Subgroups — #18081107

Order 4864.

378 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14564354	C₂	1	2	yes	maximal
2	C₂×#11926295	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926266	C₂	1	1	yes	maximal, characteristic
4	#14561476	C₂	1	2	yes	maximal
5	C₄×C₁₉×#107	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
7	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
10	#11926299	—	2	2	no	—
11	#11926295	—	2	2	no	—
12	#11926266	—	2	2	no	—
13	C₂×C₁₉×#107	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
16	C₁₉×#340	C₄	1	1	yes	characteristic
17	C₁₉×#340	C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
21	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₁₉	—	2	2	no	—
23	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
24	C₂²×C₁₉⋊C₈	—	2	1	no	—
25	C₂²×C₁₉⋊C₈	—	2	2	no	—
26	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
27	C₁₉×#107	—	2	4	no	—
28	C₄×C₈×C₁₉	D₄	1	2	yes	—
29	C₄×C₈×C₁₉	Q₈	1	2	yes	—
30	C₁₉×#95	—	2	2	no	—
31	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
32	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
33	C₄⋊(C₁₉⋊C₈)	—	4	4	no	—
34	C₄×C₁₉⋊C₈	—	2	1	no	—
35	C₄×C₁₉⋊C₈	—	2	1	no	—
36	C₄×C₁₉⋊C₈	—	2	1	no	—
37	C₄×C₁₉⋊C₈	Q₈	1	2	yes	—
38	C₂²×C₄×C₁₉	—	2	2	no	—
39	C₂²×C₄×C₁₉	—	2	2	no	—
40	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
43	C₂×C₈×C₁₉	—	2	2	no	—
44	C₂×C₈×C₁₉	—	2	2	no	—
45	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
47	C₄⋊C₄×C₁₉	—	2	2	no	—
48	C₄⋊C₄×C₁₉	—	2	2	no	—
49	C₄⋊C₄×C₁₉	—	2	2	no	—
50	C₄⋊C₄×C₁₉	M₁₆	1	4	yes	—
51	C₄²×C₁₉	—	4	2	no	—
52	C₄²×C₁₉	—	4	2	no	—
53	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
55	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₁₉	#41	1	1	yes	characteristic
57	C₂×C₁₉⋊C₈	—	2	4	no	—
58	C₂×C₁₉⋊C₈	—	2	2	no	—
59	C₂×C₁₉⋊C₈	—	2	2	no	—
60	C₂×C₁₉⋊C₈	—	2	2	no	—
61	C₂×C₁₉⋊C₈	—	2	2	no	—
62	C₂×C₁₉⋊C₈	—	4	2	no	—
63	C₂×C₁₉⋊C₈	—	4	2	no	—
64	C₂×C₁₉⋊C₈	—	4	2	no	—
65	#9599	—	19	1	no	maximal, Sylow
66	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₁₉	—	2	2	no	—
68	C₂×C₄×C₁₉	—	2	4	no	—
69	C₂×C₄×C₁₉	—	4	2	no	—
70	C₂×C₄×C₁₉	—	4	2	no	—
71	C₂×C₄×C₁₉	—	4	2	no	—
72	C₂×C₄×C₁₉	—	4	2	no	—
73	C₂×C₄×C₁₉	—	4	2	no	—
74	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
76	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
77	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₁₉	#125	1	1	yes	characteristic
79	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
81	C₂×C₄×C₁₉	#131	1	2	yes	—
82	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
83	C₂×C₄×C₁₉	#137	1	2	yes	—
84	C₈×C₁₉	—	2	2	no	—
85	C₈×C₁₉	—	2	2	no	—
86	C₁₉⋊C₈	—	2	4	no	—
87	C₁₉⋊C₈	—	4	2	no	—
88	C₁₉⋊C₈	—	8	2	no	—
89	#1939	—	19	2	no	—
90	C₄×#107	D₁₉	1	1	yes	characteristic
91	C₂×#334	—	19	1	no	—
92	C₂×#322	—	19	1	no	—
93	#1279	—	19	2	no	—
94	C₂²×C₁₉	#389	1	1	yes	characteristic
95	C₂²×C₁₉	#435	1	2	yes	—
96	C₂²×C₁₉	#441	1	1	yes	characteristic
97	C₂²×C₁₉	#442	1	1	yes	characteristic
98	C₂²×C₁₉	#443	1	2	yes	—
99	C₄×C₁₉	—	2	1	no	—
100	C₄×C₁₉	—	2	1	no	—
101	C₄×C₁₉	—	4	4	no	—
102	C₄×C₁₉	—	8	2	no	—
103	C₄×C₁₉	#423	1	1	yes	characteristic
104	C₄×C₁₉	#424	1	1	yes	characteristic
105	C₄×C₁₉	#446	1	2	yes	—
106	C₄×C₁₉	#468	1	2	yes	—
107	C₄×C₁₉	#501	1	2	yes	—
108	C₂×#107	Dic₁₉	1	2	yes	—
109	C₂×C₄⋊C₈	—	19	2	no	—
110	C₂×C₄⋊C₈	—	38	2	no	—
111	C₂×C₄×C₈	—	19	1	no	—
112	C₂×C₄×C₈	—	19	1	no	—
113	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
