Subgroups — #14286330

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⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₅×#215	C₂	1	1	yes	maximal, characteristic
3	#11746998	C₂	1	1	yes	maximal, characteristic
4	C₃×Q₈⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
5	#11745869	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745837	C₂	1	1	yes	maximal, characteristic
7	#11745832	C₂	1	1	yes	maximal, characteristic, complemented
8	SD₁₆⋊#168	—	3	1	no	maximal, Hall
9	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×#215	C₄	1	1	yes	characteristic
11	C₄×C₅×D₁₂	C₂²	1	1	yes	characteristic
12	C₂×D₃×#168	—	2	1	no	—
13	C₄⋊(C₃⋊#168)	—	2	1	no	—
14	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
15	C₄×C₃⋊#168	—	2	1	no	—
16	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
17	C₅×#835	C₄	1	1	yes	characteristic
18	C₅×#809	C₄	1	1	yes	characteristic
19	C₅×#808	C₄	1	1	yes	characteristic
20	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
22	C₃×C₄×#168	C₂²	1	1	yes	characteristic
23	(C₂×D₃)⋊#168	—	2	1	no	—
24	#68214	—	5	1	no	maximal, Hall
25	C₄×C₅×SD₁₆	—	3	1	no	—
26	Q₈×#168	—	3	1	no	—
27	D₄×#168	—	3	1	no	—
28	Q₈⋊#168	D₃	1	1	yes	characteristic, complemented
29	D₄⋊#168	—	3	1	no	—
30	#64678	—	3	1	no	—
31	#64675	—	3	1	no	—
32	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₅×D₁₂	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×C₅×D₃	—	2	1	no	—
35	C₃×C₅×C₄⋊C₄	—	2	1	no	—
36	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄²×C₅	—	2	1	no	—
38	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
39	C₂×C₃⋊#168	—	2	1	no	—
40	C₂×C₃⋊#168	D₄	1	1	yes	characteristic
41	C₂×C₃⋊#168	D₄	1	1	yes	characteristic
42	C₅×#215	C₈	1	4	yes	complemented
43	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
44	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₃×#168	—	2	1	no	—
48	C₂×C₃×#168	—	2	1	no	—
49	D₃×#168	—	4	2	no	—
50	C₄×#215	D₅	1	1	yes	characteristic
51	Q₈×C₃⋊C₈	—	5	1	no	—
52	#4834	—	5	1	no	—
53	C₃×Q₈⋊C₈	—	5	1	no	—
54	#4479	—	5	1	no	—
55	#4459	—	5	1	no	—
56	#4454	—	5	1	no	—
57	C₂×C₅×SD₁₆	—	3	1	no	—
58	C₄×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
59	C₄×C₅×D₄	—	3	1	no	—
60	C₂²×#168	—	6	1	no	—
61	C₅×#106	—	3	1	no	—
62	C₅×#103	—	3	1	no	—
63	C₅×#102	—	3	1	no	—
64	C₄×C₅×C₈	—	3	1	no	—
65	C₂²⋊#168	—	6	1	no	—
66	C₄⋊#168	—	3	1	no	—
67	C₄⋊#168	—	6	1	no	—
68	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
69	C₄×#168	—	3	1	no	—
70	C₄×#168	—	6	1	no	—
71	C₄×#168	C₂×D₃	1	1	yes	characteristic
72	SD₁₆⋊C₈	—	15	1	no	Sylow
