Subgroups — #14307381

Order 1920.

552 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#11752348	C₂	1	1	yes	maximal, characteristic
3	#11752335	C₂	1	1	yes	maximal, characteristic
4	#11746745	C₂	1	4	yes	maximal
5	#10604675	—	3	1	no	maximal, Hall
6	C₂×C₃×Q₈×D₅	—	2	1	no	—
7	C₂×C₄⋊ΓL(1,𝔽₁₆)	C₂²	1	2	yes	—
8	C₂×D₅⋊(C₃⋊C₈)	C₂²	1	2	yes	—
9	C₃×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
10	C₃×#4090	—	2	1	no	—
11	C₃×D₅×C₄⋊C₄	—	2	1	no	—
12	C₃×D₅×C₄⋊C₄	C₂²	1	2	yes	—
13	C₃×#4034	—	2	1	no	—
14	C₃×#4034	C₄	1	2	yes	complemented
15	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
16	C₃×#4015	C₄	1	1	yes	characteristic, complemented
17	#91235	—	2	1	no	—
18	#91235	—	2	1	no	—
19	#91231	—	2	1	no	—
20	#91231	—	2	1	no	—
21	#71788	—	5	1	no	maximal, Hall
22	D₅×C₄⋊Q₈	D₃	1	1	yes	characteristic
23	#65689	—	3	1	no	—
24	#65676	—	3	1	no	—
25	#64874	—	3	4	no	—
26	C₂²×ΓL(1,𝔽₁₆)	—	2	2	no	—
27	C₂×C₃×C₅×Q₈	—	2	1	no	—
28	C₃×Q₈×D₅	—	4	4	no	—
29	C₂×C₃×C₄×D₅	—	2	2	no	—
30	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
31	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
32	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
33	C₂×C₃×Dic₁₀	—	2	1	no	—
34	C₂×C₃×Dic₁₀	—	2	2	no	—
35	C₂×C₃⋊#170	—	2	2	no	—
36	C₄⋊ΓL(1,𝔽₁₆)	—	2	4	no	—
37	D₅⋊(C₃⋊C₈)	—	2	2	no	—
38	D₅⋊(C₃⋊C₈)	Q₈	1	4	yes	complemented
39	C₃×C₅×C₄⋊C₄	—	2	1	no	—
40	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
41	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
42	C₃×C₄⋊Dic₅	—	2	1	no	—
43	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
44	C₃×#683	—	2	2	no	—
45	C₃×#683	—	2	2	no	—
46	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
47	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
48	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
49	C₂×C₃×C₄⋊Q₈	—	5	1	no	—
50	#4936	—	5	1	no	—
51	C₂×#803	—	5	1	no	—
52	#4529	—	5	4	no	—
53	C₂×Q₈×D₅	—	2	1	no	—
54	C₂×C₄⋊AGL(1,𝔽₅)	—	3	2	no	—
55	C₂×D₅⋊C₈	—	3	2	no	—
56	C₅×C₄⋊Q₈	Dic₃	1	1	yes	characteristic, complemented
57	#4090	—	2	1	no	—
58	D₅×C₄⋊C₄	—	2	1	no	—
59	D₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
60	#4034	—	2	1	no	—
61	#4034	Dic₃	1	2	yes	complemented
62	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
63	#4015	Dic₃	1	1	yes	characteristic, complemented
64	(C₂×AGL(1,𝔽₅))⋊C₄	—	6	1	no	—
65	(C₂×AGL(1,𝔽₅))⋊C₄	—	6	1	no	—
66	#4002	—	6	1	no	—
67	#4002	—	6	1	no	—
68	#1993	—	15	1	no	Sylow
69	C₂²×C₃×D₅	C₂×D₄	1	1	yes	characteristic
70	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
71	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
72	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
73	C₃×C₅×Q₈	—	4	2	no	—
74	C₂×C₃×C₄×C₅	—	2	2	no	—
75	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃×Dic₅	—	2	2	no	—
