Subgroups — #14295517

Order 1920.

568 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4053	C₂	1	1	yes	maximal, characteristic
2	#11752420	C₂	1	1	yes	maximal, characteristic
3	#11752404	C₂	1	1	yes	maximal, characteristic
4	#11752377	C₂	1	2	yes	maximal, complemented
5	#11746765	C₂	1	2	yes	maximal, complemented
6	#10604764	—	3	1	no	maximal, Hall
7	C₂×C₃×#709	C₂²	1	1	yes	characteristic
8	#91989	C₂²	1	2	yes	—
9	D₅⋊#208	C₂²	1	2	yes	—
10	C₂×C₃×C₅×M₁₆	C₄	1	1	yes	characteristic, complemented
11	C₂²×C₃×#168	C₄	1	1	yes	characteristic, complemented
12	C₃×#4053	—	2	2	no	—
13	C₃×#4053	—	2	2	no	—
14	C₂×C₃×#676	C₄	1	1	yes	characteristic, complemented
15	C₂×C₃×#676	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₈×D₅	C₄	1	1	yes	characteristic, complemented
17	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
18	#91242	C₄	1	2	yes	complemented
19	#91240	C₄	1	2	yes	—
20	#91234	C₄	1	2	yes	complemented
21	#91228	C₄	1	2	yes	—
22	#91221	—	2	2	no	—
23	C₈×ΓL(1,𝔽₁₆)	—	2	2	no	—
24	#72011	—	5	1	no	maximal, Hall
25	C₂×#4053	D₃	1	1	yes	characteristic
26	#65761	—	3	1	no	—
27	#65745	—	3	1	no	—
28	#65718	—	3	2	no	—
29	#64894	—	3	2	no	—
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₂²⋊D₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₂²⋊D₅	C₂×C₄	1	1	yes	characteristic
33	C₃×#709	—	2	2	no	—
34	C₃×#709	—	2	2	no	—
35	C₂×C₃×D₂₀	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
38	C₂×C₃×Dic₁₀	C₂×C₄	1	1	yes	characteristic
39	#6763	C₂×C₄	1	4	yes	—
40	#6762	C₂×C₄	1	4	yes	—
41	C₄⋊ΓL(1,𝔽₁₆)	—	2	2	no	—
42	C₄×ΓL(1,𝔽₁₆)	—	2	2	no	—
43	#6758	—	2	2	no	—
44	D₅⋊(C₃⋊C₈)	—	2	2	no	—
45	C₃×C₅×M₁₆	—	2	2	no	—
46	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₃×#168	—	2	1	no	—
49	C₂×C₃×#168	—	2	1	no	—
50	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
51	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
52	C₃×#676	—	2	2	no	—
53	C₃×#676	—	2	2	no	—
54	C₃×C₈×D₅	—	2	1	no	—
55	C₃×C₈×D₅	—	2	1	no	—
56	C₃×C₈×D₅	D₄	1	2	yes	—
57	C₃×C₈×D₅	D₄	1	2	yes	—
58	C₂×C₃×#131	—	5	1	no	—
59	#5240	—	5	1	no	—
60	#5237	—	5	1	no	—
61	#5101	—	5	2	no	—
62	#4556	—	5	2	no	—
63	C₂×#709	C₂×D₃	1	1	yes	characteristic
64	#4130	—	3	2	no	—
65	D₅⋊M₁₆	—	3	2	no	—
66	C₂×C₅×M₁₆	Dic₃	1	1	yes	characteristic, complemented
67	C₂²×#168	Dic₃	1	1	yes	characteristic, complemented
68	#4053	—	2	2	no	—
69	#4053	—	2	2	no	—
70	C₂×#676	Dic₃	1	1	yes	characteristic, complemented
71	C₂×#676	C₂×D₃	1	1	yes	characteristic
