Subgroups — #18082348

Order 4864.

694 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926920	C₂	1	1	yes	maximal, characteristic
2	#14564442	C₂	1	1	yes	maximal, characteristic, complemented
3	#14564441	C₂	1	1	yes	maximal, characteristic, complemented
4	#14564440	C₂	1	1	yes	maximal, characteristic, complemented
5	#14563762	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11926265	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1957	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#64525	—	2	2	no	—
9	C₂×#10597320	C₂²	1	1	yes	characteristic
10	#11926920	—	2	2	no	—
11	#11926920	C₄	1	4	yes	complemented
12	C₂×#10597278	—	2	2	no	—
13	C₂×#10597289	—	2	2	no	—
14	C₂×#10597266	—	2	2	no	—
15	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
17	#11926414	C₂²	1	1	yes	characteristic
18	#11926306	—	2	2	no	—
19	#11926265	—	2	2	no	—
20	C₂×C₁₉×#127	C₂²	1	1	yes	characteristic
21	C₂×C₁₉×#102	—	2	2	no	—
22	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
23	C₁₉×#384	C₂²	1	1	yes	characteristic
24	C₂²×D₄×C₁₉	—	2	1	no	—
25	C₂²×D₄×C₁₉	—	2	1	no	—
26	C₂²×D₄×C₁₉	—	2	1	no	—
27	C₂²×D₇₆	—	2	1	no	—
28	C₂²×D₇₆	—	2	1	no	—
29	C₂²×D₇₆	—	2	1	no	—
30	C₂²×C₈×C₁₉	D₄	1	2	yes	—
31	C₁₉×#127	—	2	1	no	—
32	C₁₉×#127	—	2	1	no	—
33	C₁₉×#127	—	2	1	no	—
34	C₁₉×#127	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
36	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
37	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
38	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
39	C₂×#64525	—	2	8	no	—
40	C₂×#64525	—	4	4	no	—
41	C₂×#64525	—	4	4	no	—
42	C₂×C₄⋊Dic₁₉	—	2	1	no	—
43	C₂×C₄⋊Dic₁₉	—	2	1	no	—
44	C₂×C₄⋊Dic₁₉	—	2	1	no	—
45	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
46	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
47	#10597320	—	2	1	no	—
48	#10597320	—	2	1	no	—
49	#10597320	—	2	1	no	—
50	#10597320	C₂×C₄	1	2	yes	—
51	C₁₉×#102	—	4	4	no	—
52	C₄×C₈×C₁₉	—	2	1	no	—
53	C₄×C₈×C₁₉	—	2	1	no	—
54	#10597289	—	4	4	no	—
55	#10597278	—	4	4	no	—
56	#10597266	—	4	4	no	—
57	C₄×C₁₉⋊C₈	—	2	1	no	—
58	C₄×C₁₉⋊C₈	—	2	1	no	—
59	C₄×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
60	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
61	C₂⁴×C₁₉	—	4	1	no	—
62	C₂³×D₁₉	—	4	1	no	—
63	C₂×D₄×C₁₉	—	2	2	no	—
64	C₂×D₄×C₁₉	—	2	2	no	—
65	C₂×D₄×C₁₉	—	2	2	no	—
66	C₂×D₄×C₁₉	—	4	1	no	—
67	C₂×D₄×C₁₉	—	4	1	no	—
68	C₂×D₄×C₁₉	—	4	1	no	—
69	C₂×D₄×C₁₉	—	4	2	no	—
