Subgroups — #18091218

Order 4864.

694 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926593	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926571	C₂	1	2	yes	maximal
3	#14563345	C₂	1	2	yes	maximal
4	#14563333	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₉×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	2	yes	—
7	C₂³×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×#10597372	C₂²	1	1	yes	characteristic
9	#11926593	—	2	4	no	—
10	C₂×#10597276	—	2	2	no	—
11	#11926571	—	2	4	no	—
12	#11926571	—	2	4	no	—
13	C₂×C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
14	#11926304	—	2	4	no	—
15	#11926298	—	2	2	no	—
16	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₉×#117	C₂²	1	2	yes	—
18	C₁₉×H(3,ℤ/4)	C₄	1	4	yes	complemented
19	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₁₉	—	2	2	no	—
21	C₂×M₁₆×C₁₉	—	2	2	no	—
22	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
23	C₂×M₁₆×C₁₉	Q₈	1	2	yes	—
24	C₂²×C₈×C₁₉	D₄	1	2	yes	—
25	C₁₉×#117	—	2	2	no	—
26	C₁₉×#117	—	2	2	no	—
27	C₁₉×#117	C₂×C₄	1	4	yes	—
28	C₁₉×#117	Q₈	1	4	yes	—
29	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
30	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
31	C₂×C₄²×C₁₉	D₄	1	4	yes	—
32	C₂×C₂²⋊Dic₁₉	—	2	1	no	—
33	#10597372	—	2	2	no	—
34	#10597372	—	2	2	no	—
35	C₂×C₄⋊Dic₁₉	Q₈	1	2	yes	—
36	C₂×C₄×Dic₁₉	—	2	1	no	—
37	C₂×#64520	—	2	2	no	—
38	C₂×#64520	—	2	2	no	—
39	C₂×#64520	—	2	2	no	—
40	C₂×#64520	—	2	2	no	—
41	C₂×#64520	—	2	2	no	—
42	C₂×#64520	D₄	1	4	yes	—
43	C₂²×C₁₉⋊C₈	—	2	1	no	—
44	C₂²×C₁₉⋊C₈	—	2	2	no	—
45	C₂²×C₁₉⋊C₈	—	2	2	no	—
46	C₂²×C₁₉⋊C₈	—	2	2	no	—
47	C₂²×C₁₉⋊C₈	—	2	1	no	—
48	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
49	#10597276	—	2	8	no	—
50	C₄⋊(C₁₉⋊C₈)	—	2	16	no	—
51	C₂⁴×C₁₉	#41	1	1	yes	characteristic
52	C₂²×C₄×C₁₉	—	2	4	no	—
53	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
60	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	#41	1	2	yes	—
62	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
63	C₂²×Dic₁₉	—	2	2	no	—
64	M₁₆×C₁₉	—	2	4	no	—
65	M₁₆×C₁₉	—	2	4	no	—
66	C₂×C₈×C₁₉	—	2	2	no	—
67	C₂×C₈×C₁₉	—	2	2	no	—
68	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	—
69	C₄⋊C₄×C₁₉	—	2	4	no	—
70	C₄⋊C₄×C₁₉	—	2	4	no	—
71	C₂²⋊C₄×C₁₉	—	2	4	no	—
72	C₂²⋊C₄×C₁₉	—	2	4	no	—
73	C₄²×C₁₉	—	2	4	no	—
74	C₄²×C₁₉	C₄⋊C₄	1	8	yes	—
75	C₂²⋊Dic₁₉	—	4	2	no	—
76	C₄⋊Dic₁₉	—	2	4	no	—
77	C₄×Dic₁₉	—	4	1	no	—
78	C₄×Dic₁₉	—	4	1	no	—
79	#64520	—	2	4	no	—
80	#64520	—	2	4	no	—
81	#64520	—	4	2	no	—
82	#64520	—	4	2	no	—
83	C₂×C₁₉⋊C₈	—	2	4	no	—
84	C₂×C₁₉⋊C₈	—	2	4	no	—
85	C₂×C₁₉⋊C₈	—	2	2	no	—
86	C₂×C₁₉⋊C₈	—	4	1	no	—
87	C₂×C₁₉⋊C₈	—	4	2	no	—
88	C₂×C₁₉⋊C₈	—	4	2	no	—
89	C₂×C₁₉⋊C₈	—	4	1	no	—
90	C₂×C₁₉⋊C₈	—	4	2	no	—
91	#11336	—	19	1	no	maximal, Sylow
92	C₂³×C₁₉	—	4	2	no	—
93	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
94	C₂³×C₁₉	C₂²⋊Q₈	1	2	yes	—
95	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂³×C₁₉	#123	1	2	yes	—
97	C₂³×C₁₉	#125	1	1	yes	characteristic
98	C₂×C₄×C₁₉	—	2	8	no	—
99	C₂×C₄×C₁₉	—	4	8	no	—
