Subgroups — C₄×#11926316 #18087749

Order 4864.

906 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#64520	C₂	1	1	yes	maximal, characteristic
2	C₄×#10597372	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	C₄×#64520	—	2	1	no	—
8	C₄×#64520	—	2	1	no	—
9	C₄×#64520	—	2	1	no	—
10	C₄×#64520	—	2	1	no	—
11	C₂×#10597260	C₄	1	2	yes	complemented
12	C₂×C₄×C₁₉⋊C₈	C₄	1	2	yes	complemented
13	C₄×C₄⋊Dic₁₉	—	2	2	no	—
14	C₄×C₂²⋊Dic₁₉	C₄	1	4	yes	—
15	#11926390	—	2	1	no	—
16	C₄²×Dic₁₉	—	2	1	no	—
17	#11926316	C₄	1	16	yes	direct product factor
18	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
19	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
20	C₄×M₁₆×C₁₉	—	2	2	no	—
21	C₄×M₁₆×C₁₉	C₄	1	4	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	4	no	—
28	C₂×M₁₆×C₁₉	C₂×C₄	1	8	yes	—
29	C₂×M₁₆×C₁₉	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₁₉	—	2	2	no	—
31	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₉	D₄	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	Q₈	1	2	yes	—
40	C₂×C₂²⋊Dic₁₉	C₂×C₄	1	4	yes	—
41	#10597372	—	2	4	no	—
42	#10597372	C₂×C₄	1	8	yes	—
43	C₂×C₄⋊Dic₁₉	—	2	2	no	—
44	C₂×C₄×Dic₁₉	—	2	2	no	—
45	C₂×C₄×Dic₁₉	—	2	4	no	—
46	C₂×#64520	—	2	2	no	—
47	C₂×#64520	—	2	2	no	—
48	C₂×#64520	—	2	4	no	—
49	C₂×#64520	C₂×C₄	1	8	yes	—
50	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
51	C₁₉×#97	—	2	4	no	—
52	C₄×C₈×C₁₉	—	2	4	no	—
53	C₁₉⋊#95	—	2	4	no	—
54	#10597260	—	2	2	no	—
55	#10597260	—	2	2	no	—
56	C₄×C₁₉⋊C₈	—	2	2	no	—
57	C₄×C₁₉⋊C₈	—	2	2	no	—
58	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	—	4	4	no	—
60	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
61	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
62	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
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×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	#41	1	4	yes	—
74	C₂²×Dic₁₉	—	2	4	no	—
75	M₁₆×C₁₉	—	2	4	no	—
76	M₁₆×C₁₉	—	2	8	no	—
77	M₁₆×C₁₉	C₄²	1	8	yes	complemented
78	C₂×C₈×C₁₉	—	2	4	no	—
79	C₂×C₈×C₁₉	—	2	8	no	—
80	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
81	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
82	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
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	2	yes	—
88	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
89	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
90	C₂²⋊Dic₁₉	—	2	4	no	—
91	C₂²⋊Dic₁₉	C₄²	1	8	yes	—
92	C₄⋊Dic₁₉	—	2	8	no	—
93	C₄×Dic₁₉	—	2	8	no	—
94	C₄×Dic₁₉	—	4	8	no	—
95	#64520	—	2	8	no	—
96	#64520	—	2	8	no	—
97	C₂×C₁₉⋊C₈	—	2	4	no	—
98	C₂×C₁₉⋊C₈	—	2	4	no	—
99	C₂×C₁₉⋊C₈	C₄²	1	8	yes	complemented
100	C₄×#344	—	19	1	no	maximal, Sylow
101	C₂³×C₁₉	—	4	2	no	—
102	C₂³×C₁₉	C₂×C₄²	1	2	yes	—
103	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
104	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂³×C₁₉	#117	1	2	yes	—
107	C₂×C₄×C₁₉	—	2	4	no	—
108	C₂×C₄×C₁₉	—	2	4	no	—
109	C₂×C₄×C₁₉	—	2	4	no	—
110	C₂×C₄×C₁₉	—	4	2	no	—
111	C₂×C₄×C₁₉	—	4	2	no	—
112	C₂×C₄×C₁₉	—	4	2	no	—
113	C₂×C₄×C₁₉	#95	1	2	yes	—
114	C₂×C₄×C₁₉	#95	1	2	yes	—
115	C₂×C₄×C₁₉	#95	1	2	yes	—
116	C₂×C₄×C₁₉	#95	1	2	yes	—
117	C₂×C₄×C₁₉	C₂×C₄²	1	2	yes	—
118	C₂×C₄×C₁₉	C₂×C₄²	1	2	yes	—
119	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₉	#117	1	2	yes	—
124	C₂×C₄×C₁₉	#117	1	2	yes	—
