Subgroups — #14284892

Order 1920.

480 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#11750938	C₂	1	1	yes	maximal, characteristic
3	#11750934	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#333	C₂	1	1	yes	maximal, characteristic, complemented
5	#11746367	C₂	1	1	yes	maximal, characteristic, complemented
6	#11746302	C₂	1	1	yes	maximal, characteristic, complemented
7	#11746302	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599276	—	3	1	no	maximal, Hall
9	C₂×Q₈×D₁₅	C₄	1	2	yes	—
10	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	#91833	C₄	1	2	yes	—
12	C₂×#6705	—	2	1	no	—
13	C₄⋊C₄×D₁₅	—	2	1	no	—
14	C₄⋊C₄×D₁₅	—	2	1	no	—
15	#91777	—	2	1	no	—
16	#91777	—	2	1	no	—
17	C₄²×D₁₅	C₂²	1	1	yes	characteristic
18	#91758	C₂²	1	1	yes	characteristic
19	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	#91104	—	2	1	no	—
22	#91098	—	2	1	no	—
23	C₈×Dic₁₅	—	2	1	no	—
24	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
25	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
26	#68483	—	5	1	no	maximal, Hall
27	D₅×C₄⋊Q₈	—	3	1	no	—
28	#65126	—	3	1	no	—
29	#65122	—	3	1	no	—
30	C₅×#333	D₃	1	1	yes	characteristic, complemented
31	#64766	—	3	1	no	—
32	#64701	—	3	1	no	—
33	#64701	—	3	1	no	—
34	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
35	Q₈×D₁₅	—	4	4	no	—
36	C₂×C₄×D₁₅	—	2	2	no	—
37	C₂×C₄×D₁₅	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₁₅	D₄	1	1	yes	characteristic
39	C₂×C₄×D₁₅	D₄	1	1	yes	characteristic
40	C₂×Dic₃₀	—	2	2	no	—
41	C₂×Dic₃₀	C₂×C₄	1	2	yes	—
42	C₂×C₃×C₅×C₈	—	2	1	no	—
43	C₂×C₃×C₅×C₈	—	2	1	no	—
44	C₃×C₅×C₄⋊C₄	—	2	1	no	—
45	C₃×C₅×C₄⋊C₄	—	2	1	no	—
46	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₄⋊Dic₁₅	—	2	1	no	—
48	C₄⋊Dic₁₅	—	2	1	no	—
49	#6712	—	2	2	no	—
50	#6712	—	2	2	no	—
51	C₄×Dic₁₅	C₂×C₄	1	1	yes	characteristic
52	C₄×Dic₁₅	D₄	1	1	yes	characteristic
53	C₄×Dic₁₅	D₄	1	1	yes	characteristic
54	C₂×C₃⋊#168	—	2	1	no	—
55	C₂×C₃⋊#168	—	2	1	no	—
56	#6705	—	4	2	no	—
57	D₃×C₄⋊Q₈	—	5	1	no	—
58	#4837	—	5	1	no	—
59	#4833	—	5	1	no	—
60	C₃×#333	D₅	1	1	yes	characteristic, complemented
61	#4546	—	5	1	no	—
62	#4481	—	5	1	no	—
63	#4481	—	5	1	no	—
64	C₂×Q₈×D₅	—	3	2	no	—
65	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
66	#4090	—	3	2	no	—
67	C₂×#676	—	6	1	no	—
68	D₅×C₄⋊C₄	—	6	1	no	—
69	D₅×C₄⋊C₄	—	6	1	no	—
70	#4034	—	6	1	no	—
71	#4034	—	6	1	no	—
72	C₄²×D₅	—	3	1	no	—
73	#4015	—	3	1	no	—
74	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
75	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
76	#3952	—	6	1	no	—
77	#3946	—	6	1	no	—
78	C₈×Dic₅	—	6	1	no	—
79	C₄⋊#168	—	3	1	no	—
80	C₄⋊#168	—	3	1	no	—
81	#1548	—	15	1	no	Sylow
82	C₂²×D₁₅	C₂²⋊C₄	1	1	yes	characteristic
83	C₃×C₅×Q₈	—	4	2	no	—
84	C₂×C₃×C₄×C₅	—	2	2	no	—
