Subgroups — #11998378

Order 1280.

486 conjugacy classes of subgroups.

The subgroup lattice has 248 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×#64766	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	2	no	—
9	C₅×#391	C₄	1	4	yes	—
10	C₅×#382	—	2	1	no	—
11	C₅×#382	—	2	1	no	—
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	#64766	—	2	4	no	—
19	#64744	—	2	1	no	—
20	#9701	—	5	1	no	maximal, Sylow
21	C₂²×C₅×Q₈	—	2	1	no	—
22	C₂²×C₅×Q₈	C₂×C₄	1	2	yes	—
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	—	2	1	no	—
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×#333	—	5	2	no	—
50	C₂×#332	—	5	2	no	—
51	C₂×C₅×Q₈	—	2	4	no	—
52	C₂×C₅×Q₈	—	4	4	no	—
53	C₂×C₅×Q₈	—	4	1	no	—
54	C₂×C₅×Q₈	—	4	1	no	—
55	C₂×C₅×Q₈	—	4	2	no	—
56	C₂×C₅×Q₈	—	4	2	no	—
57	C₂²×C₄×C₅	—	2	2	no	—
58	C₂²×C₄×C₅	—	2	2	no	—
59	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
64	C₅×C₄⋊C₄	—	2	4	no	—
65	C₅×C₄⋊C₄	—	4	2	no	—
66	C₅×C₄⋊C₄	—	4	2	no	—
67	C₅×C₄⋊C₄	—	4	1	no	—
68	C₅×C₄⋊C₄	—	4	1	no	—
69	C₄²×C₅	—	2	2	no	—
70	C₄²×C₅	—	2	2	no	—
71	C₄²×C₅	—	2	2	no	—
72	C₄²×C₅	—	4	2	no	—
73	C₄²×C₅	SD₁₆	1	2	yes	—
74	C₄²×C₅	Q₁₆	1	2	yes	—
75	C₄²×C₅	C₂×D₄	1	2	yes	—
76	C₄²×C₅	#41	1	2	yes	—
77	C₂×#168	—	2	4	no	—
78	C₂×#168	—	4	2	no	—
79	C₂×#168	—	4	2	no	—
80	C₂×#168	—	4	2	no	—
81	C₂×#168	—	4	4	no	—
82	C₂×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
83	C₂×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
84	C₂×C₄⋊C₈	—	5	2	no	—
85	C₂×C₄⋊C₈	—	5	2	no	—
86	C₂×C₄⋊C₈	—	10	2	no	—
87	C₂×#97	—	10	1	no	—
88	#391	—	2	2	no	—
89	#391	Dic₅	1	4	yes	—
90	#382	—	2	1	no	—
91	#382	—	2	1	no	—
92	C₄³	C₂×D₅	1	1	yes	characteristic
93	#336	—	10	1	no	—
94	#333	—	10	4	no	—
95	#332	—	10	4	no	—
96	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₅×Q₈	—	8	4	no	—
98	C₅×Q₈	—	8	2	no	—
99	C₅×Q₈	—	8	1	no	—
100	C₅×Q₈	—	8	1	no	—
101	C₂×C₄×C₅	—	2	2	no	—
102	C₂×C₄×C₅	—	2	2	no	—
103	C₂×C₄×C₅	—	2	2	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	—	4	2	no	—
108	C₂×C₄×C₅	—	4	2	no	—
109	C₂×C₄×C₅	—	4	4	no	—
110	C₂×C₄×C₅	—	4	4	no	—
111	C₂×C₄×C₅	#103	1	4	yes	—
112	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₅	#121	1	2	yes	—
114	C₂×C₄×C₅	#124	1	1	yes	characteristic
115	C₂×C₄×C₅	#127	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
118	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
119	C₂×C₄×C₅	#137	1	2	yes	—
120	C₂×C₄×C₅	#137	1	2	yes	—
121	#168	—	4	4	no	—
122	#168	—	8	2	no	—
123	C₂²×Q₈	—	2	1	no	—
124	C₂²×Q₈	C₂×Dic₅	1	2	yes	—
125	C₂²×C₈	—	10	2	no	—
126	C₂²×C₈	—	10	2	no	—
127	C₄⋊Q₈	—	2	2	no	—
128	C₄⋊Q₈	—	2	2	no	—
129	C₄⋊Q₈	—	2	2	no	—
130	C₄⋊Q₈	—	2	2	no	—
131	C₂×C₄⋊C₄	—	2	1	no	—
132	C₂×C₄⋊C₄	—	2	1	no	—
133	C₂×C₄⋊C₄	—	2	1	no	—
134	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
135	C₂×C₄²	C₂×Dic₅	1	2	yes	—
136	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
137	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
138	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
139	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
