Subgroups — #18073673

Order 4864.

494 conjugacy classes of subgroups.

The subgroup lattice has 196 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×#11926265	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₁₉⋊#1273	C₂	1	2	yes	maximal
5	C₄×C₁₉×#97	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	C₄	1	4	yes	—
7	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
9	#11926299	C₄	1	4	yes	—
10	#11926265	—	2	2	no	—
11	C₈×C₁₉⋊C₈	—	2	2	no	—
12	C₂×C₁₉×#97	C₄	1	2	yes	—
13	C₂×C₁₉×#97	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₄³×C₁₉	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
24	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
25	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
26	C₁₉×#97	—	2	4	no	—
27	C₁₉×#97	C₈	1	8	yes	complemented
28	C₄×C₈×C₁₉	—	2	2	no	—
29	C₄×C₈×C₁₉	D₄	1	2	yes	—
30	C₄×C₈×C₁₉	Q₈	1	2	yes	—
31	C₄⋊(C₁₉⋊C₈)	—	2	8	no	—
32	C₄×C₁₉⋊C₈	—	2	2	no	—
33	C₄×C₁₉⋊C₈	—	2	2	no	—
34	C₄×C₁₉⋊C₈	—	2	2	no	—
35	C₄×C₁₉⋊C₈	—	2	2	no	—
36	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
37	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
38	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
39	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
40	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₉	—	2	2	no	—
42	C₂×C₈×C₁₉	—	2	2	no	—
43	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	—
44	C₂×C₈×C₁₉	C₂×C₈	1	4	yes	—
45	C₂×C₈×C₁₉	C₂×C₈	1	4	yes	—
46	C₂×C₈×C₁₉	M₁₆	1	4	yes	—
47	C₄²×C₁₉	—	2	4	no	—
48	C₄²×C₁₉	—	2	2	no	—
49	C₄²×C₁₉	—	2	4	no	—
50	C₄²×C₁₉	C₂×C₈	1	4	yes	—
51	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
53	C₄²×C₁₉	#41	1	2	yes	—
54	C₂×C₁₉⋊C₈	—	2	4	no	—
55	C₂×C₁₉⋊C₈	—	2	4	no	—
56	C₂×C₁₉⋊C₈	—	2	4	no	—
57	C₂×C₁₉⋊C₈	—	2	4	no	—
58	C₂×C₁₉⋊C₈	—	2	4	no	—
59	C₂×C₁₉⋊C₈	M₁₆	1	8	yes	—
60	#8262	—	19	1	no	maximal, Sylow
61	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
62	C₂×C₄×C₁₉	—	2	2	no	—
63	C₂×C₄×C₁₉	—	2	2	no	—
64	C₂×C₄×C₁₉	—	2	2	no	—
65	C₂×C₄×C₁₉	—	2	1	no	—
66	C₂×C₄×C₁₉	—	2	1	no	—
67	C₂×C₄×C₁₉	—	2	2	no	—
68	C₂×C₄×C₁₉	C₄×C₈	1	4	yes	—
69	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
71	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
72	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
73	C₂×C₄×C₁₉	C₂²×C₈	1	2	yes	—
74	C₂×C₄×C₁₉	C₂×M₁₆	1	4	yes	—
75	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
76	C₂×C₄×C₁₉	#131	1	4	yes	—
77	C₈×C₁₉	—	2	4	no	—
78	C₈×C₁₉	C₄⋊C₈	1	8	yes	complemented
79	C₁₉⋊C₈	—	2	8	no	—
80	C₁₉⋊C₈	—	4	4	no	—
81	C₄×C₄⋊C₈	—	19	2	no	—
82	C₄×#97	D₁₉	1	1	yes	characteristic
83	C₂×#322	—	19	1	no	—
84	C₂×C₈²	—	19	1	no	—
85	#1273	—	19	2	no	—
86	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
88	C₂²×C₁₉	C₄×M₁₆	1	2	yes	—
89	C₂²×C₁₉	C₄×M₁₆	1	1	yes	characteristic
90	C₂²×C₁₉	#405	1	2	yes	—
91	C₄×C₁₉	—	2	4	no	—
92	C₄×C₁₉	—	4	1	no	—
93	C₄×C₁₉	—	4	1	no	—
94	C₄×C₁₉	C₂×C₄⋊C₈	1	2	yes	—
95	C₄×C₁₉	#431	1	2	yes	—
96	C₄×C₁₉	#435	1	4	yes	—
97	C₄×C₁₉	#446	1	4	yes	—
98	C₂×C₄⋊C₈	—	19	4	no	—
99	C₂×#97	Dic₁₉	1	2	yes	—
100	C₂×#97	Dic₁₉	1	2	yes	—
101	C₂×C₄×C₈	—	19	2	no	—
102	C₂×C₄×C₈	—	19	2	no	—
103	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
104	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
105	C₄³	C₂×D₁₉	1	1	yes	characteristic
106	#336	—	19	4	no	—
