Subgroups — C₅×#7722 #11930442

Order 1280.

404 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2862	C₂	1	1	yes	maximal, characteristic
2	C₅×#1839	C₂	1	1	yes	maximal, characteristic
3	C₅×#1831	C₂	1	1	yes	maximal, characteristic
4	C₅×#1272	C₂	1	4	yes	maximal
5	C₂×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
7	C₂×C₅×C₄⋊C₈	—	2	1	no	—
8	C₅×#391	—	2	1	no	—
9	C₅×#391	—	2	1	no	—
10	C₅×#391	C₄	1	2	yes	—
11	C₅×#391	C₂²	1	2	yes	—
12	C₅×#382	—	2	1	no	—
13	C₅×#382	—	2	1	no	—
14	C₄³×C₅	C₂²	1	1	yes	characteristic
15	C₅×#336	—	2	1	no	—
16	C₅×#336	—	2	1	no	—
17	C₅×#336	—	2	1	no	—
18	C₅×#336	C₂²	1	2	yes	—
19	C₅×#336	C₂²	1	2	yes	—
20	#7722	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅×Q₈	—	2	1	no	—
22	C₂²×C₅×Q₈	C₂×C₄	1	2	yes	—
23	C₂²×C₅×C₈	—	2	2	no	—
24	C₂²×C₅×C₈	—	2	2	no	—
25	C₅×C₄⋊Q₈	—	4	2	no	—
26	C₅×C₄⋊Q₈	—	4	2	no	—
27	C₂×C₅×C₄⋊C₄	—	2	1	no	—
28	C₂×C₅×C₄⋊C₄	—	2	1	no	—
29	C₂×C₅×C₄⋊C₄	—	2	1	no	—
30	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
38	C₅×C₄⋊C₈	—	4	2	no	—
39	C₅×#95	—	2	2	no	—
40	C₅×#95	—	2	2	no	—
41	C₅×#95	—	2	2	no	—
42	C₅×#95	—	2	2	no	—
43	#2862	C₂×C₅	1	1	yes	characteristic
44	#1839	C₂×C₅	1	1	yes	characteristic
45	#1831	C₂×C₅	1	1	yes	characteristic
46	#1272	C₂×C₅	1	4	yes	—
47	C₂×C₅×Q₈	—	2	4	no	—
48	C₂×C₅×Q₈	—	4	2	no	—
49	C₂×C₅×Q₈	—	4	2	no	—
50	C₂×C₅×Q₈	—	4	2	no	—
51	C₂×C₅×Q₈	—	4	2	no	—
52	C₂×C₅×Q₈	—	4	2	no	—
53	C₂²×C₄×C₅	—	2	2	no	—
54	C₂²×C₄×C₅	—	2	2	no	—
55	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂×C₅×C₈	—	4	2	no	—
63	C₂×C₅×C₈	—	4	2	no	—
64	C₂×C₅×C₈	—	4	2	no	—
65	C₂×C₅×C₈	—	4	2	no	—
66	C₂×C₅×C₈	—	4	2	no	—
67	C₂×C₅×C₈	—	4	2	no	—
68	C₅×C₄⋊C₄	—	4	2	no	—
69	C₅×C₄⋊C₄	—	4	2	no	—
70	C₅×C₄⋊C₄	—	4	2	no	—
71	C₅×C₄⋊C₄	—	4	2	no	—
72	C₄²×C₅	—	2	2	no	—
73	C₄²×C₅	—	2	2	no	—
74	C₄²×C₅	—	2	2	no	—
75	C₄²×C₅	—	4	1	no	—
76	C₄²×C₅	—	4	1	no	—
77	C₄²×C₅	—	4	1	no	—
78	C₄²×C₅	SD₁₆	1	2	yes	—
79	C₄²×C₅	Q₁₆	1	2	yes	—
80	C₂×C₄⋊Q₈	C₄×C₅	1	1	yes	characteristic
81	C₂×C₄⋊Q₈	C₄×C₅	1	1	yes	characteristic
82	C₂×C₄⋊C₈	—	2	1	no	—
83	#391	—	2	1	no	—
84	#391	—	2	1	no	—
85	#391	C₄×C₅	1	2	yes	—
86	#391	C₂²×C₅	1	2	yes	—
87	#382	—	2	1	no	—
88	#382	—	2	1	no	—
89	C₄³	C₂²×C₅	1	1	yes	characteristic
90	#336	—	2	1	no	—
91	#336	—	2	1	no	—
92	#336	—	2	1	no	—
93	#336	C₂²×C₅	1	2	yes	—
94	#336	C₂²×C₅	1	2	yes	—
95	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₅×Q₈	—	8	4	no	—
97	C₅×Q₈	—	8	4	no	—
98	C₂×C₄×C₅	—	2	1	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	1	no	—
103	C₂×C₄×C₅	—	2	1	no	—
104	C₂×C₄×C₅	—	2	1	no	—
105	C₂×C₄×C₅	—	2	1	no	—
106	C₂×C₄×C₅	—	2	1	no	—
107	C₂×C₄×C₅	—	2	1	no	—
108	C₂×C₄×C₅	—	2	1	no	—
109	C₂×C₄×C₅	—	2	1	no	—
110	C₂×C₄×C₅	—	2	1	no	—
111	C₂×C₄×C₅	—	2	1	no	—
112	C₂×C₄×C₅	—	2	1	no	—
113	C₂×C₄×C₅	—	4	2	no	—
114	C₂×C₄×C₅	—	4	2	no	—
115	C₂×C₄×C₅	—	4	2	no	—
116	C₂×C₄×C₅	—	4	2	no	—
117	C₂×C₄×C₅	—	4	2	no	—
118	C₂×C₄×C₅	—	4	2	no	—
119	C₂×C₄×C₅	#100	1	2	yes	—
120	C₂×C₄×C₅	#103	1	2	yes	—
121	C₂×C₄×C₅	#103	1	2	yes	—
122	C₂×C₄×C₅	#131	1	2	yes	—
123	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
125	C₂×C₄×C₅	#137	1	2	yes	—
126	C₅×C₈	—	8	2	no	—
127	C₅×C₈	—	8	2	no	—
128	C₂²×Q₈	—	2	1	no	—
129	C₂²×Q₈	C₂×C₄×C₅	1	2	yes	—
130	C₂²×C₈	—	2	2	no	—
131	C₂²×C₈	—	2	2	no	—
132	C₄⋊Q₈	—	4	2	no	—
133	C₄⋊Q₈	—	4	2	no	—
134	C₂×C₄⋊C₄	—	2	1	no	—
135	C₂×C₄⋊C₄	—	2	1	no	—
136	C₂×C₄⋊C₄	—	2	1	no	—
137	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
