Subgroups — C₅×#21240 #11943960

Order 1280.

1466 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#135	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#421	C₂	1	2	yes	maximal, complemented
4	C₅×#2879	C₂	1	2	yes	maximal, complemented
5	C₂×C₅×#405	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1975	C₂	1	4	yes	maximal, complemented
7	C₅×#1974	C₂	1	4	yes	maximal, complemented
8	C₂²×C₅×#41	C₂²	1	2	yes	—
9	C₂×C₅×#135	—	2	4	no	—
10	C₂×C₅×#135	—	2	4	no	—
11	C₂×C₅×#135	C₄	1	2	yes	complemented
12	C₂×C₅×#135	C₄	1	2	yes	complemented
13	C₂×C₅×#135	C₂²	1	2	yes	—
14	C₂×C₅×#135	C₂²	1	2	yes	—
15	C₂²×C₅×Q₁₆	C₄	1	1	yes	characteristic, complemented
16	C₂²×C₅×SD₁₆	C₄	1	2	yes	complemented
17	C₂²×C₅×D₈	C₄	1	1	yes	characteristic, complemented
18	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
19	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
20	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₂×C₅×#117	C₂²	1	1	yes	characteristic
22	C₂×C₅×#108	—	2	4	no	—
23	C₂×C₅×#108	C₂²	1	2	yes	complemented
24	C₂×C₅×#108	C₂²	1	2	yes	complemented
25	C₅×#421	—	2	8	no	—
26	C₂×C₅×#104	C₂²	1	8	yes	complemented
27	C₅×#408	—	2	4	no	—
28	C₅×#408	C₂²	1	8	yes	complemented
29	C₅×#407	—	2	4	no	—
30	C₅×#405	—	2	4	no	—
31	C₂×C₅×#97	C₂²	1	2	yes	complemented
32	C₂×C₄×C₅×C₈	C₂²	1	2	yes	complemented
33	#21240	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂×C₅×#41	—	2	2	no	—
35	C₂×C₅×#41	—	2	2	no	—
36	C₂×C₅×#41	—	2	4	no	—
37	C₂×C₅×#41	—	2	4	no	—
38	C₂×C₅×#41	—	2	4	no	—
39	C₂×C₅×#41	—	2	4	no	—
40	C₂×C₅×#41	C₂×C₄	1	4	yes	—
41	C₂×C₅×#41	C₂³	1	4	yes	—
42	C₂²×C₅×Q₈	C₂×C₄	1	2	yes	—
43	C₂²×C₅×D₄	—	2	2	no	—
44	C₂²×C₅×D₄	C₂×C₄	1	2	yes	—
45	C₂³×C₄×C₅	—	2	2	no	—
46	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
47	C₅×#135	—	2	8	no	—
48	C₅×#135	—	2	8	no	—
49	C₅×#135	—	2	4	no	—
50	C₅×#135	—	2	4	no	—
51	C₅×#135	—	2	4	no	—
52	C₅×#135	—	2	4	no	—
53	C₂×C₅×Q₁₆	—	2	4	no	—
54	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
55	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
56	C₂×C₅×SD₁₆	—	2	8	no	—
57	C₂×C₅×SD₁₆	C₂×C₄	1	4	yes	—
58	C₂×C₅×SD₁₆	C₂×C₄	1	4	yes	—
59	C₂×C₅×D₈	—	2	4	no	—
60	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
61	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
62	C₂×C₅×M₁₆	—	2	2	no	—
63	C₂×C₅×M₁₆	—	2	2	no	—
64	C₂×C₅×M₁₆	—	2	2	no	—
65	C₂×C₅×M₁₆	—	2	4	no	—
66	C₂×C₅×M₁₆	—	2	4	no	—
67	C₂×C₅×M₁₆	D₄	1	2	yes	—
68	C₂×C₅×M₁₆	D₄	1	2	yes	—
69	C₂×C₅×M₁₆	C₂³	1	4	yes	—
70	C₂²×C₅×C₈	—	2	2	no	—
71	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
72	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
73	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
74	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
75	C₂²×C₅×C₈	D₄	1	4	yes	—
76	C₂²×C₅×C₈	D₄	1	4	yes	—
77	C₂²×C₅×C₈	C₂³	1	2	yes	—
78	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
