Subgroups — #18074034

Order 4864.

1030 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10597315	C₂	1	1	yes	maximal, characteristic
2	#14563615	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926292	C₂	1	4	yes	maximal, complemented
4	C₁₉×#1797	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₃₈	C₄	1	2	yes	—
6	C₂×#10597315	—	2	8	no	—
7	C₂×#10597315	—	2	1	no	—
8	C₂×#10597315	—	2	1	no	—
9	C₂×#10597315	C₄	1	2	yes	—
10	C₂×#10597315	C₂²	1	2	yes	—
11	C₂²×C₄⋊Dic₁₉	—	2	1	no	—
12	C₂²×C₄⋊Dic₁₉	—	2	1	no	—
13	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
15	#11926299	—	2	1	no	—
16	#11926292	—	2	8	no	—
17	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
18	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
19	C₂×C₁₉×C₂²⋊C₈	—	2	1	no	—
20	C₁₉×#336	—	2	1	no	—
21	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
24	C₂³×Dic₁₉	—	2	2	no	—
25	C₂²×Dic₃₈	—	2	2	no	—
26	C₂²×Dic₃₈	—	2	8	no	—
27	C₂²×Dic₃₈	—	4	4	no	—
28	C₂²×Dic₃₈	—	4	4	no	—
29	C₂²×Dic₃₈	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₁₉	—	2	2	no	—
31	C₂×C₄²×C₁₉	—	2	4	no	—
32	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
36	C₂×C₄⋊Dic₁₉	—	2	8	no	—
37	C₂×C₄⋊Dic₁₉	—	2	8	no	—
38	C₂×C₄⋊Dic₁₉	—	2	2	no	—
39	C₂×C₄⋊Dic₁₉	—	2	2	no	—
40	C₂×C₄⋊Dic₁₉	—	2	2	no	—
41	C₂×C₄⋊Dic₁₉	—	2	2	no	—
42	C₂²×C₁₉⋊C₈	—	2	2	no	—
43	#10597315	—	2	8	no	—
44	#10597315	—	2	8	no	—
45	#10597315	—	2	8	no	—
46	#10597315	—	2	4	no	—
47	#10597315	D₄	1	8	yes	—
48	C₁₉×C₄⋊C₈	—	2	4	no	—
49	C₁₉×C₂²⋊C₈	—	4	2	no	—
50	C₂²⋊(C₁₉⋊C₈)	—	4	2	no	—
51	C₄⋊(C₁₉⋊C₈)	—	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	—	2	2	no	—
56	C₂²×C₄×C₁₉	—	2	2	no	—
57	C₂²×C₄×C₁₉	—	2	2	no	—
58	C₂²×C₄×C₁₉	—	2	2	no	—
59	C₂²×C₄×C₁₉	—	2	2	no	—
60	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
64	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
65	C₂²×C₄×C₁₉	Q₁₆	1	2	yes	—
66	C₂²×C₄×C₁₉	Q₁₆	1	2	yes	—
67	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×Dic₁₉	—	2	8	no	—
71	C₂²×Dic₁₉	—	2	4	no	—
72	C₂²×Dic₁₉	—	4	4	no	—
73	C₂²×Dic₁₉	—	4	4	no	—
74	C₂×Dic₃₈	—	2	8	no	—
75	C₂×Dic₃₈	—	2	4	no	—
76	C₂×Dic₃₈	—	4	4	no	—
77	C₂×Dic₃₈	—	4	32	no	—
78	C₂×Dic₃₈	—	4	4	no	—
79	C₂×Dic₃₈	—	4	4	no	—
80	C₂×Dic₃₈	—	4	4	no	—
81	C₂×Dic₃₈	C₂²⋊C₄	1	8	yes	—
82	C₂×C₈×C₁₉	—	4	2	no	—
83	C₂×C₈×C₁₉	—	4	2	no	—
84	C₂×C₈×C₁₉	—	4	2	no	—
85	C₄²×C₁₉	—	2	2	no	—
86	C₄²×C₁₉	—	2	2	no	—
87	C₄²×C₁₉	—	2	2	no	—
88	C₄²×C₁₉	C₂×D₄	1	4	yes	—
89	C₄⋊Dic₁₉	—	2	8	no	—
