Subgroups — #14297572

Order 1920.

672 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×D₂₀	C₂	1	1	yes	maximal, characteristic
2	#11751025	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4924	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#90958	C₂	1	2	yes	maximal
5	#11745985	C₂	1	2	yes	maximal, complemented
6	#10601320	—	3	1	no	maximal, Hall
7	C₂²×C₃×D₂₀	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₄×D₅	C₄	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×#6710	C₂²	1	2	yes	—
11	C₂×C₅×#208	C₂²	1	2	yes	—
12	C₂×C₃×#685	C₄	1	2	yes	—
13	C₂×C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
14	C₃×C₄×D₂₀	—	2	1	no	—
15	C₃×C₄×D₂₀	—	2	1	no	—
16	C₃×C₄×D₂₀	—	2	1	no	—
17	C₃×C₄×D₂₀	—	2	1	no	—
18	C₄⋊(C₃⋊#168)	—	2	1	no	—
19	C₄⋊(C₃⋊#168)	—	2	1	no	—
20	C₅×C₄⋊(C₃⋊C₈)	—	2	1	no	—
21	C₅×C₄⋊(C₃⋊C₈)	—	2	1	no	—
22	#90958	—	2	4	no	—
23	#71617	—	5	1	no	maximal, Hall
24	C₂×C₄×D₂₀	D₃	1	1	yes	characteristic
25	C₅×#423	—	3	1	no	—
26	C₂×#3955	—	3	2	no	—
27	#65213	—	3	1	no	—
28	#64774	—	3	2	no	—
29	C₂³×C₃×D₅	C₂×C₄	1	2	yes	—
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	—
34	C₂×C₃×D₂₀	—	2	1	no	—
35	C₂×C₃×D₂₀	—	2	1	no	—
36	C₂×C₃×D₂₀	—	2	1	no	—
37	C₂×C₃×D₂₀	—	2	1	no	—
38	C₂×C₃×D₂₀	D₄	1	2	yes	—
39	C₂×C₃×D₂₀	Q₈	1	2	yes	—
40	C₂×C₃×C₄×D₅	—	2	2	no	—
41	C₂×C₃×C₄×D₅	—	2	2	no	—
42	C₂×C₃×C₄×D₅	—	4	4	no	—
43	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
44	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
45	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
46	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
47	#6710	—	2	2	no	—
48	#6710	—	2	2	no	—
49	C₂×C₃⋊#168	—	2	2	no	—
50	C₅×#208	—	2	2	no	—
51	C₅×#208	—	2	2	no	—
52	C₂×C₅×C₃⋊C₈	—	2	2	no	—
53	C₃×#685	—	2	2	no	—
54	C₃×#685	—	2	2	no	—
55	C₃×C₄⋊Dic₅	—	2	1	no	—
56	C₃×C₄⋊Dic₅	—	2	1	no	—
57	C₂×C₃×C₄×D₄	—	5	1	no	—
58	C₂×#833	—	5	2	no	—
59	#4924	—	5	1	no	—
60	#4924	D₅	1	1	yes	characteristic, complemented
61	#4530	—	5	2	no	—
62	C₂²×D₂₀	C₂×D₃	1	1	yes	characteristic
63	C₂²×C₄×D₅	Dic₃	1	2	yes	—
64	C₂×C₅×M₁₆	—	3	2	no	—
65	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
66	C₂×#685	Dic₃	1	2	yes	—
67	C₂×C₄⋊Dic₅	C₂×D₃	1	1	yes	characteristic
68	C₂×#681	—	3	2	no	—
69	C₄×D₂₀	—	2	1	no	—
70	C₄×D₂₀	—	2	1	no	—
71	C₄×D₂₀	—	2	1	no	—
72	C₄×D₂₀	—	2	1	no	—
73	C₅×C₄⋊C₈	—	6	1	no	—
74	C₅×C₄⋊C₈	—	6	1	no	—
75	#3955	—	6	4	no	—
76	C₄⋊#168	—	6	1	no	—
77	C₄⋊#168	—	6	1	no	—
78	#1800	—	15	1	no	Sylow
79	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₃×D₅	—	2	2	no	—
81	C₂²×C₃×D₅	—	2	2	no	—
82	C₂²×C₃×D₅	—	4	2	no	—
83	C₂²×C₃×D₅	—	4	2	no	—
84	C₂²×C₃×D₅	C₄⋊C₄	1	4	yes	—
85	C₂×C₃×C₄×C₅	—	2	2	no	—
86	C₂×C₃×C₄×C₅	—	4	1	no	—
