Subgroups — C₂×C₅×#1991 #11944037

Order 1280.

1246 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#125	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#382	C₂	1	1	yes	maximal, characteristic
4	C₅×#1991	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×#340	C₂	1	4	yes	maximal
6	C₂³×C₅×C₈	C₂²	1	2	yes	—
7	C₂×C₅×#125	—	2	2	no	—
8	C₂×C₅×#125	—	2	4	no	—
9	C₂×C₅×#125	C₄	1	8	yes	complemented
10	C₂×C₅×#125	C₂²	1	4	yes	—
11	C₂²×C₅×C₄⋊C₄	—	2	2	no	—
12	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×C₈	—	2	4	no	—
16	C₂×C₄×C₅×C₈	C₂²	1	4	yes	—
17	C₅×#382	—	2	2	no	—
18	C₅×#382	C₂²	1	4	yes	—
19	C₂×C₅×#95	—	2	2	no	—
20	C₅×#340	C₂²	1	16	yes	—
21	C₂×#1991	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂³×C₄×C₅	—	2	2	no	—
23	C₂³×C₄×C₅	—	2	2	no	—
24	C₂³×C₄×C₅	D₄	1	2	yes	—
25	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂²×C₅×C₈	—	2	4	no	—
27	C₂²×C₅×C₈	—	2	4	no	—
28	C₂²×C₅×C₈	Q₈	1	8	yes	—
29	C₂²×C₅×C₈	C₂³	1	4	yes	—
30	C₅×#125	—	2	16	no	—
31	C₅×#125	—	2	8	no	—
32	C₅×#125	C₂×C₄	1	16	yes	complemented
33	C₂×C₅×C₄⋊C₄	—	2	16	no	—
34	C₂×C₅×C₄⋊C₄	—	2	8	no	—
35	C₂×C₅×C₄⋊C₄	—	2	2	no	—
36	C₂×C₅×C₄⋊C₄	—	2	4	no	—
37	C₂×C₅×C₄⋊C₄	—	2	8	no	—
38	C₂×C₅×C₄⋊C₄	—	2	2	no	—
39	C₂×C₅×C₄⋊C₄	C₂×C₄	1	8	yes	—
40	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
41	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
42	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₅	D₄	1	2	yes	—
44	C₂×C₄²×C₅	D₄	1	4	yes	—
45	C₂×C₄²×C₅	C₂³	1	2	yes	—
46	C₄×C₅×C₈	—	2	8	no	—
47	C₅×#95	—	2	8	no	—
48	C₂²×#125	C₂×C₅	1	1	yes	characteristic
49	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic
50	C₂×#382	C₂×C₅	1	1	yes	characteristic
51	#1991	C₂×C₅	1	8	yes	direct product factor
52	C₂×#340	C₂×C₅	1	4	yes	—
53	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	—	2	8	no	—
55	C₂²×C₄×C₅	—	2	8	no	—
56	C₂²×C₄×C₅	—	2	4	no	—
57	C₂²×C₄×C₅	—	2	4	no	—
58	C₂²×C₄×C₅	—	4	8	no	—
59	C₂²×C₄×C₅	—	4	4	no	—
60	C₂²×C₄×C₅	—	4	8	no	—
61	C₂²×C₄×C₅	C₂²⋊C₄	1	8	yes	—
62	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
64	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
67	C₂²×C₄×C₅	#41	1	4	yes	—
68	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
69	C₂×C₅×C₈	—	2	16	no	—
70	C₂×C₅×C₈	—	2	4	no	—
71	C₂×C₅×C₈	—	2	4	no	—
72	C₂×C₅×C₈	C₂×Q₈	1	8	yes	—
73	C₅×C₄⋊C₄	—	2	32	no	—
74	C₅×C₄⋊C₄	—	2	8	no	—
75	C₅×C₄⋊C₄	—	2	4	no	—
76	C₅×C₄⋊C₄	—	2	8	no	—
77	C₅×C₄⋊C₄	—	4	4	no	—
78	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
79	C₄²×C₅	C₂²⋊C₄	1	8	yes	—
80	C₄²×C₅	C₂²×C₄	1	4	yes	—
81	C₄²×C₅	C₂×D₄	1	4	yes	—
82	C₂³×C₈	C₂²×C₅	1	2	yes	—
83	C₂×#125	—	2	2	no	—
84	C₂×#125	—	2	4	no	—
85	C₂×#125	C₄×C₅	1	8	yes	complemented
86	C₂×#125	C₂²×C₅	1	4	yes	—
87	C₂²×C₄⋊C₄	—	2	2	no	—
