Subgroups — C₂×C₅×#5690 #14483793

Order 1920.

2400 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#5690	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₅×#897	C₂	1	2	yes	maximal, complemented
4	C₂×C₅×#896	C₂	1	2	yes	maximal, complemented
5	C₂×C₅×#896	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#893	C₂	1	1	yes	maximal, characteristic
7	C₂×C₅×#893	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅×D₃×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
9	C₂×C₅×#890	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅×#888	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×C₅×D₁₂	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₅×#554	—	3	1	no	maximal, Hall
14	C₂²×C₅×D₁₂	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₅×D₃	C₂²	1	2	yes	—
16	C₂²×C₄×C₅×D₃	C₂²	1	2	yes	—
17	C₂²×C₄×C₅×D₃	C₂²	1	2	yes	—
18	C₂²×C₅×Dic₆	C₂²	1	1	yes	characteristic
19	C₂²×C₅×Dic₆	C₂²	1	2	yes	—
20	C₃×C₅×#125	C₂²	1	8	yes	complemented
21	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
24	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
25	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
26	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
27	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
28	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
29	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
30	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
31	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
32	C₂×C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
33	C₂×C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
34	C₂×C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
35	C₂×C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
36	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
38	C₅×#897	C₂²	1	16	yes	complemented
39	C₅×#896	C₂²	1	16	yes	complemented
40	C₅×#896	C₂²	1	16	yes	complemented
41	C₅×#893	C₂²	1	8	yes	—
42	C₅×#893	C₂²	1	8	yes	—
43	C₅×D₃×C₄⋊C₄	C₂²	1	16	yes	complemented
44	C₅×#890	C₂²	1	8	yes	complemented
45	C₅×#888	C₂²	1	8	yes	complemented
46	C₄×C₅×D₁₂	C₂²	1	8	yes	—
47	C₄×C₅×Dic₆	C₂²	1	8	yes	complemented
48	C₂×#5690	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
49	C₅×#554	—	3	16	no	—
50	C₂×C₅×C₄⋊Q₈	—	3	1	no	—
51	C₂×C₅×#125	—	3	1	no	—
52	C₂×C₅×#125	D₃	1	1	yes	characteristic, complemented
53	C₂×C₅×C₂²⋊Q₈	—	3	2	no	—
54	C₂×C₅×C₂²⋊Q₈	—	3	2	no	—
55	C₂×C₅×C₂²⋊Q₈	—	3	2	no	—
56	C₂×C₄×C₅×Q₈	—	3	1	no	—
57	C₂×C₄×C₅×D₄	—	3	1	no	—
58	C₂×C₄×C₅×D₄	—	3	1	no	—
59	C₂×C₄×C₅×D₄	—	3	1	no	—
60	C₂²×C₅×C₄⋊C₄	—	3	2	no	—
61	C₂³×C₅×D₃	C₂³	1	2	yes	—
62	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
