Subgroups — #14572300

Order 2432.

392 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#64541	C₂	1	1	yes	maximal, characteristic
2	#11926604	C₂	1	4	yes	maximal, complemented
3	#11926600	C₂	1	2	yes	maximal, complemented
4	#11926596	C₂	1	2	yes	maximal, complemented
5	#11926568	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926510	C₂	1	2	yes	maximal
7	#11926499	C₂	1	2	yes	maximal, complemented
8	C₂×C₁₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#64541	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₁₉	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
12	C₄×C₂²⋊D₁₉	C₄	1	2	yes	—
13	C₄×C₂²⋊D₁₉	C₄	1	2	yes	—
14	#10597372	C₂²	1	1	yes	characteristic
15	C₂×#64520	C₂²	1	2	yes	—
16	C₄×D₇₆	C₄	1	1	yes	characteristic
17	C₄×D₇₆	C₄	1	1	yes	characteristic
18	#10597318	C₂²	1	2	yes	—
19	C₄²×D₁₉	C₂²	1	2	yes	—
20	C₄×Dic₃₈	C₄	1	1	yes	characteristic
21	C₄×Dic₃₈	C₄	1	1	yes	characteristic
22	C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
25	(C₂×D₁₉)⋊C₈	C₂²	1	4	yes	—
26	(C₂×D₁₉)⋊C₈	C₂²	1	4	yes	—
27	#10597272	C₂²	1	4	yes	—
28	C₈×Dic₁₉	C₂²	1	4	yes	—
29	C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
30	C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
31	C₂²×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊D₁₉	C₂×C₄	1	2	yes	—
35	#64541	—	2	2	no	—
36	#64541	—	2	2	no	—
37	C₂×D₇₆	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₁₉	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₁₉	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₁₉	C₂³	1	2	yes	—
41	C₂×Dic₃₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₉	—	2	4	no	—
43	C₂×C₈×C₁₉	C₂³	1	2	yes	—
44	C₂×C₈×C₁₉	C₂³	1	2	yes	—
45	C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
46	C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
47	C₄²×C₁₉	C₂³	1	1	yes	characteristic
48	C₄²×C₁₉	C₂³	1	1	yes	characteristic
49	C₂²⋊Dic₁₉	C₂×C₄	1	2	yes	—
50	(C₂×D₁₉)⋊C₄	C₂×C₄	1	2	yes	—
51	(C₂×D₁₉)⋊C₄	C₂×C₄	1	2	yes	—
52	C₄⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
53	C₄⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
54	Dic₁₉⋊C₄	C₂×C₄	1	2	yes	—
55	Dic₁₉⋊C₄	C₂×C₄	1	2	yes	—
56	C₄×Dic₁₉	C₂×C₄	1	2	yes	—
57	C₄×Dic₁₉	C₂×C₄	1	2	yes	—
58	C₄×Dic₁₉	C₂³	1	2	yes	—
59	#64520	—	2	4	no	—
60	C₂×C₁₉⋊C₈	C₂³	1	4	yes	—
61	C₂³×C₁₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×D₁₉	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₁₉	—	2	2	no	—
64	C₂×C₄×C₁₉	—	4	1	no	—
65	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₉	#41	1	2	yes	—
71	C₂×C₄×C₁₉	#41	1	2	yes	—
72	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
73	C₂²⋊D₁₉	—	2	4	no	—
74	C₂×Dic₁₉	C₂²×C₄	1	2	yes	—
75	C₂×Dic₁₉	C₂²×C₄	1	2	yes	—
76	C₂×Dic₁₉	C₂²×C₄	1	2	yes	—
77	D₇₆	—	2	2	no	—
78	C₄×D₁₉	—	2	4	no	—
79	C₄×D₁₉	—	2	2	no	—
80	C₄×D₁₉	—	2	2	no	—
81	C₄×D₁₉	—	4	2	no	—
82	Dic₃₈	—	2	2	no	—
83	C₈×C₁₉	—	2	2	no	—
84	C₈×C₁₉	—	2	2	no	—
85	C₁₉⋊C₈	—	2	4	no	—
86	#2925	—	19	1	no	maximal, Sylow
87	C₂²×C₁₉	—	2	1	no	—
88	C₂²×C₁₉	—	2	1	no	—
89	C₂²×C₁₉	#117	1	2	yes	—
90	C₂²×C₁₉	C₂³×C₄	1	1	yes	characteristic
91	C₂×D₁₉	—	2	2	no	—
92	C₂×D₁₉	—	2	2	no	—
93	C₂×D₁₉	—	2	2	no	—
94	C₄×C₁₉	—	2	1	no	—
95	C₄×C₁₉	—	2	1	no	—
96	C₄×C₁₉	—	2	1	no	—
97	C₄×C₁₉	#117	1	2	yes	—
98	C₄×C₁₉	#131	1	2	yes	—
99	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
100	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
101	Dic₁₉	—	2	2	no	—
102	Dic₁₉	—	2	2	no	—
103	Dic₁₉	—	2	2	no	—
104	C₄×#41	—	19	1	no	—
105	#433	—	19	4	no	—
106	#432	—	19	2	no	—
107	#431	—	19	2	no	—
108	#423	—	19	1	no	—
109	C₂×C₄⋊C₈	D₁₉	1	1	yes	characteristic, complemented
110	#408	—	19	2	no	—
111	#405	—	19	2	no	—
112	C₂×C₁₉	—	2	2	no	—
