Subgroups — #14327279

Order 1920.

1104 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×ΓL(1,𝔽₁₆)	C₂	1	2	yes	maximal
2	C₃×C₄×Q₈×D₅	C₂	1	1	yes	maximal, characteristic
3	#11752371	C₂	1	6	yes	maximal
4	#11752348	C₂	1	3	yes	maximal
5	C₄×C₄⋊ΓL(1,𝔽₁₆)	C₂	1	3	yes	maximal
6	#10617212	—	3	1	no	maximal, Hall
7	C₂×C₃×Q₈×D₅	C₂²	1	1	yes	characteristic
8	Q₈×ΓL(1,𝔽₁₆)	—	2	8	no	—
9	C₂×C₄⋊ΓL(1,𝔽₁₆)	C₂²	1	6	yes	—
10	C₂×C₄⋊ΓL(1,𝔽₁₆)	C₂²	1	6	yes	—
11	C₂×C₄×ΓL(1,𝔽₁₆)	C₂²	1	6	yes	—
12	C₃×C₄×C₅×Q₈	C₄	1	1	yes	characteristic, complemented
13	C₃×Q₈×Dic₅	C₄	1	1	yes	characteristic, complemented
14	C₃×D₅×C₄⋊C₄	C₂²	1	3	yes	—
15	C₃×#4033	C₄	1	3	yes	complemented
16	C₃×C₄²×D₅	C₂²	1	3	yes	—
17	C₃×C₄×Dic₁₀	C₄	1	3	yes	complemented
18	#91235	C₂²	1	1	yes	characteristic
19	#91235	C₂²	1	3	yes	—
20	#91235	C₂²	1	6	yes	—
21	#82460	—	5	1	no	maximal, Hall
22	C₂×Q₈×AGL(1,𝔽₅)	—	3	2	no	—
23	C₄×Q₈×D₅	D₃	1	1	yes	characteristic
24	#65712	—	3	6	no	—
25	#65689	—	3	3	no	—
26	C₄×C₄⋊AGL(1,𝔽₅)	—	3	3	no	—
27	C₂²×ΓL(1,𝔽₁₆)	C₂³	1	6	yes	—
28	C₂²×ΓL(1,𝔽₁₆)	C₂³	1	2	yes	—
29	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
30	C₃×Q₈×D₅	D₄	1	4	yes	—
31	C₃×Q₈×D₅	Q₈	1	4	yes	—
32	C₂×C₃×C₄×D₅	C₂³	1	3	yes	—
33	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
34	C₂×C₃×C₄×D₅	C₂³	1	3	yes	—
35	C₂×C₃×Dic₁₀	C₂×C₄	1	3	yes	—
36	C₄⋊ΓL(1,𝔽₁₆)	—	2	12	no	—
37	C₄⋊ΓL(1,𝔽₁₆)	—	2	12	no	—
38	C₄×ΓL(1,𝔽₁₆)	—	2	12	no	—
39	C₃×C₅×C₄⋊C₄	C₂×C₄	1	3	yes	—
40	C₃×C₄²×C₅	C₂×C₄	1	3	yes	—
41	C₃×C₄⋊Dic₅	C₂×C₄	1	3	yes	—
42	C₃×#683	C₂×C₄	1	6	yes	—
43	C₃×C₄×Dic₅	C₂×C₄	1	3	yes	—
44	C₃×C₄×Dic₅	C₂×C₄	1	3	yes	—
45	C₃×C₄×Dic₅	C₂×C₄	1	3	yes	—
46	C₂×C₃×C₄×Q₈	—	5	1	no	—
47	C₂×Q₈×Dic₃	—	5	2	no	—
48	#4984	—	5	6	no	—
49	#4936	—	5	3	no	—
50	C₄×C₄⋊Dic₃	—	5	3	no	—
51	C₂×Q₈×D₅	C₂×D₃	1	1	yes	characteristic
52	Q₈×AGL(1,𝔽₅)	—	6	8	no	—
53	C₂×C₄⋊AGL(1,𝔽₅)	—	3	6	no	—
54	C₂×C₄⋊AGL(1,𝔽₅)	—	3	6	no	—
55	C₂×C₄×AGL(1,𝔽₅)	—	3	6	no	—
56	C₄×C₅×Q₈	Dic₃	1	1	yes	characteristic, complemented
57	Q₈×Dic₅	Dic₃	1	1	yes	characteristic, complemented
58	D₅×C₄⋊C₄	C₂×D₃	1	3	yes	—
59	#4033	Dic₃	1	3	yes	complemented
60	C₄²×D₅	C₂×D₃	1	3	yes	—
