Subgroups — #14282932

Order 1920.

480 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#66062	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊ΓL(1,𝔽₁₆)	C₂	1	1	yes	maximal, characteristic
3	#11752339	C₂	1	1	yes	maximal, characteristic
4	#11746797	C₂	1	4	yes	maximal
5	#10604623	—	3	1	no	maximal, Hall
6	C₂×C₄⋊ΓL(1,𝔽₁₆)	—	2	1	no	—
7	C₂×C₄×ΓL(1,𝔽₁₆)	—	2	1	no	—
8	C₃×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
9	C₃×#4092	C₂²	1	2	yes	—
10	C₃×Q₈×Dic₅	C₄	1	2	yes	complemented
11	C₃×#4043	—	2	1	no	—
12	C₃×#4038	—	2	1	no	—
13	C₃×#4036	—	2	1	no	—
14	C₃×#4033	—	2	1	no	—
15	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
16	C₃×#4015	C₄	1	1	yes	characteristic, complemented
17	#91235	C₂²	1	2	yes	—
18	#91232	C₂²	1	2	yes	—
19	C₄⋊(C₃⋊#170)	—	2	1	no	—
20	C₄×C₃⋊#170	—	2	1	no	—
21	#71835	—	5	1	no	maximal, Hall
22	#66062	D₃	1	1	yes	characteristic
23	C₄×C₄⋊AGL(1,𝔽₅)	—	3	1	no	—
24	#65680	—	3	1	no	—
25	#64926	—	3	4	no	—
26	C₂²×ΓL(1,𝔽₁₆)	—	2	2	no	—
27	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
29	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
30	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
31	C₂×C₃×Dic₁₀	—	2	1	no	—
32	C₂×C₃⋊#170	—	2	2	no	—
33	C₄⋊ΓL(1,𝔽₁₆)	—	2	4	no	—
34	C₄×ΓL(1,𝔽₁₆)	—	4	2	no	—
35	C₃×C₅×C₄⋊C₄	—	2	1	no	—
36	C₃×C₅×C₄⋊C₄	—	2	1	no	—
37	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
38	C₃×#685	—	2	2	no	—
39	C₃×C₄⋊Dic₅	—	2	1	no	—
40	C₃×C₄⋊Dic₅	—	2	2	no	—
41	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
42	C₃×#683	—	2	2	no	—
43	C₃×C₄×Dic₅	—	2	2	no	—
44	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
45	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
46	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
47	C₃×#533	—	5	1	no	—
48	C₄×C₄⋊Dic₃	—	5	1	no	—
49	#4925	—	5	1	no	—
50	#4538	—	5	4	no	—
51	C₂×C₄⋊AGL(1,𝔽₅)	—	6	1	no	—
52	C₂×C₄×AGL(1,𝔽₅)	—	6	1	no	—
53	C₅×C₄⋊Q₈	Dic₃	1	1	yes	characteristic, complemented
54	#4092	C₂×D₃	1	2	yes	—
55	Q₈×Dic₅	Dic₃	1	2	yes	complemented
56	#4043	—	2	1	no	—
57	#4038	—	2	1	no	—
58	#4036	—	2	1	no	—
59	#4033	—	2	1	no	—
60	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
61	#4015	Dic₃	1	1	yes	characteristic, complemented
62	(C₂×AGL(1,𝔽₅))⋊C₄	—	3	2	no	—
63	#4003	—	3	2	no	—
64	C₄⋊#170	—	6	1	no	—
65	C₄×#170	—	6	1	no	—
66	#1503	—	15	1	no	Sylow
67	C₂²×C₃×D₅	C₂×D₄	1	1	yes	characteristic
68	C₂×ΓL(1,𝔽₁₆)	—	2	4	no	—
69	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
70	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
