Subgroups — #14567306

Order 2432.

528 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10597342	C₂	1	1	yes	maximal, characteristic
2	C₂×#10597338	C₂	1	1	yes	maximal, characteristic
3	#11926486	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926478	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926464	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926449	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926439	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926426	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926425	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926415	C₂	1	1	yes	maximal, characteristic
11	#11926415	C₂	1	1	yes	maximal, characteristic
12	#11926406	C₂	1	1	yes	maximal, characteristic
13	#11926403	C₂	1	1	yes	maximal, characteristic
14	#95×D₁₉	C₂	1	1	yes	maximal, characteristic
15	C₁₉×#398	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₄×D₁₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₉	C₂²	1	1	yes	characteristic
18	C₂²×C₄×D₁₉	C₂²	1	1	yes	characteristic
19	C₂²×Dic₃₈	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
24	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
29	C₂×Dic₁₉⋊C₄	C₂²	1	1	yes	characteristic
30	C₂×Dic₁₉⋊C₄	C₂²	1	1	yes	characteristic
31	C₂×Dic₁₉⋊C₄	C₂²	1	1	yes	characteristic
32	C₂×Dic₁₉⋊C₄	C₂²	1	1	yes	characteristic
33	C₂×Dic₁₉⋊C₄	C₂²	1	1	yes	characteristic
34	C₂×Dic₁₉⋊C₄	C₂²	1	1	yes	characteristic
35	C₂×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
36	C₂×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
37	#10597342	—	2	4	no	—
38	#10597338	—	2	4	no	—
39	C₁₉×#95	C₂²	1	1	yes	characteristic
40	C₁₉×#95	C₂²	1	1	yes	characteristic
41	C₁₉×#95	C₂²	1	1	yes	characteristic
42	C₁₉⋊#95	C₂²	1	1	yes	characteristic
43	C₁₉⋊#95	C₂²	1	1	yes	characteristic
44	C₁₉⋊#95	C₂²	1	1	yes	characteristic
45	C₁₉⋊#95	C₂²	1	1	yes	characteristic
46	C₁₉⋊#95	C₂²	1	1	yes	characteristic
47	C₁₉⋊#95	C₂²	1	1	yes	characteristic
48	C₁₉⋊#95	C₂²	1	1	yes	characteristic
49	C₁₉⋊#95	C₂²	1	1	yes	characteristic
50	C₁₉⋊#95	C₂²	1	1	yes	characteristic
51	C₁₉⋊#95	C₂²	1	1	yes	characteristic
52	C₁₉⋊#95	C₂²	1	1	yes	characteristic
53	C₂³×D₁₉	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
61	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
62	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
63	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
64	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
65	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
66	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
67	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₁₉	—	2	2	no	—
69	C₂×C₄×D₁₉	—	2	2	no	—
70	C₂×C₄×D₁₉	—	2	2	no	—
71	C₂×C₄×D₁₉	—	2	2	no	—
72	C₂×C₄×D₁₉	—	2	2	no	—
73	C₂×C₄×D₁₉	—	2	2	no	—
74	C₂×C₄×D₁₉	—	4	1	no	—