138	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
139	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
140	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
141	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
142	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
143	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
144	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
145	C₄⋊C₈	—	4	2	no	—
146	#95	—	2	2	no	—
147	#95	—	2	2	no	—
148	#95	—	2	2	no	—
149	#95	—	2	2	no	—
150	C₂²×C₅	#408	1	1	yes	characteristic
151	C₂²×C₅	#410	1	1	yes	characteristic
152	C₂²×C₅	C₂×#100	1	1	yes	characteristic
153	C₂²×C₅	C₂×#103	1	1	yes	characteristic
154	C₂²×C₅	#419	1	1	yes	characteristic
155	C₂²×C₅	#421	1	1	yes	characteristic
156	C₂²×C₅	#421	1	1	yes	characteristic
157	C₄×C₅	—	2	2	no	—
158	C₄×C₅	—	2	2	no	—
159	C₄×C₅	—	2	2	no	—
160	C₄×C₅	—	2	2	no	—
161	C₄×C₅	—	2	1	no	—
162	C₄×C₅	—	2	1	no	—
163	C₄×C₅	—	2	2	no	—
164	C₄×C₅	—	4	1	no	—
165	C₄×C₅	—	8	2	no	—
166	C₄×C₅	—	8	2	no	—
167	C₂×Q₈	—	2	4	no	—
168	C₂×Q₈	—	4	2	no	—
169	C₂×Q₈	—	4	2	no	—
170	C₂×Q₈	—	4	2	no	—
171	C₂×Q₈	—	4	2	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
176	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
177	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
178	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
179	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	4	1	no	—
196	C₄²	—	4	1	no	—
197	C₄²	—	4	1	no	—
198	C₄²	C₅×SD₁₆	1	2	yes	—
199	C₄²	C₅×Q₁₆	1	2	yes	—
200	C₂×C₅	#1469	1	1	yes	characteristic
201	C₂×C₅	#1481	1	1	yes	characteristic
202	C₂×C₅	#1497	1	1	yes	characteristic
203	C₂×C₅	#1515	1	1	yes	characteristic
204	C₂×C₅	#1518	1	1	yes	characteristic
205	C₂×C₅	#1533	1	1	yes	characteristic
206	C₂×C₅	#1548	1	1	yes	characteristic
207	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
208	Q₈	—	8	4	no	—
209	Q₈	—	8	4	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	C₅×#100	1	2	yes	—
232	C₂×C₄	C₅×#103	1	2	yes	—
233	C₂×C₄	C₅×#103	1	2	yes	—
234	C₂×C₄	C₅×#131	1	2	yes	—
235	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
236	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
237	C₂×C₄	C₅×#137	1	2	yes	—
238	C₈	—	8	2	no	—
239	C₈	—	8	2	no	—
240	C₅	#7722	1	1	yes	characteristic, direct product factor, Sylow
241	C₂²	C₅×#408	1	1	yes	characteristic
242	C₂²	C₅×#410	1	1	yes	characteristic
243	C₂²	C₂×C₅×#100	1	1	yes	characteristic
244	C₂²	C₂×C₅×#103	1	1	yes	characteristic
245	C₂²	C₅×#419	1	1	yes	characteristic
246	C₂²	C₅×#421	1	1	yes	characteristic
247	C₂²	C₅×#421	1	1	yes	characteristic
248	C₄	—	2	2	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	2	no	—
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	4	1	no	—
256	C₄	—	8	2	no	—
257	C₄	—	8	2	no	—
258	C₂	C₅×#1469	1	1	yes	characteristic
259	C₂	C₅×#1481	1	1	yes	characteristic
260	C₂	C₅×#1497	1	1	yes	characteristic
261	C₂	C₅×#1515	1	1	yes	characteristic
262	C₂	C₅×#1518	1	1	yes	characteristic
263	C₂	C₅×#1533	1	1	yes	characteristic
264	C₂	C₅×#1548	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
20. #7722	C₅	direct (N × H)
240. C₅	#7722	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×#2862	2
C₅×#1839	2
C₅×#1831	2
C₅×#1272	2
#7722	5

Subgroups — C5×#7722 #11930442

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#7722 #11930442