Subgroups — C₅×#21372 #11944092

Order 1280.

714 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2921	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#433	C₂	1	1	yes	maximal, characteristic
3	C₅×#2873	C₂	1	1	yes	maximal, characteristic
4	C₅×#1950	C₂	1	4	yes	maximal, complemented
5	C₅×#1855	C₂	1	4	yes	maximal
6	C₅×#1778	C₂	1	2	yes	maximal, complemented
7	C₄×C₅×#108	C₂	1	2	yes	maximal, complemented
8	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
9	C₂×C₅×#117	C₂²	1	1	yes	characteristic
10	C₅×#433	—	2	4	no	—
11	C₅×#433	C₄	1	8	yes	—
12	C₂×C₅×#108	C₂²	1	4	yes	—
13	C₂×C₅×#108	C₂²	1	4	yes	—
14	C₅×#423	C₂²	1	2	yes	—
15	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
16	C₅×#407	C₂²	1	2	yes	—
17	C₂×C₅×C₂²⋊C₈	C₂²	1	2	yes	—
18	C₅×#405	—	2	4	no	—
19	C₄×C₅×M₁₆	C₂²	1	2	yes	—
20	C₄×C₅×M₁₆	C₂²	1	2	yes	—
21	C₄×C₅×M₁₆	C₂²	1	2	yes	—
22	C₂×C₅×#97	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
24	C₅×#341	C₂²	1	4	yes	—
25	C₅×#341	C₂²	1	4	yes	—
26	#21372	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂×C₅×M₁₆	—	2	2	no	—
29	C₂×C₅×M₁₆	—	2	8	no	—
30	C₂×C₅×M₁₆	C₂³	1	4	yes	—
31	C₂×C₅×M₁₆	C₂³	1	4	yes	—
32	C₂²×C₅×C₈	—	2	2	no	—
33	C₂²×C₅×C₈	C₂³	1	2	yes	—
34	C₂²×C₅×C₈	C₂³	1	2	yes	—
35	C₅×#117	C₂×C₄	1	4	yes	—
36	C₅×#117	D₄	1	4	yes	—
37	C₂×C₅×C₄⋊C₄	Q₈	1	2	yes	—
38	C₂×C₅×C₂²⋊C₄	Q₈	1	2	yes	—
39	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
41	C₅×#108	—	2	8	no	—
42	C₅×#108	—	2	8	no	—
43	C₅×C₄⋊C₈	—	2	4	no	—
44	C₅×C₄⋊C₈	C₂×C₄	1	8	yes	—
45	C₅×C₂²⋊C₈	—	2	4	no	—
46	C₅×C₂²⋊C₈	C₂×C₄	1	8	yes	—
47	C₅×#97	—	2	2	no	—
48	C₅×#97	—	2	2	no	—
49	C₅×#97	—	2	2	no	—
50	C₅×#97	C₂×C₄	1	4	yes	—
51	C₄×C₅×C₈	—	2	2	no	—
52	C₄×C₅×C₈	—	2	2	no	—
53	C₄×C₅×C₈	—	2	2	no	—
54	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
55	#2921	C₂×C₅	1	1	yes	characteristic
56	C₂×#433	C₂×C₅	1	1	yes	characteristic
57	#2873	C₂×C₅	1	1	yes	characteristic
58	#1950	C₂×C₅	1	4	yes	complemented
59	#1855	C₂×C₅	1	4	yes	—
60	#1778	C₂×C₅	1	2	yes	complemented
61	C₄×#108	C₂×C₅	1	2	yes	complemented
62	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₅	—	2	4	no	—
64	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
70	C₅×M₁₆	—	2	8	no	—
71	C₅×M₁₆	—	2	4	no	—
72	C₅×M₁₆	—	2	4	no	—
73	C₅×M₁₆	—	2	4	no	—
74	C₅×M₁₆	#41	1	8	yes	—
75	C₂×C₅×C₈	—	2	8	no	—
76	C₂×C₅×C₈	—	2	4	no	—
77	C₂×C₅×C₈	—	2	4	no	—
78	C₂×C₅×C₈	—	2	4	no	—
79	C₂×C₅×C₈	—	2	4	no	—
80	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
81	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
82	C₅×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
83	C₅×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
84	C₄²×C₅	C₂²×C₄	1	2	yes	—
85	C₄²×C₅	C₂²×C₄	1	2	yes	—
86	C₄²×C₅	C₂×D₄	1	2	yes	—
87	C₄²×C₅	C₂×D₄	1	2	yes	—
88	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
89	C₂×#117	C₂²×C₅	1	1	yes	characteristic
90	#433	—	2	4	no	—
91	#433	C₄×C₅	1	8	yes	—
92	C₂×#108	C₂²×C₅	1	4	yes	—
93	C₂×#108	C₂²×C₅	1	4	yes	—
94	#423	C₂²×C₅	1	2	yes	—
95	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
96	#407	C₂²×C₅	1	2	yes	—
97	C₂×C₂²⋊C₈	C₂²×C₅	1	2	yes	—
98	#405	—	2	4	no	—
99	C₄×M₁₆	C₂²×C₅	1	2	yes	—
100	C₄×M₁₆	C₂²×C₅	1	2	yes	—
101	C₄×M₁₆	C₂²×C₅	1	2	yes	—
102	C₂×#97	C₂²×C₅	1	1	yes	characteristic
103	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
104	#341	C₂²×C₅	1	4	yes	—
105	#341	C₂²×C₅	1	4	yes	—
106	C₂³×C₅	—	2	2	no	—
107	C₂³×C₅	—	4	2	no	—
108	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
109	C₂³×C₅	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₅	—	2	2	no	—
111	C₂×C₄×C₅	—	2	4	no	—
112	C₂×C₄×C₅	—	2	2	no	—
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	—	4	2	no	—
115	C₂×C₄×C₅	—	4	2	no	—
116	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
117	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
