Subgroups — C₂×C₅×#1728 #11943053

Order 1280.

2782 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×M₁₆	C₂	1	2	yes	maximal
2	C₂³×C₄²×C₅	C₂	1	1	yes	maximal, characteristic
3	C₅×#1728	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅×#336	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₅	C₄	1	2	yes	—
6	C₂⁴×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₂²×C₅×M₁₆	—	2	8	no	—
8	C₂²×C₅×M₁₆	—	2	8	no	—
9	C₂²×C₅×M₁₆	C₄	1	16	yes	complemented
10	C₂²×C₅×M₁₆	C₂²	1	8	yes	—
11	C₂³×C₅×C₈	C₂²	1	4	yes	—
12	C₂²×C₄²×C₅	—	2	8	no	—
13	C₂²×C₄²×C₅	C₄	1	4	yes	—
14	C₂²×C₄²×C₅	C₄	1	2	yes	—
15	C₂²×C₄²×C₅	C₂²	1	4	yes	—
16	C₂²×C₄²×C₅	C₂²	1	2	yes	—
17	C₅×#336	C₂²	1	16	yes	complemented
18	C₂×#1728	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂⁵×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₅	—	2	8	no	—
21	C₂³×C₄×C₅	—	2	16	no	—
22	C₂³×C₄×C₅	C₂×C₄	1	8	yes	—
23	C₂³×C₄×C₅	C₂×C₄	1	4	yes	—
24	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₅	C₂×C₄	1	8	yes	—
27	C₂³×C₄×C₅	D₄	1	4	yes	—
28	C₂³×C₄×C₅	D₄	1	8	yes	—
29	C₂³×C₄×C₅	Q₈	1	4	yes	—
30	C₂³×C₄×C₅	C₂³	1	2	yes	—
31	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂×C₅×M₁₆	—	2	64	no	—
33	C₂×C₅×M₁₆	—	2	16	no	—
34	C₂×C₅×M₁₆	—	2	16	no	—
35	C₂×C₅×M₁₆	C₂×C₄	1	32	yes	complemented
36	C₂²×C₅×C₈	—	2	8	no	—
37	C₂²×C₅×C₈	—	2	8	no	—
38	C₂²×C₅×C₈	C₂×C₄	1	16	yes	complemented
39	C₂²×C₅×C₈	C₂³	1	8	yes	—
40	C₂×C₄²×C₅	—	2	16	no	—
41	C₂×C₄²×C₅	—	2	8	no	—
42	C₂×C₄²×C₅	—	2	8	no	—
43	C₂×C₄²×C₅	—	2	16	no	—
44	C₂×C₄²×C₅	C₂×C₄	1	8	yes	—
45	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₅	C₂³	1	4	yes	—
47	C₂³×M₁₆	C₂×C₅	1	2	yes	—
48	C₂³×C₄²	C₂×C₅	1	1	yes	characteristic
49	#1728	C₂×C₅	1	8	yes	direct product factor
50	C₂×#336	C₂×C₅	1	4	yes	complemented
51	C₂⁴×C₅	—	2	8	no	—
52	C₂⁴×C₅	C₄²	1	4	yes	—
53	C₂⁴×C₅	C₂²⋊C₄	1	4	yes	—
54	C₂⁴×C₅	C₄⋊C₄	1	4	yes	—
55	C₂⁴×C₅	C₂²×C₄	1	2	yes	—
56	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	—	2	64	no	—
58	C₂²×C₄×C₅	—	2	16	no	—
59	C₂²×C₄×C₅	—	2	8	no	—
60	C₂²×C₄×C₅	—	2	8	no	—
61	C₂²×C₄×C₅	—	2	32	no	—
62	C₂²×C₄×C₅	C₄²	1	8	yes	—
63	C₂²×C₄×C₅	C₄²	1	4	yes	—
64	C₂²×C₄×C₅	C₂²⋊C₄	1	8	yes	—
65	C₂²×C₄×C₅	C₂²⋊C₄	1	4	yes	—
66	C₂²×C₄×C₅	C₂²⋊C₄	1	32	yes	—
67	C₂²×C₄×C₅	C₄⋊C₄	1	8	yes	—
68	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
69	C₂²×C₄×C₅	C₄⋊C₄	1	32	yes	—
70	C₂²×C₄×C₅	M₁₆	1	16	yes	complemented
71	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
72	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
76	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
78	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
79	C₂²×C₄×C₅	C₂×D₄	1	8	yes	—
80	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
81	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
83	C₅×M₁₆	—	2	64	no	—
