Subgroups — #11998340

Order 1280.

478 conjugacy classes of subgroups.

The subgroup lattice has 236 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	#10600185	C₂	1	1	yes	maximal, characteristic
5	C₄×C₄⋊#168	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	C₂²	1	2	yes	—
14	C₂×C₄⋊#168	C₂²	1	2	yes	—
15	C₂×C₄×#168	—	2	1	no	—
16	#64767	—	2	4	no	—
17	#64766	—	2	4	no	—
18	#64744	—	2	1	no	—
19	#64744	—	2	2	no	—
20	#9693	—	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	D₄	1	4	yes	—
44	C₄×#168	—	4	2	no	—
45	#2862	D₅	1	1	yes	characteristic
46	#1831	—	5	1	no	—
47	C₄×C₄⋊C₈	—	5	1	no	—
48	C₂×#333	—	5	2	no	—
49	C₂×#332	—	5	2	no	—
50	C₂×C₅×Q₈	—	2	4	no	—
51	C₂×C₅×Q₈	—	4	4	no	—
52	C₂×C₅×Q₈	—	4	1	no	—
53	C₂×C₅×Q₈	—	4	1	no	—
54	C₂×C₅×Q₈	—	4	4	no	—
55	C₂²×C₄×C₅	—	2	2	no	—
56	C₂²×C₄×C₅	—	2	2	no	—
57	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
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	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	1	no	—
66	C₅×C₄⋊C₄	—	4	1	no	—
67	C₄²×C₅	—	2	4	no	—
68	C₄²×C₅	—	2	2	no	—
69	C₄²×C₅	—	4	2	no	—
70	C₄²×C₅	SD₁₆	1	2	yes	—
71	C₄²×C₅	Q₁₆	1	2	yes	—
72	C₄²×C₅	C₂×D₄	1	2	yes	—
73	C₄²×C₅	#41	1	2	yes	—
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	4	no	—
79	C₂×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
80	C₂×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
81	C₂×C₄⋊C₈	—	5	2	no	—
82	C₂×C₄⋊C₈	—	5	2	no	—
83	C₂×C₄×C₈	—	10	1	no	—
84	#391	—	2	2	no	—
85	#391	Dic₅	1	4	yes	—
86	#382	—	2	1	no	—
87	#382	—	2	1	no	—
88	C₄³	C₂×D₅	1	1	yes	characteristic
89	#336	—	10	2	no	—
90	#336	—	10	1	no	—
91	#333	—	10	4	no	—
92	#332	—	10	4	no	—
93	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₅×Q₈	—	8	4	no	—
95	C₅×Q₈	—	8	2	no	—
96	C₅×Q₈	—	8	2	no	—
97	C₂×C₄×C₅	—	2	2	no	—
98	C₂×C₄×C₅	—	2	2	no	—
99	C₂×C₄×C₅	—	2	2	no	—
100	C₂×C₄×C₅	—	2	2	no	—
101	C₂×C₄×C₅	—	2	2	no	—
102	C₂×C₄×C₅	—	2	2	no	—
103	C₂×C₄×C₅	—	4	2	no	—
104	C₂×C₄×C₅	—	4	2	no	—
105	C₂×C₄×C₅	—	4	4	no	—
106	C₂×C₄×C₅	—	4	4	no	—
107	C₂×C₄×C₅	#103	1	4	yes	—
108	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₅	#121	1	2	yes	—
110	C₂×C₄×C₅	#124	1	1	yes	characteristic
111	C₂×C₄×C₅	#127	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
114	C₂×C₄×C₅	#135	1	4	yes	—
115	C₂×C₄×C₅	#137	1	2	yes	—
116	#168	—	4	4	no	—
117	#168	—	8	2	no	—
118	C₂²×Q₈	—	2	1	no	—
119	C₂²×Q₈	C₂×Dic₅	1	2	yes	—
120	C₂²×C₈	—	10	2	no	—
121	C₂²×C₈	—	10	2	no	—
122	C₄⋊Q₈	—	2	2	no	—
123	C₄⋊Q₈	—	2	2	no	—
124	C₄⋊Q₈	—	2	2	no	—
125	C₄⋊Q₈	—	2	2	no	—
126	C₂×C₄⋊C₄	—	2	1	no	—
127	C₂×C₄⋊C₄	—	2	1	no	—
128	C₂×C₄⋊C₄	—	2	1	no	—
129	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
130	C₂×C₄²	C₂×Dic₅	1	2	yes	—
131	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
132	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
