Subgroups — C₂×C₄×#64543 #14576813

Order 2432.

1728 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#64543	C₂	1	1	yes	maximal, characteristic
2	C₄×#64543	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×Dic₃₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#10597333	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₄×Dic₁₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×C₂²⋊D₁₉	C₂	1	2	yes	maximal, complemented
7	C₂×(C₂²⋊D₁₉)⋊C₄	C₂	1	2	yes	maximal, complemented
8	C₂×#10597338	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#10597323	C₂	1	2	yes	maximal, complemented
10	C₂²×C₄×Dic₁₉	C₂	1	2	yes	maximal, complemented
11	C₂×C₄²×D₁₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄×D₄×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×D₄×C₁₉	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₁₉	C₂²	1	2	yes	—
15	C₂²×C₂²⋊D₁₉	C₂²	1	2	yes	—
16	C₂³×Dic₁₉	C₂²	1	2	yes	—
17	C₂³×Dic₁₉	C₂²	1	2	yes	—
18	C₂×#64543	C₄	1	16	yes	direct product factor
19	C₂×#64543	C₂²	1	8	yes	—
20	C₂²×C₄×D₁₉	C₂²	1	1	yes	characteristic
21	C₂²×C₄×D₁₉	C₂²	1	1	yes	characteristic
22	C₂²×C₄×D₁₉	C₂²	1	1	yes	characteristic
23	C₂²×Dic₃₈	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₁₉	C₂²	1	8	yes	complemented
25	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₁₉	C₂²	1	2	yes	—
27	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊Dic₁₉	C₂²	1	2	yes	—
29	D₄×Dic₁₉	C₂²	1	8	yes	complemented
30	C₄×C₂²⋊D₁₉	C₂²	1	16	yes	complemented
31	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
32	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
33	C₂×Dic₁₉⋊C₄	C₂²	1	2	yes	—
34	C₂×Dic₁₉⋊C₄	C₂²	1	2	yes	—
35	C₂×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
36	C₂×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
37	C₂×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
38	C₂×C₄×Dic₁₉	C₂²	1	2	yes	—
39	C₂×C₄×Dic₁₉	C₂²	1	2	yes	—
40	C₂×C₄×Dic₁₉	C₂²	1	2	yes	—
41	C₂×C₄×Dic₁₉	C₂²	1	16	yes	complemented
42	#10597338	C₂²	1	8	yes	complemented
43	#10597333	C₂²	1	8	yes	complemented
44	(C₂²⋊D₁₉)⋊C₄	C₂²	1	16	yes	complemented
45	#10597323	C₂²	1	16	yes	complemented
46	C₄²×D₁₉	C₂²	1	8	yes	complemented
47	C₄×Dic₃₈	C₂²	1	8	yes	complemented
48	C₂⁴×C₁₉	C₂³	1	2	yes	—
49	C₂³×D₁₉	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₁₉	C₂×C₄	1	8	yes	complemented
51	C₂×D₄×C₁₉	C₂³	1	4	yes	—
52	C₂²×C₄×C₁₉	—	2	8	no	—
