Subgroups — C₅×#35278 #11957998

Order 1280.

2746 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#540	C₂	1	2	yes	maximal, complemented
2	C₅×#2867	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2859	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2856	C₂	1	2	yes	maximal, complemented
5	C₅×#2641	C₂	1	2	yes	maximal, complemented
6	C₅×#2619	C₂	1	2	yes	maximal, complemented
7	C₅×#2445	C₂	1	8	yes	maximal, complemented
8	C₅×#2428	C₂	1	8	yes	maximal, complemented
9	C₅×#2328	C₂	1	2	yes	maximal, complemented
10	C₄×C₅×#127	C₂	1	2	yes	maximal, complemented
11	C₄×C₅×#117	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#540	—	2	16	no	—
13	C₂×C₅×#127	C₂²	1	2	yes	—
14	C₂×C₅×#127	C₂²	1	2	yes	—
15	C₂×C₅×#127	C₂²	1	2	yes	—
16	C₂×C₅×#127	C₂²	1	8	yes	—
17	C₂×C₅×#125	C₂²	1	2	yes	—
18	C₂×C₅×#123	C₂²	1	8	yes	—
19	C₂×C₅×#121	C₂²	1	4	yes	—
20	C₂×C₅×#121	C₂²	1	8	yes	—
21	C₂×C₅×#121	C₂²	1	8	yes	—
22	C₂×C₄×C₅×D₄	C₂²	1	4	yes	—
23	C₂×C₄×C₅×D₄	C₂²	1	8	yes	—
24	C₂×C₅×#117	C₂²	1	1	yes	characteristic
25	C₂×C₅×#117	C₂²	1	1	yes	characteristic
26	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
27	C₅×#394	C₂²	1	8	yes	—
28	C₅×#394	C₂²	1	16	yes	—
29	C₅×#392	C₂²	1	8	yes	—
30	C₅×#390	C₂²	1	4	yes	—
31	C₅×#390	C₂²	1	8	yes	—
32	C₅×#390	C₂²	1	4	yes	—
33	C₅×#390	C₂²	1	8	yes	—
34	C₅×#389	C₂²	1	8	yes	—
35	C₅×#386	C₂²	1	16	yes	—
36	C₅×#384	C₂²	1	2	yes	—
37	C₅×#382	C₂²	1	2	yes	—
38	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
39	C₄×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
40	C₅×#376	C₂²	1	1	yes	characteristic
41	C₅×#376	C₂²	1	1	yes	characteristic
42	C₄³×C₅	C₂²	1	1	yes	characteristic
43	C₄³×C₅	C₂²	1	1	yes	characteristic
44	#35278	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂²×C₅×D₄	C₂³	1	4	yes	—
46	C₂²×C₅×D₄	C₂³	1	8	yes	—
47	C₂²×C₅×D₄	C₂³	1	8	yes	—
48	C₂²×C₅×D₄	C₂³	1	8	yes	—
49	C₂²×C₅×D₄	C₂³	1	8	yes	—
50	C₂³×C₄×C₅	C₂³	1	2	yes	—
51	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
52	C₂³×C₄×C₅	C₂³	1	8	yes	—
53	C₅×#127	—	2	8	no	—
54	C₅×#127	—	2	8	no	—
55	C₅×#127	—	2	8	no	—
56	C₅×#127	—	4	16	no	—
57	C₅×#125	—	2	8	no	—
58	C₅×#123	—	2	32	no	—
59	C₅×#121	—	2	16	no	—
60	C₅×#121	—	2	32	no	—
61	C₅×#121	—	4	16	no	—
62	C₄×C₅×D₄	—	2	16	no	—
63	C₄×C₅×D₄	—	4	16	no	—
64	C₅×#117	—	2	4	no	—
65	C₅×#117	D₄	1	8	yes	complemented
66	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
67	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
68	C₂×C₅×C₄⋊C₄	C₂³	1	8	yes	—
69	C₂×C₅×C₄⋊C₄	C₂³	1	8	yes	—
70	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
71	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
72	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
73	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
74	C₂×C₅×C₂²⋊C₄	C₂³	1	16	yes	—
75	C₂×C₅×C₂²⋊C₄	C₂³	1	16	yes	—
76	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
77	C₂×C₄²×C₅	—	2	4	no	—
