Subgroups — C₅×#34313 #11957033

Order 1280.

3276 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#550	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#520	C₂	1	2	yes	maximal, complemented
3	C₅×#2366	C₂	1	3	yes	maximal, complemented
4	C₅×#2361	C₂	1	6	yes	maximal, complemented
5	C₅×Q₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×D₄×C₄⋊C₄	C₂	1	3	yes	maximal, complemented
7	C₅×#2341	C₂	1	2	yes	maximal, complemented
8	C₅×#2324	C₂	1	6	yes	maximal, complemented
9	C₄×C₅×#127	C₂	1	3	yes	maximal, complemented
10	C₅×#2293	C₂	1	3	yes	maximal, complemented
11	C₄²×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₅×#41	C₂²	1	2	yes	—
13	C₅×#550	C₄	1	16	yes	complemented
14	C₂×C₅×#127	C₂²	1	3	yes	—
15	C₂×C₅×#121	C₂²	1	6	yes	—
16	C₅×#520	—	2	16	no	—
17	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₅×Q₈	C₂²	1	2	yes	—
19	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₅×D₄	C₂²	1	6	yes	—
22	C₂×C₄×C₅×D₄	C₂²	1	6	yes	—
23	C₂×C₄×C₅×D₄	C₂²	1	6	yes	—
24	C₂×C₄×C₅×D₄	C₂²	1	6	yes	—
25	C₂×C₄×C₅×D₄	C₂²	1	3	yes	—
26	C₂×C₅×#117	C₂²	1	6	yes	—
27	C₂²×C₅×C₄⋊C₄	C₂²	1	6	yes	—
28	C₅×#390	C₂²	1	3	yes	—
29	C₅×#390	C₂²	1	3	yes	—
30	C₅×#390	C₂²	1	6	yes	—
31	C₅×#390	C₂²	1	6	yes	—
32	C₅×#390	C₂²	1	6	yes	—
33	C₅×#389	C₂²	1	1	yes	characteristic
34	C₅×#389	C₂²	1	3	yes	—
35	C₅×#389	C₂²	1	6	yes	—
36	C₅×#388	C₂²	1	6	yes	—
37	C₅×#388	C₂²	1	6	yes	—
38	C₅×#386	C₂²	1	12	yes	—
39	C₅×#385	C₂²	1	12	yes	—
40	C₅×#384	C₂²	1	3	yes	—
41	C₅×#380	C₂²	1	6	yes	—
42	C₄×C₅×C₄⋊C₄	C₂²	1	3	yes	—
43	C₄×C₅×C₄⋊C₄	C₂²	1	3	yes	—
44	C₄×C₅×C₄⋊C₄	C₂²	1	3	yes	—
45	C₄×C₅×C₄⋊C₄	C₂²	1	3	yes	—
46	C₄×C₅×C₂²⋊C₄	C₂²	1	6	yes	—
47	C₄³×C₅	C₂²	1	3	yes	—
48	#34313	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂×C₅×#41	—	2	8	no	—
50	C₂×C₅×#41	—	2	8	no	—
51	C₂×C₅×#41	C₂×C₄	1	16	yes	complemented
52	C₂²×C₅×Q₈	C₂³	1	1	yes	characteristic
53	C₂²×C₅×D₄	C₂³	1	3	yes	—
54	C₂²×C₅×D₄	C₂³	1	6	yes	—
55	C₂²×C₅×D₄	C₂³	1	6	yes	—
56	C₂³×C₄×C₅	C₂³	1	6	yes	—
57	C₂³×C₄×C₅	C₂³	1	12	yes	—
58	C₅×#127	C₂×C₄	1	24	yes	complemented
59	C₅×#121	C₂×C₄	1	48	yes	complemented
60	C₄×C₅×Q₈	—	2	8	no	—
61	C₄×C₅×Q₈	—	2	4	no	—
62	C₄×C₅×Q₈	C₂×C₄	1	8	yes	complemented
63	C₄×C₅×Q₈	D₄	1	8	yes	complemented
64	C₄×C₅×D₄	—	2	24	no	—
65	C₄×C₅×D₄	—	2	24	no	—
66	C₄×C₅×D₄	—	4	12	no	—
67	C₄×C₅×D₄	—	4	12	no	—
