Subgroups — C₂×C₅×#5729 #14481729

Order 1920.

2160 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#946	C₂	1	2	yes	maximal
3	C₂×C₅×#944	C₂	1	2	yes	maximal, complemented
4	C₅×#5729	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₅×#896	C₂	1	2	yes	maximal
6	C₂×C₅×#892	C₂	1	1	yes	maximal, characteristic
7	C₂×C₅×D₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅×#890	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅×#889	C₂	1	2	yes	maximal, complemented
10	C₂×C₅×#888	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅×#872	C₂	1	1	yes	maximal, characteristic
12	C₂×C₅×#870	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₅×#544	—	3	1	no	maximal, Hall
14	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
15	C₂²×C₄×C₅×D₃	C₂²	1	2	yes	—
16	C₂²×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
17	C₂²×C₅×Dic₆	C₂²	1	2	yes	—
18	C₃×C₅×C₄⋊Q₈	C₂²	1	8	yes	complemented
19	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
23	C₅×#946	C₂²	1	16	yes	—
24	C₅×#944	C₂²	1	16	yes	complemented
25	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
28	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
29	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
30	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
31	C₂×C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
32	C₂×C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
33	C₂×C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
34	C₂×C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
35	C₂×C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
36	C₂×C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
37	C₂×C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
38	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₅×Dic₃	C₂²	1	2	yes	—
40	C₅×#896	C₂²	1	16	yes	—
41	C₅×#892	C₂²	1	8	yes	—
42	C₅×D₃×C₄⋊C₄	C₂²	1	8	yes	—
43	C₅×#890	C₂²	1	8	yes	complemented
44	C₅×#889	C₂²	1	16	yes	complemented
45	C₅×#888	C₂²	1	8	yes	complemented
46	C₅×#872	C₂²	1	8	yes	—
47	C₅×#870	C₂²	1	8	yes	complemented
48	C₂×#5729	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
49	C₅×#544	—	3	16	no	—
50	C₂×C₅×C₄⋊Q₈	—	3	2	no	—
51	C₂×C₅×C₄⋊Q₈	—	3	1	no	—
52	C₂×C₅×C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
53	C₂×C₅×#125	—	3	2	no	—
54	C₂×C₅×#125	—	3	1	no	—
55	C₂×C₅×#125	—	3	1	no	—
56	C₂×C₅×C₂²⋊Q₈	—	3	2	no	—
57	C₂×C₅×C₂²⋊Q₈	—	3	2	no	—
58	C₂×C₅×#117	—	3	1	no	—
59	C₂×C₅×#117	—	3	1	no	—
60	C₂²×C₅×C₄⋊C₄	—	3	1	no	—
61	C₂³×C₅×D₃	C₂³	1	1	yes	characteristic
