Subgroups — C₂×C₅×#5616 #14483919

Order 1920.

2480 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#121	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#939	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×#938	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#936	C₂	1	2	yes	maximal, complemented
5	C₂×C₅×D₄×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×C₅×#67	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#926	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#5616	C₂	1	16	yes	maximal, direct product factor
9	C₂×C₅×#892	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅×#888	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅×#885	C₂	1	2	yes	maximal, complemented
12	C₂×C₅×#877	C₂	1	2	yes	maximal, complemented
13	C₂×C₅×#555	—	3	1	no	maximal, Hall
14	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
16	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
17	C₂²×C₅×#67	C₂²	1	1	yes	characteristic
18	C₂²×C₅×#67	C₂²	1	2	yes	—
19	C₂³×C₅×Dic₃	C₂²	1	2	yes	—
20	C₂²×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
21	C₂²×C₅×Dic₆	C₂²	1	2	yes	—
22	C₃×C₅×#121	C₂²	1	8	yes	complemented
23	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂×C₅×#217	C₂²	1	2	yes	—
26	C₂×C₅×#217	C₂²	1	2	yes	—
27	C₂×C₅×#217	C₂²	1	2	yes	—
28	C₂×C₅×#217	C₂²	1	1	yes	characteristic
29	C₅×#939	C₂²	1	16	yes	complemented
30	C₅×#938	C₂²	1	8	yes	complemented
31	C₅×#936	C₂²	1	16	yes	complemented
32	C₅×D₄×Dic₃	C₂²	1	8	yes	complemented
33	C₄×C₅×#67	C₂²	1	8	yes	complemented
34	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
35	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
36	C₅×#926	C₂²	1	8	yes	complemented
37	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
38	C₂×C₅×C₄⋊Dic₃	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	2	yes	—
42	C₂×C₄×C₅×Dic₃	C₂²	1	2	yes	—
43	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
44	C₅×#892	C₂²	1	8	yes	complemented
45	C₅×#888	C₂²	1	8	yes	complemented
46	C₅×#885	C₂²	1	16	yes	complemented
47	C₅×#877	C₂²	1	16	yes	complemented
48	C₂×#5616	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
49	C₅×#555	—	3	16	no	—
50	C₂×C₅×C₄⋊Q₈	—	3	1	no	—
51	C₂×C₅×#124	—	3	2	no	—
52	C₂×C₅×#124	—	3	2	no	—
53	C₂×C₅×#121	—	3	1	no	—
54	C₂×C₅×#121	—	3	2	no	—
55	C₂×C₅×#121	D₃	1	1	yes	characteristic, complemented
56	C₂×C₄×C₅×D₄	—	3	1	no	—
57	C₂×C₄×C₅×D₄	—	3	1	no	—
58	C₂×C₅×#117	—	3	1	no	—
59	C₂×C₅×#117	—	3	1	no	—
60	C₂×C₅×#117	—	3	2	no	—
61	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
62	C₂⁴×C₃×C₅	C₂³	1	2	yes	—
63	C₂³×C₅×D₃	C₂³	1	1	yes	characteristic
64	C₂×C₃×C₅×D₄	—	2	4	no	—
65	C₂×C₃×C₅×D₄	—	4	4	no	—
66	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
67	C₂×C₃×C₅×D₄	C₂³	1	8	yes	—
68	C₂²×C₃×C₄×C₅	—	2	4	no	—