68	C₄×C₅×D₄	C₂×C₄	1	24	yes	complemented
69	C₅×#117	—	2	24	no	—
70	C₂×C₅×C₄⋊C₄	—	2	24	no	—
71	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
72	C₂×C₅×C₄⋊C₄	C₂³	1	3	yes	—
73	C₂×C₅×C₄⋊C₄	C₂³	1	3	yes	—
74	C₂×C₅×C₄⋊C₄	C₂³	1	6	yes	—
75	C₂×C₅×C₄⋊C₄	C₂³	1	6	yes	—
76	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
77	C₂×C₅×C₄⋊C₄	C₂³	1	6	yes	—
78	C₂×C₅×C₄⋊C₄	C₂³	1	3	yes	—
79	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
80	C₂×C₅×C₄⋊C₄	C₂³	1	3	yes	—
81	C₂×C₅×C₂²⋊C₄	C₂³	1	6	yes	—
82	C₂×C₅×C₂²⋊C₄	C₂³	1	6	yes	—
83	C₂×C₅×C₂²⋊C₄	C₂³	1	6	yes	—
84	C₂×C₅×C₂²⋊C₄	C₂³	1	6	yes	—
85	C₂×C₅×C₂²⋊C₄	C₂³	1	12	yes	—
86	C₂×C₅×C₂²⋊C₄	C₂³	1	6	yes	—
87	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
88	C₂×C₄²×C₅	C₂³	1	3	yes	—
89	C₂×C₄²×C₅	C₂³	1	3	yes	—
90	C₂×C₄²×C₅	C₂³	1	3	yes	—
91	C₂×C₄²×C₅	C₂³	1	6	yes	—
92	C₂×C₄²×C₅	C₂³	1	3	yes	—
93	C₂×C₄²×C₅	C₂³	1	3	yes	—
94	C₂×C₄²×C₅	C₂³	1	3	yes	—
95	C₅×#95	C₂³	1	2	yes	—
96	C₅×#95	C₂³	1	6	yes	—
97	C₅×#95	C₂³	1	6	yes	—
98	C₅×#95	C₂³	1	6	yes	—
99	C₂×#550	C₂×C₅	1	1	yes	characteristic
100	C₂×#520	C₂×C₅	1	2	yes	complemented
101	#2366	C₂×C₅	1	3	yes	complemented
102	#2361	C₂×C₅	1	6	yes	complemented
103	Q₈×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
104	D₄×C₄⋊C₄	C₂×C₅	1	3	yes	complemented
105	#2341	C₂×C₅	1	2	yes	complemented
106	#2324	C₂×C₅	1	6	yes	complemented
107	C₄×#127	C₂×C₅	1	3	yes	complemented
108	#2293	C₂×C₅	1	3	yes	complemented
109	C₄²×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
110	C₂⁴×C₅	C₂⁴	1	6	yes	—
111	C₅×#41	—	2	64	no	—
112	C₂×C₅×Q₈	C₂²×C₄	1	4	yes	—
113	C₂×C₅×Q₈	C₂×D₄	1	4	yes	—
114	C₂×C₅×Q₈	#41	1	4	yes	—
115	C₂×C₅×D₄	—	2	12	no	—
116	C₂×C₅×D₄	—	2	12	no	—
117	C₂×C₅×D₄	—	2	6	no	—
118	C₂×C₅×D₄	—	2	6	no	—
119	C₂×C₅×D₄	—	4	6	no	—
120	C₂×C₅×D₄	—	4	6	no	—
121	C₂×C₅×D₄	C₂²×C₄	1	12	yes	—
122	C₂×C₅×D₄	C₂²×C₄	1	24	yes	—
123	C₂×C₅×D₄	C₂²×C₄	1	24	yes	—
124	C₂²×C₄×C₅	—	2	12	no	—
125	C₂²×C₄×C₅	—	2	12	no	—
126	C₂²×C₄×C₅	—	2	24	no	—
127	C₂²×C₄×C₅	—	4	12	no	—
128	C₂²×C₄×C₅	—	4	12	no	—
129	C₂²×C₄×C₅	C₂²×C₄	1	24	yes	—
130	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
131	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
132	C₂²×C₄×C₅	C₂⁴	1	6	yes	—
133	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
134	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