79	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
82	C₃×C₄×D₅	—	2	2	no	—
83	C₃×C₄×D₅	—	2	2	no	—
84	C₃×C₄×D₅	—	4	4	no	—
85	C₃×C₄×D₅	C₂×Q₈	1	2	yes	—
86	C₃×C₄×D₅	#41	1	2	yes	—
87	C₃×Dic₁₀	—	4	4	no	—
88	C₃×Dic₁₀	—	4	2	no	—
89	C₃⋊#170	—	2	4	no	—
90	C₂²×C₃×Q₈	—	10	1	no	—
91	C₃×C₄⋊Q₈	—	5	2	no	—
92	C₃×C₄⋊Q₈	—	5	1	no	—
93	C₃×C₄⋊Q₈	—	10	1	no	—
94	C₃×C₄⋊Q₈	—	10	1	no	—
95	C₃×C₄⋊Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
96	C₂×C₃×C₄⋊C₄	—	5	2	no	—
97	C₂×C₃×C₄⋊C₄	—	10	1	no	—
98	C₂×C₃×C₄²	—	5	1	no	—
99	C₂×C₄⋊Dic₃	—	5	2	no	—
100	C₂²×C₃⋊C₈	—	5	2	no	—
101	C₃⋊#95	—	10	1	no	—
102	C₃⋊#95	—	10	1	no	—
103	#803	—	10	1	no	—
104	#803	—	10	1	no	—
105	C₂²×AGL(1,𝔽₅)	—	6	2	no	—
106	C₂×C₅×Q₈	—	2	1	no	—
107	Q₈×D₅	—	4	4	no	—
108	C₂×C₄×D₅	—	2	2	no	—
109	C₂×C₄×D₅	#67	1	1	yes	characteristic
110	C₂×C₄×D₅	#67	1	1	yes	characteristic
111	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
112	C₂×Dic₁₀	—	2	1	no	—
113	C₂×Dic₁₀	—	2	2	no	—
114	C₂×#170	—	6	2	no	—
115	C₄⋊AGL(1,𝔽₅)	—	6	4	no	—
116	D₅⋊C₈	—	3	4	no	—
117	D₅⋊C₈	—	6	2	no	—
118	C₅×C₄⋊C₄	—	2	1	no	—
119	C₅×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
120	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
121	C₄⋊Dic₅	—	2	1	no	—
122	C₄⋊Dic₅	C₂×Dic₃	1	2	yes	—
123	#683	—	2	2	no	—
124	#683	—	2	2	no	—
125	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
126	C₄×Dic₅	#67	1	1	yes	characteristic
127	C₄×Dic₅	#67	1	1	yes	characteristic
128	C₂×C₄⋊Q₈	—	5	1	no	—
129	C₂×#97	—	15	1	no	—
130	#382	—	15	1	no	—
131	#340	—	15	4	no	—
132	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₃×D₅	#124	1	1	yes	characteristic
134	C₂×C₃×D₅	C₄⋊Q₈	1	1	yes	characteristic
135	C₂×C₃×D₅	AGL(1,ℤ/8)	1	2	yes	—
136	C₂×C₃×D₅	#137	1	2	yes	—
137	ΓL(1,𝔽₁₆)	—	8	2	no	—
138	C₃×C₄×C₅	—	2	1	no	—
139	C₃×C₄×C₅	—	2	1	no	—
140	C₃×C₄×C₅	—	4	2	no	—
141	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
142	C₃×Dic₅	—	2	1	no	—
143	C₃×Dic₅	—	2	1	no	—
144	C₃×Dic₅	—	4	2	no	—
145	C₃×Dic₅	C₂²⋊Q₈	1	2	yes	—
146	C₂×C₃×Q₈	—	2	1	no	—
147	C₂×C₃×Q₈	—	10	2	no	—
148	C₂×C₃×Q₈	—	10	1	no	—
149	C₂×C₃×Q₈	—	20	4	no	—
150	C₂²×C₃×C₄	—	5	1	no	—
151	C₂²×C₃×C₄	—	5	1	no	—
152	C₂²×C₃×C₄	—	5	1	no	—
153	C₂²×C₃×C₄	—	10	2	no	—
154	C₂²×Dic₃	—	10	2	no	—
155	C₃×C₄⋊C₄	—	2	1	no	—
156	C₃×C₄⋊C₄	—	5	2	no	—
157	C₃×C₄⋊C₄	—	10	2	no	—
158	C₃×C₄⋊C₄	—	10	2	no	—
159	C₃×C₄⋊C₄	—	10	1	no	—
160	C₃×C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
161	C₃×C₄²	—	5	1	no	—
162	C₃×C₄²	—	5	1	no	—
163	C₃×C₄²	—	5	1	no	—
164	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
165	C₄⋊Dic₃	—	10	4	no	—
166	C₂×C₃⋊C₈	—	5	4	no	—
167	C₂×C₃⋊C₈	—	10	2	no	—
168	C₂×C₃⋊C₈	—	10	2	no	—
169	C₂²×D₅	C₂×#67	1	1	yes	characteristic
170	C₂×AGL(1,𝔽₅)	—	12	2	no	—
