Subgroups — C₂²×C₅×#451 #11983070

Order 1280.

4982 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₅×Q₁₆	C₂	1	2	yes	maximal, complemented
3	C₂²×C₅×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#451	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₅×#103	C₂	1	2	yes	maximal, complemented
6	C₂²×C₅×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₅×Q₈	—	2	16	no	—
8	C₂³×C₅×Q₈	—	2	6	no	—
9	C₂³×C₅×Q₈	—	2	1	no	—
10	C₂³×C₅×Q₈	C₂²	1	12	yes	—
11	C₂³×C₅×Q₈	C₂²	1	1	yes	characteristic
12	C₂³×C₅×Q₈	C₂²	1	2	yes	—
13	C₂⁴×C₄×C₅	—	2	1	no	—
14	C₂⁴×C₄×C₅	C₂²	1	1	yes	characteristic
15	C₂²×C₅×Q₁₆	—	2	16	no	—
16	C₂²×C₅×Q₁₆	C₂²	1	24	yes	complemented
17	C₂³×C₅×C₈	C₂²	1	2	yes	—
18	C₂×C₅×C₂²⋊Q₈	—	2	6	no	—
19	C₂×C₅×C₂²⋊Q₈	C₂²	1	12	yes	—
20	C₂²×C₅×C₄⋊C₄	—	2	1	no	—
21	C₂²×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂²×C₅×C₂²⋊C₄	—	2	1	no	—
23	C₅×#451	C₂²	1	64	yes	direct product factor
24	C₂×C₅×#103	C₂²	1	24	yes	complemented
25	C₂×C₅×C₂²⋊C₈	C₂²	1	12	yes	—
26	C₂²×#451	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂⁵×C₅	D₄	1	1	yes	characteristic
28	C₂²×C₅×Q₈	—	2	96	no	—
29	C₂²×C₅×Q₈	—	2	16	no	—
30	C₂²×C₅×Q₈	—	2	96	no	—
31	C₂²×C₅×Q₈	—	2	8	no	—
32	C₂²×C₅×Q₈	—	2	12	no	—
33	C₂²×C₅×Q₈	—	2	24	no	—
34	C₂²×C₅×Q₈	—	2	6	no	—
35	C₂²×C₅×Q₈	—	2	3	no	—
36	C₂²×C₅×Q₈	—	2	8	no	—
37	C₂²×C₅×Q₈	D₄	1	16	yes	—
38	C₂²×C₅×Q₈	C₂³	1	16	yes	—
39	C₂²×C₅×Q₈	C₂³	1	6	yes	—
40	C₂²×C₅×Q₈	C₂³	1	12	yes	—
41	C₂³×C₄×C₅	—	2	16	no	—
42	C₂³×C₄×C₅	—	2	12	no	—
43	C₂³×C₄×C₅	—	2	2	no	—
44	C₂³×C₄×C₅	—	2	8	no	—
45	C₂³×C₄×C₅	—	2	1	no	—
46	C₂³×C₄×C₅	—	2	1	no	—
47	C₂³×C₄×C₅	D₄	1	6	yes	—
48	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₅	C₂³	1	6	yes	—
50	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
51	C₂×C₅×Q₁₆	—	2	96	no	—
52	C₂×C₅×Q₁₆	C₂³	1	32	yes	complemented
53	C₂²×C₅×C₈	—	2	8	no	—
54	C₂²×C₅×C₈	C₂³	1	12	yes	—
55	C₅×C₂²⋊Q₈	—	2	24	no	—
56	C₅×C₂²⋊Q₈	C₂³	1	16	yes	—
57	C₂×C₅×C₄⋊C₄	—	2	12	no	—
58	C₂×C₅×C₄⋊C₄	—	2	3	no	—
59	C₂×C₅×C₄⋊C₄	C₂³	1	6	yes	—
60	C₂×C₅×C₂²⋊C₄	—	2	12	no	—
61	C₅×#103	C₂³	1	32	yes	complemented
62	C₅×C₂²⋊C₈	C₂³	1	16	yes	—
63	C₂⁴×Q₈	C₂×C₅	1	1	yes	characteristic
