Subgroups — #18088354

Order 4864.

894 conjugacy classes of subgroups.

The subgroup lattice has 310 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	#14566042	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926304	C₂	1	4	yes	maximal
4	C₂×C₁₉×#376	C₂	1	1	yes	maximal, characteristic
5	C₂²×#64520	C₂²	1	2	yes	—
6	C₂×#10597372	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	#11926414	C₄	1	2	yes	complemented
11	#11926410	C₄	1	2	yes	complemented
12	#11926404	—	2	2	no	—
13	C₄×C₄⋊Dic₁₉	—	2	2	no	—
14	#11926304	—	2	8	no	—
15	C₂²×C₄²×C₁₉	C₂²	1	2	yes	—
16	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#376	—	2	2	no	—
18	C₁₉×#376	C₄	1	4	yes	complemented
19	C₂×C₁₉×#95	—	2	2	no	—
20	C₂³×C₄×C₁₉	—	2	2	no	—
21	C₂³×C₄×C₁₉	D₄	1	2	yes	—
22	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₁₉	—	2	2	no	—
24	C₂×C₄²×C₁₉	—	2	2	no	—
25	C₂×C₄²×C₁₉	—	2	2	no	—
26	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₉	D₄	1	4	yes	—
32	C₂×C₄²×C₁₉	D₄	1	2	yes	—
33	C₂×C₄²×C₁₉	D₄	1	2	yes	—
34	C₂×C₄²×C₁₉	Q₈	1	4	yes	—
35	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
36	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
37	#10597372	—	2	4	no	—
38	#10597372	—	2	4	no	—
39	C₂×C₄⋊Dic₁₉	—	2	4	no	—
40	C₂×C₄⋊Dic₁₉	C₂×C₄	1	4	yes	—
41	C₂×C₄×Dic₁₉	—	2	2	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×#95	—	2	8	no	—
48	C₁₉⋊#95	—	2	4	no	—
49	C₄⋊(C₁₉⋊C₈)	—	2	8	no	—
50	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	—	2	4	no	—
52	C₂²×C₄×C₁₉	—	2	4	no	—
53	C₂²×C₄×C₁₉	—	2	4	no	—
54	C₂²×C₄×C₁₉	—	4	4	no	—
55	C₂²×C₄×C₁₉	—	4	8	no	—
56	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
59	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
60	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₁₉	#41	1	2	yes	—
69	C₂²×C₄×C₁₉	#41	1	2	yes	—
70	C₂²×Dic₁₉	—	2	4	no	—
71	C₄²×C₁₉	—	2	4	no	—
72	C₄²×C₁₉	—	2	4	no	—
73	C₄²×C₁₉	—	2	4	no	—
74	C₄²×C₁₉	C₄²	1	2	yes	—
75	C₄²×C₁₉	C₄²	1	2	yes	—
76	C₄²×C₁₉	C₂²⋊C₄	1	4	yes	—
77	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
78	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
79	C₄²×C₁₉	C₄⋊C₄	1	4	yes	—
80	C₄²×C₁₉	C₄⋊C₄	1	8	yes	complemented
81	C₂²⋊Dic₁₉	—	4	4	no	—
82	C₄⋊Dic₁₉	—	2	8	no	—
83	C₄⋊Dic₁₉	—	4	8	no	—
84	C₄×Dic₁₉	—	4	2	no	—
85	C₄×Dic₁₉	—	4	2	no	—
86	#64520	—	2	8	no	—
87	#64520	—	4	4	no	—
88	C₂×C₁₉⋊C₈	—	2	4	no	—
89	C₂×C₁₉⋊C₈	—	4	2	no	—
90	C₂×C₁₉⋊C₈	—	4	2	no	—
91	#10840	—	19	1	no	maximal, Sylow
92	C₂³×C₁₉	—	2	4	no	—
93	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂³×C₁₉	#121	1	2	yes	—
97	C₂³×C₁₉	C₂²⋊Q₈	1	2	yes	—
98	C₂×C₄×C₁₉	—	2	4	no	—
99	C₂×C₄×C₁₉	—	2	4	no	—
100	C₂×C₄×C₁₉	—	2	8	no	—
101	C₂×C₄×C₁₉	—	4	4	no	—
102	C₂×C₄×C₁₉	—	4	4	no	—
103	C₂×C₄×C₁₉	—	4	8	no	—
104	C₂×C₄×C₁₉	—	4	8	no	—
105	C₂×C₄×C₁₉	—	4	8	no	—
106	C₂×C₄×C₁₉	#95	1	4	yes	—
107	C₂×C₄×C₁₉	#95	1	4	yes	—
108	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₉	#117	1	2	yes	—
114	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
117	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
118	C₂×C₄×C₁₉	#121	1	2	yes	—
119	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
