Subgroups — #14413510

Order 1920.

952 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11756995	C₂	1	2	yes	maximal
2	C₃×D₅×#131	C₂	1	1	yes	maximal, characteristic
3	#11752386	C₂	1	6	yes	maximal, complemented
4	#11752380	C₂	1	3	yes	maximal, complemented
5	C₂×#91223	C₂	1	3	yes	maximal, complemented
6	#10617417	—	3	1	no	maximal, Hall
7	C₃×D₅×#41	C₂²	1	1	yes	characteristic
8	Q₈×ΓL(1,𝔽₁₆)	—	2	2	no	—
9	D₄×ΓL(1,𝔽₁₆)	—	2	6	no	—
10	#91989	C₂²	1	6	yes	—
11	C₂×C₄⋊ΓL(1,𝔽₁₆)	C₂²	1	6	yes	—
12	C₃×C₅×#131	C₄	1	1	yes	characteristic, complemented
13	C₃×#4094	C₄	1	1	yes	characteristic, complemented
14	C₃×#4053	C₄	1	3	yes	complemented
15	C₃×D₅×M₁₆	C₂²	1	3	yes	—
16	C₃×#4047	C₄	1	3	yes	complemented
17	C₂×C₃×C₈×D₅	C₂²	1	3	yes	—
18	#91223	C₂²	1	3	yes	—
19	#91223	C₂²	1	6	yes	—
20	#91223	C₂²	1	1	yes	characteristic
21	#91222	C₂²	1	6	yes	—
22	#83785	—	5	1	no	maximal, Hall
23	#66266	—	3	2	no	—
24	D₅×#131	D₃	1	1	yes	characteristic
25	#65727	—	3	6	no	—
26	#65721	—	3	3	no	—
27	C₂×#3994	—	3	3	no	—
28	C₂²×ΓL(1,𝔽₁₆)	—	2	6	no	—
29	C₃×C₅×#41	C₂×C₄	1	1	yes	characteristic
30	C₃×#713	Q₈	1	1	yes	characteristic
31	C₃×Q₈×D₅	D₄	1	1	yes	characteristic
32	C₃×#711	Q₈	1	3	yes	—
33	C₃×D₄×D₅	D₄	1	3	yes	—
34	C₃×#709	C₂×C₄	1	3	yes	—
35	C₂×C₃×C₄×D₅	C₂³	1	3	yes	—
36	#6763	—	2	6	no	—
37	C₄⋊ΓL(1,𝔽₁₆)	—	2	6	no	—
38	C₄⋊ΓL(1,𝔽₁₆)	C₂³	1	6	yes	—
39	C₄⋊ΓL(1,𝔽₁₆)	C₂³	1	2	yes	—
40	C₄×ΓL(1,𝔽₁₆)	—	2	6	no	—
41	C₃×C₅×M₁₆	C₂×C₄	1	3	yes	complemented
42	C₂×C₃×C₅×C₈	C₂×C₄	1	3	yes	complemented
43	C₃×#681	C₂×C₄	1	3	yes	complemented
44	C₂×C₃×#168	C₂×C₄	1	3	yes	complemented
45	C₃×#676	C₂×C₄	1	6	yes	complemented
46	C₃×C₈×D₅	C₂×C₄	1	3	yes	complemented
47	C₃×C₈×D₅	C₂×C₄	1	3	yes	complemented
48	C₃×C₈×D₅	C₂³	1	3	yes	—
49	C₃×C₈×D₅	C₂³	1	1	yes	characteristic
50	C₂×C₃×#131	—	5	1	no	—
51	#5827	—	5	2	no	—
52	#5121	—	5	6	no	—
53	#5106	—	5	3	no	—
54	C₂×#818	—	5	3	no	—
55	D₅×#41	C₂×D₃	1	1	yes	characteristic
56	Q₈×AGL(1,𝔽₅)	—	6	2	no	—
57	D₄×AGL(1,𝔽₅)	—	6	6	no	—
58	#4130	—	3	6	no	—
59	C₂×C₄⋊AGL(1,𝔽₅)	—	3	6	no	—
60	C₅×#131	Dic₃	1	1	yes	characteristic, complemented
61	#4094	Dic₃	1	1	yes	characteristic, complemented
62	#4053	Dic₃	1	3	yes	complemented
63	D₅×M₁₆	C₂×D₃	1	3	yes	—
64	#4047	Dic₃	1	3	yes	complemented
65	C₂×C₈×D₅	C₂×D₃	1	3	yes	—
66	#3994	—	3	3	no	—
67	#3994	—	3	6	no	—