63	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
64	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
65	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
66	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
67	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
68	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
69	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
70	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
71	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
72	C₂×C₅×D₁₂	Q₈	1	8	yes	complemented
73	C₂×C₅×D₁₂	C₂³	1	4	yes	—
74	C₂×C₄×C₅×D₃	—	2	8	no	—
75	C₂×C₄×C₅×D₃	—	4	4	no	—
76	C₂×C₄×C₅×D₃	—	4	4	no	—
77	C₂×C₄×C₅×D₃	C₂³	1	8	yes	—
78	C₂×C₄×C₅×D₃	C₂³	1	8	yes	—
79	C₂×C₄×C₅×D₃	C₂³	1	8	yes	—
80	C₂×C₅×Dic₆	—	2	4	no	—
81	C₂×C₅×Dic₆	—	4	4	no	—
82	C₂×C₅×Dic₆	C₂³	1	4	yes	—
83	C₂×C₅×Dic₆	C₂³	1	8	yes	—
84	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
85	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
86	C₃×C₅×C₄⋊C₄	C₂³	1	8	yes	—
87	C₃×C₅×C₄⋊C₄	C₂³	1	8	yes	—
88	C₃×C₄²×C₅	C₂³	1	4	yes	—
89	C₅×(C₂×D₃)⋊C₄	C₂³	1	8	yes	—
90	C₅×(C₂×D₃)⋊C₄	C₂³	1	8	yes	—
91	C₅×(C₂×D₃)⋊C₄	C₂³	1	8	yes	—
92	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
93	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
94	C₅×C₄⋊Dic₃	C₂³	1	8	yes	—
95	C₅×Dic₃⋊C₄	C₂³	1	4	yes	—
96	C₅×Dic₃⋊C₄	C₂³	1	4	yes	—
97	C₅×Dic₃⋊C₄	C₂³	1	8	yes	—
98	C₅×Dic₃⋊C₄	C₂³	1	8	yes	—
99	C₄×C₅×Dic₃	C₂³	1	4	yes	—
100	C₄×C₅×Dic₃	C₂³	1	4	yes	—
101	C₂×C₃×#125	C₂×C₅	1	1	yes	characteristic, complemented
102	#5690	C₂×C₅	1	16	yes	direct product factor
103	C₂×#897	C₂×C₅	1	2	yes	complemented
104	C₂×#896	C₂×C₅	1	2	yes	complemented
105	C₂×#896	C₂×C₅	1	2	yes	complemented
106	C₂×#893	C₂×C₅	1	1	yes	characteristic
107	C₂×#893	C₂×C₅	1	1	yes	characteristic
108	C₂×D₃×C₄⋊C₄	C₂×C₅	1	2	yes	complemented
109	C₂×#890	C₂×C₅	1	1	yes	characteristic, complemented
110	C₂×#888	C₂×C₅	1	1	yes	characteristic, complemented
111	C₂×C₄×D₁₂	C₂×C₅	1	1	yes	characteristic
112	C₂×C₄×Dic₆	C₂×C₅	1	1	yes	characteristic, complemented
113	C₂²×C₅×Q₈	—	3	1	no	—
114	C₂²×C₅×Q₈	—	3	2	no	—
115	C₂²×C₅×D₄	—	3	1	no	—
116	C₂³×C₄×C₅	—	3	2	no	—
117	C₂³×C₄×C₅	—	3	2	no	—
118	C₂³×C₄×C₅	—	3	2	no	—
119	C₅×C₄⋊Q₈	—	3	8	no	—
120	C₅×#125	—	3	8	no	—
121	C₅×#125	C₂×D₃	1	8	yes	complemented
122	C₅×C₂²⋊Q₈	—	3	16	no	—
123	C₅×C₂²⋊Q₈	—	3	16	no	—
124	C₅×C₂²⋊Q₈	—	3	16	no	—
125	C₄×C₅×Q₈	—	3	8	no	—
126	C₄×C₅×D₄	—	3	8	no	—
127	C₄×C₅×D₄	—	3	8	no	—
128	C₄×C₅×D₄	—	3	8	no	—
129	C₂×C₅×C₄⋊C₄	—	3	1	no	—
130	C₂×C₅×C₄⋊C₄	—	3	2	no	—
131	C₂×C₅×C₄⋊C₄	—	3	1	no	—
132	C₂×C₅×C₄⋊C₄	—	3	2	no	—
133	C₂×C₅×C₄⋊C₄	—	3	2	no	—
