Subgroups — C₂×C₅×#5494 #14480767

Order 1920.

2672 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#953	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×#930	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×C₅×#67	C₂	1	2	yes	maximal, complemented
5	C₂×C₅×#924	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#5494	C₂	1	16	yes	maximal, direct product factor
7	C₂×C₅×#885	C₂	1	2	yes	maximal, complemented
8	C₂×C₅×#884	C₂	1	2	yes	maximal, complemented
9	C₂×C₅×#883	C₂	1	2	yes	maximal, complemented
10	C₂×C₅×#879	C₂	1	2	yes	maximal, complemented
11	C₂×C₅×#538	—	3	1	no	maximal, Hall
12	C₂⁵×C₃×C₅	C₂²	1	1	yes	characteristic
13	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
14	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
15	C₂²×C₅×#67	C₂²	1	2	yes	—
16	C₂²×C₅×#67	C₂²	1	2	yes	—
17	C₂²×C₅×#67	C₂²	1	2	yes	—
18	C₂²×C₅×D₁₂	C₂²	1	1	yes	characteristic, complemented
19	C₂²×C₄×C₅×D₃	C₂²	1	2	yes	complemented
20	C₂²×C₅×Dic₆	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	8	yes	complemented
24	C₅×#953	C₂²	1	16	yes	complemented
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	C₅×#930	C₂²	1	8	yes	complemented
29	C₄×C₅×#67	C₂²	1	16	yes	complemented
30	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	complemented
31	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	complemented
32	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic, complemented
33	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic, complemented
34	C₅×#924	C₂²	1	8	yes	complemented
35	C₂×C₅×Dic₃⋊C₄	C₂²	1	2	yes	complemented
36	C₂×C₅×Dic₃⋊C₄	C₂²	1	2	yes	complemented
37	C₂×C₄×C₅×Dic₃	C₂²	1	2	yes	complemented
38	C₅×#885	C₂²	1	16	yes	complemented
39	C₅×#884	C₂²	1	16	yes	complemented
40	C₅×#883	C₂²	1	16	yes	complemented
41	C₅×#879	C₂²	1	16	yes	complemented
42	C₂×#5494	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
43	C₅×#538	—	3	16	no	—
44	C₂×C₅×#126	—	3	2	no	—
45	C₂×C₅×#124	—	3	2	no	—
46	C₂×C₅×#123	—	3	2	no	—
47	C₂×C₅×C₂²⋊Q₈	—	3	1	no	—
48	C₂×C₅×#121	—	3	1	no	—
49	C₂×C₅×#121	—	3	2	no	—
50	C₂×C₅×#120	—	3	2	no	—
51	C₂×C₄×C₅×D₄	—	3	2	no	—
52	C₂²×C₅×C₂²⋊C₄	D₃	1	1	yes	characteristic, complemented
53	C₂⁴×C₃×C₅	—	2	4	no	—
54	C₂⁴×C₃×C₅	—	4	2	no	—
55	C₂⁴×C₃×C₅	—	4	2	no	—
56	C₂⁴×C₃×C₅	C₂³	1	2	yes	—
57	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
58	C₂⁴×C₃×C₅	C₂³	1	4	yes	—
59	C₂³×C₅×D₃	C₂³	1	2	yes	—
60	C₂²×C₃×C₄×C₅	—	4	2	no	—
61	C₂²×C₃×C₄×C₅	—	4	2	no	—
62	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
63	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
64	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
65	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
66	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
67	C₂×C₅×#67	—	2	8	no	—
68	C₂×C₅×#67	—	4	4	no	—
69	C₂×C₅×#67	—	4	4	no	—
