Subgroups — C₂×C₅×#2155 #11956123

Order 1280.

658 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×#108	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#427	C₂	1	1	yes	maximal, characteristic
4	C₅×#2155	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×#365	C₂	1	4	yes	maximal, complemented
6	C₂²×C₅×M₁₆	—	2	1	no	—
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₅×#117	—	2	1	no	—
9	C₂×C₅×M₃₂	C₄	1	2	yes	complemented
10	C₂×C₅×M₃₂	C₄	1	4	yes	complemented
11	C₂×C₅×M₃₂	C₂²	1	2	yes	—
12	C₂×C₅×M₃₂	C₂²	1	4	yes	—
13	C₂²×C₅×C₁₆	C₄	1	2	yes	complemented
14	C₂×C₅×#108	—	2	1	no	—
15	C₂×C₅×#108	—	2	2	no	—
16	C₂×C₅×#108	C₂²	1	2	yes	—
17	C₂×C₅×#108	C₂²	1	4	yes	—
18	C₅×#427	—	2	2	no	—
19	C₅×#427	C₂²	1	4	yes	—
20	C₂×C₅×#107	—	2	1	no	—
21	C₂×C₅×#106	C₂²	1	2	yes	—
22	C₅×#365	C₂²	1	16	yes	complemented
23	C₂×#2155	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
25	C₂×C₅×M₁₆	—	2	2	no	—
26	C₂×C₅×M₁₆	—	2	4	no	—
27	C₂×C₅×M₁₆	—	4	1	no	—
28	C₂×C₅×M₁₆	—	4	2	no	—
29	C₂²×C₅×C₈	—	4	1	no	—
30	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
33	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	D₄	1	2	yes	—
35	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
36	C₂²×C₅×C₈	Q₈	1	2	yes	—
37	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
38	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
39	C₂²×C₅×C₈	C₂³	1	2	yes	—
40	C₅×#117	—	2	4	no	—
41	C₅×#117	—	4	2	no	—
42	C₂×C₅×C₄⋊C₄	—	2	2	no	—
43	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
44	C₂×C₄²×C₅	—	4	1	no	—
45	C₅×M₃₂	C₂×C₄	1	4	yes	—
46	C₅×M₃₂	C₂×C₄	1	8	yes	complemented
47	C₅×M₃₂	C₂³	1	4	yes	—
48	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
49	C₂×C₅×C₁₆	C₂×C₄	1	8	yes	complemented
50	C₅×#108	—	2	2	no	—
51	C₅×#108	—	2	4	no	—
52	C₅×#108	C₂³	1	4	yes	—
53	C₅×#107	—	2	4	no	—
54	C₅×#106	—	2	2	no	—
55	C₅×#106	C₂³	1	4	yes	—
56	C₂²×M₃₂	C₂×C₅	1	1	yes	characteristic
57	C₂²×#108	C₂×C₅	1	1	yes	characteristic
58	C₂×#427	C₂×C₅	1	1	yes	characteristic
59	#2155	C₂×C₅	1	8	yes	direct product factor
60	C₂×#365	C₂×C₅	1	4	yes	complemented
61	C₂⁴×C₅	SD₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₅	—	4	2	no	—
63	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
67	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
68	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
69	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
70	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
73	C₅×M₁₆	—	2	4	no	—
74	C₅×M₁₆	—	4	2	no	—
75	C₅×M₁₆	—	4	4	no	—
76	C₂×C₅×C₈	—	4	2	no	—
77	C₂×C₅×C₈	—	4	4	no	—
78	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
80	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
81	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
83	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
84	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
85	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
87	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
89	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
91	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
93	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
94	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
95	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
96	C₅×C₄⋊C₄	—	2	4	no	—
97	C₅×C₄⋊C₄	—	4	2	no	—
98	C₅×C₂²⋊C₄	—	4	4	no	—
99	C₄²×C₅	—	4	4	no	—
100	C₅×C₁₆	C₂²×C₄	1	8	yes	—
101	C₂²×M₁₆	—	2	1	no	—
102	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
103	C₂×#117	—	2	1	no	—
104	C₂×M₃₂	C₄×C₅	1	2	yes	complemented
105	C₂×M₃₂	C₄×C₅	1	4	yes	complemented
106	C₂×M₃₂	C₂²×C₅	1	2	yes	—
107	C₂×M₃₂	C₂²×C₅	1	4	yes	—
108	C₂²×C₁₆	C₄×C₅	1	2	yes	complemented
109	C₂×#108	—	2	1	no	—
110	C₂×#108	—	2	2	no	—
111	C₂×#108	C₂²×C₅	1	2	yes	—
112	C₂×#108	C₂²×C₅	1	4	yes	—
113	#427	—	2	2	no	—
114	#427	C₂²×C₅	1	4	yes	—
115	C₂×#107	—	2	1	no	—
116	C₂×#106	C₂²×C₅	1	2	yes	—
117	#365	C₂²×C₅	1	16	yes	complemented
