Subgroups — C₅×#21677 #11944397

Order 1280.

618 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2402	C₂	1	1	yes	maximal, characteristic
2	C₅×#1940	C₂	1	2	yes	maximal
3	C₅×#1917	C₂	1	4	yes	maximal
4	C₅×#1832	C₂	1	1	yes	maximal, characteristic
5	C₅×#1771	C₂	1	4	yes	maximal
6	C₅×#1770	C₂	1	2	yes	maximal
7	C₅×#1726	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
10	C₂×C₅×#97	C₂²	1	4	yes	—
11	C₂×C₅×#97	C₂²	1	2	yes	—
12	C₂×C₅×#97	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₅×#400	C₄	1	4	yes	—
16	C₅×#398	C₄	1	4	yes	—
17	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
18	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
19	C₅×#376	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	#21677	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂²×C₅×C₈	C₂³	1	4	yes	—
25	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
26	C₂²×C₅×C₈	C₂³	1	2	yes	—
27	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
28	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
29	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
30	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₅	C₂³	1	2	yes	—
32	C₂×C₄²×C₅	C₂³	1	4	yes	—
33	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
34	C₅×C₄⋊C₈	—	2	8	no	—
35	C₅×C₄⋊C₈	—	2	4	no	—
36	C₅×#97	—	2	4	no	—
37	C₅×#97	—	2	8	no	—
38	C₅×#97	—	2	2	no	—
39	C₄×C₅×C₈	—	2	4	no	—
40	C₄×C₅×C₈	Q₈	1	4	yes	—
41	C₅×#95	C₂×C₄	1	4	yes	—
42	C₅×#95	C₂×C₄	1	2	yes	—
43	C₅×#95	C₂×C₄	1	4	yes	—
44	C₅×#95	C₂×C₄	1	4	yes	—
45	C₅×#95	C₂×C₄	1	2	yes	—
46	#2402	C₂×C₅	1	1	yes	characteristic
47	#1940	C₂×C₅	1	2	yes	—
48	#1917	C₂×C₅	1	4	yes	—
49	#1832	C₂×C₅	1	1	yes	characteristic
50	#1771	C₂×C₅	1	4	yes	—
51	#1770	C₂×C₅	1	2	yes	—
52	#1726	C₂×C₅	1	1	yes	characteristic
53	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
54	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
55	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
58	C₂×C₅×C₈	—	2	2	no	—
59	C₂×C₅×C₈	—	2	2	no	—
60	C₂×C₅×C₈	—	2	2	no	—
61	C₂×C₅×C₈	—	2	4	no	—
62	C₂×C₅×C₈	—	2	4	no	—
63	C₂×C₅×C₈	—	2	4	no	—
64	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
65	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
66	C₂×C₅×C₈	#41	1	4	yes	—
67	C₂×C₅×C₈	#41	1	4	yes	—
68	C₅×C₄⋊C₄	—	4	4	no	—
69	C₅×C₄⋊C₄	—	4	4	no	—
70	C₅×C₄⋊C₄	—	4	4	no	—
71	C₄²×C₅	—	2	2	no	—
72	C₄²×C₅	—	2	4	no	—
73	C₄²×C₅	—	2	2	no	—
74	C₄²×C₅	—	2	4	no	—
75	C₄²×C₅	C₂×Q₈	1	2	yes	—
76	C₄²×C₅	#41	1	2	yes	—
77	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
78	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
79	C₂×#97	C₂²×C₅	1	2	yes	—
80	C₂×#97	C₂²×C₅	1	1	yes	characteristic
81	C₂×#97	C₂²×C₅	1	4	yes	—
82	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
83	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
84	#400	C₄×C₅	1	4	yes	—
85	#398	C₄×C₅	1	4	yes	—
86	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
87	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
88	#376	C₂²×C₅	1	1	yes	characteristic
89	#336	C₂²×C₅	1	4	yes	—
90	#336	C₂²×C₅	1	4	yes	—
91	#336	C₂²×C₅	1	4	yes	—
92	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₅	—	2	4	no	—
94	C₂×C₄×C₅	—	2	4	no	—
95	C₂×C₄×C₅	—	2	4	no	—
96	C₂×C₄×C₅	—	2	4	no	—
97	C₂×C₄×C₅	—	2	4	no	—
98	C₂×C₄×C₅	—	2	4	no	—
99	C₂×C₄×C₅	—	2	4	no	—
100	C₂×C₄×C₅	—	2	4	no	—
101	C₂×C₄×C₅	—	2	4	no	—
102	C₂×C₄×C₅	—	2	4	no	—
