Subgroups — #18079091

Order 4864.

492 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14574573	C₂	1	1	yes	maximal, characteristic
2	#14564475	C₂	1	1	yes	maximal, characteristic
3	#14562325	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11926287	C₂	1	1	yes	maximal, characteristic
5	#14561532	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561486	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#2962	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10597320	C₂²	1	1	yes	characteristic
9	#11926918	—	2	1	no	—
10	#11926892	—	2	1	no	—
11	#11926744	—	2	2	no	—
12	#11926694	C₄	1	4	yes	—
13	#11926597	C₂²	1	1	yes	characteristic
14	#11926568	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×#10597280	—	2	1	no	—
17	C₂×#10597280	—	2	1	no	—
18	#11926293	—	2	1	no	—
19	#11926290	—	2	1	no	—
20	#11926287	—	2	1	no	—
21	#11926287	—	2	1	no	—
22	C₁₉×#533	C₂²	1	1	yes	characteristic
23	C₁₉×#446	—	2	1	no	—
24	C₁₉×#446	—	2	2	no	—
25	C₈×Q₈×C₁₉	—	2	1	no	—
26	C₁₉×#424	—	2	1	no	—
27	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₁₉×#405	—	2	1	no	—
29	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
30	C₂²×D₇₆	—	2	1	no	—
31	C₂²×D₇₆	—	2	1	no	—
32	C₂²×D₇₆	C₂×C₄	1	2	yes	—
33	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
34	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₉	—	2	1	no	—
36	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
37	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
38	C₁₉×#125	—	2	1	no	—
39	C₁₉×C₂²⋊Q₈	—	2	1	no	—
40	C₁₉×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
41	C₄×Q₈×C₁₉	—	2	1	no	—
42	C₄×Q₈×C₁₉	D₄	1	2	yes	—
43	C₁₉×#117	—	2	1	no	—
44	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
45	C₂×#64527	—	2	4	no	—
46	C₂×#64520	—	2	1	no	—
47	C₂²×C₁₉⋊C₈	—	2	1	no	—
48	#10597320	—	2	1	no	—
49	#10597320	—	2	1	no	—
50	#10597320	—	2	1	no	—
51	#10597320	—	2	1	no	—
52	C₁₉×C₄⋊C₈	—	2	2	no	—
53	C₁₉×C₄⋊C₈	—	2	2	no	—
54	C₁₉×C₄⋊C₈	—	2	1	no	—
55	C₁₉×C₄⋊C₈	D₄	1	1	yes	characteristic
56	C₁₉×C₄⋊C₈	D₄	1	1	yes	characteristic
57	C₁₉×#97	—	2	2	no	—
58	C₁₉×#97	—	2	1	no	—
59	C₄×C₈×C₁₉	—	2	2	no	—
60	C₄×C₈×C₁₉	—	2	1	no	—
61	C₂²⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
62	#10597280	—	4	2	no	—
63	#10597280	—	4	2	no	—
64	#10597266	—	2	4	no	—
65	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
66	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
67	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
68	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
69	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
70	C₄×C₁₉⋊C₈	—	2	1	no	—
71	C₂³×D₁₉	—	2	2	no	—
72	C₂×Q₈×C₁₉	—	2	1	no	—
73	C₂×Q₈×C₁₉	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
77	C₂×D₇₆	—	2	2	no	—
78	C₂×D₇₆	—	2	2	no	—
79	C₂×D₇₆	—	4	1	no	—
80	C₂×D₇₆	—	4	1	no	—
81	C₂×D₇₆	—	4	1	no	—
82	C₂×D₇₆	—	4	1	no	—
83	C₂×D₇₆	—	4	1	no	—
