Subgroups — #14307135

Order 1920.

520 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	C₄×C₄⋊ΓL(1,𝔽₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₄×#6758	C₂	1	1	yes	maximal, characteristic
4	#11746746	C₂	1	4	yes	maximal
5	#10604680	—	3	1	no	maximal, Hall
6	C₂×C₄⋊ΓL(1,𝔽₁₆)	—	2	1	no	—
7	C₂×C₄×ΓL(1,𝔽₁₆)	C₂²	1	2	yes	—
8	C₂×#6758	C₂²	1	2	yes	—
9	C₃×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
10	C₃×#4092	—	2	1	no	—
11	C₃×Q₈×Dic₅	—	2	1	no	—
12	C₃×#4043	—	2	1	no	—
13	C₃×#4038	C₂²	1	2	yes	—
14	C₃×#4036	C₄	1	2	yes	complemented
15	C₃×#4033	—	2	1	no	—
16	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
17	C₃×#4015	C₄	1	1	yes	characteristic, complemented
18	#91235	—	2	1	no	—
19	#91231	—	2	1	no	—
20	C₄×C₃⋊#170	—	2	1	no	—
21	#71841	—	5	1	no	maximal, Hall
22	#66062	D₃	1	1	yes	characteristic
23	C₄×C₄⋊AGL(1,𝔽₅)	—	3	1	no	—
24	C₄×#700	—	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	#6758	—	2	2	no	—
36	#6758	Q₈	1	4	yes	complemented
37	C₃×C₅×C₄⋊C₄	—	2	1	no	—
38	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
39	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
40	C₃×#685	—	2	2	no	—
41	C₃×C₄⋊Dic₅	—	2	1	no	—
42	C₃×C₄⋊Dic₅	—	2	2	no	—
43	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
44	C₃×#683	—	2	2	no	—
45	C₃×C₄×Dic₅	—	2	2	no	—
46	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
47	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
48	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
49	C₃×#533	—	5	1	no	—
50	C₄×C₄⋊Dic₃	—	5	1	no	—
51	C₄×#208	—	5	1	no	—
52	#4532	—	5	4	no	—
53	C₂×C₄⋊AGL(1,𝔽₅)	—	6	1	no	—
54	C₂×C₄×AGL(1,𝔽₅)	—	3	2	no	—
55	C₂×#700	—	3	2	no	—
56	C₅×C₄⋊Q₈	Dic₃	1	1	yes	characteristic, complemented
57	#4092	—	2	1	no	—
58	Q₈×Dic₅	—	2	1	no	—
59	#4043	—	2	1	no	—
60	#4038	C₂×D₃	1	2	yes	—
61	#4036	Dic₃	1	2	yes	complemented
62	#4033	—	2	1	no	—
63	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
64	#4015	Dic₃	1	1	yes	characteristic, complemented
65	(C₂×AGL(1,𝔽₅))⋊C₄	—	6	1	no	—
66	#4002	—	6	1	no	—
67	C₄×#170	—	6	1	no	—
68	#1986	—	15	1	no	Sylow
69	C₂²×C₃×D₅	C₂×D₄	1	1	yes	characteristic
70	C₂×ΓL(1,𝔽₁₆)	—	2	4	no	—
71	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
72	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
73	C₃×C₅×Q₈	—	4	2	no	—
74	C₂×C₃×C₄×C₅	—	2	2	no	—
75	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
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×Dic₅	—	2	2	no	—
