Subgroups — #18088347

Order 4864.

762 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926568	C₂	1	2	yes	maximal
2	C₂×C₁₉⋊#338	C₂	1	4	yes	maximal
3	C₂×C₄×M₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×#64520	C₂²	1	4	yes	—
5	#11926568	—	2	4	no	—
6	#11926568	—	2	2	no	—
7	#11926568	—	2	2	no	—
8	C₂×C₄⋊(C₁₉⋊C₈)	C₄	1	8	yes	—
9	C₁₉⋊#338	—	2	8	no	—
10	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
11	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
12	C₄×M₁₆×C₁₉	—	2	2	no	—
13	C₄×M₁₆×C₁₉	—	2	2	no	—
14	C₂×C₁₉×#97	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₁₉	C₄	1	2	yes	—
16	C₂³×C₄×C₁₉	Q₈	1	2	yes	—
17	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
18	C₂×M₁₆×C₁₉	—	2	4	no	—
19	C₂×M₁₆×C₁₉	—	2	4	no	—
20	C₂×M₁₆×C₁₉	D₄	1	4	yes	—
21	C₂×M₁₆×C₁₉	Q₈	1	4	yes	—
22	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₉	D₄	1	2	yes	—
27	C₂×C₄²×C₁₉	D₄	1	2	yes	—
28	C₂×C₄²×C₁₉	D₄	1	2	yes	—
29	C₂×#64520	—	2	8	no	—
30	C₂×#64520	—	2	4	no	—
31	C₂×#64520	—	2	4	no	—
32	C₂×#64520	—	2	8	no	—
33	C₂²×C₁₉⋊C₈	C₂×C₄	1	8	yes	—
34	C₁₉×#97	—	2	4	no	—
35	C₄×C₈×C₁₉	—	2	2	no	—
36	C₄×C₈×C₁₉	—	2	2	no	—
37	C₄⋊(C₁₉⋊C₈)	—	2	8	no	—
38	C₄⋊(C₁₉⋊C₈)	—	2	8	no	—
39	C₂⁴×C₁₉	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₁₉	—	4	4	no	—
41	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
42	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
43	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
51	C₂²×C₄×C₁₉	#41	1	4	yes	—
52	M₁₆×C₁₉	—	2	8	no	—
53	M₁₆×C₁₉	—	2	8	no	—
54	C₂×C₈×C₁₉	—	2	4	no	—
55	C₂×C₈×C₁₉	—	2	4	no	—
56	C₂×C₈×C₁₉	C₄⋊C₄	1	8	yes	—
57	C₄²×C₁₉	C₄²	1	2	yes	—
58	C₄²×C₁₉	C₄²	1	2	yes	—
59	C₄²×C₁₉	C₂²⋊C₄	1	4	yes	—
60	C₄²×C₁₉	C₂²⋊C₄	1	4	yes	—
61	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
64	#64520	—	4	8	no	—
65	#64520	—	4	4	no	—
66	#64520	—	4	4	no	—
67	C₂×C₁₉⋊C₈	—	2	8	no	—
68	C₂×C₁₉⋊C₈	—	2	16	no	—
69	C₄⋊#338	—	19	1	no	maximal, Sylow
70	C₂³×C₁₉	—	4	2	no	—
71	C₂³×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
72	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂³×C₁₉	#125	1	2	yes	—
74	C₂³×C₁₉	C₄⋊Q₈	1	2	yes	—
75	C₂×C₄×C₁₉	—	2	8	no	—
76	C₂×C₄×C₁₉	—	2	4	no	—
77	C₂×C₄×C₁₉	—	4	4	no	—
78	C₂×C₄×C₁₉	—	4	2	no	—
79	C₂×C₄×C₁₉	#95	1	2	yes	—
80	C₂×C₄×C₁₉	#95	1	4	yes	—
81	C₂×C₄×C₁₉	#95	1	2	yes	—
82	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
83	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
85	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