62	C₂×C₃×C₅×Q₈	—	4	4	no	—
63	C₂×C₃×C₅×Q₈	C₂³	1	8	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×C₃×C₄×C₅	C₂³	1	2	yes	—
68	C₂²×C₃×C₄×C₅	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	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
73	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
74	C₂×C₄×C₅×D₃	—	4	4	no	—
75	C₂×C₄×C₅×D₃	Q₈	1	8	yes	complemented
76	C₂×C₄×C₅×D₃	C₂³	1	8	yes	—
77	C₂×C₄×C₅×D₃	C₂³	1	4	yes	—
78	C₂×C₅×Dic₆	—	4	4	no	—
79	C₂×C₅×Dic₆	C₂³	1	8	yes	—
80	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
81	C₃×C₅×C₄⋊C₄	C₂³	1	8	yes	—
82	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
83	C₃×C₄²×C₅	C₂³	1	4	yes	—
84	C₅×(C₂×D₃)⋊C₄	C₂³	1	4	yes	—
85	C₅×(C₂×D₃)⋊C₄	C₂³	1	4	yes	—
86	C₅×(C₂×D₃)⋊C₄	C₂³	1	8	yes	—
87	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
88	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
89	C₅×C₄⋊Dic₃	C₂³	1	8	yes	—
90	C₅×Dic₃⋊C₄	C₂³	1	8	yes	—
91	C₅×Dic₃⋊C₄	C₂³	1	8	yes	—
92	C₅×Dic₃⋊C₄	C₂³	1	4	yes	—
93	C₅×Dic₃⋊C₄	C₂³	1	4	yes	—
94	C₅×Dic₃⋊C₄	C₂³	1	8	yes	—
95	C₅×Dic₃⋊C₄	C₂³	1	8	yes	—
96	C₅×Dic₃⋊C₄	C₂³	1	8	yes	—
97	C₄×C₅×Dic₃	C₂³	1	4	yes	—
98	C₄×C₅×Dic₃	C₂³	1	8	yes	—
99	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic, complemented
100	C₂×#946	C₂×C₅	1	2	yes	—
101	C₂×#944	C₂×C₅	1	2	yes	complemented
102	#5729	C₂×C₅	1	16	yes	direct product factor
103	C₂×#896	C₂×C₅	1	2	yes	—
104	C₂×#892	C₂×C₅	1	1	yes	characteristic
105	C₂×D₃×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
106	C₂×#890	C₂×C₅	1	1	yes	characteristic, complemented
107	C₂×#889	C₂×C₅	1	2	yes	complemented
108	C₂×#888	C₂×C₅	1	1	yes	characteristic, complemented
109	C₂×#872	C₂×C₅	1	1	yes	characteristic
110	C₂×#870	C₂×C₅	1	1	yes	characteristic, complemented
111	C₂²×C₅×Q₈	—	3	2	no	—
112	C₂²×C₅×Q₈	C₂×D₃	1	2	yes	—
113	C₂³×C₄×C₅	—	3	1	no	—
114	C₂³×C₄×C₅	—	3	2	no	—
115	C₅×C₄⋊Q₈	—	3	16	no	—
116	C₅×C₄⋊Q₈	—	3	8	no	—
117	C₅×C₄⋊Q₈	C₂×D₃	1	8	yes	complemented
118	C₅×#125	—	3	16	no	—
119	C₅×#125	—	3	8	no	—
120	C₅×#125	—	3	8	no	—
121	C₅×C₂²⋊Q₈	—	3	16	no	—
122	C₅×C₂²⋊Q₈	—	3	16	no	—
123	C₅×#117	—	3	8	no	—
124	C₅×#117	—	3	8	no	—
125	C₂×C₅×C₄⋊C₄	—	3	2	no	—
126	C₂×C₅×C₄⋊C₄	—	3	1	no	—
127	C₂×C₅×C₄⋊C₄	—	3	1	no	—
128	C₂×C₅×C₄⋊C₄	—	3	2	no	—
129	C₂×C₅×C₄⋊C₄	—	3	2	no	—
130	C₂×C₅×C₄⋊C₄	—	3	2	no	—
131	C₂×C₅×C₄⋊C₄	—	3	2	no	—
132	C₂×C₅×C₄⋊C₄	—	3	1	no	—
133	C₂×C₅×C₄⋊C₄	—	3	1	no	—
134	C₂×C₅×C₄⋊C₄	—	3	2	no	—
135	C₂×C₅×C₄⋊C₄	—	3	8	no	—
136	C₂×C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
