Subgroups — #11987765

Order 1280.

432 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	#10600184	C₂	1	1	yes	maximal, characteristic
3	#10600181	C₂	1	1	yes	maximal, characteristic
4	#10598129	C₂	1	4	yes	maximal
5	C₂×C₅×#125	—	2	1	no	—
6	C₅×#389	—	2	1	no	—
7	C₅×#389	—	2	1	no	—
8	C₅×#389	C₄	1	2	yes	—
9	C₅×#389	C₂²	1	2	yes	—
10	C₅×#384	C₄	1	1	yes	characteristic
11	C₅×#384	C₄	1	1	yes	characteristic
12	C₅×#382	—	2	1	no	—
13	C₄³×C₅	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#168	—	2	1	no	—
15	C₂×#3935	—	2	1	no	—
16	C₂×C₄×#168	—	2	1	no	—
17	#64744	—	2	1	no	—
18	#64744	C₂²	1	2	yes	—
19	#64744	C₂²	1	2	yes	—
20	#7683	—	5	1	no	maximal, Sylow
21	C₅×#125	—	4	4	no	—
22	C₂×C₅×C₄⋊C₄	—	2	1	no	—
23	C₂×C₅×C₄⋊C₄	—	2	1	no	—
24	C₂×C₅×C₄⋊C₄	—	2	1	no	—
25	C₂×C₅×C₄⋊C₄	—	2	1	no	—
26	C₂×C₅×C₄⋊C₄	—	2	2	no	—
27	C₂×C₅×C₄⋊C₄	—	2	2	no	—
28	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
29	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅	D₄	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	C₂³	1	1	yes	characteristic
37	C₂²×#168	—	2	2	no	—
38	C₂²×#168	—	2	2	no	—
39	C₅×#95	—	2	2	no	—
40	C₅×#95	—	2	2	no	—
41	C₄⋊#168	—	2	4	no	—
42	#3935	—	4	2	no	—
43	C₄×#168	—	2	4	no	—
44	#2859	D₅	1	1	yes	characteristic
45	#1767	—	5	1	no	—
46	#1744	—	5	1	no	—
47	#1271	—	5	4	no	—
48	C₂²×C₄×C₅	—	2	2	no	—
49	C₂²×C₄×C₅	—	2	2	no	—
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	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
57	C₅×C₄⋊C₄	—	2	4	no	—
58	C₅×C₄⋊C₄	—	4	2	no	—
59	C₅×C₄⋊C₄	—	4	2	no	—
60	C₅×C₄⋊C₄	—	4	2	no	—
61	C₅×C₄⋊C₄	—	4	4	no	—
62	C₅×C₄⋊C₄	—	4	2	no	—
63	C₅×C₄⋊C₄	—	4	2	no	—
64	C₅×C₄⋊C₄	—	4	4	no	—
65	C₄²×C₅	—	2	2	no	—
66	C₄²×C₅	—	2	2	no	—
67	C₄²×C₅	—	2	2	no	—
68	C₄²×C₅	—	2	2	no	—
69	C₄²×C₅	—	2	2	no	—
70	C₄²×C₅	—	4	1	no	—
71	C₄²×C₅	SD₁₆	1	2	yes	—
72	C₄²×C₅	Q₁₆	1	2	yes	—
73	C₂×#168	—	2	4	no	—
74	C₂×#168	—	2	4	no	—
75	C₂×#168	—	4	2	no	—
76	C₂×#168	—	4	2	no	—
77	C₂×#168	—	4	2	no	—
78	C₂×#168	—	4	2	no	—
79	C₂×#125	—	2	1	no	—
80	C₂×C₄⋊C₈	—	10	1	no	—
81	C₂×#97	—	10	1	no	—
82	C₂×C₄×C₈	—	10	1	no	—
83	#389	—	2	1	no	—
84	#389	—	2	1	no	—
85	#389	Dic₅	1	2	yes	—
86	#389	C₂×D₅	1	2	yes	—
87	#384	Dic₅	1	1	yes	characteristic
88	#384	Dic₅	1	1	yes	characteristic
89	#382	—	2	1	no	—
90	C₄³	C₂×D₅	1	1	yes	characteristic
91	#336	—	5	2	no	—
92	#336	—	5	2	no	—
93	#336	—	10	1	no	—
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	—	4	2	no	—
109	C₂×C₄×C₅	—	4	2	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	#101	1	2	yes	—
115	C₂×C₄×C₅	#103	1	2	yes	—
116	C₂×C₄×C₅	#103	1	2	yes	—
117	C₂×C₄×C₅	#104	1	2	yes	—
118	C₂×C₄×C₅	#104	1	2	yes	—
119	C₂×C₄×C₅	#131	1	2	yes	—
120	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
122	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
123	#168	—	4	4	no	—
124	#168	—	4	4	no	—
125	C₂²×C₈	—	10	2	no	—
126	C₂²×C₈	—	10	2	no	—
127	#125	—	4	4	no	—
128	C₂×C₄⋊C₄	—	2	1	no	—
129	C₂×C₄⋊C₄	—	2	1	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	2	1	no	—
132	C₂×C₄⋊C₄	—	2	1	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
135	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
136	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
137	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
138	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