113	C₂×C₁₉	C₂×#117	1	1	yes	characteristic
114	C₂×C₁₉	C₂×#131	1	1	yes	characteristic
115	C₂×C₁₉	#568	1	1	yes	characteristic
116	D₁₉	—	4	2	no	—
117	C₂×#41	—	19	1	no	—
118	C₂×M₁₆	—	19	2	no	—
119	C₂²×C₈	C₂×D₁₉	1	2	yes	—
120	C₄×Q₈	—	19	1	no	—
121	C₄×Q₈	—	19	1	no	—
122	C₄×D₄	—	19	2	no	—
123	C₄×D₄	—	19	2	no	—
124	C₄×D₄	—	19	1	no	—
125	C₄×D₄	—	19	1	no	—
126	#117	—	19	1	no	—
127	#117	—	19	2	no	—
128	C₂×C₄²	—	19	2	no	—
129	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
130	C₄⋊C₈	—	19	2	no	—
131	C₄⋊C₈	—	19	2	no	—
132	C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
133	C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
134	C₄⋊C₈	C₂×D₁₉	1	2	yes	—
135	C₂²⋊C₈	—	19	4	no	—
136	C₂²⋊C₈	—	19	4	no	—
137	#97	—	19	4	no	—
138	C₄×C₈	—	19	4	no	—
139	C₁₉	#2925	1	1	yes	characteristic, complemented, Sylow
140	#41	—	38	2	no	—
141	#41	—	38	2	no	—
142	C₂×Q₈	—	19	1	no	—
143	C₂×D₄	—	19	2	no	—
144	C₂×D₄	—	19	1	no	—
145	C₂²×C₄	—	19	2	no	—
146	C₂²×C₄	—	19	2	no	—
147	C₂²×C₄	—	19	2	no	—
148	C₂²×C₄	C₄×D₁₉	1	1	yes	characteristic
149	C₂²×C₄	C₄×D₁₉	1	1	yes	characteristic
150	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
151	M₁₆	—	38	4	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	19	4	no	—
154	C₂×C₈	C₂²×D₁₉	1	2	yes	—
155	C₂×C₈	C₂²×D₁₉	1	2	yes	—
156	C₄⋊C₄	—	19	2	no	—
157	C₄⋊C₄	—	19	2	no	—
158	C₄⋊C₄	—	19	1	no	—
159	C₄⋊C₄	—	19	1	no	—
160	C₂²⋊C₄	—	19	2	no	—
161	C₂²⋊C₄	—	19	2	no	—
162	C₂²⋊C₄	—	19	2	no	—
163	C₄²	—	19	2	no	—
164	C₄²	—	19	2	no	—
165	C₄²	—	19	2	no	—
166	C₄²	C₄×D₁₉	1	1	yes	characteristic
167	C₄²	C₄×D₁₉	1	1	yes	characteristic
168	C₄²	C₂²×D₁₉	1	1	yes	characteristic
169	C₄²	C₂²×D₁₉	1	1	yes	characteristic
170	C₂³	—	19	2	no	—
171	C₂³	C₂×C₄×D₁₉	1	1	yes	characteristic
172	Q₈	—	38	2	no	—
173	D₄	—	38	4	no	—
174	D₄	—	38	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	1	no	—
177	C₂×C₄	—	19	2	no	—
178	C₂×C₄	—	19	2	no	—
179	C₂×C₄	—	19	2	no	—
180	C₂×C₄	—	38	2	no	—
181	C₂×C₄	—	38	2	no	—
182	C₂×C₄	—	38	4	no	—
183	C₂×C₄	—	76	2	no	—
184	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
185	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
187	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
188	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
189	C₂×C₄	#64543	1	2	yes	—
190	C₂×C₄	#64545	1	2	yes	—
191	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
192	C₈	—	2	2	no	—
193	C₈	—	2	2	no	—
194	C₈	—	38	4	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	38	2	no	—
198	C₂²	—	38	2	no	—
199	C₂²	—	38	2	no	—
200	C₂²	#10597338	1	2	yes	—
201	C₂²	C₂²×C₄×D₁₉	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	38	2	no	—
206	C₄	—	38	2	no	—
207	C₄	—	38	2	no	—
208	C₄	#10597338	1	2	yes	—
209	C₄	#10597347	1	2	yes	—
210	C₄	C₂×#64543	1	1	yes	characteristic
211	C₄	C₂×#64545	1	1	yes	characteristic
212	C₂	—	2	2	no	—
213	C₂	—	76	2	no	—
214	C₂	C₂×#10597338	1	1	yes	characteristic
215	C₂	C₂×#10597347	1	1	yes	characteristic
216	C₂	#11927310	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉
commutator, socle	C₂²×C₁₉	C₂³×C₄
Fitting	C₂×C₁₉×C₄⋊C₈	C₂
prime radical	C₂×#64541	C₂²
cosocle	C₂×C₄×C₁₉	C₂⁴
hypercenter	C₂×C₄⋊C₈	D₁₉
hypocenter	C₁₉	#2925
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #11926604	C₂	semidirect
3. #11926600	C₂	semidirect
4. #11926596	C₂	semidirect
5. #11926568	C₂	semidirect
7. #11926499	C₂	semidirect
8. C₂×C₁₉×C₄⋊C₈	C₂	semidirect
109. C₂×C₄⋊C₈	D₁₉	semidirect
139. C₁₉	#2925	semidirect

Maximal subgroups

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

subgroup	index
C₄×#64541	2
#11926604	2
#11926600	2
#11926596	2
#11926568	2
#11926510	2
#11926499	2
C₂×C₁₉×C₄⋊C₈	2
#2925	19