Subgroups — C₅×#8665 #11931385

Order 1280.

358 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#446	C₂	1	1	yes	maximal, characteristic
2	C₅×#1837	C₂	1	1	yes	maximal, characteristic
3	C₅×#1564	C₂	1	1	yes	maximal, characteristic
4	C₅×#1487	C₂	1	2	yes	maximal
5	C₅×#1301	C₂	1	2	yes	maximal
6	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
7	C₅×#446	—	2	1	no	—
8	C₅×#446	—	2	1	no	—
9	C₅×#446	C₄	1	2	yes	—
10	C₅×#446	C₄	1	2	yes	—
11	C₅×#431	C₂²	1	2	yes	—
12	C₅×#423	C₂²	1	2	yes	—
13	C₂×C₅×C₄⋊C₈	—	2	1	no	—
14	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₅×#97	—	2	1	no	—
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₅×#341	—	2	1	no	—
18	C₅×#341	—	2	1	no	—
19	C₅×#335	—	2	1	no	—
20	C₅×#334	—	2	1	no	—
21	C₅×Q₈⋊C₈	—	2	4	no	—
22	#8665	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂²×C₅×Q₈	Q₈	1	1	yes	characteristic
24	C₂×C₅×M₁₆	—	2	2	no	—
25	C₂²×C₅×C₈	—	2	1	no	—
26	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
28	C₄×C₅×Q₈	—	2	1	no	—
29	C₄×C₅×Q₈	—	2	1	no	—
30	C₄×C₅×Q₈	C₂×C₄	1	2	yes	—
31	C₄×C₅×Q₈	D₄	1	2	yes	—
32	C₂×C₅×C₄⋊C₄	—	2	1	no	—
33	C₂×C₅×C₄⋊C₄	Q₈	1	1	yes	characteristic
34	C₂×C₄²×C₅	—	2	1	no	—
35	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
36	C₅×C₄⋊C₈	—	2	2	no	—
37	C₅×C₄⋊C₈	—	2	2	no	—
38	C₅×C₄⋊C₈	—	2	1	no	—
39	C₅×C₄⋊C₈	—	2	2	no	—
40	C₅×C₄⋊C₈	—	2	2	no	—
41	C₅×C₄⋊C₈	—	2	2	no	—
42	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
43	C₅×C₂²⋊C₈	—	2	2	no	—
44	C₅×#97	—	2	2	no	—
45	C₅×#97	—	2	2	no	—
46	C₄×C₅×C₈	—	2	2	no	—
47	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
48	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
49	C₄×C₅×C₈	D₄	1	1	yes	characteristic
50	C₄×C₅×C₈	D₄	1	1	yes	characteristic
51	C₂×#446	C₂×C₅	1	1	yes	characteristic
52	#1837	C₂×C₅	1	1	yes	characteristic
53	#1564	C₂×C₅	1	1	yes	characteristic
54	#1487	C₂×C₅	1	2	yes	—
55	#1301	C₂×C₅	1	2	yes	—
56	C₂×C₅×Q₈	—	2	1	no	—
57	C₂×C₅×Q₈	—	4	2	no	—
58	C₂×C₅×Q₈	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅	—	2	1	no	—
60	C₂²×C₄×C₅	—	2	1	no	—
61	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₅	#41	1	1	yes	characteristic
64	C₅×M₁₆	—	4	2	no	—
65	C₅×M₁₆	—	4	2	no	—
66	C₂×C₅×C₈	—	2	2	no	—
67	C₂×C₅×C₈	—	2	2	no	—
68	C₂×C₅×C₈	—	2	2	no	—
69	C₂×C₅×C₈	—	4	2	no	—
70	C₂×C₅×C₈	—	4	2	no	—
71	C₂×C₅×C₈	—	4	2	no	—
72	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
76	C₅×C₄⋊C₄	—	2	2	no	—
77	C₅×C₄⋊C₄	—	2	2	no	—
78	C₅×C₄⋊C₄	—	2	1	no	—
79	C₅×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
80	C₄²×C₅	—	2	2	no	—
81	C₄²×C₅	—	2	2	no	—
82	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₅	#41	1	1	yes	characteristic
86	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
87	#446	—	2	1	no	—
88	#446	—	2	1	no	—
89	#446	C₄×C₅	1	2	yes	—
90	#446	C₄×C₅	1	2	yes	—
91	#431	C₂²×C₅	1	2	yes	—
92	#423	C₂²×C₅	1	2	yes	—
93	C₂×C₄⋊C₈	—	2	1	no	—
94	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
95	C₂×#97	—	2	1	no	—
96	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
97	#341	—	2	1	no	—
98	#341	—	2	1	no	—
99	#335	—	2	1	no	—
100	#334	—	2	1	no	—
101	Q₈⋊C₈	—	2	4	no	—
102	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
103	C₅×Q₈	—	2	4	no	—
104	C₅×Q₈	—	4	2	no	—
105	C₂×C₄×C₅	—	2	2	no	—
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	—	4	1	no	—
109	C₂×C₄×C₅	—	4	2	no	—
110	C₂×C₄×C₅	—	4	2	no	—
111	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₅	#117	1	1	yes	characteristic
114	C₂×C₄×C₅	#121	1	1	yes	characteristic
115	C₂×C₄×C₅	#121	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₅	#135	1	2	yes	—
118	C₂×C₄×C₅	#137	1	2	yes	—
119	C₅×C₈	—	2	1	no	—
120	C₅×C₈	—	2	1	no	—
121	C₅×C₈	—	2	1	no	—
122	C₅×C₈	—	2	1	no	—
123	C₅×C₈	—	4	2	no	—
124	C₅×C₈	—	8	2	no	—
125	C₅×C₈	—	8	2	no	—
