Subgroups — #11998409

Order 1280.

486 conjugacy classes of subgroups.

The subgroup lattice has 256 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	C₂×#64767	C₂	1	2	yes	maximal
3	C₂×#64767	C₂	1	2	yes	maximal
4	#10600187	C₂	1	1	yes	maximal, characteristic
5	#10600184	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₅×#391	—	2	1	no	—
9	C₅×#391	—	2	2	no	—
10	C₅×#391	—	2	1	no	—
11	C₅×#382	C₄	1	4	yes	—
12	C₄³×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#168	—	2	2	no	—
14	C₂×C₄⋊#168	C₂²	1	2	yes	—
15	C₂×C₄⋊#168	C₂²	1	2	yes	—
16	C₂×#3935	—	2	1	no	—
17	#64767	—	2	4	no	—
18	#64767	—	2	4	no	—
19	#64744	—	2	1	no	—
20	#9708	—	5	1	no	maximal, Sylow
21	C₂²×C₅×Q₈	—	2	1	no	—
22	C₂²×C₅×Q₈	—	2	1	no	—
23	C₅×C₄⋊Q₈	—	2	2	no	—
24	C₅×C₄⋊Q₈	—	2	2	no	—
25	C₅×C₄⋊Q₈	—	2	2	no	—
26	C₅×C₄⋊Q₈	—	2	2	no	—
27	C₂×C₅×C₄⋊C₄	—	2	1	no	—
28	C₂×C₅×C₄⋊C₄	—	2	1	no	—
29	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
30	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₅	D₄	1	2	yes	—
34	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
36	C₂²×#168	—	2	2	no	—
37	C₂²×#168	—	2	2	no	—
38	C₅×#95	—	2	4	no	—
39	C₅×#95	—	2	4	no	—
40	C₄⋊#168	—	2	2	no	—
41	C₄⋊#168	—	2	2	no	—
42	C₄⋊#168	—	2	2	no	—
43	C₄⋊#168	—	4	4	no	—
44	C₄⋊#168	D₄	1	4	yes	—
45	#3935	—	4	2	no	—
46	#2862	D₅	1	1	yes	characteristic
47	#1842	—	5	1	no	—
48	#1767	—	5	1	no	—
49	C₂×#332	—	5	2	no	—
50	C₂×#332	—	5	2	no	—
51	C₂×C₅×Q₈	—	4	2	no	—
52	C₂×C₅×Q₈	—	4	1	no	—
53	C₂×C₅×Q₈	—	4	1	no	—
54	C₂×C₅×Q₈	—	4	2	no	—
55	C₂×C₅×Q₈	—	4	1	no	—
56	C₂×C₅×Q₈	—	4	2	no	—
57	C₂×C₅×Q₈	—	4	1	no	—
58	C₂×C₅×Q₈	—	4	2	no	—
59	C₂²×C₄×C₅	—	2	2	no	—
60	C₂²×C₄×C₅	—	2	2	no	—
61	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
66	C₅×C₄⋊C₄	—	2	4	no	—
67	C₅×C₄⋊C₄	—	2	4	no	—
68	C₅×C₄⋊C₄	—	4	2	no	—
69	C₅×C₄⋊C₄	—	4	1	no	—
70	C₅×C₄⋊C₄	—	4	1	no	—
71	C₄²×C₅	—	2	4	no	—
72	C₄²×C₅	—	2	2	no	—
73	C₄²×C₅	—	2	1	no	—
74	C₄²×C₅	—	2	2	no	—
75	C₄²×C₅	—	2	2	no	—
76	C₄²×C₅	—	2	1	no	—
77	C₄²×C₅	C₂×D₄	1	2	yes	—
78	C₄²×C₅	#41	1	2	yes	—
79	C₂×#168	—	2	4	no	—
80	C₂×#168	—	4	2	no	—
81	C₂×#168	—	4	2	no	—
82	C₂×#168	—	4	2	no	—
83	C₂×#168	—	4	4	no	—
84	C₂×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
85	C₂×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
86	C₂×C₄⋊C₈	—	5	2	no	—
87	C₂×C₄⋊C₈	—	5	2	no	—
88	C₂×C₄⋊C₈	—	10	2	no	—
89	C₂×#97	—	10	1	no	—
90	#391	—	2	1	no	—
91	#391	—	2	1	no	—
92	#391	—	2	2	no	—
93	#382	Dic₅	1	4	yes	—
94	C₄³	C₂×D₅	1	1	yes	characteristic
95	#336	—	10	1	no	—
96	#332	—	10	4	no	—
97	#332	—	10	4	no	—
98	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₅×Q₈	—	8	2	no	—
100	C₅×Q₈	—	8	2	no	—
101	C₅×Q₈	—	8	2	no	—
102	C₅×Q₈	—	8	1	no	—
103	C₅×Q₈	—	8	1	no	—
104	C₂×C₄×C₅	—	2	2	no	—
105	C₂×C₄×C₅	—	2	2	no	—
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	—	2	2	no	—
109	C₂×C₄×C₅	—	2	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	4	no	—
114	C₂×C₄×C₅	—	4	2	no	—
115	C₂×C₄×C₅	#101	1	4	yes	—
116	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₅	#121	1	2	yes	—
118	C₂×C₄×C₅	#124	1	1	yes	characteristic
119	C₂×C₄×C₅	#127	1	1	yes	characteristic
120	C₂×C₄×C₅	#135	1	2	yes	—
121	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
123	C₂×C₄×C₅	#137	1	1	yes	characteristic
124	C₂×C₄×C₅	#137	1	2	yes	—
125	#168	—	4	4	no	—
126	#168	—	8	2	no	—
127	C₂²×Q₈	—	2	1	no	—
128	C₂²×Q₈	—	2	1	no	—
129	C₂²×C₈	—	10	2	no	—
130	C₂²×C₈	—	10	2	no	—
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	—	2	2	no	—
134	C₄⋊Q₈	—	2	2	no	—
135	C₂×C₄⋊C₄	—	2	1	no	—
136	C₂×C₄⋊C₄	—	2	1	no	—