70	C₂×C₅×#67	C₂³	1	8	yes	—
71	C₂×C₅×#67	C₂³	1	8	yes	—
72	C₂×C₅×#67	C₂³	1	8	yes	—
73	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
74	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
75	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
76	C₂×C₅×D₁₂	—	4	2	no	—
77	C₂×C₅×D₁₂	C₂³	1	4	yes	complemented
78	C₂×C₄×C₅×D₃	—	4	4	no	—
79	C₂×C₄×C₅×D₃	C₂³	1	8	yes	complemented
80	C₂×C₅×Dic₆	—	4	2	no	—
81	C₂×C₅×Dic₆	C₂³	1	4	yes	complemented
82	C₃×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
83	C₃×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
84	C₅×#217	C₂³	1	8	yes	—
85	C₅×#217	C₂³	1	8	yes	—
86	C₅×#217	C₂³	1	8	yes	—
87	C₅×(C₂×D₃)⋊C₄	C₂³	1	8	yes	complemented
88	C₅×(C₂×D₃)⋊C₄	C₂³	1	8	yes	complemented
89	C₅×C₄⋊Dic₃	C₂³	1	4	yes	complemented
90	C₅×C₄⋊Dic₃	C₂³	1	4	yes	complemented
91	C₅×Dic₃⋊C₄	C₂³	1	8	yes	complemented
92	C₅×Dic₃⋊C₄	C₂³	1	8	yes	complemented
93	C₄×C₅×Dic₃	C₂³	1	8	yes	complemented
94	C₂²×C₃×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
95	C₂×#953	C₂×C₅	1	2	yes	complemented
96	C₂×#930	C₂×C₅	1	1	yes	characteristic, complemented
97	C₂×C₄×#67	C₂×C₅	1	2	yes	complemented
98	C₂×#924	C₂×C₅	1	1	yes	characteristic, complemented
99	#5494	C₂×C₅	1	16	yes	direct product factor
100	C₂×#885	C₂×C₅	1	2	yes	complemented
101	C₂×#884	C₂×C₅	1	2	yes	complemented
102	C₂×#883	C₂×C₅	1	2	yes	complemented
103	C₂×#879	C₂×C₅	1	2	yes	complemented
104	C₂⁵×C₅	C₂×D₃	1	1	yes	characteristic
105	C₂²×C₅×Q₈	—	3	1	no	—
106	C₂²×C₅×D₄	—	3	2	no	—
107	C₂²×C₅×D₄	—	3	2	no	—
108	C₂²×C₅×D₄	—	3	2	no	—
109	C₂²×C₅×D₄	—	3	1	no	—
110	C₂³×C₄×C₅	—	3	2	no	—
111	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
112	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
113	C₅×#126	—	3	16	no	—
114	C₅×#124	—	3	16	no	—
115	C₅×#123	—	3	16	no	—
116	C₅×C₂²⋊Q₈	—	3	8	no	—
117	C₅×#121	—	3	8	no	—
118	C₅×#121	—	3	16	no	—
119	C₅×#120	—	3	16	no	—
120	C₄×C₅×D₄	—	3	16	no	—
121	C₂×C₅×C₄⋊C₄	—	3	2	no	—
122	C₂×C₅×C₄⋊C₄	—	3	2	no	—
123	C₂×C₅×C₄⋊C₄	—	3	1	no	—
124	C₂×C₅×C₄⋊C₄	—	3	1	no	—
125	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
126	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
127	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
128	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
129	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
130	C₂×C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
131	C₂×C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
132	C₂×C₅×C₂²⋊C₄	C₂×D₃	1	8	yes	complemented
133	C₂×C₄²×C₅	—	3	2	no	—
134	C₂×#538	—	3	1	no	Sylow
135	C₂³×C₃×C₅	—	2	2	no	—
136	C₂³×C₃×C₅	—	2	2	no	—
137	C₂³×C₃×C₅	—	2	8	no	—
138	C₂³×C₃×C₅	—	2	4	no	—
139	C₂³×C₃×C₅	—	2	4	no	—
140	C₂³×C₃×C₅	—	2	4	no	—
141	C₂³×C₃×C₅	—	4	4	no	—
142	C₂³×C₃×C₅	—	4	4	no	—
143	C₂³×C₃×C₅	—	4	16	no	—
144	C₂³×C₃×C₅	#41	1	4	yes	—
145	C₂³×C₃×C₅	C₂⁴	1	4	yes	—
