Subgroups — C₅×#9607 #11932327

Order 1280.

306 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#108	C₂	1	1	yes	maximal, characteristic
2	C₅×#1565	C₂	1	2	yes	maximal
3	C₂×C₅×#334	C₂	1	1	yes	maximal, characteristic
4	C₅×#1450	C₂	1	1	yes	maximal, characteristic
5	C₅×#1283	C₂	1	2	yes	maximal
6	C₅×#431	C₂²	1	1	yes	characteristic
7	C₅×#431	C₂²	1	1	yes	characteristic
8	C₂×C₅×#108	C₄	1	1	yes	characteristic
9	C₂×C₅×#108	C₄	1	1	yes	characteristic
10	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
11	C₄×C₅×M₁₆	C₂²	1	2	yes	—
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₅×#341	C₄	1	2	yes	—
14	C₅×#335	—	2	2	no	—
15	C₅×#334	—	2	2	no	—
16	C₅×#334	—	2	1	no	—
17	C₅×#334	—	2	1	no	—
18	C₅×#322	—	2	1	no	—
19	C₅×C₈²	—	2	1	no	—
20	#9607	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
22	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
23	C₂²×C₅×C₈	—	2	2	no	—
24	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
27	C₅×#108	—	2	2	no	—
28	C₅×#108	C₈	1	4	yes	complemented
29	C₅×C₄⋊C₈	—	2	2	no	—
30	C₅×C₄⋊C₈	—	2	2	no	—
31	C₅×C₄⋊C₈	—	2	1	no	—
32	C₅×C₄⋊C₈	D₄	1	2	yes	—
33	C₅×C₂²⋊C₈	—	2	1	no	—
34	C₅×C₂²⋊C₈	Q₈	1	2	yes	—
35	C₅×#97	—	2	2	no	—
36	C₄×C₅×C₈	—	2	1	no	—
37	C₄×C₅×C₈	—	2	1	no	—
38	C₄×C₅×C₈	—	2	2	no	—
39	C₄×C₅×C₈	D₄	1	1	yes	characteristic
40	C₄×C₅×C₈	D₄	1	1	yes	characteristic
41	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
42	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
43	C₄×#108	C₂×C₅	1	1	yes	characteristic
44	#1565	C₂×C₅	1	2	yes	—
45	C₂×#334	C₂×C₅	1	1	yes	characteristic
46	#1450	C₂×C₅	1	1	yes	characteristic
47	#1283	C₂×C₅	1	2	yes	—
48	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₅	#41	1	1	yes	characteristic
51	C₅×M₁₆	—	2	2	no	—
52	C₅×M₁₆	—	2	2	no	—
53	C₅×M₁₆	—	2	2	no	—
54	C₅×M₁₆	C₂×C₈	1	4	yes	—
55	C₂×C₅×C₈	—	2	2	no	—
56	C₂×C₅×C₈	—	2	1	no	—
57	C₂×C₅×C₈	—	2	1	no	—
58	C₂×C₅×C₈	—	2	2	no	—
59	C₂×C₅×C₈	—	2	2	no	—
60	C₂×C₅×C₈	—	2	2	no	—
61	C₂×C₅×C₈	—	4	2	no	—
62	C₂×C₅×C₈	—	8	2	no	—
63	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
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	C₂×C₈	1	2	yes	—
68	C₂×C₅×C₈	M₁₆	1	2	yes	—
69	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
72	C₄²×C₅	#41	1	1	yes	characteristic
73	#431	C₂²×C₅	1	1	yes	characteristic
74	#431	C₂²×C₅	1	1	yes	characteristic
75	C₂×#108	C₄×C₅	1	1	yes	characteristic
76	C₂×#108	C₄×C₅	1	1	yes	characteristic
77	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
78	C₄×M₁₆	C₂²×C₅	1	2	yes	—
79	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
80	#341	C₄×C₅	1	2	yes	—
81	#335	—	2	2	no	—
82	#334	—	2	2	no	—
83	#334	—	2	1	no	—
84	#334	—	2	1	no	—
85	#322	—	2	1	no	—
86	C₈²	—	2	1	no	—
87	C₂³×C₅	C₄×Q₈	1	1	yes	characteristic
88	C₂×C₄×C₅	—	2	2	no	—
89	C₂×C₄×C₅	—	4	1	no	—
90	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₅	C₄×D₄	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	C₄⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
98	C₂×C₄×C₅	#131	1	1	yes	characteristic
99	C₂×C₄×C₅	#131	1	1	yes	characteristic
100	C₅×C₈	—	2	1	no	—
101	C₅×C₈	—	2	1	no	—
102	C₅×C₈	—	2	4	no	—
103	C₅×C₈	—	4	2	no	—
104	C₅×C₈	—	4	2	no	—
105	C₅×C₈	—	4	2	no	—
106	C₅×C₈	—	8	2	no	—
107	C₅×C₈	C₄⋊C₈	1	2	yes	—
108	C₅×C₈	C₄⋊C₈	1	2	yes	—
109	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
110	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
113	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
114	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
115	#108	—	2	2	no	—
