Subgroups — C₂×C₅×#5562 #14481082

Order 1920.

2520 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#940	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#938	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#930	C₂	1	2	yes	maximal, complemented
5	C₂×C₅×#929	C₂	1	2	yes	maximal, complemented
6	C₅×#5562	C₂	1	16	yes	maximal, direct product factor
7	C₂×C₅×#895	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#890	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅×#884	C₂	1	2	yes	maximal, complemented
10	C₂×C₅×#883	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×C₅×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₅×#872	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₅×#540	—	3	1	no	maximal, Hall
14	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
15	C₂³×C₃×C₄×C₅	C₂²	1	2	yes	—
16	C₂²×C₅×#67	C₂²	1	2	yes	—
17	C₂²×C₅×#67	C₂²	1	2	yes	—
18	C₂²×C₅×#67	C₂²	1	2	yes	—
19	C₂²×C₅×D₁₂	C₂²	1	1	yes	characteristic
20	C₂²×C₅×D₁₂	C₂²	1	2	yes	—
21	C₂²×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
22	C₂²×C₄×C₅×D₃	C₂²	1	2	yes	—
23	C₃×C₄×C₅×D₄	C₂²	1	8	yes	complemented
24	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
27	C₂×C₅×#217	C₂²	1	2	yes	—
28	C₅×#940	C₂²	1	8	yes	complemented
29	C₅×#938	C₂²	1	8	yes	complemented
30	C₅×#930	C₂²	1	16	yes	complemented
31	C₅×#929	C₂²	1	16	yes	complemented
32	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
33	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
34	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
35	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
36	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
37	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
38	C₂×C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
39	C₂×C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
40	C₂×C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
41	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
42	C₅×#895	C₂²	1	8	yes	complemented
43	C₅×#890	C₂²	1	8	yes	complemented
44	C₅×#884	C₂²	1	16	yes	complemented
45	C₅×#883	C₂²	1	16	yes	complemented
46	C₄×C₅×D₁₂	C₂²	1	8	yes	complemented
47	C₅×#872	C₂²	1	8	yes	complemented
48	C₂×#5562	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
49	C₅×#540	—	3	16	no	—
50	C₂×C₅×#127	—	3	1	no	—
51	C₂×C₅×#125	—	3	1	no	—
52	C₂×C₅×#123	—	3	2	no	—
53	C₂×C₅×#123	—	3	2	no	—
54	C₂×C₅×#121	—	3	1	no	—
55	C₂×C₅×#121	—	3	2	no	—
56	C₂×C₅×#121	—	3	1	no	—
57	C₂×C₅×#121	—	3	2	no	—
58	C₂×C₄×C₅×D₄	—	3	1	no	—
59	C₂×C₄×C₅×D₄	D₃	1	1	yes	characteristic, complemented
60	C₂×C₅×#117	—	3	1	no	—
61	C₂⁴×C₃×C₅	C₂³	1	2	yes	—
62	C₂³×C₅×D₃	C₂³	1	1	yes	characteristic
63	C₂³×C₅×D₃	C₂³	1	2	yes	—
64	C₂×C₃×C₅×D₄	—	2	4	no	—
65	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
66	C₂²×C₃×C₄×C₅	—	4	4	no	—
67	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