68	#3994	—	3	1	no	—
69	#3993	—	3	6	no	—
70	#2913	—	15	1	no	Sylow
71	C₂²×C₃×D₅	C₂×D₄	1	3	yes	—
72	C₂×ΓL(1,𝔽₁₆)	—	2	2	no	—
73	C₂×ΓL(1,𝔽₁₆)	—	2	6	no	—
74	C₂×ΓL(1,𝔽₁₆)	—	8	6	no	—
75	C₃×C₅×Q₈	C₄⋊C₄	1	1	yes	characteristic
76	C₃×C₅×D₄	C₄⋊C₄	1	3	yes	—
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	3	yes	—
78	C₃×C₂²⋊D₅	C₄⋊C₄	1	6	yes	—
79	C₂×C₃×Dic₅	C₂×Q₈	1	3	yes	—
80	C₃×D₂₀	C₄⋊C₄	1	3	yes	—
81	C₃×C₄×D₅	C₂²×C₄	1	3	yes	—
82	C₃×C₄×D₅	C₂×D₄	1	3	yes	—
83	C₃×C₄×D₅	C₂×Q₈	1	3	yes	—
84	C₃×C₄×D₅	C₂⁴	1	1	yes	characteristic
85	C₃×Dic₁₀	C₄⋊C₄	1	3	yes	—
86	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₃×C₅×C₈	C₂²×C₄	1	3	yes	—
88	C₃×#168	C₂²×C₄	1	1	yes	characteristic
89	C₃×#168	C₂²×C₄	1	3	yes	—
90	C₂×C₃×#41	—	5	1	no	—
91	C₃×#131	—	5	1	no	—
92	C₃×#131	—	5	3	no	—
93	C₃×#131	—	5	3	no	—
94	C₃×#131	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
95	C₂×C₃×M₁₆	—	5	3	no	—
96	C₂²×C₃×C₈	—	5	3	no	—
97	Q₈×Dic₃	—	10	2	no	—
98	D₄×Dic₃	—	10	6	no	—
99	#926	—	5	6	no	—
100	C₂×C₄⋊Dic₃	—	5	6	no	—
101	#818	—	5	6	no	—
102	#818	—	5	3	no	—
103	#818	—	5	1	no	—
104	#817	—	5	6	no	—
105	C₂²×AGL(1,𝔽₅)	—	6	6	no	—
106	C₅×#41	C₂×Dic₃	1	1	yes	characteristic
107	#713	Dic₆	1	1	yes	characteristic
108	Q₈×D₅	D₁₂	1	1	yes	characteristic
109	#711	Dic₆	1	3	yes	—
110	D₄×D₅	D₁₂	1	3	yes	—
111	#709	C₂×Dic₃	1	3	yes	—
112	C₂×C₄×D₅	C₂²×D₃	1	3	yes	—
113	C₂²⋊AGL(1,𝔽₅)	—	6	6	no	—
114	C₄⋊AGL(1,𝔽₅)	—	3	6	no	—
115	C₄⋊AGL(1,𝔽₅)	—	3	2	no	—
116	C₄⋊AGL(1,𝔽₅)	—	6	6	no	—
117	C₄×AGL(1,𝔽₅)	—	6	6	no	—
118	C₅×M₁₆	C₂×Dic₃	1	3	yes	complemented
119	C₂×C₅×C₈	C₂×Dic₃	1	3	yes	complemented
120	#681	C₂×Dic₃	1	3	yes	complemented
121	C₂×#168	C₂×Dic₃	1	3	yes	complemented
122	#676	C₂×Dic₃	1	6	yes	complemented
123	C₈×D₅	C₂×Dic₃	1	3	yes	complemented
124	C₈×D₅	C₂×Dic₃	1	3	yes	complemented
125	C₈×D₅	C₂²×D₃	1	3	yes	—
126	C₈×D₅	C₂²×D₃	1	1	yes	characteristic
127	C₂×#131	—	5	1	no	—
128	#520	—	15	2	no	—
129	#428	—	15	6	no	—
130	#427	—	15	3	no	—
131	C₂×#107	—	15	3	no	—
132	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	3	yes	—
133	C₂×C₃×D₅	—	2	3	no	—
134	C₂×C₃×D₅	—	2	3	no	—
135	C₂×C₃×D₅	C₂×C₄⋊C₄	1	3	yes	—
136	C₂×C₃×D₅	C₂²×D₄	1	1	yes	characteristic
137	ΓL(1,𝔽₁₆)	—	4	6	no	—
138	ΓL(1,𝔽₁₆)	—	4	2	no	—
139	C₃×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