53	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
55	C₂²×C₄×C₁₉	C₂³	1	8	yes	—
56	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
58	C₂×C₂²⋊D₁₉	C₂×C₄	1	16	yes	complemented
59	C₂×C₂²⋊D₁₉	C₂³	1	8	yes	—
60	C₂²×Dic₁₉	—	2	8	no	—
61	C₂²×Dic₁₉	C₂×C₄	1	16	yes	complemented
62	C₂²×Dic₁₉	C₂³	1	8	yes	—
63	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
64	C₂²×Dic₁₉	C₂³	1	2	yes	—
65	C₂²×Dic₁₉	C₂³	1	2	yes	—
66	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
67	C₂²×Dic₁₉	C₂³	1	8	yes	—
68	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
69	#64543	C₂×C₄	1	64	yes	direct product factor
70	C₂×C₄×D₁₉	—	2	4	no	—
71	C₂×C₄×D₁₉	—	2	4	no	—
72	C₂×C₄×D₁₉	C₂×C₄	1	8	yes	complemented
73	C₂×C₄×D₁₉	C₂³	1	4	yes	—
74	C₂×C₄×D₁₉	C₂³	1	4	yes	—
75	C₂×C₄×D₁₉	C₂³	1	4	yes	—
76	C₂×Dic₃₈	C₂×C₄	1	8	yes	complemented
77	C₂×Dic₃₈	C₂³	1	4	yes	—
78	C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
79	C₂²⋊C₄×C₁₉	C₂³	1	8	yes	—
80	C₄²×C₁₉	C₂³	1	4	yes	—
81	C₂²⋊Dic₁₉	C₂³	1	8	yes	—
82	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
83	C₄⋊Dic₁₉	C₂³	1	4	yes	—
84	Dic₁₉⋊C₄	C₂³	1	8	yes	—
85	Dic₁₉⋊C₄	C₂³	1	8	yes	—
86	C₄×Dic₁₉	C₂³	1	4	yes	—
87	C₄×Dic₁₉	C₂³	1	8	yes	—
88	C₄×Dic₁₉	C₂³	1	4	yes	—
89	C₄×Dic₁₉	C₂³	1	4	yes	—
90	C₄×Dic₁₉	C₂³	1	8	yes	—
91	C₄×Dic₁₉	C₂³	1	8	yes	—
92	C₂³×C₁₉	—	2	4	no	—
93	C₂³×C₁₉	—	2	4	no	—
94	C₂³×C₁₉	C₂²×C₄	1	8	yes	—
95	C₂³×C₁₉	C₂⁴	1	4	yes	—
96	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
97	C₂²×D₁₉	—	2	2	no	—
98	C₂²×D₁₉	—	2	2	no	—
99	C₂²×D₁₉	C₂²×C₄	1	4	yes	—
100	C₂²×D₁₉	C₂⁴	1	2	yes	—
101	D₄×C₁₉	C₂²×C₄	1	16	yes	complemented
102	C₂×C₄×C₁₉	—	2	2	no	—
103	C₂×C₄×C₁₉	—	2	4	no	—
104	C₂×C₄×C₁₉	—	2	16	no	—
105	C₂×C₄×C₁₉	C₂²×C₄	1	4	yes	—
106	C₂×C₄×C₁₉	#41	1	4	yes	complemented
107	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
108	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
109	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
110	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
111	C₂²⋊D₁₉	C₂²×C₄	1	32	yes	complemented
112	C₂×Dic₁₉	—	2	2	no	—
113	C₂×Dic₁₉	—	2	4	no	—
114	C₂×Dic₁₉	—	2	16	no	—
115	C₂×Dic₁₉	C₂²×C₄	1	8	yes	—
116	C₂×Dic₁₉	C₂²×C₄	1	4	yes	—
117	C₂×Dic₁₉	C₂²×C₄	1	32	yes	complemented
118	C₂×Dic₁₉	#41	1	4	yes	complemented
119	C₂×Dic₁₉	C₂⁴	1	2	yes	—
120	C₂×Dic₁₉	C₂⁴	1	4	yes	—
121	C₂×Dic₁₉	C₂⁴	1	2	yes	—
122	C₂×Dic₁₉	C₂⁴	1	4	yes	—
123	C₂×Dic₁₉	C₂⁴	1	2	yes	—