137	C₂×C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
138	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
139	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
140	C₂×C₅×C₂²⋊C₄	—	3	1	no	—
141	C₂×C₅×C₂²⋊C₄	—	3	1	no	—
142	C₂×C₄²×C₅	—	3	2	no	—
143	C₂×C₄²×C₅	—	3	1	no	—
144	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
145	C₂×#544	—	3	1	no	Sylow
146	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
147	C₂²×C₅×D₃	—	2	2	no	—
148	C₂²×C₅×D₃	—	2	2	no	—
149	C₂²×C₅×D₃	C₂×Q₈	1	4	yes	—
150	C₂²×C₅×D₃	C₂⁴	1	2	yes	—
151	C₃×C₅×Q₈	—	4	8	no	—
152	C₂×C₃×C₄×C₅	—	2	4	no	—
153	C₂×C₃×C₄×C₅	—	2	4	no	—
154	C₂×C₃×C₄×C₅	—	2	2	no	—
155	C₂×C₃×C₄×C₅	—	2	2	no	—
156	C₂×C₃×C₄×C₅	C₂×Q₈	1	4	yes	—
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	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
161	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
162	C₂×C₅×Dic₃	—	2	2	no	—
163	C₂×C₅×Dic₃	—	2	4	no	—
164	C₂×C₅×Dic₃	—	2	2	no	—
165	C₂×C₅×Dic₃	—	2	4	no	—
166	C₂×C₅×Dic₃	C₂×Q₈	1	4	yes	—
167	C₂×C₅×Dic₃	C₂⁴	1	2	yes	—
168	C₂×C₅×Dic₃	C₂⁴	1	4	yes	—
169	C₂×C₅×Dic₃	C₂⁴	1	2	yes	—
170	C₂×C₅×Dic₃	C₂⁴	1	4	yes	—
171	C₂×C₅×Dic₃	C₂⁴	1	2	yes	—
172	C₄×C₅×D₃	—	4	8	no	—
173	C₄×C₅×D₃	C₂×Q₈	1	16	yes	complemented
174	C₅×Dic₆	—	4	8	no	—
175	C₂²×C₃×Q₈	C₂²×C₅	1	2	yes	—
176	C₂²×C₄×D₃	C₂²×C₅	1	2	yes	—
177	C₂²×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
178	C₂²×Dic₆	C₂²×C₅	1	2	yes	—
179	C₃×C₄⋊Q₈	C₂²×C₅	1	8	yes	complemented
180	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
181	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
182	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
183	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
184	#946	C₂²×C₅	1	16	yes	—
185	#944	C₂²×C₅	1	16	yes	complemented
186	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	1	yes	characteristic
187	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	1	yes	characteristic
188	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
189	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
190	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
191	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
192	C₂×Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
193	C₂×Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
194	C₂×Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
195	C₂×Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
196	C₂×Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