69	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
70	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
71	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
72	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
73	C₂×C₅×#67	—	2	4	no	—
74	C₂×C₅×#67	—	4	4	no	—
75	C₂×C₅×#67	C₂³	1	4	yes	—
76	C₂×C₅×#67	C₂³	1	8	yes	—
77	C₂²×C₅×Dic₃	—	2	8	no	—
78	C₂²×C₅×Dic₃	C₂³	1	8	yes	—
79	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
80	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
81	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
82	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
83	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
84	C₂×C₄×C₅×D₃	—	2	4	no	—
85	C₂×C₄×C₅×D₃	C₂³	1	4	yes	—
86	C₂×C₅×Dic₆	—	4	4	no	—
87	C₂×C₅×Dic₆	C₂³	1	8	yes	—
88	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
89	C₃×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
90	C₅×#217	C₂³	1	8	yes	—
91	C₅×#217	C₂³	1	8	yes	—
92	C₅×#217	C₂³	1	8	yes	—
93	C₅×#217	C₂³	1	4	yes	—
94	C₅×(C₂×D₃)⋊C₄	C₂³	1	4	yes	—
95	C₅×(C₂×D₃)⋊C₄	C₂³	1	8	yes	—
96	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
97	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
98	C₅×Dic₃⋊C₄	C₂³	1	4	yes	—
99	C₅×Dic₃⋊C₄	C₂³	1	8	yes	—
100	C₄×C₅×Dic₃	C₂³	1	8	yes	—
101	C₄×C₅×Dic₃	C₂³	1	8	yes	—
102	C₄×C₅×Dic₃	C₂³	1	4	yes	—
103	C₂×C₃×#121	C₂×C₅	1	1	yes	characteristic, complemented
104	C₂×#939	C₂×C₅	1	2	yes	complemented
105	C₂×#938	C₂×C₅	1	1	yes	characteristic, complemented
106	C₂×#936	C₂×C₅	1	2	yes	complemented
107	C₂×D₄×Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
108	C₂×C₄×#67	C₂×C₅	1	1	yes	characteristic, complemented
109	C₂×#926	C₂×C₅	1	1	yes	characteristic, complemented
110	#5616	C₂×C₅	1	16	yes	direct product factor
111	C₂×#892	C₂×C₅	1	1	yes	characteristic, complemented
112	C₂×#888	C₂×C₅	1	1	yes	characteristic, complemented
113	C₂×#885	C₂×C₅	1	2	yes	complemented
114	C₂×#877	C₂×C₅	1	2	yes	complemented
115	C₂²×C₅×Q₈	—	3	2	no	—
116	C₂²×C₅×D₄	—	3	1	no	—
117	C₂²×C₅×D₄	—	3	2	no	—
118	C₂²×C₅×D₄	C₂×D₃	1	1	yes	characteristic
119	C₂²×C₅×D₄	C₂×D₃	1	2	yes	—
120	C₂³×C₄×C₅	—	3	2	no	—
121	C₂³×C₄×C₅	—	3	1	no	—
122	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
123	C₅×C₄⋊Q₈	—	3	8	no	—
124	C₅×#124	—	3	16	no	—
125	C₅×#124	—	3	16	no	—
126	C₅×#121	—	3	8	no	—
127	C₅×#121	—	3	16	no	—
128	C₅×#121	C₂×D₃	1	8	yes	complemented
129	C₄×C₅×D₄	—	3	8	no	—
130	C₄×C₅×D₄	—	3	8	no	—
131	C₅×#117	—	3	8	no	—
132	C₅×#117	—	3	8	no	—
133	C₅×#117	—	3	16	no	—
134	C₂×C₅×C₄⋊C₄	—	3	1	no	—
135	C₂×C₅×C₄⋊C₄	—	3	2	no	—
136	C₂×C₅×C₄⋊C₄	—	3	1	no	—
137	C₂×C₅×C₄⋊C₄	—	3	1	no	—
138	C₂×C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
139	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
140	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
141	C₂×C₅×C₂²⋊C₄	—	3	1	no	—