87	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
92	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
93	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
94	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
95	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
96	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
97	C₂×C₃×Dic₅	—	2	2	no	—
98	C₂×C₃×Dic₅	—	4	2	no	—
99	C₃×D₂₀	—	2	2	no	—
100	C₃×D₂₀	—	2	2	no	—
101	C₃×D₂₀	—	2	2	no	—
102	C₃×D₂₀	—	2	2	no	—
103	C₃×C₄×D₅	—	4	4	no	—
104	C₃×C₄×D₅	—	4	4	no	—
105	C₃⋊#168	—	4	2	no	—
106	C₅×C₃⋊C₈	—	4	2	no	—
107	C₂²×C₃×D₄	—	5	1	no	—
108	C₂³×C₃×C₄	—	5	2	no	—
109	C₃×C₄×D₄	—	10	1	no	—
110	C₃×C₄×D₄	—	10	1	no	—
111	C₃×C₄×D₄	—	10	1	no	—
112	C₃×C₄×D₄	—	10	1	no	—
113	C₂×C₃×C₄⋊C₄	—	5	1	no	—
114	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
115	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
116	C₂×#208	—	5	2	no	—
117	C₂×#208	C₂×D₅	1	2	yes	—
118	#833	—	10	4	no	—
119	C₄⋊(C₃⋊C₈)	—	2	1	no	—
120	C₄⋊(C₃⋊C₈)	—	2	1	no	—
121	C₄⋊(C₃⋊C₈)	—	10	1	no	—
122	C₄⋊(C₃⋊C₈)	—	10	1	no	—
123	C₂³×D₅	C₂×Dic₃	1	2	yes	—
124	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
125	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
126	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
127	C₂²×Dic₅	C₂×Dic₃	1	2	yes	—
128	C₂×D₂₀	—	2	1	no	—
129	C₂×D₂₀	—	2	1	no	—
130	C₂×D₂₀	—	2	1	no	—
131	C₂×D₂₀	—	2	1	no	—
132	C₂×D₂₀	Dic₆	1	2	yes	—
133	C₂×D₂₀	D₁₂	1	2	yes	—
134	C₂×C₄×D₅	—	2	2	no	—
135	C₂×C₄×D₅	—	2	2	no	—
136	C₂×C₄×D₅	—	4	4	no	—
137	C₅×M₁₆	—	6	2	no	—
138	C₅×M₁₆	—	6	2	no	—
139	C₂×C₅×C₈	—	6	2	no	—
140	C₄²×C₅	#67	1	1	yes	characteristic
141	C₄²×C₅	#67	1	1	yes	characteristic
142	C₄²×C₅	#67	1	1	yes	characteristic
143	C₄²×C₅	#67	1	1	yes	characteristic
144	#685	—	2	2	no	—
145	#685	—	2	2	no	—
146	C₄⋊Dic₅	—	2	1	no	—
147	C₄⋊Dic₅	—	2	1	no	—
148	#681	—	6	2	no	—
149	#681	—	6	2	no	—
150	C₂×#168	—	6	2	no	—
151	C₂×C₄×D₄	—	5	1	no	—
152	#423	—	3	1	no	—
153	#423	—	15	1	no	—
154	C₂×#100	—	15	2	no	—
155	#343	—	15	2	no	—
156	C₂²×C₃×C₅	—	2	1	no	—
157	C₂²×C₃×C₅	—	2	1	no	—
158	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
159	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
160	C₂²×C₃×C₅	#117	1	1	yes	characteristic
161	C₂×C₃×D₅	—	2	4	no	—
162	C₂×C₃×D₅	—	4	4	no	—
163	C₂×C₃×D₅	—	4	2	no	—
164	C₂×C₃×D₅	—	4	2	no	—
165	C₂×C₃×D₅	—	4	4	no	—
166	C₃×C₄×C₅	—	2	1	no	—
167	C₃×C₄×C₅	—	2	1	no	—
168	C₃×C₄×C₅	—	2	1	no	—
169	C₃×C₄×C₅	#100	1	2	yes	complemented
170	C₃×C₄×C₅	#121	1	1	yes	characteristic
171	C₃×C₄×C₅	#121	1	1	yes	characteristic
172	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
173	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
174	C₃×Dic₅	—	4	2	no	—
