Subgroups — #14567932

Order 2432.

540 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10597372	C₂	1	1	yes	maximal, characteristic, complemented
2	#11926472	C₂	1	2	yes	maximal, complemented
3	#11926458	C₂	1	2	yes	maximal
4	#11926442	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926442	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926432	C₂	1	2	yes	maximal, complemented
7	#11926425	C₂	1	2	yes	maximal, complemented
8	#11926404	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926404	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926404	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×D₄×C₁₉	C₂	1	1	yes	maximal, characteristic
12	C₂²×D₄×C₁₉	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic
15	C₄×D₄×C₁₉	—	2	4	no	—
16	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊Dic₁₉	C₂²	1	2	yes	—
21	C₂×C₂²⋊Dic₁₉	C₂²	1	2	yes	—
22	C₂×C₂²⋊Dic₁₉	C₂²	1	2	yes	—
23	C₂×C₂²⋊Dic₁₉	C₂²	1	2	yes	—
24	#10597372	—	2	4	no	—
25	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
29	C₂×Dic₁₉⋊C₄	C₂²	1	2	yes	—
30	C₂×Dic₁₉⋊C₄	C₂²	1	2	yes	—
31	C₂×C₄×Dic₁₉	C₂²	1	2	yes	—
32	C₁₉⋊#95	C₂²	1	2	yes	—
33	C₁₉⋊#95	C₂²	1	2	yes	—
34	C₁₉⋊#95	C₂²	1	2	yes	—
35	C₁₉⋊#95	C₂²	1	2	yes	—
36	C₁₉⋊#95	C₂²	1	2	yes	—
37	C₂⁴×C₁₉	C₂³	1	1	yes	characteristic
38	C₂⁴×C₁₉	C₂³	1	1	yes	characteristic
39	C₂×D₄×C₁₉	—	2	2	no	—
40	C₂×D₄×C₁₉	—	2	2	no	—
41	C₂×D₄×C₁₉	—	2	2	no	—
42	C₂²×C₄×C₁₉	—	2	2	no	—
43	C₂²×C₄×C₁₉	—	2	2	no	—
44	C₂²×C₄×C₁₉	—	2	2	no	—
45	C₂²×C₄×C₁₉	—	2	2	no	—
46	C₂²×C₄×C₁₉	—	2	2	no	—
47	C₂²×C₄×C₁₉	D₄	1	4	yes	—
48	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
53	C₂²×Dic₁₉	C₂³	1	2	yes	—
54	C₂²×Dic₁₉	C₂³	1	2	yes	—
55	C₂²×Dic₁₉	C₂³	1	2	yes	—
56	C₂²×Dic₁₉	C₂³	1	2	yes	—
57	C₄⋊C₄×C₁₉	—	2	2	no	—
58	C₂²⋊C₄×C₁₉	—	2	2	no	—
59	C₂²⋊C₄×C₁₉	—	2	2	no	—
60	C₄²×C₁₉	—	2	2	no	—
61	C₂²⋊Dic₁₉	—	2	4	no	—
62	C₂²⋊Dic₁₉	—	2	4	no	—
63	C₂²⋊Dic₁₉	—	4	2	no	—
64	C₂²⋊Dic₁₉	—	4	2	no	—
65	C₄⋊Dic₁₉	—	2	2	no	—
66	C₄⋊Dic₁₉	—	2	2	no	—
67	C₄⋊Dic₁₉	—	2	2	no	—
68	C₄⋊Dic₁₉	—	2	2	no	—
69	Dic₁₉⋊C₄	—	2	4	no	—
70	Dic₁₉⋊C₄	—	4	2	no	—
71	C₄×Dic₁₉	—	2	4	no	—
72	C₂³×C₁₉	—	2	1	no	—
73	C₂³×C₁₉	—	2	1	no	—
74	C₂³×C₁₉	—	2	1	no	—
75	C₂³×C₁₉	—	2	1	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	—	2	1	no	—
83	C₂³×C₁₉	—	2	1	no	—
84	C₂³×C₁₉	C₂×D₄	1	2	yes	—
85	C₂³×C₁₉	#41	1	2	yes	—
86	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
87	D₄×C₁₉	—	2	8	no	—
88	C₂×C₄×C₁₉	—	2	1	no	—
89	C₂×C₄×C₁₉	—	2	1	no	—
90	C₂×C₄×C₁₉	—	2	1	no	—
91	C₂×C₄×C₁₉	—	2	1	no	—
92	C₂×C₄×C₁₉	—	2	8	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	—	4	4	no	—
100	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
101	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₁₉	#41	1	2	yes	—
103	C₂×C₄×C₁₉	#41	1	2	yes	—
104	C₂×C₄×C₁₉	#41	1	2	yes	—
105	C₂×Dic₁₉	—	2	2	no	—
106	C₂×Dic₁₉	—	2	2	no	—
107	C₂×Dic₁₉	—	2	2	no	—