64	C₂³×Q₁₆	C₂×C₅	1	2	yes	complemented
65	C₂²×C₂²⋊Q₈	C₂×C₅	1	1	yes	characteristic
66	C₂×#451	C₂×C₅	1	24	yes	direct product factor
67	C₂²×#103	C₂×C₅	1	2	yes	complemented
68	C₂²×C₂²⋊C₈	C₂×C₅	1	1	yes	characteristic
69	C₂⁴×C₅	—	2	4	no	—
70	C₂⁴×C₅	—	2	4	no	—
71	C₂⁴×C₅	Q₁₆	1	8	yes	complemented
72	C₂⁴×C₅	C₂×D₄	1	6	yes	—
73	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
74	C₂×C₅×Q₈	—	2	192	no	—
75	C₂×C₅×Q₈	—	2	96	no	—
76	C₂×C₅×Q₈	—	2	64	no	—
77	C₂×C₅×Q₈	—	2	48	no	—
78	C₂×C₅×Q₈	—	2	24	no	—
79	C₂×C₅×Q₈	—	2	16	no	—
80	C₂×C₅×Q₈	—	2	12	no	—
81	C₂×C₅×Q₈	—	2	6	no	—
82	C₂×C₅×Q₈	—	2	24	no	—
83	C₂×C₅×Q₈	—	4	12	no	—
84	C₂×C₅×Q₈	C₂×D₄	1	48	yes	—
85	C₂×C₅×Q₈	C₂⁴	1	4	yes	—
86	C₂×C₅×Q₈	C₂⁴	1	8	yes	—
87	C₂²×C₄×C₅	—	2	96	no	—
88	C₂²×C₄×C₅	—	2	16	no	—
89	C₂²×C₄×C₅	—	2	4	no	—
90	C₂²×C₄×C₅	—	2	16	no	—
91	C₂²×C₄×C₅	—	2	12	no	—
92	C₂²×C₄×C₅	—	2	24	no	—
93	C₂²×C₄×C₅	—	2	24	no	—
94	C₂²×C₄×C₅	—	2	6	no	—
95	C₂²×C₄×C₅	—	2	6	no	—
96	C₂²×C₄×C₅	—	4	4	no	—
97	C₂²×C₄×C₅	—	4	4	no	—
98	C₂²×C₄×C₅	C₂×D₄	1	6	yes	—
99	C₂²×C₄×C₅	C₂×D₄	1	12	yes	—
100	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
101	C₂²×C₄×C₅	C₂×D₄	1	8	yes	—
102	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
103	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
104	C₅×Q₁₆	—	2	64	no	—
105	C₂×C₅×C₈	—	2	24	no	—
106	C₂×C₅×C₈	C₂⁴	1	8	yes	—
107	C₅×C₄⋊C₄	—	2	16	no	—
108	C₅×C₄⋊C₄	—	2	6	no	—
109	C₅×C₄⋊C₄	C₂⁴	1	4	yes	—
110	C₅×C₂²⋊C₄	—	2	16	no	—
111	C₂³×Q₈	—	2	16	no	—
112	C₂³×Q₈	—	2	1	no	—
113	C₂³×Q₈	—	2	6	no	—
114	C₂³×Q₈	C₂²×C₅	1	12	yes	—
115	C₂³×Q₈	C₂²×C₅	1	1	yes	characteristic
116	C₂³×Q₈	C₂²×C₅	1	2	yes	—
117	C₂⁴×C₄	—	2	1	no	—
118	C₂⁴×C₄	C₂²×C₅	1	1	yes	characteristic
119	C₂²×Q₁₆	—	2	16	no	—
120	C₂²×Q₁₆	C₂²×C₅	1	24	yes	complemented
121	C₂³×C₈	C₂²×C₅	1	2	yes	—
122	C₂×C₂²⋊Q₈	—	2	6	no	—
123	C₂×C₂²⋊Q₈	C₂²×C₅	1	12	yes	—
124	C₂²×C₄⋊C₄	—	2	1	no	—
125	C₂²×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
126	C₂²×C₂²⋊C₄	—	2	1	no	—
127	#451	C₂²×C₅	1	64	yes	direct product factor
128	C₂×#103	C₂²×C₅	1	24	yes	complemented
129	C₂×C₂²⋊C₈	C₂²×C₅	1	12	yes	—
130	C₂³×C₅	—	2	12	no	—
131	C₂³×C₅	—	2	12	no	—
132	C₂³×C₅	—	2	32	no	—
