Subgroups — #14567998

Order 2432.

632 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10597315	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#10597331	C₂	1	2	yes	maximal, complemented
3	C₂×#10597376	C₂	1	1	yes	maximal, characteristic
4	#11926460	C₂	1	2	yes	maximal, complemented
5	#11926457	C₂	1	2	yes	maximal, complemented
6	#11926446	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926446	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926436	C₂	1	2	yes	maximal, complemented
9	#11926404	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₄⋊Dic₁₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂²⋊C₄×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic
13	C₂²×C₂²⋊D₁₉	C₂²	1	2	yes	—
14	C₂²×D₇₆	C₂²	1	1	yes	characteristic
15	C₂²×Dic₃₈	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊Dic₁₉	C₂²	1	2	yes	—
21	#10597376	—	2	4	no	—
22	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
23	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
24	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
25	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
26	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
27	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
29	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
31	C₂×C₄×Dic₁₉	C₂²	1	2	yes	—
32	#10597331	—	2	8	no	—
33	#10597315	—	2	4	no	—
34	C₁₉×#95	C₂²	1	1	yes	characteristic
35	C₁₉×#95	C₂²	1	1	yes	characteristic
36	C₁₉⋊#95	C₂²	1	2	yes	—
37	C₁₉⋊#95	C₂²	1	2	yes	—
38	C₂⁴×C₁₉	C₂³	1	1	yes	characteristic
39	C₂³×D₁₉	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₉	—	2	2	no	—
41	C₂²×C₄×C₁₉	—	2	2	no	—
42	C₂²×C₄×C₁₉	—	2	2	no	—
43	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
49	C₂×C₂²⋊D₁₉	—	2	4	no	—
50	C₂×C₂²⋊D₁₉	—	2	4	no	—
51	C₂×C₂²⋊D₁₉	—	4	2	no	—
52	C₂²×Dic₁₉	C₂³	1	2	yes	—
53	C₂²×Dic₁₉	C₂³	1	2	yes	—
54	C₂²×Dic₁₉	C₂³	1	2	yes	—
55	C₂×D₇₆	—	2	2	no	—
56	C₂×D₇₆	—	2	2	no	—
57	C₂×D₇₆	—	2	2	no	—
58	C₂×Dic₃₈	—	2	4	no	—
59	C₂×Dic₃₈	—	2	4	no	—
60	C₂×Dic₃₈	—	4	2	no	—
61	C₂²⋊C₄×C₁₉	—	2	2	no	—
62	C₂²⋊C₄×C₁₉	D₄	1	4	yes	complemented
63	C₄²×C₁₉	—	2	2	no	—
64	C₄²×C₁₉	—	2	2	no	—
65	C₂²⋊Dic₁₉	—	2	4	no	—
66	(C₂×D₁₉)⋊C₄	—	2	4	no	—
67	(C₂×D₁₉)⋊C₄	—	2	4	no	—
68	(C₂×D₁₉)⋊C₄	—	2	4	no	—
69	(C₂×D₁₉)⋊C₄	—	2	4	no	—
70	(C₂×D₁₉)⋊C₄	—	4	2	no	—
71	C₄⋊Dic₁₉	—	2	2	no	—
72	C₄⋊Dic₁₉	—	2	2	no	—
73	C₄⋊Dic₁₉	—	2	4	no	—
74	C₄⋊Dic₁₉	D₄	1	4	yes	—
75	C₄×Dic₁₉	—	2	4	no	—
76	C₂³×C₁₉	—	2	1	no	—