124	C₄×D₁₉	—	2	8	no	—
125	C₄×D₁₉	—	2	8	no	—
126	C₄×D₁₉	C₂²×C₄	1	16	yes	complemented
127	Dic₃₈	C₂²×C₄	1	16	yes	complemented
128	C₂×C₄×#41	—	19	1	no	maximal, Sylow
129	C₂²×C₁₉	—	2	16	no	—
130	C₂²×C₁₉	—	2	4	no	—
131	C₂²×C₁₉	—	2	4	no	—
132	C₂²×C₁₉	C₂³×C₄	1	8	yes	—
133	C₂²×C₁₉	C₂³×C₄	1	2	yes	—
134	C₂²×C₁₉	C₂×#41	1	2	yes	—
135	C₂²×C₁₉	C₂×#41	1	2	yes	—
136	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
137	C₂×D₁₉	—	2	8	no	—
138	C₂×D₁₉	—	2	2	no	—
139	C₂×D₁₉	—	2	2	no	—
140	C₂×D₁₉	C₂³×C₄	1	4	yes	—
141	C₄×C₁₉	—	2	2	no	—
142	C₄×C₁₉	—	2	4	no	—
143	C₄×C₁₉	C₂³×C₄	1	4	yes	—
144	C₄×C₁₉	C₂×#41	1	4	yes	complemented
145	Dic₁₉	—	2	2	no	—
146	Dic₁₉	—	2	4	no	—
147	Dic₁₉	C₂³×C₄	1	8	yes	—
148	Dic₁₉	C₂³×C₄	1	4	yes	—
149	Dic₁₉	C₂×#41	1	4	yes	complemented
150	C₂²×#41	—	19	1	no	—
151	C₄×#41	—	19	16	no	—
152	C₂×C₄×Q₈	—	19	1	no	—
153	C₂×C₄×Q₈	—	19	1	no	—
154	C₂×C₄×D₄	—	19	1	no	—
155	C₂×C₄×D₄	—	19	2	no	—
156	C₂×C₄×D₄	—	19	2	no	—
157	C₂×C₄×D₄	D₁₉	1	1	yes	characteristic, complemented
158	C₂×#117	—	19	1	no	—
159	C₂×#117	—	19	2	no	—
160	C₂²×C₄²	—	19	2	no	—
161	C₂²×C₄²	—	19	1	no	—
162	C₂×C₁₉	—	2	8	no	—
163	C₂×C₁₉	C₄×#41	1	4	yes	complemented
164	C₂×C₁₉	C₂⁴×C₄	1	1	yes	characteristic
165	C₂×C₁₉	C₂²×#41	1	1	yes	characteristic
166	C₂×C₁₉	C₂²×#41	1	1	yes	characteristic
167	D₁₉	—	2	4	no	—
168	C₂×#41	—	19	16	no	—
169	C₂×#41	—	19	8	no	—
170	C₂²×Q₈	—	19	1	no	—
171	C₂²×D₄	—	19	2	no	—
172	C₂²×D₄	C₂×D₁₉	1	1	yes	characteristic
173	C₂³×C₄	—	19	2	no	—
174	C₂³×C₄	—	19	1	no	—
175	C₂³×C₄	—	19	2	no	—
176	C₂³×C₄	—	19	1	no	—
177	C₂³×C₄	—	19	1	no	—
178	C₂³×C₄	C₂×D₁₉	1	2	yes	—
179	C₄×Q₈	—	19	8	no	—
180	C₄×Q₈	—	19	8	no	—
181	C₄×D₄	—	19	8	no	—
182	C₄×D₄	—	19	16	no	—
183	C₄×D₄	—	19	16	no	—
184	C₄×D₄	C₂×D₁₉	1	8	yes	complemented
185	#117	—	19	8	no	—
186	#117	—	19	16	no	—
187	C₂×C₄⋊C₄	—	19	2	no	—
188	C₂×C₄⋊C₄	—	19	2	no	—
189	C₂×C₄⋊C₄	—	19	1	no	—
190	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
191	C₂×C₂²⋊C₄	—	19	2	no	—
192	C₂×C₂²⋊C₄	—	19	2	no	—
193	C₂×C₂²⋊C₄	C₂×D₁₉	1	2	yes	—
194	C₂×C₄²	—	19	2	no	—
195	C₂×C₄²	—	19	1	no	—
196	C₂×C₄²	—	19	2	no	—
197	C₂×C₄²	—	19	1	no	—
198	C₂×C₄²	—	19	16	no	—
199	C₂×C₄²	—	19	2	no	—
200	C₂×C₄²	—	19	1	no	—
201	C₂×C₄²	—	19	8	no	—
202	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
203	C₁₉	C₂×C₄×#41	1	1	yes	characteristic, complemented, Sylow
204	C₂⁴	—	19	1	no	—
205	C₂⁴	C₂²×D₁₉	1	2	yes	—
206	#41	—	19	64	no	—
207	C₂×Q₈	—	19	8	no	—
