Subgroups — C₂×C₄×#64545 #14576816

Order 2432.

1728 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#64545	C₂	1	1	yes	maximal, characteristic
2	C₄×#64545	C₂	1	16	yes	maximal, direct product factor
3	C₂×Q₈×Dic₁₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×D₇₆	C₂	1	3	yes	maximal, complemented
5	C₂×#10597339	C₂	1	3	yes	maximal, complemented
6	C₂×#10597338	C₂	1	3	yes	maximal, complemented
7	C₂×C₄²×D₁₉	C₂	1	3	yes	maximal, complemented
8	C₂×C₄×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₈×C₁₉	C₂²	1	1	yes	characteristic
10	C₂×#64545	C₄	1	16	yes	direct product factor
11	C₂×#64545	C₂²	1	8	yes	—
12	C₂²×D₇₆	C₂²	1	3	yes	—
13	C₂²×C₄×D₁₉	C₂²	1	3	yes	—
14	C₂²×C₄×D₁₉	C₂²	1	3	yes	—
15	C₂²×C₄×D₁₉	C₂²	1	3	yes	—
16	C₄×Q₈×C₁₉	C₂²	1	8	yes	complemented
17	C₂×C₄⋊C₄×C₁₉	C₂²	1	3	yes	—
18	C₂×C₄²×C₁₉	C₂²	1	3	yes	—
19	Q₈×Dic₁₉	C₂²	1	8	yes	complemented
20	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	6	yes	—
21	C₂×C₄⋊Dic₁₉	C₂²	1	3	yes	—
22	C₂×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₁₉	C₂²	1	3	yes	—
24	C₂×C₄×Dic₁₉	C₂²	1	3	yes	—
25	#10597339	C₂²	1	24	yes	complemented
26	#10597338	C₂²	1	24	yes	complemented
27	C₄×D₇₆	C₂²	1	24	yes	complemented
28	C₄²×D₁₉	C₂²	1	24	yes	complemented
29	C₂³×D₁₉	C₂³	1	3	yes	—
30	C₂×Q₈×C₁₉	C₂×C₄	1	8	yes	complemented
31	C₂×Q₈×C₁₉	C₂³	1	4	yes	—
32	C₂²×C₄×C₁₉	C₂³	1	3	yes	—
33	C₂²×C₄×C₁₉	C₂³	1	3	yes	—
34	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
35	C₂²×Dic₁₉	C₂³	1	3	yes	—
36	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
37	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
38	#64545	C₂×C₄	1	64	yes	direct product factor
39	C₂×D₇₆	C₂×C₄	1	24	yes	complemented
40	C₂×D₇₆	C₂³	1	12	yes	—
41	C₂×C₄×D₁₉	—	2	12	no	—
42	C₂×C₄×D₁₉	—	2	12	no	—
43	C₂×C₄×D₁₉	C₂×C₄	1	24	yes	complemented
44	C₂×C₄×D₁₉	C₂³	1	12	yes	—
45	C₂×C₄×D₁₉	C₂³	1	12	yes	—
46	C₂×C₄×D₁₉	C₂³	1	12	yes	—
47	C₄⋊C₄×C₁₉	C₂³	1	12	yes	—
48	C₄²×C₁₉	C₂³	1	12	yes	—
49	(C₂×D₁₉)⋊C₄	C₂³	1	24	yes	—
50	C₄⋊Dic₁₉	C₂³	1	12	yes	—
51	C₄×Dic₁₉	C₂³	1	12	yes	—
52	C₄×Dic₁₉	C₂³	1	4	yes	—
53	C₄×Dic₁₉	C₂³	1	12	yes	—
54	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
55	C₂²×D₁₉	—	2	6	no	—
56	C₂²×D₁₉	—	2	6	no	—
57	C₂²×D₁₉	C₂²×C₄	1	12	yes	—
58	C₂²×D₁₉	C₂⁴	1	6	yes	—
59	Q₈×C₁₉	C₂²×C₄	1	16	yes	complemented
60	C₂×C₄×C₁₉	—	2	6	no	—
61	C₂×C₄×C₁₉	C₂²×C₄	1	12	yes	—
62	C₂×C₄×C₁₉	#41	1	4	yes	complemented
63	C₂×C₄×C₁₉	C₂⁴	1	6	yes	—
64	C₂×C₄×C₁₉	C₂⁴	1	6	yes	—
65	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
66	C₂×Dic₁₉	—	2	6	no	—
67	C₂×Dic₁₉	C₂²×C₄	1	4	yes	—
