Subgroups — C₅×#11664 #11934384

Order 1280.

362 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₅×#2163	C₂	1	1	yes	maximal, characteristic
3	C₅×#1566	C₂	1	1	yes	maximal, characteristic
4	C₅×#1334	C₂	1	4	yes	maximal, complemented
5	C₂²×C₅×Q₁₆	C₄	1	1	yes	characteristic
6	C₂×C₄×C₅×Q₈	—	2	1	no	—
7	C₄×C₅×Q₁₆	—	2	1	no	—
8	C₄×C₅×Q₁₆	—	2	1	no	—
9	C₄×C₅×Q₁₆	C₄	1	2	yes	—
10	C₄×C₅×Q₁₆	C₂²	1	2	yes	—
11	C₂×C₅×#107	C₄	1	1	yes	characteristic
12	C₅×#423	—	2	1	no	—
13	C₂×C₅×#103	—	2	1	no	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₅×C₄⋊C₁₆	—	2	1	no	—
16	C₅×C₂²⋊C₁₆	—	2	1	no	—
17	C₅×#346	C₂²	1	2	yes	—
18	C₅×#334	—	2	1	no	—
19	C₅×#334	C₂²	1	2	yes	—
20	#11664	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅×Q₈	—	2	1	no	—
22	C₂×C₅×Q₁₆	—	2	4	no	—
23	C₂×C₅×Q₁₆	—	2	1	no	—
24	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₅×M₁₆	—	4	1	no	—
26	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
28	C₄×C₅×Q₈	—	2	2	no	—
29	C₄×C₅×Q₈	—	2	2	no	—
30	C₄×C₅×Q₈	—	4	1	no	—
31	C₄×C₅×Q₈	—	4	1	no	—
32	C₂×C₅×C₄⋊C₄	—	2	1	no	—
33	C₂×C₅×C₄⋊C₄	—	4	1	no	—
34	C₂×C₄²×C₅	—	4	1	no	—
35	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
36	C₂×C₅×C₁₆	—	2	2	no	—
37	C₅×#107	—	2	1	no	—
38	C₅×#107	C₂×C₄	1	2	yes	—
39	C₅×C₄⋊C₈	—	2	2	no	—
40	C₅×C₄⋊C₈	—	4	1	no	—
41	C₅×#103	—	2	2	no	—
42	C₅×#103	—	2	2	no	—
43	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
44	C₄×C₅×C₈	D₄	1	1	yes	characteristic
45	C₄×C₅×C₈	D₄	1	1	yes	characteristic
46	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
47	C₂×C₄×Q₁₆	C₂×C₅	1	1	yes	characteristic
48	#2163	C₂×C₅	1	1	yes	characteristic
49	#1566	C₂×C₅	1	1	yes	characteristic
50	#1334	C₂×C₅	1	4	yes	complemented
51	C₂×C₅×Q₈	—	2	4	no	—
52	C₂×C₅×Q₈	—	2	2	no	—
53	C₂×C₅×Q₈	—	4	2	no	—
54	C₂×C₅×Q₈	—	4	1	no	—
55	C₂²×C₄×C₅	—	4	1	no	—
56	C₂²×C₄×C₅	—	4	1	no	—
57	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
60	C₅×Q₁₆	—	2	4	no	—
61	C₅×Q₁₆	—	2	2	no	—
62	C₅×Q₁₆	M₁₆	1	4	yes	complemented
63	C₅×M₁₆	—	8	2	no	—
64	C₂×C₅×C₈	—	2	2	no	—
65	C₂×C₅×C₈	—	4	1	no	—
66	C₂×C₅×C₈	—	4	2	no	—
67	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
71	C₅×C₄⋊C₄	—	2	2	no	—
72	C₅×C₄⋊C₄	—	4	1	no	—
73	C₅×C₄⋊C₄	—	4	2	no	—
74	C₅×C₄⋊C₄	—	4	2	no	—
75	C₄²×C₅	—	4	2	no	—
76	C₄²×C₅	—	4	2	no	—
77	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
78	C₄²×C₅	D₈	1	1	yes	characteristic
79	C₄²×C₅	SD₁₆	1	1	yes	characteristic
80	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
81	C₅×C₁₆	—	4	2	no	—
82	C₂²×Q₁₆	C₄×C₅	1	1	yes	characteristic
83	C₂×C₄×Q₈	—	2	1	no	—
84	C₄×Q₁₆	—	2	1	no	—
85	C₄×Q₁₆	—	2	1	no	—
86	C₄×Q₁₆	C₄×C₅	1	2	yes	—
87	C₄×Q₁₆	C₂²×C₅	1	2	yes	—
88	C₂×#107	C₄×C₅	1	1	yes	characteristic
89	#423	—	2	1	no	—
90	C₂×#103	—	2	1	no	—
91	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
92	C₄⋊C₁₆	—	2	1	no	—
93	C₂²⋊C₁₆	—	2	1	no	—
94	#346	C₂²×C₅	1	2	yes	—
95	#334	—	2	1	no	—
96	#334	C₂²×C₅	1	2	yes	—
97	C₂³×C₅	#102	1	1	yes	characteristic
98	C₅×Q₈	—	2	4	no	—
99	C₅×Q₈	—	4	4	no	—
100	C₅×Q₈	—	4	2	no	—
101	C₂×C₄×C₅	—	2	2	no	—
102	C₂×C₄×C₅	—	2	2	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	2	no	—
107	C₂×C₄×C₅	—	4	2	no	—
108	C₂×C₄×C₅	—	8	2	no	—
109	C₂×C₄×C₅	#102	1	1	yes	characteristic
110	C₂×C₄×C₅	#102	1	1	yes	characteristic
111	C₂×C₄×C₅	#102	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
115	C₅×C₈	—	2	1	no	—
116	C₅×C₈	—	2	1	no	—
117	C₅×C₈	—	2	1	no	—
118	C₅×C₈	—	8	2	no	—
119	C₅×C₈	C₂×M₁₆	1	2	yes	—
120	C₂²×Q₈	—	2	1	no	—
121	C₂×Q₁₆	—	2	4	no	—
122	C₂×Q₁₆	—	2	1	no	—
123	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
124	C₂×M₁₆	—	4	1	no	—
125	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
