Subgroups — #14572649

Order 2432.

406 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11926675	C₂	1	2	yes	maximal, complemented
2	#11926616	C₂	1	4	yes	maximal, complemented
3	D₄×C₁₉⋊C₈	C₂	1	2	yes	maximal, complemented
4	#11926601	C₂	1	2	yes	maximal, complemented
5	#11926601	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#64520	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#10597260	C₂	1	2	yes	maximal, complemented
8	C₄×#41×C₁₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#41×C₁₉	C₄	1	1	yes	characteristic
10	C₄×Q₈×C₁₉	C₄	1	1	yes	characteristic
11	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
12	C₄×D₄×C₁₉	C₄	1	2	yes	—
13	C₄×D₄×C₁₉	C₄	1	1	yes	characteristic
14	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
15	C₄×D₄×C₁₉	C₂²	1	2	yes	—
16	C₁₉×#117	C₄	1	1	yes	characteristic
17	C₁₉×#117	C₄	1	2	yes	—
18	C₂×C₄²×C₁₉	C₂²	1	2	yes	—
19	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
20	C₂×#64520	C₂²	1	2	yes	—
21	C₂²×C₁₉⋊C₈	C₂²	1	4	yes	—
22	C₂²⋊(C₁₉⋊C₈)	C₂²	1	4	yes	—
23	C₂²⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
24	C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
25	C₄⋊(C₁₉⋊C₈)	C₂²	1	4	yes	—
26	#10597260	C₂²	1	2	yes	—
27	#10597260	C₂²	1	1	yes	characteristic
28	#10597260	C₂²	1	4	yes	—
29	#10597260	C₂²	1	1	yes	characteristic
30	C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
31	#41×C₁₉	—	2	4	no	—
32	C₂×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂×D₄×C₁₉	C₂×C₄	1	2	yes	—
34	C₂×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₉	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₉	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
42	C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
43	C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
44	C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
45	C₂²⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
46	C₂²⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
47	C₂²⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
48	C₄²×C₁₉	C₂×C₄	1	2	yes	—
49	C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
50	C₄²×C₁₉	C₂×C₄	1	2	yes	—
51	C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
52	C₄²×C₁₉	C₂³	1	1	yes	characteristic
53	C₄²×C₁₉	C₂³	1	1	yes	characteristic
54	C₄²×C₁₉	C₂³	1	2	yes	—
55	#64520	—	2	4	no	—
56	C₂×C₁₉⋊C₈	—	2	8	no	—
57	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
58	C₂×C₁₉⋊C₈	C₂³	1	4	yes	—
59	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
60	C₂³×C₁₉	C₂²×C₄	1	1	yes	characteristic
61	C₂³×C₁₉	C₂²×C₄	1	2	yes	—
62	Q₈×C₁₉	—	2	2	no	—
63	D₄×C₁₉	—	2	4	no	—
64	D₄×C₁₉	M₁₆	1	4	yes	complemented
65	C₂×C₄×C₁₉	—	2	4	no	—
66	C₂×C₄×C₁₉	—	2	2	no	—
67	C₂×C₄×C₁₉	—	2	4	no	—
68	C₂×C₄×C₁₉	—	2	2	no	—
69	C₂×C₄×C₁₉	—	4	2	no	—
70	C₂×C₄×C₁₉	—	4	1	no	—
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	2	yes	—
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	2	yes	—
77	C₂×C₄×C₁₉	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
80	C₁₉⋊C₈	—	2	4	no	—
81	C₁₉⋊C₈	—	2	2	no	—
82	C₁₉⋊C₈	#41	1	4	yes	complemented
83	#2970	—	19	1	no	maximal, Sylow
84	C₂²×C₁₉	—	2	2	no	—
85	C₂²×C₁₉	—	2	2	no	—
86	C₂²×C₁₉	—	2	1	no	—
87	C₂²×C₁₉	—	2	1	no	—
88	C₂²×C₁₉	—	2	1	no	—
89	C₂²×C₁₉	C₂×M₁₆	1	4	yes	—
90	C₂²×C₁₉	C₂³×C₄	1	1	yes	characteristic
91	C₄×C₁₉	—	2	2	no	—
92	C₄×C₁₉	—	2	1	no	—
93	C₄×C₁₉	—	2	1	no	—
94	C₄×C₁₉	—	2	2	no	—
95	C₄×C₁₉	—	2	1	no	—
96	C₄×C₁₉	—	2	2	no	—
97	C₄×C₁₉	—	2	1	no	—
98	C₄×C₁₉	C₂×M₁₆	1	2	yes	—
99	C₄×C₁₉	C₂×#41	1	2	yes	—
100	C₄×#41	D₁₉	1	1	yes	characteristic
101	#446	—	19	2	no	—
102	#435	—	19	4	no	—
103	C₈×D₄	—	19	2	no	—
104	#432	—	19	1	no	—
105	#432	—	19	2	no	—
106	C₄×M₁₆	—	19	1	no	—
107	C₂×#97	—	19	2	no	—
108	C₂×C₁₉	—	2	4	no	—
109	C₂×C₁₉	—	4	1	no	—