84	C₂×D₇₆	—	4	1	no	—
85	C₂×D₇₆	—	4	1	no	—
86	C₂×D₇₆	—	4	1	no	—
87	C₂×D₇₆	—	8	2	no	—
88	C₂×D₇₆	—	8	1	no	—
89	C₂×D₇₆	—	8	1	no	—
90	M₁₆×C₁₉	—	2	1	no	—
91	M₁₆×C₁₉	—	2	1	no	—
92	M₁₆×C₁₉	—	2	1	no	—
93	M₁₆×C₁₉	SD₁₆	1	2	yes	complemented
94	C₂×C₈×C₁₉	—	2	1	no	—
95	C₂×C₈×C₁₉	—	2	1	no	—
96	C₂×C₈×C₁₉	—	2	1	no	—
97	C₂×C₈×C₁₉	—	2	1	no	—
98	C₂×C₈×C₁₉	—	4	1	no	—
99	C₂×C₈×C₁₉	—	4	1	no	—
100	C₄⋊C₄×C₁₉	—	2	2	no	—
101	C₄⋊C₄×C₁₉	—	2	1	no	—
102	C₄⋊C₄×C₁₉	—	2	1	no	—
103	C₄⋊C₄×C₁₉	—	2	1	no	—
104	C₄⋊C₄×C₁₉	—	2	2	no	—
105	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
106	C₂²⋊C₄×C₁₉	—	2	2	no	—
107	C₄²×C₁₉	—	2	2	no	—
108	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₁₉	#41	1	1	yes	characteristic
112	#64527	—	8	4	no	—
113	#64520	—	4	1	no	—
114	#64520	—	4	1	no	—
115	C₂×C₁₉⋊C₈	—	2	2	no	—
116	C₂×C₁₉⋊C₈	—	4	1	no	—
117	C₂×C₁₉⋊C₈	—	4	1	no	—
118	C₂×C₁₉⋊C₈	—	8	1	no	—
119	#9274	—	19	1	no	maximal, Sylow
120	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
121	C₂²×D₁₉	—	4	2	no	—
122	C₂²×D₁₉	—	4	2	no	—
123	C₂²×D₁₉	—	4	2	no	—
124	C₂²×D₁₉	—	8	2	no	—
125	C₂²×D₁₉	—	8	2	no	—
126	Q₈×C₁₉	—	4	2	no	—
127	Q₈×C₁₉	—	4	1	no	—
128	Q₈×C₁₉	—	4	1	no	—
129	C₂×C₄×C₁₉	—	2	1	no	—
130	C₂×C₄×C₁₉	—	2	1	no	—
131	C₂×C₄×C₁₉	—	2	2	no	—
132	C₂×C₄×C₁₉	—	2	2	no	—
133	C₂×C₄×C₁₉	—	4	1	no	—
134	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
137	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
138	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
139	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
140	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
141	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
142	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
144	D₇₆	—	4	4	no	—
145	D₇₆	—	8	2	no	—
146	D₇₆	—	8	2	no	—
147	D₇₆	—	8	1	no	—
148	D₇₆	—	8	1	no	—
149	D₇₆	—	8	1	no	—
150	D₇₆	—	8	1	no	—
151	D₇₆	—	8	1	no	—
152	D₇₆	—	8	1	no	—
153	C₈×C₁₉	—	2	2	no	—
154	C₈×C₁₉	—	4	1	no	—
155	C₈×C₁₉	—	4	1	no	—
156	C₈×C₁₉	—	4	1	no	—
157	C₁₉⋊C₈	—	4	2	no	—
158	C₁₉⋊C₈	—	8	1	no	—
159	C₁₉⋊C₈	—	8	1	no	—
160	#3208	—	19	1	no	—
161	#2962	D₁₉	1	1	yes	characteristic, complemented
162	#1965	—	19	1	no	—
163	#1550	—	19	1	no	—
164	C₂×#331	—	19	1	no	—
165	#1301	—	19	1	no	—
166	#1283	—	19	1	no	—
167	C₂²×C₁₉	—	2	1	no	—
168	C₂²×C₁₉	—	2	1	no	—
169	C₂²×C₁₉	#386	1	1	yes	characteristic
170	C₂²×C₁₉	C₄×SD₁₆	1	1	yes	characteristic
171	C₂²×C₁₉	#440	1	1	yes	characteristic
172	C₂×D₁₉	—	8	2	no	—
173	C₂×D₁₉	—	8	2	no	—
174	C₂×D₁₉	—	8	2	no	—
175	C₂×D₁₉	—	16	2	no	—
176	C₂×D₁₉	—	16	2	no	—
177	C₄×C₁₉	—	2	1	no	—
178	C₄×C₁₉	—	2	1	no	—
179	C₄×C₁₉	—	2	1	no	—
180	C₄×C₁₉	—	4	2	no	—
181	C₄×C₁₉	—	4	2	no	—
182	C₄×C₁₉	#448	1	1	yes	characteristic
183	C₄×C₁₉	#449	1	1	yes	characteristic
184	C₄×C₁₉	#460	1	1	yes	characteristic
185	C₄×C₁₉	#464	1	1	yes	characteristic
186	C₄×C₁₉	#471	1	1	yes	characteristic
187	C₄×C₁₉	#472	1	1	yes	characteristic
