Subgroups — C₅×#29820 #11952540

Order 1280.

780 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#494	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3039	C₂	1	2	yes	maximal
3	C₅×#2922	C₂	1	1	yes	maximal, characteristic
4	C₅×#2047	C₂	1	4	yes	maximal, complemented
5	C₅×#2020	C₂	1	4	yes	maximal
6	C₅×#1966	C₂	1	2	yes	maximal, complemented
7	C₅×#1837	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#137	C₂²	1	4	yes	—
9	C₂²×C₅×Q₁₆	C₂²	1	4	yes	—
10	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₄×C₅×#41	C₂²	1	1	yes	characteristic
12	C₅×#494	—	2	2	no	—
13	C₅×#494	—	2	2	no	—
14	C₅×#462	—	2	4	no	—
15	C₅×#461	—	2	4	no	—
16	C₅×#433	—	2	2	no	—
17	C₅×#431	—	2	2	no	—
18	C₅×#424	C₂²	1	2	yes	—
19	C₅×#423	C₂²	1	2	yes	—
20	C₅×#419	C₂²	1	4	yes	—
21	C₂×C₅×#101	C₂²	1	8	yes	—
22	C₅×#408	C₂²	1	2	yes	—
23	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
25	C₅×#341	C₂²	1	4	yes	—
26	#29820	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂×C₅×#41	C₂³	1	1	yes	characteristic
28	C₂²×C₅×Q₈	C₂³	1	4	yes	—
29	C₅×#137	—	4	8	no	—
30	C₂×C₅×Q₁₆	—	2	4	no	—
31	C₂×C₅×Q₁₆	—	2	8	no	—
32	C₂×C₅×Q₁₆	—	4	8	no	—
33	C₂×C₅×SD₁₆	—	2	4	no	—
34	C₂×C₅×M₁₆	C₂³	1	2	yes	—
35	C₂×C₅×M₁₆	C₂³	1	4	yes	—
36	C₂²×C₅×C₈	C₂³	1	2	yes	—
37	C₅×C₄⋊Q₈	—	2	4	no	—
38	C₅×C₄⋊Q₈	—	2	2	no	—
39	C₅×C₄⋊Q₈	—	2	2	no	—
40	C₄×C₅×Q₈	D₄	1	2	yes	—
41	C₄×C₅×D₄	—	2	2	no	—
42	C₄×C₅×D₄	D₄	1	2	yes	—
43	C₅×#117	—	2	1	no	—
44	C₅×#117	C₂³	1	1	yes	characteristic
45	C₂×C₅×C₄⋊C₄	—	2	4	no	—
46	C₂×C₄²×C₅	—	2	1	no	—
47	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
48	C₅×C₄⋊C₈	—	2	4	no	—
49	C₅×C₄⋊C₈	—	2	4	no	—
50	C₅×#103	—	2	8	no	—
51	C₅×#101	—	4	8	no	—
52	C₅×C₂²⋊C₈	—	2	4	no	—
53	C₅×#97	—	2	1	no	—
54	C₄×C₅×C₈	—	2	1	no	—
55	C₄×C₅×C₈	D₄	1	2	yes	—
56	C₄×C₅×C₈	D₄	1	2	yes	—
57	C₂×#494	C₂×C₅	1	1	yes	characteristic, complemented
58	#3039	C₂×C₅	1	2	yes	—
59	#2922	C₂×C₅	1	1	yes	characteristic
60	#2047	C₂×C₅	1	4	yes	complemented
61	#2020	C₂×C₅	1	4	yes	—
62	#1966	C₂×C₅	1	2	yes	complemented
63	#1837	C₂×C₅	1	1	yes	characteristic, complemented
64	C₅×#41	—	2	4	no	—
65	C₂×C₅×Q₈	—	2	4	no	—
66	C₂×C₅×Q₈	—	2	4	no	—
67	C₂×C₅×Q₈	—	2	8	no	—
68	C₂×C₅×Q₈	—	2	8	no	—
69	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
70	C₂×C₅×D₄	—	2	1	no	—
71	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	—	2	2	no	—
73	C₂²×C₄×C₅	—	2	1	no	—
74	C₂²×C₄×C₅	—	2	4	no	—
75	C₂²×C₄×C₅	#41	1	2	yes	—
76	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
77	C₅×Q₁₆	—	4	16	no	—
78	C₅×Q₁₆	—	4	8	no	—
