Subgroups — C₅×#21927 #11944647

Order 1280.

628 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2298	C₂	1	1	yes	maximal, characteristic
2	C₅×#1924	C₂	1	4	yes	maximal
3	C₅×#1917	C₂	1	4	yes	maximal, complemented
4	C₅×#1843	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1832	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1798	C₂	1	2	yes	maximal
7	C₅×#1767	C₂	1	2	yes	maximal, complemented
8	C₂×C₅×#117	C₂²	1	1	yes	characteristic
9	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
12	C₂×C₅×C₂²⋊C₈	C₂²	1	4	yes	—
13	C₂×C₅×#97	C₂²	1	4	yes	—
14	C₅×#388	C₄	1	4	yes	—
15	C₅×#387	C₄	1	4	yes	—
16	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
17	C₄×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
18	C₅×#376	C₂²	1	1	yes	characteristic
19	C₄³×C₅	C₂²	1	1	yes	characteristic
20	C₅×#336	C₂²	1	4	yes	—
21	C₅×#336	C₂²	1	4	yes	—
22	C₅×#336	C₂²	1	4	yes	—
23	#21927	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂²×C₅×C₈	C₂³	1	4	yes	—
26	C₂²×C₅×C₈	C₂³	1	4	yes	—
27	C₅×#117	—	2	2	no	—
28	C₅×#117	—	2	2	no	—
29	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
30	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
31	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
32	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₅	C₂³	1	2	yes	—
35	C₂×C₄²×C₅	C₂³	1	2	yes	—
36	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
38	C₅×C₄⋊C₈	—	2	4	no	—
39	C₅×C₄⋊C₈	—	2	4	no	—
40	C₅×C₄⋊C₈	—	4	4	no	—
41	C₅×C₂²⋊C₈	—	4	4	no	—
42	C₅×#97	—	4	4	no	—
43	C₅×#95	C₂×C₄	1	4	yes	—
44	C₅×#95	C₂×C₄	1	4	yes	—
45	C₅×#95	C₂×C₄	1	4	yes	—
46	#2298	C₂×C₅	1	1	yes	characteristic
47	#1924	C₂×C₅	1	4	yes	—
48	#1917	C₂×C₅	1	4	yes	complemented
49	#1843	C₂×C₅	1	1	yes	characteristic, complemented
50	#1832	C₂×C₅	1	1	yes	characteristic, complemented
51	#1798	C₂×C₅	1	2	yes	—
52	#1767	C₂×C₅	1	2	yes	complemented
53	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	—	2	1	no	—
55	C₂²×C₄×C₅	—	2	1	no	—
56	C₂²×C₄×C₅	—	2	2	no	—
57	C₂²×C₄×C₅	—	2	2	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	1	yes	characteristic
62	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
64	C₂×C₅×C₈	—	2	4	no	—
65	C₂×C₅×C₈	—	2	4	no	—
66	C₂×C₅×C₈	—	2	4	no	—
67	C₂×C₅×C₈	—	2	4	no	—
68	C₂×C₅×C₈	—	2	4	no	—
69	C₂×C₅×C₈	—	2	4	no	—
70	C₅×C₄⋊C₄	—	2	4	no	—
71	C₅×C₄⋊C₄	—	4	4	no	—
72	C₅×C₂²⋊C₄	—	2	4	no	—
73	C₅×C₂²⋊C₄	—	4	4	no	—
74	C₄²×C₅	—	2	1	no	—
75	C₄²×C₅	—	2	1	no	—
76	C₄²×C₅	—	2	2	no	—
77	C₄²×C₅	—	2	2	no	—
78	C₄²×C₅	—	2	2	no	—
79	C₄²×C₅	—	2	2	no	—
80	C₄²×C₅	—	4	4	no	—
81	C₄²×C₅	#41	1	2	yes	—
82	C₄²×C₅	#41	1	2	yes	—
83	C₂×#117	C₂²×C₅	1	1	yes	characteristic
84	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
85	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
86	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
87	C₂×C₂²⋊C₈	C₂²×C₅	1	4	yes	—
88	C₂×#97	C₂²×C₅	1	4	yes	—
89	#388	C₄×C₅	1	4	yes	—
90	#387	C₄×C₅	1	4	yes	—
91	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
92	C₄×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
93	#376	C₂²×C₅	1	1	yes	characteristic
94	C₄³	C₂²×C₅	1	1	yes	characteristic
95	#336	C₂²×C₅	1	4	yes	—
96	#336	C₂²×C₅	1	4	yes	—
97	#336	C₂²×C₅	1	4	yes	—
98	C₂³×C₅	—	2	1	no	—
99	C₂³×C₅	—	2	2	no	—
100	C₂³×C₅	—	2	4	no	—
101	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₅	—	2	4	no	—
103	C₂×C₄×C₅	—	2	2	no	—
104	C₂×C₄×C₅	—	2	1	no	—
105	C₂×C₄×C₅	—	2	4	no	—
106	C₂×C₄×C₅	—	2	4	no	—
107	C₂×C₄×C₅	—	2	2	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	—	4	1	no	—
114	C₂×C₄×C₅	—	4	1	no	—
115	C₂×C₄×C₅	—	4	2	no	—
116	C₂×C₄×C₅	#117	1	4	yes	—
117	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
119	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
120	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
121	C₅×C₈	—	4	4	no	—