140	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
141	C₃×Dic₅	C₂×C₄⋊C₄	1	3	yes	—
142	C₃×Dic₅	C₂²×Q₈	1	1	yes	characteristic
143	C₃×#41	—	5	1	no	—
144	C₃×#41	—	5	3	no	—
145	C₃×#41	—	5	3	no	—
146	C₃×#41	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
147	C₂×C₃×Q₈	—	5	1	no	—
148	C₂×C₃×D₄	—	5	3	no	—
149	C₂²×C₃×C₄	—	5	3	no	—
150	C₂²×Dic₃	—	10	6	no	—
151	C₃×M₁₆	—	5	3	no	—
152	C₃×M₁₆	—	5	6	no	—
153	C₃×M₁₆	C₂×AGL(1,𝔽₅)	1	3	yes	complemented
154	C₂×C₃×C₈	—	5	3	no	—
155	C₂×C₃×C₈	—	5	3	no	—
156	C₂×C₃×C₈	—	5	3	no	—
157	C₂×C₃×C₈	—	5	3	no	—
158	C₂×C₃×C₈	—	5	1	no	—
159	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	3	yes	complemented
160	#217	—	10	6	no	—
161	C₄⋊Dic₃	—	5	2	no	—
162	C₄⋊Dic₃	—	5	6	no	—
163	C₄⋊Dic₃	—	10	6	no	—
164	C₄×Dic₃	—	10	6	no	—
165	C₂²×D₅	C₂×D₁₂	1	3	yes	—
166	C₂×AGL(1,𝔽₅)	—	6	2	no	—
167	C₂×AGL(1,𝔽₅)	—	6	6	no	—
168	C₂×AGL(1,𝔽₅)	—	24	6	no	—
169	C₅×Q₈	C₄⋊Dic₃	1	1	yes	characteristic
170	C₅×D₄	C₄⋊Dic₃	1	3	yes	—
171	C₂×C₄×C₅	C₂²×Dic₃	1	3	yes	—
172	C₂²⋊D₅	C₄⋊Dic₃	1	6	yes	—
173	C₂×Dic₅	C₂×Dic₆	1	3	yes	—
174	D₂₀	C₄⋊Dic₃	1	3	yes	—
175	C₄×D₅	C₂×Dic₆	1	3	yes	—
176	C₄×D₅	C₂×D₁₂	1	3	yes	—
177	C₄×D₅	C₂²×Dic₃	1	3	yes	—
178	C₄×D₅	C₂³×D₃	1	1	yes	characteristic
179	Dic₁₀	C₄⋊Dic₃	1	3	yes	—
180	C₅×C₈	C₂²×Dic₃	1	1	yes	characteristic
181	C₅×C₈	C₂²×Dic₃	1	3	yes	—
182	#168	C₂²×Dic₃	1	1	yes	characteristic
183	#168	C₂²×Dic₃	1	3	yes	—
184	C₂×#41	—	5	1	no	—
185	#131	—	5	1	no	—
186	#131	—	5	3	no	—
187	#131	—	5	3	no	—
188	#131	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
189	C₂×M₁₆	—	5	3	no	—
190	C₂²×C₈	—	5	3	no	—
191	C₄×Q₈	—	30	2	no	—
192	C₄×D₄	—	30	6	no	—
193	#117	—	15	6	no	—
194	C₂×C₄⋊C₄	—	15	6	no	—
195	#107	—	15	3	no	—
196	#107	—	15	6	no	—
197	#107	—	15	1	no	—
198	#106	—	15	6	no	—
199	C₂×C₃×C₅	—	2	3	no	—
200	C₂×C₃×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
201	C₃×D₅	—	2	3	no	—
202	C₃×D₅	#576	1	1	yes	characteristic
203	C₃×D₅	#578	1	1	yes	characteristic
204	C₂³×C₃	—	5	3	no	—
205	C₃×Q₈	—	5	3	no	—
206	C₃×Q₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
207	C₃×D₄	—	5	6	no	—
208	C₃×D₄	—	5	3	no	—
209	C₃×D₄	C₄⋊AGL(1,𝔽₅)	1	3	yes	—
210	C₂×C₃×C₄	—	5	3	no	—
211	C₂×C₃×C₄	—	5	3	no	—
212	C₂×C₃×C₄	—	5	3	no	—
213	C₂×C₃×C₄	—	5	3	no	—
214	C₂×C₃×C₄	—	5	1	no	—
215	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	3	yes	—