146	C₂³×C₃×C₅	C₂⁴	1	2	yes	—
147	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
148	C₂²×C₅×D₃	—	2	4	no	—
149	C₂²×C₅×D₃	—	2	4	no	—
150	C₂²×C₅×D₃	—	2	4	no	—
151	C₂²×C₅×D₃	C₂⁴	1	4	yes	—
152	C₂×C₃×C₄×C₅	—	2	2	no	—
153	C₂×C₃×C₄×C₅	—	2	4	no	—
154	C₂×C₃×C₄×C₅	—	2	2	no	—
155	C₂×C₃×C₄×C₅	—	4	4	no	—
156	C₂×C₃×C₄×C₅	—	4	4	no	—
157	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
158	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
159	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
160	C₅×#67	—	2	16	no	—
161	C₅×#67	—	4	8	no	—
162	C₅×#67	—	4	8	no	—
163	C₂×C₅×Dic₃	—	2	4	no	—
164	C₂×C₅×Dic₃	—	2	4	no	—
165	C₂×C₅×Dic₃	—	2	4	no	—
166	C₂×C₅×Dic₃	C₂⁴	1	4	yes	—
167	C₂×C₅×Dic₃	C₂⁴	1	4	yes	—
168	C₂×C₅×Dic₃	C₂⁴	1	4	yes	—
169	C₅×D₁₂	—	4	4	no	—
170	C₄×C₅×D₃	—	4	8	no	—
171	C₅×Dic₆	—	4	4	no	—
172	C₂⁵×C₃	C₂²×C₅	1	1	yes	characteristic
173	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
174	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
175	C₂²×#67	C₂²×C₅	1	2	yes	—
176	C₂²×#67	C₂²×C₅	1	2	yes	—
177	C₂²×#67	C₂²×C₅	1	2	yes	—
178	C₂²×D₁₂	C₂²×C₅	1	1	yes	characteristic, complemented
179	C₂²×C₄×D₃	C₂²×C₅	1	2	yes	complemented
180	C₂²×Dic₆	C₂²×C₅	1	1	yes	characteristic, complemented
181	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
182	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
183	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	8	yes	complemented
184	#953	C₂²×C₅	1	16	yes	complemented
185	C₂×#217	C₂²×C₅	1	2	yes	—
186	C₂×#217	C₂²×C₅	1	2	yes	—
187	C₂×#217	C₂²×C₅	1	2	yes	—
188	#930	C₂²×C₅	1	8	yes	complemented
189	C₄×#67	C₂²×C₅	1	16	yes	complemented
190	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	complemented
191	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	complemented
192	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic, complemented
193	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic, complemented
194	#924	C₂²×C₅	1	8	yes	complemented
195	C₂×Dic₃⋊C₄	C₂²×C₅	1	2	yes	complemented
196	C₂×Dic₃⋊C₄	C₂²×C₅	1	2	yes	complemented
197	C₂×C₄×Dic₃	C₂²×C₅	1	2	yes	complemented
198	#885	C₂²×C₅	1	16	yes	complemented
199	#884	C₂²×C₅	1	16	yes	complemented
200	#883	C₂²×C₅	1	16	yes	complemented
201	#879	C₂²×C₅	1	16	yes	complemented
202	C₂⁴×C₅	—	2	4	no	—
203	C₂⁴×C₅	—	3	2	no	—
204	C₂⁴×C₅	—	4	2	no	—
205	C₂⁴×C₅	—	4	2	no	—
206	C₂⁴×C₅	C₂²×D₃	1	2	yes	—
207	C₂⁴×C₅	C₂²×D₃	1	1	yes	characteristic
208	C₂⁴×C₅	C₂²×D₃	1	4	yes	—
209	C₂×C₅×Q₈	—	3	4	no	—
210	C₂×C₅×Q₈	—	12	2	no	—
211	C₂×C₅×D₄	—	3	8	no	—
212	C₂×C₅×D₄	—	3	8	no	—
213	C₂×C₅×D₄	—	3	8	no	—
214	C₂×C₅×D₄	—	3	4	no	—
215	C₂×C₅×D₄	—	6	8	no	—
216	C₂×C₅×D₄	—	12	2	no	—
217	C₂×C₅×D₄	—	12	4	no	—
218	C₂×C₅×D₄	—	12	4	no	—
219	C₂²×C₄×C₅	—	3	2	no	—
220	C₂²×C₄×C₅	—	3	2	no	—
221	C₂²×C₄×C₅	—	3	2	no	—
222	C₂²×C₄×C₅	—	3	8	no	—
223	C₂²×C₄×C₅	—	4	2	no	—
