Subgroups — #14576386

Order 2432.

358 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927277	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927250	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926907	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926902	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926901	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926854	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926823	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926760	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926758	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926738	C₂	1	1	yes	maximal, characteristic
11	#11926671	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926655	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926643	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#64525	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#478	C₂	1	1	yes	maximal, characteristic, complemented
16	C₁₉×#125	C₂²	1	1	yes	characteristic
17	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
18	#10597386	C₂²	1	1	yes	characteristic
19	#10597382	C₂²	1	1	yes	characteristic
20	C₂×#64525	C₂²	1	1	yes	characteristic
21	#10597375	—	2	1	no	—
22	C₄×C₂²⋊D₁₉	—	2	1	no	—
23	C₂×D₁₅₂	C₂²	1	1	yes	characteristic
24	C₂×#64516	C₂²	1	1	yes	characteristic
25	#10597343	—	2	1	no	—
26	#10597342	—	2	1	no	—
27	#10597342	—	2	1	no	—
28	#10597339	C₂²	1	1	yes	characteristic
29	#10597339	C₂²	1	1	yes	characteristic
30	#10597339	C₂²	1	1	yes	characteristic
31	C₄⋊C₄×D₁₉	—	2	1	no	—
32	#10597336	C₂²	1	1	yes	characteristic
33	#10597334	C₂²	1	1	yes	characteristic
34	#10597333	C₂²	1	1	yes	characteristic
35	#10597331	—	2	1	no	—
36	#10597329	—	2	1	no	—
37	#10597321	C₂²	1	1	yes	characteristic
38	C₄×D₇₆	C₂²	1	1	yes	characteristic
39	C₄×Dic₃₈	C₂²	1	1	yes	characteristic
40	C₁₉×#107	C₂²	1	1	yes	characteristic
41	C₁₉×#106	C₂²	1	1	yes	characteristic
42	C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₁₉×#102	C₂²	1	1	yes	characteristic
44	C₁₉×#102	C₂²	1	1	yes	characteristic
45	#10597289	C₂²	1	1	yes	characteristic
46	#10597278	C₂²	1	1	yes	characteristic
47	#10597273	C₂²	1	1	yes	characteristic
48	#10597272	C₂²	1	1	yes	characteristic
49	#10597272	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	#10597267	C₂²	1	1	yes	characteristic
54	#10597266	C₂²	1	1	yes	characteristic
55	#10597266	C₂²	1	1	yes	characteristic
56	#10597266	C₂²	1	1	yes	characteristic
57	#10597264	C₂²	1	1	yes	characteristic
58	C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	—	2	1	no	—
61	C₂×C₂²⋊D₁₉	—	2	1	no	—
62	C₂×C₂²⋊D₁₉	—	2	1	no	—
63	C₂×D₇₆	C₂³	1	1	yes	characteristic
64	C₂×D₇₆	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₁₉	—	2	1	no	—
66	C₂×C₄×D₁₉	—	2	1	no	—
67	C₂×C₄×D₁₉	—	2	1	no	—
68	C₂×C₄×D₁₉	—	2	1	no	—
69	C₂×Dic₃₈	—	2	1	no	—
70	C₂×Dic₃₈	D₄	1	1	yes	characteristic
71	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₁₉	—	2	1	no	—
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	C₂²⋊Dic₁₉	—	2	1	no	—
83	#64525	—	2	2	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	C₄⋊Dic₁₉	—	2	1	no	—
91	C₄⋊Dic₁₉	D₄	1	1	yes	characteristic
92	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
93	Dic₁₉⋊C₄	—	2	1	no	—
94	Dic₁₉⋊C₄	—	2	1	no	—
95	Dic₁₉⋊C₄	—	2	1	no	—
96	Dic₁₉⋊C₄	D₄	1	1	yes	characteristic
97	Dic₁₉⋊C₄	D₄	1	1	yes	characteristic
98	C₄×Dic₁₉	—	2	1	no	—
