Subgroups — #11998166

Order 1280.

486 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₀	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#334	C₂	1	1	yes	maximal, characteristic
3	#10601265	C₂	1	2	yes	maximal
4	C₂×#64704	C₂	1	2	yes	maximal
5	C₂×#64675	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₀	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₁₀	C₂²	1	2	yes	—
8	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×#3950	C₄	1	1	yes	characteristic
11	C₂×#3948	C₄	1	2	yes	—
12	C₂×C₄×#168	C₂²	1	1	yes	characteristic
13	C₂×C₄×#168	C₂²	1	1	yes	characteristic
14	C₄×Dic₂₀	—	2	2	no	—
15	C₄×Dic₂₀	—	2	2	no	—
16	C₅×#334	—	2	2	no	—
17	#64770	—	2	4	no	—
18	#64704	—	2	4	no	—
19	#64675	—	2	2	no	—
20	#9658	—	5	1	no	maximal, Sylow
21	C₂²×Dic₁₀	C₂×C₄	1	2	yes	—
22	C₂²×C₅×C₈	—	2	2	no	—
23	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
26	C₂×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
27	C₂×#683	—	2	2	no	—
28	C₂×C₄×Dic₅	—	2	2	no	—
29	C₂²×#168	—	2	1	no	—
30	C₂²×#168	D₄	1	2	yes	—
31	C₂×Dic₂₀	—	2	2	no	—
32	C₂×Dic₂₀	—	2	2	no	—
33	C₂×Dic₂₀	C₈	1	4	yes	complemented
34	C₄×Dic₁₀	—	2	2	no	—
35	C₄×Dic₁₀	—	2	2	no	—
36	C₄×Dic₁₀	—	2	2	no	—
37	C₄×Dic₁₀	—	2	2	no	—
38	C₅×C₄⋊C₈	—	2	2	no	—
39	C₅×C₄⋊C₈	—	2	2	no	—
40	C₄×C₅×C₈	D₄	1	2	yes	—
41	C₄×C₅×C₈	D₄	1	2	yes	—
42	#3950	—	2	2	no	—
43	#3948	—	2	4	no	—
44	C₄×#168	—	2	1	no	—
45	C₄×#168	—	2	1	no	—
46	C₄×#168	—	2	1	no	—
47	C₄×#168	D₄	1	2	yes	—
48	C₂×C₄×Q₁₆	—	5	1	no	—
49	#1984	—	5	2	no	—
50	C₂×#334	D₅	1	1	yes	characteristic
51	C₂×Q₈⋊C₈	—	5	2	no	—
52	C₂×#322	—	5	1	no	—
53	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	#41	1	1	yes	characteristic
56	C₂²×Dic₅	—	2	2	no	—
57	C₂²×Dic₅	—	2	2	no	—
58	C₂×Dic₁₀	—	2	2	no	—
59	C₂×Dic₁₀	—	2	2	no	—
60	C₂×Dic₁₀	—	2	2	no	—
61	C₂×Dic₁₀	—	2	2	no	—
62	C₂×Dic₁₀	—	2	2	no	—
63	C₂×Dic₁₀	C₂×C₈	1	4	yes	—
64	C₂×C₅×C₈	—	2	1	no	—
65	C₂×C₅×C₈	—	2	1	no	—
66	C₂×C₅×C₈	—	4	2	no	—
67	C₂×C₅×C₈	—	4	4	no	—
68	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
69	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
70	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
71	C₂×C₅×C₈	M₁₆	1	2	yes	—
72	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₅	#41	1	1	yes	characteristic
76	C₄⋊Dic₅	—	2	2	no	—
77	C₄⋊Dic₅	—	2	2	no	—
78	#683	—	4	2	no	—
79	#683	—	4	2	no	—
80	C₄×Dic₅	—	4	2	no	—
81	C₄×Dic₅	—	4	2	no	—
82	C₂×#168	—	2	2	no	—
83	C₂×#168	—	2	2	no	—
84	C₂×#168	—	2	4	no	—
85	C₂×#168	—	4	2	no	—
86	Dic₂₀	—	2	8	no	—
87	C₂²×Q₁₆	—	5	1	no	—
88	C₂×C₄×Q₈	—	5	2	no	—
89	C₄×Q₁₆	—	10	2	no	—
90	C₄×Q₁₆	—	10	2	no	—
91	C₂×#107	—	5	1	no	—
92	C₂×C₄⋊C₈	C₂×D₅	1	2	yes	—