224	C₂²×C₄×C₅	—	4	2	no	—
225	C₂²×C₄×C₅	—	12	4	no	—
226	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
227	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
228	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
229	C₂²×C₄×C₅	C₂²×D₃	1	4	yes	—
230	C₂²×C₄×C₅	C₂²×D₃	1	4	yes	—
231	C₅×C₄⋊C₄	—	3	8	no	—
232	C₅×C₄⋊C₄	—	3	8	no	—
233	C₅×C₄⋊C₄	—	3	4	no	—
234	C₅×C₄⋊C₄	—	3	4	no	—
235	C₅×C₂²⋊C₄	—	3	8	no	—
236	C₅×C₂²⋊C₄	—	3	8	no	—
237	C₅×C₂²⋊C₄	—	3	8	no	—
238	C₅×C₂²⋊C₄	—	3	8	no	—
239	C₅×C₂²⋊C₄	—	3	8	no	—
240	C₅×C₂²⋊C₄	C₂²×D₃	1	8	yes	—
241	C₅×C₂²⋊C₄	C₂²×D₃	1	8	yes	—
242	C₄²×C₅	—	3	8	no	—
243	#538	—	3	16	no	—
244	C₂×#126	—	3	2	no	—
245	C₂×#124	—	3	2	no	—
246	C₂×#123	—	3	2	no	—
247	C₂×C₂²⋊Q₈	—	3	1	no	—
248	C₂×#121	—	3	1	no	—
249	C₂×#121	—	3	2	no	—
250	C₂×#120	—	3	2	no	—
251	C₂×C₄×D₄	—	3	2	no	—
252	C₂²×C₂²⋊C₄	C₅×D₃	1	1	yes	characteristic, complemented
253	C₂²×C₃×C₅	—	2	8	no	—
254	C₂²×C₃×C₅	—	2	2	no	—
255	C₂²×C₃×C₅	—	2	2	no	—
256	C₂²×C₃×C₅	—	2	4	no	—
257	C₂²×C₃×C₅	—	2	4	no	—
258	C₂²×C₃×C₅	—	2	4	no	—
259	C₂²×C₃×C₅	—	4	16	no	—
260	C₂²×C₃×C₅	—	4	8	no	—
261	C₂²×C₃×C₅	C₂×#41	1	4	yes	—
262	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
263	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
264	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
265	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
266	C₂×C₅×D₃	—	2	4	no	—
267	C₂×C₅×D₃	—	2	4	no	—
268	C₂×C₅×D₃	—	2	4	no	—
269	C₂×C₅×D₃	—	4	8	no	—
270	C₃×C₄×C₅	—	2	2	no	—
271	C₃×C₄×C₅	—	2	4	no	—
272	C₃×C₄×C₅	—	2	2	no	—
273	C₅×Dic₃	—	2	4	no	—
274	C₅×Dic₃	—	2	4	no	—
275	C₅×Dic₃	—	2	4	no	—
276	C₂⁴×C₃	—	2	4	no	—
277	C₂⁴×C₃	—	4	2	no	—
278	C₂⁴×C₃	—	4	2	no	—
279	C₂⁴×C₃	C₂³×C₅	1	2	yes	—
280	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
281	C₂⁴×C₃	C₂³×C₅	1	4	yes	—
282	C₂³×D₃	C₂³×C₅	1	2	yes	—
283	C₂²×C₃×C₄	—	4	2	no	—
284	C₂²×C₃×C₄	—	4	2	no	—
285	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
286	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
287	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
288	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
289	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
290	C₂×#67	—	2	8	no	—
291	C₂×#67	—	4	4	no	—
292	C₂×#67	—	4	4	no	—
293	C₂×#67	C₂³×C₅	1	8	yes	—
294	C₂×#67	C₂³×C₅	1	8	yes	—
295	C₂×#67	C₂³×C₅	1	8	yes	—
296	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
297	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
298	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
299	C₂×D₁₂	—	4	2	no	—
300	C₂×D₁₂	C₂³×C₅	1	4	yes	complemented
301	C₂×C₄×D₃	—	4	4	no	—
302	C₂×C₄×D₃	C₂³×C₅	1	8	yes	complemented
303	C₂×Dic₆	—	4	2	no	—
304	C₂×Dic₆	C₂³×C₅	1	4	yes	complemented
305	C₃×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
306	C₃×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
307	#217	C₂³×C₅	1	8	yes	—
