Subgroups — C₂×C₅×#5578 #14480117

Order 1920.

2336 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×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#946	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#944	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5578	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₅×#897	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×#896	C₂	1	2	yes	maximal
7	C₂×C₅×#889	C₂	1	2	yes	maximal, complemented
8	C₂×C₅×#888	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅×#872	C₂	1	2	yes	maximal
10	C₂×C₄²×C₅×D₃	C₂	1	1	yes	maximal, characteristic
11	C₂×C₄×C₅×Dic₆	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₅×#533	—	3	1	no	maximal, Hall
14	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
18	C₂²×C₅×Dic₆	C₂²	1	2	yes	—
19	C₂²×C₅×Dic₆	C₂²	1	1	yes	characteristic
20	C₃×C₄×C₅×Q₈	C₂²	1	8	yes	complemented
21	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
23	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
24	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
25	C₅×#946	C₂²	1	8	yes	—
26	C₅×#944	C₂²	1	8	yes	complemented
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	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	2	yes	—
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	2	yes	—
38	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
40	C₅×#897	C₂²	1	8	yes	—
41	C₅×#896	C₂²	1	16	yes	—
42	C₅×#889	C₂²	1	16	yes	complemented
43	C₅×#888	C₂²	1	8	yes	complemented
44	C₅×#872	C₂²	1	16	yes	—
45	C₄²×C₅×D₃	C₂²	1	8	yes	—
46	C₄×C₅×Dic₆	C₂²	1	16	yes	complemented
47	C₄×C₅×Dic₆	C₂²	1	8	yes	complemented
48	C₂×#5578	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
49	C₅×#533	—	3	16	no	—
50	C₂×C₅×C₄⋊Q₈	—	3	1	no	—
51	C₂×C₅×C₄⋊Q₈	—	3	1	no	—
52	C₂×C₅×#125	—	3	2	no	—
53	C₂×C₅×C₂²⋊Q₈	—	3	1	no	—
54	C₂×C₅×C₂²⋊Q₈	—	3	2	no	—
55	C₂×C₅×C₂²⋊Q₈	—	3	1	no	—
56	C₂×C₄×C₅×Q₈	—	3	2	no	—
57	C₂×C₄×C₅×Q₈	—	3	1	no	—
58	C₂×C₄×C₅×Q₈	D₃	1	1	yes	characteristic, complemented
59	C₂×C₅×#117	—	3	2	no	—
60	C₂²×C₄²×C₅	—	3	1	no	—
61	C₂³×C₅×D₃	C₂³	1	1	yes	characteristic
62	C₂×C₃×C₅×Q₈	—	2	4	no	—
63	C₂×C₃×C₅×Q₈	C₂³	1	4	yes	—
64	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
65	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
66	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
