Subgroups — #18081153

Order 4864.

306 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10597276	C₂	1	1	yes	maximal, characteristic
2	#14562385	C₂	1	2	yes	maximal
3	#14561918	C₂	1	1	yes	maximal, characteristic
4	#14561485	C₂	1	2	yes	maximal
5	C₂×C₁₉×#335	C₂	1	1	yes	maximal, characteristic
6	#11926597	C₂²	1	1	yes	characteristic
7	#11926597	C₂²	1	1	yes	characteristic
8	C₂×#10597276	C₄	1	1	yes	characteristic
9	C₂×#10597276	C₄	1	1	yes	characteristic
10	C₄×#64520	C₂²	1	2	yes	—
11	#11926309	C₄	1	2	yes	—
12	#11926296	—	2	2	no	—
13	#11926294	—	2	2	no	—
14	#11926265	—	2	1	no	—
15	C₈×C₁₉⋊C₈	—	2	1	no	—
16	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#335	—	2	1	no	—
19	C₁₉×#335	—	2	1	no	—
20	C₂²×C₈×C₁₉	—	2	2	no	—
21	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
24	C₂×#64520	C₂×C₄	1	2	yes	—
25	C₂×#64520	C₂×C₄	1	2	yes	—
26	C₁₉×C₄⋊C₈	—	2	2	no	—
27	C₁₉×C₄⋊C₈	—	2	2	no	—
28	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
29	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
30	C₄×C₈×C₁₉	Q₈	1	1	yes	characteristic
31	C₄×C₈×C₁₉	Q₈	1	1	yes	characteristic
32	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
33	C₂²⋊(C₁₉⋊C₈)	Q₈	1	2	yes	—
34	#10597276	—	2	2	no	—
35	#10597276	C₈	1	4	yes	complemented
36	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
37	C₄⋊(C₁₉⋊C₈)	D₄	1	1	yes	characteristic
38	C₄⋊(C₁₉⋊C₈)	D₄	1	1	yes	characteristic
39	#10597260	—	2	2	no	—
40	C₄×C₁₉⋊C₈	—	2	2	no	—
41	C₄×C₁₉⋊C₈	—	2	1	no	—
42	C₄×C₁₉⋊C₈	—	2	1	no	—
43	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
46	C₂×C₈×C₁₉	—	2	1	no	—
47	C₂×C₈×C₁₉	—	2	1	no	—
48	C₂×C₈×C₁₉	—	4	2	no	—
49	C₂×C₈×C₁₉	—	8	2	no	—
50	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₁₉	C₂×C₈	1	2	yes	—
55	C₂×C₈×C₁₉	M₁₆	1	2	yes	—
56	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₁₉	#41	1	1	yes	characteristic
60	#64520	—	2	2	no	—
61	#64520	—	2	2	no	—
62	#64520	—	2	2	no	—
63	#64520	C₂×C₈	1	4	yes	—
64	C₂×C₁₉⋊C₈	—	2	2	no	—
65	C₂×C₁₉⋊C₈	—	2	2	no	—
66	C₂×C₁₉⋊C₈	—	2	1	no	—
67	C₂×C₁₉⋊C₈	—	2	1	no	—
68	C₂×C₁₉⋊C₈	—	2	2	no	—
69	#9608	—	19	1	no	maximal, Sylow
70	C₂³×C₁₉	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₁₉	—	2	1	no	—
72	C₂×C₄×C₁₉	—	2	1	no	—
73	C₂×C₄×C₁₉	—	4	1	no	—
74	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₉	C₄×D₄	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	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
83	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
84	C₈×C₁₉	—	2	1	no	—
85	C₈×C₁₉	—	2	1	no	—
86	C₈×C₁₉	—	8	2	no	—
87	C₈×C₁₉	C₄⋊C₈	1	2	yes	—
88	C₈×C₁₉	C₄⋊C₈	1	2	yes	—
89	C₁₉⋊C₈	—	2	4	no	—
90	C₁₉⋊C₈	—	4	2	no	—
91	C₁₉⋊C₈	—	4	2	no	—
92	C₁₉⋊C₈	—	4	1	no	—
93	C₁₉⋊C₈	—	4	1	no	—
94	C₄×#108	—	19	1	no	—
95	#1565	—	19	2	no	—
96	C₂×#335	D₁₉	1	1	yes	characteristic
97	#1450	—	19	1	no	—
98	#1283	—	19	2	no	—
99	C₂²×C₁₉	—	2	1	no	—
100	C₂²×C₁₉	—	2	1	no	—
101	C₂²×C₁₉	#389	1	1	yes	characteristic
102	C₂²×C₁₉	C₈×Q₈	1	1	yes	characteristic
103	C₂²×C₁₉	#446	1	1	yes	characteristic
104	C₄×C₁₉	—	2	1	no	—
105	C₄×C₁₉	—	2	1	no	—
106	C₄×C₁₉	—	2	1	no	—
107	C₄×C₁₉	C₂×C₄⋊C₈	1	1	yes	characteristic
108	C₄×C₁₉	#424	1	1	yes	characteristic
109	C₄×C₁₉	C₈×D₄	1	1	yes	characteristic
110	C₄×C₁₉	#436	1	1	yes	characteristic
111	C₄×C₁₉	#472	1	1	yes	characteristic
112	C₄×C₁₉	#473	1	1	yes	characteristic
113	#431	—	19	1	no	—
114	#431	—	19	1	no	—
115	C₂×#108	—	19	1	no	—
116	C₂×#108	—	19	1	no	—
117	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
118	C₄×M₁₆	—	19	2	no	—
119	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
120	#341	—	19	2	no	—
121	#335	—	2	1	no	—
122	#335	—	2	1	no	—
123	#335	—	38	2	no	—
124	#334	—	38	2	no	—
125	#322	—	38	1	no	—
126	C₈²	—	38	1	no	—
127	C₂×C₁₉	—	2	1	no	—
