Subgroups — #14439861

Order 1920.

712 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	#11752506	C₂	1	2	yes	maximal
2	#11750781	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4680	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
5	#11747984	C₂	1	4	yes	maximal, complemented
6	#11747671	C₂	1	4	yes	maximal
7	#11746955	C₂	1	2	yes	maximal, complemented
8	#10613622	—	3	1	no	maximal, Hall
9	#91758	C₂²	1	2	yes	—
10	C₅×#869	C₂²	1	2	yes	—
11	C₂×C₃×C₈×D₅	C₂²	1	2	yes	—
12	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
13	#91388	C₂²	1	4	yes	—
14	#91360	—	2	2	no	—
15	#91359	—	2	2	no	—
16	#91358	C₂²	1	4	yes	—
17	#91333	—	2	2	no	—
18	Dic₅×Dic₆	—	2	2	no	—
19	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	#91101	C₂²	1	4	yes	—
21	C₅×#817	C₂²	1	4	yes	—
22	C₃×C₈×Dic₅	C₂²	1	2	yes	—
23	C₃×C₄×#168	C₂²	1	1	yes	characteristic
24	#90988	C₂²	1	8	yes	—
25	#78442	—	5	1	no	maximal, Hall
26	#66062	—	3	2	no	—
27	C₅×#498	—	3	1	no	—
28	#65367	—	3	2	no	—
29	#65149	—	3	4	no	—
30	#65147	—	3	4	no	—
31	C₄×C₈×D₅	D₃	1	1	yes	characteristic
32	#64969	—	3	1	no	—
33	C₂×Dic₃₀	—	2	2	no	—
34	C₂×C₅×Dic₆	—	2	2	no	—
35	C₂×C₃×C₄×D₅	D₄	1	2	yes	—
36	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
37	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
38	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
39	C₄⋊Dic₁₅	—	2	2	no	—
40	C₄⋊Dic₁₅	C₂³	1	4	yes	—
41	C₅×C₄⋊Dic₃	—	2	2	no	—
42	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
43	C₃×C₄×Dic₅	D₄	1	2	yes	—
44	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
45	C₂×C₃×#168	C₂³	1	2	yes	—
46	C₃×C₈×D₅	Q₈	1	8	yes	complemented
47	Dic₃⋊Dic₅	—	2	4	no	—
48	#6669	—	2	4	no	—
49	#6665	—	2	4	no	—
50	Dic₃×Dic₅	—	2	4	no	—
51	#5470	—	5	2	no	—
52	C₂×C₃×C₄×C₈	—	5	1	no	—
53	#5106	—	5	4	no	—
54	#4683	—	5	4	no	—
55	#4682	—	5	2	no	—
56	#4680	—	5	1	no	—
57	#4680	D₅	1	1	yes	characteristic, complemented
58	C₅×C₄⋊Q₈	—	3	2	no	—
59	#4092	—	6	2	no	—
60	Q₈×Dic₅	—	6	2	no	—
61	C₂×C₈×D₅	C₂×D₃	1	2	yes	—
62	#4043	—	6	2	no	—
63	#4038	—	3	4	no	—
64	#4036	—	3	4	no	—
65	#4033	—	6	2	no	—
66	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
67	#4015	—	3	2	no	—
68	C₅×#106	—	3	4	no	—
69	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
70	#3949	—	3	4	no	—
71	C₈×Dic₅	C₂×D₃	1	2	yes	—
72	#3939	—	3	8	no	—
73	C₄×#168	C₂×D₃	1	1	yes	characteristic
74	#3160	—	15	1	no	Sylow
75	C₂²×C₃×D₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
77	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
78	C₂×Dic₁₅	—	2	4	no	—
79	Dic₃₀	—	4	4	no	—
80	C₂×C₅×Dic₃	—	2	4	no	—
81	C₅×Dic₆	—	4	4	no	—
82	C₂×C₃×Dic₅	C₂×D₄	1	2	yes	—
83	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
84	C₃×C₄×D₅	—	2	4	no	—
85	C₃×C₄×D₅	C₂×Q₈	1	4	yes	—
86	C₃×C₅×C₈	C₂×Q₈	1	4	yes	—
87	C₃×#168	C₂×Q₈	1	4	yes	—
88	C₂²×C₃×C₈	—	5	2	no	—
89	C₂×C₃×C₄²	—	5	1	no	—
90	#926	—	5	4	no	—
91	#924	—	10	2	no	—
