Subgroups — #18088316

Order 4864.

678 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14565992	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926316	C₂	1	2	yes	maximal
3	C₂×#11926315	C₂	1	2	yes	maximal
4	C₂×#11926313	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₉×#343	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	2	yes	—
7	C₂×#10597372	C₂²	1	1	yes	characteristic
8	C₂×#10597372	C₂²	1	1	yes	characteristic
9	C₄⋊C₄×Dic₁₉	—	2	2	no	—
10	C₄⋊C₄×Dic₁₉	—	2	2	no	—
11	C₄×C₂²⋊Dic₁₉	C₄	1	4	yes	—
12	#11926316	C₄	1	8	yes	complemented
13	#11926315	—	2	4	no	—
14	#11926313	—	2	2	no	—
15	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
16	C₂²×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#343	—	2	2	no	—
18	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₁₉	—	2	2	no	—
21	C₂×M₁₆×C₁₉	—	2	2	no	—
22	C₂×M₁₆×C₁₉	—	2	2	no	—
23	C₂×M₁₆×C₁₉	—	2	2	no	—
24	C₂×M₁₆×C₁₉	—	2	2	no	—
25	C₂×M₁₆×C₁₉	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₁₉	—	2	2	no	—
27	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
30	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
31	C₂×C₂²⋊Dic₁₉	C₂×C₄	1	2	yes	—
32	C₂×C₂²⋊Dic₁₉	C₂×C₄	1	2	yes	—
33	#10597372	—	2	2	no	—
34	#10597372	—	2	1	no	—
35	#10597372	—	2	1	no	—
36	#10597372	C₂×C₄	1	4	yes	—
37	#10597372	D₄	1	2	yes	—
38	#10597372	Q₈	1	2	yes	—
39	C₂×C₄⋊Dic₁₉	—	2	1	no	—
40	C₂×C₄⋊Dic₁₉	—	2	1	no	—
41	C₂×C₄×Dic₁₉	—	2	1	no	—
42	C₂×C₄×Dic₁₉	—	2	1	no	—
43	C₂×C₄×Dic₁₉	—	2	4	no	—
44	C₂×#64520	—	2	2	no	—
45	C₂×#64520	—	2	2	no	—
46	C₂×#64520	—	2	2	no	—
47	C₂×#64520	—	2	2	no	—
48	C₂×#64520	—	2	2	no	—
49	C₂×#64520	C₂×C₄	1	4	yes	—
50	C₂²×C₁₉⋊C₈	—	2	2	no	—
51	C₁₉⋊#95	—	2	4	no	—
52	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	—	2	2	no	—
54	C₂²×C₄×C₁₉	—	2	2	no	—
55	C₂²×C₄×C₁₉	—	8	2	no	—
56	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
57	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	#41	1	2	yes	—
63	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	#41	1	2	yes	—
65	C₂²×C₄×C₁₉	#41	1	2	yes	—
66	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
67	C₂²×Dic₁₉	—	2	2	no	—
68	C₂²×Dic₁₉	—	2	2	no	—
69	M₁₆×C₁₉	—	2	4	no	—
70	M₁₆×C₁₉	—	2	4	no	—
71	M₁₆×C₁₉	—	4	2	no	—
72	M₁₆×C₁₉	—	4	2	no	—
73	C₂×C₈×C₁₉	—	2	4	no	—
74	C₂×C₈×C₁₉	—	4	2	no	—
75	C₂×C₈×C₁₉	—	4	2	no	—
76	C₄⋊C₄×C₁₉	—	2	4	no	—
77	C₄⋊C₄×C₁₉	—	2	4	no	—
78	C₂²⋊Dic₁₉	—	2	2	no	—
79	C₂²⋊Dic₁₉	—	2	2	no	—
80	C₂²⋊Dic₁₉	C₄²	1	4	yes	—
81	C₂²⋊Dic₁₉	C₄⋊C₄	1	4	yes	—
82	C₄⋊Dic₁₉	—	2	4	no	—
83	C₄⋊Dic₁₉	—	2	2	no	—
84	C₄⋊Dic₁₉	—	2	2	no	—
85	C₄×Dic₁₉	—	2	4	no	—
86	C₄×Dic₁₉	—	2	2	no	—
87	C₄×Dic₁₉	—	2	2	no	—
88	C₄×Dic₁₉	—	8	4	no	—
89	#64520	—	2	4	no	—
90	#64520	—	2	4	no	—
91	#64520	—	4	2	no	—
92	#64520	—	4	2	no	—
93	C₂×C₁₉⋊C₈	—	2	4	no	—
94	C₂×C₁₉⋊C₈	—	4	2	no	—
95	C₂×C₁₉⋊C₈	—	4	2	no	—
96	#10830	—	19	1	no	maximal, Sylow
97	C₂³×C₁₉	—	4	1	no	—
98	C₂³×C₁₉	—	4	1	no	—
99	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
100	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂³×C₁₉	#117	1	1	yes	characteristic
103	C₂³×C₁₉	#117	1	1	yes	characteristic
104	C₂³×C₁₉	#117	1	1	yes	characteristic
105	C₂³×C₁₉	#117	1	1	yes	characteristic
106	C₂×C₄×C₁₉	—	2	4	no	—
107	C₂×C₄×C₁₉	—	4	4	no	—
108	C₂×C₄×C₁₉	—	8	2	no	—
109	C₂×C₄×C₁₉	—	8	2	no	—