75	C₂×C₄×D₁₉	—	4	1	no	—
76	C₂×C₄×D₁₉	Q₈	1	4	yes	complemented
77	C₂×Dic₃₈	—	2	2	no	—
78	C₂×Dic₃₈	—	2	2	no	—
79	C₂×Dic₃₈	—	4	1	no	—
80	C₄⋊C₄×C₁₉	—	2	2	no	—
81	C₄⋊C₄×C₁₉	—	2	2	no	—
82	C₄⋊C₄×C₁₉	—	4	1	no	—
83	C₄⋊C₄×C₁₉	—	4	1	no	—
84	(C₂×D₁₉)⋊C₄	—	2	2	no	—
85	(C₂×D₁₉)⋊C₄	—	2	2	no	—
86	(C₂×D₁₉)⋊C₄	—	2	2	no	—
87	(C₂×D₁₉)⋊C₄	D₄	1	4	yes	—
88	C₄⋊Dic₁₉	—	2	2	no	—
89	Dic₁₉⋊C₄	—	2	2	no	—
90	Dic₁₉⋊C₄	—	2	2	no	—
91	Dic₁₉⋊C₄	—	2	2	no	—
92	Dic₁₉⋊C₄	—	2	2	no	—
93	Dic₁₉⋊C₄	—	2	2	no	—
94	Dic₁₉⋊C₄	—	4	1	no	—
95	C₄×Dic₁₉	—	2	2	no	—
96	C₄×Dic₁₉	—	2	2	no	—
97	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
98	C₂²×D₁₉	—	2	1	no	—
99	C₂²×D₁₉	—	2	1	no	—
100	C₂²×D₁₉	—	2	1	no	—
101	C₂²×D₁₉	—	2	1	no	—
102	C₂²×D₁₉	C₂×D₄	1	2	yes	—
103	C₂²×D₁₉	C₂×Q₈	1	2	yes	—
104	C₂²×D₁₉	#41	1	2	yes	—
105	C₂×C₄×C₁₉	—	2	1	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	1	no	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	2	1	no	—
118	C₂×C₄×C₁₉	—	2	1	no	—
119	C₂×C₄×C₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	—	2	1	no	—
123	C₂×C₄×C₁₉	—	2	1	no	—
124	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
125	C₂×C₄×C₁₉	C₂×Q₈	1	2	yes	—
126	C₂×Dic₁₉	—	2	1	no	—
127	C₂×Dic₁₉	—	2	1	no	—
128	C₂×Dic₁₉	—	2	1	no	—
129	C₂×Dic₁₉	—	2	1	no	—
130	C₂×Dic₁₉	—	2	1	no	—
131	C₂×Dic₁₉	—	2	1	no	—
132	C₂×Dic₁₉	—	2	1	no	—
133	C₂×Dic₁₉	—	2	1	no	—
134	C₂×Dic₁₉	—	2	1	no	—
135	C₂×Dic₁₉	—	2	1	no	—
136	C₂×Dic₁₉	—	2	1	no	—
137	C₂×Dic₁₉	—	2	1	no	—
138	C₂×Dic₁₉	—	2	1	no	—
139	C₂×Dic₁₉	—	2	1	no	—
140	C₂×Dic₁₉	—	2	1	no	—
141	C₂×Dic₁₉	—	2	1	no	—
142	C₂×Dic₁₉	—	2	1	no	—
143	C₂×Dic₁₉	C₂×D₄	1	2	yes	—
144	C₂×Dic₁₉	C₂×Q₈	1	2	yes	—
145	C₂×Dic₁₉	#41	1	2	yes	—
146	C₂×Dic₁₉	#41	1	2	yes	—
147	C₄×D₁₉	—	2	8	no	—
148	C₄×D₁₉	—	4	4	no	—
149	C₄×D₁₉	—	4	4	no	—
150	Dic₃₈	—	4	4	no	—
151	#2438	—	19	1	no	maximal, Sylow
152	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
153	C₂²×C₁₉	C₂²×Q₈	1	1	yes	characteristic
154	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
155	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
156	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
157	C₂²×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
158	C₂²×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
159	C₂×D₁₉	—	2	2	no	—
160	C₂×D₁₉	—	2	2	no	—
161	C₂×D₁₉	—	2	2	no	—
162	C₂×D₁₉	—	2	2	no	—
163	C₂×D₁₉	—	2	2	no	—
164	C₂×D₁₉	—	2	2	no	—
165	C₂×D₁₉	C₂²⋊Q₈	1	4	yes	—
166	C₄×C₁₉	—	2	2	no	—
167	C₄×C₁₉	—	2	2	no	—
168	C₄×C₁₉	—	4	1	no	—
169	C₄×C₁₉	—	4	1	no	—
170	C₄×C₁₉	—	4	1	no	—
171	C₄×C₁₉	—	4	1	no	—
172	C₄×C₁₉	—	4	1	no	—
173	Dic₁₉	—	2	2	no	—
174	Dic₁₉	—	2	2	no	—
175	Dic₁₉	—	2	2	no	—
176	Dic₁₉	—	2	2	no	—
177	Dic₁₉	—	4	1	no	—
178	Dic₁₉	—	4	1	no	—
179	Dic₁₉	—	4	1	no	—
180	C₂×C₂²⋊Q₈	—	19	1	no	—
181	C₂×#117	—	19	1	no	—
182	#400	—	19	1	no	—
183	#398	—	19	1	no	—