188	#533	C₂×D₁₉	1	1	yes	characteristic
189	C₂×#127	—	19	1	no	—
190	#492	—	38	1	no	—
191	#486	—	38	1	no	—
192	#460	—	38	2	no	—
193	#450	—	19	4	no	—
194	#446	—	2	2	no	—
195	#446	—	2	1	no	—
196	C₈×Q₈	—	2	1	no	—
197	#431	—	19	1	no	—
198	#424	—	2	1	no	—
199	#423	—	19	1	no	—
200	C₂×C₄⋊C₈	—	19	1	no	—
201	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
202	C₂×#100	—	38	1	no	—
203	C₂×#100	—	38	1	no	—
204	#405	—	2	1	no	—
205	C₄×M₁₆	C₂×D₁₉	1	1	yes	characteristic
206	#333	—	38	1	no	—
207	#332	—	38	1	no	—
208	#331	—	38	1	no	—
209	#331	—	38	1	no	—
210	C₂×C₁₉	—	2	1	no	—
211	C₂×C₁₉	—	2	1	no	—
212	C₂×C₁₉	#1880	1	1	yes	characteristic
213	C₂×C₁₉	#1907	1	1	yes	characteristic
214	C₂×C₁₉	#1914	1	1	yes	characteristic
215	D₁₉	—	16	2	no	—
216	C₂²×D₄	—	19	2	no	—
217	C₂²×D₄	—	38	1	no	—
218	C₂²×D₄	—	38	1	no	—
219	C₂×SD₁₆	—	38	4	no	—
220	C₂×M₁₆	—	38	1	no	—
221	C₂×M₁₆	C₂²⋊D₁₉	1	1	yes	characteristic
222	C₂×M₁₆	C₂²⋊D₁₉	1	1	yes	characteristic
223	C₂²×C₈	—	2	1	no	—
224	C₂²×C₈	—	38	1	no	—
225	C₄⋊Q₈	D₇₆	1	1	yes	characteristic
226	C₄⋊Q₈	D₇₆	1	1	yes	characteristic
227	#127	—	38	1	no	—
228	#127	—	38	1	no	—
229	#127	—	38	1	no	—
230	#127	—	38	1	no	—
231	#125	—	2	1	no	—
232	C₂²⋊Q₈	—	2	1	no	—
233	C₂²⋊Q₈	C₄×D₁₉	1	2	yes	—
234	C₄×Q₈	—	2	1	no	—
235	C₄×Q₈	C₂²⋊D₁₉	1	2	yes	—
236	#117	—	2	1	no	—
237	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
238	C₄⋊C₈	—	2	2	no	—
239	C₄⋊C₈	—	2	2	no	—
240	C₄⋊C₈	—	2	1	no	—
241	C₄⋊C₈	—	38	1	no	—
242	C₄⋊C₈	—	38	1	no	—
243	C₄⋊C₈	—	38	1	no	—
244	C₄⋊C₈	—	38	1	no	—
245	C₄⋊C₈	—	38	1	no	—
246	C₄⋊C₈	C₂²⋊D₁₉	1	1	yes	characteristic
247	C₄⋊C₈	C₂²⋊D₁₉	1	1	yes	characteristic
248	#102	—	38	4	no	—
249	#100	—	76	2	no	—
250	#100	—	76	2	no	—
251	C₂²⋊C₈	—	19	2	no	—
252	#97	—	2	2	no	—
253	#97	—	2	1	no	—
254	C₄×C₈	—	2	2	no	—
255	C₄×C₈	—	2	1	no	—
256	C₄×C₈	—	38	1	no	—
257	C₁₉	#9274	1	1	yes	characteristic, complemented, Sylow
258	C₂⁴	—	38	2	no	—
259	C₂×Q₈	—	2	1	no	—
260	C₂×Q₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
261	C₂×D₄	—	38	2	no	—
262	C₂×D₄	—	38	2	no	—
263	C₂×D₄	—	76	1	no	—
264	C₂×D₄	—	76	1	no	—
265	C₂×D₄	—	76	1	no	—
266	C₂×D₄	—	76	1	no	—
267	C₂×D₄	—	76	1	no	—
268	C₂×D₄	—	76	1	no	—
269	C₂×D₄	—	76	1	no	—
270	C₂×D₄	—	76	1	no	—
271	C₂×D₄	—	152	1	no	—
272	C₂×D₄	—	152	1	no	—
273	C₂×D₄	—	152	2	no	—
274	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
275	C₂²×C₄	#64541	1	1	yes	characteristic
276	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
277	SD₁₆	—	152	4	no	—
278	M₁₆	—	2	1	no	—
279	M₁₆	—	2	1	no	—
280	M₁₆	—	2	1	no	—
281	M₁₆	—	76	1	no	—
282	M₁₆	—	76	1	no	—
283	M₁₆	#64527	1	2	yes	complemented
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	4	1	no	—
289	C₂×C₈	—	4	1	no	—
290	C₂×C₈	—	38	2	no	—
291	C₂×C₈	—	76	1	no	—
292	C₂×C₈	—	76	1	no	—
293	C₂×C₈	—	152	1	no	—
294	C₄⋊C₄	—	2	1	no	—
295	C₄⋊C₄	—	2	1	no	—
