Subgroups — #14567086

Order 2432.

648 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10597370	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#10597370	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄⋊Dic₁₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926477	C₂	1	2	yes	maximal, complemented
5	#11926467	C₂	1	2	yes	maximal, complemented
6	#11926450	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926442	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926424	C₂	1	2	yes	maximal
9	#11926404	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926402	C₂	1	2	yes	maximal, 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	C₂³×Dic₁₉	C₂²	1	2	yes	—
16	C₂²×Dic₃₈	C₂²	1	1	yes	characteristic
17	C₄×D₄×C₁₉	—	2	4	no	—
18	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊Dic₁₉	C₂²	1	2	yes	—
23	C₂×C₂²⋊Dic₁₉	C₂²	1	2	yes	—
24	C₂×C₂²⋊Dic₁₉	C₂²	1	2	yes	—
25	C₂×C₄⋊Dic₁₉	—	2	4	no	—
26	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
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	#10597370	—	2	4	no	—
31	#10597370	—	2	4	no	—
32	C₂×Dic₁₉⋊C₄	C₂²	1	2	yes	—
33	C₂×Dic₁₉⋊C₄	C₂²	1	2	yes	—
34	C₂×Dic₁₉⋊C₄	C₂²	1	2	yes	—
35	C₁₉⋊#95	C₂²	1	2	yes	—
36	C₁₉⋊#95	C₂²	1	2	yes	—
37	C₁₉⋊#95	C₂²	1	2	yes	—
38	C₁₉⋊#95	C₂²	1	2	yes	—
39	C₂⁴×C₁₉	C₂³	1	1	yes	characteristic
40	C₂⁴×C₁₉	C₂³	1	1	yes	characteristic
41	C₂×D₄×C₁₉	—	2	2	no	—
42	C₂×D₄×C₁₉	—	2	2	no	—
43	C₂×D₄×C₁₉	Q₈	1	4	yes	complemented
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	—	2	2	no	—
48	C₂²×C₄×C₁₉	—	2	2	no	—
49	C₂²×C₄×C₁₉	D₄	1	4	yes	—
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×C₄×C₁₉	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
55	C₂²×Dic₁₉	—	2	4	no	—
56	C₂²×Dic₁₉	—	2	4	no	—
57	C₂²×Dic₁₉	—	4	2	no	—
58	C₂²×Dic₁₉	C₂³	1	2	yes	—
59	C₂²×Dic₁₉	C₂³	1	2	yes	—
60	C₂²×Dic₁₉	C₂³	1	2	yes	—
61	C₂²×Dic₁₉	C₂³	1	2	yes	—
62	C₂×Dic₃₈	—	2	2	no	—
63	C₂×Dic₃₈	—	2	2	no	—
64	C₂×Dic₃₈	—	2	2	no	—
65	C₄⋊C₄×C₁₉	—	2	2	no	—
66	C₂²⋊C₄×C₁₉	—	2	2	no	—
67	C₂²⋊C₄×C₁₉	—	2	2	no	—
68	C₄²×C₁₉	—	2	2	no	—
69	C₂²⋊Dic₁₉	—	2	4	no	—
70	C₂²⋊Dic₁₉	—	2	4	no	—
71	C₂²⋊Dic₁₉	—	4	2	no	—
72	C₄⋊Dic₁₉	—	2	2	no	—
73	C₄⋊Dic₁₉	—	2	2	no	—
74	C₄⋊Dic₁₉	—	2	4	no	—
75	C₄⋊Dic₁₉	D₄	1	4	yes	complemented
76	Dic₁₉⋊C₄	—	2	4	no	—
77	Dic₁₉⋊C₄	—	2	4	no	—
78	Dic₁₉⋊C₄	—	4	2	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	—	2	1	no	—
85	C₂³×C₁₉	—	2	1	no	—
86	C₂³×C₁₉	—	2	1	no	—
87	C₂³×C₁₉	—	2	1	no	—
88	C₂³×C₁₉	—	2	1	no	—
89	C₂³×C₁₉	C₂×D₄	1	2	yes	—
90	C₂³×C₁₉	C₂×Q₈	1	2	yes	—
91	C₂³×C₁₉	C₂×Q₈	1	2	yes	—
92	C₂³×C₁₉	#41	1	2	yes	—
93	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
94	D₄×C₁₉	—	2	8	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	8	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	—	2	1	no	—
104	C₂×C₄×C₁₉	—	2	1	no	—
105	C₂×C₄×C₁₉	—	2	1	no	—
106	C₂×C₄×C₁₉	—	4	4	no	—
107	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
108	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₁₉	C₂×Q₈	1	2	yes	—
111	C₂×C₄×C₁₉	#41	1	2	yes	—
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	—	2	2	no	—
