Subgroups — C₄×#11926300 #18087703

Order 4864.

954 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×C₄×#64520	C₂	1	1	yes	maximal, characteristic
2	C₄×#10597372	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926300	C₂	1	4	yes	maximal
4	C₄×C₁₉×#117	C₂	1	1	yes	maximal, characteristic
5	C₂²×#64520	C₂²	1	2	yes	—
6	C₂×#10597372	C₂²	1	2	yes	—
7	C₄×#64520	—	2	2	no	—
8	C₄×#64520	—	2	2	no	—
9	C₂×#10597260	C₄	1	2	yes	complemented
10	C₂×C₄×C₁₉⋊C₈	C₄	1	2	yes	complemented
11	C₄×C₄⋊Dic₁₉	—	2	2	no	—
12	C₄×C₂²⋊Dic₁₉	—	2	2	no	—
13	#11926390	C₄	1	2	yes	complemented
14	C₄²×Dic₁₉	C₄	1	2	yes	complemented
15	#11926300	C₄	1	16	yes	direct product factor
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	—	2	2	no	—
20	C₁₉×#376	C₄	1	2	yes	complemented
21	C₄³×C₁₉	C₄	1	2	yes	complemented
22	C₂³×C₄×C₁₉	D₄	1	2	yes	—
23	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
24	C₁₉×#117	—	2	4	no	—
25	C₁₉×#117	C₂×C₄	1	8	yes	—
26	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
27	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
28	C₂×C₄²×C₁₉	—	2	4	no	—
29	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
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	1	yes	characteristic
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	2	yes	—
37	C₂×C₄²×C₁₉	D₄	1	2	yes	—
38	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
39	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
40	#10597372	—	2	4	no	—
41	#10597372	C₂×C₄	1	8	yes	—
42	C₂×C₄⋊Dic₁₉	—	2	2	no	—
43	C₂×C₄×Dic₁₉	—	2	4	no	—
44	C₂×C₄×Dic₁₉	C₂×C₄	1	4	yes	—
45	C₂×#64520	—	2	2	no	—
46	C₂×#64520	—	2	2	no	—
47	C₂×#64520	—	2	4	no	—
48	C₂×#64520	C₂×C₄	1	8	yes	—
49	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
50	C₁₉×#95	—	2	4	no	—
51	C₁₉⋊#95	—	2	4	no	—
52	#10597260	—	2	4	no	—
53	C₄×C₁₉⋊C₈	—	2	4	no	—
54	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	—	2	4	no	—
56	C₂²×C₄×C₁₉	—	4	4	no	—
57	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
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	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	#41	1	4	yes	—
71	C₂²×Dic₁₉	—	2	4	no	—
72	C₄⋊C₄×C₁₉	—	2	8	no	—
73	C₂²⋊C₄×C₁₉	—	2	8	no	—
74	C₄²×C₁₉	—	2	4	no	—
75	C₄²×C₁₉	—	4	8	no	—
76	C₄²×C₁₉	C₄²	1	1	yes	characteristic
77	C₄²×C₁₉	C₄²	1	1	yes	characteristic
78	C₄²×C₁₉	C₄²	1	1	yes	characteristic
79	C₄²×C₁₉	C₄²	1	1	yes	characteristic
80	C₄²×C₁₉	C₄²	1	8	yes	complemented
81	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
84	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
85	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
86	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
87	C₂²⋊Dic₁₉	—	2	8	no	—
88	C₄⋊Dic₁₉	—	2	8	no	—
89	C₄×Dic₁₉	—	2	4	no	—
90	C₄×Dic₁₉	—	4	8	no	—
91	C₄×Dic₁₉	C₄²	1	8	yes	complemented
92	#64520	—	2	8	no	—
93	#64520	—	2	8	no	—
94	C₂×C₁₉⋊C₈	—	2	4	no	—
95	C₂×C₁₉⋊C₈	—	2	4	no	—
96	C₂×C₁₉⋊C₈	C₄²	1	8	yes	complemented
97	C₄×H(3,ℤ/4)	—	19	1	no	maximal, Sylow
98	C₂³×C₁₉	—	4	2	no	—
99	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂³×C₁₉	C₄×D₄	1	4	yes	—
103	C₂×C₄×C₁₉	—	2	4	no	—
104	C₂×C₄×C₁₉	—	2	4	no	—
105	C₂×C₄×C₁₉	—	2	4	no	—
106	C₂×C₄×C₁₉	—	2	8	no	—
107	C₂×C₄×C₁₉	—	4	2	no	—
108	C₂×C₄×C₁₉	—	4	2	no	—
109	C₂×C₄×C₁₉	—	4	2	no	—
110	C₂×C₄×C₁₉	—	4	8	no	—
111	C₂×C₄×C₁₉	#95	1	2	yes	—
112	C₂×C₄×C₁₉	#95	1	2	yes	—
113	C₂×C₄×C₁₉	#95	1	2	yes	—
114	C₂×C₄×C₁₉	#95	1	2	yes	—
115	C₂×C₄×C₁₉	C₂×C₄²	1	2	yes	—
116	C₂×C₄×C₁₉	C₂×C₄²	1	2	yes	—
117	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
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×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉	#117	1	2	yes	—
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×Q₈	1	4	yes	—
128	C₂×Dic₁₉	—	2	8	no	—
129	C₂×Dic₁₉	—	4	8	no	—
130	C₁₉⋊C₈	—	2	8	no	—
131	C₂×C₄×M₁₆	—	19	1	no	—
132	C₄×#117	—	19	1	no	—
133	C₄×#117	D₁₉	1	1	yes	characteristic
134	C₂×H(3,ℤ/4)	—	19	4	no	—
135	C₂²×C₁₉	—	2	2	no	—
136	C₂²×C₁₉	—	2	2	no	—
137	C₂²×C₁₉	—	2	2	no	—
138	C₂²×C₁₉	—	4	4	no	—
139	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
140	C₂²×C₁₉	C₄×C₂²⋊C₄	1	2	yes	—
141	C₂²×C₁₉	C₄×C₂²⋊C₄	1	2	yes	—
142	C₂²×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
143	C₄×C₁₉	—	2	2	no	—
144	C₄×C₁₉	—	2	2	no	—
145	C₄×C₁₉	—	2	2	no	—
146	C₄×C₁₉	—	4	8	no	—
147	C₄×C₁₉	H(3,ℤ/4)	1	4	yes	complemented
148	C₄×C₁₉	C₄³	1	1	yes	characteristic
149	C₄×C₁₉	#376	1	1	yes	characteristic
150	C₄×C₁₉	C₄×C₂²⋊C₄	1	2	yes	—
151	C₄×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
152	C₄×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
153	Dic₁₉	—	4	8	no	—
154	C₂²×M₁₆	—	19	2	no	—
155	C₂×#117	—	19	2	no	—
156	C₂×#117	C₂×D₁₉	1	2	yes	—
