Subgroups — C₅×#3098 #10616693

Order 640.

398 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#537	C₂	1	1	yes	maximal, characteristic
2	C₅×#520	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#485	C₂	1	2	yes	maximal, complemented
4	C₅×#484	C₂	1	2	yes	maximal, complemented
5	C₅×#482	C₂	1	2	yes	maximal, complemented
6	C₅×#480	C₂	1	2	yes	maximal, complemented
7	C₅×#428	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#107	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#417	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#416	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#408	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₅×#41	C₂²	1	1	yes	characteristic
13	C₂×C₅×#41	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×C₂²⋊Q₈	—	2	1	no	—
17	C₅×C₂²⋊Q₈	C₂²	1	2	yes	—
18	C₅×#121	—	2	1	no	—
19	C₅×#121	—	2	2	no	—
20	C₅×#121	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	2	yes	—
26	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
29	C₅×#107	C₂²	1	2	yes	—
30	C₅×#107	C₂²	1	2	yes	—
31	C₅×#107	C₂²	1	2	yes	—
32	C₅×#106	C₂²	1	2	yes	—
33	C₅×#103	C₂²	1	1	yes	characteristic
34	C₅×#103	C₂²	1	1	yes	characteristic
35	C₅×#103	C₂²	1	2	yes	—
36	C₅×#102	C₂²	1	1	yes	characteristic
37	C₅×#102	C₂²	1	1	yes	characteristic
38	C₅×#102	C₂²	1	2	yes	—
39	C₅×C₂²⋊C₈	C₂²	1	2	yes	—
40	C₅×C₂²⋊C₈	C₂²	1	2	yes	—
41	#3098	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₅×#41	—	2	2	no	—
43	C₅×#41	—	2	2	no	—
44	C₅×#41	—	2	2	no	—
45	C₅×#41	—	2	2	no	—
46	C₂×C₅×Q₈	D₄	1	1	yes	characteristic
47	C₂×C₅×Q₈	C₂³	1	1	yes	characteristic
48	C₂×C₅×D₄	—	2	1	no	—
49	C₂×C₅×D₄	—	2	2	no	—
50	C₂×C₅×D₄	D₄	1	2	yes	—
51	C₂×C₅×D₄	D₄	1	1	yes	characteristic
52	C₂×C₅×D₄	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₅	—	2	2	no	—
54	C₂²×C₄×C₅	—	2	1	no	—
55	C₂²×C₄×C₅	—	2	1	no	—
56	C₂²×C₄×C₅	—	2	1	no	—
57	C₂²×C₄×C₅	C₂³	1	2	yes	—
58	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
59	C₅×M₁₆	—	2	2	no	—
60	C₂×C₅×C₈	—	2	2	no	—
61	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
62	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
63	C₂×C₅×C₈	C₂³	1	2	yes	—
64	C₅×C₄⋊C₄	—	2	2	no	—
65	C₅×C₄⋊C₄	—	2	1	no	—
66	C₅×C₄⋊C₄	—	2	1	no	—
67	C₅×C₄⋊C₄	—	2	2	no	—
68	C₅×C₄⋊C₄	C₂³	1	2	yes	—
69	C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
70	C₅×C₄⋊C₄	C₂³	1	2	yes	—
71	C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
72	C₅×C₂²⋊C₄	—	2	2	no	—
73	C₅×C₂²⋊C₄	—	2	1	no	—
74	C₅×C₂²⋊C₄	—	2	2	no	—
75	C₅×C₂²⋊C₄	—	2	2	no	—
76	C₄²×C₅	—	2	2	no	—
77	C₄²×C₅	—	2	1	no	—
78	#537	C₂×C₅	1	1	yes	characteristic
79	#520	C₂×C₅	1	1	yes	characteristic, complemented
80	#485	C₂×C₅	1	2	yes	complemented
81	#484	C₂×C₅	1	2	yes	complemented
82	#482	C₂×C₅	1	2	yes	complemented
83	#480	C₂×C₅	1	2	yes	complemented
84	#428	C₂×C₅	1	1	yes	characteristic, complemented
85	C₂×#107	C₂×C₅	1	1	yes	characteristic, complemented