68	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
69	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
70	C₂²×C₃×C₄×C₅	C₂³	1	8	yes	—
71	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
72	C₂×C₅×#67	—	4	4	no	—
73	C₂×C₅×#67	—	4	4	no	—
74	C₂×C₅×#67	—	4	4	no	—
75	C₂×C₅×#67	C₂³	1	8	yes	—
76	C₂×C₅×#67	C₂³	1	8	yes	—
77	C₂×C₅×#67	C₂³	1	8	yes	—
78	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
79	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
80	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
81	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
82	C₂×C₅×D₁₂	—	2	4	no	—
83	C₂×C₅×D₁₂	—	4	4	no	—
84	C₂×C₅×D₁₂	C₂³	1	4	yes	—
85	C₂×C₅×D₁₂	C₂³	1	8	yes	—
86	C₂×C₄×C₅×D₃	—	2	4	no	—
87	C₂×C₄×C₅×D₃	—	4	4	no	—
88	C₂×C₄×C₅×D₃	C₂³	1	4	yes	—
89	C₂×C₄×C₅×D₃	C₂³	1	8	yes	—
90	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
91	C₃×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
92	C₃×C₄²×C₅	C₂³	1	4	yes	—
93	C₅×#217	C₂³	1	8	yes	—
94	C₅×(C₂×D₃)⋊C₄	C₂³	1	4	yes	—
95	C₅×(C₂×D₃)⋊C₄	C₂³	1	4	yes	—
96	C₅×(C₂×D₃)⋊C₄	C₂³	1	8	yes	—
97	C₅×(C₂×D₃)⋊C₄	C₂³	1	8	yes	—
98	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
99	C₅×C₄⋊Dic₃	C₂³	1	8	yes	—
100	C₅×Dic₃⋊C₄	C₂³	1	4	yes	—
101	C₅×Dic₃⋊C₄	C₂³	1	4	yes	—
102	C₅×Dic₃⋊C₄	C₂³	1	8	yes	—
103	C₄×C₅×Dic₃	C₂³	1	4	yes	—
104	C₂×C₃×C₄×D₄	C₂×C₅	1	1	yes	characteristic, complemented
105	C₂×#940	C₂×C₅	1	1	yes	characteristic, complemented
106	C₂×#938	C₂×C₅	1	1	yes	characteristic, complemented
107	C₂×#930	C₂×C₅	1	2	yes	complemented
108	C₂×#929	C₂×C₅	1	2	yes	complemented
109	#5562	C₂×C₅	1	16	yes	direct product factor
110	C₂×#895	C₂×C₅	1	1	yes	characteristic, complemented
111	C₂×#890	C₂×C₅	1	1	yes	characteristic, complemented
112	C₂×#884	C₂×C₅	1	2	yes	complemented
113	C₂×#883	C₂×C₅	1	2	yes	complemented
114	C₂×C₄×D₁₂	C₂×C₅	1	1	yes	characteristic, complemented
115	C₂×#872	C₂×C₅	1	1	yes	characteristic, complemented
116	C₂²×C₅×D₄	—	3	2	no	—
117	C₂²×C₅×D₄	—	3	2	no	—
118	C₂²×C₅×D₄	—	3	2	no	—
119	C₂²×C₅×D₄	—	3	1	no	—
120	C₂²×C₅×D₄	—	3	2	no	—
121	C₂²×C₅×D₄	C₂×D₃	1	1	yes	characteristic
122	C₂³×C₄×C₅	—	3	1	no	—
123	C₂³×C₄×C₅	—	3	2	no	—
124	C₂³×C₄×C₅	C₂×D₃	1	2	yes	—
125	C₅×#127	—	3	8	no	—
126	C₅×#125	—	3	8	no	—
127	C₅×#123	—	3	16	no	—
128	C₅×#123	—	3	16	no	—
129	C₅×#121	—	3	8	no	—
130	C₅×#121	—	3	16	no	—
131	C₅×#121	—	3	8	no	—
132	C₅×#121	—	3	16	no	—
133	C₄×C₅×D₄	—	3	8	no	—
134	C₄×C₅×D₄	C₂×D₃	1	8	yes	complemented
135	C₅×#117	—	3	8	no	—
136	C₂×C₅×C₄⋊C₄	—	3	1	no	—
137	C₂×C₅×C₄⋊C₄	—	3	1	no	—
138	C₂×C₅×C₄⋊C₄	—	3	2	no	—
139	C₂×C₅×C₄⋊C₄	—	3	1	no	—
140	C₂×C₅×C₄⋊C₄	—	3	2	no	—
141	C₂×C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
142	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
143	C₂×C₅×C₂²⋊C₄	—	3	1	no	—
144	C₂×C₅×C₂²⋊C₄	—	3	1	no	—
145	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
