Subgroups — #18088420

Order 4864.

830 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926568	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926410	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926312	C₂	1	4	yes	maximal
4	C₂×C₄×M₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#64520	C₂²	1	2	yes	—
6	C₂²×C₄⋊Dic₁₉	C₂²	1	2	yes	—
7	#11926568	—	2	2	no	—
8	#11926568	C₄	1	4	yes	complemented
9	C₂×C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
10	#11926410	—	2	2	no	—
11	#11926410	—	2	1	no	—
12	#11926410	—	2	1	no	—
13	C₂×C₁₉⋊#95	C₄	1	4	yes	—
14	#11926312	—	2	8	no	—
15	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
16	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
17	C₄×M₁₆×C₁₉	—	2	2	no	—
18	C₄×M₁₆×C₁₉	C₄	1	4	yes	complemented
19	C₂×C₁₉×#97	—	2	1	no	—
20	C₂×C₄×C₈×C₁₉	—	2	1	no	—
21	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
23	C₂³×Dic₁₉	C₂×C₄	1	4	yes	—
24	C₂×M₁₆×C₁₉	—	2	2	no	—
25	C₂×M₁₆×C₁₉	—	2	2	no	—
26	C₂×M₁₆×C₁₉	—	2	2	no	—
27	C₂×M₁₆×C₁₉	C₂×C₄	1	4	yes	—
28	C₂×M₁₆×C₁₉	D₄	1	4	yes	—
29	C₂×M₁₆×C₁₉	Q₈	1	4	yes	—
30	C₂²×C₈×C₁₉	—	2	2	no	—
31	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₉	D₄	1	2	yes	—
34	C₂×C₄²×C₁₉	D₄	1	2	yes	—
35	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
38	C₂×C₄⋊Dic₁₉	—	2	4	no	—
39	C₂×C₄⋊Dic₁₉	—	2	2	no	—
40	C₂×C₄⋊Dic₁₉	—	2	2	no	—
41	C₂×C₄⋊Dic₁₉	—	2	4	no	—
42	C₂×#64520	—	2	2	no	—
43	C₂×#64520	—	2	4	no	—
44	C₂×#64520	—	2	4	no	—
45	C₂×#64520	C₂×C₄	1	4	yes	—
46	C₂²×C₁₉⋊C₈	—	2	2	no	—
47	C₁₉×#97	—	2	4	no	—
48	C₄×C₈×C₁₉	—	2	4	no	—
49	C₁₉⋊#95	—	2	4	no	—
50	C₁₉⋊#95	—	2	4	no	—
51	C₄⋊(C₁₉⋊C₈)	—	2	8	no	—
52	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	—	4	4	no	—
54	C₂²×C₄×C₁₉	C₄²	1	4	yes	—
55	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
58	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₁₉	#41	1	2	yes	—
67	C₂²×C₄×C₁₉	#41	1	2	yes	—
68	C₂²×Dic₁₉	—	2	4	no	—
69	C₂²×Dic₁₉	—	2	8	no	—
70	C₂²×Dic₁₉	—	4	8	no	—
71	M₁₆×C₁₉	—	2	4	no	—
72	M₁₆×C₁₉	—	2	4	no	—
73	M₁₆×C₁₉	—	2	4	no	—
74	M₁₆×C₁₉	C₄⋊C₄	1	8	yes	—
75	C₂×C₈×C₁₉	—	2	4	no	—
76	C₂×C₈×C₁₉	—	2	4	no	—
77	C₂×C₈×C₁₉	—	2	4	no	—
78	C₄²×C₁₉	C₂²⋊C₄	1	4	yes	—
79	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
80	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
81	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
82	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
83	C₄²×C₁₉	C₄⋊C₄	1	4	yes	—
84	C₄⋊Dic₁₉	—	4	4	no	—
85	C₄⋊Dic₁₉	—	4	2	no	—
86	C₄⋊Dic₁₉	—	4	2	no	—
87	#64520	—	2	8	no	—
88	#64520	—	4	4	no	—
89	C₂×C₁₉⋊C₈	—	2	4	no	—
90	C₂×C₁₉⋊C₈	—	4	2	no	—
91	C₂×C₁₉⋊C₈	—	4	2	no	—
92	#10858	—	19	1	no	maximal, Sylow
93	C₂³×C₁₉	—	2	4	no	—
94	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
95	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂³×C₁₉	C₄×D₄	1	2	yes	—
98	C₂³×C₁₉	C₄×Q₈	1	2	yes	—
99	C₂×C₄×C₁₉	—	2	4	no	—
100	C₂×C₄×C₁₉	—	2	4	no	—
101	C₂×C₄×C₁₉	—	4	4	no	—
102	C₂×C₄×C₁₉	—	4	4	no	—
103	C₂×C₄×C₁₉	#95	1	4	yes	—
104	C₂×C₄×C₁₉	#95	1	4	yes	—
105	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₉	#117	1	2	yes	—
111	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
