Subgroups — C₅×#9624 #11932344

Order 1280.

298 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1950	C₂	1	2	yes	maximal
2	C₅×#1567	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#335	C₂	1	1	yes	maximal, characteristic
4	C₅×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
5	C₅×#1301	C₂	1	2	yes	maximal
6	C₅×#431	C₂²	1	1	yes	characteristic
7	C₂×C₅×#108	—	2	2	no	—
8	C₅×#423	C₂²	1	2	yes	—
9	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
10	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₅×#341	—	2	2	no	—
13	C₅×#335	—	2	1	no	—
14	C₅×#335	—	2	1	no	—
15	C₅×#335	C₄	1	2	yes	—
16	C₅×#335	C₄	1	2	yes	—
17	C₅×#322	—	2	1	no	—
18	C₅×C₈²	—	2	1	no	—
19	#9624	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂×C₅×M₁₆	—	2	2	no	—
21	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
22	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	—	2	2	no	—
24	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
25	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
26	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
27	C₅×#108	—	4	4	no	—
28	C₅×C₄⋊C₈	—	2	2	no	—
29	C₅×C₄⋊C₈	—	2	2	no	—
30	C₅×C₄⋊C₈	—	2	1	no	—
31	C₅×C₄⋊C₈	—	2	2	no	—
32	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	—
33	C₅×C₂²⋊C₈	Q₈	1	2	yes	—
34	C₅×#97	—	2	1	no	—
35	C₄×C₅×C₈	—	2	1	no	—
36	C₄×C₅×C₈	—	2	1	no	—
37	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
39	C₄×C₅×C₈	D₄	1	1	yes	characteristic
40	C₄×C₅×C₈	D₄	1	1	yes	characteristic
41	#1950	C₂×C₅	1	2	yes	—
42	#1567	C₂×C₅	1	1	yes	characteristic
43	C₂×#335	C₂×C₅	1	1	yes	characteristic
44	C₈×M₁₆	C₂×C₅	1	1	yes	characteristic
45	#1301	C₂×C₅	1	2	yes	—
46	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
49	C₅×M₁₆	—	2	1	no	—
50	C₅×M₁₆	—	2	1	no	—
51	C₅×M₁₆	—	4	2	no	—
52	C₅×M₁₆	—	4	2	no	—
53	C₅×M₁₆	D₈	1	2	yes	—
54	C₅×M₁₆	Q₁₆	1	2	yes	—
55	C₂×C₅×C₈	—	2	1	no	—
56	C₂×C₅×C₈	—	2	1	no	—
57	C₂×C₅×C₈	—	2	2	no	—
58	C₂×C₅×C₈	—	2	4	no	—
59	C₂×C₅×C₈	—	2	2	no	—
60	C₂×C₅×C₈	—	4	1	no	—
61	C₂×C₅×C₈	—	4	2	no	—
62	C₂×C₅×C₈	—	4	2	no	—
63	C₂×C₅×C₈	—	4	2	no	—
64	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
68	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₅	#41	1	1	yes	characteristic
71	C₄²×C₅	#41	1	1	yes	characteristic
72	#431	C₂²×C₅	1	1	yes	characteristic
73	C₂×#108	—	2	2	no	—
74	#423	C₂²×C₅	1	2	yes	—
75	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
76	C₄×M₁₆	C₂²×C₅	1	1	yes	characteristic
77	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
78	#341	—	2	2	no	—
79	#335	—	2	1	no	—
80	#335	—	2	1	no	—
81	#335	C₄×C₅	1	2	yes	—
82	#335	C₄×C₅	1	2	yes	—
83	#322	—	2	1	no	—
84	C₈²	—	2	1	no	—
85	C₂³×C₅	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₅	—	2	1	no	—
87	C₂×C₄×C₅	—	2	1	no	—
88	C₂×C₄×C₅	—	4	1	no	—
89	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
92	C₂×C₄×C₅	#121	1	1	yes	characteristic
93	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₅	#125	1	1	yes	characteristic
95	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
97	C₂×C₄×C₅	#135	1	1	yes	characteristic
98	C₂×C₄×C₅	#135	1	1	yes	characteristic
99	C₅×C₈	—	2	1	no	—
100	C₅×C₈	—	2	1	no	—
101	C₅×C₈	—	2	1	no	—
102	C₅×C₈	—	2	2	no	—
103	C₅×C₈	—	2	2	no	—
104	C₅×C₈	—	4	2	no	—
105	C₅×C₈	—	4	4	no	—
106	C₅×C₈	—	8	2	no	—
107	C₅×C₈	#108	1	2	yes	complemented
108	C₂×M₁₆	—	2	2	no	—
109	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
110	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	C₅×Q₈	1	1	yes	characteristic
113	C₂²×C₈	C₅×Q₈	1	1	yes	characteristic
114	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
