Subgroups — #14312869

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×#91265	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×D₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#90934	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746328	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×(C₂×D₃)⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745881	C₂	1	1	yes	maximal, characteristic, complemented
7	(C₂×D₃)⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599900	—	3	1	no	maximal, Hall
9	C₂×#6771	C₄	1	1	yes	characteristic
10	C₂×C₈×D₁₅	C₂²	1	1	yes	characteristic
11	C₂×C₅×D₃×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
13	C₂×#6651	C₄	1	1	yes	characteristic
14	C₂×#6650	C₄	1	1	yes	characteristic
15	C₂×#6649	C₄	1	1	yes	characteristic
16	#91265	—	2	2	no	—
17	#91265	—	2	2	no	—
18	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₃⋊#672	C₂²	1	1	yes	characteristic
20	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
21	C₂×C₃×#672	C₂²	1	1	yes	characteristic
22	C₃×D₅×C₁₆	—	2	2	no	—
23	#90934	—	2	2	no	—
24	#71409	—	5	1	no	maximal, Hall
25	C₂×#4047	—	3	1	no	—
26	C₂×#3930	—	3	1	no	—
27	C₂×D₅×C₁₆	D₃	1	1	yes	characteristic, complemented
28	C₅×C₂²⋊C₁₆	—	3	1	no	—
29	C₂²⋊#672	—	3	1	no	—
30	#64727	—	3	1	no	—
31	#64727	—	3	1	no	—
32	C₂×C₄×D₁₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
35	C₂×#1204	C₈	1	1	yes	characteristic
36	C₂×#1203	C₈	1	1	yes	characteristic
37	C₂×#1202	C₈	1	1	yes	characteristic
38	C₂×#1201	C₈	1	1	yes	characteristic
39	#6771	—	2	2	no	—
40	#6771	—	2	2	no	—
41	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
42	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
43	C₈×D₁₅	—	2	1	no	—
44	C₈×D₁₅	—	2	1	no	—
45	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₅×D₃×C₈	—	2	1	no	—
47	C₅×D₃×C₈	—	2	1	no	—
48	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
49	C₃×C₈×D₅	D₄	1	2	yes	—
50	C₃×C₈×D₅	D₄	1	2	yes	—
51	#6651	—	2	2	no	—
52	#6650	—	2	2	no	—
53	#6649	—	2	2	no	—
54	C₃×C₅×C₁₆	—	2	1	no	—
55	C₃⋊#672	—	2	1	no	—
56	C₃×#672	—	2	1	no	—
57	C₅×C₃⋊C₁₆	—	2	1	no	—
58	C₂×#901	—	5	1	no	—
59	C₂²×C₃×C₁₆	—	5	1	no	—
60	C₂×#812	—	5	1	no	—
61	(C₂×D₃)⋊C₁₆	—	5	1	no	—
62	(C₂×D₃)⋊C₁₆	—	5	1	no	—
63	(C₂×D₃)⋊C₁₆	—	5	1	no	—
64	(C₂×D₃)⋊C₁₆	D₅	1	1	yes	characteristic, complemented
