Subgroups — #18081626

Order 4864.

610 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14571993	C₂	1	1	yes	maximal, characteristic
2	#14563786	C₂	1	1	yes	maximal, characteristic, complemented
3	#14563490	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11926290	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11926287	C₂	1	1	yes	maximal, characteristic
6	C₂×#11926287	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₉×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10597320	C₂²	1	1	yes	characteristic
9	C₂×#10597393	—	2	1	no	—
10	C₂×#10597393	—	2	1	no	—
11	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×Dic₁₉⋊C₈	—	2	2	no	—
14	C₂×#10597272	—	2	1	no	—
15	#11926445	—	2	2	no	—
16	#11926445	C₄	1	4	yes	—
17	C₄²×Dic₁₉	C₂²	1	1	yes	characteristic
18	#11926298	—	2	1	no	—
19	#11926290	—	2	2	no	—
20	#11926287	—	2	2	no	—
21	#11926287	—	2	2	no	—
22	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
23	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₁₉×#332	—	2	2	no	—
26	C₂²×Q₈×C₁₉	—	2	1	no	—
27	C₂²×D₇₆	—	2	1	no	—
28	C₂²×D₇₆	—	2	1	no	—
29	C₂²×D₇₆	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₉	—	2	1	no	—
31	C₂²×C₈×C₁₉	—	2	1	no	—
32	C₁₉×C₄⋊Q₈	—	2	1	no	—
33	C₁₉×C₄⋊Q₈	—	2	1	no	—
34	C₁₉×C₄⋊Q₈	—	2	1	no	—
35	C₁₉×C₄⋊Q₈	—	2	1	no	—
36	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
37	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
39	#10597393	—	4	2	no	—
40	#10597393	—	4	2	no	—
41	#10597393	—	4	2	no	—
42	#10597393	—	4	2	no	—
43	C₂×(C₂×D₁₉)⋊C₄	—	2	4	no	—
44	C₂×(C₂×D₁₉)⋊C₄	—	2	4	no	—
45	C₂×C₄×Dic₁₉	C₂×C₄	1	2	yes	—
46	C₂×C₄×Dic₁₉	D₄	1	1	yes	characteristic
47	C₂×C₄×Dic₁₉	D₄	1	2	yes	—
48	C₂×C₄×Dic₁₉	D₄	1	1	yes	characteristic
49	C₂²×C₁₉⋊C₈	—	2	1	no	—
50	C₂²×C₁₉⋊C₈	—	2	1	no	—
51	#10597320	—	2	1	no	—
52	#10597320	—	2	1	no	—
53	#10597320	—	2	1	no	—
54	#10597320	—	2	1	no	—
55	C₁₉×C₄⋊C₈	—	2	1	no	—
56	C₁₉×C₄⋊C₈	—	2	1	no	—
57	C₁₉×C₄⋊C₈	—	2	1	no	—
58	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
59	#10597272	—	4	2	no	—
60	Dic₁₉⋊C₈	—	4	4	no	—
61	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
62	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
63	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
64	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
65	C₂³×D₁₉	—	2	2	no	—
66	C₂×Q₈×C₁₉	—	4	1	no	—
67	C₂×Q₈×C₁₉	—	4	1	no	—
68	C₂×Q₈×C₁₉	—	4	1	no	—
69	C₂×Q₈×C₁₉	—	4	1	no	—
70	C₂×Q₈×C₁₉	—	4	1	no	—
71	C₂×Q₈×C₁₉	—	4	1	no	—
72	C₂²×C₄×C₁₉	—	2	2	no	—
73	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×Dic₁₉	C₂²⋊C₄	1	2	yes	—
77	C₂²×Dic₁₉	C₂²⋊C₄	1	2	yes	—
78	C₂×D₇₆	—	2	2	no	—
79	C₂×D₇₆	—	2	2	no	—
80	C₂×D₇₆	—	4	4	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	—	4	1	no	—
88	C₂×D₇₆	—	4	1	no	—
89	C₂×D₇₆	—	4	2	no	—
90	C₂×D₇₆	—	4	2	no	—
91	C₂×C₈×C₁₉	—	2	2	no	—
92	C₂×C₈×C₁₉	—	4	1	no	—