110	C₂×C₁₉	C₄×#41	1	1	yes	characteristic
111	C₂×C₁₉	C₂²×M₁₆	1	1	yes	characteristic
112	C₂×C₁₉	#568	1	1	yes	characteristic
113	C₂×#41	Dic₁₉	1	1	yes	characteristic
114	C₂×M₁₆	—	19	2	no	—
115	C₂²×C₈	—	19	4	no	—
116	C₄×Q₈	Dic₁₉	1	1	yes	characteristic
117	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
118	C₄×D₄	Dic₁₉	1	2	yes	—
119	C₄×D₄	Dic₁₉	1	1	yes	characteristic
120	C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
121	C₄×D₄	C₂×D₁₉	1	2	yes	—
122	#117	Dic₁₉	1	1	yes	characteristic
123	#117	Dic₁₉	1	2	yes	—
124	C₂×C₄²	C₂×D₁₉	1	2	yes	—
125	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
126	C₄⋊C₈	—	19	2	no	—
127	C₄⋊C₈	—	19	4	no	—
128	C₂²⋊C₈	—	19	4	no	—
129	C₂²⋊C₈	—	19	2	no	—
130	#97	—	19	1	no	—
131	#97	—	19	2	no	—
132	#97	—	19	4	no	—
133	#97	—	19	1	no	—
134	C₄×C₈	—	19	2	no	—
135	C₁₉	#2970	1	1	yes	characteristic, complemented, Sylow
136	#41	—	2	4	no	—
137	C₂×Q₈	C₂×Dic₁₉	1	1	yes	characteristic
138	C₂×D₄	C₂×Dic₁₉	1	2	yes	—
139	C₂×D₄	C₂×Dic₁₉	1	1	yes	characteristic
140	C₂²×C₄	C₂×Dic₁₉	1	2	yes	—
141	C₂²×C₄	C₂×Dic₁₉	1	1	yes	characteristic
142	C₂²×C₄	C₂×Dic₁₉	1	1	yes	characteristic
143	C₂²×C₄	C₂×Dic₁₉	1	2	yes	—
144	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
145	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
146	M₁₆	—	38	4	no	—
147	C₂×C₈	—	19	2	no	—
148	C₂×C₈	—	19	2	no	—
149	C₂×C₈	—	19	4	no	—
150	C₂×C₈	—	38	8	no	—
151	C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
152	C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
153	C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
154	C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
155	C₂²⋊C₄	C₂×Dic₁₉	1	2	yes	—
156	C₂²⋊C₄	C₂×Dic₁₉	1	2	yes	—
157	C₂²⋊C₄	C₂×Dic₁₉	1	2	yes	—
158	C₄²	C₂×Dic₁₉	1	2	yes	—
159	C₄²	C₂×Dic₁₉	1	1	yes	characteristic
160	C₄²	C₂×Dic₁₉	1	2	yes	—
161	C₄²	C₂×Dic₁₉	1	1	yes	characteristic
162	C₄²	C₂²×D₁₉	1	1	yes	characteristic
163	C₄²	C₂²×D₁₉	1	1	yes	characteristic
164	C₄²	C₂²×D₁₉	1	2	yes	—
165	C₂³	C₂²×Dic₁₉	1	1	yes	characteristic
166	C₂³	C₂²×Dic₁₉	1	2	yes	—
167	Q₈	—	2	2	no	—
168	D₄	—	2	4	no	—
169	D₄	#64520	1	4	yes	complemented
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
177	C₂×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
178	C₂×C₄	C₂²×Dic₁₉	1	2	yes	—
179	C₂×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
180	C₂×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
181	C₂×C₄	C₂²×Dic₁₉	1	2	yes	—
182	C₂×C₄	C₂²×Dic₁₉	1	2	yes	—
183	C₂×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
184	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
185	C₈	—	19	4	no	—
186	C₈	—	38	4	no	—
187	C₈	—	38	2	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	C₂×#64520	1	4	yes	—
194	C₂²	C₂³×Dic₁₉	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	2	no	—
199	C₄	—	2	1	no	—
200	C₄	—	2	2	no	—
201	C₄	—	2	1	no	—
202	C₄	C₂×#64520	1	2	yes	—
203	C₄	#41×D₁₉	1	2	yes	—
204	C₂	—	2	4	no	—
205	C₂	—	4	1	no	—
206	C₂	#41×Dic₁₉	1	1	yes	characteristic
207	C₂	C₂²×#64520	1	1	yes	characteristic
208	C₂	#11927311	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉
commutator, socle	C₂²×C₁₉	C₂³×C₄
Fitting	C₄×#41×C₁₉	C₂
prime radical	C₂×#41×C₁₉	C₄
cosocle	C₂×C₄×C₁₉	C₂⁴
hypercenter	C₄×#41	D₁₉
hypocenter	C₁₉	#2970
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #11926675	C₂	semidirect
2. #11926616	C₂	semidirect
3. D₄×C₁₉⋊C₈	C₂	semidirect
4. #11926601	C₂	semidirect
5. #11926601	C₂	semidirect
6. C₄×#64520	C₂	semidirect
7. C₂×#10597260	C₂	semidirect
64. D₄×C₁₉	M₁₆	semidirect
82. C₁₉⋊C₈	#41	semidirect
135. C₁₉	#2970	semidirect
169. D₄	#64520	semidirect

Maximal subgroups

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

subgroup	index
#11926675	2
#11926616	2
D₄×C₁₉⋊C₈	2
#11926601	2
C₄×#64520	2
C₂×#10597260	2
C₄×#41×C₁₉	2
#2970	19