61	C₄×Dic₁₀	Dic₃	1	3	yes	complemented
62	(C₂×AGL(1,𝔽₅))⋊C₄	—	3	1	no	—
63	(C₂×AGL(1,𝔽₅))⋊C₄	—	3	3	no	—
64	(C₂×AGL(1,𝔽₅))⋊C₄	—	3	6	no	—
65	#2351	—	15	1	no	Sylow
66	C₂²×C₃×D₅	C₂⁴	1	1	yes	characteristic
67	C₂×ΓL(1,𝔽₁₆)	—	2	2	no	—
68	C₂×ΓL(1,𝔽₁₆)	—	2	2	no	—
69	C₂×ΓL(1,𝔽₁₆)	—	2	6	no	—
70	C₂×ΓL(1,𝔽₁₆)	—	2	6	no	—
71	C₂×ΓL(1,𝔽₁₆)	—	2	6	no	—
72	C₂×ΓL(1,𝔽₁₆)	#41	1	4	yes	—
73	C₃×C₅×Q₈	C₄⋊C₄	1	4	yes	complemented
74	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂²×C₄	1	3	yes	—
76	C₂×C₃×C₄×C₅	C₂²×C₄	1	3	yes	—
77	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃×Dic₅	C₂²×C₄	1	3	yes	—
79	C₂×C₃×Dic₅	C₂²×C₄	1	3	yes	—
80	C₃×C₄×D₅	—	2	6	no	—
81	C₃×C₄×D₅	—	2	2	no	—
82	C₃×C₄×D₅	C₂×D₄	1	6	yes	—
83	C₃×C₄×D₅	C₂×Q₈	1	6	yes	—
84	C₃×Dic₁₀	C₄⋊C₄	1	12	yes	complemented
85	C₂²×C₃×Q₈	—	5	1	no	—
86	C₃×C₄×Q₈	—	5	1	no	—
87	C₃×C₄×Q₈	—	5	3	no	—
88	C₃×C₄×Q₈	—	5	3	no	—
89	C₃×C₄×Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
90	C₂×C₃×C₄⋊C₄	—	5	3	no	—
91	C₂×C₃×C₄²	—	5	3	no	—
92	Q₈×Dic₃	—	10	8	no	—
93	C₂×C₄⋊Dic₃	—	5	6	no	—
94	C₂×C₄⋊Dic₃	—	5	6	no	—
95	C₂×C₄×Dic₃	—	5	6	no	—
96	C₃⋊#95	—	5	1	no	—
97	C₃⋊#95	—	5	3	no	—
98	C₃⋊#95	—	5	6	no	—
99	C₂²×AGL(1,𝔽₅)	—	3	2	no	—
100	C₂²×AGL(1,𝔽₅)	—	3	6	no	—
101	C₂×C₅×Q₈	C₂×Dic₃	1	1	yes	characteristic
102	Q₈×D₅	Dic₆	1	4	yes	—
103	Q₈×D₅	D₁₂	1	4	yes	—
104	C₂×C₄×D₅	C₂²×D₃	1	3	yes	—
105	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
106	C₂×C₄×D₅	C₂²×D₃	1	3	yes	—
107	C₂×Dic₁₀	C₂×Dic₃	1	3	yes	—
108	C₄⋊AGL(1,𝔽₅)	—	6	12	no	—
109	C₄⋊AGL(1,𝔽₅)	—	6	12	no	—
110	C₄×AGL(1,𝔽₅)	—	6	12	no	—
111	C₅×C₄⋊C₄	C₂×Dic₃	1	3	yes	—
112	C₄²×C₅	C₂×Dic₃	1	3	yes	—
113	C₄⋊Dic₅	C₂×Dic₃	1	3	yes	—
114	#683	C₂×Dic₃	1	6	yes	—
115	C₄×Dic₅	C₂×Dic₃	1	3	yes	—
116	C₄×Dic₅	C₂×Dic₃	1	3	yes	—
117	C₄×Dic₅	C₂×Dic₃	1	3	yes	—
118	C₂×C₄×Q₈	—	5	1	no	—
119	C₂×C₄×Q₈	—	15	2	no	—
120	#389	—	15	6	no	—
121	#382	—	15	3	no	—
122	C₄×C₄⋊C₄	—	15	3	no	—
123	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₃×D₅	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₃×D₅	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₃×D₅	C₂×#41	1	2	yes	—
127	C₂×C₃×D₅	2₋¹⁺⁴	1	2	yes	—
128	ΓL(1,𝔽₁₆)	—	2	4	no	—
