Subgroups — #12015222

Order 1280.

478 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#64978	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#64727	C₂	1	2	yes	maximal
4	C₂×C₄⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
5	#10598295	C₂	1	2	yes	maximal, complemented
6	C₂²×#676	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
8	C₂×#4018	C₄	1	1	yes	characteristic
9	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×#3947	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
13	C₂²×#672	C₂²	1	2	yes	—
14	C₂×#3935	C₄	1	1	yes	characteristic
15	#64978	—	2	2	no	—
16	#64978	—	2	2	no	—
17	C₅×C₄⋊C₁₆	—	2	2	no	—
18	#64727	—	2	4	no	—
19	C₄⋊#672	—	2	2	no	—
20	#12784	—	5	1	no	maximal, Sylow
21	C₂²×C₄×D₅	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂×#685	C₈	1	1	yes	characteristic
26	C₂×#685	C₈	1	1	yes	characteristic
27	C₂×#683	C₈	1	1	yes	characteristic
28	C₂×#683	C₈	1	1	yes	characteristic
29	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
30	C₂²×#168	C₂×C₄	1	1	yes	characteristic
31	C₂²×#168	C₂×C₄	1	1	yes	characteristic
32	C₂×#676	—	2	2	no	—
33	C₂×#676	—	2	2	no	—
34	C₂×#676	—	2	2	no	—
35	C₂×C₈×D₅	—	2	1	no	—
36	C₂×C₈×D₅	—	2	1	no	—
37	C₂×C₈×D₅	—	2	1	no	—
38	C₂×C₈×D₅	—	2	1	no	—
39	C₂×C₈×D₅	D₄	1	2	yes	—
40	C₂×C₈×D₅	Q₈	1	2	yes	—
41	#4018	—	2	2	no	—
42	#4018	—	2	2	no	—
43	C₂×C₅×C₁₆	—	2	2	no	—
44	C₂×C₅×C₁₆	—	2	2	no	—
45	C₂×C₅×C₁₆	—	2	2	no	—
46	C₄×C₅×C₈	D₄	1	2	yes	—
47	C₄×C₅×C₈	D₄	1	2	yes	—
48	#3947	—	2	2	no	—
49	C₈×Dic₅	—	2	1	no	—
50	C₈×Dic₅	—	2	1	no	—
51	C₂×#672	—	2	2	no	—
52	C₂×#672	—	2	2	no	—
53	C₂×#672	—	2	2	no	—
54	#3935	—	2	2	no	—
55	C₂×#405	—	5	1	no	—
56	C₂×C₄⋊C₁₆	—	5	1	no	—
57	C₂×C₄⋊C₁₆	D₅	1	1	yes	characteristic, complemented
58	C₂×C₂²⋊C₁₆	—	5	2	no	—
59	#1374	—	5	2	no	—
60	C₂³×D₅	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
64	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
65	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
66	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
67	C₂×C₄×D₅	—	2	1	no	—
68	C₂×C₄×D₅	—	2	1	no	—
69	C₂×C₄×D₅	—	2	1	no	—
70	C₂×C₄×D₅	—	2	1	no	—
71	C₂×C₄×D₅	C₄⋊C₄	1	2	yes	—
72	C₂×C₄×D₅	M₁₆	1	2	yes	—
73	C₂×C₅×C₈	—	2	1	no	—
74	C₂×C₅×C₈	—	2	1	no	—
75	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
76	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
77	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₅×C₈	#41	1	1	yes	characteristic
81	C₂×C₅×C₈	#41	1	1	yes	characteristic
82	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₅	M₁₆	1	2	yes	—
84	#685	—	2	2	no	—
85	#685	—	2	2	no	—
86	#683	—	2	2	no	—
87	#683	—	2	2	no	—
88	C₄×Dic₅	—	2	1	no	—
89	C₄×Dic₅	—	2	1	no	—
90	C₂×#168	—	2	1	no	—
91	C₂×#168	—	2	1	no	—
92	C₂×#168	—	2	1	no	—
93	C₂×#168	—	2	1	no	—
94	C₂×#168	C₄⋊C₄	1	2	yes	—
95	C₂×#168	M₁₆	1	2	yes	—
96	#676	—	2	8	no	—
97	C₈×D₅	—	2	2	no	—
98	C₈×D₅	—	2	2	no	—
99	C₈×D₅	—	2	2	no	—
100	C₈×D₅	—	2	2	no	—
101	C₅×C₁₆	—	4	2	no	—
102	#672	—	4	2	no	—
103	C₂²×M₁₆	—	5	1	no	—
104	C₂³×C₈	—	5	1	no	—
105	C₂×#117	—	5	1	no	—
106	C₂²×C₁₆	—	5	2	no	—
107	C₂²×C₁₆	C₂×D₅	1	2	yes	—
108	#405	—	10	2	no	—
109	#405	—	10	2	no	—
110	C₂×#97	—	5	1	no	—
