Subgroups — #18069067

Order 4864.

414 conjugacy classes of subgroups.

The subgroup lattice has 164 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	C₄²×C₈×C₁₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₁₉⋊C₈)	C₄	1	4	yes	—
4	C₂×C₄×C₁₉⋊C₈	—	2	2	no	—
5	#11926299	—	2	2	no	—
6	C₂×C₄×C₈×C₁₉	—	2	2	no	—
7	C₂×C₄×C₈×C₁₉	C₄	1	2	yes	—
8	C₄³×C₁₉	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₁₉	—	2	2	no	—
10	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
11	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄²×C₁₉	D₄	1	2	yes	—
13	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
14	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
15	C₂²×C₁₉⋊C₈	—	2	4	no	—
16	C₄×C₈×C₁₉	—	2	8	no	—
17	C₄×C₈×C₁₉	—	2	2	no	—
18	C₄×C₈×C₁₉	C₈	1	4	yes	complemented
19	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
20	C₄⋊(C₁₉⋊C₈)	C₈	1	8	yes	complemented
21	C₄×C₁₉⋊C₈	—	4	2	no	—
22	C₄×C₁₉⋊C₈	—	4	2	no	—
23	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
24	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
25	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
27	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₈×C₁₉	—	2	8	no	—
29	C₂×C₈×C₁₉	—	2	4	no	—
30	C₄²×C₁₉	—	2	4	no	—
31	C₄²×C₁₉	—	2	2	no	—
32	C₄²×C₁₉	C₂×C₈	1	4	yes	—
33	C₄²×C₁₉	C₂×C₈	1	2	yes	—
34	C₄²×C₁₉	M₁₆	1	4	yes	—
35	C₄²×C₁₉	M₁₆	1	2	yes	—
36	C₄²×C₁₉	D₈	1	1	yes	characteristic
37	C₄²×C₁₉	SD₁₆	1	2	yes	—
38	C₄²×C₁₉	Q₁₆	1	1	yes	characteristic
39	C₂×C₁₉⋊C₈	—	2	8	no	—
40	C₂×C₁₉⋊C₈	—	4	4	no	—
41	C₂×C₁₉⋊C₈	—	4	4	no	—
42	#7015	—	19	1	no	maximal, Sylow
43	C₂³×C₁₉	#95	1	1	yes	characteristic
44	C₂×C₄×C₁₉	—	2	2	no	—
45	C₂×C₄×C₁₉	—	2	2	no	—
46	C₂×C₄×C₁₉	—	2	2	no	—
47	C₂×C₄×C₁₉	C₄×C₈	1	1	yes	characteristic
48	C₂×C₄×C₁₉	C₄×C₈	1	2	yes	—
49	C₂×C₄×C₁₉	#97	1	1	yes	characteristic
50	C₂×C₄×C₁₉	#97	1	2	yes	—
51	C₂×C₄×C₁₉	C₂²⋊C₈	1	4	yes	—
52	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
53	C₂×C₄×C₁₉	#102	1	2	yes	—
54	C₂×C₄×C₁₉	#103	1	2	yes	—
55	C₂×C₄×C₁₉	#104	1	4	yes	—
56	C₂×C₄×C₁₉	C₄⋊C₈	1	2	yes	—
57	C₂×C₄×C₁₉	C₄⋊C₈	1	4	yes	—
58	C₂×C₄×C₁₉	#106	1	1	yes	characteristic
59	C₂×C₄×C₁₉	#107	1	1	yes	characteristic
60	C₂×C₄×C₁₉	#108	1	2	yes	—
61	C₈×C₁₉	—	2	8	no	—
62	C₁₉⋊C₈	—	4	8	no	—
63	C₄×C₄⋊C₈	—	19	2	no	—
64	C₄²×C₈	D₁₉	1	1	yes	characteristic
65	C₂²×C₁₉	#336	1	2	yes	—
66	C₂²×C₁₉	#336	1	1	yes	characteristic
67	C₂²×C₁₉	#339	1	2	yes	—
68	C₂²×C₁₉	#340	1	1	yes	characteristic
69	C₂²×C₁₉	#341	1	1	yes	characteristic
70	C₄×C₁₉	—	2	2	no	—
71	C₄×C₁₉	—	2	4	no	—
72	C₄×C₁₉	C₈²	1	1	yes	characteristic
73	C₄×C₁₉	#322	1	1	yes	characteristic
74	C₄×C₁₉	#322	1	2	yes	—
75	C₄×C₁₉	D₄⋊C₈	1	4	yes	—
76	C₄×C₁₉	Q₈⋊C₈	1	4	yes	—
77	C₄×C₁₉	#334	1	2	yes	—
78	C₄×C₁₉	#335	1	2	yes	—
79	C₂×C₄⋊C₈	—	19	4	no	—
80	C₂×C₄×C₈	—	2	2	no	—
81	C₂×C₄×C₈	—	38	2	no	—
82	C₂×C₄×C₈	Dic₁₉	1	2	yes	—
83	C₄³	C₂×D₁₉	1	1	yes	characteristic
84	#336	—	38	2	no	—
85	C₂×C₁₉	#1273	1	1	yes	characteristic
86	C₂×C₁₉	#1276	1	1	yes	characteristic