157	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
158	C₄×M₁₆	—	38	2	no	—
159	C₄×M₁₆	—	38	2	no	—
160	C₂×#97	—	19	2	no	—
161	C₂×C₄×C₈	—	19	2	no	—
162	C₄×C₄⋊C₄	—	2	2	no	—
163	C₄×C₄⋊C₄	—	38	2	no	—
164	C₄×C₂²⋊C₄	—	2	2	no	—
165	C₄×C₂²⋊C₄	—	38	2	no	—
166	#376	—	19	2	no	—
167	#376	Dic₁₉	1	2	yes	complemented
168	C₄³	—	19	2	no	—
169	C₄³	Dic₁₉	1	2	yes	complemented
170	H(3,ℤ/4)	—	19	16	no	—
171	C₂×C₁₉	—	2	2	no	—
172	C₂×C₁₉	—	2	2	no	—
173	C₂×C₁₉	—	2	2	no	—
174	C₂×C₁₉	C₄×#95	1	1	yes	characteristic
175	C₂×C₁₉	C₂×H(3,ℤ/4)	1	2	yes	—
176	C₂³×C₄	C₂²⋊D₁₉	1	2	yes	—
177	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
178	C₂×M₁₆	—	19	8	no	—
179	C₂×M₁₆	—	38	2	no	—
180	C₂×M₁₆	—	38	4	no	—
181	C₂×M₁₆	—	38	2	no	—
182	C₂²×C₈	—	19	4	no	—
183	#117	—	2	4	no	—
184	#117	—	19	8	no	—
185	#117	—	38	4	no	—
186	#117	C₄×D₁₉	1	8	yes	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	—	38	2	no	—
189	C₂×C₂²⋊C₄	—	2	2	no	—
190	C₂×C₂²⋊C₄	—	38	2	no	—
191	C₂×C₄²	—	2	4	no	—
192	C₂×C₄²	—	19	4	no	—
193	C₂×C₄²	—	38	4	no	—
194	C₂×C₄²	Dic₃₈	1	2	yes	—
195	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
196	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
197	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
198	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
199	C₂×C₄²	D₇₆	1	2	yes	—
200	C₂×C₄²	C₂×Dic₁₉	1	4	yes	—
201	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
202	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
203	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
204	#97	—	38	4	no	—
205	C₄×C₈	—	38	4	no	—
206	#95	—	2	4	no	—
207	#95	—	38	4	no	—
208	C₁₉	C₄×H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
209	C₂⁴	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	4	4	no	—
212	C₂²×C₄	—	38	4	no	—
213	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
214	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
215	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
216	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
217	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
218	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
219	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₄×D₁₉	1	4	yes	—
221	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
224	C₂²×C₄	#64541	1	4	yes	—
225	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
227	M₁₆	—	38	8	no	—
228	M₁₆	—	38	8	no	—
229	C₂×C₈	—	19	8	no	—
230	C₂×C₈	—	38	4	no	—
231	C₂×C₈	—	38	4	no	—
232	C₄⋊C₄	—	2	8	no	—
233	C₄⋊C₄	—	38	8	no	—
234	C₂²⋊C₄	—	2	8	no	—
235	C₂²⋊C₄	—	38	8	no	—
236	C₄²	—	2	4	no	—
237	C₄²	—	4	8	no	—
238	C₄²	—	19	8	no	—
239	C₄²	—	38	4	no	—
240	C₄²	—	76	8	no	—
241	C₄²	C₄×Dic₁₉	1	1	yes	characteristic
242	C₄²	C₄×Dic₁₉	1	1	yes	characteristic
243	C₄²	C₄×Dic₁₉	1	1	yes	characteristic
244	C₄²	C₄×Dic₁₉	1	1	yes	characteristic
245	C₄²	C₄×Dic₁₉	1	8	yes	complemented
246	C₄²	Dic₁₉⋊C₄	1	2	yes	—
247	C₄²	Dic₁₉⋊C₄	1	2	yes	—
248	C₄²	C₄⋊Dic₁₉	1	2	yes	—
249	C₄²	C₄⋊Dic₁₉	1	2	yes	—
250	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
251	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
252	C₂³	—	4	2	no	—
253	C₂³	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
254	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
255	C₂³	C₄×C₂²⋊D₁₉	1	4	yes	—
256	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	8	no	—
265	C₂×C₄	—	38	8	no	—
266	C₂×C₄	—	76	8	no	—
267	C₂×C₄	C₁₉⋊#95	1	2	yes	—
268	C₂×C₄	C₁₉⋊#95	1	2	yes	—
269	C₂×C₄	C₁₉⋊#95	1	2	yes	—
270	C₂×C₄	C₁₉⋊#95	1	2	yes	—
271	C₂×C₄	C₄×Dic₃₈	1	4	yes	—
272	C₂×C₄	C₄²×D₁₉	1	2	yes	—
273	C₂×C₄	C₄²×D₁₉	1	2	yes	—
274	C₂×C₄	#10597318	1	2	yes	—
275	C₂×C₄	#10597318	1	2	yes	—
276	C₂×C₄	C₄×D₇₆	1	4	yes	—
277	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
278	C₂×C₄	C₂×C₄×Dic₁₉	1	2	yes	—
279	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
280	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
281	C₂×C₄	#10597372	1	2	yes	—
282	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
283	C₂×C₄	C₄×C₂²⋊D₁₉	1	4	yes	—
284	C₈	—	38	8	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	4	4	no	—
289	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
290	C₂²	C₄×C₂²⋊Dic₁₉	1	2	yes	—
291	C₂²	C₄×(C₂×D₁₉)⋊C₄	1	2	yes	—
292	C₂²	C₄×Dic₁₉⋊C₄	1	2	yes	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	4	8	no	—
297	C₄	—	76	8	no	—
298	C₄	#11926300	1	4	yes	direct product factor
299	C₄	C₄²×Dic₁₉	1	1	yes	characteristic
300	C₄	#11926390	1	1	yes	characteristic
301	C₄	C₄×(C₂×D₁₉)⋊C₄	1	2	yes	—
302	C₄	C₄×C₄⋊Dic₁₉	1	2	yes	—
303	C₄	C₄×Dic₁₉⋊C₄	1	2	yes	—
304	C₂	—	2	2	no	—
305	C₂	—	2	2	no	—
306	C₂	—	2	2	no	—
307	C₂	C₄×C₁₉⋊#95	1	1	yes	characteristic
308	C₂	C₂×#11926300	1	2	yes	—

