Subgroups — C₅×#21691 #11944411

Order 1280.

1062 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	C₂×C₅×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#441	C₂	1	4	yes	maximal
3	C₄×C₅×C₄⋊Q₈	C₂	1	2	yes	maximal
4	C₅×#1971	C₂	1	4	yes	maximal
5	C₅×#1956	C₂	1	2	yes	maximal
6	C₅×#1844	C₂	1	1	yes	maximal, characteristic
7	C₄×C₅×#97	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₅×Q₁₆	C₂²	1	4	yes	—
9	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₄×C₅×Q₈	C₂²	1	4	yes	—
11	C₅×#494	C₄	1	8	yes	complemented
12	C₅×#441	—	2	16	no	—
13	C₂×C₅×#106	C₂²	1	4	yes	—
14	C₂×C₅×#103	C₂²	1	8	yes	—
15	C₂×C₅×#97	C₂²	1	2	yes	—
16	C₂×C₅×#97	C₂²	1	2	yes	—
17	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₅×#391	C₂²	1	4	yes	—
20	C₅×#389	—	2	4	no	—
21	C₅×#384	C₂²	1	2	yes	—
22	C₄×C₅×C₄⋊C₄	—	2	4	no	—
23	C₄³×C₅	C₂²	1	1	yes	characteristic
24	#21691	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂²×C₅×Q₈	C₂³	1	4	yes	—
26	C₂×C₅×Q₁₆	—	2	8	no	—
27	C₂×C₅×Q₁₆	—	2	8	no	—
28	C₂×C₅×Q₁₆	C₂×C₄	1	16	yes	—
29	C₂²×C₅×C₈	C₂³	1	2	yes	—
30	C₂²×C₅×C₈	C₂³	1	2	yes	—
31	C₅×C₄⋊Q₈	—	2	4	no	—
32	C₅×C₄⋊Q₈	C₂×C₄	1	8	yes	—
33	C₄×C₅×Q₈	—	2	16	no	—
34	C₂×C₅×C₄⋊C₄	—	2	4	no	—
35	C₂×C₅×C₄⋊C₄	—	2	4	no	—
36	C₂×C₅×C₄⋊C₄	—	2	4	no	—
37	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
38	C₂×C₄²×C₅	—	2	4	no	—
39	C₂×C₄²×C₅	D₄	1	4	yes	—
40	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₅×#106	—	2	8	no	—
44	C₅×#103	—	2	16	no	—
45	C₅×#97	—	2	4	no	—
46	C₅×#97	D₄	1	8	yes	—
47	C₄×C₅×C₈	—	2	2	no	—
48	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
49	C₅×#95	—	2	8	no	—
50	C₂×#494	C₂×C₅	1	1	yes	characteristic
51	C₂×#441	C₂×C₅	1	4	yes	—
52	C₄×C₄⋊Q₈	C₂×C₅	1	2	yes	—
53	#1971	C₂×C₅	1	4	yes	—
54	#1956	C₂×C₅	1	2	yes	—
55	#1844	C₂×C₅	1	1	yes	characteristic
56	C₄×#97	C₂×C₅	1	1	yes	characteristic
57	C₂×C₅×Q₈	—	2	4	no	—
58	C₂×C₅×Q₈	—	2	8	no	—
59	C₂×C₅×Q₈	—	2	8	no	—
60	C₂×C₅×Q₈	C₂²×C₄	1	8	yes	—
61	C₂²×C₄×C₅	—	2	4	no	—
62	C₂²×C₄×C₅	—	2	4	no	—
63	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
67	C₅×Q₁₆	—	2	32	no	—
68	C₂×C₅×C₈	—	2	2	no	—
69	C₂×C₅×C₈	—	2	2	no	—
70	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
71	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
72	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
73	C₂×C₅×C₈	#41	1	4	yes	—
74	C₅×C₄⋊C₄	—	2	16	no	—
75	C₅×C₄⋊C₄	—	2	8	no	—
76	C₅×C₄⋊C₄	—	4	8	no	—
77	C₅×C₄⋊C₄	—	4	8	no	—
78	C₄²×C₅	—	2	8	no	—
79	C₄²×C₅	—	2	2	no	—
80	C₄²×C₅	—	4	8	no	—
81	C₄²×C₅	C₂²×C₄	1	2	yes	—
82	C₄²×C₅	C₂×D₄	1	4	yes	—
83	C₄²×C₅	#41	1	2	yes	—
84	C₂²×Q₁₆	C₂²×C₅	1	4	yes	—
85	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
86	C₂×C₄×Q₈	C₂²×C₅	1	4	yes	—
87	#494	C₄×C₅	1	8	yes	complemented
88	#441	—	2	16	no	—
89	C₂×#106	C₂²×C₅	1	4	yes	—
90	C₂×#103	C₂²×C₅	1	8	yes	—
91	C₂×#97	C₂²×C₅	1	2	yes	—
92	C₂×#97	C₂²×C₅	1	2	yes	—
93	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
94	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
95	#391	C₂²×C₅	1	4	yes	—
96	#389	—	2	4	no	—
97	#384	C₂²×C₅	1	2	yes	—
98	C₄×C₄⋊C₄	—	2	4	no	—
99	C₄³	C₂²×C₅	1	1	yes	characteristic
100	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
101	C₅×Q₈	—	2	16	no	—
102	C₅×Q₈	—	4	8	no	—
103	C₂×C₄×C₅	—	2	4	no	—
104	C₂×C₄×C₅	—	2	8	no	—
105	C₂×C₄×C₅	—	2	2	no	—
106	C₂×C₄×C₅	—	2	4	no	—
107	C₂×C₄×C₅	—	4	4	no	—
108	C₂×C₄×C₅	—	4	4	no	—
109	C₂×C₄×C₅	—	4	4	no	—
110	C₂×C₄×C₅	—	4	8	no	—
111	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
112	C₂×C₄×C₅	#127	1	4	yes	—
113	C₂×C₄×C₅	#137	1	8	yes	—
114	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
116	C₂×C₄×C₅	C₂×#41	1	2	yes	—
117	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
