Subgroups — #14478189

Order 1920.

720 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91286	C₂	1	1	yes	maximal, characteristic
2	C₂×#91227	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3998	C₂	1	1	yes	maximal, characteristic, complemented
4	#11751738	C₂	1	4	yes	maximal, complemented
5	D₃×#3998	C₂	1	4	yes	maximal, complemented
6	#11751736	C₂	1	4	yes	maximal, complemented
7	#10617530	—	3	1	no	maximal, Hall
8	C₂×C₅×#235	C₄	1	1	yes	characteristic
9	C₂×#6710	C₄	1	1	yes	characteristic
10	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
11	C₂×#6650	C₄	1	2	yes	—
12	#91286	C₄	1	2	yes	—
13	#91286	C₄	1	2	yes	—
14	#91286	C₂²	1	2	yes	—
15	#91286	C₂²	1	2	yes	—
16	C₂×D₃×#168	C₂²	1	2	yes	—
17	#91227	C₂²	1	2	yes	—
18	#91227	C₂²	1	2	yes	complemented
19	C₂×C₃⋊#674	C₂²	1	2	yes	complemented
20	C₃×#3998	C₂²	1	2	yes	—
21	C₃×#3998	C₂²	1	2	yes	complemented
22	C₂×C₃×#674	C₂²	1	2	yes	complemented
23	#91165	C₂²	1	8	yes	complemented
24	D₃×#674	C₂²	1	8	yes	complemented
25	#82007	—	5	1	no	maximal, Hall
26	C₂×#4094	—	3	1	no	—
27	#65791	—	3	4	no	—
28	#65791	—	3	4	no	—
29	C₂×#3998	—	3	1	no	—
30	C₂×#3998	—	3	4	no	—
31	C₂×#3998	D₃	1	1	yes	characteristic, complemented
32	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₅×#67	C₈	1	2	yes	—
34	C₅×#235	C₈	1	2	yes	—
35	C₅×#235	C₈	1	2	yes	—
36	C₅×#235	C₂×C₄	1	2	yes	—
37	C₅×#235	C₂×C₄	1	2	yes	—
38	C₂×C₅×D₁₂	C₈	1	1	yes	characteristic
39	C₂×C₄×C₅×D₃	C₂×C₄	1	2	yes	—
40	C₂×C₅×Dic₆	C₈	1	1	yes	characteristic
41	#6710	C₂×C₄	1	2	yes	—
42	#6710	C₂×C₄	1	2	yes	—
43	C₂×C₃⋊#168	C₂×C₄	1	2	yes	—
44	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
45	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
47	C₂×C₃×#168	C₂³	1	1	yes	characteristic
48	C₂×C₃×#168	C₂³	1	1	yes	characteristic
49	C₂×C₃×#168	C₂³	1	1	yes	characteristic
50	#6650	C₂×C₄	1	4	yes	—
51	#6650	C₂×C₄	1	4	yes	—
52	D₃×#168	C₂×C₄	1	4	yes	—
53	D₃×#168	C₂³	1	4	yes	—
54	C₃⋊#674	C₂³	1	4	yes	—
55	C₃×#674	C₂³	1	4	yes	—
56	C₂×#901	—	5	1	no	—
57	C₂×C₃×M₃₂	—	5	1	no	—
58	C₂×#812	—	5	1	no	—
59	#4907	—	5	4	no	—
60	D₃×M₃₂	—	5	4	no	—
61	#4901	—	5	4	no	—
62	C₂×C₅×#41	—	3	1	no	—
63	#4094	—	3	2	no	—
64	#4094	—	3	2	no	—
65	#4094	—	3	2	no	—
66	#4094	—	3	2	no	—
67	C₂×#681	—	3	1	no	—
68	C₂×#681	—	3	2	no	—
69	C₂²×#168	—	3	2	no	—
70	C₂²×#168	C₂×D₃	1	1	yes	characteristic
71	#3998	—	3	2	no	—
72	#3998	—	3	2	no	—
73	#3998	—	3	8	no	—
74	#3998	C₂×D₃	1	2	yes	—
75	#3998	C₂×D₃	1	2	yes	complemented
76	C₂×#674	—	3	2	no	—
77	C₂×#674	—	3	8	no	—
78	C₂×#674	C₂×D₃	1	2	yes	complemented
79	#3407	—	15	1	no	Sylow
80	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
81	C₂²×C₅×D₃	C₂×C₈	1	2	yes	—
82	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
88	C₅×#67	C₂×C₈	1	4	yes	—
89	C₅×#67	C₂×C₈	1	4	yes	—
90	C₂×C₅×Dic₃	C₂×C₈	1	2	yes	—
91	C₅×D₁₂	C₂×C₈	1	2	yes	—
92	C₅×D₁₂	C₂×C₈	1	2	yes	—
93	C₄×C₅×D₃	C₂×C₈	1	4	yes	—
94	C₄×C₅×D₃	C₂²×C₄	1	4	yes	—
