Subgroups — C₅×#78694 #14411968

Order 1920.

1600 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#1009	C₂	1	1	yes	maximal, characteristic
2	C₅×#5521	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#410	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5240	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#5006	C₂	1	2	yes	maximal, complemented
6	C₅×D₃×#99	C₂	1	8	yes	maximal, complemented
7	C₅×#2883	—	3	1	no	maximal, Hall
8	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
9	C₅×#1009	—	2	4	no	—
10	C₅×#1009	—	2	4	no	—
11	C₂×C₅×#239	C₄	1	2	yes	complemented
12	C₂×C₅×D₃×D₄	C₄	1	2	yes	complemented
13	C₂×C₅×D₃×D₄	C₂²	1	4	yes	—
14	C₂×C₅×#235	C₄	1	2	yes	complemented
15	C₂×C₅×#235	C₂²	1	1	yes	characteristic
16	C₂²×C₅×D₁₂	C₄	1	2	yes	complemented
17	C₂²×C₅×D₁₂	C₂²	1	1	yes	characteristic
18	C₃×C₅×#117	C₂²	1	2	yes	complemented
19	C₅×#926	C₂²	1	2	yes	complemented
20	C₅×#893	—	2	4	no	—
21	C₅×D₃×C₂²⋊C₄	C₂²	1	8	yes	complemented
22	C₄×C₅×D₁₂	—	2	4	no	—
23	C₃×C₅×#99	C₂²	1	4	yes	complemented
24	C₅×#834	C₂²	1	4	yes	complemented
25	C₅×#806	C₂²	1	8	yes	complemented
26	#78694	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₂×C₅×H(5,𝔽₂)	—	3	1	no	—
28	C₅×#518	—	3	2	no	—
29	C₅×#410	—	3	1	no	—
30	C₅×#410	—	3	2	no	—
31	C₅×#410	D₃	1	1	yes	characteristic, complemented
32	C₂×C₅×#99	—	3	8	no	—
33	C₂³×C₅×D₃	C₂×C₄	1	4	yes	complemented
34	C₂³×C₅×D₃	C₂³	1	2	yes	—
35	C₃×C₅×#41	—	2	2	no	—
36	C₃×C₅×#41	—	2	2	no	—
37	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic, complemented
38	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic, complemented
39	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
40	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	complemented
41	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
42	C₂×C₅×#67	C₂×C₄	1	4	yes	complemented
43	C₂×C₅×#67	C₂³	1	2	yes	—
44	C₅×#239	—	2	8	no	—
45	C₅×D₃×D₄	—	2	16	no	—
46	C₅×D₃×D₄	—	2	8	no	—
47	C₅×#235	—	2	2	no	—
48	C₅×#235	—	2	2	no	—
49	C₅×#235	—	2	4	no	—
50	C₅×#235	—	2	4	no	—
51	C₂×C₅×D₁₂	—	2	4	no	—
52	C₂×C₅×D₁₂	—	2	4	no	—
53	C₂×C₅×D₁₂	C₂×C₄	1	2	yes	—
54	C₂×C₅×D₁₂	C₂×C₄	1	2	yes	—
55	C₂×C₅×D₁₂	C₂×C₄	1	1	yes	characteristic, complemented
56	C₂×C₅×D₁₂	C₂×C₄	1	4	yes	complemented
57	C₂×C₅×D₁₂	D₄	1	4	yes	—
58	C₂×C₅×D₁₂	D₄	1	4	yes	—
59	C₂×C₄×C₅×D₃	—	2	8	no	—
60	C₂×C₄×C₅×D₃	C₂×C₄	1	4	yes	complemented
61	C₂×C₄×C₅×D₃	C₂×C₄	1	2	yes	complemented
62	C₂×C₅×Dic₆	C₂×C₄	1	1	yes	characteristic, complemented
63	C₃×C₅×C₄⋊C₄	—	2	2	no	—
64	C₃×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
