Subgroups — #14307413

Order 1920.

568 conjugacy classes of subgroups.

The subgroup lattice has 328 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	C₄²×ΓL(1,𝔽₁₆)	C₂	1	1	yes	maximal, characteristic
3	#11752340	C₂	1	1	yes	maximal, characteristic
4	#11746746	C₂	1	4	yes	maximal
5	#10604665	—	3	1	no	maximal, Hall
6	C₂×C₄×ΓL(1,𝔽₁₆)	—	2	1	no	—
7	C₂×C₄×ΓL(1,𝔽₁₆)	—	2	1	no	—
8	C₂×C₄×ΓL(1,𝔽₁₆)	C₂²	1	2	yes	—
9	C₂×#6758	C₂²	1	2	yes	—
10	C₃×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
11	C₃×#4092	—	2	1	no	—
12	C₃×Q₈×Dic₅	—	2	1	no	—
13	C₃×#4043	—	2	1	no	—
14	C₃×#4038	C₂²	1	2	yes	—
15	C₃×#4036	C₄	1	2	yes	complemented
16	C₃×#4033	—	2	1	no	—
17	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
18	C₃×#4015	C₄	1	1	yes	characteristic, complemented
19	C₄⋊(C₃⋊#170)	—	2	1	no	—
20	C₄⋊(C₃⋊#170)	—	2	1	no	—
21	#71839	—	5	1	no	maximal, Hall
22	#66062	D₃	1	1	yes	characteristic
23	C₄²×AGL(1,𝔽₅)	—	3	1	no	—
24	#65681	—	3	1	no	—
25	#64875	—	3	4	no	—
26	C₂²×ΓL(1,𝔽₁₆)	—	2	2	no	—
27	C₂×C₃×C₅×Q₈	—	2	1	no	—
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)	—	2	4	no	—
35	C₄×ΓL(1,𝔽₁₆)	—	2	2	no	—
36	C₄×ΓL(1,𝔽₁₆)	Q₈	1	4	yes	complemented
37	#6758	—	2	4	no	—
38	C₃×C₅×C₄⋊C₄	—	2	1	no	—
39	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
40	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
41	C₃×#685	—	2	2	no	—
42	C₃×C₄⋊Dic₅	—	2	1	no	—
43	C₃×C₄⋊Dic₅	—	2	2	no	—
44	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
45	C₃×#683	—	2	2	no	—
46	C₃×C₄×Dic₅	—	2	2	no	—
47	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
48	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
49	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
50	C₃×#533	—	5	1	no	—
51	C₄²×Dic₃	—	5	1	no	—
52	#4924	—	5	1	no	—
53	#4532	—	5	4	no	—
54	C₂×C₄×AGL(1,𝔽₅)	—	3	2	no	—
55	C₂×C₄×AGL(1,𝔽₅)	—	6	1	no	—
56	C₂×C₄×AGL(1,𝔽₅)	—	6	1	no	—
57	C₂×#700	—	3	2	no	—
58	C₅×C₄⋊Q₈	Dic₃	1	1	yes	characteristic, complemented
59	#4092	—	2	1	no	—
60	Q₈×Dic₅	—	2	1	no	—
61	#4043	—	2	1	no	—
62	#4038	C₂×D₃	1	2	yes	—
63	#4036	Dic₃	1	2	yes	complemented
64	#4033	—	2	1	no	—
65	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
66	#4015	Dic₃	1	1	yes	characteristic, complemented
67	C₄⋊#170	—	6	1	no	—
68	C₄⋊#170	—	6	1	no	—
69	#1994	—	15	1	no	Sylow
70	C₂²×C₃×D₅	C₂×D₄	1	1	yes	characteristic
71	C₂×ΓL(1,𝔽₁₆)	—	2	4	no	—
72	C₂×ΓL(1,𝔽₁₆)	—	2	4	no	—
73	C₂×ΓL(1,𝔽₁₆)	—	2	4	no	—
74	C₃×C₅×Q₈	—	4	2	no	—
75	C₂×C₃×C₄×C₅	—	2	2	no	—
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×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃×Dic₅	—	2	2	no	—
