Subgroups — C₅×#11355 #11934075

Order 1280.

686 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#382	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#343	C₂	1	4	yes	maximal
3	C₅×#1728	C₂	1	1	yes	maximal, characteristic
4	C₅×#1728	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
6	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
7	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
8	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₅×#382	—	2	2	no	—
10	C₅×#382	—	2	2	no	—
11	C₂×C₅×#95	C₄	1	4	yes	—
12	C₅×#343	—	2	8	no	—
13	C₅×#336	—	2	2	no	—
14	C₅×#336	—	2	2	no	—
15	#11355	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂³×C₄×C₅	C₂×C₄	1	4	yes	—
17	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
18	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
19	C₂×C₅×M₁₆	—	2	2	no	—
20	C₂×C₅×M₁₆	—	2	2	no	—
21	C₂×C₅×M₁₆	—	2	4	no	—
22	C₂×C₅×M₁₆	—	2	4	no	—
23	C₂×C₅×M₁₆	—	2	4	no	—
24	C₂×C₅×M₁₆	—	2	4	no	—
25	C₂×C₅×M₁₆	D₄	1	4	yes	—
26	C₂×C₅×M₁₆	Q₈	1	4	yes	—
27	C₂²×C₅×C₈	—	2	2	no	—
28	C₂²×C₅×C₈	—	2	2	no	—
29	C₂×C₅×C₄⋊C₄	—	2	2	no	—
30	C₂×C₅×C₄⋊C₄	—	2	2	no	—
31	C₂×C₅×C₄⋊C₄	—	2	4	no	—
32	C₂×C₅×C₄⋊C₄	—	2	4	no	—
33	C₂×C₄²×C₅	—	2	4	no	—
34	C₂×C₄²×C₅	D₄	1	2	yes	—
35	C₂×C₄²×C₅	D₄	1	2	yes	—
36	C₅×#95	—	2	8	no	—
37	C₂×#382	C₂×C₅	1	1	yes	characteristic
38	C₂×#343	C₂×C₅	1	4	yes	—
39	#1728	C₂×C₅	1	1	yes	characteristic
40	#1728	C₂×C₅	1	1	yes	characteristic
41	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅	—	2	4	no	—
43	C₂²×C₄×C₅	—	2	4	no	—
44	C₂²×C₄×C₅	—	2	8	no	—
45	C₂²×C₄×C₅	—	4	4	no	—
46	C₂²×C₄×C₅	—	4	8	no	—
47	C₂²×C₄×C₅	C₂²⋊C₄	1	4	yes	—
48	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
52	C₂²×C₄×C₅	#41	1	2	yes	—
53	C₂²×C₄×C₅	#41	1	4	yes	—
54	C₂²×C₄×C₅	#41	1	2	yes	—
55	C₅×M₁₆	—	2	8	no	—
56	C₅×M₁₆	—	2	8	no	—
57	C₅×M₁₆	—	4	4	no	—
58	C₅×M₁₆	—	4	4	no	—
59	C₂×C₅×C₈	—	2	4	no	—
60	C₂×C₅×C₈	—	2	4	no	—
61	C₂×C₅×C₈	—	4	2	no	—
62	C₂×C₅×C₈	—	4	2	no	—
63	C₂×C₅×C₈	—	4	2	no	—
64	C₂×C₅×C₈	—	4	2	no	—
65	C₅×C₄⋊C₄	—	4	4	no	—
66	C₅×C₄⋊C₄	—	4	2	no	—
67	C₅×C₄⋊C₄	—	4	2	no	—
68	C₄²×C₅	—	2	4	no	—
69	C₄²×C₅	—	2	4	no	—
70	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
71	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
72	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
73	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
74	#382	—	2	2	no	—
75	#382	—	2	2	no	—
76	C₂×#95	C₄×C₅	1	4	yes	—
77	#343	—	2	8	no	—
78	#336	—	2	2	no	—
79	#336	—	2	2	no	—
80	C₂³×C₅	—	2	4	no	—
81	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂³×C₅	#117	1	2	yes	—
83	C₂³×C₅	C₄×D₄	1	2	yes	—
84	C₂³×C₅	C₄×Q₈	1	2	yes	—
85	C₂×C₄×C₅	—	2	4	no	—
86	C₂×C₄×C₅	—	2	8	no	—
87	C₂×C₄×C₅	—	4	8	no	—
88	C₂×C₄×C₅	—	4	4	no	—
89	C₂×C₄×C₅	—	4	8	no	—
90	C₂×C₄×C₅	—	8	8	no	—
91	C₂×C₄×C₅	#121	1	2	yes	—
92	C₂×C₄×C₅	C₂²⋊Q₈	1	4	yes	—
93	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
94	C₂×C₄×C₅	#123	1	2	yes	—
95	C₂×C₄×C₅	#123	1	4	yes	—
96	C₂×C₄×C₅	#123	1	2	yes	—
97	C₂×C₄×C₅	#124	1	1	yes	characteristic
98	C₂×C₄×C₅	#124	1	1	yes	characteristic
99	C₂×C₄×C₅	#125	1	4	yes	—
100	C₂×C₄×C₅	#126	1	4	yes	—
101	C₂×C₄×C₅	#127	1	1	yes	characteristic
102	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
103	C₅×C₈	—	4	4	no	—
104	C₅×C₈	—	4	4	no	—
105	C₂³×C₄	C₂×C₄×C₅	1	4	yes	—
106	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
107	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
