Subgroups — #18079693

Order 4864.

538 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14573115	C₂	1	1	yes	maximal, characteristic
2	#14564474	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562302	C₂	1	1	yes	maximal, characteristic, complemented
4	#14562074	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉⋊#1309	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561526	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#2991	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10597395	C₄	1	2	yes	—
9	C₂×#10597320	C₂²	1	1	yes	characteristic
10	#11926744	—	2	1	no	—
11	#11926744	—	2	1	no	—
12	#11926739	—	2	1	no	—
13	#11926739	—	2	1	no	—
14	#11926568	C₂²	1	1	yes	characteristic
15	#11926568	C₂²	1	1	yes	characteristic
16	#11926548	C₄	1	2	yes	—
17	C₂×#10597280	—	2	2	no	—
18	C₄×#64520	C₂²	1	1	yes	characteristic
19	#11926287	—	2	1	no	—
20	#11926287	—	2	1	no	—
21	Q₈⋊(C₁₉⋊C₈)	—	2	1	no	—
22	D₄⋊(C₁₉⋊C₈)	—	2	1	no	—
23	C₄×#41×C₁₉	C₂²	1	1	yes	characteristic
24	C₈×Q₈×C₁₉	—	2	1	no	—
25	C₁₉×#436	—	2	2	no	—
26	C₁₉×#436	—	2	1	no	—
27	C₁₉×#431	—	2	1	no	—
28	C₁₉×#431	C₂²	1	1	yes	characteristic
29	C₁₉×#423	—	2	1	no	—
30	C₁₉×#423	C₂²	1	1	yes	characteristic
31	C₂×#41×C₁₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×D₇₆	—	2	1	no	—
33	C₂²×D₇₆	—	2	1	no	—
34	C₂²×D₇₆	C₂×C₄	1	2	yes	—
35	C₂×M₁₆×C₁₉	—	2	2	no	—
36	C₂×M₁₆×C₁₉	D₄	1	2	yes	—
37	C₄×Q₈×C₁₉	D₄	1	1	yes	characteristic
38	C₄×Q₈×C₁₉	D₄	1	1	yes	characteristic
39	C₄×D₄×C₁₉	—	2	1	no	—
40	C₄×D₄×C₁₉	—	2	1	no	—
41	C₄×D₄×C₁₉	D₄	1	1	yes	characteristic
42	C₄×D₄×C₁₉	D₄	1	1	yes	characteristic
43	C₁₉×#117	—	2	1	no	—
44	C₁₉×#117	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₁₉	—	2	1	no	—
46	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
47	#10597395	—	4	4	no	—
48	C₂×#64527	—	2	2	no	—
49	C₂×#64525	—	2	2	no	—
50	C₂×#64520	—	2	1	no	—
51	C₂×#64520	—	2	1	no	—
52	C₂×#64520	C₂×C₄	1	2	yes	—
53	#10597320	—	2	1	no	—
54	#10597320	—	2	1	no	—
55	#10597320	—	2	1	no	—
56	#10597320	—	2	1	no	—
57	C₁₉×C₄⋊C₈	—	2	2	no	—
58	C₁₉×C₄⋊C₈	—	2	1	no	—
59	C₁₉×C₄⋊C₈	—	2	1	no	—
60	C₁₉×C₄⋊C₈	D₄	1	1	yes	characteristic
61	C₁₉×C₄⋊C₈	D₄	1	1	yes	characteristic
62	C₁₉×C₂²⋊C₈	—	2	2	no	—
63	C₁₉×C₂²⋊C₈	—	2	1	no	—
64	C₄×C₈×C₁₉	—	2	2	no	—
65	C₄×C₈×C₁₉	—	2	1	no	—
66	#10597280	—	4	4	no	—
67	#10597266	—	2	2	no	—
68	#10597266	—	2	2	no	—
69	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
70	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
