Subgroups — #14312874

Order 1920.

456 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#901	C₂	1	1	yes	maximal, characteristic
2	C₂×#90935	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₃×#672	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746370	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₂²⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
6	(C₂×D₃)⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
7	(C₂×D₃)⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
8	#10602797	—	3	1	no	maximal, Hall
9	C₂×C₅×#235	C₄	1	1	yes	characteristic
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₅×#208	C₄	1	1	yes	characteristic
12	C₅×#901	—	2	2	no	—
13	C₅×#901	—	2	2	no	—
14	C₂×C₅×ΓL(1,𝔽₂₅)	C₄	1	1	yes	characteristic
15	C₂×C₅×ΓL(1,𝔽₂₅)	C₄	1	1	yes	characteristic
16	C₂×C₅×D₃×C₈	C₂²	1	1	yes	characteristic
17	C₂×C₅×D₃×C₈	C₂²	1	1	yes	characteristic
18	C₂×C₃⋊#672	C₂²	1	1	yes	characteristic
19	C₂×C₃⋊#672	C₂²	1	1	yes	characteristic
20	C₂×C₃×#672	C₂²	1	1	yes	characteristic
21	C₂×C₃×#672	C₂²	1	1	yes	characteristic
22	#90935	—	2	2	no	—
23	D₃×#672	—	2	2	no	—
24	#69107	—	5	1	no	maximal, Hall
25	C₂×C₅×#131	—	3	1	no	—
26	C₂×#3944	—	3	1	no	—
27	C₂²×#672	—	3	1	no	—
28	C₂²⋊#672	—	3	1	no	—
29	C₂²⋊#672	—	3	1	no	—
30	C₂²⋊#672	—	3	1	no	—
31	C₂²⋊#672	D₃	1	1	yes	characteristic, complemented
32	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₅×#67	C₈	1	1	yes	characteristic
34	C₂×C₅×#67	C₈	1	1	yes	characteristic
35	C₅×#235	—	2	2	no	—
36	C₅×#235	—	2	2	no	—
37	C₂×C₅×D₁₂	C₈	1	1	yes	characteristic
38	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₅×Dic₆	C₈	1	1	yes	characteristic
41	C₂×C₃×C₅×C₈	—	2	1	no	—
42	C₂×C₃×C₅×C₈	—	2	1	no	—
43	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
45	C₅×#208	—	2	2	no	—
46	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
48	C₅×ΓL(1,𝔽₂₅)	—	2	2	no	—
49	C₅×ΓL(1,𝔽₂₅)	—	2	2	no	—
50	C₅×D₃×C₈	—	2	1	no	—
51	C₅×D₃×C₈	—	2	1	no	—
52	C₅×D₃×C₈	D₄	1	2	yes	—
53	C₅×D₃×C₈	D₄	1	2	yes	—
54	C₃⋊#672	—	2	1	no	—
55	C₃⋊#672	—	2	1	no	—
56	C₃×#672	—	2	1	no	—
57	C₃×#672	—	2	1	no	—
58	C₂×#901	D₅	1	1	yes	characteristic
59	C₂×#798	—	5	1	no	—
60	C₂×D₃×C₁₆	—	5	1	no	—
61	C₃×C₂²⋊C₁₆	—	5	1	no	—
62	#4549	—	5	1	no	—
63	(C₂×D₃)⋊C₁₆	—	5	1	no	—
64	(C₂×D₃)⋊C₁₆	—	5	1	no	—
65	C₂×C₅×#41	—	3	1	no	—
66	C₅×#131	—	6	2	no	—
67	C₅×#131	—	6	2	no	—
68	C₂×C₅×M₁₆	—	3	1	no	—
69	C₂×C₅×M₁₆	—	3	1	no	—