78	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₅	C₂³	1	4	yes	—
80	C₂×C₄²×C₅	C₂³	1	4	yes	—
81	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
82	C₂×C₄²×C₅	C₂³	1	2	yes	—
83	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
84	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
85	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
86	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
87	C₅×#95	C₂³	1	4	yes	—
88	C₅×#95	C₂³	1	4	yes	—
89	C₅×#95	C₂³	1	8	yes	—
90	C₂×#540	C₂×C₅	1	2	yes	complemented
91	#2867	C₂×C₅	1	1	yes	characteristic, complemented
92	#2859	C₂×C₅	1	1	yes	characteristic, complemented
93	#2856	C₂×C₅	1	2	yes	complemented
94	#2641	C₂×C₅	1	2	yes	complemented
95	#2619	C₂×C₅	1	2	yes	complemented
96	#2445	C₂×C₅	1	8	yes	complemented
97	#2428	C₂×C₅	1	8	yes	complemented
98	#2328	C₂×C₅	1	2	yes	complemented
99	C₄×#127	C₂×C₅	1	2	yes	complemented
100	C₄×#117	C₂×C₅	1	1	yes	characteristic, complemented
101	C₂⁴×C₅	C₂⁴	1	8	yes	—
102	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
103	C₂×C₅×D₄	—	2	16	no	—
104	C₂×C₅×D₄	—	2	8	no	—
105	C₂×C₅×D₄	—	2	16	no	—
106	C₂×C₅×D₄	—	2	8	no	—
107	C₂×C₅×D₄	—	2	8	no	—
108	C₂×C₅×D₄	—	2	16	no	—
109	C₂×C₅×D₄	—	2	16	no	—
110	C₂×C₅×D₄	—	2	16	no	—
111	C₂×C₅×D₄	—	2	16	no	—
112	C₂×C₅×D₄	—	4	16	no	—
113	C₂×C₅×D₄	—	4	16	no	—
114	C₂×C₅×D₄	—	4	16	no	—
115	C₂²×C₄×C₅	—	2	2	no	—
116	C₂²×C₄×C₅	—	2	2	no	—
117	C₂²×C₄×C₅	—	2	4	no	—
118	C₂²×C₄×C₅	—	2	8	no	—
119	C₂²×C₄×C₅	—	2	16	no	—
120	C₂²×C₄×C₅	—	4	16	no	—
121	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
122	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
123	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
124	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
125	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
126	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
127	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
128	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
129	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
130	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
131	C₅×C₄⋊C₄	—	2	8	no	—
132	C₅×C₄⋊C₄	—	2	16	no	—
133	C₅×C₄⋊C₄	—	2	4	no	—
134	C₅×C₄⋊C₄	—	2	16	no	—
135	C₅×C₄⋊C₄	C₂×D₄	1	8	yes	—
136	C₅×C₂²⋊C₄	—	2	16	no	—
137	C₅×C₂²⋊C₄	—	2	16	no	—
138	C₅×C₂²⋊C₄	—	2	4	no	—
139	C₅×C₂²⋊C₄	—	4	16	no	—
140	C₅×C₂²⋊C₄	—	4	16	no	—
141	C₅×C₂²⋊C₄	C₂×D₄	1	8	yes	—
142	C₄²×C₅	—	2	4	no	—
143	C₄²×C₅	—	2	2	no	—
144	C₄²×C₅	—	2	2	no	—
145	C₄²×C₅	—	2	8	no	—
146	C₄²×C₅	—	2	8	no	—
147	C₄²×C₅	C₂×D₄	1	4	yes	—
148	C₄²×C₅	C₂×D₄	1	4	yes	—
149	C₄²×C₅	#41	1	4	yes	—
150	C₄²×C₅	#41	1	4	yes	—
151	#540	—	2	16	no	—
152	C₂×#127	C₂²×C₅	1	2	yes	—
153	C₂×#127	C₂²×C₅	1	2	yes	—
154	C₂×#127	C₂²×C₅	1	2	yes	—
155	C₂×#127	C₂²×C₅	1	8	yes	—
156	C₂×#125	C₂²×C₅	1	2	yes	—
157	C₂×#123	C₂²×C₅	1	8	yes	—
158	C₂×#121	C₂²×C₅	1	4	yes	—
159	C₂×#121	C₂²×C₅	1	8	yes	—
160	C₂×#121	C₂²×C₅	1	8	yes	—