140	C₄⋊C₈	—	5	4	no	—
141	C₄⋊C₈	—	10	2	no	—
142	C₄⋊C₈	—	10	2	no	—
143	C₄⋊C₈	—	10	2	no	—
144	C₄⋊C₈	—	20	4	no	—
145	#97	—	20	2	no	—
146	#95	—	2	4	no	—
147	#95	—	2	4	no	—
148	C₂²×C₅	#390	1	1	yes	characteristic
149	C₂²×C₅	#413	1	2	yes	—
150	C₂²×C₅	C₂×#103	1	1	yes	characteristic
151	C₂²×C₅	#417	1	2	yes	—
152	C₂²×C₅	#419	1	1	yes	characteristic
153	C₄×C₅	—	2	4	no	—
154	C₄×C₅	—	2	2	no	—
155	C₄×C₅	—	4	2	no	—
156	C₄×C₅	—	8	2	no	—
157	C₄×C₅	—	8	2	no	—
158	C₄×C₅	#461	1	2	yes	—
159	C₄×C₅	#462	1	2	yes	—
160	C₄×C₅	#489	1	2	yes	—
161	C₄×C₅	#496	1	1	yes	characteristic
162	C₄×C₅	#497	1	1	yes	characteristic
163	C₂×Q₈	—	2	4	no	—
164	C₂×Q₈	—	4	4	no	—
165	C₂×Q₈	—	4	2	no	—
166	C₂×Q₈	—	4	1	no	—
167	C₂×Q₈	—	4	1	no	—
168	C₂×Q₈	—	4	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
172	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
173	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
174	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
175	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
176	C₂×C₈	—	10	4	no	—
177	C₂×C₈	—	20	2	no	—
178	C₂×C₈	—	20	2	no	—
179	C₂×C₈	—	20	2	no	—
180	C₂×C₈	—	20	4	no	—
181	C₄⋊C₄	—	2	4	no	—
182	C₄⋊C₄	—	4	1	no	—
183	C₄⋊C₄	—	4	2	no	—
184	C₄⋊C₄	—	4	2	no	—
185	C₄⋊C₄	—	4	1	no	—
186	C₄²	—	2	2	no	—
187	C₄²	—	2	2	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	4	2	no	—
190	C₄²	#688	1	2	yes	—
191	C₄²	#689	1	2	yes	—
192	C₄²	D₄×D₅	1	2	yes	—
193	C₄²	#711	1	2	yes	—
194	C₂×C₅	#1550	1	2	yes	—
195	C₂×C₅	#1559	1	2	yes	—
196	C₂×C₅	#1959	1	1	yes	characteristic
197	C₂×C₅	#1967	1	1	yes	characteristic
198	C₂×C₅	#1981	1	1	yes	characteristic
199	C₂³	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
200	Q₈	—	8	4	no	—
201	Q₈	—	8	2	no	—
202	Q₈	—	8	1	no	—
203	Q₈	—	8	1	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	#3967	1	4	yes	—
215	C₂×C₄	D₄×Dic₅	1	2	yes	—
216	C₂×C₄	#4082	1	1	yes	characteristic
217	C₂×C₄	#4084	1	2	yes	—
218	C₂×C₄	#4086	1	1	yes	characteristic
219	C₂×C₄	C₂×#688	1	1	yes	characteristic
220	C₂×C₄	#4088	1	2	yes	—
221	C₂×C₄	C₂×#689	1	1	yes	characteristic
222	C₂×C₄	#4095	1	2	yes	—
223	C₂×C₄	#4097	1	2	yes	—
224	C₈	—	20	4	no	—
225	C₈	—	40	2	no	—
226	C₅	#9701	1	1	yes	characteristic, complemented, Sylow
227	C₂²	#65510	1	1	yes	characteristic
228	C₂²	C₂×#3967	1	1	yes	characteristic
229	C₂²	#65514	1	1	yes	characteristic
230	C₂²	#65521	1	2	yes	—
231	C₂²	#65526	1	2	yes	—
232	C₄	—	2	2	no	—
233	C₄	—	2	4	no	—
234	C₄	—	4	2	no	—
235	C₄	—	8	2	no	—
236	C₄	—	8	2	no	—
237	C₄	#65344	1	2	yes	—
238	C₄	#65347	1	1	yes	characteristic
239	C₄	#65349	1	1	yes	characteristic
240	C₄	#65372	1	2	yes	—
241	C₄	#65379	1	2	yes	—
242	C₂	#10602519	1	1	yes	characteristic
243	C₂	#10602520	1	1	yes	characteristic
244	C₂	#10602571	1	1	yes	characteristic
245	C₂	#10602623	1	2	yes	—
246	C₂	#10602626	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	#9701
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
226. C₅	#9701	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×#64766	2
C₂×#64767	2
#10600187	2
#10600184	2
#9701	5