Subgroups — #14285677

Order 1920.

368 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11750938	C₂	1	2	yes	maximal
2	C₄×C₈×D₁₅	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#335	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746300	C₂	1	2	yes	maximal, complemented
5	#11746279	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599362	—	3	1	no	maximal, Hall
7	C₂×#6705	—	2	2	no	—
8	C₂×C₈×D₁₅	C₄	1	1	yes	characteristic
9	C₂×C₈×D₁₅	C₄	1	1	yes	characteristic
10	C₄²×D₁₅	C₂²	1	1	yes	characteristic
11	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
12	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	#91098	—	2	2	no	—
14	C₈×Dic₁₅	C₄	1	1	yes	characteristic
15	C₈×Dic₁₅	C₄	1	1	yes	characteristic
16	C₄⋊(C₃⋊#168)	C₂²	1	2	yes	—
17	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
18	#68569	—	5	1	no	maximal, Hall
19	#65126	—	3	2	no	—
20	C₄×C₈×D₅	—	3	1	no	—
21	C₅×#335	D₃	1	1	yes	characteristic, complemented
22	#64699	—	3	2	no	—
23	#64678	—	3	1	no	—
24	C₂×C₄×D₁₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₁₅	D₄	1	1	yes	characteristic
26	C₂×C₄×D₁₅	Q₈	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	—	2	2	no	—
28	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
31	C₄×Dic₁₅	C₂×C₄	1	1	yes	characteristic
32	C₄×Dic₁₅	D₄	1	1	yes	characteristic
33	C₄×Dic₁₅	Q₈	1	1	yes	characteristic
34	C₂×C₃⋊#168	—	2	2	no	—
35	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
37	#6705	—	4	4	no	—
38	C₈×D₁₅	—	2	2	no	—
39	C₈×D₁₅	—	2	2	no	—
40	#4837	—	5	2	no	—
41	C₄×D₃×C₈	—	5	1	no	—
42	C₃×#335	D₅	1	1	yes	characteristic, complemented
43	#4479	—	5	2	no	—
44	#4458	—	5	1	no	—
45	C₂×#676	—	6	2	no	—
46	C₂×C₈×D₅	—	3	1	no	—
47	C₂×C₈×D₅	—	3	1	no	—
48	C₄²×D₅	—	3	1	no	—
49	C₅×C₄⋊C₈	C₂×D₃	1	2	yes	—
50	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
51	#3946	—	6	2	no	—
52	C₈×Dic₅	—	3	1	no	—
53	C₈×Dic₅	—	3	1	no	—
54	C₄⋊#168	—	3	2	no	—
55	C₄×#168	—	3	1	no	—
56	#1567	—	15	1	no	Sylow
57	C₂²×D₁₅	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
61	C₂×Dic₁₅	C₄⋊C₄	1	1	yes	characteristic
62	C₂×Dic₁₅	C₄⋊C₄	1	1	yes	characteristic
63	C₂×Dic₁₅	C₄⋊C₄	1	1	yes	characteristic
64	C₄×D₁₅	—	2	1	no	—
65	C₄×D₁₅	—	2	1	no	—
66	C₄×D₁₅	—	2	2	no	—
67	C₄×D₁₅	D₈	1	2	yes	—
68	C₄×D₁₅	Q₁₆	1	2	yes	—
69	C₃×C₅×C₈	—	2	1	no	—
70	C₃×C₅×C₈	—	4	2	no	—
71	C₃×C₅×C₈	M₁₆	1	2	yes	complemented
72	C₃⋊#168	—	2	1	no	—
73	C₃⋊#168	—	4	2	no	—
74	C₃⋊#168	M₁₆	1	2	yes	complemented
75	C₂×ΓL(1,𝔽₂₅)	—	10	2	no	—
76	C₂×D₃×C₈	—	5	1	no	—
77	C₂×D₃×C₈	—	5	1	no	—
78	C₄²×D₃	—	5	1	no	—
79	C₃×C₄⋊C₈	C₂×D₅	1	2	yes	—
80	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
81	Dic₃⋊C₈	—	10	2	no	—
82	C₈×Dic₃	—	5	1	no	—
83	C₈×Dic₃	—	5	1	no	—
84	C₄⋊(C₃⋊C₈)	—	5	2	no	—
85	C₄×C₃⋊C₈	—	5	1	no	—
86	C₂×C₄×D₅	—	3	1	no	—
87	C₂×C₄×D₅	—	3	1	no	—
88	C₂×C₄×D₅	—	3	1	no	—
89	C₂×C₅×C₈	—	2	2	no	—
90	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
91	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
92	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
93	C₄×Dic₅	—	3	1	no	—
94	C₄×Dic₅	—	3	1	no	—
95	C₄×Dic₅	—	3	1	no	—
