Subgroups — #11998524

Order 1280.

454 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#64971	C₂	1	1	yes	maximal, characteristic
2	C₅×C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×#64701	C₂	1	2	yes	maximal
4	C₂×#64678	C₂	1	1	yes	maximal, characteristic
5	#10600282	C₂	1	2	yes	maximal
6	C₂×#4015	C₂²	1	2	yes	—
7	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₅×C₈	—	2	1	no	—
9	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₂×#3950	—	2	1	no	—
12	C₂×#3950	—	2	1	no	—
13	C₂×C₄⋊#168	C₂²	1	2	yes	—
14	#64971	—	2	2	no	—
15	#64971	C₄	1	4	yes	complemented
16	C₅×#336	—	2	1	no	—
17	#64749	—	2	4	no	—
18	#64701	—	2	4	no	—
19	#64678	—	2	2	no	—
20	#9730	—	5	1	no	maximal, Sylow
21	C₂²×Dic₁₀	—	2	2	no	—
22	C₂²×C₅×C₈	—	2	1	no	—
23	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
25	C₂×C₅×C₄⋊C₄	D₄	1	2	yes	—
26	C₂×C₄²×C₅	—	2	1	no	—
27	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊Dic₅	—	2	2	no	—
29	C₂×C₄⋊Dic₅	—	2	2	no	—
30	C₂²×#168	—	2	2	no	—
31	#4015	—	2	2	no	—
32	#4015	—	2	2	no	—
33	#4015	—	2	2	no	—
34	#4015	C₂×C₄	1	4	yes	—
35	C₅×C₄⋊C₈	—	2	1	no	—
36	C₅×C₄⋊C₈	D₄	1	2	yes	—
37	C₄×C₅×C₈	—	4	2	no	—
38	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
39	C₄×C₅×C₈	D₄	1	2	yes	—
40	C₅×#95	—	2	1	no	—
41	#3950	—	2	4	no	—
42	#3950	—	2	4	no	—
43	C₄⋊#168	—	2	2	no	—
44	C₄⋊#168	—	2	2	no	—
45	C₂×#500	—	5	1	no	—
46	#1987	—	5	2	no	—
47	C₈×C₄⋊C₄	D₅	1	1	yes	characteristic
48	C₂×#335	—	5	1	no	—
49	C₂×#333	—	5	2	no	—
50	C₂²×C₄×C₅	—	2	2	no	—
51	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅	#41	1	1	yes	characteristic
54	C₂²×Dic₅	—	4	2	no	—
55	C₂×Dic₁₀	—	2	4	no	—
56	C₂×Dic₁₀	—	4	2	no	—
57	C₂×Dic₁₀	—	8	4	no	—
58	C₂×C₅×C₈	—	2	1	no	—
59	C₂×C₅×C₈	—	2	1	no	—
60	C₂×C₅×C₈	—	2	2	no	—
61	C₂×C₅×C₈	—	4	1	no	—
62	C₂×C₅×C₈	—	4	1	no	—
63	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
64	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
65	C₂×C₅×C₈	SD₁₆	1	2	yes	—
66	C₂×C₅×C₈	Q₁₆	1	2	yes	—
67	C₅×C₄⋊C₄	—	2	2	no	—
68	C₅×C₄⋊C₄	Q₁₆	1	4	yes	—
69	C₄²×C₅	—	4	1	no	—
70	C₄²×C₅	—	4	1	no	—
71	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₅	#41	1	1	yes	characteristic
75	C₄⋊Dic₅	—	2	4	no	—
76	C₄⋊Dic₅	—	2	4	no	—
77	C₄⋊Dic₅	—	4	2	no	—
78	C₄⋊Dic₅	—	4	2	no	—
79	C₂×#168	—	4	2	no	—
80	C₂×#168	—	4	4	no	—
81	C₂×C₄⋊Q₈	—	5	2	no	—
82	#500	—	5	4	no	—
83	#500	—	10	2	no	—
84	C₂×#107	—	10	1	no	—
85	C₂×#107	—	10	1	no	—
86	C₂×C₄⋊C₈	—	5	2	no	—
87	C₂×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
88	C₂×C₄×C₈	—	2	1	no	—
