Subgroups — #14575274

Order 2432.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10597364	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#10597388	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×C₂²⋊D₁₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#11927196	C₂	1	1	yes	maximal, characteristic
5	#11926922	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926801	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926800	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926752	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926748	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926708	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926705	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926701	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926629	C₂	1	1	yes	maximal, characteristic, complemented
14	#11926619	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#451	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Q₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×#64545	C₂²	1	1	yes	characteristic
18	C₂×Q₈×D₁₉	C₂²	1	1	yes	characteristic
19	C₂×#64541	C₂²	1	1	yes	characteristic
20	C₂×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
21	C₂×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
22	C₁₉×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
23	#10597392	—	2	1	no	—
24	#10597392	C₂²	1	1	yes	characteristic
25	Q₈×Dic₁₉	C₂²	1	1	yes	characteristic
26	#10597390	—	2	1	no	—
27	#10597390	C₂²	1	1	yes	characteristic
28	#10597390	C₂²	1	1	yes	characteristic
29	C₂×#64528	C₂²	1	1	yes	characteristic
30	C₂×#64528	C₂²	1	1	yes	characteristic
31	C₂×#64528	C₂²	1	1	yes	characteristic
32	#10597388	—	2	4	no	—
33	C₂×#64527	C₂²	1	1	yes	characteristic
34	C₂×#64527	C₂²	1	1	yes	characteristic
35	C₄×C₂²⋊D₁₉	—	2	1	no	—
36	C₄×C₂²⋊D₁₉	C₂²	1	1	yes	characteristic
37	#10597370	—	2	1	no	—
38	#10597370	C₂²	1	1	yes	characteristic
39	C₂×#64520	C₂²	1	1	yes	characteristic
40	#10597364	—	2	4	no	—
41	C₂×Dic₇₆	C₂²	1	1	yes	characteristic
42	C₂×#64516	C₂²	1	1	yes	characteristic
43	C₂×#64515	C₂²	1	1	yes	characteristic
44	#10597343	C₂²	1	1	yes	characteristic
45	#10597340	—	2	1	no	—
46	#10597339	C₂²	1	1	yes	characteristic
47	C₄⋊C₄×D₁₉	—	2	1	no	—
48	#10597330	—	2	1	no	—
49	#10597329	—	2	1	no	—
50	C₁₉×#103	C₂²	1	1	yes	characteristic
51	C₁₉×#103	C₂²	1	1	yes	characteristic
52	C₁₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
53	#10597292	C₂²	1	1	yes	characteristic
54	#10597278	C₂²	1	1	yes	characteristic
55	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
