Subgroups — #14578346

Order 2432.

1424 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927380	C₂	1	1	yes	maximal, characteristic
2	#11927311	C₂	1	1	yes	maximal, characteristic, complemented
3	#11927310	C₂	1	6	yes	maximal, complemented
4	#11927309	C₂	1	8	yes	maximal, complemented
5	C₂×#10597347	C₂	1	6	yes	maximal, complemented
6	#131×D₁₉	C₂	1	8	yes	maximal, complemented
7	C₂×C₁₉×#131	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41×C₁₉	C₂²	1	1	yes	characteristic
9	#10597436	C₄	1	4	yes	—
10	#10597435	C₄	1	4	yes	—
11	#41×D₁₉	C₂²	1	8	yes	—
12	#10597433	C₄	1	2	yes	—
13	#10597430	C₄	1	6	yes	—
14	C₂×#64541	C₂²	1	6	yes	—
15	C₁₉×#131	C₂²	1	4	yes	complemented
16	C₁₉×#131	C₂²	1	4	yes	complemented
17	C₂×M₁₆×C₁₉	C₂²	1	3	yes	complemented
18	C₂²×C₈×C₁₉	C₂²	1	3	yes	complemented
19	#10597394	C₂²	1	8	yes	complemented
20	C₂×#64520	C₂²	1	6	yes	complemented
21	#10597353	C₂²	1	24	yes	complemented
22	M₁₆×D₁₉	C₂²	1	24	yes	complemented
23	#10597347	C₂²	1	12	yes	complemented
24	#10597347	C₂²	1	4	yes	complemented
25	#10597347	C₂²	1	12	yes	complemented
26	#10597347	C₂²	1	12	yes	complemented
27	C₂×#64515	C₂²	1	12	yes	complemented
28	C₂×C₈×D₁₉	C₂²	1	12	yes	complemented
29	#41×C₁₉	C₂×C₄	1	4	yes	—
30	#41×C₁₉	C₂³	1	4	yes	—
31	C₂×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
32	C₂×D₄×C₁₉	C₂×C₄	1	3	yes	—
33	C₂²×C₄×C₁₉	C₂³	1	3	yes	—
34	C₂×C₂²⋊D₁₉	C₂×C₄	1	12	yes	—
35	#64545	C₂×C₄	1	8	yes	—
36	Q₈×D₁₉	C₂×C₄	1	8	yes	—
37	#64543	C₂×C₄	1	24	yes	—
38	D₄×D₁₉	C₂×C₄	1	24	yes	—
39	#64541	C₂×C₄	1	12	yes	—
40	#64541	C₂×C₄	1	12	yes	—
41	#64541	C₂³	1	4	yes	—
42	#64541	C₂³	1	12	yes	—
43	C₂×D₇₆	C₂×C₄	1	6	yes	—
44	C₂×C₄×D₁₉	C₂³	1	12	yes	—
45	C₂×Dic₃₈	C₂×C₄	1	6	yes	—
46	M₁₆×C₁₉	C₂³	1	12	yes	—
47	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
48	C₂×C₈×C₁₉	C₂³	1	3	yes	—
49	C₂×C₈×C₁₉	C₂³	1	6	yes	—
50	C₂×C₈×C₁₉	C₂³	1	6	yes	—
51	#64520	C₂³	1	12	yes	—
52	#64520	C₂³	1	4	yes	—
53	C₂×C₁₉⋊C₈	C₂³	1	12	yes	—
54	#64515	C₂³	1	8	yes	—
55	#64515	C₂³	1	24	yes	—
56	C₈×D₁₉	C₂³	1	8	yes	—
57	C₈×D₁₉	C₂³	1	24	yes	—
58	C₂³×C₁₉	C₂²×C₄	1	3	yes	—
59	C₂²×D₁₉	C₂²×C₄	1	12	yes	—
60	Q₈×C₁₉	C₂²×C₄	1	4	yes	—
61	D₄×C₁₉	C₂²×C₄	1	12	yes	—
62	C₂×C₄×C₁₉	C₂²×C₄	1	6	yes	—
63	C₂×C₄×C₁₉	C₂²×C₄	1	3	yes	—
64	C₂×C₄×C₁₉	C₂⁴	1	6	yes	—