120	C₂×Dic₁₉	—	2	2	no	—
121	C₂×Dic₁₉	—	2	2	no	—
122	C₂×Dic₁₉	—	2	2	no	—
123	C₂×Dic₁₉	—	4	8	no	—
124	C₂×Dic₁₉	C₂×D₄	1	4	yes	—
125	Dic₃₈	—	4	4	no	—
126	#2417	—	19	1	no	maximal, Sylow
127	C₂²×C₁₉	—	2	2	no	—
128	C₂²×C₁₉	—	2	2	no	—
129	C₂²×C₁₉	—	2	2	no	—
130	C₂²×C₁₉	—	2	2	no	—
131	C₂²×C₁₉	—	2	2	no	—
132	C₂²×C₁₉	—	2	2	no	—
133	C₂²×C₁₉	—	2	2	no	—
134	C₂²×C₁₉	—	2	2	no	—
135	C₂²×C₁₉	—	2	2	no	—
136	C₂²×C₁₉	—	4	1	no	—
137	C₂²×C₁₉	—	4	1	no	—
138	C₂²×C₁₉	—	4	1	no	—
139	C₂²×C₁₉	—	4	1	no	—
140	C₂²×C₁₉	C₂²⋊Q₈	1	4	yes	complemented
141	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
142	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
143	C₂²×C₁₉	C₂²×Q₈	1	1	yes	characteristic
144	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
145	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
146	C₂²×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂²×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
148	C₄×C₁₉	—	2	2	no	—
149	C₄×C₁₉	—	2	2	no	—
150	C₄×C₁₉	—	4	1	no	—
151	C₄×C₁₉	—	4	1	no	—
152	C₄×C₁₉	#121	1	4	yes	—
153	Dic₁₉	—	2	4	no	—
154	Dic₁₉	—	4	2	no	—
155	Dic₁₉	—	4	2	no	—
156	Dic₁₉	—	4	2	no	—
157	C₂×C₂²⋊Q₈	—	19	1	no	—
158	C₂×C₂²⋊Q₈	—	19	1	no	—
159	C₂×C₄×D₄	D₁₉	1	1	yes	characteristic
160	C₂²×C₄⋊C₄	—	19	1	no	—
161	#398	—	19	2	no	—
162	#396	—	19	2	no	—
163	#391	—	19	1	no	—
164	#389	—	19	1	no	—
165	#386	—	19	2	no	—
166	#380	—	19	1	no	—
167	#380	—	19	2	no	—
168	C₂×C₁₉	—	2	4	no	—
169	C₂×C₁₉	—	4	2	no	—
170	C₂×C₁₉	C₂×#121	1	1	yes	characteristic
171	C₂×C₁₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
172	C₂×C₁₉	#537	1	1	yes	characteristic
173	C₂×C₁₉	#546	1	1	yes	characteristic
174	C₂×C₁₉	D₄×Q₈	1	1	yes	characteristic
175	C₂×C₁₉	#552	1	1	yes	characteristic
176	C₂×C₁₉	#554	1	1	yes	characteristic
177	C₂²×Q₈	—	19	1	no	—
178	C₂²×D₄	C₂×D₁₉	1	1	yes	characteristic
179	C₂³×C₄	—	19	2	no	—
180	C₂³×C₄	C₂×D₁₉	1	1	yes	characteristic
181	C₂³×C₄	C₂×D₁₉	1	1	yes	characteristic
182	C₂²⋊Q₈	—	38	4	no	—
183	C₂²⋊Q₈	—	38	4	no	—
184	C₄×D₄	—	2	4	no	—
185	C₂×C₄⋊C₄	—	19	1	no	—
186	C₂×C₄⋊C₄	—	19	1	no	—
187	C₂×C₄⋊C₄	—	19	2	no	—
188	C₂×C₄⋊C₄	—	19	1	no	—
189	C₂×C₄⋊C₄	—	19	2	no	—
190	C₂×C₄⋊C₄	—	19	2	no	—
191	C₂×C₄⋊C₄	—	19	2	no	—
192	C₂×C₄⋊C₄	—	38	4	no	—
193	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
194	C₂×C₂²⋊C₄	—	19	2	no	—
195	C₂×C₂²⋊C₄	—	19	2	no	—
196	C₂×C₂²⋊C₄	—	19	2	no	—
197	C₂×C₂²⋊C₄	C₂×D₁₉	1	1	yes	characteristic
198	C₂×C₂²⋊C₄	C₂×D₁₉	1	1	yes	characteristic
199	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
200	#95	—	19	2	no	—
201	#95	—	19	2	no	—
202	#95	—	19	2	no	—
203	#95	—	19	2	no	—
204	C₁₉	#2417	1	1	yes	characteristic, complemented, Sylow
205	C₂⁴	C₂²×D₁₉	1	1	yes	characteristic
206	C₂⁴	C₂²×D₁₉	1	1	yes	characteristic
207	C₂×Q₈	—	38	2	no	—
208	C₂×Q₈	—	38	2	no	—
209	C₂×Q₈	—	38	2	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	Dic₃₈	1	4	yes	complemented
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	19	2	no	—
219	C₂²×C₄	—	19	2	no	—
220	C₂²×C₄	—	19	2	no	—
221	C₂²×C₄	—	19	2	no	—
222	C₂²×C₄	—	38	4	no	—
223	C₂²×C₄	—	38	4	no	—