134	C₂×C₅×C₄⋊C₄	—	3	1	no	—
135	C₂×C₅×C₄⋊C₄	—	3	1	no	—
136	C₂×C₅×C₄⋊C₄	—	3	16	no	—
137	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
138	C₂×C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
139	C₂×C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
140	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
141	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
142	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
143	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
144	C₂×C₄²×C₅	—	3	1	no	—
145	C₂×C₄²×C₅	—	3	1	no	—
146	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
147	C₂×#554	—	3	1	no	Sylow
148	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
149	C₂²×C₅×D₃	—	2	4	no	—
150	C₂²×C₅×D₃	—	2	4	no	—
151	C₂²×C₅×D₃	C₂×Q₈	1	8	yes	—
152	C₂²×C₅×D₃	C₂⁴	1	4	yes	—
153	C₂×C₃×C₄×C₅	—	2	2	no	—
154	C₂×C₃×C₄×C₅	—	2	4	no	—
155	C₂×C₃×C₄×C₅	—	2	4	no	—
156	C₂×C₃×C₄×C₅	—	2	2	no	—
157	C₂×C₃×C₄×C₅	C₂×Q₈	1	4	yes	—
158	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
159	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
160	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
161	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
162	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
163	C₂×C₅×Dic₃	—	2	2	no	—
164	C₂×C₅×Dic₃	—	2	2	no	—
165	C₂×C₅×Dic₃	—	2	4	no	—
166	C₂×C₅×Dic₃	—	2	2	no	—
167	C₂×C₅×Dic₃	#41	1	4	yes	—
168	C₂×C₅×Dic₃	C₂⁴	1	2	yes	—
169	C₂×C₅×Dic₃	C₂⁴	1	2	yes	—
170	C₂×C₅×Dic₃	C₂⁴	1	2	yes	—
171	C₂×C₅×Dic₃	C₂⁴	1	4	yes	—
172	C₂×C₅×Dic₃	C₂⁴	1	2	yes	—
173	C₅×D₁₂	C₂×Q₈	1	16	yes	complemented
174	C₄×C₅×D₃	—	2	16	no	—
175	C₄×C₅×D₃	—	4	8	no	—
176	C₄×C₅×D₃	—	4	8	no	—
177	C₅×Dic₆	—	2	8	no	—
178	C₅×Dic₆	—	4	8	no	—
179	C₂²×D₁₂	C₂²×C₅	1	1	yes	characteristic
180	C₂²×C₄×D₃	C₂²×C₅	1	2	yes	—
181	C₂²×C₄×D₃	C₂²×C₅	1	2	yes	—
182	C₂²×C₄×D₃	C₂²×C₅	1	2	yes	—
183	C₂²×Dic₆	C₂²×C₅	1	1	yes	characteristic
184	C₂²×Dic₆	C₂²×C₅	1	2	yes	—
185	C₃×#125	C₂²×C₅	1	8	yes	complemented
186	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
187	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
188	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
189	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
190	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
191	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
192	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
193	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
194	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
195	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
196	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
197	C₂×Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