146	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
147	C₂×C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
148	C₂×C₄²×C₅	—	3	1	no	—
149	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
150	C₂×#540	—	3	1	no	Sylow
151	C₂³×C₃×C₅	—	2	4	no	—
152	C₂³×C₃×C₅	—	2	4	no	—
153	C₂³×C₃×C₅	—	2	4	no	—
154	C₂³×C₃×C₅	C₂⁴	1	4	yes	—
155	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
156	C₂²×C₅×D₃	—	2	2	no	—
157	C₂²×C₅×D₃	—	2	4	no	—
158	C₂²×C₅×D₃	—	2	4	no	—
159	C₂²×C₅×D₃	—	2	2	no	—
160	C₂²×C₅×D₃	—	2	4	no	—
161	C₂²×C₅×D₃	—	2	2	no	—
162	C₂²×C₅×D₃	C₂⁴	1	2	yes	—
163	C₂²×C₅×D₃	C₂⁴	1	4	yes	—
164	C₃×C₅×D₄	—	2	8	no	—
165	C₂×C₃×C₄×C₅	—	2	2	no	—
166	C₂×C₃×C₄×C₅	—	2	4	no	—
167	C₂×C₃×C₄×C₅	—	2	2	no	—
168	C₂×C₃×C₄×C₅	—	4	8	no	—
169	C₂×C₃×C₄×C₅	#41	1	4	yes	—
170	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
171	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
172	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
173	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
174	C₅×#67	—	4	8	no	—
175	C₅×#67	—	4	8	no	—
176	C₅×#67	—	4	8	no	—
177	C₂×C₅×Dic₃	—	2	2	no	—
178	C₂×C₅×Dic₃	—	2	4	no	—
179	C₂×C₅×Dic₃	—	2	2	no	—
180	C₂×C₅×Dic₃	—	2	2	no	—
181	C₂×C₅×Dic₃	C₂⁴	1	2	yes	—
182	C₂×C₅×Dic₃	C₂⁴	1	2	yes	—
183	C₂×C₅×Dic₃	C₂⁴	1	2	yes	—
184	C₂×C₅×Dic₃	C₂⁴	1	4	yes	—
185	C₅×D₁₂	—	2	8	no	—
186	C₅×D₁₂	—	4	8	no	—
187	C₄×C₅×D₃	—	2	8	no	—
188	C₄×C₅×D₃	—	4	8	no	—
189	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
190	C₂³×C₃×C₄	C₂²×C₅	1	2	yes	—
191	C₂²×#67	C₂²×C₅	1	2	yes	—
192	C₂²×#67	C₂²×C₅	1	2	yes	—
193	C₂²×#67	C₂²×C₅	1	2	yes	—
194	C₂²×D₁₂	C₂²×C₅	1	1	yes	characteristic
195	C₂²×D₁₂	C₂²×C₅	1	2	yes	—
196	C₂²×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
197	C₂²×C₄×D₃	C₂²×C₅	1	2	yes	—
198	C₃×C₄×D₄	C₂²×C₅	1	8	yes	complemented
199	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
200	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
201	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
202	C₂×#217	C₂²×C₅	1	2	yes	—
203	#940	C₂²×C₅	1	8	yes	complemented
204	#938	C₂²×C₅	1	8	yes	complemented
205	#930	C₂²×C₅	1	16	yes	complemented
206	#929	C₂²×C₅	1	16	yes	complemented
207	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	1	yes	characteristic
208	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	1	yes	characteristic
209	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
210	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
211	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
212	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
213	C₂×Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
214	C₂×Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
215	C₂×Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
216	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
217	#895	C₂²×C₅	1	8	yes	complemented
