Subgroups — #14305779

Order 1920.

664 conjugacy classes of subgroups.

The subgroup lattice has 296 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	C₂×#91238	C₂	1	4	yes	maximal
3	#11752347	C₂	1	1	yes	maximal, characteristic
4	C₄×D₅⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	#10604672	—	3	1	no	maximal, Hall
6	C₂×C₃×Q₈×D₅	C₂²	1	2	yes	—
7	C₂×C₄⋊ΓL(1,𝔽₁₆)	—	2	2	no	—
8	C₂×C₄⋊ΓL(1,𝔽₁₆)	C₂²	1	2	yes	—
9	C₂×D₅⋊(C₃⋊C₈)	C₂²	1	2	yes	—
10	C₃×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
11	C₃×#4090	C₄	1	2	yes	complemented
12	C₃×D₅×C₄⋊C₄	—	2	2	no	—
13	C₃×#4034	—	2	2	no	—
14	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
15	C₃×#4015	C₄	1	1	yes	characteristic, complemented
16	#91238	—	2	8	no	—
17	C₄×C₃⋊#170	—	2	2	no	—
18	#71785	—	5	1	no	maximal, Hall
19	D₅×C₄⋊Q₈	D₃	1	1	yes	characteristic
20	C₂×#4009	—	3	4	no	—
21	#65688	—	3	1	no	—
22	C₄×D₅⋊C₈	—	3	1	no	—
23	C₂²×ΓL(1,𝔽₁₆)	—	2	2	no	—
24	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
25	C₃×Q₈×D₅	—	2	4	no	—
26	C₃×Q₈×D₅	—	2	4	no	—
27	C₂×C₃×C₄×D₅	—	2	2	no	—
28	C₂×C₃×C₄×D₅	D₄	1	2	yes	—
29	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
30	C₂×C₃×Dic₁₀	—	2	2	no	—
31	C₂×C₃×Dic₁₀	C₂×C₄	1	2	yes	—
32	C₂×C₃⋊#170	—	2	2	no	—
33	C₄⋊ΓL(1,𝔽₁₆)	—	2	4	no	—
34	C₄⋊ΓL(1,𝔽₁₆)	—	4	4	no	—
35	D₅⋊(C₃⋊C₈)	—	2	2	no	—
36	D₅⋊(C₃⋊C₈)	D₄	1	4	yes	—
37	C₃×C₅×C₄⋊C₄	—	2	2	no	—
38	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
39	C₃×C₄⋊Dic₅	—	2	2	no	—
40	C₃×#683	—	2	4	no	—
41	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
42	C₃×C₄×Dic₅	D₄	1	2	yes	—
43	C₂×C₃×C₄⋊Q₈	—	5	1	no	—
44	C₂×#835	—	5	4	no	—
45	#4935	—	5	1	no	—
46	C₂×C₄×C₃⋊C₈	—	5	1	no	—
47	C₂×Q₈×D₅	C₂×D₃	1	2	yes	—
48	C₂×C₄⋊AGL(1,𝔽₅)	—	3	2	no	—
49	C₂×C₄⋊AGL(1,𝔽₅)	—	6	2	no	—
50	C₂×D₅⋊C₈	—	3	2	no	—
51	C₅×C₄⋊Q₈	Dic₃	1	1	yes	characteristic, complemented
52	#4090	Dic₃	1	2	yes	complemented
53	D₅×C₄⋊C₄	—	2	2	no	—
54	#4034	—	2	2	no	—
55	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
56	#4015	Dic₃	1	1	yes	characteristic, complemented
57	#4009	—	6	8	no	—
58	C₄×#170	—	6	2	no	—
59	#1956	—	15	1	no	Sylow
60	C₂²×C₃×D₅	C₂×D₄	1	1	yes	characteristic
61	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
62	C₂×ΓL(1,𝔽₁₆)	—	4	4	no	—
63	C₃×C₅×Q₈	—	2	4	no	—
64	C₂×C₃×C₄×C₅	—	2	2	no	—
65	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
66	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₃×Dic₅	—	2	2	no	—
