Subgroups — #18072176

Order 4864.

614 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572268	C₂	1	2	yes	maximal
2	#14561480	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×M₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
4	#11926601	—	2	2	no	—
5	#11926601	C₄	1	4	yes	—
6	#11926597	—	2	2	no	—
7	#11926597	C₂²	1	4	yes	—
8	#11926568	—	2	2	no	—
9	#11926508	C₄	1	4	yes	—
10	C₄×#64520	—	2	2	no	—
11	C₂²×M₁₆×C₁₉	C₄	1	2	yes	—
12	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
13	C₄×M₁₆×C₁₉	—	2	2	no	—
14	C₄×M₁₆×C₁₉	—	2	2	no	—
15	C₂×C₁₉×#97	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₉	C₄	1	2	yes	—
17	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
19	C₂×M₁₆×C₁₉	—	2	2	no	—
20	C₂×M₁₆×C₁₉	—	2	2	no	—
21	C₂×M₁₆×C₁₉	—	2	8	no	—
22	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₉	D₄	1	2	yes	—
28	C₂×C₄²×C₁₉	D₄	1	2	yes	—
29	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
30	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
31	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
32	C₂×#64520	—	2	4	no	—
33	C₁₉×#97	—	2	4	no	—
34	C₄×C₈×C₁₉	—	2	4	no	—
35	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
36	C₂²⋊(C₁₉⋊C₈)	C₂×C₄	1	8	yes	—
37	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
38	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
39	#10597260	—	2	4	no	—
40	C₄×C₁₉⋊C₈	—	2	4	no	—
41	C₂⁴×C₁₉	C₄²	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	—	2	4	no	—
43	C₂²×C₄×C₁₉	—	4	4	no	—
44	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
45	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
46	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
56	M₁₆×C₁₉	—	2	8	no	—
57	M₁₆×C₁₉	—	2	8	no	—
58	C₂×C₈×C₁₉	—	2	4	no	—
59	C₂×C₈×C₁₉	—	2	4	no	—
60	C₂×C₈×C₁₉	M₁₆	1	8	yes	complemented
61	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
62	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
65	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
66	C₄²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
67	C₄²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
68	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₁₉	C₂×D₄	1	2	yes	—
70	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
71	#64520	—	4	8	no	—
72	C₂×C₁₉⋊C₈	—	2	8	no	—
73	#7920	—	19	1	no	maximal, Sylow
74	C₂³×C₁₉	—	2	2	no	—
75	C₂³×C₁₉	—	4	2	no	—
76	C₂³×C₁₉	#95	1	1	yes	characteristic
77	C₂³×C₁₉	#95	1	1	yes	characteristic
78	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₄×C₁₉	—	2	2	no	—
80	C₂×C₄×C₁₉	—	2	4	no	—
81	C₂×C₄×C₁₉	—	2	4	no	—
82	C₂×C₄×C₁₉	—	2	4	no	—
83	C₂×C₄×C₁₉	—	2	2	no	—
84	C₂×C₄×C₁₉	—	2	2	no	—
85	C₂×C₄×C₁₉	—	2	2	no	—
86	C₂×C₄×C₁₉	—	2	2	no	—
87	C₂×C₄×C₁₉	—	4	2	no	—
88	C₂×C₄×C₁₉	—	4	2	no	—
89	C₂×C₄×C₁₉	—	4	2	no	—
90	C₂×C₄×C₁₉	#95	1	1	yes	characteristic
91	C₂×C₄×C₁₉	#95	1	1	yes	characteristic
92	C₂×C₄×C₁₉	#95	1	2	yes	—
93	C₂×C₄×C₁₉	#95	1	2	yes	—
94	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
96	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₉	C₂×M₁₆	1	4	yes	—
99	C₈×C₁₉	—	2	8	no	—
100	C₁₉⋊C₈	—	4	8	no	—
101	#2920	—	19	2	no	—
102	C₂×C₄×M₁₆	D₁₉	1	1	yes	characteristic
103	#1281	—	19	4	no	—
104	C₂²×C₁₉	—	2	2	no	—
105	C₂²×C₁₉	—	2	1	no	—
106	C₂²×C₁₉	—	2	1	no	—
107	C₂²×C₁₉	—	2	2	no	—
108	C₂²×C₁₉	—	2	2	no	—
109	C₂²×C₁₉	—	4	4	no	—
110	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
111	C₂²×C₁₉	C₄×M₁₆	1	2	yes	—
112	C₄×C₁₉	—	2	1	no	—
113	C₄×C₁₉	—	2	1	no	—
114	C₄×C₁₉	—	2	2	no	—
115	C₄×C₁₉	—	2	2	no	—
116	C₄×C₁₉	—	2	2	no	—
117	C₄×C₁₉	#344	1	4	yes	—
118	C₄×C₁₉	#407	1	2	yes	—
119	C₄×C₁₉	#423	1	2	yes	—
120	C₂²×M₁₆	Dic₁₉	1	2	yes	—
121	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
122	#432	—	19	4	no	—
123	#432	—	38	2	no	—
124	#431	—	19	4	no	—
125	#431	—	38	2	no	—
126	#423	—	38	2	no	—
127	#407	—	19	4	no	—
128	C₄×M₁₆	—	2	2	no	—
129	C₄×M₁₆	—	2	2	no	—
130	C₄×M₁₆	—	38	2	no	—
131	C₂×#97	C₂×D₁₉	1	2	yes	—
132	C₂×C₄×C₈	Dic₁₉	1	2	yes	—
133	C₂×C₁₉	—	2	2	no	—
134	C₂×C₁₉	—	4	1	no	—