71	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
72	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
73	#10597260	—	2	1	no	—
74	C₄×C₁₉⋊C₈	—	2	1	no	—
75	C₂³×D₁₉	—	2	2	no	—
76	#41×C₁₉	—	2	2	no	—
77	#41×C₁₉	—	4	1	no	—
78	C₂×Q₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×D₄×C₁₉	—	2	1	no	—
80	C₂×D₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₉	—	2	1	no	—
82	C₂²×C₄×C₁₉	—	2	1	no	—
83	C₂²×C₄×C₁₉	—	2	1	no	—
84	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
87	C₂×D₇₆	—	2	2	no	—
88	C₂×D₇₆	—	2	2	no	—
89	C₂×D₇₆	—	4	2	no	—
90	C₂×D₇₆	—	4	2	no	—
91	C₂×D₇₆	—	4	2	no	—
92	C₂×D₇₆	—	4	2	no	—
93	C₂×D₇₆	—	4	1	no	—
94	C₂×D₇₆	—	4	1	no	—
95	C₂×D₇₆	—	4	1	no	—
96	C₂×D₇₆	—	4	1	no	—
97	C₂×D₇₆	—	8	1	no	—
98	C₂×D₇₆	—	8	1	no	—
99	M₁₆×C₁₉	—	4	2	no	—
100	M₁₆×C₁₉	—	8	2	no	—
101	C₂×C₈×C₁₉	—	2	1	no	—
102	C₂×C₈×C₁₉	—	2	2	no	—
103	C₂×C₈×C₁₉	—	2	1	no	—
104	C₄⋊C₄×C₁₉	—	2	1	no	—
105	C₄⋊C₄×C₁₉	—	2	1	no	—
106	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
107	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
108	C₂²⋊C₄×C₁₉	—	2	1	no	—
109	C₂²⋊C₄×C₁₉	—	2	1	no	—
110	C₂²⋊C₄×C₁₉	—	2	1	no	—
111	C₄²×C₁₉	—	2	1	no	—
112	C₄²×C₁₉	—	2	1	no	—
113	C₄²×C₁₉	—	2	1	no	—
114	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₁₉	#41	1	1	yes	characteristic
118	#64527	—	4	4	no	—
119	#64525	—	4	4	no	—
120	#64520	—	2	2	no	—
121	#64520	—	4	1	no	—
122	#64520	—	4	2	no	—
123	#64520	—	8	1	no	—
124	C₂×C₁₉⋊C₈	—	2	2	no	—
125	C₂×C₁₉⋊C₈	—	4	1	no	—
126	C₂×C₁₉⋊C₈	—	4	1	no	—
127	#9360	—	19	1	no	maximal, Sylow
128	C₂³×C₁₉	—	2	1	no	—
129	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
130	C₂²×D₁₉	—	4	2	no	—
131	C₂²×D₁₉	—	4	4	no	—
132	C₂²×D₁₉	—	8	2	no	—
133	C₂²×D₁₉	—	8	2	no	—
134	Q₈×C₁₉	—	2	1	no	—
135	Q₈×C₁₉	#100	1	2	yes	complemented
136	D₄×C₁₉	—	2	1	no	—
137	D₄×C₁₉	—	2	1	no	—
138	D₄×C₁₉	—	4	2	no	—
139	C₂×C₄×C₁₉	—	2	1	no	—
140	C₂×C₄×C₁₉	—	2	1	no	—
141	C₂×C₄×C₁₉	—	2	1	no	—
142	C₂×C₄×C₁₉	—	2	1	no	—
143	C₂×C₄×C₁₉	—	2	1	no	—
144	C₂×C₄×C₁₉	—	2	1	no	—
145	C₂×C₄×C₁₉	—	2	1	no	—
146	C₂×C₄×C₁₉	—	2	1	no	—
147	C₂×C₄×C₁₉	—	2	1	no	—
148	C₂×C₄×C₁₉	—	4	2	no	—
149	C₂×C₄×C₁₉	—	4	1	no	—
150	C₂×C₄×C₁₉	—	4	1	no	—
151	C₂×C₄×C₁₉	—	4	1	no	—
152	C₂×C₄×C₁₉	—	4	1	no	—
153	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
155	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
156	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
157	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
