Subgroups — #14572693

Order 2432.

402 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927051	C₂	1	1	yes	maximal, characteristic
2	#11926620	C₂	1	2	yes	maximal, complemented
3	#11926619	C₂	1	2	yes	maximal, complemented
4	#11926615	C₂	1	2	yes	maximal, complemented
5	#11926612	C₂	1	2	yes	maximal, complemented
6	#11926602	C₂	1	2	yes	maximal, complemented
7	#11926511	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926506	C₂	1	2	yes	maximal, complemented
9	C₁₉×#408	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₁₉	C₂²	1	1	yes	characteristic
11	C₂²×C₄×D₁₉	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₁₉	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₈×C₁₉	C₂²	1	1	yes	characteristic, complemented
14	#10597392	C₄	1	2	yes	—
15	#10597384	C₄	1	2	yes	—
16	#10597383	C₄	1	1	yes	characteristic
17	#10597383	C₄	1	1	yes	characteristic
18	#10597381	C₄	1	1	yes	characteristic
19	#10597381	C₄	1	1	yes	characteristic
20	C₄×C₂²⋊D₁₉	C₂²	1	2	yes	—
21	C₄×C₂²⋊D₁₉	C₂²	1	2	yes	—
22	#10597372	C₂²	1	1	yes	characteristic
23	C₂×#64520	C₂²	1	1	yes	characteristic, complemented
24	C₂²×C₁₉⋊C₈	C₂²	1	1	yes	characteristic, complemented
25	C₂×#64515	C₂²	1	2	yes	—
26	C₂×#64515	C₂²	1	2	yes	—
27	C₁₉×C₂²⋊C₈	C₂²	1	2	yes	—
28	C₁₉×C₂²⋊C₈	C₂²	1	2	yes	complemented
29	C₂²⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
30	C₂²⋊(C₁₉⋊C₈)	C₂²	1	2	yes	complemented
31	(C₂×D₁₉)⋊C₈	C₂²	1	2	yes	—
32	(C₂×D₁₉)⋊C₈	C₂²	1	2	yes	—
33	#10597272	C₂²	1	2	yes	complemented
34	Dic₁₉⋊C₈	C₂²	1	2	yes	complemented
35	Dic₁₉⋊C₈	C₂²	1	2	yes	complemented
36	C₈×Dic₁₉	C₂²	1	2	yes	complemented
37	C₂³×D₁₉	C₂×C₄	1	1	yes	characteristic
38	#41×C₁₉	—	4	1	no	—
39	#41×C₁₉	—	4	1	no	—
40	C₂×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
41	C₂×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
42	C₂×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
43	C₂×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
47	C₂×C₂²⋊D₁₉	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊D₁₉	C₂×C₄	1	2	yes	—
49	C₂²×Dic₁₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₁₉	—	4	1	no	—
51	C₂×C₄×D₁₉	—	4	1	no	—
52	C₂×C₄×D₁₉	C₂×C₄	1	2	yes	—
53	C₂×C₄×D₁₉	C₂³	1	2	yes	—
54	M₁₆×C₁₉	—	4	1	no	—
55	C₂×C₈×C₁₉	—	4	1	no	—
56	C₂×C₈×C₁₉	C₂³	1	2	yes	—
57	C₂×C₈×C₁₉	C₂³	1	2	yes	—
58	C₂²⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
59	C₂²⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
60	C₂²⋊Dic₁₉	C₂×C₄	1	2	yes	—
61	C₂²⋊Dic₁₉	C₂×C₄	1	2	yes	—
62	(C₂×D₁₉)⋊C₄	C₂×C₄	1	2	yes	—
63	(C₂×D₁₉)⋊C₄	C₂×C₄	1	2	yes	—
64	C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
65	Dic₁₉⋊C₄	C₂×C₄	1	2	yes	—
66	Dic₁₉⋊C₄	C₂×C₄	1	2	yes	—
67	C₄×Dic₁₉	C₂³	1	2	yes	—
68	#64520	—	4	1	no	—
69	C₂×C₁₉⋊C₈	—	4	1	no	—
70	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
71	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
72	#64515	—	4	2	no	—
73	#64515	—	4	2	no	—
74	C₂³×C₁₉	C₂²×C₄	1	1	yes	characteristic
75	C₂³×C₁₉	C₂²×C₄	1	1	yes	characteristic
76	C₂³×C₁₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×D₁₉	—	4	1	no	—
78	C₂²×D₁₉	—	4	2	no	—
79	C₂²×D₁₉	C₂²×C₄	1	2	yes	—
80	Q₈×C₁₉	—	4	1	no	—
81	D₄×C₁₉	—	4	1	no	—
82	D₄×C₁₉	—	4	1	no	—
83	D₄×C₁₉	—	4	1	no	—
84	C₂×C₄×C₁₉	—	2	1	no	—
85	C₂×C₄×C₁₉	—	2	1	no	—
86	C₂×C₄×C₁₉	—	2	1	no	—
87	C₂×C₄×C₁₉	—	2	1	no	—
88	C₂×C₄×C₁₉	—	2	1	no	—
89	C₂×C₄×C₁₉	—	2	1	no	—
90	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
94	C₂²⋊D₁₉	—	4	2	no	—
95	C₂²⋊D₁₉	—	4	2	no	—
96	C₂×Dic₁₉	—	4	1	no	—
97	C₂×Dic₁₉	C₂²×C₄	1	2	yes	—
98	C₂×Dic₁₉	C₂²×C₄	1	2	yes	—
99	C₂×Dic₁₉	C₂²×C₄	1	2	yes	—
100	C₄×D₁₉	—	2	2	no	—
101	C₄×D₁₉	—	2	2	no	—
102	C₄×D₁₉	—	4	2	no	—
103	C₈×C₁₉	—	2	2	no	—
104	C₈×C₁₉	—	2	2	no	—
105	C₁₉⋊C₈	—	2	2	no	—
106	C₁₉⋊C₈	—	2	2	no	—
107	#2978	—	19	1	no	maximal, Sylow
108	C₂²×C₁₉	—	2	1	no	—
109	C₂²×C₁₉	—	2	1	no	—
110	C₂²×C₁₉	—	2	1	no	—
111	C₂²×C₁₉	—	2	2	no	—
112	C₂²×C₁₉	—	2	2	no	—
113	C₂²×C₁₉	—	2	2	no	—
114	C₂²×C₁₉	C₂³×C₄	1	1	yes	characteristic
115	C₂×D₁₉	—	2	2	no	—
