Subgroups — #14418019

Order 1920.

912 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11756507	C₂	1	1	yes	maximal, characteristic
2	C₂×#91796	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#430	C₂	1	1	yes	maximal, characteristic, complemented
4	#11751240	C₂	1	2	yes	maximal, complemented
5	#11751223	C₂	1	1	yes	maximal, characteristic, complemented
6	#11750996	C₂	1	4	yes	maximal, complemented
7	#11750995	C₂	1	4	yes	maximal, complemented
8	#10610988	—	3	1	no	maximal, Hall
9	C₂×#6817	C₂²	1	2	yes	—
10	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
11	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
12	#91815	C₂²	1	1	yes	characteristic
13	C₂²×C₃⋊#168	C₂²	1	1	yes	characteristic
14	#91798	C₄	1	4	yes	complemented
15	#91797	C₄	1	4	yes	complemented
16	#91796	—	2	2	no	—
17	#91796	—	2	2	no	—
18	#91796	C₂²	1	4	yes	complemented
19	#91796	C₂²	1	4	yes	complemented
20	#91793	C₂²	1	4	yes	—
21	C₂×#6705	—	2	2	no	—
22	C₂×C₈×D₁₅	—	2	2	no	—
23	C₃×C₅×#108	C₂²	1	4	yes	complemented
24	#91109	C₂²	1	8	yes	complemented
25	#91099	C₂²	1	2	yes	complemented
26	C₈×Dic₁₅	C₂²	1	2	yes	complemented
27	#79257	—	5	1	no	maximal, Hall
28	#66082	—	3	1	no	—
29	C₂×#4053	—	3	2	no	—
30	C₅×#430	D₃	1	1	yes	characteristic, complemented
31	#65428	—	3	2	no	—
32	#65411	—	3	1	no	—
33	#65184	—	3	4	no	—
34	#65183	—	3	4	no	—
35	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₂×#1220	—	2	2	no	—
37	#6817	—	2	2	no	—
38	#6817	—	2	2	no	—
39	#6817	C₂×C₄	1	4	yes	—
40	#6817	C₂³	1	4	yes	—
41	C₂×D₆₀	C₂×C₄	1	2	yes	—
42	C₂×C₄×D₁₅	—	2	2	no	—
43	C₂×Dic₃₀	C₂×C₄	1	2	yes	—
44	C₃×C₅×M₁₆	—	2	2	no	—
45	C₃×C₅×M₁₆	C₂×C₄	1	4	yes	—
46	C₃×C₅×M₁₆	C₂³	1	4	yes	—
47	C₂×C₃×C₅×C₈	—	2	2	no	—
48	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
49	#6719	—	2	1	no	—
50	C₄⋊Dic₁₅	—	2	1	no	—
51	C₄×Dic₁₅	C₂³	1	2	yes	—
52	C₂×C₃⋊#168	D₄	1	2	yes	—
53	C₂×C₃⋊#168	D₄	1	2	yes	—
54	C₂×C₃⋊#168	C₂³	1	1	yes	characteristic
55	C₂×C₃⋊#168	C₂³	1	1	yes	characteristic
56	Dic₆₀	C₂×C₄	1	4	yes	—
57	D₁₂₀	C₂×C₄	1	4	yes	—
58	#6706	C₂×C₄	1	8	yes	—
59	#6705	—	2	4	no	—
60	#6705	—	2	4	no	—
61	C₈×D₁₅	—	2	4	no	—
62	C₈×D₁₅	—	2	4	no	—
63	#5748	—	5	1	no	—
64	C₂×#907	—	5	2	no	—
65	C₃×#430	D₅	1	1	yes	characteristic, complemented
66	#5139	—	5	2	no	—
67	#5122	—	5	1	no	—
68	#4895	—	5	4	no	—
69	#4894	—	5	4	no	—
70	C₂×#709	—	3	2	no	—