68	C₂×C₃×Dic₅	C₂²⋊C₄	1	2	yes	—
69	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
70	C₃×C₄×D₅	—	4	2	no	—
71	C₃×C₄×D₅	—	4	2	no	—
72	C₃×C₄×D₅	SD₁₆	1	4	yes	—
73	C₃×C₄×D₅	Q₁₆	1	4	yes	—
74	C₃×C₄×D₅	C₂×D₄	1	2	yes	—
75	C₃×C₄×D₅	#41	1	2	yes	—
76	C₃×Dic₁₀	—	2	4	no	—
77	C₃×Dic₁₀	—	4	4	no	—
78	C₃⋊#170	—	2	4	no	—
79	C₂²×C₃×Q₈	—	5	2	no	—
80	C₃×C₄⋊Q₈	—	5	2	no	—
81	C₃×C₄⋊Q₈	—	5	1	no	—
82	C₃×C₄⋊Q₈	—	10	2	no	—
83	C₃×C₄⋊Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
84	C₂×C₃×C₄⋊C₄	—	10	2	no	—
85	C₂×C₃×C₄²	—	5	1	no	—
86	C₂×C₄⋊Dic₃	—	5	2	no	—
87	C₂×C₄⋊Dic₃	—	10	2	no	—
88	C₂²×C₃⋊C₈	—	5	2	no	—
89	#835	—	10	8	no	—
90	C₄×C₃⋊C₈	—	10	2	no	—
91	C₂²×AGL(1,𝔽₅)	—	6	2	no	—
92	C₂×C₅×Q₈	C₂×Dic₃	1	2	yes	—
93	Q₈×D₅	—	2	4	no	—
94	Q₈×D₅	—	2	4	no	—
95	C₂×C₄×D₅	—	2	2	no	—
96	C₂×C₄×D₅	#67	1	2	yes	—
97	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
98	C₂×Dic₁₀	—	2	2	no	—
99	C₂×Dic₁₀	C₂×Dic₃	1	2	yes	—
100	C₂×#170	—	6	2	no	—
101	C₄⋊AGL(1,𝔽₅)	—	6	4	no	—
102	C₄⋊AGL(1,𝔽₅)	—	12	4	no	—
103	D₅⋊C₈	—	3	4	no	—
104	D₅⋊C₈	—	6	2	no	—
105	C₅×C₄⋊C₄	—	2	2	no	—
106	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
107	C₄⋊Dic₅	—	2	2	no	—
108	#683	—	2	4	no	—
109	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
110	C₄×Dic₅	#67	1	2	yes	—
111	C₂×C₄⋊Q₈	—	5	1	no	—
112	C₂×#103	—	15	4	no	—
113	C₂×C₄×C₈	—	15	1	no	—
114	#384	—	15	1	no	—
115	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₃×D₅	#124	1	1	yes	characteristic
117	C₂×C₃×D₅	#127	1	1	yes	characteristic
118	C₂×C₃×D₅	C₂×SD₁₆	1	2	yes	—
119	C₂×C₃×D₅	C₂×Q₁₆	1	2	yes	—
120	ΓL(1,𝔽₁₆)	—	8	2	no	—
121	C₃×C₄×C₅	—	4	2	no	—
122	C₃×C₄×C₅	#103	1	4	yes	complemented
123	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
124	C₃×Dic₅	—	4	2	no	—
125	C₃×Dic₅	#103	1	4	yes	complemented
126	C₃×Dic₅	#121	1	2	yes	—
127	C₂×C₃×Q₈	—	5	2	no	—
128	C₂×C₃×Q₈	—	10	2	no	—
129	C₂×C₃×Q₈	—	10	4	no	—
130	C₂×C₃×Q₈	—	10	4	no	—
131	C₂×C₃×Q₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
132	C₂²×C₃×C₄	—	5	2	no	—
133	C₂²×C₃×C₄	—	5	1	no	—
134	C₂²×C₃×C₄	—	10	2	no	—
135	C₂²×Dic₃	—	10	2	no	—
136	C₃×C₄⋊C₄	—	2	2	no	—
137	C₃×C₄⋊C₄	—	10	4	no	—
138	C₃×C₄⋊C₄	—	10	2	no	—
139	C₃×C₄²	—	5	2	no	—
140	C₃×C₄²	—	5	1	no	—
141	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
142	C₄⋊Dic₃	—	10	4	no	—
143	C₄⋊Dic₃	—	20	4	no	—
144	C₂×C₃⋊C₈	—	5	4	no	—
145	C₂×C₃⋊C₈	—	10	2	no	—
146	C₂×C₃⋊C₈	—	10	2	no	—
147	C₂²×D₅	C₂×#67	1	1	yes	characteristic
