Subgroups — #14575045

Order 2432.

358 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927168	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927145	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926854	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926843	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926821	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926798	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926789	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926779	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926774	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926760	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926759	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926743	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926738	C₂	1	1	yes	maximal, characteristic, complemented
14	#11926569	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#476	C₂	1	1	yes	maximal, characteristic, complemented
16	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
18	#10597393	—	2	1	no	—
19	#10597392	—	2	1	no	—
20	#10597386	C₂²	1	1	yes	characteristic
21	#10597384	C₂²	1	1	yes	characteristic
22	C₂×#64526	C₂²	1	1	yes	characteristic
23	C₂×#64525	C₂²	1	1	yes	characteristic
24	#10597376	—	2	1	no	—
25	#10597375	—	2	1	no	—
26	C₂×D₁₅₂	C₂²	1	1	yes	characteristic
27	C₂×#64516	C₂²	1	1	yes	characteristic
28	C₂×#64515	C₂²	1	1	yes	characteristic
29	C₂×C₈×D₁₉	C₂²	1	1	yes	characteristic
30	#10597344	—	2	1	no	—
31	#10597343	C₂²	1	1	yes	characteristic
32	#10597341	C₂²	1	1	yes	characteristic
33	#10597341	C₂²	1	1	yes	characteristic
34	#10597340	—	2	1	no	—
35	#10597339	C₂²	1	1	yes	characteristic
36	#10597336	C₂²	1	1	yes	characteristic
37	#10597333	C₂²	1	1	yes	characteristic
38	#10597330	—	2	1	no	—
39	#10597329	—	2	1	no	—
40	#10597321	C₂²	1	1	yes	characteristic
41	C₄×D₇₆	C₂²	1	1	yes	characteristic
42	#10597318	C₂²	1	1	yes	characteristic
43	C₁₉×#106	C₂²	1	1	yes	characteristic
44	C₁₉×#106	C₂²	1	1	yes	characteristic
45	C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
46	C₁₉×#102	C₂²	1	1	yes	characteristic
47	C₁₉×#102	C₂²	1	1	yes	characteristic
48	#10597278	C₂²	1	1	yes	characteristic
49	#10597273	C₂²	1	1	yes	characteristic
50	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
51	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
52	#10597267	C₂²	1	1	yes	characteristic
53	#10597266	C₂²	1	1	yes	characteristic
54	#10597266	C₂²	1	1	yes	characteristic
55	#10597266	C₂²	1	1	yes	characteristic
56	#10597264	C₂²	1	1	yes	characteristic
57	#10597264	C₂²	1	1	yes	characteristic
58	C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₉	—	2	1	no	—
60	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	—	2	1	no	—