133	C₂³×C₅	#120	1	4	yes	—
134	C₂³×C₅	C₂×Q₁₆	1	4	yes	—
135	C₂³×C₅	C₂×Q₁₆	1	24	yes	complemented
136	C₂³×C₅	#137	1	4	yes	—
137	C₂³×C₅	C₂²×D₄	1	4	yes	—
138	C₂³×C₅	C₂²×D₄	1	3	yes	—
139	C₅×Q₈	—	2	64	no	—
140	C₅×Q₈	—	2	48	no	—
141	C₅×Q₈	—	2	8	no	—
142	C₅×Q₈	—	2	8	no	—
143	C₅×Q₈	—	4	12	no	—
144	C₅×Q₈	C₂²×D₄	1	16	yes	—
145	C₂×C₄×C₅	—	2	64	no	—
146	C₂×C₄×C₅	—	2	48	no	—
147	C₂×C₄×C₅	—	2	12	no	—
148	C₂×C₄×C₅	—	2	8	no	—
149	C₂×C₄×C₅	—	2	24	no	—
150	C₂×C₄×C₅	—	2	8	no	—
151	C₂×C₄×C₅	—	2	4	no	—
152	C₂×C₄×C₅	—	2	4	no	—
153	C₂×C₄×C₅	—	4	12	no	—
154	C₂×C₄×C₅	—	4	12	no	—
155	C₂×C₄×C₅	#120	1	12	yes	—
156	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
157	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
158	C₂×C₄×C₅	C₂²×D₄	1	12	yes	—
159	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
160	C₅×C₈	—	2	8	no	—
161	C₂⁵	C₅×D₄	1	1	yes	characteristic
162	C₂²×Q₈	—	2	96	no	—
163	C₂²×Q₈	—	2	16	no	—
164	C₂²×Q₈	—	2	96	no	—
165	C₂²×Q₈	—	2	8	no	—
166	C₂²×Q₈	—	2	12	no	—
167	C₂²×Q₈	—	2	24	no	—
168	C₂²×Q₈	—	2	6	no	—
169	C₂²×Q₈	—	2	3	no	—
170	C₂²×Q₈	—	2	8	no	—
171	C₂²×Q₈	C₅×D₄	1	16	yes	—
172	C₂²×Q₈	C₂³×C₅	1	6	yes	—
173	C₂²×Q₈	C₂³×C₅	1	16	yes	—
174	C₂²×Q₈	C₂³×C₅	1	12	yes	—
175	C₂³×C₄	—	2	16	no	—
176	C₂³×C₄	—	2	12	no	—
177	C₂³×C₄	—	2	2	no	—
178	C₂³×C₄	—	2	8	no	—
179	C₂³×C₄	—	2	1	no	—
180	C₂³×C₄	—	2	1	no	—
181	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
182	C₂³×C₄	C₅×D₄	1	6	yes	—
183	C₂³×C₄	C₂³×C₅	1	6	yes	—
184	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
185	C₂×Q₁₆	—	2	96	no	—
186	C₂×Q₁₆	C₂³×C₅	1	32	yes	complemented
187	C₂²×C₈	—	2	8	no	—
188	C₂²×C₈	C₂³×C₅	1	12	yes	—
189	C₂²⋊Q₈	—	2	24	no	—
190	C₂²⋊Q₈	C₂³×C₅	1	16	yes	—
191	C₂×C₄⋊C₄	—	2	12	no	—
192	C₂×C₄⋊C₄	—	2	3	no	—
193	C₂×C₄⋊C₄	C₂³×C₅	1	6	yes	—
194	C₂×C₂²⋊C₄	—	2	12	no	—
195	#103	C₂³×C₅	1	32	yes	complemented
196	C₂²⋊C₈	C₂³×C₅	1	16	yes	—
197	C₂²×C₅	—	2	4	no	—
198	C₂²×C₅	—	2	48	no	—
199	C₂²×C₅	—	2	4	no	—
200	C₂²×C₅	#451	1	16	yes	direct product factor
201	C₂²×C₅	C₂×#120	1	6	yes	—
202	C₂²×C₅	C₂²×Q₁₆	1	6	yes	—
203	C₂²×C₅	C₂²×Q₁₆	1	8	yes	complemented
204	C₂²×C₅	C₂×#137	1	6	yes	—
205	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