90	C₄⋊Dic₁₉	—	2	8	no	—
91	C₄⋊Dic₁₉	—	2	8	no	—
92	C₄⋊Dic₁₉	—	2	8	no	—
93	C₂×C₁₉⋊C₈	—	4	2	no	—
94	C₂×C₁₉⋊C₈	—	4	2	no	—
95	C₂×C₁₉⋊C₈	—	4	2	no	—
96	#8342	—	19	1	no	maximal, Sylow
97	C₂³×C₁₉	—	2	1	no	—
98	C₂³×C₁₉	—	2	1	no	—
99	C₂³×C₁₉	—	2	1	no	—
100	C₂³×C₁₉	—	2	1	no	—
101	C₂³×C₁₉	#100	1	2	yes	—
102	C₂³×C₁₉	#103	1	2	yes	—
103	C₂³×C₁₉	#103	1	2	yes	—
104	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₉	—	2	8	no	—
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	2	1	no	—
108	C₂×C₄×C₁₉	—	2	1	no	—
109	C₂×C₄×C₁₉	—	2	1	no	—
110	C₂×C₄×C₁₉	—	2	1	no	—
111	C₂×C₄×C₁₉	—	2	1	no	—
112	C₂×C₄×C₁₉	—	2	1	no	—
113	C₂×C₄×C₁₉	—	2	1	no	—
114	C₂×C₄×C₁₉	—	2	4	no	—
115	C₂×C₄×C₁₉	—	2	4	no	—
116	C₂×C₄×C₁₉	—	2	4	no	—
117	C₂×C₄×C₁₉	—	2	4	no	—
118	C₂×C₄×C₁₉	#100	1	2	yes	—
119	C₂×C₄×C₁₉	#103	1	2	yes	—
120	C₂×C₄×C₁₉	#103	1	2	yes	—
121	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
122	C₂×C₄×C₁₉	#120	1	2	yes	—
123	C₂×C₄×C₁₉	#120	1	2	yes	—
124	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
127	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
128	C₂×C₄×C₁₉	#137	1	2	yes	—
129	C₂×C₄×C₁₉	#137	1	2	yes	—
130	C₂×Dic₁₉	—	2	8	no	—
131	C₂×Dic₁₉	—	4	16	no	—
132	C₂×Dic₁₉	—	4	4	no	—
133	C₂×Dic₁₉	—	4	4	no	—
134	Dic₃₈	—	4	16	no	—
135	Dic₃₈	—	4	8	no	—
136	Dic₃₈	—	4	8	no	—
137	C₈×C₁₉	—	8	2	no	—
138	C₁₉⋊C₈	—	8	2	no	—
139	C₂²×C₄⋊Q₈	—	19	1	no	—
140	#1797	—	19	1	no	—
141	#1797	D₁₉	1	1	yes	characteristic
142	C₂×#333	—	19	4	no	—
143	C₂²×C₁₉	—	2	2	no	—
144	C₂²×C₁₉	—	2	2	no	—
145	C₂²×C₁₉	—	2	2	no	—
146	C₂²×C₁₉	—	2	2	no	—
147	C₂²×C₁₉	—	2	2	no	—
148	C₂²×C₁₉	—	2	2	no	—
149	C₂²×C₁₉	#333	1	4	yes	complemented
150	C₂²×C₁₉	#379	1	1	yes	characteristic
151	C₂²×C₁₉	C₂×#100	1	1	yes	characteristic
152	C₂²×C₁₉	C₂×#100	1	1	yes	characteristic
153	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
154	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
155	C₂²×C₁₉	#419	1	1	yes	characteristic
156	C₂²×C₁₉	#419	1	1	yes	characteristic
157	C₄×C₁₉	—	2	2	no	—
158	C₄×C₁₉	—	2	2	no	—
159	C₄×C₁₉	—	2	2	no	—
160	C₄×C₁₉	—	2	2	no	—
161	C₄×C₁₉	#448	1	2	yes	—
162	C₄×C₁₉	#448	1	2	yes	—
163	C₄×C₁₉	#451	1	2	yes	—
164	C₄×C₁₉	#451	1	2	yes	—
165	Dic₁₉	—	4	8	no	—
166	C₂³×Q₈	—	19	2	no	—
167	C₂×C₄⋊Q₈	—	19	2	no	—
168	C₂×C₄⋊Q₈	—	19	2	no	—
169	C₂×C₄⋊Q₈	—	38	8	no	—
170	C₂×C₄⋊Q₈	—	38	1	no	—
171	C₂×C₄⋊Q₈	—	38	1	no	—
172	C₂²×C₄⋊C₄	—	38	1	no	—
173	C₂²×C₄⋊C₄	—	38	1	no	—
174	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
175	C₂×C₄⋊C₈	—	19	2	no	—
