Subgroups — #18081700

Order 4864.

506 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14574214	C₂	1	1	yes	maximal, characteristic
2	#14564554	C₂	1	2	yes	maximal, complemented
3	C₄×#10597275	C₂	1	1	yes	maximal, characteristic, complemented
4	#14562131	C₂	1	2	yes	maximal, complemented
5	C₈×M₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#11927094	C₂²	1	2	yes	—
7	#11926949	C₄	1	1	yes	characteristic, complemented
8	#11926921	C₄	1	1	yes	characteristic, complemented
9	#11926891	C₄	1	2	yes	complemented
10	#11926780	—	2	1	no	—
11	#11926780	—	2	1	no	—
12	C₄×D₁₅₂	—	2	1	no	—
13	C₄×D₁₅₂	—	2	1	no	—
14	C₂×#10597275	—	2	1	no	—
15	C₂×#10597282	—	2	4	no	—
16	C₄×C₄⋊Dic₁₉	C₂²	1	2	yes	—
17	#11926307	—	2	1	no	—
18	#11926279	—	2	4	no	—
19	C₁₉×#431	C₂²	1	1	yes	characteristic
20	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
22	C₁₉×#322	—	2	1	no	—
23	C₈²×C₁₉	—	2	1	no	—
24	C₂×#64541	—	2	2	no	—
25	C₂×M₁₆×C₁₉	D₄	1	2	yes	—
26	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
29	#10597376	—	2	2	no	—
30	#10597376	—	2	2	no	—
31	C₂×C₄⋊Dic₁₉	—	2	2	no	—
32	C₂×C₄×Dic₁₉	—	2	2	no	—
33	C₂×D₁₅₂	—	2	1	no	—
34	C₂×D₁₅₂	—	2	1	no	—
35	#10597321	—	2	2	no	—
36	#10597320	C₂×C₄	1	2	yes	—
37	C₄×D₇₆	—	2	2	no	—
38	C₄×D₇₆	—	2	2	no	—
39	#10597315	C₂×C₄	1	2	yes	—
40	C₁₉×C₄⋊C₈	—	2	1	no	—
41	C₁₉×C₂²⋊C₈	D₄	1	2	yes	—
42	C₁₉×#97	—	2	1	no	—
43	C₄×C₈×C₁₉	—	2	1	no	—
44	C₄×C₈×C₁₉	—	2	1	no	—
45	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
48	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
49	C₁₉⋊#95	—	2	2	no	—
50	#10597282	—	4	8	no	—
51	#10597278	—	2	2	no	—
52	#10597278	—	2	2	no	—
53	#10597275	—	2	4	no	—
54	#10597275	—	2	1	no	—
55	#10597275	—	2	1	no	—
56	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₂²⋊D₁₉	—	4	2	no	—
60	C₂²×Dic₁₉	—	4	2	no	—
61	#64541	—	4	4	no	—
62	#64541	—	4	4	no	—
63	C₂×D₇₆	—	2	2	no	—
64	C₂×D₇₆	—	2	2	no	—
65	C₂×D₇₆	—	2	2	no	—
66	C₂×C₄×D₁₉	—	4	2	no	—
67	C₂×Dic₃₈	—	2	2	no	—
68	M₁₆×C₁₉	—	2	2	no	—
69	M₁₆×C₁₉	D₈	1	4	yes	complemented
70	C₂×C₈×C₁₉	—	2	2	no	—
71	C₂×C₈×C₁₉	—	2	2	no	—
72	C₂×C₈×C₁₉	—	2	2	no	—
73	C₂×C₈×C₁₉	—	4	1	no	—
74	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₁₉	#41	1	1	yes	characteristic
81	C₄²×C₁₉	#41	1	1	yes	characteristic
82	(C₂×D₁₉)⋊C₄	—	4	2	no	—
83	(C₂×D₁₉)⋊C₄	—	4	2	no	—
84	C₄⋊Dic₁₉	—	2	4	no	—
85	C₄⋊Dic₁₉	—	2	2	no	—
86	C₄⋊Dic₁₉	—	2	2	no	—
87	C₄⋊Dic₁₉	—	4	2	no	—
88	C₄×Dic₁₉	—	4	4	no	—
89	D₁₅₂	—	4	2	no	—
90	D₁₅₂	—	4	2	no	—
91	#9719	—	19	1	no	maximal, Sylow
92	C₂³×C₁₉	#127	1	1	yes	characteristic
93	C₂²×D₁₉	—	4	2	no	—
94	C₂×C₄×C₁₉	—	2	1	no	—
95	C₂×C₄×C₁₉	—	2	1	no	—
96	C₂×C₄×C₁₉	—	4	1	no	—
97	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
101	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
102	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
103	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₁₉	#135	1	2	yes	—
105	C₂²⋊D₁₉	—	8	2	no	—
106	C₂²⋊D₁₉	—	8	2	no	—
107	C₂×Dic₁₉	—	4	4	no	—