115	#108	—	4	4	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	2	1	no	—
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	C₂×C₄×C₅	1	4	yes	—
121	C₂²⋊C₈	C₅×Q₈	1	2	yes	—
122	#97	—	2	1	no	—
123	C₄×C₈	—	2	1	no	—
124	C₄×C₈	—	2	1	no	—
125	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
126	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
127	C₄×C₈	C₅×D₄	1	1	yes	characteristic
128	C₄×C₈	C₅×D₄	1	1	yes	characteristic
129	C₂²×C₅	—	2	1	no	—
130	C₂²×C₅	—	2	1	no	—
131	C₂²×C₅	#389	1	1	yes	characteristic
132	C₂²×C₅	#499	1	1	yes	characteristic
133	C₂²×C₅	#500	1	1	yes	characteristic
134	C₄×C₅	—	2	1	no	—
135	C₄×C₅	—	2	1	no	—
136	C₄×C₅	—	2	1	no	—
137	C₄×C₅	C₂×#108	1	1	yes	characteristic
138	C₄×C₅	#430	1	1	yes	characteristic
139	C₄×C₅	C₄×D₈	1	1	yes	characteristic
140	C₄×C₅	C₄×Q₁₆	1	1	yes	characteristic
141	C₄×C₅	#466	1	1	yes	characteristic
142	C₄×C₅	#467	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
144	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
146	M₁₆	—	2	1	no	—
147	M₁₆	—	2	1	no	—
148	M₁₆	—	4	2	no	—
149	M₁₆	—	4	2	no	—
150	M₁₆	C₅×D₈	1	2	yes	—
151	M₁₆	C₅×Q₁₆	1	2	yes	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	4	1	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	—	4	2	no	—
160	C₂×C₈	—	4	2	no	—
161	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
165	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
166	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
167	C₄²	C₅×#41	1	1	yes	characteristic
168	C₄²	C₅×#41	1	1	yes	characteristic
169	C₂×C₅	—	2	1	no	—
170	C₂×C₅	—	2	1	no	—
171	C₂×C₅	#1942	1	1	yes	characteristic
172	C₂×C₅	#1950	1	1	yes	characteristic
173	C₂×C₅	#1953	1	1	yes	characteristic
174	C₂³	C₅×C₄⋊Q₈	1	1	yes	characteristic
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
179	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
180	C₂×C₄	C₄×C₅×Q₈	1	1	yes	characteristic
181	C₂×C₄	C₅×#121	1	1	yes	characteristic
182	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
183	C₂×C₄	C₅×#125	1	1	yes	characteristic
184	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
185	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
186	C₂×C₄	C₅×#135	1	1	yes	characteristic
187	C₂×C₄	C₅×#135	1	1	yes	characteristic
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	2	1	no	—
191	C₈	—	2	2	no	—
192	C₈	—	2	2	no	—
193	C₈	—	4	2	no	—
194	C₈	—	4	4	no	—
195	C₈	—	8	2	no	—
196	C₈	C₅×#108	1	2	yes	complemented
197	C₅	#9624	1	1	yes	characteristic, direct product factor, Sylow
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	C₅×#389	1	1	yes	characteristic
201	C₂²	C₅×#499	1	1	yes	characteristic
202	C₂²	C₅×#500	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	C₂×C₅×#108	1	1	yes	characteristic
207	C₄	C₅×#430	1	1	yes	characteristic
208	C₄	C₄×C₅×D₈	1	1	yes	characteristic
209	C₄	C₄×C₅×Q₁₆	1	1	yes	characteristic
210	C₄	C₅×#466	1	1	yes	characteristic
211	C₄	C₅×#467	1	1	yes	characteristic
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	C₅×#1942	1	1	yes	characteristic
215	C₂	C₅×#1950	1	1	yes	characteristic
216	C₂	C₅×#1953	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	#389
prime radical	C₂³×C₅	C₄⋊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
19. #9624	C₅	direct (N × H)
107. C₅×C₈	#108	semidirect
196. C₈	C₅×#108	semidirect
197. C₅	#9624	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×#1950	2
C₅×#1567	2
C₂×C₅×#335	2
C₅×C₈×M₁₆	2
C₅×#1301	2
#9624	5

Subgroups — C5×#9624 #11932344

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#9624 #11932344