67	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
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	2	yes	—
72	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
73	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
74	C₂×C₄×C₅×D₃	—	2	4	no	—
75	C₂×C₄×C₅×D₃	—	2	4	no	—
76	C₂×C₄×C₅×D₃	Q₈	1	8	yes	complemented
77	C₂×C₄×C₅×D₃	C₂³	1	4	yes	—
78	C₂×C₄×C₅×D₃	C₂³	1	4	yes	—
79	C₂×C₄×C₅×D₃	C₂³	1	4	yes	—
80	C₂×C₅×Dic₆	—	2	4	no	—
81	C₂×C₅×Dic₆	—	2	8	no	—
82	C₂×C₅×Dic₆	C₂³	1	8	yes	—
83	C₂×C₅×Dic₆	C₂³	1	4	yes	—
84	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
85	C₃×C₅×C₄⋊C₄	C₂³	1	8	yes	—
86	C₃×C₄²×C₅	C₂³	1	4	yes	—
87	C₃×C₄²×C₅	C₂³	1	8	yes	—
88	C₅×(C₂×D₃)⋊C₄	C₂³	1	8	yes	—
89	C₅×(C₂×D₃)⋊C₄	C₂³	1	8	yes	—
90	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
91	C₅×C₄⋊Dic₃	C₂³	1	8	yes	—
92	C₅×Dic₃⋊C₄	C₂³	1	8	yes	—
93	C₅×Dic₃⋊C₄	C₂³	1	8	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	C₄×C₅×Dic₃	C₂³	1	4	yes	—
100	C₄×C₅×Dic₃	C₂³	1	4	yes	—
101	C₂×C₃×C₄×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
102	C₂×#946	C₂×C₅	1	1	yes	characteristic
103	C₂×#944	C₂×C₅	1	1	yes	characteristic, complemented
104	#5578	C₂×C₅	1	16	yes	direct product factor
105	C₂×#897	C₂×C₅	1	1	yes	characteristic
106	C₂×#896	C₂×C₅	1	2	yes	—
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	2	yes	—
110	C₂×C₄²×D₃	C₂×C₅	1	1	yes	characteristic
111	C₂×C₄×Dic₆	C₂×C₅	1	2	yes	complemented
112	C₂×C₄×Dic₆	C₂×C₅	1	1	yes	characteristic, complemented
113	C₂²×C₅×Q₈	—	3	2	no	—
114	C₂²×C₅×Q₈	—	3	1	no	—
115	C₂²×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
116	C₂³×C₄×C₅	—	3	1	no	—
117	C₂³×C₄×C₅	—	3	1	no	—
118	C₂³×C₄×C₅	—	3	1	no	—
119	C₅×C₄⋊Q₈	—	3	8	no	—
120	C₅×C₄⋊Q₈	—	3	8	no	—
121	C₅×#125	—	3	16	no	—
122	C₅×C₂²⋊Q₈	—	3	8	no	—
123	C₅×C₂²⋊Q₈	—	3	16	no	—
124	C₅×C₂²⋊Q₈	—	3	8	no	—
125	C₄×C₅×Q₈	—	3	16	no	—
126	C₄×C₅×Q₈	—	3	8	no	—
127	C₄×C₅×Q₈	C₂×D₃	1	8	yes	complemented
128	C₅×#117	—	3	16	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	2	no	—
133	C₂×C₅×C₄⋊C₄	—	3	2	no	—
134	C₂×C₅×C₄⋊C₄	—	3	2	no	—
135	C₂×C₅×C₄⋊C₄	—	3	1	no	—
136	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
137	C₂×C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
138	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
139	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
140	C₂×C₄²×C₅	—	3	1	no	—