108	C₂×Dic₁₉	—	4	2	no	—
109	C₂×Dic₁₉	—	8	2	no	—
110	C₂×Dic₁₉	—	8	2	no	—
111	D₇₆	—	4	2	no	—
112	D₇₆	—	4	2	no	—
113	D₇₆	—	4	4	no	—
114	D₇₆	—	8	2	no	—
115	C₄×D₁₉	—	8	4	no	—
116	Dic₃₈	—	4	4	no	—
117	C₈×C₁₉	—	2	1	no	—
118	C₈×C₁₉	—	2	1	no	—
119	C₈×C₁₉	—	2	1	no	—
120	C₈×C₁₉	—	2	4	no	—
121	C₈×C₁₉	—	4	2	no	—
122	C₈×C₁₉	#104	1	2	yes	complemented
123	#3169	—	19	1	no	—
124	#1980	—	19	2	no	—
125	C₄×#107	—	19	1	no	—
126	#1502	—	19	2	no	—
127	C₈×M₁₆	D₁₉	1	1	yes	characteristic, complemented
128	C₂²×C₁₉	—	2	1	no	—
129	C₂²×C₁₉	—	2	1	no	—
130	C₂²×C₁₉	#390	1	1	yes	characteristic
131	C₂²×C₁₉	#493	1	1	yes	characteristic
132	C₂²×C₁₉	#495	1	1	yes	characteristic
133	C₂×D₁₉	—	8	2	no	—
134	C₂×D₁₉	—	8	2	no	—
135	C₂×D₁₉	—	8	2	no	—
136	C₄×C₁₉	—	2	1	no	—
137	C₄×C₁₉	—	2	1	no	—
138	C₄×C₁₉	—	2	1	no	—
139	C₄×C₁₉	C₂×#104	1	1	yes	characteristic
140	C₄×C₁₉	#421	1	1	yes	characteristic
141	C₄×C₁₉	C₄×D₈	1	2	yes	—
142	C₄×C₁₉	#466	1	2	yes	—
143	Dic₁₉	—	8	4	no	—
144	Dic₁₉	—	8	2	no	—
145	#532	—	19	2	no	—
146	#500	—	19	1	no	—
147	#493	—	19	1	no	—
148	#486	—	19	2	no	—
149	#466	—	38	1	no	—
150	#466	—	38	1	no	—
151	C₄×D₈	—	38	1	no	—
152	C₄×D₈	—	38	1	no	—
153	#431	C₂×D₁₉	1	1	yes	characteristic
154	C₂×#107	—	38	1	no	—
155	C₂×#104	—	38	4	no	—
156	C₄×M₁₆	C₂×D₁₉	1	1	yes	characteristic
157	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
158	C₄×C₄⋊C₄	—	19	2	no	—
159	#340	—	38	1	no	—
160	#327	—	38	4	no	—
161	#322	—	2	1	no	—
162	C₈²	—	2	1	no	—
163	C₂×C₁₉	—	2	1	no	—
164	C₂×C₁₉	—	2	1	no	—
165	C₂×C₁₉	#1970	1	1	yes	characteristic
166	C₂×C₁₉	#1976	1	1	yes	characteristic
167	C₂×C₁₉	#1980	1	1	yes	characteristic
168	D₁₉	—	16	2	no	—
169	C₂×#41	—	38	2	no	—
170	C₂×D₈	—	38	1	no	—
171	C₂×D₈	—	38	1	no	—
172	C₂×M₁₆	D₇₆	1	2	yes	—
173	C₂²×C₈	D₇₆	1	1	yes	characteristic
174	C₂²×C₈	D₇₆	1	1	yes	characteristic
175	C₄⋊Q₈	—	19	2	no	—
176	#127	—	19	2	no	—
177	#124	—	38	2	no	—
178	#121	—	38	2	no	—
179	#121	—	38	2	no	—
180	C₄×D₄	—	38	2	no	—
181	C₄×D₄	—	38	2	no	—
182	C₂×C₄⋊C₄	—	38	2	no	—
183	C₂×C₄²	—	38	2	no	—
184	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
185	#107	—	38	4	no	—
186	#107	—	38	1	no	—
187	#107	—	38	1	no	—
188	C₄⋊C₈	—	2	1	no	—
189	#104	—	76	8	no	—
190	#102	—	38	2	no	—
191	#102	—	38	2	no	—
192	C₂²⋊C₈	D₇₆	1	2	yes	—
193	#97	—	2	1	no	—
194	C₄×C₈	—	2	1	no	—
195	C₄×C₈	—	2	1	no	—
196	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
197	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
198	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
199	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
200	#95	—	38	2	no	—
201	C₁₉	#9719	1	1	yes	characteristic, complemented, Sylow
202	#41	—	76	4	no	—
203	#41	—	76	4	no	—
204	C₂×Q₈	—	38	2	no	—
205	C₂×D₄	—	38	2	no	—
206	C₂×D₄	—	38	2	no	—
207	C₂×D₄	—	38	2	no	—
208	C₂×D₄	—	76	2	no	—
209	C₂²×C₄	—	76	2	no	—
210	C₂²×C₄	—	76	2	no	—
211	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
214	D₈	—	76	2	no	—
215	D₈	—	76	2	no	—
216	M₁₆	—	2	2	no	—
