Subgroups — #18088412

Order 4864.

810 conjugacy classes of subgroups.

The subgroup lattice has 336 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×#11926568	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926314	C₂	1	4	yes	maximal
4	C₄×C₁₉×#117	C₂	1	1	yes	maximal, characteristic
5	C₂²×#64520	C₂²	1	2	yes	—
6	C₂²×#64520	C₂²	1	2	yes	—
7	#11926568	—	2	2	no	—
8	#11926568	—	2	1	no	—
9	#11926568	—	2	1	no	—
10	#11926568	—	2	2	no	—
11	#11926568	C₄	1	2	yes	complemented
12	#11926568	C₄	1	2	yes	complemented
13	C₂×C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
14	C₂×C₄⋊(C₁₉⋊C₈)	C₄	1	4	yes	complemented
15	#11926314	—	2	8	no	—
16	C₂×C₁₉×#117	C₂²	1	2	yes	—
17	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₁₉	—	2	2	no	—
19	C₄×C₂²⋊C₄×C₁₉	C₄	1	4	yes	—
20	C₁₉×#376	—	2	1	no	—
21	C₄³×C₁₉	—	2	1	no	—
22	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
24	C₁₉×#117	—	2	2	no	—
25	C₁₉×#117	—	2	2	no	—
26	C₁₉×#117	D₄	1	4	yes	—
27	C₁₉×#117	Q₈	1	4	yes	—
28	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
29	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₁₉	—	2	2	no	—
31	C₂×C₄²×C₁₉	—	2	4	no	—
32	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₉	D₄	1	2	yes	—
36	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₉	D₄	1	2	yes	—
39	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
40	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
41	C₂×#64520	—	2	2	no	—
42	C₂×#64520	—	2	2	no	—
43	C₂×#64520	—	2	2	no	—
44	C₂×#64520	—	2	4	no	—
45	C₂×#64520	—	2	4	no	—
46	C₂×#64520	—	2	4	no	—
47	C₂×#64520	—	2	4	no	—
48	C₂×#64520	C₂×C₄	1	4	yes	—
49	C₂²×C₁₉⋊C₈	—	2	2	no	—
50	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
51	C₁₉×#95	—	2	4	no	—
52	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
53	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
54	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
55	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
56	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	—	2	4	no	—
58	C₂²×C₄×C₁₉	—	4	4	no	—
59	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
60	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
61	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₁₉	#41	1	2	yes	—
74	C₂²×C₄×C₁₉	#41	1	2	yes	—
75	C₄⋊C₄×C₁₉	—	2	4	no	—
76	C₄⋊C₄×C₁₉	—	2	4	no	—
77	C₂²⋊C₄×C₁₉	—	2	4	no	—
78	C₂²⋊C₄×C₁₉	C₄⋊C₄	1	8	yes	—
79	C₄²×C₁₉	—	2	4	no	—
80	C₄²×C₁₉	—	2	4	no	—
81	C₄²×C₁₉	—	4	8	no	—
82	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
84	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
85	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
86	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
87	C₄²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
88	C₄²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
89	C₄²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
90	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
91	C₄²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
92	#64520	—	2	8	no	—
93	#64520	—	4	4	no	—
94	#64520	—	4	4	no	—
95	#64520	—	4	2	no	—
96	#64520	—	4	2	no	—
97	C₂×C₁₉⋊C₈	—	2	4	no	—
98	C₂×C₁₉⋊C₈	—	2	4	no	—
99	C₂×C₁₉⋊C₈	—	2	8	no	—
100	C₂×C₁₉⋊C₈	—	4	2	no	—
101	C₂×C₁₉⋊C₈	—	4	2	no	—
102	#10856	—	19	1	no	maximal, Sylow
103	C₂³×C₁₉	—	4	2	no	—
104	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
105	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂³×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
108	C₂³×C₁₉	#117	1	2	yes	—
109	C₂×C₄×C₁₉	—	2	4	no	—
110	C₂×C₄×C₁₉	—	2	4	no	—
111	C₂×C₄×C₁₉	—	2	4	no	—
112	C₂×C₄×C₁₉	—	2	8	no	—
113	C₂×C₄×C₁₉	—	4	2	no	—
114	C₂×C₄×C₁₉	—	4	2	no	—
115	C₂×C₄×C₁₉	—	4	2	no	—
116	C₂×C₄×C₁₉	—	4	8	no	—
117	C₂×C₄×C₁₉	#95	1	2	yes	—