118	C₂³×C₅	—	4	2	no	—
119	C₂³×C₅	#106	1	1	yes	characteristic
120	C₂³×C₅	#106	1	1	yes	characteristic
121	C₂³×C₅	#106	1	2	yes	—
122	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
123	C₂³×C₅	C₂×SD₁₆	1	2	yes	—
124	C₂×C₄×C₅	—	4	4	no	—
125	C₂×C₄×C₅	—	8	4	no	—
126	C₂×C₄×C₅	#106	1	2	yes	—
127	C₂×C₄×C₅	#106	1	2	yes	—
128	C₂×C₄×C₅	#106	1	1	yes	characteristic
129	C₂×C₄×C₅	#106	1	2	yes	—
130	C₂×C₄×C₅	#106	1	2	yes	—
131	C₂×C₄×C₅	#106	1	2	yes	—
132	C₂×C₄×C₅	#106	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
135	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
137	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
139	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
141	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
142	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
144	C₅×C₈	—	4	4	no	—
145	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
147	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
149	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
150	C₅×C₈	C₂²×Q₈	1	1	yes	characteristic
151	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
152	C₂×M₁₆	—	2	2	no	—
153	C₂×M₁₆	—	2	4	no	—
154	C₂×M₁₆	—	4	1	no	—
155	C₂×M₁₆	—	4	2	no	—
156	C₂²×C₈	—	4	1	no	—
157	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
158	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
159	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
160	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
161	C₂²×C₈	C₅×D₄	1	2	yes	—
162	C₂²×C₈	C₅×Q₈	1	1	yes	characteristic
163	C₂²×C₈	C₅×Q₈	1	2	yes	—
164	C₂²×C₈	C₅×Q₈	1	1	yes	characteristic
165	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
166	C₂²×C₈	C₂³×C₅	1	2	yes	—
167	#117	—	2	4	no	—
168	#117	—	4	2	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₂²⋊C₄	—	4	1	no	—
171	C₂×C₄²	—	4	1	no	—
172	M₃₂	C₂×C₄×C₅	1	4	yes	—
173	M₃₂	C₂×C₄×C₅	1	8	yes	complemented
174	M₃₂	C₂³×C₅	1	4	yes	—
175	C₂×C₁₆	C₂×C₄×C₅	1	4	yes	—
176	C₂×C₁₆	C₂×C₄×C₅	1	8	yes	complemented
177	#108	—	2	2	no	—
178	#108	—	2	4	no	—
179	#108	C₂³×C₅	1	4	yes	—
180	#107	—	2	4	no	—
181	#106	—	2	2	no	—
182	#106	C₂³×C₅	1	4	yes	—
183	C₂²×C₅	—	2	2	no	—
184	C₂²×C₅	—	2	2	no	—
185	C₂²×C₅	—	2	2	no	—
186	C₂²×C₅	—	4	2	no	—
187	C₂²×C₅	—	4	2	no	—
188	C₂²×C₅	C₂×#106	1	2	yes	—
189	C₂²×C₅	C₂×#106	1	2	yes	—
190	C₂²×C₅	C₂×#106	1	1	yes	characteristic
191	C₂²×C₅	C₂×#106	1	1	yes	characteristic
192	C₂²×C₅	C₂²×SD₁₆	1	1	yes	characteristic
193	C₄×C₅	—	8	4	no	—
194	C₄×C₅	C₂×#106	1	1	yes	characteristic
195	C₄×C₅	C₂×#106	1	2	yes	—
196	C₄×C₅	C₂×#106	1	1	yes	characteristic
197	C₄×C₅	C₂×#106	1	2	yes	—
198	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
199	C₄×C₅	C₂²×SD₁₆	1	1	yes	characteristic
200	C₂⁴	C₅×SD₁₆	1	1	yes	characteristic
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
203	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
204	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
205	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
206	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
207	C₂²×C₄	C₅×SD₁₆	1	2	yes	—
208	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
209	C₂²×C₄	C₅×SD₁₆	1	2	yes	—
210	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
212	M₁₆	—	2	4	no	—
213	M₁₆	—	4	2	no	—
214	M₁₆	—	4	4	no	—
215	C₂×C₈	—	4	4	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
218	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
219	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
220	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
221	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
222	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
223	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
224	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
225	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
226	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