108	C₂×Dic₁₉	—	2	2	no	—
109	C₂×Dic₁₉	—	2	2	no	—
110	C₂×Dic₁₉	—	2	2	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	#2485	—	19	1	no	maximal, Sylow
118	C₂²×C₁₉	—	2	2	no	—
119	C₂²×C₁₉	—	2	2	no	—
120	C₂²×C₁₉	—	2	2	no	—
121	C₂²×C₁₉	—	2	2	no	—
122	C₂²×C₁₉	—	2	2	no	—
123	C₂²×C₁₉	—	2	2	no	—
124	C₂²×C₁₉	—	4	1	no	—
125	C₂²×C₁₉	—	4	1	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	—	4	1	no	—
131	C₂²×C₁₉	—	4	1	no	—
132	C₂²×C₁₉	C₂²×D₄	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	C₂×#41	1	1	yes	characteristic
136	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
137	C₂²×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
138	C₂²×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
139	C₄×C₁₉	—	2	2	no	—
140	C₄×C₁₉	—	2	2	no	—
141	C₄×C₁₉	—	4	1	no	—
142	C₄×C₁₉	—	4	1	no	—
143	C₄×C₁₉	#123	1	4	yes	—
144	Dic₁₉	—	4	2	no	—
145	Dic₁₉	—	4	2	no	—
146	Dic₁₉	—	4	2	no	—
147	Dic₁₉	—	4	2	no	—
148	C₂×C₄×D₄	D₁₉	1	1	yes	characteristic
149	C₂×#117	—	19	1	no	—
150	#397	—	19	2	no	—
151	#394	—	19	2	no	—
152	#389	—	19	1	no	—
153	#389	—	19	1	no	—
154	#388	—	19	2	no	—
155	#387	—	19	2	no	—
156	#380	—	19	1	no	—
157	#380	—	19	1	no	—
158	#380	—	19	1	no	—
159	C₂×C₁₉	—	4	2	no	—
160	C₂×C₁₉	—	4	2	no	—
161	C₂×C₁₉	C₂×#123	1	1	yes	characteristic
162	C₂×C₁₉	#525	1	1	yes	characteristic
163	C₂×C₁₉	#537	1	1	yes	characteristic
164	C₂×C₁₉	#552	1	1	yes	characteristic
165	C₂×C₁₉	#553	1	1	yes	characteristic
166	C₂×C₁₉	#555	1	1	yes	characteristic
167	C₂×C₁₉	#556	1	1	yes	characteristic
168	C₂²×D₄	C₂×D₁₉	1	1	yes	characteristic
169	C₂³×C₄	C₂×D₁₉	1	1	yes	characteristic
170	C₂³×C₄	C₂×D₁₉	1	1	yes	characteristic
171	C₄×D₄	—	2	4	no	—
172	#117	—	38	4	no	—
173	C₂×C₄⋊C₄	—	19	1	no	—
174	C₂×C₄⋊C₄	—	19	1	no	—
175	C₂×C₄⋊C₄	—	19	1	no	—
176	C₂×C₄⋊C₄	—	19	1	no	—
177	C₂×C₄⋊C₄	—	19	2	no	—
178	C₂×C₄⋊C₄	—	19	2	no	—
179	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
180	C₂×C₂²⋊C₄	—	19	2	no	—
181	C₂×C₂²⋊C₄	—	19	2	no	—
182	C₂×C₂²⋊C₄	—	19	2	no	—
183	C₂×C₂²⋊C₄	—	19	2	no	—
184	C₂×C₂²⋊C₄	C₂×D₁₉	1	1	yes	characteristic
185	C₂×C₂²⋊C₄	C₂×D₁₉	1	1	yes	characteristic
186	C₂×C₄²	—	19	2	no	—
187	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
188	#95	—	19	2	no	—
189	#95	—	19	2	no	—
190	#95	—	19	2	no	—
191	#95	—	19	2	no	—
192	#95	—	19	2	no	—
193	C₁₉	#2485	1	1	yes	characteristic, complemented, Sylow
194	C₂⁴	C₂²×D₁₉	1	1	yes	characteristic
195	C₂⁴	C₂²×D₁₉	1	1	yes	characteristic
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	19	2	no	—
205	C₂²×C₄	—	19	2	no	—
206	C₂²×C₄	—	19	2	no	—
207	C₂²×C₄	—	19	2	no	—
208	C₂²×C₄	C₂²⋊D₁₉	1	4	yes	—
209	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
210	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
211	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
212	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