79	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
82	C₃×C₄×D₅	—	2	2	no	—
83	C₃×C₄×D₅	—	4	1	no	—
84	C₃×C₄×D₅	C₂×Q₈	1	2	yes	—
85	C₃×C₄×D₅	#41	1	2	yes	—
86	C₃×Dic₁₀	—	4	2	no	—
87	C₃⋊#170	—	2	4	no	—
88	C₃×C₄⋊Q₈	—	5	1	no	—
89	C₃×C₄⋊Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
90	C₃×#125	—	5	2	no	—
91	C₃×C₂²⋊Q₈	—	10	1	no	—
92	C₃×C₂²⋊Q₈	—	10	1	no	—
93	C₃×C₄×Q₈	—	10	1	no	—
94	C₃×C₄×Q₈	—	10	1	no	—
95	C₃×#117	—	5	2	no	—
96	C₂×C₃×C₄²	—	5	1	no	—
97	C₂×C₄⋊Dic₃	—	10	1	no	—
98	C₂×C₄×Dic₃	—	5	2	no	—
99	C₂×#208	—	5	2	no	—
100	C₃⋊#95	—	10	1	no	—
101	#803	—	10	1	no	—
102	C₄×C₃⋊C₈	—	10	1	no	—
103	C₂²×AGL(1,𝔽₅)	—	6	2	no	—
104	C₂×C₅×Q₈	—	2	1	no	—
105	C₂×C₄×D₅	#67	1	1	yes	characteristic
106	C₂×C₄×D₅	#67	1	1	yes	characteristic
107	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
108	C₂×Dic₁₀	—	2	1	no	—
109	C₂×#170	—	6	2	no	—
110	C₄⋊AGL(1,𝔽₅)	—	6	4	no	—
111	C₄×AGL(1,𝔽₅)	—	6	4	no	—
112	#700	—	3	4	no	—
113	#700	—	6	2	no	—
114	C₅×C₄⋊C₄	—	2	1	no	—
115	C₅×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
116	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
117	#685	—	2	2	no	—
118	C₄⋊Dic₅	—	2	1	no	—
119	C₄⋊Dic₅	—	2	2	no	—
120	C₄⋊Dic₅	C₂×Dic₃	1	2	yes	—
121	#683	—	2	2	no	—
122	C₄×Dic₅	—	2	2	no	—
123	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
124	C₄×Dic₅	#67	1	1	yes	characteristic
125	C₄×Dic₅	#67	1	1	yes	characteristic
126	#533	—	5	1	no	—
127	C₄×M₁₆	—	15	1	no	—
128	C₄×C₄⋊C₄	—	15	1	no	—
129	#339	—	15	4	no	—
130	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₃×D₅	—	2	1	no	—
132	C₂×C₃×D₅	—	2	1	no	—
133	C₂×C₃×D₅	#124	1	1	yes	characteristic
134	C₂×C₃×D₅	C₄⋊Q₈	1	1	yes	characteristic
135	ΓL(1,𝔽₁₆)	—	4	4	no	—
136	C₃×C₄×C₅	—	2	1	no	—
137	C₃×C₄×C₅	—	4	2	no	—
138	C₃×C₄×C₅	#104	1	2	yes	—
139	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
140	C₃×Dic₅	—	2	1	no	—
141	C₃×Dic₅	—	2	1	no	—
142	C₃×Dic₅	—	4	2	no	—
143	C₃×Dic₅	C₂²⋊Q₈	1	2	yes	—
144	C₂×C₃×Q₈	—	2	1	no	—
145	C₂×C₃×Q₈	—	10	1	no	—
146	C₂²×C₃×C₄	—	5	1	no	—
147	C₂²×C₃×C₄	—	5	1	no	—
148	C₂²×C₃×C₄	—	5	1	no	—
149	C₂²×Dic₃	—	10	2	no	—
150	C₃×C₄⋊C₄	—	2	1	no	—
151	C₃×C₄⋊C₄	—	5	2	no	—
152	C₃×C₄⋊C₄	—	10	2	no	—
153	C₃×C₄⋊C₄	—	10	2	no	—
154	C₃×C₄⋊C₄	—	10	1	no	—
155	C₃×C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
156	C₃×C₂²⋊C₄	—	10	2	no	—
157	C₃×C₄²	—	5	1	no	—
158	C₃×C₄²	—	5	1	no	—
159	C₃×C₄²	—	5	1	no	—
160	C₃×C₄²	—	10	2	no	—
161	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