296	C₄⋊C₄	—	2	1	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
300	C₂²⋊C₄	—	2	2	no	—
301	C₄²	—	2	2	no	—
302	C₄²	C₂×D₇₆	1	1	yes	characteristic
303	C₄²	#64541	1	1	yes	characteristic
304	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
305	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
306	C₂³	—	76	2	no	—
307	C₂³	—	76	2	no	—
308	C₂³	—	76	2	no	—
309	C₂³	—	152	2	no	—
310	C₂³	—	152	2	no	—
311	C₂³	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
312	Q₈	—	4	2	no	—
313	Q₈	—	4	1	no	—
314	Q₈	—	4	1	no	—
315	D₄	—	76	4	no	—
316	D₄	—	152	2	no	—
317	D₄	—	152	2	no	—
318	D₄	—	152	1	no	—
319	D₄	—	152	1	no	—
320	D₄	—	152	1	no	—
321	D₄	—	152	1	no	—
322	D₄	—	152	1	no	—
323	D₄	—	152	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
330	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
331	C₂×C₄	#10597375	1	1	yes	characteristic
332	C₂×C₄	#10597376	1	1	yes	characteristic
333	C₂×C₄	#10597376	1	1	yes	characteristic
334	C₂×C₄	C₂×#64527	1	1	yes	characteristic
335	C₂×C₄	#10597388	1	1	yes	characteristic
336	C₂×C₄	#10597395	1	1	yes	characteristic
337	C₂×C₄	#10597396	1	1	yes	characteristic
338	C₂×C₄	#10597399	1	1	yes	characteristic
339	C₈	—	2	2	no	—
340	C₈	—	4	1	no	—
341	C₈	—	4	1	no	—
342	C₈	—	4	1	no	—
343	C₈	—	76	2	no	—
344	C₈	—	152	1	no	—
345	C₈	—	152	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	304	2	no	—
349	C₂²	—	304	2	no	—
350	C₂²	—	152	2	no	—
351	C₂²	—	152	2	no	—
352	C₂²	—	152	2	no	—
353	C₂²	#11926423	1	1	yes	characteristic
354	C₂²	C₄×#64527	1	1	yes	characteristic
355	C₂²	#11926642	1	1	yes	characteristic
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	4	2	no	—
360	C₄	—	4	2	no	—
361	C₄	#11926683	1	1	yes	characteristic
362	C₄	#11926689	1	1	yes	characteristic
363	C₄	#11926744	1	1	yes	characteristic
364	C₄	#11926766	1	1	yes	characteristic
365	C₄	#11926807	1	1	yes	characteristic
366	C₄	#11926814	1	1	yes	characteristic
367	C₂	—	2	1	no	—
368	C₂	—	2	1	no	—
369	C₂	—	304	2	no	—
370	C₂	#14563990	1	1	yes	characteristic
371	C₂	#14564143	1	1	yes	characteristic
372	C₂	#14564182	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926423
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2962	C₂
socle	C₂²×C₁₉	#386
prime radical	C₂×#10597320	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2962	D₁₉
hypocenter	C₁₉	#9274
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2962

Complements

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

N	H ≅ G/N	type
3. #14562325	C₂	semidirect
5. #14561532	C₂	semidirect
6. #14561486	C₂	semidirect
7. C₁₉×#2962	C₂	semidirect
93. M₁₆×C₁₉	SD₁₆	semidirect
161. #2962	D₁₉	semidirect
257. C₁₉	#9274	semidirect
283. M₁₆	#64527	semidirect

Maximal subgroups

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

subgroup	index
#14574573	2
#14564475	2
#14562325	2
C₂×#11926287	2
#14561532	2
#14561486	2
C₁₉×#2962	2
#9274	19