95	C₅×Dic₆	C₂×C₈	1	2	yes	—
96	C₅×Dic₆	C₂×C₈	1	2	yes	—
97	C₃⋊#168	C₂²×C₄	1	4	yes	—
98	C₃×#168	C₂²×C₄	1	1	yes	characteristic
99	C₃×#168	C₂²×C₄	1	1	yes	characteristic
100	C₃×#168	C₂²×C₄	1	1	yes	characteristic
101	C₃×#168	C₂⁴	1	1	yes	characteristic
102	C₂×#235	AGL(1,𝔽₅)	1	1	yes	characteristic
103	C₂²×C₃×C₈	—	5	1	no	—
104	C₂×#208	—	5	1	no	—
105	#901	—	5	2	no	—
106	#901	—	5	2	no	—
107	#901	—	5	2	no	—
108	#901	—	5	2	no	—
109	C₂×ΓL(1,𝔽₂₅)	—	5	2	no	—
110	C₂×D₃×C₈	—	5	2	no	—
111	C₃×M₃₂	—	5	2	no	—
112	C₃×M₃₂	—	5	2	no	—
113	C₂×C₃×C₁₆	—	5	2	no	—
114	#812	—	5	2	no	—
115	#812	—	5	2	no	—
116	C₂×C₃⋊C₁₆	—	5	2	no	—
117	#798	—	5	8	no	—
118	D₃×C₁₆	—	5	8	no	—
119	C₅×#41	—	3	2	no	—
120	C₅×#41	—	3	2	no	—
121	C₅×#41	—	3	2	no	—
122	C₅×#41	—	3	2	no	—
123	C₂×C₅×Q₈	—	3	1	no	—
124	C₂×C₅×D₄	—	3	2	no	—
125	C₂×C₅×D₄	—	3	1	no	—
126	C₂²×C₄×C₅	—	3	2	no	—
127	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
128	#681	—	3	2	no	—
129	#681	—	3	2	no	—
130	#681	—	3	4	no	—
131	#681	—	3	4	no	—
132	C₂×#168	—	3	2	no	—
133	C₂×#168	—	3	4	no	—
134	C₂×#168	—	3	4	no	—
135	C₂×#168	C₄×D₃	1	1	yes	characteristic
136	C₂×#168	C₄×D₃	1	1	yes	characteristic
137	C₂×#168	C₄×D₃	1	1	yes	characteristic
138	C₂×#168	C₂²×D₃	1	1	yes	characteristic
139	C₂×#168	C₂²×D₃	1	1	yes	characteristic
140	C₂×#168	C₂²×D₃	1	1	yes	characteristic
141	#674	—	3	4	no	—
142	#674	C₂²×D₃	1	4	yes	—
143	C₂×#131	—	15	1	no	—
144	#504	—	15	4	no	—
145	#504	—	15	4	no	—
146	C₂×M₃₂	—	5	1	no	—
147	C₂×M₃₂	—	15	4	no	—
148	C₂×M₃₂	—	15	1	no	—
149	C₂²×C₃×C₅	—	4	1	no	—
150	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
151	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
152	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
153	C₂×C₅×D₃	—	4	2	no	—
154	C₂×C₅×D₃	C₂²×C₈	1	4	yes	—
155	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
156	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
157	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
158	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
159	C₅×Dic₃	C₂²×C₈	1	4	yes	—
160	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
161	C₂×#67	#170	1	2	yes	—
162	#235	#170	1	2	yes	—
163	#235	#170	1	2	yes	—
164	#235	C₂×AGL(1,𝔽₅)	1	2	yes	—
165	#235	C₂×AGL(1,𝔽₅)	1	2	yes	—
166	C₂×D₁₂	#170	1	1	yes	characteristic
167	C₂×C₄×D₃	C₂×AGL(1,𝔽₅)	1	2	yes	—
168	C₂×Dic₆	#170	1	1	yes	characteristic
169	C₂×C₃×C₈	—	5	1	no	—
170	C₂×C₃×C₈	—	5	1	no	—
171	C₂×C₃×C₈	—	5	1	no	—
172	C₂×C₃×C₈	—	5	1	no	—
173	C₂×C₃×C₈	—	5	1	no	—
174	C₂×C₃×C₈	—	5	1	no	—
175	#208	—	5	2	no	—
176	#208	—	5	2	no	—
177	C₂×C₃⋊C₈	—	5	2	no	—
178	ΓL(1,𝔽₂₅)	—	5	4	no	—
179	ΓL(1,𝔽₂₅)	—	5	4	no	—
180	D₃×C₈	—	5	4	no	—
181	D₃×C₈	—	5	4	no	—
182	C₃×C₁₆	—	5	4	no	—
183	C₃⋊C₁₆	—	5	4	no	—
184	C₂³×C₅	—	3	2	no	—
185	C₂³×C₅	D₃×C₈	1	1	yes	characteristic
186	C₅×Q₈	—	3	2	no	—
187	C₅×Q₈	—	3	2	no	—
188	C₅×D₄	—	3	4	no	—
189	C₅×D₄	—	3	4	no	—
190	C₅×D₄	—	3	2	no	—
191	C₅×D₄	—	3	2	no	—
192	C₂×C₄×C₅	—	3	2	no	—
