Subgroups — #14566720

Order 2432.

548 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11926451	C₂	1	2	yes	maximal, complemented
2	#11926430	C₂	1	2	yes	maximal, complemented
3	#11926430	C₂	1	2	yes	maximal, complemented
4	#11926419	C₂	1	2	yes	maximal
5	C₄⋊C₄×Dic₁₉	C₂	1	2	yes	maximal, complemented
6	C₂²⋊C₄×Dic₁₉	C₂	1	2	yes	maximal
7	C₄×C₂²⋊Dic₁₉	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₉⋊#95	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₉×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₈×C₁₉	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic
12	C₂³×Dic₁₉	C₂²	1	2	yes	—
13	C₁₉×C₂²⋊Q₈	C₄	1	8	yes	complemented
14	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊C₄×C₁₉	C₂²	1	2	yes	—
17	C₂×C₂²⋊Dic₁₉	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊Dic₁₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
20	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
21	C₂×C₄×Dic₁₉	C₂²	1	2	yes	—
22	C₂×C₄×Dic₁₉	C₂²	1	2	yes	—
23	C₂×C₄×Dic₁₉	C₂²	1	2	yes	—
24	C₁₉⋊#95	C₂²	1	1	yes	characteristic
25	C₁₉⋊#95	C₂²	1	1	yes	characteristic
26	C₁₉⋊#95	C₂²	1	2	yes	—
27	C₁₉⋊#95	C₂²	1	2	yes	—
28	C₁₉⋊#95	C₂²	1	2	yes	—
29	C₁₉⋊#95	C₂²	1	2	yes	—
30	C₁₉⋊#95	C₂²	1	2	yes	—
31	C₁₉⋊#95	C₂²	1	2	yes	—
32	C₂⁴×C₁₉	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₁₉	—	4	2	no	—
34	C₂×Q₈×C₁₉	C₂×C₄	1	4	yes	complemented
35	C₂²×C₄×C₁₉	—	4	2	no	—
36	C₂²×C₄×C₁₉	C₂×C₄	1	4	yes	—
37	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
41	C₂²×Dic₁₉	—	2	4	no	—
42	C₂²×Dic₁₉	—	2	4	no	—
43	C₂²×Dic₁₉	—	4	2	no	—
44	C₂²×Dic₁₉	C₂³	1	2	yes	—
45	C₂²×Dic₁₉	C₂³	1	2	yes	—
46	C₂²×Dic₁₉	C₂³	1	2	yes	—
47	C₂²×Dic₁₉	C₂³	1	2	yes	—
48	C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	complemented
49	C₄⋊C₄×C₁₉	C₂×C₄	1	8	yes	complemented
50	C₂²⋊C₄×C₁₉	C₂×C₄	1	8	yes	—
51	C₂²⋊Dic₁₉	—	2	2	no	—
52	C₂²⋊Dic₁₉	Q₈	1	4	yes	complemented
53	C₄⋊Dic₁₉	—	2	4	no	—
54	C₄⋊Dic₁₉	—	4	2	no	—
55	C₄×Dic₁₉	—	2	4	no	—
56	C₄×Dic₁₉	—	4	2	no	—
57	C₄×Dic₁₉	—	4	2	no	—
58	C₂³×C₁₉	—	2	2	no	—
59	C₂³×C₁₉	—	2	2	no	—
60	C₂³×C₁₉	C₂²×C₄	1	2	yes	—
61	C₂³×C₁₉	C₂×Q₈	1	2	yes	—
62	C₂³×C₁₉	#41	1	2	yes	—
63	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
64	Q₈×C₁₉	—	4	4	no	—
65	C₂×C₄×C₁₉	—	2	2	no	—
66	C₂×C₄×C₁₉	—	2	2	no	—
67	C₂×C₄×C₁₉	—	2	2	no	—
68	C₂×C₄×C₁₉	—	2	2	no	—
69	C₂×C₄×C₁₉	—	2	2	no	—
70	C₂×C₄×C₁₉	—	2	2	no	—
71	C₂×C₄×C₁₉	—	4	4	no	—
72	C₂×C₄×C₁₉	C₂²×C₄	1	4	yes	—
73	C₂×C₄×C₁₉	C₂²×C₄	1	4	yes	—
74	C₂×C₄×C₁₉	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₁₉	C₂²×C₄	1	2	yes	—