113	C₂×C₄×C₁₉	#121	1	2	yes	—
114	C₂×C₄×C₁₉	#121	1	2	yes	—
115	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
118	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
119	C₂×C₄×C₁₉	#125	1	2	yes	—
120	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
122	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
123	C₂×Dic₁₉	—	4	4	no	—
124	C₂×Dic₁₉	—	4	8	no	—
125	C₂×Dic₁₉	—	8	8	no	—
126	C₈×C₁₉	—	2	8	no	—
127	C₁₉⋊C₈	—	4	4	no	—
128	C₂×#423	—	19	1	no	—
129	C₂×C₄×M₁₆	D₁₉	1	1	yes	characteristic
130	C₂×#382	—	19	1	no	—
131	C₂×#343	—	19	4	no	—
132	C₂²×C₁₉	—	2	4	no	—
133	C₂²×C₁₉	—	2	2	no	—
134	C₂²×C₁₉	—	2	2	no	—
135	C₂²×C₁₉	—	2	4	no	—
136	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
137	C₂²×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
138	C₂²×C₁₉	#389	1	2	yes	—
139	C₂²×C₁₉	#390	1	1	yes	characteristic
140	C₂²×C₁₉	#391	1	1	yes	characteristic
141	C₂²×C₁₉	#413	1	4	yes	—
142	C₄×C₁₉	—	2	4	no	—
143	C₄×C₁₉	—	2	4	no	—
144	C₄×C₁₉	#380	1	2	yes	—
145	C₄×C₁₉	#382	1	1	yes	characteristic
146	C₄×C₁₉	#384	1	1	yes	characteristic
147	C₄×C₁₉	#389	1	2	yes	—
148	C₄×C₁₉	#390	1	1	yes	characteristic
149	C₄×C₁₉	#391	1	1	yes	characteristic
150	Dic₁₉	—	8	4	no	—
151	C₂²×M₁₆	—	19	2	no	—
152	C₂²×M₁₆	C₂×D₁₉	1	2	yes	—
153	C₂²×C₄⋊C₄	—	19	2	no	—
154	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
155	#423	—	19	4	no	—
156	#423	—	38	2	no	—
157	C₂×C₄⋊C₈	—	38	2	no	—
158	C₄×M₁₆	—	2	2	no	—
159	C₄×M₁₆	Dic₁₉	1	4	yes	complemented
160	C₂×#97	—	2	1	no	—
161	C₂×C₄×C₈	—	2	1	no	—
162	#382	—	38	2	no	—
163	#382	—	38	1	no	—
164	#382	—	38	1	no	—
165	C₂×#95	—	19	4	no	—
166	#343	—	38	8	no	—
167	C₂×C₁₉	—	2	2	no	—
168	C₂×C₁₉	—	2	2	no	—
169	C₂×C₁₉	C₄⋊#95	1	1	yes	characteristic
170	C₂×C₁₉	#1742	1	2	yes	—
171	C₂×C₁₉	#1967	1	2	yes	—
172	C₂×C₁₉	#1998	1	2	yes	—
173	C₂³×C₄	—	19	4	no	—
174	C₂³×C₄	C₄×D₁₉	1	2	yes	—
175	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	19	4	no	—
180	C₂×M₁₆	—	38	2	no	—
181	C₂×M₁₆	—	38	4	no	—
182	C₂×M₁₆	—	38	4	no	—
183	C₂×M₁₆	Dic₃₈	1	4	yes	—
184	C₂×M₁₆	D₇₆	1	4	yes	—
185	C₂×M₁₆	C₂×Dic₁₉	1	4	yes	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	38	2	no	—
188	C₂×C₄⋊C₄	—	38	4	no	—
189	C₂×C₄⋊C₄	—	38	2	no	—
190	C₂×C₄⋊C₄	—	38	4	no	—
191	C₂×C₄⋊C₄	—	38	2	no	—
192	C₂×C₄²	Dic₃₈	1	2	yes	—
193	C₂×C₄²	C₄×D₁₉	1	2	yes	—
194	C₂×C₄²	D₇₆	1	2	yes	—
195	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
196	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
197	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
198	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
199	C₄⋊C₈	—	38	8	no	—
200	#97	—	2	4	no	—
201	C₄×C₈	—	2	4	no	—
202	#95	—	38	4	no	—
203	#95	—	38	4	no	—
204	C₁₉	#10858	1	1	yes	characteristic, complemented, Sylow
205	C₂⁴	C₂×C₄×D₁₉	1	1	yes	characteristic
206	C₂²×C₄	—	4	4	no	—
207	C₂²×C₄	—	38	4	no	—
208	C₂²×C₄	—	38	8	no	—
209	C₂²×C₄	—	76	8	no	—
210	C₂²×C₄	C₄×Dic₁₉	1	4	yes	—
211	C₂²×C₄	Dic₁₉⋊C₄	1	4	yes	—
212	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
213	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
214	C₂²×C₄	C₂×Dic₃₈	1	2	yes	—
215	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₇₆	1	2	yes	—