79	C₅×SD₁₆	—	4	8	no	—
80	C₅×M₁₆	—	2	8	no	—
81	C₅×M₁₆	—	2	4	no	—
82	C₂×C₅×C₈	—	2	4	no	—
83	C₂×C₅×C₈	—	2	4	no	—
84	C₂×C₅×C₈	—	2	2	no	—
85	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
86	C₅×C₄⋊C₄	—	2	2	no	—
87	C₅×C₄⋊C₄	—	4	4	no	—
88	C₅×C₄⋊C₄	—	4	4	no	—
89	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
90	C₅×C₂²⋊C₄	—	2	1	no	—
91	C₅×C₂²⋊C₄	—	2	2	no	—
92	C₄²×C₅	—	2	2	no	—
93	C₄²×C₅	—	2	1	no	—
94	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₅	C₂×D₄	1	2	yes	—
97	C₂×#137	C₂²×C₅	1	4	yes	—
98	C₂²×Q₁₆	C₂²×C₅	1	4	yes	—
99	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
100	C₄×#41	C₂²×C₅	1	1	yes	characteristic
101	#494	—	2	2	no	—
102	#494	—	2	2	no	—
103	#462	—	2	4	no	—
104	#461	—	2	4	no	—
105	#433	—	2	2	no	—
106	#431	—	2	2	no	—
107	#424	C₂²×C₅	1	2	yes	—
108	#423	C₂²×C₅	1	2	yes	—
109	#419	C₂²×C₅	1	4	yes	—
110	C₂×#101	C₂²×C₅	1	8	yes	—
111	#408	C₂²×C₅	1	2	yes	—
112	C₄×M₁₆	C₂²×C₅	1	1	yes	characteristic
113	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
114	#341	C₂²×C₅	1	4	yes	—
115	C₂³×C₅	—	2	1	no	—
116	C₂³×C₅	C₂×#41	1	1	yes	characteristic
117	C₅×Q₈	—	2	2	no	—
118	C₅×Q₈	—	4	4	no	—
119	C₅×Q₈	—	4	8	no	—
120	C₅×Q₈	—	4	4	no	—
121	C₅×D₄	—	2	2	no	—
122	C₅×D₄	—	4	2	no	—
123	C₂×C₄×C₅	—	2	4	no	—
124	C₂×C₄×C₅	—	2	2	no	—
125	C₂×C₄×C₅	—	2	1	no	—
126	C₂×C₄×C₅	—	4	2	no	—
127	C₂×C₄×C₅	—	4	4	no	—
128	C₂×C₄×C₅	—	4	4	no	—
129	C₂×C₄×C₅	—	4	8	no	—
130	C₂×C₄×C₅	#121	1	2	yes	—
131	C₂×C₄×C₅	#121	1	2	yes	—
132	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₅	H(5,𝔽₂)	1	4	yes	—
135	C₅×C₈	—	2	4	no	—
136	C₅×C₈	—	4	2	no	—
137	C₅×C₈	—	4	4	no	—
138	C₂×#41	C₂³×C₅	1	1	yes	characteristic
139	C₂²×Q₈	C₂³×C₅	1	4	yes	—
140	#137	—	4	8	no	—
141	C₂×Q₁₆	—	2	4	no	—
142	C₂×Q₁₆	—	2	8	no	—
143	C₂×Q₁₆	—	4	8	no	—
144	C₂×SD₁₆	—	2	4	no	—
145	C₂×M₁₆	C₂³×C₅	1	4	yes	—
146	C₂×M₁₆	C₂³×C₅	1	2	yes	—
147	C₂²×C₈	C₂³×C₅	1	2	yes	—
148	C₄⋊Q₈	—	2	4	no	—
149	C₄⋊Q₈	—	2	2	no	—
150	C₄⋊Q₈	—	2	2	no	—
151	C₄×Q₈	C₅×D₄	1	2	yes	—
152	C₄×D₄	—	2	2	no	—
153	C₄×D₄	C₅×D₄	1	2	yes	—
154	#117	—	2	1	no	—
155	#117	C₂³×C₅	1	1	yes	characteristic
156	C₂×C₄⋊C₄	—	2	4	no	—
157	C₂×C₄²	—	2	1	no	—
158	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
159	C₄⋊C₈	—	2	4	no	—
160	C₄⋊C₈	—	2	4	no	—
161	#103	—	2	8	no	—
162	#101	—	4	8	no	—
163	C₂²⋊C₈	—	2	4	no	—
164	#97	—	2	1	no	—
165	C₄×C₈	—	2	1	no	—
166	C₄×C₈	C₅×D₄	1	2	yes	—
167	C₄×C₈	C₅×D₄	1	2	yes	—
168	C₂²×C₅	—	2	2	no	—
169	C₂²×C₅	—	4	1	no	—
170	C₂²×C₅	—	4	2	no	—
171	C₂²×C₅	C₂×#121	1	1	yes	characteristic
172	C₂²×C₅	#540	1	2	yes	—