198	C₂×Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
199	C₂×Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
200	C₂×Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
201	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
202	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
203	#897	C₂²×C₅	1	16	yes	complemented
204	#896	C₂²×C₅	1	16	yes	complemented
205	#896	C₂²×C₅	1	16	yes	complemented
206	#893	C₂²×C₅	1	8	yes	—
207	#893	C₂²×C₅	1	8	yes	—
208	D₃×C₄⋊C₄	C₂²×C₅	1	16	yes	complemented
209	#890	C₂²×C₅	1	8	yes	complemented
210	#888	C₂²×C₅	1	8	yes	complemented
211	C₄×D₁₂	C₂²×C₅	1	8	yes	—
212	C₄×Dic₆	C₂²×C₅	1	8	yes	complemented
213	C₂⁴×C₅	—	3	2	no	—
214	C₂×C₅×Q₈	—	3	4	no	—
215	C₂×C₅×Q₈	—	3	8	no	—
216	C₂×C₅×Q₈	—	6	4	no	—
217	C₂×C₅×Q₈	—	12	4	no	—
218	C₂×C₅×D₄	—	3	8	no	—
219	C₂×C₅×D₄	—	3	4	no	—
220	C₂²×C₄×C₅	—	3	1	no	—
221	C₂²×C₄×C₅	—	3	1	no	—
222	C₂²×C₄×C₅	—	3	1	no	—
223	C₂²×C₄×C₅	—	3	1	no	—
224	C₂²×C₄×C₅	—	3	2	no	—
225	C₂²×C₄×C₅	—	3	8	no	—
226	C₂²×C₄×C₅	—	3	8	no	—
227	C₂²×C₄×C₅	—	3	8	no	—
228	C₂²×C₄×C₅	—	6	8	no	—
229	C₂²×C₄×C₅	—	12	4	no	—
230	C₂²×C₄×C₅	—	12	4	no	—
231	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
232	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
233	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
234	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
235	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
236	C₅×C₄⋊C₄	—	3	4	no	—
237	C₅×C₄⋊C₄	—	3	8	no	—
238	C₅×C₄⋊C₄	—	3	8	no	—
239	C₅×C₄⋊C₄	—	3	4	no	—
240	C₅×C₄⋊C₄	—	3	4	no	—
241	C₅×C₄⋊C₄	—	3	4	no	—
242	C₅×C₄⋊C₄	—	3	8	no	—
243	C₅×C₄⋊C₄	C₂²×D₃	1	8	yes	—
244	C₅×C₄⋊C₄	C₂²×D₃	1	4	yes	—
245	C₅×C₄⋊C₄	C₂²×D₃	1	4	yes	—
246	C₅×C₄⋊C₄	C₂²×D₃	1	8	yes	—
247	C₅×C₂²⋊C₄	—	3	8	no	—
248	C₅×C₂²⋊C₄	—	3	8	no	—
249	C₅×C₂²⋊C₄	—	3	8	no	—
250	C₄²×C₅	—	3	4	no	—
251	C₄²×C₅	—	3	4	no	—
252	C₄²×C₅	C₂²×D₃	1	4	yes	—
253	#554	—	3	16	no	—
254	C₂×C₄⋊Q₈	—	3	1	no	—
255	C₂×#125	—	3	1	no	—
256	C₂×#125	C₅×D₃	1	1	yes	characteristic, complemented
257	C₂×C₂²⋊Q₈	—	3	2	no	—
258	C₂×C₂²⋊Q₈	—	3	2	no	—
259	C₂×C₂²⋊Q₈	—	3	2	no	—
260	C₂×C₄×Q₈	—	3	1	no	—
261	C₂×C₄×D₄	—	3	1	no	—
262	C₂×C₄×D₄	—	3	1	no	—
263	C₂×C₄×D₄	—	3	1	no	—
264	C₂²×C₄⋊C₄	—	3	2	no	—
265	C₂²×C₃×C₅	C₂²×Q₈	1	2	yes	—
266	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
267	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
268	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
269	C₂×C₅×D₃	—	2	16	no	—
270	C₂×C₅×D₃	—	2	4	no	—
271	C₂×C₅×D₃	—	2	4	no	—
272	C₂×C₅×D₃	C₂²×Q₈	1	8	yes	—
273	C₃×C₄×C₅	—	2	2	no	—
274	C₃×C₄×C₅	—	2	4	no	—
275	C₃×C₄×C₅	—	2	4	no	—
276	C₃×C₄×C₅	—	2	2	no	—