161	C₂×C₄×D₄	C₂²×C₅	1	4	yes	—
162	C₂×C₄×D₄	C₂²×C₅	1	8	yes	—
163	C₂×#117	C₂²×C₅	1	1	yes	characteristic
164	C₂×#117	C₂²×C₅	1	1	yes	characteristic
165	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
166	#394	C₂²×C₅	1	8	yes	—
167	#394	C₂²×C₅	1	16	yes	—
168	#392	C₂²×C₅	1	8	yes	—
169	#390	C₂²×C₅	1	4	yes	—
170	#390	C₂²×C₅	1	8	yes	—
171	#390	C₂²×C₅	1	4	yes	—
172	#390	C₂²×C₅	1	8	yes	—
173	#389	C₂²×C₅	1	8	yes	—
174	#386	C₂²×C₅	1	16	yes	—
175	#384	C₂²×C₅	1	2	yes	—
176	#382	C₂²×C₅	1	2	yes	—
177	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
178	C₄×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
179	#376	C₂²×C₅	1	1	yes	characteristic
180	#376	C₂²×C₅	1	1	yes	characteristic
181	C₄³	C₂²×C₅	1	1	yes	characteristic
182	C₄³	C₂²×C₅	1	1	yes	characteristic
183	C₂³×C₅	—	2	1	no	—
184	C₂³×C₅	—	2	1	no	—
185	C₂³×C₅	—	2	4	no	—
186	C₂³×C₅	—	2	8	no	—
187	C₂³×C₅	—	2	8	no	—
188	C₂³×C₅	—	2	8	no	—
189	C₂³×C₅	—	2	16	no	—
190	C₂³×C₅	—	2	16	no	—
191	C₂³×C₅	C₂²×D₄	1	2	yes	—
192	C₂³×C₅	C₂⁵	1	1	yes	characteristic
193	C₅×D₄	—	4	16	no	—
194	C₅×D₄	—	4	32	no	—
195	C₅×D₄	—	8	16	no	—
196	C₅×D₄	—	8	16	no	—
197	C₅×D₄	—	8	16	no	—
198	C₂×C₄×C₅	—	2	1	no	—
199	C₂×C₄×C₅	—	2	1	no	—
200	C₂×C₄×C₅	—	2	4	no	—
201	C₂×C₄×C₅	—	2	4	no	—
202	C₂×C₄×C₅	—	2	4	no	—
203	C₂×C₄×C₅	—	2	8	no	—
204	C₂×C₄×C₅	—	2	8	no	—
205	C₂×C₄×C₅	—	2	2	no	—
206	C₂×C₄×C₅	—	2	2	no	—
207	C₂×C₄×C₅	—	2	4	no	—
208	C₂×C₄×C₅	—	2	4	no	—
209	C₂×C₄×C₅	—	2	16	no	—
210	C₂×C₄×C₅	—	4	8	no	—
211	C₂×C₄×C₅	—	8	16	no	—
212	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
213	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
214	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
215	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
216	C₂×C₄×C₅	C₂×#41	1	4	yes	—
217	C₂×C₄×C₅	C₂×#41	1	2	yes	—
218	C₂×C₄×C₅	C₂×#41	1	2	yes	—
219	C₂×C₄×C₅	C₂×#41	1	4	yes	—
220	C₂×C₄×C₅	H(5,𝔽₂)	1	8	yes	—
221	C₂²×D₄	C₂³×C₅	1	4	yes	—
222	C₂²×D₄	C₂³×C₅	1	8	yes	—
223	C₂²×D₄	C₂³×C₅	1	8	yes	—
224	C₂²×D₄	C₂³×C₅	1	8	yes	—
225	C₂²×D₄	C₂³×C₅	1	8	yes	—
226	C₂³×C₄	C₂³×C₅	1	2	yes	—
227	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
228	C₂³×C₄	C₂³×C₅	1	8	yes	—
229	#127	—	2	8	no	—
230	#127	—	2	8	no	—
231	#127	—	2	8	no	—
232	#127	—	4	16	no	—
233	#125	—	2	8	no	—
234	#123	—	2	32	no	—
235	#121	—	2	16	no	—
236	#121	—	2	32	no	—
237	#121	—	4	16	no	—
238	C₄×D₄	—	2	16	no	—
239	C₄×D₄	—	4	16	no	—
240	#117	—	2	4	no	—
241	#117	C₅×D₄	1	8	yes	complemented
242	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
243	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
244	C₂×C₄⋊C₄	C₂³×C₅	1	8	yes	—
245	C₂×C₄⋊C₄	C₂³×C₅	1	8	yes	—
246	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
247	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
248	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
249	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
250	C₂×C₂²⋊C₄	C₂³×C₅	1	16	yes	—