70	C₂×C₅×M₁₆	—	3	1	no	—
71	C₂²×C₅×C₈	—	3	1	no	—
72	C₂²×C₅×C₈	—	3	1	no	—
73	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
74	#3944	—	6	2	no	—
75	C₂×#672	—	3	1	no	—
76	C₂×#672	—	3	1	no	—
77	C₂×#672	—	6	2	no	—
78	C₂×#672	C₂×D₃	1	1	yes	characteristic
79	C₂×#672	C₂×D₃	1	1	yes	characteristic
80	#2113	—	15	1	no	Sylow
81	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
82	C₂²×C₅×D₃	C₂×C₈	1	1	yes	characteristic
83	C₂²×C₅×D₃	C₂×C₈	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	—	2	1	no	—
85	C₂×C₃×C₄×C₅	—	2	1	no	—
86	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
88	C₅×#67	—	2	2	no	—
89	C₅×#67	—	2	2	no	—
90	C₂×C₅×Dic₃	C₂×C₈	1	1	yes	characteristic
91	C₂×C₅×Dic₃	C₂×C₈	1	1	yes	characteristic
92	C₅×D₁₂	—	2	2	no	—
93	C₄×C₅×D₃	—	2	1	no	—
94	C₄×C₅×D₃	—	2	1	no	—
95	C₄×C₅×D₃	C₂²⋊C₄	1	2	yes	—
96	C₄×C₅×D₃	M₁₆	1	2	yes	—
97	C₅×Dic₆	—	2	2	no	—
98	C₃×C₅×C₈	—	2	1	no	—
99	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
100	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
101	C₅×C₃⋊C₈	—	2	1	no	—
102	C₅×C₃⋊C₈	C₂²⋊C₄	1	2	yes	—
103	C₂×#235	Dic₅	1	1	yes	characteristic
104	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
105	C₂×#208	Dic₅	1	1	yes	characteristic
106	#901	—	2	2	no	—
107	#901	—	2	2	no	—
108	C₂×ΓL(1,𝔽₂₅)	Dic₅	1	1	yes	characteristic
109	C₂×ΓL(1,𝔽₂₅)	Dic₅	1	1	yes	characteristic
110	C₂×D₃×C₈	C₂×D₅	1	1	yes	characteristic
111	C₂×D₃×C₈	C₂×D₅	1	1	yes	characteristic
112	C₂×C₃×C₁₆	—	5	1	no	—
113	C₂×C₃×C₁₆	—	5	1	no	—
114	C₂×C₃⋊C₁₆	—	5	1	no	—
115	C₂×C₃⋊C₁₆	—	5	1	no	—
116	#798	—	10	2	no	—
117	D₃×C₁₆	—	10	2	no	—
118	C₅×#41	—	6	2	no	—
119	C₅×#41	—	6	2	no	—
120	C₂×C₅×Q₈	—	3	1	no	—
121	C₂×C₅×D₄	—	3	1	no	—
122	C₂×C₅×D₄	—	3	1	no	—
123	C₂×C₅×D₄	—	3	1	no	—
124	C₂²×C₄×C₅	—	3	1	no	—
125	C₂²×C₄×C₅	—	3	1	no	—
126	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
127	C₅×M₁₆	—	6	2	no	—
128	C₅×M₁₆	—	6	2	no	—
129	C₅×M₁₆	—	6	2	no	—
130	C₂×C₅×C₈	—	2	1	no	—
131	C₂×C₅×C₈	—	2	1	no	—
132	C₂×C₅×C₈	—	3	1	no	—
133	C₂×C₅×C₈	—	3	2	no	—
134	C₂×C₅×C₈	—	3	2	no	—
135	C₂×C₅×C₈	—	3	1	no	—
136	C₂×C₅×C₈	—	6	1	no	—
137	C₂×C₅×C₈	—	6	1	no	—
138	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
139	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
140	#672	—	2	1	no	—
141	#672	—	2	1	no	—
142	#672	—	6	1	no	—
143	#672	—	6	1	no	—
144	C₂×#131	—	3	1	no	—
145	C₂×M₃₂	—	15	1	no	—
146	C₂²×C₁₆	—	15	1	no	—
147	C₂²⋊C₁₆	—	5	1	no	—