142	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
143	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
144	C₂×C₅×C₂²⋊C₄	—	3	1	no	—
145	C₂×C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
146	C₂×C₄²×C₅	—	3	2	no	—
147	C₂×C₄²×C₅	—	3	2	no	—
148	C₂×C₄²×C₅	—	3	1	no	—
149	C₂×#555	—	3	1	no	Sylow
150	C₂³×C₃×C₅	—	2	4	no	—
151	C₂³×C₃×C₅	—	2	2	no	—
152	C₂³×C₃×C₅	—	2	2	no	—
153	C₂³×C₃×C₅	—	2	4	no	—
154	C₂³×C₃×C₅	—	2	4	no	—
155	C₂³×C₃×C₅	—	2	2	no	—
156	C₂³×C₃×C₅	C₂⁴	1	2	yes	—
157	C₂³×C₃×C₅	C₂⁴	1	4	yes	—
158	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
159	C₂²×C₅×D₃	—	2	2	no	—
160	C₂²×C₅×D₃	—	2	2	no	—
161	C₂²×C₅×D₃	—	2	2	no	—
162	C₂²×C₅×D₃	C₂⁴	1	2	yes	—
163	C₃×C₅×D₄	—	2	8	no	—
164	C₃×C₅×D₄	—	4	8	no	—
165	C₂×C₃×C₄×C₅	—	2	2	no	—
166	C₂×C₃×C₄×C₅	—	2	4	no	—
167	C₂×C₃×C₄×C₅	—	2	8	no	—
168	C₂×C₃×C₄×C₅	#41	1	4	yes	—
169	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
170	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
171	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
172	C₅×#67	—	2	8	no	—
173	C₅×#67	—	4	8	no	—
174	C₂×C₅×Dic₃	—	2	2	no	—
175	C₂×C₅×Dic₃	—	2	4	no	—
176	C₂×C₅×Dic₃	—	2	2	no	—
177	C₂×C₅×Dic₃	—	2	4	no	—
178	C₂×C₅×Dic₃	—	2	16	no	—
179	C₂×C₅×Dic₃	#41	1	4	yes	—
180	C₂×C₅×Dic₃	C₂⁴	1	2	yes	—
181	C₂×C₅×Dic₃	C₂⁴	1	4	yes	—
182	C₂×C₅×Dic₃	C₂⁴	1	4	yes	—
183	C₂×C₅×Dic₃	C₂⁴	1	2	yes	—
184	C₂×C₅×Dic₃	C₂⁴	1	2	yes	—
185	C₄×C₅×D₃	—	2	8	no	—
186	C₅×Dic₆	—	4	8	no	—
187	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
188	C₂²×C₃×D₄	C₂²×C₅	1	2	yes	—
189	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
190	C₂²×#67	C₂²×C₅	1	1	yes	characteristic
191	C₂²×#67	C₂²×C₅	1	2	yes	—
192	C₂³×Dic₃	C₂²×C₅	1	2	yes	—
193	C₂²×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
194	C₂²×Dic₆	C₂²×C₅	1	2	yes	—
195	C₃×#121	C₂²×C₅	1	8	yes	complemented
196	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
197	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
198	C₂×#217	C₂²×C₅	1	2	yes	—
199	C₂×#217	C₂²×C₅	1	2	yes	—
200	C₂×#217	C₂²×C₅	1	2	yes	—
201	C₂×#217	C₂²×C₅	1	1	yes	characteristic
202	#939	C₂²×C₅	1	16	yes	complemented
203	#938	C₂²×C₅	1	8	yes	complemented
204	#936	C₂²×C₅	1	16	yes	complemented
205	D₄×Dic₃	C₂²×C₅	1	8	yes	complemented
206	C₄×#67	C₂²×C₅	1	8	yes	complemented
207	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	1	yes	characteristic
208	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
209	#926	C₂²×C₅	1	8	yes	complemented
210	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
211	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
212	C₂×Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
213	C₂×Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
214	C₂×C₄×Dic₃	C₂²×C₅	1	2	yes	—
215	C₂×C₄×Dic₃	C₂²×C₅	1	2	yes	—
216	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