135	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
136	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
137	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
138	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
139	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
140	C₅×C₄⋊C₄	—	2	12	no	—
141	C₅×C₄⋊C₄	—	2	12	no	—
142	C₅×C₄⋊C₄	—	2	4	no	—
143	C₅×C₄⋊C₄	—	2	6	no	—
144	C₅×C₄⋊C₄	—	2	6	no	—
145	C₅×C₄⋊C₄	—	2	2	no	—
146	C₅×C₄⋊C₄	—	4	6	no	—
147	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
148	C₅×C₄⋊C₄	C₂²×C₄	1	12	yes	—
149	C₅×C₄⋊C₄	C₂×D₄	1	12	yes	—
150	C₅×C₂²⋊C₄	—	2	12	no	—
151	C₅×C₂²⋊C₄	—	2	12	no	—
152	C₅×C₂²⋊C₄	—	2	12	no	—
153	C₅×C₂²⋊C₄	—	4	6	no	—
154	C₅×C₂²⋊C₄	—	4	12	no	—
155	C₅×C₂²⋊C₄	C₂²×C₄	1	24	yes	—
156	C₄²×C₅	—	2	12	no	—
157	C₄²×C₅	—	2	6	no	—
158	C₄²×C₅	—	2	6	no	—
159	C₄²×C₅	—	2	6	no	—
160	C₄²×C₅	—	2	2	no	—
161	C₄²×C₅	—	4	3	no	—
162	C₄²×C₅	C₂²×C₄	1	12	yes	—
163	C₄²×C₅	C₂×D₄	1	12	yes	—
164	C₂²×#41	C₂²×C₅	1	2	yes	—
165	#550	C₄×C₅	1	16	yes	complemented
166	C₂×#127	C₂²×C₅	1	3	yes	—
167	C₂×#121	C₂²×C₅	1	6	yes	—
168	#520	—	2	16	no	—
169	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
170	C₂×C₄×Q₈	C₂²×C₅	1	2	yes	—
171	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
172	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
173	C₂×C₄×D₄	C₂²×C₅	1	6	yes	—
174	C₂×C₄×D₄	C₂²×C₅	1	6	yes	—
175	C₂×C₄×D₄	C₂²×C₅	1	6	yes	—
176	C₂×C₄×D₄	C₂²×C₅	1	6	yes	—
177	C₂×C₄×D₄	C₂²×C₅	1	3	yes	—
178	C₂×#117	C₂²×C₅	1	6	yes	—
179	C₂²×C₄⋊C₄	C₂²×C₅	1	6	yes	—
180	#390	C₂²×C₅	1	3	yes	—
181	#390	C₂²×C₅	1	6	yes	—
182	#390	C₂²×C₅	1	6	yes	—
183	#390	C₂²×C₅	1	6	yes	—
184	#390	C₂²×C₅	1	3	yes	—
185	#389	C₂²×C₅	1	6	yes	—
186	#389	C₂²×C₅	1	3	yes	—
187	#389	C₂²×C₅	1	1	yes	characteristic
188	#388	C₂²×C₅	1	6	yes	—
189	#388	C₂²×C₅	1	6	yes	—
190	#386	C₂²×C₅	1	12	yes	—
191	#385	C₂²×C₅	1	12	yes	—
192	#384	C₂²×C₅	1	3	yes	—
193	#380	C₂²×C₅	1	6	yes	—
194	C₄×C₄⋊C₄	C₂²×C₅	1	3	yes	—
195	C₄×C₄⋊C₄	C₂²×C₅	1	3	yes	—
196	C₄×C₄⋊C₄	C₂²×C₅	1	3	yes	—
197	C₄×C₄⋊C₄	C₂²×C₅	1	3	yes	—
198	C₄×C₂²⋊C₄	C₂²×C₅	1	6	yes	—
199	C₄³	C₂²×C₅	1	3	yes	—
200	C₂³×C₅	—	2	6	no	—
201	C₂³×C₅	—	2	6	no	—
202	C₂³×C₅	—	2	6	no	—
203	C₂³×C₅	—	2	6	no	—
204	C₂³×C₅	—	2	6	no	—
205	C₂³×C₅	—	2	6	no	—
206	C₂³×C₅	C₂³×C₄	1	12	yes	—
207	C₂³×C₅	C₂⁵	1	1	yes	characteristic
208	C₅×Q₈	C₄×D₄	1	16	yes	complemented