308	#217	C₂³×C₅	1	8	yes	—
309	#217	C₂³×C₅	1	8	yes	—
310	(C₂×D₃)⋊C₄	C₂³×C₅	1	8	yes	complemented
311	(C₂×D₃)⋊C₄	C₂³×C₅	1	8	yes	complemented
312	C₄⋊Dic₃	C₂³×C₅	1	4	yes	complemented
313	C₄⋊Dic₃	C₂³×C₅	1	4	yes	complemented
314	Dic₃⋊C₄	C₂³×C₅	1	8	yes	complemented
315	Dic₃⋊C₄	C₂³×C₅	1	8	yes	complemented
316	C₄×Dic₃	C₂³×C₅	1	8	yes	complemented
317	C₂³×C₅	—	2	2	no	—
318	C₂³×C₅	—	2	2	no	—
319	C₂³×C₅	—	2	8	no	—
320	C₂³×C₅	—	2	4	no	—
321	C₂³×C₅	—	2	4	no	—
322	C₂³×C₅	—	2	4	no	—
323	C₂³×C₅	—	3	4	no	—
324	C₂³×C₅	—	4	4	no	—
325	C₂³×C₅	—	4	4	no	—
326	C₂³×C₅	—	4	16	no	—
327	C₂³×C₅	—	6	4	no	—
328	C₂³×C₅	—	6	4	no	—
329	C₂³×C₅	—	6	4	no	—
330	C₂³×C₅	#235	1	4	yes	—
331	C₂³×C₅	C₂³×D₃	1	4	yes	—
332	C₂³×C₅	C₂³×D₃	1	2	yes	—
333	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
334	C₅×Q₈	—	12	4	no	—
335	C₅×D₄	—	6	16	no	—
336	C₅×D₄	—	12	4	no	—
337	C₅×D₄	—	12	8	no	—
338	C₅×D₄	—	12	8	no	—
339	C₂×C₄×C₅	—	2	4	no	—
340	C₂×C₄×C₅	—	2	2	no	—
341	C₂×C₄×C₅	—	2	2	no	—
342	C₂×C₄×C₅	—	3	4	no	—
343	C₂×C₄×C₅	—	3	4	no	—
344	C₂×C₄×C₅	—	3	4	no	—
345	C₂×C₄×C₅	—	4	4	no	—
346	C₂×C₄×C₅	—	4	4	no	—
347	C₂×C₄×C₅	—	6	4	no	—
348	C₂×C₄×C₅	—	6	4	no	—
349	C₂×C₄×C₅	—	6	4	no	—
350	C₂×C₄×C₅	—	12	8	no	—
351	C₂×C₄×C₅	C₂³×D₃	1	4	yes	—
352	C₂×C₄×C₅	C₂³×D₃	1	2	yes	—
353	C₂×C₄×C₅	C₂³×D₃	1	2	yes	—
354	C₂⁵	C₂×C₅×D₃	1	1	yes	characteristic
355	C₂²×Q₈	—	3	1	no	—
356	C₂²×D₄	—	3	2	no	—
357	C₂²×D₄	—	3	2	no	—
358	C₂²×D₄	—	3	2	no	—
359	C₂²×D₄	—	3	1	no	—
360	C₂³×C₄	—	3	2	no	—
361	C₂³×C₄	C₂×C₅×D₃	1	1	yes	characteristic
362	C₂³×C₄	C₂×C₅×D₃	1	1	yes	characteristic
363	#126	—	3	16	no	—
364	#124	—	3	16	no	—
365	#123	—	3	16	no	—
366	C₂²⋊Q₈	—	3	8	no	—
367	#121	—	3	8	no	—
368	#121	—	3	16	no	—
369	#120	—	3	16	no	—
370	C₄×D₄	—	3	16	no	—
371	C₂×C₄⋊C₄	—	3	2	no	—
372	C₂×C₄⋊C₄	—	3	2	no	—
373	C₂×C₄⋊C₄	—	3	1	no	—
374	C₂×C₄⋊C₄	—	3	1	no	—
375	C₂×C₂²⋊C₄	—	3	2	no	—
376	C₂×C₂²⋊C₄	—	3	2	no	—
377	C₂×C₂²⋊C₄	—	3	2	no	—
378	C₂×C₂²⋊C₄	—	3	2	no	—
379	C₂×C₂²⋊C₄	—	3	2	no	—
380	C₂×C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
381	C₂×C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
382	C₂×C₂²⋊C₄	C₂×C₅×D₃	1	8	yes	complemented
383	C₂×C₄²	—	3	2	no	—
384	C₂×C₃×C₅	—	2	4	no	—
385	C₂×C₃×C₅	—	4	4	no	—
386	C₂×C₃×C₅	#538	1	4	yes	complemented
387	C₂×C₃×C₅	C₂²×#41	1	1	yes	characteristic
388	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
389	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
390	C₅×D₃	—	4	4	no	—
391	C₂³×C₃	—	2	2	no	—
392	C₂³×C₃	—	2	2	no	—
393	C₂³×C₃	—	2	8	no	—
394	C₂³×C₃	—	2	4	no	—
395	C₂³×C₃	—	2	4	no	—
396	C₂³×C₃	—	2	4	no	—
397	C₂³×C₃	—	4	4	no	—
398	C₂³×C₃	—	4	4	no	—
399	C₂³×C₃	—	4	16	no	—
400	C₂³×C₃	C₅×#41	1	4	yes	—
401	C₂³×C₃	C₂⁴×C₅	1	4	yes	—
402	C₂³×C₃	C₂⁴×C₅	1	2	yes	—
403	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
404	C₂²×D₃	—	2	4	no	—
405	C₂²×D₃	—	2	4	no	—
