Subgroups — #12016161

Order 1280.

554 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₈×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×#3944	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂²⋊#672	C₂	1	2	yes	maximal, complemented
4	C₂×C₂²⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
5	#10598148	C₂	1	2	yes	maximal, complemented
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₅×D₄	C₄	1	1	yes	characteristic
9	C₅×C₈×D₄	—	2	2	no	—
10	C₅×C₈×D₄	—	2	2	no	—
11	C₂×C₅×C₄⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₅×C₂²⋊C₈	C₄	1	1	yes	characteristic
13	C₂×C₅×C₂²⋊C₈	C₄	1	1	yes	characteristic
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₂×#3944	—	2	4	no	—
16	C₂×#3944	—	2	2	no	—
17	C₂²×#672	C₂²	1	2	yes	—
18	C₂²×#672	C₂²	1	2	yes	—
19	C₂²⋊#672	—	2	4	no	—
20	C₂²⋊#672	—	2	2	no	—
21	#12977	—	5	1	no	maximal, Sylow
22	C₂²×C₅×D₄	C₈	1	1	yes	characteristic
23	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	—	2	2	no	—
26	C₂²×C₅×C₈	—	2	2	no	—
27	C₂²×C₅×C₈	—	2	2	no	—
28	C₂²×C₅×C₈	—	2	1	no	—
29	C₂²×C₅×C₈	—	2	1	no	—
30	C₂²×C₅×C₈	—	2	2	no	—
31	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	D₄	1	2	yes	—
35	C₂²×C₅×C₈	D₄	1	2	yes	—
36	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
37	C₄×C₅×D₄	—	2	2	no	—
38	C₄×C₅×D₄	—	2	2	no	—
39	C₂×C₅×C₄⋊C₄	C₈	1	1	yes	characteristic
40	C₂×C₅×C₂²⋊C₄	C₈	1	1	yes	characteristic
41	C₂×C₅×C₂²⋊C₄	C₈	1	1	yes	characteristic
42	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
43	C₅×C₄⋊C₈	—	2	2	no	—
44	C₅×C₂²⋊C₈	—	2	2	no	—
45	C₅×C₂²⋊C₈	—	2	2	no	—
46	C₄×C₅×C₈	—	2	1	no	—
47	C₄×C₅×C₈	—	2	1	no	—
48	#3944	—	2	8	no	—
49	C₂×#672	—	2	2	no	—
50	C₂×#672	—	2	2	no	—
51	C₂×#672	—	2	2	no	—
52	C₂×#672	—	2	2	no	—
53	C₂×#672	—	2	2	no	—
54	C₂×#672	D₄	1	4	yes	complemented
55	C₂²×M₃₂	—	5	1	no	—
56	C₂×C₈×D₄	D₅	1	1	yes	characteristic
57	C₂×C₂²⋊C₁₆	—	5	2	no	—
58	C₂×C₂²⋊C₁₆	—	5	1	no	—
59	#1374	—	5	2	no	—
60	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
61	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
62	C₂×C₅×D₄	—	2	2	no	—
63	C₂×C₅×D₄	—	2	2	no	—
64	C₂×C₅×D₄	—	2	2	no	—
65	C₂²×C₄×C₅	—	2	2	no	—
66	C₂²×C₄×C₅	—	2	2	no	—
67	C₂²×C₄×C₅	—	2	1	no	—
68	C₂²×C₄×C₅	—	2	2	no	—
69	C₂²×C₄×C₅	—	2	1	no	—
70	C₂²×C₄×C₅	—	2	2	no	—
71	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
72	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₄×C₅	M₁₆	1	2	yes	—
77	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₅×C₈	—	2	2	no	—
79	C₂×C₅×C₈	—	2	2	no	—
80	C₂×C₅×C₈	—	2	4	no	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	—	2	1	no	—
83	C₂×C₅×C₈	—	2	1	no	—
84	C₂×C₅×C₈	—	2	2	no	—
85	C₂×C₅×C₈	—	2	2	no	—
86	C₂×C₅×C₈	—	4	1	no	—
87	C₂×C₅×C₈	—	4	2	no	—
88	C₂×C₅×C₈	—	4	1	no	—
89	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
90	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
91	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂×C₅×C₈	#41	1	2	yes	—
94	C₅×C₄⋊C₄	—	2	2	no	—
95	C₅×C₂²⋊C₄	—	2	2	no	—
96	C₅×C₂²⋊C₄	—	2	2	no	—
97	C₄²×C₅	—	2	1	no	—
98	C₄²×C₅	—	2	1	no	—
99	#672	—	2	4	no	—
100	#672	—	4	2	no	—
101	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
102	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
