Subgroups — #12069145

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	C₂×C₄×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	#10614167	C₂	1	2	yes	maximal
3	#10602414	C₂	1	2	yes	maximal
4	#10601699	C₂	1	4	yes	maximal, complemented
5	#10601698	C₂	1	4	yes	maximal, complemented
6	#10600522	C₂	1	1	yes	maximal, characteristic
7	#10600517	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₅×Q₁₆	C₄	1	1	yes	characteristic
9	C₂×C₄×C₅×Q₈	C₂²	1	2	yes	—
10	C₄×C₅×Q₁₆	C₄	1	2	yes	—
11	C₄×C₅×Q₁₆	C₄	1	2	yes	—
12	C₄×C₅×Q₁₆	C₂²	1	2	yes	—
13	C₄×C₅×Q₁₆	C₂²	1	2	yes	—
14	C₂×C₅×#107	C₄	1	1	yes	characteristic
15	C₂×C₅×#103	C₄	1	2	yes	—
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	#65313	—	2	4	no	—
18	#65313	C₂²	1	8	yes	—
19	#65215	—	2	4	no	—
20	#65214	C₂²	1	2	yes	—
21	#65214	C₂²	1	1	yes	characteristic
22	#65214	C₂²	1	1	yes	characteristic
23	Q₈⋊#168	C₂²	1	8	yes	—
24	#64678	C₂²	1	2	yes	—
25	#64677	C₂²	1	2	yes	—
26	#64675	C₂²	1	2	yes	—
27	C₈×#168	C₂²	1	2	yes	—
28	#17897	—	5	1	no	maximal, Sylow
29	C₂²×C₅×Q₈	C₂×C₄	1	2	yes	—
30	C₂×C₅×Q₁₆	—	2	4	no	—
31	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
32	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
33	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
35	C₄×C₅×Q₈	—	2	2	no	—
36	C₄×C₅×Q₈	—	2	2	no	—
37	C₄×C₅×Q₈	C₂×C₄	1	4	yes	—
38	C₄×C₅×Q₈	C₂³	1	4	yes	—
39	C₂×C₅×C₄⋊C₄	—	2	2	no	—
40	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅	—	2	2	no	—
42	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₅×#107	C₂×C₄	1	2	yes	—
44	C₅×#107	C₂×C₄	1	2	yes	—
45	C₅×#103	C₂×C₄	1	4	yes	—
46	C₅×#103	C₂×C₄	1	4	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	1	yes	characteristic
50	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
51	C₂²⋊#168	—	2	2	no	—
52	C₂²⋊#168	—	2	1	no	—
53	C₂²⋊#168	D₄	1	2	yes	—
54	C₄⋊#168	—	2	1	no	—
55	C₄⋊#168	—	2	8	no	—
56	C₄⋊#168	D₄	1	2	yes	—
57	C₄⋊#168	C₂³	1	4	yes	—
58	#3935	—	2	8	no	—
59	C₄×#168	—	2	2	no	—
60	C₄×#168	C₂³	1	2	yes	—
61	C₄×#168	C₂³	1	2	yes	—
62	#2983	—	5	2	no	—
63	C₂×C₄×Q₁₆	D₅	1	1	yes	characteristic
64	#1593	—	5	4	no	—
65	#1591	—	5	4	no	—
66	#1564	—	5	1	no	—
67	#1479	—	5	2	no	—
68	#1450	—	5	1	no	—
69	C₂×C₅×Q₈	—	2	2	no	—
70	C₂×C₅×Q₈	—	2	4	no	—
71	C₂×C₅×Q₈	—	2	4	no	—
72	C₂×C₅×Q₈	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₅	—	2	2	no	—
74	C₂²×C₄×C₅	—	2	2	no	—
75	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₅	#41	1	1	yes	characteristic
78	C₅×Q₁₆	—	2	4	no	—
79	C₅×Q₁₆	—	2	4	no	—
80	C₂×C₅×C₈	—	2	2	no	—
81	C₂×C₅×C₈	M₁₆	1	4	yes	—
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	C₂²×C₄	1	1	yes	characteristic
86	C₅×C₄⋊C₄	—	2	4	no	—
87	C₅×C₄⋊C₄	—	2	2	no	—
88	C₅×C₄⋊C₄	—	2	4	no	—
89	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
90	C₄²×C₅	—	2	4	no	—
91	C₄²×C₅	—	2	4	no	—
92	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₅	#41	1	1	yes	characteristic
95	C₄²×C₅	C₂⁴	1	1	yes	characteristic
96	C₂×#168	—	2	4	no	—
97	C₂×#168	—	2	2	no	—
98	C₂×#168	C₂×D₄	1	4	yes	—
99	C₂²×Q₁₆	Dic₅	1	1	yes	characteristic
100	C₂×C₄×Q₈	C₂×D₅	1	2	yes	—
101	#446	—	5	8	no	—
102	#446	—	10	4	no	—
103	C₄×Q₁₆	Dic₅	1	2	yes	—
104	C₄×Q₁₆	Dic₅	1	2	yes	—
105	C₄×Q₁₆	C₂×D₅	1	2	yes	—
106	C₄×Q₁₆	C₂×D₅	1	2	yes	—
107	#432	—	10	4	no	—
108	#431	—	5	1	no	—
109	#431	—	5	1	no	—
110	#431	—	5	2	no	—
111	C₂×#107	Dic₅	1	1	yes	characteristic
112	C₂×#103	Dic₅	1	2	yes	—
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	—	5	2	no	—
118	C₈²	—	5	2	no	—
119	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
120	C₅×Q₈	—	2	4	no	—
121	C₅×Q₈	—	2	4	no	—
122	C₅×Q₈	—	4	4	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	2	4	no	—
125	C₂×C₄×C₅	—	2	4	no	—
126	C₂×C₄×C₅	—	4	1	no	—
127	C₂×C₄×C₅	—	4	4	no	—
128	C₂×C₄×C₅	—	8	2	no	—
129	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
133	C₂×C₄×C₅	#131	1	2	yes	—
134	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
137	C₅×C₈	—	2	1	no	—
138	C₅×C₈	—	2	1	no	—