148	C₂×AGL(1,𝔽₅)	—	12	2	no	—
149	C₂×AGL(1,𝔽₅)	—	12	4	no	—
150	C₅×Q₈	—	2	4	no	—
151	C₂×C₄×C₅	—	2	2	no	—
152	C₂×C₄×C₅	#217	1	2	yes	—
153	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
154	C₂×Dic₅	—	2	2	no	—
155	C₂×Dic₅	#217	1	2	yes	—
156	C₂×Dic₅	C₂×#67	1	1	yes	characteristic
157	C₄×D₅	—	4	2	no	—
158	C₄×D₅	—	4	2	no	—
159	C₄×D₅	#215	1	4	yes	—
160	C₄×D₅	#216	1	4	yes	—
161	C₄×D₅	D₃×D₄	1	2	yes	—
162	C₄×D₅	#237	1	2	yes	—
163	Dic₁₀	—	2	4	no	—
164	Dic₁₀	—	4	4	no	—
165	#170	—	6	4	no	—
166	C₂²×Q₈	—	5	2	no	—
167	C₂²×C₈	—	15	2	no	—
168	C₄⋊Q₈	—	5	2	no	—
169	C₄⋊Q₈	—	5	1	no	—
170	C₄⋊Q₈	—	10	2	no	—
171	C₄⋊Q₈	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
172	C₂×C₄⋊C₄	—	10	2	no	—
173	C₂×C₄⋊C₄	—	15	2	no	—
174	C₂×C₄⋊C₄	—	30	2	no	—
175	C₂×C₄²	—	5	1	no	—
176	#103	—	30	8	no	—
177	C₄×C₈	—	30	2	no	—
178	C₂×C₃×C₅	#390	1	1	yes	characteristic
179	C₂×C₃×C₅	C₂×#103	1	2	yes	—
180	C₃×D₅	#487	1	2	yes	—
181	C₃×D₅	#492	1	1	yes	characteristic
182	C₃×D₅	#494	1	1	yes	characteristic
183	C₂³×C₃	—	5	1	no	—
184	C₃×Q₈	—	2	4	no	—
185	C₃×Q₈	—	10	4	no	—
186	C₃×Q₈	—	20	4	no	—
187	C₂×C₃×C₄	—	2	2	no	—
188	C₂×C₃×C₄	—	5	2	no	—
189	C₂×C₃×C₄	—	5	1	no	—
190	C₂×C₃×C₄	—	5	2	no	—
191	C₂×C₃×C₄	—	5	2	no	—
192	C₂×C₃×C₄	—	5	4	no	—
193	C₂×C₃×C₄	—	5	4	no	—
194	C₂×C₃×C₄	—	10	2	no	—
195	C₂×C₃×C₄	—	20	2	no	—
196	C₂×C₃×C₄	—	20	2	no	—
197	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
198	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
199	C₂×Dic₃	—	20	2	no	—
200	C₂×Dic₃	—	20	4	no	—
201	C₃⋊C₈	—	10	4	no	—
202	C₂²×C₅	C₂×#217	1	1	yes	characteristic
203	C₂×D₅	#936	1	1	yes	characteristic
204	C₂×D₅	#940	1	1	yes	characteristic
205	C₂×D₅	C₂×#215	1	2	yes	—
206	C₂×D₅	C₂×#216	1	2	yes	—
207	AGL(1,𝔽₅)	—	24	2	no	—
208	C₄×C₅	—	4	2	no	—
209	C₄×C₅	#835	1	4	yes	complemented
210	C₄×C₅	D₄×Dic₃	1	2	yes	—
211	Dic₅	—	4	2	no	—
212	Dic₅	#835	1	4	yes	complemented
213	Dic₅	#938	1	2	yes	—
214	C₂×Q₈	—	5	2	no	—
215	C₂×Q₈	—	10	2	no	—
216	C₂×Q₈	—	10	4	no	—
217	C₂×Q₈	—	10	4	no	—
218	C₂×Q₈	C₂×ΓL(1,𝔽₁₆)	1	2	yes	—
219	C₂²×C₄	—	5	2	no	—
220	C₂²×C₄	—	5	1	no	—
221	C₂²×C₄	—	10	2	no	—
222	C₂²×C₄	—	30	2	no	—
223	C₂×C₈	—	15	4	no	—
224	C₂×C₈	—	30	2	no	—
225	C₂×C₈	—	30	2	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	10	4	no	—
228	C₄⋊C₄	—	10	2	no	—
229	C₄⋊C₄	—	30	4	no	—
230	C₄⋊C₄	—	60	4	no	—
231	C₄²	—	5	2	no	—
232	C₄²	—	5	1	no	—
233	C₄²	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