62	C₂×C₂²⋊D₁₉	—	2	1	no	—
63	C₂×C₂²⋊D₁₉	—	2	1	no	—
64	C₂×C₂²⋊D₁₉	—	2	1	no	—
65	C₂×D₇₆	—	2	1	no	—
66	C₂×D₇₆	—	2	1	no	—
67	C₂×D₇₆	C₂³	1	1	yes	characteristic
68	C₂×D₇₆	C₂³	1	1	yes	characteristic
69	C₂×C₄×D₁₉	—	2	1	no	—
70	C₂×C₄×D₁₉	—	2	1	no	—
71	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
72	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₁₉	—	2	1	no	—
76	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
79	C₂²⋊C₄×C₁₉	—	2	1	no	—
80	C₄²×C₁₉	C₂³	1	1	yes	characteristic
81	C₂²⋊Dic₁₉	—	2	1	no	—
82	#64526	—	4	1	no	—
83	#64525	—	4	1	no	—
84	(C₂×D₁₉)⋊C₄	—	2	1	no	—
85	(C₂×D₁₉)⋊C₄	—	2	1	no	—
86	(C₂×D₁₉)⋊C₄	—	2	1	no	—
87	(C₂×D₁₉)⋊C₄	—	2	1	no	—
88	(C₂×D₁₉)⋊C₄	—	2	1	no	—
89	(C₂×D₁₉)⋊C₄	—	2	1	no	—
90	(C₂×D₁₉)⋊C₄	—	2	1	no	—
91	(C₂×D₁₉)⋊C₄	D₄	1	1	yes	characteristic
92	(C₂×D₁₉)⋊C₄	D₄	1	1	yes	characteristic
93	C₄⋊Dic₁₉	—	2	1	no	—
94	C₄⋊Dic₁₉	—	2	1	no	—
95	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
96	Dic₁₉⋊C₄	—	2	1	no	—
97	Dic₁₉⋊C₄	—	2	1	no	—
98	Dic₁₉⋊C₄	D₄	1	1	yes	characteristic
99	Dic₁₉⋊C₄	D₄	1	1	yes	characteristic
100	C₄×Dic₁₉	—	2	1	no	—
101	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
102	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
103	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
104	D₁₅₂	—	4	1	no	—
105	#64516	—	4	1	no	—
106	#64515	—	4	1	no	—
107	C₈×D₁₉	—	4	1	no	—
108	C₂³×C₁₉	—	2	1	no	—
109	C₂²×D₁₉	—	2	1	no	—
110	C₂²×D₁₉	—	2	1	no	—
111	C₂²×D₁₉	C₂×D₄	1	1	yes	characteristic
112	Q₈×C₁₉	—	8	1	no	—
113	D₄×C₁₉	—	2	1	no	—
114	D₄×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	—	8	1	no	—
119	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₁₉	—	8	1	no	—
123	C₂²⋊D₁₉	—	8	1	no	—
124	C₂²⋊D₁₉	—	8	1	no	—
125	C₂×Dic₁₉	—	2	1	no	—
126	C₂×Dic₁₉	—	2	1	no	—
127	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
130	D₇₆	—	2	1	no	—
131	D₇₆	—	2	1	no	—
132	D₇₆	—	2	1	no	—
133	D₇₆	—	2	1	no	—
134	D₇₆	—	8	1	no	—
135	D₇₆	—	8	1	no	—
136	C₄×D₁₉	—	2	1	no	—
137	C₄×D₁₉	—	2	1	no	—
138	C₄×D₁₉	—	8	1	no	—
139	C₄×D₁₉	—	8	1	no	—
140	Dic₃₈	—	2	1	no	—
141	Dic₃₈	—	2	1	no	—
142	C₈×C₁₉	—	2	1	no	—
143	C₈×C₁₉	—	2	1	no	—
144	C₁₉⋊C₈	—	2	1	no	—
145	C₁₉⋊C₈	—	2	1	no	—
146	#3257	—	19	1	no	maximal, Sylow
147	C₂²×C₁₉	—	4	1	no	—
148	C₂²×C₁₉	—	4	1	no	—
149	C₂²×C₁₉	—	4	1	no	—
150	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
151	C₂×D₁₉	—	2	1	no	—
152	C₂×D₁₉	—	2	1	no	—
153	C₂×D₁₉	—	2	1	no	—
154	C₂×D₁₉	—	4	1	no	—
155	C₂×D₁₉	—	4	1	no	—
156	C₂×D₁₉	—	4	1	no	—
157	C₂×D₁₉	—	4	1	no	—
158	C₂×D₁₉	—	4	1	no	—
159	C₂×D₁₉	—	4	1	no	—
160	C₄×C₁₉	—	2	1	no	—
161	C₄×C₁₉	—	2	1	no	—
162	C₄×C₁₉	—	4	1	no	—
163	C₄×C₁₉	—	4	1	no	—
164	C₄×C₁₉	—	4	1	no	—
165	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
166	C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