218	#890	C₂²×C₅	1	8	yes	complemented
219	#884	C₂²×C₅	1	16	yes	complemented
220	#883	C₂²×C₅	1	16	yes	complemented
221	C₄×D₁₂	C₂²×C₅	1	8	yes	complemented
222	#872	C₂²×C₅	1	8	yes	complemented
223	C₂⁴×C₅	—	3	1	no	—
224	C₂⁴×C₅	—	3	2	no	—
225	C₂⁴×C₅	C₂²×D₃	1	2	yes	—
226	C₂×C₅×D₄	—	2	4	no	—
227	C₂×C₅×D₄	—	3	8	no	—
228	C₂×C₅×D₄	—	3	8	no	—
229	C₂×C₅×D₄	—	3	8	no	—
230	C₂×C₅×D₄	—	3	4	no	—
231	C₂×C₅×D₄	—	3	8	no	—
232	C₂×C₅×D₄	—	6	4	no	—
233	C₂×C₅×D₄	—	12	4	no	—
234	C₂×C₅×D₄	—	12	4	no	—
235	C₂×C₅×D₄	—	12	4	no	—
236	C₂×C₅×D₄	—	12	4	no	—
237	C₂×C₅×D₄	C₂²×D₃	1	4	yes	—
238	C₂²×C₄×C₅	—	3	1	no	—
239	C₂²×C₄×C₅	—	3	1	no	—
240	C₂²×C₄×C₅	—	3	1	no	—
241	C₂²×C₄×C₅	—	3	2	no	—
242	C₂²×C₄×C₅	—	3	4	no	—
243	C₂²×C₄×C₅	—	3	8	no	—
244	C₂²×C₄×C₅	—	4	4	no	—
245	C₂²×C₄×C₅	—	6	4	no	—
246	C₂²×C₄×C₅	—	12	4	no	—
247	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
248	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
249	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
250	C₂²×C₄×C₅	C₂²×D₃	1	8	yes	—
251	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
252	C₅×C₄⋊C₄	—	3	4	no	—
253	C₅×C₄⋊C₄	—	3	4	no	—
254	C₅×C₄⋊C₄	—	3	4	no	—
255	C₅×C₄⋊C₄	—	3	8	no	—
256	C₅×C₄⋊C₄	—	3	8	no	—
257	C₅×C₄⋊C₄	C₂²×D₃	1	4	yes	—
258	C₅×C₂²⋊C₄	—	3	8	no	—
259	C₅×C₂²⋊C₄	—	3	4	no	—
260	C₅×C₂²⋊C₄	—	3	4	no	—
261	C₅×C₂²⋊C₄	—	3	8	no	—
262	C₅×C₂²⋊C₄	—	3	8	no	—
263	C₅×C₂²⋊C₄	C₂²×D₃	1	8	yes	—
264	C₄²×C₅	—	3	4	no	—
265	C₄²×C₅	C₂²×D₃	1	4	yes	—
266	#540	—	3	16	no	—
267	C₂×#127	—	3	1	no	—
268	C₂×#125	—	3	1	no	—
269	C₂×#123	—	3	2	no	—
270	C₂×#123	—	3	2	no	—
271	C₂×#121	—	3	1	no	—
272	C₂×#121	—	3	2	no	—
273	C₂×#121	—	3	1	no	—
274	C₂×#121	—	3	2	no	—
275	C₂×C₄×D₄	—	3	1	no	—
276	C₂×C₄×D₄	C₅×D₃	1	1	yes	characteristic, complemented
277	C₂×#117	—	3	1	no	—
278	C₂²×C₃×C₅	—	2	4	no	—
279	C₂²×C₃×C₅	—	2	4	no	—
280	C₂²×C₃×C₅	—	2	4	no	—
281	C₂²×C₃×C₅	—	4	8	no	—
282	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
283	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
284	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
285	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
286	C₂×C₅×D₃	—	2	2	no	—
287	C₂×C₅×D₃	—	2	4	no	—
288	C₂×C₅×D₃	—	2	4	no	—
289	C₂×C₅×D₃	—	2	2	no	—
290	C₂×C₅×D₃	—	2	4	no	—
291	C₂×C₅×D₃	—	2	2	no	—
292	C₂×C₅×D₃	—	4	4	no	—
293	C₂×C₅×D₃	—	4	8	no	—
294	C₃×C₄×C₅	—	2	2	no	—
295	C₃×C₄×C₅	—	2	4	no	—
296	C₃×C₄×C₅	—	2	2	no	—
297	C₃×C₄×C₅	C₂×#41	1	4	yes	—
298	C₅×Dic₃	—	2	2	no	—
299	C₅×Dic₃	—	2	4	no	—
300	C₅×Dic₃	—	2	2	no	—
301	C₅×Dic₃	—	2	2	no	—
302	C₂⁴×C₃	C₂³×C₅	1	2	yes	—
303	C₂³×D₃	C₂³×C₅	1	1	yes	characteristic
304	C₂³×D₃	C₂³×C₅	1	2	yes	—
305	C₂×C₃×D₄	—	2	4	no	—
306	C₂×C₃×D₄	C₂³×C₅	1	4	yes	—
307	C₂²×C₃×C₄	—	4	4	no	—
308	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
309	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
310	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