100	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
104	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
105	C₂×C₄×C₁₉	#120	1	2	yes	—
106	C₂×C₄×C₁₉	#121	1	2	yes	—
107	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
108	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
110	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
111	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
112	C₂×C₄×C₁₉	#125	1	2	yes	—
113	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
115	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
117	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
118	C₂×Dic₁₉	—	4	4	no	—
119	C₂×Dic₁₉	—	4	2	no	—
120	C₈×C₁₉	—	2	4	no	—
121	C₁₉⋊C₈	—	4	2	no	—
122	C₁₉⋊C₈	—	4	4	no	—
123	C₂×#430	—	19	1	no	—
124	C₂×#424	—	19	2	no	—
125	#1733	—	19	2	no	—
126	C₂×H(3,ℤ/4)	D₁₉	1	1	yes	characteristic
127	#1729	—	19	1	no	—
128	C₂²×C₁₉	—	2	2	no	—
129	C₂²×C₁₉	—	2	4	no	—
130	C₂²×C₁₉	—	4	1	no	—
131	C₂²×C₁₉	—	4	2	no	—
132	C₂²×C₁₉	—	4	1	no	—
133	C₂²×C₁₉	#385	1	2	yes	—
134	C₂²×C₁₉	#389	1	1	yes	characteristic
135	C₂²×C₁₉	#395	1	1	yes	characteristic
136	C₂²×C₁₉	#398	1	2	yes	—
137	C₂²×C₁₉	#399	1	1	yes	characteristic
138	C₄×C₁₉	—	4	8	no	—
139	C₄×C₁₉	#384	1	1	yes	characteristic
140	C₄×C₁₉	#385	1	1	yes	characteristic
141	C₄×C₁₉	#389	1	2	yes	—
142	C₄×C₁₉	#392	1	1	yes	characteristic
143	C₄×C₁₉	#395	1	2	yes	—
144	C₄×C₁₉	#398	1	1	yes	characteristic
145	Dic₁₉	—	8	2	no	—
146	C₂²×M₁₆	—	19	2	no	—
147	C₂²×M₁₆	C₂×D₁₉	1	1	yes	characteristic
148	C₂³×C₈	—	19	1	no	—
149	C₂×#117	—	19	1	no	—
150	C₂×#117	C₂×D₁₉	1	2	yes	—
151	#430	—	38	4	no	—
152	C₂×#108	—	38	2	no	—
153	#424	—	38	4	no	—
154	#424	—	38	4	no	—
155	C₂×C₄⋊C₈	—	38	4	no	—
156	#339	—	38	4	no	—
157	H(3,ℤ/4)	Dic₁₉	1	4	yes	complemented
158	#336	—	38	2	no	—
159	C₂×C₁₉	—	2	2	no	—
160	C₂×C₁₉	—	2	1	no	—
161	C₂×C₁₉	—	2	1	no	—
162	C₂×C₁₉	—	2	2	no	—
163	C₂×C₁₉	#1444	1	1	yes	characteristic
164	C₂×C₁₉	#2098	1	2	yes	—
165	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	19	4	no	—
169	C₂×M₁₆	—	38	2	no	—
170	C₂×M₁₆	—	38	2	no	—
171	C₂×M₁₆	—	38	2	no	—
172	C₂×M₁₆	—	38	2	no	—
173	C₂×M₁₆	—	38	2	no	—
174	C₂×M₁₆	Dic₃₈	1	2	yes	—
175	C₂×M₁₆	C₂×Dic₁₉	1	2	yes	—
176	C₂²×C₈	—	19	2	no	—
177	C₂²×C₈	—	38	1	no	—
178	C₂²×C₈	—	38	2	no	—
179	C₂²×C₈	—	38	2	no	—
180	C₂²×C₈	—	38	2	no	—
181	C₂²×C₈	—	38	1	no	—
182	C₂²×C₈	D₇₆	1	2	yes	—
183	#117	—	2	2	no	—
184	#117	—	2	2	no	—
185	#117	—	38	2	no	—
186	#117	—	38	2	no	—
187	#117	Dic₃₈	1	4	yes	—
188	#117	C₂×Dic₁₉	1	4	yes	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	19	2	no	—
191	C₂×C₂²⋊C₄	—	2	2	no	—
192	C₂×C₂²⋊C₄	—	38	1	no	—
193	C₂×C₄²	—	38	1	no	—
194	C₂×C₄²	D₇₆	1	4	yes	—
195	#108	—	38	8	no	—
196	C₄⋊C₈	—	38	16	no	—
197	C₁₉	#11336	1	1	yes	characteristic, complemented, Sylow
198	C₂⁴	#64543	1	1	yes	characteristic
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	38	2	no	—
201	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
202	C₂²×C₄	C₂×Dic₃₈	1	2	yes	—
203	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₇₆	1	2	yes	—
205	C₂²×C₄	D₄×D₁₉	1	2	yes	—
206	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
