Subgroups — #14576337

Order 2432.

396 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927253	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927214	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926908	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926907	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926889	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926845	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926842	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926832	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926737	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926737	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926673	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926656	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926654	C₂	1	1	yes	maximal, characteristic, complemented
14	D₈×Dic₁₉	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#459	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#64543	—	2	1	no	—
17	C₂×D₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₂×D₈×C₁₉	C₂²	1	1	yes	characteristic
19	C₁₉×#127	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
21	#10597386	C₂²	1	1	yes	characteristic
22	#10597386	C₂²	1	1	yes	characteristic
23	#10597385	—	2	1	no	—
24	#10597385	—	2	1	no	—
25	#10597384	—	2	1	no	—
26	D₄×Dic₁₉	C₂²	1	1	yes	characteristic
27	D₄×Dic₁₉	C₂²	1	1	yes	characteristic
28	D₄×Dic₁₉	C₂²	1	1	yes	characteristic
29	D₄×Dic₁₉	C₂²	1	1	yes	characteristic
30	C₂×#64525	C₂²	1	1	yes	characteristic
31	#10597370	—	2	1	no	—
32	C₂×Dic₁₉⋊C₄	—	2	1	no	—
33	#10597336	C₂²	1	1	yes	characteristic
34	#10597334	C₂²	1	1	yes	characteristic
35	#10597324	—	2	1	no	—
36	#10597324	—	2	1	no	—
37	C₄×D₇₆	C₂²	1	1	yes	characteristic
38	#10597316	C₂²	1	1	yes	characteristic
39	C₄×Dic₃₈	C₂²	1	1	yes	characteristic
40	C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
41	C₁₉×#102	C₂²	1	1	yes	characteristic
42	C₁₉×#102	C₂²	1	1	yes	characteristic
43	#10597289	C₂²	1	1	yes	characteristic
44	#10597289	C₂²	1	1	yes	characteristic
45	#10597289	C₂²	1	1	yes	characteristic
46	#10597289	C₂²	1	1	yes	characteristic
47	#10597289	C₂²	1	1	yes	characteristic
48	#10597278	C₂²	1	1	yes	characteristic
49	#10597278	C₂²	1	1	yes	characteristic
50	#10597275	C₂²	1	1	yes	characteristic
51	#10597274	C₂²	1	1	yes	characteristic
52	#10597272	C₂²	1	1	yes	characteristic
53	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
54	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
55	C₈×Dic₁₉	C₂²	1	1	yes	characteristic
56	#10597266	C₂²	1	1	yes	characteristic
57	#10597265	C₂²	1	1	yes	characteristic
58	#10597260	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₁₉	—	2	1	no	—
60	C₂×D₄×C₁₉	—	2	1	no	—
61	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