216	C₂×Dic₃	—	10	2	no	—
217	C₂×Dic₃	—	10	6	no	—
218	C₂×Dic₃	—	40	6	no	—
219	C₃×C₈	—	5	1	no	—
220	C₃×C₈	—	5	3	no	—
221	C₃×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
222	C₃×C₈	C₂²×AGL(1,𝔽₅)	1	3	yes	—
223	C₂²×C₅	C₂×C₄⋊Dic₃	1	3	yes	—
224	C₂×D₅	—	2	3	no	—
225	C₂×D₅	—	2	3	no	—
226	C₂×D₅	C₂×C₄⋊Dic₃	1	3	yes	—
227	C₂×D₅	C₂²×D₁₂	1	1	yes	characteristic
228	AGL(1,𝔽₅)	—	12	6	no	—
229	AGL(1,𝔽₅)	—	12	2	no	—
230	C₄×C₅	C₂×C₄⋊Dic₃	1	3	yes	—
231	C₄×C₅	C₂³×Dic₃	1	1	yes	characteristic
232	Dic₅	C₂×C₄⋊Dic₃	1	3	yes	—
233	Dic₅	C₂²×Dic₆	1	1	yes	characteristic
234	#41	—	5	1	no	—
235	#41	—	5	3	no	—
236	#41	—	5	3	no	—
237	#41	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
238	C₂×Q₈	—	5	1	no	—
239	C₂×D₄	—	5	3	no	—
240	C₂²×C₄	—	5	3	no	—
241	C₂²×C₄	—	30	6	no	—
242	M₁₆	—	5	3	no	—
243	M₁₆	—	5	6	no	—
244	M₁₆	C₂×ΓL(1,𝔽₁₆)	1	3	yes	complemented
245	C₂×C₈	—	5	3	no	—
246	C₂×C₈	—	5	3	no	—
247	C₂×C₈	—	5	1	no	—
248	C₂×C₈	—	5	3	no	—
249	C₂×C₈	—	5	3	no	—
250	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	3	yes	complemented
251	C₄⋊C₄	—	15	2	no	—
252	C₄⋊C₄	—	15	6	no	—
253	C₄⋊C₄	—	30	6	no	—
254	C₂²⋊C₄	—	30	6	no	—
255	C₄²	—	30	6	no	—
256	C₃×C₅	#2913	1	1	yes	characteristic, complemented, Hall
257	C₂²×C₃	—	5	1	no	—
258	C₂²×C₃	—	5	3	no	—
259	C₂²×C₃	—	10	3	no	—
260	C₂²×C₃	—	10	3	no	—
261	C₂²×C₃	C₂×C₄⋊AGL(1,𝔽₅)	1	3	yes	—
262	C₃×C₄	—	5	3	no	—
263	C₃×C₄	—	5	1	no	—
264	C₃×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	3	yes	—
265	C₃×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
266	Dic₃	—	20	2	no	—
267	Dic₃	—	20	6	no	—
268	C₂×C₅	—	2	3	no	—
269	C₂×C₅	C₂²×C₄⋊Dic₃	1	1	yes	characteristic
270	D₅	—	2	3	no	—
271	D₅	#5752	1	1	yes	characteristic
272	D₅	#5753	1	1	yes	characteristic
273	C₂³	—	5	3	no	—
274	Q₈	—	5	3	no	—
275	Q₈	C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
276	D₄	—	5	6	no	—
277	D₄	—	5	3	no	—
278	D₄	C₄⋊ΓL(1,𝔽₁₆)	1	3	yes	—
279	C₂×C₄	—	5	3	no	—
280	C₂×C₄	—	5	3	no	—
281	C₂×C₄	—	5	3	no	—
282	C₂×C₄	—	5	1	no	—
283	C₂×C₄	—	5	3	no	—
284	C₂×C₄	—	30	2	no	—
285	C₂×C₄	—	30	6	no	—
286	C₂×C₄	—	120	6	no	—
287	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	3	yes	—
288	C₈	—	5	1	no	—
289	C₈	—	5	3	no	—
290	C₈	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
291	C₈	C₂²×ΓL(1,𝔽₁₆)	1	3	yes	—