133	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
134	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
135	C₄⋊C₈	—	5	4	no	—
136	C₄⋊C₈	—	10	2	no	—
137	C₄⋊C₈	—	10	2	no	—
138	C₄⋊C₈	—	10	2	no	—
139	C₄×C₈	—	20	2	no	—
140	#95	—	2	4	no	—
141	#95	—	2	4	no	—
142	C₂²×C₅	#390	1	1	yes	characteristic
143	C₂²×C₅	#413	1	2	yes	—
144	C₂²×C₅	C₂×#103	1	1	yes	characteristic
145	C₂²×C₅	#416	1	2	yes	—
146	C₂²×C₅	#419	1	1	yes	characteristic
147	C₄×C₅	—	2	4	no	—
148	C₄×C₅	—	2	2	no	—
149	C₄×C₅	—	4	2	no	—
150	C₄×C₅	—	8	2	no	—
151	C₄×C₅	—	8	2	no	—
152	C₄×C₅	#461	1	2	yes	—
153	C₄×C₅	#462	1	2	yes	—
154	C₄×C₅	#488	1	2	yes	—
155	C₄×C₅	#495	1	2	yes	—
156	C₂×Q₈	—	2	4	no	—
157	C₂×Q₈	—	4	4	no	—
158	C₂×Q₈	—	4	4	no	—
159	C₂×Q₈	—	4	1	no	—
160	C₂×Q₈	—	4	1	no	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
164	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
165	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
168	C₂×C₈	—	10	4	no	—
169	C₂×C₈	—	20	2	no	—
170	C₂×C₈	—	20	2	no	—
171	C₂×C₈	—	20	4	no	—
172	C₂×C₈	—	20	2	no	—
173	C₄⋊C₄	—	2	4	no	—
174	C₄⋊C₄	—	4	1	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	1	no	—
178	C₄²	—	2	4	no	—
179	C₄²	—	2	2	no	—
180	C₄²	—	4	2	no	—
181	C₄²	#688	1	2	yes	—
182	C₄²	#689	1	2	yes	—
183	C₄²	D₄×D₅	1	2	yes	—
184	C₄²	#711	1	2	yes	—
185	C₂×C₅	#1542	1	2	yes	—
186	C₂×C₅	#1554	1	2	yes	—
187	C₂×C₅	#1958	1	1	yes	characteristic
188	C₂×C₅	#1967	1	1	yes	characteristic
189	C₂×C₅	#1981	1	1	yes	characteristic
190	C₂³	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
191	Q₈	—	8	4	no	—
192	Q₈	—	8	2	no	—
193	Q₈	—	8	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	#3967	1	4	yes	—
205	C₂×C₄	D₄×Dic₅	1	2	yes	—
206	C₂×C₄	#4082	1	1	yes	characteristic
207	C₂×C₄	#4084	1	2	yes	—
208	C₂×C₄	#4086	1	1	yes	characteristic
209	C₂×C₄	C₂×#688	1	1	yes	characteristic
210	C₂×C₄	#4088	1	2	yes	—
211	C₂×C₄	C₂×#689	1	1	yes	characteristic
212	C₂×C₄	#4096	1	4	yes	—
213	C₈	—	20	4	no	—
214	C₈	—	40	2	no	—
215	C₅	#9693	1	1	yes	characteristic, complemented, Sylow
216	C₂²	#65510	1	1	yes	characteristic
217	C₂²	C₂×#3967	1	1	yes	characteristic
218	C₂²	#65514	1	1	yes	characteristic
219	C₂²	#65522	1	2	yes	—
220	C₂²	#65526	1	2	yes	—
221	C₄	—	2	2	no	—
222	C₄	—	2	4	no	—
223	C₄	—	4	2	no	—
224	C₄	—	8	2	no	—
225	C₄	—	8	2	no	—
226	C₄	#65345	1	2	yes	—
227	C₄	#65348	1	2	yes	—
228	C₄	#65372	1	2	yes	—
229	C₄	#65379	1	2	yes	—
230	C₂	#10602518	1	1	yes	characteristic
231	C₂	#10602520	1	1	yes	characteristic
232	C₂	#10602571	1	1	yes	characteristic
233	C₂	#10602621	1	2	yes	—
234	C₂	#10602625	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	#9693
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
215. C₅	#9693	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
#10600185	2
C₄×C₄⋊#168	2
#9693	5