77	C₂³×C₁₉	—	2	1	no	—
78	C₂³×C₁₉	—	2	1	no	—
79	C₂³×C₁₉	—	2	1	no	—
80	C₂³×C₁₉	—	2	1	no	—
81	C₂³×C₁₉	—	2	1	no	—
82	C₂³×C₁₉	C₂×D₄	1	2	yes	—
83	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
84	C₂²×D₁₉	—	2	2	no	—
85	C₂²×D₁₉	—	2	2	no	—
86	C₂²×D₁₉	—	2	2	no	—
87	C₂²×D₁₉	—	2	2	no	—
88	C₂²×D₁₉	—	2	2	no	—
89	C₂²×D₁₉	—	2	2	no	—
90	C₂²×D₁₉	—	2	2	no	—
91	C₂×C₄×C₁₉	—	2	1	no	—
92	C₂×C₄×C₁₉	—	2	1	no	—
93	C₂×C₄×C₁₉	—	2	1	no	—
94	C₂×C₄×C₁₉	—	2	1	no	—
95	C₂×C₄×C₁₉	—	2	1	no	—
96	C₂×C₄×C₁₉	—	2	1	no	—
97	C₂×C₄×C₁₉	—	2	1	no	—
98	C₂×C₄×C₁₉	—	2	1	no	—
99	C₂×C₄×C₁₉	—	2	1	no	—
100	C₂×C₄×C₁₉	—	2	1	no	—
101	C₂×C₄×C₁₉	—	2	1	no	—
102	C₂×C₄×C₁₉	—	2	1	no	—
103	C₂×C₄×C₁₉	—	4	4	no	—
104	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
107	C₂×C₄×C₁₉	#41	1	2	yes	—
108	C₂×C₄×C₁₉	#41	1	2	yes	—
109	C₂×C₄×C₁₉	#41	1	2	yes	—
110	C₂²⋊D₁₉	—	4	8	no	—
111	C₂×Dic₁₉	—	2	2	no	—
112	C₂×Dic₁₉	—	2	2	no	—
113	C₂×Dic₁₉	—	2	2	no	—
114	C₂×Dic₁₉	—	2	2	no	—
115	C₂×Dic₁₉	—	2	2	no	—
116	C₂×Dic₁₉	—	2	2	no	—
117	C₂×Dic₁₉	—	2	2	no	—
118	C₂×Dic₁₉	—	2	2	no	—
119	C₂×Dic₁₉	C₂×D₄	1	4	yes	—
120	D₇₆	—	4	4	no	—
121	Dic₃₈	—	4	8	no	—
122	#2491	—	19	1	no	maximal, Sylow
123	C₂²×C₁₉	—	2	2	no	—
124	C₂²×C₁₉	—	2	2	no	—
125	C₂²×C₁₉	—	2	2	no	—
126	C₂²×C₁₉	—	4	1	no	—
127	C₂²×C₁₉	—	4	1	no	—
128	C₂²×C₁₉	—	4	1	no	—
129	C₂²×C₁₉	—	4	1	no	—
130	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
131	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
132	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
133	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
134	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
135	C₂²×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
136	C₂²×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
137	C₂×D₁₉	—	4	2	no	—
138	C₂×D₁₉	—	4	2	no	—
139	C₂×D₁₉	—	4	2	no	—
140	C₂×D₁₉	—	4	2	no	—
141	C₂×D₁₉	—	4	2	no	—
142	C₂×D₁₉	—	4	2	no	—
143	C₂×D₁₉	—	4	2	no	—
144	C₄×C₁₉	—	2	2	no	—
145	C₄×C₁₉	—	2	2	no	—
146	C₄×C₁₉	—	2	2	no	—
147	C₄×C₁₉	—	4	1	no	—
148	C₄×C₁₉	—	4	1	no	—
149	C₄×C₁₉	#124	1	4	yes	complemented
150	Dic₁₉	—	2	4	no	—
151	Dic₁₉	—	4	2	no	—
152	Dic₁₉	—	4	2	no	—
153	C₂×C₄⋊Q₈	—	19	1	no	—
154	C₂×#124	—	19	2	no	—
155	C₂×#121	—	19	1	no	—
156	#394	—	19	2	no	—
157	#393	—	19	2	no	—
158	#390	—	19	1	no	—
159	#390	—	19	1	no	—
160	#388	—	19	2	no	—
161	#380	—	19	1	no	—
162	C₄×C₄⋊C₄	—	19	1	no	—
163	C₄×C₂²⋊C₄	D₁₉	1	1	yes	characteristic, complemented
164	C₂×C₁₉	—	4	2	no	—