72	C₂×C₈×D₅	Dic₃	1	1	yes	characteristic, complemented
73	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
74	#4013	—	3	2	no	—
75	#4011	—	3	2	no	—
76	#4005	—	3	2	no	—
77	#3999	—	3	2	no	—
78	#3992	—	6	2	no	—
79	C₈×AGL(1,𝔽₅)	—	6	2	no	—
80	#1757	—	15	1	no	Sylow
81	C₂³×C₃×C₅	C₄²	1	1	yes	characteristic
82	C₂²×C₃×D₅	C₄²	1	1	yes	characteristic
83	C₂²×C₃×D₅	C₂²×C₄	1	1	yes	characteristic
84	C₂×ΓL(1,𝔽₁₆)	—	2	4	no	—
85	C₂×C₃×C₄×C₅	—	2	1	no	—
86	C₂×C₃×C₄×C₅	—	2	1	no	—
87	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
89	C₃×C₂²⋊D₅	—	2	2	no	—
90	C₃×C₂²⋊D₅	—	2	2	no	—
91	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
92	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
93	C₃×D₂₀	—	2	2	no	—
94	C₃×C₄×D₅	—	2	1	no	—
95	C₃×C₄×D₅	—	2	1	no	—
96	C₃×C₄×D₅	C₂×D₄	1	2	yes	—
97	C₃×C₄×D₅	#41	1	2	yes	—
98	C₃×Dic₁₀	—	2	2	no	—
99	C₃⋊#170	—	2	4	no	—
100	C₃×C₅×C₈	—	2	1	no	—
101	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
102	C₃×#168	—	2	1	no	—
103	C₃×#168	C₂²⋊C₄	1	1	yes	characteristic
104	C₃×#168	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₃×#41	—	5	1	no	—
106	C₃×#131	—	10	2	no	—
107	C₃×#131	—	10	2	no	—
108	C₂×C₃×M₁₆	—	5	1	no	—
109	C₂×C₃×M₁₆	—	5	1	no	—
110	C₂×C₃×M₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
111	C₂²×C₃×C₈	—	5	1	no	—
112	C₂²×C₃×C₈	—	5	1	no	—
113	C₂²×C₃×C₈	—	5	1	no	—
114	#926	—	5	2	no	—
115	C₂×#208	—	5	2	no	—
116	#836	—	5	2	no	—
117	#834	—	5	2	no	—
118	#834	—	5	2	no	—
119	#833	—	5	2	no	—
120	#815	—	10	2	no	—
121	C₈×Dic₃	—	10	2	no	—
122	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
123	C₂×C₂²⋊D₅	C₄×D₃	1	1	yes	characteristic
124	C₂×C₂²⋊D₅	C₄×D₃	1	1	yes	characteristic
125	#709	—	2	2	no	—
126	#709	—	2	2	no	—
127	C₂×D₂₀	C₄×D₃	1	1	yes	characteristic
128	C₂×C₄×D₅	C₂×Dic₃	1	1	yes	characteristic
129	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
130	C₂×Dic₁₀	C₄×D₃	1	1	yes	characteristic
131	C₂²⋊AGL(1,𝔽₅)	—	3	4	no	—
132	#704	—	3	4	no	—
133	C₄⋊AGL(1,𝔽₅)	—	6	2	no	—
134	C₄×AGL(1,𝔽₅)	—	6	2	no	—
135	#700	—	6	2	no	—
136	D₅⋊C₈	—	6	2	no	—
137	C₅×M₁₆	—	2	2	no	—
138	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
139	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
140	C₂×#168	—	2	1	no	—
141	C₂×#168	—	2	1	no	—
142	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
143	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
144	#676	—	2	2	no	—
145	#676	—	2	2	no	—
146	C₈×D₅	—	2	1	no	—
147	C₈×D₅	—	2	1	no	—
148	C₈×D₅	#67	1	2	yes	—
149	C₈×D₅	#67	1	2	yes	—