110	C₂×C₄×C₁₉	#117	1	2	yes	—
111	C₂×C₄×C₁₉	#117	1	2	yes	—
112	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
116	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
117	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₁₉	#123	1	2	yes	—
121	C₂×C₄×C₁₉	#123	1	2	yes	—
122	C₂×C₄×C₁₉	#124	1	2	yes	—
123	C₂×C₄×C₁₉	#125	1	2	yes	—
124	C₂×C₄×C₁₉	#126	1	2	yes	—
125	C₂×C₄×C₁₉	#126	1	2	yes	—
126	C₂×Dic₁₉	—	2	4	no	—
127	C₂×Dic₁₉	—	2	4	no	—
128	C₂×Dic₁₉	—	4	2	no	—
129	C₂×Dic₁₉	—	4	2	no	—
130	C₂×Dic₁₉	—	4	2	no	—
131	C₂×Dic₁₉	—	4	2	no	—
132	C₈×C₁₉	—	4	4	no	—
133	C₁₉⋊C₈	—	4	4	no	—
134	#2274	—	19	1	no	—
135	C₂×#344	—	19	2	no	—
136	C₂×#344	—	19	2	no	—
137	C₂×#343	—	19	1	no	—
138	C₂×#343	D₁₉	1	1	yes	characteristic
139	C₂²×C₁₉	—	2	1	no	—
140	C₂²×C₁₉	—	2	1	no	—
141	C₂²×C₁₉	—	2	1	no	—
142	C₂²×C₁₉	—	2	1	no	—
143	C₂²×C₁₉	—	2	1	no	—
144	C₂²×C₁₉	—	2	1	no	—
145	C₂²×C₁₉	—	4	1	no	—
146	C₂²×C₁₉	—	4	1	no	—
147	C₂²×C₁₉	—	4	1	no	—
148	C₂²×C₁₉	—	4	1	no	—
149	C₂²×C₁₉	#376	1	1	yes	characteristic
150	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
151	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
152	C₂²×C₁₉	#380	1	1	yes	characteristic
153	C₂²×C₁₉	#381	1	1	yes	characteristic
154	C₂²×C₁₉	#382	1	1	yes	characteristic
155	C₂²×C₁₉	#383	1	1	yes	characteristic
156	C₄×C₁₉	—	4	4	no	—
157	C₄×C₁₉	#387	1	2	yes	—
158	C₄×C₁₉	#387	1	2	yes	—
159	C₄×C₁₉	#388	1	2	yes	—
160	C₄×C₁₉	#388	1	2	yes	—
161	Dic₁₉	—	4	4	no	—
162	Dic₁₉	—	4	4	no	—
163	C₂²×M₁₆	—	19	2	no	—
164	C₂²×M₁₆	C₂×D₁₉	1	2	yes	—
165	C₂×#117	—	19	1	no	—
166	C₂×#117	—	19	1	no	—
167	C₂²×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
168	C₄×C₄⋊C₄	—	38	2	no	—
169	C₄×C₄⋊C₄	—	38	2	no	—
170	C₄×C₂²⋊C₄	—	19	4	no	—
171	#344	—	19	8	no	—
172	#344	—	38	4	no	—
173	#343	—	2	2	no	—
174	#343	—	38	2	no	—
175	C₂×C₁₉	—	2	1	no	—
176	C₂×C₁₉	—	2	1	no	—
177	C₂×C₁₉	—	2	1	no	—
178	C₂×C₁₉	—	2	1	no	—
179	C₂×C₁₉	—	2	1	no	—
180	C₂×C₁₉	—	2	1	no	—
181	C₂×C₁₉	#1434	1	1	yes	characteristic
182	C₂×C₁₉	#1829	1	1	yes	characteristic
183	C₂×C₁₉	#1830	1	1	yes	characteristic
184	C₂³×C₄	C₄×D₁₉	1	2	yes	—
185	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	19	4	no	—
192	C₂×M₁₆	—	38	2	no	—
193	C₂×M₁₆	—	38	2	no	—
194	C₂×M₁₆	—	38	2	no	—
195	C₂×M₁₆	—	38	2	no	—
196	C₂×M₁₆	—	38	2	no	—
197	C₂×M₁₆	C₄×D₁₉	1	4	yes	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	—	38	2	no	—
200	#117	—	19	4	no	—
201	#117	—	19	2	no	—
202	#117	—	19	2	no	—
203	#117	—	38	2	no	—
204	#117	—	38	1	no	—
205	#117	—	38	1	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	38	1	no	—
209	C₂×C₄⋊C₄	—	38	1	no	—
210	C₂×C₄⋊C₄	D₇₆	1	2	yes	—
211	C₂×C₄⋊C₄	C₂²⋊D₁₉	1	2	yes	—
212	C₂×C₂²⋊C₄	—	19	2	no	—
213	C₂×C₂²⋊C₄	—	19	2	no	—
214	C₂×C₄²	—	38	1	no	—
215	C₂×C₄²	—	38	1	no	—
216	C₂×C₄²	—	38	4	no	—
217	#95	—	38	4	no	—
218	C₁₉	#10830	1	1	yes	characteristic, complemented, Sylow
219	C₂⁴	C₂×C₄×D₁₉	1	1	yes	characteristic
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	8	2	no	—
223	C₂²×C₄	—	38	2	no	—
224	C₂²×C₄	—	38	2	no	—
225	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	4	yes	—
226	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
227	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
228	C₂²×C₄	#64541	1	2	yes	—
229	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