197	C₂×Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
198	C₂×Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
199	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
200	C₂×C₄×Dic₃	C₂²×C₅	1	2	yes	—
201	#896	C₂²×C₅	1	16	yes	—
202	#892	C₂²×C₅	1	8	yes	—
203	D₃×C₄⋊C₄	C₂²×C₅	1	8	yes	—
204	#890	C₂²×C₅	1	8	yes	complemented
205	#889	C₂²×C₅	1	16	yes	complemented
206	#888	C₂²×C₅	1	8	yes	complemented
207	#872	C₂²×C₅	1	8	yes	—
208	#870	C₂²×C₅	1	8	yes	complemented
209	C₂⁴×C₅	—	3	1	no	—
210	C₂×C₅×Q₈	—	3	8	no	—
211	C₂×C₅×Q₈	—	4	4	no	—
212	C₂×C₅×Q₈	—	12	4	no	—
213	C₂×C₅×Q₈	C₂²×D₃	1	8	yes	—
214	C₂²×C₄×C₅	—	3	1	no	—
215	C₂²×C₄×C₅	—	3	8	no	—
216	C₂²×C₄×C₅	—	3	4	no	—
217	C₂²×C₄×C₅	—	3	1	no	—
218	C₂²×C₄×C₅	—	3	2	no	—
219	C₂²×C₄×C₅	—	3	1	no	—
220	C₂²×C₄×C₅	—	3	2	no	—
221	C₂²×C₄×C₅	—	3	8	no	—
222	C₂²×C₄×C₅	—	12	4	no	—
223	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
224	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
225	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
226	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
227	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
228	C₅×C₄⋊C₄	—	3	8	no	—
229	C₅×C₄⋊C₄	—	3	4	no	—
230	C₅×C₄⋊C₄	—	3	4	no	—
231	C₅×C₄⋊C₄	—	3	4	no	—
232	C₅×C₄⋊C₄	—	3	4	no	—
233	C₅×C₄⋊C₄	—	3	8	no	—
234	C₅×C₄⋊C₄	—	3	8	no	—
235	C₅×C₄⋊C₄	—	3	8	no	—
236	C₅×C₄⋊C₄	—	3	8	no	—
237	C₅×C₄⋊C₄	—	3	8	no	—
238	C₅×C₄⋊C₄	C₂²×D₃	1	4	yes	—
239	C₅×C₄⋊C₄	C₂²×D₃	1	4	yes	—
240	C₅×C₄⋊C₄	C₂²×D₃	1	8	yes	—
241	C₅×C₂²⋊C₄	—	3	8	no	—
242	C₅×C₂²⋊C₄	—	3	4	no	—
243	C₅×C₂²⋊C₄	—	3	4	no	—
244	C₄²×C₅	—	3	8	no	—
245	C₄²×C₅	—	3	4	no	—
246	C₄²×C₅	C₂²×D₃	1	4	yes	—
247	#544	—	3	16	no	—
248	C₂×C₄⋊Q₈	—	3	2	no	—
249	C₂×C₄⋊Q₈	—	3	1	no	—
250	C₂×C₄⋊Q₈	C₅×D₃	1	1	yes	characteristic, complemented
251	C₂×#125	—	3	2	no	—
252	C₂×#125	—	3	1	no	—
253	C₂×#125	—	3	1	no	—
254	C₂×C₂²⋊Q₈	—	3	2	no	—
255	C₂×C₂²⋊Q₈	—	3	2	no	—
256	C₂×#117	—	3	1	no	—
257	C₂×#117	—	3	1	no	—
258	C₂²×C₄⋊C₄	—	3	1	no	—
259	C₂²×C₃×C₅	C₂²×Q₈	1	2	yes	—
260	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
261	C₂²×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
262	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
263	C₂×C₅×D₃	—	2	8	no	—
264	C₂×C₅×D₃	—	2	2	no	—
265	C₂×C₅×D₃	—	2	2	no	—
266	C₂×C₅×D₃	C₂²×Q₈	1	4	yes	—
267	C₃×C₄×C₅	—	2	4	no	—
268	C₃×C₄×C₅	—	2	4	no	—
269	C₃×C₄×C₅	—	2	2	no	—
270	C₃×C₄×C₅	—	2	2	no	—
271	C₃×C₄×C₅	C₂²×Q₈	1	4	yes	—
272	C₅×Dic₃	—	2	2	no	—
273	C₅×Dic₃	—	2	4	no	—
274	C₅×Dic₃	—	2	2	no	—
275	C₅×Dic₃	—	2	4	no	—
276	C₅×Dic₃	C₂²×Q₈	1	4	yes	—
