Subgroups — #14418939

Order 1920.

912 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#6821	C₂	1	1	yes	maximal, characteristic
2	C₅×Q₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×Q₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746994	C₂	1	3	yes	maximal, complemented
5	#11746988	C₂	1	3	yes	maximal, complemented
6	C₄×#6648	C₂	1	3	yes	maximal, complemented
7	#11746859	C₂	1	3	yes	maximal, complemented
8	C₈×#713	—	3	1	no	maximal, Hall
9	C₂×#6821	C₄	1	1	yes	characteristic
10	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
11	Q₈×Dic₁₅	C₄	1	1	yes	characteristic
12	#91782	C₄	1	3	yes	—
13	#91781	C₄	1	3	yes	—
14	C₄×D₆₀	C₂²	1	3	yes	—
15	C₄²×D₁₅	C₂²	1	3	yes	—
16	C₂×#6648	C₂²	1	3	yes	—
17	C₂×#6648	C₂²	1	3	yes	—
18	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	3	yes	—
19	C₄×C₅×C₃⋊C₈	C₂²	1	3	yes	—
20	C₃×C₄⋊#168	C₂²	1	3	yes	—
21	C₃×C₄×#168	C₂²	1	3	yes	—
22	#90957	C₂²	1	6	yes	—
23	#90951	C₂²	1	3	yes	—
24	#90951	C₂²	1	1	yes	characteristic
25	C₈×#239	—	5	1	no	maximal, Hall
26	C₄×#713	—	3	1	no	—
27	C₅×C₈×Q₈	—	3	1	no	—
28	Q₈×#168	D₃	1	1	yes	characteristic, complemented
29	#65123	—	3	3	no	—
30	#65122	—	3	3	no	—
31	C₈×D₂₀	—	3	3	no	—
32	C₄×C₈×D₅	—	3	3	no	—
33	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
34	#6821	C₈	1	8	yes	complemented
35	C₂×D₆₀	C₂×C₄	1	3	yes	—
36	C₂×C₄×D₁₅	C₂×C₄	1	3	yes	—
37	C₂×C₄×D₁₅	C₂×C₄	1	3	yes	—
38	C₂×C₄×D₁₅	C₂³	1	3	yes	—
39	C₃×C₅×C₄⋊C₄	C₂×C₄	1	3	yes	—
40	C₃×C₄²×C₅	C₂³	1	3	yes	—
41	#6714	C₂×C₄	1	6	yes	—
42	C₄⋊Dic₁₅	C₂×C₄	1	3	yes	—
43	C₄×Dic₁₅	C₂×C₄	1	3	yes	—
44	C₄×Dic₁₅	C₂×C₄	1	3	yes	—
45	C₄×Dic₁₅	C₂³	1	1	yes	characteristic
46	C₂×C₅×C₃⋊C₈	C₂³	1	3	yes	—
47	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
48	C₂×C₃×#168	C₂³	1	3	yes	—
49	C₂×C₃×#168	C₂³	1	1	yes	characteristic
50	#6648	—	2	6	no	—
51	#6648	—	2	6	no	—
52	C₄×#239	—	5	1	no	—
53	C₃×C₈×Q₈	—	5	1	no	—
54	Q₈×C₃⋊C₈	D₅	1	1	yes	characteristic, complemented
55	#4834	—	5	3	no	—
56	#4833	—	5	3	no	—
57	C₈×D₁₂	—	5	3	no	—
58	C₄×D₃×C₈	—	5	3	no	—
59	C₂×#713	—	3	1	no	—
60	C₄×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
61	Q₈×Dic₅	—	3	1	no	—
62	C₂×C₈×D₅	—	3	3	no	—
63	C₂×C₈×D₅	—	3	3	no	—
64	#4039	—	3	3	no	—
65	#4038	—	3	3	no	—
66	C₄×D₂₀	—	3	3	no	—
67	C₄²×D₅	—	3	3	no	—
68	C₅×C₄⋊C₈	—	3	3	no	—
69	C₄×C₅×C₈	—	3	3	no	—
70	#3952	—	3	6	no	—
71	C₈×Dic₅	—	3	3	no	—
72	C₈×Dic₅	—	3	1	no	—
73	C₄⋊#168	C₂×D₃	1	3	yes	—
74	C₄×#168	C₂×D₃	1	3	yes	—
75	C₈×#41	—	15	1	no	Sylow
76	C₂²×D₁₅	C₂²×C₄	1	3	yes	—