217	M₁₆	D₁₅₂	1	4	yes	complemented
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
223	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
224	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
225	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
226	C₄⋊C₄	—	38	4	no	—
227	C₄⋊C₄	—	38	2	no	—
228	C₄⋊C₄	—	38	2	no	—
229	C₄⋊C₄	—	76	2	no	—
230	C₂²⋊C₄	—	76	2	no	—
231	C₂²⋊C₄	—	76	2	no	—
232	C₄²	—	76	4	no	—
233	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
234	C₄²	#64541	1	1	yes	characteristic
235	C₄²	#64541	1	1	yes	characteristic
236	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
237	C₂³	—	76	2	no	—
238	C₂³	#10597320	1	1	yes	characteristic
239	Q₈	—	76	4	no	—
240	D₄	—	76	2	no	—
241	D₄	—	76	2	no	—
242	D₄	—	76	4	no	—
243	D₄	—	152	2	no	—
244	D₄	—	152	2	no	—
245	D₄	—	152	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	76	4	no	—
250	C₂×C₄	—	76	2	no	—
251	C₂×C₄	—	152	2	no	—
252	C₂×C₄	—	152	2	no	—
253	C₂×C₄	—	152	4	no	—
254	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
255	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
256	C₂×C₄	#10597321	1	1	yes	characteristic
257	C₂×C₄	C₂×D₁₅₂	1	2	yes	—
258	C₂×C₄	#10597350	1	2	yes	—
259	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
260	C₂×C₄	#10597376	1	1	yes	characteristic
261	C₂×C₄	#10597376	1	1	yes	characteristic
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	2	1	no	—
265	C₈	—	2	4	no	—
266	C₈	—	4	2	no	—
267	C₈	#10597282	1	2	yes	complemented
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	152	2	no	—
271	C₂²	—	152	2	no	—
272	C₂²	—	152	2	no	—
273	C₂²	#11926446	1	1	yes	characteristic
274	C₂²	#11926921	1	1	yes	characteristic
275	C₂²	#11926928	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	152	4	no	—
280	C₄	—	152	2	no	—
281	C₄	C₂×#10597282	1	1	yes	characteristic
282	C₄	#11926560	1	1	yes	characteristic
283	C₄	C₄×D₁₅₂	1	2	yes	—
284	C₄	#11926780	1	2	yes	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	304	2	no	—
288	C₂	#14564501	1	1	yes	characteristic
289	C₂	#14564533	1	1	yes	characteristic
290	C₂	#14564554	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597376
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₈×M₁₆×C₁₉	C₂
socle	C₂²×C₁₉	#390
prime radical	#14574214	C₂
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₈×M₁₆	D₁₉
hypocenter	C₁₉	#9719
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
2. #14564554	C₂	semidirect
3. C₄×#10597275	C₂	semidirect
4. #14562131	C₂	semidirect
5. C₈×M₁₆×C₁₉	C₂	semidirect
7. #11926949	C₄	semidirect
8. #11926921	C₄	semidirect
9. #11926891	C₄	semidirect
69. M₁₆×C₁₉	D₈	semidirect
122. C₈×C₁₉	#104	semidirect
127. C₈×M₁₆	D₁₉	semidirect
201. C₁₉	#9719	semidirect
217. M₁₆	D₁₅₂	semidirect
267. C₈	#10597282	semidirect

Maximal subgroups

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

subgroup	index
#14574214	2
#14564554	2
C₄×#10597275	2
#14562131	2
C₈×M₁₆×C₁₉	2
#9719	19