184	#398	D₁₉	1	1	yes	characteristic, complemented
185	#395	—	19	1	no	—
186	#391	—	19	1	no	—
187	#389	—	19	1	no	—
188	#387	—	19	1	no	—
189	#387	—	19	1	no	—
190	#385	—	19	1	no	—
191	#385	—	19	1	no	—
192	#381	—	19	1	no	—
193	#380	—	19	1	no	—
194	C₂×#95	—	19	1	no	—
195	C₂×C₁₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
196	C₂×C₁₉	#533	1	1	yes	characteristic
197	C₂×C₁₉	#544	1	1	yes	characteristic
198	C₂×C₁₉	#546	1	1	yes	characteristic
199	C₂×C₁₉	#547	1	1	yes	characteristic
200	C₂×C₁₉	#551	1	1	yes	characteristic
201	C₂×C₁₉	#552	1	1	yes	characteristic
202	D₁₉	—	2	4	no	—
203	C₂²×Q₈	—	19	1	no	—
204	C₂³×C₄	—	19	1	no	—
205	C₂³×C₄	—	19	1	no	—
206	C₂³×C₄	—	19	1	no	—
207	C₂²⋊Q₈	—	38	4	no	—
208	#117	—	38	4	no	—
209	C₂×C₄⋊C₄	—	19	1	no	—
210	C₂×C₄⋊C₄	—	19	1	no	—
211	C₂×C₄⋊C₄	—	19	1	no	—
212	C₂×C₄⋊C₄	—	19	1	no	—
213	C₂×C₄⋊C₄	—	19	1	no	—
214	C₂×C₄⋊C₄	—	19	1	no	—
215	C₂×C₄⋊C₄	—	19	1	no	—
216	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
217	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
218	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
219	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
220	C₂×C₂²⋊C₄	—	19	1	no	—
221	C₂×C₂²⋊C₄	—	19	1	no	—
222	C₂×C₂²⋊C₄	—	19	1	no	—
223	C₂×C₂²⋊C₄	—	19	1	no	—
224	C₂×C₄²	—	19	1	no	—
225	C₂×C₄²	—	19	1	no	—
226	#95	—	19	1	no	—
227	#95	—	19	1	no	—
228	#95	—	19	1	no	—
229	#95	—	19	1	no	—
230	#95	—	19	1	no	—
231	#95	—	19	1	no	—
232	#95	—	19	1	no	—
233	#95	—	19	1	no	—
234	#95	—	19	1	no	—
235	#95	—	19	1	no	—
236	#95	—	19	1	no	—
237	#95	C₂×D₁₉	1	1	yes	characteristic
238	#95	C₂×D₁₉	1	1	yes	characteristic
239	#95	C₂×D₁₉	1	1	yes	characteristic
240	C₁₉	#2438	1	1	yes	characteristic, complemented, Sylow
241	C₂⁴	—	19	1	no	—
242	C₂×Q₈	—	38	2	no	—
243	C₂×Q₈	—	38	2	no	—
244	C₂×Q₈	—	76	1	no	—
245	C₂²×C₄	—	19	1	no	—
246	C₂²×C₄	—	19	1	no	—
247	C₂²×C₄	—	19	1	no	—
248	C₂²×C₄	—	19	1	no	—
249	C₂²×C₄	—	19	4	no	—
250	C₂²×C₄	—	19	1	no	—
251	C₂²×C₄	—	19	1	no	—
252	C₂²×C₄	—	19	1	no	—
253	C₂²×C₄	—	38	2	no	—
254	C₂²×C₄	—	38	2	no	—
255	C₂²×C₄	—	38	2	no	—
256	C₂²×C₄	—	38	2	no	—
257	C₂²×C₄	—	38	2	no	—
258	C₂²×C₄	—	38	2	no	—
259	C₂²×C₄	—	76	1	no	—
260	C₂²×C₄	—	76	1	no	—
261	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
262	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
263	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
264	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
265	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
266	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
267	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	38	2	no	—
273	C₄⋊C₄	—	38	2	no	—
274	C₄⋊C₄	—	38	2	no	—
275	C₄⋊C₄	—	38	2	no	—
276	C₄⋊C₄	—	38	2	no	—
277	C₄⋊C₄	—	38	2	no	—
278	C₄⋊C₄	—	76	1	no	—
279	C₂²⋊C₄	—	19	4	no	—
280	C₂²⋊C₄	—	38	2	no	—
