Subgroups — #18081151

Order 4864.

306 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	#14562387	C₂	1	2	yes	maximal
2	C₂×#11926295	C₂	1	1	yes	maximal, characteristic
3	#14561918	C₂	1	1	yes	maximal, characteristic
4	#14561486	C₂	1	2	yes	maximal
5	C₄×C₁₉×#108	C₂	1	1	yes	maximal, characteristic
6	#11926597	C₂²	1	1	yes	characteristic
7	#11926597	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
9	#11926297	—	2	2	no	—
10	#11926295	—	2	1	no	—
11	#11926295	—	2	1	no	—
12	#11926295	—	2	2	no	—
13	#11926265	—	2	1	no	—
14	C₈×C₁₉⋊C₈	—	2	1	no	—
15	C₂×C₁₉×#108	C₄	1	1	yes	characteristic
16	C₂×C₁₉×#108	C₄	1	1	yes	characteristic
17	C₄×M₁₆×C₁₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
19	C₁₉×#341	C₄	1	2	yes	—
20	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
21	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
25	C₂²×C₁₉⋊C₈	—	2	2	no	—
26	C₁₉×#108	—	2	2	no	—
27	C₁₉×#108	C₈	1	4	yes	complemented
28	C₁₉×#97	—	2	2	no	—
29	C₄×C₈×C₁₉	—	2	2	no	—
30	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
31	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
32	C₄×C₈×C₁₉	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₁₉	Q₈	1	1	yes	characteristic
34	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
35	C₂²⋊(C₁₉⋊C₈)	Q₈	1	2	yes	—
36	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
37	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
38	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
39	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
40	C₄×C₁₉⋊C₈	—	2	1	no	—
41	C₄×C₁₉⋊C₈	—	2	1	no	—
42	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
45	M₁₆×C₁₉	—	2	2	no	—
46	M₁₆×C₁₉	—	2	2	no	—
47	M₁₆×C₁₉	—	2	2	no	—
48	M₁₆×C₁₉	C₂×C₈	1	4	yes	—
49	C₂×C₈×C₁₉	—	2	2	no	—
50	C₂×C₈×C₁₉	—	2	1	no	—
51	C₂×C₈×C₁₉	—	2	1	no	—
52	C₂×C₈×C₁₉	—	2	2	no	—
53	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₉	C₂×C₈	1	2	yes	—
58	C₂×C₈×C₁₉	M₁₆	1	2	yes	—
59	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
62	C₄²×C₁₉	#41	1	1	yes	characteristic
63	C₂×C₁₉⋊C₈	—	2	2	no	—
64	C₂×C₁₉⋊C₈	—	2	2	no	—
65	C₂×C₁₉⋊C₈	—	4	2	no	—
66	C₂×C₁₉⋊C₈	—	8	2	no	—
67	#9607	—	19	1	no	maximal, Sylow
68	C₂³×C₁₉	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₁₉	—	2	2	no	—
70	C₂×C₄×C₁₉	—	4	1	no	—
71	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
74	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₁₉	#125	1	1	yes	characteristic
76	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
80	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
81	C₈×C₁₉	—	2	1	no	—
82	C₈×C₁₉	—	2	1	no	—
83	C₈×C₁₉	—	2	4	no	—
84	C₈×C₁₉	—	4	2	no	—
85	C₈×C₁₉	C₄⋊C₈	1	2	yes	—
86	C₈×C₁₉	C₄⋊C₈	1	2	yes	—
87	C₁₉⋊C₈	—	4	2	no	—
88	C₁₉⋊C₈	—	4	2	no	—
89	C₁₉⋊C₈	—	8	2	no	—
90	C₄×#108	D₁₉	1	1	yes	characteristic
91	#1565	—	19	2	no	—
92	C₂×#334	—	19	1	no	—
93	#1450	—	19	1	no	—
94	#1283	—	19	2	no	—
95	C₂²×C₁₉	—	2	1	no	—
96	C₂²×C₁₉	—	2	1	no	—
97	C₂²×C₁₉	#389	1	1	yes	characteristic
98	C₂²×C₁₉	C₈×Q₈	1	1	yes	characteristic
99	C₂²×C₁₉	#446	1	1	yes	characteristic
100	C₄×C₁₉	—	2	1	no	—
101	C₄×C₁₉	—	2	1	no	—
102	C₄×C₁₉	—	2	1	no	—
103	C₄×C₁₉	C₂×C₄⋊C₈	1	1	yes	characteristic
104	C₄×C₁₉	#424	1	1	yes	characteristic
105	C₄×C₁₉	C₈×D₄	1	1	yes	characteristic
106	C₄×C₁₉	#436	1	1	yes	characteristic
107	C₄×C₁₉	#471	1	2	yes	—
108	#431	—	19	1	no	—
109	#431	—	19	1	no	—
110	C₂×#108	Dic₁₉	1	1	yes	characteristic
111	C₂×#108	Dic₁₉	1	1	yes	characteristic
112	C₂×C₄⋊C₈	—	19	2	no	—
113	C₄×M₁₆	C₂×D₁₉	1	2	yes	—
114	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
115	#341	Dic₁₉	1	2	yes	—
116	#335	—	38	2	no	—
117	#334	—	38	2	no	—
118	#334	—	38	1	no	—