89	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
90	C₄×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
91	#340	—	10	4	no	—
92	#336	—	2	1	no	—
93	#335	—	10	2	no	—
94	#333	—	10	4	no	—
95	C₂³×C₅	#121	1	1	yes	characteristic
96	C₂×C₄×C₅	—	2	2	no	—
97	C₂×C₄×C₅	—	2	4	no	—
98	C₂×C₄×C₅	—	4	2	no	—
99	C₂×C₄×C₅	—	4	2	no	—
100	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₅	#121	1	1	yes	characteristic
104	C₂×C₄×C₅	#124	1	1	yes	characteristic
105	C₂×C₄×C₅	#127	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
108	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
109	C₂×C₄×C₅	#135	1	2	yes	—
110	C₂×C₄×C₅	#137	1	1	yes	characteristic
111	C₂×C₄×C₅	#137	1	1	yes	characteristic
112	C₂×Dic₅	—	4	4	no	—
113	C₂×Dic₅	—	8	4	no	—
114	Dic₁₀	—	8	4	no	—
115	Dic₁₀	—	8	4	no	—
116	C₅×C₈	—	2	2	no	—
117	C₅×C₈	—	4	2	no	—
118	C₅×C₈	#103	1	4	yes	—
119	#168	—	8	2	no	—
120	C₂²×Q₈	—	10	2	no	—
121	C₂²×C₈	—	2	1	no	—
122	C₂²×C₈	—	10	2	no	—
123	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
124	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
125	C₄⋊Q₈	—	5	4	no	—
126	C₄⋊Q₈	—	10	2	no	—
127	C₄⋊Q₈	—	10	2	no	—
128	C₄⋊Q₈	—	10	2	no	—
129	C₂×C₄⋊C₄	—	10	2	no	—
130	C₂×C₄⋊C₄	—	10	2	no	—
131	C₂×C₄⋊C₄	D₂₀	1	2	yes	—
132	C₂×C₄²	—	2	1	no	—
133	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
134	#107	—	10	4	no	—
135	#107	—	10	4	no	—
136	C₄⋊C₈	—	2	1	no	—
137	C₄⋊C₈	—	10	2	no	—
138	C₄⋊C₈	—	10	2	no	—
139	C₄⋊C₈	D₂₀	1	2	yes	—
140	C₄×C₈	—	4	2	no	—
141	C₄×C₈	C₄×D₅	1	2	yes	—
142	C₄×C₈	D₂₀	1	2	yes	—
143	#95	—	2	1	no	—
144	C₂²×C₅	#390	1	1	yes	characteristic
145	C₂²×C₅	#461	1	1	yes	characteristic
146	C₂²×C₅	#462	1	1	yes	characteristic
147	C₂²×C₅	#467	1	2	yes	—
148	C₂²×C₅	#472	1	2	yes	—
149	C₄×C₅	—	2	1	no	—
150	C₄×C₅	—	2	1	no	—
151	C₄×C₅	—	4	4	no	—
152	C₄×C₅	C₂×#103	1	1	yes	characteristic
153	C₄×C₅	#419	1	1	yes	characteristic
154	C₄×C₅	C₄×Q₁₆	1	2	yes	—
155	C₄×C₅	#461	1	1	yes	characteristic
156	C₄×C₅	#462	1	1	yes	characteristic
157	C₄×C₅	#494	1	1	yes	characteristic
158	C₄×C₅	#495	1	1	yes	characteristic
159	Dic₅	—	8	4	no	—
160	C₂×Q₈	—	10	4	no	—
161	C₂×Q₈	—	20	2	no	—
162	C₂×Q₈	—	40	4	no	—
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	—	20	2	no	—
165	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
166	C₂²×C₄	#709	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	4	1	no	—
172	C₂×C₈	—	4	1	no	—
173	C₂×C₈	—	20	2	no	—
174	C₂×C₈	—	20	4	no	—
175	C₂×C₈	#685	1	2	yes	—
176	C₂×C₈	#685	1	2	yes	—
177	C₂×C₈	#687	1	2	yes	—
