Subgroups — C₄×#11926297 #18073724

Order 4864.

470 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	C₄×C₄⋊(C₁₉⋊C₈)	C₂	1	2	yes	maximal
2	C₂×#11926297	C₂	1	2	yes	maximal
3	#14561470	C₂	1	2	yes	maximal
4	C₄²×C₈×C₁₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	4	yes	—
6	C₂×C₄×C₁₉⋊C₈	—	2	2	no	—
7	#11926299	—	2	2	no	—
8	#11926297	C₄	1	8	yes	direct product factor
9	C₂×C₄×C₈×C₁₉	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₁₉	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
13	C₄³×C₁₉	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
21	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
22	C₂²×C₁₉⋊C₈	—	2	4	no	—
23	C₄×C₈×C₁₉	—	2	4	no	—
24	C₄×C₈×C₁₉	—	2	2	no	—
25	C₄×C₈×C₁₉	C₈	1	8	yes	complemented
26	C₄×C₈×C₁₉	C₂×C₄	1	4	yes	—
27	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
28	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	8	yes	—
29	C₄×C₁₉⋊C₈	—	4	4	no	—
30	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
31	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₈×C₁₉	—	2	2	no	—
36	C₂×C₈×C₁₉	—	2	2	no	—
37	C₂×C₈×C₁₉	—	2	2	no	—
38	C₂×C₈×C₁₉	C₄²	1	2	yes	—
39	C₂×C₈×C₁₉	C₄²	1	2	yes	—
40	C₂×C₈×C₁₉	C₂×C₈	1	4	yes	—
41	C₂×C₈×C₁₉	C₂×C₈	1	4	yes	—
42	C₄²×C₁₉	—	2	4	no	—
43	C₄²×C₁₉	—	2	2	no	—
44	C₄²×C₁₉	C₂×C₈	1	4	yes	—
45	C₄²×C₁₉	M₁₆	1	4	yes	—
46	C₄²×C₁₉	D₈	1	2	yes	—
47	C₄²×C₁₉	Q₁₆	1	2	yes	—
48	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
49	C₄²×C₁₉	#41	1	2	yes	—
50	C₂×C₁₉⋊C₈	—	2	8	no	—
51	C₂×C₁₉⋊C₈	—	4	4	no	—
52	C₂×C₁₉⋊C₈	—	4	4	no	—
53	C₄×#335	—	19	1	no	maximal, Sylow
54	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₉	—	2	2	no	—
56	C₂×C₄×C₁₉	—	2	2	no	—
57	C₂×C₄×C₁₉	—	2	2	no	—
58	C₂×C₄×C₁₉	C₄⋊C₈	1	4	yes	—
59	C₂×C₄×C₁₉	C₄⋊C₈	1	4	yes	—
60	C₂×C₄×C₁₉	#107	1	4	yes	—
61	C₂×C₄×C₁₉	#108	1	4	yes	—
62	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
64	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
65	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
66	C₂×C₄×C₁₉	C₂²×C₈	1	2	yes	—
67	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
68	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×C₄×C₁₉	#135	1	2	yes	—
71	C₈×C₁₉	—	2	2	no	—
72	C₈×C₁₉	—	2	4	no	—
73	C₈×C₁₉	C₄×C₈	1	4	yes	complemented
74	C₁₉⋊C₈	—	4	8	no	—
75	C₄×C₄⋊C₈	—	19	2	no	—
76	C₄²×C₈	D₁₉	1	1	yes	characteristic
77	C₂×#335	—	19	2	no	—
78	#1277	—	19	2	no	—
79	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₁₉	C₂×C₄⋊C₈	1	2	yes	—
81	C₂²×C₁₉	C₂×#107	1	1	yes	characteristic
82	C₂²×C₁₉	#427	1	1	yes	characteristic
83	C₂²×C₁₉	C₂×#108	1	2	yes	—
84	C₄×C₁₉	—	2	4	no	—
85	C₄×C₁₉	—	2	2	no	—
86	C₄×C₁₉	#335	1	8	yes	complemented
87	C₄×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
88	C₄×C₁₉	C₄×M₁₆	1	1	yes	characteristic
89	C₄×C₁₉	#431	1	2	yes	—
90	C₄×C₁₉	C₄×D₈	1	2	yes	—
91	C₄×C₁₉	C₄×Q₁₆	1	2	yes	—
92	C₂×C₄⋊C₈	—	19	4	no	—
93	C₂×C₄×C₈	—	38	2	no	—
94	C₂×C₄×C₈	Dic₁₉	1	2	yes	—
95	C₂×C₄×C₈	Dic₁₉	1	2	yes	—
96	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
97	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
98	C₄³	C₂×D₁₉	1	1	yes	characteristic
99	#336	—	38	2	no	—
100	#335	—	19	8	no	—
101	C₂×C₁₉	C₂×#335	1	2	yes	—
102	C₂×C₁₉	#1564	1	2	yes	—
103	C₂×C₁₉	C₄×C₄⋊C₈	1	1	yes	characteristic
104	C₂×C₁₉	C₄×#107	1	1	yes	characteristic
105	C₂×C₁₉	C₄×#108	1	1	yes	characteristic
106	C₂²×C₈	—	38	4	no	—
