Subgroups — C₅×#7723 #11930443

Order 1280.

388 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2859	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×#1271	C₂	1	4	yes	maximal
5	C₂×C₅×#125	—	2	1	no	—
6	C₂×C₅×C₄⋊C₈	—	2	1	no	—
7	C₅×#389	—	2	1	no	—
8	C₅×#389	—	2	1	no	—
9	C₅×#389	C₄	1	2	yes	—
10	C₅×#389	C₂²	1	2	yes	—
11	C₅×#384	C₄	1	1	yes	characteristic
12	C₅×#384	C₄	1	1	yes	characteristic
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	#7723	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅×C₈	—	2	2	no	—
22	C₂²×C₅×C₈	—	2	2	no	—
23	C₅×#125	—	4	2	no	—
24	C₅×#125	—	4	2	no	—
25	C₂×C₅×C₄⋊C₄	—	2	1	no	—
26	C₂×C₅×C₄⋊C₄	—	2	1	no	—
27	C₂×C₅×C₄⋊C₄	—	2	1	no	—
28	C₂×C₅×C₄⋊C₄	—	2	1	no	—
29	C₂×C₅×C₄⋊C₄	—	2	2	no	—
30	C₂×C₅×C₄⋊C₄	—	2	2	no	—
31	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
32	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅	C₂×C₄	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	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₅×C₄⋊C₈	—	4	2	no	—
41	C₅×#95	—	2	2	no	—
42	C₅×#95	—	2	2	no	—
43	#2859	C₂×C₅	1	1	yes	characteristic
44	#1839	C₂×C₅	1	1	yes	characteristic
45	#1831	C₂×C₅	1	1	yes	characteristic
46	#1271	C₂×C₅	1	4	yes	—
47	C₂²×C₄×C₅	—	2	2	no	—
48	C₂²×C₄×C₅	—	2	2	no	—
49	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
56	C₂×C₅×C₈	—	4	2	no	—
57	C₂×C₅×C₈	—	4	2	no	—
58	C₂×C₅×C₈	—	4	2	no	—
59	C₂×C₅×C₈	—	4	2	no	—
60	C₂×C₅×C₈	—	4	2	no	—
61	C₂×C₅×C₈	—	4	2	no	—
62	C₅×C₄⋊C₄	—	2	4	no	—
63	C₅×C₄⋊C₄	—	4	2	no	—
64	C₅×C₄⋊C₄	—	4	2	no	—
65	C₅×C₄⋊C₄	—	4	2	no	—
66	C₅×C₄⋊C₄	—	4	2	no	—
67	C₅×C₄⋊C₄	—	4	2	no	—
68	C₅×C₄⋊C₄	—	4	4	no	—
69	C₅×C₄⋊C₄	—	4	2	no	—
70	C₅×C₄⋊C₄	—	4	2	no	—
71	C₄²×C₅	—	2	2	no	—
72	C₄²×C₅	—	2	2	no	—
73	C₄²×C₅	—	2	2	no	—
74	C₄²×C₅	—	4	1	no	—
75	C₄²×C₅	—	4	1	no	—
76	C₄²×C₅	—	4	1	no	—
77	C₄²×C₅	D₈	1	2	yes	—
78	C₄²×C₅	SD₁₆	1	2	yes	—
79	C₂×#125	—	2	1	no	—
80	C₂×C₄⋊C₈	—	2	1	no	—
81	#389	—	2	1	no	—
82	#389	—	2	1	no	—
83	#389	C₄×C₅	1	2	yes	—
84	#389	C₂²×C₅	1	2	yes	—
85	#384	C₄×C₅	1	1	yes	characteristic
86	#384	C₄×C₅	1	1	yes	characteristic
87	#382	—	2	1	no	—
88	C₄³	C₂²×C₅	1	1	yes	characteristic
89	#336	—	2	1	no	—
90	#336	—	2	1	no	—
91	#336	—	2	1	no	—
92	#336	C₂²×C₅	1	2	yes	—
93	#336	C₂²×C₅	1	2	yes	—
94	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅	—	2	1	no	—
96	C₂×C₄×C₅	—	2	1	no	—
97	C₂×C₄×C₅	—	2	1	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	—	4	2	no	—
111	C₂×C₄×C₅	—	4	2	no	—
112	C₂×C₄×C₅	—	4	2	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	#101	1	2	yes	—
117	C₂×C₄×C₅	#102	1	2	yes	—
118	C₂×C₄×C₅	#102	1	2	yes	—
119	C₂×C₄×C₅	#131	1	2	yes	—
120	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
122	C₂×C₄×C₅	#137	1	2	yes	—
123	C₅×C₈	—	8	2	no	—
124	C₅×C₈	—	8	2	no	—
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	—	2	2	no	—
127	#125	—	4	2	no	—
128	#125	—	4	2	no	—
129	C₂×C₄⋊C₄	—	2	1	no	—
130	C₂×C₄⋊C₄	—	2	1	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	—	2	1	no	—
133	C₂×C₄⋊C₄	—	2	1	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
136	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
137	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
138	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