70	C₂×D₄×C₁₉	—	4	2	no	—
71	C₂×D₄×C₁₉	—	4	2	no	—
72	C₂×D₄×C₁₉	—	4	2	no	—
73	C₂×D₄×C₁₉	—	4	2	no	—
74	C₂×D₄×C₁₉	—	4	2	no	—
75	C₂²×C₄×C₁₉	—	4	1	no	—
76	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×Dic₁₉	—	4	1	no	—
80	C₂×D₇₆	—	2	2	no	—
81	C₂×D₇₆	—	2	2	no	—
82	C₂×D₇₆	—	2	2	no	—
83	C₂×D₇₆	—	4	2	no	—
84	C₂×D₇₆	—	4	2	no	—
85	C₂×D₇₆	—	4	2	no	—
86	C₂×D₇₆	—	4	2	no	—
87	C₂×D₇₆	—	4	2	no	—
88	C₂×D₇₆	—	4	2	no	—
89	C₂×D₇₆	—	4	1	no	—
90	C₂×D₇₆	—	4	1	no	—
91	C₂×D₇₆	—	4	1	no	—
92	C₂×C₈×C₁₉	—	2	2	no	—
93	C₂×C₈×C₁₉	—	2	2	no	—
94	C₂×C₈×C₁₉	D₈	1	4	yes	complemented
95	C₄⋊C₄×C₁₉	—	4	2	no	—
96	C₄⋊C₄×C₁₉	—	4	2	no	—
97	C₄⋊C₄×C₁₉	—	4	2	no	—
98	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₁₉	#41	1	1	yes	characteristic
101	C₄²×C₁₉	#41	1	1	yes	characteristic
102	#64525	—	4	16	no	—
103	C₄⋊Dic₁₉	—	4	2	no	—
104	C₄⋊Dic₁₉	—	4	2	no	—
105	C₄⋊Dic₁₉	—	4	2	no	—
106	C₂×C₁₉⋊C₈	—	2	2	no	—
107	C₂×C₁₉⋊C₈	—	2	2	no	—
108	C₂×C₁₉⋊C₈	—	2	2	no	—
109	C₂×C₁₉⋊C₈	—	2	2	no	—
110	C₂×C₁₉⋊C₈	—	2	2	no	—
111	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	4	yes	—
112	#9832	—	19	1	no	maximal, Sylow
113	C₂³×C₁₉	—	4	2	no	—
114	C₂³×C₁₉	—	8	1	no	—
115	C₂³×C₁₉	—	8	1	no	—
116	C₂³×C₁₉	—	8	1	no	—
117	C₂³×C₁₉	—	8	1	no	—
118	C₂³×C₁₉	—	8	1	no	—
119	C₂³×C₁₉	—	8	1	no	—
120	C₂³×C₁₉	#127	1	1	yes	characteristic
121	C₂²×D₁₉	—	4	2	no	—
122	C₂²×D₁₉	—	8	1	no	—
123	C₂²×D₁₉	—	8	1	no	—
124	C₂²×D₁₉	—	8	1	no	—
125	C₂²×D₁₉	—	8	1	no	—
126	C₂²×D₁₉	—	8	1	no	—
127	C₂²×D₁₉	—	8	1	no	—
128	D₄×C₁₉	—	4	4	no	—
129	D₄×C₁₉	—	4	4	no	—
130	D₄×C₁₉	—	4	4	no	—
131	D₄×C₁₉	—	8	2	no	—
132	D₄×C₁₉	—	8	2	no	—
133	D₄×C₁₉	—	8	2	no	—
134	C₂×C₄×C₁₉	—	8	1	no	—
135	C₂×C₄×C₁₉	—	8	1	no	—
136	C₂×C₄×C₁₉	—	8	1	no	—
137	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
141	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
142	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
143	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
144	C₂×C₄×C₁₉	#135	1	2	yes	—
145	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
146	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
147	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
148	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
149	C₂×Dic₁₉	—	8	1	no	—
150	C₂×Dic₁₉	—	8	1	no	—
151	C₂×Dic₁₉	—	8	1	no	—
152	D₇₆	—	4	4	no	—
153	D₇₆	—	4	4	no	—
154	D₇₆	—	4	4	no	—
155	D₇₆	—	8	2	no	—
156	D₇₆	—	8	2	no	—
157	D₇₆	—	8	2	no	—
158	C₈×C₁₉	—	2	4	no	—
159	C₁₉⋊C₈	—	2	4	no	—