65	C₃×C₄²×C₅	—	2	2	no	—
66	C₅×#217	C₂³	1	4	yes	—
67	C₅×(C₂×D₃)⋊C₄	—	2	8	no	—
68	C₅×C₄⋊Dic₃	—	2	2	no	—
69	C₄×C₅×Dic₃	—	2	2	no	—
70	C₂×#1009	C₂×C₅	1	1	yes	characteristic
71	#5521	C₂×C₅	1	2	yes	complemented
72	C₃×#410	C₂×C₅	1	1	yes	characteristic, complemented
73	#5240	C₂×C₅	1	1	yes	characteristic, complemented
74	#5006	C₂×C₅	1	2	yes	complemented
75	D₃×#99	C₂×C₅	1	8	yes	complemented
76	C₅×H(5,𝔽₂)	—	6	4	no	—
77	C₅×H(5,𝔽₂)	—	6	4	no	—
78	C₂×C₅×#41	—	3	2	no	—
79	C₂×C₅×#41	—	3	2	no	—
80	C₂×C₅×#41	—	3	1	no	—
81	C₂×C₅×#41	C₂×D₃	1	1	yes	characteristic
82	C₂²×C₅×D₄	—	3	2	no	—
83	C₂²×C₅×D₄	—	3	4	no	—
84	C₂²×C₅×D₄	—	3	2	no	—
85	C₂²×C₅×D₄	—	3	1	no	—
86	C₄×C₅×D₄	—	6	4	no	—
87	C₄×C₅×D₄	—	6	4	no	—
88	C₅×#117	—	3	2	no	—
89	C₅×#117	C₂×D₃	1	2	yes	complemented
90	C₂×C₅×C₂²⋊C₄	—	3	8	no	—
91	C₅×#99	—	3	4	no	—
92	C₅×#99	—	3	8	no	—
93	C₅×#99	C₂×D₃	1	4	yes	complemented
94	#2883	—	3	1	no	Sylow
95	C₂³×C₃×C₅	C₂²×C₄	1	2	yes	—
96	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
97	C₂²×C₅×D₃	—	2	8	no	—
98	C₂²×C₅×D₃	—	8	2	no	—
99	C₂²×C₅×D₃	—	8	4	no	—
100	C₂²×C₅×D₃	C₂²×C₄	1	2	yes	—
101	C₂²×C₅×D₃	C₂²×C₄	1	4	yes	—
102	C₂²×C₅×D₃	C₂×D₄	1	8	yes	—
103	C₃×C₅×Q₈	—	2	2	no	—
104	C₃×C₅×D₄	—	2	4	no	—
105	C₃×C₅×D₄	—	2	2	no	—
106	C₂×C₃×C₄×C₅	—	2	2	no	—
107	C₂×C₃×C₄×C₅	—	2	2	no	—
108	C₂×C₃×C₄×C₅	—	2	4	no	—
109	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
110	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
113	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
114	C₅×#67	—	2	4	no	—
115	C₅×#67	—	2	8	no	—
116	C₂×C₅×Dic₃	—	2	4	no	—
117	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
118	C₅×D₁₂	—	2	8	no	—
119	C₅×D₁₂	—	2	1	no	—
120	C₅×D₁₂	—	2	1	no	—
121	C₅×D₁₂	C₂²⋊C₄	1	4	yes	—
122	C₅×D₁₂	C₂²⋊C₄	1	4	yes	—
123	C₅×D₁₂	C₂²⋊C₄	1	8	yes	—
124	C₄×C₅×D₃	—	2	8	no	—
125	C₄×C₅×D₃	—	2	4	no	—
126	C₄×C₅×D₃	—	8	8	no	—
127	C₅×Dic₆	—	2	1	no	—
128	C₅×Dic₆	—	2	1	no	—
129	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
130	#1009	—	2	4	no	—
131	#1009	—	2	4	no	—
132	C₂×#239	C₄×C₅	1	2	yes	complemented
133	C₂×D₃×D₄	C₄×C₅	1	2	yes	complemented
134	C₂×D₃×D₄	C₂²×C₅	1	4	yes	—
135	C₂×#235	C₄×C₅	1	2	yes	complemented
136	C₂×#235	C₂²×C₅	1	1	yes	characteristic
137	C₂²×D₁₂	C₄×C₅	1	2	yes	complemented
138	C₂²×D₁₂	C₂²×C₅	1	1	yes	characteristic
139	C₃×#117	C₂²×C₅	1	2	yes	complemented
140	#926	C₂²×C₅	1	2	yes	complemented
141	#893	—	2	4	no	—
142	D₃×C₂²⋊C₄	C₂²×C₅	1	8	yes	complemented
143	C₄×D₁₂	—	2	4	no	—
144	C₃×#99	C₂²×C₅	1	4	yes	complemented