213	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	38	4	no	—
216	C₄⋊C₄	—	38	2	no	—
217	C₄⋊C₄	—	38	2	no	—
218	C₄⋊C₄	—	38	2	no	—
219	C₄⋊C₄	—	38	2	no	—
220	C₄⋊C₄	—	76	2	no	—
221	C₂²⋊C₄	—	2	2	no	—
222	C₂²⋊C₄	—	2	2	no	—
223	C₂²⋊C₄	—	38	4	no	—
224	C₂²⋊C₄	—	38	4	no	—
225	C₂²⋊C₄	—	76	2	no	—
226	C₂²⋊C₄	—	76	2	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	38	4	no	—
229	C₂³	—	2	1	no	—
230	C₂³	—	2	1	no	—
231	C₂³	—	2	1	no	—
232	C₂³	—	2	1	no	—
233	C₂³	—	2	1	no	—
234	C₂³	—	2	1	no	—
235	C₂³	—	2	1	no	—
236	C₂³	—	2	1	no	—
237	C₂³	—	2	1	no	—
238	C₂³	—	2	1	no	—
239	C₂³	—	2	1	no	—
240	C₂³	—	2	1	no	—
241	C₂³	#64541	1	2	yes	—
242	C₂³	C₂×C₂²⋊D₁₉	1	2	yes	—
243	C₂³	C₂³×D₁₉	1	1	yes	characteristic
244	D₄	—	2	8	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	8	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	38	2	no	—
258	C₂×C₄	—	38	2	no	—
259	C₂×C₄	—	38	2	no	—
260	C₂×C₄	—	38	2	no	—
261	C₂×C₄	—	38	2	no	—
262	C₂×C₄	—	38	2	no	—
263	C₂×C₄	—	38	2	no	—
264	C₂×C₄	—	38	2	no	—
265	C₂×C₄	—	38	2	no	—
266	C₂×C₄	—	38	2	no	—
267	C₂×C₄	—	38	2	no	—
268	C₂×C₄	—	38	2	no	—
269	C₂×C₄	#64541	1	2	yes	—
270	C₂×C₄	#64543	1	2	yes	—
271	C₂×C₄	#64543	1	2	yes	—
272	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
273	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	1	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	4	1	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	1	no	—
287	C₂²	—	4	1	no	—
288	C₂²	C₂×#64541	1	1	yes	characteristic
289	C₂²	C₂×#64541	1	1	yes	characteristic
290	C₂²	C₂×#64543	1	1	yes	characteristic
291	C₂²	C₂×#64543	1	1	yes	characteristic
292	C₂²	#10597435	1	1	yes	characteristic
293	C₂²	#10597436	1	1	yes	characteristic
294	C₂²	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	4	1	no	—
298	C₄	—	4	1	no	—
299	C₄	—	76	2	no	—
300	C₄	—	76	2	no	—
301	C₄	—	76	2	no	—
302	C₄	—	76	2	no	—
303	C₄	#10597381	1	4	yes	—
304	C₂	—	4	2	no	—
305	C₂	—	4	2	no	—
306	C₂	C₂×#10597381	1	1	yes	characteristic
307	C₂	#11927055	1	1	yes	characteristic
308	C₂	#11927122	1	1	yes	characteristic
309	C₂	#11927234	1	1	yes	characteristic
310	C₂	#11927240	1	1	yes	characteristic
311	C₂	#11927259	1	1	yes	characteristic
312	C₂	#11927264	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉
commutator, socle, cosocle	C₂³×C₁₉	C₂⁴
Fitting	C₂×C₄×D₄×C₁₉	C₂
prime radical	C₂²×D₄×C₁₉	C₂²
hypercenter	C₂×C₄×D₄	D₁₉
hypocenter	C₁₉	#2485
LCS	G ▷ C₂³×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#10597372	C₂	semidirect
2. #11926472	C₂	semidirect
4. #11926442	C₂	semidirect
5. #11926442	C₂	semidirect
6. #11926432	C₂	semidirect
7. #11926425	C₂	semidirect
8. #11926404	C₂	semidirect
9. #11926404	C₂	semidirect
10. #11926404	C₂	semidirect
193. C₁₉	#2485	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10597372	2
#11926472	2
#11926458	2
#11926442	2
#11926432	2
#11926425	2
#11926404	2
C₂×C₄×D₄×C₁₉	2
#2485	19