76	C₂×Dic₁₉	—	2	2	no	—
77	C₂×Dic₁₉	—	2	2	no	—
78	C₂×Dic₁₉	—	2	2	no	—
79	C₂×Dic₁₉	—	2	2	no	—
80	C₂×Dic₁₉	—	2	2	no	—
81	C₂×Dic₁₉	—	2	2	no	—
82	C₂×Dic₁₉	—	2	2	no	—
83	C₂×Dic₁₉	—	2	2	no	—
84	C₂×Dic₁₉	—	2	2	no	—
85	C₂×Dic₁₉	—	2	2	no	—
86	C₂×Dic₁₉	—	4	8	no	—
87	C₂×Dic₁₉	C₂×Q₈	1	4	yes	—
88	C₂×Dic₁₉	#41	1	4	yes	—
89	#2368	—	19	1	no	maximal, Sylow
90	C₂²×C₁₉	—	2	2	no	—
91	C₂²×C₁₉	—	2	2	no	—
92	C₂²×C₁₉	—	2	4	no	—
93	C₂²×C₁₉	—	2	4	no	—
94	C₂²×C₁₉	C₄×Q₈	1	4	yes	—
95	C₂²×C₁₉	C₂³×C₄	1	1	yes	characteristic
96	C₂²×C₁₉	C₂²×Q₈	1	1	yes	characteristic
97	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
98	C₂²×C₁₉	C₂×#41	1	2	yes	—
99	C₂²×C₁₉	H(5,𝔽₂)	1	2	yes	—
100	C₄×C₁₉	—	2	4	no	—
101	C₄×C₁₉	—	2	2	no	—
102	C₄×C₁₉	—	2	2	no	—
103	C₄×C₁₉	—	2	4	no	—
104	Dic₁₉	—	2	4	no	—
105	Dic₁₉	—	2	4	no	—
106	Dic₁₉	—	4	2	no	—
107	Dic₁₉	—	4	2	no	—
108	C₂×C₂²⋊Q₈	D₁₉	1	1	yes	characteristic
109	#391	—	19	2	no	—
110	#387	—	19	2	no	—
111	#387	—	19	2	no	—
112	#385	—	19	2	no	—
113	C₄×C₄⋊C₄	—	19	2	no	—
114	C₄×C₂²⋊C₄	—	19	1	no	—
115	C₄×C₂²⋊C₄	—	19	2	no	—
116	C₂×#95	—	19	1	no	—
117	C₂×C₁₉	—	2	4	no	—
118	C₂×C₁₉	C₂×C₄×Q₈	1	1	yes	characteristic
119	C₂×C₁₉	C₄×#41	1	1	yes	characteristic
120	C₂×C₁₉	#518	1	1	yes	characteristic
121	C₂×C₁₉	#551	1	2	yes	—
122	C₂×C₁₉	#554	1	2	yes	—
123	C₂²×Q₈	C₂×D₁₉	1	1	yes	characteristic
124	C₂³×C₄	—	19	2	no	—
125	C₂³×C₄	C₂×D₁₉	1	1	yes	characteristic
126	C₂²⋊Q₈	Dic₁₉	1	8	yes	complemented
127	C₂×C₄⋊C₄	—	19	2	no	—
128	C₂×C₄⋊C₄	—	19	2	no	—
129	C₂×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
130	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
131	C₂×C₂²⋊C₄	—	19	1	no	—
132	C₂×C₂²⋊C₄	—	19	1	no	—
133	C₂×C₂²⋊C₄	C₂×D₁₉	1	2	yes	—
134	C₂×C₄²	—	19	2	no	—
135	C₂×C₄²	—	19	2	no	—
136	C₂×C₄²	—	19	2	no	—
137	#95	—	19	1	no	—
138	#95	—	19	1	no	—
139	#95	—	19	2	no	—
140	#95	—	19	2	no	—
141	#95	—	19	2	no	—
142	#95	—	19	2	no	—
143	#95	—	19	2	no	—
144	#95	—	19	2	no	—
145	C₁₉	#2368	1	1	yes	characteristic, complemented, Sylow
146	C₂⁴	C₂²×D₁₉	1	1	yes	characteristic
147	C₂×Q₈	—	4	2	no	—
148	C₂×Q₈	C₂×Dic₁₉	1	4	yes	complemented
149	C₂²×C₄	—	4	2	no	—
150	C₂²×C₄	—	19	2	no	—
151	C₂²×C₄	—	19	2	no	—
152	C₂²×C₄	—	19	2	no	—
153	C₂²×C₄	—	19	2	no	—
154	C₂²×C₄	—	38	4	no	—
155	C₂²×C₄	—	38	4	no	—
156	C₂²×C₄	—	76	2	no	—
157	C₂²×C₄	C₂×Dic₁₉	1	4	yes	—
158	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
159	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
160	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