71	C₃×C₅×Q₈	—	4	2	no	—
72	C₂×C₃×C₄×C₅	—	2	2	no	—
73	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₃×Dic₅	—	2	2	no	—
77	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
80	C₃×C₄×D₅	—	2	2	no	—
81	C₃×C₄×D₅	—	4	1	no	—
82	C₃×C₄×D₅	SD₁₆	1	2	yes	—
83	C₃×C₄×D₅	Q₁₆	1	2	yes	—
84	C₃×Dic₁₀	—	4	2	no	—
85	C₃⋊#170	—	4	2	no	—
86	C₃×C₄⋊Q₈	—	5	1	no	—
87	C₃×C₄⋊Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
88	C₃×#125	—	10	1	no	—
89	C₃×C₂²⋊Q₈	—	5	2	no	—
90	C₃×C₂²⋊Q₈	—	10	1	no	—
91	C₃×C₄×Q₈	—	5	2	no	—
92	C₃×C₄×Q₈	—	10	1	no	—
93	C₃×#117	—	10	1	no	—
94	C₂×C₃×C₄²	—	5	1	no	—
95	C₂×C₄⋊Dic₃	—	10	1	no	—
96	C₂×C₄×Dic₃	—	10	1	no	—
97	C₃⋊#95	—	5	2	no	—
98	#830	—	5	2	no	—
99	C₄⋊(C₃⋊C₈)	—	10	1	no	—
100	C₄×C₃⋊C₈	—	10	1	no	—
101	C₂²×AGL(1,𝔽₅)	—	6	2	no	—
102	C₂×C₅×Q₈	C₂×Dic₃	1	2	yes	—
103	C₂×C₄×D₅	#67	1	1	yes	characteristic
104	C₂×C₄×D₅	#67	1	1	yes	characteristic
105	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
106	C₂×Dic₁₀	—	2	1	no	—
107	C₂×#170	—	6	2	no	—
108	C₄⋊AGL(1,𝔽₅)	—	6	4	no	—
109	C₄×AGL(1,𝔽₅)	—	12	2	no	—
110	C₅×C₄⋊C₄	—	2	1	no	—
111	C₅×C₄⋊C₄	—	2	1	no	—
112	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
113	#685	—	2	2	no	—
114	C₄⋊Dic₅	—	2	1	no	—
115	C₄⋊Dic₅	—	2	2	no	—
116	C₄⋊Dic₅	C₂×Dic₃	1	2	yes	—
117	#683	—	2	2	no	—
118	C₄×Dic₅	—	2	2	no	—
119	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
120	C₄×Dic₅	#67	1	1	yes	characteristic
121	C₄×Dic₅	#67	1	1	yes	characteristic
122	#533	—	5	1	no	—
123	#431	—	15	1	no	—
124	C₄×C₄⋊C₄	—	15	1	no	—
125	#328	—	15	4	no	—
126	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₃×D₅	—	2	1	no	—
128	C₂×C₃×D₅	—	2	1	no	—
129	C₂×C₃×D₅	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₃×D₅	C₂×Q₁₆	1	1	yes	characteristic
131	ΓL(1,𝔽₁₆)	—	4	4	no	—
132	C₃×C₄×C₅	—	2	1	no	—
133	C₃×C₄×C₅	—	4	2	no	—
134	C₃×C₄×C₅	#103	1	2	yes	—
135	C₃×C₄×C₅	#104	1	2	yes	—
136	C₃×Dic₅	—	2	1	no	—
137	C₃×Dic₅	—	2	1	no	—
138	C₃×Dic₅	—	4	2	no	—
139	C₃×Dic₅	#103	1	2	yes	—
140	C₂×C₃×Q₈	—	10	1	no	—
141	C₂×C₃×Q₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
142	C₂²×C₃×C₄	—	5	1	no	—
143	C₂²×C₃×C₄	—	5	1	no	—
144	C₂²×C₃×C₄	—	5	1	no	—
145	C₂²×Dic₃	—	10	2	no	—
146	C₃×C₄⋊C₄	—	2	1	no	—
147	C₃×C₄⋊C₄	—	2	1	no	—
148	C₃×C₄⋊C₄	—	5	2	no	—
149	C₃×C₄⋊C₄	—	10	2	no	—
150	C₃×C₄⋊C₄	—	10	2	no	—
151	C₃×C₄⋊C₄	—	10	1	no	—
152	C₃×C₂²⋊C₄	—	10	2	no	—
153	C₃×C₄²	—	5	1	no	—
154	C₃×C₄²	—	5	1	no	—
155	C₃×C₄²	—	5	1	no	—
156	C₃×C₄²	—	10	2	no	—
157	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
158	C₄⋊Dic₃	—	10	4	no	—