84	C₂×C₅×C₈	—	2	32	no	—
85	C₂×C₅×C₈	—	2	8	no	—
86	C₂×C₅×C₈	—	2	8	no	—
87	C₂×C₅×C₈	C₂²×C₄	1	16	yes	complemented
88	C₄²×C₅	—	2	16	no	—
89	C₄²×C₅	—	2	8	no	—
90	C₄²×C₅	—	2	8	no	—
91	C₂⁴×C₄	C₄×C₅	1	2	yes	—
92	C₂⁴×C₄	C₂²×C₅	1	1	yes	characteristic
93	C₂²×M₁₆	—	2	8	no	—
94	C₂²×M₁₆	—	2	8	no	—
95	C₂²×M₁₆	C₄×C₅	1	16	yes	complemented
96	C₂²×M₁₆	C₂²×C₅	1	8	yes	—
97	C₂³×C₈	C₂²×C₅	1	4	yes	—
98	C₂²×C₄²	—	2	8	no	—
99	C₂²×C₄²	C₄×C₅	1	4	yes	—
100	C₂²×C₄²	C₄×C₅	1	2	yes	—
101	C₂²×C₄²	C₂²×C₅	1	2	yes	—
102	C₂²×C₄²	C₂²×C₅	1	4	yes	—
103	#336	C₂²×C₅	1	16	yes	complemented
104	C₂³×C₅	—	2	8	no	—
105	C₂³×C₅	—	2	16	no	—
106	C₂³×C₅	—	2	16	no	—
107	C₂³×C₅	—	2	16	no	—
108	C₂³×C₅	#95	1	16	yes	—
109	C₂³×C₅	C₂×C₄²	1	4	yes	—
110	C₂³×C₅	C₂×C₄²	1	2	yes	—
111	C₂³×C₅	C₂×C₂²⋊C₄	1	4	yes	—
112	C₂³×C₅	C₂×C₂²⋊C₄	1	2	yes	—
113	C₂³×C₅	C₂×C₄⋊C₄	1	4	yes	—
114	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
115	C₂³×C₅	C₂×M₁₆	1	8	yes	—
116	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₅	—	2	64	no	—
118	C₂×C₄×C₅	—	2	8	no	—
119	C₂×C₄×C₅	—	2	8	no	—
120	C₂×C₄×C₅	—	2	32	no	—
121	C₂×C₄×C₅	#95	1	32	yes	—
122	C₂×C₄×C₅	#95	1	16	yes	—
123	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
124	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
125	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	8	yes	—
126	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	16	yes	—
127	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	4	yes	—
128	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
129	C₂×C₄×C₅	C₂×C₄⋊C₄	1	16	yes	—
130	C₂×C₄×C₅	C₂×C₄⋊C₄	1	8	yes	—
131	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
132	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
133	C₂×C₄×C₅	C₂×M₁₆	1	16	yes	complemented
134	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
135	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
137	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
138	C₅×C₈	—	2	16	no	—
139	C₂⁵	C₂×C₄×C₅	1	1	yes	characteristic
140	C₂³×C₄	—	2	8	no	—
141	C₂³×C₄	—	2	16	no	—
142	C₂³×C₄	C₂×C₄×C₅	1	8	yes	—
143	C₂³×C₄	C₂×C₄×C₅	1	4	yes	—
144	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
145	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
146	C₂³×C₄	C₂×C₄×C₅	1	8	yes	—
147	C₂³×C₄	C₅×D₄	1	4	yes	—
148	C₂³×C₄	C₅×D₄	1	8	yes	—
149	C₂³×C₄	C₅×Q₈	1	4	yes	—
150	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
151	C₂³×C₄	C₂³×C₅	1	2	yes	—
152	C₂×M₁₆	—	2	64	no	—
153	C₂×M₁₆	—	2	16	no	—
154	C₂×M₁₆	—	2	16	no	—
155	C₂×M₁₆	C₂×C₄×C₅	1	32	yes	complemented
156	C₂²×C₈	—	2	8	no	—
157	C₂²×C₈	—	2	8	no	—
158	C₂²×C₈	C₂×C₄×C₅	1	16	yes	complemented
159	C₂²×C₈	C₂³×C₅	1	8	yes	—
160	C₂×C₄²	—	2	16	no	—
161	C₂×C₄²	—	2	8	no	—
162	C₂×C₄²	—	2	8	no	—
163	C₂×C₄²	—	2	16	no	—
164	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