234	C₃×C₅	#1956	1	1	yes	characteristic, complemented, Hall
235	C₂²×C₃	—	5	1	no	—
236	C₂²×C₃	—	5	1	no	—
237	C₂²×C₃	—	5	2	no	—
238	C₂²×C₃	—	5	2	no	—
239	C₂²×C₃	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
240	C₃×C₄	—	4	2	no	—
241	C₃×C₄	—	5	2	no	—
242	C₃×C₄	—	5	4	no	—
243	C₃×C₄	—	20	2	no	—
244	C₃×C₄	#4009	1	4	yes	complemented
245	C₃×C₄	D₄×AGL(1,𝔽₅)	1	2	yes	—
246	Dic₃	—	40	2	no	—
247	C₂×C₅	#5221	1	1	yes	characteristic
248	C₂×C₅	C₂×#835	1	2	yes	—
249	D₅	#5079	1	2	yes	—
250	D₅	#5085	1	1	yes	characteristic
251	D₅	#5092	1	1	yes	characteristic
252	C₂³	—	5	1	no	—
253	Q₈	—	2	4	no	—
254	Q₈	—	10	4	no	—
255	Q₈	—	20	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	5	2	no	—
258	C₂×C₄	—	5	1	no	—
259	C₂×C₄	—	5	2	no	—
260	C₂×C₄	—	5	2	no	—
261	C₂×C₄	—	5	4	no	—
262	C₂×C₄	—	5	4	no	—
263	C₂×C₄	—	10	2	no	—
264	C₂×C₄	—	20	2	no	—
265	C₂×C₄	—	20	2	no	—
266	C₂×C₄	—	60	2	no	—
267	C₂×C₄	—	60	4	no	—
268	C₂×C₄	#6763	1	2	yes	—
269	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
270	C₈	—	30	4	no	—
271	C₂×C₃	—	5	1	no	—
272	C₂×C₃	—	5	2	no	—
273	C₂×C₃	—	5	1	no	—
274	C₂×C₃	#65780	1	1	yes	characteristic
275	C₂×C₃	C₂×#4009	1	2	yes	—
276	C₅	#71785	1	1	yes	characteristic, complemented, Sylow
277	C₂²	—	5	1	no	—
278	C₂²	—	5	1	no	—
279	C₂²	—	5	2	no	—
280	C₂²	—	5	2	no	—
281	C₂²	C₂×#6763	1	1	yes	characteristic
282	C₄	—	4	2	no	—
283	C₄	—	5	2	no	—
284	C₄	—	5	4	no	—
285	C₄	—	20	2	no	—
286	C₄	—	120	2	no	—
287	C₄	#91238	1	4	yes	complemented
288	C₄	D₄×ΓL(1,𝔽₁₆)	1	2	yes	—
289	C₃	#10604672	1	1	yes	characteristic, complemented, Sylow
290	C₂	—	5	1	no	—
291	C₂	—	5	2	no	—
292	C₂	—	5	1	no	—
293	C₂	#11752439	1	1	yes	characteristic
294	C₂	C₂×#91238	1	2	yes	—

	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	#1956
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
10. C₃×C₅×C₄⋊Q₈	C₄	semidirect
11. C₃×#4090	C₄	semidirect
15. C₃×#4015	C₄	semidirect
51. C₅×C₄⋊Q₈	Dic₃	semidirect
52. #4090	Dic₃	semidirect
56. #4015	Dic₃	semidirect
83. C₃×C₄⋊Q₈	AGL(1,𝔽₅)	semidirect
122. C₃×C₄×C₅	#103	semidirect
125. C₃×Dic₅	#103	semidirect
171. C₄⋊Q₈	ΓL(1,𝔽₁₆)	semidirect
209. C₄×C₅	#835	semidirect
212. Dic₅	#835	semidirect
234. C₃×C₅	#1956	semidirect
244. C₃×C₄	#4009	semidirect
276. C₅	#71785	semidirect
287. C₄	#91238	semidirect
289. C₃	#10604672	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
C₂×#91238	2
#11752347	2
C₄×D₅⋊(C₃⋊C₈)	2
#10604672	3
#71785	5