251	C₂×C₂²⋊C₄	C₂³×C₅	1	16	yes	—
252	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
253	C₂×C₄²	—	2	4	no	—
254	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
255	C₂×C₄²	C₂³×C₅	1	4	yes	—
256	C₂×C₄²	C₂³×C₅	1	4	yes	—
257	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
258	C₂×C₄²	C₂³×C₅	1	2	yes	—
259	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
260	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
261	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
262	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
263	#95	C₂³×C₅	1	4	yes	—
264	#95	C₂³×C₅	1	4	yes	—
265	#95	C₂³×C₅	1	8	yes	—
266	C₂²×C₅	—	2	2	no	—
267	C₂²×C₅	—	2	4	no	—
268	C₂²×C₅	—	4	2	no	—
269	C₂²×C₅	—	4	2	no	—
270	C₂²×C₅	—	4	8	no	—
271	C₂²×C₅	—	4	16	no	—
272	C₂²×C₅	—	4	8	no	—
273	C₂²×C₅	—	4	8	no	—
274	C₂²×C₅	—	4	16	no	—
275	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
276	C₂²×C₅	C₂²×#41	1	1	yes	characteristic
277	C₂²×C₅	C₂²×#41	1	1	yes	characteristic
278	C₂²×C₅	C₂×H(5,𝔽₂)	1	4	yes	—
279	C₄×C₅	—	2	2	no	—
280	C₄×C₅	—	2	4	no	—
281	C₄×C₅	—	2	4	no	—
282	C₄×C₅	—	2	4	no	—
283	C₄×C₅	—	4	2	no	—
284	C₄×C₅	—	4	8	no	—
285	C₄×C₅	—	4	2	no	—
286	C₄×C₅	#532	1	4	yes	—
287	C₄×C₅	#540	1	8	yes	complemented
288	C₂⁴	C₂⁴×C₅	1	8	yes	—
289	C₂⁴	C₂⁴×C₅	1	1	yes	characteristic
290	C₂×D₄	—	2	16	no	—
291	C₂×D₄	—	2	8	no	—
292	C₂×D₄	—	2	16	no	—
293	C₂×D₄	—	2	8	no	—
294	C₂×D₄	—	2	8	no	—
295	C₂×D₄	—	2	16	no	—
296	C₂×D₄	—	2	16	no	—
297	C₂×D₄	—	2	16	no	—
298	C₂×D₄	—	2	16	no	—
299	C₂×D₄	—	4	16	no	—
300	C₂×D₄	—	4	16	no	—
301	C₂×D₄	—	4	16	no	—
302	C₂²×C₄	—	2	2	no	—
303	C₂²×C₄	—	2	2	no	—
304	C₂²×C₄	—	2	4	no	—
305	C₂²×C₄	—	2	8	no	—
306	C₂²×C₄	—	2	16	no	—
307	C₂²×C₄	—	4	16	no	—
308	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
309	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
310	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
311	C₂²×C₄	C₂⁴×C₅	1	8	yes	—
312	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
313	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
314	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
315	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
316	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
317	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
318	C₄⋊C₄	—	2	8	no	—
319	C₄⋊C₄	—	2	16	no	—
320	C₄⋊C₄	—	2	4	no	—
321	C₄⋊C₄	—	2	16	no	—
322	C₄⋊C₄	C₂×C₅×D₄	1	8	yes	—
323	C₂²⋊C₄	—	2	16	no	—
324	C₂²⋊C₄	—	2	16	no	—
325	C₂²⋊C₄	—	2	4	no	—
326	C₂²⋊C₄	—	4	16	no	—
327	C₂²⋊C₄	—	4	16	no	—
328	C₂²⋊C₄	C₂×C₅×D₄	1	8	yes	—
329	C₄²	—	2	4	no	—
330	C₄²	—	2	2	no	—
331	C₄²	—	2	2	no	—
332	C₄²	—	2	8	no	—
333	C₄²	—	2	8	no	—
334	C₄²	C₂×C₅×D₄	1	4	yes	—
335	C₄²	C₂×C₅×D₄	1	4	yes	—
336	C₄²	C₅×#41	1	4	yes	—
337	C₄²	C₅×#41	1	4	yes	—
338	C₂×C₅	—	4	2	no	—
339	C₂×C₅	—	8	8	no	—
340	C₂×C₅	C₂×#532	1	1	yes	characteristic
341	C₂×C₅	C₂×#540	1	2	yes	—
342	C₂×C₅	#3498	1	2	yes	—