277	C₃×C₄×C₅	C₂²×Q₈	1	4	yes	—
278	C₅×Dic₃	—	2	2	no	—
279	C₅×Dic₃	—	2	2	no	—
280	C₅×Dic₃	—	2	4	no	—
281	C₅×Dic₃	—	2	2	no	—
282	C₅×Dic₃	C₂×#41	1	4	yes	—
283	C₂³×D₃	C₂³×C₅	1	2	yes	—
284	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
285	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
286	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
287	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
288	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
289	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
290	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
291	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
292	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
293	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
294	C₂×D₁₂	C₅×Q₈	1	8	yes	complemented
295	C₂×D₁₂	C₂³×C₅	1	4	yes	—
296	C₂×C₄×D₃	—	2	8	no	—
297	C₂×C₄×D₃	—	4	4	no	—
298	C₂×C₄×D₃	—	4	4	no	—
299	C₂×C₄×D₃	C₂³×C₅	1	8	yes	—
300	C₂×C₄×D₃	C₂³×C₅	1	8	yes	—
301	C₂×C₄×D₃	C₂³×C₅	1	8	yes	—
302	C₂×Dic₆	—	2	4	no	—
303	C₂×Dic₆	—	4	4	no	—
304	C₂×Dic₆	C₂³×C₅	1	4	yes	—
305	C₂×Dic₆	C₂³×C₅	1	8	yes	—
306	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
307	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
308	C₃×C₄⋊C₄	C₂³×C₅	1	8	yes	—
309	C₃×C₄⋊C₄	C₂³×C₅	1	8	yes	—
310	C₃×C₄²	C₂³×C₅	1	4	yes	—
311	(C₂×D₃)⋊C₄	C₂³×C₅	1	8	yes	—
312	(C₂×D₃)⋊C₄	C₂³×C₅	1	8	yes	—
313	(C₂×D₃)⋊C₄	C₂³×C₅	1	8	yes	—
314	C₄⋊Dic₃	C₂³×C₅	1	4	yes	—
315	C₄⋊Dic₃	C₂³×C₅	1	4	yes	—
316	C₄⋊Dic₃	C₂³×C₅	1	8	yes	—
317	Dic₃⋊C₄	C₂³×C₅	1	4	yes	—
318	Dic₃⋊C₄	C₂³×C₅	1	4	yes	—
319	Dic₃⋊C₄	C₂³×C₅	1	8	yes	—
320	Dic₃⋊C₄	C₂³×C₅	1	8	yes	—
321	C₄×Dic₃	C₂³×C₅	1	4	yes	—
322	C₄×Dic₃	C₂³×C₅	1	4	yes	—
323	C₂³×C₅	—	3	8	no	—
324	C₂³×C₅	—	3	4	no	—
325	C₂³×C₅	—	6	4	no	—
326	C₂³×C₅	—	6	4	no	—
327	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
328	C₅×Q₈	—	6	8	no	—
329	C₅×Q₈	—	12	8	no	—
330	C₅×D₄	—	3	16	no	—
331	C₂×C₄×C₅	—	2	4	no	—
332	C₂×C₄×C₅	—	2	2	no	—
333	C₂×C₄×C₅	—	2	2	no	—
334	C₂×C₄×C₅	—	2	4	no	—
335	C₂×C₄×C₅	—	3	4	no	—
336	C₂×C₄×C₅	—	3	2	no	—
337	C₂×C₄×C₅	—	3	2	no	—
338	C₂×C₄×C₅	—	3	2	no	—
339	C₂×C₄×C₅	—	3	2	no	—
340	C₂×C₄×C₅	—	3	4	no	—
341	C₂×C₄×C₅	—	6	16	no	—
342	C₂×C₄×C₅	—	6	2	no	—
343	C₂×C₄×C₅	—	6	2	no	—
344	C₂×C₄×C₅	—	6	4	no	—
345	C₂×C₄×C₅	—	6	2	no	—
346	C₂×C₄×C₅	—	12	8	no	—
347	C₂×C₄×C₅	—	12	8	no	—
348	C₂×C₄×C₅	D₃×Q₈	1	4	yes	—
349	C₂×C₄×C₅	C₂³×D₃	1	2	yes	—
350	C₂×C₄×C₅	C₂³×D₃	1	2	yes	—
351	C₂×C₄×C₅	C₂³×D₃	1	4	yes	—
352	C₂×C₄×C₅	C₂³×D₃	1	4	yes	—
353	C₂×C₄×C₅	C₂³×D₃	1	2	yes	—
354	C₂²×Q₈	—	3	1	no	—
355	C₂²×Q₈	—	3	2	no	—
356	C₂²×D₄	—	3	1	no	—
357	C₂³×C₄	—	3	2	no	—
358	C₂³×C₄	—	3	2	no	—
359	C₂³×C₄	—	3	2	no	—