121	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
122	C₂×C₄×C₁₉	#125	1	2	yes	—
123	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
125	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
126	C₂×Dic₁₉	—	4	4	no	—
127	C₂×Dic₁₉	—	4	8	no	—
128	C₁₉⋊C₈	—	4	4	no	—
129	C₂×#423	—	19	1	no	—
130	#2285	—	19	1	no	—
131	C₂×#376	D₁₉	1	1	yes	characteristic
132	C₂×#339	—	19	4	no	—
133	C₂²×C₁₉	—	2	4	no	—
134	C₂²×C₁₉	—	2	2	no	—
135	C₂²×C₁₉	—	2	2	no	—
136	C₂²×C₁₉	—	2	4	no	—
137	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
138	C₂²×C₁₉	#380	1	2	yes	—
139	C₂²×C₁₉	#389	1	2	yes	—
140	C₂²×C₁₉	#390	1	1	yes	characteristic
141	C₂²×C₁₉	#391	1	1	yes	characteristic
142	C₂²×C₁₉	#421	1	4	yes	—
143	C₄×C₁₉	—	2	4	no	—
144	C₄×C₁₉	—	2	4	no	—
145	C₄×C₁₉	—	4	8	no	—
146	C₄×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
147	C₄×C₁₉	#382	1	1	yes	characteristic
148	C₄×C₁₉	#384	1	1	yes	characteristic
149	C₄×C₁₉	#389	1	2	yes	—
150	C₄×C₁₉	#390	1	1	yes	characteristic
151	C₄×C₁₉	#391	1	1	yes	characteristic
152	Dic₁₉	—	8	4	no	—
153	C₂²×M₁₆	—	19	2	no	—
154	C₂×#117	—	19	2	no	—
155	C₂²×C₄²	C₂×D₁₉	1	2	yes	—
156	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
157	#423	—	19	4	no	—
158	#423	—	38	2	no	—
159	C₂×C₄⋊C₈	—	38	2	no	—
160	#384	—	19	2	no	—
161	#382	—	19	2	no	—
162	#380	—	38	2	no	—
163	C₄×C₄⋊C₄	—	38	2	no	—
164	#376	—	2	2	no	—
165	#376	Dic₁₉	1	4	yes	complemented
166	C₂×#95	—	2	2	no	—
167	#339	—	38	8	no	—
168	C₂×C₁₉	—	2	2	no	—
169	C₂×C₁₉	—	2	2	no	—
170	C₂×C₁₉	C₄⋊#95	1	1	yes	characteristic
171	C₂×C₁₉	#1733	1	2	yes	—
172	C₂×C₁₉	#1963	1	2	yes	—
173	C₂×C₁₉	#1995	1	2	yes	—
174	C₂³×C₄	—	2	2	no	—
175	C₂³×C₄	C₂²⋊D₁₉	1	2	yes	—
176	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
177	C₂×M₁₆	—	19	4	no	—
178	C₂×M₁₆	—	38	2	no	—
179	C₂×M₁₆	—	38	4	no	—
180	C₂×M₁₆	—	38	4	no	—
181	C₂²×C₈	—	38	2	no	—
182	#117	—	38	4	no	—
183	#117	—	38	4	no	—
184	C₂×C₄⋊C₄	—	19	4	no	—
185	C₂×C₄⋊C₄	—	38	4	no	—
186	C₂×C₂²⋊C₄	—	38	2	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	—	2	2	no	—
190	C₂×C₄²	—	38	2	no	—
191	C₂×C₄²	Dic₃₈	1	4	yes	—
192	C₂×C₄²	Dic₃₈	1	2	yes	—
193	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
194	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
195	C₂×C₄²	C₄×D₁₉	1	2	yes	—
196	C₂×C₄²	D₇₆	1	4	yes	—
197	C₂×C₄²	D₇₆	1	2	yes	—
198	C₂×C₄²	C₂×Dic₁₉	1	4	yes	—
199	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
200	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
201	C₄⋊C₈	—	38	8	no	—
202	#95	—	2	8	no	—
203	#95	—	38	4	no	—
204	C₁₉	#10840	1	1	yes	characteristic, complemented, Sylow
205	C₂⁴	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	4	4	no	—
210	C₂²×C₄	—	4	8	no	—
211	C₂²×C₄	—	38	4	no	—
212	C₂²×C₄	Dic₁₉⋊C₄	1	4	yes	—
213	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
214	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
215	C₂²×C₄	C₂²⋊Dic₁₉	1	4	yes	—
216	C₂²×C₄	C₂×Dic₃₈	1	2	yes	—
217	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₇₆	1	2	yes	—
221	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
222	C₂²×C₄	#64541	1	2	yes	—
223	C₂²×C₄	#64541	1	2	yes	—