103	C₂×C₄×D₄	Dic₅	1	1	yes	characteristic
104	C₂×M₃₂	—	10	4	no	—
105	C₂×M₃₂	—	10	2	no	—
106	C₂²×C₁₆	—	5	2	no	—
107	C₂²×C₁₆	—	5	2	no	—
108	C₈×D₄	—	2	2	no	—
109	C₈×D₄	—	2	2	no	—
110	C₂×C₄⋊C₈	Dic₅	1	1	yes	characteristic
111	C₂×C₂²⋊C₈	Dic₅	1	1	yes	characteristic
112	C₂×C₂²⋊C₈	Dic₅	1	1	yes	characteristic
113	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
114	C₂²⋊C₁₆	—	10	4	no	—
115	C₂²⋊C₁₆	—	10	2	no	—
116	C₂³×C₅	—	2	1	no	—
117	C₂³×C₅	—	2	2	no	—
118	C₂³×C₅	—	2	2	no	—
119	C₂³×C₅	—	2	1	no	—
120	C₂³×C₅	—	2	1	no	—
121	C₂³×C₅	—	2	1	no	—
122	C₂³×C₅	—	2	2	no	—
123	C₂³×C₅	—	2	2	no	—
124	C₂³×C₅	C₂²⋊C₈	1	2	yes	—
125	C₂³×C₅	M₃₂	1	2	yes	complemented
126	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
127	C₅×D₄	—	2	8	no	—
128	C₂×C₄×C₅	—	2	1	no	—
129	C₂×C₄×C₅	—	2	2	no	—
130	C₂×C₄×C₅	—	2	1	no	—
131	C₂×C₄×C₅	—	2	2	no	—
132	C₂×C₄×C₅	—	2	2	no	—
133	C₂×C₄×C₅	—	2	4	no	—
134	C₂×C₄×C₅	—	2	1	no	—
135	C₂×C₄×C₅	—	2	1	no	—
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	—	4	1	no	—
139	C₂×C₄×C₅	—	4	1	no	—
140	C₂×C₄×C₅	—	4	2	no	—
141	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
142	C₂×C₄×C₅	M₃₂	1	2	yes	complemented
143	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
145	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
146	C₂×C₄×C₅	#131	1	2	yes	—
147	C₅×C₈	—	2	2	no	—
148	C₅×C₈	—	4	1	no	—
149	C₅×C₈	—	4	1	no	—
150	C₅×C₈	#120	1	1	yes	characteristic
151	C₅×C₈	#121	1	2	yes	—
152	C₅×C₈	#123	1	1	yes	characteristic
153	C₂²×D₄	#168	1	1	yes	characteristic
154	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
155	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	2	2	no	—
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	—	2	1	no	—
161	C₂²×C₈	—	2	1	no	—
162	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
163	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
164	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
165	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
166	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
167	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
168	C₄×D₄	—	2	2	no	—
169	C₄×D₄	—	2	2	no	—
170	C₂×C₄⋊C₄	#168	1	1	yes	characteristic
171	C₂×C₂²⋊C₄	#168	1	1	yes	characteristic
172	C₂×C₂²⋊C₄	#168	1	1	yes	characteristic
173	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
174	M₃₂	—	10	8	no	—
175	C₂×C₁₆	—	5	4	no	—
176	C₂×C₁₆	—	10	2	no	—
177	C₂×C₁₆	—	10	2	no	—
178	C₂×C₁₆	—	10	2	no	—
179	C₂×C₁₆	—	10	2	no	—
180	C₂×C₁₆	—	10	2	no	—
181	C₄⋊C₈	—	2	2	no	—
182	C₂²⋊C₈	—	2	2	no	—
183	C₂²⋊C₈	—	2	2	no	—
184	C₄×C₈	—	2	1	no	—
185	C₄×C₈	—	2	1	no	—
186	C₂²×C₅	—	2	2	no	—
187	C₂²×C₅	—	2	2	no	—
188	C₂²×C₅	—	2	2	no	—
189	C₂²×C₅	—	2	2	no	—
190	C₂²×C₅	—	2	4	no	—
191	C₂²×C₅	—	4	1	no	—
192	C₂²×C₅	—	4	1	no	—
193	C₂²×C₅	—	4	2	no	—
194	C₂²×C₅	—	4	2	no	—
195	C₂²×C₅	—	4	2	no	—
196	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
197	C₂²×C₅	C₈×D₄	1	2	yes	—
198	C₂²×C₅	C₂×M₃₂	1	1	yes	characteristic
199	C₂²×C₅	C₂×M₃₂	1	1	yes	characteristic
200	C₂²×C₅	#504	1	2	yes	—
201	C₄×C₅	—	2	2	no	—
202	C₄×C₅	—	2	2	no	—
203	C₄×C₅	—	4	1	no	—
204	C₄×C₅	—	4	1	no	—
205	C₄×C₅	#386	1	1	yes	characteristic
206	C₄×C₅	#408	1	1	yes	characteristic
207	C₄×C₅	#435	1	2	yes	—