207	C₂²×C₄	#64543	1	2	yes	—
208	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
209	C₂²×C₄	#64545	1	1	yes	characteristic
210	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
211	M₁₆	—	2	4	no	—
212	M₁₆	—	2	4	no	—
213	M₁₆	—	38	4	no	—
214	M₁₆	—	38	4	no	—
215	M₁₆	—	76	2	no	—
216	M₁₆	—	76	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	38	4	no	—
220	C₂×C₈	—	38	4	no	—
221	C₂×C₈	—	38	2	no	—
222	C₂×C₈	—	76	1	no	—
223	C₂×C₈	—	76	2	no	—
224	C₂×C₈	—	76	1	no	—
225	C₂×C₈	—	76	2	no	—
226	C₂×C₈	—	76	2	no	—
227	C₂×C₈	C₄⋊Dic₁₉	1	4	yes	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	38	4	no	—
231	C₂²⋊C₄	—	2	4	no	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₂²⋊C₄	—	76	2	no	—
234	C₄²	—	2	4	no	—
235	C₄²	—	76	1	no	—
236	C₄²	—	76	1	no	—
237	C₄²	C₄⋊Dic₁₉	1	8	yes	—
238	C₂³	—	4	2	no	—
239	C₂³	#10597324	1	2	yes	—
240	C₂³	#10597332	1	2	yes	—
241	C₂³	#10597336	1	1	yes	characteristic
242	C₂³	#10597343	1	1	yes	characteristic
243	C₂³	D₄×Dic₁₉	1	1	yes	characteristic
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	4	8	no	—
246	C₂×C₄	—	76	4	no	—
247	C₂×C₄	—	76	2	no	—
248	C₂×C₄	#10597315	1	2	yes	—
249	C₂×C₄	#10597315	1	1	yes	characteristic
250	C₂×C₄	#10597320	1	1	yes	characteristic
251	C₂×C₄	#10597324	1	2	yes	—
252	C₂×C₄	#10597324	1	2	yes	—
253	C₂×C₄	#10597328	1	1	yes	characteristic
254	C₂×C₄	#10597328	1	2	yes	—
255	C₂×C₄	#10597332	1	1	yes	characteristic
256	C₂×C₄	#10597334	1	2	yes	—
257	C₂×C₄	#10597334	1	1	yes	characteristic
258	C₂×C₄	#10597336	1	2	yes	—
259	C₂×C₄	#10597341	1	2	yes	—
260	C₂×C₄	#10597341	1	1	yes	characteristic
261	C₂×C₄	#10597343	1	2	yes	—
262	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
263	C₂×C₄	C₂×C₄⋊Dic₁₉	1	2	yes	—
264	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
265	C₂×C₄	Q₈×Dic₁₉	1	1	yes	characteristic
266	C₈	—	2	4	no	—
267	C₈	—	76	2	no	—
268	C₈	—	76	4	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	4	2	no	—
273	C₂²	—	4	1	no	—
274	C₂²	#11926419	1	2	yes	—
275	C₂²	#11926443	1	1	yes	characteristic
276	C₂²	#11926465	1	1	yes	characteristic
277	C₂²	#11926476	1	2	yes	—
278	C₂²	#11926481	1	1	yes	characteristic
279	C₄	—	4	8	no	—
280	C₄	—	152	2	no	—
281	C₄	#11926414	1	1	yes	characteristic
282	C₄	#11926419	1	1	yes	characteristic
283	C₄	#11926443	1	2	yes	—
284	C₄	#11926454	1	1	yes	characteristic
285	C₄	#11926465	1	2	yes	—
286	C₄	#11926476	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	2	2	no	—
291	C₂	#14561900	1	1	yes	characteristic
292	C₂	#14565189	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#10597320
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×H(3,ℤ/4)	C₂
socle	C₂²×C₁₉	#399
prime radical	C₂⁴×C₁₉	#41
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×H(3,ℤ/4)	D₁₉
hypocenter	C₁₉	#11336
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
18. C₁₉×H(3,ℤ/4)	C₄	semidirect
157. H(3,ℤ/4)	Dic₁₉	semidirect
197. C₁₉	#11336	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926593	2
C₂×#11926571	2
#14563345	2
#14563333	2
C₂×C₁₉×H(3,ℤ/4)	2
#11336	19