135	C₂×C₁₉	—	4	1	no	—
136	C₂×C₁₉	#1728	1	1	yes	characteristic
137	C₂×C₁₉	C₂×#344	1	2	yes	—
138	C₂³×C₄	C₄×D₁₉	1	2	yes	—
139	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
140	C₂×M₁₆	—	2	2	no	—
141	C₂×M₁₆	—	2	2	no	—
142	C₂×M₁₆	—	2	8	no	—
143	C₂×M₁₆	—	38	4	no	—
144	C₂²×C₈	C₂×Dic₁₉	1	4	yes	—
145	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
146	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
147	C₂×C₄²	C₄×D₁₉	1	2	yes	—
148	C₂×C₄²	D₇₆	1	1	yes	characteristic
149	C₂×C₄²	D₇₆	1	1	yes	characteristic
150	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
151	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
152	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
153	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
154	C₄⋊C₈	—	38	4	no	—
155	C₄⋊C₈	—	38	4	no	—
156	C₂²⋊C₈	—	19	8	no	—
157	C₂²⋊C₈	—	38	4	no	—
158	#97	—	2	4	no	—
159	#97	—	38	4	no	—
160	C₄×C₈	—	2	4	no	—
161	C₄×C₈	—	38	4	no	—
162	C₁₉	#7920	1	1	yes	characteristic, complemented, Sylow
163	C₂⁴	C₄×Dic₁₉	1	1	yes	characteristic
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	—	4	4	no	—
166	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
167	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
168	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
169	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
170	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
171	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
172	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
173	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
174	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
175	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
177	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
178	M₁₆	—	2	8	no	—
179	M₁₆	—	2	8	no	—
180	M₁₆	—	76	8	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	38	8	no	—
184	C₂×C₈	#64520	1	8	yes	complemented
185	C₄²	Dic₁₉⋊C₄	1	2	yes	—
186	C₄²	Dic₁₉⋊C₄	1	2	yes	—
187	C₄²	C₄⋊Dic₁₉	1	1	yes	characteristic
188	C₄²	C₄⋊Dic₁₉	1	1	yes	characteristic
189	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
190	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
191	C₄²	C₂²⋊Dic₁₉	1	2	yes	—
192	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
193	C₄²	C₂×D₇₆	1	1	yes	characteristic
194	C₄²	C₂×C₂²⋊D₁₉	1	2	yes	—
195	C₂³	—	2	2	no	—
196	C₂³	—	4	2	no	—
197	C₂³	C₁₉⋊#95	1	1	yes	characteristic
198	C₂³	C₁₉⋊#95	1	1	yes	characteristic
199	C₂³	C₂×C₄×Dic₁₉	1	1	yes	characteristic
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	C₁₉⋊#95	1	1	yes	characteristic
212	C₂×C₄	C₁₉⋊#95	1	1	yes	characteristic
213	C₂×C₄	C₁₉⋊#95	1	2	yes	—
214	C₂×C₄	C₁₉⋊#95	1	2	yes	—
215	C₂×C₄	C₂×#64520	1	4	yes	—
216	C₂×C₄	C₂×Dic₁₉⋊C₄	1	2	yes	—
217	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
218	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	2	yes	—
219	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
220	C₈	—	2	8	no	—
221	C₈	—	76	8	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	4	4	no	—
228	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
229	C₂²	C₄×#64520	1	2	yes	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	#11926315	1	4	yes	—
236	C₄	#11926508	1	2	yes	—
237	C₄	#11926568	1	2	yes	—
238	C₂	—	2	2	no	—
239	C₂	—	4	1	no	—
240	C₂	—	4	1	no	—
241	C₂	#14563329	1	1	yes	characteristic
242	C₂	C₂×#11926315	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₁₉
commutator	C₂³×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₄×M₁₆×C₁₉	C₂
socle	C₂²×C₁₉	C₂×#95
prime radical	C₂⁴×C₁₉	C₄²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×C₄×M₁₆	D₁₉
hypocenter	C₁₉	#7920
LCS	G ▷ C₂³×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #14561480	C₂	semidirect
60. C₂×C₈×C₁₉	M₁₆	semidirect
162. C₁₉	#7920	semidirect
184. C₂×C₈	#64520	semidirect

Maximal subgroups

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

subgroup	index
#14572268	2
#14561480	2
C₂×C₄×M₁₆×C₁₉	2
#7920	19