92	#924	—	10	2	no	—
93	#870	—	5	4	no	—
94	#869	—	5	2	no	—
95	#869	C₂×D₅	1	2	yes	—
96	C₄×Dic₆	—	10	2	no	—
97	C₄×Dic₆	—	10	2	no	—
98	C₃×C₄×C₈	—	5	2	no	—
99	C₃×C₄×C₈	—	5	1	no	—
100	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
101	#818	—	5	8	no	—
102	#817	—	5	4	no	—
103	#817	C₂×D₅	1	4	yes	—
104	C₂×C₅×Q₈	—	6	2	no	—
105	C₂×C₄×D₅	D₁₂	1	2	yes	—
106	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
107	C₂×Dic₁₀	—	6	2	no	—
108	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
109	C₅×C₄⋊C₄	—	3	4	no	—
110	C₅×C₄⋊C₄	—	6	2	no	—
111	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
112	#685	—	6	4	no	—
113	C₄⋊Dic₅	—	3	4	no	—
114	C₄⋊Dic₅	—	6	2	no	—
115	C₄⋊Dic₅	—	6	4	no	—
116	#683	—	6	4	no	—
117	C₄×Dic₅	—	6	4	no	—
118	C₄×Dic₅	D₁₂	1	2	yes	—
119	C₄×Dic₅	C₂²×D₃	1	1	yes	characteristic
120	C₂×#168	C₂²×D₃	1	2	yes	—
121	C₈×D₅	Dic₆	1	8	yes	complemented
122	#533	—	15	2	no	—
123	#500	—	15	2	no	—
124	#499	—	15	4	no	—
125	#498	—	3	1	no	—
126	#498	—	15	1	no	—
127	#427	—	15	4	no	—
128	C₂×C₄×C₈	—	5	1	no	—
129	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₃×D₅	—	2	2	no	—
131	C₂×C₃×D₅	C₄⋊Q₈	1	2	yes	—
132	C₃×C₄×C₅	#135	1	4	yes	—
133	C₃×C₄×C₅	C₂²×Q₈	1	2	yes	—
134	Dic₁₅	—	4	4	no	—
135	C₃×Dic₅	—	2	2	no	—
136	C₃×Dic₅	C₄⋊Q₈	1	2	yes	—
137	C₅×Dic₃	—	4	4	no	—
138	C₂²×C₃×C₄	—	5	1	no	—
139	C₂²×C₃×C₄	—	5	2	no	—
140	C₂×Dic₆	—	2	2	no	—
141	C₂×Dic₆	—	10	2	no	—
142	C₂×C₃×C₈	—	5	2	no	—
143	C₂×C₃×C₈	—	5	8	no	—
144	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
145	C₃×C₄²	—	5	2	no	—
146	C₃×C₄²	—	5	1	no	—
147	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
148	#217	—	10	4	no	—
149	C₄⋊Dic₃	—	2	2	no	—
150	C₄⋊Dic₃	—	5	4	no	—
151	C₄⋊Dic₃	—	10	2	no	—
152	C₄⋊Dic₃	C₂²×D₅	1	4	yes	—
153	Dic₃⋊C₄	—	10	4	no	—
154	Dic₃⋊C₄	—	10	4	no	—
155	C₄×Dic₃	—	10	4	no	—
156	C₂²×D₅	C₂×D₁₂	1	1	yes	characteristic
157	C₅×Q₈	—	12	4	no	—
158	C₂×C₄×C₅	—	6	4	no	—
159	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
160	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
161	C₂×Dic₅	—	6	4	no	—
162	C₂×Dic₅	C₂×D₁₂	1	2	yes	—
163	C₂×Dic₅	C₂×D₁₂	1	1	yes	characteristic
164	C₄×D₅	—	2	4	no	—
165	C₄×D₅	C₂×Dic₆	1	4	yes	—
166	Dic₁₀	—	12	4	no	—
167	C₅×C₈	C₂×Dic₆	1	4	yes	—
168	#168	C₂×Dic₆	1	4	yes	—
169	C₂²×C₈	—	5	2	no	—
170	C₄⋊Q₈	—	3	2	no	—
171	C₄⋊Q₈	—	15	2	no	—
172	#125	—	15	4	no	—
173	C₂²⋊Q₈	—	30	2	no	—
174	C₂²⋊Q₈	—	30	2	no	—
175	C₄×Q₈	—	30	2	no	—
176	C₄×Q₈	—	30	2	no	—
177	#117	—	15	4	no	—
178	C₂×C₄²	—	5	1	no	—
179	#107	—	15	8	no	—
180	#106	—	3	4	no	—
181	#106	—	15	4	no	—
182	C₄×C₈	—	5	2	no	—
183	C₄×C₈	—	5	1	no	—
184	C₄×C₈	D₃×D₅	1	1	yes	characteristic
185	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
186	C₂×C₃×C₅	C₂×#135	1	2	yes	—
187	C₃×D₅	—	2	2	no	—
188	C₂³×C₃	—	5	1	no	—
189	C₂×C₃×C₄	—	5	1	no	—
190	C₂×C₃×C₄	—	5	4	no	—
191	C₂×C₃×C₄	—	5	2	no	—
192	C₂×C₃×C₄	—	10	4	no	—
193	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
