Subgroups — #14572709

Order 2432.

402 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927091	C₂	1	1	yes	maximal, characteristic
2	#11926622	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926621	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926618	C₂	1	2	yes	maximal, complemented
5	#11926615	C₂	1	2	yes	maximal, complemented
6	C₈×D₇₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926600	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926595	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926510	C₂	1	2	yes	maximal, complemented
10	#11926509	C₂	1	2	yes	maximal, complemented
11	C₈×D₄×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#64541	C₂²	1	2	yes	—
13	C₂²×C₈×C₁₉	C₂²	1	2	yes	—
14	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
15	#10597386	C₄	1	1	yes	characteristic
16	#10597384	C₄	1	1	yes	characteristic
17	C₄×C₂²⋊D₁₉	C₂²	1	2	yes	—
18	C₂×#64520	C₂²	1	2	yes	—
19	C₂×#64515	C₂²	1	2	yes	—
20	C₂×C₈×D₁₉	C₂²	1	2	yes	—
21	#10597341	C₄	1	1	yes	characteristic
22	#10597334	C₄	1	1	yes	characteristic
23	#10597331	C₄	1	2	yes	—
24	#10597330	C₄	1	2	yes	—
25	C₄×D₇₆	C₂²	1	1	yes	characteristic
26	C₄²×D₁₉	C₂²	1	1	yes	characteristic
27	C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₁₉×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
30	C₂²⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
31	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
32	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
33	(C₂×D₁₉)⋊C₈	C₂²	1	2	yes	—
34	(C₂×D₁₉)⋊C₈	C₂²	1	2	yes	—
35	(C₂×D₁₉)⋊C₈	C₂²	1	2	yes	—
36	#10597272	C₂²	1	2	yes	—
37	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
38	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
39	C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
40	C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
41	C₂×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
43	C₂×C₂²⋊D₁₉	C₂×C₄	1	2	yes	—
44	C₂×C₂²⋊D₁₉	C₂×C₄	1	2	yes	—
45	C₂×C₂²⋊D₁₉	C₂×C₄	1	2	yes	—
46	#64541	—	4	2	no	—
47	#64541	—	4	2	no	—
48	C₂×D₇₆	C₂×C₄	1	1	yes	characteristic
49	C₂×D₇₆	C₂×C₄	1	2	yes	—
50	C₂×C₄×D₁₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₁₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₁₉	C₂³	1	2	yes	—
54	C₂×Dic₃₈	C₂×C₄	1	2	yes	—
55	C₂×C₈×C₁₉	—	4	2	no	—
56	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₁₉	C₂³	1	2	yes	—
58	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
59	C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
60	C₂²⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
61	C₄²×C₁₉	C₂³	1	1	yes	characteristic
62	C₂²⋊Dic₁₉	C₂×C₄	1	2	yes	—
63	(C₂×D₁₉)⋊C₄	C₂×C₄	1	2	yes	—
64	(C₂×D₁₉)⋊C₄	C₂×C₄	1	2	yes	—
65	C₄⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
66	Dic₁₉⋊C₄	C₂×C₄	1	2	yes	—
67	C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
68	C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
70	#64520	—	4	2	no	—