111	C₂×#97	—	5	1	no	—
112	C₂×C₄×C₈	—	5	1	no	—
113	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
114	C₄⋊C₁₆	—	2	2	no	—
115	C₄⋊C₁₆	—	10	2	no	—
116	C₂²⋊C₁₆	—	10	4	no	—
117	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
118	C₂²×D₅	—	2	1	no	—
119	C₂²×D₅	—	2	1	no	—
120	C₂²×D₅	—	2	2	no	—
121	C₂²×D₅	—	2	2	no	—
122	C₂²×D₅	C₄⋊C₈	1	2	yes	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	2	2	no	—
125	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
126	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
127	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₅	#117	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
133	C₂×Dic₅	—	2	1	no	—
134	C₂×Dic₅	—	2	1	no	—
135	C₂×Dic₅	—	2	1	no	—
136	C₂×Dic₅	—	2	1	no	—
137	C₂×Dic₅	—	2	2	no	—
138	C₂×Dic₅	—	2	2	no	—
139	C₂×Dic₅	C₄⋊C₈	1	2	yes	—
140	C₄×D₅	—	2	2	no	—
141	C₄×D₅	—	2	2	no	—
142	C₄×D₅	—	2	2	no	—
143	C₄×D₅	—	2	2	no	—
144	C₅×C₈	—	2	2	no	—
145	C₅×C₈	#121	1	1	yes	characteristic
146	C₅×C₈	#121	1	1	yes	characteristic
147	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
148	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
149	#168	—	2	2	no	—
150	#168	—	2	2	no	—
151	C₂³×C₄	—	5	1	no	—
152	C₂×M₁₆	—	10	2	no	—
153	C₂×M₁₆	—	10	2	no	—
154	C₂×M₁₆	—	10	2	no	—
155	C₂²×C₈	—	5	1	no	—
156	C₂²×C₈	—	5	1	no	—
157	C₂²×C₈	—	5	2	no	—
158	C₂²×C₈	—	5	2	no	—
159	C₂²×C₈	—	10	1	no	—
160	C₂²×C₈	—	10	1	no	—
161	C₂²×C₈	—	10	1	no	—
162	C₂²×C₈	—	10	1	no	—
163	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
164	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
165	#117	—	10	2	no	—
166	#117	—	10	2	no	—
167	C₂×C₄⋊C₄	—	5	1	no	—
168	C₂×C₄⋊C₄	—	5	1	no	—
169	C₂×C₂²⋊C₄	—	5	1	no	—
170	C₂×C₂²⋊C₄	—	5	1	no	—
171	C₂×C₄²	—	5	1	no	—
172	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	—	2	2	no	—
175	C₂×C₁₆	—	2	2	no	—
176	C₂×C₁₆	—	10	2	no	—
177	C₂×C₁₆	—	10	2	no	—
178	C₂×C₁₆	—	10	2	no	—
179	#97	—	10	2	no	—
180	#97	—	10	2	no	—
181	C₄×C₈	—	10	1	no	—
182	C₄×C₈	—	10	1	no	—
183	C₄×C₈	D₂₀	1	2	yes	—
184	C₄×C₈	C₂²⋊D₅	1	2	yes	—
185	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
186	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
187	C₂²×C₅	#431	1	1	yes	characteristic
188	C₂²×C₅	#504	1	2	yes	—
189	C₂²×C₅	#504	1	2	yes	—
190	C₂×D₅	—	2	2	no	—
191	C₂×D₅	—	2	4	no	—
192	C₂×D₅	—	4	1	no	—
193	C₂×D₅	—	4	1	no	—
194	C₂×D₅	—	4	2	no	—
195	C₄×C₅	—	2	2	no	—
196	C₄×C₅	—	2	2	no	—
197	C₄×C₅	#380	1	1	yes	characteristic
198	C₄×C₅	#407	1	1	yes	characteristic
199	C₄×C₅	#423	1	1	yes	characteristic
200	C₄×C₅	#432	1	1	yes	characteristic
201	Dic₅	—	2	2	no	—
202	Dic₅	—	4	1	no	—
203	Dic₅	—	4	1	no	—
204	C₂⁴	—	5	1	no	—
205	C₂²×C₄	—	5	1	no	—
206	C₂²×C₄	—	5	2	no	—
207	C₂²×C₄	—	5	2	no	—
208	C₂²×C₄	—	5	1	no	—
209	C₂²×C₄	—	5	1	no	—
210	C₂²×C₄	—	10	1	no	—
211	C₂²×C₄	—	10	1	no	—
212	C₂²×C₄	—	10	1	no	—
213	C₂²×C₄	—	10	1	no	—
214	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
215	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
217	M₁₆	—	10	8	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	5	2	no	—
221	C₂×C₈	—	5	2	no	—
222	C₂×C₈	—	10	1	no	—
223	C₂×C₈	—	10	1	no	—
224	C₂×C₈	—	10	1	no	—