175	C₂⁴×C₃	—	5	2	no	—
176	C₂×C₃×D₄	—	5	2	no	—
177	C₂×C₃×D₄	—	5	2	no	—
178	C₂×C₃×D₄	—	10	1	no	—
179	C₂×C₃×D₄	—	10	1	no	—
180	C₂×C₃×D₄	—	10	1	no	—
181	C₂×C₃×D₄	—	10	1	no	—
182	C₂²×C₃×C₄	—	5	2	no	—
183	C₂²×C₃×C₄	—	10	2	no	—
184	C₂²×C₃×C₄	—	10	2	no	—
185	C₂²×C₃×C₄	—	20	4	no	—
186	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
187	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
188	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
189	C₃×C₄⋊C₄	—	10	1	no	—
190	C₃×C₄⋊C₄	—	10	1	no	—
191	C₃×C₂²⋊C₄	—	10	2	no	—
192	C₃×C₂²⋊C₄	—	10	2	no	—
193	C₃×C₄²	D₂₀	1	1	yes	characteristic
194	C₃×C₄²	D₂₀	1	1	yes	characteristic
195	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
196	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
197	#208	—	2	2	no	—
198	#208	—	2	2	no	—
199	#208	—	10	2	no	—
200	#208	—	10	2	no	—
201	C₂×C₃⋊C₈	—	2	2	no	—
202	C₂×C₃⋊C₈	—	10	2	no	—
203	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
204	C₂²×D₅	—	2	2	no	—
205	C₂²×D₅	—	2	2	no	—
206	C₂²×D₅	—	4	2	no	—
207	C₂²×D₅	—	4	2	no	—
208	C₂²×D₅	C₄⋊Dic₃	1	4	yes	—
209	C₂×C₄×C₅	—	2	2	no	—
210	C₂×C₄×C₅	—	4	1	no	—
211	C₂×C₄×C₅	#217	1	1	yes	characteristic
212	C₂×C₄×C₅	#217	1	1	yes	characteristic
213	C₂×C₄×C₅	#217	1	1	yes	characteristic
214	C₂×C₄×C₅	#217	1	1	yes	characteristic
215	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
216	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
217	C₂×C₄×C₅	#235	1	1	yes	characteristic
218	C₂×C₄×C₅	#235	1	1	yes	characteristic
219	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
220	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
221	C₂×Dic₅	—	2	2	no	—
222	C₂×Dic₅	—	4	2	no	—
223	D₂₀	—	2	2	no	—
224	D₂₀	—	2	2	no	—
225	D₂₀	—	2	2	no	—
226	D₂₀	—	2	2	no	—
227	C₄×D₅	—	4	4	no	—
228	C₄×D₅	—	4	4	no	—
229	C₅×C₈	—	12	2	no	—
230	#168	—	12	2	no	—
231	C₂²×D₄	—	5	1	no	—
232	C₂³×C₄	—	5	2	no	—
233	C₂×M₁₆	—	3	2	no	—
234	C₂×M₁₆	—	15	2	no	—
235	C₄×D₄	—	10	1	no	—
236	C₄×D₄	—	10	1	no	—
237	C₄×D₄	—	10	1	no	—
238	C₄×D₄	—	10	1	no	—
239	C₂×C₄⋊C₄	—	5	1	no	—
240	C₂×C₂²⋊C₄	—	5	2	no	—
241	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
242	C₄⋊C₈	—	6	1	no	—
243	C₄⋊C₈	—	6	1	no	—
244	C₄⋊C₈	—	30	1	no	—
245	C₄⋊C₈	—	30	1	no	—
246	#100	—	30	4	no	—
247	C₂×C₃×C₅	—	2	1	no	—
248	C₂×C₃×C₅	—	2	1	no	—
249	C₂×C₃×C₅	#380	1	1	yes	characteristic
250	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
251	C₂×C₃×C₅	#413	1	1	yes	characteristic
252	C₃×D₅	—	4	4	no	—
253	C₂³×C₃	—	5	4	no	—
254	C₂³×C₃	—	10	2	no	—
255	C₂³×C₃	—	10	2	no	—
256	C₂³×C₃	—	20	2	no	—
257	C₂³×C₃	—	20	2	no	—
258	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
259	C₃×D₄	—	10	2	no	—
260	C₃×D₄	—	10	2	no	—
261	C₃×D₄	—	10	2	no	—