139	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
140	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
141	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
142	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
143	C₄⋊C₈	—	10	4	no	—
144	#97	—	20	2	no	—
145	C₄×C₈	—	10	4	no	—
146	#95	—	2	2	no	—
147	#95	—	2	2	no	—
148	C₂²×C₅	#408	1	1	yes	characteristic
149	C₂²×C₅	#410	1	1	yes	characteristic
150	C₂²×C₅	C₂×#101	1	1	yes	characteristic
151	C₂²×C₅	C₂×#103	1	1	yes	characteristic
152	C₂²×C₅	#418	1	1	yes	characteristic
153	C₂²×C₅	C₂×#104	1	1	yes	characteristic
154	C₂²×C₅	C₂×#104	1	1	yes	characteristic
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	C₄×C₅	#330	1	4	yes	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
168	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
169	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
170	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
171	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
174	C₂×C₈	—	10	4	no	—
175	C₂×C₈	—	10	4	no	—
176	C₂×C₈	—	20	2	no	—
177	C₂×C₈	—	20	2	no	—
178	C₂×C₈	—	20	2	no	—
179	C₂×C₈	—	20	2	no	—
180	C₄⋊C₄	—	2	4	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₄⋊C₄	—	4	2	no	—
184	C₄⋊C₄	—	4	4	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	—	2	2	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	4	1	no	—
194	C₄²	#688	1	2	yes	—
195	C₄²	#689	1	2	yes	—
196	C₂×C₅	#1470	1	1	yes	characteristic
197	C₂×C₅	#1484	1	1	yes	characteristic
198	C₂×C₅	#1493	1	1	yes	characteristic
199	C₂×C₅	#1503	1	1	yes	characteristic
200	C₂×C₅	#1505	1	1	yes	characteristic
201	C₂×C₅	C₂×#330	1	1	yes	characteristic
202	C₂×C₅	#1549	1	1	yes	characteristic
203	C₂³	C₂×C₂²⋊Dic₅	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	—	4	2	no	—
218	C₂×C₄	—	4	2	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	#3967	1	2	yes	—
224	C₂×C₄	#3967	1	2	yes	—
225	C₂×C₄	#3968	1	2	yes	—
226	C₂×C₄	#3969	1	2	yes	—
227	C₂×C₄	#3969	1	2	yes	—
228	C₂×C₄	#4078	1	2	yes	—
229	C₂×C₄	C₂×#688	1	1	yes	characteristic
230	C₂×C₄	C₂×#689	1	1	yes	characteristic
231	C₂×C₄	#4094	1	2	yes	—
232	C₈	—	20	4	no	—
233	C₈	—	20	4	no	—
234	C₅	#7683	1	1	yes	characteristic, complemented, Sylow
235	C₂²	#65504	1	1	yes	characteristic
236	C₂²	C₂×#3967	1	1	yes	characteristic
237	C₂²	C₂×#3968	1	1	yes	characteristic
238	C₂²	#65523	1	1	yes	characteristic
239	C₂²	C₂×#3969	1	1	yes	characteristic
240	C₂²	C₂×#3969	1	1	yes	characteristic
241	C₂²	#65529	1	1	yes	characteristic
242	C₄	—	2	2	no	—
243	C₄	—	2	2	no	—
244	C₄	—	2	2	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	4	1	no	—
249	C₄	—	8	2	no	—
250	C₄	—	8	2	no	—
251	C₄	#64763	1	4	yes	—
252	C₂	C₂×#64763	1	1	yes	characteristic
253	C₂	#10602530	1	1	yes	characteristic
254	C₂	#10602568	1	1	yes	characteristic
255	C₂	#10602594	1	1	yes	characteristic
256	C₂	#10602605	1	1	yes	characteristic
257	C₂	#10602619	1	1	yes	characteristic
258	C₂	#10602628	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#2859	C₂
socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#2859	D₅
hypocenter	C₅	#7683
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

Complements

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

N	H ≅ G/N	type
234. C₅	#7683	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅×#2859	2
#10600184	2
#10600181	2
#10598129	2
#7683	5