71	C₂×C₅×M₁₆	C₂×D₃	1	2	yes	—
72	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
73	#4072	—	3	1	no	—
74	C₂²×#168	—	3	1	no	—
75	#4055	—	3	4	no	—
76	#4054	—	3	4	no	—
77	#4053	—	3	4	no	—
78	#4053	—	3	4	no	—
79	#4053	—	6	2	no	—
80	#4053	—	6	2	no	—
81	#4050	—	3	4	no	—
82	C₂×#676	—	6	2	no	—
83	C₂×C₈×D₅	—	6	2	no	—
84	C₅×#108	C₂×D₃	1	4	yes	complemented
85	#3957	—	3	8	no	—
86	#3947	—	3	2	no	—
87	C₈×Dic₅	—	3	2	no	—
88	#2955	—	15	1	no	Sylow
89	C₂³×C₃×C₅	#41	1	1	yes	characteristic
90	C₂²×D₁₅	—	2	2	no	—
91	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
92	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
93	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
94	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
95	#1220	—	2	4	no	—
96	#1220	—	2	4	no	—
97	C₂×Dic₁₅	—	2	2	no	—
98	C₂×Dic₁₅	—	2	2	no	—
99	D₆₀	—	2	2	no	—
100	D₆₀	C₂²×C₄	1	4	yes	—
101	C₄×D₁₅	—	2	4	no	—
102	C₄×D₁₅	—	2	4	no	—
103	Dic₃₀	—	2	2	no	—
104	Dic₃₀	C₂²×C₄	1	4	yes	—
105	C₃×C₅×C₈	—	2	4	no	—
106	C₃×C₅×C₈	C₂²×C₄	1	4	yes	—
107	C₃⋊#168	C₂×D₄	1	1	yes	characteristic
108	C₃⋊#168	C₂×D₄	1	2	yes	—
109	C₃⋊#168	C₂×D₄	1	1	yes	characteristic
110	C₂×#235	—	5	2	no	—
111	C₂×C₃×M₁₆	C₂×D₅	1	2	yes	—
112	C₂×C₃×M₁₆	C₂×D₅	1	1	yes	characteristic
113	#926	—	5	1	no	—
114	C₂²×C₃⋊C₈	—	5	1	no	—
115	#909	—	5	4	no	—
116	#908	—	5	4	no	—
117	#907	—	5	4	no	—
118	#907	—	5	4	no	—
119	#907	—	10	2	no	—
120	#907	—	10	2	no	—
121	#904	—	5	4	no	—
122	C₂×ΓL(1,𝔽₂₅)	—	10	2	no	—
123	C₂×D₃×C₈	—	10	2	no	—
124	C₃×#108	C₂×D₅	1	4	yes	complemented
125	#825	—	5	8	no	—
126	#815	—	5	2	no	—
127	C₈×Dic₃	—	5	2	no	—
128	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
129	C₂×C₂²⋊D₅	—	6	2	no	—
130	#709	—	3	4	no	—
131	#709	—	3	4	no	—
132	#709	—	6	2	no	—
133	#709	—	6	2	no	—
134	C₂×D₂₀	—	3	2	no	—
135	C₂×C₄×D₅	—	6	2	no	—
136	C₂×Dic₁₀	—	3	2	no	—
137	C₅×M₁₆	—	2	2	no	—
138	C₅×M₁₆	C₄×D₃	1	4	yes	—
139	C₅×M₁₆	C₂²×D₃	1	4	yes	—
140	C₂×C₅×C₈	—	2	2	no	—
141	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
142	C₂²⋊Dic₅	—	6	1	no	—
143	C₄⋊Dic₅	—	6	1	no	—
144	C₄×Dic₅	—	3	2	no	—
145	C₂×#168	—	3	2	no	—
146	C₂×#168	—	3	2	no	—
147	C₂×#168	—	3	1	no	—
148	C₂×#168	—	3	1	no	—
149	Dic₂₀	—	3	4	no	—
150	D₄₀	—	3	4	no	—
151	#677	—	3	8	no	—
152	#676	—	6	4	no	—
153	#676	—	6	4	no	—
154	C₈×D₅	—	6	4	no	—
155	C₈×D₅	—	6	4	no	—