107	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
108	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
109	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
110	C₂×C₄²	Dic₃₈	1	2	yes	—
111	C₂×C₄²	D₇₆	1	1	yes	characteristic
112	C₂×C₄²	D₇₆	1	1	yes	characteristic
113	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
114	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
115	C₄⋊C₈	—	19	8	no	—
116	C₄⋊C₈	—	38	4	no	—
117	C₄×C₈	—	2	4	no	—
118	C₄×C₈	—	2	2	no	—
119	C₄×C₈	—	76	4	no	—
120	C₄×C₈	C₁₉⋊C₈	1	8	yes	complemented
121	C₄×C₈	C₄×D₁₉	1	4	yes	—
122	C₁₉	C₄×#335	1	1	yes	characteristic, complemented, Sylow
123	C₂²×C₄	C₄⋊Dic₁₉	1	2	yes	—
124	C₂²×C₄	C₄⋊Dic₁₉	1	2	yes	—
125	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
126	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
127	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	38	8	no	—
132	C₂×C₈	—	76	4	no	—
133	C₂×C₈	—	76	4	no	—
134	C₂×C₈	C₂×C₁₉⋊C₈	1	4	yes	—
135	C₂×C₈	C₂×C₁₉⋊C₈	1	4	yes	—
136	C₂×C₈	C₄×Dic₁₉	1	2	yes	—
137	C₂×C₈	C₄×Dic₁₉	1	2	yes	—
138	C₄²	—	2	4	no	—
139	C₄²	—	2	2	no	—
140	C₄²	D₁₅₂	1	2	yes	—
141	C₄²	Dic₇₆	1	2	yes	—
142	C₄²	C₂×C₁₉⋊C₈	1	4	yes	—
143	C₄²	#64520	1	4	yes	—
144	C₄²	C₂×C₄×D₁₉	1	2	yes	—
145	C₄²	#64541	1	2	yes	—
146	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	C₄⋊(C₁₉⋊C₈)	1	4	yes	—
151	C₂×C₄	C₄⋊(C₁₉⋊C₈)	1	4	yes	—
152	C₂×C₄	#10597275	1	4	yes	—
153	C₂×C₄	#10597276	1	4	yes	—
154	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
155	C₂×C₄	C₄×D₇₆	1	2	yes	—
156	C₂×C₄	C₂×D₁₅₂	1	1	yes	characteristic
157	C₂×C₄	#10597350	1	2	yes	—
158	C₂×C₄	C₂×Dic₇₆	1	1	yes	characteristic
159	C₂×C₄	C₂²×C₁₉⋊C₈	1	2	yes	—
160	C₂×C₄	C₂×#64520	1	2	yes	—
161	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
162	C₂×C₄	#10597372	1	1	yes	characteristic
163	C₈	—	2	4	no	—
164	C₈	—	2	2	no	—
165	C₈	—	76	8	no	—
166	C₈	C₄×C₁₉⋊C₈	1	4	yes	complemented
167	C₂²	C₄×C₄⋊Dic₁₉	1	1	yes	characteristic
168	C₂²	C₂×C₄⋊(C₁₉⋊C₈)	1	2	yes	—
169	C₂²	C₂×#10597275	1	1	yes	characteristic
170	C₂²	#11926582	1	1	yes	characteristic
171	C₂²	C₂×#10597276	1	2	yes	—
172	C₄	—	2	4	no	—
173	C₄	—	2	2	no	—
174	C₄	#11926297	1	8	yes	direct product factor
175	C₄	C₂×C₄×C₁₉⋊C₈	1	1	yes	characteristic
176	C₄	C₄×#64520	1	1	yes	characteristic
177	C₄	#11926597	1	2	yes	—
178	C₄	C₄×D₁₅₂	1	2	yes	—
179	C₄	C₄×Dic₇₆	1	2	yes	—
180	C₂	C₂×#11926297	1	2	yes	—
181	C₂	#14562383	1	2	yes	—
182	C₂	C₄×C₄⋊(C₁₉⋊C₈)	1	1	yes	characteristic
183	C₂	C₄×#10597275	1	1	yes	characteristic
184	C₂	C₄×#10597276	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	D₇₆
commutator	C₄×C₁₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₄²×C₈×C₁₉	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄²×C₈	D₁₉
hypocenter	C₁₉	C₄×#335
LCS	G ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄² ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
8. #11926297	C₄	direct (N × H)
25. C₄×C₈×C₁₉	C₈	semidirect
73. C₈×C₁₉	C₄×C₈	semidirect
86. C₄×C₁₉	#335	semidirect
120. C₄×C₈	C₁₉⋊C₈	semidirect
122. C₁₉	C₄×#335	semidirect
166. C₈	C₄×C₁₉⋊C₈	semidirect
174. C₄	#11926297	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₄⋊(C₁₉⋊C₈)	2
C₂×#11926297	2
#14561470	2
C₄²×C₈×C₁₉	2
C₄×#335	19

Subgroups — C4×#11926297 #18073724

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11926297 #18073724