79	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
80	C₅×#117	—	2	2	no	—
81	C₅×#117	D₄	1	4	yes	—
82	C₂×C₅×C₄⋊C₄	—	2	1	no	—
83	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
84	C₂×C₄²×C₅	C₂³	1	2	yes	—
85	C₅×#108	—	2	4	no	—
86	C₅×#108	—	2	4	no	—
87	C₅×#104	—	2	16	no	—
88	C₅×C₂²⋊C₈	—	2	16	no	—
89	C₅×#97	—	2	4	no	—
90	C₄×C₅×C₈	—	2	4	no	—
91	C₂²×#135	C₂×C₅	1	1	yes	characteristic
92	C₂²×#108	C₂×C₅	1	1	yes	characteristic, complemented
93	C₂×#421	C₂×C₅	1	2	yes	complemented
94	#2879	C₂×C₅	1	2	yes	complemented
95	C₂×#405	C₂×C₅	1	1	yes	characteristic, complemented
96	#1975	C₂×C₅	1	4	yes	complemented
97	#1974	C₂×C₅	1	4	yes	complemented
98	C₂⁴×C₅	—	2	2	no	—
99	C₂⁴×C₅	#41	1	1	yes	characteristic
100	C₅×#41	—	2	8	no	—
101	C₅×#41	—	2	8	no	—
102	C₅×#41	—	2	4	no	—
103	C₅×#41	—	2	4	no	—
104	C₅×#41	—	2	4	no	—
105	C₅×#41	—	2	4	no	—
106	C₅×#41	—	4	4	no	—
107	C₅×#41	—	4	4	no	—
108	C₅×#41	—	4	4	no	—
109	C₅×#41	—	4	4	no	—
110	C₂×C₅×Q₈	—	2	2	no	—
111	C₂×C₅×Q₈	—	2	4	no	—
112	C₂×C₅×Q₈	—	2	4	no	—
113	C₂×C₅×Q₈	C₂²×C₄	1	4	yes	—
114	C₂×C₅×D₄	—	2	4	no	—
115	C₂×C₅×D₄	—	2	4	no	—
116	C₂×C₅×D₄	—	2	4	no	—
117	C₂×C₅×D₄	—	2	4	no	—
118	C₂×C₅×D₄	—	2	2	no	—
119	C₂×C₅×D₄	—	4	4	no	—
120	C₂×C₅×D₄	—	4	4	no	—
121	C₂×C₅×D₄	C₂²×C₄	1	4	yes	—
122	C₂²×C₄×C₅	—	2	2	no	—
123	C₂²×C₄×C₅	—	2	2	no	—
124	C₂²×C₄×C₅	—	2	4	no	—
125	C₂²×C₄×C₅	—	2	4	no	—
126	C₂²×C₄×C₅	—	4	4	no	—
127	C₂²×C₄×C₅	—	4	4	no	—
128	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
131	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
132	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
133	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₅	#41	1	1	yes	characteristic
136	C₂²×C₄×C₅	#41	1	2	yes	—
137	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
138	C₅×Q₁₆	—	2	4	no	—
139	C₅×Q₁₆	—	2	4	no	—
140	C₅×SD₁₆	—	2	8	no	—
141	C₅×SD₁₆	—	2	8	no	—
142	C₅×D₈	—	2	4	no	—
143	C₅×D₈	—	2	4	no	—
144	C₅×M₁₆	—	2	4	no	—
145	C₅×M₁₆	—	2	4	no	—
146	C₅×M₁₆	—	2	4	no	—
147	C₅×M₁₆	—	2	4	no	—
148	C₅×M₁₆	—	4	4	no	—
149	C₂×C₅×C₈	—	2	2	no	—
150	C₂×C₅×C₈	—	2	2	no	—
151	C₂×C₅×C₈	—	2	4	no	—
152	C₂×C₅×C₈	—	4	4	no	—
153	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
154	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
155	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
156	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
157	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
158	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
159	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
160	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
161	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
162	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
163	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
164	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
165	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
166	C₅×C₄⋊C₄	—	2	4	no	—
167	C₅×C₂²⋊C₄	—	2	4	no	—