118	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×#41	1	4	yes	—
122	C₅×C₈	—	2	4	no	—
123	C₅×C₈	—	2	4	no	—
124	C₅×C₈	—	2	8	no	—
125	C₅×C₈	—	4	4	no	—
126	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
127	C₂×M₁₆	—	2	2	no	—
128	C₂×M₁₆	—	2	8	no	—
129	C₂×M₁₆	C₂³×C₅	1	4	yes	—
130	C₂×M₁₆	C₂³×C₅	1	4	yes	—
131	C₂²×C₈	—	2	2	no	—
132	C₂²×C₈	C₂³×C₅	1	2	yes	—
133	C₂²×C₈	C₂³×C₅	1	2	yes	—
134	#117	C₂×C₄×C₅	1	4	yes	—
135	#117	C₅×D₄	1	4	yes	—
136	C₂×C₄⋊C₄	C₅×Q₈	1	2	yes	—
137	C₂×C₂²⋊C₄	C₅×Q₈	1	2	yes	—
138	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
139	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
140	#108	—	2	8	no	—
141	#108	—	2	8	no	—
142	C₄⋊C₈	—	2	4	no	—
143	C₄⋊C₈	C₂×C₄×C₅	1	8	yes	—
144	C₂²⋊C₈	—	2	4	no	—
145	C₂²⋊C₈	C₂×C₄×C₅	1	8	yes	—
146	#97	—	2	2	no	—
147	#97	—	2	2	no	—
148	#97	—	2	2	no	—
149	#97	C₂×C₄×C₅	1	4	yes	—
150	C₄×C₈	—	2	2	no	—
151	C₄×C₈	—	2	2	no	—
152	C₄×C₈	—	2	2	no	—
153	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
154	C₂²×C₅	—	2	2	no	—
155	C₂²×C₅	—	2	4	no	—
156	C₂²×C₅	—	2	2	no	—
157	C₂²×C₅	—	8	2	no	—
158	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
159	C₂²×C₅	#533	1	2	yes	—
160	C₄×C₅	—	2	4	no	—
161	C₄×C₅	—	2	4	no	—
162	C₄×C₅	—	2	2	no	—
163	C₄×C₅	C₄×#41	1	2	yes	—
164	C₄×C₅	#525	1	2	yes	—
165	C₂⁴	C₂×C₅×Q₈	1	1	yes	characteristic
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
168	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
169	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
170	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
171	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
172	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
173	M₁₆	—	2	8	no	—
174	M₁₆	—	2	4	no	—
175	M₁₆	—	2	4	no	—
176	M₁₆	—	2	4	no	—
177	M₁₆	C₅×#41	1	8	yes	—
178	C₂×C₈	—	2	8	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
184	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
185	C₄⋊C₄	C₅×C₄⋊C₄	1	8	yes	—
186	C₂²⋊C₄	C₅×C₄⋊C₄	1	8	yes	—
187	C₄²	C₂²×C₄×C₅	1	2	yes	—
188	C₄²	C₂²×C₄×C₅	1	2	yes	—
189	C₄²	C₂×C₅×D₄	1	2	yes	—
190	C₄²	C₂×C₅×D₄	1	2	yes	—
191	C₂×C₅	—	2	2	no	—
192	C₂×C₅	—	4	2	no	—
193	C₂×C₅	#2296	1	1	yes	characteristic
194	C₂×C₅	#2916	1	2	yes	—
195	C₂³	—	2	2	no	—
196	C₂³	—	4	2	no	—
197	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
198	C₂³	C₂²×C₅×Q₈	1	1	yes	characteristic
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
206	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
207	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
208	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
209	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
210	C₂×C₄	C₂×C₅×#41	1	4	yes	—
211	C₈	—	2	4	no	—
212	C₈	—	2	4	no	—
213	C₈	—	2	8	no	—
214	C₈	—	4	4	no	—
215	C₅	#21372	1	1	yes	characteristic, direct product factor, Sylow
216	C₂²	—	2	2	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	4	no	—
219	C₂²	—	8	2	no	—
220	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
221	C₂²	C₅×#533	1	2	yes	—
222	C₄	—	2	2	no	—
223	C₄	—	2	4	no	—
224	C₄	—	2	4	no	—
225	C₄	C₄×C₅×#41	1	2	yes	—
226	C₄	C₅×#525	1	2	yes	—
227	C₂	—	2	2	no	—
228	C₂	—	4	2	no	—
229	C₂	C₅×#2296	1	1	yes	characteristic
230	C₂	C₅×#2916	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂²×C₅	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#117 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1950	C₂	semidirect
6. C₅×#1778	C₂	semidirect
7. C₄×C₅×#108	C₂	semidirect
26. #21372	C₅	direct (N × H)
58. #1950	C₂×C₅	semidirect
60. #1778	C₂×C₅	semidirect
61. C₄×#108	C₂×C₅	semidirect
215. C₅	#21372	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×#2921	2
C₂×C₅×#433	2
C₅×#2873	2
C₅×#1950	2
C₅×#1855	2
C₅×#1778	2
C₄×C₅×#108	2
#21372	5

Subgroups — C5×#21372 #11944092

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21372 #11944092