126	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
127	C₄×Q₈	—	2	2	no	—
128	C₄×Q₈	—	2	2	no	—
129	C₄×Q₈	—	4	1	no	—
130	C₄×Q₈	—	4	1	no	—
131	C₂×C₄⋊C₄	—	2	1	no	—
132	C₂×C₄⋊C₄	—	4	1	no	—
133	C₂×C₄²	—	4	1	no	—
134	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
135	C₂×C₁₆	—	2	2	no	—
136	#107	—	2	1	no	—
137	#107	C₂×C₄×C₅	1	2	yes	—
138	C₄⋊C₈	—	2	2	no	—
139	C₄⋊C₈	—	4	1	no	—
140	#103	—	2	2	no	—
141	#103	—	2	2	no	—
142	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
143	C₄×C₈	C₅×D₄	1	1	yes	characteristic
144	C₄×C₈	C₅×D₄	1	1	yes	characteristic
145	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
146	C₂²×C₅	—	2	1	no	—
147	C₂²×C₅	—	2	1	no	—
148	C₂²×C₅	#360	1	2	yes	—
149	C₂²×C₅	C₂×#102	1	1	yes	characteristic
150	C₄×C₅	—	2	1	no	—
151	C₄×C₅	—	2	1	no	—
152	C₄×C₅	—	2	1	no	—
153	C₄×C₅	—	2	1	no	—
154	C₄×C₅	—	4	4	no	—
155	C₄×C₅	—	8	2	no	—
156	C₄×C₅	#362	1	2	yes	—
157	C₄×C₅	#407	1	1	yes	characteristic
158	C₄×C₅	#421	1	1	yes	characteristic
159	C₂×Q₈	—	2	4	no	—
160	C₂×Q₈	—	2	2	no	—
161	C₂×Q₈	—	4	2	no	—
162	C₂×Q₈	—	4	1	no	—
163	C₂²×C₄	—	4	1	no	—
164	C₂²×C₄	—	4	1	no	—
165	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
166	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
167	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
168	Q₁₆	—	2	4	no	—
169	Q₁₆	—	2	2	no	—
170	Q₁₆	C₅×M₁₆	1	4	yes	complemented
171	M₁₆	—	8	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	4	1	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
176	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
177	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
178	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	1	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₄²	—	4	2	no	—
184	C₄²	—	4	2	no	—
185	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
186	C₄²	C₅×D₈	1	1	yes	characteristic
187	C₄²	C₅×SD₁₆	1	1	yes	characteristic
188	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
189	C₁₆	—	4	2	no	—
190	C₂×C₅	—	2	2	no	—
191	C₂×C₅	#1486	1	1	yes	characteristic
192	C₂×C₅	C₂×#360	1	1	yes	characteristic
193	C₂×C₅	C₂×#362	1	1	yes	characteristic
194	C₂³	C₅×#102	1	1	yes	characteristic
195	Q₈	—	2	4	no	—
196	Q₈	—	4	4	no	—
197	Q₈	—	4	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	8	2	no	—
206	C₂×C₄	C₅×#102	1	1	yes	characteristic
207	C₂×C₄	C₅×#102	1	1	yes	characteristic
208	C₂×C₄	C₅×#102	1	1	yes	characteristic
209	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
210	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
211	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
212	C₈	—	2	1	no	—
213	C₈	—	2	1	no	—
214	C₈	—	2	1	no	—
215	C₈	—	8	2	no	—
216	C₈	C₂×C₅×M₁₆	1	2	yes	—
217	C₅	#11664	1	1	yes	characteristic, direct product factor, Sylow
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	C₅×#360	1	2	yes	—
221	C₂²	C₂×C₅×#102	1	1	yes	characteristic
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	4	4	no	—
227	C₄	—	8	2	no	—
228	C₄	C₅×#362	1	2	yes	—
229	C₄	C₅×#407	1	1	yes	characteristic
230	C₄	C₅×#421	1	1	yes	characteristic
231	C₂	—	2	2	no	—
232	C₂	C₅×#1486	1	1	yes	characteristic
233	C₂	C₂×C₅×#360	1	1	yes	characteristic
234	C₂	C₂×C₅×#362	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₂³×C₅	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1334	C₂	semidirect
20. #11664	C₅	direct (N × H)
50. #1334	C₂×C₅	semidirect
62. C₅×Q₁₆	M₁₆	semidirect
170. Q₁₆	C₅×M₁₆	semidirect
217. C₅	#11664	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×C₄×C₅×Q₁₆	2
C₅×#2163	2
C₅×#1566	2
C₅×#1334	2
#11664	5

Subgroups — C5×#11664 #11934384

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#11664 #11934384