Subgroups — #14575464

Order 2432.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927190	C₂	1	1	yes	maximal, characteristic
2	#11926901	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926874	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926868	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926841	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926783	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926775	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926766	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926704	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926694	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#64527	C₂	1	1	yes	maximal, characteristic, complemented
12	D₄×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926538	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#10597266	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#552	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#64541	C₂²	1	1	yes	characteristic
17	C₂²×D₇₆	C₂²	1	1	yes	characteristic
18	C₁₉×#126	—	2	1	no	—
19	C₁₉×#125	—	2	1	no	—
20	C₁₉×#125	C₂²	1	1	yes	characteristic, complemented
21	C₁₉×#123	—	2	1	no	—
22	C₁₉×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
23	C₁₉×C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
24	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic, complemented
25	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
26	C₁₉×#117	—	2	1	no	—
27	C₁₉×#117	C₂²	1	1	yes	characteristic, complemented
28	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
29	C₂×#64528	C₂²	1	1	yes	characteristic, complemented
30	C₂×#64527	C₂²	1	1	yes	characteristic
31	C₂×#64527	C₂²	1	1	yes	characteristic
32	#10597383	—	2	1	no	—
33	#10597376	C₂²	1	1	yes	characteristic
34	#10597370	C₂²	1	1	yes	characteristic
35	C₂²×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
36	C₂²×C₁₉⋊C₈	C₂²	1	1	yes	characteristic, complemented
37	#10597340	—	2	1	no	—
38	#10597330	—	2	1	no	—
39	C₂²⋊C₄×D₁₉	—	2	1	no	—
40	#10597321	C₂²	1	1	yes	characteristic
41	C₄×D₇₆	C₂²	1	1	yes	characteristic
42	#10597292	C₂²	1	1	yes	characteristic, complemented
43	C₂²⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
44	C₂²⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic, complemented
45	#10597267	C₂²	1	1	yes	characteristic, complemented
46	#10597267	C₂²	1	1	yes	characteristic, complemented
47	#10597267	C₂²	1	1	yes	characteristic, complemented
48	#10597266	C₂²	1	1	yes	characteristic
49	#10597266	C₂²	1	1	yes	characteristic
50	#10597266	C₂²	1	1	yes	characteristic
51	#10597266	C₂²	1	1	yes	characteristic
52	#10597266	C₂²	1	1	yes	characteristic
53	#10597266	C₂²	1	1	yes	characteristic
54	#10597265	C₂²	1	1	yes	characteristic, complemented
55	#10597265	C₂²	1	1	yes	characteristic, complemented
56	#10597264	C₂²	1	1	yes	characteristic, complemented
57	C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic, complemented
58	C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₁₉	—	2	1	no	—
60	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₁₉	D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	—	2	1	no	—
63	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊D₁₉	—	2	1	no	—
66	C₂×C₂²⋊D₁₉	—	2	1	no	—
67	#64541	—	2	2	no	—
68	#64541	—	4	1	no	—
69	C₂×D₇₆	—	2	2	no	—
70	C₂×D₇₆	—	2	1	no	—
71	C₂×D₇₆	—	4	1	no	—
72	C₂×D₇₆	C₂³	1	1	yes	characteristic
73	C₂×D₇₆	C₂³	1	1	yes	characteristic
74	C₂×C₄×D₁₉	—	2	1	no	—
75	C₂×C₄×D₁₉	—	2	1	no	—
76	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₁₉	—	2	1	no	—
78	C₄⋊C₄×C₁₉	—	2	1	no	—
79	C₄⋊C₄×C₁₉	—	2	1	no	—
80	C₄⋊C₄×C₁₉	—	2	1	no	—
81	C₄⋊C₄×C₁₉	—	2	1	no	—
82	C₄⋊C₄×C₁₉	—	2	1	no	—
83	C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
84	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
85	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