148	C₂²⋊C₁₆	—	15	1	no	—
149	C₂²⋊C₁₆	—	15	1	no	—
150	C₂²⋊C₁₆	—	15	1	no	—
151	C₂²×C₃×C₅	—	2	1	no	—
152	C₂²×C₃×C₅	—	2	1	no	—
153	C₂²×C₃×C₅	—	2	1	no	—
154	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
155	C₂×C₅×D₃	—	2	1	no	—
156	C₂×C₅×D₃	—	2	1	no	—
157	C₂×C₅×D₃	—	2	1	no	—
158	C₂×C₅×D₃	—	2	1	no	—
159	C₂×C₅×D₃	—	2	1	no	—
160	C₂×C₅×D₃	C₂²⋊C₈	1	2	yes	—
161	C₃×C₄×C₅	—	2	1	no	—
162	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
163	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
164	C₅×Dic₃	—	2	1	no	—
165	C₅×Dic₃	C₂²⋊C₈	1	2	yes	—
166	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
167	C₂×#67	#168	1	1	yes	characteristic
168	C₂×#67	#168	1	1	yes	characteristic
169	#235	—	2	2	no	—
170	#235	—	2	2	no	—
171	C₂×D₁₂	#168	1	1	yes	characteristic
172	C₂×C₄×D₃	C₂×Dic₅	1	1	yes	characteristic
173	C₂×C₄×D₃	C₂×Dic₅	1	1	yes	characteristic
174	C₂×Dic₆	#168	1	1	yes	characteristic
175	C₂×C₃×C₈	—	2	1	no	—
176	C₂×C₃×C₈	—	2	1	no	—
177	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
178	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
179	#208	—	2	2	no	—
180	C₂×C₃⋊C₈	C₂×Dic₅	1	1	yes	characteristic
181	C₂×C₃⋊C₈	C₂×Dic₅	1	1	yes	characteristic
182	ΓL(1,𝔽₂₅)	—	2	2	no	—
183	ΓL(1,𝔽₂₅)	—	2	2	no	—
184	D₃×C₈	—	2	1	no	—
185	D₃×C₈	—	2	1	no	—
186	D₃×C₈	C₂²⋊D₅	1	2	yes	—
187	D₃×C₈	C₂²⋊D₅	1	2	yes	—
188	C₃×C₁₆	—	10	1	no	—
189	C₃×C₁₆	—	10	1	no	—
190	C₃⋊C₁₆	—	10	1	no	—
191	C₃⋊C₁₆	—	10	1	no	—
192	C₂³×C₅	—	3	1	no	—
193	C₂³×C₅	—	3	1	no	—
194	C₂³×C₅	D₃×C₈	1	1	yes	characteristic
195	C₅×Q₈	—	6	2	no	—
196	C₅×D₄	—	6	2	no	—
197	C₅×D₄	—	6	2	no	—
198	C₅×D₄	—	6	2	no	—
199	C₂×C₄×C₅	—	2	1	no	—
200	C₂×C₄×C₅	—	2	1	no	—
201	C₂×C₄×C₅	—	3	1	no	—
202	C₂×C₄×C₅	—	3	2	no	—
203	C₂×C₄×C₅	—	3	2	no	—
204	C₂×C₄×C₅	—	3	1	no	—
205	C₂×C₄×C₅	—	6	1	no	—
206	C₂×C₄×C₅	—	6	1	no	—
207	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
208	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
209	C₅×C₈	—	2	1	no	—
210	C₅×C₈	—	3	2	no	—
211	C₅×C₈	—	6	1	no	—
212	C₅×C₈	D₃×D₄	1	1	yes	characteristic
213	C₅×C₈	D₃×D₄	1	1	yes	characteristic
214	C₂×#41	—	3	1	no	—
215	#131	—	6	2	no	—
216	#131	—	6	2	no	—
217	C₂×M₁₆	—	3	1	no	—
218	C₂×M₁₆	—	3	1	no	—
219	C₂×M₁₆	—	3	1	no	—
220	C₂²×C₈	—	3	1	no	—
221	C₂²×C₈	—	3	1	no	—
222	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
223	M₃₂	—	30	2	no	—
224	C₂×C₁₆	—	5	1	no	—
225	C₂×C₁₆	—	5	1	no	—
226	C₂×C₁₆	—	15	1	no	—
227	C₂×C₁₆	—	15	1	no	—