80	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
83	C₃×C₄×D₅	—	2	2	no	—
84	C₃×C₄×D₅	—	2	2	no	—
85	C₃×C₄×D₅	C₂×Q₈	1	2	yes	—
86	C₃×C₄×D₅	#41	1	2	yes	—
87	C₃×Dic₁₀	—	4	2	no	—
88	C₃⋊#170	—	4	2	no	—
89	C₃×C₄⋊Q₈	—	5	1	no	—
90	C₃×C₄⋊Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
91	C₃×#125	—	5	2	no	—
92	C₃×C₂²⋊Q₈	—	10	1	no	—
93	C₃×C₂²⋊Q₈	—	10	1	no	—
94	C₃×C₄×Q₈	—	10	1	no	—
95	C₃×C₄×Q₈	—	10	1	no	—
96	C₃×#117	—	5	2	no	—
97	C₂×C₃×C₄²	—	5	1	no	—
98	C₂×C₄×Dic₃	—	5	2	no	—
99	C₂×C₄×Dic₃	—	10	1	no	—
100	C₂×C₄×Dic₃	—	10	1	no	—
101	C₂×#208	—	5	2	no	—
102	C₄⋊(C₃⋊C₈)	—	10	1	no	—
103	C₄⋊(C₃⋊C₈)	—	10	1	no	—
104	C₂²×AGL(1,𝔽₅)	—	6	2	no	—
105	C₂×C₅×Q₈	—	2	1	no	—
106	C₂×C₄×D₅	#67	1	1	yes	characteristic
107	C₂×C₄×D₅	#67	1	1	yes	characteristic
108	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
109	C₂×Dic₁₀	—	2	1	no	—
110	C₂×#170	—	6	2	no	—
111	C₄×AGL(1,𝔽₅)	—	3	4	no	—
112	C₄×AGL(1,𝔽₅)	—	6	4	no	—
113	C₄×AGL(1,𝔽₅)	—	6	4	no	—
114	C₄×AGL(1,𝔽₅)	—	6	2	no	—
115	#700	—	6	4	no	—
116	C₅×C₄⋊C₄	—	2	1	no	—
117	C₅×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
118	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
119	#685	—	2	2	no	—
120	C₄⋊Dic₅	—	2	1	no	—
121	C₄⋊Dic₅	—	2	2	no	—
122	C₄⋊Dic₅	C₂×Dic₃	1	2	yes	—
123	#683	—	2	2	no	—
124	C₄×Dic₅	—	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	#533	—	5	1	no	—
129	#423	—	15	1	no	—
130	C₄³	—	15	1	no	—
131	#339	—	15	4	no	—
132	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₃×D₅	—	2	1	no	—
134	C₂×C₃×D₅	—	2	1	no	—
135	C₂×C₃×D₅	C₂²⋊Q₈	1	2	yes	—
136	ΓL(1,𝔽₁₆)	—	2	8	no	—
137	C₃×C₄×C₅	—	4	2	no	—
138	C₃×C₄×C₅	#104	1	2	yes	—
139	C₃×C₄×C₅	#104	1	2	yes	—
140	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
141	C₃×Dic₅	—	2	1	no	—
142	C₃×Dic₅	—	2	1	no	—
143	C₃×Dic₅	—	4	2	no	—
144	C₃×Dic₅	#124	1	1	yes	characteristic
145	C₃×Dic₅	C₄⋊Q₈	1	1	yes	characteristic
146	C₂×C₃×Q₈	—	2	1	no	—
147	C₂×C₃×Q₈	—	10	1	no	—
148	C₂²×C₃×C₄	—	5	1	no	—
149	C₂²×C₃×C₄	—	5	1	no	—
150	C₂²×C₃×C₄	—	5	1	no	—
151	C₂²×Dic₃	—	10	2	no	—
152	C₃×C₄⋊C₄	—	2	1	no	—
153	C₃×C₄⋊C₄	—	5	2	no	—
154	C₃×C₄⋊C₄	—	10	2	no	—
155	C₃×C₄⋊C₄	—	10	2	no	—
156	C₃×C₄⋊C₄	—	10	1	no	—
157	C₃×C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
158	C₃×C₂²⋊C₄	—	10	2	no	—
159	C₃×C₄²	—	5	1	no	—
160	C₃×C₄²	—	5	1	no	—
161	C₃×C₄²	—	5	1	no	—
162	C₃×C₄²	—	10	2	no	—
163	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
164	C₄×Dic₃	—	5	4	no	—
165	C₄×Dic₃	—	10	2	no	—
166	C₄×Dic₃	—	10	4	no	—