160	C₁₉⋊C₈	—	4	2	no	—
161	C₂×#493	—	19	1	no	—
162	#1957	—	19	1	no	—
163	#1957	—	19	1	no	—
164	#1957	—	19	1	no	—
165	#1957	D₁₉	1	1	yes	characteristic, complemented
166	#1834	—	19	1	no	—
167	C₂×#322	—	19	1	no	—
168	C₂²×C₁₉	—	8	2	no	—
169	C₂²×C₁₉	—	8	2	no	—
170	C₂²×C₁₉	—	8	2	no	—
171	C₂²×C₁₉	—	16	1	no	—
172	C₂²×C₁₉	—	16	1	no	—
173	C₂²×C₁₉	—	16	1	no	—
174	C₂²×C₁₉	—	16	1	no	—
175	C₂²×C₁₉	#390	1	1	yes	characteristic
176	C₂²×C₁₉	#493	1	1	yes	characteristic
177	C₂²×C₁₉	#495	1	1	yes	characteristic
178	C₂²×C₁₉	#496	1	1	yes	characteristic
179	C₂²×C₁₉	#496	1	1	yes	characteristic
180	C₂²×C₁₉	#496	1	1	yes	characteristic
181	C₂²×C₁₉	#496	1	1	yes	characteristic
182	C₂×D₁₉	—	8	2	no	—
183	C₂×D₁₉	—	8	2	no	—
184	C₂×D₁₉	—	8	2	no	—
185	C₂×D₁₉	—	16	1	no	—
186	C₂×D₁₉	—	16	1	no	—
187	C₂×D₁₉	—	16	1	no	—
188	C₂×D₁₉	—	16	1	no	—
189	C₄×C₁₉	—	16	1	no	—
190	C₄×C₁₉	#418	1	2	yes	—
191	C₄×C₁₉	C₄×D₈	1	2	yes	—
192	C₄×C₁₉	#442	1	2	yes	—
193	C₄×C₁₉	#466	1	2	yes	—
194	C₄×C₁₉	#469	1	2	yes	—
195	C₄×C₁₉	#490	1	2	yes	—
196	Dic₁₉	—	16	1	no	—
197	C₂²×D₈	—	38	2	no	—
198	C₂×#127	—	19	1	no	—
199	C₂×#127	C₂×D₁₉	1	1	yes	characteristic
200	#493	—	19	4	no	—
201	#493	—	38	2	no	—
202	C₂×#102	—	2	2	no	—
203	C₂×#102	—	38	2	no	—
204	C₂×#102	—	38	2	no	—
205	C₂×#102	—	38	2	no	—
206	C₂×C₄×C₈	—	19	1	no	—
207	C₂×C₄×C₈	—	19	1	no	—
208	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
209	#384	—	19	1	no	—
210	#384	C₂×D₁₉	1	1	yes	characteristic
211	#340	—	38	2	no	—
212	#322	—	38	2	no	—
213	C₂×C₁₉	—	16	2	no	—
214	C₂×C₁₉	#1717	1	1	yes	characteristic
215	C₂×C₁₉	#1717	1	1	yes	characteristic
216	C₂×C₁₉	#1721	1	1	yes	characteristic
217	C₂×C₁₉	#1721	1	1	yes	characteristic
218	C₂×C₁₉	#1961	1	1	yes	characteristic
219	C₂×C₁₉	#1970	1	1	yes	characteristic
220	C₂×C₁₉	#1972	1	1	yes	characteristic
221	D₁₉	—	16	2	no	—
222	C₂²×D₄	—	2	1	no	—
223	C₂²×D₄	—	2	1	no	—
224	C₂²×D₄	—	2	1	no	—
225	C₂²×D₄	—	38	1	no	—
226	C₂²×D₄	—	38	1	no	—
227	C₂²×D₄	—	38	1	no	—
228	C₂×D₈	—	38	8	no	—
229	C₂×D₈	—	76	4	no	—
230	C₂×D₈	—	76	4	no	—
231	C₂²×C₈	—	19	2	no	—
232	C₂²×C₈	—	19	2	no	—
233	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
234	#127	—	2	1	no	—
235	#127	—	2	1	no	—
236	#127	—	2	1	no	—
237	#127	—	19	2	no	—
238	#127	—	38	1	no	—
239	#127	—	38	1	no	—
240	#127	—	38	1	no	—
241	#127	C₄×D₁₉	1	2	yes	—
242	C₂×C₄⋊C₄	—	2	1	no	—
243	C₂×C₄⋊C₄	—	2	1	no	—
244	C₂×C₄⋊C₄	—	2	1	no	—
245	C₂×C₄⋊C₄	—	38	1	no	—
246	C₂×C₄⋊C₄	—	38	1	no	—