77	C₃×C₅×Q₈	C₂×C₈	1	4	yes	complemented
78	C₂×C₃×C₄×C₅	C₂²×C₄	1	3	yes	—
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	3	yes	—
80	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
81	C₂×Dic₁₅	C₂²×C₄	1	3	yes	—
82	C₂×Dic₁₅	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₁₅	C₂²×C₄	1	1	yes	characteristic
84	D₆₀	C₂×C₈	1	12	yes	complemented
85	C₄×D₁₅	—	2	3	no	—
86	C₄×D₁₅	—	2	3	no	—
87	C₄×D₁₅	—	2	6	no	—
88	C₄×D₁₅	C₂×C₈	1	12	yes	complemented
89	C₃×#168	—	2	3	no	—
90	C₃×#168	#41	1	2	yes	complemented
91	C₅×C₃⋊C₈	—	2	3	no	—
92	C₅×C₃⋊C₈	#41	1	2	yes	complemented
93	C₂×#239	—	5	1	no	—
94	C₃×C₄×Q₈	C₂×D₅	1	1	yes	characteristic
95	Q₈×Dic₃	—	5	1	no	—
96	C₂×D₃×C₈	—	5	3	no	—
97	C₂×D₃×C₈	—	5	3	no	—
98	#893	—	5	3	no	—
99	#892	—	5	3	no	—
100	C₄×D₁₂	—	5	3	no	—
101	C₄²×D₃	—	5	3	no	—
102	C₃×C₄⋊C₈	—	5	3	no	—
103	C₃×C₄×C₈	—	5	3	no	—
104	(C₂×D₃)⋊C₈	—	5	6	no	—
105	C₈×Dic₃	—	5	1	no	—
106	C₈×Dic₃	—	5	3	no	—
107	C₄⋊(C₃⋊C₈)	C₂×D₅	1	3	yes	—
108	C₄×C₃⋊C₈	C₂×D₅	1	3	yes	—
109	C₂×C₅×Q₈	C₄×D₃	1	1	yes	characteristic
110	#713	—	3	8	no	—
111	C₂×D₂₀	—	3	3	no	—
112	C₂×C₄×D₅	—	3	3	no	—
113	C₂×C₄×D₅	—	3	3	no	—
114	C₂×C₄×D₅	—	3	3	no	—
115	C₂×C₅×C₈	—	3	1	no	—
116	C₂×C₅×C₈	—	3	3	no	—
117	C₅×C₄⋊C₄	C₄×D₃	1	3	yes	—
118	C₄²×C₅	C₂²×D₃	1	3	yes	—
119	#685	—	3	6	no	—
120	C₄⋊Dic₅	—	3	3	no	—
121	C₄×Dic₅	—	3	3	no	—
122	C₄×Dic₅	—	3	3	no	—
123	C₄×Dic₅	—	3	1	no	—
124	C₂×#168	C₂²×D₃	1	3	yes	—
125	C₂×#168	C₂²×D₃	1	1	yes	characteristic
126	C₈×D₅	—	6	6	no	—
127	C₈×D₅	—	6	6	no	—
128	C₄×#41	—	15	1	no	—
129	C₈×Q₈	—	3	1	no	—
130	C₈×Q₈	—	5	1	no	—
131	C₈×D₄	—	15	3	no	—
132	C₈×D₄	—	15	3	no	—
133	#431	—	15	3	no	—
134	C₂×C₄×C₈	—	15	3	no	—
135	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
136	C₂×D₁₅	—	2	3	no	—
137	C₂×D₁₅	—	2	3	no	—
138	C₂×D₁₅	C₂²×C₈	1	6	yes	—
139	C₃×C₄×C₅	—	2	3	no	—
140	C₃×C₄×C₅	C₂²×C₈	1	6	yes	—
141	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
142	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
143	Dic₁₅	—	2	3	no	—
144	Dic₁₅	C₂²×C₈	1	2	yes	—
145	Dic₁₅	#131	1	2	yes	—
146	C₂×C₃×Q₈	C₄×D₅	1	1	yes	characteristic
147	#239	—	5	8	no	—
148	C₂×D₁₂	—	5	3	no	—
149	C₂×C₄×D₃	—	5	3	no	—
150	C₂×C₄×D₃	—	5	3	no	—
151	C₂×C₄×D₃	—	5	3	no	—
152	C₂×C₃×C₈	—	5	1	no	—
153	C₂×C₃×C₈	—	5	3	no	—
154	C₃×C₄⋊C₄	C₄×D₅	1	3	yes	—
155	C₃×C₄²	C₂²×D₅	1	3	yes	—
156	(C₂×D₃)⋊C₄	—	5	6	no	—
157	C₄⋊Dic₃	—	5	3	no	—
158	C₄×Dic₃	—	5	3	no	—
159	C₄×Dic₃	—	5	3	no	—
160	C₄×Dic₃	—	5	1	no	—
161	C₂×C₃⋊C₈	C₂²×D₅	1	3	yes	—
162	C₂×C₃⋊C₈	C₂²×D₅	1	1	yes	characteristic
163	D₃×C₈	—	10	6	no	—