86	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
87	C₂²⋊C₄×C₁₉	—	2	1	no	—
88	C₂²⋊C₄×C₁₉	—	2	1	no	—
89	C₂²⋊C₄×C₁₉	—	2	1	no	—
90	C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
91	C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
92	C₄²×C₁₉	—	2	1	no	—
93	C₄²×C₁₉	—	2	1	no	—
94	C₄²×C₁₉	C₂³	1	1	yes	characteristic
95	C₂²⋊Dic₁₉	—	2	1	no	—
96	#64528	—	4	1	no	—
97	#64527	—	2	2	no	—
98	#64527	—	4	1	no	—
99	(C₂×D₁₉)⋊C₄	—	2	1	no	—
100	(C₂×D₁₉)⋊C₄	—	2	1	no	—
101	(C₂×D₁₉)⋊C₄	—	2	1	no	—
102	(C₂×D₁₉)⋊C₄	—	2	1	no	—
103	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
104	Dic₁₉⋊C₄	—	2	1	no	—
105	C₂×C₁₉⋊C₈	—	4	1	no	—
106	C₂×C₁₉⋊C₈	—	4	1	no	—
107	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
108	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
109	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
110	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
111	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
112	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₁₉	—	2	1	no	—
114	C₂²×D₁₉	—	2	1	no	—
115	C₂²×D₁₉	—	4	2	no	—
116	C₂²×D₁₉	—	4	2	no	—
117	C₂²×D₁₉	—	8	1	no	—
118	Q₈×C₁₉	—	2	1	no	—
119	Q₈×C₁₉	—	2	1	no	—
120	D₄×C₁₉	—	2	2	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	—	2	1	no	—
125	C₂×C₄×C₁₉	—	2	1	no	—
126	C₂×C₄×C₁₉	—	2	1	no	—
127	C₂×C₄×C₁₉	—	2	1	no	—
128	C₂×C₄×C₁₉	—	2	1	no	—
129	C₂×C₄×C₁₉	—	4	2	no	—
130	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
135	C₂²⋊D₁₉	—	4	2	no	—
136	C₂²⋊D₁₉	—	8	1	no	—
137	C₂×Dic₁₉	—	2	1	no	—
138	C₂×Dic₁₉	—	2	1	no	—
139	D₇₆	—	2	1	no	—
140	D₇₆	—	2	1	no	—
141	D₇₆	—	2	1	no	—
142	D₇₆	—	4	2	no	—
143	D₇₆	#41	1	2	yes	—
144	C₄×D₁₉	—	4	2	no	—
145	C₄×D₁₉	—	8	1	no	—
146	Dic₃₈	—	2	1	no	—
147	Dic₃₈	—	2	1	no	—
148	C₁₉⋊C₈	—	2	1	no	—
149	C₁₉⋊C₈	—	2	1	no	—
150	C₁₉⋊C₈	—	2	1	no	—
151	C₁₉⋊C₈	—	2	1	no	—
152	#3296	—	19	1	no	maximal, Sylow
153	C₂²×C₁₉	—	2	1	no	—
154	C₂²×C₁₉	—	2	1	no	—
155	C₂²×C₁₉	—	2	1	no	—
156	C₂²×C₁₉	—	2	1	no	—
157	C₂²×C₁₉	—	2	1	no	—
158	C₂²×C₁₉	#135	1	2	yes	—
159	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
160	C₂×D₁₉	—	2	2	no	—
161	C₂×D₁₉	—	4	1	no	—
162	C₂×D₁₉	—	4	1	no	—
163	C₂×D₁₉	—	4	1	no	—
164	C₂×D₁₉	—	4	1	no	—
165	C₂×D₁₉	—	4	1	no	—
166	C₂×D₁₉	—	8	4	no	—
167	C₄×C₁₉	—	2	1	no	—
168	C₄×C₁₉	—	2	1	no	—
169	C₄×C₁₉	—	2	1	no	—
170	C₄×C₁₉	—	2	1	no	—
171	C₄×C₁₉	—	4	1	no	—
172	C₄×C₁₉	—	4	1	no	—
173	C₄×C₁₉	—	4	1	no	—
174	C₄×C₁₉	—	4	1	no	—
175	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
176	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
177	Dic₁₉	—	4	1	no	—
178	Dic₁₉	—	4	1	no	—
179	#552	D₁₉	1	1	yes	characteristic, complemented
180	#546	—	19	1	no	—
181	#488	—	19	1	no	—
182	#483	—	19	1	no	—
183	#482	—	19	1	no	—
184	#478	—	19	1	no	—
185	#467	—	19	1	no	—
186	#465	—	19	1	no	—
187	#464	—	19	1	no	—
188	#452	—	19	1	no	—
189	#450	—	19	1	no	—
190	C₄×SD₁₆	—	19	1	no	—
191	C₈×D₄	—	19	1	no	—
192	#416	—	19	1	no	—
193	C₂×#102	—	19	1	no	—
194	C₂×C₁₉	—	2	2	no	—
195	C₂×C₁₉	—	4	1	no	—
196	C₂×C₁₉	#546	1	1	yes	characteristic
197	C₂×C₁₉	C₂×#135	1	1	yes	characteristic
198	C₂×C₁₉	#577	1	1	yes	characteristic
199	D₁₉	—	4	2	no	—
200	D₁₉	—	8	1	no	—
201	C₂×#41	—	19	1	no	—
202	C₂²×D₄	—	19	1	no	—
203	C₂×Q₁₆	—	19	1	no	—
204	C₂×SD₁₆	—	19	1	no	—
205	C₂×SD₁₆	—	19	1	no	—
206	C₂²×C₈	—	19	1	no	—
207	C₂²×C₈	—	19	1	no	—
208	#126	—	2	1	no	—
209	#125	—	2	1	no	—
210	#125	C₂×D₁₉	1	1	yes	characteristic, complemented
211	#124	—	19	1	no	—
212	#123	—	2	1	no	—
213	#123	—	38	1	no	—
214	C₂²⋊Q₈	—	19	1	no	—
215	C₂²⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
216	C₂²⋊Q₈	C₂×D₁₉	1	1	yes	characteristic, complemented
217	#121	—	19	1	no	—
218	#121	—	38	1	no	—
219	#120	—	38	1	no	—
220	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic, complemented
221	C₄×D₄	—	19	1	no	—
222	C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
223	#117	—	2	1	no	—
224	#117	C₂×D₁₉	1	1	yes	characteristic, complemented