247	C₂×C₄⋊C₄	—	38	1	no	—
248	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
249	#102	—	4	4	no	—
250	#102	—	76	4	no	—
251	#102	—	76	4	no	—
252	#102	—	76	4	no	—
253	C₄×C₈	—	2	1	no	—
254	C₄×C₈	—	2	1	no	—
255	C₄×C₈	—	19	2	no	—
256	C₄×C₈	—	19	2	no	—
257	C₄×C₈	—	38	1	no	—
258	C₄×C₈	—	38	1	no	—
259	C₁₉	#9832	1	1	yes	characteristic, complemented, Sylow
260	C₂⁴	—	4	1	no	—
261	C₂⁴	—	76	1	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	4	2	no	—
266	C₂×D₄	—	4	1	no	—
267	C₂×D₄	—	4	2	no	—
268	C₂×D₄	—	4	1	no	—
269	C₂×D₄	—	4	1	no	—
270	C₂×D₄	—	4	2	no	—
271	C₂×D₄	—	4	2	no	—
272	C₂×D₄	—	4	2	no	—
273	C₂×D₄	—	4	2	no	—
274	C₂×D₄	—	38	2	no	—
275	C₂×D₄	—	38	2	no	—
276	C₂×D₄	—	38	2	no	—
277	C₂×D₄	—	76	2	no	—
278	C₂×D₄	—	76	1	no	—
279	C₂×D₄	—	76	2	no	—
280	C₂×D₄	—	76	1	no	—
281	C₂×D₄	—	76	1	no	—
282	C₂×D₄	—	76	2	no	—
283	C₂×D₄	—	76	2	no	—
284	C₂×D₄	—	76	2	no	—
285	C₂×D₄	—	76	2	no	—
286	C₂²×C₄	—	4	1	no	—
287	C₂²×C₄	—	76	1	no	—
288	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
289	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
290	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
291	D₈	—	76	16	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	19	4	no	—
295	C₂×C₈	—	38	2	no	—
296	C₂×C₈	—	38	2	no	—
297	C₂×C₈	—	38	2	no	—
298	C₂×C₈	—	38	2	no	—
299	C₂×C₈	—	38	2	no	—
300	C₂×C₈	#64525	1	4	yes	complemented
301	C₄⋊C₄	—	4	2	no	—
302	C₄⋊C₄	—	4	2	no	—
303	C₄⋊C₄	—	4	2	no	—
304	C₄⋊C₄	—	76	2	no	—
305	C₄⋊C₄	—	76	2	no	—
306	C₄⋊C₄	—	76	2	no	—
307	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
308	C₄²	#64541	1	1	yes	characteristic
309	C₄²	D₄×D₁₉	1	1	yes	characteristic
310	C₄²	#64543	1	1	yes	characteristic
311	C₂³	—	4	2	no	—
312	C₂³	—	8	1	no	—
313	C₂³	—	8	1	no	—
314	C₂³	—	8	1	no	—
315	C₂³	—	8	1	no	—
316	C₂³	—	8	1	no	—
317	C₂³	—	8	1	no	—
318	C₂³	—	76	2	no	—
319	C₂³	—	152	1	no	—
320	C₂³	—	152	1	no	—
321	C₂³	—	152	1	no	—
322	C₂³	—	152	1	no	—
323	C₂³	—	152	1	no	—
324	C₂³	—	152	1	no	—
325	C₂³	#10597386	1	1	yes	characteristic
326	D₄	—	4	4	no	—
327	D₄	—	4	4	no	—
328	D₄	—	4	4	no	—
329	D₄	—	8	2	no	—
330	D₄	—	8	2	no	—
331	D₄	—	8	2	no	—
332	D₄	—	76	4	no	—
333	D₄	—	76	4	no	—
334	D₄	—	76	4	no	—
335	D₄	—	152	2	no	—
336	D₄	—	152	2	no	—
337	D₄	—	152	2	no	—
338	C₂×C₄	—	8	1	no	—
339	C₂×C₄	—	8	1	no	—
340	C₂×C₄	—	8	1	no	—
341	C₂×C₄	—	152	1	no	—
342	C₂×C₄	—	152	1	no	—
343	C₂×C₄	—	152	1	no	—
344	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