164	D₃×C₈	—	10	6	no	—
165	C₂²×D₅	—	3	3	no	—
166	C₅×Q₈	D₃×C₈	1	4	yes	complemented
167	C₂×C₄×C₅	C₂×C₄×D₃	1	3	yes	—
168	C₂×C₄×C₅	C₂×C₄×D₃	1	3	yes	—
169	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
170	C₂×Dic₅	—	3	1	no	—
171	C₂×Dic₅	—	3	1	no	—
172	C₂×Dic₅	—	3	3	no	—
173	D₂₀	—	3	12	no	—
174	C₄×D₅	—	3	12	no	—
175	C₄×D₅	—	6	6	no	—
176	C₄×D₅	—	6	3	no	—
177	C₄×D₅	—	6	3	no	—
178	C₅×C₈	—	3	2	no	—
179	C₅×C₈	—	6	3	no	—
180	#168	—	2	3	no	—
181	#168	#239	1	2	yes	complemented
182	C₂×#41	—	15	1	no	—
183	C₂²×C₈	—	15	3	no	—
184	C₂²×C₈	—	15	3	no	—
185	C₄×Q₈	—	15	1	no	—
186	C₄×Q₈	D₃×D₅	1	1	yes	characteristic
187	C₄×D₄	—	15	3	no	—
188	C₄×D₄	—	15	3	no	—
189	#117	—	15	3	no	—
190	C₂×C₄²	—	15	3	no	—
191	C₄⋊C₈	—	3	3	no	—
192	C₄⋊C₈	—	5	3	no	—
193	C₂²⋊C₈	—	15	6	no	—
194	C₄×C₈	—	3	3	no	—
195	C₄×C₈	—	5	3	no	—
196	C₄×C₈	—	15	1	no	—
197	C₄×C₈	—	15	3	no	—
198	C₂×C₃×C₅	C₄×#41	1	1	yes	characteristic
199	C₂×C₃×C₅	C₂³×C₈	1	1	yes	characteristic
200	C₂×C₃×C₅	C₂×#131	1	1	yes	characteristic
201	D₁₅	—	2	6	no	—
202	C₂²×D₃	—	5	3	no	—
203	C₃×Q₈	C₈×D₅	1	4	yes	complemented
204	C₂×C₃×C₄	C₂×C₄×D₅	1	3	yes	—
205	C₂×C₃×C₄	C₂×C₄×D₅	1	3	yes	—
206	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
207	C₂×Dic₃	—	5	1	no	—
208	C₂×Dic₃	—	5	1	no	—
209	C₂×Dic₃	—	5	3	no	—
210	D₁₂	—	5	12	no	—
211	C₄×D₃	—	5	12	no	—
212	C₄×D₃	—	10	6	no	—
213	C₄×D₃	—	10	3	no	—
214	C₄×D₃	—	10	3	no	—
215	C₃×C₈	—	5	2	no	—
216	C₃×C₈	—	10	3	no	—
217	C₃⋊C₈	—	2	3	no	—
218	C₃⋊C₈	#713	1	2	yes	complemented
219	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
220	C₂×D₅	—	3	6	no	—
221	C₂×D₅	—	6	3	no	—
222	C₂×D₅	—	6	3	no	—
223	C₄×C₅	—	2	3	no	—
224	C₄×C₅	C₂×D₃×C₈	1	6	yes	—
225	C₄×C₅	C₂×#239	1	1	yes	characteristic
226	C₄×C₅	D₃×#41	1	1	yes	characteristic
227	Dic₅	—	3	2	no	—
228	Dic₅	—	3	2	no	—
229	Dic₅	—	6	3	no	—
230	#41	—	15	8	no	—
231	C₂×Q₈	#1200	1	1	yes	characteristic
232	C₂×D₄	—	15	3	no	—
233	C₂²×C₄	—	15	3	no	—
234	C₂²×C₄	—	15	3	no	—
235	C₂²×C₄	—	15	3	no	—
236	C₂×C₈	—	3	1	no	—
237	C₂×C₈	—	3	3	no	—
238	C₂×C₈	—	5	1	no	—
239	C₂×C₈	—	5	3	no	—
240	C₂×C₈	—	30	6	no	—
241	C₂×C₈	—	30	6	no	—
242	C₄⋊C₄	—	15	3	no	—
243	C₄⋊C₄	#1200	1	3	yes	—
244	C₂²⋊C₄	—	15	6	no	—
245	C₄²	—	15	3	no	—
246	C₄²	—	15	3	no	—
247	C₄²	—	15	1	no	—
248	C₄²	C₂×D₃×D₅	1	3	yes	—
249	C₃×C₅	C₈×#41	1	1	yes	characteristic, complemented, Hall
250	C₂²×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
251	C₂×D₃	—	5	6	no	—
252	C₂×D₃	—	10	3	no	—
253	C₂×D₃	—	10	3	no	—
254	C₃×C₄	—	2	3	no	—
255	C₃×C₄	C₂×C₈×D₅	1	6	yes	—