193	C₂×C₄×C₅	—	3	4	no	—
194	C₂×C₄×C₅	—	3	4	no	—
195	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
196	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
197	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
198	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
199	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
200	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
201	#168	—	3	4	no	—
202	#168	C₂×C₄×D₃	1	1	yes	characteristic
203	#168	C₂×C₄×D₃	1	1	yes	characteristic
204	#168	C₂×C₄×D₃	1	1	yes	characteristic
205	#168	C₂³×D₃	1	1	yes	characteristic
206	C₂×#41	—	3	1	no	—
207	#131	—	15	2	no	—
208	#131	—	15	2	no	—
209	#131	—	15	2	no	—
210	#131	—	15	2	no	—
211	C₂×M₁₆	—	15	2	no	—
212	C₂×M₁₆	—	15	1	no	—
213	C₂²×C₈	—	5	1	no	—
214	C₂²×C₈	—	15	2	no	—
215	M₃₂	—	5	2	no	—
216	M₃₂	—	5	2	no	—
217	M₃₂	—	15	8	no	—
218	M₃₂	—	15	2	no	—
219	M₃₂	—	15	2	no	—
220	C₂×C₁₆	—	5	2	no	—
221	C₂×C₁₆	—	15	8	no	—
222	C₂×C₁₆	—	15	2	no	—
223	C₂×C₃×C₅	—	2	1	no	—
224	C₂×C₃×C₅	—	2	1	no	—
225	C₂×C₃×C₅	—	2	1	no	—
226	C₂×C₃×C₅	C₂³×C₈	1	1	yes	characteristic
227	C₅×D₃	—	2	4	no	—
228	C₂³×C₃	C₂×#170	1	1	yes	characteristic
229	C₂²×D₃	C₂×#170	1	2	yes	—
230	C₂×C₃×C₄	C₂×#170	1	1	yes	characteristic
231	C₂×C₃×C₄	C₂×#170	1	1	yes	characteristic
232	C₂×C₃×C₄	C₂×#170	1	1	yes	characteristic
233	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
234	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
235	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
236	#67	C₂×#170	1	4	yes	—
237	#67	C₂×#170	1	4	yes	—
238	C₂×Dic₃	C₂×#170	1	2	yes	—
239	D₁₂	C₂×#170	1	2	yes	—
240	D₁₂	C₂×#170	1	2	yes	—
241	C₄×D₃	C₂×#170	1	4	yes	—
242	C₄×D₃	C₂²×AGL(1,𝔽₅)	1	4	yes	—
243	Dic₆	C₂×#170	1	2	yes	—
244	Dic₆	C₂×#170	1	2	yes	—
245	C₃×C₈	—	5	1	no	—
246	C₃×C₈	—	5	1	no	—
247	C₃×C₈	—	5	1	no	—
248	C₃×C₈	—	5	1	no	—
249	C₃⋊C₈	—	5	4	no	—
250	C₂²×C₅	—	3	4	no	—
251	C₂²×C₅	—	4	1	no	—
252	C₂²×C₅	—	12	2	no	—
253	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
254	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
255	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
256	C₄×C₅	—	3	4	no	—
257	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
258	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
259	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
260	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
261	#41	—	3	2	no	—
262	#41	—	3	2	no	—
263	#41	—	3	2	no	—
264	#41	—	3	2	no	—
265	C₂×Q₈	—	3	1	no	—
266	C₂×D₄	—	3	2	no	—
267	C₂×D₄	—	3	1	no	—
268	C₂²×C₄	—	3	2	no	—
269	C₂²×C₄	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
270	M₁₆	—	15	4	no	—
271	M₁₆	—	15	4	no	—
272	M₁₆	—	15	2	no	—
273	M₁₆	—	15	2	no	—
274	C₂×C₈	—	5	1	no	—
275	C₂×C₈	—	5	1	no	—
276	C₂×C₈	—	5	1	no	—
277	C₂×C₈	—	5	1	no	—
278	C₂×C₈	—	5	1	no	—
279	C₂×C₈	—	5	1	no	—
280	C₂×C₈	—	15	4	no	—
281	C₂×C₈	—	15	4	no	—
282	C₂×C₈	—	15	2	no	—
283	C₁₆	—	5	4	no	—
284	C₁₆	—	15	4	no	—