141	C₂×C₄²×C₅	—	3	1	no	—
142	C₂×C₄²×C₅	—	3	1	no	—
143	C₂×C₄²×C₅	—	3	2	no	—
144	C₂×C₄²×C₅	—	3	8	no	—
145	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
146	C₂×C₄²×C₅	C₂×D₃	1	2	yes	—
147	C₂×#533	—	3	1	no	Sylow
148	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
149	C₂²×C₅×D₃	—	2	2	no	—
150	C₂²×C₅×D₃	—	2	2	no	—
151	C₂²×C₅×D₃	C₂×Q₈	1	4	yes	—
152	C₂²×C₅×D₃	C₂⁴	1	2	yes	—
153	C₃×C₅×Q₈	—	2	8	no	—
154	C₂×C₃×C₄×C₅	—	2	4	no	—
155	C₂×C₃×C₄×C₅	—	2	2	no	—
156	C₂×C₃×C₄×C₅	—	2	4	no	—
157	C₂×C₃×C₄×C₅	C₂×Q₈	1	4	yes	—
158	C₂×C₃×C₄×C₅	#41	1	4	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	4	yes	—
162	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
163	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
164	C₂×C₅×Dic₃	—	2	4	no	—
165	C₂×C₅×Dic₃	—	2	4	no	—
166	C₂×C₅×Dic₃	—	2	2	no	—
167	C₂×C₅×Dic₃	C₂×Q₈	1	4	yes	—
168	C₂×C₅×Dic₃	#41	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	4	yes	—
172	C₂×C₅×Dic₃	C₂⁴	1	2	yes	—
173	C₂×C₅×Dic₃	C₂⁴	1	2	yes	—
174	C₄×C₅×D₃	—	2	8	no	—
175	C₄×C₅×D₃	—	2	8	no	—
176	C₄×C₅×D₃	C₂×Q₈	1	16	yes	complemented
177	C₅×Dic₆	—	2	8	no	—
178	C₅×Dic₆	—	2	16	no	—
179	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
180	C₂²×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
181	C₂²×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
182	C₂²×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
183	C₂²×Dic₆	C₂²×C₅	1	2	yes	—
184	C₂²×Dic₆	C₂²×C₅	1	1	yes	characteristic
185	C₃×C₄×Q₈	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	2	yes	—
188	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
189	C₂×C₃×C₄²	C₂²×C₅	1	2	yes	—
190	#946	C₂²×C₅	1	8	yes	—
191	#944	C₂²×C₅	1	8	yes	complemented
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	2	yes	—
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×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	2	yes	—
203	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
204	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
205	#897	C₂²×C₅	1	8	yes	—
206	#896	C₂²×C₅	1	16	yes	—
207	#889	C₂²×C₅	1	16	yes	complemented
208	#888	C₂²×C₅	1	8	yes	complemented
209	#872	C₂²×C₅	1	16	yes	—
210	C₄²×D₃	C₂²×C₅	1	8	yes	—
211	C₄×Dic₆	C₂²×C₅	1	16	yes	complemented
212	C₄×Dic₆	C₂²×C₅	1	8	yes	complemented
213	C₂⁴×C₅	—	3	1	no	—
214	C₂×C₅×Q₈	—	2	4	no	—
215	C₂×C₅×Q₈	—	3	8	no	—