119	#334	—	38	1	no	—
120	#322	—	38	1	no	—
121	C₈²	—	38	1	no	—
122	C₂×C₁₉	—	2	1	no	—
123	C₂×C₁₉	—	2	1	no	—
124	C₂×C₁₉	#1939	1	1	yes	characteristic
125	C₂×C₁₉	#1951	1	1	yes	characteristic
126	C₂×C₁₉	#1953	1	1	yes	characteristic
127	C₂×M₁₆	C₂×Dic₁₉	1	2	yes	—
128	C₂×M₁₆	C₂×Dic₁₉	1	2	yes	—
129	C₂²×C₈	—	38	2	no	—
130	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
131	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
132	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
133	#108	—	2	2	no	—
134	#108	C₁₉⋊C₈	1	4	yes	complemented
135	C₄⋊C₈	—	19	2	no	—
136	C₄⋊C₈	—	38	2	no	—
137	C₄⋊C₈	—	38	2	no	—
138	C₄⋊C₈	—	38	1	no	—
139	C₂²⋊C₈	—	19	2	no	—
140	C₂²⋊C₈	—	38	1	no	—
141	#97	—	2	2	no	—
142	C₄×C₈	—	2	2	no	—
143	C₄×C₈	—	38	1	no	—
144	C₄×C₈	—	38	1	no	—
145	C₄×C₈	Dic₃₈	1	1	yes	characteristic
146	C₄×C₈	Dic₃₈	1	1	yes	characteristic
147	C₄×C₈	D₇₆	1	1	yes	characteristic
148	C₄×C₈	D₇₆	1	1	yes	characteristic
149	C₁₉	#9607	1	1	yes	characteristic, complemented, Sylow
150	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
151	C₂²×C₄	#64545	1	1	yes	characteristic
152	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
153	M₁₆	—	2	2	no	—
154	M₁₆	—	2	2	no	—
155	M₁₆	—	2	2	no	—
156	M₁₆	C₂×C₁₉⋊C₈	1	4	yes	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	38	2	no	—
162	C₂×C₈	—	38	2	no	—
163	C₂×C₈	—	76	2	no	—
164	C₂×C₈	—	152	2	no	—
165	C₂×C₈	C₂×C₁₉⋊C₈	1	2	yes	—
166	C₂×C₈	#64520	1	2	yes	—
167	C₂×C₈	C₄⋊Dic₁₉	1	1	yes	characteristic
168	C₂×C₈	C₄⋊Dic₁₉	1	1	yes	characteristic
169	C₂×C₈	C₄⋊Dic₁₉	1	1	yes	characteristic
170	C₂×C₈	C₄⋊Dic₁₉	1	1	yes	characteristic
171	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
172	C₄²	C₂×D₇₆	1	1	yes	characteristic
173	C₄²	D₄×D₁₉	1	1	yes	characteristic
174	C₄²	#64543	1	1	yes	characteristic
175	C₂³	Q₈×Dic₁₉	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	#10597334	1	1	yes	characteristic
179	C₂×C₄	#10597336	1	1	yes	characteristic
180	C₂×C₄	#10597341	1	1	yes	characteristic
181	C₂×C₄	#10597343	1	1	yes	characteristic
182	C₂×C₄	C₂²×C₁₉⋊C₈	1	1	yes	characteristic
183	C₂×C₄	C₂×#64520	1	1	yes	characteristic
184	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
185	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
186	C₂×C₄	#10597394	1	1	yes	characteristic
187	C₂×C₄	#10597394	1	1	yes	characteristic
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	2	4	no	—
191	C₈	—	4	2	no	—
192	C₈	—	76	2	no	—
193	C₈	—	76	2	no	—
194	C₈	—	152	2	no	—
195	C₈	C₄⋊(C₁₉⋊C₈)	1	2	yes	—
196	C₈	C₄⋊(C₁₉⋊C₈)	1	2	yes	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	#11926443	1	1	yes	characteristic
200	C₂²	Q₈×C₁₉⋊C₈	1	1	yes	characteristic
201	C₂²	#11926675	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	C₂×C₄⋊(C₁₉⋊C₈)	1	1	yes	characteristic
206	C₄	#11926571	1	1	yes	characteristic
207	C₄	D₄×C₁₉⋊C₈	1	1	yes	characteristic
208	C₄	#11926622	1	1	yes	characteristic
209	C₄	#11926808	1	2	yes	—
210	C₂	—	2	1	no	—
211	C₂	—	2	1	no	—
212	C₂	#14564354	1	1	yes	characteristic
213	C₂	#14564416	1	1	yes	characteristic
214	C₂	#14564426	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597341
commutator	C₂×C₄×C₁₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₄×C₁₉×#108	C₂
socle	C₂²×C₁₉	#389
prime radical	C₂³×C₁₉	C₄×Q₈
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄×#108	D₁₉
hypocenter	C₁₉	#9607
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
27. C₁₉×#108	C₈	semidirect
134. #108	C₁₉⋊C₈	semidirect
149. C₁₉	#9607	semidirect

Maximal subgroups

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

subgroup	index
#14562387	2
C₂×#11926295	2
#14561918	2
#14561486	2
C₄×C₁₉×#108	2
#9607	19