208	C₂⁴	C₂×#168	1	1	yes	characteristic
209	C₂⁴	C₂×#168	1	1	yes	characteristic
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	C₂×#168	1	1	yes	characteristic
220	C₂²×C₄	C₂×#168	1	1	yes	characteristic
221	C₂²×C₄	C₂×#168	1	1	yes	characteristic
222	C₂²×C₄	C₂×#168	1	1	yes	characteristic
223	C₂²×C₄	#681	1	2	yes	—
224	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
225	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	4	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
238	C₂×C₈	D₄×D₅	1	2	yes	—
239	C₂×C₈	#711	1	2	yes	—
240	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
241	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
242	C₄⋊C₄	—	2	2	no	—
243	C₂²⋊C₄	—	2	2	no	—
244	C₂²⋊C₄	—	2	2	no	—
245	C₄²	—	2	1	no	—
246	C₄²	—	2	1	no	—
247	C₁₆	—	10	4	no	—
248	C₁₆	—	20	2	no	—
249	C₂×C₅	—	4	2	no	—
250	C₂×C₅	—	4	2	no	—
251	C₂×C₅	#1869	1	1	yes	characteristic
252	C₂×C₅	#2112	1	1	yes	characteristic
253	C₂×C₅	#2113	1	1	yes	characteristic
254	C₂×C₅	#2168	1	2	yes	—
255	C₂×C₅	#2169	1	2	yes	—
256	C₂³	—	2	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	1	no	—
259	C₂³	—	2	1	no	—
260	C₂³	—	2	1	no	—
261	C₂³	—	2	2	no	—
262	C₂³	—	2	2	no	—
263	C₂³	—	2	1	no	—
264	C₂³	#3944	1	2	yes	complemented
265	C₂³	C₂²⋊#168	1	2	yes	—
266	C₂³	C₂²×#168	1	1	yes	characteristic
267	D₄	—	2	8	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	#3944	1	2	yes	complemented
282	C₂×C₄	C₂²⋊#168	1	2	yes	—
283	C₂×C₄	C₂×#681	1	1	yes	characteristic
284	C₂×C₄	D₄×Dic₅	1	2	yes	—
285	C₂×C₄	#4094	1	2	yes	—
286	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
287	C₈	—	2	2	no	—
288	C₈	—	4	1	no	—
289	C₈	—	4	1	no	—
290	C₈	#4081	1	1	yes	characteristic
291	C₈	#4083	1	1	yes	characteristic
292	C₈	#4085	1	2	yes	—
293	C₅	#12977	1	1	yes	characteristic, complemented, Sylow
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	4	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	1	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	—	4	2	no	—
304	C₂²	D₄×#168	1	2	yes	—
305	C₂²	C₂×#3944	1	1	yes	characteristic
306	C₂²	C₂×#3944	1	1	yes	characteristic
307	C₂²	#65429	1	2	yes	—
308	C₂²	C₂×C₂²⋊#168	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	4	1	no	—
312	C₄	—	4	1	no	—
313	C₄	#65300	1	2	yes	—
314	C₄	#65509	1	1	yes	characteristic
315	C₄	#65523	1	1	yes	characteristic
316	C₂	—	4	2	no	—
317	C₂	—	4	2	no	—
318	C₂	#10601650	1	2	yes	—
319	C₂	#10601651	1	2	yes	—
320	C₂	#10602403	1	1	yes	characteristic
321	C₂	#10602756	1	1	yes	characteristic
322	C₂	#10602797	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅
commutator, socle	C₂³×C₅	C₂²×C₈
Fitting	C₂×C₅×C₈×D₄	C₂
prime radical	C₂²×C₅×D₄	C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×C₈×D₄	D₅
hypocenter	C₅	#12977
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
2. C₂²×#3944	C₂	semidirect
3. C₂×C₂²⋊#672	C₂	semidirect
4. C₂×C₂²⋊#672	C₂	semidirect
5. #10598148	C₂	semidirect
54. C₂×#672	D₄	semidirect
125. C₂³×C₅	M₃₂	semidirect
142. C₂×C₄×C₅	M₃₂	semidirect
264. C₂³	#3944	semidirect
281. C₂×C₄	#3944	semidirect
293. C₅	#12977	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×C₈×D₄	2
C₂²×#3944	2
C₂×C₂²⋊#672	2
#10598148	2
#12977	5