209	C₅×D₄	—	2	48	no	—
210	C₅×D₄	—	2	24	no	—
211	C₅×D₄	—	4	24	no	—
212	C₂×C₄×C₅	—	2	2	no	—
213	C₂×C₄×C₅	—	2	1	no	—
214	C₂×C₄×C₅	—	2	2	no	—
215	C₂×C₄×C₅	—	2	3	no	—
216	C₂×C₄×C₅	—	2	3	no	—
217	C₂×C₄×C₅	—	2	3	no	—
218	C₂×C₄×C₅	—	2	1	no	—
219	C₂×C₄×C₅	—	2	1	no	—
220	C₂×C₄×C₅	—	2	1	no	—
221	C₂×C₄×C₅	—	2	3	no	—
222	C₂×C₄×C₅	—	2	48	no	—
223	C₂×C₄×C₅	—	2	2	no	—
224	C₂×C₄×C₅	—	2	6	no	—
225	C₂×C₄×C₅	—	2	3	no	—
226	C₂×C₄×C₅	—	2	3	no	—
227	C₂×C₄×C₅	—	2	6	no	—
228	C₂×C₄×C₅	—	2	6	no	—
229	C₂×C₄×C₅	—	2	2	no	—
230	C₂×C₄×C₅	—	2	3	no	—
231	C₂×C₄×C₅	—	2	2	no	—
232	C₂×C₄×C₅	—	8	24	no	—
233	C₂×C₄×C₅	C₂³×C₄	1	6	yes	—
234	C₂×C₄×C₅	C₂³×C₄	1	4	yes	—
235	C₂×C₄×C₅	C₂³×C₄	1	6	yes	—
236	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
237	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
238	C₂×C₄×C₅	C₂²×D₄	1	6	yes	—
239	C₂×C₄×C₅	C₂²×D₄	1	6	yes	—
240	C₂×C₄×C₅	C₂×#41	1	6	yes	—
241	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
242	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
243	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
244	C₂×#41	—	2	8	no	—
245	C₂×#41	—	2	8	no	—
246	C₂×#41	C₂×C₄×C₅	1	16	yes	complemented
247	C₂²×Q₈	C₂³×C₅	1	1	yes	characteristic
248	C₂²×D₄	C₂³×C₅	1	3	yes	—
249	C₂²×D₄	C₂³×C₅	1	6	yes	—
250	C₂²×D₄	C₂³×C₅	1	6	yes	—
251	C₂³×C₄	C₂³×C₅	1	6	yes	—
252	C₂³×C₄	C₂³×C₅	1	12	yes	—
253	#127	C₂×C₄×C₅	1	24	yes	complemented
254	#121	C₂×C₄×C₅	1	48	yes	complemented
255	C₄×Q₈	—	2	8	no	—
256	C₄×Q₈	—	2	4	no	—
257	C₄×Q₈	C₂×C₄×C₅	1	8	yes	complemented
258	C₄×Q₈	C₅×D₄	1	8	yes	complemented
259	C₄×D₄	—	2	24	no	—
260	C₄×D₄	—	2	24	no	—
261	C₄×D₄	—	4	12	no	—
262	C₄×D₄	—	4	12	no	—
263	C₄×D₄	C₂×C₄×C₅	1	24	yes	complemented
264	#117	—	2	24	no	—
265	C₂×C₄⋊C₄	—	2	24	no	—
266	C₂×C₄⋊C₄	C₂³×C₅	1	6	yes	—
267	C₂×C₄⋊C₄	C₂³×C₅	1	6	yes	—
268	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
269	C₂×C₄⋊C₄	C₂³×C₅	1	3	yes	—
270	C₂×C₄⋊C₄	C₂³×C₅	1	3	yes	—
271	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
272	C₂×C₄⋊C₄	C₂³×C₅	1	6	yes	—
273	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
274	C₂×C₄⋊C₄	C₂³×C₅	1	3	yes	—
275	C₂×C₄⋊C₄	C₂³×C₅	1	3	yes	—
276	C₂×C₂²⋊C₄	C₂³×C₅	1	6	yes	—
277	C₂×C₂²⋊C₄	C₂³×C₅	1	6	yes	—
278	C₂×C₂²⋊C₄	C₂³×C₅	1	6	yes	—
279	C₂×C₂²⋊C₄	C₂³×C₅	1	12	yes	—
280	C₂×C₂²⋊C₄	C₂³×C₅	1	6	yes	—