311	C₂²×C₃×C₄	C₂³×C₅	1	8	yes	—
312	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
313	C₂×#67	—	4	4	no	—
314	C₂×#67	—	4	4	no	—
315	C₂×#67	—	4	4	no	—
316	C₂×#67	C₂³×C₅	1	8	yes	—
317	C₂×#67	C₂³×C₅	1	8	yes	—
318	C₂×#67	C₂³×C₅	1	8	yes	—
319	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
320	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
321	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
322	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
323	C₂×D₁₂	—	2	4	no	—
324	C₂×D₁₂	—	4	4	no	—
325	C₂×D₁₂	C₂³×C₅	1	4	yes	—
326	C₂×D₁₂	C₂³×C₅	1	8	yes	—
327	C₂×C₄×D₃	—	2	4	no	—
328	C₂×C₄×D₃	—	4	4	no	—
329	C₂×C₄×D₃	C₂³×C₅	1	4	yes	—
330	C₂×C₄×D₃	C₂³×C₅	1	8	yes	—
331	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
332	C₃×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
333	C₃×C₄²	C₂³×C₅	1	4	yes	—
334	#217	C₂³×C₅	1	8	yes	—
335	(C₂×D₃)⋊C₄	C₂³×C₅	1	4	yes	—
336	(C₂×D₃)⋊C₄	C₂³×C₅	1	4	yes	—
337	(C₂×D₃)⋊C₄	C₂³×C₅	1	8	yes	—
338	(C₂×D₃)⋊C₄	C₂³×C₅	1	8	yes	—
339	C₄⋊Dic₃	C₂³×C₅	1	4	yes	—
340	C₄⋊Dic₃	C₂³×C₅	1	8	yes	—
341	Dic₃⋊C₄	C₂³×C₅	1	4	yes	—
342	Dic₃⋊C₄	C₂³×C₅	1	4	yes	—
343	Dic₃⋊C₄	C₂³×C₅	1	8	yes	—
344	C₄×Dic₃	C₂³×C₅	1	4	yes	—
345	C₂³×C₅	—	2	4	no	—
346	C₂³×C₅	—	2	4	no	—
347	C₂³×C₅	—	2	4	no	—
348	C₂³×C₅	—	3	2	no	—
349	C₂³×C₅	—	3	4	no	—
350	C₂³×C₅	—	6	2	no	—
351	C₂³×C₅	—	6	4	no	—
352	C₂³×C₅	—	6	2	no	—
353	C₂³×C₅	—	6	4	no	—
354	C₂³×C₅	—	6	4	no	—
355	C₂³×C₅	—	6	2	no	—
356	C₂³×C₅	C₂³×D₃	1	4	yes	—
357	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
358	C₅×D₄	—	2	8	no	—
359	C₅×D₄	—	6	8	no	—
360	C₅×D₄	—	12	8	no	—
361	C₅×D₄	—	12	8	no	—
362	C₅×D₄	—	12	8	no	—
363	C₅×D₄	—	12	8	no	—
364	C₂×C₄×C₅	—	2	2	no	—
365	C₂×C₄×C₅	—	2	4	no	—
366	C₂×C₄×C₅	—	2	2	no	—
367	C₂×C₄×C₅	—	3	2	no	—
368	C₂×C₄×C₅	—	3	2	no	—
369	C₂×C₄×C₅	—	3	2	no	—
370	C₂×C₄×C₅	—	3	4	no	—
371	C₂×C₄×C₅	—	4	8	no	—
372	C₂×C₄×C₅	—	6	2	no	—
373	C₂×C₄×C₅	—	6	2	no	—
374	C₂×C₄×C₅	—	6	4	no	—
375	C₂×C₄×C₅	—	6	2	no	—
376	C₂×C₄×C₅	—	6	8	no	—
377	C₂×C₄×C₅	—	12	8	no	—
378	C₂×C₄×C₅	#235	1	4	yes	—
379	C₂×C₄×C₅	C₂³×D₃	1	2	yes	—
380	C₂×C₄×C₅	C₂³×D₃	1	2	yes	—
381	C₂×C₄×C₅	C₂³×D₃	1	4	yes	—
382	C₂×C₄×C₅	C₂³×D₃	1	2	yes	—
383	C₂²×D₄	—	3	2	no	—
384	C₂²×D₄	—	3	2	no	—
385	C₂²×D₄	—	3	2	no	—
386	C₂²×D₄	—	3	1	no	—
387	C₂²×D₄	—	3	2	no	—
388	C₂²×D₄	C₂×C₅×D₃	1	1	yes	characteristic
389	C₂³×C₄	—	3	1	no	—
390	C₂³×C₄	—	3	2	no	—
391	C₂³×C₄	C₂×C₅×D₃	1	2	yes	—
392	#127	—	3	8	no	—
393	#125	—	3	8	no	—
394	#123	—	3	16	no	—
395	#123	—	3	16	no	—
396	#121	—	3	8	no	—
397	#121	—	3	16	no	—
398	#121	—	3	8	no	—
399	#121	—	3	16	no	—
400	C₄×D₄	—	3	8	no	—
401	C₄×D₄	C₂×C₅×D₃	1	8	yes	complemented
402	#117	—	3	8	no	—
403	C₂×C₄⋊C₄	—	3	1	no	—
404	C₂×C₄⋊C₄	—	3	1	no	—
405	C₂×C₄⋊C₄	—	3	1	no	—
406	C₂×C₄⋊C₄	—	3	2	no	—
407	C₂×C₄⋊C₄	—	3	2	no	—
408	C₂×C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
409	C₂×C₂²⋊C₄	—	3	2	no	—