56	#10597273	C₂²	1	1	yes	characteristic
57	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
58	C₈×Dic₁₉	C₂²	1	1	yes	characteristic
59	#10597267	C₂²	1	1	yes	characteristic
60	#10597265	C₂²	1	1	yes	characteristic
61	C₂×Q₈×C₁₉	—	2	1	no	—
62	C₂×Q₈×C₁₉	—	2	2	no	—
63	C₂×Q₈×C₁₉	—	4	1	no	—
64	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
65	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
66	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	—	2	1	no	—
68	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
69	C₂×C₂²⋊D₁₉	—	2	1	no	—
70	C₂×C₂²⋊D₁₉	D₄	1	1	yes	characteristic
71	#64545	—	2	2	no	—
72	#64545	—	4	1	no	—
73	Q₈×D₁₉	—	2	2	no	—
74	Q₈×D₁₉	—	4	1	no	—
75	#64541	—	2	2	no	—
76	#64541	—	4	1	no	—
77	C₂×D₇₆	—	2	1	no	—
78	C₂×D₇₆	C₂³	1	1	yes	characteristic
79	C₂×C₄×D₁₉	—	2	1	no	—
80	C₂×C₄×D₁₉	—	2	1	no	—
81	C₂×C₄×D₁₉	—	2	1	no	—
82	C₂×C₄×D₁₉	—	2	1	no	—
83	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
84	C₂×Dic₃₈	—	2	1	no	—
85	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
86	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
87	Q₁₆×C₁₉	—	2	2	no	—
88	Q₁₆×C₁₉	—	4	1	no	—
89	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
90	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
91	C₄⋊C₄×C₁₉	—	2	1	no	—
92	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
93	C₂²⋊C₄×C₁₉	—	2	1	no	—
94	C₂²⋊Dic₁₉	—	2	1	no	—
95	C₂²⋊Dic₁₉	D₄	1	1	yes	characteristic
96	#64528	—	2	2	no	—
97	#64528	—	2	2	no	—
98	#64528	—	2	2	no	—
99	#64527	—	2	2	no	—
100	#64527	D₄	1	4	yes	complemented
101	(C₂×D₁₉)⋊C₄	—	2	1	no	—
102	(C₂×D₁₉)⋊C₄	—	2	1	no	—
103	(C₂×D₁₉)⋊C₄	—	2	1	no	—
104	(C₂×D₁₉)⋊C₄	D₄	1	1	yes	characteristic
105	C₄⋊Dic₁₉	—	2	1	no	—
106	C₄⋊Dic₁₉	—	2	1	no	—
107	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
108	Dic₁₉⋊C₄	—	2	1	no	—
109	Dic₁₉⋊C₄	—	2	1	no	—
110	Dic₁₉⋊C₄	—	2	1	no	—
111	Dic₁₉⋊C₄	—	2	1	no	—
112	Dic₁₉⋊C₄	—	2	1	no	—
113	Dic₁₉⋊C₄	—	2	1	no	—
114	Dic₁₉⋊C₄	D₄	1	1	yes	characteristic
115	C₄×Dic₁₉	—	2	1	no	—
116	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
117	#64520	—	2	2	no	—
118	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
119	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
120	Dic₇₆	—	4	1	no	—
121	#64516	—	2	2	no	—
122	#64515	—	2	2	no	—
123	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
124	C₂²×D₁₉	—	2	1	no	—
125	C₂²×D₁₉	C₂×D₄	1	1	yes	characteristic
126	Q₈×C₁₉	—	2	1	no	—
127	Q₈×C₁₉	—	2	1	no	—
128	Q₈×C₁₉	—	2	1	no	—
129	Q₈×C₁₉	—	2	1	no	—
130	Q₈×C₁₉	—	2	1	no	—
131	Q₈×C₁₉	—	4	2	no	—
132	Q₈×C₁₉	C₂×D₄	1	2	yes	—
133	C₂×C₄×C₁₉	—	2	1	no	—