173	C₄×C₅	—	2	2	no	—
174	C₄×C₅	—	4	1	no	—
175	C₄×C₅	—	4	2	no	—
176	C₄×C₅	—	8	4	no	—
177	C₄×C₅	#473	1	4	yes	—
178	C₄×C₅	#535	1	2	yes	—
179	C₄×C₅	#535	1	2	yes	—
180	#41	—	2	4	no	—
181	C₂×Q₈	—	2	4	no	—
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	2	8	no	—
184	C₂×Q₈	—	2	8	no	—
185	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	C₅×#41	1	2	yes	—
192	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
193	Q₁₆	—	4	16	no	—
194	Q₁₆	—	4	8	no	—
195	SD₁₆	—	4	8	no	—
196	M₁₆	—	2	4	no	—
197	M₁₆	—	2	8	no	—
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	4	4	no	—
204	C₄⋊C₄	—	4	4	no	—
205	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	1	no	—
210	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
211	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
212	C₄²	C₂×C₅×D₄	1	2	yes	—
213	C₂×C₅	—	2	2	no	—
214	C₂×C₅	—	8	1	no	—
215	C₂×C₅	#2619	1	1	yes	characteristic
216	C₂×C₅	C₂×#473	1	2	yes	—
217	C₂³	—	2	1	no	—
218	C₂³	C₂×C₅×#41	1	1	yes	characteristic
219	Q₈	—	2	2	no	—
220	Q₈	—	4	4	no	—
221	Q₈	—	4	8	no	—
222	Q₈	—	4	4	no	—
223	D₄	—	2	2	no	—
224	D₄	—	4	2	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	8	no	—
232	C₂×C₄	C₅×#121	1	2	yes	—
233	C₂×C₄	C₅×#121	1	2	yes	—
234	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
235	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
236	C₂×C₄	C₅×H(5,𝔽₂)	1	4	yes	—
237	C₈	—	2	4	no	—
238	C₈	—	4	4	no	—
239	C₈	—	4	2	no	—
240	C₅	#29820	1	1	yes	characteristic, direct product factor, Sylow
241	C₂²	—	2	2	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	4	1	no	—
244	C₂²	C₂×C₅×#121	1	1	yes	characteristic
245	C₂²	C₅×#540	1	2	yes	—
246	C₄	—	2	2	no	—
247	C₄	—	4	2	no	—
248	C₄	—	4	1	no	—
249	C₄	—	8	4	no	—
250	C₄	C₅×#473	1	4	yes	—
251	C₄	C₅×#535	1	2	yes	—
252	C₄	C₅×#535	1	2	yes	—
253	C₂	—	2	2	no	—
254	C₂	—	8	1	no	—
255	C₂	C₅×#2619	1	1	yes	characteristic
256	C₂	C₂×C₅×#473	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂×C₅×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#494	C₂	semidirect
4. C₅×#2047	C₂	semidirect
6. C₅×#1966	C₂	semidirect
7. C₅×#1837	C₂	semidirect
26. #29820	C₅	direct (N × H)
57. C₂×#494	C₂×C₅	semidirect
60. #2047	C₂×C₅	semidirect
62. #1966	C₂×C₅	semidirect
63. #1837	C₂×C₅	semidirect
240. C₅	#29820	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×C₅×#494	2
C₅×#3039	2
C₅×#2922	2
C₅×#2047	2
C₅×#2020	2
C₅×#1966	2
C₅×#1837	2
#29820	5

Subgroups — C5×#29820 #11952540

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#29820 #11952540