139	C₂×C₄²	C₅×D₄	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	C₂³×C₅	1	1	yes	characteristic
144	C₄⋊C₈	—	4	2	no	—
145	#95	—	2	2	no	—
146	#95	—	2	2	no	—
147	C₂²×C₅	#408	1	1	yes	characteristic
148	C₂²×C₅	#410	1	1	yes	characteristic
149	C₂²×C₅	C₂×#101	1	1	yes	characteristic
150	C₂²×C₅	C₂×#102	1	1	yes	characteristic
151	C₂²×C₅	#419	1	1	yes	characteristic
152	C₂²×C₅	#421	1	1	yes	characteristic
153	C₂²×C₅	#421	1	1	yes	characteristic
154	C₄×C₅	—	2	2	no	—
155	C₄×C₅	—	2	2	no	—
156	C₄×C₅	—	2	2	no	—
157	C₄×C₅	—	2	2	no	—
158	C₄×C₅	—	2	1	no	—
159	C₄×C₅	—	2	1	no	—
160	C₄×C₅	—	2	2	no	—
161	C₄×C₅	—	4	1	no	—
162	C₄×C₅	—	8	2	no	—
163	C₄×C₅	—	8	2	no	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
167	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
168	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
169	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
170	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
171	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	—	4	2	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	—	4	2	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₄⋊C₄	—	4	2	no	—
184	C₄⋊C₄	—	4	2	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	—	4	4	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	—	2	2	no	—
191	C₄²	—	4	1	no	—
192	C₄²	—	4	1	no	—
193	C₄²	—	4	1	no	—
194	C₄²	C₅×D₈	1	2	yes	—
195	C₄²	C₅×SD₁₆	1	2	yes	—
196	C₂×C₅	#1470	1	1	yes	characteristic
197	C₂×C₅	#1480	1	1	yes	characteristic
198	C₂×C₅	#1497	1	1	yes	characteristic
199	C₂×C₅	#1515	1	1	yes	characteristic
200	C₂×C₅	#1517	1	1	yes	characteristic
201	C₂×C₅	#1534	1	1	yes	characteristic
202	C₂×C₅	#1546	1	1	yes	characteristic
203	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	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	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	C₅×#101	1	2	yes	—
226	C₂×C₄	C₅×#102	1	2	yes	—
227	C₂×C₄	C₅×#102	1	2	yes	—
228	C₂×C₄	C₅×#131	1	2	yes	—
229	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
230	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
231	C₂×C₄	C₅×#137	1	2	yes	—
232	C₈	—	8	2	no	—
233	C₈	—	8	2	no	—
234	C₅	#7723	1	1	yes	characteristic, direct product factor, Sylow
235	C₂²	C₅×#408	1	1	yes	characteristic
236	C₂²	C₅×#410	1	1	yes	characteristic
237	C₂²	C₂×C₅×#101	1	1	yes	characteristic
238	C₂²	C₂×C₅×#102	1	1	yes	characteristic
239	C₂²	C₅×#419	1	1	yes	characteristic
240	C₂²	C₅×#421	1	1	yes	characteristic
241	C₂²	C₅×#421	1	1	yes	characteristic
242	C₄	—	2	2	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	4	1	no	—
250	C₄	—	8	2	no	—
251	C₄	—	8	2	no	—
252	C₂	C₅×#1470	1	1	yes	characteristic
253	C₂	C₅×#1480	1	1	yes	characteristic
254	C₂	C₅×#1497	1	1	yes	characteristic
255	C₂	C₅×#1515	1	1	yes	characteristic
256	C₂	C₅×#1517	1	1	yes	characteristic
257	C₂	C₅×#1534	1	1	yes	characteristic
258	C₂	C₅×#1546	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. #7723	C₅	direct (N × H)
234. C₅	#7723	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×#2859	2
C₅×#1839	2
C₅×#1831	2
C₅×#1271	2
#7723	5

Subgroups — C5×#7723 #11930443

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#7723 #11930443