167	Dic₁₉	—	2	1	no	—
168	Dic₁₉	—	2	1	no	—
169	Dic₁₉	—	2	1	no	—
170	Dic₁₉	—	4	1	no	—
171	Dic₁₉	—	4	1	no	—
172	#542	—	19	1	no	—
173	#540	—	19	1	no	—
174	#479	—	19	1	no	—
175	#478	—	19	1	no	—
176	#476	D₁₉	1	1	yes	characteristic, complemented
177	#474	—	19	1	no	—
178	#469	—	19	1	no	—
179	#468	—	19	1	no	—
180	#466	—	19	1	no	—
181	#465	—	19	1	no	—
182	#463	—	19	1	no	—
183	#463	—	19	1	no	—
184	#460	—	19	1	no	—
185	#459	—	19	1	no	—
186	#424	—	19	1	no	—
187	C₂×C₁₉	—	8	1	no	—
188	C₂×C₁₉	#537	1	1	yes	characteristic
189	C₂×C₁₉	#576	1	1	yes	characteristic
190	C₂×C₁₉	#577	1	1	yes	characteristic
191	D₁₉	—	4	1	no	—
192	D₁₉	—	8	1	no	—
193	D₁₉	—	8	1	no	—
194	C₂×SD₁₆	—	19	1	no	—
195	C₂×SD₁₆	—	19	1	no	—
196	C₂×D₈	—	19	1	no	—
197	C₂×D₈	—	19	1	no	—
198	C₂×M₁₆	—	19	1	no	—
199	C₂²×C₈	—	19	1	no	—
200	C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
201	#127	—	19	1	no	—
202	#126	—	38	1	no	—
203	#125	—	19	1	no	—
204	#124	—	19	1	no	—
205	#124	—	38	1	no	—
206	#123	—	38	1	no	—
207	#123	—	38	1	no	—
208	#123	—	38	1	no	—
209	C₂²⋊Q₈	—	19	1	no	—
210	C₂²⋊Q₈	—	38	1	no	—
211	#121	—	19	1	no	—
212	#121	—	19	1	no	—
213	#121	—	19	1	no	—
214	#121	—	38	1	no	—
215	#121	—	38	1	no	—
216	C₄×Q₈	—	19	1	no	—
217	C₄×D₄	—	19	1	no	—
218	C₄×D₄	—	19	1	no	—
219	C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
220	#117	—	19	1	no	—
221	#107	—	19	1	no	—
222	#107	—	19	1	no	—
223	#106	C₂×D₁₉	1	1	yes	characteristic
224	#106	C₂×D₁₉	1	1	yes	characteristic
225	C₄⋊C₈	—	19	1	no	—
226	C₄⋊C₈	—	19	1	no	—
227	C₄⋊C₈	—	19	1	no	—
228	C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
229	#103	—	19	1	no	—
230	#103	—	19	1	no	—
231	#102	—	19	1	no	—
232	#102	—	19	1	no	—
233	#102	—	19	1	no	—
234	#102	—	19	1	no	—
235	#102	C₂×D₁₉	1	1	yes	characteristic
236	#102	C₂×D₁₉	1	1	yes	characteristic
237	C₁₉	#3257	1	1	yes	characteristic, complemented, Sylow
238	C₂×Q₈	—	2	1	no	—
239	C₂×Q₈	—	19	1	no	—
240	C₂×D₄	—	19	1	no	—
241	C₂×D₄	—	19	1	no	—
242	C₂×D₄	—	38	1	no	—
243	C₂×D₄	—	38	1	no	—
244	C₂×D₄	—	38	1	no	—
245	C₂×D₄	—	38	1	no	—
246	C₂×D₄	—	38	1	no	—
247	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	19	1	no	—
250	C₂²×C₄	—	38	1	no	—
251	C₂²×C₄	—	38	1	no	—
252	SD₁₆	—	76	1	no	—
253	SD₁₆	—	76	1	no	—
254	D₈	—	76	1	no	—
255	D₈	—	76	1	no	—
256	M₁₆	—	76	1	no	—
257	C₂×C₈	—	19	1	no	—
258	C₂×C₈	—	19	1	no	—
259	C₂×C₈	—	76	1	no	—
260	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	19	1	no	—
264	C₄⋊C₄	—	19	1	no	—
265	C₄⋊C₄	—	19	1	no	—
266	C₄⋊C₄	—	38	1	no	—
267	C₄⋊C₄	—	38	1	no	—
268	C₄⋊C₄	—	38	1	no	—
269	C₄⋊C₄	—	38	1	no	—
270	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
271	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
272	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
273	C₂²⋊C₄	—	2	1	no	—