208	C₂×Q₈	—	19	4	no	—
209	C₂×D₄	—	19	16	no	—
210	C₂×D₄	—	19	8	no	—
211	C₂×D₄	C₄×D₁₉	1	8	yes	complemented
212	C₂×D₄	C₂²×D₁₉	1	4	yes	—
213	C₂²×C₄	—	2	8	no	—
214	C₂²×C₄	—	19	16	no	—
215	C₂²×C₄	—	19	1	no	—
216	C₂²×C₄	—	19	1	no	—
217	C₂²×C₄	—	19	2	no	—
218	C₂²×C₄	—	19	8	no	—
219	C₂²×C₄	—	19	8	no	—
220	C₂²×C₄	—	19	4	no	—
221	C₂²×C₄	—	19	1	no	—
222	C₂²×C₄	—	19	2	no	—
223	C₂²×C₄	—	19	8	no	—
224	C₂²×C₄	—	19	4	no	—
225	C₂²×C₄	—	19	4	no	—
226	C₂²×C₄	—	38	4	no	—
227	C₂²×C₄	—	38	8	no	—
228	C₂²×C₄	—	38	4	no	—
229	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
230	C₂²×C₄	C₂²×D₁₉	1	8	yes	—
231	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
232	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
233	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
234	C₄⋊C₄	—	19	8	no	—
235	C₄⋊C₄	—	19	8	no	—
236	C₄⋊C₄	—	19	4	no	—
237	C₄⋊C₄	C₂²×D₁₉	1	4	yes	—
238	C₂²⋊C₄	—	19	8	no	—
239	C₂²⋊C₄	—	19	8	no	—
240	C₂²⋊C₄	C₂²×D₁₉	1	8	yes	—
241	C₄²	—	19	4	no	—
242	C₄²	—	19	8	no	—
243	C₄²	—	19	4	no	—
244	C₄²	—	19	8	no	—
245	C₄²	—	19	4	no	—
246	C₄²	—	19	8	no	—
247	C₄²	C₂²×D₁₉	1	4	yes	—
248	C₂³	—	2	4	no	—
249	C₂³	—	2	4	no	—
250	C₂³	—	19	4	no	—
251	C₂³	—	19	2	no	—
252	C₂³	—	38	2	no	—
253	C₂³	—	38	2	no	—
254	C₂³	C₂×C₄×D₁₉	1	8	yes	—
255	C₂³	C₂³×D₁₉	1	4	yes	—
256	C₂³	C₂³×D₁₉	1	1	yes	characteristic
257	Q₈	—	19	16	no	—
258	D₄	—	19	32	no	—
259	D₄	C₂×C₄×D₁₉	1	16	yes	complemented
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	16	no	—
263	C₂×C₄	—	19	4	no	—
264	C₂×C₄	—	19	8	no	—
265	C₂×C₄	—	19	32	no	—
266	C₂×C₄	—	19	4	no	—
267	C₂×C₄	—	19	2	no	—
268	C₂×C₄	—	19	2	no	—
269	C₂×C₄	—	19	4	no	—
270	C₂×C₄	—	19	16	no	—
271	C₂×C₄	—	19	4	no	—
272	C₂×C₄	—	19	2	no	—
273	C₂×C₄	—	38	8	no	—
274	C₂×C₄	—	38	16	no	—
275	C₂×C₄	—	38	2	no	—
276	C₂×C₄	—	38	4	no	—
277	C₂×C₄	—	38	8	no	—
278	C₂×C₄	C₂×C₄×D₁₉	1	4	yes	—
279	C₂×C₄	#64543	1	4	yes	direct product factor
280	C₂×C₄	C₂³×D₁₉	1	4	yes	—
281	C₂×C₄	C₂³×D₁₉	1	2	yes	—
282	C₂×C₄	C₂³×D₁₉	1	2	yes	—
283	C₂×C₄	C₂³×D₁₉	1	2	yes	—
284	C₂²	—	2	4	no	—
285	C₂²	—	2	4	no	—
286	C₂²	—	2	16	no	—
287	C₂²	—	19	4	no	—
288	C₂²	—	38	2	no	—
289	C₂²	—	38	2	no	—
290	C₂²	—	38	8	no	—
291	C₂²	C₂²×C₄×D₁₉	1	8	yes	—
292	C₂²	C₂²×C₄×D₁₉	1	2	yes	—
293	C₂²	C₂×#64543	1	2	yes	—
294	C₂²	#41×D₁₉	1	2	yes	—
295	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	4	no	—
298	C₄	—	19	4	no	—
299	C₄	—	19	8	no	—
300	C₄	—	19	4	no	—
