Subgroups — #14437784

Order 1920.

800 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#532	C₂	1	1	yes	maximal, characteristic
2	#11756229	C₂	1	1	yes	maximal, characteristic, complemented
3	#11751334	C₂	1	4	yes	maximal, complemented
4	#11751180	C₂	1	2	yes	maximal, complemented
5	#11751171	C₂	1	2	yes	maximal, complemented
6	#11751157	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×C₃⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
8	#10615012	—	3	1	no	maximal, Hall
9	C₂×C₃×C₅×#41	C₂²	1	2	yes	—
10	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₃×C₅×#127	C₂²	1	1	yes	characteristic
12	C₃×C₅×#124	C₂²	1	2	yes	—
13	C₃×C₅×#121	—	2	2	no	—
14	C₃×C₄×C₅×D₄	—	2	2	no	—
15	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
16	#91815	C₂²	1	2	yes	—
17	#91813	—	2	2	no	—
18	C₂²×C₃⋊#168	C₂²	1	2	yes	—
19	#91759	C₂²	1	2	yes	—
20	#91758	C₂²	1	2	yes	—
21	C₄×Dic₃₀	—	2	2	no	—
22	#91119	C₂²	1	8	yes	—
23	#91116	C₂²	1	8	yes	—
24	C₄×C₃⋊#168	C₂²	1	2	yes	—
25	C₄×C₃⋊#168	C₂²	1	2	yes	—
26	#83281	—	5	1	no	maximal, Hall
27	C₅×#532	D₃	1	1	yes	characteristic
28	#65804	—	3	1	no	—
29	#65522	—	3	4	no	—
30	#65370	—	3	2	no	—
31	#65359	—	3	2	no	—
32	#65345	—	3	4	no	—
33	C₂×C₄×#168	—	3	1	no	—
34	C₃×C₅×#41	—	2	8	no	—
35	C₂×C₃×C₅×Q₈	C₂³	1	2	yes	—
36	C₂×C₃×C₅×D₄	—	2	2	no	—
37	C₂×C₃×C₅×D₄	—	2	2	no	—
38	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
39	C₂²×C₃×C₄×C₅	—	2	2	no	—
40	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
41	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
42	C₂×Dic₃₀	—	2	2	no	—
43	C₃×C₅×C₄⋊C₄	—	2	2	no	—
44	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
45	C₃×C₄²×C₅	D₄	1	2	yes	—
46	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
48	#6719	—	2	2	no	—
49	C₄⋊Dic₁₅	—	2	2	no	—
50	C₄⋊Dic₁₅	C₂³	1	4	yes	—
51	#6712	—	2	4	no	—
52	C₄×Dic₁₅	—	2	2	no	—
53	C₂×C₃⋊#168	—	2	4	no	—
54	C₂×C₃⋊#168	C₂³	1	4	yes	—
55	C₃×#532	D₅	1	1	yes	characteristic
56	#5470	—	5	1	no	—
57	#5233	—	5	4	no	—
58	#5079	—	5	2	no	—
59	#5070	—	5	2	no	—
60	#5056	—	5	4	no	—
61	C₂×C₄×C₃⋊C₈	—	5	1	no	—
62	C₂×C₅×#41	C₂×D₃	1	2	yes	—
63	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
64	C₅×#127	C₂×D₃	1	1	yes	characteristic
65	C₅×#124	C₂×D₃	1	2	yes	—
66	C₅×#121	—	2	2	no	—
67	C₄×C₅×D₄	—	2	2	no	—
68	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
69	#4072	—	3	2	no	—
70	#4070	—	6	2	no	—
71	C₂²×#168	—	3	2	no	—
72	#4016	—	3	2	no	—
73	#4015	—	3	2	no	—
74	C₄×Dic₁₀	—	6	2	no	—
75	#3967	—	3	8	no	—
76	#3964	—	3	8	no	—
77	C₄×#168	—	3	2	no	—
78	C₄×#168	—	3	2	no	—
79	#3131	—	15	1	no	Sylow