217	#892	C₂²×C₅	1	8	yes	complemented
218	#888	C₂²×C₅	1	8	yes	complemented
219	#885	C₂²×C₅	1	16	yes	complemented
220	#877	C₂²×C₅	1	16	yes	complemented
221	C₂⁴×C₅	—	3	1	no	—
222	C₂⁴×C₅	C₂²×D₃	1	1	yes	characteristic
223	C₂⁴×C₅	C₂²×D₃	1	2	yes	—
224	C₂×C₅×Q₈	—	3	8	no	—
225	C₂×C₅×Q₈	—	12	4	no	—
226	C₂×C₅×D₄	—	2	4	no	—
227	C₂×C₅×D₄	—	3	4	no	—
228	C₂×C₅×D₄	—	3	8	no	—
229	C₂×C₅×D₄	—	4	4	no	—
230	C₂×C₅×D₄	—	6	4	no	—
231	C₂×C₅×D₄	—	12	4	no	—
232	C₂×C₅×D₄	C₂²×D₃	1	4	yes	—
233	C₂×C₅×D₄	C₂²×D₃	1	8	yes	—
234	C₂²×C₄×C₅	—	2	4	no	—
235	C₂²×C₄×C₅	—	3	1	no	—
236	C₂²×C₄×C₅	—	3	8	no	—
237	C₂²×C₄×C₅	—	3	1	no	—
238	C₂²×C₄×C₅	—	3	2	no	—
239	C₂²×C₄×C₅	—	3	1	no	—
240	C₂²×C₄×C₅	—	3	2	no	—
241	C₂²×C₄×C₅	—	3	4	no	—
242	C₂²×C₄×C₅	—	6	8	no	—
243	C₂²×C₄×C₅	—	6	4	no	—
244	C₂²×C₄×C₅	C₂²×D₃	1	4	yes	—
245	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
246	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
247	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
248	C₅×C₄⋊C₄	—	3	4	no	—
249	C₅×C₄⋊C₄	—	3	4	no	—
250	C₅×C₄⋊C₄	—	3	8	no	—
251	C₅×C₄⋊C₄	—	3	4	no	—
252	C₅×C₄⋊C₄	C₂²×D₃	1	4	yes	—
253	C₅×C₂²⋊C₄	—	3	8	no	—
254	C₅×C₂²⋊C₄	—	3	8	no	—
255	C₅×C₂²⋊C₄	—	3	4	no	—
256	C₅×C₂²⋊C₄	—	3	8	no	—
257	C₅×C₂²⋊C₄	—	3	8	no	—
258	C₅×C₂²⋊C₄	—	3	4	no	—
259	C₅×C₂²⋊C₄	C₂²×D₃	1	8	yes	—
260	C₄²×C₅	—	3	8	no	—
261	C₄²×C₅	—	3	8	no	—
262	C₄²×C₅	—	3	4	no	—
263	#555	—	3	16	no	—
264	C₂×C₄⋊Q₈	—	3	1	no	—
265	C₂×#124	—	3	2	no	—
266	C₂×#124	—	3	2	no	—
267	C₂×#121	—	3	1	no	—
268	C₂×#121	—	3	2	no	—
269	C₂×#121	C₅×D₃	1	1	yes	characteristic, complemented
270	C₂×C₄×D₄	—	3	1	no	—
271	C₂×C₄×D₄	—	3	1	no	—
272	C₂×#117	—	3	1	no	—
273	C₂×#117	—	3	1	no	—
274	C₂×#117	—	3	2	no	—
275	C₂²×C₃×C₅	—	2	4	no	—
276	C₂²×C₃×C₅	—	2	2	no	—
277	C₂²×C₃×C₅	—	2	2	no	—
278	C₂²×C₃×C₅	—	2	4	no	—
279	C₂²×C₃×C₅	—	2	4	no	—
280	C₂²×C₃×C₅	—	2	2	no	—
281	C₂²×C₃×C₅	—	4	8	no	—
282	C₂²×C₃×C₅	—	4	4	no	—
283	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
284	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
285	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
286	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
287	C₂×C₅×D₃	—	2	2	no	—
288	C₂×C₅×D₃	—	2	2	no	—
289	C₂×C₅×D₃	—	2	2	no	—
290	C₂×C₅×D₃	—	4	4	no	—
291	C₃×C₄×C₅	—	2	2	no	—
292	C₃×C₄×C₅	—	2	4	no	—
293	C₃×C₄×C₅	C₂×#41	1	4	yes	—
294	C₅×Dic₃	—	2	2	no	—
295	C₅×Dic₃	—	2	4	no	—
296	C₅×Dic₃	—	2	2	no	—
297	C₅×Dic₃	—	2	4	no	—
298	C₅×Dic₃	C₂×#41	1	4	yes	—
299	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
300	C₂⁴×C₃	C₂³×C₅	1	2	yes	—
301	C₂³×D₃	C₂³×C₅	1	1	yes	characteristic
302	C₂×C₃×D₄	—	2	4	no	—
303	C₂×C₃×D₄	—	4	4	no	—
304	C₂×C₃×D₄	C₂³×C₅	1	4	yes	—
305	C₂×C₃×D₄	C₂³×C₅	1	8	yes	—
306	C₂²×C₃×C₄	—	2	4	no	—
307	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
308	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