301	C₄	—	38	4	no	—
302	C₄	—	38	2	no	—
303	C₄	C₂²×C₄×D₁₉	1	4	yes	—
304	C₄	C₂×#64543	1	4	yes	direct product factor
305	C₂	—	2	8	no	—
306	C₂	—	38	4	no	—
307	C₂	C₄×#64543	1	4	yes	direct product factor
308	C₂	C₂³×C₄×D₁₉	1	1	yes	characteristic
309	C₂	C₂×#41×D₁₉	1	1	yes	characteristic
310	C₂	C₂²×#64543	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₉
commutator	C₂×C₁₉	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₉
Fitting	C₂×C₄×D₄×C₁₉	C₂
socle	C₂³×C₁₉	C₂⁴
prime radical	C₂²×#64543	C₂
cosocle	C₂²×C₁₉	C₂⁵
hypercenter	C₂×C₄×D₄	D₁₉
hypocenter	C₁₉	C₂×C₄×#41
LCS	G ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₄×#64543	C₂	direct (N × H)
3. C₂×C₄×Dic₃₈	C₂	semidirect
4. C₂×#10597333	C₂	semidirect
5. C₂×D₄×Dic₁₉	C₂	semidirect
6. C₂×C₄×C₂²⋊D₁₉	C₂	semidirect
7. C₂×(C₂²⋊D₁₉)⋊C₄	C₂	semidirect
8. C₂×#10597338	C₂	semidirect
9. C₂×#10597323	C₂	semidirect
10. C₂²×C₄×Dic₁₉	C₂	semidirect
11. C₂×C₄²×D₁₉	C₂	semidirect
12. C₂×C₄×D₄×C₁₉	C₂	semidirect
18. C₂×#64543	C₄	direct (N × H)
24. C₄×D₄×C₁₉	C₂²	semidirect
29. D₄×Dic₁₉	C₂²	semidirect
30. C₄×C₂²⋊D₁₉	C₂²	semidirect
41. C₂×C₄×Dic₁₉	C₂²	semidirect
42. #10597338	C₂²	semidirect
43. #10597333	C₂²	semidirect
44. (C₂²⋊D₁₉)⋊C₄	C₂²	semidirect
45. #10597323	C₂²	semidirect
46. C₄²×D₁₉	C₂²	semidirect
47. C₄×Dic₃₈	C₂²	semidirect
50. C₂×D₄×C₁₉	C₂×C₄	semidirect
58. C₂×C₂²⋊D₁₉	C₂×C₄	semidirect
61. C₂²×Dic₁₉	C₂×C₄	semidirect
69. #64543	C₂×C₄	direct (N × H)
72. C₂×C₄×D₁₉	C₂×C₄	semidirect
76. C₂×Dic₃₈	C₂×C₄	semidirect
101. D₄×C₁₉	C₂²×C₄	semidirect
106. C₂×C₄×C₁₉	#41	semidirect
111. C₂²⋊D₁₉	C₂²×C₄	semidirect
117. C₂×Dic₁₉	C₂²×C₄	semidirect
118. C₂×Dic₁₉	#41	semidirect
126. C₄×D₁₉	C₂²×C₄	semidirect
127. Dic₃₈	C₂²×C₄	semidirect
144. C₄×C₁₉	C₂×#41	semidirect
149. Dic₁₉	C₂×#41	semidirect
157. C₂×C₄×D₄	D₁₉	semidirect
163. C₂×C₁₉	C₄×#41	semidirect
184. C₄×D₄	C₂×D₁₉	semidirect
203. C₁₉	C₂×C₄×#41	semidirect
211. C₂×D₄	C₄×D₁₉	semidirect
259. D₄	C₂×C₄×D₁₉	semidirect
279. C₂×C₄	#64543	direct (N × H)
304. C₄	C₂×#64543	direct (N × H)
307. C₂	C₄×#64543	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#64543	2
C₄×#64543	2
C₂×C₄×Dic₃₈	2
C₂×#10597333	2
C₂×D₄×Dic₁₉	2
C₂×C₄×C₂²⋊D₁₉	2
C₂×(C₂²⋊D₁₉)⋊C₄	2
C₂×#10597338	2
C₂×#10597323	2
C₂²×C₄×Dic₁₉	2
C₂×C₄²×D₁₉	2
C₂×C₄×D₄×C₁₉	2
C₂×C₄×#41	19

Subgroups — C2×C4×#64543 #14576813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#64543 #14576813