Subgroups — #14572783

Order 2432.

354 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927081	C₂	1	1	yes	maximal, characteristic
2	#11926675	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926623	C₂	1	2	yes	maximal, complemented
4	#11926621	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926614	C₂	1	2	yes	maximal, complemented
6	C₈×D₇₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926603	C₂	1	2	yes	maximal, complemented
8	#11926595	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926569	C₂	1	2	yes	maximal, complemented
10	#11926566	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
13	#10597393	C₄	1	1	yes	characteristic
14	#10597392	C₄	1	1	yes	characteristic
15	C₂×#64515	C₂²	1	2	yes	—
16	C₂×#64515	C₂²	1	2	yes	—
17	C₂×C₈×D₁₉	C₂²	1	2	yes	—
18	#10597344	C₄	1	2	yes	—
19	#10597341	C₄	1	1	yes	characteristic
20	#10597340	C₄	1	2	yes	—
21	#10597336	C₄	1	1	yes	characteristic
22	C₄×D₇₆	C₂²	1	1	yes	characteristic
23	C₄×D₇₆	C₂²	1	2	yes	—
24	#10597318	C₂²	1	2	yes	—
25	C₄²×D₁₉	C₂²	1	1	yes	characteristic
26	C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
28	C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
29	C₄×C₈×C₁₉	C₂²	1	2	yes	—
30	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
31	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
32	(C₂×D₁₉)⋊C₈	C₂²	1	2	yes	—
33	(C₂×D₁₉)⋊C₈	C₂²	1	2	yes	—
34	Dic₁₉⋊C₈	C₂²	1	2	yes	—
35	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
36	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
37	C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
38	C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
39	#10597260	C₂²	1	2	yes	—
40	C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
42	C₂×D₇₆	C₂×C₄	1	1	yes	characteristic
43	C₂×D₇₆	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₁₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
47	C₂×C₄×D₁₉	C₂³	1	2	yes	—
48	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₁₉	C₂³	1	2	yes	—
51	C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
52	C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
53	C₄²×C₁₉	C₂³	1	1	yes	characteristic
54	C₄²×C₁₉	C₂³	1	2	yes	—
55	(C₂×D₁₉)⋊C₄	C₂×C₄	1	2	yes	—
56	(C₂×D₁₉)⋊C₄	C₂×C₄	1	2	yes	—
57	(C₂×D₁₉)⋊C₄	C₂×C₄	1	2	yes	—
58	(C₂×D₁₉)⋊C₄	C₂×C₄	1	2	yes	—
59	(C₂×D₁₉)⋊C₄	C₂×C₄	1	2	yes	—
60	C₄⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
61	C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
62	Dic₁₉⋊C₄	C₂×C₄	1	2	yes	—
63	Dic₁₉⋊C₄	C₂×C₄	1	2	yes	—
64	C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
67	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
68	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
70	#64515	—	4	2	no	—