216	C₂×C₅×Q₈	—	3	4	no	—
217	C₂×C₅×Q₈	—	6	8	no	—
218	C₂×C₅×Q₈	—	6	4	no	—
219	C₂×C₅×Q₈	C₂²×D₃	1	4	yes	—
220	C₂²×C₄×C₅	—	3	1	no	—
221	C₂²×C₄×C₅	—	3	1	no	—
222	C₂²×C₄×C₅	—	3	8	no	—
223	C₂²×C₄×C₅	—	3	4	no	—
224	C₂²×C₄×C₅	—	3	1	no	—
225	C₂²×C₄×C₅	—	3	2	no	—
226	C₂²×C₄×C₅	—	3	2	no	—
227	C₂²×C₄×C₅	—	3	4	no	—
228	C₂²×C₄×C₅	—	3	4	no	—
229	C₂²×C₄×C₅	—	6	4	no	—
230	C₂²×C₄×C₅	—	6	4	no	—
231	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
232	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
233	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
234	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
235	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
236	C₅×C₄⋊C₄	—	3	8	no	—
237	C₅×C₄⋊C₄	—	3	8	no	—
238	C₅×C₄⋊C₄	—	3	8	no	—
239	C₅×C₄⋊C₄	—	3	8	no	—
240	C₅×C₄⋊C₄	—	3	8	no	—
241	C₅×C₄⋊C₄	—	3	8	no	—
242	C₅×C₄⋊C₄	—	3	4	no	—
243	C₅×C₄⋊C₄	C₂²×D₃	1	8	yes	—
244	C₅×C₄⋊C₄	C₂²×D₃	1	4	yes	—
245	C₅×C₂²⋊C₄	—	3	8	no	—
246	C₅×C₂²⋊C₄	—	3	8	no	—
247	C₄²×C₅	—	3	4	no	—
248	C₄²×C₅	—	3	4	no	—
249	C₄²×C₅	—	3	4	no	—
250	C₄²×C₅	—	3	8	no	—
251	C₄²×C₅	C₂²×D₃	1	4	yes	—
252	C₄²×C₅	C₂²×D₃	1	8	yes	—
253	#533	—	3	16	no	—
254	C₂×C₄⋊Q₈	—	3	1	no	—
255	C₂×C₄⋊Q₈	—	3	1	no	—
256	C₂×#125	—	3	2	no	—
257	C₂×C₂²⋊Q₈	—	3	1	no	—
258	C₂×C₂²⋊Q₈	—	3	2	no	—
259	C₂×C₂²⋊Q₈	—	3	1	no	—
260	C₂×C₄×Q₈	—	3	1	no	—
261	C₂×C₄×Q₈	—	3	2	no	—
262	C₂×C₄×Q₈	C₅×D₃	1	1	yes	characteristic, complemented
263	C₂×#117	—	3	2	no	—
264	C₂²×C₄²	—	3	1	no	—
265	C₂²×C₃×C₅	C₂²×Q₈	1	2	yes	—
266	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
267	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
268	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
269	C₂×C₅×D₃	—	2	8	no	—
270	C₂×C₅×D₃	—	2	2	no	—
271	C₂×C₅×D₃	—	2	2	no	—
272	C₂×C₅×D₃	C₂²×Q₈	1	4	yes	—
273	C₃×C₄×C₅	—	2	4	no	—
274	C₃×C₄×C₅	—	2	2	no	—
275	C₃×C₄×C₅	—	2	4	no	—
276	C₃×C₄×C₅	C₂²×Q₈	1	4	yes	—
277	C₃×C₄×C₅	C₂×#41	1	4	yes	—
278	C₅×Dic₃	—	2	4	no	—
279	C₅×Dic₃	—	2	4	no	—
280	C₅×Dic₃	—	2	2	no	—
281	C₅×Dic₃	C₂²×Q₈	1	4	yes	—
282	C₅×Dic₃	C₂×#41	1	4	yes	—
283	C₂³×D₃	C₂³×C₅	1	1	yes	characteristic
284	C₂×C₃×Q₈	—	2	4	no	—
285	C₂×C₃×Q₈	C₂³×C₅	1	4	yes	—
286	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
287	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
288	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
289	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