156	#575	—	15	1	no	—
157	C₂×#131	—	15	2	no	—
158	#444	—	15	4	no	—
159	#443	—	15	4	no	—
160	#430	D₁₅	1	1	yes	characteristic, complemented
161	#421	—	15	2	no	—
162	#405	—	15	1	no	—
163	C₂²×C₃×C₅	—	4	1	no	—
164	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
165	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
166	C₂×D₁₅	—	2	4	no	—
167	C₂×D₁₅	—	4	4	no	—
168	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
169	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
170	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
171	Dic₁₅	—	2	4	no	—
172	Dic₁₅	—	4	2	no	—
173	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
174	C₂×#67	—	10	2	no	—
175	#235	—	5	4	no	—
176	#235	—	5	4	no	—
177	#235	—	10	2	no	—
178	#235	—	10	2	no	—
179	C₂×D₁₂	—	5	2	no	—
180	C₂×C₄×D₃	—	10	2	no	—
181	C₂×Dic₆	—	5	2	no	—
182	C₃×M₁₆	—	2	2	no	—
183	C₃×M₁₆	C₄×D₅	1	4	yes	—
184	C₃×M₁₆	C₂²×D₅	1	4	yes	—
185	C₂×C₃×C₈	—	2	2	no	—
186	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
187	#217	—	10	1	no	—
188	C₄⋊Dic₃	—	10	1	no	—
189	C₄×Dic₃	—	5	2	no	—
190	C₂×C₃⋊C₈	—	5	2	no	—
191	C₂×C₃⋊C₈	—	5	2	no	—
192	C₂×C₃⋊C₈	—	5	1	no	—
193	C₂×C₃⋊C₈	—	5	1	no	—
194	Dic₁₂	—	5	4	no	—
195	D₂₄	—	5	4	no	—
196	#204	—	5	8	no	—
197	ΓL(1,𝔽₂₅)	—	10	4	no	—
198	ΓL(1,𝔽₂₅)	—	10	4	no	—
199	D₃×C₈	—	10	4	no	—
200	D₃×C₈	—	10	4	no	—
201	C₂³×C₅	#239	1	1	yes	characteristic
202	C₂²×D₅	—	6	2	no	—
203	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
204	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
205	C₂×C₄×C₅	#239	1	1	yes	characteristic
206	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
207	C₂²⋊D₅	—	6	4	no	—
208	C₂²⋊D₅	—	6	4	no	—
209	C₂×Dic₅	—	6	2	no	—
210	C₂×Dic₅	—	6	2	no	—
211	D₂₀	—	3	4	no	—
212	D₂₀	—	6	2	no	—
213	C₄×D₅	—	6	4	no	—
214	C₄×D₅	—	6	4	no	—
215	Dic₁₀	—	3	4	no	—
216	Dic₁₀	—	6	2	no	—
217	C₅×C₈	—	2	4	no	—
218	C₅×C₈	C₂×C₄×D₃	1	4	yes	—
219	#168	—	3	1	no	—
220	#168	—	3	2	no	—
221	#168	—	3	1	no	—
222	C₂×#41	—	15	2	no	—
223	#137	—	15	4	no	—
224	AGL(1,ℤ/8)	—	15	4	no	—
225	#135	—	15	4	no	—
226	#131	—	15	4	no	—
227	#131	—	15	4	no	—
228	#131	—	30	2	no	—
229	#131	—	30	2	no	—
230	C₂×M₁₆	—	30	2	no	—
231	C₂×M₁₆	C₂×D₁₅	1	2	yes	—
232	C₂×M₁₆	C₂×D₁₅	1	1	yes	characteristic
233	C₂²×C₈	—	15	1	no	—
234	C₂²×C₈	—	30	2	no	—
235	#117	—	15	1	no	—
236	#108	C₂×D₁₅	1	4	yes	complemented