88	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
89	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
90	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
91	C₂×C₄×C₈	—	2	4	no	—
92	C₂×C₄×C₈	C₂²×C₅	1	4	yes	—
93	#382	—	2	2	no	—
94	#382	C₂²×C₅	1	4	yes	—
95	C₂×#95	—	2	2	no	—
96	#340	C₂²×C₅	1	16	yes	—
97	C₂³×C₅	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂³×C₅	#124	1	2	yes	—
99	C₂³×C₅	C₄⋊Q₈	1	2	yes	—
100	C₂³×C₅	#135	1	8	yes	—
101	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₅	—	2	8	no	—
103	C₂×C₄×C₅	—	2	16	no	—
104	C₂×C₄×C₅	—	2	8	no	—
105	C₂×C₄×C₅	—	4	8	no	—
106	C₂×C₄×C₅	—	4	16	no	—
107	C₂×C₄×C₅	—	4	4	no	—
108	C₂×C₄×C₅	—	4	8	no	—
109	C₂×C₄×C₅	—	8	8	no	—
110	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	8	yes	—
111	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
112	C₂×C₄×C₅	C₄×Q₈	1	8	yes	—
113	C₂×C₄×C₅	C₂²⋊Q₈	1	8	yes	—
114	C₂×C₄×C₅	#124	1	2	yes	—
115	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
116	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
119	C₂×C₄×C₅	C₂×#41	1	2	yes	—
120	C₅×C₈	—	2	8	no	—
121	C₂³×C₄	—	2	2	no	—
122	C₂³×C₄	—	2	2	no	—
123	C₂³×C₄	C₅×D₄	1	2	yes	—
124	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
125	C₂²×C₈	—	2	4	no	—
126	C₂²×C₈	—	2	4	no	—
127	C₂²×C₈	C₅×Q₈	1	8	yes	—
128	C₂²×C₈	C₂³×C₅	1	4	yes	—
129	#125	—	2	16	no	—
130	#125	—	2	8	no	—
131	#125	C₂×C₄×C₅	1	16	yes	complemented
132	C₂×C₄⋊C₄	—	2	16	no	—
133	C₂×C₄⋊C₄	—	2	8	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	2	4	no	—
136	C₂×C₄⋊C₄	—	2	8	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	C₂×C₄×C₅	1	8	yes	—
139	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
140	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
141	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
142	C₂×C₄²	C₅×D₄	1	2	yes	—
143	C₂×C₄²	C₅×D₄	1	4	yes	—
144	C₂×C₄²	C₂³×C₅	1	2	yes	—
145	C₄×C₈	—	2	8	no	—
146	#95	—	2	8	no	—
147	C₂²×C₅	#391	1	4	yes	—
148	C₂²×C₅	#416	1	8	yes	—
149	C₂²×C₅	#488	1	8	yes	—
150	C₂²×C₅	#499	1	8	yes	—
151	C₂²×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₅	C₂×#124	1	1	yes	characteristic
153	C₂²×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
154	C₂²×C₅	C₂×#135	1	4	yes	—
155	C₄×C₅	—	2	8	no	—
156	C₄×C₅	—	4	8	no	—
157	C₄×C₅	—	8	4	no	—
158	C₄×C₅	#391	1	4	yes	—
159	C₄×C₅	C₂×C₄×Q₈	1	2	yes	—
160	C₄×C₅	C₂×C₂²⋊Q₈	1	2	yes	—
161	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
162	C₂²×C₄	—	2	8	no	—
163	C₂²×C₄	—	2	8	no	—
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	—	4	8	no	—
167	C₂²×C₄	—	4	4	no	—
168	C₂²×C₄	—	4	8	no	—
169	C₂²×C₄	C₅×C₂²⋊C₄	1	8	yes	—
170	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
171	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
172	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