139	C₅×C₈	C₂×M₁₆	1	2	yes	—
140	C₅×C₈	C₂×M₁₆	1	2	yes	—
141	#168	—	2	4	no	—
142	#168	—	4	2	no	—
143	#168	—	4	4	no	—
144	C₂²×Q₈	C₂×Dic₅	1	2	yes	—
145	C₂×Q₁₆	—	2	4	no	—
146	C₂×Q₁₆	C₂×Dic₅	1	2	yes	—
147	C₂×Q₁₆	C₂×Dic₅	1	2	yes	—
148	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
149	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
150	C₄×Q₈	—	2	2	no	—
151	C₄×Q₈	—	2	2	no	—
152	C₄×Q₈	C₂×Dic₅	1	4	yes	—
153	C₄×Q₈	C₂²×D₅	1	4	yes	—
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
156	C₂×C₄²	—	2	2	no	—
157	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
158	#107	C₂×Dic₅	1	2	yes	—
159	#107	C₂×Dic₅	1	2	yes	—
160	C₄⋊C₈	—	5	2	no	—
161	C₄⋊C₈	—	5	4	no	—
162	C₄⋊C₈	—	10	8	no	—
163	C₄⋊C₈	—	10	1	no	—
164	#103	C₂×Dic₅	1	4	yes	—
165	#103	C₂×Dic₅	1	4	yes	—
166	C₂²⋊C₈	—	5	2	no	—
167	C₂²⋊C₈	—	10	2	no	—
168	C₂²⋊C₈	—	10	1	no	—
169	#97	—	10	8	no	—
170	C₄×C₈	—	5	2	no	—
171	C₄×C₈	—	5	2	no	—
172	C₄×C₈	—	10	2	no	—
173	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
174	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
175	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
176	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
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	—	4	4	no	—
185	C₄×C₅	—	4	4	no	—
186	C₄×C₅	#436	1	2	yes	—
187	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
188	C₄×C₅	C₂×#131	1	1	yes	characteristic
189	C₄×C₅	#578	1	2	yes	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	—	2	4	no	—
192	C₂×Q₈	—	2	4	no	—
193	C₂×Q₈	C₂²×Dic₅	1	4	yes	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
197	C₂²×C₄	#711	1	1	yes	characteristic
198	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
199	Q₁₆	—	2	4	no	—
200	Q₁₆	—	2	4	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	5	4	no	—
203	C₂×C₈	—	10	4	no	—
204	C₂×C₈	—	10	2	no	—
205	C₂×C₈	#681	1	4	yes	—
206	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
207	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
208	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
209	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	C₂²×Dic₅	1	4	yes	—
214	C₄²	—	2	4	no	—
215	C₄²	—	2	4	no	—
216	C₄²	D₄×D₅	1	1	yes	characteristic
217	C₄²	#711	1	1	yes	characteristic
218	C₄²	C₂²×Dic₅	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	D₄×Dic₅	1	1	yes	characteristic
225	Q₈	—	2	4	no	—
226	Q₈	—	2	4	no	—
227	Q₈	—	4	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	8	2	no	—
234	C₂×C₄	C₂×#681	1	2	yes	—
235	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
236	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
237	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
238	C₂×C₄	#4094	1	2	yes	—
239	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
240	C₂×C₄	C₂×#711	1	1	yes	characteristic
241	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
242	C₈	—	2	1	no	—
243	C₈	—	2	1	no	—
244	C₈	—	10	4	no	—
245	C₈	—	20	2	no	—
246	C₈	—	20	4	no	—
247	C₈	C₂×#681	1	2	yes	—
248	C₈	C₂×#681	1	2	yes	—
249	C₅	#17897	1	1	yes	characteristic, complemented, Sylow
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	#65301	1	2	yes	—
253	C₂²	C₂×D₄×Dic₅	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	4	4	no	—
258	C₄	—	4	4	no	—
259	C₄	#65301	1	2	yes	—
260	C₄	#66100	1	2	yes	—
261	C₄	C₂²×#681	1	1	yes	characteristic
262	C₄	C₂×#4094	1	1	yes	characteristic
263	C₂	—	2	2	no	—
264	C₂	C₂×#65301	1	1	yes	characteristic
265	C₂	#10615605	1	1	yes	characteristic
266	C₂	#10615641	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₂×C₄×C₅×Q₁₆	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂³×C₅	C₄×D₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₅
hypocenter	C₅	#17897
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
4. #10601699	C₂	semidirect
5. #10601698	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
C₂×C₄×C₅×Q₁₆	2
#10614167	2
#10602414	2
#10601699	2
#10601698	2
#10600522	2
#10600517	2
#17897	5