65	C₂×#709	—	3	1	no	—
66	C₂²×C₅×C₈	—	3	1	no	—
67	C₂×#681	—	3	1	no	—
68	#4047	—	6	2	no	—
69	#4047	—	6	2	no	—
70	C₂×#676	—	3	1	no	—
71	C₂×#676	—	3	1	no	—
72	C₂×C₈×D₅	—	3	1	no	—
73	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
74	C₂×C₅×C₁₆	—	3	1	no	—
75	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
76	C₂×#672	—	3	1	no	—
77	C₂×#672	C₂×D₃	1	1	yes	characteristic
78	#3930	—	6	2	no	—
79	D₅×C₁₆	—	2	2	no	—
80	#2113	—	15	1	no	Sylow
81	C₂²×D₁₅	C₂×C₈	1	1	yes	characteristic
82	C₂²×C₅×D₃	C₂×C₈	1	1	yes	characteristic
83	C₂²×C₃×D₅	C₂×C₈	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₁₅	C₂×C₈	1	1	yes	characteristic
86	C₄×D₁₅	—	2	1	no	—
87	C₄×D₁₅	—	2	1	no	—
88	C₂×C₅×Dic₃	C₂×C₈	1	1	yes	characteristic
89	C₄×C₅×D₃	—	2	1	no	—
90	C₄×C₅×D₃	—	2	1	no	—
91	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
92	C₃×C₄×D₅	C₂²⋊C₄	1	2	yes	—
93	C₃×C₄×D₅	M₁₆	1	2	yes	—
94	#1204	—	2	2	no	—
95	#1203	—	2	2	no	—
96	#1202	—	2	2	no	—
97	#1201	—	2	2	no	—
98	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
99	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
100	C₃⋊#168	—	2	1	no	—
101	C₃×#168	C₂²⋊C₄	1	2	yes	—
102	C₅×C₃⋊C₈	—	2	1	no	—
103	C₂×#235	—	5	1	no	—
104	C₂²×C₃×C₈	—	5	1	no	—
105	C₂×#208	—	5	1	no	—
106	#901	—	10	2	no	—
107	#901	—	10	2	no	—
108	C₂×ΓL(1,𝔽₂₅)	—	5	1	no	—
109	C₂×ΓL(1,𝔽₂₅)	—	5	1	no	—
110	C₂×D₃×C₈	—	5	1	no	—
111	C₂×D₃×C₈	C₂×D₅	1	1	yes	characteristic
112	C₂×C₃×C₁₆	—	5	1	no	—
113	C₂×C₃×C₁₆	—	10	2	no	—
114	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic
115	#812	—	10	2	no	—
116	C₂×C₃⋊C₁₆	—	5	1	no	—
117	C₂×C₃⋊C₁₆	C₂×D₅	1	1	yes	characteristic
118	C₂²×C₄×C₅	—	3	1	no	—
119	C₂×C₂²⋊D₅	—	3	1	no	—
120	C₂×C₂²⋊D₅	—	3	1	no	—
121	#709	—	6	2	no	—
122	#709	—	6	2	no	—
123	C₂×D₂₀	—	3	1	no	—
124	C₂×C₄×D₅	—	3	1	no	—
125	C₂×C₄×D₅	C₄×D₃	1	1	yes	characteristic
126	C₂×Dic₁₀	—	3	1	no	—
127	C₂×C₅×C₈	—	3	1	no	—
128	C₂×C₅×C₈	—	6	1	no	—
129	C₂×C₅×C₈	—	6	1	no	—
130	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
131	#681	—	6	2	no	—
132	C₂×#168	—	3	1	no	—
133	C₂×#168	C₄×D₃	1	1	yes	characteristic
134	#676	—	6	2	no	—
135	#676	—	6	2	no	—
136	C₈×D₅	—	6	1	no	—
137	C₈×D₅	—	6	1	no	—
138	C₈×D₅	D₁₂	1	2	yes	—
139	C₈×D₅	#67	1	2	yes	—
140	C₅×C₁₆	—	2	1	no	—
141	C₅×C₁₆	—	6	1	no	—
142	#672	—	2	1	no	—
143	#672	—	6	1	no	—
144	C₂×#131	—	15	1	no	—
