Subgroups — #14568557

Order 2432.

620 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10597370	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×Dic₃₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926467	C₂	1	4	yes	maximal, complemented
4	#11926447	C₂	1	2	yes	maximal
5	#11926440	C₂	1	2	yes	maximal, complemented
6	#11926414	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926404	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×D₄×C₁₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×D₄×C₁₉	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₁₉	C₂²	1	2	yes	—
11	C₂²×Dic₃₈	C₂²	1	1	yes	characteristic
12	C₄×D₄×C₁₉	—	2	4	no	—
13	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊C₄×C₁₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊Dic₁₉	C₂²	1	4	yes	—
17	C₂×C₂²⋊Dic₁₉	C₂²	1	4	yes	—
18	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
21	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
22	#10597370	—	2	8	no	—
23	C₂×Dic₁₉⋊C₄	C₂²	1	4	yes	—
24	C₂×Dic₁₉⋊C₄	C₂²	1	2	yes	—
25	C₂×C₄×Dic₁₉	C₂²	1	2	yes	—
26	C₄×Dic₃₈	—	2	4	no	—
27	C₁₉⋊#95	C₂²	1	4	yes	—
28	C₂⁴×C₁₉	C₂³	1	2	yes	—
29	C₂×D₄×C₁₉	—	2	2	no	—
30	C₂×D₄×C₁₉	—	2	2	no	—
31	C₂×D₄×C₁₉	Q₈	1	4	yes	complemented
32	C₂²×C₄×C₁₉	—	2	4	no	—
33	C₂²×C₄×C₁₉	—	2	4	no	—
34	C₂²×C₄×C₁₉	—	2	4	no	—
35	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
36	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
39	C₂²×Dic₁₉	C₂³	1	2	yes	—
40	C₂²×Dic₁₉	C₂³	1	4	yes	—
41	C₂²×Dic₁₉	C₂³	1	2	yes	—
42	C₂×Dic₃₈	—	2	2	no	—
43	C₂×Dic₃₈	—	2	2	no	—
44	C₂×Dic₃₈	D₄	1	4	yes	complemented
45	C₄⋊C₄×C₁₉	—	2	2	no	—
46	C₂²⋊C₄×C₁₉	—	2	4	no	—
47	C₄²×C₁₉	D₄	1	4	yes	—
48	C₂²⋊Dic₁₉	—	2	8	no	—
49	C₂²⋊Dic₁₉	—	4	4	no	—
50	C₄⋊Dic₁₉	—	2	2	no	—
51	C₄⋊Dic₁₉	—	2	4	no	—
52	C₄⋊Dic₁₉	—	2	4	no	—
53	C₄⋊Dic₁₉	—	4	2	no	—
54	Dic₁₉⋊C₄	—	2	4	no	—
55	Dic₁₉⋊C₄	—	2	8	no	—
56	C₄×Dic₁₉	—	2	4	no	—
57	C₂³×C₁₉	—	2	2	no	—
58	C₂³×C₁₉	—	2	2	no	—
59	C₂³×C₁₉	—	2	2	no	—
60	C₂³×C₁₉	—	2	2	no	—
61	C₂³×C₁₉	—	2	2	no	—
62	C₂³×C₁₉	—	2	2	no	—
63	C₂³×C₁₉	C₂×Q₈	1	4	yes	—
64	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
65	D₄×C₁₉	—	2	8	no	—
66	C₂×C₄×C₁₉	—	2	1	no	—
67	C₂×C₄×C₁₉	—	2	1	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	8	no	—
72	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
73	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
74	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
75	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
76	C₂×C₄×C₁₉	C₂×Q₈	1	2	yes	—
77	C₂×C₄×C₁₉	#41	1	2	yes	—
78	C₂×C₄×C₁₉	#41	1	2	yes	—
79	C₂×Dic₁₉	—	2	2	no	—
80	C₂×Dic₁₉	—	2	2	no	—
81	C₂×Dic₁₉	—	2	2	no	—
82	C₂×Dic₁₉	—	2	4	no	—
83	C₂×Dic₁₉	—	2	4	no	—