162	C₄⋊Dic₃	—	10	4	no	—
163	C₄×Dic₃	—	10	4	no	—
164	#208	—	5	4	no	—
165	#208	—	10	2	no	—
166	C₂×C₃⋊C₈	—	10	2	no	—
167	C₂²×D₅	C₂×#67	1	1	yes	characteristic
168	C₂×AGL(1,𝔽₅)	—	6	4	no	—
169	C₂×AGL(1,𝔽₅)	—	12	2	no	—
170	C₂×AGL(1,𝔽₅)	—	12	2	no	—
171	C₅×Q₈	—	4	2	no	—
172	C₂×C₄×C₅	—	2	2	no	—
173	C₂×C₄×C₅	#217	1	1	yes	characteristic
174	C₂×C₄×C₅	#217	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
176	C₂×Dic₅	—	2	2	no	—
177	C₂×Dic₅	#217	1	1	yes	characteristic
178	C₂×Dic₅	#217	1	1	yes	characteristic
179	C₂×Dic₅	C₂×#67	1	1	yes	characteristic
180	C₄×D₅	—	2	2	no	—
181	C₄×D₅	—	4	1	no	—
182	C₄×D₅	D₃×Q₈	1	2	yes	—
183	C₄×D₅	#239	1	2	yes	—
184	Dic₁₀	—	4	2	no	—
185	#170	—	6	4	no	—
186	C₂×M₁₆	—	15	2	no	—
187	C₄⋊Q₈	—	5	1	no	—
188	C₄⋊Q₈	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
189	#125	—	5	2	no	—
190	C₂²⋊Q₈	—	10	1	no	—
191	C₂²⋊Q₈	—	10	1	no	—
192	C₄×Q₈	—	10	1	no	—
193	C₄×Q₈	—	10	1	no	—
194	#117	—	5	2	no	—
195	C₂×C₄⋊C₄	—	30	1	no	—
196	C₂×C₄²	—	5	1	no	—
197	C₂×C₄²	—	15	2	no	—
198	#97	—	30	1	no	—
199	C₄×C₈	—	30	1	no	—
200	#95	—	30	1	no	—
201	C₂×C₃×C₅	#391	1	1	yes	characteristic
202	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
203	C₂×C₃×C₅	#421	1	1	yes	characteristic
204	C₃×D₅	—	2	1	no	—
205	C₃×D₅	—	2	1	no	—
206	C₂³×C₃	—	5	1	no	—
207	C₃×Q₈	—	4	2	no	—
208	C₃×Q₈	—	20	2	no	—
209	C₂×C₃×C₄	—	2	2	no	—
210	C₂×C₃×C₄	—	5	1	no	—
211	C₂×C₃×C₄	—	5	2	no	—
212	C₂×C₃×C₄	—	5	2	no	—
213	C₂×C₃×C₄	—	5	1	no	—
214	C₂×C₃×C₄	—	5	1	no	—
215	C₂×C₃×C₄	—	10	2	no	—
216	C₂×C₃×C₄	—	10	2	no	—
217	C₂×C₃×C₄	—	20	1	no	—
218	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
221	C₂×Dic₃	—	10	4	no	—
222	C₂×Dic₃	—	20	2	no	—
223	C₂×Dic₃	—	20	2	no	—
224	C₃⋊C₈	—	10	4	no	—
225	C₂²×C₅	C₂×#217	1	1	yes	characteristic
226	C₂×D₅	—	2	1	no	—
227	C₂×D₅	—	2	1	no	—
228	C₂×D₅	#944	1	1	yes	characteristic
229	C₂×D₅	#947	1	1	yes	characteristic
230	AGL(1,𝔽₅)	—	12	4	no	—
231	C₄×C₅	—	2	1	no	—
232	C₄×C₅	—	4	2	no	—
233	C₄×C₅	#837	1	2	yes	—
234	C₄×C₅	Q₈×Dic₃	1	2	yes	—
235	Dic₅	—	2	1	no	—
236	Dic₅	—	2	1	no	—
237	Dic₅	—	4	2	no	—
238	Dic₅	#946	1	2	yes	—
239	C₂×Q₈	—	2	1	no	—
240	C₂×Q₈	—	10	1	no	—
241	C₂²×C₄	—	5	1	no	—
242	C₂²×C₄	—	5	1	no	—
243	C₂²×C₄	—	5	1	no	—
244	C₂²×C₄	—	30	2	no	—
245	M₁₆	—	15	4	no	—
246	M₁₆	—	30	2	no	—
247	C₂×C₈	—	30	2	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	5	2	no	—
250	C₄⋊C₄	—	10	2	no	—
251	C₄⋊C₄	—	10	2	no	—
252	C₄⋊C₄	—	10	1	no	—