281	C₂×C₂²⋊C₄	C₂³×C₅	1	6	yes	—
282	C₂×C₄²	C₂³×C₅	1	6	yes	—
283	C₂×C₄²	C₂³×C₅	1	3	yes	—
284	C₂×C₄²	C₂³×C₅	1	3	yes	—
285	C₂×C₄²	C₂³×C₅	1	3	yes	—
286	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
287	C₂×C₄²	C₂³×C₅	1	3	yes	—
288	C₂×C₄²	C₂³×C₅	1	3	yes	—
289	C₂×C₄²	C₂³×C₅	1	3	yes	—
290	#95	C₂³×C₅	1	2	yes	—
291	#95	C₂³×C₅	1	6	yes	—
292	#95	C₂³×C₅	1	6	yes	—
293	#95	C₂³×C₅	1	6	yes	—
294	C₂²×C₅	—	2	12	no	—
295	C₂²×C₅	—	2	12	no	—
296	C₂²×C₅	—	2	12	no	—
297	C₂²×C₅	—	4	6	no	—
298	C₂²×C₅	—	4	6	no	—
299	C₂²×C₅	—	4	6	no	—
300	C₂²×C₅	—	4	6	no	—
301	C₂²×C₅	C₂⁴×C₄	1	1	yes	characteristic
302	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
303	C₂²×C₅	C₂²×#41	1	1	yes	characteristic
304	C₂²×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
305	C₂²×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
306	C₂²×C₅	C₂×2₋¹⁺⁴	1	1	yes	characteristic
307	C₂²×C₅	#585	1	1	yes	characteristic
308	C₄×C₅	—	2	4	no	—
309	C₄×C₅	—	2	6	no	—
310	C₄×C₅	—	2	2	no	—
311	C₄×C₅	—	2	6	no	—
312	C₄×C₅	—	2	2	no	—
313	C₄×C₅	—	4	2	no	—
314	C₄×C₅	—	4	6	no	—
315	C₄×C₅	—	4	3	no	—
316	C₄×C₅	C₂×C₄×D₄	1	12	yes	—
317	C₄×C₅	#520	1	4	yes	complemented
318	C₂⁴	C₂⁴×C₅	1	6	yes	—
319	#41	—	2	64	no	—
320	C₂×Q₈	C₂²×C₄×C₅	1	4	yes	—
321	C₂×Q₈	C₂×C₅×D₄	1	4	yes	—
322	C₂×Q₈	C₅×#41	1	4	yes	—
323	C₂×D₄	—	2	12	no	—
324	C₂×D₄	—	2	12	no	—
325	C₂×D₄	—	2	6	no	—
326	C₂×D₄	—	2	6	no	—
327	C₂×D₄	—	4	6	no	—
328	C₂×D₄	—	4	6	no	—
329	C₂×D₄	C₂²×C₄×C₅	1	12	yes	—
330	C₂×D₄	C₂²×C₄×C₅	1	24	yes	—
331	C₂×D₄	C₂²×C₄×C₅	1	24	yes	—
332	C₂²×C₄	—	2	12	no	—
333	C₂²×C₄	—	2	12	no	—
334	C₂²×C₄	—	2	24	no	—
335	C₂²×C₄	—	4	12	no	—
336	C₂²×C₄	—	4	12	no	—
337	C₂²×C₄	C₂²×C₄×C₅	1	24	yes	—
338	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
339	C₂²×C₄	C₂⁴×C₅	1	3	yes	—
340	C₂²×C₄	C₂⁴×C₅	1	6	yes	—
341	C₂²×C₄	C₂⁴×C₅	1	3	yes	—
342	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
343	C₂²×C₄	C₂⁴×C₅	1	3	yes	—
344	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
345	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
346	C₂²×C₄	C₂⁴×C₅	1	3	yes	—
347	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
348	C₄⋊C₄	—	2	12	no	—
349	C₄⋊C₄	—	2	12	no	—
350	C₄⋊C₄	—	2	4	no	—
351	C₄⋊C₄	—	2	6	no	—
352	C₄⋊C₄	—	2	6	no	—
353	C₄⋊C₄	—	2	2	no	—
354	C₄⋊C₄	—	4	6	no	—
355	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