410	C₂×C₂²⋊C₄	—	3	1	no	—
411	C₂×C₂²⋊C₄	—	3	1	no	—
412	C₂×C₂²⋊C₄	—	3	2	no	—
413	C₂×C₂²⋊C₄	—	3	2	no	—
414	C₂×C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
415	C₂×C₄²	—	3	1	no	—
416	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
417	C₂×C₃×C₅	—	4	4	no	—
418	C₂×C₃×C₅	#540	1	4	yes	complemented
419	C₂×C₃×C₅	C₂²×#41	1	1	yes	characteristic
420	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
421	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
422	C₅×D₃	—	4	2	no	—
423	C₅×D₃	—	4	4	no	—
424	C₂³×C₃	—	2	4	no	—
425	C₂³×C₃	—	2	4	no	—
426	C₂³×C₃	—	2	4	no	—
427	C₂³×C₃	C₂⁴×C₅	1	4	yes	—
428	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
429	C₂²×D₃	—	2	2	no	—
430	C₂²×D₃	—	2	4	no	—
431	C₂²×D₃	—	2	4	no	—
432	C₂²×D₃	—	2	2	no	—
433	C₂²×D₃	—	2	4	no	—
434	C₂²×D₃	—	2	2	no	—
435	C₂²×D₃	C₂⁴×C₅	1	2	yes	—
436	C₂²×D₃	C₂⁴×C₅	1	4	yes	—
437	C₃×D₄	—	2	8	no	—
438	C₂×C₃×C₄	—	2	2	no	—
439	C₂×C₃×C₄	—	2	4	no	—
440	C₂×C₃×C₄	—	2	2	no	—
441	C₂×C₃×C₄	—	4	8	no	—
442	C₂×C₃×C₄	C₅×#41	1	4	yes	—
443	C₂×C₃×C₄	C₂⁴×C₅	1	4	yes	—
444	C₂×C₃×C₄	C₂⁴×C₅	1	2	yes	—
445	C₂×C₃×C₄	C₂⁴×C₅	1	2	yes	—
446	C₂×C₃×C₄	C₂⁴×C₅	1	2	yes	—
447	#67	—	4	8	no	—
448	#67	—	4	8	no	—
449	#67	—	4	8	no	—
450	C₂×Dic₃	—	2	2	no	—
451	C₂×Dic₃	—	2	4	no	—
452	C₂×Dic₃	—	2	2	no	—
453	C₂×Dic₃	—	2	2	no	—
454	C₂×Dic₃	C₂⁴×C₅	1	2	yes	—
455	C₂×Dic₃	C₂⁴×C₅	1	2	yes	—
456	C₂×Dic₃	C₂⁴×C₅	1	2	yes	—
457	C₂×Dic₃	C₂⁴×C₅	1	4	yes	—
458	D₁₂	—	2	8	no	—
459	D₁₂	—	4	8	no	—
460	C₄×D₃	—	2	8	no	—
461	C₄×D₃	—	4	8	no	—
462	C₂²×C₅	—	2	4	no	—
463	C₂²×C₅	—	2	4	no	—
464	C₂²×C₅	—	2	4	no	—
465	C₂²×C₅	—	4	8	no	—
466	C₂²×C₅	—	6	2	no	—
467	C₂²×C₅	—	6	4	no	—
468	C₂²×C₅	—	6	2	no	—
469	C₂²×C₅	—	6	4	no	—
470	C₂²×C₅	—	6	4	no	—
471	C₂²×C₅	—	6	2	no	—
472	C₂²×C₅	—	12	4	no	—
473	C₂²×C₅	—	12	8	no	—
474	C₂²×C₅	C₂×#235	1	2	yes	—
475	C₂²×C₅	#1004	1	2	yes	—
476	C₂²×C₅	#1009	1	2	yes	—
477	C₂²×C₅	C₂⁴×D₃	1	1	yes	characteristic
478	C₄×C₅	—	2	2	no	—
479	C₄×C₅	—	2	4	no	—
480	C₄×C₅	—	2	2	no	—
481	C₄×C₅	—	6	2	no	—
482	C₄×C₅	—	6	2	no	—
483	C₄×C₅	—	6	4	no	—
484	C₄×C₅	—	6	2	no	—
485	C₄×C₅	C₂×#235	1	4	yes	—
486	C₂⁴	—	3	1	no	—
487	C₂⁴	—	3	2	no	—
488	C₂⁴	C₂²×C₅×D₃	1	2	yes	—
489	C₂×D₄	—	2	4	no	—
490	C₂×D₄	—	3	8	no	—
491	C₂×D₄	—	3	8	no	—
492	C₂×D₄	—	3	8	no	—
493	C₂×D₄	—	3	4	no	—
494	C₂×D₄	—	3	8	no	—
495	C₂×D₄	—	6	4	no	—
496	C₂×D₄	—	12	4	no	—
497	C₂×D₄	—	12	4	no	—
498	C₂×D₄	—	12	4	no	—
499	C₂×D₄	—	12	4	no	—
500	C₂×D₄	C₂²×C₅×D₃	1	4	yes	—
501	C₂²×C₄	—	3	1	no	—
502	C₂²×C₄	—	3	1	no	—
503	C₂²×C₄	—	3	1	no	—
504	C₂²×C₄	—	3	2	no	—
505	C₂²×C₄	—	3	4	no	—
506	C₂²×C₄	—	3	8	no	—
507	C₂²×C₄	—	4	4	no	—
508	C₂²×C₄	—	6	4	no	—
509	C₂²×C₄	—	12	4	no	—
510	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
511	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
512	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
513	C₂²×C₄	C₂²×C₅×D₃	1	8	yes	—