134	C₂×C₄×C₁₉	—	2	1	no	—
135	C₂×C₄×C₁₉	—	2	1	no	—
136	C₂×C₄×C₁₉	—	2	1	no	—
137	C₂×C₄×C₁₉	—	2	1	no	—
138	C₂×C₄×C₁₉	—	2	1	no	—
139	C₂×C₄×C₁₉	—	4	2	no	—
140	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
142	C₂²⋊D₁₉	—	2	2	no	—
143	C₂²⋊D₁₉	—	4	2	no	—
144	C₂×Dic₁₉	—	2	1	no	—
145	C₂×Dic₁₉	—	2	1	no	—
146	C₂×Dic₁₉	—	2	1	no	—
147	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
148	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
149	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
150	D₇₆	—	2	1	no	—
151	D₇₆	—	4	2	no	—
152	D₇₆	C₂×D₄	1	2	yes	—
153	C₄×D₁₉	—	2	1	no	—
154	C₄×D₁₉	—	2	1	no	—
155	C₄×D₁₉	—	4	2	no	—
156	C₄×D₁₉	—	4	2	no	—
157	C₄×D₁₉	—	4	2	no	—
158	C₄×D₁₉	—	4	2	no	—
159	Dic₃₈	—	2	1	no	—
160	Dic₃₈	—	2	1	no	—
161	Dic₃₈	—	2	1	no	—
162	Dic₃₈	—	2	1	no	—
163	Dic₃₈	—	4	2	no	—
164	C₈×C₁₉	—	2	1	no	—
165	C₈×C₁₉	—	2	1	no	—
166	C₁₉⋊C₈	—	2	1	no	—
167	C₁₉⋊C₈	C₂×D₄	1	2	yes	—
168	#3276	—	19	1	no	maximal, Sylow
169	C₂²×C₁₉	—	2	1	no	—
170	C₂²×C₁₉	—	2	1	no	—
171	C₂²×C₁₉	—	2	1	no	—
172	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
173	C₂×D₁₉	—	2	1	no	—
174	C₂×D₁₉	—	2	2	no	—
175	C₂×D₁₉	—	2	1	no	—
176	C₂×D₁₉	—	2	1	no	—
177	C₂×D₁₉	—	4	1	no	—
178	C₂×D₁₉	—	4	1	no	—
179	C₄×C₁₉	—	2	2	no	—
180	C₄×C₁₉	—	2	1	no	—
181	C₄×C₁₉	—	4	1	no	—
182	C₄×C₁₉	—	4	1	no	—
183	C₄×C₁₉	—	4	1	no	—
184	C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
185	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
186	Dic₁₉	—	2	1	no	—
187	Dic₁₉	—	2	1	no	—
188	Dic₁₉	—	4	1	no	—
189	Dic₁₉	—	4	1	no	—
190	Dic₁₉	—	4	1	no	—
191	Dic₁₉	#137	1	2	yes	complemented
192	C₂×#137	—	19	1	no	—
193	C₂×#137	—	19	1	no	—
194	D₄×Q₈	—	19	1	no	—
195	#547	—	19	1	no	—
196	#494	—	19	1	no	—
197	#470	—	19	1	no	—
198	#470	—	19	1	no	—
199	#462	—	19	1	no	—
200	#461	—	19	1	no	—
201	#452	—	19	1	no	—
202	#452	—	19	1	no	—
203	#451	—	19	1	no	—
204	#451	D₁₉	1	1	yes	characteristic, complemented
205	C₄×SD₁₆	—	19	1	no	—
206	#436	—	19	1	no	—
207	C₂×C₁₉	—	4	1	no	—
208	C₂×C₁₉	D₄²	1	1	yes	characteristic
209	C₂×C₁₉	C₂×#137	1	1	yes	characteristic
210	C₂×C₁₉	#578	1	1	yes	characteristic
211	D₁₉	—	4	1	no	—
212	D₁₉	—	4	2	no	—
213	C₂×#41	—	19	1	no	—
214	C₂×#41	—	19	1	no	—
215	C₂²×Q₈	—	19	1	no	—
216	C₂²×Q₈	C₂×D₁₉	1	1	yes	characteristic
217	#137	—	38	4	no	—
218	#137	—	38	4	no	—
219	C₂×Q₁₆	—	19	1	no	—
220	C₂×Q₁₆	—	19	1	no	—
221	C₂×Q₁₆	—	19	1	no	—
222	C₂×Q₁₆	—	19	1	no	—
223	C₂×Q₁₆	C₂×D₁₉	1	1	yes	characteristic
224	C₂×Q₁₆	C₂×D₁₉	1	1	yes	characteristic
225	C₂×SD₁₆	—	19	1	no	—
226	C₂×SD₁₆	—	19	1	no	—
227	C₂×SD₁₆	—	19	1	no	—
228	C₂×M₁₆	—	19	1	no	—
229	C₂×M₁₆	—	19	1	no	—