93	C₂×C₈×C₁₉	—	4	1	no	—
94	C₂×C₈×C₁₉	—	4	1	no	—
95	C₂×C₈×C₁₉	—	4	2	no	—
96	C₄⋊C₄×C₁₉	—	2	2	no	—
97	C₄⋊C₄×C₁₉	—	2	2	no	—
98	C₄⋊C₄×C₁₉	—	4	1	no	—
99	C₄⋊C₄×C₁₉	—	4	1	no	—
100	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₁₉	#41	1	1	yes	characteristic
103	C₄²×C₁₉	#41	1	1	yes	characteristic
104	(C₂×D₁₉)⋊C₄	—	4	8	no	—
105	(C₂×D₁₉)⋊C₄	—	4	8	no	—
106	C₄×Dic₁₉	—	2	1	no	—
107	C₄×Dic₁₉	—	2	1	no	—
108	C₄×Dic₁₉	—	2	2	no	—
109	C₄×Dic₁₉	—	2	2	no	—
110	C₄×Dic₁₉	—	4	2	no	—
111	C₄×Dic₁₉	D₈	1	2	yes	—
112	C₄×Dic₁₉	SD₁₆	1	2	yes	—
113	C₂×C₁₉⋊C₈	—	2	2	no	—
114	C₂×C₁₉⋊C₈	—	4	1	no	—
115	C₂×C₁₉⋊C₈	—	4	1	no	—
116	C₂×C₁₉⋊C₈	—	4	1	no	—
117	C₂×C₁₉⋊C₈	—	4	2	no	—
118	#9704	—	19	1	no	maximal, Sylow
119	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂²×D₁₉	—	4	2	no	—
121	C₂²×D₁₉	—	4	4	no	—
122	C₂²×D₁₉	—	4	2	no	—
123	C₂²×D₁₉	—	4	4	no	—
124	C₂²×D₁₉	—	4	2	no	—
125	Q₈×C₁₉	—	8	1	no	—
126	Q₈×C₁₉	—	8	1	no	—
127	Q₈×C₁₉	—	8	1	no	—
128	Q₈×C₁₉	—	8	1	no	—
129	C₂×C₄×C₁₉	—	2	2	no	—
130	C₂×C₄×C₁₉	—	4	2	no	—
131	C₂×C₄×C₁₉	—	4	2	no	—
132	C₂×C₄×C₁₉	—	4	2	no	—
133	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
136	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
137	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
138	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
139	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
141	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
144	C₂×C₄×C₁₉	#137	1	2	yes	—
145	C₂×Dic₁₉	—	2	2	no	—
146	C₂×Dic₁₉	—	2	2	no	—
147	C₂×Dic₁₉	—	2	2	no	—
148	C₂×Dic₁₉	—	2	2	no	—
149	C₂×Dic₁₉	—	2	2	no	—
150	C₂×Dic₁₉	#102	1	4	yes	—
151	D₇₆	—	8	2	no	—
152	D₇₆	—	8	2	no	—
153	D₇₆	—	8	1	no	—
154	D₇₆	—	8	1	no	—
155	D₇₆	—	8	1	no	—
156	D₇₆	—	8	1	no	—
157	D₇₆	—	8	2	no	—
158	D₇₆	—	8	1	no	—
159	D₇₆	—	8	1	no	—
160	C₈×C₁₉	—	4	2	no	—
161	C₈×C₁₉	—	8	1	no	—
162	C₁₉⋊C₈	—	4	2	no	—
163	C₁₉⋊C₈	—	8	1	no	—
164	#2858	—	19	1	no	—
165	#1842	—	19	1	no	—
166	#1767	—	19	1	no	—
167	C₂×#332	—	19	1	no	—
168	C₂×#332	D₁₉	1	1	yes	characteristic, complemented
169	C₂×#331	—	19	1	no	—
170	C₂×#331	—	19	1	no	—
171	C₂²×C₁₉	#390	1	1	yes	characteristic
172	C₂²×C₁₉	#413	1	2	yes	—
173	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
174	C₂²×C₁₉	#417	1	2	yes	—
175	C₂²×C₁₉	#418	1	1	yes	characteristic
176	C₂×D₁₉	—	8	4	no	—
177	C₂×D₁₉	—	8	4	no	—
178	C₂×D₁₉	—	8	2	no	—
179	C₂×D₁₉	—	8	2	no	—
180	C₂×D₁₉	—	8	2	no	—
181	C₄×C₁₉	—	2	1	no	—
182	C₄×C₁₉	—	2	1	no	—
183	C₄×C₁₉	—	8	2	no	—
184	C₄×C₁₉	#459	1	1	yes	characteristic
185	C₄×C₁₉	#459	1	1	yes	characteristic
186	C₄×C₁₉	#460	1	1	yes	characteristic
187	C₄×C₁₉	#460	1	1	yes	characteristic
188	C₄×C₁₉	#489	1	2	yes	—
189	C₄×C₁₉	#496	1	1	yes	characteristic
190	C₄×C₁₉	#497	1	1	yes	characteristic
191	Dic₁₉	—	2	4	no	—
192	Dic₁₉	—	4	2	no	—
193	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
194	C₂×#127	—	19	1	no	—