145	#834	C₂²×C₅	1	4	yes	complemented
146	#806	C₂²×C₅	1	8	yes	complemented
147	C₂⁴×C₅	—	3	2	no	—
148	C₂⁴×C₅	—	3	4	no	—
149	C₅×#41	—	2	2	no	—
150	C₅×#41	—	2	2	no	—
151	C₅×#41	—	6	4	no	—
152	C₅×#41	—	6	4	no	—
153	C₅×#41	—	6	8	no	—
154	C₅×#41	—	6	2	no	—
155	C₅×#41	—	6	2	no	—
156	C₂×C₅×Q₈	—	3	1	no	—
157	C₂×C₅×Q₈	C₄×D₃	1	1	yes	characteristic, complemented
158	C₂×C₅×D₄	—	3	4	no	—
159	C₂×C₅×D₄	—	3	2	no	—
160	C₂×C₅×D₄	—	3	2	no	—
161	C₂×C₅×D₄	—	3	2	no	—
162	C₂×C₅×D₄	—	3	4	no	—
163	C₂×C₅×D₄	—	3	1	no	—
164	C₂×C₅×D₄	—	3	4	no	—
165	C₂×C₅×D₄	—	3	4	no	—
166	C₂×C₅×D₄	—	6	4	no	—
167	C₂×C₅×D₄	—	6	4	no	—
168	C₂×C₅×D₄	—	6	8	no	—
169	C₂×C₅×D₄	—	6	16	no	—
170	C₂×C₅×D₄	C₄×D₃	1	1	yes	characteristic, complemented
171	C₂×C₅×D₄	C₂²×D₃	1	2	yes	—
172	C₂²×C₄×C₅	—	3	4	no	—
173	C₂²×C₄×C₅	—	3	2	no	—
174	C₂²×C₄×C₅	—	6	8	no	—
175	C₂²×C₄×C₅	C₄×D₃	1	2	yes	complemented
176	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
177	C₅×C₄⋊C₄	—	2	2	no	—
178	C₅×C₄⋊C₄	—	6	2	no	—
179	C₅×C₂²⋊C₄	—	3	4	no	—
180	C₅×C₂²⋊C₄	—	6	8	no	—
181	C₅×C₂²⋊C₄	C₂²×D₃	1	4	yes	—
182	C₄²×C₅	—	2	2	no	—
183	C₄²×C₅	—	6	2	no	—
184	C₂×H(5,𝔽₂)	—	3	1	no	—
185	#518	—	3	2	no	—
186	#410	—	3	1	no	—
187	#410	—	3	2	no	—
188	#410	C₅×D₃	1	1	yes	characteristic, complemented
189	C₂×#99	—	3	8	no	—
190	C₂²×C₃×C₅	—	2	2	no	—
191	C₂²×C₃×C₅	—	4	1	no	—
192	C₂²×C₃×C₅	—	4	2	no	—
193	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
194	C₂²×C₃×C₅	C₂²×D₄	1	2	yes	—
195	C₂×C₅×D₃	—	2	2	no	—
196	C₂×C₅×D₃	—	4	8	no	—
197	C₂×C₅×D₃	—	4	4	no	—
198	C₂×C₅×D₃	—	4	2	no	—
199	C₂×C₅×D₃	—	8	8	no	—
200	C₂×C₅×D₃	C₂×C₂²⋊C₄	1	8	yes	—
201	C₃×C₄×C₅	—	2	2	no	—
202	C₃×C₄×C₅	—	4	4	no	—
203	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
204	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
205	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
206	C₅×Dic₃	—	2	2	no	—
207	C₅×Dic₃	—	4	4	no	—
208	C₂³×D₃	C₂×C₄×C₅	1	4	yes	complemented
209	C₂³×D₃	C₂³×C₅	1	2	yes	—
210	C₃×#41	—	2	2	no	—
211	C₃×#41	—	2	2	no	—
212	C₂×C₃×Q₈	C₂×C₄×C₅	1	1	yes	characteristic, complemented
213	C₂×C₃×D₄	C₂×C₄×C₅	1	1	yes	characteristic, complemented
214	C₂×C₃×D₄	C₂³×C₅	1	2	yes	—
215	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	complemented
216	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
217	C₂×#67	C₂×C₄×C₅	1	4	yes	complemented
218	C₂×#67	C₂³×C₅	1	2	yes	—
219	#239	—	2	8	no	—
220	D₃×D₄	—	2	16	no	—
221	D₃×D₄	—	2	8	no	—
222	#235	—	2	2	no	—
223	#235	—	2	2	no	—
224	#235	—	2	4	no	—
225	#235	—	2	4	no	—
226	C₂×D₁₂	—	2	4	no	—
227	C₂×D₁₂	—	2	4	no	—
