Subgroups — #14573172

Order 2432.

504 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×#10597395	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927108	C₂	1	1	yes	maximal, characteristic, complemented
3	#11927027	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926768	C₂	1	2	yes	maximal, complemented
5	#11926761	C₂	1	2	yes	maximal, complemented
6	#11926742	C₂	1	2	yes	maximal, complemented
7	#11926737	C₂	1	2	yes	maximal, complemented
8	#11926570	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926545	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10597278	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₁₉×#135	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₁₉	C₂²	1	1	yes	characteristic
13	C₂×#41×C₁₉	C₂²	1	1	yes	characteristic
14	C₂²×D₇₆	C₂²	1	1	yes	characteristic
15	C₁₉×#135	—	2	4	no	—
16	C₂×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic, complemented
17	C₂×SD₁₆×C₁₉	C₂²	1	1	yes	characteristic, complemented
18	C₂×SD₁₆×C₁₉	C₂²	1	1	yes	characteristic, complemented
19	C₂×D₈×C₁₉	C₂²	1	1	yes	characteristic, complemented
20	C₂²×C₈×C₁₉	C₂²	1	1	yes	characteristic
21	#10597395	—	2	4	no	—
22	#10597393	C₂²	1	2	yes	—
23	Q₈×Dic₁₉	C₂²	1	2	yes	complemented
24	C₂×#64527	C₂²	1	2	yes	—
25	#10597386	C₂²	1	2	yes	—
26	#10597383	—	2	2	no	—
27	D₄×Dic₁₉	—	2	2	no	—
28	D₄×Dic₁₉	C₂²	1	2	yes	complemented
29	C₂×#64525	C₂²	1	2	yes	—
30	#10597376	—	2	1	no	—
31	#10597376	—	2	1	no	—
32	#10597372	—	2	1	no	—
33	#10597372	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊Dic₁₉	—	2	1	no	—
35	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
36	C₂×#64520	C₂²	1	1	yes	characteristic
37	#10597292	C₂²	1	2	yes	complemented
38	#10597289	C₂²	1	2	yes	complemented
39	#10597278	C₂²	1	2	yes	—
40	#10597278	C₂²	1	2	yes	—
41	Dic₁₉⋊C₈	C₂²	1	2	yes	complemented
42	Dic₁₉⋊C₈	C₂²	1	2	yes	complemented
43	C₂³×D₁₉	—	2	1	no	—
44	#41×C₁₉	—	2	2	no	—
45	#41×C₁₉	—	2	2	no	—
46	#41×C₁₉	—	2	2	no	—
47	#41×C₁₉	D₄	1	4	yes	complemented
48	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
49	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₁₉	—	2	1	no	—
51	C₂×D₄×C₁₉	—	2	1	no	—
52	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
53	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	—	2	1	no	—
55	C₂²×C₄×C₁₉	—	2	1	no	—
56	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
57	C₂×C₂²⋊D₁₉	—	2	2	no	—
58	C₂×C₂²⋊D₁₉	—	2	2	no	—
59	C₂²×Dic₁₉	—	2	1	no	—
60	C₂²×Dic₁₉	—	2	2	no	—
61	C₂×D₇₆	—	2	2	no	—
62	C₂×D₇₆	—	2	2	no	—
63	C₂×D₇₆	—	2	1	no	—
64	C₂×D₇₆	C₂³	1	2	yes	—