71	#64515	—	4	2	no	—
72	C₈×D₁₉	—	4	2	no	—
73	C₂²×D₁₉	C₂²×C₄	1	2	yes	—
74	C₂²×D₁₉	C₂²×C₄	1	1	yes	characteristic
75	Q₈×C₁₉	—	2	2	no	—
76	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₉	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₁₉	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
81	C₂×Dic₁₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×Dic₁₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₁₉	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₁₉	C₂²×C₄	1	2	yes	—
85	D₇₆	—	2	2	no	—
86	D₇₆	—	4	2	no	—
87	C₄×D₁₉	—	2	2	no	—
88	C₄×D₁₉	—	2	2	no	—
89	C₄×D₁₉	—	2	1	no	—
90	C₄×D₁₉	—	2	1	no	—
91	C₄×D₁₉	—	2	2	no	—
92	C₄×D₁₉	—	4	1	no	—
93	C₈×C₁₉	—	2	1	no	—
94	C₈×C₁₉	—	2	2	no	—
95	C₈×C₁₉	—	2	1	no	—
96	C₁₉⋊C₈	—	2	2	no	—
97	C₁₉⋊C₈	—	2	1	no	—
98	C₁₉⋊C₈	—	2	1	no	—
99	#2993	—	19	1	no	maximal, Sylow
100	C₂²×C₁₉	C₂³×C₄	1	1	yes	characteristic
101	C₂×D₁₉	—	2	2	no	—
102	C₂×D₁₉	—	2	1	no	—
103	C₂×D₁₉	—	2	1	no	—
104	C₂×D₁₉	—	2	2	no	—
105	C₂×D₁₉	—	2	1	no	—
106	C₂×D₁₉	—	2	2	no	—
107	C₄×C₁₉	—	2	2	no	—
108	C₄×C₁₉	—	2	1	no	—
109	C₄×C₁₉	—	2	2	no	—
110	C₄×C₁₉	#131	1	2	yes	—
111	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
112	C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
113	Dic₁₉	—	2	2	no	—
114	Dic₁₉	—	2	1	no	—
115	Dic₁₉	—	2	1	no	—
116	Dic₁₉	—	2	1	no	—
117	#529	—	19	1	no	—
118	#446	—	19	1	no	—
119	C₈×Q₈	D₁₉	1	1	yes	characteristic, complemented
120	#436	—	19	1	no	—
121	#436	—	19	2	no	—
122	#435	—	19	2	no	—
123	C₈×D₄	—	19	1	no	—
124	#433	—	19	2	no	—
125	#431	—	19	1	no	—
126	#424	—	19	2	no	—
127	#423	—	19	1	no	—
128	C₂×C₁₉	#520	1	1	yes	characteristic
129	C₂×C₁₉	C₂×#131	1	1	yes	characteristic
130	C₂×C₁₉	#568	1	1	yes	characteristic
131	D₁₉	—	4	1	no	—
132	D₁₉	—	4	2	no	—
133	C₂×M₁₆	—	19	2	no	—
134	C₂×M₁₆	—	19	2	no	—
135	C₂²×C₈	—	19	2	no	—
136	#126	—	19	2	no	—
137	#125	—	19	1	no	—
138	#124	—	19	1	no	—
139	#123	—	19	2	no	—
140	C₂²⋊Q₈	—	19	1	no	—
141	#121	—	19	1	no	—
142	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
143	C₄×D₄	—	19	1	no	—
144	C₄×D₄	—	19	2	no	—
145	#117	—	19	2	no	—
146	C₂×C₄²	—	19	1	no	—
147	C₄⋊C₈	—	19	2	no	—
148	C₄⋊C₈	—	19	1	no	—
149	C₄⋊C₈	—	19	1	no	—
150	C₄⋊C₈	—	19	1	no	—
151	C₄⋊C₈	—	19	2	no	—
152	C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
153	C₄⋊C₈	C₂×D₁₉	1	2	yes	—
154	C₂²⋊C₈	—	19	1	no	—
155	C₂²⋊C₈	—	19	1	no	—
156	C₂²⋊C₈	—	19	2	no	—
157	C₂²⋊C₈	—	19	2	no	—
158	#97	—	19	2	no	—
159	C₄×C₈	—	19	1	no	—
160	C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
161	C₄×C₈	C₂×D₁₉	1	2	yes	—
162	C₁₉	#2993	1	1	yes	characteristic, complemented, Sylow
163	C₂×Q₈	C₄×D₁₉	1	1	yes	characteristic
164	C₂×D₄	—	19	1	no	—
165	C₂×D₄	—	19	2	no	—
166	C₂²×C₄	—	19	1	no	—
167	C₂²×C₄	—	19	1	no	—
