Subgroups — C₅×#9903 #11932623

Order 1280.

506 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2868	C₂	1	1	yes	maximal, characteristic
2	C₅×#1831	C₂	1	1	yes	maximal, characteristic
3	C₅×#1767	C₂	1	1	yes	maximal, characteristic
4	C₅×#1296	C₂	1	4	yes	maximal
5	C₂×C₅×C₄⋊Q₈	—	2	1	no	—
6	C₂×C₅×C₄⋊Q₈	—	2	1	no	—
7	C₂×C₅×C₄⋊Q₈	C₄	1	2	yes	—
8	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₅×#97	—	2	1	no	—
11	C₅×#384	—	2	1	no	—
12	C₅×#384	—	2	1	no	—
13	C₅×#384	C₄	1	2	yes	—
14	C₅×#384	C₂²	1	2	yes	—
15	C₄³×C₅	C₂²	1	1	yes	characteristic
16	C₅×#336	—	2	1	no	—
17	C₅×#336	—	2	1	no	—
18	C₅×#336	—	2	1	no	—
19	#9903	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂²×C₅×Q₈	—	2	2	no	—
21	C₂²×C₅×C₈	—	2	2	no	—
22	C₂²×C₅×C₈	—	2	2	no	—
23	C₅×C₄⋊Q₈	—	4	4	no	—
24	C₅×C₄⋊Q₈	—	4	4	no	—
25	C₅×C₄⋊Q₈	—	4	4	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	—	2	2	no	—
30	C₂×C₅×C₄⋊C₄	—	2	2	no	—
31	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
39	C₅×C₄⋊C₈	—	2	2	no	—
40	C₅×C₄⋊C₈	—	2	4	no	—
41	C₅×C₄⋊C₈	Q₈	1	4	yes	complemented
42	C₅×#97	—	4	2	no	—
43	#2868	C₂×C₅	1	1	yes	characteristic
44	#1831	C₂×C₅	1	1	yes	characteristic
45	#1767	C₂×C₅	1	1	yes	characteristic
46	#1296	C₂×C₅	1	4	yes	—
47	C₂×C₅×Q₈	—	4	4	no	—
48	C₂×C₅×Q₈	—	4	4	no	—
49	C₂×C₅×Q₈	—	4	4	no	—
50	C₂²×C₄×C₅	—	2	4	no	—
51	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
58	C₂×C₅×C₈	—	2	4	no	—
59	C₂×C₅×C₈	—	4	2	no	—
60	C₂×C₅×C₈	—	4	2	no	—
61	C₂×C₅×C₈	—	4	2	no	—
62	C₂×C₅×C₈	—	4	2	no	—
63	C₂×C₅×C₈	—	4	2	no	—
64	C₅×C₄⋊C₄	—	4	4	no	—
65	C₅×C₄⋊C₄	—	4	4	no	—
66	C₅×C₄⋊C₄	—	4	4	no	—
67	C₅×C₄⋊C₄	—	4	4	no	—
68	C₅×C₄⋊C₄	—	4	4	no	—
69	C₅×C₄⋊C₄	—	4	4	no	—
70	C₄²×C₅	—	2	2	no	—
71	C₄²×C₅	—	2	2	no	—
72	C₄²×C₅	—	2	2	no	—
73	C₄²×C₅	—	2	2	no	—
74	C₄²×C₅	—	2	2	no	—
75	C₄²×C₅	SD₁₆	1	2	yes	—
76	C₄²×C₅	Q₁₆	1	2	yes	—
77	C₄²×C₅	C₂×Q₈	1	2	yes	—
78	C₄²×C₅	#41	1	2	yes	—
79	C₂×C₄⋊Q₈	—	2	1	no	—
80	C₂×C₄⋊Q₈	—	2	1	no	—
81	C₂×C₄⋊Q₈	C₄×C₅	1	2	yes	—
82	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
83	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
84	C₂×#97	—	2	1	no	—
85	#384	—	2	1	no	—
86	#384	—	2	1	no	—
87	#384	C₄×C₅	1	2	yes	—
88	#384	C₂²×C₅	1	2	yes	—
89	C₄³	C₂²×C₅	1	1	yes	characteristic
90	#336	—	2	1	no	—
91	#336	—	2	1	no	—
92	#336	—	2	1	no	—
93	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₅×Q₈	—	8	8	no	—
95	C₂×C₄×C₅	—	2	1	no	—
96	C₂×C₄×C₅	—	2	1	no	—
97	C₂×C₄×C₅	—	2	1	no	—
98	C₂×C₄×C₅	—	2	1	no	—
99	C₂×C₄×C₅	—	2	1	no	—
100	C₂×C₄×C₅	—	2	1	no	—
101	C₂×C₄×C₅	—	2	1	no	—
102	C₂×C₄×C₅	—	2	1	no	—
103	C₂×C₄×C₅	—	2	1	no	—
104	C₂×C₄×C₅	—	2	1	no	—
105	C₂×C₄×C₅	—	4	4	no	—
106	C₂×C₄×C₅	—	4	4	no	—
107	C₂×C₄×C₅	—	4	4	no	—
108	C₂×C₄×C₅	#100	1	2	yes	—
109	C₂×C₄×C₅	#101	1	2	yes	—
110	C₂×C₄×C₅	#103	1	2	yes	—
111	C₂×C₄×C₅	#103	1	2	yes	—
112	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
113	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₅	#124	1	1	yes	characteristic
115	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
118	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
119	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
120	C₂×C₄×C₅	#137	1	2	yes	—
121	C₅×C₈	—	4	4	no	—
122	C₅×C₈	—	8	2	no	—
123	C₂²×Q₈	—	2	2	no	—
124	C₂²×C₈	—	2	2	no	—
125	C₂²×C₈	—	2	2	no	—
126	C₄⋊Q₈	—	4	4	no	—
127	C₄⋊Q₈	—	4	4	no	—
128	C₄⋊Q₈	—	4	4	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
135	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
136	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
137	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
138	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
139	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
140	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