206	C₄×C₅	—	2	16	no	—
207	C₄×C₅	—	2	4	no	—
208	C₄×C₅	—	4	4	no	—
209	C₄×C₅	—	4	4	no	—
210	C₄×C₅	C₂×#120	1	6	yes	—
211	C₄×C₅	C₂³×D₄	1	2	yes	—
212	C₂⁴	—	2	4	no	—
213	C₂⁴	—	2	4	no	—
214	C₂⁴	C₅×Q₁₆	1	8	yes	complemented
215	C₂⁴	C₂×C₅×D₄	1	6	yes	—
216	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
217	C₂×Q₈	—	2	192	no	—
218	C₂×Q₈	—	2	96	no	—
219	C₂×Q₈	—	2	64	no	—
220	C₂×Q₈	—	2	48	no	—
221	C₂×Q₈	—	2	24	no	—
222	C₂×Q₈	—	2	16	no	—
223	C₂×Q₈	—	2	12	no	—
224	C₂×Q₈	—	2	6	no	—
225	C₂×Q₈	—	2	24	no	—
226	C₂×Q₈	—	4	12	no	—
227	C₂×Q₈	C₂×C₅×D₄	1	48	yes	—
228	C₂×Q₈	C₂⁴×C₅	1	4	yes	—
229	C₂×Q₈	C₂⁴×C₅	1	8	yes	—
230	C₂²×C₄	—	2	96	no	—
231	C₂²×C₄	—	2	16	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	16	no	—
234	C₂²×C₄	—	2	12	no	—
235	C₂²×C₄	—	2	24	no	—
236	C₂²×C₄	—	2	24	no	—
237	C₂²×C₄	—	2	6	no	—
238	C₂²×C₄	—	2	6	no	—
239	C₂²×C₄	—	4	4	no	—
240	C₂²×C₄	—	4	4	no	—
241	C₂²×C₄	C₂×C₅×D₄	1	6	yes	—
242	C₂²×C₄	C₂×C₅×D₄	1	12	yes	—
243	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
244	C₂²×C₄	C₂×C₅×D₄	1	8	yes	—
245	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
246	C₂²×C₄	C₂⁴×C₅	1	3	yes	—
247	Q₁₆	—	2	64	no	—
248	C₂×C₈	—	2	24	no	—
249	C₂×C₈	C₂⁴×C₅	1	8	yes	—
250	C₄⋊C₄	—	2	16	no	—
251	C₄⋊C₄	—	2	6	no	—
252	C₄⋊C₄	C₂⁴×C₅	1	4	yes	—
253	C₂²⋊C₄	—	2	16	no	—
254	C₂×C₅	—	2	8	no	—
255	C₂×C₅	C₂×#451	1	12	yes	direct product factor
256	C₂×C₅	C₂²×#120	1	1	yes	characteristic
257	C₂×C₅	C₂³×Q₁₆	1	1	yes	characteristic
258	C₂×C₅	C₂²×#137	1	1	yes	characteristic
259	C₂³	—	2	32	no	—
260	C₂³	—	2	12	no	—
261	C₂³	—	2	12	no	—
262	C₂³	C₅×#120	1	4	yes	—
263	C₂³	C₂×C₅×Q₁₆	1	4	yes	—
264	C₂³	C₂×C₅×Q₁₆	1	24	yes	complemented
265	C₂³	C₅×#137	1	4	yes	—
266	C₂³	C₂²×C₅×D₄	1	4	yes	—
267	C₂³	C₂²×C₅×D₄	1	3	yes	—
268	Q₈	—	2	64	no	—
269	Q₈	—	2	48	no	—
270	Q₈	—	2	8	no	—
271	Q₈	—	2	8	no	—
272	Q₈	—	4	12	no	—
273	Q₈	C₂²×C₅×D₄	1	16	yes	—
274	C₂×C₄	—	2	64	no	—
275	C₂×C₄	—	2	48	no	—
276	C₂×C₄	—	2	12	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	2	24	no	—
279	C₂×C₄	—	2	8	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	4	12	no	—
283	C₂×C₄	—	4	12	no	—
284	C₂×C₄	C₅×#120	1	12	yes	—