65	Q₁₆×C₁₉	—	2	2	no	—
66	SD₁₆×C₁₉	—	2	2	no	—
67	SD₁₆×C₁₉	—	2	2	no	—
68	D₈×C₁₉	—	2	2	no	—
69	C₂×C₈×C₁₉	—	2	2	no	—
70	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
72	C₂²⋊Dic₁₉	—	2	2	no	—
73	C₂²⋊Dic₁₉	D₄	1	2	yes	—
74	#64527	—	2	4	no	—
75	#64525	—	2	4	no	—
76	(C₂×D₁₉)⋊C₄	—	2	2	no	—
77	(C₂×D₁₉)⋊C₄	—	2	2	no	—
78	C₄⋊Dic₁₉	—	2	1	no	—
79	C₄⋊Dic₁₉	—	2	2	no	—
80	C₄⋊Dic₁₉	D₄	1	1	yes	characteristic
81	C₄⋊Dic₁₉	D₄	1	1	yes	characteristic
82	C₄⋊Dic₁₉	C₂³	1	2	yes	—
83	C₄×Dic₁₉	—	2	2	no	—
84	C₄×Dic₁₉	C₂³	1	2	yes	—
85	#64520	—	2	2	no	—
86	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
87	C₂³×C₁₉	—	2	1	no	—
88	C₂³×C₁₉	—	2	1	no	—
89	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
90	C₂²×D₁₉	—	2	2	no	—
91	C₂²×D₁₉	—	4	2	no	—
92	C₂²×D₁₉	—	8	1	no	—
93	C₂²×D₁₉	—	8	1	no	—
94	Q₈×C₁₉	—	2	1	no	—
95	Q₈×C₁₉	—	2	1	no	—
96	Q₈×C₁₉	—	2	1	no	—
97	Q₈×C₁₉	C₂×D₄	1	2	yes	—
98	D₄×C₁₉	—	2	4	no	—
99	D₄×C₁₉	—	2	1	no	—
100	D₄×C₁₉	—	2	1	no	—
101	D₄×C₁₉	—	2	1	no	—
102	D₄×C₁₉	—	4	2	no	—
103	D₄×C₁₉	C₂×D₄	1	2	yes	—
104	C₂×C₄×C₁₉	—	2	1	no	—
105	C₂×C₄×C₁₉	—	2	4	no	—
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	4	2	no	—
108	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₁₉	#41	1	2	yes	—
111	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
112	C₂²⋊D₁₉	—	8	2	no	—
113	C₂²⋊D₁₉	—	8	2	no	—
114	C₂×Dic₁₉	—	2	2	no	—
115	C₂×Dic₁₉	—	8	2	no	—
116	C₂×Dic₁₉	—	8	1	no	—
117	C₂×Dic₁₉	C₂×D₄	1	2	yes	—
118	C₂×Dic₁₉	C₂×D₄	1	2	yes	—
119	D₇₆	—	2	2	no	—
120	D₇₆	—	2	2	no	—
121	D₇₆	—	4	2	no	—
122	C₈×C₁₉	—	2	1	no	—
123	C₈×C₁₉	—	2	1	no	—
124	C₁₉⋊C₈	—	2	2	no	—
125	#3045	—	19	1	no	maximal, Sylow
126	C₂²×C₁₉	—	2	1	no	—
127	C₂²×C₁₉	—	2	1	no	—
128	C₂²×C₁₉	—	2	2	no	—
129	C₂²×C₁₉	—	4	1	no	—
130	C₂²×C₁₉	—	4	1	no	—
131	C₂²×C₁₉	—	4	1	no	—
132	C₂²×C₁₉	—	4	1	no	—
133	C₂²×C₁₉	—	4	1	no	—
134	C₂²×C₁₉	#121	1	2	yes	—
135	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
136	C₂×D₁₉	—	4	2	no	—
137	C₂×D₁₉	—	4	2	no	—
138	C₂×D₁₉	—	4	2	no	—
139	C₂×D₁₉	—	8	4	no	—
140	C₄×C₁₉	—	2	2	no	—
141	C₄×C₁₉	—	4	1	no	—
142	C₄×C₁₉	#121	1	2	yes	—
143	C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
144	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
145	Dic₁₉	—	2	2	no	—
146	Dic₁₉	—	2	2	no	—
147	Dic₁₉	—	4	2	no	—
148	C₂×AGL(1,ℤ/8)	—	19	1	no	—
149	C₂×#135	D₁₉	1	1	yes	characteristic, complemented
150	#535	—	19	1	no	—
151	#520	—	19	1	no	—
152	#464	—	19	2	no	—
153	#463	—	19	2	no	—
154	#460	—	19	2	no	—
155	#459	—	19	2	no	—
156	#424	—	19	1	no	—
157	#417	—	19	1	no	—
158	C₂×#102	—	19	1	no	—
159	C₂×C₁₉	—	2	2	no	—
160	C₂×C₁₉	—	4	2	no	—