224	C₂²×C₄	—	76	2	no	—
225	C₂²×C₄	C₂²⋊D₁₉	1	4	yes	—
226	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
227	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
228	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
229	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
230	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	19	4	no	—
233	C₄⋊C₄	—	38	4	no	—
234	C₄⋊C₄	—	38	4	no	—
235	C₄⋊C₄	—	38	2	no	—
236	C₄⋊C₄	—	38	4	no	—
237	C₄⋊C₄	—	38	2	no	—
238	C₄⋊C₄	—	76	2	no	—
239	C₂²⋊C₄	—	2	2	no	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₂²⋊C₄	—	38	4	no	—
242	C₂²⋊C₄	—	38	4	no	—
243	C₂²⋊C₄	—	76	2	no	—
244	C₄²	—	2	2	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	1	no	—
250	C₂³	—	2	1	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	C₂×Dic₃₈	1	2	yes	—
256	C₂³	C₂×Dic₃₈	1	2	yes	—
257	C₂³	#64541	1	2	yes	—
258	C₂³	C₂×C₂²⋊D₁₉	1	2	yes	—
259	C₂³	C₂³×D₁₉	1	1	yes	characteristic
260	Q₈	—	76	4	no	—
261	D₄	—	2	8	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	8	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	19	4	no	—
275	C₂×C₄	—	38	2	no	—
276	C₂×C₄	—	38	2	no	—
277	C₂×C₄	—	38	2	no	—
278	C₂×C₄	—	38	2	no	—
279	C₂×C₄	—	38	2	no	—
280	C₂×C₄	—	38	2	no	—
281	C₂×C₄	—	38	2	no	—
282	C₂×C₄	—	38	2	no	—
283	C₂×C₄	—	38	2	no	—
284	C₂×C₄	—	38	2	no	—
285	C₂×C₄	—	38	2	no	—
286	C₂×C₄	—	76	8	no	—
287	C₂×C₄	C₂×Dic₃₈	1	2	yes	—
288	C₂×C₄	D₄×D₁₉	1	2	yes	—
289	C₂×C₄	#64543	1	2	yes	—
290	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
291	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	1	no	—
305	C₂²	#10597370	1	4	yes	complemented
306	C₂²	C₂²×Dic₃₈	1	1	yes	characteristic
307	C₂²	C₂×#64541	1	1	yes	characteristic
308	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
309	C₂²	C₂×#64543	1	1	yes	characteristic
310	C₂²	#10597435	1	1	yes	characteristic
311	C₂²	#10597436	1	1	yes	characteristic
312	C₂²	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	38	4	no	—
318	C₄	—	76	2	no	—
319	C₄	—	76	2	no	—
320	C₄	—	76	2	no	—
321	C₄	#10597385	1	4	yes	—
322	C₂	—	2	4	no	—
323	C₂	—	4	2	no	—
324	C₂	C₂×#10597385	1	1	yes	characteristic
325	C₂	C₂×#10597370	1	1	yes	characteristic
326	C₂	#11927122	1	1	yes	characteristic
327	C₂	#11927190	1	1	yes	characteristic
328	C₂	D₄×Dic₃₈	1	1	yes	characteristic
329	C₂	#11927234	1	1	yes	characteristic
330	C₂	#11927251	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	#2417
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×#10597370	C₂	semidirect
2. C₂×#10597370	C₂	semidirect
3. C₂²×C₄⋊Dic₁₉	C₂	semidirect
4. #11926477	C₂	semidirect
5. #11926467	C₂	semidirect
6. #11926450	C₂	semidirect
7. #11926442	C₂	semidirect
9. #11926404	C₂	semidirect
10. #11926402	C₂	semidirect
43. C₂×D₄×C₁₉	Q₈	semidirect
75. C₄⋊Dic₁₉	D₄	semidirect
140. C₂²×C₁₉	C₂²⋊Q₈	semidirect
204. C₁₉	#2417	semidirect
212. C₂×D₄	Dic₃₈	semidirect
305. C₂²	#10597370	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10597370	2
C₂²×C₄⋊Dic₁₉	2
#11926477	2
#11926467	2
#11926450	2
#11926442	2
#11926424	2
#11926404	2
#11926402	2
C₂×C₄×D₄×C₁₉	2
#2417	19