262	C₃×D₄	—	10	2	no	—
263	C₂×C₃×C₄	—	2	2	no	—
264	C₂×C₃×C₄	—	4	1	no	—
265	C₂×C₃×C₄	—	10	2	no	—
266	C₂×C₃×C₄	—	20	4	no	—
267	C₂×C₃×C₄	—	20	4	no	—
268	C₂×C₃×C₄	—	20	2	no	—
269	C₂×C₃×C₄	#685	1	1	yes	characteristic
270	C₂×C₃×C₄	#685	1	1	yes	characteristic
271	C₂×C₃×C₄	#685	1	1	yes	characteristic
272	C₂×C₃×C₄	#685	1	1	yes	characteristic
273	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
274	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
275	C₂×C₃×C₄	#711	1	1	yes	characteristic
276	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
277	C₂×C₃×C₄	#713	1	1	yes	characteristic
278	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
279	C₃⋊C₈	—	4	2	no	—
280	C₃⋊C₈	—	20	2	no	—
281	C₂²×C₅	—	2	1	no	—
282	C₂²×C₅	—	2	1	no	—
283	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
284	C₂²×C₅	#926	1	1	yes	characteristic
285	C₂²×C₅	C₂×#217	1	1	yes	characteristic
286	C₂×D₅	—	2	4	no	—
287	C₂×D₅	—	4	4	no	—
288	C₂×D₅	—	4	2	no	—
289	C₂×D₅	—	4	2	no	—
290	C₂×D₅	—	4	4	no	—
291	C₄×C₅	—	2	1	no	—
292	C₄×C₅	—	2	1	no	—
293	C₄×C₅	—	2	1	no	—
294	C₄×C₅	#833	1	2	yes	complemented
295	C₄×C₅	#924	1	1	yes	characteristic
296	C₄×C₅	#924	1	1	yes	characteristic
297	C₄×C₅	#930	1	1	yes	characteristic
298	C₄×C₅	#930	1	1	yes	characteristic
299	Dic₅	—	4	2	no	—
300	C₂⁴	—	5	2	no	—
301	C₂×D₄	—	5	2	no	—
302	C₂×D₄	—	5	2	no	—
303	C₂×D₄	—	10	1	no	—
304	C₂×D₄	—	10	1	no	—
305	C₂×D₄	—	10	1	no	—
306	C₂×D₄	—	10	1	no	—
307	C₂²×C₄	—	5	2	no	—
308	C₂²×C₄	—	10	2	no	—
309	C₂²×C₄	—	10	2	no	—
310	C₂²×C₄	—	20	4	no	—
311	C₂²×C₄	D₅×Dic₃	1	1	yes	characteristic
312	C₂²×C₄	D₅×Dic₃	1	1	yes	characteristic
313	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
314	M₁₆	—	6	2	no	—
315	M₁₆	—	6	2	no	—
316	M₁₆	—	30	2	no	—
317	M₁₆	—	30	2	no	—
318	C₂×C₈	—	6	2	no	—
319	C₂×C₈	—	30	2	no	—
320	C₄⋊C₄	—	10	1	no	—
321	C₄⋊C₄	—	10	1	no	—
322	C₂²⋊C₄	—	10	2	no	—
323	C₂²⋊C₄	—	10	2	no	—
324	C₄²	#1201	1	1	yes	characteristic
325	C₄²	#1201	1	1	yes	characteristic
326	C₄²	#1202	1	1	yes	characteristic
327	C₄²	#1202	1	1	yes	characteristic
328	C₃×C₅	#1800	1	1	yes	characteristic, complemented, Hall
329	C₂²×C₃	—	2	1	no	—
330	C₂²×C₃	—	2	1	no	—
331	C₂²×C₃	—	10	4	no	—
332	C₂²×C₃	—	20	4	no	—
333	C₂²×C₃	—	20	4	no	—
334	C₂²×C₃	—	20	2	no	—
335	C₂²×C₃	—	20	2	no	—
336	C₂²×C₃	D₅×C₄⋊C₄	1	1	yes	characteristic
337	C₂²×C₃	#4038	1	1	yes	characteristic
338	C₂²×C₃	C₂×#685	1	1	yes	characteristic
339	C₃×C₄	—	2	1	no	—
340	C₃×C₄	—	2	1	no	—
341	C₃×C₄	—	2	1	no	—
342	C₃×C₄	—	20	2	no	—
343	C₃×C₄	#3955	1	2	yes	complemented
344	C₃×C₄	#4041	1	1	yes	characteristic
345	C₃×C₄	#4043	1	1	yes	characteristic
346	C₃×C₄	#4084	1	1	yes	characteristic
347	C₃×C₄	#4092	1	1	yes	characteristic