290	C₂²×C₃×C₄	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	2	yes	—
294	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
295	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
296	C₂×C₄×D₃	—	2	4	no	—
297	C₂×C₄×D₃	—	2	4	no	—
298	C₂×C₄×D₃	C₅×Q₈	1	8	yes	complemented
299	C₂×C₄×D₃	C₂³×C₅	1	4	yes	—
300	C₂×C₄×D₃	C₂³×C₅	1	4	yes	—
301	C₂×C₄×D₃	C₂³×C₅	1	4	yes	—
302	C₂×Dic₆	—	2	4	no	—
303	C₂×Dic₆	—	2	8	no	—
304	C₂×Dic₆	C₂³×C₅	1	8	yes	—
305	C₂×Dic₆	C₂³×C₅	1	4	yes	—
306	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
307	C₃×C₄⋊C₄	C₂³×C₅	1	8	yes	—
308	C₃×C₄²	C₂³×C₅	1	4	yes	—
309	C₃×C₄²	C₂³×C₅	1	8	yes	—
310	(C₂×D₃)⋊C₄	C₂³×C₅	1	8	yes	—
311	(C₂×D₃)⋊C₄	C₂³×C₅	1	8	yes	—
312	C₄⋊Dic₃	C₂³×C₅	1	4	yes	—
313	C₄⋊Dic₃	C₂³×C₅	1	8	yes	—
314	Dic₃⋊C₄	C₂³×C₅	1	8	yes	—
315	Dic₃⋊C₄	C₂³×C₅	1	8	yes	—
316	Dic₃⋊C₄	C₂³×C₅	1	8	yes	—
317	Dic₃⋊C₄	C₂³×C₅	1	8	yes	—
318	Dic₃⋊C₄	C₂³×C₅	1	8	yes	—
319	C₄×Dic₃	C₂³×C₅	1	4	yes	—
320	C₄×Dic₃	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	4	no	—
324	C₂³×C₅	—	3	2	no	—
325	C₂³×C₅	—	6	2	no	—
326	C₂³×C₅	—	6	2	no	—
327	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
328	C₅×Q₈	—	2	8	no	—
329	C₅×Q₈	—	6	16	no	—
330	C₅×Q₈	—	6	8	no	—
331	C₂×C₄×C₅	—	2	4	no	—
332	C₂×C₄×C₅	—	2	2	no	—
333	C₂×C₄×C₅	—	2	4	no	—
334	C₂×C₄×C₅	—	3	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	16	no	—
339	C₂×C₄×C₅	—	3	2	no	—
340	C₂×C₄×C₅	—	3	4	no	—
341	C₂×C₄×C₅	—	3	4	no	—
342	C₂×C₄×C₅	—	6	8	no	—
343	C₂×C₄×C₅	—	6	2	no	—
344	C₂×C₄×C₅	—	6	4	no	—
345	C₂×C₄×C₅	—	6	4	no	—
346	C₂×C₄×C₅	—	6	8	no	—
347	C₂×C₄×C₅	#235	1	4	yes	—
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	4	yes	—
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	C₂²×Q₈	—	3	2	no	—
355	C₂²×Q₈	—	3	1	no	—
356	C₂²×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
357	C₂³×C₄	—	3	1	no	—
358	C₂³×C₄	—	3	1	no	—
359	C₂³×C₄	—	3	1	no	—
360	C₄⋊Q₈	—	3	8	no	—
361	C₄⋊Q₈	—	3	8	no	—
362	#125	—	3	16	no	—
363	C₂²⋊Q₈	—	3	8	no	—
364	C₂²⋊Q₈	—	3	16	no	—
365	C₂²⋊Q₈	—	3	8	no	—
366	C₄×Q₈	—	3	16	no	—
367	C₄×Q₈	—	3	8	no	—
368	C₄×Q₈	C₂×C₅×D₃	1	8	yes	complemented
369	#117	—	3	16	no	—
370	C₂×C₄⋊C₄	—	3	2	no	—
371	C₂×C₄⋊C₄	—	3	2	no	—
372	C₂×C₄⋊C₄	—	3	2	no	—
373	C₂×C₄⋊C₄	—	3	2	no	—
374	C₂×C₄⋊C₄	—	3	2	no	—
375	C₂×C₄⋊C₄	—	3	2	no	—
376	C₂×C₄⋊C₄	—	3	1	no	—
377	C₂×C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