225	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
226	C₂×C₂²⋊C₄	—	38	1	no	—
227	#107	—	19	1	no	—
228	#106	—	19	1	no	—
229	#106	—	19	1	no	—
230	C₄⋊C₈	—	19	1	no	—
231	#103	—	19	1	no	—
232	#103	—	19	1	no	—
233	#103	—	19	1	no	—
234	#103	—	19	1	no	—
235	#102	—	19	1	no	—
236	#102	—	19	1	no	—
237	#102	—	19	1	no	—
238	#102	—	19	1	no	—
239	#102	—	19	1	no	—
240	#102	—	19	1	no	—
241	C₂²⋊C₈	—	19	1	no	—
242	C₂²⋊C₈	—	19	1	no	—
243	C₄×C₈	—	19	1	no	—
244	C₁₉	#3296	1	1	yes	characteristic, complemented, Sylow
245	C₂⁴	—	38	1	no	—
246	#41	—	38	2	no	—
247	#41	—	76	1	no	—
248	C₂×Q₈	—	19	1	no	—
249	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
250	C₂×D₄	—	19	1	no	—
251	C₂×D₄	—	19	1	no	—
252	C₂×D₄	—	38	2	no	—
253	C₂×D₄	—	38	1	no	—
254	C₂×D₄	—	38	1	no	—
255	C₂×D₄	—	38	1	no	—
256	C₂×D₄	—	76	1	no	—
257	C₂×D₄	C₂²⋊D₁₉	1	1	yes	characteristic
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	38	1	no	—
260	C₂²×C₄	—	38	1	no	—
261	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
262	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
263	Q₁₆	—	76	1	no	—
264	SD₁₆	—	38	2	no	—
265	SD₁₆	—	76	1	no	—
266	C₂×C₈	—	19	1	no	—
267	C₂×C₈	—	19	1	no	—
268	C₂×C₈	—	19	1	no	—
269	C₂×C₈	—	19	1	no	—
270	C₂×C₈	—	76	1	no	—
271	C₂×C₈	—	76	1	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	19	1	no	—
279	C₄⋊C₄	—	38	1	no	—
280	C₄⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
281	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
282	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
283	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
284	C₂²⋊C₄	—	2	1	no	—
285	C₂²⋊C₄	—	2	1	no	—
286	C₂²⋊C₄	—	2	1	no	—
287	C₂²⋊C₄	—	38	1	no	—
288	C₂²⋊C₄	—	38	1	no	—
289	C₂²⋊C₄	—	38	1	no	—
290	C₂²⋊C₄	—	38	1	no	—
291	C₂²⋊C₄	—	38	1	no	—
292	C₂²⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
293	C₂²⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
294	C₄²	—	2	1	no	—
295	C₄²	—	2	1	no	—
296	C₄²	C₂²×D₁₉	1	1	yes	characteristic
297	C₂³	—	38	1	no	—
298	C₂³	—	38	1	no	—
299	C₂³	—	76	2	no	—
300	C₂³	—	76	2	no	—
301	C₂³	—	152	1	no	—
302	C₂³	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
303	C₂³	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
304	Q₈	—	2	1	no	—
305	Q₈	—	2	1	no	—
306	Q₈	—	38	1	no	—
307	Q₈	—	38	1	no	—
308	D₄	—	2	2	no	—
309	D₄	—	19	2	no	—
310	D₄	—	38	1	no	—
311	D₄	—	38	1	no	—
312	D₄	—	38	1	no	—
313	D₄	—	76	2	no	—
314	D₄	—	76	2	no	—
315	D₄	—	152	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	38	1	no	—
326	C₂×C₄	—	38	1	no	—
327	C₂×C₄	—	76	2	no	—
328	C₂×C₄	—	152	1	no	—
329	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
330	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
331	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
332	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
333	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
334	C₈	—	38	1	no	—
335	C₈	—	38	1	no	—
336	C₈	—	38	1	no	—
337	C₈	—	38	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	38	2	no	—
344	C₂²	—	76	1	no	—
345	C₂²	—	76	1	no	—
346	C₂²	—	76	1	no	—
347	C₂²	—	76	1	no	—
348	C₂²	—	76	1	no	—
349	C₂²	—	152	4	no	—
350	C₂²	#10597396	1	2	yes	—
351	C₂²	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	4	1	no	—
358	C₄	—	4	1	no	—
359	C₄	—	4	1	no	—
360	C₄	—	76	1	no	—
361	C₄	—	76	1	no	—
362	C₄	#41×D₁₉	1	1	yes	characteristic
363	C₄	#10597435	1	1	yes	characteristic
364	C₂	—	2	2	no	—
365	C₂	—	4	1	no	—
366	C₂	—	76	2	no	—
367	C₂	—	152	1	no	—
368	C₂	#11927182	1	1	yes	characteristic
369	C₂	C₂×#10597396	1	1	yes	characteristic
370	C₂	#11927352	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₉
commutator, cosocle	C₂×C₄×C₁₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉
Fitting	C₁₉×#552	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	#11927190	C₂
hypercenter	#552	D₁₉
hypocenter	C₁₉	#3296
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #552

