Subgroups — #14572629

Order 2432.

372 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927078	C₂	1	1	yes	maximal, characteristic
2	#11926672	C₂	1	2	yes	maximal, complemented
3	#11926617	C₂	1	2	yes	maximal, complemented
4	#11926615	C₂	1	2	yes	maximal, complemented
5	C₈×C₂²⋊D₁₉	C₂	1	2	yes	maximal, complemented
6	#11926606	C₂	1	2	yes	maximal, complemented
7	#11926597	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926501	C₂	1	2	yes	maximal, complemented
9	C₂×C₁₉×#97	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₈×C₁₉	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
12	#10597376	C₄	1	1	yes	characteristic
13	#10597375	C₄	1	2	yes	—
14	C₄×C₂²⋊D₁₉	C₂²	1	2	yes	—
15	#10597370	C₄	1	1	yes	characteristic
16	C₂×#64515	C₂²	1	2	yes	—
17	C₂×C₈×D₁₉	C₂²	1	2	yes	—
18	#10597322	C₄	1	2	yes	—
19	#10597321	C₄	1	1	yes	characteristic
20	C₄×D₇₆	C₂²	1	1	yes	characteristic
21	#10597318	C₂²	1	2	yes	—
22	#10597316	C₄	1	1	yes	characteristic
23	C₄×Dic₃₈	C₂²	1	1	yes	characteristic
24	C₁₉×#97	C₂²	1	1	yes	characteristic
25	C₁₉×#97	C₂²	1	2	yes	—
26	C₁₉×#97	C₂²	1	1	yes	characteristic
27	C₂²⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
28	C₂²⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
29	(C₂×D₁₉)⋊C₈	C₂²	1	2	yes	—
30	(C₂×D₁₉)⋊C₈	C₂²	1	2	yes	—
31	#10597272	C₂²	1	2	yes	—
32	Dic₁₉⋊C₈	C₂²	1	2	yes	—
33	Dic₁₉⋊C₈	C₂²	1	2	yes	—
34	C₈×Dic₁₉	C₂²	1	2	yes	—
35	C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
36	C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
37	C₂²×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
40	C₂×C₂²⋊D₁₉	C₂×C₄	1	2	yes	—
41	C₂×D₇₆	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₁₉	C₂³	1	2	yes	—
43	C₂×Dic₃₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₉	—	2	4	no	—
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	C₂²⋊Dic₁₉	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	(C₂×D₁₉)⋊C₄	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
58	Dic₁₉⋊C₄	C₂×C₄	1	2	yes	—
59	Dic₁₉⋊C₄	C₂×C₄	1	2	yes	—
60	Dic₁₉⋊C₄	C₂×C₄	1	2	yes	—
61	Dic₁₉⋊C₄	C₂×C₄	1	2	yes	—
62	C₄×Dic₁₉	C₂³	1	2	yes	—
63	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
64	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
65	#64515	—	2	4	no	—
66	C₈×D₁₉	—	2	4	no	—
67	C₂³×C₁₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×D₁₉	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₁₉	—	2	2	no	—
70	C₂×C₄×C₁₉	—	2	2	no	—
71	C₂×C₄×C₁₉	—	4	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
78	C₂²⋊D₁₉	—	2	4	no	—
79	C₂×Dic₁₉	C₂²×C₄	1	2	yes	—
80	C₂×Dic₁₉	C₂²×C₄	1	2	yes	—
81	C₂×Dic₁₉	C₂²×C₄	1	2	yes	—
82	D₇₆	—	4	1	no	—
83	C₄×D₁₉	—	2	2	no	—
84	C₄×D₁₉	—	2	2	no	—
85	Dic₃₈	—	4	1	no	—
86	C₈×C₁₉	—	2	2	no	—
87	C₈×C₁₉	#41	1	4	yes	—
88	C₁₉⋊C₈	—	2	2	no	—
89	C₁₉⋊C₈	—	2	2	no	—
90	#2968	—	19	1	no	maximal, Sylow
91	C₂²×C₁₉	—	2	1	no	—
92	C₂²×C₁₉	—	2	1	no	—
93	C₂²×C₁₉	#131	1	2	yes	—
94	C₂²×C₁₉	C₂³×C₄	1	1	yes	characteristic
95	C₂×D₁₉	—	2	2	no	—
96	C₂×D₁₉	—	2	2	no	—
97	C₂×D₁₉	—	2	2	no	—
98	C₄×C₁₉	—	2	1	no	—
99	C₄×C₁₉	—	2	1	no	—
100	C₄×C₁₉	—	2	1	no	—
101	C₄×C₁₉	—	2	1	no	—
102	C₄×C₁₉	—	2	1	no	—
103	C₄×C₁₉	C₂×#41	1	2	yes	—
104	Dic₁₉	—	2	2	no	—
105	Dic₁₉	—	2	2	no	—
106	Dic₁₉	—	2	2	no	—
107	#529	—	19	1	no	—
108	#446	—	19	2	no	—
109	#435	—	19	2	no	—
110	#435	—	19	2	no	—
111	C₈×D₄	—	19	2	no	—
112	#433	—	19	2	no	—
113	#431	—	19	1	no	—
114	#405	—	19	2	no	—
115	C₂×#97	D₁₉	1	1	yes	characteristic, complemented
116	C₂×C₁₉	—	2	2	no	—
117	C₂×C₁₉	C₄×#41	1	1	yes	characteristic