141	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
142	C₄⋊C₈	—	2	2	no	—
143	C₄⋊C₈	—	2	4	no	—
144	C₄⋊C₈	C₅×Q₈	1	4	yes	complemented
145	#97	—	4	2	no	—
146	C₂²×C₅	#391	1	1	yes	characteristic
147	C₂²×C₅	C₂×#100	1	1	yes	characteristic
148	C₂²×C₅	C₂×#101	1	1	yes	characteristic
149	C₂²×C₅	C₂×#103	1	1	yes	characteristic
150	C₂²×C₅	#418	1	1	yes	characteristic
151	C₂²×C₅	#418	1	1	yes	characteristic
152	C₂²×C₅	#419	1	1	yes	characteristic
153	C₄×C₅	—	2	2	no	—
154	C₄×C₅	—	2	2	no	—
155	C₄×C₅	—	2	2	no	—
156	C₄×C₅	—	2	2	no	—
157	C₄×C₅	—	2	2	no	—
158	C₄×C₅	—	8	4	no	—
159	C₄×C₅	#476	1	2	yes	—
160	C₄×C₅	#477	1	2	yes	—
161	C₄×C₅	#489	1	2	yes	—
162	C₄×C₅	#501	1	2	yes	—
163	C₂×Q₈	—	4	4	no	—
164	C₂×Q₈	—	4	4	no	—
165	C₂×Q₈	—	4	4	no	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
168	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
169	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
170	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
171	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	4	2	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	—	4	2	no	—
179	C₂×C₈	—	4	2	no	—
180	C₄⋊C₄	—	4	4	no	—
181	C₄⋊C₄	—	4	4	no	—
182	C₄⋊C₄	—	4	4	no	—
183	C₄⋊C₄	—	4	4	no	—
184	C₄⋊C₄	—	4	4	no	—
185	C₄⋊C₄	—	4	4	no	—
186	C₄²	—	2	2	no	—
187	C₄²	—	2	2	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	—	2	2	no	—
191	C₄²	C₅×SD₁₆	1	2	yes	—
192	C₄²	C₅×Q₁₆	1	2	yes	—
193	C₄²	C₂×C₅×Q₈	1	2	yes	—
194	C₄²	C₅×#41	1	2	yes	—
195	C₂×C₅	#1548	1	1	yes	characteristic
196	C₂×C₅	#1549	1	1	yes	characteristic
197	C₂×C₅	#1555	1	1	yes	characteristic
198	C₂×C₅	#1556	1	1	yes	characteristic
199	C₂×C₅	#1987	1	1	yes	characteristic
200	C₂×C₅	#1993	1	1	yes	characteristic
201	C₂×C₅	#1997	1	1	yes	characteristic
202	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
203	Q₈	—	8	8	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	C₅×#100	1	2	yes	—
218	C₂×C₄	C₅×#101	1	2	yes	—
219	C₂×C₄	C₅×#103	1	2	yes	—
220	C₂×C₄	C₅×#103	1	2	yes	—
221	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
222	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
223	C₂×C₄	C₅×#124	1	1	yes	characteristic
224	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
225	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
226	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
227	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
228	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
229	C₂×C₄	C₅×#137	1	2	yes	—
230	C₈	—	4	4	no	—
231	C₈	—	8	2	no	—
232	C₅	#9903	1	1	yes	characteristic, direct product factor, Sylow
233	C₂²	C₅×#391	1	1	yes	characteristic
234	C₂²	C₂×C₅×#100	1	1	yes	characteristic
235	C₂²	C₂×C₅×#101	1	1	yes	characteristic
236	C₂²	C₂×C₅×#103	1	1	yes	characteristic
237	C₂²	C₅×#418	1	1	yes	characteristic
238	C₂²	C₅×#418	1	1	yes	characteristic
239	C₂²	C₅×#419	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	2	2	no	—
243	C₄	—	2	2	no	—
244	C₄	—	2	2	no	—
245	C₄	—	8	4	no	—
246	C₄	C₅×#476	1	2	yes	—
247	C₄	C₅×#477	1	2	yes	—
248	C₄	C₅×#489	1	2	yes	—
249	C₄	C₅×#501	1	2	yes	—
250	C₂	C₅×#1548	1	1	yes	characteristic
251	C₂	C₅×#1549	1	1	yes	characteristic
252	C₂	C₅×#1555	1	1	yes	characteristic
253	C₂	C₅×#1556	1	1	yes	characteristic
254	C₂	C₅×#1987	1	1	yes	characteristic
255	C₂	C₅×#1993	1	1	yes	characteristic
256	C₂	C₅×#1997	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
19. #9903	C₅	direct (N × H)
41. C₅×C₄⋊C₈	Q₈	semidirect
144. C₄⋊C₈	C₅×Q₈	semidirect
232. C₅	#9903	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×#2868	2
C₅×#1831	2
C₅×#1767	2
C₅×#1296	2
#9903	5

Subgroups — C5×#9903 #11932623

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#9903 #11932623