218	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
219	C₂²×C₄	#64541	1	2	yes	—
220	C₂²×C₄	#64541	1	2	yes	—
221	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
224	M₁₆	—	2	4	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	2	4	no	—
227	M₁₆	—	38	8	no	—
228	M₁₆	—	76	4	no	—
229	M₁₆	C₄⋊Dic₁₉	1	8	yes	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	38	4	no	—
234	C₂×C₈	—	76	2	no	—
235	C₂×C₈	—	76	2	no	—
236	C₄⋊C₄	—	76	4	no	—
237	C₄⋊C₄	—	76	2	no	—
238	C₄⋊C₄	—	76	2	no	—
239	C₄²	Dic₁₉⋊C₄	1	2	yes	—
240	C₄²	Dic₁₉⋊C₄	1	2	yes	—
241	C₄²	C₄⋊Dic₁₉	1	4	yes	—
242	C₄²	(C₂×D₁₉)⋊C₄	1	4	yes	—
243	C₄²	C₂²⋊Dic₁₉	1	2	yes	—
244	C₄²	C₂²⋊Dic₁₉	1	2	yes	—
245	C₂³	—	2	4	no	—
246	C₂³	C₄×Dic₃₈	1	2	yes	—
247	C₂³	C₄×D₇₆	1	2	yes	—
248	C₂³	C₂×C₄×Dic₁₉	1	1	yes	characteristic
249	C₂³	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
250	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	76	4	no	—
256	C₂×C₄	—	76	8	no	—
257	C₂×C₄	—	152	8	no	—
258	C₂×C₄	C₁₉⋊#95	1	4	yes	—
259	C₂×C₄	C₁₉⋊#95	1	4	yes	—
260	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
261	C₂×C₄	#10597315	1	2	yes	—
262	C₂×C₄	#10597315	1	1	yes	characteristic
263	C₂×C₄	#10597316	1	2	yes	—
264	C₂×C₄	C₄×D₇₆	1	2	yes	—
265	C₂×C₄	#10597320	1	1	yes	characteristic
266	C₂×C₄	#10597321	1	1	yes	characteristic
267	C₂×C₄	#10597321	1	1	yes	characteristic
268	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
269	C₂×C₄	#10597370	1	2	yes	—
270	C₂×C₄	#10597370	1	2	yes	—
271	C₂×C₄	C₂×C₄⋊Dic₁₉	1	2	yes	—
272	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
273	C₂×C₄	#10597372	1	2	yes	—
274	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
275	C₂×C₄	#10597376	1	2	yes	—
276	C₂×C₄	#10597376	1	2	yes	—
277	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
278	C₈	—	2	8	no	—
279	C₈	—	76	4	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	4	no	—
284	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
285	C₂²	C₄×C₄⋊Dic₁₉	1	2	yes	—
286	C₂²	#11926442	1	2	yes	—
287	C₂²	#11926446	1	1	yes	characteristic
288	C₂²	#11926450	1	1	yes	characteristic
289	C₂²	#11926526	1	4	yes	—
290	C₄	—	2	4	no	—
291	C₄	—	2	4	no	—
292	C₄	—	152	4	no	—
293	C₄	#11926404	1	2	yes	—
294	C₄	#11926410	1	1	yes	characteristic
295	C₄	#11926414	1	1	yes	characteristic
296	C₄	#11926442	1	2	yes	—
297	C₄	#11926446	1	1	yes	characteristic
298	C₄	#11926450	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	2	2	no	—
301	C₂	#14561865	1	1	yes	characteristic
302	C₂	#14563373	1	2	yes	—
303	C₂	#14564484	1	2	yes	—
304	C₂	#14564628	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×(C₂×D₁₉)⋊C₄
commutator	C₂³×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₄×M₁₆×C₁₉	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂²×C₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×C₄×M₁₆	D₁₉
hypocenter	C₁₉	#10858
LCS	G ▷ C₂³×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
8. #11926568	C₄	semidirect
18. C₄×M₁₆×C₁₉	C₄	semidirect
159. C₄×M₁₆	Dic₁₉	semidirect
204. C₁₉	#10858	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926568	2
C₂×#11926410	2
C₂×#11926312	2
C₂×C₄×M₁₆×C₁₉	2
#10858	19