108	C₂×M₁₆	—	2	2	no	—
109	C₂×M₁₆	—	2	2	no	—
110	C₂×M₁₆	—	2	4	no	—
111	C₂×M₁₆	—	2	4	no	—
112	C₂×M₁₆	—	2	4	no	—
113	C₂×M₁₆	—	2	4	no	—
114	C₂×M₁₆	C₅×D₄	1	4	yes	—
115	C₂×M₁₆	C₅×Q₈	1	4	yes	—
116	C₂²×C₈	—	2	2	no	—
117	C₂²×C₈	—	2	2	no	—
118	C₂×C₄⋊C₄	—	2	4	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	4	no	—
122	C₂×C₄²	—	2	4	no	—
123	C₂×C₄²	C₅×D₄	1	2	yes	—
124	C₂×C₄²	C₅×D₄	1	2	yes	—
125	#95	—	2	8	no	—
126	C₂²×C₅	—	2	4	no	—
127	C₂²×C₅	—	2	4	no	—
128	C₂²×C₅	—	2	4	no	—
129	C₂²×C₅	#381	1	1	yes	characteristic
130	C₂²×C₅	#387	1	4	yes	—
131	C₂²×C₅	#390	1	1	yes	characteristic
132	C₂²×C₅	#391	1	1	yes	characteristic
133	C₂²×C₅	#413	1	4	yes	—
134	C₄×C₅	—	4	4	no	—
135	C₄×C₅	—	8	4	no	—
136	C₄×C₅	#397	1	2	yes	—
137	C₄×C₅	#398	1	2	yes	—
138	C₄×C₅	#399	1	2	yes	—
139	C₄×C₅	#400	1	2	yes	—
140	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	—	2	8	no	—
144	C₂²×C₄	—	4	4	no	—
145	C₂²×C₄	—	4	8	no	—
146	C₂²×C₄	C₅×C₂²⋊C₄	1	4	yes	—
147	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
149	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
150	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
151	C₂²×C₄	C₅×#41	1	2	yes	—
152	C₂²×C₄	C₅×#41	1	4	yes	—
153	C₂²×C₄	C₅×#41	1	2	yes	—
154	M₁₆	—	2	8	no	—
155	M₁₆	—	2	8	no	—
156	M₁₆	—	4	4	no	—
157	M₁₆	—	4	4	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	4	2	no	—
161	C₂×C₈	—	4	2	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	—	4	2	no	—
164	C₄⋊C₄	—	4	4	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄²	—	2	4	no	—
168	C₄²	—	2	4	no	—
169	C₂×C₅	—	2	4	no	—
170	C₂×C₅	#1445	1	1	yes	characteristic
171	C₂×C₅	#1822	1	2	yes	—
172	C₂×C₅	#1967	1	2	yes	—
173	C₂×C₅	#1998	1	2	yes	—
174	C₂³	—	2	4	no	—
175	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
176	C₂³	C₅×#117	1	2	yes	—
177	C₂³	C₄×C₅×D₄	1	2	yes	—
178	C₂³	C₄×C₅×Q₈	1	2	yes	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	8	no	—
181	C₂×C₄	—	4	8	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	4	8	no	—
184	C₂×C₄	—	8	8	no	—
185	C₂×C₄	C₅×#121	1	2	yes	—
186	C₂×C₄	C₅×C₂²⋊Q₈	1	4	yes	—
187	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
188	C₂×C₄	C₅×#123	1	2	yes	—
189	C₂×C₄	C₅×#123	1	4	yes	—
190	C₂×C₄	C₅×#123	1	2	yes	—
191	C₂×C₄	C₅×#124	1	1	yes	characteristic
192	C₂×C₄	C₅×#124	1	1	yes	characteristic
193	C₂×C₄	C₅×#125	1	4	yes	—
194	C₂×C₄	C₅×#126	1	4	yes	—
195	C₂×C₄	C₅×#127	1	1	yes	characteristic
196	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
197	C₈	—	4	4	no	—
198	C₈	—	4	4	no	—
199	C₅	#11355	1	1	yes	characteristic, direct product factor, Sylow
200	C₂²	—	2	4	no	—
201	C₂²	—	2	4	no	—
202	C₂²	—	2	4	no	—
203	C₂²	C₅×#381	1	1	yes	characteristic
204	C₂²	C₅×#387	1	4	yes	—
205	C₂²	C₅×#390	1	1	yes	characteristic
206	C₂²	C₅×#391	1	1	yes	characteristic
207	C₂²	C₅×#413	1	4	yes	—
208	C₄	—	4	4	no	—
209	C₄	—	8	4	no	—
210	C₄	C₅×#397	1	2	yes	—
211	C₄	C₅×#398	1	2	yes	—
212	C₄	C₅×#399	1	2	yes	—
213	C₄	C₅×#400	1	2	yes	—
214	C₂	—	2	4	no	—
215	C₂	C₅×#1445	1	1	yes	characteristic
216	C₂	C₅×#1822	1	2	yes	—
217	C₂	C₅×#1967	1	2	yes	—
218	C₂	C₅×#1998	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
15. #11355	C₅	direct (N × H)
199. C₅	#11355	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×#382	2
C₂×C₅×#343	2
C₅×#1728	2
#11355	5

Subgroups — C5×#11355 #11934075

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#11355 #11934075