277	C₂³×D₃	C₂³×C₅	1	1	yes	characteristic
278	C₂×C₃×Q₈	—	4	4	no	—
279	C₂×C₃×Q₈	C₂³×C₅	1	8	yes	—
280	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
281	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
282	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
283	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
284	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
285	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
286	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
287	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
288	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
289	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
290	C₂×C₄×D₃	—	4	4	no	—
291	C₂×C₄×D₃	C₅×Q₈	1	8	yes	complemented
292	C₂×C₄×D₃	C₂³×C₅	1	8	yes	—
293	C₂×C₄×D₃	C₂³×C₅	1	4	yes	—
294	C₂×Dic₆	—	4	4	no	—
295	C₂×Dic₆	C₂³×C₅	1	8	yes	—
296	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
297	C₃×C₄⋊C₄	C₂³×C₅	1	8	yes	—
298	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
299	C₃×C₄²	C₂³×C₅	1	4	yes	—
300	(C₂×D₃)⋊C₄	C₂³×C₅	1	4	yes	—
301	(C₂×D₃)⋊C₄	C₂³×C₅	1	4	yes	—
302	(C₂×D₃)⋊C₄	C₂³×C₅	1	8	yes	—
303	C₄⋊Dic₃	C₂³×C₅	1	4	yes	—
304	C₄⋊Dic₃	C₂³×C₅	1	4	yes	—
305	C₄⋊Dic₃	C₂³×C₅	1	8	yes	—
306	Dic₃⋊C₄	C₂³×C₅	1	8	yes	—
307	Dic₃⋊C₄	C₂³×C₅	1	8	yes	—
308	Dic₃⋊C₄	C₂³×C₅	1	4	yes	—
309	Dic₃⋊C₄	C₂³×C₅	1	4	yes	—
310	Dic₃⋊C₄	C₂³×C₅	1	8	yes	—
311	Dic₃⋊C₄	C₂³×C₅	1	8	yes	—
312	Dic₃⋊C₄	C₂³×C₅	1	8	yes	—
313	C₄×Dic₃	C₂³×C₅	1	4	yes	—
314	C₄×Dic₃	C₂³×C₅	1	8	yes	—
315	C₂³×C₅	—	3	4	no	—
316	C₂³×C₅	—	3	2	no	—
317	C₂³×C₅	—	6	2	no	—
318	C₂³×C₅	—	6	2	no	—
319	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
320	C₅×Q₈	—	4	8	no	—
321	C₅×Q₈	—	12	8	no	—
322	C₂×C₄×C₅	—	2	4	no	—
323	C₂×C₄×C₅	—	2	2	no	—
324	C₂×C₄×C₅	—	2	4	no	—
325	C₂×C₄×C₅	—	2	2	no	—
326	C₂×C₄×C₅	—	3	4	no	—
327	C₂×C₄×C₅	—	3	2	no	—
328	C₂×C₄×C₅	—	3	16	no	—
329	C₂×C₄×C₅	—	3	2	no	—
330	C₂×C₄×C₅	—	3	4	no	—
331	C₂×C₄×C₅	—	3	2	no	—
332	C₂×C₄×C₅	—	3	4	no	—
333	C₂×C₄×C₅	—	6	4	no	—
334	C₂×C₄×C₅	—	6	2	no	—
335	C₂×C₄×C₅	—	6	2	no	—
336	C₂×C₄×C₅	—	6	4	no	—
337	C₂×C₄×C₅	—	12	8	no	—
338	C₂×C₄×C₅	D₃×Q₈	1	4	yes	—
339	C₂×C₄×C₅	C₂³×D₃	1	4	yes	—
340	C₂×C₄×C₅	C₂³×D₃	1	2	yes	—
341	C₂×C₄×C₅	C₂³×D₃	1	2	yes	—
342	C₂×C₄×C₅	C₂³×D₃	1	4	yes	—
343	C₂×C₄×C₅	C₂³×D₃	1	2	yes	—
344	C₂²×Q₈	—	3	2	no	—
345	C₂²×Q₈	C₂×C₅×D₃	1	2	yes	—
346	C₂³×C₄	—	3	1	no	—
347	C₂³×C₄	—	3	2	no	—
348	C₄⋊Q₈	—	3	16	no	—
349	C₄⋊Q₈	—	3	8	no	—
350	C₄⋊Q₈	C₂×C₅×D₃	1	8	yes	complemented
351	#125	—	3	16	no	—
352	#125	—	3	8	no	—