228	C₂×D₁₂	C₂×C₄×C₅	1	2	yes	—
229	C₂×D₁₂	C₂×C₄×C₅	1	2	yes	—
230	C₂×D₁₂	C₂×C₄×C₅	1	1	yes	characteristic, complemented
231	C₂×D₁₂	C₂×C₄×C₅	1	4	yes	complemented
232	C₂×D₁₂	C₅×D₄	1	4	yes	—
233	C₂×D₁₂	C₅×D₄	1	4	yes	—
234	C₂×C₄×D₃	—	2	8	no	—
235	C₂×C₄×D₃	C₂×C₄×C₅	1	4	yes	complemented
236	C₂×C₄×D₃	C₂×C₄×C₅	1	2	yes	complemented
237	C₂×Dic₆	C₂×C₄×C₅	1	1	yes	characteristic, complemented
238	C₃×C₄⋊C₄	—	2	2	no	—
239	C₃×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
240	C₃×C₄²	—	2	2	no	—
241	#217	C₂³×C₅	1	4	yes	—
242	(C₂×D₃)⋊C₄	—	2	8	no	—
243	C₄⋊Dic₃	—	2	2	no	—
244	C₄×Dic₃	—	2	2	no	—
245	C₂³×C₅	—	3	4	no	—
246	C₂³×C₅	—	3	8	no	—
247	C₂³×C₅	—	3	2	no	—
248	C₂³×C₅	—	6	8	no	—
249	C₂³×C₅	—	24	2	no	—
250	C₂³×C₅	—	24	4	no	—
251	C₂³×C₅	C₂×C₄×D₃	1	2	yes	—
252	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
253	C₅×Q₈	—	2	2	no	—
254	C₅×Q₈	—	6	1	no	—
255	C₅×Q₈	—	6	1	no	—
256	C₅×D₄	—	2	4	no	—
257	C₅×D₄	—	2	2	no	—
258	C₅×D₄	—	3	4	no	—
259	C₅×D₄	—	3	4	no	—
260	C₅×D₄	—	3	8	no	—
261	C₅×D₄	—	6	8	no	—
262	C₅×D₄	—	6	8	no	—
263	C₅×D₄	—	6	4	no	—
264	C₅×D₄	—	6	1	no	—
265	C₅×D₄	—	6	1	no	—
266	C₂×C₄×C₅	—	2	2	no	—
267	C₂×C₄×C₅	—	2	2	no	—
268	C₂×C₄×C₅	—	2	4	no	—
269	C₂×C₄×C₅	—	3	2	no	—
270	C₂×C₄×C₅	—	6	8	no	—
271	C₂×C₄×C₅	—	6	4	no	—
272	C₂×C₄×C₅	—	6	4	no	—
273	C₂×C₄×C₅	—	24	8	no	—
274	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
275	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
276	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
277	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
278	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
279	H(5,𝔽₂)	—	6	4	no	—
280	H(5,𝔽₂)	—	6	4	no	—
281	C₂×#41	—	3	2	no	—
282	C₂×#41	—	3	2	no	—
283	C₂×#41	—	3	1	no	—
284	C₂×#41	C₂×C₅×D₃	1	1	yes	characteristic
285	C₂²×D₄	—	3	2	no	—
286	C₂²×D₄	—	3	4	no	—
287	C₂²×D₄	—	3	2	no	—
288	C₂²×D₄	—	3	1	no	—
289	C₄×D₄	—	6	4	no	—
290	C₄×D₄	—	6	4	no	—
291	#117	—	3	2	no	—
292	#117	C₂×C₅×D₃	1	2	yes	complemented
293	C₂×C₂²⋊C₄	—	3	8	no	—
294	#99	—	3	4	no	—
295	#99	—	3	8	no	—
296	#99	C₂×C₅×D₃	1	4	yes	complemented
297	C₂×C₃×C₅	—	2	2	no	—
298	C₂×C₃×C₅	—	2	1	no	—
299	C₂×C₃×C₅	—	4	2	no	—
300	C₂×C₃×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
301	C₅×D₃	—	2	8	no	—
302	C₅×D₃	—	4	2	no	—
303	C₂³×C₃	C₂²×C₄×C₅	1	2	yes	—
304	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
305	C₂²×D₃	—	2	8	no	—
306	C₂²×D₃	—	8	2	no	—
307	C₂²×D₃	—	8	4	no	—
308	C₂²×D₃	C₂²×C₄×C₅	1	2	yes	—
309	C₂²×D₃	C₂²×C₄×C₅	1	4	yes	—
310	C₂²×D₃	C₂×C₅×D₄	1	8	yes	—
311	C₃×Q₈	—	2	2	no	—