195	C₂×#124	—	38	1	no	—
196	C₂×#124	—	38	1	no	—
197	C₂×C₄⋊C₈	—	19	1	no	—
198	C₂×C₄⋊C₈	—	19	1	no	—
199	C₂×C₄⋊C₈	—	38	2	no	—
200	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
201	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
202	C₂×#97	—	38	1	no	—
203	#390	—	19	4	no	—
204	#390	—	38	2	no	—
205	C₄³	—	19	1	no	—
206	#336	—	38	1	no	—
207	#332	—	2	2	no	—
208	#332	—	38	2	no	—
209	#331	—	38	2	no	—
210	#331	—	38	2	no	—
211	C₂×C₁₉	#1547	1	2	yes	—
212	C₂×C₁₉	#1557	1	2	yes	—
213	C₂×C₁₉	#1959	1	1	yes	characteristic
214	C₂×C₁₉	#1967	1	1	yes	characteristic
215	C₂×C₁₉	#1982	1	1	yes	characteristic
216	D₁₉	—	16	2	no	—
217	C₂²×Q₈	—	2	1	no	—
218	C₂²×D₄	—	19	2	no	—
219	C₂²×D₄	—	38	1	no	—
220	C₂²×D₄	—	38	1	no	—
221	C₂²×C₈	—	2	1	no	—
222	C₂²×C₈	—	2	1	no	—
223	C₂²×C₈	—	38	1	no	—
224	C₂²×C₈	—	38	1	no	—
225	C₄⋊Q₈	—	2	1	no	—
226	C₄⋊Q₈	—	2	1	no	—
227	C₄⋊Q₈	—	2	1	no	—
228	C₄⋊Q₈	—	2	1	no	—
229	#127	—	38	1	no	—
230	#127	—	38	1	no	—
231	#127	—	38	1	no	—
232	#127	—	38	1	no	—
233	#124	—	76	2	no	—
234	#124	—	76	2	no	—
235	#124	—	76	2	no	—
236	#124	—	76	2	no	—
237	C₂×C₄⋊C₄	—	2	1	no	—
238	C₂×C₄⋊C₄	C₄×D₁₉	1	2	yes	—
239	C₂×C₂²⋊C₄	—	38	4	no	—
240	C₂×C₂²⋊C₄	—	38	4	no	—
241	C₂×C₄²	—	19	1	no	—
242	C₂×C₄²	—	19	1	no	—
243	C₂×C₄²	—	19	2	no	—
244	C₂×C₄²	—	19	2	no	—
245	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
246	C₄⋊C₈	—	2	1	no	—
247	C₄⋊C₈	—	2	1	no	—
248	C₄⋊C₈	—	2	1	no	—
249	C₄⋊C₈	—	19	2	no	—
250	C₄⋊C₈	—	38	1	no	—
251	C₄⋊C₈	—	38	1	no	—
252	C₄⋊C₈	—	38	1	no	—
253	C₄⋊C₈	—	76	4	no	—
254	C₄⋊C₈	C₂²⋊D₁₉	1	2	yes	—
255	#97	—	76	2	no	—
256	C₁₉	#9704	1	1	yes	characteristic, complemented, Sylow
257	C₂⁴	—	38	2	no	—
258	C₂×Q₈	—	4	1	no	—
259	C₂×Q₈	—	4	1	no	—
260	C₂×Q₈	—	4	1	no	—
261	C₂×Q₈	—	4	1	no	—
262	C₂×Q₈	—	4	1	no	—
263	C₂×Q₈	—	4	1	no	—
264	C₂×D₄	—	38	2	no	—
265	C₂×D₄	—	38	2	no	—
266	C₂×D₄	—	76	4	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	—	76	1	no	—
272	C₂×D₄	—	76	1	no	—
273	C₂×D₄	—	76	1	no	—
274	C₂×D₄	—	76	1	no	—
275	C₂×D₄	—	76	2	no	—
276	C₂×D₄	—	76	2	no	—
277	C₂²×C₄	—	2	2	no	—
278	C₂²×C₄	—	19	2	no	—
279	C₂²×C₄	—	19	2	no	—
280	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
281	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
282	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	4	1	no	—
286	C₂×C₈	—	4	1	no	—
287	C₂×C₈	—	4	2	no	—
288	C₂×C₈	—	38	2	no	—
289	C₂×C₈	—	76	1	no	—
290	C₂×C₈	—	76	1	no	—
291	C₂×C₈	—	76	1	no	—
292	C₂×C₈	—	76	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₂²⋊C₄	—	76	8	no	—
298	C₂²⋊C₄	—	76	8	no	—
299	C₄²	—	19	2	no	—
300	C₄²	—	19	2	no	—
301	C₄²	—	38	1	no	—
302	C₄²	—	38	2	no	—
303	C₄²	—	38	1	no	—
304	C₄²	—	38	2	no	—
305	C₄²	—	76	2	no	—
306	C₄²	#64541	1	1	yes	characteristic
307	C₄²	D₄×D₁₉	1	1	yes	characteristic
308	C₄²	#64543	1	1	yes	characteristic
309	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