353	#125	—	3	8	no	—
354	C₂²⋊Q₈	—	3	16	no	—
355	C₂²⋊Q₈	—	3	16	no	—
356	#117	—	3	8	no	—
357	#117	—	3	8	no	—
358	C₂×C₄⋊C₄	—	3	2	no	—
359	C₂×C₄⋊C₄	—	3	1	no	—
360	C₂×C₄⋊C₄	—	3	1	no	—
361	C₂×C₄⋊C₄	—	3	1	no	—
362	C₂×C₄⋊C₄	—	3	1	no	—
363	C₂×C₄⋊C₄	—	3	2	no	—
364	C₂×C₄⋊C₄	—	3	2	no	—
365	C₂×C₄⋊C₄	—	3	2	no	—
366	C₂×C₄⋊C₄	—	3	2	no	—
367	C₂×C₄⋊C₄	—	3	2	no	—
368	C₂×C₄⋊C₄	—	3	8	no	—
369	C₂×C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
370	C₂×C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
371	C₂×C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
372	C₂×C₂²⋊C₄	—	3	2	no	—
373	C₂×C₂²⋊C₄	—	3	1	no	—
374	C₂×C₂²⋊C₄	—	3	1	no	—
375	C₂×C₄²	—	3	2	no	—
376	C₂×C₄²	—	3	1	no	—
377	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
378	C₂×C₃×C₅	#544	1	4	yes	complemented
379	C₂×C₃×C₅	C₂³×Q₈	1	1	yes	characteristic
380	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
381	C₂×C₃×C₅	C₂×2₋¹⁺⁴	1	1	yes	characteristic
382	C₅×D₃	—	2	4	no	—
383	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
384	C₂²×D₃	—	2	2	no	—
385	C₂²×D₃	—	2	2	no	—
386	C₂²×D₃	C₂×C₅×Q₈	1	4	yes	—
387	C₂²×D₃	C₂⁴×C₅	1	2	yes	—
388	C₃×Q₈	—	4	8	no	—
389	C₂×C₃×C₄	—	2	4	no	—
390	C₂×C₃×C₄	—	2	4	no	—
391	C₂×C₃×C₄	—	2	2	no	—
392	C₂×C₃×C₄	—	2	2	no	—
393	C₂×C₃×C₄	C₂×C₅×Q₈	1	4	yes	—
394	C₂×C₃×C₄	C₂⁴×C₅	1	4	yes	—
395	C₂×C₃×C₄	C₂⁴×C₅	1	2	yes	—
396	C₂×C₃×C₄	C₂⁴×C₅	1	2	yes	—
397	C₂×C₃×C₄	C₂⁴×C₅	1	4	yes	—
398	C₂×C₃×C₄	C₂⁴×C₅	1	2	yes	—
399	C₂×Dic₃	—	2	2	no	—
400	C₂×Dic₃	—	2	4	no	—
401	C₂×Dic₃	—	2	2	no	—
402	C₂×Dic₃	—	2	4	no	—
403	C₂×Dic₃	C₂×C₅×Q₈	1	4	yes	—
404	C₂×Dic₃	C₂⁴×C₅	1	2	yes	—
405	C₂×Dic₃	C₂⁴×C₅	1	4	yes	—
406	C₂×Dic₃	C₂⁴×C₅	1	2	yes	—
407	C₂×Dic₃	C₂⁴×C₅	1	4	yes	—
408	C₂×Dic₃	C₂⁴×C₅	1	2	yes	—
409	C₄×D₃	—	4	8	no	—
410	C₄×D₃	C₂×C₅×Q₈	1	16	yes	complemented
411	Dic₆	—	4	8	no	—
412	C₂²×C₅	—	3	4	no	—
413	C₂²×C₅	—	6	2	no	—
414	C₂²×C₅	—	6	2	no	—
415	C₂²×C₅	—	6	8	no	—
416	C₂²×C₅	#1004	1	2	yes	—
417	C₂²×C₅	C₂×D₃×Q₈	1	2	yes	—
418	C₂²×C₅	#1007	1	2	yes	—
419	C₂²×C₅	C₂⁴×D₃	1	1	yes	characteristic
420	C₄×C₅	—	2	4	no	—
421	C₄×C₅	—	2	2	no	—
422	C₄×C₅	—	2	4	no	—
423	C₄×C₅	—	2	2	no	—
424	C₄×C₅	—	3	4	no	—
425	C₄×C₅	—	6	4	no	—
426	C₄×C₅	—	6	2	no	—
427	C₄×C₅	—	6	2	no	—
428	C₄×C₅	—	6	4	no	—
429	C₄×C₅	C₂×D₃×Q₈	1	4	yes	—
430	C₂⁴	—	3	1	no	—
431	C₂×Q₈	—	3	8	no	—
432	C₂×Q₈	—	4	4	no	—
433	C₂×Q₈	—	12	4	no	—
434	C₂×Q₈	C₂²×C₅×D₃	1	8	yes	—
435	C₂²×C₄	—	3	1	no	—
436	C₂²×C₄	—	3	8	no	—
437	C₂²×C₄	—	3	4	no	—
438	C₂²×C₄	—	3	1	no	—
439	C₂²×C₄	—	3	2	no	—
440	C₂²×C₄	—	3	1	no	—
441	C₂²×C₄	—	3	2	no	—