168	C₄²×C₅	—	2	2	no	—
169	C₄²×C₅	C₂×D₄	1	4	yes	—
170	C₂²×#41	C₂²×C₅	1	2	yes	—
171	C₂×#135	—	2	4	no	—
172	C₂×#135	—	2	4	no	—
173	C₂×#135	C₄×C₅	1	2	yes	complemented
174	C₂×#135	C₄×C₅	1	2	yes	complemented
175	C₂×#135	C₂²×C₅	1	2	yes	—
176	C₂×#135	C₂²×C₅	1	2	yes	—
177	C₂²×Q₁₆	C₄×C₅	1	1	yes	characteristic, complemented
178	C₂²×SD₁₆	C₄×C₅	1	2	yes	complemented
179	C₂²×D₈	C₄×C₅	1	1	yes	characteristic, complemented
180	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
181	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
182	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
183	C₂×#117	C₂²×C₅	1	1	yes	characteristic
184	C₂×#108	—	2	4	no	—
185	C₂×#108	C₂²×C₅	1	2	yes	complemented
186	C₂×#108	C₂²×C₅	1	2	yes	complemented
187	#421	—	2	8	no	—
188	C₂×#104	C₂²×C₅	1	8	yes	complemented
189	#408	—	2	4	no	—
190	#408	C₂²×C₅	1	8	yes	complemented
191	#407	—	2	4	no	—
192	#405	—	2	4	no	—
193	C₂×#97	C₂²×C₅	1	2	yes	complemented
194	C₂×C₄×C₈	C₂²×C₅	1	2	yes	complemented
195	C₂³×C₅	—	2	4	no	—
196	C₂³×C₅	—	4	2	no	—
197	C₂³×C₅	—	4	4	no	—
198	C₂³×C₅	—	8	4	no	—
199	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
200	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
201	C₂³×C₅	#121	1	2	yes	—
202	C₂³×C₅	#124	1	2	yes	—
203	C₂³×C₅	C₂×#41	1	1	yes	characteristic
204	C₅×Q₈	—	2	4	no	—
205	C₅×Q₈	—	2	4	no	—
206	C₅×Q₈	—	4	4	no	—
207	C₅×D₄	—	2	4	no	—
208	C₅×D₄	—	2	4	no	—
209	C₅×D₄	—	4	4	no	—
210	C₅×D₄	—	4	4	no	—
211	C₅×D₄	—	4	8	no	—
212	C₅×D₄	—	4	4	no	—
213	C₂×C₄×C₅	—	2	4	no	—
214	C₂×C₄×C₅	—	2	4	no	—
215	C₂×C₄×C₅	—	4	4	no	—
216	C₂×C₄×C₅	—	4	4	no	—
217	C₂×C₄×C₅	—	4	4	no	—
218	C₂×C₄×C₅	—	4	4	no	—
219	C₂×C₄×C₅	—	4	8	no	—
220	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
221	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
222	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
223	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
224	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
225	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
226	C₂×C₄×C₅	#121	1	2	yes	—
227	C₂×C₄×C₅	#121	1	2	yes	—
228	C₂×C₄×C₅	#121	1	2	yes	—
229	C₂×C₄×C₅	#124	1	2	yes	—
230	C₂×C₄×C₅	#127	1	2	yes	—
231	C₂×C₄×C₅	#127	1	2	yes	—
232	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
233	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
234	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
235	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
236	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
238	C₅×C₈	—	2	4	no	—
239	C₅×C₈	—	4	4	no	—
240	C₅×C₈	C₂×C₂²⋊C₄	1	2	yes	—
241	C₅×C₈	C₂×C₂²⋊C₄	1	2	yes	—
242	C₅×C₈	C₂×C₂²⋊C₄	1	2	yes	—
243	C₅×C₈	C₂×C₂²⋊C₄	1	2	yes	—
244	C₂×#41	—	2	2	no	—
245	C₂×#41	—	2	2	no	—
246	C₂×#41	—	2	4	no	—
247	C₂×#41	—	2	4	no	—
248	C₂×#41	—	2	4	no	—
249	C₂×#41	—	2	4	no	—
250	C₂×#41	C₂×C₄×C₅	1	4	yes	—
251	C₂×#41	C₂³×C₅	1	4	yes	—
252	C₂²×Q₈	C₂×C₄×C₅	1	2	yes	—