93	C₂×#103	—	5	2	no	—
94	C₂×C₄×C₈	—	5	1	no	—
95	C₂×C₄×C₈	—	5	1	no	—
96	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
97	#336	—	10	4	no	—
98	#334	—	2	2	no	—
99	Q₈⋊C₈	—	10	4	no	—
100	#322	—	10	2	no	—
101	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₅	—	2	2	no	—
103	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅	#121	1	1	yes	characteristic
106	C₂×C₄×C₅	#121	1	1	yes	characteristic
107	C₂×C₄×C₅	#124	1	1	yes	characteristic
108	C₂×C₄×C₅	#127	1	1	yes	characteristic
109	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₄×C₅	#131	1	1	yes	characteristic
112	C₂×C₄×C₅	#131	1	1	yes	characteristic
113	C₂×C₄×C₅	#137	1	4	yes	—
114	C₂×Dic₅	—	2	4	no	—
115	C₂×Dic₅	—	4	2	no	—
116	C₂×Dic₅	—	4	2	no	—
117	C₂×Dic₅	—	4	2	no	—
118	C₂×Dic₅	—	4	4	no	—
119	Dic₁₀	—	2	4	no	—
120	Dic₁₀	—	2	4	no	—
121	Dic₁₀	—	4	2	no	—
122	Dic₁₀	—	4	2	no	—
123	C₅×C₈	—	2	2	no	—
124	C₅×C₈	—	8	2	no	—
125	C₅×C₈	C₂²⋊C₈	1	4	yes	—
126	#168	—	4	2	no	—
127	#168	—	4	2	no	—
128	C₂²×Q₈	—	5	2	no	—
129	C₂×Q₁₆	—	5	4	no	—
130	C₂×Q₁₆	—	10	2	no	—
131	C₂×Q₁₆	—	10	2	no	—
132	C₂²×C₈	—	2	2	no	—
133	C₂²×C₈	—	5	2	no	—
134	C₂²×C₈	—	10	1	no	—
135	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
136	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
137	C₄×Q₈	—	10	2	no	—
138	C₄×Q₈	—	10	2	no	—
139	C₄×Q₈	—	10	2	no	—
140	C₄×Q₈	—	10	2	no	—
141	C₂×C₄⋊C₄	—	5	2	no	—
142	C₂×C₄⋊C₄	—	10	2	no	—
143	C₂×C₄²	—	10	2	no	—
144	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
145	#107	—	10	2	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	2	2	no	—
148	#103	—	10	4	no	—
149	C₄×C₈	—	5	2	no	—
150	C₄×C₈	—	10	1	no	—
151	C₄×C₈	—	10	1	no	—
152	C₄×C₈	—	10	1	no	—
153	C₄×C₈	D₂₀	1	2	yes	—
154	C₄×C₈	C₂²⋊D₅	1	2	yes	—
155	C₂²×C₅	#390	1	1	yes	characteristic
156	C₂²×C₅	C₈×D₄	1	1	yes	characteristic
157	C₂²×C₅	#436	1	1	yes	characteristic
158	C₂²×C₅	#441	1	2	yes	—
159	C₂²×C₅	#444	1	2	yes	—
160	C₄×C₅	—	2	1	no	—
161	C₄×C₅	—	2	1	no	—
162	C₄×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
163	C₄×C₅	#408	1	1	yes	characteristic
164	C₄×C₅	C₈×D₄	1	1	yes	characteristic
165	C₄×C₅	#436	1	1	yes	characteristic
166	C₄×C₅	#470	1	2	yes	—
167	C₄×C₅	#497	1	2	yes	—
168	Dic₅	—	4	4	no	—
169	Dic₅	—	8	2	no	—
170	C₂×Q₈	—	5	4	no	—
171	C₂×Q₈	—	10	2	no	—
172	C₂×Q₈	—	10	2	no	—
173	C₂×Q₈	—	10	2	no	—
174	C₂×Q₈	—	10	2	no	—
175	C₂×Q₈	—	10	2	no	—
176	C₂²×C₄	—	10	2	no	—
177	C₂²×C₄	—	10	2	no	—
178	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
179	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
180	C₂²×C₄	#713	1	1	yes	characteristic