253	C₄⋊C₄	—	30	4	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	—	30	4	no	—
261	C₄²	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
262	C₃×C₅	#1986	1	1	yes	characteristic, complemented, Hall
263	C₂²×C₃	—	5	1	no	—
264	C₂²×C₃	—	5	1	no	—
265	C₂²×C₃	—	10	1	no	—
266	C₂²×C₃	—	10	1	no	—
267	C₂²×C₃	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
268	C₃×C₄	—	2	1	no	—
269	C₃×C₄	—	4	2	no	—
270	C₃×C₄	—	5	2	no	—
271	C₃×C₄	—	10	1	no	—
272	C₃×C₄	—	10	1	no	—
273	C₃×C₄	—	20	2	no	—
274	C₃×C₄	#4010	1	2	yes	—
275	C₃×C₄	Q₈×AGL(1,𝔽₅)	1	2	yes	—
276	Dic₃	—	20	4	no	—
277	C₂×C₅	#5229	1	1	yes	characteristic
278	C₂×C₅	C₂×#837	1	1	yes	characteristic
279	C₂×C₅	#5236	1	1	yes	characteristic
280	D₅	—	2	1	no	—
281	D₅	—	2	1	no	—
282	C₂³	—	5	1	no	—
283	Q₈	—	4	2	no	—
284	Q₈	—	20	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	5	1	no	—
287	C₂×C₄	—	5	2	no	—
288	C₂×C₄	—	5	2	no	—
289	C₂×C₄	—	5	1	no	—
290	C₂×C₄	—	5	1	no	—
291	C₂×C₄	—	10	2	no	—
292	C₂×C₄	—	10	2	no	—
293	C₂×C₄	—	20	1	no	—
294	C₂×C₄	—	30	4	no	—
295	C₂×C₄	—	60	2	no	—
296	C₂×C₄	—	60	2	no	—
297	C₂×C₄	#6763	1	1	yes	characteristic
298	C₂×C₄	#6763	1	1	yes	characteristic
299	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
300	C₈	—	30	4	no	—
301	C₂×C₃	—	10	1	no	—
302	C₂×C₃	—	10	1	no	—
303	C₂×C₃	#65769	1	1	yes	characteristic
304	C₂×C₃	C₂×#4010	1	1	yes	characteristic
305	C₂×C₃	(C₂×AGL(1,𝔽₅))⋊Q₈	1	1	yes	characteristic
306	C₅	#71841	1	1	yes	characteristic, complemented, Sylow
307	C₂²	—	5	1	no	—
308	C₂²	—	5	1	no	—
309	C₂²	—	10	1	no	—
310	C₂²	—	10	1	no	—
311	C₂²	C₂×#6763	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	4	2	no	—
314	C₄	—	5	2	no	—
315	C₄	—	10	1	no	—
316	C₄	—	10	1	no	—
317	C₄	—	20	2	no	—
318	C₄	—	60	4	no	—
319	C₄	#91239	1	2	yes	—
320	C₄	Q₈×ΓL(1,𝔽₁₆)	1	2	yes	—
321	C₃	#10604680	1	1	yes	characteristic, complemented, Sylow
322	C₂	—	10	1	no	—
323	C₂	—	10	1	no	—
324	C₂	#11752428	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	#1986
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
9. C₃×C₅×C₄⋊Q₈	C₄	semidirect
14. C₃×#4036	C₄	semidirect
17. C₃×#4015	C₄	semidirect
36. #6758	Q₈	semidirect
56. C₅×C₄⋊Q₈	Dic₃	semidirect
61. #4036	Dic₃	semidirect
64. #4015	Dic₃	semidirect
89. C₃×C₄⋊Q₈	AGL(1,𝔽₅)	semidirect
188. C₄⋊Q₈	ΓL(1,𝔽₁₆)	semidirect
262. C₃×C₅	#1986	semidirect
306. C₅	#71841	semidirect
321. C₃	#10604680	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
C₄×#6758	2
#11746746	2
#10604680	3
#71841	5