Subgroups — C₅×#22312 #11945032

Order 1280.

610 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2399	C₂	1	1	yes	maximal, characteristic
2	C₅×#1985	C₂	1	4	yes	maximal
3	C₅×#1940	C₂	1	2	yes	maximal
4	C₅×#1917	C₂	1	4	yes	maximal
5	C₅×#1832	C₂	1	1	yes	maximal, characteristic
6	C₅×#1770	C₂	1	2	yes	maximal
7	C₅×#1746	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×C₅×Q₈	C₂²	1	2	yes	—
9	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
11	C₂×C₅×#97	C₂²	1	1	yes	characteristic
12	C₂×C₅×#97	C₂²	1	4	yes	—
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₅×#398	C₄	1	4	yes	—
15	C₅×#395	C₄	1	4	yes	—
16	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
17	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
18	C₅×#376	C₂²	1	1	yes	characteristic
19	C₅×#336	C₂²	1	4	yes	—
20	C₅×#336	C₂²	1	2	yes	—
21	C₅×#336	C₂²	1	2	yes	—
22	C₅×#336	C₂²	1	2	yes	—
23	C₅×#336	C₂²	1	4	yes	—
24	C₅×#336	C₂²	1	2	yes	—
25	#22312	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂²×C₅×Q₈	C₂×C₄	1	2	yes	—
27	C₂²×C₅×C₈	C₂³	1	2	yes	—
28	C₂²×C₅×C₈	C₂³	1	4	yes	—
29	C₂²×C₅×C₈	C₂³	1	2	yes	—
30	C₄×C₅×Q₈	—	4	4	no	—
31	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
32	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
33	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
34	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₅	C₂³	1	4	yes	—
38	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
39	C₅×C₄⋊C₈	—	2	4	no	—
40	C₅×C₄⋊C₈	—	4	4	no	—
41	C₅×#97	—	2	2	no	—
42	C₅×#97	—	4	4	no	—
43	C₄×C₅×C₈	Q₈	1	4	yes	complemented
44	C₅×#95	C₂×C₄	1	4	yes	—
45	C₅×#95	C₂×C₄	1	2	yes	—
46	C₅×#95	C₂×C₄	1	2	yes	—
47	C₅×#95	C₂×C₄	1	2	yes	—
48	C₅×#95	C₂×C₄	1	2	yes	—
49	#2399	C₂×C₅	1	1	yes	characteristic
50	#1985	C₂×C₅	1	4	yes	—
51	#1940	C₂×C₅	1	2	yes	—
52	#1917	C₂×C₅	1	4	yes	—
53	#1832	C₂×C₅	1	1	yes	characteristic
54	#1770	C₂×C₅	1	2	yes	—
55	#1746	C₂×C₅	1	1	yes	characteristic
56	C₂×C₅×Q₈	—	4	2	no	—
57	C₂×C₅×Q₈	—	4	4	no	—
58	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
64	C₂×C₅×C₈	—	2	2	no	—
65	C₂×C₅×C₈	—	2	2	no	—
66	C₂×C₅×C₈	—	2	2	no	—
67	C₂×C₅×C₈	—	2	4	no	—
68	C₂×C₅×C₈	—	2	4	no	—
69	C₂×C₅×C₈	—	2	2	no	—
70	C₂×C₅×C₈	—	2	2	no	—
71	C₂×C₅×C₈	—	2	4	no	—
72	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
73	C₅×C₄⋊C₄	—	4	4	no	—
74	C₅×C₄⋊C₄	—	4	4	no	—
75	C₅×C₄⋊C₄	—	4	2	no	—
76	C₅×C₄⋊C₄	—	4	4	no	—
77	C₄²×C₅	—	2	2	no	—
78	C₄²×C₅	—	2	4	no	—
79	C₄²×C₅	—	2	4	no	—
80	C₄²×C₅	—	2	2	no	—
81	C₄²×C₅	C₂×Q₈	1	2	yes	—
82	C₄²×C₅	#41	1	2	yes	—
83	C₂×C₄×Q₈	C₂²×C₅	1	2	yes	—
84	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
85	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
86	C₂×#97	C₂²×C₅	1	1	yes	characteristic
87	C₂×#97	C₂²×C₅	1	4	yes	—
88	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
89	#398	C₄×C₅	1	4	yes	—
90	#395	C₄×C₅	1	4	yes	—
91	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
92	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
93	#376	C₂²×C₅	1	1	yes	characteristic
94	#336	C₂²×C₅	1	2	yes	—
95	#336	C₂²×C₅	1	2	yes	—
96	#336	C₂²×C₅	1	2	yes	—
97	#336	C₂²×C₅	1	4	yes	—
98	#336	C₂²×C₅	1	4	yes	—
99	#336	C₂²×C₅	1	2	yes	—
100	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
101	C₅×Q₈	—	4	8	no	—
102	C₂×C₄×C₅	—	2	4	no	—
103	C₂×C₄×C₅	—	2	2	no	—
104	C₂×C₄×C₅	—	2	4	no	—
105	C₂×C₄×C₅	—	2	4	no	—
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	—	2	4	no	—
108	C₂×C₄×C₅	—	2	4	no	—
109	C₂×C₄×C₅	—	2	4	no	—
110	C₂×C₄×C₅	—	2	4	no	—
111	C₂×C₄×C₅	—	2	4	no	—
112	C₂×C₄×C₅	—	2	4	no	—
113	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
114	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