165	C₂×C₄²	C₂×C₄×C₅	1	8	yes	—
166	C₂×C₄²	C₂³×C₅	1	4	yes	—
167	C₂²×C₅	—	2	8	no	—
168	C₂²×C₅	—	2	8	no	—
169	C₂²×C₅	—	2	8	no	—
170	C₂²×C₅	—	2	32	no	—
171	C₂²×C₅	C₂×#95	1	8	yes	—
172	C₂²×C₅	C₂×#95	1	4	yes	—
173	C₂²×C₅	C₄×M₁₆	1	8	yes	—
174	C₂²×C₅	#407	1	8	yes	—
175	C₂²×C₅	#423	1	8	yes	—
176	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
177	C₂²×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
178	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
179	C₂²×C₅	C₂²×M₁₆	1	4	yes	—
180	C₄×C₅	—	2	16	no	—
181	C₄×C₅	C₂×#95	1	4	yes	—
182	C₄×C₅	C₂×#95	1	8	yes	—
183	C₄×C₅	C₂²×C₂²⋊C₄	1	2	yes	—
184	C₄×C₅	C₂²×C₄⋊C₄	1	2	yes	—
185	C₂⁴	—	2	8	no	—
186	C₂⁴	C₄²×C₅	1	4	yes	—
187	C₂⁴	C₅×C₂²⋊C₄	1	4	yes	—
188	C₂⁴	C₅×C₄⋊C₄	1	4	yes	—
189	C₂⁴	C₂²×C₄×C₅	1	2	yes	—
190	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
191	C₂²×C₄	—	2	64	no	—
192	C₂²×C₄	—	2	16	no	—
193	C₂²×C₄	—	2	8	no	—
194	C₂²×C₄	—	2	8	no	—
195	C₂²×C₄	—	2	32	no	—
196	C₂²×C₄	C₄²×C₅	1	8	yes	—
197	C₂²×C₄	C₄²×C₅	1	4	yes	—
198	C₂²×C₄	C₅×C₂²⋊C₄	1	32	yes	—
199	C₂²×C₄	C₅×C₂²⋊C₄	1	8	yes	—
200	C₂²×C₄	C₅×C₂²⋊C₄	1	4	yes	—
201	C₂²×C₄	C₅×C₄⋊C₄	1	32	yes	—
202	C₂²×C₄	C₅×C₄⋊C₄	1	8	yes	—
203	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
204	C₂²×C₄	C₅×M₁₆	1	16	yes	complemented
205	C₂²×C₄	C₂²×C₄×C₅	1	8	yes	—
206	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
207	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
208	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
209	C₂²×C₄	C₂²×C₄×C₅	1	8	yes	—
210	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
211	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
212	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
213	C₂²×C₄	C₂×C₅×D₄	1	8	yes	—
214	C₂²×C₄	C₂×C₅×Q₈	1	4	yes	—
215	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
216	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
217	M₁₆	—	2	64	no	—
218	C₂×C₈	—	2	32	no	—
219	C₂×C₈	—	2	8	no	—
220	C₂×C₈	—	2	8	no	—
221	C₂×C₈	C₂²×C₄×C₅	1	16	yes	complemented
222	C₄²	—	2	8	no	—
223	C₄²	—	2	16	no	—
224	C₄²	—	2	8	no	—
225	C₂×C₅	—	2	8	no	—
226	C₂×C₅	#1728	1	8	yes	direct product factor
227	C₂×C₅	C₂²×#95	1	1	yes	characteristic
228	C₂×C₅	C₂×C₄×M₁₆	1	2	yes	—
229	C₂×C₅	C₂×#407	1	2	yes	—
230	C₂×C₅	C₂×#423	1	2	yes	—
231	C₂³	—	2	16	no	—
232	C₂³	—	2	16	no	—
233	C₂³	—	2	16	no	—
234	C₂³	—	2	8	no	—
235	C₂³	C₅×#95	1	16	yes	—
236	C₂³	C₂×C₄²×C₅	1	4	yes	—
237	C₂³	C₂×C₄²×C₅	1	2	yes	—
238	C₂³	C₂×C₅×C₂²⋊C₄	1	4	yes	—
239	C₂³	C₂×C₅×C₂²⋊C₄	1	2	yes	—
240	C₂³	C₂×C₅×C₄⋊C₄	1	4	yes	—
241	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
242	C₂³	C₂×C₅×M₁₆	1	8	yes	—
243	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
244	C₂×C₄	—	2	64	no	—
245	C₂×C₄	—	2	32	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	C₅×#95	1	32	yes	—