167	C₄×Dic₃	—	10	4	no	—
168	#208	—	10	4	no	—
169	C₂×C₃⋊C₈	—	10	2	no	—
170	C₂²×D₅	C₂×#67	1	1	yes	characteristic
171	C₂×AGL(1,𝔽₅)	—	6	4	no	—
172	C₂×AGL(1,𝔽₅)	—	6	4	no	—
173	C₂×AGL(1,𝔽₅)	—	6	4	no	—
174	C₅×Q₈	—	4	2	no	—
175	C₂×C₄×C₅	—	2	2	no	—
176	C₂×C₄×C₅	#217	1	1	yes	characteristic
177	C₂×C₄×C₅	#217	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
179	C₂×Dic₅	—	2	2	no	—
180	C₂×Dic₅	#217	1	1	yes	characteristic
181	C₂×Dic₅	#217	1	1	yes	characteristic
182	C₂×Dic₅	C₂×#67	1	1	yes	characteristic
183	C₄×D₅	—	2	2	no	—
184	C₄×D₅	—	2	2	no	—
185	C₄×D₅	D₃×Q₈	1	2	yes	—
186	C₄×D₅	#239	1	2	yes	—
187	Dic₁₀	—	4	2	no	—
188	#170	—	12	2	no	—
189	C₂×M₁₆	—	15	2	no	—
190	C₄⋊Q₈	—	5	1	no	—
191	C₄⋊Q₈	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
192	#125	—	5	2	no	—
193	C₂²⋊Q₈	—	10	1	no	—
194	C₂²⋊Q₈	—	10	1	no	—
195	C₄×Q₈	—	10	1	no	—
196	C₄×Q₈	—	10	1	no	—
197	#117	—	5	2	no	—
198	C₂×C₄²	—	5	1	no	—
199	C₂×C₄²	—	15	2	no	—
200	C₂×C₄²	—	30	1	no	—
201	C₂×C₄²	—	30	1	no	—
202	C₄⋊C₈	—	30	1	no	—
203	C₄⋊C₈	—	30	1	no	—
204	C₂×C₃×C₅	#391	1	1	yes	characteristic
205	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
206	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
207	C₃×D₅	—	2	2	no	—
208	C₂³×C₃	—	5	1	no	—
209	C₃×Q₈	—	4	2	no	—
210	C₃×Q₈	—	20	2	no	—
211	C₂×C₃×C₄	—	2	2	no	—
212	C₂×C₃×C₄	—	5	1	no	—
213	C₂×C₃×C₄	—	5	2	no	—
214	C₂×C₃×C₄	—	5	2	no	—
215	C₂×C₃×C₄	—	5	1	no	—
216	C₂×C₃×C₄	—	5	1	no	—
217	C₂×C₃×C₄	—	10	2	no	—
218	C₂×C₃×C₄	—	10	2	no	—
219	C₂×C₃×C₄	—	10	2	no	—
220	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
222	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
223	C₂×Dic₃	—	10	4	no	—
224	C₂×Dic₃	—	10	4	no	—
225	C₂×Dic₃	—	10	4	no	—
226	C₃⋊C₈	—	20	2	no	—
227	C₂²×C₅	C₂×#217	1	1	yes	characteristic
228	C₂×D₅	—	2	1	no	—
229	C₂×D₅	—	2	1	no	—
230	C₂×D₅	#946	1	2	yes	—
231	AGL(1,𝔽₅)	—	6	8	no	—
232	C₄×C₅	—	4	2	no	—
233	C₄×C₅	#837	1	2	yes	—
234	C₄×C₅	#837	1	2	yes	—
235	C₄×C₅	Q₈×Dic₃	1	2	yes	—
236	Dic₅	—	2	1	no	—
237	Dic₅	—	2	1	no	—
238	Dic₅	—	4	2	no	—
239	Dic₅	#944	1	1	yes	characteristic
240	Dic₅	#947	1	1	yes	characteristic
241	C₂×Q₈	—	2	1	no	—
242	C₂×Q₈	—	10	1	no	—
243	C₂²×C₄	—	5	1	no	—
244	C₂²×C₄	—	5	1	no	—
245	C₂²×C₄	—	5	1	no	—
246	C₂²×C₄	—	30	2	no	—
247	M₁₆	—	30	4	no	—
248	C₂×C₈	—	30	2	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	5	2	no	—
251	C₄⋊C₄	—	10	2	no	—
252	C₄⋊C₄	—	10	2	no	—
253	C₄⋊C₄	—	10	1	no	—
254	C₄⋊C₄	C₂×ΓL(1,𝔽₁₆)	1	2	yes	—
255	C₂²⋊C₄	—	10	2	no	—
256	C₄²	—	5	1	no	—
257	C₄²	—	5	1	no	—
258	C₄²	—	5	1	no	—