230	C₄⋊Q₈	—	19	1	no	—
231	C₄⋊Q₈	—	19	1	no	—
232	C₄⋊Q₈	—	38	1	no	—
233	#123	—	38	1	no	—
234	#123	—	38	1	no	—
235	C₂²⋊Q₈	—	19	1	no	—
236	C₂²⋊Q₈	—	19	1	no	—
237	C₂²⋊Q₈	—	19	1	no	—
238	C₂²⋊Q₈	—	38	1	no	—
239	C₂²⋊Q₈	—	38	1	no	—
240	C₂²⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
241	#121	—	38	1	no	—
242	C₄×Q₈	—	19	1	no	—
243	C₄×D₄	—	19	1	no	—
244	C₄×D₄	—	19	1	no	—
245	C₄×D₄	—	38	1	no	—
246	C₂×C₄⋊C₄	—	38	1	no	—
247	#106	—	19	1	no	—
248	C₄⋊C₈	—	19	1	no	—
249	#103	—	19	1	no	—
250	#103	—	19	1	no	—
251	#103	—	19	1	no	—
252	#103	C₂×D₁₉	1	1	yes	characteristic
253	#103	C₂×D₁₉	1	1	yes	characteristic
254	#102	—	19	1	no	—
255	C₂²⋊C₈	—	19	1	no	—
256	C₂²⋊C₈	C₂×D₁₉	1	1	yes	characteristic
257	C₄×C₈	—	19	1	no	—
258	C₁₉	#3276	1	1	yes	characteristic, complemented, Sylow
259	#41	—	38	2	no	—
260	#41	—	38	2	no	—
261	#41	—	76	1	no	—
262	#41	—	76	1	no	—
263	C₂×Q₈	—	2	1	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	—	4	1	no	—
266	C₂×Q₈	—	19	1	no	—
267	C₂×Q₈	—	19	1	no	—
268	C₂×Q₈	—	38	1	no	—
269	C₂×Q₈	—	38	2	no	—
270	C₂×Q₈	—	76	1	no	—
271	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
272	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
273	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
274	C₂×D₄	—	19	1	no	—
275	C₂×D₄	—	19	1	no	—
276	C₂×D₄	—	38	1	no	—
277	C₂×D₄	—	38	1	no	—
278	C₂²×C₄	—	2	1	no	—
279	C₂²×C₄	—	19	1	no	—
280	C₂²×C₄	—	38	1	no	—
281	C₂²×C₄	—	38	1	no	—
282	C₂²×C₄	—	38	1	no	—
283	C₂²×C₄	—	38	1	no	—
284	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
285	Q₁₆	—	2	2	no	—
286	Q₁₆	—	4	1	no	—
287	Q₁₆	—	38	2	no	—
288	Q₁₆	—	38	2	no	—
289	Q₁₆	—	38	2	no	—
290	Q₁₆	—	76	1	no	—
291	SD₁₆	—	19	4	no	—
292	SD₁₆	—	38	2	no	—
293	SD₁₆	—	38	2	no	—
294	M₁₆	—	38	2	no	—
295	M₁₆	—	38	2	no	—
296	C₂×C₈	—	19	1	no	—
297	C₂×C₈	—	19	1	no	—
298	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
299	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	19	1	no	—
302	C₄⋊C₄	—	19	1	no	—
303	C₄⋊C₄	—	38	1	no	—
304	C₄⋊C₄	—	38	1	no	—
305	C₄⋊C₄	—	38	1	no	—
306	C₄⋊C₄	—	38	1	no	—
307	C₄⋊C₄	—	38	1	no	—
308	C₄⋊C₄	—	38	1	no	—
309	C₄⋊C₄	—	38	1	no	—
310	C₄⋊C₄	—	38	1	no	—
311	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
312	C₂²⋊C₄	—	2	1	no	—
313	C₂²⋊C₄	—	19	1	no	—
314	C₂²⋊C₄	—	19	1	no	—
315	C₂²⋊C₄	—	38	1	no	—
316	C₂²⋊C₄	—	38	1	no	—
317	C₂²⋊C₄	—	38	1	no	—
318	C₂²⋊C₄	—	38	1	no	—
319	C₄²	—	19	1	no	—
320	C₄²	—	38	1	no	—
321	C₂³	—	19	1	no	—
322	C₂³	—	38	1	no	—
323	C₂³	D₄×D₁₉	1	1	yes	characteristic
324	Q₈	—	2	1	no	—
325	Q₈	—	2	1	no	—
326	Q₈	—	2	1	no	—
327	Q₈	—	2	1	no	—
328	Q₈	—	2	1	no	—