145	C₂×M₃₂	—	15	1	no	—
146	C₂²×C₁₆	—	5	1	no	—
147	C₂²⋊C₁₆	—	3	1	no	—
148	C₂²⋊C₁₆	—	15	1	no	—
149	C₂²⋊C₁₆	—	15	1	no	—
150	C₂²⋊C₁₆	—	15	1	no	—
151	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
152	C₂×D₁₅	—	2	1	no	—
153	C₂×D₁₅	—	2	1	no	—
154	C₂×D₁₅	—	2	1	no	—
155	C₂×C₅×D₃	—	2	1	no	—
156	C₂×C₅×D₃	—	2	1	no	—
157	C₂×C₅×D₃	—	2	1	no	—
158	C₂×C₃×D₅	—	2	1	no	—
159	C₂×C₃×D₅	—	2	1	no	—
160	C₂×C₃×D₅	C₂²⋊C₈	1	2	yes	—
161	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
162	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
163	Dic₁₅	—	2	1	no	—
164	C₃×Dic₅	C₂²⋊C₈	1	2	yes	—
165	C₅×Dic₃	—	2	1	no	—
166	C₂²×C₃×C₄	—	5	1	no	—
167	C₂×#67	—	5	1	no	—
168	C₂×#67	—	5	1	no	—
169	#235	—	10	2	no	—
170	#235	—	10	2	no	—
171	C₂×D₁₂	—	5	1	no	—
172	C₂×C₄×D₃	—	5	1	no	—
173	C₂×C₄×D₃	C₄×D₅	1	1	yes	characteristic
174	C₂×Dic₆	—	5	1	no	—
175	C₂×C₃×C₈	—	5	1	no	—
176	C₂×C₃×C₈	—	5	2	no	—
177	C₂×C₃×C₈	—	5	2	no	—
178	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
179	#208	—	10	2	no	—
180	C₂×C₃⋊C₈	—	5	1	no	—
181	C₂×C₃⋊C₈	C₄×D₅	1	1	yes	characteristic
182	ΓL(1,𝔽₂₅)	—	10	2	no	—
183	ΓL(1,𝔽₂₅)	—	10	2	no	—
184	D₃×C₈	—	2	1	no	—
185	D₃×C₈	—	2	1	no	—
186	D₃×C₈	—	10	1	no	—
187	D₃×C₈	—	10	1	no	—
188	C₃×C₁₆	—	2	1	no	—
189	C₃×C₁₆	—	10	1	no	—
190	C₃⋊C₁₆	—	2	1	no	—
191	C₃⋊C₁₆	—	10	1	no	—
192	C₂³×C₅	—	3	1	no	—
193	C₂²×D₅	—	3	1	no	—
194	C₂²×D₅	D₃×C₈	1	1	yes	characteristic
195	C₂×C₄×C₅	—	3	1	no	—
196	C₂×C₄×C₅	—	6	1	no	—
197	C₂×C₄×C₅	—	6	1	no	—
198	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
199	C₂²⋊D₅	—	6	2	no	—
200	C₂²⋊D₅	—	6	2	no	—
201	C₂×Dic₅	—	3	1	no	—
202	C₂×Dic₅	D₃×C₈	1	1	yes	characteristic
203	D₂₀	—	6	2	no	—
204	C₄×D₅	—	6	1	no	—
205	C₄×D₅	—	6	1	no	—
206	C₄×D₅	ΓL(1,𝔽₂₅)	1	2	yes	—
207	C₄×D₅	(C₂×D₃)⋊C₄	1	2	yes	—
208	Dic₁₀	—	6	2	no	—
209	C₅×C₈	—	6	1	no	—
210	C₅×C₈	C₂×D₁₂	1	1	yes	characteristic
211	C₅×C₈	C₂×#67	1	1	yes	characteristic
212	#168	—	6	1	no	—
213	#168	(C₂×D₃)⋊C₄	1	2	yes	—
214	C₂×#41	—	15	1	no	—
215	#131	—	30	2	no	—
216	#131	—	30	2	no	—
217	C₂×M₁₆	—	15	1	no	—
218	C₂×M₁₆	—	15	1	no	—
219	C₂×M₁₆	—	15	1	no	—
220	C₂²×C₈	—	3	1	no	—
221	C₂²×C₈	—	5	1	no	—
222	C₂²×C₈	—	15	1	no	—
223	M₃₂	—	30	2	no	—
224	C₂×C₁₆	—	3	1	no	—
225	C₂×C₁₆	—	5	1	no	—
226	C₂×C₁₆	—	10	2	no	—
227	C₂×C₁₆	—	15	1	no	—
228	C₂×C₁₆	D₃×D₅	1	1	yes	characteristic
