Subgroups — #18088408

Order 4864.

802 conjugacy classes of subgroups.

The subgroup lattice has 344 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×#11926316	C₂	1	4	yes	maximal
4	C₂×C₄×M₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#64520	C₂²	1	2	yes	—
6	C₂×#10597372	C₂²	1	2	yes	—
7	#11926568	—	2	1	no	—
8	#11926568	—	2	1	no	—
9	#11926568	—	2	2	no	—
10	C₂×C₄⋊(C₁₉⋊C₈)	C₄	1	4	yes	complemented
11	#11926414	—	2	1	no	—
12	#11926410	—	2	1	no	—
13	#11926404	C₄	1	4	yes	—
14	C₄×C₄⋊Dic₁₉	—	2	2	no	—
15	#11926316	—	2	8	no	—
16	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
17	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
18	C₄×M₁₆×C₁₉	—	2	1	no	—
19	C₄×M₁₆×C₁₉	—	2	1	no	—
20	C₄×M₁₆×C₁₉	C₄	1	2	yes	complemented
21	C₄×M₁₆×C₁₉	C₄	1	2	yes	complemented
22	C₂×C₁₉×#97	—	2	1	no	—
23	C₂×C₄×C₈×C₁₉	—	2	1	no	—
24	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₁₉	—	2	2	no	—
27	C₂×M₁₆×C₁₉	—	2	2	no	—
28	C₂×M₁₆×C₁₉	—	2	2	no	—
29	C₂×M₁₆×C₁₉	C₂×C₄	1	4	yes	—
30	C₂×M₁₆×C₁₉	D₄	1	4	yes	—
31	C₂×M₁₆×C₁₉	Q₈	1	4	yes	—
32	C₂²×C₈×C₁₉	—	2	2	no	—
33	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
42	C₂×C₂²⋊Dic₁₉	C₂×C₄	1	4	yes	—
43	#10597372	—	2	4	no	—
44	#10597372	—	2	4	no	—
45	C₂×C₄⋊Dic₁₉	—	2	4	no	—
46	C₂×C₄⋊Dic₁₉	—	2	2	no	—
47	C₂×C₄×Dic₁₉	—	2	2	no	—
48	C₂×#64520	—	2	2	no	—
49	C₂×#64520	—	2	2	no	—
50	C₂×#64520	—	2	4	no	—
51	C₂×#64520	—	2	4	no	—
52	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
53	C₁₉×#97	—	2	2	no	—
54	C₁₉×#97	—	2	2	no	—
55	C₄×C₈×C₁₉	—	2	2	no	—
56	C₄×C₈×C₁₉	—	2	2	no	—
57	C₁₉⋊#95	—	2	4	no	—
58	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
59	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
60	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	—	4	4	no	—
62	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
63	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
64	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₁₉	#41	1	2	yes	—
77	C₂²×C₄×C₁₉	#41	1	2	yes	—
78	C₂²×Dic₁₉	—	2	4	no	—
79	M₁₆×C₁₉	—	2	4	no	—
80	M₁₆×C₁₉	—	2	4	no	—
81	M₁₆×C₁₉	—	2	4	no	—
82	M₁₆×C₁₉	C₄⋊C₄	1	8	yes	complemented
83	C₂×C₈×C₁₉	—	2	4	no	—
84	C₂×C₈×C₁₉	—	2	4	no	—
85	C₂×C₈×C₁₉	—	2	4	no	—
86	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
87	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
88	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
89	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
90	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
91	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
92	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
93	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
94	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
95	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
