Subgroups — #12069146

Order 1280.

614 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10610995	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₀	C₂	1	1	yes	maximal, characteristic
3	C₅×#1450	C₂	1	1	yes	maximal, characteristic
4	#10601560	C₂	1	2	yes	maximal
5	#10600522	C₂	1	1	yes	maximal, characteristic
6	#10598882	C₂	1	4	yes	maximal, complemented
7	#10598849	C₂	1	4	yes	maximal, complemented
8	C₂²×Dic₂₀	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₁₀	C₂²	1	2	yes	—
10	C₅×#431	C₂²	1	1	yes	characteristic
11	C₅×#431	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂×#3950	C₄	1	1	yes	characteristic
14	C₂×#3948	C₄	1	2	yes	—
15	#65214	C₂²	1	2	yes	—
16	#65120	—	2	4	no	—
17	#65008	—	2	4	no	—
18	C₄×Dic₂₀	C₄	1	2	yes	—
19	C₄×Dic₂₀	C₄	1	2	yes	—
20	C₄×Dic₂₀	C₂²	1	2	yes	—
21	C₄×Dic₂₀	C₂²	1	2	yes	—
22	#64968	C₂²	1	8	yes	—
23	C₅×#322	C₂²	1	2	yes	—
24	C₅×C₈²	C₂²	1	2	yes	—
25	#64678	C₂²	1	2	yes	—
26	#64677	C₂²	1	2	yes	—
27	#64676	C₂²	1	8	yes	—
28	#17897	—	5	1	no	maximal, Sylow
29	C₂²×Dic₁₀	C₂×C₄	1	2	yes	—
30	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
34	C₂×#683	—	2	2	no	—
35	C₂×C₄×Dic₅	—	2	2	no	—
36	C₂×Dic₂₀	—	2	4	no	—
37	C₂×Dic₂₀	C₂×C₄	1	2	yes	—
38	C₂×Dic₂₀	C₂×C₄	1	2	yes	—
39	C₄×Dic₁₀	—	2	2	no	—
40	C₄×Dic₁₀	—	2	2	no	—
41	C₄×Dic₁₀	C₂×C₄	1	4	yes	—
42	C₄×Dic₁₀	C₂³	1	4	yes	—
43	C₅×C₄⋊C₈	—	2	1	no	—
44	C₅×C₄⋊C₈	D₄	1	2	yes	—
45	C₅×C₂²⋊C₈	—	2	1	no	—
46	C₅×C₂²⋊C₈	D₄	1	2	yes	—
47	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
48	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
49	C₄×C₅×C₈	C₂³	1	2	yes	—
50	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
51	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
52	C₄×C₅×C₈	C₂³	1	2	yes	—
53	C₂²⋊#168	—	2	2	no	—
54	#3950	C₂×C₄	1	2	yes	—
55	#3950	C₂×C₄	1	2	yes	—
56	#3948	C₂×C₄	1	4	yes	—
57	#3948	C₂×C₄	1	4	yes	—
58	#3947	—	2	8	no	—
59	#3946	—	2	8	no	—
60	C₄⋊#168	C₂³	1	4	yes	—
61	C₄×#168	—	2	2	no	—
62	#2983	—	5	2	no	—
63	C₂×C₄×Q₁₆	—	5	1	no	—
64	#1593	—	5	4	no	—
65	#1591	—	5	4	no	—
66	#1564	—	5	1	no	—
67	#1479	—	5	2	no	—
68	#1450	D₅	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	#41	1	1	yes	characteristic
72	C₂²×Dic₅	—	2	2	no	—
73	C₂²×Dic₅	—	2	2	no	—
74	C₂×Dic₁₀	—	2	4	no	—
75	C₂×Dic₁₀	—	2	4	no	—
76	C₂×Dic₁₀	—	2	2	no	—
77	C₂×Dic₁₀	C₂²×C₄	1	4	yes	—
78	C₂×C₅×C₈	—	2	2	no	—
79	C₂×C₅×C₈	—	2	2	no	—
80	C₂×C₅×C₈	M₁₆	1	4	yes	—
81	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
86	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₅	#41	1	1	yes	characteristic
89	C₄²×C₅	C₂⁴	1	1	yes	characteristic
90	C₄⋊Dic₅	—	2	2	no	—
91	C₄⋊Dic₅	C₂²×C₄	1	4	yes	—
92	#683	—	2	4	no	—
93	#683	—	2	4	no	—
94	C₄×Dic₅	—	2	4	no	—
95	C₄×Dic₅	—	2	4	no	—
96	C₂×#168	—	2	4	no	—
97	Dic₂₀	—	2	4	no	—