116	C₂×D₁₉	—	2	2	no	—
117	C₂×D₁₉	—	2	2	no	—
118	C₂×D₁₉	—	4	4	no	—
119	C₄×C₁₉	—	2	1	no	—
120	C₄×C₁₉	—	2	1	no	—
121	C₄×C₁₉	—	2	1	no	—
122	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
123	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
124	Dic₁₉	—	2	2	no	—
125	Dic₁₉	—	2	2	no	—
126	Dic₁₉	—	2	2	no	—
127	#525	—	19	1	no	—
128	#436	—	19	2	no	—
129	#436	—	19	2	no	—
130	#435	—	19	2	no	—
131	#435	—	19	2	no	—
132	#433	—	19	2	no	—
133	#408	—	19	1	no	—
134	#408	D₁₉	1	1	yes	characteristic, complemented
135	#407	—	19	2	no	—
136	C₂×C₁₉	—	4	1	no	—
137	C₂×C₁₉	—	4	1	no	—
138	C₂×C₁₉	—	4	1	no	—
139	C₂×C₁₉	#518	1	1	yes	characteristic
140	C₂×C₁₉	#568	1	2	yes	—
141	D₁₉	—	4	2	no	—
142	C₂×#41	C₂×D₁₉	1	1	yes	characteristic
143	C₂³×C₄	—	19	1	no	—
144	C₂×M₁₆	—	19	1	no	—
145	C₂×M₁₆	—	19	2	no	—
146	C₂×M₁₆	—	19	2	no	—
147	C₂×M₁₆	C₂×D₁₉	1	1	yes	characteristic, complemented
148	C₂²×C₈	—	19	1	no	—
149	C₂²×C₈	C₂×D₁₉	1	1	yes	characteristic, complemented
150	#123	—	19	1	no	—
151	#123	—	19	1	no	—
152	C₂²⋊Q₈	—	19	2	no	—
153	#121	—	19	2	no	—
154	#120	—	19	1	no	—
155	#120	—	19	1	no	—
156	C₄×D₄	—	19	2	no	—
157	C₄×D₄	—	19	2	no	—
158	#117	—	19	1	no	—
159	C₄⋊C₈	—	19	2	no	—
160	C₄⋊C₈	—	19	2	no	—
161	C₂²⋊C₈	—	19	2	no	—
162	C₂²⋊C₈	—	19	2	no	—
163	C₂²⋊C₈	—	19	2	no	—
164	C₂²⋊C₈	—	19	2	no	—
165	C₂²⋊C₈	C₂×D₁₉	1	2	yes	—
166	C₂²⋊C₈	C₂×D₁₉	1	2	yes	complemented
167	#97	—	19	2	no	—
168	C₄×C₈	—	19	2	no	—
169	C₁₉	#2978	1	1	yes	characteristic, complemented, Sylow
170	C₂⁴	—	19	1	no	—
171	#41	—	4	1	no	—
172	#41	—	4	1	no	—
173	C₂×Q₈	C₄×D₁₉	1	1	yes	characteristic
174	C₂×D₄	—	19	2	no	—
175	C₂×D₄	—	19	2	no	—
176	C₂×D₄	C₄×D₁₉	1	1	yes	characteristic
177	C₂×D₄	C₄×D₁₉	1	1	yes	characteristic
178	C₂×D₄	C₄×D₁₉	1	1	yes	characteristic
179	C₂²×C₄	—	19	1	no	—
180	C₂²×C₄	—	19	2	no	—
181	C₂²×C₄	—	19	2	no	—
182	C₂²×C₄	—	76	1	no	—
183	C₂²×C₄	—	76	1	no	—
184	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
185	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
186	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
187	M₁₆	—	4	1	no	—
188	M₁₆	—	76	2	no	—
189	M₁₆	—	76	2	no	—
190	M₁₆	—	76	1	no	—
191	C₂×C₈	—	4	1	no	—
192	C₂×C₈	—	19	2	no	—
193	C₂×C₈	—	19	2	no	—
194	C₂×C₈	—	76	1	no	—
195	C₂×C₈	C₂²×D₁₉	1	2	yes	—
196	C₂×C₈	C₂²×D₁₉	1	2	yes	—
197	C₄⋊C₄	—	19	2	no	—
198	C₄⋊C₄	—	19	2	no	—
199	C₄⋊C₄	—	19	2	no	—
200	C₂²⋊C₄	—	19	1	no	—
201	C₂²⋊C₄	—	19	1	no	—
202	C₂²⋊C₄	—	19	2	no	—
203	C₂²⋊C₄	—	19	2	no	—
204	C₂²⋊C₄	—	19	2	no	—
205	C₂²⋊C₄	—	19	2	no	—
206	C₄²	—	19	2	no	—
207	C₂³	—	19	2	no	—
208	C₂³	—	76	1	no	—
209	C₂³	—	76	2	no	—
210	C₂³	C₂×C₄×D₁₉	1	1	yes	characteristic
211	C₂³	C₂×C₄×D₁₉	1	1	yes	characteristic
212	C₂³	C₂×C₄×D₁₉	1	1	yes	characteristic
213	Q₈	—	4	1	no	—
214	D₄	—	4	1	no	—
215	D₄	—	4	1	no	—
216	D₄	—	4	1	no	—
217	D₄	—	76	2	no	—
218	D₄	—	76	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	19	2	no	—
226	C₂×C₄	—	19	2	no	—
227	C₂×C₄	—	19	2	no	—
228	C₂×C₄	—	38	2	no	—
229	C₂×C₄	—	38	2	no	—
230	C₂×C₄	—	76	2	no	—
231	C₂×C₄	—	76	1	no	—
232	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
234	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
235	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	—	2	2	no	—
238	C₈	—	38	2	no	—
239	C₈	—	38	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	38	2	no	—
247	C₂²	—	38	2	no	—
248	C₂²	—	38	2	no	—
249	C₂²	—	76	4	no	—
250	C₂²	C₂²×C₄×D₁₉	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	38	2	no	—
255	C₄	—	38	2	no	—
256	C₄	—	38	2	no	—
257	C₄	#10597430	1	1	yes	characteristic
258	C₄	#10597430	1	1	yes	characteristic
259	C₂	—	4	1	no	—
260	C₂	—	4	1	no	—
261	C₂	—	4	1	no	—
262	C₂	—	76	2	no	—
263	C₂	#11927017	1	1	yes	characteristic
264	C₂	#11927309	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉
commutator, socle	C₂²×C₁₉	C₂³×C₄
Fitting	C₁₉×#408	C₂
prime radical	#11927051	C₂
cosocle	C₂×C₄×C₁₉	C₂⁴
hypercenter	#408	D₁₉
hypocenter	C₁₉	#2978
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ #408