96	C₂×#168	—	3	1	no	—
97	C₂×#168	—	3	1	no	—
98	C₂×#168	—	6	2	no	—
99	#676	—	12	4	no	—
100	C₈×D₅	—	6	2	no	—
101	C₈×D₅	—	6	2	no	—
102	#423	—	15	2	no	—
103	C₂×C₄×C₈	—	15	1	no	—
104	#335	—	15	2	no	—
105	#335	—	15	1	no	—
106	#335	D₁₅	1	1	yes	characteristic, complemented
107	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×D₁₅	—	2	1	no	—
109	C₂×D₁₅	—	2	1	no	—
110	C₂×D₁₅	#107	1	2	yes	—
111	C₃×C₄×C₅	—	2	1	no	—
112	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
113	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
114	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
115	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
116	Dic₁₅	—	2	1	no	—
117	Dic₁₅	#107	1	2	yes	—
118	Dic₁₅	#108	1	2	yes	—
119	C₂×C₄×D₃	—	5	1	no	—
120	C₂×C₄×D₃	—	5	1	no	—
121	C₂×C₄×D₃	—	5	1	no	—
122	C₂×C₃×C₈	—	2	2	no	—
123	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
124	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
125	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
126	C₄×Dic₃	—	5	1	no	—
127	C₄×Dic₃	—	5	1	no	—
128	C₄×Dic₃	—	5	1	no	—
129	C₂×C₃⋊C₈	—	5	1	no	—
130	C₂×C₃⋊C₈	—	5	1	no	—
131	C₂×C₃⋊C₈	—	10	2	no	—
132	ΓL(1,𝔽₂₅)	—	20	4	no	—
133	D₃×C₈	—	10	2	no	—
134	D₃×C₈	—	10	2	no	—
135	C₂²×D₅	—	3	1	no	—
136	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
137	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
139	C₂×Dic₅	—	3	1	no	—
140	C₂×Dic₅	—	3	1	no	—
141	C₂×Dic₅	—	3	1	no	—
142	C₄×D₅	—	3	2	no	—
143	C₄×D₅	—	3	2	no	—
144	C₄×D₅	—	6	2	no	—
145	C₄×D₅	—	6	1	no	—
146	C₄×D₅	—	6	1	no	—
147	C₅×C₈	—	2	1	no	—
148	C₅×C₈	—	4	2	no	—
149	C₅×C₈	ΓL(1,𝔽₂₅)	1	2	yes	complemented
150	#168	—	3	2	no	—
151	#168	—	6	1	no	—
152	#168	—	12	2	no	—
153	C₂×M₁₆	—	30	2	no	—
154	C₂²×C₈	—	15	1	no	—
155	C₂²×C₈	—	15	1	no	—
156	C₂×C₄²	—	15	1	no	—
157	C₄⋊C₈	—	15	2	no	—
158	C₄⋊C₈	—	30	2	no	—
159	C₄⋊C₈	C₂×D₁₅	1	2	yes	—
160	C₄×C₈	—	15	1	no	—
161	C₄×C₈	—	15	1	no	—
162	C₄×C₈	—	15	1	no	—
163	C₄×C₈	C₂×D₁₅	1	1	yes	characteristic
164	C₂×C₃×C₅	#423	1	1	yes	characteristic
165	C₂×C₃×C₅	C₂×#107	1	1	yes	characteristic
166	C₂×C₃×C₅	C₂×#108	1	1	yes	characteristic
167	D₁₅	—	2	2	no	—
168	C₂²×D₃	—	5	1	no	—
169	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
170	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
171	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
172	C₂×Dic₃	—	5	1	no	—
173	C₂×Dic₃	—	5	1	no	—
174	C₂×Dic₃	—	5	1	no	—
175	C₄×D₃	—	5	2	no	—
176	C₄×D₃	—	5	2	no	—
177	C₄×D₃	—	10	2	no	—
178	C₄×D₃	—	10	1	no	—
179	C₄×D₃	—	10	1	no	—
180	C₃×C₈	—	2	1	no	—
181	C₃×C₈	—	4	2	no	—
182	C₃×C₈	#676	1	2	yes	complemented
183	C₃⋊C₈	—	5	2	no	—
184	C₃⋊C₈	—	10	1	no	—
185	C₃⋊C₈	—	20	2	no	—
186	C₂²×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
187	C₂×D₅	—	3	2	no	—
188	C₂×D₅	—	6	1	no	—
189	C₂×D₅	—	6	1	no	—
190	C₄×C₅	—	2	1	no	—
191	C₄×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
192	C₄×C₅	D₃×M₁₆	1	1	yes	characteristic
193	C₄×C₅	D₃×D₈	1	1	yes	characteristic
194	C₄×C₅	D₃×Q₁₆	1	1	yes	characteristic
195	Dic₅	—	3	2	no	—
196	Dic₅	—	3	2	no	—
197	Dic₅	—	6	1	no	—
198	C₂²×C₄	—	15	1	no	—
199	C₂²×C₄	—	15	1	no	—