150	C₂×#131	—	5	1	no	—
151	#413	—	15	1	no	—
152	#410	—	15	1	no	—
153	#405	—	15	2	no	—
154	#344	—	15	2	no	—
155	C₂²×C₃×C₅	—	2	1	no	—
156	C₂²×C₃×C₅	—	4	1	no	—
157	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
158	C₂×C₃×D₅	—	2	1	no	—
159	C₂×C₃×D₅	—	4	1	no	—
160	C₂×C₃×D₅	—	4	1	no	—
161	C₂×C₃×D₅	C₄×D₄	1	2	yes	—
162	ΓL(1,𝔽₁₆)	—	4	4	no	—
163	C₃×C₄×C₅	—	2	1	no	—
164	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
165	C₃×C₄×C₅	#117	1	1	yes	characteristic
166	C₃×Dic₅	—	2	1	no	—
167	C₃×Dic₅	C₄×D₄	1	2	yes	—
168	C₃×#41	—	10	2	no	—
169	C₃×#41	—	10	2	no	—
170	C₂×C₃×Q₈	—	5	1	no	—
171	C₂×C₃×D₄	—	5	1	no	—
172	C₂×C₃×D₄	—	5	1	no	—
173	C₂×C₃×D₄	—	5	1	no	—
174	C₂²×C₃×C₄	—	5	1	no	—
175	C₂²×C₃×C₄	—	5	1	no	—
176	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
177	C₃×M₁₆	—	2	2	no	—
178	C₃×M₁₆	—	10	2	no	—
179	C₃×M₁₆	—	10	2	no	—
180	C₂×C₃×C₈	—	5	1	no	—
181	C₂×C₃×C₈	—	5	2	no	—
182	C₂×C₃×C₈	—	5	2	no	—
183	C₂×C₃×C₈	—	5	1	no	—
184	C₂×C₃×C₈	—	10	1	no	—
185	C₂×C₃×C₈	—	10	1	no	—
186	C₂×C₃×C₈	—	10	1	no	—
187	C₂×C₃×C₈	—	10	1	no	—
188	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
189	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
190	#217	—	5	4	no	—
191	C₄⋊Dic₃	—	10	2	no	—
192	C₄×Dic₃	—	10	2	no	—
193	#208	—	5	4	no	—
194	#208	—	10	2	no	—
195	C₂×C₃⋊C₈	—	10	2	no	—
196	C₂³×C₅	C₄×Dic₃	1	1	yes	characteristic
197	C₂²×D₅	C₄×Dic₃	1	1	yes	characteristic
198	C₂²×D₅	C₂×C₄×D₃	1	1	yes	characteristic
199	C₂×AGL(1,𝔽₅)	—	6	4	no	—
200	C₂×C₄×C₅	—	2	1	no	—
201	C₂×C₄×C₅	—	2	1	no	—
202	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
203	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
204	C₂²⋊D₅	—	2	2	no	—
205	C₂²⋊D₅	—	2	2	no	—
206	C₂×Dic₅	C₄×Dic₃	1	1	yes	characteristic
207	C₂×Dic₅	C₂×C₄×D₃	1	1	yes	characteristic
208	D₂₀	—	2	2	no	—
209	C₄×D₅	—	2	1	no	—
210	C₄×D₅	—	2	1	no	—
211	C₄×D₅	#235	1	2	yes	—
212	C₄×D₅	C₂×#67	1	2	yes	—
213	Dic₁₀	—	2	2	no	—
214	#170	—	6	4	no	—
215	C₅×C₈	—	2	1	no	—
216	C₅×C₈	#217	1	2	yes	—
217	#168	—	2	1	no	—
218	#168	#217	1	1	yes	characteristic
219	#168	#217	1	1	yes	characteristic
220	C₂×#41	—	5	1	no	—
221	#131	—	10	2	no	—
222	#131	—	10	2	no	—
223	C₂×M₁₆	—	5	1	no	—
224	C₂×M₁₆	—	5	1	no	—
225	C₂×M₁₆	—	15	2	no	—
226	C₂×M₁₆	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
227	C₂²×C₈	—	5	1	no	—
228	C₂²×C₈	—	5	1	no	—
229	C₂²×C₈	—	5	1	no	—
230	#117	—	15	2	no	—
231	#101	—	15	2	no	—
232	#100	—	15	2	no	—
233	#99	—	15	2	no	—
234	#99	—	15	2	no	—
235	#97	—	30	2	no	—