85	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂×Dic₁₅	—	2	2	no	—
89	C₂×Dic₁₅	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×Dic₁₅	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×Dic₁₅	C₂²⋊C₄	1	1	yes	characteristic
92	C₄×D₁₅	—	2	1	no	—
93	C₄×D₁₅	—	2	1	no	—
94	C₄×D₁₅	—	2	2	no	—
95	C₄×D₁₅	—	4	4	no	—
96	C₄×D₁₅	SD₁₆	1	2	yes	—
97	C₄×D₁₅	Q₁₆	1	2	yes	—
98	Dic₃₀	—	4	4	no	—
99	Dic₃₀	—	4	2	no	—
100	C₃×C₅×C₈	—	4	1	no	—
101	C₃×C₅×C₈	—	4	1	no	—
102	C₃⋊#168	—	4	1	no	—
103	C₃⋊#168	—	4	1	no	—
104	C₂×D₃×Q₈	—	5	2	no	—
105	C₃×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
106	#944	—	5	2	no	—
107	C₂×ΓL(1,𝔽₂₅)	—	10	1	no	—
108	D₃×C₄⋊C₄	—	10	1	no	—
109	D₃×C₄⋊C₄	—	10	1	no	—
110	#888	—	10	1	no	—
111	#888	—	10	1	no	—
112	C₄²×D₃	—	5	1	no	—
113	#869	—	5	1	no	—
114	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
115	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
116	(C₂×D₃)⋊C₈	—	10	1	no	—
117	Dic₃⋊C₈	—	10	1	no	—
118	C₈×Dic₃	—	10	1	no	—
119	C₄⋊(C₃⋊C₈)	—	5	1	no	—
120	C₄⋊(C₃⋊C₈)	—	5	1	no	—
121	C₂×C₅×Q₈	C₄×D₃	1	2	yes	—
122	Q₈×D₅	—	12	4	no	—
123	C₂×C₄×D₅	—	3	1	no	—
124	C₂×C₄×D₅	—	3	1	no	—
125	C₂×C₄×D₅	—	3	1	no	—
126	C₂×C₄×D₅	—	6	2	no	—
127	C₂×Dic₁₀	—	3	2	no	—
128	C₂×Dic₁₀	—	6	2	no	—
129	C₂×C₅×C₈	—	2	1	no	—
130	C₂×C₅×C₈	—	2	1	no	—
131	C₅×C₄⋊C₄	—	2	1	no	—
132	C₅×C₄⋊C₄	—	2	1	no	—
133	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
134	C₄⋊Dic₅	—	6	1	no	—
135	C₄⋊Dic₅	—	6	1	no	—
136	#683	—	6	2	no	—
137	#683	—	6	2	no	—
138	C₄×Dic₅	—	3	1	no	—
139	C₄×Dic₅	—	3	1	no	—
140	C₄×Dic₅	—	3	1	no	—
141	C₂×#168	—	6	1	no	—
142	C₂×#168	—	6	1	no	—
143	#676	—	12	2	no	—
144	C₂×C₄⋊Q₈	—	15	1	no	—
145	#431	—	15	1	no	—
146	#423	—	15	1	no	—
147	#333	—	15	1	no	—
148	#333	—	15	1	no	—
149	#333	—	15	1	no	—
150	#333	D₁₅	1	1	yes	characteristic, complemented
151	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	C₂×D₁₅	—	2	1	no	—
153	C₂×D₁₅	—	2	1	no	—
154	C₂×D₁₅	#103	1	2	yes	—
155	C₃×C₄×C₅	—	2	1	no	—
156	C₃×C₄×C₅	—	4	2	no	—
157	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
158	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
159	C₃×C₄×C₅	#137	1	1	yes	characteristic
160	C₃×C₄×C₅	#137	1	1	yes	characteristic
161	Dic₁₅	—	2	1	no	—
162	Dic₁₅	—	4	2	no	—
163	Dic₁₅	#100	1	2	yes	—
164	Dic₁₅	#103	1	2	yes	—
165	C₂×C₃×Q₈	C₄×D₅	1	2	yes	—
166	D₃×Q₈	—	20	4	no	—
167	C₂×C₄×D₃	—	5	1	no	—
168	C₂×C₄×D₃	—	5	1	no	—
169	C₂×C₄×D₃	—	5	1	no	—
170	C₂×C₄×D₃	—	10	2	no	—
171	C₂×Dic₆	—	5	2	no	—
172	C₂×Dic₆	—	10	2	no	—
173	C₂×C₃×C₈	—	2	1	no	—
174	C₂×C₃×C₈	—	2	1	no	—
175	C₃×C₄⋊C₄	—	2	1	no	—
176	C₃×C₄⋊C₄	—	2	1	no	—
177	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
178	C₄⋊Dic₃	—	10	1	no	—
179	C₄⋊Dic₃	—	10	1	no	—
180	Dic₃⋊C₄	—	10	2	no	—
181	Dic₃⋊C₄	—	10	2	no	—
182	C₄×Dic₃	—	5	1	no	—
183	C₄×Dic₃	—	5	1	no	—
184	C₄×Dic₃	—	5	1	no	—