99	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
100	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
101	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
102	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
103	D₁₅₂	—	2	2	no	—
104	#64516	—	2	2	no	—
105	C₂³×C₁₉	—	2	1	no	—
106	C₂²×D₁₉	—	2	1	no	—
107	C₂²×D₁₉	—	2	1	no	—
108	D₄×C₁₉	—	2	1	no	—
109	D₄×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	—	8	1	no	—
114	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
117	C₂²⋊D₁₉	—	4	2	no	—
118	C₂²⋊D₁₉	—	8	1	no	—
119	C₂×Dic₁₉	—	2	1	no	—
120	C₂×Dic₁₉	—	2	1	no	—
121	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
125	D₇₆	—	2	1	no	—
126	D₇₆	—	2	1	no	—
127	D₇₆	—	2	1	no	—
128	D₇₆	#41	1	2	yes	—
129	C₄×D₁₉	—	4	2	no	—
130	C₄×D₁₉	—	8	1	no	—
131	C₄×D₁₉	—	8	1	no	—
132	C₄×D₁₉	—	8	1	no	—
133	Dic₃₈	—	2	1	no	—
134	Dic₃₈	—	2	1	no	—
135	Dic₃₈	—	4	1	no	—
136	Dic₃₈	—	8	1	no	—
137	C₈×C₁₉	—	2	1	no	—
138	C₈×C₁₉	—	2	1	no	—
139	C₁₉⋊C₈	—	2	1	no	—
140	C₁₉⋊C₈	—	2	1	no	—
141	#3370	—	19	1	no	maximal, Sylow
142	C₂²×C₁₉	—	4	1	no	—
143	C₂²×C₁₉	—	4	1	no	—
144	C₂²×C₁₉	—	4	1	no	—
145	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
146	C₂×D₁₉	—	2	2	no	—
147	C₂×D₁₉	—	4	1	no	—
148	C₂×D₁₉	—	4	1	no	—
149	C₂×D₁₉	—	4	1	no	—
150	C₂×D₁₉	—	4	1	no	—
151	C₂×D₁₉	—	4	1	no	—
152	C₄×C₁₉	—	2	1	no	—
153	C₄×C₁₉	—	2	1	no	—
154	C₄×C₁₉	—	4	1	no	—
155	C₄×C₁₉	—	4	1	no	—
156	C₄×C₁₉	—	4	1	no	—
157	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
158	C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
159	Dic₁₉	—	2	1	no	—
160	Dic₁₉	—	2	1	no	—
161	Dic₁₉	—	2	1	no	—
162	Dic₁₉	—	2	1	no	—
163	Dic₁₉	—	4	1	no	—
164	Dic₁₉	—	4	1	no	—
165	#559	—	19	1	no	—
166	#554	—	19	1	no	—
167	#490	—	19	1	no	—
168	#489	—	19	1	no	—
169	#488	—	19	1	no	—
170	#479	—	19	1	no	—
171	#478	D₁₉	1	1	yes	characteristic, complemented
172	#474	—	19	1	no	—
173	#463	—	19	1	no	—
174	#463	—	19	1	no	—
175	#459	—	19	1	no	—
176	#446	—	19	1	no	—
177	#442	—	19	1	no	—
178	#440	—	19	1	no	—
179	C₄×D₈	—	19	1	no	—
180	C₂×C₁₉	—	8	1	no	—
181	C₂×C₁₉	#548	1	1	yes	characteristic
182	C₂×C₁₉	#575	1	1	yes	characteristic
183	C₂×C₁₉	#576	1	1	yes	characteristic
184	D₁₉	—	4	2	no	—
185	D₁₉	—	8	1	no	—
186	C₂×SD₁₆	—	19	1	no	—
187	C₂×D₈	—	19	1	no	—
188	C₂×D₈	—	19	1	no	—
189	C₄⋊Q₈	—	19	1	no	—
190	#127	—	19	1	no	—
191	#125	—	19	1	no	—
192	#125	C₂×D₁₉	1	1	yes	characteristic
193	#124	—	19	1	no	—
194	#124	—	19	1	no	—
195	#124	—	38	1	no	—
196	#123	—	38	1	no	—
197	#123	—	38	1	no	—
198	C₂²⋊Q₈	—	38	1	no	—
199	C₂²⋊Q₈	—	38	1	no	—
200	C₂²⋊Q₈	—	38	1	no	—
201	C₄×Q₈	—	19	1	no	—
202	C₄×Q₈	—	19	1	no	—
203	C₄×D₄	—	19	1	no	—
204	C₄×D₄	—	19	1	no	—
205	C₄×D₄	—	19	1	no	—
206	C₄×D₄	—	19	1	no	—
207	C₄×D₄	—	38	1	no	—
208	C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
209	C₂×C₄⋊C₄	—	38	1	no	—
210	#107	—	19	1	no	—
211	#107	C₂×D₁₉	1	1	yes	characteristic
212	#106	C₂×D₁₉	1	1	yes	characteristic
213	C₄⋊C₈	—	19	1	no	—
214	C₄⋊C₈	—	19	1	no	—
215	C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
216	#103	—	19	1	no	—
217	#103	—	19	1	no	—
218	#103	—	19	1	no	—
219	#102	—	19	1	no	—
220	#102	—	19	1	no	—
221	#102	—	19	1	no	—
222	#102	—	19	1	no	—
223	#102	—	19	1	no	—
224	#102	C₂×D₁₉	1	1	yes	characteristic
225	#102	C₂×D₁₉	1	1	yes	characteristic
226	#97	—	19	1	no	—
227	#97	—	19	1	no	—
228	C₄×C₈	—	19	1	no	—
229	C₁₉	#3370	1	1	yes	characteristic, complemented, Sylow