84	C₂×Dic₁₉	—	2	4	no	—
85	C₂×Dic₁₉	—	2	2	no	—
86	C₂×Dic₁₉	—	2	2	no	—
87	C₂×Dic₁₉	C₂×D₄	1	4	yes	—
88	Dic₃₈	—	2	8	no	—
89	#2538	—	19	1	no	maximal, Sylow
90	C₂²×C₁₉	—	2	4	no	—
91	C₂²×C₁₉	—	2	4	no	—
92	C₂²×C₁₉	—	2	4	no	—
93	C₂²×C₁₉	—	4	2	no	—
94	C₂²×C₁₉	—	4	2	no	—
95	C₂²×C₁₉	—	4	2	no	—
96	C₂²×C₁₉	—	4	2	no	—
97	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
98	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
99	C₂²×C₁₉	C₂²×Q₈	1	1	yes	characteristic
100	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
101	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
102	C₂²×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
103	C₂²×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
104	C₄×C₁₉	—	2	2	no	—
105	C₄×C₁₉	—	4	2	no	—
106	C₄×C₁₉	#121	1	4	yes	—
107	C₄×C₁₉	C₂²⋊Q₈	1	4	yes	complemented
108	Dic₁₉	—	2	4	no	—
109	Dic₁₉	—	4	2	no	—
110	Dic₁₉	—	4	4	no	—
111	C₂×C₂²⋊Q₈	—	19	2	no	—
112	C₂×C₄×Q₈	—	19	1	no	—
113	C₂×C₄×D₄	D₁₉	1	1	yes	characteristic
114	#396	—	19	4	no	—
115	#390	—	19	2	no	—
116	#389	—	19	2	no	—
117	#384	—	19	1	no	—
118	#380	—	19	2	no	—
119	C₂×C₁₉	—	4	4	no	—
120	C₂×C₁₉	C₂×#121	1	1	yes	characteristic
121	C₂×C₁₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₁₉	#537	1	1	yes	characteristic
123	C₂×C₁₉	D₄×Q₈	1	1	yes	characteristic
124	C₂×C₁₉	#550	1	1	yes	characteristic
125	C₂×C₁₉	#554	1	1	yes	characteristic
126	C₂×C₁₉	#556	1	1	yes	characteristic
127	C₂²×Q₈	—	19	1	no	—
128	C₂²×D₄	C₂×D₁₉	1	1	yes	characteristic
129	C₂³×C₄	C₂×D₁₉	1	2	yes	—
130	C₂²⋊Q₈	—	38	8	no	—
131	C₄×Q₈	—	38	4	no	—
132	C₄×D₄	—	2	4	no	—
133	C₂×C₄⋊C₄	—	19	1	no	—
134	C₂×C₄⋊C₄	—	19	2	no	—
135	C₂×C₄⋊C₄	—	19	2	no	—
136	C₂×C₄⋊C₄	—	19	4	no	—
137	C₂×C₄⋊C₄	—	19	2	no	—
138	C₂×C₄⋊C₄	—	19	2	no	—
139	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
140	C₂×C₂²⋊C₄	—	19	4	no	—
141	C₂×C₂²⋊C₄	—	19	4	no	—
142	C₂×C₂²⋊C₄	C₂×D₁₉	1	2	yes	—
143	C₂×C₄²	—	19	2	no	—
144	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
145	#95	—	19	4	no	—
146	C₁₉	#2538	1	1	yes	characteristic, complemented, Sylow
147	C₂⁴	C₂²×D₁₉	1	2	yes	—
148	C₂×Q₈	—	19	4	no	—
149	C₂×Q₈	—	38	2	no	—
150	C₂×Q₈	—	38	2	no	—
151	C₂×D₄	—	2	2	no	—
152	C₂×D₄	—	2	2	no	—
153	C₂×D₄	Dic₃₈	1	4	yes	complemented
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	—	19	2	no	—
158	C₂²×C₄	—	19	2	no	—
159	C₂²×C₄	—	19	4	no	—
160	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
161	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
162	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
163	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
164	C₄⋊C₄	—	2	2	no	—
165	C₄⋊C₄	—	38	8	no	—
166	C₄⋊C₄	—	38	4	no	—
167	C₄⋊C₄	—	38	4	no	—