514	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
515	C₄⋊C₄	—	3	4	no	—
516	C₄⋊C₄	—	3	4	no	—
517	C₄⋊C₄	—	3	4	no	—
518	C₄⋊C₄	—	3	8	no	—
519	C₄⋊C₄	—	3	8	no	—
520	C₄⋊C₄	C₂²×C₅×D₃	1	4	yes	—
521	C₂²⋊C₄	—	3	8	no	—
522	C₂²⋊C₄	—	3	4	no	—
523	C₂²⋊C₄	—	3	4	no	—
524	C₂²⋊C₄	—	3	8	no	—
525	C₂²⋊C₄	—	3	8	no	—
526	C₂²⋊C₄	C₂²×C₅×D₃	1	8	yes	—
527	C₄²	—	3	4	no	—
528	C₄²	C₂²×C₅×D₃	1	4	yes	—
529	C₃×C₅	C₂×#540	1	1	yes	characteristic, complemented, Hall
530	C₂²×C₃	—	2	4	no	—
531	C₂²×C₃	—	2	4	no	—
532	C₂²×C₃	—	2	4	no	—
533	C₂²×C₃	—	4	8	no	—
534	C₂²×C₃	C₂×C₅×#41	1	2	yes	—
535	C₂²×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
536	C₂²×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
537	C₂²×C₃	C₂⁵×C₅	1	1	yes	characteristic
538	C₂×D₃	—	2	2	no	—
539	C₂×D₃	—	2	4	no	—
540	C₂×D₃	—	2	4	no	—
541	C₂×D₃	—	2	2	no	—
542	C₂×D₃	—	2	4	no	—
543	C₂×D₃	—	2	2	no	—
544	C₂×D₃	—	4	4	no	—
545	C₂×D₃	—	4	8	no	—
546	C₃×C₄	—	2	2	no	—
547	C₃×C₄	—	2	4	no	—
548	C₃×C₄	—	2	2	no	—
549	C₃×C₄	C₂×C₅×#41	1	4	yes	—
550	Dic₃	—	2	2	no	—
551	Dic₃	—	2	4	no	—
552	Dic₃	—	2	2	no	—
553	Dic₃	—	2	2	no	—
554	C₂×C₅	—	4	4	no	—
555	C₂×C₅	—	12	2	no	—
556	C₂×C₅	—	12	4	no	—
557	C₂×C₅	#5562	1	4	yes	direct product factor
558	C₂×C₅	C₂²×#235	1	1	yes	characteristic
559	C₂×C₅	C₂×#1004	1	1	yes	characteristic
560	C₂×C₅	C₂×#1009	1	1	yes	characteristic
561	C₂³	—	2	4	no	—
562	C₂³	—	2	4	no	—
563	C₂³	—	2	4	no	—
564	C₂³	—	3	2	no	—
565	C₂³	—	3	4	no	—
566	C₂³	—	6	2	no	—
567	C₂³	—	6	4	no	—
568	C₂³	—	6	2	no	—
569	C₂³	—	6	4	no	—
570	C₂³	—	6	4	no	—
571	C₂³	—	6	2	no	—
572	C₂³	C₂³×C₅×D₃	1	4	yes	—
573	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
574	D₄	—	2	8	no	—
575	D₄	—	6	8	no	—
576	D₄	—	12	8	no	—
577	D₄	—	12	8	no	—
578	D₄	—	12	8	no	—
579	D₄	—	12	8	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	2	4	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	3	2	no	—
584	C₂×C₄	—	3	2	no	—
585	C₂×C₄	—	3	2	no	—
586	C₂×C₄	—	3	4	no	—
587	C₂×C₄	—	4	8	no	—
588	C₂×C₄	—	6	2	no	—
589	C₂×C₄	—	6	2	no	—
590	C₂×C₄	—	6	4	no	—
591	C₂×C₄	—	6	2	no	—
592	C₂×C₄	—	6	8	no	—
593	C₂×C₄	—	12	8	no	—
594	C₂×C₄	C₅×#235	1	4	yes	—
595	C₂×C₄	C₂³×C₅×D₃	1	2	yes	—
596	C₂×C₄	C₂³×C₅×D₃	1	2	yes	—
597	C₂×C₄	C₂³×C₅×D₃	1	4	yes	—
598	C₂×C₄	C₂³×C₅×D₃	1	2	yes	—
599	C₂×C₃	—	4	4	no	—
600	C₂×C₃	C₅×#540	1	4	yes	complemented
601	C₂×C₃	C₂²×C₅×#41	1	1	yes	characteristic
602	C₂×C₃	C₂×C₅×H(5,𝔽₂)	1	1	yes	characteristic
603	C₂×C₃	C₂×C₅×H(5,𝔽₂)	1	1	yes	characteristic
604	D₃	—	4	2	no	—
605	D₃	—	4	4	no	—
606	C₅	C₂×#5562	1	1	yes	characteristic, direct product factor, Sylow
607	C₂²	—	2	4	no	—
608	C₂²	—	2	4	no	—
609	C₂²	—	2	4	no	—
610	C₂²	—	4	8	no	—
611	C₂²	—	6	2	no	—
612	C₂²	—	6	4	no	—
613	C₂²	—	6	2	no	—
614	C₂²	—	6	4	no	—
615	C₂²	—	6	4	no	—
616	C₂²	—	6	2	no	—
617	C₂²	—	12	4	no	—
618	C₂²	—	12	8	no	—
619	C₂²	C₂×C₅×#235	1	2	yes	—
620	C₂²	C₅×#1004	1	2	yes	—
