Subgroups — #18081111

Order 4864.

378 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14564353	C₂	1	2	yes	maximal
2	C₄×#10597274	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926297	C₂	1	1	yes	maximal, characteristic
4	#14561475	C₂	1	2	yes	maximal
5	C₂×C₁₉×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597274	C₄	1	2	yes	—
7	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×Dic₁₉⋊C₈	—	2	2	no	—
9	C₄×C₄⋊Dic₁₉	C₂²	1	2	yes	—
10	#11926307	C₄	1	1	yes	characteristic
11	#11926307	C₄	1	1	yes	characteristic
12	#11926298	—	2	2	no	—
13	#11926297	—	2	2	no	—
14	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#322	—	2	2	no	—
18	C₂²×C₈×C₁₉	—	2	1	no	—
19	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
25	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
26	C₂×C₄×Dic₁₉	—	2	2	no	—
27	C₂²×C₁₉⋊C₈	—	2	2	no	—
28	C₄×C₈×C₁₉	—	2	1	no	—
29	C₄×C₈×C₁₉	—	2	1	no	—
30	C₄×C₈×C₁₉	—	2	1	no	—
31	C₄×C₈×C₁₉	D₄	1	2	yes	—
32	C₄×C₈×C₁₉	Q₈	1	2	yes	—
33	C₄×C₈×C₁₉	Q₈	1	2	yes	—
34	C₁₉⋊#95	—	2	2	no	—
35	#10597274	—	2	4	no	—
36	Dic₁₉⋊C₈	—	4	4	no	—
37	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
38	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
39	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
42	C₂²×Dic₁₉	—	2	2	no	—
43	C₂²×Dic₁₉	—	2	2	no	—
44	C₂×C₈×C₁₉	—	2	2	no	—
45	C₂×C₈×C₁₉	—	2	2	no	—
46	C₂×C₈×C₁₉	—	2	2	no	—
47	C₂×C₈×C₁₉	—	2	2	no	—
48	C₂×C₈×C₁₉	—	2	2	no	—
49	C₂×C₈×C₁₉	—	2	1	no	—
50	C₂×C₈×C₁₉	—	2	1	no	—
51	C₂×C₈×C₁₉	—	2	2	no	—
52	C₂×C₈×C₁₉	—	2	2	no	—
53	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
54	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
55	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₁₉	#41	1	1	yes	characteristic
59	C₄⋊Dic₁₉	—	2	2	no	—
60	C₄⋊Dic₁₉	—	2	2	no	—
61	C₄⋊Dic₁₉	—	2	2	no	—
62	C₄⋊Dic₁₉	M₁₆	1	4	yes	—
63	C₄×Dic₁₉	—	4	2	no	—
64	C₄×Dic₁₉	—	4	2	no	—
65	C₂×C₁₉⋊C₈	—	4	2	no	—
66	C₂×C₁₉⋊C₈	—	4	2	no	—
67	C₂×C₁₉⋊C₈	—	4	2	no	—
68	#9600	—	19	1	no	maximal, Sylow
69	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₁₉	—	2	1	no	—
71	C₂×C₄×C₁₉	—	2	1	no	—
72	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
75	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₁₉	#125	1	1	yes	characteristic
77	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
79	C₂×C₄×C₁₉	#131	1	2	yes	—
80	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
81	C₂×C₄×C₁₉	#137	1	2	yes	—
82	C₂×Dic₁₉	—	2	4	no	—
83	C₂×Dic₁₉	—	4	2	no	—
84	C₂×Dic₁₉	—	4	2	no	—
85	C₂×Dic₁₉	—	4	2	no	—
86	C₂×Dic₁₉	—	4	2	no	—
87	C₂×Dic₁₉	—	4	2	no	—
88	C₈×C₁₉	—	2	4	no	—
89	C₈×C₁₉	—	2	2	no	—
90	C₈×C₁₉	—	2	2	no	—
91	C₈×C₁₉	—	4	1	no	—
92	C₈×C₁₉	—	4	1	no	—
93	C₁₉⋊C₈	—	8	2	no	—
94	#1939	—	19	2	no	—
95	C₄×#106	—	19	1	no	—
96	C₂×#335	—	19	1	no	—
97	C₂×#322	D₁₉	1	1	yes	characteristic
98	#1279	—	19	2	no	—
99	C₂²×C₁₉	#389	1	1	yes	characteristic
100	C₂²×C₁₉	#435	1	2	yes	—
101	C₂²×C₁₉	#440	1	2	yes	—
102	C₂²×C₁₉	#443	1	1	yes	characteristic
103	C₂²×C₁₉	#443	1	1	yes	characteristic
104	C₄×C₁₉	—	2	1	no	—
105	C₄×C₁₉	—	2	1	no	—
106	C₄×C₁₉	#423	1	1	yes	characteristic
107	C₄×C₁₉	#424	1	1	yes	characteristic
108	C₄×C₁₉	#446	1	2	yes	—
109	C₄×C₁₉	#469	1	1	yes	characteristic
110	C₄×C₁₉	#470	1	1	yes	characteristic
111	C₄×C₁₉	#501	1	2	yes	—
112	Dic₁₉	—	4	4	no	—
113	Dic₁₉	—	8	2	no	—
114	C₂×#106	—	19	2	no	—
115	C₂×C₄⋊C₈	—	19	2	no	—
116	C₂×C₄⋊C₈	—	38	2	no	—
117	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
120	C₄×C₄⋊C₄	—	19	2	no	—
121	#340	—	19	1	no	—
122	#340	—	19	1	no	—
123	#336	—	38	2	no	—
