Subgroups — C₅×#3097 #10616692

Order 640.

386 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#536	C₂	1	1	yes	maximal, characteristic
2	C₅×#520	C₂	1	1	yes	maximal, characteristic
3	C₅×#485	C₂	1	2	yes	maximal, complemented
4	C₅×#485	C₂	1	2	yes	maximal, complemented
5	C₅×#484	C₂	1	2	yes	maximal, complemented
6	C₅×#483	C₂	1	2	yes	maximal, complemented
7	C₅×#428	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#427	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#419	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅×#103	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#408	C₂	1	1	yes	maximal, characteristic
12	C₂×C₅×#41	C₂²	1	1	yes	characteristic
13	C₂²×C₅×Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₅×M₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₅×C₄⋊Q₈	—	2	1	no	—
17	C₅×#124	—	2	1	no	—
18	C₅×#123	—	2	2	no	—
19	C₅×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₅×C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₄×C₅×Q₈	—	2	1	no	—
22	C₄×C₅×D₄	—	2	2	no	—
23	C₄×C₅×D₄	—	2	1	no	—
24	C₅×#117	C₂²	1	1	yes	characteristic
25	C₅×#117	C₂²	1	1	yes	characteristic
26	C₅×#117	C₂²	1	2	yes	—
27	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
29	C₅×#107	C₂²	1	1	yes	characteristic
30	C₅×#107	C₂²	1	1	yes	characteristic
31	C₅×#107	C₂²	1	2	yes	—
32	C₅×#106	C₂²	1	1	yes	characteristic
33	C₅×#106	C₂²	1	1	yes	characteristic
34	C₅×#106	C₂²	1	2	yes	—
35	C₅×#103	C₂²	1	2	yes	—
36	C₅×#103	C₂²	1	2	yes	—
37	C₅×#103	C₂²	1	2	yes	—
38	C₅×#103	C₂²	1	2	yes	—
39	C₅×C₂²⋊C₈	C₂²	1	2	yes	—
40	C₅×C₂²⋊C₈	C₂²	1	2	yes	—
41	#3097	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₅×#41	—	2	2	no	—
43	C₅×#41	—	2	2	no	—
44	C₂×C₅×Q₈	—	2	1	no	—
45	C₂×C₅×Q₈	—	2	2	no	—
46	C₂×C₅×Q₈	—	2	2	no	—
47	C₂×C₅×Q₈	D₄	1	1	yes	characteristic
48	C₂×C₅×Q₈	C₂³	1	2	yes	—
49	C₂×C₅×D₄	D₄	1	2	yes	—
50	C₂×C₅×D₄	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅	—	2	1	no	—
52	C₂²×C₄×C₅	—	2	2	no	—
53	C₂²×C₄×C₅	—	2	1	no	—
54	C₂²×C₄×C₅	C₂³	1	2	yes	—
55	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
56	C₅×M₁₆	—	2	2	no	—
57	C₂×C₅×C₈	—	2	2	no	—
58	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
59	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
60	C₂×C₅×C₈	C₂³	1	2	yes	—
61	C₅×C₄⋊C₄	—	2	2	no	—
62	C₅×C₄⋊C₄	—	2	1	no	—
63	C₅×C₄⋊C₄	—	2	2	no	—
64	C₅×C₄⋊C₄	—	2	2	no	—
65	C₅×C₄⋊C₄	C₂³	1	2	yes	—
66	C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
67	C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
68	C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
69	C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
70	C₅×C₂²⋊C₄	—	2	2	no	—
71	C₅×C₂²⋊C₄	—	2	1	no	—
72	C₅×C₂²⋊C₄	—	2	1	no	—
73	C₅×C₂²⋊C₄	—	2	2	no	—
74	C₅×C₂²⋊C₄	—	2	2	no	—
75	C₄²×C₅	—	2	2	no	—
76	C₄²×C₅	—	2	1	no	—
77	C₄²×C₅	—	2	1	no	—
78	#536	C₂×C₅	1	1	yes	characteristic
79	#520	C₂×C₅	1	1	yes	characteristic