309	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
310	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
311	C₂×#67	—	2	4	no	—
312	C₂×#67	—	4	4	no	—
313	C₂×#67	C₂³×C₅	1	4	yes	—
314	C₂×#67	C₂³×C₅	1	8	yes	—
315	C₂²×Dic₃	—	2	8	no	—
316	C₂²×Dic₃	C₂³×C₅	1	8	yes	—
317	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
318	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
319	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
320	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
321	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
322	C₂×C₄×D₃	—	2	4	no	—
323	C₂×C₄×D₃	C₂³×C₅	1	4	yes	—
324	C₂×Dic₆	—	4	4	no	—
325	C₂×Dic₆	C₂³×C₅	1	8	yes	—
326	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
327	C₃×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
328	#217	C₂³×C₅	1	8	yes	—
329	#217	C₂³×C₅	1	8	yes	—
330	#217	C₂³×C₅	1	8	yes	—
331	#217	C₂³×C₅	1	4	yes	—
332	(C₂×D₃)⋊C₄	C₂³×C₅	1	4	yes	—
333	(C₂×D₃)⋊C₄	C₂³×C₅	1	8	yes	—
334	C₄⋊Dic₃	C₂³×C₅	1	4	yes	—
335	C₄⋊Dic₃	C₂³×C₅	1	4	yes	—
336	Dic₃⋊C₄	C₂³×C₅	1	4	yes	—
337	Dic₃⋊C₄	C₂³×C₅	1	8	yes	—
338	C₄×Dic₃	C₂³×C₅	1	8	yes	—
339	C₄×Dic₃	C₂³×C₅	1	8	yes	—
340	C₄×Dic₃	C₂³×C₅	1	4	yes	—
341	C₂³×C₅	—	2	2	no	—
342	C₂³×C₅	—	2	4	no	—
343	C₂³×C₅	—	2	2	no	—
344	C₂³×C₅	—	2	4	no	—
345	C₂³×C₅	—	2	4	no	—
346	C₂³×C₅	—	2	2	no	—
347	C₂³×C₅	—	3	2	no	—
348	C₂³×C₅	—	6	2	no	—
349	C₂³×C₅	—	6	2	no	—
350	C₂³×C₅	—	6	2	no	—
351	C₂³×C₅	C₂³×D₃	1	2	yes	—
352	C₂³×C₅	C₂³×D₃	1	4	yes	—
353	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
354	C₅×Q₈	—	12	8	no	—
355	C₅×D₄	—	2	8	no	—
356	C₅×D₄	—	4	8	no	—
357	C₅×D₄	—	6	8	no	—
358	C₅×D₄	—	12	8	no	—
359	C₂×C₄×C₅	—	2	2	no	—
360	C₂×C₄×C₅	—	2	4	no	—
361	C₂×C₄×C₅	—	2	8	no	—
362	C₂×C₄×C₅	—	3	4	no	—
363	C₂×C₄×C₅	—	3	2	no	—
364	C₂×C₄×C₅	—	3	2	no	—
365	C₂×C₄×C₅	—	3	4	no	—
366	C₂×C₄×C₅	—	3	2	no	—
367	C₂×C₄×C₅	—	3	4	no	—
368	C₂×C₄×C₅	—	6	16	no	—
369	C₂×C₄×C₅	—	6	2	no	—
370	C₂×C₄×C₅	—	6	4	no	—
371	C₂×C₄×C₅	—	6	2	no	—
372	C₂×C₄×C₅	—	6	4	no	—
373	C₂×C₄×C₅	—	6	8	no	—
374	C₂×C₄×C₅	#237	1	4	yes	—
375	C₂×C₄×C₅	C₂³×D₃	1	2	yes	—
376	C₂×C₄×C₅	C₂³×D₃	1	4	yes	—
377	C₂×C₄×C₅	C₂³×D₃	1	2	yes	—
378	C₂²×Q₈	—	3	2	no	—
379	C₂²×D₄	—	3	1	no	—
380	C₂²×D₄	—	3	2	no	—
381	C₂²×D₄	C₂×C₅×D₃	1	1	yes	characteristic
382	C₂²×D₄	C₂×C₅×D₃	1	2	yes	—
383	C₂³×C₄	—	3	2	no	—
384	C₂³×C₄	—	3	1	no	—
385	C₂³×C₄	C₂×C₅×D₃	1	1	yes	characteristic
386	C₄⋊Q₈	—	3	8	no	—
387	#124	—	3	16	no	—
388	#124	—	3	16	no	—
389	#121	—	3	8	no	—
390	#121	—	3	16	no	—
391	#121	C₂×C₅×D₃	1	8	yes	complemented
392	C₄×D₄	—	3	8	no	—
393	C₄×D₄	—	3	8	no	—
394	#117	—	3	8	no	—
395	#117	—	3	8	no	—
396	#117	—	3	16	no	—
397	C₂×C₄⋊C₄	—	3	1	no	—
398	C₂×C₄⋊C₄	—	3	1	no	—
399	C₂×C₄⋊C₄	—	3	2	no	—
400	C₂×C₄⋊C₄	—	3	1	no	—
401	C₂×C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