103	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
104	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₅	C₂×#41	1	2	yes	—
106	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂×#41	1	2	yes	—
108	C₅×C₈	—	2	4	no	—
109	C₅×C₈	—	2	2	no	—
110	C₅×C₈	—	4	4	no	—
111	C₅×C₈	#125	1	4	yes	—
112	C₂²×C₈	C₂³×C₅	1	4	yes	—
113	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
114	C₂²×C₈	C₂³×C₅	1	2	yes	—
115	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
116	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
117	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
118	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
119	C₂×C₄²	C₂³×C₅	1	2	yes	—
120	C₂×C₄²	C₂³×C₅	1	4	yes	—
121	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
122	C₄⋊C₈	—	2	8	no	—
123	C₄⋊C₈	—	2	4	no	—
124	#97	—	2	4	no	—
125	#97	—	2	8	no	—
126	#97	—	2	2	no	—
127	C₄×C₈	—	2	4	no	—
128	C₄×C₈	C₅×Q₈	1	4	yes	—
129	#95	C₂×C₄×C₅	1	4	yes	—
130	#95	C₂×C₄×C₅	1	2	yes	—
131	#95	C₂×C₄×C₅	1	4	yes	—
132	#95	C₂×C₄×C₅	1	4	yes	—
133	#95	C₂×C₄×C₅	1	2	yes	—
134	C₂²×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
135	C₂²×C₅	C₄×#41	1	2	yes	—
136	C₂²×C₅	#568	1	4	yes	—
137	C₄×C₅	—	2	4	no	—
138	C₄×C₅	—	4	4	no	—
139	C₄×C₅	—	4	4	no	—
140	C₄×C₅	—	4	4	no	—
141	C₄×C₅	C₂×#125	1	1	yes	characteristic
142	C₄×C₅	#529	1	2	yes	—
143	C₄×C₅	#533	1	1	yes	characteristic
144	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
145	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
146	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
147	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
148	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
156	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
157	C₂×C₈	C₅×#41	1	4	yes	—
158	C₂×C₈	C₅×#41	1	4	yes	—
159	C₄⋊C₄	—	4	4	no	—
160	C₄⋊C₄	—	4	4	no	—
161	C₄⋊C₄	—	4	4	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	4	no	—
164	C₄²	—	2	2	no	—
165	C₄²	—	2	4	no	—
166	C₄²	C₂×C₅×Q₈	1	2	yes	—
167	C₄²	C₅×#41	1	2	yes	—
168	C₂×C₅	C₄×#125	1	1	yes	characteristic
169	C₂×C₅	#2961	1	2	yes	—
170	C₂×C₅	#2967	1	4	yes	—
171	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	C₄×C₅×Q₈	1	4	yes	—
183	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
184	C₂×C₄	C₂×C₅×#41	1	2	yes	—
185	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
186	C₂×C₄	C₂×C₅×#41	1	2	yes	—
187	C₈	—	2	4	no	—
188	C₈	—	2	2	no	—
189	C₈	—	4	4	no	—
190	C₈	C₅×#125	1	4	yes	—
191	C₅	#21677	1	1	yes	characteristic, direct product factor, Sylow
192	C₂²	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
193	C₂²	C₄×C₅×#41	1	2	yes	—
194	C₂²	C₅×#568	1	4	yes	—
195	C₄	—	2	4	no	—
196	C₄	—	4	4	no	—
197	C₄	—	4	4	no	—
198	C₄	—	4	4	no	—
199	C₄	C₂×C₅×#125	1	1	yes	characteristic
200	C₄	C₅×#529	1	2	yes	—
201	C₄	C₅×#533	1	1	yes	characteristic
202	C₂	C₄×C₅×#125	1	1	yes	characteristic
203	C₂	C₅×#2961	1	2	yes	—
204	C₂	C₅×#2967	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂⁴
commutator	C₂³	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle, prime radical	C₂³×C₅	C₂³×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
23. #21677	C₅	direct (N × H)
191. C₅	#21677	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×#2402	2
C₅×#1940	2
C₅×#1917	2
C₅×#1832	2
C₅×#1771	2
C₅×#1770	2
C₅×#1726	2
#21677	5

Subgroups — C5×#21677 #11944397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21677 #11944397