126	C₂²×Q₈	C₅×Q₈	1	1	yes	characteristic
127	C₂×M₁₆	—	2	2	no	—
128	C₂²×C₈	—	2	1	no	—
129	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
130	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
131	C₄×Q₈	—	2	1	no	—
132	C₄×Q₈	—	2	1	no	—
133	C₄×Q₈	C₂×C₄×C₅	1	2	yes	—
134	C₄×Q₈	C₅×D₄	1	2	yes	—
135	C₂×C₄⋊C₄	—	2	1	no	—
136	C₂×C₄⋊C₄	C₅×Q₈	1	1	yes	characteristic
137	C₂×C₄²	—	2	1	no	—
138	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
139	C₄⋊C₈	—	2	2	no	—
140	C₄⋊C₈	—	2	2	no	—
141	C₄⋊C₈	—	2	1	no	—
142	C₄⋊C₈	—	2	2	no	—
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	2	2	no	—
145	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
146	C₂²⋊C₈	—	2	2	no	—
147	#97	—	2	2	no	—
148	#97	—	2	2	no	—
149	C₄×C₈	—	2	2	no	—
150	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
151	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
152	C₄×C₈	C₅×D₄	1	1	yes	characteristic
153	C₄×C₈	C₅×D₄	1	1	yes	characteristic
154	C₂²×C₅	—	2	1	no	—
155	C₂²×C₅	—	2	1	no	—
156	C₂²×C₅	#380	1	1	yes	characteristic
157	C₂²×C₅	#479	1	2	yes	—
158	C₄×C₅	—	2	1	no	—
159	C₄×C₅	—	2	1	no	—
160	C₄×C₅	—	2	1	no	—
161	C₄×C₅	—	4	2	no	—
162	C₄×C₅	—	4	2	no	—
163	C₄×C₅	#416	1	1	yes	characteristic
164	C₄×C₅	#419	1	1	yes	characteristic
165	C₄×C₅	#430	1	2	yes	—
166	C₄×C₅	#464	1	2	yes	—
167	C₂×Q₈	—	2	1	no	—
168	C₂×Q₈	—	4	2	no	—
169	C₂×Q₈	C₅×C₄⋊C₄	1	2	yes	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
174	C₂²×C₄	C₅×#41	1	1	yes	characteristic
175	M₁₆	—	4	2	no	—
176	M₁₆	—	4	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	4	2	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
184	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
185	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
186	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	C₅×C₄⋊C₄	1	2	yes	—
191	C₄²	—	2	2	no	—
192	C₄²	—	2	2	no	—
193	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
194	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
195	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
196	C₄²	C₅×#41	1	1	yes	characteristic
197	C₂×C₅	—	2	2	no	—
198	C₂×C₅	#1805	1	1	yes	characteristic
199	C₂×C₅	#1811	1	1	yes	characteristic
200	C₂×C₅	#1814	1	1	yes	characteristic
201	C₂³	C₅×C₂²⋊Q₈	1	1	yes	characteristic
202	Q₈	—	2	4	no	—
203	Q₈	—	4	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
211	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
212	C₂×C₄	C₅×#117	1	1	yes	characteristic
213	C₂×C₄	C₅×#121	1	1	yes	characteristic
214	C₂×C₄	C₅×#121	1	1	yes	characteristic
215	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
216	C₂×C₄	C₅×#135	1	2	yes	—
217	C₂×C₄	C₅×#137	1	2	yes	—
218	C₈	—	2	1	no	—
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₈	—	2	1	no	—
222	C₈	—	4	2	no	—
223	C₈	—	8	2	no	—
224	C₈	—	8	2	no	—
225	C₅	#8665	1	1	yes	characteristic, direct product factor, Sylow
226	C₂²	—	2	1	no	—
227	C₂²	—	2	1	no	—
228	C₂²	C₅×#380	1	1	yes	characteristic
229	C₂²	C₅×#479	1	2	yes	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	2	no	—
235	C₄	C₅×#416	1	1	yes	characteristic
236	C₄	C₅×#419	1	1	yes	characteristic
237	C₄	C₅×#430	1	2	yes	—
238	C₄	C₅×#464	1	2	yes	—
239	C₂	—	2	2	no	—
240	C₂	C₅×#1805	1	1	yes	characteristic
241	C₂	C₅×#1811	1	1	yes	characteristic
242	C₂	C₅×#1814	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#121
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#380
prime radical	C₂³×C₅	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
22. #8665	C₅	direct (N × H)
225. C₅	#8665	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×#446	2
C₅×#1837	2
C₅×#1564	2
C₅×#1487	2
C₅×#1301	2
#8665	5

Subgroups — C5×#8665 #11931385

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8665 #11931385