Subgroups — #12015229

Order 1280.

486 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#64773	C₂	1	2	yes	maximal
3	#10600536	C₂	1	2	yes	maximal, complemented
4	#10598155	C₂	1	2	yes	maximal, complemented
5	C₂²×C₅×M₁₆	C₄	1	1	yes	characteristic
6	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₅	C₄	1	1	yes	characteristic
8	C₄×C₅×M₁₆	—	2	4	no	—
9	C₂×C₅×#97	C₄	1	1	yes	characteristic
10	C₂×C₅×#97	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂×#3944	C₂²	1	4	yes	—
14	#64773	—	2	4	no	—
15	C₄⋊#672	—	2	4	no	—
16	#12785	—	5	1	no	maximal, Sylow
17	C₂³×C₄×C₅	C₈	1	2	yes	—
18	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂×C₅×M₁₆	—	2	2	no	—
20	C₂×C₅×M₁₆	—	2	2	no	—
21	C₂×C₅×M₁₆	—	2	2	no	—
22	C₂×C₅×M₁₆	—	2	2	no	—
23	C₂×C₅×M₁₆	—	2	2	no	—
24	C₂×C₅×M₁₆	D₄	1	2	yes	—
25	C₂×C₅×M₁₆	Q₈	1	2	yes	—
26	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₅	—	2	2	no	—
31	C₂×C₄²×C₅	—	2	1	no	—
32	C₂×C₄²×C₅	—	2	1	no	—
33	C₂×C₄²×C₅	C₈	1	2	yes	—
34	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₅×#97	—	2	2	no	—
37	C₅×#97	—	2	2	no	—
38	C₄×C₅×C₈	—	2	2	no	—
39	C₄×C₅×C₈	D₄	1	4	yes	—
40	#3944	—	2	4	no	—
41	#3944	—	2	4	no	—
42	C₂×#672	—	2	4	no	—
43	C₂×C₄×M₁₆	D₅	1	1	yes	characteristic
44	#2150	—	5	2	no	—
45	C₂×#349	—	5	2	no	—
46	#1377	—	5	2	no	—
47	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₅	—	2	1	no	—
49	C₂²×C₄×C₅	—	2	1	no	—
50	C₂²×C₄×C₅	—	2	1	no	—
51	C₂²×C₄×C₅	—	2	1	no	—
52	C₂²×C₄×C₅	—	2	2	no	—
53	C₂²×C₄×C₅	—	2	2	no	—
54	C₂²×C₄×C₅	—	4	4	no	—
55	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
59	C₂²×C₄×C₅	M₁₆	1	2	yes	—
60	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₅×M₁₆	—	2	8	no	—
62	C₅×M₁₆	—	2	4	no	—
63	C₅×M₁₆	—	2	4	no	—
64	C₂×C₅×C₈	—	2	1	no	—
65	C₂×C₅×C₈	—	2	1	no	—
66	C₂×C₅×C₈	—	2	1	no	—
67	C₂×C₅×C₈	—	2	1	no	—
68	C₂×C₅×C₈	—	2	1	no	—
69	C₂×C₅×C₈	—	2	1	no	—
70	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
71	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
72	C₂×C₅×C₈	M₁₆	1	2	yes	—
73	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
75	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₅×C₈	#41	1	2	yes	—
77	C₄²×C₅	—	2	2	no	—
78	C₄²×C₅	—	2	2	no	—
79	C₄²×C₅	—	2	1	no	—
80	C₄²×C₅	—	2	1	no	—
81	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₅	M₁₆	1	1	yes	characteristic
83	C₄²×C₅	M₁₆	1	1	yes	characteristic
84	#672	—	4	4	no	—
85	C₂²×M₁₆	Dic₅	1	1	yes	characteristic
86	C₂²×M₁₆	C₂×D₅	1	1	yes	characteristic
87	C₂²×C₄²	Dic₅	1	1	yes	characteristic
88	C₂×M₃₂	—	5	4	no	—
89	C₄×M₁₆	—	2	4	no	—
90	C₂×#97	Dic₅	1	1	yes	characteristic
91	C₂×#97	C₂×D₅	1	1	yes	characteristic
92	C₂×C₄×C₈	Dic₅	1	1	yes	characteristic
93	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
94	C₄⋊C₁₆	—	10	4	no	—
95	#349	—	10	4	no	—
96	C₂³×C₅	—	2	1	no	—
97	C₂³×C₅	—	2	1	no	—
98	C₂³×C₅	—	4	2	no	—
99	C₂³×C₅	C₄⋊C₈	1	2	yes	—
100	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₅	—	2	1	no	—
102	C₂×C₄×C₅	—	2	1	no	—
103	C₂×C₄×C₅	—	2	4	no	—
104	C₂×C₄×C₅	—	2	2	no	—
105	C₂×C₄×C₅	—	2	2	no	—
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	—	2	2	no	—
109	C₂×C₄×C₅	—	4	4	no	—
110	C₂×C₄×C₅	—	4	4	no	—
111	C₂×C₄×C₅	—	4	2	no	—
112	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
113	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
114	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
115	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅	#117	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
121	C₅×C₈	—	2	2	no	—
122	C₅×C₈	—	2	2	no	—
123	C₅×C₈	—	2	2	no	—
124	C₅×C₈	#121	1	2	yes	—
125	C₅×C₈	C₂²⋊Q₈	1	2	yes	—
126	C₂³×C₄	#168	1	2	yes	—
127	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
128	C₂×M₁₆	—	2	2	no	—
129	C₂×M₁₆	—	2	2	no	—
130	C₂×M₁₆	—	2	2	no	—
131	C₂×M₁₆	—	2	2	no	—
132	C₂×M₁₆	—	2	2	no	—
133	C₂×M₁₆	Dic₁₀	1	2	yes	—
134	C₂×M₁₆	D₂₀	1	2	yes	—
135	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
136	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