176	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
177	C₂×C₂²⋊C₈	—	2	1	no	—
178	C₂×C₂²⋊C₈	—	38	1	no	—
179	#336	—	2	1	no	—
180	#336	—	38	1	no	—
181	#333	—	38	8	no	—
182	C₂×C₁₉	—	2	4	no	—
183	C₂×C₁₉	C₂×#333	1	1	yes	characteristic
184	C₂×C₁₉	#1548	1	1	yes	characteristic
185	C₂×C₁₉	#1548	1	1	yes	characteristic
186	C₂×C₁₉	#1558	1	1	yes	characteristic
187	C₂×C₁₉	#1783	1	1	yes	characteristic
188	C₂×C₁₉	#1787	1	1	yes	characteristic
189	C₂×C₁₉	#1787	1	1	yes	characteristic
190	C₂²×Q₈	—	19	4	no	—
191	C₂²×Q₈	—	38	8	no	—
192	C₂²×Q₈	—	38	2	no	—
193	C₂²×Q₈	—	76	4	no	—
194	C₂²×Q₈	—	76	4	no	—
195	C₂³×C₄	—	38	2	no	—
196	C₂³×C₄	C₄×D₁₉	1	1	yes	characteristic
197	C₂³×C₄	D₇₆	1	1	yes	characteristic
198	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
199	C₂²×C₈	—	2	2	no	—
200	C₂²×C₈	—	38	2	no	—
201	C₄⋊Q₈	—	19	8	no	—
202	C₄⋊Q₈	—	38	8	no	—
203	C₄⋊Q₈	—	38	8	no	—
204	C₄⋊Q₈	—	38	8	no	—
205	C₄⋊Q₈	—	38	4	no	—
206	C₂×C₄⋊C₄	—	38	8	no	—
207	C₂×C₄⋊C₄	—	38	8	no	—
208	C₂×C₄⋊C₄	—	38	2	no	—
209	C₂×C₄⋊C₄	—	38	2	no	—
210	C₂×C₄⋊C₄	—	38	2	no	—
211	C₂×C₄⋊C₄	—	38	2	no	—
212	C₂×C₄²	—	2	4	no	—
213	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
214	C₂×C₄²	D₇₆	1	1	yes	characteristic
215	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
216	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
217	C₄⋊C₈	—	2	4	no	—
218	C₄⋊C₈	—	38	4	no	—
219	C₂²⋊C₈	—	4	2	no	—
220	C₂²⋊C₈	—	76	2	no	—
221	C₁₉	#8342	1	1	yes	characteristic, complemented, Sylow
222	C₂⁴	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
223	C₂×Q₈	—	19	8	no	—
224	C₂×Q₈	—	38	8	no	—
225	C₂×Q₈	—	38	4	no	—
226	C₂×Q₈	—	76	32	no	—
227	C₂×Q₈	—	76	4	no	—
228	C₂×Q₈	—	76	4	no	—
229	C₂×Q₈	—	76	4	no	—
230	C₂×Q₈	—	76	4	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	38	8	no	—
239	C₂²×C₄	—	38	4	no	—
240	C₂²×C₄	—	76	4	no	—
241	C₂²×C₄	—	76	4	no	—
242	C₂²×C₄	#64516	1	2	yes	—
243	C₂²×C₄	Dic₇₆	1	2	yes	—
244	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
245	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
246	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
247	C₂²×C₄	#64526	1	2	yes	—
248	C₂²×C₄	#64528	1	2	yes	—
249	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
250	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
252	C₂×C₈	—	4	2	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	—	4	2	no	—
255	C₂×C₈	—	76	2	no	—
256	C₂×C₈	—	76	2	no	—
257	C₂×C₈	—	76	2	no	—
258	C₄⋊C₄	—	38	8	no	—