292	C₂×C₃	—	2	3	no	—
293	C₂×C₃	—	5	1	no	—
294	C₂×C₃	—	5	1	no	—
295	C₂×C₃	—	10	3	no	—
296	C₂×C₃	C₂²×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
297	C₅	#83785	1	1	yes	characteristic, complemented, Sylow
298	C₂²	—	5	1	no	—
299	C₂²	—	5	3	no	—
300	C₂²	—	10	3	no	—
301	C₂²	—	10	3	no	—
302	C₂²	C₂×C₄⋊ΓL(1,𝔽₁₆)	1	3	yes	—
303	C₄	—	5	1	no	—
304	C₄	—	5	3	no	—
305	C₄	—	60	6	no	—
306	C₄	—	60	2	no	—
307	C₄	C₂×C₄⋊ΓL(1,𝔽₁₆)	1	3	yes	—
308	C₄	C₂³×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
309	C₃	#10617417	1	1	yes	characteristic, complemented, Sylow
310	C₂	—	2	3	no	—
311	C₂	—	5	1	no	—
312	C₂	—	5	1	no	—
313	C₂	—	10	3	no	—
314	C₂	C₂²×C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×C₄⋊ΓL(1,𝔽₁₆)
commutator	C₃×C₄×C₅	C₂³×C₄
Frattini	C₄	C₂³×ΓL(1,𝔽₁₆)
Fitting	C₃×C₅×#131	C₄
socle	C₂×C₃×C₅	C₂²×C₄⋊C₄
prime radical	C₃×D₅×#41	C₂²
cosocle	C₃×C₄×D₅	C₂⁴
hypercenter	#131	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#2913
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
3. #11752386	C₂	semidirect
4. #11752380	C₂	semidirect
5. C₂×#91223	C₂	semidirect
12. C₃×C₅×#131	C₄	semidirect
13. C₃×#4094	C₄	semidirect
14. C₃×#4053	C₄	semidirect
16. C₃×#4047	C₄	semidirect
41. C₃×C₅×M₁₆	C₂×C₄	semidirect
42. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
43. C₃×#681	C₂×C₄	semidirect
44. C₂×C₃×#168	C₂×C₄	semidirect
45. C₃×#676	C₂×C₄	semidirect
46. C₃×C₈×D₅	C₂×C₄	semidirect
47. C₃×C₈×D₅	C₂×C₄	semidirect
60. C₅×#131	Dic₃	semidirect
61. #4094	Dic₃	semidirect
62. #4053	Dic₃	semidirect
64. #4047	Dic₃	semidirect
94. C₃×#131	AGL(1,𝔽₅)	semidirect
118. C₅×M₁₆	C₂×Dic₃	semidirect
119. C₂×C₅×C₈	C₂×Dic₃	semidirect
120. #681	C₂×Dic₃	semidirect
121. C₂×#168	C₂×Dic₃	semidirect
122. #676	C₂×Dic₃	semidirect
123. C₈×D₅	C₂×Dic₃	semidirect
124. C₈×D₅	C₂×Dic₃	semidirect
153. C₃×M₁₆	C₂×AGL(1,𝔽₅)	semidirect
159. C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	semidirect
188. #131	ΓL(1,𝔽₁₆)	semidirect
244. M₁₆	C₂×ΓL(1,𝔽₁₆)	semidirect
250. C₂×C₈	C₂×ΓL(1,𝔽₁₆)	semidirect
256. C₃×C₅	#2913	semidirect
297. C₅	#83785	semidirect
309. C₃	#10617417	semidirect

Maximal subgroups

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

subgroup	index
#11756995	2
C₃×D₅×#131	2
#11752386	2
#11752380	2
C₂×#91223	2
#10617417	3
#83785	5