87	C₂×C₁₉	#1277	1	1	yes	characteristic
88	C₂×C₁₉	#1278	1	2	yes	—
89	C₂×C₁₉	#1279	1	2	yes	—
90	C₂²×C₈	—	2	2	no	—
91	C₂²×C₈	—	38	4	no	—
92	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
93	C₂×C₄²	C₄×D₁₉	1	2	yes	—
94	C₂×C₄²	D₇₆	1	1	yes	characteristic
95	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
96	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
97	C₄⋊C₈	—	19	8	no	—
98	C₄⋊C₈	—	38	4	no	—
99	C₄×C₈	—	2	8	no	—
100	C₄×C₈	—	2	2	no	—
101	C₄×C₈	—	76	2	no	—
102	C₄×C₈	—	76	2	no	—
103	C₄×C₈	C₁₉⋊C₈	1	4	yes	complemented
104	C₁₉	#7015	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
106	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
107	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
108	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
109	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
110	C₂×C₈	—	2	8	no	—
111	C₂×C₈	—	2	4	no	—
112	C₂×C₈	—	38	8	no	—
113	C₂×C₈	—	76	4	no	—
114	C₂×C₈	—	76	4	no	—
115	C₄²	—	2	4	no	—
116	C₄²	—	2	2	no	—
117	C₄²	C₈×D₁₉	1	4	yes	—
118	C₄²	#64515	1	4	yes	—
119	C₄²	#64516	1	2	yes	—
120	C₄²	D₁₅₂	1	1	yes	characteristic
121	C₄²	Dic₇₆	1	1	yes	characteristic
122	C₄²	C₂×C₁₉⋊C₈	1	2	yes	—
123	C₄²	#64520	1	2	yes	—
124	C₂³	C₁₉⋊#95	1	1	yes	characteristic
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	C₄×C₁₉⋊C₈	1	1	yes	characteristic
129	C₂×C₄	#10597260	1	1	yes	characteristic
130	C₂×C₄	C₄⋊(C₁₉⋊C₈)	1	2	yes	—
131	C₂×C₄	#10597262	1	4	yes	—
132	C₂×C₄	C₈×Dic₁₉	1	2	yes	—
133	C₂×C₄	Dic₁₉⋊C₈	1	4	yes	—
134	C₂×C₄	#10597272	1	2	yes	—
135	C₂×C₄	#10597273	1	2	yes	—
136	C₂×C₄	#10597274	1	1	yes	characteristic
137	C₂×C₄	#10597275	1	1	yes	characteristic
138	C₂×C₄	#10597276	1	2	yes	—
139	C₂×C₄	(C₂×D₁₉)⋊C₈	1	4	yes	—
140	C₂×C₄	#10597278	1	2	yes	—
141	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
142	C₈	—	2	8	no	—
143	C₈	—	76	8	no	—
144	C₂²	#11926298	1	2	yes	—
145	C₂²	#11926299	1	1	yes	characteristic
146	C₂²	#11926304	1	2	yes	—
147	C₂²	#11926307	1	1	yes	characteristic
148	C₂²	#11926309	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	4	no	—
151	C₄	C₈×C₁₉⋊C₈	1	1	yes	characteristic
152	C₄	#11926265	1	1	yes	characteristic
153	C₄	#11926266	1	2	yes	—
154	C₄	#11926273	1	4	yes	—
155	C₄	#11926276	1	4	yes	—
156	C₄	#11926295	1	2	yes	—
157	C₄	#11926297	1	2	yes	—
158	C₂	C₁₉⋊#1273	1	1	yes	characteristic
159	C₂	#14561468	1	1	yes	characteristic
160	C₂	#14561470	1	1	yes	characteristic
161	C₂	#14561472	1	2	yes	—
162	C₂	#14561475	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	D₇₆
commutator	C₄×C₁₉	C₈²
Frattini	C₄³	C₂×D₁₉
Fitting	C₄²×C₈×C₁₉	C₂
socle, prime radical	C₂³×C₁₉	#95
cosocle	C₄³×C₁₉	C₂²
hypercenter	C₄²×C₈	D₁₉
hypocenter	C₁₉	#7015
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
18. C₄×C₈×C₁₉	C₈	semidirect
20. C₄⋊(C₁₉⋊C₈)	C₈	semidirect
103. C₄×C₈	C₁₉⋊C₈	semidirect
104. C₁₉	#7015	semidirect

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
C₄²×C₈×C₁₉	2
#7015	19