68	C₂×Dic₁₉	#41	1	4	yes	complemented
69	C₂×Dic₁₉	C₂⁴	1	6	yes	—
70	C₂×Dic₁₉	C₂⁴	1	2	yes	—
71	C₂×Dic₁₉	C₂⁴	1	2	yes	—
72	D₇₆	C₂²×C₄	1	48	yes	complemented
73	C₄×D₁₉	—	2	24	no	—
74	C₄×D₁₉	—	2	24	no	—
75	C₄×D₁₉	C₂²×C₄	1	48	yes	complemented
76	C₂×C₄×#41	—	19	1	no	maximal, Sylow
77	C₂²×C₁₉	C₂³×C₄	1	2	yes	—
78	C₂²×C₁₉	C₂×#41	1	2	yes	—
79	C₂²×C₁₉	C₂×#41	1	2	yes	—
80	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
81	C₂×D₁₉	—	2	24	no	—
82	C₂×D₁₉	—	2	6	no	—
83	C₂×D₁₉	—	2	6	no	—
84	C₂×D₁₉	C₂³×C₄	1	12	yes	—
85	C₄×C₁₉	—	2	6	no	—
86	C₄×C₁₉	C₂³×C₄	1	12	yes	—
87	C₄×C₁₉	C₂×#41	1	4	yes	complemented
88	Dic₁₉	—	2	6	no	—
89	Dic₁₉	C₂³×C₄	1	4	yes	—
90	Dic₁₉	C₂×#41	1	4	yes	complemented
91	C₂²×#41	—	19	1	no	—
92	C₄×#41	—	19	16	no	—
93	C₂×C₄×Q₈	—	19	1	no	—
94	C₂×C₄×Q₈	D₁₉	1	1	yes	characteristic, complemented
95	C₂×C₄×D₄	—	19	3	no	—
96	C₂×C₄×D₄	—	19	3	no	—
97	C₂×#117	—	19	3	no	—
98	C₂²×C₄²	—	19	3	no	—
99	C₂×C₁₉	C₄×#41	1	4	yes	complemented
100	C₂×C₁₉	C₂⁴×C₄	1	1	yes	characteristic
101	C₂×C₁₉	C₂²×#41	1	1	yes	characteristic
102	C₂×C₁₉	C₂²×#41	1	1	yes	characteristic
103	D₁₉	—	2	12	no	—
104	C₂×#41	—	19	16	no	—
105	C₂×#41	—	19	8	no	—
106	C₂²×Q₈	C₂×D₁₉	1	1	yes	characteristic
107	C₂²×D₄	—	19	3	no	—
108	C₂³×C₄	—	19	3	no	—
109	C₂³×C₄	—	19	3	no	—
110	C₂³×C₄	—	19	3	no	—
111	C₄×Q₈	—	19	8	no	—
112	C₄×Q₈	C₂×D₁₉	1	8	yes	complemented
113	C₄×D₄	—	19	24	no	—
114	C₄×D₄	—	19	24	no	—
115	#117	—	19	24	no	—
116	C₂×C₄⋊C₄	—	19	3	no	—
117	C₂×C₄⋊C₄	C₂×D₁₉	1	3	yes	—
118	C₂×C₂²⋊C₄	—	19	6	no	—
119	C₂×C₄²	—	19	3	no	—
120	C₂×C₄²	—	19	3	no	—
121	C₂×C₄²	—	19	1	no	—
122	C₂×C₄²	—	19	24	no	—
123	C₂×C₄²	C₂×D₁₉	1	3	yes	—
124	C₁₉	C₂×C₄×#41	1	1	yes	characteristic, complemented, Sylow
125	C₂⁴	—	19	3	no	—
126	#41	—	19	64	no	—
127	C₂×Q₈	C₄×D₁₉	1	8	yes	complemented
128	C₂×Q₈	C₂²×D₁₉	1	4	yes	—
129	C₂×D₄	—	19	24	no	—
130	C₂×D₄	—	19	12	no	—
131	C₂²×C₄	—	19	1	no	—
132	C₂²×C₄	—	19	1	no	—
133	C₂²×C₄	—	19	24	no	—
134	C₂²×C₄	—	19	12	no	—
135	C₂²×C₄	—	19	3	no	—
136	C₂²×C₄	—	19	12	no	—
137	C₂²×C₄	—	19	12	no	—
138	C₂²×C₄	—	38	12	no	—
139	C₂²×C₄	—	38	12	no	—
140	C₂²×C₄	C₂²×D₁₉	1	3	yes	—
141	C₂²×C₄	C₂²×D₁₉	1	3	yes	—
142	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
143	C₄⋊C₄	—	19	12	no	—
144	C₄⋊C₄	C₂²×D₁₉	1	12	yes	—
145	C₂²⋊C₄	—	19	24	no	—
146	C₄²	—	19	4	no	—
147	C₄²	—	19	12	no	—
148	C₄²	—	19	12	no	—
149	C₄²	C₂²×D₁₉	1	12	yes	—
150	C₂³	—	19	12	no	—
151	C₂³	—	19	6	no	—
152	C₂³	—	38	6	no	—
153	C₂³	—	38	6	no	—
154	C₂³	C₂³×D₁₉	1	1	yes	characteristic
155	Q₈	C₂×C₄×D₁₉	1	16	yes	complemented
156	D₄	—	19	48	no	—
157	C₂×C₄	—	2	6	no	—