107	#322	—	38	2	no	—
108	C₈²	—	38	2	no	—
109	C₂×C₁₉	C₈×M₁₆	1	2	yes	—
110	C₂×C₁₉	#1453	1	1	yes	characteristic
111	C₂×C₁₉	#1455	1	1	yes	characteristic
112	C₂×C₁₉	C₄×C₄⋊C₈	1	1	yes	characteristic
113	C₂×C₁₉	#1771	1	2	yes	—
114	C₂²×C₈	—	19	4	no	—
115	C₂²×C₈	—	19	4	no	—
116	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
117	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
118	C₂×C₄²	C₄×D₁₉	1	2	yes	—
119	C₂×C₄²	C₄×D₁₉	1	2	yes	—
120	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
121	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
122	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
123	C₄⋊C₈	—	38	8	no	—
124	#97	—	2	4	no	—
125	#97	C₁₉⋊C₈	1	8	yes	complemented
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	—	38	2	no	—
128	C₄×C₈	—	38	2	no	—
129	C₄×C₈	—	38	2	no	—
130	C₄×C₈	—	38	2	no	—
131	C₄×C₈	Dic₃₈	1	2	yes	—
132	C₄×C₈	D₇₆	1	2	yes	—
133	C₁₉	#8262	1	1	yes	characteristic, complemented, Sylow
134	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
135	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
136	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
137	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
138	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	19	8	no	—
142	C₂×C₈	—	38	4	no	—
143	C₂×C₈	—	38	4	no	—
144	C₂×C₈	—	38	4	no	—
145	C₂×C₈	—	38	4	no	—
146	C₂×C₈	—	38	4	no	—
147	C₂×C₈	C₂×C₁₉⋊C₈	1	4	yes	—
148	C₂×C₈	C₂×C₁₉⋊C₈	1	4	yes	—
149	C₂×C₈	#64520	1	4	yes	—
150	C₂×C₈	C₄⋊Dic₁₉	1	4	yes	—
151	C₄²	—	2	4	no	—
152	C₄²	—	2	4	no	—
153	C₄²	—	2	2	no	—
154	C₄²	C₂×C₁₉⋊C₈	1	4	yes	—
155	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
156	C₄²	C₂×D₇₆	1	1	yes	characteristic
157	C₄²	#64541	1	2	yes	—
158	C₂³	C₂×C₄×Dic₁₉	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	C₄×C₁₉⋊C₈	1	4	yes	—
166	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
167	C₂×C₄	C₄×D₇₆	1	2	yes	—
168	C₂×C₄	M₁₆×D₁₉	1	4	yes	—
169	C₂×C₄	#10597353	1	4	yes	—
170	C₂×C₄	C₂²×C₁₉⋊C₈	1	2	yes	—
171	C₂×C₄	C₂×#64520	1	2	yes	—
172	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
173	C₂×C₄	#10597372	1	1	yes	characteristic
174	C₈	—	2	4	no	—
175	C₈	—	38	8	no	—
176	C₈	—	76	4	no	—
177	C₈	C₄⋊(C₁₉⋊C₈)	1	8	yes	complemented
178	C₂²	C₄×C₄⋊Dic₁₉	1	1	yes	characteristic
179	C₂²	C₂×C₄×C₁₉⋊C₈	1	1	yes	characteristic
180	C₂²	M₁₆×Dic₁₉	1	2	yes	—
181	C₂²	C₄×#64520	1	1	yes	characteristic
182	C₂²	#11926502	1	2	yes	—
183	C₄	—	2	4	no	—
184	C₄	—	4	1	no	—
185	C₄	—	4	1	no	—
186	C₄	C₂×C₄⋊(C₁₉⋊C₈)	1	2	yes	—
187	C₄	#11926597	1	2	yes	—
188	C₄	#11926617	1	4	yes	—
189	C₄	#11926672	1	4	yes	—
190	C₂	M₁₆×C₁₉⋊C₈	1	2	yes	—
191	C₂	#14561932	1	1	yes	characteristic
192	C₂	#14561938	1	1	yes	characteristic
193	C₂	C₄×C₄⋊(C₁₉⋊C₈)	1	1	yes	characteristic
194	C₂	#14563507	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₁₉
commutator	C₂²×C₁₉	C₂×C₄×C₈
Fitting	C₄×C₁₉×#97	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₄²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄×#97	D₁₉
hypocenter	C₁₉	#8262
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
27. C₁₉×#97	C₈	semidirect
78. C₈×C₁₉	C₄⋊C₈	semidirect
125. #97	C₁₉⋊C₈	semidirect
133. C₁₉	#8262	semidirect
177. C₈	C₄⋊(C₁₉⋊C₈)	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
C₂×#11926265	2
C₂×C₈×C₁₉⋊C₈	2
C₁₉⋊#1273	2
C₄×C₁₉×#97	2
#8262	19