406	C₂²×D₃	—	2	4	no	—
407	C₂²×D₃	C₂⁴×C₅	1	4	yes	—
408	C₂×C₃×C₄	—	2	2	no	—
409	C₂×C₃×C₄	—	2	4	no	—
410	C₂×C₃×C₄	—	2	2	no	—
411	C₂×C₃×C₄	—	4	4	no	—
412	C₂×C₃×C₄	—	4	4	no	—
413	C₂×C₃×C₄	C₂⁴×C₅	1	4	yes	—
414	C₂×C₃×C₄	C₂⁴×C₅	1	2	yes	—
415	C₂×C₃×C₄	C₂⁴×C₅	1	2	yes	—
416	#67	—	2	16	no	—
417	#67	—	4	8	no	—
418	#67	—	4	8	no	—
419	C₂×Dic₃	—	2	4	no	—
420	C₂×Dic₃	—	2	4	no	—
421	C₂×Dic₃	—	2	4	no	—
422	C₂×Dic₃	C₂⁴×C₅	1	4	yes	—
423	C₂×Dic₃	C₂⁴×C₅	1	4	yes	—
424	C₂×Dic₃	C₂⁴×C₅	1	4	yes	—
425	D₁₂	—	4	4	no	—
426	C₄×D₃	—	4	8	no	—
427	Dic₆	—	4	4	no	—
428	C₂²×C₅	—	2	2	no	—
429	C₂²×C₅	—	2	2	no	—
430	C₂²×C₅	—	2	8	no	—
431	C₂²×C₅	—	2	4	no	—
432	C₂²×C₅	—	2	4	no	—
433	C₂²×C₅	—	2	4	no	—
434	C₂²×C₅	—	4	16	no	—
435	C₂²×C₅	—	4	8	no	—
436	C₂²×C₅	—	6	4	no	—
437	C₂²×C₅	—	6	4	no	—
438	C₂²×C₅	—	6	4	no	—
439	C₂²×C₅	—	12	8	no	—
440	C₂²×C₅	C₂×#235	1	4	yes	—
441	C₂²×C₅	C₂×#235	1	2	yes	—
442	C₂²×C₅	#1004	1	2	yes	—
443	C₂²×C₅	#1004	1	2	yes	—
444	C₂²×C₅	C₂⁴×D₃	1	1	yes	characteristic
445	C₄×C₅	—	2	4	no	—
446	C₄×C₅	—	2	2	no	—
447	C₄×C₅	—	2	2	no	—
448	C₄×C₅	—	6	4	no	—
449	C₄×C₅	—	6	4	no	—
450	C₄×C₅	—	6	4	no	—
451	C₂⁴	—	2	4	no	—
452	C₂⁴	—	3	2	no	—
453	C₂⁴	—	4	2	no	—
454	C₂⁴	—	4	2	no	—
455	C₂⁴	C₂²×C₅×D₃	1	2	yes	—
456	C₂⁴	C₂²×C₅×D₃	1	1	yes	characteristic
457	C₂⁴	C₂²×C₅×D₃	1	4	yes	—
458	C₂×Q₈	—	3	4	no	—
459	C₂×Q₈	—	12	2	no	—
460	C₂×D₄	—	3	8	no	—
461	C₂×D₄	—	3	8	no	—
462	C₂×D₄	—	3	8	no	—
463	C₂×D₄	—	3	4	no	—
464	C₂×D₄	—	6	8	no	—
465	C₂×D₄	—	12	2	no	—
466	C₂×D₄	—	12	4	no	—
467	C₂×D₄	—	12	4	no	—
468	C₂²×C₄	—	3	2	no	—
469	C₂²×C₄	—	3	2	no	—
470	C₂²×C₄	—	3	2	no	—
471	C₂²×C₄	—	3	8	no	—
472	C₂²×C₄	—	4	2	no	—
473	C₂²×C₄	—	4	2	no	—
474	C₂²×C₄	—	12	4	no	—
475	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
476	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
477	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
478	C₂²×C₄	C₂²×C₅×D₃	1	4	yes	—
479	C₂²×C₄	C₂²×C₅×D₃	1	4	yes	—
480	C₄⋊C₄	—	3	8	no	—
481	C₄⋊C₄	—	3	8	no	—
482	C₄⋊C₄	—	3	4	no	—
483	C₄⋊C₄	—	3	4	no	—
484	C₂²⋊C₄	—	3	8	no	—
485	C₂²⋊C₄	—	3	8	no	—
486	C₂²⋊C₄	—	3	8	no	—
487	C₂²⋊C₄	—	3	8	no	—
488	C₂²⋊C₄	—	3	8	no	—
489	C₂²⋊C₄	C₂²×C₅×D₃	1	8	yes	—
490	C₂²⋊C₄	C₂²×C₅×D₃	1	8	yes	—
491	C₄²	—	3	8	no	—
492	C₃×C₅	C₂×#538	1	1	yes	characteristic, complemented, Hall
493	C₂²×C₃	—	2	8	no	—
494	C₂²×C₃	—	2	2	no	—
495	C₂²×C₃	—	2	2	no	—
496	C₂²×C₃	—	2	4	no	—
497	C₂²×C₃	—	2	4	no	—
498	C₂²×C₃	—	2	4	no	—
499	C₂²×C₃	—	4	16	no	—
500	C₂²×C₃	—	4	8	no	—
501	C₂²×C₃	C₂×C₅×#41	1	4	yes	—
502	C₂²×C₃	C₂×C₅×#41	1	2	yes	—
503	C₂²×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
504	C₂²×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
505	C₂²×C₃	C₂⁵×C₅	1	1	yes	characteristic
506	C₂×D₃	—	2	4	no	—
507	C₂×D₃	—	2	4	no	—
508	C₂×D₃	—	2	4	no	—
509	C₂×D₃	—	4	8	no	—
510	C₃×C₄	—	2	2	no	—
511	C₃×C₄	—	2	4	no	—
512	C₃×C₄	—	2	2	no	—
513	Dic₃	—	2	4	no	—