63	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	—	2	1	no	—
65	C₂×C₂²⋊D₁₉	—	2	1	no	—
66	C₂×C₂²⋊D₁₉	—	2	1	no	—
67	C₂×C₂²⋊D₁₉	—	2	1	no	—
68	C₂×C₂²⋊D₁₉	—	2	1	no	—
69	C₂²×Dic₁₉	—	2	1	no	—
70	C₂²×Dic₁₉	—	2	1	no	—
71	C₂²×Dic₁₉	—	2	1	no	—
72	C₂²×Dic₁₉	—	2	1	no	—
73	#64543	—	4	4	no	—
74	C₂×D₇₆	C₂³	1	1	yes	characteristic
75	C₂×C₄×D₁₉	—	2	1	no	—
76	C₂×Dic₃₈	—	2	1	no	—
77	C₂×Dic₃₈	D₄	1	1	yes	characteristic
78	D₈×C₁₉	—	2	2	no	—
79	D₈×C₁₉	—	2	2	no	—
80	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
81	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
83	C₂²⋊C₄×C₁₉	—	2	1	no	—
84	C₄²×C₁₉	C₂³	1	1	yes	characteristic
85	C₂²⋊Dic₁₉	—	2	1	no	—
86	C₂²⋊Dic₁₉	—	2	1	no	—
87	C₂²⋊Dic₁₉	—	2	1	no	—
88	C₂²⋊Dic₁₉	—	2	1	no	—
89	#64525	—	2	2	no	—
90	(C₂×D₁₉)⋊C₄	—	2	1	no	—
91	C₄⋊Dic₁₉	—	2	1	no	—
92	C₄⋊Dic₁₉	D₄	1	1	yes	characteristic
93	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
94	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
95	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
96	Dic₁₉⋊C₄	—	2	1	no	—
97	Dic₁₉⋊C₄	—	2	1	no	—
98	Dic₁₉⋊C₄	—	2	1	no	—
99	Dic₁₉⋊C₄	—	2	1	no	—
100	Dic₁₉⋊C₄	D₄	1	1	yes	characteristic
101	Dic₁₉⋊C₄	D₄	1	1	yes	characteristic
102	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
103	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
104	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
105	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
106	C₂³×C₁₉	—	2	1	no	—
107	C₂³×C₁₉	—	2	1	no	—
108	C₂³×C₁₉	—	2	1	no	—
109	C₂²×D₁₉	—	2	1	no	—
110	D₄×C₁₉	—	2	1	no	—
111	D₄×C₁₉	—	2	1	no	—
112	D₄×C₁₉	—	2	1	no	—
113	D₄×C₁₉	—	2	1	no	—
114	D₄×C₁₉	—	2	1	no	—
115	D₄×C₁₉	—	4	2	no	—
116	D₄×C₁₉	—	8	1	no	—
117	D₄×C₁₉	#41	1	2	yes	complemented
118	C₂×C₄×C₁₉	—	2	1	no	—
119	C₂×C₄×C₁₉	—	8	1	no	—
120	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
123	C₂²⋊D₁₉	—	4	2	no	—
124	C₂²⋊D₁₉	—	4	2	no	—
125	C₂²⋊D₁₉	—	8	1	no	—
126	C₂²⋊D₁₉	—	8	1	no	—
127	C₂×Dic₁₉	—	2	1	no	—
128	C₂×Dic₁₉	—	2	1	no	—
129	C₂×Dic₁₉	—	2	1	no	—
130	C₂×Dic₁₉	—	4	2	no	—
131	C₂×Dic₁₉	—	4	2	no	—
132	C₂×Dic₁₉	—	4	2	no	—
133	C₂×Dic₁₉	—	8	1	no	—
134	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
138	D₇₆	—	2	1	no	—
139	D₇₆	—	2	1	no	—
140	C₄×D₁₉	—	4	2	no	—
141	Dic₃₈	—	2	2	no	—
142	Dic₃₈	—	8	1	no	—
143	C₈×C₁₉	—	2	1	no	—
144	C₈×C₁₉	—	2	1	no	—
145	C₁₉⋊C₈	—	2	1	no	—
146	C₁₉⋊C₈	—	2	1	no	—
147	#3366	—	19	1	no	maximal, Sylow
148	C₂²×C₁₉	—	2	2	no	—
149	C₂²×C₁₉	—	4	1	no	—
150	C₂²×C₁₉	—	4	1	no	—
151	C₂²×C₁₉	—	4	1	no	—
152	C₂²×C₁₉	—	4	1	no	—
153	C₂²×C₁₉	—	4	1	no	—
154	C₂²×C₁₉	—	4	1	no	—
155	C₂²×C₁₉	—	4	1	no	—
156	C₂²×C₁₉	—	4	1	no	—
157	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
158	C₂×D₁₉	—	4	1	no	—
159	C₂×D₁₉	—	4	1	no	—
160	C₂×D₁₉	—	4	1	no	—
161	C₄×C₁₉	—	2	1	no	—
162	C₄×C₁₉	—	2	1	no	—
163	C₄×C₁₉	—	4	1	no	—
164	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