402	C₂×C₂²⋊C₄	—	3	2	no	—
403	C₂×C₂²⋊C₄	—	3	2	no	—
404	C₂×C₂²⋊C₄	—	3	1	no	—
405	C₂×C₂²⋊C₄	—	3	2	no	—
406	C₂×C₂²⋊C₄	—	3	2	no	—
407	C₂×C₂²⋊C₄	—	3	1	no	—
408	C₂×C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
409	C₂×C₄²	—	3	2	no	—
410	C₂×C₄²	—	3	2	no	—
411	C₂×C₄²	—	3	1	no	—
412	C₂×C₃×C₅	—	4	4	no	—
413	C₂×C₃×C₅	—	4	2	no	—
414	C₂×C₃×C₅	#555	1	4	yes	complemented
415	C₂×C₃×C₅	C₂²×#41	1	1	yes	characteristic
416	C₂×C₃×C₅	C₂²×#41	1	1	yes	characteristic
417	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
418	C₅×D₃	—	4	2	no	—
419	C₂³×C₃	—	2	4	no	—
420	C₂³×C₃	—	2	2	no	—
421	C₂³×C₃	—	2	2	no	—
422	C₂³×C₃	—	2	4	no	—
423	C₂³×C₃	—	2	4	no	—
424	C₂³×C₃	—	2	2	no	—
425	C₂³×C₃	C₂⁴×C₅	1	2	yes	—
426	C₂³×C₃	C₂⁴×C₅	1	4	yes	—
427	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
428	C₂²×D₃	—	2	2	no	—
429	C₂²×D₃	—	2	2	no	—
430	C₂²×D₃	—	2	2	no	—
431	C₂²×D₃	C₂⁴×C₅	1	2	yes	—
432	C₃×D₄	—	2	8	no	—
433	C₃×D₄	—	4	8	no	—
434	C₂×C₃×C₄	—	2	2	no	—
435	C₂×C₃×C₄	—	2	4	no	—
436	C₂×C₃×C₄	—	2	8	no	—
437	C₂×C₃×C₄	C₅×#41	1	4	yes	—
438	C₂×C₃×C₄	C₂⁴×C₅	1	2	yes	—
439	C₂×C₃×C₄	C₂⁴×C₅	1	4	yes	—
440	C₂×C₃×C₄	C₂⁴×C₅	1	2	yes	—
441	#67	—	2	8	no	—
442	#67	—	4	8	no	—
443	C₂×Dic₃	—	2	2	no	—
444	C₂×Dic₃	—	2	4	no	—
445	C₂×Dic₃	—	2	2	no	—
446	C₂×Dic₃	—	2	4	no	—
447	C₂×Dic₃	—	2	16	no	—
448	C₂×Dic₃	C₅×#41	1	4	yes	—
449	C₂×Dic₃	C₂⁴×C₅	1	2	yes	—
450	C₂×Dic₃	C₂⁴×C₅	1	4	yes	—
451	C₂×Dic₃	C₂⁴×C₅	1	4	yes	—
452	C₂×Dic₃	C₂⁴×C₅	1	2	yes	—
453	C₂×Dic₃	C₂⁴×C₅	1	2	yes	—
454	C₄×D₃	—	2	8	no	—
455	Dic₆	—	4	8	no	—
456	C₂²×C₅	—	2	2	no	—
457	C₂²×C₅	—	2	4	no	—
458	C₂²×C₅	—	2	2	no	—
459	C₂²×C₅	—	2	4	no	—
460	C₂²×C₅	—	2	4	no	—
461	C₂²×C₅	—	2	2	no	—
462	C₂²×C₅	—	4	4	no	—
463	C₂²×C₅	—	4	8	no	—
464	C₂²×C₅	—	6	2	no	—
465	C₂²×C₅	—	6	2	no	—
466	C₂²×C₅	—	6	2	no	—
467	C₂²×C₅	—	12	4	no	—
468	C₂²×C₅	C₂×#237	1	2	yes	—
469	C₂²×C₅	#1004	1	2	yes	—
470	C₂²×C₅	D₃×#41	1	2	yes	—
471	C₂²×C₅	C₂⁴×D₃	1	1	yes	characteristic
472	C₄×C₅	—	2	2	no	—
473	C₄×C₅	—	2	4	no	—
474	C₄×C₅	—	3	4	no	—
475	C₄×C₅	—	6	2	no	—
476	C₄×C₅	—	6	4	no	—
477	C₄×C₅	—	6	2	no	—
478	C₄×C₅	—	6	4	no	—
479	C₄×C₅	C₂×#237	1	4	yes	—
480	C₂⁴	—	3	1	no	—
481	C₂⁴	C₂²×C₅×D₃	1	1	yes	characteristic
482	C₂⁴	C₂²×C₅×D₃	1	2	yes	—
483	C₂×Q₈	—	3	8	no	—
484	C₂×Q₈	—	12	4	no	—
485	C₂×D₄	—	2	4	no	—
486	C₂×D₄	—	3	4	no	—
487	C₂×D₄	—	3	8	no	—
488	C₂×D₄	—	4	4	no	—
489	C₂×D₄	—	6	4	no	—
490	C₂×D₄	—	12	4	no	—
491	C₂×D₄	C₂²×C₅×D₃	1	4	yes	—
492	C₂×D₄	C₂²×C₅×D₃	1	8	yes	—
493	C₂²×C₄	—	2	4	no	—
494	C₂²×C₄	—	3	1	no	—
495	C₂²×C₄	—	3	1	no	—
496	C₂²×C₄	—	3	2	no	—
497	C₂²×C₄	—	3	1	no	—
498	C₂²×C₄	—	3	2	no	—
499	C₂²×C₄	—	3	8	no	—
500	C₂²×C₄	—	3	4	no	—
501	C₂²×C₄	—	6	8	no	—
502	C₂²×C₄	—	6	4	no	—
503	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
504	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
505	C₂²×C₄	C₂²×C₅×D₃	1	4	yes	—