185	C₂×C₃⋊C₈	—	10	1	no	—
186	C₂×C₃⋊C₈	—	10	1	no	—
187	ΓL(1,𝔽₂₅)	—	20	2	no	—
188	C₂²×D₅	—	3	1	no	—
189	C₅×Q₈	—	4	2	no	—
190	C₂×C₄×C₅	—	2	2	no	—
191	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
192	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
193	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
194	C₂×Dic₅	—	3	1	no	—
195	C₂×Dic₅	—	3	1	no	—
196	C₂×Dic₅	—	3	1	no	—
197	C₂×Dic₅	—	6	2	no	—
198	C₄×D₅	—	3	2	no	—
199	C₄×D₅	—	3	2	no	—
200	C₄×D₅	—	6	2	no	—
201	C₄×D₅	—	6	1	no	—
202	C₄×D₅	—	6	1	no	—
203	C₄×D₅	—	12	4	no	—
204	Dic₁₀	—	12	4	no	—
205	Dic₁₀	—	12	2	no	—
206	C₅×C₈	—	4	1	no	—
207	C₅×C₈	—	4	1	no	—
208	#168	—	12	1	no	—
209	#168	—	12	1	no	—
210	C₂²×Q₈	—	15	2	no	—
211	C₂×M₁₆	—	30	1	no	—
212	C₄⋊Q₈	—	15	2	no	—
213	C₄⋊Q₈	—	15	1	no	—
214	C₄⋊Q₈	—	30	1	no	—
215	C₄⋊Q₈	—	30	1	no	—
216	C₄⋊Q₈	C₂×D₁₅	1	1	yes	characteristic
217	C₂×C₄⋊C₄	—	30	1	no	—
218	C₂×C₄⋊C₄	—	30	1	no	—
219	C₂×C₄²	—	15	1	no	—
220	C₄⋊C₈	—	15	1	no	—
221	C₄⋊C₈	—	15	1	no	—
222	C₄⋊C₈	—	30	1	no	—
223	C₄⋊C₈	C₂×D₁₅	1	1	yes	characteristic
224	C₄⋊C₈	C₂×D₁₅	1	1	yes	characteristic
225	C₂²⋊C₈	—	30	1	no	—
226	C₄×C₈	—	30	1	no	—
227	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
228	C₂×C₃×C₅	C₂×#103	1	1	yes	characteristic
229	C₂×C₃×C₅	#419	1	1	yes	characteristic
230	D₁₅	—	2	2	no	—
231	C₂²×D₃	—	5	1	no	—
232	C₃×Q₈	—	4	2	no	—
233	C₂×C₃×C₄	—	2	2	no	—
234	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
235	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
236	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
237	C₂×Dic₃	—	5	1	no	—
238	C₂×Dic₃	—	5	1	no	—
239	C₂×Dic₃	—	5	1	no	—
240	C₂×Dic₃	—	10	2	no	—
241	C₄×D₃	—	5	2	no	—
242	C₄×D₃	—	5	2	no	—
243	C₄×D₃	—	10	2	no	—
244	C₄×D₃	—	10	1	no	—
245	C₄×D₃	—	10	1	no	—
246	C₄×D₃	—	20	4	no	—
247	Dic₆	—	20	4	no	—
248	Dic₆	—	20	2	no	—
249	C₃×C₈	—	4	1	no	—
250	C₃×C₈	—	4	1	no	—
251	C₃⋊C₈	—	20	1	no	—
252	C₃⋊C₈	—	20	1	no	—
253	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
254	C₂×D₅	—	3	2	no	—
255	C₂×D₅	—	6	1	no	—
256	C₂×D₅	—	6	1	no	—
257	C₄×C₅	—	2	1	no	—
258	C₄×C₅	—	4	2	no	—
259	C₄×C₅	D₃×SD₁₆	1	1	yes	characteristic
260	C₄×C₅	#915	1	1	yes	characteristic
261	C₄×C₅	D₃×Q₁₆	1	1	yes	characteristic
262	C₄×C₅	#918	1	1	yes	characteristic
263	Dic₅	—	3	2	no	—
264	Dic₅	—	3	2	no	—
265	Dic₅	—	6	1	no	—
266	Dic₅	—	12	2	no	—
267	C₂×Q₈	—	15	2	no	—
268	C₂×Q₈	—	30	2	no	—
269	C₂×Q₈	—	60	4	no	—
270	C₂×Q₈	C₄×D₁₅	1	2	yes	—
271	C₂²×C₄	—	15	1	no	—
272	C₂²×C₄	—	15	1	no	—
273	C₂²×C₄	—	15	1	no	—
274	C₂²×C₄	—	30	2	no	—
275	M₁₆	—	60	2	no	—
276	C₂×C₈	—	2	1	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	30	1	no	—
279	C₂×C₈	—	30	1	no	—
280	C₄⋊C₄	—	2	1	no	—
281	C₄⋊C₄	—	2	1	no	—
282	C₄⋊C₄	—	30	2	no	—
283	C₄⋊C₄	—	30	2	no	—
284	C₄⋊C₄	—	30	1	no	—
285	C₄⋊C₄	—	30	1	no	—
286	C₄²	—	15	1	no	—
287	C₄²	—	15	1	no	—