161	C₂×C₁₉	—	8	1	no	—
162	C₂×C₁₉	C₂×#121	1	1	yes	characteristic
163	C₂×C₁₉	#576	1	1	yes	characteristic
164	C₂×C₁₉	#577	1	1	yes	characteristic
165	D₁₉	—	8	2	no	—
166	C₂×#41	C₂×D₁₉	1	1	yes	characteristic
167	C₂×#41	C₂×D₁₉	1	1	yes	characteristic
168	C₂²×D₄	—	19	1	no	—
169	AGL(1,ℤ/8)	—	38	4	no	—
170	#135	—	2	4	no	—
171	C₂×Q₁₆	C₂×D₁₉	1	1	yes	characteristic, complemented
172	C₂×SD₁₆	—	19	2	no	—
173	C₂×SD₁₆	C₂×D₁₉	1	1	yes	characteristic, complemented
174	C₂×SD₁₆	C₂×D₁₉	1	1	yes	characteristic, complemented
175	C₂×D₈	—	19	2	no	—
176	C₂×D₈	C₂×D₁₉	1	1	yes	characteristic, complemented
177	C₂×M₁₆	—	19	1	no	—
178	C₂²×C₈	C₂×D₁₉	1	1	yes	characteristic
179	#127	—	19	2	no	—
180	#124	—	19	2	no	—
181	#121	—	38	1	no	—
182	#121	—	38	1	no	—
183	#120	—	38	2	no	—
184	C₄×Q₈	—	19	2	no	—
185	C₄×D₄	—	19	2	no	—
186	C₄×D₄	—	38	2	no	—
187	#117	—	19	1	no	—
188	#117	—	38	1	no	—
189	C₂×C₄⋊C₄	—	19	1	no	—
190	C₂×C₄⋊C₄	—	38	1	no	—
191	C₄⋊C₈	—	19	2	no	—
192	C₄⋊C₈	—	19	2	no	—
193	#103	—	19	2	no	—
194	#102	—	19	2	no	—
195	#102	—	19	2	no	—
196	#102	—	19	2	no	—
197	C₁₉	#3045	1	1	yes	characteristic, complemented, Sylow
198	C₂⁴	—	38	1	no	—
199	#41	—	2	2	no	—
200	#41	—	2	2	no	—
201	#41	—	2	2	no	—
202	#41	C₂²⋊D₁₉	1	4	yes	complemented
203	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
204	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
205	C₂×D₄	—	2	1	no	—
206	C₂×D₄	—	2	1	no	—
207	C₂×D₄	—	19	2	no	—
208	C₂×D₄	—	38	2	no	—
209	C₂×D₄	—	38	2	no	—
210	C₂×D₄	—	38	1	no	—
211	C₂×D₄	—	38	2	no	—
212	C₂×D₄	—	38	2	no	—
213	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
214	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	38	2	no	—
218	C₂²×C₄	—	38	1	no	—
219	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
220	Q₁₆	—	2	2	no	—
221	SD₁₆	—	2	2	no	—
222	SD₁₆	—	2	2	no	—
223	SD₁₆	—	38	4	no	—
224	D₈	—	2	2	no	—
225	D₈	—	38	4	no	—
226	M₁₆	—	38	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	19	2	no	—
229	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
230	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
231	C₄⋊C₄	—	19	1	no	—
232	C₄⋊C₄	—	19	1	no	—
233	C₄⋊C₄	—	19	2	no	—
234	C₄⋊C₄	—	38	2	no	—
235	C₄⋊C₄	—	38	1	no	—
236	C₂²⋊C₄	—	19	2	no	—
237	C₂²⋊C₄	—	38	2	no	—
238	C₂²⋊C₄	—	38	2	no	—
239	C₂²⋊C₄	—	38	2	no	—
240	C₄²	—	19	2	no	—
241	C₄²	—	38	2	no	—
242	C₂³	—	2	1	no	—
243	C₂³	—	2	1	no	—
244	C₂³	—	38	2	no	—
245	C₂³	—	76	2	no	—
246	C₂³	—	152	1	no	—
247	C₂³	—	152	1	no	—
248	C₂³	D₄×D₁₉	1	1	yes	characteristic
249	Q₈	—	2	1	no	—
250	Q₈	—	2	1	no	—
251	Q₈	—	2	1	no	—
252	Q₈	C₂×C₂²⋊D₁₉	1	2	yes	—