114	C₄×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
115	#340	Dic₁₉	1	1	yes	characteristic
116	#340	Dic₁₉	1	1	yes	characteristic
117	#336	—	38	2	no	—
118	#334	—	38	2	no	—
119	#322	—	38	2	no	—
120	C₂×C₁₉	#1586	1	2	yes	—
121	C₂×C₁₉	#1587	1	2	yes	—
122	C₂×C₁₉	#1940	1	1	yes	characteristic
123	C₂×C₁₉	#1943	1	1	yes	characteristic
124	C₂×C₁₉	#1945	1	1	yes	characteristic
125	C₂²×C₈	—	19	2	no	—
126	C₂²×C₈	—	38	1	no	—
127	C₂²×C₈	—	38	2	no	—
128	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
129	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
130	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
131	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
132	C₂×C₄²	—	2	2	no	—
133	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
134	#107	—	2	4	no	—
135	C₄⋊C₈	—	38	2	no	—
136	C₄⋊C₈	—	38	2	no	—
137	C₄⋊C₈	—	76	4	no	—
138	C₄×C₈	—	19	2	no	—
139	C₄×C₈	—	38	1	no	—
140	C₄×C₈	—	38	1	no	—
141	C₄×C₈	—	38	1	no	—
142	C₄×C₈	Dic₃₈	1	2	yes	—
143	C₄×C₈	D₇₆	1	2	yes	—
144	#95	—	2	2	no	—
145	C₁₉	#9599	1	1	yes	characteristic, complemented, Sylow
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
149	C₂²×C₄	#64543	1	1	yes	characteristic
150	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	38	4	no	—
154	C₂×C₈	—	38	2	no	—
155	C₂×C₈	—	38	2	no	—
156	C₂×C₈	—	38	2	no	—
157	C₂×C₈	—	38	2	no	—
158	C₂×C₈	—	76	2	no	—
159	C₂×C₈	—	76	2	no	—
160	C₂×C₈	—	76	2	no	—
161	C₂×C₈	C₄⋊Dic₁₉	1	2	yes	—
162	C₂×C₈	C₄⋊Dic₁₉	1	2	yes	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	2	2	no	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	#64520	1	4	yes	—
167	C₄²	—	4	2	no	—
168	C₄²	—	4	2	no	—
169	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
170	C₄²	C₂×D₇₆	1	1	yes	characteristic
171	C₄²	Q₈×D₁₉	1	1	yes	characteristic
172	C₄²	#64545	1	1	yes	characteristic
173	C₂³	D₄×Dic₁₉	1	1	yes	characteristic
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	#10597334	1	1	yes	characteristic
182	C₂×C₄	#10597336	1	1	yes	characteristic
183	C₂×C₄	#10597341	1	1	yes	characteristic
184	C₂×C₄	#10597343	1	1	yes	characteristic
185	C₂×C₄	#10597357	1	2	yes	—
186	C₂×C₄	#10597364	1	2	yes	—
187	C₂×C₄	C₂×#64520	1	2	yes	—
188	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
189	C₂×C₄	Q₈×Dic₁₉	1	1	yes	characteristic
190	C₂×C₄	#10597394	1	2	yes	—
191	C₈	—	2	2	no	—
192	C₈	—	2	2	no	—
193	C₈	—	38	4	no	—
194	C₈	—	76	2	no	—
195	C₈	—	152	2	no	—
196	C₂²	#11926443	1	1	yes	characteristic
197	C₂²	#11926616	1	2	yes	—
198	C₂²	#11926651	1	1	yes	characteristic
199	C₂²	#11926656	1	1	yes	characteristic
200	C₂²	#11926661	1	2	yes	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	4	4	no	—
204	C₄	—	8	2	no	—
205	C₄	#11926568	1	1	yes	characteristic
206	C₄	#11926571	1	1	yes	characteristic
207	C₄	#11926675	1	2	yes	—
208	C₄	#11926791	1	2	yes	—
209	C₄	#11926952	1	2	yes	—
210	C₂	#14562500	1	2	yes	—
211	C₂	#14562507	1	2	yes	—
212	C₂	#14564359	1	1	yes	characteristic
213	C₂	#14564374	1	1	yes	characteristic
214	C₂	#14564384	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₄×C₁₉×#107	C₂
socle, prime radical	C₂³×C₁₉	C₄×D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄×#107	D₁₉
hypocenter	C₁₉	#9599
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
145. C₁₉	#9599	semidirect

Maximal subgroups

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

subgroup	index
#14564354	2
C₂×#11926295	2
C₂×#11926266	2
#14561476	2
C₄×C₁₉×#107	2
#9599	19