118	C₂×C₁₉	C₂×#131	1	1	yes	characteristic
119	C₂×C₁₉	#568	1	1	yes	characteristic
120	D₁₉	—	4	2	no	—
121	C₂×M₁₆	—	19	2	no	—
122	C₂²×C₈	—	19	2	no	—
123	C₂²×C₈	C₂×D₁₉	1	2	yes	—
124	#126	—	19	2	no	—
125	#125	—	19	1	no	—
126	#124	—	19	1	no	—
127	#123	—	19	2	no	—
128	C₂²⋊Q₈	—	19	1	no	—
129	#121	—	19	1	no	—
130	C₄×Q₈	—	19	1	no	—
131	C₄×D₄	—	19	2	no	—
132	C₄×D₄	—	19	1	no	—
133	#117	—	19	2	no	—
134	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
135	C₄⋊C₈	—	19	2	no	—
136	C₄⋊C₈	—	19	2	no	—
137	C₄⋊C₈	—	19	2	no	—
138	C₂²⋊C₈	—	19	1	no	—
139	C₂²⋊C₈	—	19	1	no	—
140	C₂²⋊C₈	—	19	2	no	—
141	C₂²⋊C₈	—	19	2	no	—
142	#97	—	19	2	no	—
143	#97	C₂×D₁₉	1	1	yes	characteristic
144	#97	C₂×D₁₉	1	1	yes	characteristic
145	#97	C₂×D₁₉	1	2	yes	—
146	C₄×C₈	—	19	2	no	—
147	C₄×C₈	—	19	2	no	—
148	C₁₉	#2968	1	1	yes	characteristic, complemented, Sylow
149	C₂×Q₈	—	19	1	no	—
150	C₂×D₄	—	19	2	no	—
151	C₂×D₄	—	19	1	no	—
152	C₂²×C₄	—	19	2	no	—
153	C₂²×C₄	C₄×D₁₉	1	1	yes	characteristic
154	C₂²×C₄	C₄×D₁₉	1	1	yes	characteristic
155	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
156	M₁₆	—	38	4	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	19	2	no	—
159	C₂×C₈	—	19	2	no	—
160	C₂×C₈	—	38	4	no	—
161	C₂×C₈	C₂²×D₁₉	1	2	yes	—
162	C₂×C₈	C₂²×D₁₉	1	2	yes	—
163	C₄⋊C₄	—	19	2	no	—
164	C₄⋊C₄	—	19	2	no	—
165	C₄⋊C₄	—	19	2	no	—
166	C₄⋊C₄	—	19	2	no	—
167	C₄⋊C₄	—	19	1	no	—
168	C₄⋊C₄	—	19	1	no	—
169	C₂²⋊C₄	—	19	2	no	—
170	C₂²⋊C₄	—	19	2	no	—
171	C₂²⋊C₄	—	19	2	no	—
172	C₂²⋊C₄	—	19	2	no	—
173	C₂²⋊C₄	—	19	2	no	—
174	C₄²	—	19	2	no	—
175	C₄²	C₄×D₁₉	1	1	yes	characteristic
176	C₄²	C₄×D₁₉	1	1	yes	characteristic
177	C₄²	C₂²×D₁₉	1	1	yes	characteristic
178	C₄²	C₂²×D₁₉	1	1	yes	characteristic
179	C₂³	—	19	2	no	—
180	C₂³	C₂×C₄×D₁₉	1	1	yes	characteristic
181	Q₈	—	76	1	no	—
182	D₄	—	38	4	no	—
183	D₄	—	76	1	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	4	1	no	—
187	C₂×C₄	—	19	2	no	—
188	C₂×C₄	—	19	2	no	—
189	C₂×C₄	—	19	2	no	—
190	C₂×C₄	—	38	2	no	—
191	C₂×C₄	—	38	2	no	—
192	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
195	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
196	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
197	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	—	38	2	no	—
200	C₈	—	38	2	no	—
201	C₈	#64541	1	4	yes	—
202	C₂²	—	2	1	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	38	2	no	—
205	C₂²	—	38	2	no	—
206	C₂²	—	38	2	no	—
207	C₂²	#10597353	1	2	yes	—
208	C₂²	C₂²×C₄×D₁₉	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	38	2	no	—
215	C₄	—	38	2	no	—
216	C₄	—	38	2	no	—
217	C₄	C₂×#64541	1	2	yes	—
218	C₂	—	2	2	no	—
219	C₂	—	76	2	no	—
220	C₂	C₄×#64541	1	1	yes	characteristic
221	C₂	C₂×#10597353	1	1	yes	characteristic
222	C₂	#11927310	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉
commutator, socle	C₂²×C₁₉	C₂³×C₄
Fitting	C₂×C₁₉×#97	C₂
prime radical	#10597376	C₄
cosocle	C₂×C₄×C₁₉	C₂⁴
hypercenter	C₂×#97	D₁₉
hypocenter	C₁₉	#2968
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
2. #11926672	C₂	semidirect
3. #11926617	C₂	semidirect
4. #11926615	C₂	semidirect
5. C₈×C₂²⋊D₁₉	C₂	semidirect
6. #11926606	C₂	semidirect
7. #11926597	C₂	semidirect
8. #11926501	C₂	semidirect
9. C₂×C₁₉×#97	C₂	semidirect
115. C₂×#97	D₁₉	semidirect
148. C₁₉	#2968	semidirect

Maximal subgroups

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

subgroup	index
#11927078	2
#11926672	2
#11926617	2
#11926615	2
C₈×C₂²⋊D₁₉	2
#11926606	2
#11926597	2
#11926501	2
C₂×C₁₉×#97	2
#2968	19