360	C₄⋊Q₈	—	3	8	no	—
361	#125	—	3	8	no	—
362	#125	C₂×C₅×D₃	1	8	yes	complemented
363	C₂²⋊Q₈	—	3	16	no	—
364	C₂²⋊Q₈	—	3	16	no	—
365	C₂²⋊Q₈	—	3	16	no	—
366	C₄×Q₈	—	3	8	no	—
367	C₄×D₄	—	3	8	no	—
368	C₄×D₄	—	3	8	no	—
369	C₄×D₄	—	3	8	no	—
370	C₂×C₄⋊C₄	—	3	1	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×C₄⋊C₄	—	3	1	no	—
376	C₂×C₄⋊C₄	—	3	2	no	—
377	C₂×C₄⋊C₄	—	3	16	no	—
378	C₂×C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
379	C₂×C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
380	C₂×C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
381	C₂×C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
382	C₂×C₂²⋊C₄	—	3	2	no	—
383	C₂×C₂²⋊C₄	—	3	2	no	—
384	C₂×C₂²⋊C₄	—	3	2	no	—
385	C₂×C₄²	—	3	1	no	—
386	C₂×C₄²	—	3	1	no	—
387	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
388	C₂×C₃×C₅	#554	1	4	yes	complemented
389	C₂×C₃×C₅	C₂³×Q₈	1	1	yes	characteristic
390	C₂×C₃×C₅	C₂²×#41	1	1	yes	characteristic
391	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
392	C₅×D₃	—	2	8	no	—
393	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
394	C₂²×D₃	—	2	4	no	—
395	C₂²×D₃	—	2	4	no	—
396	C₂²×D₃	C₂×C₅×Q₈	1	8	yes	—
397	C₂²×D₃	C₂⁴×C₅	1	4	yes	—
398	C₂×C₃×C₄	—	2	2	no	—
399	C₂×C₃×C₄	—	2	4	no	—
400	C₂×C₃×C₄	—	2	4	no	—
401	C₂×C₃×C₄	—	2	2	no	—
402	C₂×C₃×C₄	C₂×C₅×Q₈	1	4	yes	—
403	C₂×C₃×C₄	C₂⁴×C₅	1	2	yes	—
404	C₂×C₃×C₄	C₂⁴×C₅	1	4	yes	—
405	C₂×C₃×C₄	C₂⁴×C₅	1	4	yes	—
406	C₂×C₃×C₄	C₂⁴×C₅	1	2	yes	—
407	C₂×C₃×C₄	C₂⁴×C₅	1	2	yes	—
408	C₂×Dic₃	—	2	2	no	—
409	C₂×Dic₃	—	2	2	no	—
410	C₂×Dic₃	—	2	4	no	—
411	C₂×Dic₃	—	2	2	no	—
412	C₂×Dic₃	C₅×#41	1	4	yes	—
413	C₂×Dic₃	C₂⁴×C₅	1	2	yes	—
414	C₂×Dic₃	C₂⁴×C₅	1	2	yes	—
415	C₂×Dic₃	C₂⁴×C₅	1	2	yes	—
416	C₂×Dic₃	C₂⁴×C₅	1	4	yes	—
417	C₂×Dic₃	C₂⁴×C₅	1	2	yes	—
418	D₁₂	C₂×C₅×Q₈	1	16	yes	complemented
419	C₄×D₃	—	2	16	no	—
420	C₄×D₃	—	4	8	no	—
421	C₄×D₃	—	4	8	no	—
422	Dic₆	—	2	8	no	—
423	Dic₆	—	4	8	no	—
424	C₂²×C₅	—	3	8	no	—
425	C₂²×C₅	—	6	16	no	—
426	C₂²×C₅	—	6	4	no	—
427	C₂²×C₅	—	6	4	no	—
428	C₂²×C₅	C₂×D₃×Q₈	1	2	yes	—
429	C₂²×C₅	D₃×#41	1	2	yes	—
430	C₂²×C₅	#1009	1	2	yes	—
431	C₂²×C₅	C₂⁴×D₃	1	1	yes	characteristic
432	C₄×C₅	—	2	4	no	—
433	C₄×C₅	—	2	2	no	—
434	C₄×C₅	—	2	2	no	—
435	C₄×C₅	—	2	4	no	—
436	C₄×C₅	—	3	4	no	—
437	C₄×C₅	—	6	2	no	—
438	C₄×C₅	—	6	2	no	—
439	C₄×C₅	—	6	4	no	—
440	C₄×C₅	—	6	2	no	—
441	C₄×C₅	C₂×D₃×Q₈	1	4	yes	—
442	C₂⁴	—	3	2	no	—
443	C₂×Q₈	—	3	4	no	—
444	C₂×Q₈	—	3	8	no	—
445	C₂×Q₈	—	6	4	no	—
446	C₂×Q₈	—	12	4	no	—
447	C₂×D₄	—	3	8	no	—
448	C₂×D₄	—	3	4	no	—
449	C₂²×C₄	—	3	1	no	—
450	C₂²×C₄	—	3	1	no	—
451	C₂²×C₄	—	3	1	no	—
452	C₂²×C₄	—	3	1	no	—
453	C₂²×C₄	—	3	2	no	—
454	C₂²×C₄	—	3	8	no	—
455	C₂²×C₄	—	3	8	no	—