128	C₂×C₁₉	—	2	1	no	—
129	C₂×C₁₉	#1939	1	1	yes	characteristic
130	C₂×C₁₉	#1952	1	1	yes	characteristic
131	C₂×C₁₉	#1953	1	1	yes	characteristic
132	C₂×M₁₆	—	19	2	no	—
133	C₂×M₁₆	—	19	2	no	—
134	C₂²×C₈	—	2	2	no	—
135	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
136	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
137	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
138	#108	—	19	4	no	—
139	#108	—	38	2	no	—
140	C₄⋊C₈	—	2	2	no	—
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	19	1	no	—
143	C₄⋊C₈	—	19	1	no	—
144	C₄⋊C₈	—	38	1	no	—
145	C₂²⋊C₈	—	19	2	no	—
146	C₂²⋊C₈	—	38	1	no	—
147	#97	—	38	2	no	—
148	C₄×C₈	—	38	2	no	—
149	C₄×C₈	—	38	1	no	—
150	C₄×C₈	—	38	1	no	—
151	C₄×C₈	Dic₃₈	1	1	yes	characteristic
152	C₄×C₈	Dic₃₈	1	1	yes	characteristic
153	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
154	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
155	C₁₉	#9608	1	1	yes	characteristic, complemented, Sylow
156	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
157	C₂²×C₄	#64543	1	1	yes	characteristic
158	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
159	M₁₆	—	19	4	no	—
160	M₁₆	—	38	2	no	—
161	M₁₆	—	38	2	no	—
162	M₁₆	—	38	2	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	—	8	2	no	—
167	C₂×C₈	—	38	2	no	—
168	C₂×C₈	—	38	1	no	—
169	C₂×C₈	—	38	1	no	—
170	C₂×C₈	—	38	2	no	—
171	C₂×C₈	—	38	2	no	—
172	C₂×C₈	C₈×D₁₉	1	2	yes	—
173	C₂×C₈	#64515	1	2	yes	—
174	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
175	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
176	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
177	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
178	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
179	C₄²	D₄×D₁₉	1	1	yes	characteristic
180	C₄²	#64545	1	1	yes	characteristic
181	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
182	C₂³	#10597333	1	1	yes	characteristic
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	#10597334	1	1	yes	characteristic
187	C₂×C₄	#10597336	1	1	yes	characteristic
188	C₂×C₄	#10597339	1	1	yes	characteristic
189	C₂×C₄	C₂×C₈×D₁₉	1	1	yes	characteristic
190	C₂×C₄	C₂×#64515	1	1	yes	characteristic
191	C₂×C₄	#10597353	1	1	yes	characteristic
192	C₂×C₄	#10597353	1	1	yes	characteristic
193	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
194	C₂×C₄	#10597384	1	1	yes	characteristic
195	C₂×C₄	#10597392	1	1	yes	characteristic
196	C₈	—	2	1	no	—
197	C₈	—	2	1	no	—
198	C₈	—	8	2	no	—
199	C₈	—	38	4	no	—
200	C₈	—	76	2	no	—
201	C₈	—	76	1	no	—
202	C₈	—	76	1	no	—
203	C₈	—	76	2	no	—
204	C₈	Dic₁₉⋊C₈	1	2	yes	—
205	C₈	Dic₁₉⋊C₈	1	2	yes	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#11926440	1	1	yes	characteristic
209	C₂²	#11926668	1	1	yes	characteristic
210	C₂²	#11926671	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	C₂×Dic₁₉⋊C₈	1	1	yes	characteristic
215	C₄	#11926572	1	1	yes	characteristic
216	C₄	#11926609	1	1	yes	characteristic
217	C₄	#11926621	1	1	yes	characteristic
218	C₄	#11926811	1	1	yes	characteristic
219	C₄	#11926816	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	#14564351	1	1	yes	characteristic
223	C₂	#14564418	1	1	yes	characteristic
224	C₂	#14564423	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597384
commutator	C₂×C₄×C₁₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#335	C₂
socle	C₂²×C₁₉	#389
prime radical	C₂³×C₁₉	C₄×Q₈
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#335	D₁₉
hypocenter	C₁₉	#9608
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
35. #10597276	C₈	semidirect
155. C₁₉	#9608	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10597276	2
#14562385	2
#14561918	2
#14561485	2
C₂×C₁₉×#335	2
#9608	19