378	C₂×C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
379	C₂×C₂²⋊C₄	—	3	2	no	—
380	C₂×C₂²⋊C₄	—	3	2	no	—
381	C₂×C₄²	—	3	2	no	—
382	C₂×C₄²	—	3	1	no	—
383	C₂×C₄²	—	3	1	no	—
384	C₂×C₄²	—	3	1	no	—
385	C₂×C₄²	—	3	8	no	—
386	C₂×C₄²	C₂×C₅×D₃	1	2	yes	—
387	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
388	C₂×C₃×C₅	#533	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	C₂²×#41	1	1	yes	characteristic
392	C₅×D₃	—	2	4	no	—
393	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
394	C₂²×D₃	—	2	2	no	—
395	C₂²×D₃	—	2	2	no	—
396	C₂²×D₃	C₂×C₅×Q₈	1	4	yes	—
397	C₂²×D₃	C₂⁴×C₅	1	2	yes	—
398	C₃×Q₈	—	2	8	no	—
399	C₂×C₃×C₄	—	2	4	no	—
400	C₂×C₃×C₄	—	2	2	no	—
401	C₂×C₃×C₄	—	2	4	no	—
402	C₂×C₃×C₄	C₂×C₅×Q₈	1	4	yes	—
403	C₂×C₃×C₄	C₅×#41	1	4	yes	—
404	C₂×C₃×C₄	C₂⁴×C₅	1	2	yes	—
405	C₂×C₃×C₄	C₂⁴×C₅	1	4	yes	—
406	C₂×C₃×C₄	C₂⁴×C₅	1	4	yes	—
407	C₂×C₃×C₄	C₂⁴×C₅	1	2	yes	—
408	C₂×C₃×C₄	C₂⁴×C₅	1	2	yes	—
409	C₂×Dic₃	—	2	4	no	—
410	C₂×Dic₃	—	2	4	no	—
411	C₂×Dic₃	—	2	2	no	—
412	C₂×Dic₃	C₂×C₅×Q₈	1	4	yes	—
413	C₂×Dic₃	C₅×#41	1	4	yes	—
414	C₂×Dic₃	C₂⁴×C₅	1	2	yes	—
415	C₂×Dic₃	C₂⁴×C₅	1	4	yes	—
416	C₂×Dic₃	C₂⁴×C₅	1	4	yes	—
417	C₂×Dic₃	C₂⁴×C₅	1	2	yes	—
418	C₂×Dic₃	C₂⁴×C₅	1	2	yes	—
419	C₄×D₃	—	2	8	no	—
420	C₄×D₃	—	2	8	no	—
421	C₄×D₃	C₂×C₅×Q₈	1	16	yes	complemented
422	Dic₆	—	2	8	no	—
423	Dic₆	—	2	16	no	—
424	C₂²×C₅	—	3	4	no	—
425	C₂²×C₅	—	6	2	no	—
426	C₂²×C₅	—	6	2	no	—
427	C₂²×C₅	—	6	8	no	—
428	C₂²×C₅	C₂×#235	1	2	yes	—
429	C₂²×C₅	C₂×D₃×Q₈	1	2	yes	—
430	C₂²×C₅	D₃×#41	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	4	no	—
435	C₄×C₅	—	3	4	no	—
436	C₄×C₅	—	3	4	no	—
437	C₄×C₅	—	6	2	no	—
438	C₄×C₅	—	6	4	no	—
439	C₄×C₅	—	6	4	no	—
440	C₄×C₅	C₂×#235	1	4	yes	—
441	C₄×C₅	C₂×D₃×Q₈	1	4	yes	—
442	C₂⁴	—	3	1	no	—
443	C₂×Q₈	—	2	4	no	—
444	C₂×Q₈	—	3	8	no	—
445	C₂×Q₈	—	3	4	no	—
446	C₂×Q₈	—	6	8	no	—
447	C₂×Q₈	—	6	4	no	—
448	C₂×Q₈	C₂²×C₅×D₃	1	4	yes	—
449	C₂²×C₄	—	3	1	no	—
450	C₂²×C₄	—	3	1	no	—
451	C₂²×C₄	—	3	8	no	—
452	C₂²×C₄	—	3	4	no	—
453	C₂²×C₄	—	3	1	no	—
454	C₂²×C₄	—	3	2	no	—
455	C₂²×C₄	—	3	2	no	—
456	C₂²×C₄	—	3	4	no	—
457	C₂²×C₄	—	3	4	no	—
458	C₂²×C₄	—	6	4	no	—
459	C₂²×C₄	—	6	4	no	—
460	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
461	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
462	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