456	C₂²×C₄	—	3	8	no	—
457	C₂²×C₄	—	6	8	no	—
458	C₂²×C₄	—	12	4	no	—
459	C₂²×C₄	—	12	4	no	—
460	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
461	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
462	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
463	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
464	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
465	C₄⋊C₄	—	3	4	no	—
466	C₄⋊C₄	—	3	8	no	—
467	C₄⋊C₄	—	3	8	no	—
468	C₄⋊C₄	—	3	4	no	—
469	C₄⋊C₄	—	3	4	no	—
470	C₄⋊C₄	—	3	4	no	—
471	C₄⋊C₄	—	3	8	no	—
472	C₄⋊C₄	C₂²×C₅×D₃	1	8	yes	—
473	C₄⋊C₄	C₂²×C₅×D₃	1	4	yes	—
474	C₄⋊C₄	C₂²×C₅×D₃	1	4	yes	—
475	C₄⋊C₄	C₂²×C₅×D₃	1	8	yes	—
476	C₂²⋊C₄	—	3	8	no	—
477	C₂²⋊C₄	—	3	8	no	—
478	C₂²⋊C₄	—	3	8	no	—
479	C₄²	—	3	4	no	—
480	C₄²	—	3	4	no	—
481	C₄²	C₂²×C₅×D₃	1	4	yes	—
482	C₃×C₅	C₂×#554	1	1	yes	characteristic, complemented, Hall
483	C₂²×C₃	C₂²×C₅×Q₈	1	2	yes	—
484	C₂²×C₃	C₂×C₅×#41	1	2	yes	—
485	C₂²×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
486	C₂²×C₃	C₂⁵×C₅	1	1	yes	characteristic
487	C₂×D₃	—	2	16	no	—
488	C₂×D₃	—	2	4	no	—
489	C₂×D₃	—	2	4	no	—
490	C₂×D₃	C₂²×C₅×Q₈	1	8	yes	—
491	C₃×C₄	—	2	2	no	—
492	C₃×C₄	—	2	4	no	—
493	C₃×C₄	—	2	4	no	—
494	C₃×C₄	—	2	2	no	—
495	C₃×C₄	C₂²×C₅×Q₈	1	4	yes	—
496	Dic₃	—	2	2	no	—
497	Dic₃	—	2	2	no	—
498	Dic₃	—	2	4	no	—
499	Dic₃	—	2	2	no	—
500	Dic₃	C₂×C₅×#41	1	4	yes	—
501	C₂×C₅	—	6	8	no	—
502	C₂×C₅	#5690	1	4	yes	direct product factor
503	C₂×C₅	C₂²×D₃×Q₈	1	1	yes	characteristic
504	C₂×C₅	C₂×D₃×#41	1	1	yes	characteristic
505	C₂×C₅	C₂×#1009	1	1	yes	characteristic
506	C₂³	—	3	8	no	—
507	C₂³	—	3	4	no	—
508	C₂³	—	6	4	no	—
509	C₂³	—	6	4	no	—
510	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
511	Q₈	—	6	8	no	—
512	Q₈	—	12	8	no	—
513	D₄	—	3	16	no	—
514	C₂×C₄	—	2	4	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	4	no	—
518	C₂×C₄	—	3	4	no	—
519	C₂×C₄	—	3	2	no	—
520	C₂×C₄	—	3	2	no	—
521	C₂×C₄	—	3	2	no	—
522	C₂×C₄	—	3	2	no	—
523	C₂×C₄	—	3	4	no	—
524	C₂×C₄	—	6	16	no	—
525	C₂×C₄	—	6	2	no	—
526	C₂×C₄	—	6	2	no	—
527	C₂×C₄	—	6	4	no	—
528	C₂×C₄	—	6	2	no	—
529	C₂×C₄	—	12	8	no	—
530	C₂×C₄	—	12	8	no	—
531	C₂×C₄	C₅×D₃×Q₈	1	4	yes	—
532	C₂×C₄	C₂³×C₅×D₃	1	2	yes	—
533	C₂×C₄	C₂³×C₅×D₃	1	2	yes	—
534	C₂×C₄	C₂³×C₅×D₃	1	4	yes	—
535	C₂×C₄	C₂³×C₅×D₃	1	4	yes	—
536	C₂×C₄	C₂³×C₅×D₃	1	2	yes	—
537	C₂×C₃	C₅×#554	1	4	yes	complemented
538	C₂×C₃	C₂³×C₅×Q₈	1	1	yes	characteristic
539	C₂×C₃	C₂²×C₅×#41	1	1	yes	characteristic
540	C₂×C₃	C₂×C₅×H(5,𝔽₂)	1	1	yes	characteristic
541	D₃	—	2	8	no	—
542	C₅	C₂×#5690	1	1	yes	characteristic, direct product factor, Sylow
543	C₂²	—	3	8	no	—
544	C₂²	—	6	16	no	—
545	C₂²	—	6	4	no	—
546	C₂²	—	6	4	no	—