230	C₂²×C₄	#64543	1	2	yes	—
231	C₂²×C₄	#64543	1	2	yes	—
232	C₂²×C₄	#64543	1	1	yes	characteristic
233	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
234	C₂²×C₄	#64545	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	4	2	no	—
239	M₁₆	—	4	2	no	—
240	M₁₆	—	38	4	no	—
241	M₁₆	—	38	4	no	—
242	M₁₆	—	76	2	no	—
243	M₁₆	—	76	2	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	38	4	no	—
248	C₂×C₈	—	76	2	no	—
249	C₂×C₈	—	76	2	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	38	2	no	—
253	C₄⋊C₄	—	38	2	no	—
254	C₄⋊C₄	—	38	4	no	—
255	C₂²⋊C₄	—	19	4	no	—
256	C₂²⋊C₄	—	19	4	no	—
257	C₂²⋊C₄	—	38	2	no	—
258	C₂²⋊C₄	—	38	2	no	—
259	C₄²	—	38	2	no	—
260	C₄²	—	38	2	no	—
261	C₄²	—	38	4	no	—
262	C₄²	—	152	4	no	—
263	C₂³	—	4	1	no	—
264	C₂³	—	4	1	no	—
265	C₂³	C₄²×D₁₉	1	1	yes	characteristic
266	C₂³	#10597318	1	1	yes	characteristic
267	C₂³	#10597323	1	1	yes	characteristic
268	C₂³	#10597323	1	1	yes	characteristic
269	C₂³	C₄⋊C₄×D₁₉	1	1	yes	characteristic
270	C₂³	#10597338	1	1	yes	characteristic
271	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	8	2	no	—
275	C₂×C₄	—	8	2	no	—
276	C₂×C₄	—	38	4	no	—
277	C₂×C₄	—	38	4	no	—
278	C₂×C₄	—	76	2	no	—
279	C₂×C₄	—	76	2	no	—
280	C₂×C₄	—	76	2	no	—
281	C₂×C₄	—	76	2	no	—
282	C₂×C₄	C₄×D₇₆	1	2	yes	—
283	C₂×C₄	#10597323	1	2	yes	—
284	C₂×C₄	#10597325	1	2	yes	—
285	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
286	C₂×C₄	#10597331	1	2	yes	—
287	C₂×C₄	#10597332	1	2	yes	—
288	C₂×C₄	#10597333	1	2	yes	—
289	C₂×C₄	#10597335	1	2	yes	—
290	C₂×C₄	#10597338	1	2	yes	—
291	C₂×C₄	#10597341	1	1	yes	characteristic
292	C₂×C₄	#10597343	1	1	yes	characteristic
293	C₂×C₄	#10597344	1	2	yes	—
294	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
295	C₂×C₄	#10597381	1	2	yes	—
296	C₂×C₄	#10597384	1	1	yes	characteristic
297	C₂×C₄	#10597392	1	1	yes	characteristic
298	C₈	—	4	4	no	—
299	C₈	—	76	4	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	#11926389	1	1	yes	characteristic
311	C₂²	C₄×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
312	C₂²	#11926395	1	1	yes	characteristic
313	C₂²	#11926401	1	1	yes	characteristic
314	C₂²	#11926405	1	1	yes	characteristic
315	C₂²	#11926408	1	1	yes	characteristic
316	C₂²	#11926411	1	1	yes	characteristic
317	C₄	—	4	4	no	—
318	C₄	—	76	4	no	—
319	C₄	—	76	4	no	—
320	C₄	#11926425	1	2	yes	—
321	C₄	#11926429	1	2	yes	—
322	C₄	#11926432	1	2	yes	—
323	C₄	#11926436	1	2	yes	—
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	2	1	no	—
329	C₂	—	2	1	no	—
330	C₂	#14561859	1	1	yes	characteristic
331	C₂	#14563744	1	1	yes	characteristic
332	C₂	#14563747	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926405
commutator	C₂³×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×#343	C₂
socle	C₂²×C₁₉	#381
prime radical	C₂⁴×C₁₉	C₂²×C₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×#343	D₁₉
hypocenter	C₁₉	#10830
LCS	G ▷ C₂³×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#343

Complements

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

N	H ≅ G/N	type
12. #11926316	C₄	semidirect
218. C₁₉	#10830	semidirect

Maximal subgroups

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

subgroup	index
#14565992	2
C₂×#11926316	2
C₂×#11926315	2
C₂×#11926313	2
C₂×C₁₉×#343	2
#10830	19