463	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
464	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
465	C₄⋊C₄	—	3	8	no	—
466	C₄⋊C₄	—	3	8	no	—
467	C₄⋊C₄	—	3	8	no	—
468	C₄⋊C₄	—	3	8	no	—
469	C₄⋊C₄	—	3	8	no	—
470	C₄⋊C₄	—	3	8	no	—
471	C₄⋊C₄	—	3	4	no	—
472	C₄⋊C₄	C₂²×C₅×D₃	1	8	yes	—
473	C₄⋊C₄	C₂²×C₅×D₃	1	4	yes	—
474	C₂²⋊C₄	—	3	8	no	—
475	C₂²⋊C₄	—	3	8	no	—
476	C₄²	—	3	4	no	—
477	C₄²	—	3	4	no	—
478	C₄²	—	3	4	no	—
479	C₄²	—	3	8	no	—
480	C₄²	C₂²×C₅×D₃	1	4	yes	—
481	C₄²	C₂²×C₅×D₃	1	8	yes	—
482	C₃×C₅	C₂×#533	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	C₂×C₅×#41	1	2	yes	—
486	C₂²×C₃	C₂⁵×C₅	1	1	yes	characteristic
487	C₂×D₃	—	2	8	no	—
488	C₂×D₃	—	2	2	no	—
489	C₂×D₃	—	2	2	no	—
490	C₂×D₃	C₂²×C₅×Q₈	1	4	yes	—
491	C₃×C₄	—	2	4	no	—
492	C₃×C₄	—	2	2	no	—
493	C₃×C₄	—	2	4	no	—
494	C₃×C₄	C₂²×C₅×Q₈	1	4	yes	—
495	C₃×C₄	C₂×C₅×#41	1	4	yes	—
496	Dic₃	—	2	4	no	—
497	Dic₃	—	2	4	no	—
498	Dic₃	—	2	2	no	—
499	Dic₃	C₂²×C₅×Q₈	1	4	yes	—
500	Dic₃	C₂×C₅×#41	1	4	yes	—
501	C₂×C₅	—	6	4	no	—
502	C₂×C₅	#5578	1	4	yes	direct product factor
503	C₂×C₅	C₂²×#235	1	1	yes	characteristic
504	C₂×C₅	C₂²×D₃×Q₈	1	1	yes	characteristic
505	C₂×C₅	C₂×D₃×#41	1	1	yes	characteristic
506	C₂³	—	3	4	no	—
507	C₂³	—	3	2	no	—
508	C₂³	—	6	2	no	—
509	C₂³	—	6	2	no	—
510	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
511	Q₈	—	2	8	no	—
512	Q₈	—	6	16	no	—
513	Q₈	—	6	8	no	—
514	C₂×C₄	—	2	4	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	4	no	—
517	C₂×C₄	—	3	4	no	—
518	C₂×C₄	—	3	4	no	—
519	C₂×C₄	—	3	2	no	—
520	C₂×C₄	—	3	2	no	—
521	C₂×C₄	—	3	16	no	—
522	C₂×C₄	—	3	2	no	—
523	C₂×C₄	—	3	4	no	—
524	C₂×C₄	—	3	4	no	—
525	C₂×C₄	—	6	8	no	—
526	C₂×C₄	—	6	2	no	—
527	C₂×C₄	—	6	4	no	—
528	C₂×C₄	—	6	4	no	—
529	C₂×C₄	—	6	8	no	—
530	C₂×C₄	C₅×#235	1	4	yes	—
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	4	yes	—
534	C₂×C₄	C₂³×C₅×D₃	1	4	yes	—
535	C₂×C₄	C₂³×C₅×D₃	1	2	yes	—
536	C₂×C₄	C₂³×C₅×D₃	1	2	yes	—
537	C₂×C₃	C₅×#533	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	C₂²×C₅×#41	1	1	yes	characteristic
541	D₃	—	2	4	no	—
542	C₅	C₂×#5578	1	1	yes	characteristic, direct product factor, Sylow
543	C₂²	—	3	4	no	—
544	C₂²	—	6	2	no	—
545	C₂²	—	6	2	no	—
546	C₂²	—	6	8	no	—
547	C₂²	C₂×C₅×#235	1	2	yes	—
548	C₂²	C₂×C₅×D₃×Q₈	1	2	yes	—
549	C₂²	C₅×D₃×#41	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	4	no	—