253	C₂²×D₄	—	2	2	no	—
254	C₂²×D₄	C₂×C₄×C₅	1	2	yes	—
255	C₂³×C₄	—	2	2	no	—
256	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
257	#135	—	2	8	no	—
258	#135	—	2	8	no	—
259	#135	—	2	4	no	—
260	#135	—	2	4	no	—
261	#135	—	2	4	no	—
262	#135	—	2	4	no	—
263	C₂×Q₁₆	—	2	4	no	—
264	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
265	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
266	C₂×SD₁₆	—	2	8	no	—
267	C₂×SD₁₆	C₂×C₄×C₅	1	4	yes	—
268	C₂×SD₁₆	C₂×C₄×C₅	1	4	yes	—
269	C₂×D₈	—	2	4	no	—
270	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
271	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
272	C₂×M₁₆	—	2	2	no	—
273	C₂×M₁₆	—	2	2	no	—
274	C₂×M₁₆	—	2	2	no	—
275	C₂×M₁₆	—	2	4	no	—
276	C₂×M₁₆	—	2	4	no	—
277	C₂×M₁₆	C₅×D₄	1	2	yes	—
278	C₂×M₁₆	C₅×D₄	1	2	yes	—
279	C₂×M₁₆	C₂³×C₅	1	4	yes	—
280	C₂²×C₈	—	2	2	no	—
281	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
282	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
283	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
284	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
285	C₂²×C₈	C₅×D₄	1	4	yes	—
286	C₂²×C₈	C₅×D₄	1	4	yes	—
287	C₂²×C₈	C₂³×C₅	1	2	yes	—
288	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
289	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
290	#117	—	2	2	no	—
291	#117	C₅×D₄	1	4	yes	—
292	C₂×C₄⋊C₄	—	2	1	no	—
293	C₂×C₂²⋊C₄	—	2	1	no	—
294	C₂×C₄²	C₂³×C₅	1	2	yes	—
295	#108	—	2	4	no	—
296	#108	—	2	4	no	—
297	#104	—	2	16	no	—
298	C₂²⋊C₈	—	2	16	no	—
299	#97	—	2	4	no	—
300	C₄×C₈	—	2	4	no	—
301	C₂²×C₅	—	2	2	no	—
302	C₂²×C₅	—	2	2	no	—
303	C₂²×C₅	—	2	2	no	—
304	C₂²×C₅	—	4	2	no	—
305	C₂²×C₅	—	4	2	no	—
306	C₂²×C₅	—	4	8	no	—
307	C₂²×C₅	—	4	4	no	—
308	C₂²×C₅	—	8	8	no	—
309	C₂²×C₅	#390	1	2	yes	—
310	C₂²×C₅	#390	1	2	yes	—
311	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
312	C₂²×C₅	C₂×#121	1	1	yes	characteristic
313	C₂²×C₅	C₂×#124	1	1	yes	characteristic
314	C₄×C₅	—	4	4	no	—
315	C₄×C₅	—	4	4	no	—
316	C₄×C₅	#390	1	2	yes	—
317	C₄×C₅	#390	1	2	yes	—
318	C₄×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
319	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
320	C₄×C₅	C₂×#121	1	1	yes	characteristic
321	C₄×C₅	C₂×#127	1	1	yes	characteristic
322	C₂⁴	—	2	2	no	—
323	C₂⁴	C₅×#41	1	1	yes	characteristic
324	#41	—	2	8	no	—
325	#41	—	2	8	no	—
326	#41	—	2	4	no	—
327	#41	—	2	4	no	—
328	#41	—	2	4	no	—
329	#41	—	2	4	no	—
330	#41	—	4	4	no	—
331	#41	—	4	4	no	—
332	#41	—	4	4	no	—
333	#41	—	4	4	no	—
334	C₂×Q₈	—	2	2	no	—
335	C₂×Q₈	—	2	4	no	—
336	C₂×Q₈	—	2	4	no	—
337	C₂×Q₈	C₂²×C₄×C₅	1	4	yes	—
338	C₂×D₄	—	2	4	no	—
339	C₂×D₄	—	2	4	no	—
340	C₂×D₄	—	2	4	no	—
341	C₂×D₄	—	2	4	no	—
342	C₂×D₄	—	2	2	no	—
343	C₂×D₄	—	4	4	no	—
344	C₂×D₄	—	4	4	no	—
345	C₂×D₄	C₂²×C₄×C₅	1	4	yes	—
346	C₂²×C₄	—	2	2	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	2	4	no	—
349	C₂²×C₄	—	2	4	no	—
350	C₂²×C₄	—	4	4	no	—
351	C₂²×C₄	—	4	4	no	—