118	C₂×C₄×C₁₉	#95	1	2	yes	—
119	C₂×C₄×C₁₉	#95	1	2	yes	—
120	C₂×C₄×C₁₉	#95	1	2	yes	—
121	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
128	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
129	C₂×C₄×C₁₉	#121	1	2	yes	—
130	C₂×C₄×C₁₉	#121	1	2	yes	—
131	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
132	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
133	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
134	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
135	C₂×C₄×C₁₉	#125	1	2	yes	—
136	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
137	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
138	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
139	C₁₉⋊C₈	—	4	4	no	—
140	C₁₉⋊C₈	—	4	4	no	—
141	C₂×#423	—	19	1	no	—
142	C₂×#423	—	19	1	no	—
143	C₄×#117	D₁₉	1	1	yes	characteristic
144	C₂×#344	—	19	4	no	—
145	C₂²×C₁₉	—	2	2	no	—
146	C₂²×C₁₉	—	2	2	no	—
147	C₂²×C₁₉	—	2	2	no	—
148	C₂²×C₁₉	—	4	2	no	—
149	C₂²×C₁₉	—	4	2	no	—
150	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
151	C₂²×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
152	C₂²×C₁₉	#380	1	2	yes	—
153	C₂²×C₁₉	#382	1	1	yes	characteristic
154	C₂²×C₁₉	#384	1	1	yes	characteristic
155	C₄×C₁₉	—	2	2	no	—
156	C₄×C₁₉	—	2	2	no	—
157	C₄×C₁₉	—	2	2	no	—
158	C₄×C₁₉	—	4	8	no	—
159	C₄×C₁₉	#344	1	4	yes	complemented
160	C₄×C₁₉	#389	1	2	yes	—
161	C₄×C₁₉	#389	1	2	yes	—
162	C₄×C₁₉	#390	1	1	yes	characteristic
163	C₄×C₁₉	#390	1	1	yes	characteristic
164	C₄×C₁₉	#391	1	1	yes	characteristic
165	C₄×C₁₉	#391	1	1	yes	characteristic
166	C₂²×M₁₆	—	19	2	no	—
167	C₂²×M₁₆	—	19	2	no	—
168	C₂×#117	C₂×D₁₉	1	2	yes	—
169	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
170	#423	—	19	2	no	—
171	#423	—	19	2	no	—
172	#423	—	38	2	no	—
173	#423	—	38	1	no	—
174	#423	—	38	1	no	—
175	#423	—	38	2	no	—
176	C₂×C₄⋊C₈	—	19	4	no	—
177	C₂×C₄⋊C₈	—	38	2	no	—
178	C₄×C₄⋊C₄	—	2	2	no	—
179	C₄×C₂²⋊C₄	Dic₁₉	1	4	yes	—
180	#376	—	2	1	no	—
181	C₄³	—	2	1	no	—
182	#344	—	38	8	no	—
183	C₂×C₁₉	—	2	2	no	—
184	C₂×C₁₉	—	2	2	no	—
185	C₂×C₁₉	—	2	1	no	—
186	C₂×C₁₉	—	2	1	no	—
187	C₂×C₁₉	C₄⋊#95	1	1	yes	characteristic
188	C₂×C₁₉	C₂×#344	1	2	yes	—
189	C₂³×C₄	C₂×Dic₁₉	1	2	yes	—
190	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
191	C₂×M₁₆	—	19	4	no	—
192	C₂×M₁₆	—	38	4	no	—
193	C₂×M₁₆	—	38	2	no	—
194	C₂×M₁₆	—	38	4	no	—
195	C₂×M₁₆	—	38	2	no	—
196	C₂×M₁₆	—	38	2	no	—
197	C₂×M₁₆	—	38	4	no	—
198	C₂×M₁₆	—	38	4	no	—
199	C₂²×C₈	—	19	4	no	—
200	C₂²×C₈	—	38	2	no	—
201	#117	—	2	2	no	—
202	#117	—	2	2	no	—
203	#117	Dic₃₈	1	4	yes	—
204	#117	D₇₆	1	4	yes	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₂²⋊C₄	C₂×Dic₁₉	1	4	yes	—
207	C₂×C₄²	—	2	2	no	—
208	C₂×C₄²	—	2	4	no	—
209	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
210	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
211	C₂×C₄²	C₄×D₁₉	1	2	yes	—
212	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
213	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
214	C₂×C₄²	D₇₆	1	2	yes	—
215	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
216	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
217	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
218	C₄⋊C₈	—	38	4	no	—
219	C₄⋊C₈	—	38	4	no	—
220	C₄⋊C₈	—	38	4	no	—
221	C₄⋊C₈	—	38	4	no	—
222	#95	—	2	4	no	—
223	C₁₉	#10856	1	1	yes	characteristic, complemented, Sylow
224	C₂⁴	C₂²×Dic₁₉	1	1	yes	characteristic
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	4	4	no	—
227	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
228	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
229	C₂²×C₄	C₄⋊Dic₁₉	1	2	yes	—
230	C₂²×C₄	C₄⋊Dic₁₉	1	2	yes	—
231	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