506	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
507	C₄⋊C₄	—	3	4	no	—
508	C₄⋊C₄	—	3	4	no	—
509	C₄⋊C₄	—	3	8	no	—
510	C₄⋊C₄	—	3	4	no	—
511	C₄⋊C₄	C₂²×C₅×D₃	1	4	yes	—
512	C₂²⋊C₄	—	3	8	no	—
513	C₂²⋊C₄	—	3	8	no	—
514	C₂²⋊C₄	—	3	4	no	—
515	C₂²⋊C₄	—	3	8	no	—
516	C₂²⋊C₄	—	3	8	no	—
517	C₂²⋊C₄	—	3	4	no	—
518	C₂²⋊C₄	C₂²×C₅×D₃	1	8	yes	—
519	C₄²	—	3	8	no	—
520	C₄²	—	3	8	no	—
521	C₄²	—	3	4	no	—
522	C₃×C₅	C₂×#555	1	1	yes	characteristic, complemented, Hall
523	C₂²×C₃	—	2	4	no	—
524	C₂²×C₃	—	2	2	no	—
525	C₂²×C₃	—	2	2	no	—
526	C₂²×C₃	—	2	4	no	—
527	C₂²×C₃	—	2	4	no	—
528	C₂²×C₃	—	2	2	no	—
529	C₂²×C₃	—	4	8	no	—
530	C₂²×C₃	—	4	4	no	—
531	C₂²×C₃	C₂×C₅×#41	1	2	yes	—
532	C₂²×C₃	C₂×C₅×#41	1	2	yes	—
533	C₂²×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
534	C₂²×C₃	C₂⁵×C₅	1	1	yes	characteristic
535	C₂×D₃	—	2	2	no	—
536	C₂×D₃	—	2	2	no	—
537	C₂×D₃	—	2	2	no	—
538	C₂×D₃	—	4	4	no	—
539	C₃×C₄	—	2	2	no	—
540	C₃×C₄	—	2	4	no	—
541	C₃×C₄	C₂×C₅×#41	1	4	yes	—
542	Dic₃	—	2	2	no	—
543	Dic₃	—	2	4	no	—
544	Dic₃	—	2	2	no	—
545	Dic₃	—	2	4	no	—
546	Dic₃	C₂×C₅×#41	1	4	yes	—
547	C₂×C₅	—	4	2	no	—
548	C₂×C₅	—	4	4	no	—
549	C₂×C₅	—	12	2	no	—
550	C₂×C₅	#5616	1	4	yes	direct product factor
551	C₂×C₅	C₂²×#237	1	1	yes	characteristic
552	C₂×C₅	C₂×#1004	1	1	yes	characteristic
553	C₂×C₅	C₂×D₃×#41	1	1	yes	characteristic
554	C₂³	—	2	2	no	—
555	C₂³	—	2	4	no	—
556	C₂³	—	2	2	no	—
557	C₂³	—	2	4	no	—
558	C₂³	—	2	4	no	—
559	C₂³	—	2	2	no	—
560	C₂³	—	3	2	no	—
561	C₂³	—	6	2	no	—
562	C₂³	—	6	2	no	—
563	C₂³	—	6	2	no	—
564	C₂³	C₂³×C₅×D₃	1	2	yes	—
565	C₂³	C₂³×C₅×D₃	1	4	yes	—
566	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
567	Q₈	—	12	8	no	—
568	D₄	—	2	8	no	—
569	D₄	—	4	8	no	—
570	D₄	—	6	8	no	—
571	D₄	—	12	8	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	4	no	—
574	C₂×C₄	—	2	8	no	—
575	C₂×C₄	—	3	4	no	—
576	C₂×C₄	—	3	2	no	—
577	C₂×C₄	—	3	2	no	—
578	C₂×C₄	—	3	4	no	—
579	C₂×C₄	—	3	2	no	—
580	C₂×C₄	—	3	4	no	—
581	C₂×C₄	—	6	16	no	—
582	C₂×C₄	—	6	2	no	—
583	C₂×C₄	—	6	4	no	—
584	C₂×C₄	—	6	2	no	—
585	C₂×C₄	—	6	4	no	—
586	C₂×C₄	—	6	8	no	—
587	C₂×C₄	C₅×#237	1	4	yes	—
588	C₂×C₄	C₂³×C₅×D₃	1	2	yes	—
589	C₂×C₄	C₂³×C₅×D₃	1	4	yes	—
590	C₂×C₄	C₂³×C₅×D₃	1	2	yes	—
591	C₂×C₃	—	4	4	no	—
592	C₂×C₃	—	4	2	no	—
593	C₂×C₃	C₅×#555	1	4	yes	complemented
594	C₂×C₃	C₂²×C₅×#41	1	1	yes	characteristic
595	C₂×C₃	C₂²×C₅×#41	1	1	yes	characteristic
596	C₂×C₃	C₂×C₅×H(5,𝔽₂)	1	1	yes	characteristic
597	D₃	—	4	2	no	—
598	C₅	C₂×#5616	1	1	yes	characteristic, direct product factor, Sylow
599	C₂²	—	2	2	no	—
600	C₂²	—	2	4	no	—
601	C₂²	—	2	2	no	—
602	C₂²	—	2	4	no	—
603	C₂²	—	2	4	no	—
604	C₂²	—	2	2	no	—
605	C₂²	—	4	4	no	—
606	C₂²	—	4	8	no	—
607	C₂²	—	6	2	no	—
608	C₂²	—	6	2	no	—
609	C₂²	—	6	2	no	—
610	C₂²	—	12	4	no	—
611	C₂²	C₂×C₅×#237	1	2	yes	—
612	C₂²	C₅×#1004	1	2	yes	—
613	C₂²	C₅×D₃×#41	1	2	yes	—