159	C₄×Dic₃	—	20	2	no	—
160	C₂×C₃⋊C₈	—	10	2	no	—
161	C₂²×D₅	C₂×#67	1	1	yes	characteristic
162	C₂×AGL(1,𝔽₅)	—	6	4	no	—
163	C₂×AGL(1,𝔽₅)	—	12	2	no	—
164	C₂×AGL(1,𝔽₅)	—	12	2	no	—
165	C₅×Q₈	—	4	2	no	—
166	C₂×C₄×C₅	—	2	2	no	—
167	C₂×C₄×C₅	#217	1	1	yes	characteristic
168	C₂×C₄×C₅	#217	1	1	yes	characteristic
169	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
170	C₂×Dic₅	—	2	2	no	—
171	C₂×Dic₅	#217	1	1	yes	characteristic
172	C₂×Dic₅	#217	1	1	yes	characteristic
173	C₂×Dic₅	C₂×#67	1	1	yes	characteristic
174	C₄×D₅	—	2	2	no	—
175	C₄×D₅	—	4	1	no	—
176	C₄×D₅	#215	1	2	yes	—
177	C₄×D₅	#216	1	2	yes	—
178	Dic₁₀	—	4	2	no	—
179	#170	—	12	2	no	—
180	C₄⋊Q₈	—	5	1	no	—
181	C₄⋊Q₈	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
182	#125	—	10	1	no	—
183	C₂²⋊Q₈	—	5	2	no	—
184	C₂²⋊Q₈	—	10	1	no	—
185	C₄×Q₈	—	5	2	no	—
186	C₄×Q₈	—	10	1	no	—
187	#117	—	10	1	no	—
188	C₂×C₄⋊C₄	—	30	1	no	—
189	C₂×C₄²	—	5	1	no	—
190	C₂×C₄²	—	30	1	no	—
191	C₄⋊C₈	—	30	1	no	—
192	C₂²⋊C₈	—	15	2	no	—
193	C₄×C₈	—	30	1	no	—
194	#95	—	15	2	no	—
195	C₂×C₃×C₅	#410	1	1	yes	characteristic
196	C₂×C₃×C₅	C₂×#103	1	1	yes	characteristic
197	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
198	C₃×D₅	—	2	1	no	—
199	C₃×D₅	—	2	1	no	—
200	C₂³×C₃	—	5	1	no	—
201	C₃×Q₈	—	4	2	no	—
202	C₃×Q₈	—	20	2	no	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	5	1	no	—
205	C₂×C₃×C₄	—	5	2	no	—
206	C₂×C₃×C₄	—	5	2	no	—
207	C₂×C₃×C₄	—	5	1	no	—
208	C₂×C₃×C₄	—	5	1	no	—
209	C₂×C₃×C₄	—	10	2	no	—
210	C₂×C₃×C₄	—	10	2	no	—
211	C₂×C₃×C₄	—	20	1	no	—
212	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
215	C₂×Dic₃	—	10	4	no	—
216	C₂×Dic₃	—	20	2	no	—
217	C₂×Dic₃	—	20	2	no	—
218	C₃⋊C₈	—	20	2	no	—
219	C₂²×C₅	C₂×#217	1	1	yes	characteristic
220	C₂×D₅	—	2	1	no	—
221	C₂×D₅	—	2	1	no	—
222	C₂×D₅	C₂×#215	1	1	yes	characteristic
223	C₂×D₅	C₂×#216	1	1	yes	characteristic
224	AGL(1,𝔽₅)	—	12	4	no	—
225	C₄×C₅	—	2	1	no	—
226	C₄×C₅	—	4	2	no	—
227	C₄×C₅	#835	1	2	yes	—
228	C₄×C₅	#837	1	2	yes	—
229	Dic₅	—	2	1	no	—
230	Dic₅	—	2	1	no	—
231	Dic₅	—	4	2	no	—
232	Dic₅	#835	1	2	yes	—
233	C₂×Q₈	—	10	1	no	—
234	C₂×Q₈	C₂×ΓL(1,𝔽₁₆)	1	2	yes	—
235	C₂²×C₄	—	5	1	no	—
236	C₂²×C₄	—	5	1	no	—
237	C₂²×C₄	—	5	1	no	—
238	C₂²×C₄	—	30	2	no	—
239	C₂×C₈	—	30	2	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	5	2	no	—
243	C₄⋊C₄	—	10	2	no	—
244	C₄⋊C₄	—	10	2	no	—
245	C₄⋊C₄	—	10	1	no	—
246	C₄⋊C₄	—	30	4	no	—
247	C₂²⋊C₄	—	10	2	no	—
248	C₄²	—	5	1	no	—
249	C₄²	—	5	1	no	—
250	C₄²	—	5	1	no	—
251	C₄²	—	10	2	no	—
252	C₄²	—	60	2	no	—