65	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
66	C₂²⋊D₁₉	C₂²×C₄	1	8	yes	—
67	C₂²⋊D₁₉	C₂²×C₄	1	24	yes	—
68	C₂×Dic₁₉	C₂²×C₄	1	12	yes	—
69	D₇₆	C₂²×C₄	1	4	yes	—
70	D₇₆	C₂²×C₄	1	12	yes	—
71	C₄×D₁₉	C₂²×C₄	1	24	yes	—
72	C₄×D₁₉	C₂⁴	1	8	yes	—
73	Dic₃₈	C₂²×C₄	1	4	yes	—
74	Dic₃₈	C₂²×C₄	1	12	yes	—
75	C₈×C₁₉	C₂⁴	1	6	yes	—
76	C₈×C₁₉	C₂⁴	1	1	yes	characteristic
77	C₈×C₁₉	C₂⁴	1	1	yes	characteristic
78	C₁₉⋊C₈	C₂⁴	1	8	yes	—
79	#3573	—	19	1	no	maximal, Sylow
80	C₂²×C₁₉	—	4	3	no	—
81	C₂²×C₁₉	C₂³×C₄	1	6	yes	—
82	C₂²×C₁₉	C₂³×C₄	1	1	yes	characteristic
83	C₂×D₁₉	—	4	12	no	—
84	C₂×D₁₉	C₂³×C₄	1	8	yes	—
85	C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
86	C₄×C₁₉	C₂³×C₄	1	6	yes	—
87	C₄×C₁₉	C₂⁵	1	1	yes	characteristic
88	Dic₁₉	C₂³×C₄	1	8	yes	—
89	#585	—	19	1	no	—
90	#568	—	19	1	no	—
91	#568	—	19	8	no	—
92	#568	—	19	6	no	—
93	C₂×#131	—	19	8	no	—
94	C₂×#131	—	19	6	no	—
95	C₂×#131	D₁₉	1	1	yes	characteristic, complemented
96	C₂×C₁₉	—	2	6	no	—
97	C₂×C₁₉	—	2	1	no	—
98	C₂×C₁₉	C₂⁴×C₄	1	1	yes	characteristic
99	D₁₉	—	2	8	no	—
100	2₋¹⁺⁴	—	19	4	no	—
101	2₋¹⁺⁴	—	19	2	no	—
102	H(5,𝔽₂)	—	19	4	no	—
103	H(5,𝔽₂)	—	19	6	no	—
104	C₂×#41	—	19	8	no	—
105	C₂×#41	—	19	6	no	—
106	C₂×#41	C₂×D₁₉	1	1	yes	characteristic
107	#131	—	19	8	no	—
108	#131	—	19	24	no	—
109	#131	—	19	12	no	—
110	#131	—	19	4	no	—
111	#131	—	19	12	no	—
112	#131	—	19	12	no	—
113	#131	C₂×D₁₉	1	4	yes	complemented
114	#131	C₂×D₁₉	1	4	yes	complemented
115	C₂×M₁₆	—	19	6	no	—
116	C₂×M₁₆	—	19	24	no	—
117	C₂×M₁₆	—	19	12	no	—
118	C₂×M₁₆	C₂×D₁₉	1	3	yes	complemented
119	C₂²×C₈	—	19	12	no	—
120	C₂²×C₈	C₂×D₁₉	1	3	yes	complemented
121	C₁₉	#3573	1	1	yes	characteristic, complemented, Sylow
122	#41	—	19	8	no	—
123	#41	—	19	24	no	—
124	#41	—	19	12	no	—
125	#41	—	19	12	no	—
126	#41	—	19	12	no	—
127	#41	—	19	4	no	—
128	#41	C₄×D₁₉	1	4	yes	—
129	#41	C₂²×D₁₉	1	4	yes	—
130	C₂×Q₈	—	19	8	no	—
131	C₂×Q₈	—	19	6	no	—
132	C₂×Q₈	C₄×D₁₉	1	1	yes	characteristic
133	C₂×D₄	—	19	12	no	—
134	C₂×D₄	—	19	24	no	—
135	C₂×D₄	—	19	6	no	—
136	C₂×D₄	C₄×D₁₉	1	3	yes	—
137	C₂²×C₄	—	19	12	no	—
138	C₂²×C₄	C₂²×D₁₉	1	3	yes	—
139	M₁₆	—	19	12	no	—
140	M₁₆	—	19	4	no	—
141	M₁₆	—	19	24	no	—
142	M₁₆	—	19	8	no	—
143	M₁₆	C₂²×D₁₉	1	12	yes	—
144	C₂×C₈	—	19	12	no	—
145	C₂×C₈	—	19	24	no	—
146	C₂×C₈	—	19	8	no	—
147	C₂×C₈	C₂²×D₁₉	1	6	yes	—