181	Q₁₆	—	10	8	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	4	4	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	10	2	no	—
187	C₂×C₈	—	10	2	no	—
188	C₂×C₈	—	10	4	no	—
189	C₂×C₈	—	20	2	no	—
190	C₂×C₈	C₈×D₅	1	2	yes	—
191	C₂×C₈	#676	1	2	yes	—
192	C₂×C₈	#685	1	2	yes	—
193	C₂×C₈	#685	1	2	yes	—
194	C₄⋊C₄	—	10	2	no	—
195	C₄⋊C₄	—	10	2	no	—
196	C₄⋊C₄	—	20	2	no	—
197	C₄⋊C₄	—	20	2	no	—
198	C₄²	—	20	2	no	—
199	C₄²	—	20	2	no	—
200	C₄²	C₂×D₂₀	1	1	yes	characteristic
201	C₄²	D₄×D₅	1	1	yes	characteristic
202	C₄²	#713	1	1	yes	characteristic
203	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
204	C₂×C₅	#1579	1	2	yes	—
205	C₂×C₅	#1593	1	2	yes	—
206	C₂×C₅	#1954	1	1	yes	characteristic
207	C₂×C₅	#1971	1	1	yes	characteristic
208	C₂×C₅	#1975	1	1	yes	characteristic
209	C₂³	#4039	1	1	yes	characteristic
210	Q₈	—	10	4	no	—
211	Q₈	—	10	4	no	—
212	Q₈	—	20	2	no	—
213	Q₈	—	20	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	10	4	no	—
216	C₂×C₄	—	20	2	no	—
217	C₂×C₄	—	20	2	no	—
218	C₂×C₄	—	20	4	no	—
219	C₂×C₄	—	20	2	no	—
220	C₂×C₄	#4039	1	1	yes	characteristic
221	C₂×C₄	#4041	1	1	yes	characteristic
222	C₂×C₄	#4041	1	1	yes	characteristic
223	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
224	C₂×C₄	C₂×#676	1	1	yes	characteristic
225	C₂×C₄	#4053	1	1	yes	characteristic
226	C₂×C₄	#4053	1	1	yes	characteristic
227	C₂×C₄	#4061	1	4	yes	—
228	C₂×C₄	C₂×#685	1	1	yes	characteristic
229	C₂×C₄	#4086	1	1	yes	characteristic
230	C₂×C₄	#4093	1	1	yes	characteristic
231	C₈	—	2	2	no	—
232	C₈	—	8	2	no	—
233	C₈	—	20	2	no	—
234	C₈	—	20	2	no	—
235	C₈	#3952	1	4	yes	—
236	C₅	#9658	1	1	yes	characteristic, complemented, Sylow
237	C₂²	#65123	1	1	yes	characteristic
238	C₂²	#65127	1	1	yes	characteristic
239	C₂²	#65142	1	2	yes	—
240	C₂²	#65183	1	2	yes	—
241	C₂²	#65277	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	20	4	no	—
245	C₄	—	40	2	no	—
246	C₄	#65123	1	1	yes	characteristic
247	C₄	#65127	1	1	yes	characteristic
248	C₄	#65157	1	2	yes	—
249	C₄	C₂×#3952	1	1	yes	characteristic
250	C₄	#65418	1	1	yes	characteristic
251	C₄	#65456	1	2	yes	—
252	C₂	#10599310	1	2	yes	—
253	C₂	#10599317	1	2	yes	—
254	C₂	#10601263	1	1	yes	characteristic
255	C₂	#10601368	1	1	yes	characteristic
256	C₂	#10601490	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂×C₅×#334	C₂
socle, prime radical	C₂³×C₅	C₄×D₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×#334	D₅
hypocenter	C₅	#9658
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
33. C₂×Dic₂₀	C₈	semidirect
236. C₅	#9658	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₀	2
C₂×C₅×#334	2
#10601265	2
C₂×#64704	2
C₂×#64675	2
#9658	5