237	#104	—	15	8	no	—
238	#97	—	15	2	no	—
239	C₄×C₈	—	15	2	no	—
240	C₂×C₃×C₅	—	2	1	no	—
241	C₂×C₃×C₅	—	2	2	no	—
242	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
243	D₁₅	—	4	4	no	—
244	C₂³×C₃	#713	1	1	yes	characteristic
245	C₂²×D₃	—	10	2	no	—
246	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
247	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
248	C₂×C₃×C₄	#713	1	1	yes	characteristic
249	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
250	#67	—	10	4	no	—
251	#67	—	10	4	no	—
252	C₂×Dic₃	—	10	2	no	—
253	C₂×Dic₃	—	10	2	no	—
254	D₁₂	—	5	4	no	—
255	D₁₂	—	10	2	no	—
256	C₄×D₃	—	10	4	no	—
257	C₄×D₃	—	10	4	no	—
258	Dic₆	—	5	4	no	—
259	Dic₆	—	10	2	no	—
260	C₃×C₈	—	2	4	no	—
261	C₃×C₈	C₂×C₄×D₅	1	4	yes	—
262	C₃⋊C₈	—	5	1	no	—
263	C₃⋊C₈	—	5	2	no	—
264	C₃⋊C₈	—	5	1	no	—
265	C₂²×C₅	—	4	1	no	—
266	C₂²×C₅	#893	1	2	yes	—
267	C₂²×C₅	C₂×#239	1	1	yes	characteristic
268	C₂×D₅	—	6	4	no	—
269	C₂×D₅	—	12	4	no	—
270	C₄×C₅	#893	1	2	yes	—
271	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
272	C₄×C₅	C₂×D₃×D₄	1	1	yes	characteristic
273	Dic₅	—	6	4	no	—
274	Dic₅	—	12	2	no	—
275	#41	—	15	4	no	—
276	#41	—	15	4	no	—
277	#41	—	30	2	no	—
278	#41	—	30	2	no	—
279	C₂×Q₈	—	15	2	no	—
280	C₂×D₄	—	15	2	no	—
281	C₂×D₄	—	30	2	no	—
282	C₂²×C₄	—	30	2	no	—
283	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
284	Q₁₆	—	15	4	no	—
285	SD₁₆	—	15	8	no	—
286	D₈	—	15	4	no	—
287	M₁₆	—	2	2	no	—
288	M₁₆	—	30	4	no	—
289	M₁₆	—	30	4	no	—
290	M₁₆	C₄×D₁₅	1	4	yes	—
291	M₁₆	C₂²×D₁₅	1	4	yes	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	15	2	no	—
294	C₂×C₈	—	15	2	no	—
295	C₂×C₈	—	15	1	no	—
296	C₂×C₈	—	15	1	no	—
297	C₂×C₈	—	30	4	no	—
298	C₂×C₈	—	30	4	no	—
299	C₂×C₈	C₂²×D₁₅	1	2	yes	—
300	C₄⋊C₄	—	30	1	no	—
301	C₂²⋊C₄	—	30	1	no	—
302	C₄²	—	15	2	no	—
303	C₃×C₅	#2955	1	1	yes	characteristic, complemented, Hall
304	C₂²×C₃	—	4	1	no	—
305	C₂²×C₃	#4039	1	2	yes	—
306	C₂²×C₃	C₂×#713	1	1	yes	characteristic
307	C₂×D₃	—	10	4	no	—
308	C₂×D₃	—	20	4	no	—
309	C₃×C₄	#4039	1	2	yes	—
310	C₃×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
311	C₃×C₄	C₂×D₄×D₅	1	1	yes	characteristic
312	Dic₃	—	10	4	no	—
313	Dic₃	—	20	2	no	—
314	C₂×C₅	—	2	1	no	—
315	C₂×C₅	—	2	2	no	—
316	C₂×C₅	C₂×#893	1	1	yes	characteristic
317	D₅	—	12	4	no	—
318	C₂³	—	30	2	no	—
319	C₂³	#6821	1	1	yes	characteristic