228	C₂×C₁₆	—	30	2	no	—
229	C₂×C₃×C₅	—	4	1	no	—
230	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
231	C₂×C₃×C₅	#504	1	1	yes	characteristic
232	C₂×C₃×C₅	#504	1	1	yes	characteristic
233	C₅×D₃	—	2	2	no	—
234	C₅×D₃	—	4	1	no	—
235	C₂³×C₃	C₂×#168	1	1	yes	characteristic
236	C₂²×D₃	C₂×#168	1	1	yes	characteristic
237	C₂²×D₃	C₂×#168	1	1	yes	characteristic
238	C₂×C₃×C₄	—	2	1	no	—
239	C₂×C₃×C₄	—	2	1	no	—
240	C₂×C₃×C₄	C₂×#168	1	1	yes	characteristic
241	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
242	#67	—	2	2	no	—
243	#67	—	2	2	no	—
244	C₂×Dic₃	C₂×#168	1	1	yes	characteristic
245	C₂×Dic₃	C₂×#168	1	1	yes	characteristic
246	D₁₂	—	2	2	no	—
247	C₄×D₃	—	2	1	no	—
248	C₄×D₃	—	2	1	no	—
249	C₄×D₃	#681	1	2	yes	—
250	C₄×D₃	C₂²⋊Dic₅	1	2	yes	—
251	Dic₆	—	2	2	no	—
252	C₃×C₈	—	2	1	no	—
253	C₃×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
254	C₃×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
255	C₃⋊C₈	—	2	1	no	—
256	C₃⋊C₈	C₂²⋊Dic₅	1	2	yes	—
257	C₂²×C₅	—	2	1	no	—
258	C₂²×C₅	—	2	1	no	—
259	C₂²×C₅	—	2	1	no	—
260	C₂²×C₅	—	3	2	no	—
261	C₂²×C₅	—	6	1	no	—
262	C₂²×C₅	—	6	1	no	—
263	C₂²×C₅	—	6	1	no	—
264	C₂²×C₅	—	6	1	no	—
265	C₂²×C₅	—	6	1	no	—
266	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
267	C₄×C₅	—	2	1	no	—
268	C₄×C₅	—	3	2	no	—
269	C₄×C₅	—	6	1	no	—
270	C₄×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
271	C₄×C₅	D₃×M₁₆	1	1	yes	characteristic
272	#41	—	6	2	no	—
273	#41	—	6	2	no	—
274	C₂×Q₈	—	3	1	no	—
275	C₂×D₄	—	3	1	no	—
276	C₂×D₄	—	3	1	no	—
277	C₂×D₄	—	3	1	no	—
278	C₂²×C₄	—	3	1	no	—
279	C₂²×C₄	—	3	1	no	—
280	C₂²×C₄	D₃×Dic₅	1	1	yes	characteristic
281	M₁₆	—	6	2	no	—
282	M₁₆	—	6	2	no	—
283	M₁₆	—	6	2	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	3	1	no	—
287	C₂×C₈	—	3	2	no	—
288	C₂×C₈	—	3	2	no	—
289	C₂×C₈	—	3	1	no	—
290	C₂×C₈	—	6	1	no	—
291	C₂×C₈	—	6	1	no	—
292	C₂×C₈	D₃×Dic₅	1	1	yes	characteristic
293	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
294	C₁₆	—	10	1	no	—
295	C₁₆	—	10	1	no	—
296	C₁₆	—	30	1	no	—
297	C₁₆	—	30	1	no	—
298	C₃×C₅	#2113	1	1	yes	characteristic, complemented, Hall
299	C₂²×C₃	—	2	1	no	—
300	C₂²×C₃	—	2	1	no	—
301	C₂²×C₃	—	2	1	no	—
302	C₂²×C₃	C₂²×#168	1	1	yes	characteristic
303	C₂×D₃	—	2	1	no	—
304	C₂×D₃	—	2	1	no	—
305	C₂×D₃	—	2	1	no	—
306	C₂×D₃	—	2	1	no	—
307	C₂×D₃	—	2	1	no	—
308	C₂×D₃	C₂²⋊#168	1	2	yes	—
309	C₃×C₄	—	2	1	no	—
310	C₃×C₄	C₂×#681	1	1	yes	characteristic