129	ΓL(1,𝔽₁₆)	—	4	6	no	—
130	C₃×C₄×C₅	—	2	3	no	—
131	C₃×C₄×C₅	—	2	1	no	—
132	C₃×C₄×C₅	C₂×C₄⋊C₄	1	6	yes	—
133	C₃×Dic₅	—	2	3	no	—
134	C₃×Dic₅	—	2	1	no	—
135	C₃×Dic₅	C₂×C₄⋊C₄	1	6	yes	—
136	C₂×C₃×Q₈	—	5	4	no	—
137	C₂×C₃×Q₈	—	5	4	no	—
138	C₂×C₃×Q₈	—	5	3	no	—
139	C₂×C₃×Q₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
140	C₂²×C₃×C₄	—	5	3	no	—
141	C₂²×C₃×C₄	—	5	1	no	—
142	C₂²×C₃×C₄	—	5	3	no	—
143	C₂²×Dic₃	—	5	2	no	—
144	C₂²×Dic₃	—	5	6	no	—
145	C₃×C₄⋊C₄	—	5	6	no	—
146	C₃×C₄⋊C₄	—	5	3	no	—
147	C₃×C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	3	yes	—
148	C₃×C₄²	—	5	3	no	—
149	C₃×C₄²	—	5	3	no	—
150	C₃×C₄²	—	5	3	no	—
151	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	3	yes	—
152	C₄⋊Dic₃	—	10	12	no	—
153	C₄⋊Dic₃	—	10	12	no	—
154	C₄×Dic₃	—	10	12	no	—
155	C₂²×D₅	C₂³×D₃	1	1	yes	characteristic
156	C₂×AGL(1,𝔽₅)	—	3	4	no	—
157	C₂×AGL(1,𝔽₅)	—	6	2	no	—
158	C₂×AGL(1,𝔽₅)	—	6	2	no	—
159	C₂×AGL(1,𝔽₅)	—	6	6	no	—
160	C₂×AGL(1,𝔽₅)	—	6	6	no	—
161	C₂×AGL(1,𝔽₅)	—	6	6	no	—
162	C₅×Q₈	C₄⋊Dic₃	1	4	yes	complemented
163	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
164	C₂×C₄×C₅	C₂²×Dic₃	1	3	yes	—
165	C₂×C₄×C₅	C₂²×Dic₃	1	3	yes	—
166	C₂×Dic₅	C₂²×Dic₃	1	1	yes	characteristic
167	C₂×Dic₅	C₂²×Dic₃	1	3	yes	—
168	C₂×Dic₅	C₂²×Dic₃	1	3	yes	—
169	C₄×D₅	—	2	2	no	—
170	C₄×D₅	—	2	6	no	—
171	C₄×D₅	C₂×Dic₆	1	6	yes	—
172	C₄×D₅	C₂×D₁₂	1	6	yes	—
173	Dic₁₀	C₄⋊Dic₃	1	12	yes	complemented
174	C₂²×Q₈	—	5	1	no	—
175	C₄×Q₈	—	5	1	no	—
176	C₄×Q₈	—	5	3	no	—
177	C₄×Q₈	—	5	3	no	—
178	C₄×Q₈	—	30	8	no	—
179	C₄×Q₈	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
180	C₂×C₄⋊C₄	—	5	3	no	—
181	C₂×C₄⋊C₄	—	15	6	no	—
182	C₂×C₄⋊C₄	—	15	6	no	—
183	C₂×C₄²	—	5	3	no	—
184	C₂×C₄²	—	15	6	no	—
185	#95	—	15	1	no	—
186	#95	—	15	3	no	—
187	#95	—	15	6	no	—
188	C₂×C₃×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
189	C₂×C₃×C₅	C₄×#41	1	1	yes	characteristic
190	C₂×C₃×C₅	#519	1	1	yes	characteristic
191	C₃×D₅	#548	1	2	yes	—
192	C₃×D₅	#557	1	2	yes	—
193	C₂³×C₃	—	5	1	no	—
194	C₃×Q₈	—	5	12	no	—
195	C₃×Q₈	C₄⋊AGL(1,𝔽₅)	1	4	yes	complemented
196	C₂×C₃×C₄	—	5	3	no	—
197	C₂×C₃×C₄	—	5	6	no	—