86	C₂×C₄×C₁₉	#117	1	2	yes	—
87	C₂×C₄×C₁₉	C₄×D₄	1	4	yes	—
88	C₂×C₄×C₁₉	C₄×Q₈	1	4	yes	—
89	C₂×C₄×C₁₉	#121	1	4	yes	—
90	C₂×C₄×C₁₉	C₂²⋊Q₈	1	4	yes	—
91	C₂×C₄×C₁₉	#124	1	2	yes	—
92	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
93	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
94	C₈×C₁₉	—	2	8	no	—
95	C₁₉⋊C₈	—	4	8	no	—
96	C₂×#423	—	19	2	no	—
97	C₂×C₄×M₁₆	D₁₉	1	1	yes	characteristic
98	C₂×#338	—	19	4	no	—
99	C₂²×C₁₉	—	2	4	no	—
100	C₂²×C₁₉	—	2	2	no	—
101	C₂²×C₁₉	—	4	2	no	—
102	C₂²×C₁₉	—	4	2	no	—
103	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
104	C₂²×C₁₉	#382	1	1	yes	characteristic
105	C₂²×C₁₉	#384	1	1	yes	characteristic
106	C₂²×C₁₉	#389	1	4	yes	—
107	C₄×C₁₉	—	2	4	no	—
108	C₄×C₁₉	—	2	2	no	—
109	C₄×C₁₉	#338	1	4	yes	complemented
110	C₄×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
111	C₄×C₁₉	#380	1	2	yes	—
112	C₄×C₁₉	#390	1	2	yes	—
113	C₄×C₁₉	#391	1	2	yes	—
114	C₂²×M₁₆	—	19	4	no	—
115	C₂²×M₁₆	C₂×D₁₉	1	2	yes	—
116	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
117	#423	—	38	4	no	—
118	#423	—	38	2	no	—
119	#423	—	38	2	no	—
120	C₂×C₄⋊C₈	—	19	8	no	—
121	C₄×M₁₆	—	2	2	no	—
122	C₄×M₁₆	—	2	2	no	—
123	C₂×#97	Dic₁₉	1	2	yes	—
124	C₂×C₄×C₈	Dic₁₉	1	2	yes	—
125	#338	—	38	8	no	—
126	C₂×C₁₉	—	2	1	no	—
127	C₂×C₁₉	—	2	1	no	—
128	C₂×C₁₉	—	2	4	no	—
129	C₂×C₁₉	C₄⋊#95	1	1	yes	characteristic
130	C₂×C₁₉	C₂×#338	1	2	yes	—
131	C₂³×C₄	Dic₃₈	1	2	yes	—
132	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
133	C₂×M₁₆	—	2	4	no	—
134	C₂×M₁₆	—	2	4	no	—
135	C₂×M₁₆	—	38	8	no	—
136	C₂×M₁₆	—	38	4	no	—
137	C₂×M₁₆	—	38	8	no	—
138	C₂×M₁₆	—	38	4	no	—
139	C₂×M₁₆	Dic₃₈	1	4	yes	—
140	C₂×M₁₆	D₇₆	1	4	yes	—
141	C₂²×C₈	—	19	8	no	—
142	C₂²×C₈	C₂×Dic₁₉	1	4	yes	—
143	C₂×C₄²	C₄×D₁₉	1	4	yes	—
144	C₂×C₄²	D₇₆	1	2	yes	—
145	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
146	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
147	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
148	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
149	C₄⋊C₈	—	38	8	no	—
150	C₄⋊C₈	—	38	8	no	—
151	#97	—	2	4	no	—
152	C₄×C₈	—	2	2	no	—
153	C₄×C₈	—	2	2	no	—
154	C₁₉	C₄⋊#338	1	1	yes	characteristic, complemented, Sylow
155	C₂⁴	C₂×Dic₃₈	1	1	yes	characteristic
156	C₂²×C₄	—	4	4	no	—
157	C₂²×C₄	Dic₁₉⋊C₄	1	4	yes	—
158	C₂²×C₄	Dic₁₉⋊C₄	1	4	yes	—
159	C₂²×C₄	C₄⋊Dic₁₉	1	2	yes	—
160	C₂²×C₄	C₄⋊Dic₁₉	1	2	yes	—
161	C₂²×C₄	C₂×Dic₃₈	1	2	yes	—
162	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