614	C₂²	C₂⁴×C₅×D₃	1	1	yes	characteristic
615	C₄	—	2	2	no	—
616	C₄	—	2	4	no	—
617	C₄	—	3	4	no	—
618	C₄	—	6	2	no	—
619	C₄	—	6	4	no	—
620	C₄	—	6	2	no	—
621	C₄	—	6	4	no	—
622	C₄	C₂×C₅×#237	1	4	yes	—
623	C₃	C₂×C₅×#555	1	1	yes	characteristic, complemented, Sylow
624	C₂	—	4	2	no	—
625	C₂	—	4	4	no	—
626	C₂	—	12	2	no	—
627	C₂	C₅×#5616	1	4	yes	direct product factor
628	C₂	C₂²×C₅×#237	1	1	yes	characteristic
629	C₂	C₂×C₅×#1004	1	1	yes	characteristic
630	C₂	C₂×C₅×D₃×#41	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×C₅×#121	C₂
socle	C₂³×C₃×C₅	C₂⁴
hypercenter	C₂×C₅×#121	D₃
hypocenter	C₃	C₂×C₅×#555
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₅×#121

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×#121	C₂	semidirect
2. C₂×C₅×#939	C₂	semidirect
3. C₂×C₅×#938	C₂	semidirect
4. C₂×C₅×#936	C₂	semidirect
5. C₂×C₅×D₄×Dic₃	C₂	semidirect
6. C₂×C₄×C₅×#67	C₂	semidirect
7. C₂×C₅×#926	C₂	semidirect
8. C₅×#5616	C₂	direct (N × H)
9. C₂×C₅×#892	C₂	semidirect
10. C₂×C₅×#888	C₂	semidirect
11. C₂×C₅×#885	C₂	semidirect
12. C₂×C₅×#877	C₂	semidirect
22. C₃×C₅×#121	C₂²	semidirect
29. C₅×#939	C₂²	semidirect
30. C₅×#938	C₂²	semidirect
31. C₅×#936	C₂²	semidirect
32. C₅×D₄×Dic₃	C₂²	semidirect
33. C₄×C₅×#67	C₂²	semidirect
36. C₅×#926	C₂²	semidirect
44. C₅×#892	C₂²	semidirect
45. C₅×#888	C₂²	semidirect
46. C₅×#885	C₂²	semidirect
47. C₅×#877	C₂²	semidirect
48. C₂×#5616	C₅	direct (N × H)
55. C₂×C₅×#121	D₃	semidirect
103. C₂×C₃×#121	C₂×C₅	semidirect
104. C₂×#939	C₂×C₅	semidirect
105. C₂×#938	C₂×C₅	semidirect
106. C₂×#936	C₂×C₅	semidirect
107. C₂×D₄×Dic₃	C₂×C₅	semidirect
108. C₂×C₄×#67	C₂×C₅	semidirect
109. C₂×#926	C₂×C₅	semidirect
110. #5616	C₂×C₅	direct (N × H)
111. C₂×#892	C₂×C₅	semidirect
112. C₂×#888	C₂×C₅	semidirect
113. C₂×#885	C₂×C₅	semidirect
114. C₂×#877	C₂×C₅	semidirect
128. C₅×#121	C₂×D₃	semidirect
195. C₃×#121	C₂²×C₅	semidirect
202. #939	C₂²×C₅	semidirect
203. #938	C₂²×C₅	semidirect
204. #936	C₂²×C₅	semidirect
205. D₄×Dic₃	C₂²×C₅	semidirect
206. C₄×#67	C₂²×C₅	semidirect
209. #926	C₂²×C₅	semidirect
217. #892	C₂²×C₅	semidirect
218. #888	C₂²×C₅	semidirect
219. #885	C₂²×C₅	semidirect
220. #877	C₂²×C₅	semidirect
269. C₂×#121	C₅×D₃	semidirect
391. #121	C₂×C₅×D₃	semidirect
414. C₂×C₃×C₅	#555	semidirect
522. C₃×C₅	C₂×#555	semidirect
550. C₂×C₅	#5616	direct (N × H)
593. C₂×C₃	C₅×#555	semidirect
598. C₅	C₂×#5616	direct (N × H)
623. C₃	C₂×C₅×#555	semidirect
627. C₂	C₅×#5616	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×C₅×#121	2
C₂×C₅×#939	2
C₂×C₅×#938	2
C₂×C₅×#936	2
C₂×C₅×D₄×Dic₃	2
C₂×C₄×C₅×#67	2
C₂×C₅×#926	2
C₅×#5616	2
C₂×C₅×#892	2
C₂×C₅×#888	2
C₂×C₅×#885	2
C₂×C₅×#877	2
C₂×C₅×#555	3
C₂×#5616	5

Subgroups — C2×C5×#5616 #14483919

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#5616 #14483919