Subgroups — #14286304

Order 1920.

352 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₃⋊#170	C₂	1	1	yes	maximal, characteristic, complemented
2	#11752361	C₂	1	1	yes	maximal, characteristic
3	C₃×#65077	C₂	1	1	yes	maximal, characteristic
4	#11746801	C₂	1	1	yes	maximal, characteristic, complemented
5	#11746755	C₂	1	1	yes	maximal, characteristic, complemented
6	#11746750	C₂	1	1	yes	maximal, characteristic, complemented
7	#11746742	C₂	1	1	yes	maximal, characteristic
8	#10603969	—	3	1	no	maximal, Hall
9	C₂×D₅⋊(C₃⋊C₈)	—	2	1	no	—
10	C₃×Q₈×Dic₅	C₂²	1	1	yes	characteristic
11	C₂×C₃×#688	C₄	1	1	yes	characteristic
12	C₃×#4039	C₂²	1	1	yes	characteristic
13	#91232	—	2	1	no	—
14	C₄⋊(C₃⋊#170)	—	2	1	no	—
15	C₄⋊(C₃⋊#170)	C₂²	1	1	yes	characteristic
16	C₄⋊(C₃⋊#170)	C₂²	1	1	yes	characteristic
17	C₄×C₃⋊#170	—	2	1	no	—
18	C₄×C₃⋊#170	C₂²	1	1	yes	characteristic
19	C₄×C₃⋊#170	C₂²	1	1	yes	characteristic
20	C₃×C₅×#103	C₄	1	1	yes	characteristic
21	C₃×#3953	C₄	1	1	yes	characteristic
22	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
23	C₃×#3940	C₄	1	1	yes	characteristic
24	SD₁₆⋊(C₃⋊C₈)	—	5	1	no	maximal, Hall
25	Q₈×#170	—	3	1	no	—
26	#65702	—	3	1	no	—
27	#65077	D₃	1	1	yes	characteristic
28	Q₈⋊#170	—	3	1	no	—
29	#64884	—	3	1	no	—
30	#64879	—	3	1	no	—
31	#64871	—	3	1	no	—
32	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×D₂₀	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃×C₄×D₅	—	2	1	no	—
35	C₂×C₃⋊#170	—	2	1	no	—
36	C₂×C₃⋊#170	—	2	1	no	—
37	C₂×C₃⋊#170	—	2	1	no	—
38	C₂×C₃⋊#170	D₄	1	1	yes	characteristic
39	C₂×C₃⋊#170	D₄	1	1	yes	characteristic
40	D₅⋊(C₃⋊C₈)	—	4	2	no	—
41	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
42	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
43	C₃×#688	C₈	1	4	yes	complemented
44	C₃×#685	—	2	1	no	—
45	C₃×C₄⋊Dic₅	—	2	1	no	—
46	C₃×C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
47	C₃×C₄×Dic₅	—	2	1	no	—
48	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
49	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
50	C₃×C₄×SD₁₆	—	5	1	no	—
51	Q₈×C₃⋊C₈	—	5	1	no	—
52	D₄×C₃⋊C₈	—	5	1	no	—
53	#4535	—	5	1	no	—
54	D₄⋊(C₃⋊C₈)	—	5	1	no	—
55	#4458	—	5	1	no	—
56	#4455	—	5	1	no	—
57	C₂×D₅⋊C₈	—	6	1	no	—
58	Q₈×Dic₅	C₂×D₃	1	1	yes	characteristic
59	C₂×#688	Dic₃	1	1	yes	characteristic
60	#4039	C₂×D₃	1	1	yes	characteristic
61	#4003	—	6	1	no	—
62	C₄⋊#170	—	3	1	no	—
63	C₄⋊#170	—	3	1	no	—
64	C₄⋊#170	—	6	1	no	—
65	C₄×#170	—	3	1	no	—
66	C₄×#170	—	3	1	no	—
67	C₄×#170	—	6	1	no	—
68	C₅×#103	Dic₃	1	1	yes	characteristic
69	#3953	Dic₃	1	1	yes	characteristic
70	C₈×Dic₅	C₂×D₃	1	1	yes	characteristic
71	#3940	Dic₃	1	1	yes	characteristic