352	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
353	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
354	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
355	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
356	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
357	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
358	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
359	C₂²×C₄	C₅×#41	1	1	yes	characteristic
360	C₂²×C₄	C₅×#41	1	2	yes	—
361	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
362	Q₁₆	—	2	4	no	—
363	Q₁₆	—	2	4	no	—
364	SD₁₆	—	2	8	no	—
365	SD₁₆	—	2	8	no	—
366	D₈	—	2	4	no	—
367	D₈	—	2	4	no	—
368	M₁₆	—	2	4	no	—
369	M₁₆	—	2	4	no	—
370	M₁₆	—	2	4	no	—
371	M₁₆	—	2	4	no	—
372	M₁₆	—	4	4	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	2	2	no	—
375	C₂×C₈	—	2	4	no	—
376	C₂×C₈	—	4	4	no	—
377	C₂×C₈	C₅×C₂²⋊C₄	1	4	yes	—
378	C₂×C₈	C₅×C₂²⋊C₄	1	4	yes	—
379	C₂×C₈	C₅×C₂²⋊C₄	1	4	yes	—
380	C₂×C₈	C₅×C₂²⋊C₄	1	4	yes	—
381	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
382	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
383	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
384	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
385	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
386	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
387	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
388	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
389	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
390	C₄⋊C₄	—	2	4	no	—
391	C₂²⋊C₄	—	2	4	no	—
392	C₄²	—	2	2	no	—
393	C₄²	C₂×C₅×D₄	1	4	yes	—
394	C₂×C₅	—	2	1	no	—
395	C₂×C₅	—	2	2	no	—
396	C₂×C₅	—	2	1	no	—
397	C₂×C₅	—	2	2	no	—
398	C₂×C₅	—	8	4	no	—
399	C₂×C₅	C₂×#390	1	1	yes	characteristic
400	C₂×C₅	#2957	1	2	yes	—
401	C₂³	—	2	4	no	—
402	C₂³	—	4	2	no	—
403	C₂³	—	4	4	no	—
404	C₂³	—	8	4	no	—
405	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
406	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
407	C₂³	C₅×#121	1	2	yes	—
408	C₂³	C₅×#124	1	2	yes	—
409	C₂³	C₂×C₅×#41	1	1	yes	characteristic
410	Q₈	—	2	4	no	—
411	Q₈	—	2	4	no	—
412	Q₈	—	4	4	no	—
413	D₄	—	2	4	no	—
414	D₄	—	2	4	no	—
415	D₄	—	4	8	no	—
416	D₄	—	4	4	no	—
417	D₄	—	4	4	no	—
418	D₄	—	4	4	no	—
419	C₂×C₄	—	2	4	no	—
420	C₂×C₄	—	2	4	no	—
421	C₂×C₄	—	4	4	no	—
422	C₂×C₄	—	4	4	no	—
423	C₂×C₄	—	4	8	no	—
424	C₂×C₄	—	4	4	no	—
425	C₂×C₄	—	4	4	no	—
426	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
427	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
428	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
429	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
430	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
431	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
432	C₂×C₄	C₅×#121	1	2	yes	—
433	C₂×C₄	C₅×#121	1	2	yes	—
434	C₂×C₄	C₅×#121	1	2	yes	—
435	C₂×C₄	C₅×#124	1	2	yes	—
436	C₂×C₄	C₅×#127	1	2	yes	—