348	C₂×C₅	—	2	1	no	—
349	C₂×C₅	—	2	1	no	—
350	C₂×C₅	#5212	1	1	yes	characteristic
351	C₂×C₅	C₂×#833	1	1	yes	characteristic
352	C₂×C₅	#5237	1	1	yes	characteristic
353	D₅	—	4	4	no	—
354	C₂³	—	5	4	no	—
355	C₂³	—	10	2	no	—
356	C₂³	—	10	2	no	—
357	C₂³	—	20	2	no	—
358	C₂³	—	20	2	no	—
359	C₂³	C₂×D₅×Dic₃	1	1	yes	characteristic
360	D₄	—	10	2	no	—
361	D₄	—	10	2	no	—
362	D₄	—	10	2	no	—
363	D₄	—	10	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	4	1	no	—
366	C₂×C₄	—	10	2	no	—
367	C₂×C₄	—	20	4	no	—
368	C₂×C₄	—	20	4	no	—
369	C₂×C₄	—	20	2	no	—
370	C₂×C₄	#6665	1	1	yes	characteristic
371	C₂×C₄	#6665	1	1	yes	characteristic
372	C₂×C₄	#6665	1	1	yes	characteristic
373	C₂×C₄	#6665	1	1	yes	characteristic
374	C₂×C₄	D₅×Dic₆	1	1	yes	characteristic
375	C₂×C₄	#6772	1	1	yes	characteristic
376	C₂×C₄	#6773	1	1	yes	characteristic
377	C₂×C₄	D₅×D₁₂	1	1	yes	characteristic
378	C₂×C₄	C₂×#1201	1	1	yes	characteristic
379	C₂×C₄	C₂×#1202	1	1	yes	characteristic
380	C₈	—	12	2	no	—
381	C₈	—	60	2	no	—
382	C₂×C₃	—	2	1	no	—
383	C₂×C₃	—	2	1	no	—
384	C₂×C₃	—	20	4	no	—
385	C₂×C₃	#65276	1	1	yes	characteristic
386	C₂×C₃	C₂×#3955	1	1	yes	characteristic
387	C₂×C₃	#65421	1	1	yes	characteristic
388	C₅	#71617	1	1	yes	characteristic, complemented, Sylow
389	C₂²	—	2	1	no	—
390	C₂²	—	2	1	no	—
391	C₂²	—	10	4	no	—
392	C₂²	—	20	4	no	—
393	C₂²	—	20	4	no	—
394	C₂²	—	20	2	no	—
395	C₂²	—	20	2	no	—
396	C₂²	#91358	1	1	yes	characteristic
397	C₂²	D₅×C₄⋊Dic₃	1	1	yes	characteristic
398	C₂²	C₂×#6665	1	1	yes	characteristic
399	C₄	—	2	1	no	—
400	C₄	—	2	1	no	—
401	C₄	—	2	1	no	—
402	C₄	—	20	2	no	—
403	C₄	#90958	1	2	yes	complemented
404	C₄	#91359	1	1	yes	characteristic
405	C₄	#91360	1	1	yes	characteristic
406	C₄	#91449	1	1	yes	characteristic
407	C₄	#91450	1	1	yes	characteristic
408	C₃	#10601320	1	1	yes	characteristic, complemented, Sylow
409	C₂	—	2	1	no	—
410	C₂	—	2	1	no	—
411	C₂	—	20	4	no	—
412	C₂	C₂×#90958	1	1	yes	characteristic
413	C₂	#11748353	1	1	yes	characteristic
414	C₂	#11748759	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6665
commutator	C₂³×C₃×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₃×D₂₀	C₂²
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1800
LCS	G ▷ C₂³×C₃×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. #11751025	C₂	semidirect
3. C₅×#4924	C₂	semidirect
5. #11745985	C₂	semidirect
60. #4924	D₅	semidirect
169. C₃×C₄×C₅	#100	semidirect
294. C₄×C₅	#833	semidirect
328. C₃×C₅	#1800	semidirect
343. C₃×C₄	#3955	semidirect
388. C₅	#71617	semidirect
403. C₄	#90958	semidirect
408. C₃	#10601320	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×D₂₀	2
#11751025	2
C₅×#4924	2
C₂×#90958	2
#11745985	2
#10601320	3
#71617	5