288	C₄²	—	15	1	no	—
289	C₄²	C₂²×D₁₅	1	1	yes	characteristic
290	C₃×C₅	#1548	1	1	yes	characteristic, complemented, Hall
291	C₂²×C₃	D₅×C₂²⋊C₄	1	1	yes	characteristic
292	C₂×D₃	—	5	2	no	—
293	C₂×D₃	—	10	1	no	—
294	C₂×D₃	—	10	1	no	—
295	C₃×C₄	—	2	1	no	—
296	C₃×C₄	—	4	2	no	—
297	C₃×C₄	D₅×SD₁₆	1	1	yes	characteristic
298	C₃×C₄	#4061	1	1	yes	characteristic
299	C₃×C₄	D₅×Q₁₆	1	1	yes	characteristic
300	C₃×C₄	#4064	1	1	yes	characteristic
301	Dic₃	—	5	2	no	—
302	Dic₃	—	5	2	no	—
303	Dic₃	—	10	1	no	—
304	Dic₃	—	20	2	no	—
305	C₂×C₅	D₃×#100	1	1	yes	characteristic
306	C₂×C₅	D₃×#103	1	1	yes	characteristic
307	C₂×C₅	#4802	1	1	yes	characteristic
308	D₅	—	6	2	no	—
309	C₂³	—	15	1	no	—
310	Q₈	—	4	2	no	—
311	Q₈	—	60	2	no	—
312	Q₈	—	60	4	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	15	1	no	—
315	C₂×C₄	—	15	1	no	—
316	C₂×C₄	—	15	2	no	—
317	C₂×C₄	—	15	2	no	—
318	C₂×C₄	—	15	1	no	—
319	C₂×C₄	—	30	2	no	—
320	C₂×C₄	—	30	1	no	—
321	C₂×C₄	—	30	1	no	—
322	C₂×C₄	—	30	2	no	—
323	C₂×C₄	—	60	4	no	—
324	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
325	C₂×C₄	D₄×D₁₅	1	1	yes	characteristic
326	C₂×C₄	D₄×D₁₅	1	1	yes	characteristic
327	C₈	—	4	1	no	—
328	C₈	—	4	1	no	—
329	C₈	—	60	1	no	—
330	C₈	—	60	1	no	—
331	C₂×C₃	D₅×#100	1	1	yes	characteristic
332	C₂×C₃	D₅×#103	1	1	yes	characteristic
333	C₂×C₃	#65091	1	1	yes	characteristic
334	D₃	—	10	2	no	—
335	C₅	#68483	1	1	yes	characteristic, complemented, Sylow
336	C₂²	—	15	2	no	—
337	C₂²	—	30	1	no	—
338	C₂²	—	30	1	no	—
339	C₂²	C₂²⋊C₄×D₁₅	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	4	2	no	—
342	C₄	—	15	2	no	—
343	C₄	—	15	2	no	—
344	C₄	—	30	1	no	—
345	C₄	—	60	2	no	—
346	C₄	SD₁₆×D₁₅	1	1	yes	characteristic
347	C₄	#91804	1	1	yes	characteristic
348	C₄	Q₁₆×D₁₅	1	1	yes	characteristic
349	C₄	#91807	1	1	yes	characteristic
350	C₃	#10599276	1	1	yes	characteristic, complemented, Sylow
351	C₂	—	30	2	no	—
352	C₂	D₁₅×#100	1	1	yes	characteristic
353	C₂	D₁₅×#103	1	1	yes	characteristic
354	C₂	#11750903	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₅
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂²×D₁₅
Fitting	C₃×C₅×#333	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₅	C₂²⋊C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	#333	D₁₅
hypocenter	C₃×C₅	#1548
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×#333	C₂	semidirect
5. #11746367	C₂	semidirect
6. #11746302	C₂	semidirect
7. #11746302	C₂	semidirect
30. C₅×#333	D₃	semidirect
60. C₃×#333	D₅	semidirect
150. #333	D₁₅	semidirect
290. C₃×C₅	#1548	semidirect
335. C₅	#68483	semidirect
350. C₃	#10599276	semidirect

Maximal subgroups

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

subgroup	index
D₁₅×C₄⋊Q₈	2
#11750938	2
#11750934	2
C₃×C₅×#333	2
#11746367	2
#11746302	2
#10599276	3
#68483	5