96	C₂²⋊Dic₁₉	—	2	8	no	—
97	C₄⋊Dic₁₉	—	4	8	no	—
98	C₄⋊Dic₁₉	—	4	4	no	—
99	C₄×Dic₁₉	—	4	2	no	—
100	C₄×Dic₁₉	—	4	2	no	—
101	#64520	—	4	4	no	—
102	#64520	—	4	2	no	—
103	#64520	—	4	2	no	—
104	C₂×C₁₉⋊C₈	—	2	4	no	—
105	C₂×C₁₉⋊C₈	—	2	8	no	—
106	#10855	—	19	1	no	maximal, Sylow
107	C₂³×C₁₉	—	4	2	no	—
108	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
109	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂³×C₁₉	C₄×D₄	1	2	yes	—
112	C₂³×C₁₉	C₄×Q₈	1	2	yes	—
113	C₂×C₄×C₁₉	—	2	4	no	—
114	C₂×C₄×C₁₉	—	2	4	no	—
115	C₂×C₄×C₁₉	—	2	4	no	—
116	C₂×C₄×C₁₉	—	4	2	no	—
117	C₂×C₄×C₁₉	—	4	2	no	—
118	C₂×C₄×C₁₉	—	4	2	no	—
119	C₂×C₄×C₁₉	#95	1	2	yes	—
120	C₂×C₄×C₁₉	#95	1	2	yes	—
121	C₂×C₄×C₁₉	#95	1	2	yes	—
122	C₂×C₄×C₁₉	#95	1	2	yes	—
123	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₉	#117	1	2	yes	—
129	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
130	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
131	C₂×C₄×C₁₉	#121	1	2	yes	—
132	C₂×C₄×C₁₉	#121	1	2	yes	—
133	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
134	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
135	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
136	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
137	C₂×C₄×C₁₉	#125	1	2	yes	—
138	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
139	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
140	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
141	C₂×Dic₁₉	—	4	8	no	—
142	C₂×Dic₁₉	—	4	4	no	—
143	C₈×C₁₉	—	2	8	no	—
144	C₁₉⋊C₈	—	4	4	no	—
145	C₂×#423	—	19	1	no	—
146	C₂×C₄×M₁₆	D₁₉	1	1	yes	characteristic
147	#2285	—	19	1	no	—
148	C₂×#344	—	19	4	no	—
149	C₂²×C₁₉	—	2	2	no	—
150	C₂²×C₁₉	—	2	2	no	—
151	C₂²×C₁₉	—	2	2	no	—
152	C₂²×C₁₉	—	4	2	no	—
153	C₂²×C₁₉	—	4	2	no	—
154	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
155	C₂²×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
156	C₂²×C₁₉	#389	1	2	yes	—
157	C₂²×C₁₉	#390	1	1	yes	characteristic
158	C₂²×C₁₉	#391	1	1	yes	characteristic
159	C₄×C₁₉	—	2	2	no	—
160	C₄×C₁₉	—	2	2	no	—
161	C₄×C₁₉	—	2	2	no	—
162	C₄×C₁₉	#344	1	4	yes	complemented
163	C₄×C₁₉	#380	1	2	yes	—
164	C₄×C₁₉	#382	1	1	yes	characteristic
165	C₄×C₁₉	#384	1	1	yes	characteristic
166	C₄×C₁₉	#389	1	2	yes	—
167	C₄×C₁₉	#390	1	1	yes	characteristic
168	C₄×C₁₉	#391	1	1	yes	characteristic
169	Dic₁₉	—	8	4	no	—
170	C₂²×M₁₆	—	19	2	no	—
171	C₂²×M₁₆	C₂×D₁₉	1	2	yes	—
172	C₂×#117	—	19	2	no	—
173	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
174	#423	—	38	2	no	—
175	#423	—	38	1	no	—
176	#423	—	38	1	no	—
177	C₂×C₄⋊C₈	—	19	4	no	—
178	C₄×M₁₆	—	2	1	no	—
179	C₄×M₁₆	—	2	1	no	—
180	C₄×M₁₆	Dic₁₉	1	2	yes	complemented
181	C₄×M₁₆	Dic₁₉	1	2	yes	complemented
182	C₂×#97	—	2	1	no	—