125	C₂×C₄×C₁₉	C₄×D₄	1	4	yes	—
126	C₂×C₄×C₁₉	C₄×D₄	1	4	yes	—
127	C₂×C₄×C₁₉	C₄×D₄	1	4	yes	—
128	C₂×C₄×C₁₉	C₄×Q₈	1	4	yes	—
129	C₂×Dic₁₉	—	2	8	no	—
130	C₂×Dic₁₉	—	4	8	no	—
131	C₈×C₁₉	—	2	8	no	—
132	C₁₉⋊C₈	—	2	8	no	—
133	C₂×C₄×M₁₆	—	19	1	no	—
134	C₂×C₄×M₁₆	D₁₉	1	1	yes	characteristic
135	C₄×#117	—	19	1	no	—
136	C₂×#344	—	19	4	no	—
137	C₂²×C₁₉	—	2	2	no	—
138	C₂²×C₁₉	—	2	2	no	—
139	C₂²×C₁₉	—	2	2	no	—
140	C₂²×C₁₉	—	4	2	no	—
141	C₂²×C₁₉	—	4	2	no	—
142	C₂²×C₁₉	C₄³	1	1	yes	characteristic
143	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
144	C₂²×C₁₉	#376	1	1	yes	characteristic
145	C₂²×C₁₉	C₄×C₂²⋊C₄	1	2	yes	—
146	C₂²×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
147	C₄×C₁₉	—	2	2	no	—
148	C₄×C₁₉	—	2	2	no	—
149	C₄×C₁₉	—	2	2	no	—
150	C₄×C₁₉	#344	1	4	yes	complemented
151	C₄×C₁₉	C₄×C₂²⋊C₄	1	2	yes	—
152	C₄×C₁₉	C₄×C₂²⋊C₄	1	2	yes	—
153	C₄×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
154	C₄×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
155	Dic₁₉	—	4	8	no	—
156	C₂²×M₁₆	—	19	2	no	—
157	C₂²×M₁₆	C₂×D₁₉	1	2	yes	—
158	C₂×#117	—	19	2	no	—
159	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
160	C₄×M₁₆	—	2	2	no	—
161	C₄×M₁₆	—	38	1	no	—
162	C₄×M₁₆	—	38	1	no	—
163	C₄×M₁₆	—	38	1	no	—
164	C₄×M₁₆	—	38	1	no	—
165	C₄×M₁₆	Dic₁₉	1	4	yes	complemented
166	C₂×#97	—	2	1	no	—
167	C₂×#97	—	19	2	no	—
168	C₂×C₄×C₈	—	2	1	no	—
169	C₂×C₄×C₈	—	19	2	no	—
170	C₄×C₄⋊C₄	—	38	2	no	—
171	C₄×C₂²⋊C₄	—	19	4	no	—
172	#376	—	38	1	no	—
173	C₄³	—	38	1	no	—
174	#344	—	19	16	no	—
175	C₂×C₁₉	—	2	2	no	—
176	C₂×C₁₉	—	2	1	no	—
177	C₂×C₁₉	—	2	1	no	—
178	C₂×C₁₉	—	2	2	no	—
179	C₂×C₁₉	C₄×#95	1	1	yes	characteristic
180	C₂×C₁₉	C₂×#344	1	2	yes	—
181	C₂³×C₄	C₄×D₁₉	1	2	yes	—
182	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
183	C₂×M₁₆	—	2	4	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	19	8	no	—
186	C₂×M₁₆	—	38	2	no	—
187	C₂×M₁₆	—	38	4	no	—
188	C₂×M₁₆	—	38	2	no	—
189	C₂×M₁₆	C₄×D₁₉	1	8	yes	—
190	C₂×M₁₆	C₂×Dic₁₉	1	4	yes	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	—	19	4	no	—
193	#117	—	19	8	no	—
194	#117	—	38	4	no	—
195	C₂×C₄⋊C₄	—	38	2	no	—
196	C₂×C₂²⋊C₄	—	19	4	no	—
197	C₂×C₄²	—	38	2	no	—
198	C₂×C₄²	—	38	4	no	—
199	C₂×C₄²	Dic₃₈	1	2	yes	—
200	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
201	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
202	C₂×C₄²	D₇₆	1	1	yes	characteristic
203	C₂×C₄²	D₇₆	1	1	yes	characteristic
204	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
205	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
206	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
207	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
208	#97	—	2	4	no	—
209	#97	—	38	2	no	—
210	#97	—	38	2	no	—
211	C₄×C₈	—	2	4	no	—
212	C₄×C₈	—	38	2	no	—
213	C₄×C₈	—	38	2	no	—
214	#95	—	38	4	no	—
215	C₁₉	C₄×#344	1	1	yes	characteristic, complemented, Sylow
216	C₂⁴	C₂×C₄×D₁₉	1	1	yes	characteristic
217	C₂²×C₄	—	4	4	no	—
218	C₂²×C₄	—	38	4	no	—
219	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
220	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
221	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
222	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
223	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
224	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