86	#417	C₂×C₅	1	1	yes	characteristic, complemented
87	#416	C₂×C₅	1	1	yes	characteristic, complemented
88	#408	C₂×C₅	1	1	yes	characteristic, complemented
89	C₂³×C₅	—	2	1	no	—
90	C₂³×C₅	C₂×D₄	1	2	yes	—
91	C₂³×C₅	C₂×D₄	1	1	yes	characteristic
92	C₅×Q₈	—	2	2	no	—
93	C₅×Q₈	—	2	1	no	—
94	C₅×Q₈	—	2	1	no	—
95	C₅×D₄	—	2	4	no	—
96	C₅×D₄	—	2	2	no	—
97	C₅×D₄	—	2	1	no	—
98	C₅×D₄	—	2	1	no	—
99	C₅×D₄	—	4	2	no	—
100	C₅×D₄	—	8	2	no	—
101	C₂×C₄×C₅	—	2	2	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	—	2	1	no	—
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	—	2	1	no	—
108	C₂×C₄×C₅	—	4	2	no	—
109	C₂×C₄×C₅	—	8	1	no	—
110	C₂×C₄×C₅	—	8	2	no	—
111	C₂×C₄×C₅	—	8	1	no	—
112	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
113	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅	#41	1	2	yes	—
115	C₂×C₄×C₅	#41	1	2	yes	—
116	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
117	C₅×C₈	—	2	1	no	—
118	C₅×C₈	—	2	1	no	—
119	C₅×C₈	—	2	2	no	—
120	C₂×#41	C₂²×C₅	1	1	yes	characteristic
121	C₂×#41	C₂²×C₅	1	1	yes	characteristic
122	C₂×M₁₆	C₂²×C₅	1	1	yes	characteristic
123	C₂²×C₈	C₂²×C₅	1	1	yes	characteristic
124	C₂²⋊Q₈	—	2	1	no	—
125	C₂²⋊Q₈	C₂²×C₅	1	2	yes	—
126	#121	—	2	1	no	—
127	#121	—	2	2	no	—
128	#121	C₂²×C₅	1	2	yes	—
129	C₄×Q₈	—	2	1	no	—
130	C₄×D₄	—	2	2	no	—
131	C₄×D₄	—	2	1	no	—
132	#117	C₂²×C₅	1	1	yes	characteristic
133	#117	C₂²×C₅	1	2	yes	—
134	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
135	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
136	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
137	#107	C₂²×C₅	1	2	yes	—
138	#107	C₂²×C₅	1	2	yes	—
139	#107	C₂²×C₅	1	2	yes	—
140	#106	C₂²×C₅	1	2	yes	—
141	#103	C₂²×C₅	1	1	yes	characteristic
142	#103	C₂²×C₅	1	1	yes	characteristic
143	#103	C₂²×C₅	1	2	yes	—
144	#102	C₂²×C₅	1	1	yes	characteristic
145	#102	C₂²×C₅	1	1	yes	characteristic
146	#102	C₂²×C₅	1	2	yes	—
147	C₂²⋊C₈	C₂²×C₅	1	2	yes	—
148	C₂²⋊C₈	C₂²×C₅	1	2	yes	—
149	C₂²×C₅	—	2	1	no	—
150	C₂²×C₅	—	2	1	no	—
151	C₂²×C₅	—	2	2	no	—
152	C₂²×C₅	—	2	2	no	—
153	C₂²×C₅	—	2	2	no	—
154	C₂²×C₅	—	4	1	no	—
155	C₂²×C₅	—	4	1	no	—
156	C₂²×C₅	—	4	1	no	—
157	C₂²×C₅	#123	1	2	yes	—
158	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
159	C₄×C₅	—	2	2	no	—
160	C₄×C₅	—	4	1	no	—
161	C₄×C₅	—	4	2	no	—
162	C₄×C₅	—	4	1	no	—
163	C₄×C₅	—	4	2	no	—
164	C₄×C₅	—	4	1	no	—
165	C₄×C₅	#123	1	2	yes	—
166	C₄×C₅	C₂×#41	1	1	yes	characteristic
167	C₄×C₅	C₂×#41	1	1	yes	characteristic
168	#41	—	2	2	no	—
169	#41	—	2	2	no	—
170	#41	—	2	2	no	—
171	#41	—	2	2	no	—
172	C₂×Q₈	C₅×D₄	1	1	yes	characteristic
173	C₂×Q₈	C₂³×C₅	1	1	yes	characteristic
174	C₂×D₄	—	2	1	no	—
175	C₂×D₄	—	2	2	no	—
176	C₂×D₄	C₅×D₄	1	2	yes	—
177	C₂×D₄	C₅×D₄	1	1	yes	characteristic
178	C₂×D₄	C₂³×C₅	1	1	yes	characteristic
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	C₂³×C₅	1	2	yes	—
184	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