	subgroup	G/H
center	C₄²	C₄×Dic₁₉
commutator	C₄×C₁₉	C₄³
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₄×C₁₉×#117	C₂
socle	C₂²×C₁₉	C₂×#95
prime radical	C₂⁴×C₁₉	C₂×D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₄×#117	D₁₉
hypocenter	C₁₉	C₄×H(3,ℤ/4)
LCS	G ▷ C₄×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
9. C₂×#10597260	C₄	semidirect
10. C₂×C₄×C₁₉⋊C₈	C₄	semidirect
13. #11926390	C₄	semidirect
14. C₄²×Dic₁₉	C₄	semidirect
15. #11926300	C₄	direct (N × H)
20. C₁₉×#376	C₄	semidirect
21. C₄³×C₁₉	C₄	semidirect
80. C₄²×C₁₉	C₄²	semidirect
91. C₄×Dic₁₉	C₄²	semidirect
96. C₂×C₁₉⋊C₈	C₄²	semidirect
147. C₄×C₁₉	H(3,ℤ/4)	semidirect
167. #376	Dic₁₉	semidirect
169. C₄³	Dic₁₉	semidirect
208. C₁₉	C₄×H(3,ℤ/4)	semidirect
245. C₄²	C₄×Dic₁₉	semidirect
298. C₄	#11926300	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×#11926300	2
C₄×C₁₉×#117	2
C₄×H(3,ℤ/4)	19

Subgroups — C4×#11926300 #18087703

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11926300 #18087703