253	D₄	—	2	4	no	—
254	D₄	—	2	1	no	—
255	D₄	—	2	1	no	—
256	D₄	—	2	1	no	—
257	D₄	—	4	2	no	—
258	D₄	—	38	2	no	—
259	D₄	—	38	2	no	—
260	D₄	—	76	2	no	—
261	D₄	—	152	2	no	—
262	D₄	—	152	2	no	—
263	D₄	C₂×C₂²⋊D₁₉	1	2	yes	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	19	2	no	—
269	C₂×C₄	—	19	2	no	—
270	C₂×C₄	—	38	2	no	—
271	C₂×C₄	—	152	2	no	—
272	C₂×C₄	—	152	1	no	—
273	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
274	C₂×C₄	#64543	1	2	yes	—
275	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
276	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	38	2	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	1	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	76	2	no	—
289	C₂²	—	76	2	no	—
290	C₂²	—	76	2	no	—
291	C₂²	—	152	4	no	—
292	C₂²	#10597385	1	2	yes	—
293	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	4	1	no	—
296	C₄	—	38	2	no	—
297	C₄	—	38	2	no	—
298	C₄	—	76	2	no	—
299	C₄	#10597385	1	2	yes	—
300	C₄	C₂×#64543	1	1	yes	characteristic
301	C₄	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
302	C₂	—	2	2	no	—
303	C₂	—	4	2	no	—
304	C₂	—	8	1	no	—
305	C₂	—	152	2	no	—
306	C₂	C₂×#10597385	1	1	yes	characteristic
307	C₂	#11927346	1	1	yes	characteristic
308	C₂	#11927351	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₉
commutator, cosocle	C₂×C₄×C₁₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉
Fitting	C₂×C₁₉×#135	C₂
socle	C₂²×C₁₉	C₂²×D₄
hypercenter	C₂×#135	D₁₉
hypocenter	C₁₉	#3045
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#10597395	C₂	semidirect
2. #11927108	C₂	semidirect
3. #11927027	C₂	semidirect
4. #11926768	C₂	semidirect
5. #11926761	C₂	semidirect
6. #11926742	C₂	semidirect
7. #11926737	C₂	semidirect
8. #11926570	C₂	semidirect
9. #11926545	C₂	semidirect
10. C₂×#10597278	C₂	semidirect
11. C₂×C₁₉×#135	C₂	semidirect
16. C₂×Q₁₆×C₁₉	C₂²	semidirect
17. C₂×SD₁₆×C₁₉	C₂²	semidirect
18. C₂×SD₁₆×C₁₉	C₂²	semidirect
19. C₂×D₈×C₁₉	C₂²	semidirect
23. Q₈×Dic₁₉	C₂²	semidirect
28. D₄×Dic₁₉	C₂²	semidirect
37. #10597292	C₂²	semidirect
38. #10597289	C₂²	semidirect
41. Dic₁₉⋊C₈	C₂²	semidirect
42. Dic₁₉⋊C₈	C₂²	semidirect
47. #41×C₁₉	D₄	semidirect
149. C₂×#135	D₁₉	semidirect
171. C₂×Q₁₆	C₂×D₁₉	semidirect
173. C₂×SD₁₆	C₂×D₁₉	semidirect
174. C₂×SD₁₆	C₂×D₁₉	semidirect
176. C₂×D₈	C₂×D₁₉	semidirect
197. C₁₉	#3045	semidirect
202. #41	C₂²⋊D₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10597395	2
#11927108	2
#11927027	2
#11926768	2
#11926761	2
#11926742	2
#11926737	2
#11926570	2
#11926545	2
C₂×#10597278	2
C₂×C₁₉×#135	2
#3045	19