514	Dic₃	—	2	4	no	—
515	Dic₃	—	2	4	no	—
516	C₂×C₅	—	2	4	no	—
517	C₂×C₅	—	4	4	no	—
518	C₂×C₅	—	12	4	no	—
519	C₂×C₅	#5494	1	4	yes	direct product factor
520	C₂×C₅	C₂²×#235	1	1	yes	characteristic
521	C₂×C₅	C₂×#1004	1	1	yes	characteristic
522	C₂×C₅	C₂×#1004	1	1	yes	characteristic
523	C₂³	—	2	2	no	—
524	C₂³	—	2	2	no	—
525	C₂³	—	2	8	no	—
526	C₂³	—	2	4	no	—
527	C₂³	—	2	4	no	—
528	C₂³	—	2	4	no	—
529	C₂³	—	3	4	no	—
530	C₂³	—	4	4	no	—
531	C₂³	—	4	4	no	—
532	C₂³	—	4	16	no	—
533	C₂³	—	6	4	no	—
534	C₂³	—	6	4	no	—
535	C₂³	—	6	4	no	—
536	C₂³	C₅×#235	1	4	yes	—
537	C₂³	C₂³×C₅×D₃	1	4	yes	—
538	C₂³	C₂³×C₅×D₃	1	2	yes	—
539	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
540	Q₈	—	12	4	no	—
541	D₄	—	6	16	no	—
542	D₄	—	12	4	no	—
543	D₄	—	12	8	no	—
544	D₄	—	12	8	no	—
545	C₂×C₄	—	2	4	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	3	4	no	—
549	C₂×C₄	—	3	4	no	—
550	C₂×C₄	—	3	4	no	—
551	C₂×C₄	—	4	4	no	—
552	C₂×C₄	—	4	4	no	—
553	C₂×C₄	—	6	4	no	—
554	C₂×C₄	—	6	4	no	—
555	C₂×C₄	—	6	4	no	—
556	C₂×C₄	—	12	8	no	—
557	C₂×C₄	C₂³×C₅×D₃	1	4	yes	—
558	C₂×C₄	C₂³×C₅×D₃	1	2	yes	—
559	C₂×C₄	C₂³×C₅×D₃	1	2	yes	—
560	C₂×C₃	—	2	4	no	—
561	C₂×C₃	—	4	4	no	—
562	C₂×C₃	C₅×#538	1	4	yes	complemented
563	C₂×C₃	C₂²×C₅×#41	1	1	yes	characteristic
564	C₂×C₃	C₂×C₅×H(5,𝔽₂)	1	1	yes	characteristic
565	C₂×C₃	C₂×C₅×H(5,𝔽₂)	1	1	yes	characteristic
566	D₃	—	4	4	no	—
567	C₅	C₂×#5494	1	1	yes	characteristic, direct product factor, Sylow
568	C₂²	—	2	2	no	—
569	C₂²	—	2	2	no	—
570	C₂²	—	2	8	no	—
571	C₂²	—	2	4	no	—
572	C₂²	—	2	4	no	—
573	C₂²	—	2	4	no	—
574	C₂²	—	4	16	no	—
575	C₂²	—	4	8	no	—
576	C₂²	—	6	4	no	—
577	C₂²	—	6	4	no	—
578	C₂²	—	6	4	no	—
579	C₂²	—	12	8	no	—
580	C₂²	C₂×C₅×#235	1	4	yes	—
581	C₂²	C₂×C₅×#235	1	2	yes	—
582	C₂²	C₅×#1004	1	2	yes	—
583	C₂²	C₅×#1004	1	2	yes	—
584	C₂²	C₂⁴×C₅×D₃	1	1	yes	characteristic
585	C₄	—	2	4	no	—
586	C₄	—	2	2	no	—
587	C₄	—	2	2	no	—
588	C₄	—	6	4	no	—
589	C₄	—	6	4	no	—
590	C₄	—	6	4	no	—
591	C₃	C₂×C₅×#538	1	1	yes	characteristic, complemented, Sylow
592	C₂	—	2	4	no	—
593	C₂	—	4	4	no	—
594	C₂	—	12	4	no	—
595	C₂	C₅×#5494	1	4	yes	direct product factor
596	C₂	C₂²×C₅×#235	1	1	yes	characteristic
597	C₂	C₂×C₅×#1004	1	1	yes	characteristic
598	C₂	C₂×C₅×#1004	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁵×C₅
Frattini	C₂²	C₂⁴×C₅×D₃
Fitting	C₂²×C₃×C₅×C₂²⋊C₄	C₂
socle	C₂³×C₃×C₅	C₂⁴
hypercenter	C₂²×C₅×C₂²⋊C₄	D₃
hypocenter	C₃	C₂×C₅×#538
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₅×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₅×C₂²⋊C₄	C₂	semidirect
2. C₂×C₅×#953	C₂	semidirect
3. C₂×C₅×#930	C₂	semidirect
4. C₂×C₄×C₅×#67	C₂	semidirect
5. C₂×C₅×#924	C₂	semidirect
6. C₅×#5494	C₂	direct (N × H)
7. C₂×C₅×#885	C₂	semidirect
8. C₂×C₅×#884	C₂	semidirect
9. C₂×C₅×#883	C₂	semidirect
10. C₂×C₅×#879	C₂	semidirect
18. C₂²×C₅×D₁₂	C₂²	semidirect