161	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
162	C₄⋊C₄	—	38	4	no	—
163	C₄⋊C₄	—	76	2	no	—
164	C₄⋊C₄	C₂×Dic₁₉	1	4	yes	complemented
165	C₄⋊C₄	C₂×Dic₁₉	1	8	yes	complemented
166	C₂²⋊C₄	—	19	4	no	—
167	C₂²⋊C₄	—	38	2	no	—
168	C₂²⋊C₄	C₂×Dic₁₉	1	8	yes	—
169	C₄²	—	38	4	no	—
170	C₄²	—	76	2	no	—
171	C₄²	—	76	2	no	—
172	C₂³	—	2	2	no	—
173	C₂³	—	2	2	no	—
174	C₂³	Q₈×D₁₉	1	2	yes	—
175	C₂³	#64545	1	2	yes	—
176	C₂³	C₂²×Dic₁₉	1	2	yes	—
177	C₂³	C₂³×D₁₉	1	1	yes	characteristic
178	Q₈	—	4	4	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	19	4	no	—
187	C₂×C₄	—	19	4	no	—
188	C₂×C₄	—	38	2	no	—
189	C₂×C₄	—	38	2	no	—
190	C₂×C₄	—	38	2	no	—
191	C₂×C₄	—	38	2	no	—
192	C₂×C₄	—	38	2	no	—
193	C₂×C₄	—	38	2	no	—
194	C₂×C₄	—	38	2	no	—
195	C₂×C₄	—	38	2	no	—
196	C₂×C₄	—	38	2	no	—
197	C₂×C₄	—	38	2	no	—
198	C₂×C₄	—	76	8	no	—
199	C₂×C₄	C₂²×Dic₁₉	1	4	yes	—
200	C₂×C₄	C₂²×Dic₁₉	1	4	yes	—
201	C₂×C₄	C₂²×Dic₁₉	1	2	yes	—
202	C₂×C₄	C₂²×Dic₁₉	1	2	yes	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	4	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	Q₈×Dic₁₉	1	4	yes	—
208	C₂²	#10597430	1	2	yes	—
209	C₂²	C₂×Q₈×D₁₉	1	1	yes	characteristic
210	C₂²	C₂×#64545	1	1	yes	characteristic
211	C₂²	#41×D₁₉	1	2	yes	—
212	C₂²	C₂³×Dic₁₉	1	1	yes	characteristic
213	C₄	—	2	4	no	—
214	C₄	—	2	2	no	—
215	C₄	—	2	2	no	—
216	C₄	—	2	4	no	—
217	C₄	—	38	4	no	—
218	C₄	—	38	4	no	—
219	C₄	—	76	2	no	—
220	C₄	—	76	2	no	—
221	C₂	—	2	4	no	—
222	C₂	C₂×Q₈×Dic₁₉	1	1	yes	characteristic
223	C₂	#41×Dic₁₉	1	1	yes	characteristic
224	C₂	#11927019	1	1	yes	characteristic
225	C₂	#11927221	1	2	yes	—
226	C₂	#11927248	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉
commutator	C₂²×C₁₉	C₂³×C₄
Fitting	C₂×C₁₉×C₂²⋊Q₈	C₂
socle, cosocle	C₂³×C₁₉	C₂⁴
prime radical	C₂⁴×C₁₉	C₂³
hypercenter	C₂×C₂²⋊Q₈	D₁₉
hypocenter	C₁₉	#2368
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #11926451	C₂	semidirect
2. #11926430	C₂	semidirect
3. #11926430	C₂	semidirect
5. C₄⋊C₄×Dic₁₉	C₂	semidirect
13. C₁₉×C₂²⋊Q₈	C₄	semidirect
34. C₂×Q₈×C₁₉	C₂×C₄	semidirect
48. C₄⋊C₄×C₁₉	C₂×C₄	semidirect
49. C₄⋊C₄×C₁₉	C₂×C₄	semidirect
52. C₂²⋊Dic₁₉	Q₈	semidirect
126. C₂²⋊Q₈	Dic₁₉	semidirect
145. C₁₉	#2368	semidirect
148. C₂×Q₈	C₂×Dic₁₉	semidirect
164. C₄⋊C₄	C₂×Dic₁₉	semidirect
165. C₄⋊C₄	C₂×Dic₁₉	semidirect

Maximal subgroups

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

subgroup	index
#11926451	2
#11926430	2
#11926419	2
C₄⋊C₄×Dic₁₉	2
C₂²⋊C₄×Dic₁₉	2
C₄×C₂²⋊Dic₁₉	2
C₂×C₁₉⋊#95	2
C₂×C₁₉×C₂²⋊Q₈	2
#2368	19