148	C₂×C₈	C₂²×D₁₉	1	3	yes	—
149	C₂×C₈	C₂²×D₁₉	1	6	yes	—
150	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
151	C₂³	—	19	12	no	—
152	C₂³	C₂×C₄×D₁₉	1	3	yes	—
153	Q₈	—	19	12	no	—
154	Q₈	—	19	4	no	—
155	Q₈	C₂×C₄×D₁₉	1	4	yes	—
156	D₄	—	19	24	no	—
157	D₄	—	19	8	no	—
158	D₄	—	19	12	no	—
159	D₄	—	19	4	no	—
160	D₄	C₂×C₄×D₁₉	1	12	yes	—
161	C₂×C₄	—	19	12	no	—
162	C₂×C₄	—	19	24	no	—
163	C₂×C₄	—	19	8	no	—
164	C₂×C₄	C₂×C₄×D₁₉	1	6	yes	—
165	C₂×C₄	C₂×C₄×D₁₉	1	3	yes	—
166	C₂×C₄	C₂³×D₁₉	1	6	yes	—
167	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
168	C₈	—	19	8	no	—
169	C₈	C₂³×D₁₉	1	6	yes	—
170	C₈	C₂³×D₁₉	1	1	yes	characteristic
171	C₈	C₂³×D₁₉	1	1	yes	characteristic
172	C₂²	—	4	3	no	—
173	C₂²	—	19	8	no	—
174	C₂²	—	76	12	no	—
175	C₂²	C₂²×C₄×D₁₉	1	6	yes	—
176	C₂²	C₂²×C₄×D₁₉	1	1	yes	characteristic
177	C₄	—	19	8	no	—
178	C₄	C₂²×C₄×D₁₉	1	6	yes	—
179	C₄	C₂²×C₄×D₁₉	1	1	yes	characteristic
180	C₄	C₂⁴×D₁₉	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	6	no	—
183	C₂	—	38	8	no	—
184	C₂	C₂³×C₄×D₁₉	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂³×D₁₉
commutator, socle	C₂×C₁₉	C₂⁴×C₄
Frattini	C₄	C₂⁴×D₁₉
Fitting	C₂×C₁₉×#131	C₂
prime radical	#11927380	C₂
cosocle	C₄×C₁₉	C₂⁵
hypercenter	C₂×#131	D₁₉
hypocenter	C₁₉	#3573
LCS	G ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. #11927311	C₂	semidirect
3. #11927310	C₂	semidirect
4. #11927309	C₂	semidirect
5. C₂×#10597347	C₂	semidirect
6. #131×D₁₉	C₂	semidirect
7. C₂×C₁₉×#131	C₂	semidirect
15. C₁₉×#131	C₂²	semidirect
16. C₁₉×#131	C₂²	semidirect
17. C₂×M₁₆×C₁₉	C₂²	semidirect
18. C₂²×C₈×C₁₉	C₂²	semidirect
19. #10597394	C₂²	semidirect
20. C₂×#64520	C₂²	semidirect
21. #10597353	C₂²	semidirect
22. M₁₆×D₁₉	C₂²	semidirect
23. #10597347	C₂²	semidirect
24. #10597347	C₂²	semidirect
25. #10597347	C₂²	semidirect
26. #10597347	C₂²	semidirect
27. C₂×#64515	C₂²	semidirect
28. C₂×C₈×D₁₉	C₂²	semidirect
95. C₂×#131	D₁₉	semidirect
113. #131	C₂×D₁₉	semidirect
114. #131	C₂×D₁₉	semidirect
118. C₂×M₁₆	C₂×D₁₉	semidirect
120. C₂²×C₈	C₂×D₁₉	semidirect
121. C₁₉	#3573	semidirect

Maximal subgroups

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

subgroup	index
#11927380	2
#11927311	2
#11927310	2
#11927309	2
C₂×#10597347	2
#131×D₁₉	2
C₂×C₁₉×#131	2
#3573	19