72	SD₁₆⋊C₈	—	15	1	no	Sylow
73	C₂²×C₃×D₅	—	2	1	no	—
74	C₃×C₅×Q₈	—	2	1	no	—
75	C₃×C₅×Q₈	C₂×C₈	1	2	yes	complemented
76	C₂×C₃×C₄×C₅	—	2	1	no	—
77	C₂×C₃×C₄×C₅	—	2	1	no	—
78	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃×Dic₅	—	2	1	no	—
80	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×Dic₅	#41	1	1	yes	characteristic
82	C₃×D₂₀	—	2	1	no	—
83	C₃×D₂₀	C₂×C₈	1	2	yes	—
84	C₃×C₄×D₅	—	4	1	no	—
85	C₃×C₄×D₅	—	4	1	no	—
86	C₃⋊#170	—	2	1	no	—
87	C₃⋊#170	—	2	1	no	—
88	C₃⋊#170	—	4	1	no	—
89	C₃⋊#170	—	4	1	no	—
90	C₃⋊#170	—	4	1	no	—
91	C₃×C₅×C₈	—	2	1	no	—
92	C₃×#168	C₂×C₈	1	2	yes	complemented
93	C₂×C₃×SD₁₆	—	5	1	no	—
94	C₃×C₄×Q₈	—	5	1	no	—
95	C₃×C₄×D₄	—	5	1	no	—
96	C₂²×C₃⋊C₈	—	10	1	no	—
97	C₃×#106	—	5	1	no	—
98	C₃×#103	AGL(1,𝔽₅)	1	1	yes	characteristic
99	C₃×#102	—	5	1	no	—
100	C₃×C₄×C₈	—	5	1	no	—
101	#830	—	10	1	no	—
102	C₄⋊(C₃⋊C₈)	—	5	1	no	—
103	C₄⋊(C₃⋊C₈)	—	5	1	no	—
104	C₄⋊(C₃⋊C₈)	—	10	1	no	—
105	C₄×C₃⋊C₈	—	5	1	no	—
106	C₄×C₃⋊C₈	—	5	1	no	—
107	C₄×C₃⋊C₈	—	10	1	no	—
108	C₂×C₅×Q₈	C₂×Dic₃	1	1	yes	characteristic
109	C₂×D₂₀	C₂×Dic₃	1	1	yes	characteristic
110	C₂×C₄×D₅	—	2	1	no	—
111	C₂×#170	—	3	1	no	—
112	C₂×#170	—	3	1	no	—
113	C₂×#170	—	6	1	no	—
114	C₂×#170	—	6	1	no	—
115	C₂×#170	—	6	1	no	—
116	D₅⋊C₈	—	12	2	no	—
117	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
118	C₅×C₄⋊C₄	C₂×Dic₃	1	1	yes	characteristic
119	#688	C₃⋊C₈	1	4	yes	complemented
120	#685	—	2	1	no	—
121	C₄⋊Dic₅	—	2	1	no	—
122	C₄⋊Dic₅	C₂×Dic₃	1	1	yes	characteristic
123	C₄×Dic₅	—	2	1	no	—
124	C₄×Dic₅	C₂²×D₃	1	1	yes	characteristic
125	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
126	C₈×Q₈	—	15	1	no	—
127	C₄×SD₁₆	—	5	1	no	—
128	C₈×D₄	—	15	1	no	—
129	#335	—	15	1	no	—
130	Q₈⋊C₈	—	15	1	no	—
131	D₄⋊C₈	—	15	1	no	—
132	#322	—	15	1	no	—
133	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
134	C₂×C₃×D₅	—	2	2	no	—
135	C₂×C₃×D₅	—	4	1	no	—
136	C₂×C₃×D₅	—	4	1	no	—
137	C₃×C₄×C₅	—	2	2	no	—
138	C₃×C₄×C₅	—	4	1	no	—
139	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
140	C₃×C₄×C₅	#131	1	1	yes	characteristic
141	C₃×Dic₅	—	2	1	no	—
142	C₃×Dic₅	—	4	1	no	—
143	C₃×Dic₅	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₃×Dic₅	#137	1	1	yes	characteristic
145	C₂×C₃×Q₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
146	C₂×C₃×D₄	—	5	1	no	—
147	C₂²×C₃×C₄	—	10	1	no	—
148	C₃×SD₁₆	—	5	4	no	—
149	C₂×C₃×C₈	—	5	1	no	—
150	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
151	C₃×C₄⋊C₄	—	5	1	no	—
152	C₃×C₄⋊C₄	—	10	1	no	—
153	C₃×C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
154	C₃×C₂²⋊C₄	—	10	1	no	—
155	C₃×C₄²	—	5	1	no	—
156	C₃×C₄²	—	10	1	no	—
157	C₂×C₃⋊C₈	—	5	1	no	—
158	C₂×C₃⋊C₈	—	5	1	no	—
159	C₂×C₃⋊C₈	—	10	1	no	—