158	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
159	D₇₆	—	4	2	no	—
160	D₇₆	—	4	2	no	—
161	D₇₆	—	8	2	no	—
162	D₇₆	—	8	2	no	—
163	D₇₆	—	8	2	no	—
164	D₇₆	—	8	1	no	—
165	D₇₆	—	8	1	no	—
166	D₇₆	—	8	1	no	—
167	D₇₆	—	8	1	no	—
168	C₈×C₁₉	—	4	2	no	—
169	C₈×C₁₉	—	4	1	no	—
170	C₈×C₁₉	—	4	1	no	—
171	C₁₉⋊C₈	—	4	2	no	—
172	C₁₉⋊C₈	—	4	2	no	—
173	C₁₉⋊C₈	—	8	1	no	—
174	#3038	—	19	1	no	—
175	#2991	D₁₉	1	1	yes	characteristic, complemented
176	#1965	—	19	1	no	—
177	#1545	—	19	1	no	—
178	#1489	—	19	1	no	—
179	#1309	—	19	1	no	—
180	#1299	—	19	1	no	—
181	C₂²×C₁₉	—	2	1	no	—
182	C₂²×C₁₉	—	2	1	no	—
183	C₂²×C₁₉	—	2	1	no	—
184	C₂²×C₁₉	—	4	1	no	—
185	C₂²×C₁₉	—	4	1	no	—
186	C₂²×C₁₉	—	4	1	no	—
187	C₂²×C₁₉	#386	1	1	yes	characteristic
188	C₂×D₁₉	—	8	2	no	—
189	C₂×D₁₉	—	8	2	no	—
190	C₂×D₁₉	—	8	2	no	—
191	C₂×D₁₉	—	16	4	no	—
192	C₄×C₁₉	—	2	1	no	—
193	C₄×C₁₉	—	2	1	no	—
194	C₄×C₁₉	—	2	2	no	—
195	C₄×C₁₉	—	2	1	no	—
196	C₄×C₁₉	—	4	1	no	—
197	C₄×C₁₉	—	4	1	no	—
198	C₄×C₁₉	C₂×#100	1	1	yes	characteristic
199	C₄×C₁₉	#413	1	1	yes	characteristic
200	C₄×C₁₉	#453	1	1	yes	characteristic
201	C₄×C₁₉	#454	1	1	yes	characteristic
202	C₄×C₁₉	#471	1	1	yes	characteristic
203	C₄×C₁₉	#472	1	1	yes	characteristic
204	C₂×AGL(1,ℤ/8)	—	19	2	no	—
205	C₂×#127	—	19	1	no	—
206	C₄×#41	C₂×D₁₉	1	1	yes	characteristic
207	#460	—	38	1	no	—
208	#460	—	38	1	no	—
209	#459	—	38	1	no	—
210	#459	—	38	1	no	—
211	C₈×Q₈	—	2	1	no	—
212	#436	—	2	2	no	—
213	#436	—	2	1	no	—
214	#431	—	2	1	no	—
215	#431	C₂×D₁₉	1	1	yes	characteristic
216	#423	—	2	1	no	—
217	#423	—	19	1	no	—
218	#423	—	19	1	no	—
219	#423	C₂×D₁₉	1	1	yes	characteristic
220	#418	—	19	2	no	—
221	C₂×#100	—	38	2	no	—
222	C₄×M₁₆	—	19	1	no	—
223	#331	—	38	1	no	—
224	#331	—	38	1	no	—
225	Q₈⋊C₈	—	38	1	no	—
226	D₄⋊C₈	—	38	1	no	—
227	C₂×C₁₉	—	4	1	no	—
228	C₂×C₁₉	—	8	1	no	—
229	C₂×C₁₉	#1883	1	1	yes	characteristic
230	C₂×C₁₉	#1900	1	1	yes	characteristic
231	C₂×C₁₉	#1907	1	1	yes	characteristic
232	D₁₉	—	16	2	no	—
233	C₂×#41	C₄×D₁₉	1	1	yes	characteristic
234	C₂²×D₄	—	19	2	no	—
235	C₂²×D₄	—	38	1	no	—
236	C₂²×D₄	—	38	1	no	—
237	AGL(1,ℤ/8)	—	76	4	no	—
238	C₂×SD₁₆	—	38	2	no	—
239	C₂×D₈	—	38	2	no	—
240	C₂×M₁₆	—	2	2	no	—
241	C₂×M₁₆	—	19	2	no	—
242	C₂×M₁₆	—	38	1	no	—
243	C₂×M₁₆	—	38	1	no	—
244	C₂×M₁₆	C₂²⋊D₁₉	1	2	yes	—
245	#127	—	38	1	no	—
246	#127	—	38	1	no	—
247	#127	—	38	1	no	—
248	#127	—	38	1	no	—
249	C₄×Q₈	D₇₆	1	1	yes	characteristic
250	C₄×Q₈	C₂²⋊D₁₉	1	1	yes	characteristic