168	C₂²×C₄	—	19	2	no	—
169	C₂²×C₄	—	19	1	no	—
170	M₁₆	—	76	2	no	—
171	M₁₆	—	76	2	no	—
172	C₂×C₈	—	19	1	no	—
173	C₂×C₈	—	19	1	no	—
174	C₂×C₈	—	19	2	no	—
175	C₂×C₈	—	76	2	no	—
176	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
177	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₁₉	1	2	yes	—
179	C₄⋊C₄	—	19	2	no	—
180	C₄⋊C₄	—	19	2	no	—
181	C₄⋊C₄	—	19	1	no	—
182	C₄⋊C₄	—	19	2	no	—
183	C₄⋊C₄	C₄×D₁₉	1	1	yes	characteristic
184	C₄⋊C₄	C₄×D₁₉	1	2	yes	—
185	C₂²⋊C₄	—	19	2	no	—
186	C₂²⋊C₄	—	19	2	no	—
187	C₂²⋊C₄	—	19	2	no	—
188	C₂²⋊C₄	—	19	2	no	—
189	C₂²⋊C₄	—	19	2	no	—
190	C₄²	—	19	1	no	—
191	C₄²	—	19	1	no	—
192	C₄²	—	19	1	no	—
193	C₄²	C₂²×D₁₉	1	1	yes	characteristic
194	C₄²	C₂²×D₁₉	1	2	yes	—
195	C₂³	—	19	2	no	—
196	C₂³	—	19	1	no	—
197	Q₈	—	2	2	no	—
198	D₄	—	38	2	no	—
199	D₄	—	76	2	no	—
200	C₂×C₄	—	19	1	no	—
201	C₂×C₄	—	19	1	no	—
202	C₂×C₄	—	19	1	no	—
203	C₂×C₄	—	19	2	no	—
204	C₂×C₄	—	38	1	no	—
205	C₂×C₄	—	38	2	no	—
206	C₂×C₄	—	38	2	no	—
207	C₂×C₄	—	38	1	no	—
208	C₂×C₄	—	38	2	no	—
209	C₂×C₄	—	76	1	no	—
210	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×D₁₉	1	2	yes	—
212	C₂×C₄	C₂×C₄×D₁₉	1	2	yes	—
213	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
214	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	1	no	—
217	C₈	—	2	2	no	—
218	C₈	—	38	1	no	—
219	C₈	—	38	2	no	—
220	C₈	—	38	1	no	—
221	C₂²	—	38	1	no	—
222	C₂²	—	38	2	no	—
223	C₂²	—	38	1	no	—
224	C₂²	—	38	2	no	—
225	C₂²	—	38	2	no	—
226	C₂²	—	38	1	no	—
227	C₂²	C₂²×C₄×D₁₉	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	2	no	—
231	C₄	—	38	1	no	—
232	C₄	—	38	1	no	—
233	C₄	—	38	1	no	—
234	C₄	—	38	2	no	—
235	C₄	#10597347	1	2	yes	—
236	C₄	#10597433	1	1	yes	characteristic
237	C₄	#10597435	1	1	yes	characteristic
238	C₂	—	76	1	no	—
239	C₂	—	76	2	no	—
240	C₂	#11927026	1	1	yes	characteristic
241	C₂	C₂×#10597347	1	1	yes	characteristic
242	C₂	#11927309	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉
commutator, socle	C₂²×C₁₉	C₂³×C₄
Fitting	C₈×Q₈×C₁₉	C₂
prime radical	#10597341	C₄
cosocle	C₂×C₄×C₁₉	C₂⁴
hypercenter	C₈×Q₈	D₁₉
hypocenter	C₁₉	#2993
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. #11926675	C₂	semidirect
3. #11926623	C₂	semidirect
4. #11926621	C₂	semidirect
5. #11926614	C₂	semidirect
6. C₈×D₇₆	C₂	semidirect
7. #11926603	C₂	semidirect
8. #11926595	C₂	semidirect
9. #11926569	C₂	semidirect
10. #11926566	C₂	semidirect
11. C₈×Q₈×C₁₉	C₂	semidirect
119. C₈×Q₈	D₁₉	semidirect
162. C₁₉	#2993	semidirect

Maximal subgroups

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

subgroup	index
#11927081	2
#11926675	2
#11926623	2
#11926621	2
#11926614	2
C₈×D₇₆	2
#11926603	2
#11926595	2
#11926569	2
#11926566	2
C₈×Q₈×C₁₉	2
#2993	19