98	Dic₂₀	—	2	4	no	—
99	C₂²×Q₁₆	—	5	1	no	—
100	C₂×C₄×Q₈	—	5	2	no	—
101	#446	—	5	8	no	—
102	#446	—	10	4	no	—
103	C₄×Q₁₆	—	5	2	no	—
104	C₄×Q₁₆	—	5	2	no	—
105	C₄×Q₁₆	—	5	2	no	—
106	C₄×Q₁₆	—	5	2	no	—
107	#432	—	10	4	no	—
108	#431	—	5	2	no	—
109	#431	C₂×D₅	1	1	yes	characteristic
110	#431	C₂×D₅	1	1	yes	characteristic
111	C₂×#107	—	5	1	no	—
112	C₂×#103	—	5	2	no	—
113	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
114	#335	—	5	2	no	—
115	#334	—	5	2	no	—
116	Q₈⋊C₈	—	5	8	no	—
117	#322	C₂×D₅	1	2	yes	—
118	C₈²	C₂×D₅	1	2	yes	—
119	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	—	4	1	no	—
122	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
126	C₂×C₄×C₅	#131	1	2	yes	—
127	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
130	C₂×Dic₅	—	2	4	no	—
131	C₂×Dic₅	—	2	4	no	—
132	C₂×Dic₅	—	4	4	no	—
133	C₂×Dic₅	—	8	2	no	—
134	Dic₁₀	—	2	4	no	—
135	Dic₁₀	—	2	4	no	—
136	Dic₁₀	—	4	4	no	—
137	C₅×C₈	—	2	1	no	—
138	C₅×C₈	—	2	1	no	—
139	C₅×C₈	—	2	4	no	—
140	C₅×C₈	—	4	2	no	—
141	C₅×C₈	C₂×M₁₆	1	2	yes	—
142	C₅×C₈	C₂×M₁₆	1	2	yes	—
143	#168	—	4	4	no	—
144	C₂²×Q₈	—	5	2	no	—
145	C₂×Q₁₆	—	5	2	no	—
146	C₂×Q₁₆	—	5	2	no	—
147	C₂×Q₁₆	—	10	4	no	—
148	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
149	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
150	C₄×Q₈	—	5	4	no	—
151	C₄×Q₈	—	5	4	no	—
152	C₄×Q₈	—	10	2	no	—
153	C₄×Q₈	—	10	2	no	—
154	C₂×C₄⋊C₄	—	5	2	no	—
155	C₂×C₄⋊C₄	—	10	2	no	—
156	C₂×C₄²	—	10	2	no	—
157	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
158	#107	—	5	2	no	—
159	#107	—	5	2	no	—
160	C₄⋊C₈	—	2	1	no	—
161	C₄⋊C₈	—	5	4	no	—
162	C₄⋊C₈	—	10	8	no	—
163	C₄⋊C₈	D₂₀	1	2	yes	—
164	#103	—	5	4	no	—
165	#103	—	5	4	no	—
166	C₂²⋊C₈	—	2	1	no	—
167	C₂²⋊C₈	—	10	2	no	—
168	C₂²⋊C₈	D₂₀	1	2	yes	—
169	#97	—	10	8	no	—
170	C₄×C₈	—	10	2	no	—
171	C₄×C₈	C₄×D₅	1	1	yes	characteristic
172	C₄×C₈	C₄×D₅	1	1	yes	characteristic
173	C₄×C₈	C₂²×D₅	1	2	yes	—
174	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
175	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
176	C₄×C₈	C₂²×D₅	1	2	yes	—
177	C₂²×C₅	—	2	1	no	—
178	C₂²×C₅	—	2	1	no	—
179	C₂²×C₅	#436	1	2	yes	—
180	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
181	C₄×C₅	—	2	1	no	—
182	C₄×C₅	—	2	1	no	—
183	C₄×C₅	—	2	1	no	—
184	C₄×C₅	#436	1	2	yes	—
185	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
186	C₄×C₅	C₂×#131	1	1	yes	characteristic
187	C₄×C₅	#578	1	2	yes	—
188	Dic₅	—	4	4	no	—
189	Dic₅	—	4	4	no	—
190	C₂×Q₈	—	5	4	no	—
191	C₂×Q₈	—	10	2	no	—
192	C₂×Q₈	—	10	4	no	—
193	C₂×Q₈	—	10	4	no	—
194	C₂²×C₄	—	10	2	no	—
195	C₂²×C₄	—	10	2	no	—
196	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
198	C₂²×C₄	#709	1	1	yes	characteristic
199	Q₁₆	—	10	4	no	—