312	C₃×D₄	—	2	4	no	—
313	C₃×D₄	—	2	2	no	—
314	C₂×C₃×C₄	—	2	2	no	—
315	C₂×C₃×C₄	—	2	2	no	—
316	C₂×C₃×C₄	—	2	4	no	—
317	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
318	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
319	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
320	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
321	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
322	#67	—	2	4	no	—
323	#67	—	2	8	no	—
324	C₂×Dic₃	—	2	4	no	—
325	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
326	D₁₂	—	2	8	no	—
327	D₁₂	—	2	1	no	—
328	D₁₂	—	2	1	no	—
329	D₁₂	C₅×C₂²⋊C₄	1	4	yes	—
330	D₁₂	C₅×C₂²⋊C₄	1	4	yes	—
331	D₁₂	C₅×C₂²⋊C₄	1	8	yes	—
332	C₄×D₃	—	2	8	no	—
333	C₄×D₃	—	2	4	no	—
334	C₄×D₃	—	8	8	no	—
335	Dic₆	—	2	1	no	—
336	Dic₆	—	2	1	no	—
337	C₂²×C₅	—	2	2	no	—
338	C₂²×C₅	—	3	8	no	—
339	C₂²×C₅	—	4	1	no	—
340	C₂²×C₅	—	4	2	no	—
341	C₂²×C₅	—	6	2	no	—
342	C₂²×C₅	—	12	8	no	—
343	C₂²×C₅	—	12	4	no	—
344	C₂²×C₅	—	12	2	no	—
345	C₂²×C₅	—	24	8	no	—
346	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
347	C₂²×C₅	C₂×D₃×D₄	1	2	yes	—
348	C₄×C₅	—	2	2	no	—
349	C₄×C₅	—	4	4	no	—
350	C₄×C₅	—	6	2	no	—
351	C₄×C₅	—	12	4	no	—
352	C₄×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
353	C₄×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
354	C₄×C₅	D₃×C₂²⋊C₄	1	2	yes	—
355	C₂⁴	—	3	2	no	—
356	C₂⁴	—	3	4	no	—
357	#41	—	2	2	no	—
358	#41	—	2	2	no	—
359	#41	—	6	4	no	—
360	#41	—	6	4	no	—
361	#41	—	6	8	no	—
362	#41	—	6	2	no	—
363	#41	—	6	2	no	—
364	C₂×Q₈	—	3	1	no	—
365	C₂×Q₈	C₄×C₅×D₃	1	1	yes	characteristic, complemented
366	C₂×D₄	—	3	4	no	—
367	C₂×D₄	—	3	2	no	—
368	C₂×D₄	—	3	2	no	—
369	C₂×D₄	—	3	2	no	—
370	C₂×D₄	—	3	4	no	—
371	C₂×D₄	—	3	1	no	—
372	C₂×D₄	—	3	4	no	—
373	C₂×D₄	—	3	4	no	—
374	C₂×D₄	—	6	4	no	—
375	C₂×D₄	—	6	4	no	—
376	C₂×D₄	—	6	8	no	—
377	C₂×D₄	—	6	16	no	—
378	C₂×D₄	C₄×C₅×D₃	1	1	yes	characteristic, complemented
379	C₂×D₄	C₂²×C₅×D₃	1	2	yes	—
380	C₂²×C₄	—	3	4	no	—
381	C₂²×C₄	—	3	2	no	—
382	C₂²×C₄	—	6	8	no	—
383	C₂²×C₄	C₄×C₅×D₃	1	2	yes	complemented
384	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
385	C₄⋊C₄	—	2	2	no	—
386	C₄⋊C₄	—	6	2	no	—
387	C₂²⋊C₄	—	3	4	no	—
388	C₂²⋊C₄	—	6	8	no	—
389	C₂²⋊C₄	C₂²×C₅×D₃	1	4	yes	—
390	C₄²	—	2	2	no	—
391	C₄²	—	6	2	no	—
392	C₃×C₅	#2883	1	1	yes	characteristic, complemented, Hall
393	C₂²×C₃	—	2	2	no	—
394	C₂²×C₃	—	4	1	no	—
395	C₂²×C₃	—	4	2	no	—
396	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
397	C₂²×C₃	C₂²×C₅×D₄	1	2	yes	—
398	C₂×D₃	—	2	2	no	—
399	C₂×D₃	—	4	8	no	—