19. C₂²×C₄×C₅×D₃	C₂²	semidirect
20. C₂²×C₅×Dic₆	C₂²	semidirect
23. C₂×C₃×C₅×C₂²⋊C₄	C₂²	semidirect
24. C₅×#953	C₂²	semidirect
28. C₅×#930	C₂²	semidirect
29. C₄×C₅×#67	C₂²	semidirect
30. C₂×C₅×(C₂×D₃)⋊C₄	C₂²	semidirect
31. C₂×C₅×(C₂×D₃)⋊C₄	C₂²	semidirect
32. C₂×C₅×C₄⋊Dic₃	C₂²	semidirect
33. C₂×C₅×C₄⋊Dic₃	C₂²	semidirect
34. C₅×#924	C₂²	semidirect
35. C₂×C₅×Dic₃⋊C₄	C₂²	semidirect
36. C₂×C₅×Dic₃⋊C₄	C₂²	semidirect
37. C₂×C₄×C₅×Dic₃	C₂²	semidirect
38. C₅×#885	C₂²	semidirect
39. C₅×#884	C₂²	semidirect
40. C₅×#883	C₂²	semidirect
41. C₅×#879	C₂²	semidirect
42. C₂×#5494	C₅	direct (N × H)
52. C₂²×C₅×C₂²⋊C₄	D₃	semidirect
77. C₂×C₅×D₁₂	C₂³	semidirect
79. C₂×C₄×C₅×D₃	C₂³	semidirect
81. C₂×C₅×Dic₆	C₂³	semidirect
87. C₅×(C₂×D₃)⋊C₄	C₂³	semidirect
88. C₅×(C₂×D₃)⋊C₄	C₂³	semidirect
89. C₅×C₄⋊Dic₃	C₂³	semidirect
90. C₅×C₄⋊Dic₃	C₂³	semidirect
91. C₅×Dic₃⋊C₄	C₂³	semidirect
92. C₅×Dic₃⋊C₄	C₂³	semidirect
93. C₄×C₅×Dic₃	C₂³	semidirect
94. C₂²×C₃×C₂²⋊C₄	C₂×C₅	semidirect
95. C₂×#953	C₂×C₅	semidirect
96. C₂×#930	C₂×C₅	semidirect
97. C₂×C₄×#67	C₂×C₅	semidirect
98. C₂×#924	C₂×C₅	semidirect
99. #5494	C₂×C₅	direct (N × H)
100. C₂×#885	C₂×C₅	semidirect
101. C₂×#884	C₂×C₅	semidirect
102. C₂×#883	C₂×C₅	semidirect
103. C₂×#879	C₂×C₅	semidirect
132. C₂×C₅×C₂²⋊C₄	C₂×D₃	semidirect
178. C₂²×D₁₂	C₂²×C₅	semidirect
179. C₂²×C₄×D₃	C₂²×C₅	semidirect
180. C₂²×Dic₆	C₂²×C₅	semidirect
183. C₂×C₃×C₂²⋊C₄	C₂²×C₅	semidirect
184. #953	C₂²×C₅	semidirect
188. #930	C₂²×C₅	semidirect
189. C₄×#67	C₂²×C₅	semidirect
190. C₂×(C₂×D₃)⋊C₄	C₂²×C₅	semidirect
191. C₂×(C₂×D₃)⋊C₄	C₂²×C₅	semidirect
192. C₂×C₄⋊Dic₃	C₂²×C₅	semidirect
193. C₂×C₄⋊Dic₃	C₂²×C₅	semidirect
194. #924	C₂²×C₅	semidirect
195. C₂×Dic₃⋊C₄	C₂²×C₅	semidirect
196. C₂×Dic₃⋊C₄	C₂²×C₅	semidirect
197. C₂×C₄×Dic₃	C₂²×C₅	semidirect
198. #885	C₂²×C₅	semidirect
199. #884	C₂²×C₅	semidirect
200. #883	C₂²×C₅	semidirect
201. #879	C₂²×C₅	semidirect
252. C₂²×C₂²⋊C₄	C₅×D₃	semidirect
300. C₂×D₁₂	C₂³×C₅	semidirect
302. C₂×C₄×D₃	C₂³×C₅	semidirect
304. C₂×Dic₆	C₂³×C₅	semidirect
310. (C₂×D₃)⋊C₄	C₂³×C₅	semidirect
311. (C₂×D₃)⋊C₄	C₂³×C₅	semidirect
312. C₄⋊Dic₃	C₂³×C₅	semidirect
313. C₄⋊Dic₃	C₂³×C₅	semidirect
314. Dic₃⋊C₄	C₂³×C₅	semidirect
315. Dic₃⋊C₄	C₂³×C₅	semidirect
316. C₄×Dic₃	C₂³×C₅	semidirect
382. C₂×C₂²⋊C₄	C₂×C₅×D₃	semidirect
386. C₂×C₃×C₅	#538	semidirect
492. C₃×C₅	C₂×#538	semidirect
519. C₂×C₅	#5494	direct (N × H)
562. C₂×C₃	C₅×#538	semidirect
567. C₅	C₂×#5494	direct (N × H)
591. C₃	C₂×C₅×#538	semidirect
595. C₂	C₅×#5494	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×C₃×C₅×C₂²⋊C₄	2
C₂×C₅×#953	2
C₂×C₅×#930	2
C₂×C₄×C₅×#67	2
C₂×C₅×#924	2
C₅×#5494	2
C₂×C₅×#885	2
C₂×C₅×#884	2
C₂×C₅×#883	2
C₂×C₅×#879	2
C₂×C₅×#538	3
C₂×#5494	5

Subgroups — C2×C5×#5494 #14480767

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#5494 #14480767