225	C₂×C₈	—	10	1	no	—
226	C₂×C₈	—	10	2	no	—
227	C₂×C₈	—	10	2	no	—
228	C₂×C₈	—	10	2	no	—
229	C₂×C₈	—	10	2	no	—
230	C₂×C₈	#685	1	2	yes	—
231	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
232	C₂×C₈	D₄×D₅	1	1	yes	characteristic
233	C₂×C₈	#711	1	1	yes	characteristic
234	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
235	C₂×C₈	#713	1	1	yes	characteristic
236	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
237	C₄⋊C₄	—	10	2	no	—
238	C₄⋊C₄	—	10	2	no	—
239	C₂²⋊C₄	—	10	2	no	—
240	C₂²⋊C₄	—	10	2	no	—
241	C₄²	—	10	1	no	—
242	C₄²	—	10	1	no	—
243	C₄²	#676	1	2	yes	—
244	C₄²	#685	1	2	yes	—
245	C₁₆	—	4	2	no	—
246	C₁₆	—	20	2	no	—
247	C₂×C₅	#1797	1	1	yes	characteristic
248	C₂×C₅	#2113	1	1	yes	characteristic
249	C₂×C₅	#2113	1	1	yes	characteristic
250	C₂×C₅	#2151	1	2	yes	—
251	C₂×C₅	#2164	1	2	yes	—
252	D₅	—	4	2	no	—
253	C₂³	—	5	2	no	—
254	C₂³	—	10	1	no	—
255	C₂³	—	10	1	no	—
256	C₂³	—	10	2	no	—
257	C₂³	—	10	2	no	—
258	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	5	2	no	—
262	C₂×C₄	—	10	1	no	—
263	C₂×C₄	—	10	1	no	—
264	C₂×C₄	—	10	2	no	—
265	C₂×C₄	—	10	2	no	—
266	C₂×C₄	—	10	2	no	—
267	C₂×C₄	—	10	2	no	—
268	C₂×C₄	—	10	1	no	—
269	C₂×C₄	—	10	1	no	—
270	C₂×C₄	—	10	2	no	—
271	C₂×C₄	—	10	2	no	—
272	C₂×C₄	#3952	1	2	yes	—
273	C₂×C₄	#3952	1	2	yes	—
274	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
275	C₂×C₄	#4038	1	1	yes	characteristic
276	C₂×C₄	C₂×#676	1	1	yes	characteristic
277	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
278	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
279	C₂×C₄	C₂×#685	1	1	yes	characteristic
280	C₈	—	2	2	no	—
281	C₈	—	10	2	no	—
282	C₈	—	10	2	no	—
283	C₈	#4041	1	1	yes	characteristic
284	C₈	#4043	1	1	yes	characteristic
285	C₈	#4084	1	1	yes	characteristic
286	C₈	#4092	1	1	yes	characteristic
287	C₅	#12784	1	1	yes	characteristic, complemented, Sylow
288	C₂²	—	10	2	no	—
289	C₂²	—	10	4	no	—
290	C₂²	—	20	2	no	—
291	C₂²	—	20	1	no	—
292	C₂²	—	20	1	no	—
293	C₂²	D₅×C₄⋊C₈	1	1	yes	characteristic
294	C₂²	#65122	1	1	yes	characteristic
295	C₂²	#65190	1	2	yes	—
296	C₂²	#65196	1	2	yes	—
297	C₂²	C₂×#3952	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	10	2	no	—
301	C₄	—	20	1	no	—
302	C₄	—	20	1	no	—
303	C₄	#65124	1	1	yes	characteristic
304	C₄	#65126	1	1	yes	characteristic
305	C₄	#65276	1	1	yes	characteristic
306	C₄	#65415	1	1	yes	characteristic
307	C₂	—	20	2	no	—
308	C₂	#10600069	1	2	yes	—
309	C₂	#10600078	1	2	yes	—
310	C₂	#10601268	1	1	yes	characteristic
311	C₂	#10602202	1	1	yes	characteristic
312	C₂	#10602225	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅
commutator, socle	C₂³×C₅	C₂²×C₈
Fitting	C₂×C₅×C₄⋊C₁₆	C₂
prime radical	C₂³×D₅	C₂×C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₅
hypocenter	C₅	#12784
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×C₄⋊C₁₆	C₂	semidirect
4. C₂×C₄⋊#672	C₂	semidirect
5. #10598295	C₂	semidirect
57. C₂×C₄⋊C₁₆	D₅	semidirect
287. C₅	#12784	semidirect

Maximal subgroups

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

subgroup	index
C₂×#64978	2
C₂×C₅×C₄⋊C₁₆	2
C₂×#64727	2
C₂×C₄⋊#672	2
#10598295	2
#12784	5