160	C₂×C₃⋊C₈	—	10	1	no	—
161	C₂×C₃⋊C₈	—	10	1	no	—
162	C₂×C₃⋊C₈	—	20	2	no	—
163	C₂²×D₅	—	2	1	no	—
164	C₅×Q₈	—	2	1	no	—
165	C₅×Q₈	C₂×C₃⋊C₈	1	2	yes	complemented
166	C₂×C₄×C₅	—	2	1	no	—
167	C₂×C₄×C₅	—	2	1	no	—
168	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
169	C₂×Dic₅	—	2	1	no	—
170	C₂×Dic₅	D₃×D₄	1	1	yes	characteristic
171	C₂×Dic₅	#237	1	1	yes	characteristic
172	D₂₀	—	2	1	no	—
173	D₂₀	C₂×C₃⋊C₈	1	2	yes	—
174	C₄×D₅	—	4	1	no	—
175	C₄×D₅	—	4	1	no	—
176	#170	—	6	1	no	—
177	#170	—	6	1	no	—
178	#170	—	12	1	no	—
179	#170	—	12	1	no	—
180	#170	—	12	1	no	—
181	C₅×C₈	—	2	1	no	—
182	#168	C₂×C₃⋊C₈	1	2	yes	complemented
183	C₂×SD₁₆	—	5	1	no	—
184	C₂²×C₈	—	30	1	no	—
185	C₄×Q₈	—	5	1	no	—
186	C₄×D₄	—	5	1	no	—
187	#106	—	5	1	no	—
188	C₄⋊C₈	—	15	1	no	—
189	C₄⋊C₈	—	15	1	no	—
190	C₄⋊C₈	—	30	1	no	—
191	#103	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
192	#102	—	5	1	no	—
193	C₂²⋊C₈	—	30	1	no	—
194	C₄×C₈	—	5	1	no	—
195	C₄×C₈	—	15	1	no	—
196	C₄×C₈	—	15	1	no	—
197	C₄×C₈	—	30	1	no	—
198	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
199	C₂×C₃×C₅	#440	1	1	yes	characteristic
200	C₂×C₃×C₅	#444	1	1	yes	characteristic
201	C₃×D₅	—	4	2	no	—
202	C₂³×C₃	—	10	1	no	—
203	C₃×Q₈	—	2	1	no	—
204	C₃×Q₈	D₅⋊C₈	1	2	yes	complemented
205	C₃×D₄	—	5	2	no	—
206	C₃×D₄	—	10	1	no	—
207	C₂×C₃×C₄	—	2	1	no	—
208	C₂×C₃×C₄	—	2	1	no	—
209	C₂×C₃×C₄	—	5	1	no	—
210	C₂×C₃×C₄	—	5	1	no	—
211	C₂×C₃×C₄	—	10	1	no	—
212	C₂×C₃×C₄	—	20	1	no	—
213	C₂×C₃×C₄	—	20	1	no	—
214	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
215	C₃×C₈	—	2	1	no	—
216	C₃×C₈	—	5	2	no	—
217	C₃⋊C₈	—	10	1	no	—
218	C₃⋊C₈	—	10	1	no	—
219	C₃⋊C₈	—	20	1	no	—
220	C₃⋊C₈	—	20	1	no	—
221	C₃⋊C₈	—	20	1	no	—
222	C₂²×C₅	D₄×Dic₃	1	1	yes	characteristic
223	C₂×D₅	—	2	2	no	—
224	C₂×D₅	—	4	1	no	—
225	C₂×D₅	—	4	1	no	—
226	C₄×C₅	—	2	2	no	—
227	C₄×C₅	—	4	1	no	—
228	C₄×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
229	C₄×C₅	#948	1	1	yes	characteristic
230	Dic₅	—	2	1	no	—
231	Dic₅	—	4	1	no	—
232	Dic₅	#914	1	1	yes	characteristic
233	Dic₅	#915	1	1	yes	characteristic
234	C₂×Q₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
235	C₂×D₄	—	5	1	no	—
236	C₂²×C₄	—	10	1	no	—
237	SD₁₆	—	5	4	no	—
238	C₂×C₈	—	5	1	no	—
239	C₂×C₈	—	15	1	no	—
240	C₂×C₈	—	15	1	no	—
241	C₂×C₈	—	30	1	no	—
242	C₂×C₈	—	30	1	no	—
243	C₂×C₈	—	30	1	no	—
244	C₂×C₈	—	60	2	no	—
245	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
246	C₄⋊C₄	—	5	1	no	—
247	C₄⋊C₄	—	10	1	no	—
248	C₄⋊C₄	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
249	C₂²⋊C₄	—	10	1	no	—
250	C₄²	—	5	1	no	—
251	C₄²	—	10	1	no	—
252	C₃×C₅	SD₁₆⋊C₈	1	1	yes	characteristic, complemented, Hall