343	C₂×C₅	#3508	1	2	yes	—
344	C₂³	—	2	1	no	—
345	C₂³	—	2	1	no	—
346	C₂³	—	2	4	no	—
347	C₂³	—	2	8	no	—
348	C₂³	—	2	8	no	—
349	C₂³	—	2	8	no	—
350	C₂³	—	2	16	no	—
351	C₂³	—	2	16	no	—
352	C₂³	C₂²×C₅×D₄	1	2	yes	—
353	C₂³	C₂⁵×C₅	1	1	yes	characteristic
354	D₄	—	4	16	no	—
355	D₄	—	4	32	no	—
356	D₄	—	8	16	no	—
357	D₄	—	8	16	no	—
358	D₄	—	8	16	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	4	no	—
362	C₂×C₄	—	2	4	no	—
363	C₂×C₄	—	2	4	no	—
364	C₂×C₄	—	2	8	no	—
365	C₂×C₄	—	2	8	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	4	no	—
369	C₂×C₄	—	2	4	no	—
370	C₂×C₄	—	2	16	no	—
371	C₂×C₄	—	4	8	no	—
372	C₂×C₄	—	8	16	no	—
373	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
374	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
375	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
376	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
377	C₂×C₄	C₂×C₅×#41	1	4	yes	—
378	C₂×C₄	C₂×C₅×#41	1	2	yes	—
379	C₂×C₄	C₂×C₅×#41	1	2	yes	—
380	C₂×C₄	C₂×C₅×#41	1	4	yes	—
381	C₂×C₄	C₅×H(5,𝔽₂)	1	8	yes	—
382	C₅	#35278	1	1	yes	characteristic, direct product factor, Sylow
383	C₂²	—	2	2	no	—
384	C₂²	—	2	4	no	—
385	C₂²	—	4	2	no	—
386	C₂²	—	4	2	no	—
387	C₂²	—	4	8	no	—
388	C₂²	—	4	16	no	—
389	C₂²	—	4	8	no	—
390	C₂²	—	4	8	no	—
391	C₂²	—	4	16	no	—
392	C₂²	C₂³×C₅×D₄	1	1	yes	characteristic
393	C₂²	C₂²×C₅×#41	1	1	yes	characteristic
394	C₂²	C₂²×C₅×#41	1	1	yes	characteristic
395	C₂²	C₂×C₅×H(5,𝔽₂)	1	4	yes	—
396	C₄	—	2	2	no	—
397	C₄	—	2	4	no	—
398	C₄	—	2	4	no	—
399	C₄	—	2	4	no	—
400	C₄	—	4	2	no	—
401	C₄	—	4	8	no	—
402	C₄	—	4	2	no	—
403	C₄	C₅×#532	1	4	yes	—
404	C₄	C₅×#540	1	8	yes	complemented
405	C₂	—	4	2	no	—
406	C₂	—	8	8	no	—
407	C₂	C₂×C₅×#532	1	1	yes	characteristic
408	C₂	C₂×C₅×#540	1	2	yes	—
409	C₂	C₅×#3498	1	2	yes	—
410	C₂	C₅×#3508	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#540	C₂	semidirect
2. C₅×#2867	C₂	semidirect
3. C₅×#2859	C₂	semidirect
4. C₅×#2856	C₂	semidirect
5. C₅×#2641	C₂	semidirect
6. C₅×#2619	C₂	semidirect
7. C₅×#2445	C₂	semidirect
8. C₅×#2428	C₂	semidirect
9. C₅×#2328	C₂	semidirect
10. C₄×C₅×#127	C₂	semidirect
11. C₄×C₅×#117	C₂	semidirect
44. #35278	C₅	direct (N × H)
65. C₅×#117	D₄	semidirect
90. C₂×#540	C₂×C₅	semidirect
91. #2867	C₂×C₅	semidirect
92. #2859	C₂×C₅	semidirect
93. #2856	C₂×C₅	semidirect
94. #2641	C₂×C₅	semidirect
95. #2619	C₂×C₅	semidirect
96. #2445	C₂×C₅	semidirect
97. #2428	C₂×C₅	semidirect
98. #2328	C₂×C₅	semidirect
99. C₄×#127	C₂×C₅	semidirect
100. C₄×#117	C₂×C₅	semidirect
241. #117	C₅×D₄	semidirect
287. C₄×C₅	#540	semidirect
382. C₅	#35278	direct (N × H)
404. C₄	C₅×#540	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#540	2
C₅×#2867	2
C₅×#2859	2
C₅×#2856	2
C₅×#2641	2
C₅×#2619	2
C₅×#2445	2
C₅×#2428	2
C₅×#2328	2
C₄×C₅×#127	2
C₄×C₅×#117	2
#35278	5

Subgroups — C5×#35278 #11957998

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#35278 #11957998