Subgroups — #14572669

Order 2432.

464 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₇₆	C₂	1	1	yes	maximal, characteristic
2	#11926623	C₂	1	2	yes	maximal, complemented
3	#11926614	C₂	1	4	yes	maximal, complemented
4	C₈×D₇₆	C₂	1	2	yes	maximal, complemented
5	#11926601	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926506	C₂	1	2	yes	maximal, complemented
7	C₂²⋊C₈×D₁₉	C₂	1	2	yes	maximal, complemented
8	C₁₉×#431	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₇₆	C₄	1	1	yes	characteristic
10	C₂²×C₄×D₁₉	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
12	C₂×(C₂×D₁₉)⋊C₄	C₄	1	2	yes	—
13	C₂×C₄⋊Dic₁₉	C₄	1	1	yes	characteristic
14	C₂×#64515	C₂²	1	4	yes	complemented
15	C₂×C₈×D₁₉	C₂²	1	4	yes	complemented
16	C₄×D₇₆	C₄	1	2	yes	—
17	C₄×D₇₆	C₄	1	1	yes	characteristic
18	C₄×D₇₆	C₄	1	1	yes	characteristic
19	C₄×D₇₆	C₂²	1	1	yes	characteristic
20	C₄×D₇₆	C₂²	1	2	yes	—
21	C₄×D₇₆	C₂²	1	1	yes	characteristic
22	C₁₉×C₄⋊C₈	C₂²	1	2	yes	complemented
23	C₁₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
24	C₁₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
25	C₄×C₈×C₁₉	C₂²	1	2	yes	complemented
26	C₂²⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
27	(C₂×D₁₉)⋊C₈	C₂²	1	4	yes	complemented
28	(C₂×D₁₉)⋊C₈	C₂²	1	4	yes	complemented
29	C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	complemented
30	#10597260	C₂²	1	2	yes	complemented
31	C₂³×D₁₉	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
35	C₂²×Dic₁₉	C₂×C₄	1	2	yes	—
36	C₂×D₇₆	—	2	4	no	—
37	C₂×D₇₆	C₂×C₄	1	2	yes	—
38	C₂×D₇₆	C₂×C₄	1	1	yes	characteristic
39	C₂×D₇₆	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×D₁₉	—	4	4	no	—
41	C₂×C₄×D₁₉	C₂×C₄	1	2	yes	—
42	C₂×C₄×D₁₉	C₂×C₄	1	2	yes	—
43	C₂×C₄×D₁₉	C₂³	1	2	yes	—
44	C₂×C₄×D₁₉	C₂³	1	2	yes	—
45	C₂×C₈×C₁₉	C₂³	1	2	yes	—
46	C₂×C₈×C₁₉	C₂³	1	2	yes	—
47	C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
48	C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
49	C₄²×C₁₉	C₂³	1	1	yes	characteristic
50	C₄²×C₁₉	C₂³	1	1	yes	characteristic
51	(C₂×D₁₉)⋊C₄	C₂×C₄	1	2	yes	—
52	(C₂×D₁₉)⋊C₄	C₂×C₄	1	2	yes	—
53	(C₂×D₁₉)⋊C₄	C₂×C₄	1	2	yes	—
54	(C₂×D₁₉)⋊C₄	C₂×C₄	1	2	yes	—
55	C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
58	C₂×C₁₉⋊C₈	C₂³	1	4	yes	—
59	#64515	—	4	4	no	—
60	C₈×D₁₉	—	4	4	no	—
61	C₂³×C₁₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×D₁₉	—	2	4	no	—
63	C₂²×D₁₉	—	2	4	no	—
64	C₂²×D₁₉	—	4	2	no	—
65	C₂²×D₁₉	C₂²×C₄	1	2	yes	—
66	C₂²×D₁₉	C₂²×C₄	1	2	yes	—