259	C₄²	—	10	2	no	—
260	C₄²	—	15	4	no	—
261	C₄²	—	30	2	no	—
262	C₄²	—	30	4	no	—
263	C₄²	—	30	4	no	—
264	C₄²	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
265	C₃×C₅	#1994	1	1	yes	characteristic, complemented, Hall
266	C₂²×C₃	—	5	2	no	—
267	C₂²×C₃	—	10	1	no	—
268	C₂²×C₃	—	10	1	no	—
269	C₂²×C₃	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
270	C₃×C₄	—	4	2	no	—
271	C₃×C₄	—	5	1	no	—
272	C₃×C₄	—	5	1	no	—
273	C₃×C₄	—	10	1	no	—
274	C₃×C₄	—	10	1	no	—
275	C₃×C₄	—	20	2	no	—
276	C₃×C₄	#4008	1	2	yes	—
277	C₃×C₄	#4010	1	2	yes	—
278	C₃×C₄	Q₈×AGL(1,𝔽₅)	1	2	yes	—
279	Dic₃	—	10	8	no	—
280	C₂×C₅	#5229	1	1	yes	characteristic
281	C₂×C₅	C₂×#837	1	1	yes	characteristic
282	C₂×C₅	C₂×#837	1	1	yes	characteristic
283	D₅	—	2	2	no	—
284	C₂³	—	5	1	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	20	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	5	1	no	—
289	C₂×C₄	—	5	2	no	—
290	C₂×C₄	—	5	2	no	—
291	C₂×C₄	—	5	1	no	—
292	C₂×C₄	—	5	1	no	—
293	C₂×C₄	—	10	2	no	—
294	C₂×C₄	—	10	2	no	—
295	C₂×C₄	—	10	2	no	—
296	C₂×C₄	—	30	4	no	—
297	C₂×C₄	—	30	4	no	—
298	C₂×C₄	—	30	4	no	—
299	C₂×C₄	#6763	1	1	yes	characteristic
300	C₂×C₄	#6763	1	1	yes	characteristic
301	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
302	C₈	—	60	2	no	—
303	C₂×C₃	—	10	2	no	—
304	C₂×C₃	C₂×#4008	1	1	yes	characteristic
305	C₂×C₃	C₂×#4010	1	1	yes	characteristic
306	C₂×C₃	(C₂×AGL(1,𝔽₅))⋊Q₈	1	1	yes	characteristic
307	C₅	#71839	1	1	yes	characteristic, complemented, Sylow
308	C₂²	—	5	2	no	—
309	C₂²	—	10	1	no	—
310	C₂²	—	10	1	no	—
311	C₂²	C₂×#6763	1	1	yes	characteristic
312	C₄	—	4	2	no	—
313	C₄	—	5	1	no	—
314	C₄	—	5	1	no	—
315	C₄	—	10	1	no	—
316	C₄	—	10	1	no	—
317	C₄	—	20	2	no	—
318	C₄	—	30	8	no	—
319	C₄	#91237	1	2	yes	—
320	C₄	#91239	1	2	yes	—
321	C₄	Q₈×ΓL(1,𝔽₁₆)	1	2	yes	—
322	C₃	#10604665	1	1	yes	characteristic, complemented, Sylow
323	C₂	—	10	2	no	—
324	C₂	C₂×#91237	1	1	yes	characteristic
325	C₂	C₂×#91239	1	1	yes	characteristic
326	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	#1994
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
15. C₃×#4036	C₄	semidirect
18. C₃×#4015	C₄	semidirect
36. C₄×ΓL(1,𝔽₁₆)	Q₈	semidirect
58. C₅×C₄⋊Q₈	Dic₃	semidirect
63. #4036	Dic₃	semidirect
66. #4015	Dic₃	semidirect
90. C₃×C₄⋊Q₈	AGL(1,𝔽₅)	semidirect
191. C₄⋊Q₈	ΓL(1,𝔽₁₆)	semidirect
265. C₃×C₅	#1994	semidirect
307. C₅	#71839	semidirect
322. C₃	#10604665	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
C₄²×ΓL(1,𝔽₁₆)	2
#11752340	2
#11746746	2
#10604665	3
#71839	5