400	C₂×D₃	—	4	4	no	—
401	C₂×D₃	—	4	2	no	—
402	C₂×D₃	—	8	8	no	—
403	C₂×D₃	C₂×C₅×C₂²⋊C₄	1	8	yes	—
404	C₃×C₄	—	2	2	no	—
405	C₃×C₄	—	4	4	no	—
406	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
407	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
408	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
409	Dic₃	—	2	2	no	—
410	Dic₃	—	4	4	no	—
411	C₂×C₅	—	2	2	no	—
412	C₂×C₅	—	2	1	no	—
413	C₂×C₅	—	4	2	no	—
414	C₂×C₅	—	6	8	no	—
415	C₂×C₅	—	12	2	no	—
416	C₂×C₅	C₂×D₃×C₂²⋊C₄	1	1	yes	characteristic
417	C₂³	—	3	4	no	—
418	C₂³	—	3	8	no	—
419	C₂³	—	3	2	no	—
420	C₂³	—	6	8	no	—
421	C₂³	—	24	2	no	—
422	C₂³	—	24	4	no	—
423	C₂³	C₂×C₄×C₅×D₃	1	2	yes	—
424	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
425	Q₈	—	2	2	no	—
426	Q₈	—	6	1	no	—
427	Q₈	—	6	1	no	—
428	D₄	—	2	4	no	—
429	D₄	—	2	2	no	—
430	D₄	—	3	4	no	—
431	D₄	—	3	4	no	—
432	D₄	—	3	8	no	—
433	D₄	—	6	8	no	—
434	D₄	—	6	8	no	—
435	D₄	—	6	4	no	—
436	D₄	—	6	1	no	—
437	D₄	—	6	1	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	4	no	—
441	C₂×C₄	—	3	2	no	—
442	C₂×C₄	—	6	8	no	—
443	C₂×C₄	—	6	4	no	—
444	C₂×C₄	—	6	4	no	—
445	C₂×C₄	—	24	8	no	—
446	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
447	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
448	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
449	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
450	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
451	C₂×C₃	—	2	2	no	—
452	C₂×C₃	—	2	1	no	—
453	C₂×C₃	—	4	2	no	—
454	C₂×C₃	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
455	D₃	—	2	8	no	—
456	D₃	—	4	2	no	—
457	C₅	#78694	1	1	yes	characteristic, direct product factor, Sylow
458	C₂²	—	2	2	no	—
459	C₂²	—	3	8	no	—
460	C₂²	—	4	1	no	—
461	C₂²	—	4	2	no	—
462	C₂²	—	6	2	no	—
463	C₂²	—	12	8	no	—
464	C₂²	—	12	4	no	—
465	C₂²	—	12	2	no	—
466	C₂²	—	24	8	no	—
467	C₂²	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
468	C₂²	C₂×C₅×D₃×D₄	1	2	yes	—
469	C₄	—	2	2	no	—
470	C₄	—	4	4	no	—
471	C₄	—	6	2	no	—
472	C₄	—	12	4	no	—
473	C₄	C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic
474	C₄	C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic
475	C₄	C₅×D₃×C₂²⋊C₄	1	2	yes	—
476	C₃	C₅×#2883	1	1	yes	characteristic, complemented, Sylow
477	C₂	—	2	2	no	—
478	C₂	—	2	1	no	—
479	C₂	—	4	2	no	—
480	C₂	—	6	8	no	—
481	C₂	—	12	2	no	—
482	C₂	C₂×C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	C₂×D₃×C₂²⋊C₄
commutator	C₂²×C₃	C₂³×C₄×C₅
Frattini	C₂³	C₂³×C₅×D₃
Fitting	C₃×C₅×#410	C₂
socle	C₂×C₃×C₅	C₂²×C₂²⋊C₄