229	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
230	C₂×C₃×C₅	#504	1	1	yes	characteristic
231	C₂×C₃×C₅	#504	1	1	yes	characteristic
232	D₁₅	—	4	1	no	—
233	C₃×D₅	—	2	2	no	—
234	C₅×D₃	—	4	1	no	—
235	C₂³×C₃	—	5	1	no	—
236	C₂²×D₃	—	5	1	no	—
237	C₂²×D₃	C₈×D₅	1	1	yes	characteristic
238	C₂×C₃×C₄	—	5	1	no	—
239	C₂×C₃×C₄	—	5	2	no	—
240	C₂×C₃×C₄	—	5	2	no	—
241	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
242	#67	—	10	2	no	—
243	#67	—	10	2	no	—
244	C₂×Dic₃	—	5	1	no	—
245	C₂×Dic₃	C₈×D₅	1	1	yes	characteristic
246	D₁₂	—	10	2	no	—
247	C₄×D₃	—	2	1	no	—
248	C₄×D₃	—	2	1	no	—
249	C₄×D₃	—	10	1	no	—
250	C₄×D₃	—	10	1	no	—
251	Dic₆	—	10	2	no	—
252	C₃×C₈	—	5	2	no	—
253	C₃×C₈	D₄×D₅	1	1	yes	characteristic
254	C₃×C₈	D₄×D₅	1	1	yes	characteristic
255	C₃⋊C₈	—	2	1	no	—
256	C₃⋊C₈	—	10	1	no	—
257	C₂²×C₅	—	6	1	no	—
258	C₂²×C₅	—	6	1	no	—
259	C₂²×C₅	—	6	1	no	—
260	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
261	C₂×D₅	—	2	1	no	—
262	C₂×D₅	—	2	1	no	—
263	C₂×D₅	—	6	1	no	—
264	C₂×D₅	—	6	1	no	—
265	C₂×D₅	—	6	1	no	—
266	C₂×D₅	(C₂×D₃)⋊C₈	1	2	yes	—
267	C₄×C₅	—	6	1	no	—
268	C₄×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
269	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
270	Dic₅	—	6	1	no	—
271	Dic₅	(C₂×D₃)⋊C₈	1	2	yes	—
272	#41	—	30	2	no	—
273	#41	—	30	2	no	—
274	C₂×Q₈	—	15	1	no	—
275	C₂×D₄	—	15	1	no	—
276	C₂×D₄	—	15	1	no	—
277	C₂×D₄	—	15	1	no	—
278	C₂²×C₄	—	3	1	no	—
279	C₂²×C₄	—	5	1	no	—
280	C₂²×C₄	—	15	1	no	—
281	M₁₆	—	30	2	no	—
282	M₁₆	—	30	2	no	—
283	M₁₆	—	30	2	no	—
284	C₂×C₈	—	3	1	no	—
285	C₂×C₈	—	5	1	no	—
286	C₂×C₈	—	5	2	no	—
287	C₂×C₈	—	5	2	no	—
288	C₂×C₈	—	6	1	no	—
289	C₂×C₈	—	6	1	no	—
290	C₂×C₈	—	15	1	no	—
291	C₂×C₈	—	30	1	no	—
292	C₂×C₈	—	30	1	no	—
293	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
294	C₁₆	—	2	1	no	—
295	C₁₆	—	6	1	no	—
296	C₁₆	—	10	1	no	—
297	C₁₆	—	30	1	no	—
298	C₃×C₅	#2113	1	1	yes	characteristic, complemented, Hall
299	C₂²×C₃	—	5	2	no	—
300	C₂²×C₃	—	10	1	no	—
301	C₂²×C₃	—	10	1	no	—
302	C₂²×C₃	C₂×C₈×D₅	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	—	10	1	no	—
307	C₂×D₃	—	10	1	no	—
308	C₂×D₃	—	10	1	no	—
309	C₃×C₄	—	5	2	no	—
310	C₃×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
311	C₃×C₄	D₅×M₁₆	1	1	yes	characteristic
312	Dic₃	—	2	1	no	—
313	Dic₃	—	10	1	no	—
314	C₂×C₅	—	12	1	no	—