329	Q₈	—	4	2	no	—
330	Q₈	—	38	1	no	—
331	Q₈	—	38	1	no	—
332	Q₈	—	38	1	no	—
333	Q₈	—	38	1	no	—
334	Q₈	—	76	2	no	—
335	Q₈	D₄×D₁₉	1	2	yes	—
336	D₄	—	19	2	no	—
337	D₄	—	38	2	no	—
338	D₄	—	38	1	no	—
339	D₄	—	76	2	no	—
340	D₄	—	76	2	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	19	1	no	—
349	C₂×C₄	—	19	1	no	—
350	C₂×C₄	—	19	1	no	—
351	C₂×C₄	—	38	1	no	—
352	C₂×C₄	—	38	1	no	—
353	C₂×C₄	—	38	1	no	—
354	C₂×C₄	—	38	1	no	—
355	C₂×C₄	—	38	1	no	—
356	C₂×C₄	—	76	2	no	—
357	C₂×C₄	—	76	2	no	—
358	C₂×C₄	—	76	2	no	—
359	C₂×C₄	—	76	2	no	—
360	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
361	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
362	C₈	—	2	1	no	—
363	C₈	—	2	1	no	—
364	C₈	—	19	2	no	—
365	C₈	—	38	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	2	1	no	—
369	C₂²	—	38	1	no	—
370	C₂²	—	38	1	no	—
371	C₂²	—	38	1	no	—
372	C₂²	—	38	2	no	—
373	C₂²	—	76	1	no	—
374	C₂²	—	76	1	no	—
375	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
376	C₄	—	2	1	no	—
377	C₄	—	2	2	no	—
378	C₄	—	4	1	no	—
379	C₄	—	4	1	no	—
380	C₄	—	4	1	no	—
381	C₄	—	19	2	no	—
382	C₄	—	38	1	no	—
383	C₄	—	38	1	no	—
384	C₄	—	76	1	no	—
385	C₄	—	76	1	no	—
386	C₄	—	76	1	no	—
387	C₄	C₂×D₄×D₁₉	1	1	yes	characteristic
388	C₄	#10597430	1	1	yes	characteristic
389	C₂	—	4	1	no	—
390	C₂	—	76	1	no	—
391	C₂	—	76	2	no	—
392	C₂	#11927176	1	1	yes	characteristic
393	C₂	#137×D₁₉	1	1	yes	characteristic
394	C₂	#11927357	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₉
commutator, cosocle	C₂×C₄×C₁₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉
Fitting	C₁₉×#451	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	#11927196	C₂
hypercenter	#451	D₁₉
hypocenter	C₁₉	#3276
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #451

Complements

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

N	H ≅ G/N	type
1. C₂×#10597364	C₂	semidirect
2. C₂×#10597388	C₂	semidirect
3. Q₈×C₂²⋊D₁₉	C₂	semidirect
5. #11926922	C₂	semidirect
6. #11926801	C₂	semidirect
7. #11926800	C₂	semidirect
8. #11926752	C₂	semidirect
9. #11926748	C₂	semidirect
10. #11926708	C₂	semidirect
11. #11926705	C₂	semidirect
12. #11926701	C₂	semidirect
13. #11926629	C₂	semidirect
14. #11926619	C₂	semidirect
15. C₁₉×#451	C₂	semidirect
100. #64527	D₄	semidirect
191. Dic₁₉	#137	semidirect
204. #451	D₁₉	semidirect
258. C₁₉	#3276	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10597364	2
C₂×#10597388	2
Q₈×C₂²⋊D₁₉	2
#11927196	2
#11926922	2
#11926801	2
#11926800	2
#11926752	2
#11926748	2
#11926708	2
#11926705	2
#11926701	2
#11926629	2
#11926619	2
C₁₉×#451	2
#3276	19