122	C₅×C₈	—	4	4	no	—
123	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
124	C₂²×C₈	C₂³×C₅	1	4	yes	—
125	C₂²×C₈	C₂³×C₅	1	4	yes	—
126	#117	—	2	2	no	—
127	#117	—	2	2	no	—
128	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
129	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
130	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	2	yes	—
131	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	4	yes	—
132	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
133	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
134	C₂×C₄²	C₂³×C₅	1	2	yes	—
135	C₂×C₄²	C₂³×C₅	1	2	yes	—
136	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
137	C₄⋊C₈	—	2	4	no	—
138	C₄⋊C₈	—	2	4	no	—
139	C₄⋊C₈	—	4	4	no	—
140	C₂²⋊C₈	—	4	4	no	—
141	#97	—	4	4	no	—
142	#95	C₂×C₄×C₅	1	4	yes	—
143	#95	C₂×C₄×C₅	1	4	yes	—
144	#95	C₂×C₄×C₅	1	4	yes	—
145	C₂²×C₅	—	4	1	no	—
146	C₂²×C₅	—	4	2	no	—
147	C₂²×C₅	—	4	4	no	—
148	C₂²×C₅	C₂×#117	1	1	yes	characteristic
149	C₂²×C₅	#520	1	2	yes	—
150	C₂²×C₅	#568	1	4	yes	—
151	C₄×C₅	—	2	4	no	—
152	C₄×C₅	—	4	1	no	—
153	C₄×C₅	—	4	2	no	—
154	C₄×C₅	—	4	4	no	—
155	C₄×C₅	—	4	4	no	—
156	C₄×C₅	#540	1	1	yes	characteristic
157	C₄×C₅	#541	1	1	yes	characteristic
158	C₄×C₅	#542	1	2	yes	—
159	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
165	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
166	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
167	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
168	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
169	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
170	C₂×C₈	—	2	4	no	—
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	2	4	no	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	4	4	no	—
178	C₂²⋊C₄	—	2	4	no	—
179	C₂²⋊C₄	—	4	4	no	—
180	C₄²	—	2	1	no	—
181	C₄²	—	2	1	no	—
182	C₄²	—	2	2	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	4	4	no	—
187	C₄²	C₅×#41	1	2	yes	—
188	C₄²	C₅×#41	1	2	yes	—
189	C₂×C₅	—	8	1	no	—
190	C₂×C₅	#2314	1	1	yes	characteristic
191	C₂×C₅	#2923	1	2	yes	—
192	C₂×C₅	#2995	1	4	yes	—
193	C₂³	—	2	1	no	—
194	C₂³	—	2	2	no	—
195	C₂³	—	2	4	no	—
196	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	4	1	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	C₅×#117	1	4	yes	—
212	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
213	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
214	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
215	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
216	C₈	—	4	4	no	—
217	C₈	—	4	4	no	—
218	C₅	#21927	1	1	yes	characteristic, direct product factor, Sylow
219	C₂²	—	4	4	no	—
220	C₂²	—	4	1	no	—
221	C₂²	—	4	2	no	—
222	C₂²	C₂×C₅×#117	1	1	yes	characteristic
223	C₂²	C₅×#520	1	2	yes	—
224	C₂²	C₅×#568	1	4	yes	—
225	C₄	—	2	4	no	—
226	C₄	—	4	4	no	—
227	C₄	—	4	4	no	—
228	C₄	—	4	1	no	—
229	C₄	—	4	2	no	—
230	C₄	C₅×#540	1	1	yes	characteristic
231	C₄	C₅×#541	1	1	yes	characteristic
232	C₄	C₅×#542	1	2	yes	—
233	C₂	—	8	1	no	—
234	C₂	C₅×#2314	1	1	yes	characteristic
235	C₂	C₅×#2923	1	2	yes	—
236	C₂	C₅×#2995	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂⁴
commutator	C₂³	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂³×C₅	C₂³×C₄
prime radical	C₂⁴×C₅	C₂²×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
3. C₅×#1917	C₂	semidirect
4. C₅×#1843	C₂	semidirect
5. C₅×#1832	C₂	semidirect
7. C₅×#1767	C₂	semidirect
23. #21927	C₅	direct (N × H)
48. #1917	C₂×C₅	semidirect
49. #1843	C₂×C₅	semidirect
50. #1832	C₂×C₅	semidirect
52. #1767	C₂×C₅	semidirect
218. C₅	#21927	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×#2298	2
C₅×#1924	2
C₅×#1917	2
C₅×#1843	2
C₅×#1832	2
C₅×#1798	2
C₅×#1767	2
#21927	5

Subgroups — C5×#21927 #11944647

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21927 #11944647