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	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
76	C₂×Dic₁₉	—	4	2	no	—
77	C₂×Dic₁₉	C₂²×C₄	1	2	yes	—
78	C₂×Dic₁₉	C₂²×C₄	1	2	yes	—
79	D₇₆	—	2	4	no	—
80	D₇₆	—	2	2	no	—
81	D₇₆	M₁₆	1	4	yes	complemented
82	C₄×D₁₉	—	2	4	no	—
83	C₄×D₁₉	—	2	4	no	—
84	C₄×D₁₉	—	4	2	no	—
85	C₄×D₁₉	—	4	2	no	—
86	C₈×C₁₉	—	2	2	no	—
87	C₈×C₁₉	—	2	2	no	—
88	C₁₉⋊C₈	—	2	4	no	—
89	#2974	—	19	1	no	maximal, Sylow
90	C₂²×C₁₉	—	2	1	no	—
91	C₂²×C₁₉	—	2	1	no	—
92	C₂²×C₁₉	#131	1	2	yes	—
93	C₂²×C₁₉	C₂³×C₄	1	1	yes	characteristic
94	C₂×D₁₉	—	2	2	no	—
95	C₂×D₁₉	—	2	2	no	—
96	C₂×D₁₉	—	2	2	no	—
97	C₂×D₁₉	—	2	2	no	—
98	C₂×D₁₉	—	2	2	no	—
99	C₂×D₁₉	—	4	8	no	—
100	C₂×D₁₉	C₂×M₁₆	1	4	yes	—
101	C₄×C₁₉	—	2	1	no	—
102	C₄×C₁₉	—	2	1	no	—
103	C₄×C₁₉	—	2	1	no	—
104	C₄×C₁₉	—	2	1	no	—
105	C₄×C₁₉	C₂×M₁₆	1	2	yes	—
106	C₄×C₁₉	H(5,𝔽₂)	1	2	yes	—
107	Dic₁₉	—	2	2	no	—
108	Dic₁₉	—	2	2	no	—
109	C₂×C₄×D₄	—	19	1	no	—
110	#436	—	19	2	no	—
111	#435	—	19	4	no	—
112	C₈×D₄	—	19	2	no	—
113	#432	—	19	1	no	—
114	#431	D₁₉	1	1	yes	characteristic, complemented
115	#407	—	19	2	no	—
116	C₂×C₂²⋊C₈	—	19	2	no	—
117	C₂×C₁₉	—	2	2	no	—
118	C₂×C₁₉	#518	1	1	yes	characteristic
119	C₂×C₁₉	C₂²×M₁₆	1	1	yes	characteristic
120	C₂×C₁₉	C₂×#131	1	1	yes	characteristic
121	D₁₉	—	2	4	no	—
122	D₁₉	—	4	2	no	—
123	C₂²×D₄	—	19	1	no	—
124	C₂³×C₄	—	19	2	no	—
125	C₂×M₁₆	—	19	4	no	—
126	C₂²×C₈	—	19	4	no	—
127	C₄×D₄	—	19	1	no	—
128	C₄×D₄	—	19	2	no	—
129	C₄×D₄	—	19	1	no	—
130	C₄×D₄	—	19	1	no	—
131	C₄×D₄	—	19	2	no	—
132	C₄×D₄	—	19	1	no	—
133	C₂×C₄⋊C₄	—	19	1	no	—
134	C₂×C₂²⋊C₄	—	19	2	no	—
135	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
136	C₄⋊C₈	—	19	2	no	—
137	C₄⋊C₈	C₂×D₁₉	1	2	yes	complemented
138	C₂²⋊C₈	—	19	2	no	—
139	C₂²⋊C₈	—	19	4	no	—
140	C₂²⋊C₈	—	19	4	no	—
141	C₂²⋊C₈	C₂×D₁₉	1	1	yes	characteristic
142	C₂²⋊C₈	C₂×D₁₉	1	1	yes	characteristic
143	#97	—	19	2	no	—
144	C₄×C₈	C₂×D₁₉	1	2	yes	complemented
145	C₁₉	#2974	1	1	yes	characteristic, complemented, Sylow
146	C₂⁴	—	19	2	no	—
147	C₂×D₄	—	19	1	no	—
148	C₂×D₄	—	19	2	no	—
149	C₂×D₄	—	19	1	no	—
150	C₂×D₄	—	38	4	no	—
151	C₂²×C₄	—	19	2	no	—
152	C₂²×C₄	—	19	2	no	—
153	C₂²×C₄	—	19	2	no	—
154	C₂²×C₄	—	19	2	no	—
155	C₂²×C₄	—	19	2	no	—
156	C₂²×C₄	—	76	4	no	—
157	C₂²×C₄	C₄×D₁₉	1	1	yes	characteristic
158	C₂²×C₄	C₄×D₁₉	1	1	yes	characteristic
159	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
160	M₁₆	—	76	4	no	—
161	C₂×C₈	—	19	4	no	—
162	C₂×C₈	—	76	4	no	—
163	C₂×C₈	C₂²×D₁₉	1	2	yes	—
164	C₂×C₈	C₂²×D₁₉	1	2	yes	—
165	C₄⋊C₄	—	19	2	no	—
166	C₄⋊C₄	—	19	1	no	—
167	C₄⋊C₄	—	19	1	no	—