198	C₂×C₃×C₄	—	5	6	no	—
199	C₂×C₃×C₄	—	5	1	no	—
200	C₂×C₃×C₄	—	5	3	no	—
201	C₂×C₃×C₄	—	10	6	no	—
202	C₂×C₃×C₄	—	10	2	no	—
203	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	3	yes	—
205	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	3	yes	—
206	C₂×Dic₃	—	5	4	no	—
207	C₂×Dic₃	—	10	2	no	—
208	C₂×Dic₃	—	10	2	no	—
209	C₂×Dic₃	—	10	6	no	—
210	C₂×Dic₃	—	10	6	no	—
211	C₂×Dic₃	—	10	6	no	—
212	C₂²×C₅	C₂³×Dic₃	1	1	yes	characteristic
213	C₂×D₅	C₂²×Dic₆	1	1	yes	characteristic
214	C₂×D₅	C₂²×D₁₂	1	1	yes	characteristic
215	C₂×D₅	#1007	1	2	yes	—
216	C₂×D₅	D₃×#41	1	2	yes	—
217	AGL(1,𝔽₅)	—	6	4	no	—
218	AGL(1,𝔽₅)	—	12	6	no	—
219	C₄×C₅	—	2	1	no	—
220	C₄×C₅	—	2	3	no	—
221	C₄×C₅	C₂×C₄⋊Dic₃	1	6	yes	—
222	Dic₅	—	2	1	no	—
223	Dic₅	—	2	3	no	—
224	Dic₅	C₂×C₄⋊Dic₃	1	6	yes	—
225	C₂×Q₈	—	5	4	no	—
226	C₂×Q₈	—	5	4	no	—
227	C₂×Q₈	—	5	3	no	—
228	C₂×Q₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
229	C₂²×C₄	—	5	3	no	—
230	C₂²×C₄	—	5	1	no	—
231	C₂²×C₄	—	5	3	no	—
232	C₂²×C₄	—	15	2	no	—
233	C₂²×C₄	—	15	6	no	—
234	C₄⋊C₄	—	5	6	no	—
235	C₄⋊C₄	—	5	3	no	—
236	C₄⋊C₄	—	30	12	no	—
237	C₄⋊C₄	—	30	12	no	—
238	C₄⋊C₄	C₂×ΓL(1,𝔽₁₆)	1	3	yes	—
239	C₄²	—	5	3	no	—
240	C₄²	—	5	3	no	—
241	C₄²	—	5	3	no	—
242	C₄²	—	30	12	no	—
243	C₄²	C₂×ΓL(1,𝔽₁₆)	1	3	yes	—
244	C₃×C₅	#2351	1	1	yes	characteristic, complemented, Hall
245	C₂²×C₃	—	5	1	no	—
246	C₂²×C₃	—	5	1	no	—
247	C₂²×C₃	—	5	2	no	—
248	C₂²×C₃	—	5	2	no	—
249	C₂²×C₃	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	—	2	3	no	—
252	C₃×C₄	—	5	6	no	—
253	C₃×C₄	—	10	1	no	—
254	C₃×C₄	—	10	3	no	—
255	C₃×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	6	yes	—
256	Dic₃	—	10	4	no	—
257	Dic₃	—	20	6	no	—
258	C₂×C₅	C₂²×C₄⋊Dic₃	1	1	yes	characteristic
259	C₂×C₅	#5812	1	1	yes	characteristic
260	C₂×C₅	Dic₃×#41	1	1	yes	characteristic
261	D₅	#5569	1	2	yes	—
262	D₅	#5576	1	2	yes	—
263	C₂³	—	5	1	no	—
264	Q₈	—	5	12	no	—
265	Q₈	C₄⋊ΓL(1,𝔽₁₆)	1	4	yes	complemented
266	C₂×C₄	—	5	3	no	—
267	C₂×C₄	—	5	6	no	—
268	C₂×C₄	—	5	6	no	—
269	C₂×C₄	—	5	1	no	—
270	C₂×C₄	—	5	3	no	—
271	C₂×C₄	—	10	6	no	—
272	C₂×C₄	—	10	2	no	—
273	C₂×C₄	—	15	4	no	—
