Subgroups — #18088344

Order 4864.

842 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14566042	C₂	1	2	yes	maximal
2	C₂×#11926301	C₂	1	4	yes	maximal
3	C₂×C₄×M₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#10597372	C₂²	1	4	yes	—
5	#11926414	—	2	2	no	—
6	#11926410	—	2	2	no	—
7	#11926404	—	2	4	no	—
8	C₄×C₄⋊Dic₁₉	C₄	1	8	yes	complemented
9	#11926301	—	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	complemented
15	C₂×C₄×C₈×C₁₉	C₄	1	2	yes	complemented
16	C₂³×C₄×C₁₉	D₄	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	1	yes	characteristic
24	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₁₉	D₄	1	2	yes	—
27	C₂×C₄²×C₁₉	D₄	1	2	yes	—
28	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
29	C₂×C₂²⋊Dic₁₉	—	2	4	no	—
30	#10597372	—	2	8	no	—
31	#10597372	—	2	8	no	—
32	C₂×C₄⋊Dic₁₉	—	2	8	no	—
33	C₂×C₄⋊Dic₁₉	—	2	4	no	—
34	C₂×C₄×Dic₁₉	C₂×C₄	1	8	yes	—
35	C₁₉×#97	—	2	4	no	—
36	C₄×C₈×C₁₉	—	2	2	no	—
37	C₄×C₈×C₁₉	—	2	2	no	—
38	C₁₉⋊#95	—	2	8	no	—
39	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₉	—	4	4	no	—
41	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
42	C₂²×C₄×C₁₉	C₂²⋊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	2	yes	—
48	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
51	C₂²×C₄×C₁₉	#41	1	4	yes	—
52	C₂²×Dic₁₉	—	2	8	no	—
53	M₁₆×C₁₉	—	2	8	no	—
54	M₁₆×C₁₉	—	2	8	no	—
55	C₂×C₈×C₁₉	—	2	4	no	—
56	C₂×C₈×C₁₉	—	2	4	no	—
57	C₂×C₈×C₁₉	C₄⋊C₄	1	8	yes	complemented
58	C₄²×C₁₉	C₄²	1	2	yes	—
59	C₄²×C₁₉	C₄²	1	2	yes	—
60	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₁₉	C₄⋊C₄	1	4	yes	—
64	C₄²×C₁₉	C₄⋊C₄	1	4	yes	—
65	C₂²⋊Dic₁₉	—	4	4	no	—
66	C₂²⋊Dic₁₉	—	4	4	no	—
67	C₄⋊Dic₁₉	—	4	16	no	—
68	C₄⋊Dic₁₉	—	4	8	no	—
69	C₄×Dic₁₉	—	2	16	no	—
70	#10837	—	19	1	no	maximal, Sylow
71	C₂³×C₁₉	—	4	2	no	—
72	C₂³×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
73	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂³×C₁₉	#124	1	2	yes	—
75	C₂³×C₁₉	#127	1	1	yes	characteristic
76	C₂³×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₁₉	—	2	8	no	—
78	C₂×C₄×C₁₉	—	2	4	no	—
79	C₂×C₄×C₁₉	—	4	4	no	—
80	C₂×C₄×C₁₉	—	4	2	no	—
81	C₂×C₄×C₁₉	#95	1	2	yes	—
82	C₂×C₄×C₁₉	#95	1	4	yes	—
83	C₂×C₄×C₁₉	#95	1	2	yes	—
84	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
85	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
87	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
88	C₂×C₄×C₁₉	#117	1	2	yes	—
89	C₂×C₄×C₁₉	C₄×D₄	1	4	yes	—
90	C₂×C₄×C₁₉	C₄×Q₈	1	4	yes	—
91	C₂×C₄×C₁₉	#121	1	4	yes	—
92	C₂×C₄×C₁₉	C₂²⋊Q₈	1	4	yes	—
93	C₂×C₄×C₁₉	#125	1	2	yes	—
94	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