118	C₅×C₈	—	2	4	no	—
119	C₅×C₈	C₄×D₄	1	8	yes	—
120	C₂²×Q₈	C₂³×C₅	1	4	yes	—
121	C₂×Q₁₆	—	2	8	no	—
122	C₂×Q₁₆	—	2	8	no	—
123	C₂×Q₁₆	C₂×C₄×C₅	1	16	yes	—
124	C₂²×C₈	C₂³×C₅	1	2	yes	—
125	C₂²×C₈	C₂³×C₅	1	2	yes	—
126	C₄⋊Q₈	—	2	4	no	—
127	C₄⋊Q₈	C₂×C₄×C₅	1	8	yes	—
128	C₄×Q₈	—	2	16	no	—
129	C₂×C₄⋊C₄	—	2	4	no	—
130	C₂×C₄⋊C₄	—	2	4	no	—
131	C₂×C₄⋊C₄	—	2	4	no	—
132	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
133	C₂×C₄²	—	2	4	no	—
134	C₂×C₄²	C₅×D₄	1	4	yes	—
135	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
136	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
137	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
138	#106	—	2	8	no	—
139	#103	—	2	16	no	—
140	#97	—	2	4	no	—
141	#97	C₅×D₄	1	8	yes	—
142	C₄×C₈	—	2	2	no	—
143	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
144	#95	—	2	8	no	—
145	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
146	C₂²×C₅	C₂×#127	1	1	yes	characteristic
147	C₂²×C₅	#532	1	1	yes	characteristic
148	C₂²×C₅	C₂×#137	1	4	yes	—
149	C₄×C₅	—	2	4	no	—
150	C₄×C₅	—	4	8	no	—
151	C₄×C₅	—	4	2	no	—
152	C₄×C₅	—	8	4	no	—
153	C₄×C₅	#441	1	8	yes	complemented
154	C₄×C₅	C₂×C₄×D₄	1	2	yes	—
155	C₄×C₅	#532	1	2	yes	—
156	C₂×Q₈	—	2	4	no	—
157	C₂×Q₈	—	2	8	no	—
158	C₂×Q₈	—	2	8	no	—
159	C₂×Q₈	C₂²×C₄×C₅	1	8	yes	—
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
163	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
164	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
165	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
166	Q₁₆	—	2	32	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
170	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
171	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
172	C₂×C₈	C₅×#41	1	4	yes	—
173	C₄⋊C₄	—	2	16	no	—
174	C₄⋊C₄	—	2	8	no	—
175	C₄⋊C₄	—	4	8	no	—
176	C₄⋊C₄	—	4	8	no	—
177	C₄²	—	2	8	no	—
178	C₄²	—	2	2	no	—
179	C₄²	—	4	8	no	—
180	C₄²	C₂²×C₄×C₅	1	2	yes	—
181	C₄²	C₂×C₅×D₄	1	4	yes	—
182	C₄²	C₅×#41	1	2	yes	—
183	C₂×C₅	C₄×#127	1	1	yes	characteristic
184	C₂×C₅	C₂×#441	1	2	yes	—
185	C₂×C₅	#3152	1	2	yes	—
186	C₂×C₅	#3185	1	2	yes	—
187	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
188	Q₈	—	2	16	no	—
189	Q₈	—	4	8	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	4	8	no	—
198	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
199	C₂×C₄	C₅×#127	1	4	yes	—
200	C₂×C₄	C₅×#137	1	8	yes	—
201	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
202	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
203	C₂×C₄	C₂×C₅×#41	1	2	yes	—
204	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
205	C₈	—	2	4	no	—
206	C₈	C₄×C₅×D₄	1	8	yes	—
207	C₅	#21691	1	1	yes	characteristic, direct product factor, Sylow
208	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
209	C₂²	C₂×C₅×#127	1	1	yes	characteristic
210	C₂²	C₅×#532	1	1	yes	characteristic
211	C₂²	C₂×C₅×#137	1	4	yes	—
212	C₄	—	2	4	no	—
213	C₄	—	4	8	no	—
214	C₄	—	4	2	no	—
215	C₄	—	8	4	no	—
216	C₄	C₅×#441	1	8	yes	complemented
217	C₄	C₂×C₄×C₅×D₄	1	2	yes	—
218	C₄	C₅×#532	1	2	yes	—
219	C₂	C₄×C₅×#127	1	1	yes	characteristic
220	C₂	C₂×C₅×#441	1	2	yes	—
221	C₂	C₅×#3152	1	2	yes	—
222	C₂	C₅×#3185	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₅×#494	C₄	semidirect
24. #21691	C₅	direct (N × H)
87. #494	C₄×C₅	semidirect
153. C₄×C₅	#441	semidirect
207. C₅	#21691	direct (N × H)
216. C₄	C₅×#441	semidirect

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
C₂×C₅×#441	2
C₄×C₅×C₄⋊Q₈	2
C₅×#1971	2
C₅×#1956	2
C₅×#1844	2
C₄×C₅×#97	2
#21691	5

Subgroups — C5×#21691 #11944411

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21691 #11944411