200	Q₁₆	—	10	4	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	10	4	no	—
204	C₂×C₈	#676	1	4	yes	—
205	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
206	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
207	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
208	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
209	C₂×C₈	C₂×D₂₀	1	4	yes	—
210	C₄⋊C₄	—	5	4	no	—
211	C₄⋊C₄	—	10	4	no	—
212	C₄⋊C₄	—	10	2	no	—
213	C₄⋊C₄	—	10	4	no	—
214	C₄²	—	10	4	no	—
215	C₄²	—	10	4	no	—
216	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
217	C₄²	C₂×D₂₀	1	1	yes	characteristic
218	C₄²	#709	1	1	yes	characteristic
219	C₄²	C₂³×D₅	1	1	yes	characteristic
220	C₂×C₅	—	2	2	no	—
221	C₂×C₅	C₂×#436	1	1	yes	characteristic
222	C₂×C₅	#2950	1	1	yes	characteristic
223	C₂×C₅	#2957	1	1	yes	characteristic
224	C₂³	C₄×D₂₀	1	1	yes	characteristic
225	Q₈	—	10	4	no	—
226	Q₈	—	10	4	no	—
227	Q₈	—	20	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	4	1	no	—
230	C₂×C₄	—	10	4	no	—
231	C₂×C₄	—	10	4	no	—
232	C₂×C₄	—	20	4	no	—
233	C₂×C₄	—	40	2	no	—
234	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
235	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
236	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
237	C₂×C₄	C₂×#676	1	2	yes	—
238	C₂×C₄	#4047	1	2	yes	—
239	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
240	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
241	C₂×C₄	C₂×#709	1	1	yes	characteristic
242	C₈	—	2	1	no	—
243	C₈	—	2	1	no	—
244	C₈	—	2	4	no	—
245	C₈	—	4	2	no	—
246	C₈	—	20	4	no	—
247	C₈	C₂×#676	1	2	yes	—
248	C₈	C₂×#676	1	2	yes	—
249	C₅	#17897	1	1	yes	characteristic, complemented, Sylow
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	#64977	1	2	yes	—
253	C₂²	C₂×C₄×D₂₀	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	20	4	no	—
258	C₄	—	20	4	no	—
259	C₄	#64977	1	2	yes	—
260	C₄	C₂²×#676	1	1	yes	characteristic
261	C₄	C₂×#4047	1	1	yes	characteristic
262	C₄	#66087	1	2	yes	—
263	C₂	—	2	2	no	—
264	C₂	C₂×#64977	1	1	yes	characteristic
265	C₂	#10610304	1	1	yes	characteristic
266	C₂	#10610311	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₀
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₅×#1450	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂³×C₅	C₄×D₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	#1450	D₅
hypocenter	C₅	#17897
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1450

Complements

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

N	H ≅ G/N	type
6. #10598882	C₂	semidirect
7. #10598849	C₂	semidirect
249. C₅	#17897	semidirect

Maximal subgroups

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

subgroup	index
#10610995	2
C₂×C₄×Dic₂₀	2
C₅×#1450	2
#10601560	2
#10600522	2
#10598882	2
#10598849	2
#17897	5