95	C₂×Dic₁₉	—	4	16	no	—
96	C₂×Dic₁₉	—	4	8	no	—
97	C₈×C₁₉	—	2	8	no	—
98	C₂×C₄×M₁₆	D₁₉	1	1	yes	characteristic
99	#2285	—	19	2	no	—
100	C₂×H(3,ℤ/4)	—	19	4	no	—
101	C₂²×C₁₉	—	2	4	no	—
102	C₂²×C₁₉	—	2	2	no	—
103	C₂²×C₁₉	—	4	2	no	—
104	C₂²×C₁₉	—	4	1	no	—
105	C₂²×C₁₉	—	4	1	no	—
106	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
107	C₂²×C₁₉	#382	1	1	yes	characteristic
108	C₂²×C₁₉	#384	1	1	yes	characteristic
109	C₂²×C₁₉	#390	1	2	yes	—
110	C₂²×C₁₉	#391	1	2	yes	—
111	C₄×C₁₉	—	2	4	no	—
112	C₄×C₁₉	—	2	2	no	—
113	C₄×C₁₉	H(3,ℤ/4)	1	4	yes	complemented
114	C₄×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
115	C₄×C₁₉	#380	1	2	yes	—
116	C₄×C₁₉	#389	1	4	yes	—
117	Dic₁₉	—	8	8	no	—
118	C₂²×M₁₆	C₂×D₁₉	1	2	yes	—
119	C₂×#117	—	19	4	no	—
120	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
121	C₄×M₁₆	—	2	2	no	—
122	C₄×M₁₆	—	2	2	no	—
123	C₂×#97	Dic₁₉	1	2	yes	complemented
124	C₂×C₄×C₈	Dic₁₉	1	2	yes	complemented
125	#384	—	38	2	no	—
126	#382	—	38	2	no	—
127	#380	—	38	4	no	—
128	C₄×C₄⋊C₄	—	19	8	no	—
129	H(3,ℤ/4)	—	38	8	no	—
130	C₂×C₁₉	—	2	2	no	—
131	C₂×C₁₉	—	2	1	no	—
132	C₂×C₁₉	—	2	1	no	—
133	C₂×C₁₉	—	2	2	no	—
134	C₂×C₁₉	C₄⋊#95	1	1	yes	characteristic
135	C₂×C₁₉	C₂×H(3,ℤ/4)	1	2	yes	—
136	C₂³×C₄	D₇₆	1	2	yes	—
137	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
138	C₂×M₁₆	—	2	4	no	—
139	C₂×M₁₆	—	2	4	no	—
140	C₂×M₁₆	Dic₃₈	1	4	yes	—
141	C₂×M₁₆	D₇₆	1	4	yes	—
142	C₂²×C₈	C₂×Dic₁₉	1	4	yes	—
143	#117	—	38	8	no	—
144	#117	—	38	8	no	—
145	C₂×C₄⋊C₄	—	38	8	no	—
146	C₂×C₄⋊C₄	—	38	4	no	—
147	C₂×C₂²⋊C₄	—	38	4	no	—
148	C₂×C₄²	—	19	8	no	—
149	C₂×C₄²	Dic₃₈	1	2	yes	—
150	C₂×C₄²	C₄×D₁₉	1	4	yes	—
151	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
152	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
153	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
154	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
155	#97	—	2	4	no	—
156	C₄×C₈	—	2	2	no	—
157	C₄×C₈	—	2	2	no	—
158	#95	—	38	8	no	—
159	C₁₉	#10837	1	1	yes	characteristic, complemented, Sylow
160	C₂⁴	C₂×D₇₆	1	1	yes	characteristic
161	C₂²×C₄	—	4	4	no	—
162	C₂²×C₄	—	38	8	no	—
163	C₂²×C₄	C₄⋊Dic₁₉	1	2	yes	—
164	C₂²×C₄	C₄⋊Dic₁₉	1	2	yes	—
165	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	4	yes	—
166	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	4	yes	—
167	C₂²×C₄	C₂×Dic₃₈	1	2	yes	—
168	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
169	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
170	C₂²×C₄	C₂×D₇₆	1	2	yes	—
171	C₂²×C₄	#64541	1	4	yes	—
172	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
174	M₁₆	—	2	8	no	—