prime radical	C₂×C₅×#1009	C₂
cosocle	C₂³×C₃	C₂⁴×C₅
hypercenter	C₅×#410	D₃
hypocenter	C₃	C₅×#2883
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#410

Complements

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

N	H ≅ G/N	type
2. C₅×#5521	C₂	semidirect
3. C₃×C₅×#410	C₂	semidirect
4. C₅×#5240	C₂	semidirect
5. C₅×#5006	C₂	semidirect
6. C₅×D₃×#99	C₂	semidirect
11. C₂×C₅×#239	C₄	semidirect
12. C₂×C₅×D₃×D₄	C₄	semidirect
14. C₂×C₅×#235	C₄	semidirect
16. C₂²×C₅×D₁₂	C₄	semidirect
18. C₃×C₅×#117	C₂²	semidirect
19. C₅×#926	C₂²	semidirect
21. C₅×D₃×C₂²⋊C₄	C₂²	semidirect
23. C₃×C₅×#99	C₂²	semidirect
24. C₅×#834	C₂²	semidirect
25. C₅×#806	C₂²	semidirect
26. #78694	C₅	direct (N × H)
31. C₅×#410	D₃	semidirect
33. C₂³×C₅×D₃	C₂×C₄	semidirect
37. C₂×C₃×C₅×Q₈	C₂×C₄	semidirect
38. C₂×C₃×C₅×D₄	C₂×C₄	semidirect
40. C₂²×C₃×C₄×C₅	C₂×C₄	semidirect
42. C₂×C₅×#67	C₂×C₄	semidirect
55. C₂×C₅×D₁₂	C₂×C₄	semidirect
56. C₂×C₅×D₁₂	C₂×C₄	semidirect
60. C₂×C₄×C₅×D₃	C₂×C₄	semidirect
61. C₂×C₄×C₅×D₃	C₂×C₄	semidirect
62. C₂×C₅×Dic₆	C₂×C₄	semidirect
71. #5521	C₂×C₅	semidirect
72. C₃×#410	C₂×C₅	semidirect
73. #5240	C₂×C₅	semidirect
74. #5006	C₂×C₅	semidirect
75. D₃×#99	C₂×C₅	semidirect
89. C₅×#117	C₂×D₃	semidirect
93. C₅×#99	C₂×D₃	semidirect
132. C₂×#239	C₄×C₅	semidirect
133. C₂×D₃×D₄	C₄×C₅	semidirect
135. C₂×#235	C₄×C₅	semidirect
137. C₂²×D₁₂	C₄×C₅	semidirect
139. C₃×#117	C₂²×C₅	semidirect
140. #926	C₂²×C₅	semidirect
142. D₃×C₂²⋊C₄	C₂²×C₅	semidirect
144. C₃×#99	C₂²×C₅	semidirect
145. #834	C₂²×C₅	semidirect
146. #806	C₂²×C₅	semidirect
157. C₂×C₅×Q₈	C₄×D₃	semidirect
170. C₂×C₅×D₄	C₄×D₃	semidirect
175. C₂²×C₄×C₅	C₄×D₃	semidirect
188. #410	C₅×D₃	semidirect
208. C₂³×D₃	C₂×C₄×C₅	semidirect
212. C₂×C₃×Q₈	C₂×C₄×C₅	semidirect
213. C₂×C₃×D₄	C₂×C₄×C₅	semidirect
215. C₂²×C₃×C₄	C₂×C₄×C₅	semidirect
217. C₂×#67	C₂×C₄×C₅	semidirect
230. C₂×D₁₂	C₂×C₄×C₅	semidirect
231. C₂×D₁₂	C₂×C₄×C₅	semidirect
235. C₂×C₄×D₃	C₂×C₄×C₅	semidirect
236. C₂×C₄×D₃	C₂×C₄×C₅	semidirect
237. C₂×Dic₆	C₂×C₄×C₅	semidirect
292. #117	C₂×C₅×D₃	semidirect
296. #99	C₂×C₅×D₃	semidirect
365. C₂×Q₈	C₄×C₅×D₃	semidirect
378. C₂×D₄	C₄×C₅×D₃	semidirect
383. C₂²×C₄	C₄×C₅×D₃	semidirect
392. C₃×C₅	#2883	semidirect
457. C₅	#78694	direct (N × H)
476. C₃	C₅×#2883	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#1009	2
C₅×#5521	2
C₃×C₅×#410	2
C₅×#5240	2
C₅×#5006	2
C₅×D₃×#99	2
C₅×#2883	3
#78694	5

Subgroups — C5×#78694 #14411968

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#78694 #14411968