Complements

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

N	H ≅ G/N	type
2. #11926620	C₂	semidirect
3. #11926619	C₂	semidirect
4. #11926615	C₂	semidirect
5. #11926612	C₂	semidirect
6. #11926602	C₂	semidirect
7. #11926511	C₂	semidirect
8. #11926506	C₂	semidirect
9. C₁₉×#408	C₂	semidirect
12. C₂×M₁₆×C₁₉	C₂²	semidirect
13. C₂²×C₈×C₁₉	C₂²	semidirect
23. C₂×#64520	C₂²	semidirect
24. C₂²×C₁₉⋊C₈	C₂²	semidirect
28. C₁₉×C₂²⋊C₈	C₂²	semidirect
30. C₂²⋊(C₁₉⋊C₈)	C₂²	semidirect
33. #10597272	C₂²	semidirect
34. Dic₁₉⋊C₈	C₂²	semidirect
35. Dic₁₉⋊C₈	C₂²	semidirect
36. C₈×Dic₁₉	C₂²	semidirect
134. #408	D₁₉	semidirect
147. C₂×M₁₆	C₂×D₁₉	semidirect
149. C₂²×C₈	C₂×D₁₉	semidirect
166. C₂²⋊C₈	C₂×D₁₉	semidirect
169. C₁₉	#2978	semidirect

Maximal subgroups

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

subgroup	index
#11927051	2
#11926620	2
#11926619	2
#11926615	2
#11926612	2
#11926602	2
#11926511	2
#11926506	2
C₁₉×#408	2
#2978	19