116	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
117	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
118	C₅×C₈	—	2	4	no	—
119	C₅×C₈	—	4	2	no	—
120	C₅×C₈	—	4	4	no	—
121	C₂²×Q₈	C₂×C₄×C₅	1	2	yes	—
122	C₂²×C₈	C₂³×C₅	1	2	yes	—
123	C₂²×C₈	C₂³×C₅	1	4	yes	—
124	C₂²×C₈	C₂³×C₅	1	2	yes	—
125	C₄×Q₈	—	4	4	no	—
126	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
127	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
128	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
129	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
130	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
131	C₂×C₄²	C₂³×C₅	1	4	yes	—
132	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
133	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
134	C₄⋊C₈	—	2	4	no	—
135	C₄⋊C₈	—	4	4	no	—
136	#97	—	2	2	no	—
137	#97	—	4	4	no	—
138	C₄×C₈	C₅×Q₈	1	4	yes	complemented
139	#95	C₂×C₄×C₅	1	4	yes	—
140	#95	C₂×C₄×C₅	1	2	yes	—
141	#95	C₂×C₄×C₅	1	2	yes	—
142	#95	C₂×C₄×C₅	1	2	yes	—
143	#95	C₂×C₄×C₅	1	2	yes	—
144	C₂²×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
145	C₂²×C₅	#518	1	1	yes	characteristic
146	C₂²×C₅	#519	1	1	yes	characteristic
147	C₂²×C₅	#568	1	4	yes	—
148	C₄×C₅	—	2	4	no	—
149	C₄×C₅	—	4	4	no	—
150	C₄×C₅	—	4	2	no	—
151	C₄×C₅	—	4	4	no	—
152	C₄×C₅	—	4	2	no	—
153	C₄×C₅	#542	1	2	yes	—
154	C₄×C₅	#544	1	2	yes	—
155	C₂×Q₈	—	4	2	no	—
156	C₂×Q₈	—	4	4	no	—
157	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
158	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
159	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
160	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
161	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
162	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	4	no	—
171	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	4	4	no	—
174	C₄⋊C₄	—	4	4	no	—
175	C₄⋊C₄	—	4	4	no	—
176	C₄²	—	2	4	no	—
177	C₄²	—	2	4	no	—
178	C₄²	—	2	2	no	—
179	C₄²	—	2	2	no	—
180	C₄²	C₂×C₅×Q₈	1	2	yes	—
181	C₄²	C₅×#41	1	2	yes	—
182	C₂×C₅	#2331	1	1	yes	characteristic
183	C₂×C₅	#2961	1	2	yes	—
184	C₂×C₅	#2978	1	2	yes	—
185	C₂×C₅	#2987	1	2	yes	—
186	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
187	Q₈	—	4	8	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	C₄×C₅×Q₈	1	4	yes	—
200	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
201	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
202	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
203	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
204	C₈	—	2	4	no	—
205	C₈	—	4	2	no	—
206	C₈	—	4	4	no	—
207	C₅	#22312	1	1	yes	characteristic, direct product factor, Sylow
208	C₂²	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
209	C₂²	C₅×#518	1	1	yes	characteristic
210	C₂²	C₅×#519	1	1	yes	characteristic
211	C₂²	C₅×#568	1	4	yes	—
212	C₄	—	2	4	no	—
213	C₄	—	4	2	no	—
214	C₄	—	4	2	no	—
215	C₄	—	4	4	no	—
216	C₄	—	4	4	no	—
217	C₄	C₅×#542	1	2	yes	—
218	C₄	C₅×#544	1	2	yes	—
219	C₂	C₅×#2331	1	1	yes	characteristic
220	C₂	C₅×#2961	1	2	yes	—
221	C₂	C₅×#2978	1	2	yes	—
222	C₂	C₅×#2987	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂⁴
commutator	C₂³	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle, prime radical	C₂³×C₅	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
25. #22312	C₅	direct (N × H)
43. C₄×C₅×C₈	Q₈	semidirect
138. C₄×C₈	C₅×Q₈	semidirect
207. C₅	#22312	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×#2399	2
C₅×#1985	2
C₅×#1940	2
C₅×#1917	2
C₅×#1832	2
C₅×#1770	2
C₅×#1746	2
#22312	5

Subgroups — C5×#22312 #11945032

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22312 #11945032