168	C₄⋊C₄	—	38	4	no	—
169	C₄⋊C₄	—	38	2	no	—
170	C₄⋊C₄	—	76	2	no	—
171	C₂²⋊C₄	—	2	4	no	—
172	C₂²⋊C₄	—	38	8	no	—
173	C₂²⋊C₄	—	76	4	no	—
174	C₄²	—	38	4	no	—
175	C₄²	C₂²⋊D₁₉	1	4	yes	—
176	C₂³	—	2	2	no	—
177	C₂³	—	2	2	no	—
178	C₂³	—	2	2	no	—
179	C₂³	—	2	2	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	C₂×Dic₃₈	1	4	yes	—
183	C₂³	C₂³×D₁₉	1	1	yes	characteristic
184	Q₈	—	38	8	no	—
185	D₄	—	2	8	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	8	no	—
192	C₂×C₄	—	19	4	no	—
193	C₂×C₄	—	38	2	no	—
194	C₂×C₄	—	38	2	no	—
195	C₂×C₄	—	38	4	no	—
196	C₂×C₄	—	38	4	no	—
197	C₂×C₄	—	38	2	no	—
198	C₂×C₄	—	38	4	no	—
199	C₂×C₄	—	38	2	no	—
200	C₂×C₄	—	38	2	no	—
201	C₂×C₄	C₂×Dic₃₈	1	2	yes	—
202	C₂×C₄	#64541	1	2	yes	—
203	C₂×C₄	D₄×D₁₉	1	2	yes	—
204	C₂×C₄	#64543	1	2	yes	—
205	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
206	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
207	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
208	C₂²	—	2	4	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	2	4	no	—
211	C₂²	—	4	2	no	—
212	C₂²	—	4	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	—	4	2	no	—
215	C₂²	C₂²×Dic₃₈	1	1	yes	characteristic
216	C₂²	C₂×#64541	1	1	yes	characteristic
217	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
218	C₂²	C₂×#64543	1	1	yes	characteristic
219	C₂²	#10597435	1	1	yes	characteristic
220	C₂²	#10597436	1	1	yes	characteristic
221	C₂²	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
222	C₄	—	2	2	no	—
223	C₄	—	4	2	no	—
224	C₄	—	38	4	no	—
225	C₄	—	76	2	no	—
226	C₄	—	76	4	no	—
227	C₄	#10597370	1	4	yes	complemented
228	C₄	#10597384	1	4	yes	—
229	C₂	—	4	4	no	—
230	C₂	C₂×#10597384	1	1	yes	characteristic
231	C₂	C₂×#10597370	1	1	yes	characteristic
232	C₂	#11927122	1	1	yes	characteristic
233	C₂	D₄×Dic₃₈	1	1	yes	characteristic
234	C₂	#11927215	1	1	yes	characteristic
235	C₂	#11927251	1	1	yes	characteristic
236	C₂	#11927264	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉
commutator, socle, cosocle	C₂³×C₁₉	C₂⁴
Fitting	C₂×C₄×D₄×C₁₉	C₂
prime radical	C₂²×D₄×C₁₉	C₂²
hypercenter	C₂×C₄×D₄	D₁₉
hypocenter	C₁₉	#2538
LCS	G ▷ C₂³×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#10597370	C₂	semidirect
2. C₂×C₄×Dic₃₈	C₂	semidirect
3. #11926467	C₂	semidirect
5. #11926440	C₂	semidirect
6. #11926414	C₂	semidirect
7. #11926404	C₂	semidirect
31. C₂×D₄×C₁₉	Q₈	semidirect
44. C₂×Dic₃₈	D₄	semidirect
107. C₄×C₁₉	C₂²⋊Q₈	semidirect
146. C₁₉	#2538	semidirect
153. C₂×D₄	Dic₃₈	semidirect
227. C₄	#10597370	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10597370	2
C₂×C₄×Dic₃₈	2
#11926467	2
#11926447	2
#11926440	2
#11926414	2
#11926404	2
C₂×C₄×D₄×C₁₉	2
#2538	19