256	C₃×C₄	C₂×#713	1	1	yes	characteristic
257	C₃×C₄	D₅×#41	1	1	yes	characteristic
258	Dic₃	—	5	2	no	—
259	Dic₃	—	5	2	no	—
260	Dic₃	—	10	3	no	—
261	C₂×C₅	C₄×#239	1	1	yes	characteristic
262	C₂×C₅	C₂²×D₃×C₈	1	1	yes	characteristic
263	C₂×C₅	D₃×#131	1	1	yes	characteristic
264	D₅	—	6	6	no	—
265	C₂³	—	15	3	no	—
266	Q₈	#6648	1	4	yes	complemented
267	D₄	—	15	12	no	—
268	C₂×C₄	—	15	1	no	—
269	C₂×C₄	—	15	1	no	—
270	C₂×C₄	—	15	12	no	—
271	C₂×C₄	—	15	3	no	—
272	C₂×C₄	—	30	6	no	—
273	C₂×C₄	—	30	3	no	—
274	C₂×C₄	—	30	3	no	—
275	C₂×C₄	C₂×#1200	1	3	yes	—
276	C₂×C₄	C₂×#1200	1	3	yes	—
277	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
278	C₈	—	3	2	no	—
279	C₈	—	5	2	no	—
280	C₈	—	6	3	no	—
281	C₈	—	10	3	no	—
282	C₂×C₃	C₄×#713	1	1	yes	characteristic
283	C₂×C₃	C₂²×C₈×D₅	1	1	yes	characteristic
284	C₂×C₃	D₅×#131	1	1	yes	characteristic
285	D₃	—	10	6	no	—
286	C₅	C₈×#239	1	1	yes	characteristic, complemented, Sylow
287	C₂²	—	15	6	no	—
288	C₂²	—	30	3	no	—
289	C₂²	—	30	3	no	—
290	C₂²	C₂²×#1200	1	1	yes	characteristic
291	C₄	—	2	3	no	—
292	C₄	—	15	2	no	—
293	C₄	—	15	2	no	—
294	C₄	—	30	3	no	—
295	C₄	C₂×#6648	1	6	yes	—
296	C₄	D₅×#239	1	1	yes	characteristic
297	C₄	D₃×#713	1	1	yes	characteristic
298	C₃	C₈×#713	1	1	yes	characteristic, complemented, Sylow
299	C₂	—	30	6	no	—
300	C₂	C₂²×#6648	1	1	yes	characteristic
301	C₂	#11755200	1	1	yes	characteristic
302	C₂	#11755299	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃×D₅
commutator	C₂×C₃×C₅	C₂³×C₈
Fitting	C₃×C₄×C₅×Q₈	C₂²
socle	C₂²×C₃×C₅	C₂³×C₄
prime radical	C₂×#6821	C₄
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₄×Q₈	D₃×D₅
hypocenter	C₃×C₅	C₈×#41
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₅×Q₈×C₃⋊C₈	C₂	semidirect
3. C₃×Q₈×#168	C₂	semidirect
4. #11746994	C₂	semidirect
5. #11746988	C₂	semidirect
6. C₄×#6648	C₂	semidirect
7. #11746859	C₂	semidirect
28. Q₈×#168	D₃	semidirect
34. #6821	C₈	semidirect
54. Q₈×C₃⋊C₈	D₅	semidirect
77. C₃×C₅×Q₈	C₂×C₈	semidirect
84. D₆₀	C₂×C₈	semidirect
88. C₄×D₁₅	C₂×C₈	semidirect
90. C₃×#168	#41	semidirect
92. C₅×C₃⋊C₈	#41	semidirect
166. C₅×Q₈	D₃×C₈	semidirect
181. #168	#239	semidirect
203. C₃×Q₈	C₈×D₅	semidirect
218. C₃⋊C₈	#713	semidirect
249. C₃×C₅	C₈×#41	semidirect
266. Q₈	#6648	semidirect
286. C₅	C₈×#239	semidirect
298. C₃	C₈×#713	semidirect

Maximal subgroups

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

subgroup	index
C₄×#6821	2
C₅×Q₈×C₃⋊C₈	2
C₃×Q₈×#168	2
#11746994	2
#11746988	2
C₄×#6648	2
#11746859	2
C₈×#713	3
C₈×#239	5