124	#335	—	38	2	no	—
125	#322	—	2	2	no	—
126	C₂×C₁₉	#1585	1	2	yes	—
127	C₂×C₁₉	#1588	1	2	yes	—
128	C₂×C₁₉	#1940	1	1	yes	characteristic
129	C₂×C₁₉	#1944	1	1	yes	characteristic
130	C₂×C₁₉	#1945	1	1	yes	characteristic
131	C₂²×C₈	—	2	1	no	—
132	C₂²×C₈	—	38	2	no	—
133	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
134	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
135	C₂²×C₈	D₇₆	1	1	yes	characteristic
136	C₂²×C₈	D₇₆	1	1	yes	characteristic
137	C₂×C₄⋊C₄	—	19	2	no	—
138	C₂×C₄⋊C₄	—	19	2	no	—
139	C₂×C₄²	—	38	2	no	—
140	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
141	#106	—	38	4	no	—
142	C₄⋊C₈	—	38	2	no	—
143	C₄⋊C₈	—	38	2	no	—
144	C₄⋊C₈	—	76	4	no	—
145	C₄×C₈	—	2	1	no	—
146	C₄×C₈	—	2	1	no	—
147	C₄×C₈	—	2	1	no	—
148	C₄×C₈	Dic₃₈	1	2	yes	—
149	C₄×C₈	Dic₃₈	1	2	yes	—
150	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
151	#95	—	38	2	no	—
152	C₁₉	#9600	1	1	yes	characteristic, complemented, Sylow
153	C₂²×C₄	—	38	2	no	—
154	C₂²×C₄	—	38	2	no	—
155	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
156	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
157	C₂²×C₄	#64541	1	1	yes	characteristic
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	76	2	no	—
168	C₂×C₈	—	76	2	no	—
169	C₂×C₈	—	76	2	no	—
170	C₂×C₈	Dic₁₉⋊C₄	1	2	yes	—
171	C₂×C₈	Dic₁₉⋊C₄	1	2	yes	—
172	C₄⋊C₄	—	19	4	no	—
173	C₄⋊C₄	—	38	2	no	—
174	C₄⋊C₄	—	38	2	no	—
175	C₄⋊C₄	—	38	2	no	—
176	C₄²	—	76	2	no	—
177	C₄²	—	76	2	no	—
178	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
179	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
180	C₄²	#64541	1	1	yes	characteristic
181	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
182	C₂³	C₄×D₇₆	1	1	yes	characteristic
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	38	4	no	—
186	C₂×C₄	—	76	2	no	—
187	C₂×C₄	—	76	2	no	—
188	C₂×C₄	—	76	2	no	—
189	C₂×C₄	—	76	2	no	—
190	C₂×C₄	—	76	2	no	—
191	C₂×C₄	C₄×Dic₃₈	1	1	yes	characteristic
192	C₂×C₄	#10597315	1	1	yes	characteristic
193	C₂×C₄	#10597316	1	1	yes	characteristic
194	C₂×C₄	M₁₆×D₁₉	1	2	yes	—
195	C₂×C₄	#10597353	1	2	yes	—
196	C₂×C₄	#10597354	1	2	yes	—
197	C₂×C₄	#10597355	1	2	yes	—
198	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
199	C₂×C₄	#10597370	1	1	yes	characteristic
200	C₂×C₄	#10597376	1	1	yes	characteristic
201	C₈	—	2	4	no	—
202	C₈	—	2	2	no	—
203	C₈	—	2	2	no	—
204	C₈	—	4	1	no	—
205	C₈	—	4	1	no	—
206	C₈	—	152	2	no	—
207	C₂²	#11926442	1	1	yes	characteristic
208	C₂²	#11926617	1	2	yes	—
209	C₂²	#11926646	1	2	yes	—
210	C₂²	#11926662	1	1	yes	characteristic
211	C₂²	#11926662	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	76	4	no	—
215	C₄	—	152	2	no	—
216	C₄	#11926567	1	1	yes	characteristic
217	C₄	#11926572	1	1	yes	characteristic
218	C₄	#11926672	1	2	yes	—
219	C₄	#11926797	1	1	yes	characteristic
220	C₄	#11926802	1	1	yes	characteristic
221	C₄	#11926953	1	2	yes	—
222	C₂	#14562494	1	2	yes	—
223	C₂	#14562515	1	2	yes	—
224	C₂	#14564358	1	1	yes	characteristic
225	C₂	#14564378	1	1	yes	characteristic
226	C₂	#14564383	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₇₆
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#322	C₂
socle, prime radical	C₂³×C₁₉	C₄×D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#322	D₁₉
hypocenter	C₁₉	#9600
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
152. C₁₉	#9600	semidirect

Maximal subgroups

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

subgroup	index
#14564353	2
C₄×#10597274	2
C₂×#11926297	2
#14561475	2
C₂×C₁₉×#322	2
#9600	19