—
92	C₂×C₄×C₅	—	2	1	no	—
93	C₂×C₄×C₅	—	2	2	no	—
94	C₂×C₄×C₅	—	4	1	no	—
95	C₂×C₄×C₅	—	4	1	no	—
96	C₂×C₄×C₅	—	4	1	no	—
97	C₂×C₄×C₅	—	4	1	no	—
98	C₂×C₄×C₅	—	4	1	no	—
99	C₂×C₄×C₅	—	8	2	no	—
100	C₂×C₄×C₅	—	8	2	no	—
101	C₂×C₄×C₅	—	8	2	no	—
102	C₂×C₄×C₅	#123	1	2	yes	—
103	C₂×C₄×C₅	#123	1	1	yes	characteristic
104	C₂×C₄×C₅	#124	1	1	yes	characteristic
105	C₂×C₄×C₅	#126	1	2	yes	—
106	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
108	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
109	C₂×C₄×C₅	#135	1	1	yes	characteristic
110	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
111	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₂×C₄×C₅	#137	1	1	yes	characteristic
113	C₅×C₈	—	2	2	no	—
114	C₅×C₈	—	2	2	no	—
115	C₅×C₈	—	8	2	no	—
116	C₅×C₈	—	8	1	no	—
117	C₂²×Q₈	C₅×D₄	1	1	yes	characteristic
118	C₂²×C₈	—	2	1	no	—
119	C₂²×C₈	—	2	2	no	—
120	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
121	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
122	C₄×Q₈	—	2	2	no	—
123	C₄×Q₈	—	2	2	no	—
124	C₂×C₄⋊C₄	—	2	1	no	—
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
129	C₂×C₄²	—	2	1	no	—
130	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
131	#107	—	4	2	no	—
132	#107	—	4	2	no	—
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	2	2	no	—
136	#103	—	4	4	no	—
137	C₄×C₈	—	2	2	no	—
138	C₂²×C₅	#397	1	1	yes	characteristic
139	C₂²×C₅	#460	1	1	yes	characteristic
140	C₂²×C₅	#462	1	1	yes	characteristic
141	C₂²×C₅	#467	1	1	yes	characteristic
142	C₂²×C₅	#469	1	1	yes	characteristic
143	C₂²×C₅	#472	1	1	yes	characteristic
144	C₂²×C₅	#473	1	1	yes	characteristic
145	C₄×C₅	—	2	2	no	—
146	C₄×C₅	—	4	2	no	—
147	C₄×C₅	—	8	1	no	—
148	C₄×C₅	—	16	2	no	—
149	C₄×C₅	#482	1	2	yes	—
150	C₄×C₅	#483	1	1	yes	characteristic
151	C₄×C₅	#484	1	1	yes	characteristic
152	C₄×C₅	#484	1	2	yes	—
153	C₄×C₅	#487	1	1	yes	characteristic
154	C₄×C₅	#489	1	1	yes	characteristic
155	C₂×Q₈	—	2	1	no	—
156	C₂×Q₈	—	2	2	no	—
157	C₂×Q₈	—	2	2	no	—
158	C₂×Q₈	C₅×Q₁₆	1	2	yes	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	4	2	no	—
162	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
164	C₂²×C₄	C₅×#41	1	1	yes	characteristic
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	—	4	1	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	C₅×SD₁₆	1	2	yes	—
176	C₂×C₈	C₅×Q₁₆	1	2	yes	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₄⋊C₄	—	8	2	no	—
184	C₄²	—	4	1	no	—
185	C₄²	—	4	1	no	—
186	C₄²	C₅×#41	1	2	yes	—
187	C₄²	C₅×#41	1	2	yes	—
188	C₂×C₅	#1670	1	1	yes	characteristic
189	C₂×C₅	#1676	1	1	yes	characteristic
190	C₂×C₅	#1688	1	1	yes	characteristic
191	C₂×C₅	#1694	1	1	yes	characteristic
192	C₂×C₅	#2072	1	1	yes	characteristic
193	C₂×C₅	#2076	1	1	yes	characteristic
194	C₂×C₅	#2078	1	1	yes	characteristic
195	C₂³	C₅×#121	1	1	yes	characteristic
196	Q₈	—	2	4	no	—
197	Q₈	—	4	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	8	2	no	—
207	C₂×C₄	—	8	2	no	—
208	C₂×C₄	—	8	2	no	—
209	C₂×C₄	C₅×#123	1	2	yes	—
210	C₂×C₄	C₅×#123	1	1	yes	characteristic
211	C₂×C₄	C₅×#124	1	1	yes	characteristic
212	C₂×C₄	C₅×#126	1	2	yes	—
213	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
214	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
215	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
216	C₂×C₄	C₅×#135	1	1	yes	characteristic
217	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
218	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
219	C₂×C₄	C₅×#137	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	—	2	2	no	—
222	C₈	—	8	2	no	—
223	C₈	—	8	1	no	—
224	C₅	#10618	1	1	yes	characteristic, direct product factor, Sylow
225	C₂²	C₅×#397	1	1	yes	characteristic
226	C₂²	C₅×#460	1	1	yes	characteristic
227	C₂²	C₅×#462	1	1	yes	characteristic
228	C₂²	C₅×#467	1	1	yes	characteristic
229	C₂²	C₅×#469	1	1	yes	characteristic
230	C₂²	C₅×#472	1	1	yes	characteristic
231	C₂²	C₅×#473	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	4	2	no	—
234	C₄	—	8	1	no	—
235	C₄	—	16	2	no	—
236	C₄	C₅×#482	1	2	yes	—
237	C₄	C₅×#483	1	1	yes	characteristic
238	C₄	C₅×#484	1	1	yes	characteristic
239	C₄	C₅×#484	1	2	yes	—
240	C₄	C₅×#487	1	1	yes	characteristic
241	C₄	C₅×#489	1	1	yes	characteristic
242	C₂	C₅×#1670	1	1	yes	characteristic
243	C₂	C₅×#1676	1	1	yes	characteristic
244	C₂	C₅×#1688	1	1	yes	characteristic
245	C₂	C₅×#1694	1	1	yes	characteristic
246	C₂	C₅×#2072	1	1	yes	characteristic
247	C₂	C₅×#2076	1	1	yes	characteristic
248	C₂	C₅×#2078	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
17. #10618	C₅	direct (N × H)
224. C₅	#10618	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×#1984	2
C₅×#1845	2
C₅×#1803	2
C₅×#1296	2
C₅×#1277	2
#10618	5

Subgroups — C5×#10618 #11933338

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10618 #11933338