345	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
346	C₂×C₄	#10597330	1	1	yes	characteristic
347	C₂×C₄	#10597331	1	1	yes	characteristic
348	C₂×C₄	#10597357	1	2	yes	—
349	C₂×C₄	#10597357	1	2	yes	—
350	C₂×C₄	#10597360	1	2	yes	—
351	C₂×C₄	#10597360	1	2	yes	—
352	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
353	C₂×C₄	C₂×#64525	1	2	yes	—
354	C₂×C₄	#10597385	1	1	yes	characteristic
355	C₂×C₄	#10597396	1	2	yes	—
356	C₈	—	2	4	no	—
357	C₈	—	38	4	no	—
358	C₈	—	76	2	no	—
359	C₂²	—	8	2	no	—
360	C₂²	—	8	2	no	—
361	C₂²	—	8	2	no	—
362	C₂²	—	16	1	no	—
363	C₂²	—	16	1	no	—
364	C₂²	—	16	1	no	—
365	C₂²	—	16	1	no	—
366	C₂²	—	304	1	no	—
367	C₂²	—	304	1	no	—
368	C₂²	—	304	1	no	—
369	C₂²	—	304	1	no	—
370	C₂²	—	152	2	no	—
371	C₂²	—	152	2	no	—
372	C₂²	—	152	2	no	—
373	C₂²	#11926444	1	1	yes	characteristic
374	C₂²	#11926920	1	1	yes	characteristic
375	C₂²	#11926926	1	1	yes	characteristic
376	C₂²	#11926930	1	1	yes	characteristic
377	C₂²	#11926930	1	1	yes	characteristic
378	C₂²	#11926932	1	1	yes	characteristic
379	C₂²	#11926932	1	1	yes	characteristic
380	C₄	—	16	1	no	—
381	C₄	—	304	1	no	—
382	C₄	#11926547	1	2	yes	—
383	C₄	C₄×#64525	1	2	yes	—
384	C₄	#11926653	1	2	yes	—
385	C₄	#11926778	1	2	yes	—
386	C₄	#11926796	1	2	yes	—
387	C₄	#11926907	1	2	yes	—
388	C₂	—	16	2	no	—
389	C₂	—	304	2	no	—
390	C₂	#14563271	1	1	yes	characteristic
391	C₂	#14563275	1	1	yes	characteristic
392	C₂	#14563299	1	1	yes	characteristic
393	C₂	#14563302	1	1	yes	characteristic
394	C₂	#14564457	1	1	yes	characteristic
395	C₂	#14564500	1	1	yes	characteristic
396	C₂	#14564508	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597386
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1957	C₂
socle	C₂³×C₁₉	#127
prime radical	C₂×#11926920	C₂
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1957	D₁₉
hypocenter	C₁₉	#9832
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1957

Complements

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

N	H ≅ G/N	type
2. #14564442	C₂	semidirect
3. #14564441	C₂	semidirect
4. #14564440	C₂	semidirect
5. #14563762	C₂	semidirect
6. C₂×#11926265	C₂	semidirect
7. C₁₉×#1957	C₂	semidirect
11. #11926920	C₄	semidirect
94. C₂×C₈×C₁₉	D₈	semidirect
165. #1957	D₁₉	semidirect
259. C₁₉	#9832	semidirect
300. C₂×C₈	#64525	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926920	2
#14564442	2
#14564441	2
#14564440	2
#14563762	2
C₂×#11926265	2
C₁₉×#1957	2
#9832	19