80	#485	C₂×C₅	1	2	yes	complemented
81	#485	C₂×C₅	1	2	yes	complemented
82	#484	C₂×C₅	1	2	yes	complemented
83	#483	C₂×C₅	1	2	yes	complemented
84	#428	C₂×C₅	1	1	yes	characteristic, complemented
85	#427	C₂×C₅	1	1	yes	characteristic, complemented
86	#419	C₂×C₅	1	1	yes	characteristic, complemented
87	C₂×#103	C₂×C₅	1	1	yes	characteristic, complemented
88	#408	C₂×C₅	1	1	yes	characteristic
89	C₂³×C₅	C₂×D₄	1	2	yes	—
90	C₂³×C₅	C₂×D₄	1	1	yes	characteristic
91	C₅×Q₈	—	2	2	no	—
92	C₅×Q₈	—	2	2	no	—
93	C₅×Q₈	—	2	2	no	—
94	C₅×Q₈	—	4	2	no	—
95	C₅×D₄	—	2	4	no	—
96	C₅×D₄	—	2	2	no	—
97	C₂×C₄×C₅	—	2	2	no	—
98	C₂×C₄×C₅	—	2	2	no	—
99	C₂×C₄×C₅	—	2	1	no	—
100	C₂×C₄×C₅	—	2	1	no	—
101	C₂×C₄×C₅	—	2	1	no	—
102	C₂×C₄×C₅	—	2	2	no	—
103	C₂×C₄×C₅	—	2	1	no	—
104	C₂×C₄×C₅	—	2	2	no	—
105	C₂×C₄×C₅	—	4	2	no	—
106	C₂×C₄×C₅	—	8	1	no	—
107	C₂×C₄×C₅	—	8	2	no	—
108	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₅	#41	1	2	yes	—
111	C₂×C₄×C₅	#41	1	2	yes	—
112	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
113	C₅×C₈	—	2	1	no	—
114	C₅×C₈	—	2	1	no	—
115	C₅×C₈	—	2	2	no	—
116	C₂×#41	C₂²×C₅	1	1	yes	characteristic
117	C₂²×Q₈	C₂²×C₅	1	1	yes	characteristic
118	C₂×M₁₆	C₂²×C₅	1	1	yes	characteristic
119	C₂²×C₈	C₂²×C₅	1	1	yes	characteristic
120	C₄⋊Q₈	—	2	1	no	—
121	#124	—	2	1	no	—
122	#123	—	2	2	no	—
123	C₂²⋊Q₈	C₂²×C₅	1	2	yes	—
124	C₂²⋊Q₈	C₂²×C₅	1	2	yes	—
125	C₄×Q₈	—	2	1	no	—
126	C₄×D₄	—	2	2	no	—
127	C₄×D₄	—	2	1	no	—
128	#117	C₂²×C₅	1	1	yes	characteristic
129	#117	C₂²×C₅	1	2	yes	—
130	#117	C₂²×C₅	1	1	yes	characteristic
131	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
132	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
133	#107	C₂²×C₅	1	1	yes	characteristic
134	#107	C₂²×C₅	1	1	yes	characteristic
135	#107	C₂²×C₅	1	2	yes	—
136	#106	C₂²×C₅	1	1	yes	characteristic
137	#106	C₂²×C₅	1	1	yes	characteristic
138	#106	C₂²×C₅	1	2	yes	—
139	#103	C₂²×C₅	1	2	yes	—
140	#103	C₂²×C₅	1	2	yes	—
141	#103	C₂²×C₅	1	2	yes	—
142	#103	C₂²×C₅	1	2	yes	—
143	C₂²⋊C₈	C₂²×C₅	1	2	yes	—
144	C₂²⋊C₈	C₂²×C₅	1	2	yes	—
145	C₂²×C₅	—	2	1	no	—
146	C₂²×C₅	—	2	1	no	—
147	C₂²×C₅	—	2	2	no	—
148	C₂²×C₅	—	2	2	no	—
149	C₂²×C₅	—	2	2	no	—
150	C₂²×C₅	#123	1	2	yes	—
151	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
152	C₄×C₅	—	2	2	no	—
153	C₄×C₅	—	4	2	no	—
154	C₄×C₅	—	4	1	no	—
155	C₄×C₅	—	4	2	no	—
156	C₄×C₅	—	4	1	no	—
157	C₄×C₅	—	4	1	no	—
158	C₄×C₅	—	4	1	no	—
159	C₄×C₅	#123	1	2	yes	—
160	C₄×C₅	C₂×#41	1	1	yes	characteristic
161	C₄×C₅	C₂×#41	1	1	yes	characteristic
162	#41	—	2	2	no	—
163	#41	—	2	2	no	—
164	C₂×Q₈	—	2	1	no	—
165	C₂×Q₈	—	2	2	no	—
166	C₂×Q₈	—	2	2	no	—
167	C₂×Q₈	C₅×D₄	1	1	yes	characteristic
168	C₂×Q₈	C₂³×C₅	1	2	yes	—
169	C₂×D₄	C₅×D₄	1	2	yes	—
170	C₂×D₄	C₅×D₄	1	1	yes	characteristic
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	C₂³×C₅	1	2	yes	—
175	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