259	C₄⋊C₄	—	38	8	no	—
260	C₄⋊C₄	—	38	8	no	—
261	C₄⋊C₄	—	38	8	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	2	no	—
265	C₄²	D₄×D₁₉	1	4	yes	—
266	C₂³	—	2	1	no	—
267	C₂³	—	2	1	no	—
268	C₂³	—	2	1	no	—
269	C₂³	—	2	1	no	—
270	C₂³	#10597267	1	2	yes	—
271	C₂³	#10597273	1	2	yes	—
272	C₂³	#10597280	1	2	yes	—
273	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
274	Q₈	—	76	8	no	—
275	Q₈	—	76	16	no	—
276	Q₈	—	76	8	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	38	8	no	—
291	C₂×C₄	—	76	16	no	—
292	C₂×C₄	—	76	4	no	—
293	C₂×C₄	—	76	4	no	—
294	C₂×C₄	#10597267	1	2	yes	—
295	C₂×C₄	#10597273	1	2	yes	—
296	C₂×C₄	#10597280	1	2	yes	—
297	C₂×C₄	C₂²⋊C₄×D₁₉	1	2	yes	—
298	C₂×C₄	#10597328	1	2	yes	—
299	C₂×C₄	C₂×#64516	1	1	yes	characteristic
300	C₂×C₄	C₂×Dic₇₆	1	1	yes	characteristic
301	C₂×C₄	#10597355	1	2	yes	—
302	C₂×C₄	C₂×#64526	1	1	yes	characteristic
303	C₂×C₄	#10597383	1	2	yes	—
304	C₂×C₄	C₂×#64528	1	1	yes	characteristic
305	C₂×C₄	#10597397	1	2	yes	—
306	C₈	—	8	2	no	—
307	C₈	—	152	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	#11926292	1	4	yes	complemented
315	C₂²	#11926399	1	1	yes	characteristic
316	C₂²	C₂×#10597280	1	1	yes	characteristic
317	C₂²	C₂×#10597280	1	1	yes	characteristic
318	C₂²	C₂×#10597267	1	1	yes	characteristic
319	C₂²	C₂×#10597273	1	1	yes	characteristic
320	C₂²	#11926552	1	1	yes	characteristic
321	C₂²	#11926554	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	2	no	—
325	C₄	—	2	2	no	—
326	C₄	—	76	8	no	—
327	C₄	#11926682	1	2	yes	—
328	C₄	#11926685	1	2	yes	—
329	C₄	#11926699	1	2	yes	—
330	C₄	#11926702	1	2	yes	—
331	C₂	—	2	4	no	—
332	C₂	C₂×#11926292	1	1	yes	characteristic
333	C₂	#14562315	1	1	yes	characteristic
334	C₂	#14562317	1	1	yes	characteristic
335	C₂	#14562360	1	1	yes	characteristic
336	C₂	#14563555	1	1	yes	characteristic
337	C₂	#14563569	1	1	yes	characteristic
338	C₂	#14563571	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₁₉)⋊C₄
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1797	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1797	D₁₉
hypocenter	C₁₉	#8342
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1797

Complements

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

N	H ≅ G/N	type
3. C₂×#11926292	C₂	semidirect
149. C₂²×C₁₉	#333	semidirect
221. C₁₉	#8342	semidirect
314. C₂²	#11926292	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#10597315	2
#14563615	2
C₂×#11926292	2
C₁₉×#1797	2
#8342	19