165	C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
166	Dic₁₉	—	2	1	no	—
167	Dic₁₉	—	2	1	no	—
168	Dic₁₉	—	2	1	no	—
169	Dic₁₉	—	4	1	no	—
170	Dic₁₉	—	4	1	no	—
171	Dic₁₉	—	4	1	no	—
172	Dic₁₉	AGL(1,ℤ/8)	1	2	yes	—
173	#554	—	19	1	no	—
174	#550	—	19	1	no	—
175	#490	—	19	1	no	—
176	#490	—	19	1	no	—
177	#486	—	19	1	no	—
178	#478	—	19	1	no	—
179	#478	—	19	1	no	—
180	#476	—	19	1	no	—
181	#459	—	19	1	no	—
182	#459	—	19	1	no	—
183	#459	D₁₉	1	1	yes	characteristic, complemented
184	#446	—	19	1	no	—
185	#442	—	19	1	no	—
186	#442	—	19	1	no	—
187	C₄×D₈	—	19	1	no	—
188	C₂×C₁₉	—	4	2	no	—
189	C₂×C₁₉	—	8	1	no	—
190	C₂×C₁₉	—	8	1	no	—
191	C₂×C₁₉	#548	1	1	yes	characteristic
192	C₂×C₁₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
193	C₂×C₁₉	#576	1	1	yes	characteristic
194	D₁₉	—	8	1	no	—
195	C₂×#41	—	38	1	no	—
196	C₂×D₈	—	19	1	no	—
197	C₂×D₈	C₂×D₁₉	1	1	yes	characteristic
198	C₂×D₈	C₂×D₁₉	1	1	yes	characteristic
199	C₄⋊Q₈	—	19	1	no	—
200	#127	—	19	1	no	—
201	#127	—	19	1	no	—
202	#127	C₂×D₁₉	1	1	yes	characteristic
203	#125	—	19	1	no	—
204	#125	—	19	1	no	—
205	C₂²⋊Q₈	—	38	1	no	—
206	C₂²⋊Q₈	—	38	1	no	—
207	C₂²⋊Q₈	—	38	1	no	—
208	#121	—	38	1	no	—
209	#121	—	38	1	no	—
210	#121	—	38	1	no	—
211	C₄×Q₈	—	19	1	no	—
212	C₄×D₄	—	19	1	no	—
213	C₄×D₄	—	19	1	no	—
214	C₄×D₄	—	19	1	no	—
215	C₄×D₄	—	19	1	no	—
216	C₄×D₄	—	19	1	no	—
217	C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
218	C₂×C₄⋊C₄	—	38	1	no	—
219	#107	—	19	1	no	—
220	#106	—	19	1	no	—
221	#106	—	19	1	no	—
222	C₄⋊C₈	—	19	1	no	—
223	C₄⋊C₈	—	19	1	no	—
224	C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
225	#102	—	19	1	no	—
226	#102	—	19	1	no	—
227	#102	—	19	1	no	—
228	#102	—	19	1	no	—
229	#102	—	19	1	no	—
230	#102	—	19	1	no	—
231	#102	—	19	1	no	—
232	#102	—	19	1	no	—
233	#102	C₂×D₁₉	1	1	yes	characteristic
234	#102	C₂×D₁₉	1	1	yes	characteristic
235	#97	—	19	1	no	—
236	#97	—	19	1	no	—
237	C₄×C₈	—	19	1	no	—
238	C₁₉	#3366	1	1	yes	characteristic, complemented, Sylow
239	#41	—	76	4	no	—
240	C₂×Q₈	—	19	1	no	—
241	C₂×Q₈	—	38	1	no	—
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	—	19	1	no	—
245	C₂×D₄	—	38	1	no	—
246	C₂×D₄	—	38	1	no	—
247	C₂×D₄	—	38	1	no	—
248	C₂×D₄	—	38	1	no	—
249	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
250	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
251	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	38	1	no	—
254	C₂²×C₄	—	38	1	no	—
255	C₂²×C₄	—	38	1	no	—
256	C₂²×C₄	—	38	1	no	—
257	C₂²×C₄	—	38	1	no	—
258	D₈	—	2	2	no	—
259	D₈	—	2	2	no	—
260	D₈	—	38	2	no	—
261	C₂×C₈	—	19	1	no	—
262	C₂×C₈	—	19	1	no	—
263	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
264	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
265	C₄⋊C₄	—	19	1	no	—
266	C₄⋊C₄	—	19	1	no	—
267	C₄⋊C₄	—	19	1	no	—
268	C₄⋊C₄	—	19	1	no	—