232	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
233	C₂²×C₄	C₂×Dic₃₈	1	2	yes	—
234	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₇₆	1	2	yes	—
238	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
239	C₂²×C₄	#64541	1	2	yes	—
240	C₂²×C₄	#64541	1	2	yes	—
241	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
243	M₁₆	—	38	8	no	—
244	M₁₆	—	76	4	no	—
245	M₁₆	—	76	4	no	—
246	M₁₆	—	76	2	no	—
247	M₁₆	—	76	2	no	—
248	C₂×C₈	—	38	8	no	—
249	C₂×C₈	—	38	4	no	—
250	C₂×C₈	—	38	4	no	—
251	C₂×C₈	—	76	2	no	—
252	C₂×C₈	—	76	2	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₂²⋊C₄	—	2	4	no	—
256	C₂²⋊C₄	C₄⋊Dic₁₉	1	8	yes	—
257	C₄²	—	2	4	no	—
258	C₄²	—	2	4	no	—
259	C₄²	—	4	8	no	—
260	C₄²	Dic₁₉⋊C₄	1	2	yes	—
261	C₄²	Dic₁₉⋊C₄	1	1	yes	characteristic
262	C₄²	Dic₁₉⋊C₄	1	1	yes	characteristic
263	C₄²	Dic₁₉⋊C₄	1	1	yes	characteristic
264	C₄²	Dic₁₉⋊C₄	1	2	yes	—
265	C₄²	Dic₁₉⋊C₄	1	1	yes	characteristic
266	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
267	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
268	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
269	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
270	C₂³	—	4	2	no	—
271	C₂³	C₂×C₄×Dic₁₉	1	1	yes	characteristic
272	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
273	C₂³	C₂×C₄⋊Dic₁₉	1	2	yes	—
274	C₂³	#10597372	1	2	yes	—
275	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	8	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	8	no	—
284	C₂×C₄	C₁₉⋊#95	1	2	yes	—
285	C₂×C₄	C₁₉⋊#95	1	2	yes	—
286	C₂×C₄	C₁₉⋊#95	1	2	yes	—
287	C₂×C₄	C₁₉⋊#95	1	2	yes	—
288	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
289	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
290	C₂×C₄	#10597315	1	2	yes	—
291	C₂×C₄	#10597315	1	1	yes	characteristic
292	C₂×C₄	#10597316	1	2	yes	—
293	C₂×C₄	C₄×D₇₆	1	2	yes	—
294	C₂×C₄	C₄×D₇₆	1	2	yes	—
295	C₂×C₄	#10597320	1	1	yes	characteristic
296	C₂×C₄	#10597321	1	1	yes	characteristic
297	C₂×C₄	#10597321	1	1	yes	characteristic
298	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
299	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
300	C₂×C₄	#10597370	1	2	yes	—
301	C₂×C₄	#10597370	1	2	yes	—
302	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
303	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
304	C₂×C₄	#10597376	1	2	yes	—
305	C₂×C₄	#10597376	1	2	yes	—
306	C₈	—	76	4	no	—
307	C₈	—	76	4	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	4	2	no	—
313	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
314	C₂²	C₄×C₄⋊Dic₁₉	1	2	yes	—
315	C₂²	#11926404	1	2	yes	—
316	C₂²	#11926410	1	1	yes	characteristic
317	C₂²	#11926414	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	4	8	no	—
322	C₄	#11926314	1	4	yes	complemented
323	C₄	#11926442	1	2	yes	—
324	C₄	#11926442	1	2	yes	—
325	C₄	#11926446	1	1	yes	characteristic
326	C₄	#11926446	1	1	yes	characteristic
327	C₄	#11926450	1	1	yes	characteristic
328	C₄	#11926450	1	1	yes	characteristic
329	C₂	—	2	2	no	—
330	C₂	—	2	2	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	#14561865	1	1	yes	characteristic
334	C₂	C₂×#11926314	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄
commutator	C₂³×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₄×C₁₉×#117	C₂
socle	C₂²×C₁₉	C₂×#95
prime radical	C₂⁴×C₁₉	C₂²×C₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₄×#117	D₁₉
hypocenter	C₁₉	#10856
LCS	G ▷ C₂³×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
11. #11926568	C₄	semidirect
12. #11926568	C₄	semidirect
14. C₂×C₄⋊(C₁₉⋊C₈)	C₄	semidirect
159. C₄×C₁₉	#344	semidirect
223. C₁₉	#10856	semidirect
322. C₄	#11926314	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×#11926314	2
C₄×C₁₉×#117	2
#10856	19