80	C₂³×C₃×C₅	—	2	2	no	—
81	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
82	C₃×C₅×Q₈	—	2	4	no	—
83	C₃×C₅×D₄	—	2	4	no	—
84	C₃×C₅×D₄	—	4	4	no	—
85	C₃×C₅×D₄	—	4	4	no	—
86	C₂×C₃×C₄×C₅	—	2	4	no	—
87	C₂×C₃×C₄×C₅	—	2	2	no	—
88	C₂×C₃×C₄×C₅	—	4	4	no	—
89	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
90	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
92	C₂×C₃×C₄×C₅	#41	1	4	yes	—
93	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
94	C₂×Dic₁₅	—	2	4	no	—
95	Dic₃₀	—	4	4	no	—
96	C₃⋊#168	—	2	4	no	—
97	C₂×C₃×#41	C₂×D₅	1	2	yes	—
98	C₃×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
99	C₃×#127	C₂×D₅	1	1	yes	characteristic
100	C₃×#124	C₂×D₅	1	2	yes	—
101	C₃×#121	—	2	2	no	—
102	C₃×C₄×D₄	—	2	2	no	—
103	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
104	#926	—	5	2	no	—
105	#924	—	10	2	no	—
106	C₂²×C₃⋊C₈	—	5	2	no	—
107	#870	—	5	2	no	—
108	#869	—	5	2	no	—
109	C₄×Dic₆	—	10	2	no	—
110	#835	—	5	8	no	—
111	#832	—	5	8	no	—
112	C₄×C₃⋊C₈	—	5	2	no	—
113	C₄×C₃⋊C₈	—	5	2	no	—
114	C₅×#41	—	2	8	no	—
115	C₂×C₅×Q₈	C₂²×D₃	1	2	yes	—
116	C₂×C₅×D₄	—	2	2	no	—
117	C₂×C₅×D₄	—	2	2	no	—
118	C₂×C₅×D₄	C₂²×D₃	1	2	yes	—
119	C₂²×C₄×C₅	—	2	2	no	—
120	C₂²×C₄×C₅	#67	1	2	yes	—
121	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
122	C₂×Dic₁₀	—	6	2	no	—
123	C₅×C₄⋊C₄	—	2	2	no	—
124	C₅×C₂²⋊C₄	—	2	4	no	—
125	C₄²×C₅	#67	1	2	yes	—
126	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
127	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
128	C₂²⋊Dic₅	—	6	2	no	—
129	C₄⋊Dic₅	—	3	4	no	—
130	C₄⋊Dic₅	—	6	2	no	—
131	#683	—	6	4	no	—
132	C₄×Dic₅	—	6	2	no	—
133	C₂×#168	—	3	4	no	—
134	C₂×#168	—	6	4	no	—
135	#533	—	15	1	no	—
136	#532	D₁₅	1	1	yes	characteristic
137	#488	—	15	4	no	—
138	#487	—	15	2	no	—
139	#486	—	15	2	no	—
140	#416	—	15	4	no	—
141	C₂×C₄×C₈	—	15	1	no	—
142	C₂²×C₃×C₅	—	2	2	no	—
143	C₂²×C₃×C₅	—	4	2	no	—
144	C₂²×C₃×C₅	—	4	4	no	—
145	C₂²×C₃×C₅	#124	1	2	yes	—
146	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
147	C₃×C₄×C₅	—	2	2	no	—
148	C₃×C₄×C₅	—	4	2	no	—
149	C₃×C₄×C₅	#124	1	2	yes	—
150	C₃×C₄×C₅	#135	1	4	yes	—
151	C₃×C₄×C₅	C₂×#41	1	2	yes	—
152	Dic₁₅	—	4	4	no	—
153	C₃×#41	—	2	8	no	—
154	C₂×C₃×Q₈	C₂²×D₅	1	2	yes	—
155	C₂×C₃×D₄	—	2	2	no	—
156	C₂×C₃×D₄	—	2	2	no	—
157	C₂×C₃×D₄	C₂²×D₅	1	2	yes	—
158	C₂²×C₃×C₄	—	2	2	no	—
159	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
160	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
161	C₂×Dic₆	—	10	2	no	—
162	C₃×C₄⋊C₄	—	2	2	no	—
163	C₃×C₂²⋊C₄	—	2	4	no	—
164	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