320	Q₈	—	15	4	no	—
321	Q₈	—	30	2	no	—
322	D₄	—	15	4	no	—
323	D₄	—	30	4	no	—
324	D₄	—	30	4	no	—
325	D₄	—	30	2	no	—
326	C₂×C₄	—	30	2	no	—
327	C₂×C₄	—	30	4	no	—
328	C₂×C₄	—	30	4	no	—
329	C₂×C₄	—	30	2	no	—
330	C₂×C₄	C₂×C₄×D₁₅	1	2	yes	—
331	C₂×C₄	D₄×D₁₅	1	2	yes	—
332	C₂×C₄	#6821	1	1	yes	characteristic
333	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
334	C₈	—	2	4	no	—
335	C₈	—	15	1	no	—
336	C₈	—	15	2	no	—
337	C₈	—	15	1	no	—
338	C₈	C₂×C₄×D₁₅	1	4	yes	—
339	C₂×C₃	—	2	1	no	—
340	C₂×C₃	—	2	2	no	—
341	C₂×C₃	C₂×#4039	1	1	yes	characteristic
342	D₃	—	20	4	no	—
343	C₅	#79257	1	1	yes	characteristic, complemented, Sylow
344	C₂²	—	4	1	no	—
345	C₂²	—	30	4	no	—
346	C₂²	—	60	4	no	—
347	C₂²	#91782	1	2	yes	—
348	C₂²	C₂×#6821	1	1	yes	characteristic
349	C₄	—	30	4	no	—
350	C₄	—	60	2	no	—
351	C₄	#91782	1	2	yes	—
352	C₄	C₂²×C₄×D₁₅	1	1	yes	characteristic
353	C₄	C₂×D₄×D₁₅	1	1	yes	characteristic
354	C₃	#10610988	1	1	yes	characteristic, complemented, Sylow
355	C₂	—	2	1	no	—
356	C₂	—	2	2	no	—
357	C₂	—	60	4	no	—
358	C₂	C₂×#91782	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₁₅
commutator	C₃×C₄×C₅	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₅
Fitting	C₃×C₅×#430	C₂
socle	C₂×C₃×C₅	C₂×C₄×D₄
prime radical	#11756507	C₂
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	#430	D₁₅
hypocenter	C₃×C₅	#2955
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

Complements

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

N	H ≅ G/N	type
2. C₂×#91796	C₂	semidirect
3. C₃×C₅×#430	C₂	semidirect
4. #11751240	C₂	semidirect
5. #11751223	C₂	semidirect
6. #11750996	C₂	semidirect
7. #11750995	C₂	semidirect
14. #91798	C₄	semidirect
15. #91797	C₄	semidirect
18. #91796	C₂²	semidirect
19. #91796	C₂²	semidirect
23. C₃×C₅×#108	C₂²	semidirect
24. #91109	C₂²	semidirect
25. #91099	C₂²	semidirect
26. C₈×Dic₁₅	C₂²	semidirect
30. C₅×#430	D₃	semidirect
65. C₃×#430	D₅	semidirect
84. C₅×#108	C₂×D₃	semidirect
124. C₃×#108	C₂×D₅	semidirect
160. #430	D₁₅	semidirect
236. #108	C₂×D₁₅	semidirect
303. C₃×C₅	#2955	semidirect
343. C₅	#79257	semidirect
354. C₃	#10610988	semidirect

Maximal subgroups

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

subgroup	index
#11756507	2
C₂×#91796	2
C₃×C₅×#430	2
#11751240	2
#11751223	2
#11750996	2
#11750995	2
#10610988	3
#79257	5