178	C₂×C₈	#689	1	2	yes	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	10	4	no	—
181	C₄⋊C₄	—	10	4	no	—
182	C₄⋊C₄	—	20	2	no	—
183	C₄⋊C₄	—	20	2	no	—
184	C₄⋊C₄	Dic₂₀	1	4	yes	—
185	C₄²	—	4	1	no	—
186	C₄²	—	4	1	no	—
187	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
188	C₄²	C₂×D₂₀	1	1	yes	characteristic
189	C₄²	C₂×D₂₀	1	1	yes	characteristic
190	C₄²	#709	1	1	yes	characteristic
191	C₂×C₅	#1675	1	2	yes	—
192	C₂×C₅	#1676	1	2	yes	—
193	C₂×C₅	#1969	1	1	yes	characteristic
194	C₂×C₅	#1978	1	1	yes	characteristic
195	C₂×C₅	#1981	1	1	yes	characteristic
196	C₂³	#4076	1	1	yes	characteristic
197	Q₈	—	40	4	no	—
198	Q₈	—	40	4	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	20	4	no	—
204	C₂×C₄	—	40	4	no	—
205	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
206	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
207	C₂×C₄	#4020	1	1	yes	characteristic
208	C₂×C₄	#4021	1	1	yes	characteristic
209	C₂×C₄	#4050	1	2	yes	—
210	C₂×C₄	C₂×Dic₂₀	1	2	yes	—
211	C₂×C₄	C₂×#685	1	1	yes	characteristic
212	C₂×C₄	#4076	1	1	yes	characteristic
213	C₂×C₄	C₂×#687	1	1	yes	characteristic
214	C₂×C₄	C₂×#689	1	1	yes	characteristic
215	C₂×C₄	#4097	1	1	yes	characteristic
216	C₂×C₄	#4097	1	1	yes	characteristic
217	C₈	—	2	2	no	—
218	C₈	—	4	2	no	—
219	C₈	—	40	2	no	—
220	C₈	#3942	1	4	yes	—
221	C₅	#9730	1	1	yes	characteristic, complemented, Sylow
222	C₂²	#65228	1	1	yes	characteristic
223	C₂²	#65308	1	1	yes	characteristic
224	C₂²	#65320	1	1	yes	characteristic
225	C₂²	#65405	1	2	yes	—
226	C₂²	#65431	1	2	yes	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	4	4	no	—
230	C₄	—	40	4	no	—
231	C₄	#64985	1	1	yes	characteristic
232	C₄	C₄×Dic₂₀	1	2	yes	—
233	C₄	#64988	1	1	yes	characteristic
234	C₄	C₂×#3942	1	1	yes	characteristic
235	C₄	#65293	1	1	yes	characteristic
236	C₄	#65308	1	1	yes	characteristic
237	C₄	#65320	1	1	yes	characteristic
238	C₂	#10600279	1	1	yes	characteristic
239	C₂	#10600568	1	1	yes	characteristic
240	C₂	#10600696	1	1	yes	characteristic
241	C₂	#10601646	1	2	yes	—
242	C₂	#10601738	1	2	yes	—

	subgroup	G/H
center	C₂³	#4076
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×C₈×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₅	#121
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₈×C₄⋊C₄	D₅
hypocenter	C₅	#9730
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
15. #64971	C₄	semidirect
221. C₅	#9730	semidirect

Maximal subgroups

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

subgroup	index
C₂×#64971	2
C₅×C₈×C₄⋊C₄	2
C₂×#64701	2
C₂×#64678	2
#10600282	2
#9730	5