236	C₄×C₈	—	30	2	no	—
237	C₂×C₃×C₅	—	2	1	no	—
238	C₂×C₃×C₅	—	4	1	no	—
239	C₂×C₃×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
240	C₃×D₅	—	2	2	no	—
241	C₃×D₅	—	4	1	no	—
242	C₂³×C₃	—	5	1	no	—
243	C₂³×C₃	—	5	1	no	—
244	C₂³×C₃	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
245	C₃×Q₈	—	10	2	no	—
246	C₃×D₄	—	10	2	no	—
247	C₃×D₄	—	10	2	no	—
248	C₃×D₄	—	10	2	no	—
249	C₂×C₃×C₄	—	2	1	no	—
250	C₂×C₃×C₄	—	2	1	no	—
251	C₂×C₃×C₄	—	5	1	no	—
252	C₂×C₃×C₄	—	5	2	no	—
253	C₂×C₃×C₄	—	5	2	no	—
254	C₂×C₃×C₄	—	5	1	no	—
255	C₂×C₃×C₄	—	10	1	no	—
256	C₂×C₃×C₄	—	10	1	no	—
257	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
258	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
259	C₂×Dic₃	—	10	4	no	—
260	C₃×C₈	—	2	1	no	—
261	C₃×C₈	—	5	1	no	—
262	C₃×C₈	—	5	1	no	—
263	C₃×C₈	—	10	1	no	—
264	C₃×C₈	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
265	C₃⋊C₈	—	10	4	no	—
266	C₂²×C₅	—	2	1	no	—
267	C₂²×C₅	—	4	1	no	—
268	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
269	C₂×D₅	—	2	1	no	—
270	C₂×D₅	—	4	1	no	—
271	C₂×D₅	—	4	1	no	—
272	C₂×D₅	C₄×#67	1	2	yes	—
273	AGL(1,𝔽₅)	—	12	4	no	—
274	C₄×C₅	—	2	1	no	—
275	C₄×C₅	#926	1	1	yes	characteristic
276	C₄×C₅	C₂×#217	1	1	yes	characteristic
277	Dic₅	—	2	1	no	—
278	Dic₅	C₄×#67	1	2	yes	—
279	#41	—	10	2	no	—
280	#41	—	10	2	no	—
281	C₂×Q₈	—	5	1	no	—
282	C₂×D₄	—	5	1	no	—
283	C₂×D₄	—	5	1	no	—
284	C₂×D₄	—	5	1	no	—
285	C₂²×C₄	—	5	1	no	—
286	C₂²×C₄	—	5	1	no	—
287	C₂²×C₄	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
288	M₁₆	—	2	2	no	—
289	M₁₆	—	10	2	no	—
290	M₁₆	—	10	2	no	—
291	M₁₆	—	15	4	no	—
292	M₁₆	—	30	2	no	—
293	C₂×C₈	—	5	1	no	—
294	C₂×C₈	—	5	2	no	—
295	C₂×C₈	—	5	2	no	—
296	C₂×C₈	—	5	1	no	—
297	C₂×C₈	—	10	1	no	—
298	C₂×C₈	—	10	1	no	—
299	C₂×C₈	—	10	1	no	—
300	C₂×C₈	—	10	1	no	—
301	C₂×C₈	—	30	2	no	—
302	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
303	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
304	C₄⋊C₄	—	30	2	no	—
305	C₂²⋊C₄	—	15	4	no	—
306	C₄²	—	30	2	no	—
307	C₃×C₅	#1757	1	1	yes	characteristic, complemented, Hall
308	C₂²×C₃	—	2	1	no	—
309	C₂²×C₃	—	4	1	no	—
310	C₂²×C₃	—	5	2	no	—
311	C₂²×C₃	—	10	1	no	—
312	C₂²×C₃	—	20	1	no	—
313	C₂²×C₃	—	20	1	no	—
314	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
315	C₃×C₄	—	2	1	no	—
316	C₃×C₄	—	5	2	no	—
317	C₃×C₄	—	10	1	no	—
318	C₃×C₄	#4130	1	1	yes	characteristic
319	C₃×C₄	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
320	Dic₃	—	20	4	no	—
321	C₂×C₅	—	2	1	no	—