185	M₁₆	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	C₂³×C₅	1	2	yes	—
188	C₂×C₈	C₂³×C₅	1	1	yes	characteristic
189	C₂×C₈	C₂³×C₅	1	1	yes	characteristic
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
195	C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
196	C₄⋊C₄	C₂³×C₅	1	2	yes	—
197	C₄⋊C₄	C₂³×C₅	1	2	yes	—
198	C₂²⋊C₄	—	2	2	no	—
199	C₂²⋊C₄	—	2	1	no	—
200	C₂²⋊C₄	—	2	2	no	—
201	C₂²⋊C₄	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	1	no	—
204	C₂×C₅	—	2	2	no	—
205	C₂×C₅	—	4	2	no	—
206	C₂×C₅	—	8	1	no	—
207	C₂×C₅	C₂×#123	1	1	yes	characteristic
208	C₂×C₅	#576	1	1	yes	characteristic
209	C₂×C₅	#578	1	1	yes	characteristic
210	C₂³	—	2	1	no	—
211	C₂³	C₂×C₅×D₄	1	2	yes	—
212	C₂³	C₂×C₅×D₄	1	1	yes	characteristic
213	Q₈	—	2	2	no	—
214	Q₈	—	2	1	no	—
215	Q₈	—	2	1	no	—
216	D₄	—	2	4	no	—
217	D₄	—	2	2	no	—
218	D₄	—	2	1	no	—
219	D₄	—	2	1	no	—
220	D₄	—	4	2	no	—
221	D₄	—	8	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	8	1	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	—	8	1	no	—
233	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
234	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
235	C₂×C₄	C₅×#41	1	2	yes	—
236	C₂×C₄	C₅×#41	1	2	yes	—
237	C₂×C₄	C₂⁴×C₅	1	1	yes	characteristic
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	2	2	no	—
241	C₅	#3098	1	1	yes	characteristic, direct product factor, Sylow
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	4	1	no	—
248	C₂²	—	4	1	no	—
249	C₂²	—	4	1	no	—
250	C₂²	C₅×#123	1	2	yes	—
251	C₂²	C₂²×C₅×D₄	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	4	1	no	—
254	C₄	—	4	2	no	—
255	C₄	—	4	1	no	—
256	C₄	—	4	2	no	—
257	C₄	—	4	1	no	—
258	C₄	C₅×#123	1	2	yes	—
259	C₄	C₂×C₅×#41	1	1	yes	characteristic
260	C₄	C₂×C₅×#41	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	4	2	no	—
263	C₂	—	8	1	no	—
264	C₂	C₂×C₅×#123	1	1	yes	characteristic
265	C₂	C₅×#576	1	1	yes	characteristic
266	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	C₅×#537	C₂
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
2. C₅×#520	C₂	semidirect
3. C₅×#485	C₂	semidirect
4. C₅×#484	C₂	semidirect
5. C₅×#482	C₂	semidirect
6. C₅×#480	C₂	semidirect
7. C₅×#428	C₂	semidirect
8. C₂×C₅×#107	C₂	semidirect
9. C₅×#417	C₂	semidirect
10. C₅×#416	C₂	semidirect
11. C₅×#408	C₂	semidirect
41. #3098	C₅	direct (N × H)
79. #520	C₂×C₅	semidirect
80. #485	C₂×C₅	semidirect
81. #484	C₂×C₅	semidirect
82. #482	C₂×C₅	semidirect
83. #480	C₂×C₅	semidirect
84. #428	C₂×C₅	semidirect
85. C₂×#107	C₂×C₅	semidirect
86. #417	C₂×C₅	semidirect
87. #416	C₂×C₅	semidirect
88. #408	C₂×C₅	semidirect
241. C₅	#3098	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×#537	2
C₅×#520	2
C₅×#485	2
C₅×#484	2
C₅×#482	2
C₅×#480	2
C₅×#428	2
C₂×C₅×#107	2
C₅×#417	2
C₅×#416	2
C₅×#408	2
#3098	5

Subgroups — C5×#3098 #10616693

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#3098 #10616693