442	C₂²×C₄	—	3	8	no	—
443	C₂²×C₄	—	12	4	no	—
444	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
445	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
446	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
447	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
448	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
449	C₄⋊C₄	—	3	8	no	—
450	C₄⋊C₄	—	3	4	no	—
451	C₄⋊C₄	—	3	4	no	—
452	C₄⋊C₄	—	3	4	no	—
453	C₄⋊C₄	—	3	4	no	—
454	C₄⋊C₄	—	3	8	no	—
455	C₄⋊C₄	—	3	8	no	—
456	C₄⋊C₄	—	3	8	no	—
457	C₄⋊C₄	—	3	8	no	—
458	C₄⋊C₄	—	3	8	no	—
459	C₄⋊C₄	C₂²×C₅×D₃	1	4	yes	—
460	C₄⋊C₄	C₂²×C₅×D₃	1	4	yes	—
461	C₄⋊C₄	C₂²×C₅×D₃	1	8	yes	—
462	C₂²⋊C₄	—	3	8	no	—
463	C₂²⋊C₄	—	3	4	no	—
464	C₂²⋊C₄	—	3	4	no	—
465	C₄²	—	3	8	no	—
466	C₄²	—	3	4	no	—
467	C₄²	C₂²×C₅×D₃	1	4	yes	—
468	C₃×C₅	C₂×#544	1	1	yes	characteristic, complemented, Hall
469	C₂²×C₃	C₂²×C₅×Q₈	1	2	yes	—
470	C₂²×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
471	C₂²×C₃	C₅×2₋¹⁺⁴	1	2	yes	—
472	C₂²×C₃	C₂⁵×C₅	1	1	yes	characteristic
473	C₂×D₃	—	2	8	no	—
474	C₂×D₃	—	2	2	no	—
475	C₂×D₃	—	2	2	no	—
476	C₂×D₃	C₂²×C₅×Q₈	1	4	yes	—
477	C₃×C₄	—	2	4	no	—
478	C₃×C₄	—	2	4	no	—
479	C₃×C₄	—	2	2	no	—
480	C₃×C₄	—	2	2	no	—
481	C₃×C₄	C₂²×C₅×Q₈	1	4	yes	—
482	Dic₃	—	2	2	no	—
483	Dic₃	—	2	4	no	—
484	Dic₃	—	2	2	no	—
485	Dic₃	—	2	4	no	—
486	Dic₃	C₂²×C₅×Q₈	1	4	yes	—
487	C₂×C₅	—	6	4	no	—
488	C₂×C₅	#5729	1	4	yes	direct product factor
489	C₂×C₅	C₂×#1004	1	1	yes	characteristic
490	C₂×C₅	C₂²×D₃×Q₈	1	1	yes	characteristic
491	C₂×C₅	C₂×#1007	1	1	yes	characteristic
492	C₂³	—	3	4	no	—
493	C₂³	—	3	2	no	—
494	C₂³	—	6	2	no	—
495	C₂³	—	6	2	no	—
496	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
497	Q₈	—	4	8	no	—
498	Q₈	—	12	8	no	—
499	C₂×C₄	—	2	4	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	3	4	no	—
504	C₂×C₄	—	3	2	no	—
505	C₂×C₄	—	3	16	no	—
506	C₂×C₄	—	3	2	no	—
507	C₂×C₄	—	3	4	no	—
508	C₂×C₄	—	3	2	no	—
509	C₂×C₄	—	3	4	no	—
510	C₂×C₄	—	6	4	no	—
511	C₂×C₄	—	6	2	no	—
512	C₂×C₄	—	6	2	no	—
513	C₂×C₄	—	6	4	no	—
514	C₂×C₄	—	12	8	no	—
515	C₂×C₄	C₅×D₃×Q₈	1	4	yes	—
516	C₂×C₄	C₂³×C₅×D₃	1	4	yes	—
517	C₂×C₄	C₂³×C₅×D₃	1	2	yes	—
518	C₂×C₄	C₂³×C₅×D₃	1	2	yes	—
519	C₂×C₄	C₂³×C₅×D₃	1	4	yes	—
520	C₂×C₄	C₂³×C₅×D₃	1	2	yes	—
521	C₂×C₃	C₅×#544	1	4	yes	complemented
522	C₂×C₃	C₂³×C₅×Q₈	1	1	yes	characteristic
523	C₂×C₃	C₂×C₅×H(5,𝔽₂)	1	1	yes	characteristic
524	C₂×C₃	C₂×C₅×2₋¹⁺⁴	1	1	yes	characteristic
525	D₃	—	2	4	no	—
526	C₅	C₂×#5729	1	1	yes	characteristic, direct product factor, Sylow
527	C₂²	—	3	4	no	—