315	C₂×C₅	#4901	1	1	yes	characteristic
316	C₂×C₅	#4901	1	1	yes	characteristic
317	C₂×C₅	C₂×(C₂×D₃)⋊C₈	1	1	yes	characteristic
318	D₅	—	2	2	no	—
319	D₅	—	12	1	no	—
320	C₂³	—	3	1	no	—
321	C₂³	—	5	1	no	—
322	C₂³	—	15	1	no	—
323	Q₈	—	30	2	no	—
324	D₄	—	30	2	no	—
325	D₄	—	30	2	no	—
326	D₄	—	30	2	no	—
327	C₂×C₄	—	3	1	no	—
328	C₂×C₄	—	5	1	no	—
329	C₂×C₄	—	5	2	no	—
330	C₂×C₄	—	5	2	no	—
331	C₂×C₄	—	6	1	no	—
332	C₂×C₄	—	6	1	no	—
333	C₂×C₄	—	15	1	no	—
334	C₂×C₄	—	30	1	no	—
335	C₂×C₄	—	30	1	no	—
336	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
337	C₈	—	5	2	no	—
338	C₈	—	6	1	no	—
339	C₈	—	30	1	no	—
340	C₈	D₅×D₁₂	1	1	yes	characteristic
341	C₈	D₅×#67	1	1	yes	characteristic
342	C₂×C₃	—	10	2	no	—
343	C₂×C₃	D₅×C₂²⋊C₈	1	1	yes	characteristic
344	C₂×C₃	#65190	1	1	yes	characteristic
345	C₂×C₃	#65196	1	1	yes	characteristic
346	D₃	—	4	1	no	—
347	D₃	—	20	1	no	—
348	C₅	#71409	1	1	yes	characteristic, complemented, Sylow
349	C₂²	—	5	2	no	—
350	C₂²	—	6	1	no	—
351	C₂²	—	6	1	no	—
352	C₂²	—	6	1	no	—
353	C₂²	—	10	1	no	—
354	C₂²	—	10	1	no	—
355	C₂²	—	30	1	no	—
356	C₂²	—	30	1	no	—
357	C₂²	—	30	1	no	—
358	C₂²	D₃×C₈×D₅	1	1	yes	characteristic
359	C₄	—	5	2	no	—
360	C₄	—	6	1	no	—
361	C₄	—	30	1	no	—
362	C₄	D₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
363	C₄	D₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
364	C₃	#10599900	1	1	yes	characteristic, complemented, Sylow
365	C₂	—	10	2	no	—
366	C₂	—	12	1	no	—
367	C₂	—	60	1	no	—
368	C₂	#11746810	1	1	yes	characteristic
369	C₂	#11746811	1	1	yes	characteristic
370	C₂	D₅×(C₂×D₃)⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₃×D₅
commutator, socle	C₂²×C₃×C₅	C₂²×C₈
Fitting	C₂×C₃×C₅×C₁₆	C₂²
prime radical	C₂×#6771	C₄
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₂×C₁₆	D₃×D₅
hypocenter	C₃×C₅	#2113
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×D₅×C₁₆	C₂	semidirect
3. C₂×#90934	C₂	semidirect
4. #11746328	C₂	semidirect
5. C₅×(C₂×D₃)⋊C₁₆	C₂	semidirect
6. #11745881	C₂	semidirect
7. (C₂×D₃)⋊#672	C₂	semidirect
27. C₂×D₅×C₁₆	D₃	semidirect
64. (C₂×D₃)⋊C₁₆	D₅	semidirect
298. C₃×C₅	#2113	semidirect
348. C₅	#71409	semidirect
364. C₃	#10599900	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91265	2
C₂×C₃×D₅×C₁₆	2
C₂×#90934	2
#11746328	2
C₅×(C₂×D₃)⋊C₁₆	2
#11745881	2
(C₂×D₃)⋊#672	2
#10599900	3
#71409	5