176	M₁₆	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	C₂³×C₅	1	1	yes	characteristic
179	C₂×C₈	C₂³×C₅	1	2	yes	—
180	C₂×C₈	C₂³×C₅	1	1	yes	characteristic
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	2	1	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	C₂³×C₅	1	2	yes	—
186	C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
187	C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
188	C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
189	C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	2	1	no	—
192	C₂²⋊C₄	—	2	1	no	—
193	C₂²⋊C₄	—	2	2	no	—
194	C₂²⋊C₄	—	2	2	no	—
195	C₄²	—	2	2	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	2	1	no	—
198	C₂×C₅	—	2	2	no	—
199	C₂×C₅	—	4	2	no	—
200	C₂×C₅	C₂×#123	1	1	yes	characteristic
201	C₂×C₅	#577	1	1	yes	characteristic
202	C₂×C₅	#578	1	1	yes	characteristic
203	C₂³	C₂×C₅×D₄	1	2	yes	—
204	C₂³	C₂×C₅×D₄	1	1	yes	characteristic
205	Q₈	—	2	2	no	—
206	Q₈	—	2	2	no	—
207	Q₈	—	2	2	no	—
208	Q₈	—	4	2	no	—
209	D₄	—	2	4	no	—
210	D₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	8	1	no	—
221	C₂×C₄	—	8	2	no	—
222	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
223	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
224	C₂×C₄	C₅×#41	1	2	yes	—
225	C₂×C₄	C₅×#41	1	2	yes	—
226	C₂×C₄	C₂⁴×C₅	1	1	yes	characteristic
227	C₈	—	2	1	no	—
228	C₈	—	2	1	no	—
229	C₈	—	2	2	no	—
230	C₅	#3097	1	1	yes	characteristic, direct product factor, Sylow
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	C₅×#123	1	2	yes	—
237	C₂²	C₂²×C₅×D₄	1	1	yes	characteristic
238	C₄	—	2	2	no	—
239	C₄	—	4	2	no	—
240	C₄	—	4	2	no	—
241	C₄	—	4	1	no	—
242	C₄	—	4	1	no	—
243	C₄	—	4	1	no	—
244	C₄	—	4	1	no	—
245	C₄	C₅×#123	1	2	yes	—
246	C₄	C₂×C₅×#41	1	1	yes	characteristic
247	C₄	C₂×C₅×#41	1	1	yes	characteristic
248	C₂	—	2	2	no	—
249	C₂	—	4	2	no	—
250	C₂	C₂×C₅×#123	1	1	yes	characteristic
251	C₂	C₅×#577	1	1	yes	characteristic
252	C₂	C₅×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅
prime radical	C₂×C₅×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₅×#41 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#485	C₂	semidirect
4. C₅×#485	C₂	semidirect
5. C₅×#484	C₂	semidirect
6. C₅×#483	C₂	semidirect
7. C₅×#428	C₂	semidirect
8. C₅×#427	C₂	semidirect
9. C₅×#419	C₂	semidirect
10. C₂×C₅×#103	C₂	semidirect
41. #3097	C₅	direct (N × H)
80. #485	C₂×C₅	semidirect
81. #485	C₂×C₅	semidirect
82. #484	C₂×C₅	semidirect
83. #483	C₂×C₅	semidirect
84. #428	C₂×C₅	semidirect
85. #427	C₂×C₅	semidirect
86. #419	C₂×C₅	semidirect
87. C₂×#103	C₂×C₅	semidirect
230. C₅	#3097	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×#536	2
C₅×#520	2
C₅×#485	2
C₅×#484	2
C₅×#483	2
C₅×#428	2
C₅×#427	2
C₅×#419	2
C₂×C₅×#103	2
C₅×#408	2
#3097	5

Subgroups — C5×#3097 #10616692

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#3097 #10616692