73	C₂²×C₅×D₃	—	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×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
81	C₂×C₅×Dic₃	—	2	1	no	—
82	C₅×D₁₂	—	2	1	no	—
83	C₅×D₁₂	C₂×C₈	1	2	yes	—
84	C₄×C₅×D₃	—	4	1	no	—
85	C₄×C₅×D₃	—	4	1	no	—
86	C₃⋊#168	—	2	1	no	—
87	C₃⋊#168	—	2	1	no	—
88	C₃⋊#168	—	4	1	no	—
89	C₃×#168	—	4	1	no	—
90	C₃×#168	—	4	1	no	—
91	C₅×C₃⋊C₈	—	2	1	no	—
92	C₅×C₃⋊C₈	C₂×C₈	1	2	yes	complemented
93	C₃×C₄×Q₈	C₂×D₅	1	1	yes	characteristic
94	C₂×#215	Dic₅	1	1	yes	characteristic
95	C₂×D₃×C₈	—	10	1	no	—
96	C₄×D₁₂	C₂×D₅	1	1	yes	characteristic
97	C₃×C₄⋊C₈	—	5	1	no	—
98	C₃×C₄×C₈	—	5	1	no	—
99	#835	Dic₅	1	1	yes	characteristic
100	(C₂×D₃)⋊C₈	—	10	1	no	—
101	#809	Dic₅	1	1	yes	characteristic
102	#808	Dic₅	1	1	yes	characteristic
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	—	10	1	no	—
107	C₄×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
108	C₂×C₅×Q₈	C₄×D₃	1	1	yes	characteristic
109	C₂×C₅×D₄	—	3	1	no	—
110	C₂²×C₄×C₅	—	6	1	no	—
111	C₅×SD₁₆	—	3	4	no	—
112	C₂×C₅×C₈	—	3	1	no	—
113	C₂×C₅×C₈	—	3	1	no	—
114	C₅×C₄⋊C₄	—	2	1	no	—
115	C₅×C₄⋊C₄	—	3	1	no	—
116	C₅×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
117	C₅×C₂²⋊C₄	—	6	1	no	—
118	C₄²×C₅	—	2	1	no	—
119	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
120	C₂×#168	—	2	1	no	—
121	C₂×#168	—	2	1	no	—
122	C₂×#168	—	3	1	no	—
123	C₂×#168	—	3	1	no	—
124	C₂×#168	—	6	1	no	—
125	C₂×#168	—	12	2	no	—
126	C₈×Q₈	—	15	1	no	—
127	C₄×SD₁₆	—	3	1	no	—
128	C₈×D₄	—	15	1	no	—
129	#335	—	15	1	no	—
130	Q₈⋊C₈	—	5	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×C₅×D₃	—	2	2	no	—
135	C₂×C₅×D₃	—	4	1	no	—
136	C₂×C₅×D₃	—	4	1	no	—
137	C₃×C₄×C₅	—	2	2	no	—
138	C₃×C₄×C₅	—	2	1	no	—
139	C₃×C₄×C₅	—	4	1	no	—
140	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
141	C₃×C₄×C₅	#131	1	1	yes	characteristic
142	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₃×C₄×C₅	#137	1	1	yes	characteristic
144	C₅×Dic₃	—	4	1	no	—
145	C₂×C₃×Q₈	C₂×Dic₅	1	1	yes	characteristic
146	C₂×D₁₂	C₂×Dic₅	1	1	yes	characteristic
147	C₂×C₄×D₃	—	2	1	no	—
148	C₂×C₃×C₈	—	10	1	no	—
149	C₂×C₃×C₈	—	10	1	no	—
150	C₃×C₄⋊C₄	—	2	1	no	—
151	C₃×C₄⋊C₄	C₂×Dic₅	1	1	yes	characteristic
152	C₃×C₄²	—	2	1	no	—
153	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
154	#215	#168	1	4	yes	complemented
155	(C₂×D₃)⋊C₄	—	2	1	no	—
156	C₄⋊Dic₃	C₂×Dic₅	1	1	yes	characteristic
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	C₂×Dic₅	1	1	yes	characteristic
161	C₂×C₃⋊C₈	C₂×Dic₅	1	1	yes	characteristic
162	D₃×C₈	—	20	2	no	—