171	C₂×AGL(1,𝔽₅)	—	12	2	no	—
172	C₂×AGL(1,𝔽₅)	—	12	2	no	—
173	C₅×Q₈	—	4	2	no	—
174	C₂×C₄×C₅	—	2	2	no	—
175	C₂×C₄×C₅	#217	1	1	yes	characteristic
176	C₂×C₄×C₅	#217	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
178	C₂×Dic₅	—	2	2	no	—
179	C₂×Dic₅	#217	1	1	yes	characteristic
180	C₂×Dic₅	#217	1	1	yes	characteristic
181	C₂×Dic₅	C₂×#67	1	1	yes	characteristic
182	C₄×D₅	—	2	2	no	—
183	C₄×D₅	—	2	2	no	—
184	C₄×D₅	—	4	4	no	—
185	C₄×D₅	D₃×Q₈	1	2	yes	—
186	C₄×D₅	#239	1	2	yes	—
187	Dic₁₀	—	4	4	no	—
188	Dic₁₀	—	4	2	no	—
189	#170	—	6	4	no	—
190	C₂²×Q₈	—	10	1	no	—
191	C₂²×C₈	—	15	2	no	—
192	C₄⋊Q₈	—	5	2	no	—
193	C₄⋊Q₈	—	5	1	no	—
194	C₄⋊Q₈	—	10	1	no	—
195	C₄⋊Q₈	—	10	1	no	—
196	C₄⋊Q₈	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
197	C₂×C₄⋊C₄	—	5	2	no	—
198	C₂×C₄⋊C₄	—	10	1	no	—
199	C₂×C₄⋊C₄	—	15	2	no	—
200	C₂×C₄²	—	5	1	no	—
201	#97	—	30	1	no	—
202	#97	—	30	1	no	—
203	#95	—	30	1	no	—
204	#95	—	30	1	no	—
205	C₂×C₃×C₅	#391	1	1	yes	characteristic
206	C₂×C₃×C₅	#418	1	1	yes	characteristic
207	C₂×C₃×C₅	#419	1	1	yes	characteristic
208	C₃×D₅	#489	1	2	yes	—
209	C₃×D₅	#501	1	2	yes	—
210	C₂³×C₃	—	5	1	no	—
211	C₃×Q₈	—	4	2	no	—
212	C₃×Q₈	—	20	2	no	—
213	C₃×Q₈	—	20	4	no	—
214	C₂×C₃×C₄	—	2	2	no	—
215	C₂×C₃×C₄	—	5	1	no	—
216	C₂×C₃×C₄	—	5	2	no	—
217	C₂×C₃×C₄	—	5	2	no	—
218	C₂×C₃×C₄	—	5	1	no	—
219	C₂×C₃×C₄	—	5	1	no	—
220	C₂×C₃×C₄	—	10	2	no	—
221	C₂×C₃×C₄	—	10	2	no	—
222	C₂×C₃×C₄	—	10	2	no	—
223	C₂×C₃×C₄	—	20	4	no	—
224	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
225	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
226	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
227	C₂×Dic₃	—	20	2	no	—
228	C₂×Dic₃	—	20	2	no	—
229	C₂×Dic₃	—	20	2	no	—
230	C₃⋊C₈	—	10	4	no	—
231	C₂²×C₅	C₂×#217	1	1	yes	characteristic
232	C₂×D₅	#932	1	2	yes	—
233	C₂×D₅	#942	1	2	yes	—
234	C₂×D₅	#944	1	1	yes	characteristic
235	C₂×D₅	#947	1	1	yes	characteristic
236	AGL(1,𝔽₅)	—	24	2	no	—
237	C₄×C₅	—	2	1	no	—
238	C₄×C₅	—	2	1	no	—
239	C₄×C₅	—	4	2	no	—
240	C₄×C₅	Q₈×Dic₃	1	2	yes	—
241	Dic₅	—	2	1	no	—
242	Dic₅	—	2	1	no	—
243	Dic₅	—	4	2	no	—
244	Dic₅	#946	1	2	yes	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	10	2	no	—
247	C₂×Q₈	—	10	1	no	—
248	C₂×Q₈	—	20	4	no	—
249	C₂²×C₄	—	5	1	no	—
250	C₂²×C₄	—	5	1	no	—
251	C₂²×C₄	—	5	1	no	—
252	C₂²×C₄	—	10	2	no	—
253	C₂²×C₄	—	30	2	no	—
254	C₂×C₈	—	15	4	no	—
255	C₂×C₈	—	30	2	no	—
256	C₂×C₈	—	30	2	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	5	2	no	—
259	C₄⋊C₄	—	10	2	no	—
260	C₄⋊C₄	—	10	2	no	—
261	C₄⋊C₄	—	10	1	no	—
262	C₄⋊C₄	—	30	4	no	—
263	C₄⋊C₄	C₂×ΓL(1,𝔽₁₆)	1	2	yes	—
264	C₄²	—	5	1	no	—
265	C₄²	—	5	1	no	—
266	C₄²	—	5	1	no	—
267	C₄²	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
268	C₃×C₅	#1993	1	1	yes	characteristic, complemented, Hall