249	C₂×C₄	C₅×#95	1	16	yes	—
250	C₂×C₄	C₂×C₄²×C₅	1	2	yes	—
251	C₂×C₄	C₂×C₄²×C₅	1	4	yes	—
252	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
253	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	4	yes	—
254	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	8	yes	—
255	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	16	yes	—
256	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
257	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
258	C₂×C₄	C₂×C₅×C₄⋊C₄	1	16	yes	—
259	C₂×C₄	C₂×C₅×C₄⋊C₄	1	8	yes	—
260	C₂×C₄	C₂×C₅×M₁₆	1	16	yes	complemented
261	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
262	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
263	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
264	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
265	C₈	—	2	16	no	—
266	C₅	C₂×#1728	1	1	yes	characteristic, direct product factor, Sylow
267	C₂²	—	2	8	no	—
268	C₂²	—	2	8	no	—
269	C₂²	—	2	8	no	—
270	C₂²	—	2	32	no	—
271	C₂²	C₂×C₅×#95	1	8	yes	—
272	C₂²	C₂×C₅×#95	1	4	yes	—
273	C₂²	C₄×C₅×M₁₆	1	8	yes	—
274	C₂²	C₅×#407	1	8	yes	—
275	C₂²	C₅×#423	1	8	yes	—
276	C₂²	C₂²×C₄²×C₅	1	1	yes	characteristic
277	C₂²	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
278	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
279	C₂²	C₂²×C₅×M₁₆	1	4	yes	—
280	C₄	—	2	16	no	—
281	C₄	C₂×C₅×#95	1	4	yes	—
282	C₄	C₂×C₅×#95	1	8	yes	—
283	C₄	C₂²×C₅×C₂²⋊C₄	1	2	yes	—
284	C₄	C₂²×C₅×C₄⋊C₄	1	2	yes	—
285	C₂	—	2	8	no	—
286	C₂	C₅×#1728	1	8	yes	direct product factor
287	C₂	C₂²×C₅×#95	1	1	yes	characteristic
288	C₂	C₂×C₄×C₅×M₁₆	1	2	yes	—
289	C₂	C₂×C₅×#407	1	2	yes	—
290	C₂	C₂×C₅×#423	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₅	C₂³
commutator	C₂²	C₂²×C₄²×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂⁴×C₅	C₂²×C₄
prime radical	C₂⁵×C₅	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1728	C₂	direct (N × H)
4. C₂×C₅×#336	C₂	semidirect
9. C₂²×C₅×M₁₆	C₄	semidirect
17. C₅×#336	C₂²	semidirect
18. C₂×#1728	C₅	direct (N × H)
35. C₂×C₅×M₁₆	C₂×C₄	semidirect
38. C₂²×C₅×C₈	C₂×C₄	semidirect
49. #1728	C₂×C₅	direct (N × H)
50. C₂×#336	C₂×C₅	semidirect
70. C₂²×C₄×C₅	M₁₆	semidirect
87. C₂×C₅×C₈	C₂²×C₄	semidirect
95. C₂²×M₁₆	C₄×C₅	semidirect
103. #336	C₂²×C₅	semidirect
133. C₂×C₄×C₅	C₂×M₁₆	semidirect
155. C₂×M₁₆	C₂×C₄×C₅	semidirect
158. C₂²×C₈	C₂×C₄×C₅	semidirect
204. C₂²×C₄	C₅×M₁₆	semidirect
221. C₂×C₈	C₂²×C₄×C₅	semidirect
226. C₂×C₅	#1728	direct (N × H)
260. C₂×C₄	C₂×C₅×M₁₆	semidirect
266. C₅	C₂×#1728	direct (N × H)
286. C₂	C₅×#1728	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₅×M₁₆	2
C₂³×C₄²×C₅	2
C₅×#1728	2
C₂×C₅×#336	2
C₂×#1728	5

Subgroups — C2×C5×#1728 #11943053

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1728 #11943053