230	C₂×Q₈	—	19	1	no	—
231	C₂×Q₈	—	19	1	no	—
232	C₂×Q₈	—	38	1	no	—
233	C₂×D₄	—	19	1	no	—
234	C₂×D₄	—	19	1	no	—
235	C₂×D₄	—	38	1	no	—
236	C₂×D₄	—	38	1	no	—
237	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	38	1	no	—
240	C₂²×C₄	—	38	1	no	—
241	C₂²×C₄	—	38	1	no	—
242	C₂²×C₄	—	38	1	no	—
243	SD₁₆	—	38	2	no	—
244	D₈	—	38	2	no	—
245	D₈	—	38	2	no	—
246	C₂×C₈	—	19	1	no	—
247	C₂×C₈	—	19	1	no	—
248	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
249	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	19	1	no	—
253	C₄⋊C₄	—	19	1	no	—
254	C₄⋊C₄	—	19	1	no	—
255	C₄⋊C₄	—	19	1	no	—
256	C₄⋊C₄	—	38	1	no	—
257	C₄⋊C₄	—	38	1	no	—
258	C₄⋊C₄	—	38	1	no	—
259	C₄⋊C₄	—	38	1	no	—
260	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
261	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
262	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	38	1	no	—
265	C₂²⋊C₄	—	38	1	no	—
266	C₂²⋊C₄	—	38	1	no	—
267	C₂²⋊C₄	—	38	1	no	—
268	C₂²⋊C₄	—	38	1	no	—
269	C₂²⋊C₄	—	38	1	no	—
270	C₂²⋊C₄	—	38	1	no	—
271	C₂²⋊C₄	—	38	1	no	—
272	C₄²	—	19	1	no	—
273	C₄²	—	19	1	no	—
274	C₄²	—	38	1	no	—
275	C₄²	C₂²×D₁₉	1	1	yes	characteristic
276	C₂³	—	2	1	no	—
277	C₂³	—	38	1	no	—
278	C₂³	—	38	1	no	—
279	Q₈	—	38	1	no	—
280	Q₈	—	38	1	no	—
281	Q₈	—	76	1	no	—
282	Q₈	—	152	1	no	—
283	D₄	—	2	1	no	—
284	D₄	—	2	1	no	—
285	D₄	—	19	2	no	—
286	D₄	—	38	1	no	—
287	D₄	—	38	1	no	—
288	D₄	—	38	1	no	—
289	D₄	—	76	2	no	—
290	D₄	—	152	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	8	1	no	—
295	C₂×C₄	—	19	1	no	—
296	C₂×C₄	—	19	1	no	—
297	C₂×C₄	—	19	1	no	—
298	C₂×C₄	—	19	1	no	—
299	C₂×C₄	—	38	1	no	—
300	C₂×C₄	—	38	1	no	—
301	C₂×C₄	—	76	2	no	—
302	C₂×C₄	—	152	1	no	—
303	C₂×C₄	—	152	1	no	—
304	C₂×C₄	—	152	1	no	—
305	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
306	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	38	1	no	—
311	C₈	—	38	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	38	2	no	—
316	C₂²	—	76	1	no	—
317	C₂²	—	76	1	no	—
318	C₂²	—	76	1	no	—
319	C₂²	—	76	1	no	—
320	C₂²	—	76	1	no	—
321	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	38	1	no	—
328	C₄	—	38	1	no	—
329	C₄	—	38	1	no	—
330	C₄	—	38	1	no	—
331	C₄	—	76	1	no	—
332	C₄	—	76	1	no	—
333	C₄	#10597433	1	1	yes	characteristic
334	C₄	#41×D₁₉	1	1	yes	characteristic
335	C₂	—	8	1	no	—
336	C₂	—	76	2	no	—
337	C₂	—	152	1	no	—
338	C₂	#11927204	1	1	yes	characteristic
339	C₂	#11927341	1	1	yes	characteristic
340	C₂	#11927348	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×#478	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	#11926738	C₂
hypercenter	#478	D₁₉
hypocenter	C₁₉	#3370
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #11927277	C₂	semidirect
2. #11927250	C₂	semidirect
3. #11926907	C₂	semidirect
4. #11926902	C₂	semidirect
5. #11926901	C₂	semidirect
6. #11926854	C₂	semidirect
7. #11926823	C₂	semidirect
8. #11926760	C₂	semidirect
9. #11926758	C₂	semidirect
11. #11926671	C₂	semidirect
12. #11926655	C₂	semidirect
13. #11926643	C₂	semidirect
14. C₄×#64525	C₂	semidirect
15. C₁₉×#478	C₂	semidirect
171. #478	D₁₉	semidirect
229. C₁₉	#3370	semidirect

Maximal subgroups

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

subgroup	index
#11927277	2
#11927250	2
#11926907	2
#11926902	2
#11926901	2
#11926854	2
#11926823	2
#11926760	2
#11926758	2
#11926738	2
#11926671	2
#11926655	2
#11926643	2
C₄×#64525	2
C₁₉×#478	2
#3370	19