163	C₂³×C₅	—	6	1	no	—
164	C₅×Q₈	—	2	1	no	—
165	C₅×Q₈	D₃×C₈	1	2	yes	complemented
166	C₅×D₄	—	3	2	no	—
167	C₅×D₄	—	6	1	no	—
168	C₂×C₄×C₅	—	2	1	no	—
169	C₂×C₄×C₅	—	2	1	no	—
170	C₂×C₄×C₅	—	6	1	no	—
171	C₂×C₄×C₅	—	12	1	no	—
172	C₂×C₄×C₅	—	12	1	no	—
173	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
174	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
175	C₂×C₄×C₅	#237	1	1	yes	characteristic
176	C₅×C₈	—	3	2	no	—
177	C₅×C₈	—	6	1	no	—
178	#168	—	4	1	no	—
179	#168	—	4	1	no	—
180	#168	—	6	1	no	—
181	#168	—	6	1	no	—
182	#168	—	12	1	no	—
183	C₂×SD₁₆	—	3	1	no	—
184	C₂²×C₈	—	30	1	no	—
185	C₄×Q₈	D₃×D₅	1	1	yes	characteristic
186	C₄×D₄	—	3	1	no	—
187	#106	—	3	1	no	—
188	C₄⋊C₈	—	5	1	no	—
189	C₄⋊C₈	—	15	1	no	—
190	C₄⋊C₈	—	30	1	no	—
191	#103	—	3	1	no	—
192	#102	—	3	1	no	—
193	C₂²⋊C₈	—	30	1	no	—
194	C₄×C₈	—	3	1	no	—
195	C₄×C₈	—	5	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	C₅×D₃	—	4	2	no	—
202	C₂²×D₃	—	2	1	no	—
203	C₃×Q₈	—	2	1	no	—
204	C₃×Q₈	C₂×#168	1	2	yes	complemented
205	C₂×C₃×C₄	—	2	1	no	—
206	C₂×C₃×C₄	—	2	1	no	—
207	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
208	C₂×C₃×C₄	#711	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
210	C₂×Dic₃	—	2	1	no	—
211	D₁₂	—	2	1	no	—
212	D₁₂	C₂×#168	1	2	yes	—
213	C₄×D₃	—	4	1	no	—
214	C₄×D₃	—	4	1	no	—
215	C₃×C₈	—	20	1	no	—
216	C₃×C₈	—	20	1	no	—
217	C₃⋊C₈	—	2	1	no	—
218	C₃⋊C₈	—	10	1	no	—
219	C₃⋊C₈	—	10	1	no	—
220	C₃⋊C₈	—	20	1	no	—
221	C₃⋊C₈	C₂×#168	1	2	yes	complemented
222	C₂²×C₅	—	6	2	no	—
223	C₂²×C₅	—	12	1	no	—
224	C₂²×C₅	—	12	1	no	—
225	C₂²×C₅	#67⋊C₄	1	1	yes	characteristic
226	C₄×C₅	—	2	2	no	—
227	C₄×C₅	—	2	1	no	—
228	C₄×C₅	—	4	1	no	—
229	C₄×C₅	—	12	1	no	—
230	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
231	C₄×C₅	#907	1	1	yes	characteristic
232	C₄×C₅	#914	1	1	yes	characteristic
233	C₄×C₅	#918	1	1	yes	characteristic
234	C₂×Q₈	D₃×Dic₅	1	1	yes	characteristic
235	C₂×D₄	—	3	1	no	—
236	C₂²×C₄	—	6	1	no	—
237	SD₁₆	—	3	4	no	—
238	C₂×C₈	—	3	1	no	—
239	C₂×C₈	—	3	1	no	—
240	C₂×C₈	—	10	1	no	—
241	C₂×C₈	—	10	1	no	—
242	C₂×C₈	—	15	1	no	—
243	C₂×C₈	—	15	1	no	—
244	C₂×C₈	—	30	1	no	—
245	C₂×C₈	—	60	2	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	3	1	no	—
248	C₄⋊C₄	D₃×Dic₅	1	1	yes	characteristic
249	C₂²⋊C₄	—	6	1	no	—
250	C₄²	—	2	1	no	—
251	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
252	C₃×C₅	SD₁₆⋊C₈	1	1	yes	characteristic, complemented, Hall
253	C₂²×C₃	D₄×Dic₅	1	1	yes	characteristic
254	C₂×D₃	—	2	2	no	—
255	C₂×D₃	—	4	1	no	—
256	C₂×D₃	—	4	1	no	—
257	C₃×C₄	—	2	2	no	—