224	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
226	M₁₆	—	38	8	no	—
227	M₁₆	—	76	4	no	—
228	C₂×C₈	—	38	4	no	—
229	C₂×C₈	—	76	2	no	—
230	C₂×C₈	—	76	2	no	—
231	C₄⋊C₄	—	38	8	no	—
232	C₄⋊C₄	—	76	8	no	—
233	C₂²⋊C₄	—	76	4	no	—
234	C₄²	—	2	4	no	—
235	C₄²	—	2	4	no	—
236	C₄²	—	2	4	no	—
237	C₄²	—	76	2	no	—
238	C₄²	—	76	2	no	—
239	C₄²	C₄×Dic₁₉	1	2	yes	—
240	C₄²	C₄×Dic₁₉	1	2	yes	—
241	C₄²	Dic₁₉⋊C₄	1	2	yes	—
242	C₄²	Dic₁₉⋊C₄	1	2	yes	—
243	C₄²	C₄⋊Dic₁₉	1	4	yes	—
244	C₄²	C₄⋊Dic₁₉	1	8	yes	complemented
245	C₄²	(C₂×D₁₉)⋊C₄	1	4	yes	—
246	C₂³	—	2	4	no	—
247	C₂³	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
248	C₂³	#10597370	1	2	yes	—
249	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
250	C₂³	#10597376	1	2	yes	—
251	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	8	no	—
258	C₂×C₄	—	4	8	no	—
259	C₂×C₄	—	4	8	no	—
260	C₂×C₄	—	76	8	no	—
261	C₂×C₄	—	76	4	no	—
262	C₂×C₄	C₁₉⋊#95	1	4	yes	—
263	C₂×C₄	C₁₉⋊#95	1	4	yes	—
264	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
265	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
266	C₂×C₄	#10597315	1	2	yes	—
267	C₂×C₄	#10597315	1	1	yes	characteristic
268	C₂×C₄	#10597316	1	2	yes	—
269	C₂×C₄	C₄×D₇₆	1	2	yes	—
270	C₂×C₄	C₄×D₇₆	1	2	yes	—
271	C₂×C₄	#10597320	1	1	yes	characteristic
272	C₂×C₄	#10597321	1	1	yes	characteristic
273	C₂×C₄	#10597321	1	1	yes	characteristic
274	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
275	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
276	C₂×C₄	#10597370	1	2	yes	—
277	C₂×C₄	C₂×C₄⋊Dic₁₉	1	2	yes	—
278	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
279	C₂×C₄	#10597372	1	2	yes	—
280	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
281	C₂×C₄	#10597376	1	2	yes	—
282	C₈	—	76	4	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	4	no	—
287	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
288	C₂²	#11926404	1	2	yes	—
289	C₂²	#11926442	1	2	yes	—
290	C₂²	#11926446	1	1	yes	characteristic
291	C₂²	#11926450	1	1	yes	characteristic
292	C₂²	#11926562	1	4	yes	—
293	C₄	—	2	4	no	—
294	C₄	—	2	4	no	—
295	C₄	—	4	8	no	—
296	C₄	—	152	4	no	—
297	C₄	C₄×C₄⋊Dic₁₉	1	2	yes	—
298	C₄	#11926410	1	1	yes	characteristic
299	C₄	#11926414	1	1	yes	characteristic
300	C₄	#11926442	1	2	yes	—
301	C₄	#11926446	1	1	yes	characteristic
302	C₄	#11926450	1	1	yes	characteristic
303	C₂	—	2	2	no	—
304	C₂	—	2	2	no	—
305	C₂	#14561865	1	1	yes	characteristic
306	C₂	#14563345	1	2	yes	—
307	C₂	#14564467	1	2	yes	—
308	C₂	#14564616	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₇₆
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×#376	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂×D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×#376	D₁₉
hypocenter	C₁₉	#10840
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
8. #11926568	C₄	semidirect
10. #11926414	C₄	semidirect
11. #11926410	C₄	semidirect
18. C₁₉×#376	C₄	semidirect
80. C₄²×C₁₉	C₄⋊C₄	semidirect
165. #376	Dic₁₉	semidirect
204. C₁₉	#10840	semidirect
244. C₄²	C₄⋊Dic₁₉	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
#14566042	2
C₂×#11926304	2
C₂×C₁₉×#376	2
#10840	19