285	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
286	C₂×C₄	C₂²×C₅×D₄	1	3	yes	—
287	C₂×C₄	C₂²×C₅×D₄	1	12	yes	—
288	C₂×C₄	C₂⁵×C₅	1	1	yes	characteristic
289	C₈	—	2	8	no	—
290	C₅	C₂²×#451	1	1	yes	characteristic, direct product factor, Sylow
291	C₂²	—	2	48	no	—
292	C₂²	—	2	4	no	—
293	C₂²	—	2	4	no	—
294	C₂²	C₅×#451	1	16	yes	direct product factor
295	C₂²	C₂×C₅×#120	1	6	yes	—
296	C₂²	C₂²×C₅×Q₁₆	1	6	yes	—
297	C₂²	C₂²×C₅×Q₁₆	1	8	yes	complemented
298	C₂²	C₂×C₅×#137	1	6	yes	—
299	C₂²	C₂³×C₅×D₄	1	1	yes	characteristic
300	C₄	—	2	16	no	—
301	C₄	—	2	4	no	—
302	C₄	—	4	4	no	—
303	C₄	—	4	4	no	—
304	C₄	C₂×C₅×#120	1	6	yes	—
305	C₄	C₂³×C₅×D₄	1	2	yes	—
306	C₂	—	2	8	no	—
307	C₂	C₂×C₅×#451	1	12	yes	direct product factor
308	C₂	C₂²×C₅×#120	1	1	yes	characteristic
309	C₂	C₂³×C₅×Q₁₆	1	1	yes	characteristic
310	C₂	C₂²×C₅×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅
prime radical	C₂⁵×C₅	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ C₂⁴×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₅×Q₁₆	C₂	semidirect
4. C₂×C₅×#451	C₂	direct (N × H)
5. C₂²×C₅×#103	C₂	semidirect
16. C₂²×C₅×Q₁₆	C₂²	semidirect
23. C₅×#451	C₂²	direct (N × H)
24. C₂×C₅×#103	C₂²	semidirect
26. C₂²×#451	C₅	direct (N × H)
52. C₂×C₅×Q₁₆	C₂³	semidirect
61. C₅×#103	C₂³	semidirect
64. C₂³×Q₁₆	C₂×C₅	semidirect
66. C₂×#451	C₂×C₅	direct (N × H)
67. C₂²×#103	C₂×C₅	semidirect
71. C₂⁴×C₅	Q₁₆	semidirect
120. C₂²×Q₁₆	C₂²×C₅	semidirect
127. #451	C₂²×C₅	direct (N × H)
128. C₂×#103	C₂²×C₅	semidirect
135. C₂³×C₅	C₂×Q₁₆	semidirect
186. C₂×Q₁₆	C₂³×C₅	semidirect
195. #103	C₂³×C₅	semidirect
200. C₂²×C₅	#451	direct (N × H)
203. C₂²×C₅	C₂²×Q₁₆	semidirect
214. C₂⁴	C₅×Q₁₆	semidirect
255. C₂×C₅	C₂×#451	direct (N × H)
264. C₂³	C₂×C₅×Q₁₆	semidirect
290. C₅	C₂²×#451	direct (N × H)
294. C₂²	C₅×#451	direct (N × H)
297. C₂²	C₂²×C₅×Q₁₆	semidirect
307. C₂	C₂×C₅×#451	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₅×Q₈	2
C₂³×C₅×Q₁₆	2
C₂²×C₅×C₂²⋊Q₈	2
C₂×C₅×#451	2
C₂²×C₅×#103	2
C₂²×C₅×C₂²⋊C₈	2
C₂²×#451	5

Subgroups — C22×C5×#451 #11983070

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#451 #11983070