258	C₃×C₄	—	2	1	no	—
259	C₃×C₄	—	4	1	no	—
260	C₃×C₄	#4060	1	1	yes	characteristic
261	C₃×C₄	#4061	1	1	yes	characteristic
262	C₃×C₄	C₂²×#168	1	1	yes	characteristic
263	C₃×C₄	#4094	1	1	yes	characteristic
264	Dic₃	—	4	1	no	—
265	C₂×C₅	—	12	2	no	—
266	C₂×C₅	#67⋊C₈	1	1	yes	characteristic
267	C₂×C₅	#4817	1	1	yes	characteristic
268	C₂×C₅	#4823	1	1	yes	characteristic
269	C₂³	—	6	1	no	—
270	Q₈	—	2	1	no	—
271	Q₈	D₃×#168	1	2	yes	complemented
272	D₄	—	3	2	no	—
273	D₄	—	6	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	6	1	no	—
277	C₂×C₄	—	12	1	no	—
278	C₂×C₄	—	12	1	no	—
279	C₂×C₄	#6780	1	1	yes	characteristic
280	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
281	C₂×C₄	#6790	1	1	yes	characteristic
282	C₈	—	3	2	no	—
283	C₈	—	6	1	no	—
284	C₈	—	20	1	no	—
285	C₈	—	20	1	no	—
286	C₈	—	30	1	no	—
287	C₈	—	30	1	no	—
288	C₈	—	60	1	no	—
289	C₂×C₃	D₄×#168	1	1	yes	characteristic
290	C₂×C₃	#65453	1	1	yes	characteristic
291	C₂×C₃	#65486	1	1	yes	characteristic
292	D₃	—	4	2	no	—
293	C₅	#68214	1	1	yes	characteristic, complemented, Sylow
294	C₂²	—	6	2	no	—
295	C₂²	—	12	1	no	—
296	C₂²	—	12	1	no	—
297	C₂²	#67⋊Dic₅	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	2	no	—
300	C₄	—	4	1	no	—
301	C₄	—	12	1	no	—
302	C₄	C₂×D₃×#168	1	1	yes	characteristic
303	C₄	#91288	1	1	yes	characteristic
304	C₄	#91506	1	1	yes	characteristic
305	C₄	#91524	1	1	yes	characteristic
306	C₃	SD₁₆⋊#168	1	1	yes	characteristic, complemented, Sylow
307	C₂	—	12	2	no	—
308	C₂	#67⋊#168	1	1	yes	characteristic
309	C₂	#11749242	1	1	yes	characteristic
310	C₂	#11749443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6790
commutator	C₃×C₄×C₅	C₂²×C₈
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₄×C₅×Q₈	C₂²
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×C₅×D₁₂	C₂×C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄×Q₈	D₃×D₅
hypocenter	C₃×C₅	SD₁₆⋊C₈
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×C₃⋊#168	C₂	semidirect
4. C₃×Q₈⋊#168	C₂	semidirect
5. #11745869	C₂	semidirect
7. #11745832	C₂	semidirect
28. Q₈⋊#168	D₃	semidirect
42. C₅×#215	C₈	semidirect
75. C₃×C₅×Q₈	C₂×C₈	semidirect
92. C₅×C₃⋊C₈	C₂×C₈	semidirect
154. #215	#168	semidirect
165. C₅×Q₈	D₃×C₈	semidirect
204. C₃×Q₈	C₂×#168	semidirect
221. C₃⋊C₈	C₂×#168	semidirect
252. C₃×C₅	SD₁₆⋊C₈	semidirect
271. Q₈	D₃×#168	semidirect
293. C₅	#68214	semidirect
306. C₃	SD₁₆⋊#168	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₃⋊#168	2
C₄×C₅×#215	2
#11746998	2
C₃×Q₈⋊#168	2
#11745869	2
#11745837	2
#11745832	2
SD₁₆⋊#168	3
#68214	5