274	C₂²⋊C₄	—	19	1	no	—
275	C₂²⋊C₄	—	19	1	no	—
276	C₂²⋊C₄	—	38	1	no	—
277	C₂²⋊C₄	—	38	1	no	—
278	C₂²⋊C₄	—	38	1	no	—
279	C₂²⋊C₄	—	38	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	C₄²	—	19	1	no	—
285	C₄²	—	38	1	no	—
286	C₄²	C₂²×D₁₉	1	1	yes	characteristic
287	C₂³	—	2	1	no	—
288	C₂³	—	19	1	no	—
289	C₂³	—	38	1	no	—
290	C₂³	—	38	1	no	—
291	Q₈	—	8	1	no	—
292	Q₈	—	38	1	no	—
293	Q₈	—	38	1	no	—
294	D₄	—	2	1	no	—
295	D₄	—	2	1	no	—
296	D₄	—	38	1	no	—
297	D₄	—	38	1	no	—
298	D₄	—	38	1	no	—
299	D₄	—	38	1	no	—
300	D₄	—	152	1	no	—
301	D₄	—	152	1	no	—
302	D₄	—	152	1	no	—
303	D₄	—	152	1	no	—
304	D₄	—	152	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	8	1	no	—
309	C₂×C₄	—	19	1	no	—
310	C₂×C₄	—	19	1	no	—
311	C₂×C₄	—	19	1	no	—
312	C₂×C₄	—	38	1	no	—
313	C₂×C₄	—	38	1	no	—
314	C₂×C₄	—	38	1	no	—
315	C₂×C₄	—	38	1	no	—
316	C₂×C₄	—	152	1	no	—
317	C₂×C₄	—	152	1	no	—
318	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
319	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
320	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	2	1	no	—
323	C₈	—	38	1	no	—
324	C₈	—	38	1	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	38	1	no	—
329	C₂²	—	38	1	no	—
330	C₂²	—	38	1	no	—
331	C₂²	—	76	1	no	—
332	C₂²	—	76	1	no	—
333	C₂²	—	76	1	no	—
334	C₂²	—	76	1	no	—
335	C₂²	—	76	1	no	—
336	C₂²	—	76	1	no	—
337	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	38	1	no	—
344	C₄	—	38	1	no	—
345	C₄	—	38	1	no	—
346	C₄	—	76	1	no	—
347	C₄	—	76	1	no	—
348	C₄	#10597433	1	1	yes	characteristic
349	C₄	#10597435	1	1	yes	characteristic
350	C₂	—	8	1	no	—
351	C₂	—	76	1	no	—
352	C₂	—	152	1	no	—
353	C₂	—	152	1	no	—
354	C₂	#11927124	1	1	yes	characteristic
355	C₂	#11927348	1	1	yes	characteristic
356	C₂	#11927350	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×#476	C₂
socle	C₂²×C₁₉	C₂²×D₄
hypercenter	#476	D₁₉
hypocenter	C₁₉	#3257
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #476

Complements

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

N	H ≅ G/N	type
1. #11927168	C₂	semidirect
2. #11927145	C₂	semidirect
3. #11926854	C₂	semidirect
4. #11926843	C₂	semidirect
5. #11926821	C₂	semidirect
6. #11926798	C₂	semidirect
7. #11926789	C₂	semidirect
8. #11926779	C₂	semidirect
9. #11926774	C₂	semidirect
10. #11926760	C₂	semidirect
11. #11926759	C₂	semidirect
12. #11926743	C₂	semidirect
13. #11926738	C₂	semidirect
14. #11926569	C₂	semidirect
15. C₁₉×#476	C₂	semidirect
176. #476	D₁₉	semidirect
237. C₁₉	#3257	semidirect

Maximal subgroups

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

subgroup	index
#11927168	2
#11927145	2
#11926854	2
#11926843	2
#11926821	2
#11926798	2
#11926789	2
#11926779	2
#11926774	2
#11926760	2
#11926759	2
#11926743	2
#11926738	2
#11926569	2
C₁₉×#476	2
#3257	19