253	C₂²×C₃	—	10	2	no	—
254	C₂²×C₃	—	20	1	no	—
255	C₂²×C₃	—	20	1	no	—
256	C₂²×C₃	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
257	C₃×C₄	—	2	2	no	—
258	C₃×C₄	—	4	1	no	—
259	C₃×C₄	—	5	1	no	—
260	C₃×C₄	—	5	1	no	—
261	C₃×C₄	—	10	1	no	—
262	C₃×C₄	—	20	1	no	—
263	C₃×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
264	C₃×C₄	#4131	1	1	yes	characteristic
265	C₂×C₅	D₄×C₃⋊C₈	1	1	yes	characteristic
266	C₂×C₅	#5164	1	1	yes	characteristic
267	C₂×C₅	#5197	1	1	yes	characteristic
268	D₅	—	4	2	no	—
269	C₂³	—	10	1	no	—
270	Q₈	—	2	1	no	—
271	Q₈	D₅⋊(C₃⋊C₈)	1	2	yes	complemented
272	D₄	—	5	2	no	—
273	D₄	—	10	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	5	1	no	—
277	C₂×C₄	—	5	1	no	—
278	C₂×C₄	—	10	1	no	—
279	C₂×C₄	—	20	1	no	—
280	C₂×C₄	—	20	1	no	—
281	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
282	C₈	—	2	1	no	—
283	C₈	—	5	2	no	—
284	C₈	—	30	1	no	—
285	C₈	—	30	1	no	—
286	C₈	—	60	1	no	—
287	C₈	—	60	1	no	—
288	C₈	—	60	1	no	—
289	C₂×C₃	—	20	2	no	—
290	C₂×C₃	#65692	1	1	yes	characteristic
291	C₂×C₃	#65735	1	1	yes	characteristic
292	C₂×C₃	#65741	1	1	yes	characteristic
293	C₅	SD₁₆⋊(C₃⋊C₈)	1	1	yes	characteristic, complemented, Sylow
294	C₂²	—	10	2	no	—
295	C₂²	—	20	1	no	—
296	C₂²	—	20	1	no	—
297	C₂²	D₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	4	1	no	—
300	C₄	—	5	1	no	—
301	C₄	—	5	1	no	—
302	C₄	—	10	1	no	—
303	C₄	—	20	1	no	—
304	C₄	C₂×D₅⋊(C₃⋊C₈)	1	1	yes	characteristic
305	C₄	#91990	1	1	yes	characteristic
306	C₃	#10603969	1	1	yes	characteristic, complemented, Sylow
307	C₂	—	20	2	no	—
308	C₂	#11752351	1	1	yes	characteristic
309	C₂	#11752394	1	1	yes	characteristic
310	C₂	#11752400	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×ΓL(1,𝔽₁₆)
commutator	C₃×C₄×C₅	C₂²×C₈
Frattini	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)
Fitting	C₃×C₅×#103	C₄
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×C₃×D₂₀	C₂×C₄
cosocle	C₃×C₄×Dic₅	C₂³
hypercenter	#103	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	SD₁₆⋊C₈
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
1. Q₈×C₃⋊#170	C₂	semidirect
4. #11746801	C₂	semidirect
5. #11746755	C₂	semidirect
6. #11746750	C₂	semidirect
43. C₃×#688	C₈	semidirect
75. C₃×C₅×Q₈	C₂×C₈	semidirect
92. C₃×#168	C₂×C₈	semidirect
119. #688	C₃⋊C₈	semidirect
165. C₅×Q₈	C₂×C₃⋊C₈	semidirect
182. #168	C₂×C₃⋊C₈	semidirect
204. C₃×Q₈	D₅⋊C₈	semidirect
252. C₃×C₅	SD₁₆⋊C₈	semidirect
271. Q₈	D₅⋊(C₃⋊C₈)	semidirect
293. C₅	SD₁₆⋊(C₃⋊C₈)	semidirect
306. C₃	#10603969	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₃⋊#170	2
#11752361	2
C₃×#65077	2
#11746801	2
#11746755	2
#11746750	2
#11746742	2
#10603969	3
SD₁₆⋊(C₃⋊C₈)	5