285	C₃×C₅	#3407	1	1	yes	characteristic, complemented, Hall
286	C₂²×C₃	—	4	1	no	—
287	C₂²×C₃	C₂²×#170	1	1	yes	characteristic
288	C₂²×C₃	C₂²×#170	1	1	yes	characteristic
289	C₂²×C₃	C₂²×#170	1	1	yes	characteristic
290	C₂×D₃	—	4	2	no	—
291	C₂×D₃	C₂²×#170	1	4	yes	—
292	C₃×C₄	C₂²×#170	1	1	yes	characteristic
293	C₃×C₄	C₂²×#170	1	1	yes	characteristic
294	C₃×C₄	C₂²×#170	1	1	yes	characteristic
295	C₃×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
296	Dic₃	C₂²×#170	1	4	yes	—
297	C₂×C₅	—	2	1	no	—
298	C₂×C₅	—	2	1	no	—
299	C₂×C₅	—	2	1	no	—
300	C₂×C₅	—	6	4	no	—
301	C₂×C₅	C₂²×D₃×C₈	1	1	yes	characteristic
302	C₂³	—	3	2	no	—
303	C₂³	D₃×#170	1	1	yes	characteristic
304	Q₈	—	3	2	no	—
305	Q₈	—	3	2	no	—
306	D₄	—	3	4	no	—
307	D₄	—	3	4	no	—
308	D₄	—	3	2	no	—
309	D₄	—	3	2	no	—
310	C₂×C₄	—	3	2	no	—
311	C₂×C₄	—	3	4	no	—
312	C₂×C₄	—	3	4	no	—
313	C₂×C₄	D₃×#170	1	1	yes	characteristic
314	C₂×C₄	D₃×#170	1	1	yes	characteristic
315	C₂×C₄	D₃×#170	1	1	yes	characteristic
316	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
317	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
318	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
319	C₈	—	5	1	no	—
320	C₈	—	5	1	no	—
321	C₈	—	5	1	no	—
322	C₈	—	5	1	no	—
323	C₈	—	15	4	no	—
324	C₂×C₃	—	2	1	no	—
325	C₂×C₃	—	2	1	no	—
326	C₂×C₃	—	2	1	no	—
327	C₂×C₃	C₂³×#170	1	1	yes	characteristic
328	D₃	—	2	4	no	—
329	C₅	#82007	1	1	yes	characteristic, complemented, Sylow
330	C₂²	—	3	4	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	12	2	no	—
333	C₂²	C₂×D₃×#170	1	1	yes	characteristic
334	C₂²	C₂×D₃×#170	1	1	yes	characteristic
335	C₂²	C₂×D₃×#170	1	1	yes	characteristic
336	C₄	—	3	4	no	—
337	C₄	C₂×D₃×#170	1	1	yes	characteristic
338	C₄	C₂×D₃×#170	1	1	yes	characteristic
339	C₄	C₂×D₃×#170	1	1	yes	characteristic
340	C₄	C₂²×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
341	C₃	#10617530	1	1	yes	characteristic, complemented, Sylow
342	C₂	—	2	1	no	—
343	C₂	—	2	1	no	—
344	C₂	—	2	1	no	—
345	C₂	—	6	4	no	—
346	C₂	C₂²×D₃×#170	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×D₃×AGL(1,𝔽₅)
commutator, socle	C₂×C₃×C₅	C₂³×C₈
Fitting	C₂²×C₃×C₄×C₅	C₂×C₄
prime radical	C₂×C₅×#235	C₄
cosocle	C₃×#168	C₂⁴
hypercenter	C₂²×C₄	D₃×AGL(1,𝔽₅)
hypocenter	C₃×C₅	#3407
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#91227	C₂	semidirect
3. C₂×C₃×#3998	C₂	semidirect
4. #11751738	C₂	semidirect
5. D₃×#3998	C₂	semidirect
6. #11751736	C₂	semidirect
18. #91227	C₂²	semidirect
19. C₂×C₃⋊#674	C₂²	semidirect
21. C₃×#3998	C₂²	semidirect
22. C₂×C₃×#674	C₂²	semidirect
23. #91165	C₂²	semidirect
24. D₃×#674	C₂²	semidirect
31. C₂×#3998	D₃	semidirect
75. #3998	C₂×D₃	semidirect
78. C₂×#674	C₂×D₃	semidirect
285. C₃×C₅	#3407	semidirect
329. C₅	#82007	semidirect
341. C₃	#10617530	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91286	2
C₂×#91227	2
C₂×C₃×#3998	2
#11751738	2
D₃×#3998	2
#11751736	2
#10617530	3
#82007	5