356	C₄⋊C₄	C₂²×C₄×C₅	1	12	yes	—
357	C₄⋊C₄	C₂×C₅×D₄	1	12	yes	—
358	C₂²⋊C₄	—	2	12	no	—
359	C₂²⋊C₄	—	2	12	no	—
360	C₂²⋊C₄	—	2	12	no	—
361	C₂²⋊C₄	—	4	12	no	—
362	C₂²⋊C₄	—	4	6	no	—
363	C₂²⋊C₄	C₂²×C₄×C₅	1	24	yes	—
364	C₄²	—	2	12	no	—
365	C₄²	—	2	6	no	—
366	C₄²	—	2	6	no	—
367	C₄²	—	2	6	no	—
368	C₄²	—	2	2	no	—
369	C₄²	—	4	3	no	—
370	C₄²	C₂²×C₄×C₅	1	12	yes	—
371	C₄²	C₂×C₅×D₄	1	12	yes	—
372	C₂×C₅	—	4	12	no	—
373	C₂×C₅	C₂²×C₄×D₄	1	1	yes	characteristic
374	C₂×C₅	C₂×#520	1	1	yes	characteristic
375	C₂×C₅	C₄×H(5,𝔽₂)	1	1	yes	characteristic
376	C₂×C₅	#3497	1	1	yes	characteristic
377	C₂×C₅	#3498	1	1	yes	characteristic
378	C₂×C₅	#3517	1	1	yes	characteristic
379	C₂×C₅	#3522	1	1	yes	characteristic
380	C₂³	—	2	6	no	—
381	C₂³	—	2	6	no	—
382	C₂³	—	2	6	no	—
383	C₂³	—	2	6	no	—
384	C₂³	—	2	6	no	—
385	C₂³	—	2	6	no	—
386	C₂³	C₂³×C₄×C₅	1	12	yes	—
387	C₂³	C₂⁵×C₅	1	1	yes	characteristic
388	Q₈	C₄×C₅×D₄	1	16	yes	complemented
389	D₄	—	2	48	no	—
390	D₄	—	2	24	no	—
391	D₄	—	4	24	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	1	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	3	no	—
396	C₂×C₄	—	2	3	no	—
397	C₂×C₄	—	2	3	no	—
398	C₂×C₄	—	2	1	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	2	3	no	—
402	C₂×C₄	—	2	48	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	6	no	—
405	C₂×C₄	—	2	3	no	—
406	C₂×C₄	—	2	3	no	—
407	C₂×C₄	—	2	6	no	—
408	C₂×C₄	—	2	6	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	3	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	8	24	no	—
413	C₂×C₄	C₂³×C₄×C₅	1	6	yes	—
414	C₂×C₄	C₂³×C₄×C₅	1	4	yes	—
415	C₂×C₄	C₂³×C₄×C₅	1	6	yes	—
416	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
417	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
418	C₂×C₄	C₂²×C₅×D₄	1	6	yes	—
419	C₂×C₄	C₂²×C₅×D₄	1	6	yes	—
420	C₂×C₄	C₂×C₅×#41	1	6	yes	—
421	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
422	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
423	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
424	C₅	#34313	1	1	yes	characteristic, direct product factor, Sylow
425	C₂²	—	2	12	no	—
426	C₂²	—	2	12	no	—
427	C₂²	—	2	12	no	—
428	C₂²	—	4	6	no	—
429	C₂²	—	4	6	no	—
430	C₂²	—	4	6	no	—
431	C₂²	—	4	6	no	—
432	C₂²	C₂⁴×C₄×C₅	1	1	yes	characteristic
433	C₂²	C₂³×C₅×D₄	1	1	yes	characteristic
434	C₂²	C₂²×C₅×#41	1	1	yes	characteristic