71	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
74	#64515	—	4	2	no	—
75	C₈×D₁₉	—	4	2	no	—
76	C₂³×C₁₉	C₂²×C₄	1	2	yes	—
77	C₂²×D₁₉	C₂²×C₄	1	2	yes	—
78	C₂²×D₁₉	C₂²×C₄	1	1	yes	characteristic
79	D₄×C₁₉	—	2	2	no	—
80	C₂×C₄×C₁₉	—	2	2	no	—
81	C₂×C₄×C₁₉	—	2	2	no	—
82	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₉	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
86	C₂²⋊D₁₉	—	4	2	no	—
87	C₂²⋊D₁₉	—	4	2	no	—
88	C₂²⋊D₁₉	—	4	2	no	—
89	C₂×Dic₁₉	C₂²×C₄	1	1	yes	characteristic
90	C₂×Dic₁₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×Dic₁₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×Dic₁₉	C₂²×C₄	1	2	yes	—
93	D₇₆	—	2	2	no	—
94	D₇₆	—	4	2	no	—
95	C₄×D₁₉	—	2	2	no	—
96	C₄×D₁₉	—	2	2	no	—
97	C₄×D₁₉	—	2	2	no	—
98	C₄×D₁₉	—	2	1	no	—
99	C₄×D₁₉	—	2	1	no	—
100	C₄×D₁₉	—	4	1	no	—
101	Dic₃₈	—	4	2	no	—
102	C₈×C₁₉	—	2	1	no	—
103	C₈×C₁₉	—	2	2	no	—
104	C₈×C₁₉	—	2	1	no	—
105	C₁₉⋊C₈	—	2	2	no	—
106	C₁₉⋊C₈	—	2	1	no	—
107	C₁₉⋊C₈	—	2	1	no	—
108	#2981	—	19	1	no	maximal, Sylow
109	C₂²×C₁₉	—	2	2	no	—
110	C₂²×C₁₉	—	2	2	no	—
111	C₂²×C₁₉	—	2	2	no	—
112	C₂²×C₁₉	C₂³×C₄	1	1	yes	characteristic
113	C₂×D₁₉	—	2	1	no	—
114	C₂×D₁₉	—	2	2	no	—
115	C₂×D₁₉	—	2	1	no	—
116	C₂×D₁₉	—	2	2	no	—
117	C₂×D₁₉	—	2	1	no	—
118	C₂×D₁₉	—	2	2	no	—
119	C₄×C₁₉	—	2	2	no	—
120	C₄×C₁₉	—	2	1	no	—
121	C₄×C₁₉	#131	1	2	yes	—
122	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
123	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
124	Dic₁₉	—	2	1	no	—
125	Dic₁₉	—	2	2	no	—
126	Dic₁₉	—	2	1	no	—
127	Dic₁₉	—	2	1	no	—
128	#532	—	19	1	no	—
129	#436	—	19	1	no	—
130	#436	—	19	1	no	—
131	#435	—	19	2	no	—
132	#435	—	19	2	no	—
133	C₈×D₄	—	19	1	no	—
134	C₈×D₄	D₁₉	1	1	yes	characteristic, complemented
135	#432	—	19	1	no	—
136	#431	—	19	1	no	—
137	#408	—	19	2	no	—
138	#408	—	19	2	no	—
139	C₂×C₁₉	—	4	2	no	—
140	C₂×C₁₉	#518	1	1	yes	characteristic
141	C₂×C₁₉	C₂×#131	1	1	yes	characteristic
142	C₂×C₁₉	#568	1	1	yes	characteristic
143	D₁₉	—	4	1	no	—
144	D₁₉	—	4	2	no	—
145	C₂×#41	—	19	2	no	—
146	C₂×M₁₆	—	19	2	no	—
147	C₂×M₁₆	—	19	2	no	—
148	C₂²×C₈	—	19	2	no	—
149	C₂²×C₈	C₂×D₁₉	1	2	yes	—
150	C₄⋊Q₈	—	19	1	no	—
151	#127	—	19	1	no	—
152	#124	—	19	2	no	—
153	#121	—	19	1	no	—
154	#121	—	19	1	no	—
155	#121	—	19	2	no	—
156	C₄×D₄	—	19	2	no	—
157	C₄×D₄	—	19	1	no	—
158	C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
159	C₂×C₄²	—	19	1	no	—
160	C₄⋊C₈	—	19	1	no	—
161	C₄⋊C₈	—	19	1	no	—
162	C₄⋊C₈	—	19	1	no	—
163	C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
164	C₂²⋊C₈	—	19	2	no	—
165	C₂²⋊C₈	—	19	1	no	—
166	C₂²⋊C₈	—	19	1	no	—
167	C₂²⋊C₈	—	19	2	no	—
168	C₂²⋊C₈	—	19	2	no	—
169	C₂²⋊C₈	—	19	2	no	—
170	C₂²⋊C₈	C₂×D₁₉	1	2	yes	—
171	#97	—	19	2	no	—
172	C₄×C₈	—	19	1	no	—
173	C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
174	C₁₉	#2981	1	1	yes	characteristic, complemented, Sylow
175	#41	—	76	2	no	—
176	#41	—	76	2	no	—
177	C₂×Q₈	—	19	2	no	—
178	C₂×D₄	—	19	2	no	—
179	C₂×D₄	—	19	2	no	—
180	C₂×D₄	—	19	2	no	—