251	C₄×D₄	—	2	1	no	—
252	C₄×D₄	—	2	1	no	—
253	C₄×D₄	D₇₆	1	1	yes	characteristic
254	C₄×D₄	C₂²⋊D₁₉	1	1	yes	characteristic
255	#117	—	2	1	no	—
256	#117	C₄×D₁₉	1	1	yes	characteristic
257	C₂×C₄²	—	2	1	no	—
258	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
259	C₄⋊C₈	—	2	2	no	—
260	C₄⋊C₈	—	2	1	no	—
261	C₄⋊C₈	—	2	1	no	—
262	C₄⋊C₈	—	38	1	no	—
263	C₄⋊C₈	—	38	1	no	—
264	C₄⋊C₈	—	38	1	no	—
265	C₄⋊C₈	—	38	1	no	—
266	C₄⋊C₈	C₂²⋊D₁₉	1	1	yes	characteristic
267	C₄⋊C₈	C₂²⋊D₁₉	1	1	yes	characteristic
268	#102	—	38	2	no	—
269	#102	—	38	2	no	—
270	#100	—	76	4	no	—
271	C₂²⋊C₈	—	2	2	no	—
272	C₂²⋊C₈	—	2	1	no	—
273	#97	—	38	1	no	—
274	C₄×C₈	—	2	2	no	—
275	C₄×C₈	—	2	1	no	—
276	C₄×C₈	—	38	1	no	—
277	C₁₉	#9360	1	1	yes	characteristic, complemented, Sylow
278	C₂⁴	—	38	2	no	—
279	#41	—	2	2	no	—
280	#41	—	4	1	no	—
281	C₂×Q₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
282	C₂×D₄	—	2	1	no	—
283	C₂×D₄	—	38	2	no	—
284	C₂×D₄	—	38	2	no	—
285	C₂×D₄	—	76	2	no	—
286	C₂×D₄	—	76	2	no	—
287	C₂×D₄	—	76	2	no	—
288	C₂×D₄	—	76	2	no	—
289	C₂×D₄	—	76	1	no	—
290	C₂×D₄	—	76	1	no	—
291	C₂×D₄	—	76	1	no	—
292	C₂×D₄	—	76	1	no	—
293	C₂×D₄	—	152	1	no	—
294	C₂×D₄	—	152	1	no	—
295	C₂×D₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
296	C₂²×C₄	—	2	1	no	—
297	C₂²×C₄	—	2	1	no	—
298	C₂²×C₄	—	2	1	no	—
299	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
300	C₂²×C₄	#64541	1	1	yes	characteristic
301	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
302	SD₁₆	—	76	4	no	—
303	D₈	—	76	4	no	—
304	M₁₆	—	4	2	no	—
305	M₁₆	—	8	2	no	—
306	M₁₆	—	38	2	no	—
307	M₁₆	—	76	1	no	—
308	M₁₆	—	76	2	no	—
309	M₁₆	—	152	1	no	—
310	C₂×C₈	—	2	1	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	2	1	no	—
313	C₂×C₈	—	38	2	no	—
314	C₂×C₈	—	76	1	no	—
315	C₂×C₈	—	76	1	no	—
316	C₄⋊C₄	—	2	1	no	—
317	C₄⋊C₄	—	2	1	no	—
318	C₄⋊C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
319	C₄⋊C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
320	C₂²⋊C₄	—	2	1	no	—
321	C₂²⋊C₄	—	2	1	no	—
322	C₂²⋊C₄	—	2	1	no	—
323	C₄²	—	2	1	no	—
324	C₄²	—	2	1	no	—
325	C₄²	—	2	1	no	—
326	C₄²	C₂×D₇₆	1	1	yes	characteristic
327	C₄²	#64541	1	1	yes	characteristic
328	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
329	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
330	C₂³	—	2	1	no	—
331	C₂³	—	76	4	no	—
332	C₂³	—	76	2	no	—
333	C₂³	—	152	2	no	—
334	C₂³	—	152	2	no	—
335	C₂³	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