225	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₄×D₁₉	1	4	yes	—
227	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
229	C₂²×C₄	#64541	1	4	yes	—
230	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
233	M₁₆	—	2	4	no	—
234	M₁₆	—	2	8	no	—
235	M₁₆	—	38	8	no	—
236	M₁₆	—	38	8	no	—
237	M₁₆	C₄×Dic₁₉	1	8	yes	complemented
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	8	no	—
240	C₂×C₈	—	19	8	no	—
241	C₂×C₈	—	38	4	no	—
242	C₂×C₈	—	38	4	no	—
243	C₄⋊C₄	—	38	8	no	—
244	C₂²⋊C₄	—	19	8	no	—
245	C₂²⋊C₄	—	38	4	no	—
246	C₄²	—	38	8	no	—
247	C₄²	—	76	8	no	—
248	C₄²	Dic₁₉⋊C₄	1	2	yes	—
249	C₄²	Dic₁₉⋊C₄	1	2	yes	—
250	C₄²	C₄⋊Dic₁₉	1	2	yes	—
251	C₄²	C₄⋊Dic₁₉	1	2	yes	—
252	C₄²	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
253	C₄²	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
254	C₄²	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
255	C₄²	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
256	C₄²	C₂²⋊Dic₁₉	1	2	yes	—
257	C₄²	C₂²⋊Dic₁₉	1	2	yes	—
258	C₂³	—	4	2	no	—
259	C₂³	C₄²×D₁₉	1	2	yes	—
260	C₂³	#10597318	1	2	yes	—
261	C₂³	C₂×C₄×Dic₁₉	1	1	yes	characteristic
262	C₂³	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
263	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	38	8	no	—
271	C₂×C₄	—	76	8	no	—
272	C₂×C₄	C₁₉⋊#95	1	2	yes	—
273	C₂×C₄	C₁₉⋊#95	1	2	yes	—
274	C₂×C₄	C₁₉⋊#95	1	2	yes	—
275	C₂×C₄	C₁₉⋊#95	1	2	yes	—
276	C₂×C₄	C₄×Dic₃₈	1	4	yes	—
277	C₂×C₄	C₄²×D₁₉	1	2	yes	—
278	C₂×C₄	#10597318	1	2	yes	—
279	C₂×C₄	C₄×D₇₆	1	4	yes	—
280	C₂×C₄	C₂×C₄×Dic₁₉	1	2	yes	—
281	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
282	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
283	C₂×C₄	#10597372	1	2	yes	—
284	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
285	C₂×C₄	C₄×C₂²⋊D₁₉	1	4	yes	—
286	C₂×C₄	C₄×C₂²⋊D₁₉	1	4	yes	—
287	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
288	C₈	—	2	8	no	—
289	C₈	—	38	8	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	4	2	no	—
295	C₂²	C₄²×Dic₁₉	1	1	yes	characteristic
296	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
297	C₂²	#11926390	1	1	yes	characteristic
298	C₂²	C₄×(C₂×D₁₉)⋊C₄	1	2	yes	—
299	C₂²	C₄×Dic₁₉⋊C₄	1	2	yes	—
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	2	2	no	—
303	C₄	—	76	8	no	—
304	C₄	#11926316	1	4	yes	direct product factor
305	C₄	C₄×C₂²⋊Dic₁₉	1	2	yes	—
306	C₄	C₄×(C₂×D₁₉)⋊C₄	1	2	yes	—
307	C₄	C₄×C₄⋊Dic₁₉	1	2	yes	—
308	C₄	C₄×Dic₁₉⋊C₄	1	2	yes	—
309	C₂	—	2	2	no	—
310	C₂	—	2	1	no	—
311	C₂	—	2	1	no	—
312	C₂	—	2	2	no	—
313	C₂	C₄×C₁₉⋊#95	1	1	yes	characteristic
314	C₂	C₂×#11926316	1	2	yes	—

	subgroup	G/H
center	C₄²	(C₂×D₁₉)⋊C₄
commutator	C₂²×C₁₉	C₄³
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	C₄×#344
LCS	G ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
11. C₂×#10597260	C₄	semidirect
12. C₂×C₄×C₁₉⋊C₈	C₄	semidirect
17. #11926316	C₄	direct (N × H)
21. C₄×M₁₆×C₁₉	C₄	semidirect
77. M₁₆×C₁₉	C₄²	semidirect
99. C₂×C₁₉⋊C₈	C₄²	semidirect
150. C₄×C₁₉	#344	semidirect
165. C₄×M₁₆	Dic₁₉	semidirect
215. C₁₉	C₄×#344	semidirect
237. M₁₆	C₄×Dic₁₉	semidirect
304. C₄	#11926316	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×#64520	2
C₄×#10597372	2
C₂×#11926316	2
C₂×C₄×M₁₆×C₁₉	2
C₄×#344	19

Subgroups — C4×#11926316 #18087749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11926316 #18087749