137	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
138	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
139	C₂×C₄²	C₂×Dic₅	1	2	yes	—
140	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
141	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
142	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
143	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
144	C₄⋊C₈	—	5	4	no	—
145	C₄⋊C₈	—	10	2	no	—
146	C₄⋊C₈	—	10	2	no	—
147	C₄⋊C₈	—	10	2	no	—
148	C₄⋊C₈	—	20	4	no	—
149	#97	—	20	2	no	—
150	#95	—	2	4	no	—
151	#95	—	2	4	no	—
152	C₂²×C₅	#390	1	1	yes	characteristic
153	C₂²×C₅	C₂×#101	1	2	yes	—
154	C₂²×C₅	#416	1	1	yes	characteristic
155	C₂²×C₅	#417	1	2	yes	—
156	C₂²×C₅	#417	1	1	yes	characteristic
157	C₄×C₅	—	2	4	no	—
158	C₄×C₅	—	2	2	no	—
159	C₄×C₅	—	4	2	no	—
160	C₄×C₅	—	8	2	no	—
161	C₄×C₅	—	8	2	no	—
162	C₄×C₅	#463	1	2	yes	—
163	C₄×C₅	#464	1	2	yes	—
164	C₄×C₅	#489	1	2	yes	—
165	C₄×C₅	#496	1	1	yes	characteristic
166	C₄×C₅	#497	1	1	yes	characteristic
167	C₂×Q₈	—	4	2	no	—
168	C₂×Q₈	—	4	1	no	—
169	C₂×Q₈	—	4	2	no	—
170	C₂×Q₈	—	4	1	no	—
171	C₂×Q₈	—	4	2	no	—
172	C₂×Q₈	—	4	1	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	1	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
178	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
179	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
182	C₂×C₈	—	10	4	no	—
183	C₂×C₈	—	20	2	no	—
184	C₂×C₈	—	20	2	no	—
185	C₂×C₈	—	20	2	no	—
186	C₂×C₈	—	20	4	no	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	4	1	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₄⋊C₄	—	4	1	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	2	1	no	—
195	C₄²	—	2	4	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	2	2	no	—
198	C₄²	D₄×D₅	1	2	yes	—
199	C₄²	#711	1	2	yes	—
200	C₂×C₅	#1552	1	2	yes	—
201	C₂×C₅	#1561	1	2	yes	—
202	C₂×C₅	#1959	1	1	yes	characteristic
203	C₂×C₅	#1966	1	1	yes	characteristic
204	C₂×C₅	#1983	1	1	yes	characteristic
205	C₂³	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
206	Q₈	—	8	2	no	—
207	Q₈	—	8	2	no	—
208	Q₈	—	8	2	no	—
209	Q₈	—	8	1	no	—
210	Q₈	—	8	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	#3968	1	4	yes	—
223	C₂×C₄	D₄×Dic₅	1	2	yes	—
224	C₂×C₄	#4082	1	1	yes	characteristic
225	C₂×C₄	#4084	1	2	yes	—
226	C₂×C₄	#4086	1	1	yes	characteristic
227	C₂×C₄	#4095	1	2	yes	—
228	C₂×C₄	#4095	1	1	yes	characteristic
229	C₂×C₄	#4096	1	2	yes	—
230	C₂×C₄	#4097	1	1	yes	characteristic
231	C₂×C₄	#4097	1	2	yes	—
232	C₈	—	20	4	no	—
233	C₈	—	40	2	no	—
234	C₅	#9708	1	1	yes	characteristic, complemented, Sylow
235	C₂²	#65510	1	1	yes	characteristic
236	C₂²	C₂×#3968	1	2	yes	—
237	C₂²	#65521	1	1	yes	characteristic
238	C₂²	#65521	1	2	yes	—
239	C₂²	#65522	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	2	4	no	—
242	C₄	—	4	2	no	—
243	C₄	—	8	2	no	—
244	C₄	—	8	2	no	—
245	C₄	#65343	1	2	yes	—
246	C₄	#65344	1	2	yes	—
247	C₄	#65346	1	2	yes	—
248	C₄	#65347	1	1	yes	characteristic
249	C₄	#65349	1	1	yes	characteristic
250	C₂	#10602514	1	1	yes	characteristic
251	C₂	#10602519	1	1	yes	characteristic
252	C₂	#10602582	1	1	yes	characteristic
253	C₂	#10602624	1	2	yes	—
254	C₂	#10602627	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#2862	C₂
socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#2862	D₅
hypocenter	C₅	#9708
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2862

Complements

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

N	H ≅ G/N	type
234. C₅	#9708	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2862	2
C₂×#64767	2
#10600187	2
#10600184	2
#9708	5