336	Q₈	—	2	1	no	—
337	Q₈	#10597280	1	2	yes	complemented
338	D₄	—	2	1	no	—
339	D₄	—	2	1	no	—
340	D₄	—	4	2	no	—
341	D₄	—	76	2	no	—
342	D₄	—	76	2	no	—
343	D₄	—	152	2	no	—
344	D₄	—	152	2	no	—
345	D₄	—	152	2	no	—
346	D₄	—	152	1	no	—
347	D₄	—	152	1	no	—
348	D₄	—	152	1	no	—
349	D₄	—	152	1	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	4	1	no	—
361	C₂×C₄	—	4	1	no	—
362	C₂×C₄	—	4	1	no	—
363	C₂×C₄	—	4	1	no	—
364	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
365	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
366	C₂×C₄	#10597375	1	1	yes	characteristic
367	C₂×C₄	#10597376	1	1	yes	characteristic
368	C₂×C₄	#10597376	1	1	yes	characteristic
369	C₂×C₄	#10597399	1	1	yes	characteristic
370	C₈	—	4	2	no	—
371	C₈	—	4	1	no	—
372	C₈	—	4	1	no	—
373	C₈	—	76	2	no	—
374	C₈	—	76	2	no	—
375	C₈	—	152	1	no	—
376	C₂²	—	2	1	no	—
377	C₂²	—	2	1	no	—
378	C₂²	—	2	1	no	—
379	C₂²	—	4	1	no	—
380	C₂²	—	4	1	no	—
381	C₂²	—	4	1	no	—
382	C₂²	—	304	4	no	—
383	C₂²	—	152	2	no	—
384	C₂²	—	152	2	no	—
385	C₂²	—	152	2	no	—
386	C₂²	#11926423	1	1	yes	characteristic
387	C₄	—	2	1	no	—
388	C₄	—	2	2	no	—
389	C₄	—	2	1	no	—
390	C₄	—	2	1	no	—
391	C₄	—	4	1	no	—
392	C₄	—	4	1	no	—
393	C₄	C₂×#10597280	1	1	yes	characteristic
394	C₄	#11926526	1	1	yes	characteristic
395	C₄	#11926712	1	1	yes	characteristic
396	C₄	#11926717	1	1	yes	characteristic
397	C₄	#11926807	1	1	yes	characteristic
398	C₄	#11926814	1	1	yes	characteristic
399	C₂	—	4	1	no	—
400	C₂	—	8	1	no	—
401	C₂	—	304	2	no	—
402	C₂	#14564011	1	1	yes	characteristic
403	C₂	#14564098	1	1	yes	characteristic
404	C₂	#14564143	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926423
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2991	C₂
socle	C₂²×C₁₉	#386
prime radical	#14573115	C₂
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2991	D₁₉
hypocenter	C₁₉	#9360
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2991

Complements

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

N	H ≅ G/N	type
2. #14564474	C₂	semidirect
3. #14562302	C₂	semidirect
4. #14562074	C₂	semidirect
5. C₁₉⋊#1309	C₂	semidirect
6. #14561526	C₂	semidirect
7. C₁₉×#2991	C₂	semidirect
135. Q₈×C₁₉	#100	semidirect
175. #2991	D₁₉	semidirect
277. C₁₉	#9360	semidirect
337. Q₈	#10597280	semidirect

Maximal subgroups

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

subgroup	index
#14573115	2
#14564474	2
#14562302	2
#14562074	2
C₁₉⋊#1309	2
#14561526	2
C₁₉×#2991	2
#9360	19