Complements

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

N	H ≅ G/N	type
2. #11926901	C₂	semidirect
3. #11926874	C₂	semidirect
4. #11926868	C₂	semidirect
5. #11926841	C₂	semidirect
6. #11926783	C₂	semidirect
7. #11926775	C₂	semidirect
8. #11926766	C₂	semidirect
9. #11926704	C₂	semidirect
10. #11926694	C₂	semidirect
11. C₄×#64527	C₂	semidirect
12. D₄×C₁₉⋊C₈	C₂	semidirect
13. #11926538	C₂	semidirect
14. C₂×#10597266	C₂	semidirect
15. C₁₉×#552	C₂	semidirect
20. C₁₉×#125	C₂²	semidirect
23. C₁₉×C₂²⋊Q₈	C₂²	semidirect
24. C₄×Q₈×C₁₉	C₂²	semidirect
27. C₁₉×#117	C₂²	semidirect
29. C₂×#64528	C₂²	semidirect
36. C₂²×C₁₉⋊C₈	C₂²	semidirect
42. #10597292	C₂²	semidirect
44. C₂²⋊(C₁₉⋊C₈)	C₂²	semidirect
45. #10597267	C₂²	semidirect
46. #10597267	C₂²	semidirect
47. #10597267	C₂²	semidirect
54. #10597265	C₂²	semidirect
55. #10597265	C₂²	semidirect
56. #10597264	C₂²	semidirect
57. C₄⋊(C₁₉⋊C₈)	C₂²	semidirect
58. C₄×C₁₉⋊C₈	C₂²	semidirect
179. #552	D₁₉	semidirect
210. #125	C₂×D₁₉	semidirect
216. C₂²⋊Q₈	C₂×D₁₉	semidirect
220. C₄×Q₈	C₂×D₁₉	semidirect
224. #117	C₂×D₁₉	semidirect
244. C₁₉	#3296	semidirect

Maximal subgroups

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

subgroup	index
#11927190	2
#11926901	2
#11926874	2
#11926868	2
#11926841	2
#11926783	2
#11926775	2
#11926766	2
#11926704	2
#11926694	2
C₄×#64527	2
D₄×C₁₉⋊C₈	2
#11926538	2
C₂×#10597266	2
C₁₉×#552	2
#3296	19