163	C₂²×C₄	C₂×D₇₆	1	2	yes	—
164	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
165	C₂²×C₄	#64541	1	4	yes	—
166	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
168	M₁₆	—	2	8	no	—
169	M₁₆	—	2	8	no	—
170	M₁₆	—	76	8	no	—
171	M₁₆	—	76	4	no	—
172	M₁₆	—	76	4	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	38	16	no	—
176	C₂×C₈	—	38	8	no	—
177	C₂×C₈	C₄⋊Dic₁₉	1	8	yes	—
178	C₄²	C₄×Dic₁₉	1	2	yes	—
179	C₄²	C₄×Dic₁₉	1	2	yes	—
180	C₄²	(C₂×D₁₉)⋊C₄	1	4	yes	—
181	C₄²	(C₂×D₁₉)⋊C₄	1	4	yes	—
182	C₄²	C₂²⋊Dic₁₉	1	1	yes	characteristic
183	C₄²	C₂²⋊Dic₁₉	1	1	yes	characteristic
184	C₄²	C₂²⋊Dic₁₉	1	2	yes	—
185	C₂³	—	4	2	no	—
186	C₂³	#10597315	1	2	yes	—
187	C₂³	#10597316	1	2	yes	—
188	C₂³	C₂×Dic₁₉⋊C₄	1	2	yes	—
189	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
190	C₂×C₄	—	2	8	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	C₁₉⋊#95	1	2	yes	—
195	C₂×C₄	C₁₉⋊#95	1	4	yes	—
196	C₂×C₄	C₁₉⋊#95	1	2	yes	—
197	C₂×C₄	C₄×Dic₃₈	1	4	yes	—
198	C₂×C₄	#10597315	1	1	yes	characteristic
199	C₂×C₄	C₄×D₇₆	1	4	yes	—
200	C₂×C₄	#10597320	1	1	yes	characteristic
201	C₂×C₄	#10597321	1	2	yes	—
202	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
203	C₂×C₄	#10597370	1	4	yes	—
204	C₂×C₄	C₂×C₄⋊Dic₁₉	1	2	yes	—
205	C₂×C₄	#10597372	1	2	yes	—
206	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	2	yes	—
207	C₂×C₄	#10597376	1	4	yes	—
208	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
209	C₈	—	2	8	no	—
210	C₈	—	76	8	no	—
211	C₂²	—	2	4	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	—	4	2	no	—
215	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
216	C₂²	#11926410	1	1	yes	characteristic
217	C₂²	#11926414	1	1	yes	characteristic
218	C₂²	#11926442	1	4	yes	—
219	C₄	—	2	4	no	—
220	C₄	—	2	2	no	—
221	C₄	C₁₉⋊#338	1	4	yes	complemented
222	C₄	C₄×C₄⋊Dic₁₉	1	2	yes	—
223	C₄	#11926404	1	2	yes	—
224	C₄	#11926446	1	2	yes	—
225	C₄	#11926450	1	2	yes	—
226	C₂	—	2	1	no	—
227	C₂	—	2	1	no	—
228	C₂	—	2	4	no	—
229	C₂	#14561865	1	1	yes	characteristic
230	C₂	C₂×C₁₉⋊#338	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₄×M₁₆×C₁₉	C₂
socle	C₂²×C₁₉	C₂×#95
prime radical	C₂⁴×C₁₉	C₂×Q₈
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×C₄×M₁₆	D₁₉
hypocenter	C₁₉	C₄⋊#338
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
109. C₄×C₁₉	#338	semidirect
154. C₁₉	C₄⋊#338	semidirect
221. C₄	C₁₉⋊#338	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926568	2
C₂×C₁₉⋊#338	2
C₂×C₄×M₁₆×C₁₉	2
C₄⋊#338	19