269	C₂²×C₃	—	5	2	no	—
270	C₂²×C₃	—	5	2	no	—
271	C₂²×C₃	—	5	1	no	—
272	C₂²×C₃	—	5	1	no	—
273	C₂²×C₃	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
274	C₃×C₄	—	2	1	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	4	2	no	—
277	C₃×C₄	—	5	2	no	—
278	C₃×C₄	—	10	1	no	—
279	C₃×C₄	—	10	1	no	—
280	C₃×C₄	—	20	2	no	—
281	C₃×C₄	Q₈×AGL(1,𝔽₅)	1	2	yes	—
282	Dic₃	—	40	2	no	—
283	C₂×C₅	#5215	1	1	yes	characteristic
284	C₂×C₅	#5225	1	1	yes	characteristic
285	C₂×C₅	#5229	1	1	yes	characteristic
286	D₅	#5081	1	2	yes	—
287	D₅	#5089	1	2	yes	—
288	C₂³	—	5	1	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	20	2	no	—
291	Q₈	—	20	4	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	5	1	no	—
294	C₂×C₄	—	5	2	no	—
295	C₂×C₄	—	5	2	no	—
296	C₂×C₄	—	5	1	no	—
297	C₂×C₄	—	5	1	no	—
298	C₂×C₄	—	10	2	no	—
299	C₂×C₄	—	10	2	no	—
300	C₂×C₄	—	10	2	no	—
301	C₂×C₄	—	20	4	no	—
302	C₂×C₄	—	60	2	no	—
303	C₂×C₄	—	60	2	no	—
304	C₂×C₄	—	60	2	no	—
305	C₂×C₄	#6763	1	1	yes	characteristic
306	C₂×C₄	#6763	1	1	yes	characteristic
307	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
308	C₈	—	30	4	no	—
309	C₂×C₃	—	5	2	no	—
310	C₂×C₃	—	5	2	no	—
311	C₂×C₃	#65767	1	1	yes	characteristic
312	C₂×C₃	#65782	1	1	yes	characteristic
313	C₂×C₃	(C₂×AGL(1,𝔽₅))⋊Q₈	1	1	yes	characteristic
314	C₅	#71788	1	1	yes	characteristic, complemented, Sylow
315	C₂²	—	5	1	no	—
316	C₂²	—	5	1	no	—
317	C₂²	—	5	2	no	—
318	C₂²	—	5	2	no	—
319	C₂²	C₂×#6763	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	4	2	no	—
323	C₄	—	5	2	no	—
324	C₄	—	10	1	no	—
325	C₄	—	10	1	no	—
326	C₄	—	20	2	no	—
327	C₄	—	120	2	no	—
328	C₄	Q₈×ΓL(1,𝔽₁₆)	1	2	yes	—
329	C₃	#10604675	1	1	yes	characteristic, complemented, Sylow
330	C₂	—	5	2	no	—
331	C₂	—	5	2	no	—
332	C₂	#11752426	1	1	yes	characteristic
333	C₂	#11752441	1	1	yes	characteristic
334	C₂	#11752449	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6763
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)
Fitting	C₃×C₅×C₄⋊Q₈	C₄
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₃×D₅	C₂×D₄
cosocle	C₂×C₃×C₄×D₅	C₂³
hypercenter	C₄⋊Q₈	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#1993
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
9. C₃×C₅×C₄⋊Q₈	C₄	semidirect
14. C₃×#4034	C₄	semidirect
16. C₃×#4015	C₄	semidirect
38. D₅⋊(C₃⋊C₈)	Q₈	semidirect
56. C₅×C₄⋊Q₈	Dic₃	semidirect
61. #4034	Dic₃	semidirect
63. #4015	Dic₃	semidirect
95. C₃×C₄⋊Q₈	AGL(1,𝔽₅)	semidirect
196. C₄⋊Q₈	ΓL(1,𝔽₁₆)	semidirect
268. C₃×C₅	#1993	semidirect
314. C₅	#71788	semidirect
329. C₃	#10604675	semidirect

Maximal subgroups

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

subgroup	index
C₃×D₅×C₄⋊Q₈	2
#11752348	2
#11752335	2
#11746745	2
#10604675	3
#71788	5