311	C₃×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
312	Dic₃	—	2	1	no	—
313	Dic₃	C₂²⋊#168	1	2	yes	—
314	C₂×C₅	—	4	1	no	—
315	C₂×C₅	—	6	2	no	—
316	C₂×C₅	—	12	1	no	—
317	C₂×C₅	D₃×C₂²⋊C₈	1	1	yes	characteristic
318	C₂×C₅	#4901	1	1	yes	characteristic
319	C₂×C₅	#4907	1	1	yes	characteristic
320	C₂³	—	3	1	no	—
321	C₂³	—	3	1	no	—
322	C₂³	D₃×#168	1	1	yes	characteristic
323	Q₈	—	6	2	no	—
324	D₄	—	6	2	no	—
325	D₄	—	6	2	no	—
326	D₄	—	6	2	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	3	1	no	—
330	C₂×C₄	—	3	2	no	—
331	C₂×C₄	—	3	2	no	—
332	C₂×C₄	—	3	1	no	—
333	C₂×C₄	—	6	1	no	—
334	C₂×C₄	—	6	1	no	—
335	C₂×C₄	D₃×#168	1	1	yes	characteristic
336	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
337	C₈	—	2	1	no	—
338	C₈	—	3	2	no	—
339	C₈	—	6	1	no	—
340	C₈	D₃×C₂²⋊D₅	1	1	yes	characteristic
341	C₈	D₃×C₂²⋊D₅	1	1	yes	characteristic
342	C₂×C₃	—	4	1	no	—
343	C₂×C₃	#65429	1	1	yes	characteristic
344	C₂×C₃	#65429	1	1	yes	characteristic
345	C₂×C₃	C₂×C₂²⋊#168	1	1	yes	characteristic
346	D₃	—	2	2	no	—
347	D₃	—	4	1	no	—
348	C₅	#69107	1	1	yes	characteristic, complemented, Sylow
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	3	2	no	—
353	C₂²	—	6	1	no	—
354	C₂²	—	6	1	no	—
355	C₂²	—	6	1	no	—
356	C₂²	—	6	1	no	—
357	C₂²	—	6	1	no	—
358	C₂²	C₂×D₃×#168	1	1	yes	characteristic
359	C₄	—	2	1	no	—
360	C₄	—	3	2	no	—
361	C₄	—	6	1	no	—
362	C₄	D₃×#681	1	1	yes	characteristic
363	C₄	D₃×C₂²⋊Dic₅	1	1	yes	characteristic
364	C₃	#10602797	1	1	yes	characteristic, complemented, Sylow
365	C₂	—	4	1	no	—
366	C₂	—	6	2	no	—
367	C₂	—	12	1	no	—
368	C₂	#11747527	1	1	yes	characteristic
369	C₂	#11747529	1	1	yes	characteristic
370	C₂	D₃×C₂²⋊#168	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₃×D₅
commutator, socle	C₂²×C₃×C₅	C₂²×C₈
Fitting	C₂²×C₃×C₅×C₈	C₂²
prime radical	C₂×C₅×#235	C₄
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₂²×C₈	D₃×D₅
hypocenter	C₃×C₅	#2113
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#90935	C₂	semidirect
3. C₂×D₃×#672	C₂	semidirect
4. #11746370	C₂	semidirect
5. C₃×C₂²⋊#672	C₂	semidirect
6. (C₂×D₃)⋊#672	C₂	semidirect
7. (C₂×D₃)⋊#672	C₂	semidirect
31. C₂²⋊#672	D₃	semidirect
298. C₃×C₅	#2113	semidirect
348. C₅	#69107	semidirect
364. C₃	#10602797	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#901	2
C₂×#90935	2
C₂×D₃×#672	2
#11746370	2
C₃×C₂²⋊#672	2
(C₂×D₃)⋊#672	2
#10602797	3
#69107	5