621	C₂²	C₅×#1009	1	2	yes	—
622	C₂²	C₂⁴×C₅×D₃	1	1	yes	characteristic
623	C₄	—	2	2	no	—
624	C₄	—	2	4	no	—
625	C₄	—	2	2	no	—
626	C₄	—	6	2	no	—
627	C₄	—	6	2	no	—
628	C₄	—	6	4	no	—
629	C₄	—	6	2	no	—
630	C₄	C₂×C₅×#235	1	4	yes	—
631	C₃	C₂×C₅×#540	1	1	yes	characteristic, complemented, Sylow
632	C₂	—	4	4	no	—
633	C₂	—	12	2	no	—
634	C₂	—	12	4	no	—
635	C₂	C₅×#5562	1	4	yes	direct product factor
636	C₂	C₂²×C₅×#235	1	1	yes	characteristic
637	C₂	C₂×C₅×#1004	1	1	yes	characteristic
638	C₂	C₂×C₅×#1009	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁵×C₅
Frattini	C₂²	C₂⁴×C₅×D₃
Fitting	C₂×C₃×C₄×C₅×D₄	C₂
socle	C₂³×C₃×C₅	C₂⁴
hypercenter	C₂×C₄×C₅×D₄	D₃
hypocenter	C₃	C₂×C₅×#540
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄×C₅×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₄×C₅×D₄	C₂	semidirect
2. C₂×C₅×#940	C₂	semidirect
3. C₂×C₅×#938	C₂	semidirect
4. C₂×C₅×#930	C₂	semidirect
5. C₂×C₅×#929	C₂	semidirect
6. C₅×#5562	C₂	direct (N × H)
7. C₂×C₅×#895	C₂	semidirect
8. C₂×C₅×#890	C₂	semidirect
9. C₂×C₅×#884	C₂	semidirect
10. C₂×C₅×#883	C₂	semidirect
11. C₂×C₄×C₅×D₁₂	C₂	semidirect
12. C₂×C₅×#872	C₂	semidirect
23. C₃×C₄×C₅×D₄	C₂²	semidirect
28. C₅×#940	C₂²	semidirect
29. C₅×#938	C₂²	semidirect
30. C₅×#930	C₂²	semidirect
31. C₅×#929	C₂²	semidirect
42. C₅×#895	C₂²	semidirect
43. C₅×#890	C₂²	semidirect
44. C₅×#884	C₂²	semidirect
45. C₅×#883	C₂²	semidirect
46. C₄×C₅×D₁₂	C₂²	semidirect
47. C₅×#872	C₂²	semidirect
48. C₂×#5562	C₅	direct (N × H)
59. C₂×C₄×C₅×D₄	D₃	semidirect
104. C₂×C₃×C₄×D₄	C₂×C₅	semidirect
105. C₂×#940	C₂×C₅	semidirect
106. C₂×#938	C₂×C₅	semidirect
107. C₂×#930	C₂×C₅	semidirect
108. C₂×#929	C₂×C₅	semidirect
109. #5562	C₂×C₅	direct (N × H)
110. C₂×#895	C₂×C₅	semidirect
111. C₂×#890	C₂×C₅	semidirect
112. C₂×#884	C₂×C₅	semidirect
113. C₂×#883	C₂×C₅	semidirect
114. C₂×C₄×D₁₂	C₂×C₅	semidirect
115. C₂×#872	C₂×C₅	semidirect
134. C₄×C₅×D₄	C₂×D₃	semidirect
198. C₃×C₄×D₄	C₂²×C₅	semidirect
203. #940	C₂²×C₅	semidirect
204. #938	C₂²×C₅	semidirect
205. #930	C₂²×C₅	semidirect
206. #929	C₂²×C₅	semidirect
217. #895	C₂²×C₅	semidirect
218. #890	C₂²×C₅	semidirect
219. #884	C₂²×C₅	semidirect
220. #883	C₂²×C₅	semidirect
221. C₄×D₁₂	C₂²×C₅	semidirect
222. #872	C₂²×C₅	semidirect
276. C₂×C₄×D₄	C₅×D₃	semidirect
401. C₄×D₄	C₂×C₅×D₃	semidirect
418. C₂×C₃×C₅	#540	semidirect
529. C₃×C₅	C₂×#540	semidirect
557. C₂×C₅	#5562	direct (N × H)
600. C₂×C₃	C₅×#540	semidirect
606. C₅	C₂×#5562	direct (N × H)
631. C₃	C₂×C₅×#540	semidirect
635. C₂	C₅×#5562	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₅×D₄	2
C₂×C₅×#940	2
C₂×C₅×#938	2
C₂×C₅×#930	2
C₂×C₅×#929	2
C₅×#5562	2
C₂×C₅×#895	2
C₂×C₅×#890	2
C₂×C₅×#884	2
C₂×C₅×#883	2
C₂×C₄×C₅×D₁₂	2
C₂×C₅×#872	2
C₂×C₅×#540	3
C₂×#5562	5

Subgroups — C2×C5×#5562 #14481082

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#5562 #14481082