322	C₂×C₅	—	4	1	no	—
323	C₂×C₅	C₄×#217	1	1	yes	characteristic
324	D₅	—	2	2	no	—
325	D₅	—	4	1	no	—
326	C₂³	—	5	1	no	—
327	C₂³	—	5	1	no	—
328	C₂³	C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
329	Q₈	—	10	2	no	—
330	D₄	—	10	2	no	—
331	D₄	—	10	2	no	—
332	D₄	—	10	2	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	5	1	no	—
336	C₂×C₄	—	5	2	no	—
337	C₂×C₄	—	5	2	no	—
338	C₂×C₄	—	5	1	no	—
339	C₂×C₄	—	10	1	no	—
340	C₂×C₄	—	10	1	no	—
341	C₂×C₄	—	30	4	no	—
342	C₂×C₄	C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
343	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
344	C₈	—	2	1	no	—
345	C₈	—	5	1	no	—
346	C₈	—	5	1	no	—
347	C₈	—	10	1	no	—
348	C₈	—	30	4	no	—
349	C₈	#6763	1	2	yes	—
350	C₂×C₃	—	2	1	no	—
351	C₂×C₃	—	4	1	no	—
352	C₂×C₃	—	10	2	no	—
353	C₂×C₃	—	20	1	no	—
354	C₂×C₃	C₄×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
355	C₅	#72011	1	1	yes	characteristic, complemented, Sylow
356	C₂²	—	2	1	no	—
357	C₂²	—	4	1	no	—
358	C₂²	—	5	2	no	—
359	C₂²	—	10	1	no	—
360	C₂²	—	20	1	no	—
361	C₂²	—	20	1	no	—
362	C₂²	C₂×C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
363	C₄	—	2	1	no	—
364	C₄	—	5	2	no	—
365	C₄	—	10	1	no	—
366	C₄	—	60	4	no	—
367	C₄	#91989	1	1	yes	characteristic
368	C₄	C₂×#6763	1	1	yes	characteristic
369	C₃	#10604764	1	1	yes	characteristic, complemented, Sylow
370	C₂	—	2	1	no	—
371	C₂	—	4	1	no	—
372	C₂	—	10	2	no	—
373	C₂	—	20	1	no	—
374	C₂	C₄×#6763	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#6763
commutator	C₂²×C₃×C₅	C₂×C₄²
Frattini	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)
Fitting	C₂×C₃×C₅×M₁₆	C₄
socle	C₂×C₃×C₅	C₄×C₂²⋊C₄
prime radical	C₂×C₃×#709	C₂²
cosocle	C₂×C₃×C₄×D₅	C₂³
hypercenter	C₂×M₁₆	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#1757
LCS	G ▷ C₂²×C₃×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
4. #11752377	C₂	semidirect
5. #11746765	C₂	semidirect
10. C₂×C₃×C₅×M₁₆	C₄	semidirect
11. C₂²×C₃×#168	C₄	semidirect
14. C₂×C₃×#676	C₄	semidirect
16. C₂×C₃×C₈×D₅	C₄	semidirect
18. #91242	C₄	semidirect
20. #91234	C₄	semidirect
66. C₂×C₅×M₁₆	Dic₃	semidirect
67. C₂²×#168	Dic₃	semidirect
70. C₂×#676	Dic₃	semidirect
72. C₂×C₈×D₅	Dic₃	semidirect
110. C₂×C₃×M₁₆	AGL(1,𝔽₅)	semidirect
226. C₂×M₁₆	ΓL(1,𝔽₁₆)	semidirect
307. C₃×C₅	#1757	semidirect
355. C₅	#72011	semidirect
369. C₃	#10604764	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#4053	2
#11752420	2
#11752404	2
#11752377	2
#11746765	2
#10604764	3
#72011	5