116	#108	C₅×C₈	1	4	yes	complemented
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	2	1	no	—
120	C₄⋊C₈	C₅×D₄	1	2	yes	—
121	C₂²⋊C₈	—	2	1	no	—
122	C₂²⋊C₈	C₅×Q₈	1	2	yes	—
123	#97	—	2	2	no	—
124	C₄×C₈	—	2	1	no	—
125	C₄×C₈	—	2	1	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	C₅×D₄	1	1	yes	characteristic
128	C₄×C₈	C₅×D₄	1	1	yes	characteristic
129	C₄×C₈	C₅×Q₈	1	1	yes	characteristic
130	C₄×C₈	C₅×Q₈	1	1	yes	characteristic
131	C₂²×C₅	—	2	1	no	—
132	C₂²×C₅	—	2	1	no	—
133	C₂²×C₅	#389	1	1	yes	characteristic
134	C₂²×C₅	C₈×Q₈	1	1	yes	characteristic
135	C₂²×C₅	#446	1	1	yes	characteristic
136	C₄×C₅	—	2	1	no	—
137	C₄×C₅	—	2	1	no	—
138	C₄×C₅	—	2	1	no	—
139	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
140	C₄×C₅	#424	1	1	yes	characteristic
141	C₄×C₅	C₈×D₄	1	1	yes	characteristic
142	C₄×C₅	#436	1	1	yes	characteristic
143	C₄×C₅	#471	1	2	yes	—
144	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
146	C₂²×C₄	C₅×#41	1	1	yes	characteristic
147	M₁₆	—	2	2	no	—
148	M₁₆	—	2	2	no	—
149	M₁₆	—	2	2	no	—
150	M₁₆	C₂×C₅×C₈	1	4	yes	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	8	2	no	—
159	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
161	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₈	C₂×C₅×C₈	1	2	yes	—
164	C₂×C₈	C₅×M₁₆	1	2	yes	—
165	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
166	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
167	C₄²	C₂×C₅×Q₈	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	#1939	1	1	yes	characteristic
172	C₂×C₅	#1951	1	1	yes	characteristic
173	C₂×C₅	#1953	1	1	yes	characteristic
174	C₂³	C₄×C₅×Q₈	1	1	yes	characteristic
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	1	no	—
177	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
178	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
179	C₂×C₄	C₅×#121	1	1	yes	characteristic
180	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
181	C₂×C₄	C₅×#125	1	1	yes	characteristic
182	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
183	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
184	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
185	C₂×C₄	C₅×#131	1	1	yes	characteristic
186	C₂×C₄	C₅×#131	1	1	yes	characteristic
187	C₈	—	2	1	no	—
188	C₈	—	2	1	no	—
189	C₈	—	2	4	no	—
190	C₈	—	4	2	no	—
191	C₈	—	4	2	no	—
192	C₈	—	4	2	no	—
193	C₈	—	8	2	no	—
194	C₈	C₅×C₄⋊C₈	1	2	yes	—
195	C₈	C₅×C₄⋊C₈	1	2	yes	—
196	C₅	#9607	1	1	yes	characteristic, direct product factor, Sylow
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	C₅×#389	1	1	yes	characteristic
200	C₂²	C₅×C₈×Q₈	1	1	yes	characteristic
201	C₂²	C₅×#446	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
206	C₄	C₅×#424	1	1	yes	characteristic
207	C₄	C₅×C₈×D₄	1	1	yes	characteristic
208	C₄	C₅×#436	1	1	yes	characteristic
209	C₄	C₅×#471	1	2	yes	—
210	C₂	—	2	1	no	—
211	C₂	—	2	1	no	—
212	C₂	C₅×#1939	1	1	yes	characteristic
213	C₂	C₅×#1951	1	1	yes	characteristic
214	C₂	C₅×#1953	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#121
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#389
prime radical	C₂³×C₅	C₄×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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
20. #9607	C₅	direct (N × H)
28. C₅×#108	C₈	semidirect
116. #108	C₅×C₈	semidirect
196. C₅	#9607	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×C₅×#108	2
C₅×#1565	2
C₂×C₅×#334	2
C₅×#1450	2
C₅×#1283	2
#9607	5

Subgroups — C5×#9607 #11932327

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#9607 #11932327