165	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
166	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
167	#217	—	10	2	no	—
168	C₄⋊Dic₃	—	5	4	no	—
169	C₄⋊Dic₃	—	10	2	no	—
170	Dic₃⋊C₄	—	10	4	no	—
171	C₄×Dic₃	—	10	2	no	—
172	C₂×C₃⋊C₈	—	5	4	no	—
173	C₂×C₃⋊C₈	—	10	4	no	—
174	C₂³×C₅	—	2	2	no	—
175	C₂³×C₅	C₂×#67	1	1	yes	characteristic
176	C₅×Q₈	—	2	4	no	—
177	C₅×D₄	—	2	4	no	—
178	C₅×D₄	—	4	4	no	—
179	C₅×D₄	—	4	4	no	—
180	C₂×C₄×C₅	—	2	4	no	—
181	C₂×C₄×C₅	—	2	2	no	—
182	C₂×C₄×C₅	—	4	4	no	—
183	C₂×C₄×C₅	#237	1	4	yes	—
184	C₂×C₄×C₅	C₂×#67	1	2	yes	—
185	C₂×C₄×C₅	C₂×#67	1	2	yes	—
186	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
187	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
188	C₂×Dic₅	—	6	4	no	—
189	Dic₁₀	—	12	4	no	—
190	#168	—	6	4	no	—
191	C₂×#41	C₂×D₁₅	1	2	yes	—
192	C₂²×C₈	—	15	2	no	—
193	C₄⋊Q₈	—	15	2	no	—
194	C₄⋊Q₈	C₂×D₁₅	1	1	yes	characteristic
195	#127	C₂×D₁₅	1	1	yes	characteristic
196	#125	—	15	2	no	—
197	#124	C₂×D₁₅	1	2	yes	—
198	C₂²⋊Q₈	—	30	2	no	—
199	#121	—	2	2	no	—
200	C₄×Q₈	—	30	2	no	—
201	C₄×D₄	—	2	2	no	—
202	#117	—	15	2	no	—
203	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
204	#103	—	15	8	no	—
205	#102	—	15	8	no	—
206	C₄×C₈	—	15	2	no	—
207	C₄×C₈	—	15	2	no	—
208	C₂×C₃×C₅	—	2	2	no	—
209	C₂×C₃×C₅	—	8	2	no	—
210	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
211	C₂×C₃×C₅	C₂×#135	1	2	yes	—
212	C₂³×C₃	—	2	2	no	—
213	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
214	C₃×Q₈	—	2	4	no	—
215	C₃×D₄	—	2	4	no	—
216	C₃×D₄	—	4	4	no	—
217	C₃×D₄	—	4	4	no	—
218	C₂×C₃×C₄	—	2	4	no	—
219	C₂×C₃×C₄	—	2	2	no	—
220	C₂×C₃×C₄	—	4	4	no	—
221	C₂×C₃×C₄	#711	1	4	yes	—
222	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
223	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
224	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
225	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
226	C₂×Dic₃	—	10	4	no	—
227	Dic₆	—	20	4	no	—
228	C₃⋊C₈	—	10	4	no	—
229	C₂²×C₅	—	2	2	no	—
230	C₂²×C₅	—	4	2	no	—
231	C₂²×C₅	—	4	4	no	—
232	C₂²×C₅	#936	1	2	yes	—
233	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
234	C₄×C₅	—	2	2	no	—
235	C₄×C₅	—	4	2	no	—
236	C₄×C₅	#936	1	2	yes	—
237	C₄×C₅	#950	1	4	yes	—
238	C₄×C₅	C₂×#237	1	2	yes	—
239	Dic₅	—	12	4	no	—
240	#41	—	2	8	no	—
241	C₂×Q₈	—	30	2	no	—
242	C₂×Q₈	C₂²×D₁₅	1	2	yes	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	C₂²×D₁₅	1	2	yes	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	#1220	1	2	yes	—
248	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
249	C₂×C₈	—	15	4	no	—