165	C₂×C₁₉	C₂×#124	1	1	yes	characteristic
166	C₂×C₁₉	#532	1	1	yes	characteristic
167	C₂×C₁₉	#542	1	1	yes	characteristic
168	C₂×C₁₉	D₄²	1	1	yes	characteristic
169	C₂×C₁₉	#547	1	1	yes	characteristic
170	C₂×C₁₉	#548	1	1	yes	characteristic
171	C₂×C₁₉	#555	1	1	yes	characteristic
172	D₁₉	—	8	2	no	—
173	C₂²×Q₈	—	19	2	no	—
174	C₂²×D₄	—	19	2	no	—
175	C₂²×D₄	—	19	1	no	—
176	C₂³×C₄	C₂×D₁₉	1	1	yes	characteristic
177	C₄⋊Q₈	—	38	4	no	—
178	#124	—	38	8	no	—
179	#121	—	38	4	no	—
180	C₂×C₄⋊C₄	—	19	1	no	—
181	C₂×C₄⋊C₄	—	19	1	no	—
182	C₂×C₄⋊C₄	—	19	2	no	—
183	C₂×C₄⋊C₄	—	19	1	no	—
184	C₂×C₂²⋊C₄	—	19	2	no	—
185	C₂×C₂²⋊C₄	—	19	2	no	—
186	C₂×C₂²⋊C₄	—	19	2	no	—
187	C₂×C₂²⋊C₄	—	19	2	no	—
188	C₂×C₂²⋊C₄	—	19	2	no	—
189	C₂×C₂²⋊C₄	—	19	2	no	—
190	C₂×C₂²⋊C₄	C₂×D₁₉	1	1	yes	characteristic
191	C₂×C₂²⋊C₄	C₂×D₁₉	1	1	yes	characteristic
192	C₂×C₄²	—	19	2	no	—
193	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
194	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
195	#95	—	19	2	no	—
196	#95	—	19	2	no	—
197	#95	C₂×D₁₉	1	1	yes	characteristic
198	#95	C₂×D₁₉	1	1	yes	characteristic
199	C₁₉	#2491	1	1	yes	characteristic, complemented, Sylow
200	C₂⁴	—	19	2	no	—
201	C₂⁴	C₂²×D₁₉	1	1	yes	characteristic
202	C₂×Q₈	—	38	4	no	—
203	C₂×Q₈	—	38	4	no	—
204	C₂×Q₈	—	76	2	no	—
205	C₂×D₄	—	38	4	no	—
206	C₂×D₄	—	38	4	no	—
207	C₂×D₄	—	38	2	no	—
208	C₂×D₄	—	38	2	no	—
209	C₂×D₄	—	38	2	no	—
210	C₂×D₄	—	76	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	19	2	no	—
215	C₂²×C₄	—	19	2	no	—
216	C₂²×C₄	—	19	2	no	—
217	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
218	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
219	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
220	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
221	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
222	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
223	C₄⋊C₄	—	19	4	no	—
224	C₄⋊C₄	—	38	4	no	—
225	C₄⋊C₄	—	38	2	no	—
226	C₄⋊C₄	—	38	2	no	—
227	C₂²⋊C₄	—	2	2	no	—
228	C₂²⋊C₄	—	38	4	no	—
229	C₂²⋊C₄	—	38	4	no	—
230	C₂²⋊C₄	—	38	4	no	—
231	C₂²⋊C₄	—	38	4	no	—
232	C₂²⋊C₄	—	38	4	no	—
233	C₂²⋊C₄	—	76	2	no	—
234	C₂²⋊C₄	D₇₆	1	4	yes	complemented
235	C₄²	—	2	2	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	38	4	no	—
238	C₂³	—	2	1	no	—
239	C₂³	—	2	1	no	—
240	C₂³	—	2	1	no	—
241	C₂³	—	2	1	no	—
242	C₂³	—	2	1	no	—
243	C₂³	—	2	1	no	—
244	C₂³	—	38	2	no	—
245	C₂³	—	38	2	no	—
246	C₂³	—	38	2	no	—
247	C₂³	—	38	2	no	—