183	C₂×C₄×C₈	—	2	1	no	—
184	#384	—	38	1	no	—
185	#382	—	38	1	no	—
186	#380	—	19	4	no	—
187	C₄×C₄⋊C₄	—	38	2	no	—
188	#344	—	38	8	no	—
189	C₂×C₁₉	—	2	1	no	—
190	C₂×C₁₉	—	2	2	no	—
191	C₂×C₁₉	—	2	2	no	—
192	C₂×C₁₉	—	2	1	no	—
193	C₂×C₁₉	C₄⋊#95	1	1	yes	characteristic
194	C₂×C₁₉	C₂×#344	1	2	yes	—
195	C₂³×C₄	C₄×D₁₉	1	2	yes	—
196	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
197	C₂×M₁₆	—	2	2	no	—
198	C₂×M₁₆	—	2	2	no	—
199	C₂×M₁₆	—	2	2	no	—
200	C₂×M₁₆	—	38	4	no	—
201	C₂×M₁₆	—	38	2	no	—
202	C₂×M₁₆	—	38	4	no	—
203	C₂×M₁₆	—	38	2	no	—
204	C₂×M₁₆	Dic₃₈	1	4	yes	—
205	C₂×M₁₆	D₇₆	1	4	yes	—
206	C₂×M₁₆	C₂×Dic₁₉	1	4	yes	—
207	C₂²×C₈	—	2	2	no	—
208	C₂²×C₈	—	19	4	no	—
209	#117	—	38	4	no	—
210	#117	—	38	4	no	—
211	C₂×C₄⋊C₄	—	38	4	no	—
212	C₂×C₄⋊C₄	—	38	2	no	—
213	C₂×C₂²⋊C₄	—	19	4	no	—
214	C₂×C₄²	—	38	2	no	—
215	C₂×C₄²	Dic₃₈	1	2	yes	—
216	C₂×C₄²	C₄×D₁₉	1	2	yes	—
217	C₂×C₄²	D₇₆	1	2	yes	—
218	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
219	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
220	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
221	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
222	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
223	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
224	C₄⋊C₈	—	38	4	no	—
225	C₄⋊C₈	—	38	4	no	—
226	#97	—	2	2	no	—
227	#97	—	2	2	no	—
228	C₄×C₈	—	2	2	no	—
229	C₄×C₈	—	2	2	no	—
230	#95	—	38	4	no	—
231	C₁₉	#10855	1	1	yes	characteristic, complemented, Sylow
232	C₂⁴	C₂×C₄×D₁₉	1	1	yes	characteristic
233	C₂²×C₄	—	4	4	no	—
234	C₂²×C₄	—	38	4	no	—
235	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
236	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
237	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
238	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
239	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
240	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
241	C₂²×C₄	C₂×Dic₃₈	1	2	yes	—
242	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₇₆	1	2	yes	—
245	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
246	C₂²×C₄	#64541	1	2	yes	—
247	C₂²×C₄	#64541	1	2	yes	—
248	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
251	M₁₆	—	2	4	no	—
252	M₁₆	—	2	4	no	—
253	M₁₆	—	2	4	no	—
254	M₁₆	—	76	4	no	—
255	M₁₆	—	76	2	no	—
256	M₁₆	—	76	2	no	—
257	M₁₆	C₄⋊Dic₁₉	1	8	yes	complemented
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	2	4	no	—
261	C₂×C₈	—	38	8	no	—
262	C₂×C₈	—	38	4	no	—
263	C₄⋊C₄	—	76	8	no	—
264	C₄⋊C₄	—	76	4	no	—
265	C₂²⋊C₄	—	38	8	no	—
266	C₄²	—	76	2	no	—
267	C₄²	—	76	2	no	—
268	C₄²	Dic₁₉⋊C₄	1	2	yes	—
269	C₄²	Dic₁₉⋊C₄	1	2	yes	—
270	C₄²	C₄⋊Dic₁₉	1	2	yes	—
271	C₄²	C₄⋊Dic₁₉	1	2	yes	—
272	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