554	C₄	—	3	4	no	—
555	C₄	—	3	4	no	—
556	C₄	—	6	2	no	—
557	C₄	—	6	4	no	—
558	C₄	—	6	4	no	—
559	C₄	C₂×C₅×#235	1	4	yes	—
560	C₄	C₂×C₅×D₃×Q₈	1	4	yes	—
561	C₃	C₂×C₅×#533	1	1	yes	characteristic, complemented, Sylow
562	C₂	—	6	4	no	—
563	C₂	C₅×#5578	1	4	yes	direct product factor
564	C₂	C₂²×C₅×#235	1	1	yes	characteristic
565	C₂	C₂²×C₅×D₃×Q₈	1	1	yes	characteristic
566	C₂	C₂×C₅×D₃×#41	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂⁵×C₅
Frattini	C₂²	C₂⁴×C₅×D₃
Fitting	C₂×C₃×C₄×C₅×Q₈	C₂
socle	C₂³×C₃×C₅	C₂⁴
prime radical	C₂³×C₅×D₃	C₂³
hypercenter	C₂×C₄×C₅×Q₈	D₃
hypocenter	C₃	C₂×C₅×#533
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂×C₄×C₅×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₄×C₅×Q₈	C₂	semidirect
3. C₂×C₅×#944	C₂	semidirect
4. C₅×#5578	C₂	direct (N × H)
7. C₂×C₅×#889	C₂	semidirect
8. C₂×C₅×#888	C₂	semidirect
11. C₂×C₄×C₅×Dic₆	C₂	semidirect
12. C₂×C₄×C₅×Dic₆	C₂	semidirect
20. C₃×C₄×C₅×Q₈	C₂²	semidirect
26. C₅×#944	C₂²	semidirect
42. C₅×#889	C₂²	semidirect
43. C₅×#888	C₂²	semidirect
46. C₄×C₅×Dic₆	C₂²	semidirect
47. C₄×C₅×Dic₆	C₂²	semidirect
48. C₂×#5578	C₅	direct (N × H)
58. C₂×C₄×C₅×Q₈	D₃	semidirect
76. C₂×C₄×C₅×D₃	Q₈	semidirect
101. C₂×C₃×C₄×Q₈	C₂×C₅	semidirect
103. C₂×#944	C₂×C₅	semidirect
104. #5578	C₂×C₅	direct (N × H)
107. C₂×#889	C₂×C₅	semidirect
108. C₂×#888	C₂×C₅	semidirect
111. C₂×C₄×Dic₆	C₂×C₅	semidirect
112. C₂×C₄×Dic₆	C₂×C₅	semidirect
127. C₄×C₅×Q₈	C₂×D₃	semidirect
176. C₄×C₅×D₃	C₂×Q₈	semidirect
185. C₃×C₄×Q₈	C₂²×C₅	semidirect
191. #944	C₂²×C₅	semidirect
207. #889	C₂²×C₅	semidirect
208. #888	C₂²×C₅	semidirect
211. C₄×Dic₆	C₂²×C₅	semidirect
212. C₄×Dic₆	C₂²×C₅	semidirect
262. C₂×C₄×Q₈	C₅×D₃	semidirect
298. C₂×C₄×D₃	C₅×Q₈	semidirect
368. C₄×Q₈	C₂×C₅×D₃	semidirect
388. C₂×C₃×C₅	#533	semidirect
421. C₄×D₃	C₂×C₅×Q₈	semidirect
482. C₃×C₅	C₂×#533	semidirect
502. C₂×C₅	#5578	direct (N × H)
537. C₂×C₃	C₅×#533	semidirect
542. C₅	C₂×#5578	direct (N × H)
561. C₃	C₂×C₅×#533	semidirect
563. C₂	C₅×#5578	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×Q₈	2
C₂×C₅×#946	2
C₂×C₅×#944	2
C₅×#5578	2
C₂×C₅×#897	2
C₂×C₅×#896	2
C₂×C₅×#889	2
C₂×C₅×#888	2
C₂×C₅×#872	2
C₂×C₄²×C₅×D₃	2
C₂×C₄×C₅×Dic₆	2
C₂×C₅×#533	3
C₂×#5578	5

Subgroups — C2×C5×#5578 #14480117

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#5578 #14480117