200	C₂²×C₄	—	15	1	no	—
201	M₁₆	—	60	4	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	15	1	no	—
204	C₂×C₈	—	15	1	no	—
205	C₂×C₈	—	30	2	no	—
206	C₂×C₈	—	30	2	no	—
207	C₂×C₈	—	30	2	no	—
208	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
209	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
210	C₄²	—	15	1	no	—
211	C₄²	—	15	1	no	—
212	C₄²	—	15	1	no	—
213	C₄²	C₂²×D₁₅	1	1	yes	characteristic
214	C₃×C₅	#1567	1	1	yes	characteristic, complemented, Hall
215	C₂²×C₃	D₅×C₄⋊C₄	1	1	yes	characteristic
216	C₂×D₃	—	5	2	no	—
217	C₂×D₃	—	10	1	no	—
218	C₂×D₃	—	10	1	no	—
219	C₃×C₄	—	2	1	no	—
220	C₃×C₄	C₂×#676	1	1	yes	characteristic
221	C₃×C₄	D₅×M₁₆	1	1	yes	characteristic
222	C₃×C₄	D₅×D₈	1	1	yes	characteristic
223	C₃×C₄	D₅×Q₁₆	1	1	yes	characteristic
224	Dic₃	—	5	2	no	—
225	Dic₃	—	5	2	no	—
226	Dic₃	—	10	1	no	—
227	C₂×C₅	#4837	1	1	yes	characteristic
228	C₂×C₅	D₃×#107	1	1	yes	characteristic
229	C₂×C₅	D₃×#108	1	1	yes	characteristic
230	D₅	—	6	2	no	—
231	C₂³	—	15	1	no	—
232	C₂×C₄	—	15	1	no	—
233	C₂×C₄	—	15	1	no	—
234	C₂×C₄	—	15	2	no	—
235	C₂×C₄	—	15	2	no	—
236	C₂×C₄	—	15	1	no	—
237	C₂×C₄	—	30	2	no	—
238	C₂×C₄	—	30	1	no	—
239	C₂×C₄	—	30	1	no	—
240	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
241	C₂×C₄	D₄×D₁₅	1	1	yes	characteristic
242	C₂×C₄	Q₈×D₁₅	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	4	2	no	—
245	C₈	—	15	2	no	—
246	C₈	—	30	1	no	—
247	C₈	—	60	2	no	—
248	C₈	#6705	1	2	yes	complemented
249	C₂×C₃	#65126	1	1	yes	characteristic
250	C₂×C₃	D₅×#107	1	1	yes	characteristic
251	C₂×C₃	D₅×#108	1	1	yes	characteristic
252	D₃	—	10	2	no	—
253	C₅	#68569	1	1	yes	characteristic, complemented, Sylow
254	C₂²	—	15	2	no	—
255	C₂²	—	30	1	no	—
256	C₂²	—	30	1	no	—
257	C₂²	C₄⋊C₄×D₁₅	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	15	2	no	—
260	C₄	—	15	2	no	—
261	C₄	—	30	1	no	—
262	C₄	C₂×#6705	1	1	yes	characteristic
263	C₄	M₁₆×D₁₅	1	1	yes	characteristic
264	C₄	D₈×D₁₅	1	1	yes	characteristic
265	C₄	Q₁₆×D₁₅	1	1	yes	characteristic
266	C₃	#10599362	1	1	yes	characteristic, complemented, Sylow
267	C₂	—	30	2	no	—
268	C₂	#11750938	1	1	yes	characteristic
269	C₂	D₁₅×#107	1	1	yes	characteristic
270	C₂	D₁₅×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₅
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂²×D₁₅
Fitting	C₃×C₅×#335	C₂
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂²×D₁₅	C₄⋊C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	#335	D₁₅
hypocenter	C₃×C₅	#1567
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#335	C₂	semidirect
4. #11746300	C₂	semidirect
5. #11746279	C₂	semidirect
21. C₅×#335	D₃	semidirect
42. C₃×#335	D₅	semidirect
71. C₃×C₅×C₈	M₁₆	semidirect
74. C₃⋊#168	M₁₆	semidirect
106. #335	D₁₅	semidirect
149. C₅×C₈	ΓL(1,𝔽₂₅)	semidirect
182. C₃×C₈	#676	semidirect
214. C₃×C₅	#1567	semidirect
248. C₈	#6705	semidirect
253. C₅	#68569	semidirect
266. C₃	#10599362	semidirect

Maximal subgroups

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

subgroup	index
#11750938	2
C₄×C₈×D₁₅	2
C₃×C₅×#335	2
#11746300	2
#11746279	2
#10599362	3
#68569	5