273	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
274	C₄²	C₂²⋊Dic₁₉	1	1	yes	characteristic
275	C₄²	C₂²⋊Dic₁₉	1	1	yes	characteristic
276	C₄²	C₂²⋊Dic₁₉	1	1	yes	characteristic
277	C₄²	C₂²⋊Dic₁₉	1	1	yes	characteristic
278	C₂³	—	4	2	no	—
279	C₂³	C₄×Dic₃₈	1	2	yes	—
280	C₂³	C₄×D₇₆	1	2	yes	—
281	C₂³	C₂×C₄×Dic₁₉	1	1	yes	characteristic
282	C₂³	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
283	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	76	8	no	—
291	C₂×C₄	—	76	4	no	—
292	C₂×C₄	C₁₉⋊#95	1	2	yes	—
293	C₂×C₄	C₁₉⋊#95	1	2	yes	—
294	C₂×C₄	C₁₉⋊#95	1	2	yes	—
295	C₂×C₄	C₁₉⋊#95	1	2	yes	—
296	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
297	C₂×C₄	#10597315	1	2	yes	—
298	C₂×C₄	#10597315	1	1	yes	characteristic
299	C₂×C₄	#10597316	1	2	yes	—
300	C₂×C₄	C₄×D₇₆	1	2	yes	—
301	C₂×C₄	#10597320	1	1	yes	characteristic
302	C₂×C₄	#10597321	1	1	yes	characteristic
303	C₂×C₄	#10597321	1	1	yes	characteristic
304	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
305	C₂×C₄	#10597370	1	2	yes	—
306	C₂×C₄	#10597370	1	2	yes	—
307	C₂×C₄	C₂×C₄⋊Dic₁₉	1	2	yes	—
308	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
309	C₂×C₄	#10597372	1	2	yes	—
310	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
311	C₂×C₄	#10597376	1	2	yes	—
312	C₂×C₄	#10597376	1	2	yes	—
313	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
314	C₈	—	2	8	no	—
315	C₈	—	76	4	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
322	C₂²	C₄×C₄⋊Dic₁₉	1	2	yes	—
323	C₂²	#11926442	1	2	yes	—
324	C₂²	#11926446	1	1	yes	characteristic
325	C₂²	#11926450	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	152	4	no	—
330	C₄	#11926316	1	4	yes	complemented
331	C₄	#11926404	1	2	yes	—
332	C₄	#11926410	1	1	yes	characteristic
333	C₄	#11926414	1	1	yes	characteristic
334	C₄	#11926442	1	2	yes	—
335	C₄	#11926446	1	1	yes	characteristic
336	C₄	#11926450	1	1	yes	characteristic
337	C₂	—	2	1	no	—
338	C₂	—	2	2	no	—
339	C₂	—	2	2	no	—
340	C₂	—	2	1	no	—
341	C₂	#14561865	1	1	yes	characteristic
342	C₂	C₂×#11926316	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄
commutator	C₂³×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₄×M₁₆×C₁₉	C₂
socle	C₂²×C₁₉	C₂×#95
prime radical	C₂⁴×C₁₉	C₂²×C₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×C₄×M₁₆	D₁₉
hypocenter	C₁₉	#10855
LCS	G ▷ C₂³×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
10. C₂×C₄⋊(C₁₉⋊C₈)	C₄	semidirect
20. C₄×M₁₆×C₁₉	C₄	semidirect
21. C₄×M₁₆×C₁₉	C₄	semidirect
82. M₁₆×C₁₉	C₄⋊C₄	semidirect
162. C₄×C₁₉	#344	semidirect
180. C₄×M₁₆	Dic₁₉	semidirect
181. C₄×M₁₆	Dic₁₉	semidirect
231. C₁₉	#10855	semidirect
257. M₁₆	C₄⋊Dic₁₉	semidirect
330. C₄	#11926316	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×#11926316	2
C₂×C₄×M₁₆×C₁₉	2
#10855	19