168	C₂²⋊C₄	—	19	2	no	—
169	C₂²⋊C₄	—	19	2	no	—
170	C₂²⋊C₄	—	19	2	no	—
171	C₂²⋊C₄	—	19	2	no	—
172	C₄²	C₄×D₁₉	1	1	yes	characteristic
173	C₄²	C₄×D₁₉	1	1	yes	characteristic
174	C₄²	C₂²×D₁₉	1	1	yes	characteristic
175	C₄²	C₂²×D₁₉	1	1	yes	characteristic
176	C₂³	—	19	2	no	—
177	C₂³	—	19	2	no	—
178	C₂³	—	38	4	no	—
179	C₂³	—	38	4	no	—
180	C₂³	—	76	2	no	—
181	C₂³	C₂×C₄×D₁₉	1	1	yes	characteristic
182	D₄	—	19	4	no	—
183	D₄	—	38	4	no	—
184	D₄	—	38	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	19	2	no	—
189	C₂×C₄	—	19	2	no	—
190	C₂×C₄	—	38	4	no	—
191	C₂×C₄	—	38	4	no	—
192	C₂×C₄	—	76	2	no	—
193	C₂×C₄	—	76	2	no	—
194	C₂×C₄	—	76	2	no	—
195	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
196	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
197	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
199	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
200	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
201	C₈	—	2	2	no	—
202	C₈	—	2	2	no	—
203	C₈	—	38	4	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	19	4	no	—
207	C₂²	—	38	2	no	—
208	C₂²	—	38	2	no	—
209	C₂²	—	38	2	no	—
210	C₂²	—	38	2	no	—
211	C₂²	—	38	2	no	—
212	C₂²	—	76	8	no	—
213	C₂²	#10597347	1	2	yes	—
214	C₂²	C₂²×C₄×D₁₉	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	38	2	no	—
220	C₄	—	38	2	no	—
221	C₄	M₁₆×D₁₉	1	2	yes	—
222	C₄	#10597435	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	38	4	no	—
225	C₂	—	76	2	no	—
226	C₂	#11927018	1	1	yes	characteristic
227	C₂	C₂×M₁₆×D₁₉	1	1	yes	characteristic
228	C₂	C₂×#10597347	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉
commutator, socle	C₂²×C₁₉	C₂³×C₄
Fitting	C₁₉×#431	C₂
prime radical	C₂²×D₇₆	C₄
cosocle	C₂×C₄×C₁₉	C₂⁴
hypercenter	#431	D₁₉
hypocenter	C₁₉	#2974
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
2. #11926623	C₂	semidirect
3. #11926614	C₂	semidirect
4. C₈×D₇₆	C₂	semidirect
5. #11926601	C₂	semidirect
6. #11926506	C₂	semidirect
7. C₂²⋊C₈×D₁₉	C₂	semidirect
8. C₁₉×#431	C₂	semidirect
14. C₂×#64515	C₂²	semidirect
15. C₂×C₈×D₁₉	C₂²	semidirect
22. C₁₉×C₄⋊C₈	C₂²	semidirect
25. C₄×C₈×C₁₉	C₂²	semidirect
27. (C₂×D₁₉)⋊C₈	C₂²	semidirect
28. (C₂×D₁₉)⋊C₈	C₂²	semidirect
29. C₄⋊(C₁₉⋊C₈)	C₂²	semidirect
30. #10597260	C₂²	semidirect
81. D₇₆	M₁₆	semidirect
114. #431	D₁₉	semidirect
137. C₄⋊C₈	C₂×D₁₉	semidirect
144. C₄×C₈	C₂×D₁₉	semidirect
145. C₁₉	#2974	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₇₆	2
#11926623	2
#11926614	2
C₈×D₇₆	2
#11926601	2
#11926506	2
C₂²⋊C₈×D₁₉	2
C₁₉×#431	2
#2974	19