269	C₄⋊C₄	—	19	1	no	—
270	C₄⋊C₄	—	19	1	no	—
271	C₄⋊C₄	—	38	1	no	—
272	C₄⋊C₄	—	38	1	no	—
273	C₄⋊C₄	—	38	1	no	—
274	C₄⋊C₄	—	38	1	no	—
275	C₄⋊C₄	—	38	1	no	—
276	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	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	C₄²	—	19	1	no	—
284	C₄²	—	19	1	no	—
285	C₄²	C₂²×D₁₉	1	1	yes	characteristic
286	C₂³	—	2	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	—	38	1	no	—
290	Q₈	—	38	2	no	—
291	Q₈	—	152	1	no	—
292	D₄	—	2	1	no	—
293	D₄	—	2	1	no	—
294	D₄	—	2	1	no	—
295	D₄	—	2	1	no	—
296	D₄	—	2	1	no	—
297	D₄	—	4	2	no	—
298	D₄	—	8	1	no	—
299	D₄	—	38	1	no	—
300	D₄	—	38	1	no	—
301	D₄	—	76	2	no	—
302	D₄	—	76	2	no	—
303	D₄	—	152	1	no	—
304	D₄	—	152	1	no	—
305	D₄	#64543	1	2	yes	complemented
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	8	1	no	—
308	C₂×C₄	—	19	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	—	76	2	no	—
316	C₂×C₄	—	76	2	no	—
317	C₂×C₄	—	76	2	no	—
318	C₂×C₄	—	76	2	no	—
319	C₂×C₄	—	152	1	no	—
320	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
321	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
322	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	38	1	no	—
326	C₈	—	38	1	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	76	1	no	—
337	C₂²	—	76	1	no	—
338	C₂²	—	76	1	no	—
339	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	19	2	no	—
344	C₄	—	38	1	no	—
345	C₄	—	38	1	no	—
346	C₄	—	38	1	no	—
347	C₄	—	76	1	no	—
348	C₄	—	76	1	no	—
349	C₄	—	76	1	no	—
350	C₄	C₂×#64543	1	1	yes	characteristic
351	C₄	#10597436	1	1	yes	characteristic
352	C₂	—	4	2	no	—
353	C₂	—	8	1	no	—
354	C₂	—	8	1	no	—
355	C₂	—	152	1	no	—
356	C₂	#11927200	1	1	yes	characteristic
357	C₂	AGL(1,ℤ/8)×D₁₉	1	1	yes	characteristic
358	C₂	#11927345	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×#459	C₂
socle	C₂²×C₁₉	C₂²×D₄
hypercenter	#459	D₁₉
hypocenter	C₁₉	#3366
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #459

Complements

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

N	H ≅ G/N	type
1. #11927253	C₂	semidirect
2. #11927214	C₂	semidirect
3. #11926908	C₂	semidirect
4. #11926907	C₂	semidirect
5. #11926889	C₂	semidirect
6. #11926845	C₂	semidirect
7. #11926842	C₂	semidirect
8. #11926832	C₂	semidirect
9. #11926737	C₂	semidirect
10. #11926737	C₂	semidirect
11. #11926673	C₂	semidirect
12. #11926656	C₂	semidirect
13. #11926654	C₂	semidirect
14. D₈×Dic₁₉	C₂	semidirect
15. C₁₉×#459	C₂	semidirect
117. D₄×C₁₉	#41	semidirect
183. #459	D₁₉	semidirect
238. C₁₉	#3366	semidirect
305. D₄	#64543	semidirect

Maximal subgroups

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

subgroup	index
#11927253	2
#11927214	2
#11926908	2
#11926907	2
#11926889	2
#11926845	2
#11926842	2
#11926832	2
#11926737	2
#11926673	2
#11926656	2
#11926654	2
D₈×Dic₁₉	2
C₁₉×#459	2
#3366	19