248	C₂³	—	38	2	no	—
249	C₂³	—	38	2	no	—
250	C₂³	—	38	2	no	—
251	C₂³	C₂×D₇₆	1	2	yes	—
252	C₂³	C₂³×D₁₉	1	1	yes	characteristic
253	Q₈	—	76	8	no	—
254	D₄	—	76	8	no	—
255	D₄	—	76	4	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	19	4	no	—
270	C₂×C₄	—	38	2	no	—
271	C₂×C₄	—	38	2	no	—
272	C₂×C₄	—	38	2	no	—
273	C₂×C₄	—	38	2	no	—
274	C₂×C₄	—	38	2	no	—
275	C₂×C₄	—	38	2	no	—
276	C₂×C₄	—	38	2	no	—
277	C₂×C₄	—	38	2	no	—
278	C₂×C₄	C₂×D₇₆	1	2	yes	—
279	C₂×C₄	C₂×D₇₆	1	2	yes	—
280	C₂×C₄	#64541	1	2	yes	—
281	C₂×C₄	#64541	1	2	yes	—
282	C₂×C₄	D₄×D₁₉	1	2	yes	—
283	C₂×C₄	#64543	1	2	yes	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	76	2	no	—
292	C₂²	—	76	2	no	—
293	C₂²	—	76	2	no	—
294	C₂²	—	76	2	no	—
295	C₂²	—	76	2	no	—
296	C₂²	—	76	2	no	—
297	C₂²	—	76	2	no	—
298	C₂²	C₂²×D₇₆	1	1	yes	characteristic
299	C₂²	C₂×#64541	1	1	yes	characteristic
300	C₂²	C₂×#64541	1	1	yes	characteristic
301	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
302	C₂²	C₂×#64543	1	1	yes	characteristic
303	C₂²	#10597435	1	1	yes	characteristic
304	C₂²	#10597436	1	1	yes	characteristic
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	38	4	no	—
311	C₄	—	76	2	no	—
312	C₄	—	76	2	no	—
313	C₄	#10597331	1	4	yes	complemented
314	C₂	—	4	2	no	—
315	C₂	—	152	2	no	—
316	C₂	C₂×#10597331	1	1	yes	characteristic
317	C₂	#11927094	1	1	yes	characteristic
318	C₂	#11927165	1	1	yes	characteristic
319	C₂	D₄×D₇₆	1	1	yes	characteristic
320	C₂	#11927198	1	1	yes	characteristic
321	C₂	#11927203	1	1	yes	characteristic
322	C₂	#11927259	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉
commutator, socle, cosocle	C₂³×C₁₉	C₂⁴
Fitting	C₄×C₂²⋊C₄×C₁₉	C₂
prime radical	C₂×#10597376	C₂
hypercenter	C₄×C₂²⋊C₄	D₁₉
hypocenter	C₁₉	#2491
LCS	G ▷ C₂³×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#10597315	C₂	semidirect
2. C₂×#10597331	C₂	semidirect
4. #11926460	C₂	semidirect
5. #11926457	C₂	semidirect
6. #11926446	C₂	semidirect
7. #11926446	C₂	semidirect
8. #11926436	C₂	semidirect
9. #11926404	C₂	semidirect
10. C₄×C₄⋊Dic₁₉	C₂	semidirect
11. C₄×C₂²⋊C₄×C₁₉	C₂	semidirect
62. C₂²⋊C₄×C₁₉	D₄	semidirect
149. C₄×C₁₉	#124	semidirect
163. C₄×C₂²⋊C₄	D₁₉	semidirect
199. C₁₉	#2491	semidirect
234. C₂²⋊C₄	D₇₆	semidirect
313. C₄	#10597331	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10597315	2
C₂×#10597331	2
C₂×#10597376	2
#11926460	2
#11926457	2
#11926446	2
#11926436	2
#11926404	2
C₄×C₄⋊Dic₁₉	2
C₄×C₂²⋊C₄×C₁₉	2
#2491	19