435	C₂²	C₂×C₅×H(5,𝔽₂)	1	1	yes	characteristic
436	C₂²	C₂×C₅×H(5,𝔽₂)	1	1	yes	characteristic
437	C₂²	C₂×C₅×2₋¹⁺⁴	1	1	yes	characteristic
438	C₂²	C₅×#585	1	1	yes	characteristic
439	C₄	—	2	4	no	—
440	C₄	—	2	6	no	—
441	C₄	—	2	2	no	—
442	C₄	—	2	6	no	—
443	C₄	—	2	2	no	—
444	C₄	—	4	2	no	—
445	C₄	—	4	6	no	—
446	C₄	—	4	3	no	—
447	C₄	C₂×C₄×C₅×D₄	1	12	yes	—
448	C₄	C₅×#520	1	4	yes	complemented
449	C₂	—	4	12	no	—
450	C₂	C₂²×C₄×C₅×D₄	1	1	yes	characteristic
451	C₂	C₂×C₅×#520	1	1	yes	characteristic
452	C₂	C₄×C₅×H(5,𝔽₂)	1	1	yes	characteristic
453	C₂	C₅×#3497	1	1	yes	characteristic
454	C₂	C₅×#3498	1	1	yes	characteristic
455	C₂	C₅×#3517	1	1	yes	characteristic
456	C₂	C₅×#3522	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₅
Frattini, cosocle	C₂³	C₂⁵×C₅
prime radical	C₂×C₅×#550	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#520	C₂	semidirect
3. C₅×#2366	C₂	semidirect
4. C₅×#2361	C₂	semidirect
5. C₅×Q₈×C₄⋊C₄	C₂	semidirect
6. C₅×D₄×C₄⋊C₄	C₂	semidirect
7. C₅×#2341	C₂	semidirect
8. C₅×#2324	C₂	semidirect
9. C₄×C₅×#127	C₂	semidirect
10. C₅×#2293	C₂	semidirect
11. C₄²×C₅×Q₈	C₂	semidirect
13. C₅×#550	C₄	semidirect
48. #34313	C₅	direct (N × H)
51. C₂×C₅×#41	C₂×C₄	semidirect
58. C₅×#127	C₂×C₄	semidirect
59. C₅×#121	C₂×C₄	semidirect
62. C₄×C₅×Q₈	C₂×C₄	semidirect
63. C₄×C₅×Q₈	D₄	semidirect
68. C₄×C₅×D₄	C₂×C₄	semidirect
100. C₂×#520	C₂×C₅	semidirect
101. #2366	C₂×C₅	semidirect
102. #2361	C₂×C₅	semidirect
103. Q₈×C₄⋊C₄	C₂×C₅	semidirect
104. D₄×C₄⋊C₄	C₂×C₅	semidirect
105. #2341	C₂×C₅	semidirect
106. #2324	C₂×C₅	semidirect
107. C₄×#127	C₂×C₅	semidirect
108. #2293	C₂×C₅	semidirect
109. C₄²×Q₈	C₂×C₅	semidirect
165. #550	C₄×C₅	semidirect
208. C₅×Q₈	C₄×D₄	semidirect
246. C₂×#41	C₂×C₄×C₅	semidirect
253. #127	C₂×C₄×C₅	semidirect
254. #121	C₂×C₄×C₅	semidirect
257. C₄×Q₈	C₂×C₄×C₅	semidirect
258. C₄×Q₈	C₅×D₄	semidirect
263. C₄×D₄	C₂×C₄×C₅	semidirect
317. C₄×C₅	#520	semidirect
388. Q₈	C₄×C₅×D₄	semidirect
424. C₅	#34313	direct (N × H)
448. C₄	C₅×#520	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₅×#550	2
C₂×C₅×#520	2
C₅×#2366	2
C₅×#2361	2
C₅×Q₈×C₄⋊C₄	2
C₅×D₄×C₄⋊C₄	2
C₅×#2341	2
C₅×#2324	2
C₄×C₅×#127	2
C₅×#2293	2
C₄²×C₅×Q₈	2
#34313	5

Subgroups — C5×#34313 #11957033

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#34313 #11957033