274	C₂×C₄	—	30	2	no	—
275	C₂×C₄	—	30	2	no	—
276	C₂×C₄	—	30	6	no	—
277	C₂×C₄	—	30	6	no	—
278	C₂×C₄	—	30	6	no	—
279	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
280	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	3	yes	—
281	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	3	yes	—
282	C₂×C₃	—	5	2	no	—
283	C₂×C₃	—	5	2	no	—
284	C₂×C₃	C₂²×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
285	C₂×C₃	#66262	1	1	yes	characteristic
286	C₂×C₃	#41×AGL(1,𝔽₅)	1	1	yes	characteristic
287	C₅	#82460	1	1	yes	characteristic, complemented, Sylow
288	C₂²	—	5	1	no	—
289	C₂²	—	5	1	no	—
290	C₂²	—	5	2	no	—
291	C₂²	—	5	2	no	—
292	C₂²	C₂³×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
293	C₄	—	2	1	no	—
294	C₄	—	2	3	no	—
295	C₄	—	5	6	no	—
296	C₄	—	10	1	no	—
297	C₄	—	10	3	no	—
298	C₄	—	30	4	no	—
299	C₄	—	60	6	no	—
300	C₄	C₂×C₄⋊ΓL(1,𝔽₁₆)	1	6	yes	—
301	C₃	#10617212	1	1	yes	characteristic, complemented, Sylow
302	C₂	—	5	2	no	—
303	C₂	—	5	2	no	—
304	C₂	C₂²×C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
305	C₂	#11756991	1	1	yes	characteristic
306	C₂	#41×ΓL(1,𝔽₁₆)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×ΓL(1,𝔽₁₆)
commutator, socle	C₂²×C₃×C₅	C₂³×C₄
Fitting	C₃×C₄×C₅×Q₈	C₄
prime radical, cosocle	C₂²×C₃×D₅	C₂⁴
hypercenter	C₄×Q₈	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#2351
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
12. C₃×C₄×C₅×Q₈	C₄	semidirect
13. C₃×Q₈×Dic₅	C₄	semidirect
15. C₃×#4033	C₄	semidirect
17. C₃×C₄×Dic₁₀	C₄	semidirect
56. C₄×C₅×Q₈	Dic₃	semidirect
57. Q₈×Dic₅	Dic₃	semidirect
59. #4033	Dic₃	semidirect
61. C₄×Dic₁₀	Dic₃	semidirect
73. C₃×C₅×Q₈	C₄⋊C₄	semidirect
84. C₃×Dic₁₀	C₄⋊C₄	semidirect
89. C₃×C₄×Q₈	AGL(1,𝔽₅)	semidirect
162. C₅×Q₈	C₄⋊Dic₃	semidirect
173. Dic₁₀	C₄⋊Dic₃	semidirect
179. C₄×Q₈	ΓL(1,𝔽₁₆)	semidirect
195. C₃×Q₈	C₄⋊AGL(1,𝔽₅)	semidirect
244. C₃×C₅	#2351	semidirect
265. Q₈	C₄⋊ΓL(1,𝔽₁₆)	semidirect
287. C₅	#82460	semidirect
301. C₃	#10617212	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×ΓL(1,𝔽₁₆)	2
C₃×C₄×Q₈×D₅	2
#11752371	2
#11752348	2
C₄×C₄⋊ΓL(1,𝔽₁₆)	2
#10617212	3
#82460	5