253	C₄²	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
254	C₃×C₅	#1503	1	1	yes	characteristic, complemented, Hall
255	C₂²×C₃	—	5	1	no	—
256	C₂²×C₃	—	5	1	no	—
257	C₂²×C₃	—	10	1	no	—
258	C₂²×C₃	—	10	1	no	—
259	C₂²×C₃	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	—	4	2	no	—
262	C₃×C₄	—	5	2	no	—
263	C₃×C₄	—	10	1	no	—
264	C₃×C₄	—	10	1	no	—
265	C₃×C₄	—	20	2	no	—
266	C₃×C₄	#4009	1	2	yes	—
267	C₃×C₄	#4010	1	2	yes	—
268	Dic₃	—	20	4	no	—
269	C₂×C₅	C₂×#835	1	1	yes	characteristic
270	C₂×C₅	C₂×#837	1	1	yes	characteristic
271	C₂×C₅	#5240	1	1	yes	characteristic
272	D₅	—	2	1	no	—
273	D₅	—	2	1	no	—
274	C₂³	—	5	1	no	—
275	Q₈	—	4	2	no	—
276	Q₈	—	20	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	5	1	no	—
279	C₂×C₄	—	5	2	no	—
280	C₂×C₄	—	5	2	no	—
281	C₂×C₄	—	5	1	no	—
282	C₂×C₄	—	5	1	no	—
283	C₂×C₄	—	10	2	no	—
284	C₂×C₄	—	10	2	no	—
285	C₂×C₄	—	20	1	no	—
286	C₂×C₄	—	30	4	no	—
287	C₂×C₄	—	60	2	no	—
288	C₂×C₄	—	60	2	no	—
289	C₂×C₄	#6763	1	1	yes	characteristic
290	C₂×C₄	#6763	1	1	yes	characteristic
291	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
292	C₈	—	60	2	no	—
293	C₂×C₃	—	10	1	no	—
294	C₂×C₃	—	10	1	no	—
295	C₂×C₃	#65771	1	1	yes	characteristic
296	C₂×C₃	C₂×#4009	1	1	yes	characteristic
297	C₂×C₃	C₂×#4010	1	1	yes	characteristic
298	C₅	#71835	1	1	yes	characteristic, complemented, Sylow
299	C₂²	—	5	1	no	—
300	C₂²	—	5	1	no	—
301	C₂²	—	10	1	no	—
302	C₂²	—	10	1	no	—
303	C₂²	C₂×#6763	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	4	2	no	—
306	C₄	—	5	2	no	—
307	C₄	—	10	1	no	—
308	C₄	—	10	1	no	—
309	C₄	—	20	2	no	—
310	C₄	—	60	4	no	—
311	C₄	#91238	1	2	yes	—
312	C₄	#91239	1	2	yes	—
313	C₃	#10604623	1	1	yes	characteristic, complemented, Sylow
314	C₂	—	10	1	no	—
315	C₂	—	10	1	no	—
316	C₂	#11752430	1	1	yes	characteristic
317	C₂	C₂×#91238	1	1	yes	characteristic
318	C₂	C₂×#91239	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	#1503
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×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
8. C₃×C₅×C₄⋊Q₈	C₄	semidirect
10. C₃×Q₈×Dic₅	C₄	semidirect
16. C₃×#4015	C₄	semidirect
53. C₅×C₄⋊Q₈	Dic₃	semidirect
55. Q₈×Dic₅	Dic₃	semidirect
61. #4015	Dic₃	semidirect
87. C₃×C₄⋊Q₈	AGL(1,𝔽₅)	semidirect
181. C₄⋊Q₈	ΓL(1,𝔽₁₆)	semidirect
254. C₃×C₅	#1503	semidirect
298. C₅	#71835	semidirect
313. C₃	#10604623	semidirect

Maximal subgroups

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

subgroup	index
C₃×#66062	2
C₄×C₄⋊ΓL(1,𝔽₁₆)	2
#11752339	2
#11746797	2
#10604623	3
#71835	5