310	C₂³	—	76	2	no	—
311	C₂³	—	76	4	no	—
312	C₂³	—	76	2	no	—
313	C₂³	—	76	2	no	—
314	C₂³	—	76	4	no	—
315	C₂³	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
316	Q₈	—	8	1	no	—
317	Q₈	—	8	1	no	—
318	Q₈	—	8	1	no	—
319	Q₈	—	8	1	no	—
320	D₄	—	152	2	no	—
321	D₄	—	152	2	no	—
322	D₄	—	152	2	no	—
323	D₄	—	152	1	no	—
324	D₄	—	152	1	no	—
325	D₄	—	152	1	no	—
326	D₄	—	152	1	no	—
327	D₄	—	152	1	no	—
328	D₄	—	152	1	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	19	4	no	—
334	C₂×C₄	—	38	2	no	—
335	C₂×C₄	—	38	2	no	—
336	C₂×C₄	—	38	2	no	—
337	C₂×C₄	—	38	2	no	—
338	C₂×C₄	—	38	2	no	—
339	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
340	C₂×C₄	#10597330	1	1	yes	characteristic
341	C₂×C₄	#10597331	1	1	yes	characteristic
342	C₂×C₄	D₈×D₁₉	1	1	yes	characteristic
343	C₂×C₄	#10597357	1	1	yes	characteristic
344	C₂×C₄	SD₁₆×D₁₉	1	1	yes	characteristic
345	C₂×C₄	#10597360	1	2	yes	—
346	C₂×C₄	#10597360	1	1	yes	characteristic
347	C₂×C₄	#10597364	1	2	yes	—
348	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
349	C₂×C₄	#10597385	1	1	yes	characteristic
350	C₂×C₄	#10597386	1	1	yes	characteristic
351	C₈	—	4	2	no	—
352	C₈	—	8	1	no	—
353	C₈	—	76	2	no	—
354	C₈	—	152	1	no	—
355	C₂²	—	152	4	no	—
356	C₂²	—	152	4	no	—
357	C₂²	—	152	2	no	—
358	C₂²	—	152	2	no	—
359	C₂²	—	152	2	no	—
360	C₂²	#11926444	1	1	yes	characteristic
361	C₂²	#11926527	1	2	yes	—
362	C₂²	#102×D₁₉	1	1	yes	characteristic
363	C₂²	#11926544	1	2	yes	—
364	C₂²	#11926547	1	1	yes	characteristic
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	8	2	no	—
368	C₄	—	38	4	no	—
369	C₄	—	76	2	no	—
370	C₄	#11926737	1	1	yes	characteristic
371	C₄	#11926738	1	1	yes	characteristic
372	C₄	#11926742	1	1	yes	characteristic
373	C₄	#11926743	1	1	yes	characteristic
374	C₄	#11926905	1	2	yes	—
375	C₄	#11926930	1	1	yes	characteristic
376	C₄	#11926937	1	1	yes	characteristic
377	C₂	—	304	2	no	—
378	C₂	#14562311	1	2	yes	—
379	C₂	#14562356	1	2	yes	—
380	C₂	#14564448	1	1	yes	characteristic
381	C₂	#14564483	1	1	yes	characteristic
382	C₂	#14564560	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊C₄×D₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#332	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂×#10597320	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#332	D₁₉
hypocenter	C₁₉	#9704
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#332

Complements

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

N	H ≅ G/N	type
2. #14563786	C₂	semidirect
3. #14563490	C₂	semidirect
4. C₂×#11926290	C₂	semidirect
7. C₂×C₁₉×#332	C₂	semidirect
168. C₂×#332	D₁₉	semidirect
256. C₁₉	#9704	semidirect

Maximal subgroups

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

subgroup	index
#14571993	2
#14563786	2
#14563490	2
C₂×#11926290	2
C₂×#11926287	2
C₂×C₁₉×#332	2
#9704	19