175	M₁₆	—	2	8	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	C₄⋊Dic₁₉	1	8	yes	complemented
179	C₄⋊C₄	—	76	16	no	—
180	C₄⋊C₄	—	76	8	no	—
181	C₂²⋊C₄	—	76	4	no	—
182	C₂²⋊C₄	—	76	4	no	—
183	C₄²	—	38	16	no	—
184	C₄²	C₄×Dic₁₉	1	2	yes	—
185	C₄²	C₄×Dic₁₉	1	2	yes	—
186	C₄²	Dic₁₉⋊C₄	1	4	yes	—
187	C₄²	Dic₁₉⋊C₄	1	4	yes	—
188	C₄²	C₂²⋊Dic₁₉	1	1	yes	characteristic
189	C₄²	C₂²⋊Dic₁₉	1	1	yes	characteristic
190	C₄²	C₂²⋊Dic₁₉	1	2	yes	—
191	C₂³	—	4	2	no	—
192	C₂³	#10597315	1	1	yes	characteristic
193	C₂³	#10597320	1	1	yes	characteristic
194	C₂³	#10597321	1	2	yes	—
195	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
196	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	2	yes	—
197	C₂×C₄	—	2	8	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	76	16	no	—
202	C₂×C₄	—	76	8	no	—
203	C₂×C₄	C₁₉⋊#95	1	2	yes	—
204	C₂×C₄	C₁₉⋊#95	1	4	yes	—
205	C₂×C₄	C₁₉⋊#95	1	2	yes	—
206	C₂×C₄	C₄×Dic₃₈	1	4	yes	—
207	C₂×C₄	#10597315	1	2	yes	—
208	C₂×C₄	#10597316	1	2	yes	—
209	C₂×C₄	C₄×D₇₆	1	4	yes	—
210	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
211	C₂×C₄	C₂×Dic₁₉⋊C₄	1	2	yes	—
212	C₂×C₄	#10597370	1	4	yes	—
213	C₂×C₄	C₂×C₄⋊Dic₁₉	1	2	yes	—
214	C₂×C₄	#10597372	1	2	yes	—
215	C₂×C₄	#10597376	1	4	yes	—
216	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
217	C₈	—	2	8	no	—
218	C₂²	—	2	4	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	4	1	no	—
221	C₂²	—	4	1	no	—
222	C₂²	—	4	2	no	—
223	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
224	C₂²	#11926410	1	1	yes	characteristic
225	C₂²	#11926414	1	1	yes	characteristic
226	C₂²	#11926446	1	2	yes	—
227	C₂²	#11926450	1	2	yes	—
228	C₄	—	2	4	no	—
229	C₄	—	2	2	no	—
230	C₄	—	152	8	no	—
231	C₄	#11926301	1	4	yes	complemented
232	C₄	C₄×C₄⋊Dic₁₉	1	2	yes	—
233	C₄	#11926404	1	2	yes	—
234	C₄	#11926442	1	4	yes	—
235	C₂	—	2	2	no	—
236	C₂	—	2	1	no	—
237	C₂	—	2	1	no	—
238	C₂	—	2	2	no	—
239	C₂	#14561865	1	1	yes	characteristic
240	C₂	C₂×#11926301	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×D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×C₄×M₁₆	D₁₉
hypocenter	C₁₉	#10837
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
8. C₄×C₄⋊Dic₁₉	C₄	semidirect
14. C₂×C₁₉×#97	C₄	semidirect
15. C₂×C₄×C₈×C₁₉	C₄	semidirect
57. C₂×C₈×C₁₉	C₄⋊C₄	semidirect
113. C₄×C₁₉	H(3,ℤ/4)	semidirect
123. C₂×#97	Dic₁₉	semidirect
124. C₂×C₄×C₈	Dic₁₉	semidirect
159. C₁₉	#10837	semidirect
178. C₂×C₈	C₄⋊Dic₁₉	semidirect
231. C₄	#11926301	semidirect

Maximal subgroups

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

subgroup	index
#14566042	2
C₂×#11926301	2
C₂×C₄×M₁₆×C₁₉	2
#10837	19