174	C₂²×C₄	C₂×C₅×Q₈	1	4	yes	—
175	C₂²×C₄	C₅×#41	1	4	yes	—
176	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
177	C₂×C₈	—	2	16	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	C₂×C₅×Q₈	1	8	yes	—
181	C₄⋊C₄	—	2	32	no	—
182	C₄⋊C₄	—	2	8	no	—
183	C₄⋊C₄	—	2	4	no	—
184	C₄⋊C₄	—	2	8	no	—
185	C₄⋊C₄	—	4	4	no	—
186	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
187	C₄²	C₅×C₂²⋊C₄	1	8	yes	—
188	C₄²	C₂²×C₄×C₅	1	4	yes	—
189	C₄²	C₂×C₅×D₄	1	4	yes	—
190	C₂×C₅	#1991	1	8	yes	direct product factor
191	C₂×C₅	C₂×#391	1	1	yes	characteristic
192	C₂×C₅	C₂×#416	1	2	yes	—
193	C₂×C₅	C₂×#488	1	2	yes	—
194	C₂×C₅	C₂×#499	1	2	yes	—
195	C₂³	C₂×C₅×C₂²⋊C₄	1	2	yes	—
196	C₂³	C₅×#124	1	2	yes	—
197	C₂³	C₅×C₄⋊Q₈	1	2	yes	—
198	C₂³	C₅×#135	1	8	yes	—
199	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
200	C₂×C₄	—	2	16	no	—
201	C₂×C₄	—	2	8	no	—
202	C₂×C₄	—	2	8	no	—
203	C₂×C₄	—	4	16	no	—
204	C₂×C₄	—	4	8	no	—
205	C₂×C₄	—	4	4	no	—
206	C₂×C₄	—	4	8	no	—
207	C₂×C₄	—	8	8	no	—
208	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	8	yes	—
209	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
210	C₂×C₄	C₄×C₅×Q₈	1	8	yes	—
211	C₂×C₄	C₅×C₂²⋊Q₈	1	8	yes	—
212	C₂×C₄	C₅×#124	1	2	yes	—
213	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
214	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
215	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
216	C₂×C₄	C₂²×C₅×Q₈	1	2	yes	—
217	C₂×C₄	C₂×C₅×#41	1	2	yes	—
218	C₈	—	2	8	no	—
219	C₅	C₂×#1991	1	1	yes	characteristic, direct product factor, Sylow
220	C₂²	C₅×#391	1	4	yes	—
221	C₂²	C₅×#416	1	8	yes	—
222	C₂²	C₅×#488	1	8	yes	—
223	C₂²	C₅×#499	1	8	yes	—
224	C₂²	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
225	C₂²	C₂×C₅×#124	1	1	yes	characteristic
226	C₂²	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
227	C₂²	C₂×C₅×#135	1	4	yes	—
228	C₄	—	2	8	no	—
229	C₄	—	4	8	no	—
230	C₄	—	8	4	no	—
231	C₄	C₅×#391	1	4	yes	—
232	C₄	C₂×C₄×C₅×Q₈	1	2	yes	—
233	C₄	C₂×C₅×C₂²⋊Q₈	1	2	yes	—
234	C₂	C₅×#1991	1	8	yes	direct product factor
235	C₂	C₂×C₅×#391	1	1	yes	characteristic
236	C₂	C₂×C₅×#416	1	2	yes	—
237	C₂	C₂×C₅×#488	1	2	yes	—
238	C₂	C₂×C₅×#499	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1991	C₂	direct (N × H)
9. C₂×C₅×#125	C₄	semidirect
21. C₂×#1991	C₅	direct (N × H)
32. C₅×#125	C₂×C₄	semidirect
51. #1991	C₂×C₅	direct (N × H)
85. C₂×#125	C₄×C₅	semidirect
131. #125	C₂×C₄×C₅	semidirect
190. C₂×C₅	#1991	direct (N × H)
219. C₅	C₂×#1991	direct (N × H)
234. C₂	C₅×#1991	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×#125	2
C₂²×C₄×C₅×C₈	2
C₂×C₅×#382	2
C₅×#1991	2
C₂×C₅×#340	2
C₂×#1991	5

Subgroups — C2×C5×#1991 #11944037

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1991 #11944037