194	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
195	C₂×Dic₃	—	2	4	no	—
196	C₂×Dic₃	—	10	4	no	—
197	Dic₆	—	4	4	no	—
198	Dic₆	—	20	4	no	—
199	C₃×C₈	—	5	4	no	—
200	C₃×C₈	Q₈×D₅	1	4	yes	—
201	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
202	C₂×D₅	—	2	2	no	—
203	C₂×D₅	#869	1	2	yes	—
204	C₄×C₅	—	12	4	no	—
205	C₄×C₅	#904	1	4	yes	—
206	C₄×C₅	C₂²×Dic₆	1	2	yes	—
207	Dic₅	—	2	2	no	—
208	Dic₅	—	12	4	no	—
209	Dic₅	#869	1	2	yes	—
210	C₂×Q₈	—	6	2	no	—
211	C₂×Q₈	—	30	2	no	—
212	C₂²×C₄	—	5	1	no	—
213	C₂²×C₄	—	5	2	no	—
214	C₂×C₈	—	5	2	no	—
215	C₂×C₈	—	5	8	no	—
216	C₂×C₈	C₂×D₃×D₅	1	2	yes	—
217	C₄⋊C₄	—	3	4	no	—
218	C₄⋊C₄	—	6	2	no	—
219	C₄⋊C₄	—	15	4	no	—
220	C₄⋊C₄	—	30	4	no	—
221	C₄⋊C₄	—	30	4	no	—
222	C₄⋊C₄	—	30	2	no	—
223	C₂²⋊C₄	—	30	4	no	—
224	C₄²	—	5	2	no	—
225	C₄²	—	5	1	no	—
226	C₄²	—	30	4	no	—
227	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
228	C₃×C₅	#3160	1	1	yes	characteristic, complemented, Hall
229	C₂²×C₃	—	5	2	no	—
230	C₂²×C₃	—	10	2	no	—
231	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
232	C₃×C₄	—	5	2	no	—
233	C₃×C₄	—	10	2	no	—
234	C₃×C₄	#4062	1	4	yes	—
235	C₃×C₄	C₂×Q₈×D₅	1	2	yes	—
236	Dic₃	—	4	4	no	—
237	Dic₃	—	20	4	no	—
238	C₂×C₅	C₂×#869	1	1	yes	characteristic
239	C₂×C₅	C₂×#904	1	2	yes	—
240	D₅	—	2	2	no	—
241	C₂³	—	5	1	no	—
242	Q₈	—	12	4	no	—
243	Q₈	—	60	4	no	—
244	C₂×C₄	—	5	1	no	—
245	C₂×C₄	—	5	4	no	—
246	C₂×C₄	—	5	2	no	—
247	C₂×C₄	—	6	4	no	—
248	C₂×C₄	—	10	4	no	—
249	C₂×C₄	—	30	4	no	—
250	C₂×C₄	D₅×D₁₂	1	2	yes	—
251	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
252	C₈	—	5	4	no	—
253	C₈	D₅×Dic₆	1	4	yes	—
254	C₂×C₃	—	10	2	no	—
255	C₂×C₃	D₅×C₄⋊Q₈	1	1	yes	characteristic
256	C₂×C₃	C₂×#4062	1	2	yes	—
257	C₅	#78442	1	1	yes	characteristic, complemented, Sylow
258	C₂²	—	5	2	no	—
259	C₂²	—	10	2	no	—
260	C₂²	C₂×D₅×D₁₂	1	1	yes	characteristic
261	C₄	—	5	2	no	—
262	C₄	—	10	2	no	—
263	C₄	—	12	4	no	—
264	C₄	—	60	4	no	—
265	C₄	#91269	1	4	yes	—
266	C₄	C₂×D₅×Dic₆	1	2	yes	—
267	C₃	#10613622	1	1	yes	characteristic, complemented, Sylow
268	C₂	—	10	2	no	—
269	C₂	D₅×#869	1	1	yes	characteristic
270	C₂	C₂×#91269	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₅×D₁₂
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₃×C₄×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂²×C₃×D₅	C₂×D₄
hypercenter	C₄×C₈	D₃×D₅
hypocenter	C₃×C₅	#3160
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #11750781	C₂	semidirect
3. C₅×#4680	C₂	semidirect
5. #11747984	C₂	semidirect
7. #11746955	C₂	semidirect
46. C₃×C₈×D₅	Q₈	semidirect
57. #4680	D₅	semidirect
121. C₈×D₅	Dic₆	semidirect
228. C₃×C₅	#3160	semidirect
257. C₅	#78442	semidirect
267. C₃	#10613622	semidirect

Maximal subgroups

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

subgroup	index
#11752506	2
#11750781	2
C₅×#4680	2
C₃×C₄×C₈×D₅	2
#11747984	2
#11747671	2
#11746955	2
#10613622	3
#78442	5