181	C₂×D₄	—	19	1	no	—
182	C₂×D₄	—	19	2	no	—
183	C₂×D₄	C₄×D₁₉	1	1	yes	characteristic
184	C₂²×C₄	—	19	1	no	—
185	C₂²×C₄	—	19	1	no	—
186	C₂²×C₄	—	19	2	no	—
187	C₂²×C₄	—	19	1	no	—
188	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
189	M₁₆	—	76	2	no	—
190	M₁₆	—	76	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	19	1	no	—
193	C₂×C₈	—	19	1	no	—
194	C₂×C₈	—	19	2	no	—
195	C₂×C₈	—	76	2	no	—
196	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
197	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
198	C₂×C₈	C₂²×D₁₉	1	2	yes	—
199	C₄⋊C₄	—	19	2	no	—
200	C₄⋊C₄	—	19	1	no	—
201	C₄⋊C₄	C₄×D₁₉	1	1	yes	characteristic
202	C₂²⋊C₄	—	19	2	no	—
203	C₂²⋊C₄	—	19	2	no	—
204	C₂²⋊C₄	—	19	2	no	—
205	C₂²⋊C₄	C₄×D₁₉	1	2	yes	—
206	C₄²	—	19	1	no	—
207	C₄²	—	19	1	no	—
208	C₄²	—	19	1	no	—
209	C₄²	C₂²×D₁₉	1	1	yes	characteristic
210	C₂³	—	19	2	no	—
211	C₂³	—	19	1	no	—
212	C₂³	C₂×C₄×D₁₉	1	2	yes	—
213	Q₈	—	76	2	no	—
214	D₄	—	2	2	no	—
215	D₄	—	38	2	no	—
216	D₄	—	76	2	no	—
217	D₄	—	76	2	no	—
218	D₄	—	76	2	no	—
219	D₄	—	76	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	19	1	no	—
223	C₂×C₄	—	19	1	no	—
224	C₂×C₄	—	19	1	no	—
225	C₂×C₄	—	19	2	no	—
226	C₂×C₄	—	38	2	no	—
227	C₂×C₄	—	38	1	no	—
228	C₂×C₄	—	38	2	no	—
229	C₂×C₄	—	38	1	no	—
230	C₂×C₄	—	38	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	2	yes	—
234	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
235	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	2	no	—
238	C₈	—	2	1	no	—
239	C₈	—	38	1	no	—
240	C₈	—	38	2	no	—
241	C₈	—	38	1	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	38	1	no	—
246	C₂²	—	38	2	no	—
247	C₂²	—	38	1	no	—
248	C₂²	—	38	2	no	—
249	C₂²	—	38	1	no	—
250	C₂²	—	38	2	no	—
251	C₂²	C₂²×C₄×D₁₉	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	2	2	no	—
254	C₄	—	38	1	no	—
255	C₄	—	38	1	no	—
256	C₄	—	38	1	no	—
257	C₄	—	38	2	no	—
258	C₄	#10597347	1	2	yes	—
259	C₄	#10597430	1	1	yes	characteristic
260	C₄	#10597435	1	1	yes	characteristic
261	C₂	—	4	2	no	—
262	C₂	—	76	1	no	—
263	C₂	—	76	2	no	—
264	C₂	#11927016	1	1	yes	characteristic
265	C₂	C₂×#10597347	1	1	yes	characteristic
266	C₂	#11927309	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉
commutator, socle	C₂²×C₁₉	C₂³×C₄
Fitting	C₈×D₄×C₁₉	C₂
prime radical	#11927091	C₂
cosocle	C₂×C₄×C₁₉	C₂⁴
hypercenter	C₈×D₄	D₁₉
hypocenter	C₁₉	#2981
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. #11926622	C₂	semidirect
3. #11926621	C₂	semidirect
4. #11926618	C₂	semidirect
5. #11926615	C₂	semidirect
6. C₈×D₇₆	C₂	semidirect
7. #11926600	C₂	semidirect
8. #11926595	C₂	semidirect
9. #11926510	C₂	semidirect
10. #11926509	C₂	semidirect
11. C₈×D₄×C₁₉	C₂	semidirect
134. C₈×D₄	D₁₉	semidirect
174. C₁₉	#2981	semidirect

Maximal subgroups

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

subgroup	index
#11927091	2
#11926622	2
#11926621	2
#11926618	2
#11926615	2
C₈×D₇₆	2
#11926600	2
#11926595	2
#11926510	2
#11926509	2
C₈×D₄×C₁₉	2
#2981	19