137	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
138	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
139	C₂×C₄²	—	2	1	no	—
140	C₂×C₄²	—	2	1	no	—
141	C₂×C₄²	—	2	2	no	—
142	C₂×C₄²	#168	1	2	yes	—
143	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
144	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
145	M₃₂	—	10	4	no	—
146	M₃₂	—	10	4	no	—
147	C₂×C₁₆	—	10	4	no	—
148	#97	—	2	2	no	—
149	#97	—	2	2	no	—
150	C₄×C₈	—	2	2	no	—
151	C₄×C₈	C₂²⋊D₅	1	4	yes	—
152	C₂²×C₅	—	2	2	no	—
153	C₂²×C₅	—	2	2	no	—
154	C₂²×C₅	—	4	2	no	—
155	C₂²×C₅	—	4	2	no	—
156	C₂²×C₅	—	4	2	no	—
157	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
158	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
159	C₂²×C₅	#431	1	1	yes	characteristic
160	C₄×C₅	—	2	2	no	—
161	C₄×C₅	—	2	2	no	—
162	C₄×C₅	—	4	2	no	—
163	C₄×C₅	#349	1	4	yes	complemented
164	C₄×C₅	#380	1	1	yes	characteristic
165	C₄×C₅	#407	1	1	yes	characteristic
166	C₄×C₅	#423	1	1	yes	characteristic
167	C₄×C₅	#432	1	1	yes	characteristic
168	C₂⁴	C₂×#168	1	1	yes	characteristic
169	C₂²×C₄	—	2	1	no	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	4	4	no	—
176	C₂²×C₄	C₂×#168	1	1	yes	characteristic
177	C₂²×C₄	C₂×#168	1	2	yes	—
178	C₂²×C₄	C₂×#168	1	2	yes	—
179	C₂²×C₄	#681	1	2	yes	—
180	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
181	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
182	M₁₆	—	2	8	no	—
183	M₁₆	—	2	4	no	—
184	M₁₆	—	2	4	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	#681	1	2	yes	—
192	C₂×C₈	C₄⋊Dic₅	1	2	yes	—
193	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
194	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
195	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
196	C₂×C₈	#709	1	2	yes	—
197	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
198	C₄²	—	2	1	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	1	no	—
202	C₄²	#681	1	1	yes	characteristic
203	C₄²	#681	1	1	yes	characteristic
204	C₄²	C₂²⋊Dic₅	1	2	yes	—
205	C₁₆	—	20	4	no	—
206	C₂×C₅	—	2	1	no	—
207	C₂×C₅	—	2	1	no	—
208	C₂×C₅	—	4	2	no	—
209	C₂×C₅	#1797	1	1	yes	characteristic
210	C₂×C₅	C₂×#349	1	2	yes	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	4	2	no	—
214	C₂³	C₄⋊#168	1	2	yes	—
215	C₂³	C₂²×#168	1	1	yes	characteristic
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	C₄⋊#168	1	2	yes	—
228	C₂×C₄	C₂²⋊#168	1	2	yes	—
229	C₂×C₄	C₂²⋊#168	1	2	yes	—
230	C₂×C₄	C₂×#681	1	1	yes	characteristic
231	C₂×C₄	C₂×#681	1	1	yes	characteristic
232	C₂×C₄	C₂×#681	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
234	C₂×C₄	#4072	1	1	yes	characteristic
235	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	—	2	2	no	—
238	C₈	—	2	2	no	—
239	C₈	#4070	1	2	yes	—
240	C₈	#4076	1	2	yes	—
241	C₅	#12785	1	1	yes	characteristic, complemented, Sylow
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	C₂×C₄⋊#168	1	1	yes	characteristic
248	C₂²	#65214	1	1	yes	characteristic
249	C₂²	C₂×C₂²⋊#168	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	4	2	no	—
253	C₄	#64773	1	4	yes	complemented
254	C₄	#65213	1	1	yes	characteristic
255	C₄	#65215	1	1	yes	characteristic
256	C₄	#65496	1	1	yes	characteristic
257	C₄	#65501	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	4	2	no	—
261	C₂	#10602323	1	1	yes	characteristic
262	C₂	C₂×#64773	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₅
commutator	C₂³×C₅	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₄×C₅×M₁₆	C₂
socle	C₂²×C₅	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₅	C₂×C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×C₄×M₁₆	D₅
hypocenter	C₅	#12785
LCS	G ▷ C₂³×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #10600536	C₂	semidirect
4. #10598155	C₂	semidirect
163. C₄×C₅	#349	semidirect
241. C₅	#12785	semidirect
253. C₄	#64773	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×M₁₆	2
C₂×#64773	2
#10600536	2
#10598155	2
#12785	5