547	C₂²	C₂×C₅×D₃×Q₈	1	2	yes	—
548	C₂²	C₅×D₃×#41	1	2	yes	—
549	C₂²	C₅×#1009	1	2	yes	—
550	C₂²	C₂⁴×C₅×D₃	1	1	yes	characteristic
551	C₄	—	2	4	no	—
552	C₄	—	2	2	no	—
553	C₄	—	2	2	no	—
554	C₄	—	2	4	no	—
555	C₄	—	3	4	no	—
556	C₄	—	6	2	no	—
557	C₄	—	6	2	no	—
558	C₄	—	6	4	no	—
559	C₄	—	6	2	no	—
560	C₄	C₂×C₅×D₃×Q₈	1	4	yes	—
561	C₃	C₂×C₅×#554	1	1	yes	characteristic, complemented, Sylow
562	C₂	—	6	8	no	—
563	C₂	C₅×#5690	1	4	yes	direct product factor
564	C₂	C₂²×C₅×D₃×Q₈	1	1	yes	characteristic
565	C₂	C₂×C₅×D₃×#41	1	1	yes	characteristic
566	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×C₅×#125	C₂
socle	C₂³×C₃×C₅	C₂⁴
prime radical	C₂²×C₅×D₁₂	C₂²
hypercenter	C₂×C₅×#125	D₃
hypocenter	C₃	C₂×C₅×#554
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₅×#125

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×#125	C₂	semidirect
2. C₅×#5690	C₂	direct (N × H)
3. C₂×C₅×#897	C₂	semidirect
4. C₂×C₅×#896	C₂	semidirect
5. C₂×C₅×#896	C₂	semidirect
8. C₂×C₅×D₃×C₄⋊C₄	C₂	semidirect
9. C₂×C₅×#890	C₂	semidirect
10. C₂×C₅×#888	C₂	semidirect
12. C₂×C₄×C₅×Dic₆	C₂	semidirect
20. C₃×C₅×#125	C₂²	semidirect
38. C₅×#897	C₂²	semidirect
39. C₅×#896	C₂²	semidirect
40. C₅×#896	C₂²	semidirect
43. C₅×D₃×C₄⋊C₄	C₂²	semidirect
44. C₅×#890	C₂²	semidirect
45. C₅×#888	C₂²	semidirect
47. C₄×C₅×Dic₆	C₂²	semidirect
48. C₂×#5690	C₅	direct (N × H)
52. C₂×C₅×#125	D₃	semidirect
72. C₂×C₅×D₁₂	Q₈	semidirect
101. C₂×C₃×#125	C₂×C₅	semidirect
102. #5690	C₂×C₅	direct (N × H)
103. C₂×#897	C₂×C₅	semidirect
104. C₂×#896	C₂×C₅	semidirect
105. C₂×#896	C₂×C₅	semidirect
108. C₂×D₃×C₄⋊C₄	C₂×C₅	semidirect
109. C₂×#890	C₂×C₅	semidirect
110. C₂×#888	C₂×C₅	semidirect
112. C₂×C₄×Dic₆	C₂×C₅	semidirect
121. C₅×#125	C₂×D₃	semidirect
173. C₅×D₁₂	C₂×Q₈	semidirect
185. C₃×#125	C₂²×C₅	semidirect
203. #897	C₂²×C₅	semidirect
204. #896	C₂²×C₅	semidirect
205. #896	C₂²×C₅	semidirect
208. D₃×C₄⋊C₄	C₂²×C₅	semidirect
209. #890	C₂²×C₅	semidirect
210. #888	C₂²×C₅	semidirect
212. C₄×Dic₆	C₂²×C₅	semidirect
256. C₂×#125	C₅×D₃	semidirect
294. C₂×D₁₂	C₅×Q₈	semidirect
362. #125	C₂×C₅×D₃	semidirect
388. C₂×C₃×C₅	#554	semidirect
418. D₁₂	C₂×C₅×Q₈	semidirect
482. C₃×C₅	C₂×#554	semidirect
502. C₂×C₅	#5690	direct (N × H)
537. C₂×C₃	C₅×#554	semidirect
542. C₅	C₂×#5690	direct (N × H)
561. C₃	C₂×C₅×#554	semidirect
563. C₂	C₅×#5690	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×#125	2
C₅×#5690	2
C₂×C₅×#897	2
C₂×C₅×#896	2
C₂×C₅×#893	2
C₂×C₅×D₃×C₄⋊C₄	2
C₂×C₅×#890	2
C₂×C₅×#888	2
C₂×C₄×C₅×D₁₂	2
C₂×C₄×C₅×Dic₆	2
C₂×C₅×#554	3
C₂×#5690	5

Subgroups — C2×C5×#5690 #14483793

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#5690 #14483793