227	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
228	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
229	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
230	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
231	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
232	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
233	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
234	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₂²⋊C₄	—	4	4	no	—
238	C₄²	—	4	4	no	—
239	C₁₆	C₂²×C₄×C₅	1	8	yes	—
240	C₂×C₅	—	2	1	no	—
241	C₂×C₅	—	2	1	no	—
242	C₂×C₅	—	2	2	no	—
243	C₂×C₅	—	2	2	no	—
244	C₂×C₅	#2155	1	2	yes	direct product factor
245	C₂×C₅	C₂²×#106	1	1	yes	characteristic
246	C₂³	—	4	2	no	—
247	C₂³	C₅×#106	1	1	yes	characteristic
248	C₂³	C₅×#106	1	1	yes	characteristic
249	C₂³	C₅×#106	1	2	yes	—
250	C₂³	C₂×C₅×SD₁₆	1	1	yes	characteristic
251	C₂³	C₂×C₅×SD₁₆	1	2	yes	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	8	4	no	—
254	C₂×C₄	C₅×#106	1	2	yes	—
255	C₂×C₄	C₅×#106	1	2	yes	—
256	C₂×C₄	C₅×#106	1	2	yes	—
257	C₂×C₄	C₅×#106	1	2	yes	—
258	C₂×C₄	C₅×#106	1	2	yes	—
259	C₂×C₄	C₅×#106	1	1	yes	characteristic
260	C₂×C₄	C₅×#106	1	1	yes	characteristic
261	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
262	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
263	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
264	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
265	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
266	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
267	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
268	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
269	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
270	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
271	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
272	C₈	—	4	4	no	—
273	C₈	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
274	C₈	C₂×C₅×C₄⋊C₄	1	2	yes	—
275	C₈	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
276	C₈	C₂×C₅×C₄⋊C₄	1	2	yes	—
277	C₈	C₂³×C₄×C₅	1	1	yes	characteristic
278	C₈	C₂²×C₅×Q₈	1	1	yes	characteristic
279	C₅	C₂×#2155	1	1	yes	characteristic, direct product factor, Sylow
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	C₂×C₅×#106	1	2	yes	—
286	C₂²	C₂×C₅×#106	1	1	yes	characteristic
287	C₂²	C₂×C₅×#106	1	2	yes	—
288	C₂²	C₂×C₅×#106	1	1	yes	characteristic
289	C₂²	C₂²×C₅×SD₁₆	1	1	yes	characteristic
290	C₄	—	8	4	no	—
291	C₄	C₂×C₅×#106	1	2	yes	—
292	C₄	C₂×C₅×#106	1	2	yes	—
293	C₄	C₂×C₅×#106	1	1	yes	characteristic
294	C₄	C₂×C₅×#106	1	1	yes	characteristic
295	C₄	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
296	C₄	C₂²×C₅×SD₁₆	1	1	yes	characteristic
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	—	2	2	no	—
301	C₂	C₅×#2155	1	2	yes	direct product factor
302	C₂	C₂²×C₅×#106	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#106
commutator	C₈	C₂³×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂²×C₅	C₂×#106
prime radical	C₂⁴×C₅	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#2155	C₂	direct (N × H)
5. C₂×C₅×#365	C₂	semidirect
9. C₂×C₅×M₃₂	C₄	semidirect
10. C₂×C₅×M₃₂	C₄	semidirect
13. C₂²×C₅×C₁₆	C₄	semidirect
22. C₅×#365	C₂²	semidirect
23. C₂×#2155	C₅	direct (N × H)
46. C₅×M₃₂	C₂×C₄	semidirect
49. C₂×C₅×C₁₆	C₂×C₄	semidirect
59. #2155	C₂×C₅	direct (N × H)
60. C₂×#365	C₂×C₅	semidirect
104. C₂×M₃₂	C₄×C₅	semidirect
105. C₂×M₃₂	C₄×C₅	semidirect
108. C₂²×C₁₆	C₄×C₅	semidirect
117. #365	C₂²×C₅	semidirect
173. M₃₂	C₂×C₄×C₅	semidirect
176. C₂×C₁₆	C₂×C₄×C₅	semidirect
244. C₂×C₅	#2155	direct (N × H)
279. C₅	C₂×#2155	direct (N × H)
301. C₂	C₅×#2155	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×M₃₂	2
C₂²×C₅×#108	2
C₂×C₅×#427	2
C₅×#2155	2
C₂×C₅×#365	2
C₂×#2155	5

Subgroups — C2×C5×#2155 #11956123

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2155 #11956123