158	C₂×C₄	—	19	4	no	—
159	C₂×C₄	—	19	4	no	—
160	C₂×C₄	—	19	2	no	—
161	C₂×C₄	—	19	2	no	—
162	C₂×C₄	—	19	48	no	—
163	C₂×C₄	—	19	6	no	—
164	C₂×C₄	—	38	24	no	—
165	C₂×C₄	—	38	6	no	—
166	C₂×C₄	—	38	24	no	—
167	C₂×C₄	C₂×C₄×D₁₉	1	12	yes	—
168	C₂×C₄	#64545	1	4	yes	direct product factor
169	C₂×C₄	C₂³×D₁₉	1	6	yes	—
170	C₂×C₄	C₂³×D₁₉	1	6	yes	—
171	C₂×C₄	C₂³×D₁₉	1	2	yes	—
172	C₂²	—	19	12	no	—
173	C₂²	—	38	6	no	—
174	C₂²	—	38	6	no	—
175	C₂²	—	38	24	no	—
176	C₂²	C₂²×C₄×D₁₉	1	2	yes	—
177	C₂²	C₂×#64545	1	2	yes	—
178	C₂²	#41×D₁₉	1	2	yes	—
179	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
180	C₄	—	2	6	no	—
181	C₄	—	19	4	no	—
182	C₄	—	19	4	no	—
183	C₄	—	38	6	no	—
184	C₄	C₂²×C₄×D₁₉	1	12	yes	—
185	C₄	C₂×#64545	1	4	yes	direct product factor
186	C₂	—	38	12	no	—
187	C₂	C₄×#64545	1	4	yes	direct product factor
188	C₂	C₂³×C₄×D₁₉	1	1	yes	characteristic
189	C₂	C₂×#41×D₁₉	1	1	yes	characteristic
190	C₂	C₂²×#64545	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₉
commutator	C₂×C₁₉	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₉
Fitting	C₂×C₄×Q₈×C₁₉	C₂
socle	C₂³×C₁₉	C₂⁴
prime radical	C₂²×#64545	C₂
cosocle	C₂²×C₁₉	C₂⁵
hypercenter	C₂×C₄×Q₈	D₁₉
hypocenter	C₁₉	C₂×C₄×#41
LCS	G ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₄×#64545	C₂	direct (N × H)
3. C₂×Q₈×Dic₁₉	C₂	semidirect
4. C₂×C₄×D₇₆	C₂	semidirect
5. C₂×#10597339	C₂	semidirect
6. C₂×#10597338	C₂	semidirect
7. C₂×C₄²×D₁₉	C₂	semidirect
8. C₂×C₄×Q₈×C₁₉	C₂	semidirect
10. C₂×#64545	C₄	direct (N × H)
16. C₄×Q₈×C₁₉	C₂²	semidirect
19. Q₈×Dic₁₉	C₂²	semidirect
25. #10597339	C₂²	semidirect
26. #10597338	C₂²	semidirect
27. C₄×D₇₆	C₂²	semidirect
28. C₄²×D₁₉	C₂²	semidirect
30. C₂×Q₈×C₁₉	C₂×C₄	semidirect
38. #64545	C₂×C₄	direct (N × H)
39. C₂×D₇₆	C₂×C₄	semidirect
43. C₂×C₄×D₁₉	C₂×C₄	semidirect
59. Q₈×C₁₉	C₂²×C₄	semidirect
62. C₂×C₄×C₁₉	#41	semidirect
68. C₂×Dic₁₉	#41	semidirect
72. D₇₆	C₂²×C₄	semidirect
75. C₄×D₁₉	C₂²×C₄	semidirect
87. C₄×C₁₉	C₂×#41	semidirect
90. Dic₁₉	C₂×#41	semidirect
94. C₂×C₄×Q₈	D₁₉	semidirect
99. C₂×C₁₉	C₄×#41	semidirect
112. C₄×Q₈	C₂×D₁₉	semidirect
124. C₁₉	C₂×C₄×#41	semidirect
127. C₂×Q₈	C₄×D₁₉	semidirect
155. Q₈	C₂×C₄×D₁₉	semidirect
168. C₂×C₄	#64545	direct (N × H)
185. C₄	C₂×#64545	direct (N × H)
187. C₂	C₄×#64545	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#64545	2
C₄×#64545	2
C₂×Q₈×Dic₁₉	2
C₂×C₄×D₇₆	2
C₂×#10597339	2
C₂×#10597338	2
C₂×C₄²×D₁₉	2
C₂×C₄×Q₈×C₁₉	2
C₂×C₄×#41	19

Subgroups — C2×C4×#64545 #14576816

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#64545 #14576816