528	C₂²	—	6	2	no	—
529	C₂²	—	6	2	no	—
530	C₂²	—	6	8	no	—
531	C₂²	C₅×#1004	1	2	yes	—
532	C₂²	C₂×C₅×D₃×Q₈	1	2	yes	—
533	C₂²	C₅×#1007	1	2	yes	—
534	C₂²	C₂⁴×C₅×D₃	1	1	yes	characteristic
535	C₄	—	2	4	no	—
536	C₄	—	2	2	no	—
537	C₄	—	2	4	no	—
538	C₄	—	2	2	no	—
539	C₄	—	3	4	no	—
540	C₄	—	6	4	no	—
541	C₄	—	6	2	no	—
542	C₄	—	6	2	no	—
543	C₄	—	6	4	no	—
544	C₄	C₂×C₅×D₃×Q₈	1	4	yes	—
545	C₃	C₂×C₅×#544	1	1	yes	characteristic, complemented, Sylow
546	C₂	—	6	4	no	—
547	C₂	C₅×#5729	1	4	yes	direct product factor
548	C₂	C₂×C₅×#1004	1	1	yes	characteristic
549	C₂	C₂²×C₅×D₃×Q₈	1	1	yes	characteristic
550	C₂	C₂×C₅×#1007	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×C₄⋊Q₈	C₂
socle	C₂³×C₃×C₅	C₂⁴
prime radical	C₂³×C₅×D₃	C₂³
hypercenter	C₂×C₅×C₄⋊Q₈	D₃
hypocenter	C₃	C₂×C₅×#544
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₅×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×C₄⋊Q₈	C₂	semidirect
3. C₂×C₅×#944	C₂	semidirect
4. C₅×#5729	C₂	direct (N × H)
8. C₂×C₅×#890	C₂	semidirect
9. C₂×C₅×#889	C₂	semidirect
10. C₂×C₅×#888	C₂	semidirect
12. C₂×C₅×#870	C₂	semidirect
18. C₃×C₅×C₄⋊Q₈	C₂²	semidirect
24. C₅×#944	C₂²	semidirect
43. C₅×#890	C₂²	semidirect
44. C₅×#889	C₂²	semidirect
45. C₅×#888	C₂²	semidirect
47. C₅×#870	C₂²	semidirect
48. C₂×#5729	C₅	direct (N × H)
52. C₂×C₅×C₄⋊Q₈	D₃	semidirect
75. C₂×C₄×C₅×D₃	Q₈	semidirect
99. C₂×C₃×C₄⋊Q₈	C₂×C₅	semidirect
101. C₂×#944	C₂×C₅	semidirect
102. #5729	C₂×C₅	direct (N × H)
106. C₂×#890	C₂×C₅	semidirect
107. C₂×#889	C₂×C₅	semidirect
108. C₂×#888	C₂×C₅	semidirect
110. C₂×#870	C₂×C₅	semidirect
117. C₅×C₄⋊Q₈	C₂×D₃	semidirect
173. C₄×C₅×D₃	C₂×Q₈	semidirect
179. C₃×C₄⋊Q₈	C₂²×C₅	semidirect
185. #944	C₂²×C₅	semidirect
204. #890	C₂²×C₅	semidirect
205. #889	C₂²×C₅	semidirect
206. #888	C₂²×C₅	semidirect
208. #870	C₂²×C₅	semidirect
250. C₂×C₄⋊Q₈	C₅×D₃	semidirect
291. C₂×C₄×D₃	C₅×Q₈	semidirect
350. C₄⋊Q₈	C₂×C₅×D₃	semidirect
378. C₂×C₃×C₅	#544	semidirect
410. C₄×D₃	C₂×C₅×Q₈	semidirect
468. C₃×C₅	C₂×#544	semidirect
488. C₂×C₅	#5729	direct (N × H)
521. C₂×C₃	C₅×#544	semidirect
526. C₅	C₂×#5729	direct (N × H)
545. C₃	C₂×C₅×#544	semidirect
547. C₂	C₅×#5729	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×C₄⋊Q₈	2
C₂×C₅×#946	2
C₂×C₅×#944	2
C₅×#5729	2
C₂×C₅×#896	2
C₂×C₅×#892	2
C₂×C₅×D₃×C₄⋊C₄	2
C₂×C₅×#890	2
C₂×C₅×#889	2
C₂×C₅×#888	2
C₂×C₅×#872	2
C₂×C₅×#870	2
C₂×C₅×#544	3
C₂×#5729	5

Subgroups — C2×C5×#5729 #14481729

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#5729 #14481729