281	C₂²⋊C₄	—	38	2	no	—
282	C₂²⋊C₄	—	38	2	no	—
283	C₄²	—	38	2	no	—
284	C₄²	—	38	2	no	—
285	C₂³	—	19	2	no	—
286	C₂³	—	19	2	no	—
287	C₂³	—	19	2	no	—
288	C₂³	—	38	1	no	—
289	C₂³	—	38	1	no	—
290	C₂³	—	38	1	no	—
291	C₂³	—	38	1	no	—
292	C₂³	C₂³×D₁₉	1	1	yes	characteristic
293	Q₈	—	76	4	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	19	2	no	—
314	C₂×C₄	—	19	2	no	—
315	C₂×C₄	—	19	2	no	—
316	C₂×C₄	—	19	2	no	—
317	C₂×C₄	—	38	8	no	—
318	C₂×C₄	—	38	1	no	—
319	C₂×C₄	—	38	1	no	—
320	C₂×C₄	—	38	1	no	—
321	C₂×C₄	—	38	1	no	—
322	C₂×C₄	—	38	1	no	—
323	C₂×C₄	—	38	1	no	—
324	C₂×C₄	—	38	1	no	—
325	C₂×C₄	—	38	1	no	—
326	C₂×C₄	—	38	1	no	—
327	C₂×C₄	—	38	1	no	—
328	C₂×C₄	—	38	1	no	—
329	C₂×C₄	—	38	1	no	—
330	C₂×C₄	—	38	1	no	—
331	C₂×C₄	—	38	1	no	—
332	C₂×C₄	—	38	1	no	—
333	C₂×C₄	—	38	1	no	—
334	C₂×C₄	—	38	1	no	—
335	C₂×C₄	—	76	4	no	—
336	C₂×C₄	—	76	4	no	—
337	C₂×C₄	D₄×D₁₉	1	2	yes	—
338	C₂×C₄	Q₈×D₁₉	1	2	yes	—
339	C₂²	—	19	4	no	—
340	C₂²	—	38	2	no	—
341	C₂²	—	38	2	no	—
342	C₂²	—	38	2	no	—
343	C₂²	—	38	2	no	—
344	C₂²	—	38	2	no	—
345	C₂²	—	38	2	no	—
346	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
347	C₂²	#10597430	1	1	yes	characteristic
348	C₂²	C₂×Q₈×D₁₉	1	1	yes	characteristic
349	C₂²	#10597433	1	1	yes	characteristic
350	C₂²	#41×D₁₉	1	1	yes	characteristic
351	C₂²	#41×D₁₉	1	1	yes	characteristic
352	C₂²	#41×D₁₉	1	1	yes	characteristic
353	C₄	—	2	2	no	—
354	C₄	—	2	2	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	4	1	no	—
358	C₄	—	4	1	no	—
359	C₄	—	4	1	no	—
360	C₄	—	38	2	no	—
361	C₄	—	38	2	no	—
362	C₄	—	38	2	no	—
363	C₄	—	38	2	no	—
364	C₄	—	76	1	no	—
365	C₄	—	76	1	no	—
366	C₄	—	76	1	no	—
367	C₂	—	38	4	no	—
368	C₂	C₂²⋊Q₈×D₁₉	1	1	yes	characteristic
369	C₂	#11927100	1	1	yes	characteristic
370	C₂	#11927171	1	1	yes	characteristic
371	C₂	#11927183	1	1	yes	characteristic
372	C₂	#11927196	1	1	yes	characteristic
373	C₂	#11927222	1	1	yes	characteristic
374	C₂	#11927226	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉
commutator, socle, cosocle	C₂³×C₁₉	C₂⁴
Fitting	C₁₉×#398	C₂
prime radical	C₂³×D₁₉	C₂³
hypercenter	#398	D₁₉
hypocenter	C₁₉	#2438
LCS	G ▷ C₂³×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ #398

Complements

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

N	H ≅ G/N	type
3. #11926486	C₂	semidirect
4. #11926478	C₂	semidirect
5. #11926464	C₂	semidirect
6. #11926449	C₂	semidirect
7. #11926439	C₂	semidirect
8. #11926426	C₂	semidirect
9. #11926425	C₂	semidirect
15. C₁₉×#398	C₂	semidirect
76. C₂×C₄×D₁₉	Q₈	semidirect
184. #398	D₁₉	semidirect
240. C₁₉	#2438	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10597342	2
C₂×#10597338	2
#11926486	2
#11926478	2
#11926464	2
#11926449	2
#11926439	2
#11926426	2
#11926425	2
#11926415	2
#11926406	2
#11926403	2
#95×D₁₉	2
C₁₉×#398	2
#2438	19