437	C₂×C₄	C₅×#127	1	2	yes	—
438	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
439	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
440	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
441	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
442	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
443	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
444	C₈	—	2	4	no	—
445	C₈	—	4	4	no	—
446	C₈	C₂×C₅×C₂²⋊C₄	1	2	yes	—
447	C₈	C₂×C₅×C₂²⋊C₄	1	2	yes	—
448	C₈	C₂×C₅×C₂²⋊C₄	1	2	yes	—
449	C₈	C₂×C₅×C₂²⋊C₄	1	2	yes	—
450	C₅	#21240	1	1	yes	characteristic, direct product factor, Sylow
451	C₂²	—	2	2	no	—
452	C₂²	—	2	2	no	—
453	C₂²	—	2	2	no	—
454	C₂²	—	4	2	no	—
455	C₂²	—	4	2	no	—
456	C₂²	—	4	4	no	—
457	C₂²	—	4	8	no	—
458	C₂²	—	8	8	no	—
459	C₂²	C₅×#390	1	2	yes	—
460	C₂²	C₅×#390	1	2	yes	—
461	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
462	C₂²	C₂×C₅×#121	1	1	yes	characteristic
463	C₂²	C₂×C₅×#124	1	1	yes	characteristic
464	C₄	—	4	4	no	—
465	C₄	—	4	4	no	—
466	C₄	C₅×#390	1	2	yes	—
467	C₄	C₅×#390	1	2	yes	—
468	C₄	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
469	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
470	C₄	C₂×C₅×#121	1	1	yes	characteristic
471	C₄	C₂×C₅×#127	1	1	yes	characteristic
472	C₂	—	2	1	no	—
473	C₂	—	2	2	no	—
474	C₂	—	2	2	no	—
475	C₂	—	2	1	no	—
476	C₂	—	8	4	no	—
477	C₂	C₂×C₅×#390	1	1	yes	characteristic
478	C₂	C₅×#2957	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂²×C₅×#135	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₅×M₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×#108	C₂	semidirect
3. C₂×C₅×#421	C₂	semidirect
4. C₅×#2879	C₂	semidirect
5. C₂×C₅×#405	C₂	semidirect
6. C₅×#1975	C₂	semidirect
7. C₅×#1974	C₂	semidirect
11. C₂×C₅×#135	C₄	semidirect
12. C₂×C₅×#135	C₄	semidirect
15. C₂²×C₅×Q₁₆	C₄	semidirect
16. C₂²×C₅×SD₁₆	C₄	semidirect
17. C₂²×C₅×D₈	C₄	semidirect
23. C₂×C₅×#108	C₂²	semidirect
24. C₂×C₅×#108	C₂²	semidirect
26. C₂×C₅×#104	C₂²	semidirect
28. C₅×#408	C₂²	semidirect
31. C₂×C₅×#97	C₂²	semidirect
32. C₂×C₄×C₅×C₈	C₂²	semidirect
33. #21240	C₅	direct (N × H)
92. C₂²×#108	C₂×C₅	semidirect
93. C₂×#421	C₂×C₅	semidirect
94. #2879	C₂×C₅	semidirect
95. C₂×#405	C₂×C₅	semidirect
96. #1975	C₂×C₅	semidirect
97. #1974	C₂×C₅	semidirect
173. C₂×#135	C₄×C₅	semidirect
174. C₂×#135	C₄×C₅	semidirect
177. C₂²×Q₁₆	C₄×C₅	semidirect
178. C₂²×SD₁₆	C₄×C₅	semidirect
179. C₂²×D₈	C₄×C₅	semidirect
185. C₂×#108	C₂²×C₅	semidirect
186. C₂×#108	C₂²×C₅	semidirect
188. C₂×#104	C₂²×C₅	semidirect
190. #408	C₂²×C₅	semidirect
193. C₂×#97	C₂²×C₅	semidirect
194. C₂×C₄×C₈	C₂²×C₅	semidirect
450. C₅	#21240	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×#135	2
C₂²×C₅×#108	2
C₂×C₅×#421	2
C₅×#2879	2
C₂×C₅×#405	2
C₅×#1975	2
C₅×#1974	2
#21240	5

Subgroups — C5×#21240 #11943960

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21240 #11943960