250	C₂×C₈	—	30	4	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	15	4	no	—
253	C₄⋊C₄	—	30	4	no	—
254	C₄⋊C₄	—	30	2	no	—
255	C₂²⋊C₄	—	2	4	no	—
256	C₂²⋊C₄	—	30	2	no	—
257	C₄²	—	30	2	no	—
258	C₄²	#1220	1	2	yes	—
259	C₄²	C₂²×D₁₅	1	1	yes	characteristic
260	C₄²	C₂²×D₁₅	1	1	yes	characteristic
261	C₃×C₅	#3131	1	1	yes	characteristic, complemented, Hall
262	C₂²×C₃	—	2	2	no	—
263	C₂²×C₃	—	4	2	no	—
264	C₂²×C₃	—	4	4	no	—
265	C₂²×C₃	#4082	1	2	yes	—
266	C₂²×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
267	C₃×C₄	—	2	2	no	—
268	C₃×C₄	—	4	2	no	—
269	C₃×C₄	#4082	1	2	yes	—
270	C₃×C₄	#4096	1	4	yes	—
271	C₃×C₄	C₂×#711	1	2	yes	—
272	Dic₃	—	20	4	no	—
273	C₂×C₅	—	2	2	no	—
274	C₂×C₅	—	8	2	no	—
275	C₂×C₅	C₂×#936	1	1	yes	characteristic
276	C₂×C₅	C₂×#950	1	2	yes	—
277	C₂³	—	2	2	no	—
278	C₂³	C₂×#1220	1	1	yes	characteristic
279	Q₈	—	2	4	no	—
280	Q₈	—	60	4	no	—
281	D₄	—	2	4	no	—
282	D₄	—	4	4	no	—
283	D₄	—	4	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	30	4	no	—
288	C₂×C₄	#6819	1	4	yes	—
289	C₂×C₄	C₂×#1220	1	2	yes	—
290	C₂×C₄	C₂×#1220	1	2	yes	—
291	C₂×C₄	C₂×#1220	1	1	yes	characteristic
292	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
293	C₈	—	30	4	no	—
294	C₂×C₃	—	2	2	no	—
295	C₂×C₃	—	8	2	no	—
296	C₂×C₃	C₂×#4082	1	1	yes	characteristic
297	C₂×C₃	C₂×#4096	1	2	yes	—
298	C₅	#83281	1	1	yes	characteristic, complemented, Sylow
299	C₂²	—	2	2	no	—
300	C₂²	—	4	2	no	—
301	C₂²	—	4	4	no	—
302	C₂²	#91825	1	2	yes	—
303	C₂²	C₂²×#1220	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	4	2	no	—
306	C₄	—	60	4	no	—
307	C₄	#91825	1	2	yes	—
308	C₄	#91839	1	4	yes	—
309	C₄	C₂×#6819	1	2	yes	—
310	C₃	#10615012	1	1	yes	characteristic, complemented, Sylow
311	C₂	—	2	2	no	—
312	C₂	—	8	2	no	—
313	C₂	C₂×#91825	1	1	yes	characteristic
314	C₂	C₂×#91839	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#1220
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅
Fitting, prime radical	C₃×C₅×#532	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
hypercenter	#532	D₁₅
hypocenter	C₃×C₅	#3131
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

Complements

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

N	H ≅ G/N	type
2. #11756229	C₂	semidirect
3. #11751334	C₂	semidirect
4. #11751180	C₂	semidirect
5. #11751171	C₂	semidirect
6. #11751157	C₂	semidirect
7. C₂×C₄×C₃⋊#168	C₂	semidirect
261. C₃×C₅	#3131	semidirect
298. C₅	#83281	semidirect
310. C₃	#10615012	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#532	2
#11756229	2
#11751334	2
#11751180	2
#11751171	2
#11751157	2
C₂×C₄×C₃⋊#168	2
#10615012	3
#83281	5