Subgroups — #18072170

Order 4864.

666 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926597	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	#11926597	—	2	2	no	—
5	#11926597	—	2	2	no	—
6	#11926597	C₄	1	4	yes	—
7	#11926597	C₂²	1	4	yes	—
8	C₂×C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
9	C₂×C₂²⋊(C₁₉⋊C₈)	C₄	1	4	yes	—
10	C₂×C₄×C₁₉⋊C₈	—	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	2	yes	—
18	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
19	C₂×M₁₆×C₁₉	—	2	4	no	—
20	C₂×M₁₆×C₁₉	—	2	4	no	—
21	C₂×M₁₆×C₁₉	C₈	1	8	yes	complemented
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	2	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×C₄²×C₁₉	C₂³	1	1	yes	characteristic
30	C₂²×C₁₉⋊C₈	—	2	4	no	—
31	C₁₉×#97	—	2	4	no	—
32	C₄×C₈×C₁₉	—	2	2	no	—
33	C₄×C₈×C₁₉	—	2	2	no	—
34	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
35	C₂²⋊(C₁₉⋊C₈)	C₂×C₄	1	8	yes	—
36	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
37	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
38	C₄×C₁₉⋊C₈	—	2	4	no	—
39	C₄×C₁₉⋊C₈	—	2	4	no	—
40	C₂⁴×C₁₉	C₄²	1	1	yes	characteristic
41	C₂²×C₄×C₁₉	—	4	4	no	—
42	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
43	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₁₉	C₂×C₈	1	4	yes	—
51	C₂²×C₄×C₁₉	M₁₆	1	4	yes	—
52	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
54	M₁₆×C₁₉	—	2	16	no	—
55	C₂×C₈×C₁₉	—	2	4	no	—
56	C₂×C₈×C₁₉	—	2	4	no	—
57	C₂×C₈×C₁₉	C₂×C₈	1	8	yes	complemented
58	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
59	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
60	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
63	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
64	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
65	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₁₉	C₂×D₄	1	2	yes	—
67	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₁₉⋊C₈	—	2	8	no	—
69	C₂×C₁₉⋊C₈	—	4	8	no	—
70	#7918	—	19	1	no	maximal, Sylow
71	C₂³×C₁₉	—	4	2	no	—
72	C₂³×C₁₉	#95	1	2	yes	—
73	C₂³×C₁₉	C₄×C₈	1	2	yes	—
74	C₂³×C₁₉	#97	1	2	yes	—
75	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₁₉	—	2	4	no	—
77	C₂×C₄×C₁₉	—	2	4	no	—
78	C₂×C₄×C₁₉	—	2	4	no	—
79	C₂×C₄×C₁₉	—	4	2	no	—
80	C₂×C₄×C₁₉	—	4	2	no	—
81	C₂×C₄×C₁₉	—	4	2	no	—
82	C₂×C₄×C₁₉	#95	1	2	yes	—
83	C₂×C₄×C₁₉	#95	1	2	yes	—
84	C₂×C₄×C₁₉	#95	1	2	yes	—
85	C₂×C₄×C₁₉	C₄×C₈	1	2	yes	—
86	C₂×C₄×C₁₉	#97	1	2	yes	—
87	C₂×C₄×C₁₉	C₂²⋊C₈	1	8	yes	—
88	C₂×C₄×C₁₉	C₄⋊C₈	1	8	yes	—
89	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
91	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₉	C₂²×C₈	1	2	yes	—
94	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
95	C₈×C₁₉	—	2	8	no	—
96	C₁₉⋊C₈	—	4	8	no	—
97	C₂×#431	—	19	2	no	—
98	C₂×C₄×M₁₆	D₁₉	1	1	yes	characteristic
99	#1281	—	19	4	no	—
100	C₂²×C₁₉	—	2	2	no	—
101	C₂²×C₁₉	—	2	2	no	—
102	C₂²×C₁₉	—	2	2	no	—
103	C₂²×C₁₉	—	4	2	no	—
104	C₂²×C₁₉	—	4	2	no	—
105	C₂²×C₁₉	#336	1	4	yes	—
106	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
107	C₂²×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
108	C₂²×C₁₉	C₂×#97	1	1	yes	characteristic
109	C₄×C₁₉	—	2	2	no	—
110	C₄×C₁₉	—	2	2	no	—
111	C₄×C₁₉	—	2	2	no	—
112	C₄×C₁₉	#336	1	4	yes	—
113	C₄×C₁₉	#344	1	4	yes	—
114	C₄×C₁₉	C₂×C₂²⋊C₈	1	2	yes	—
115	C₄×C₁₉	C₂×C₄⋊C₈	1	2	yes	—
116	C₂²×M₁₆	Dic₁₉	1	2	yes	—
117	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
118	#431	—	19	4	no	—
119	#431	—	19	4	no	—
120	#431	—	38	2	no	—
121	#431	—	38	2	no	—
122	C₂×C₄⋊C₈	—	38	2	no	—
123	C₂×C₂²⋊C₈	—	19	4	no	—
124	C₄×M₁₆	—	2	2	no	—
125	C₄×M₁₆	—	2	2	no	—
126	C₂×#97	C₂×D₁₉	1	2	yes	—
127	C₂×C₄×C₈	—	38	2	no	—
128	C₂×C₄×C₈	Dic₁₉	1	2	yes	—
129	C₂×C₁₉	—	2	4	no	—
130	C₂×C₁₉	—	2	1	no	—
131	C₂×C₁₉	—	2	1	no	—
132	C₂×C₁₉	C₂×#336	1	1	yes	characteristic
133	C₂×C₁₉	C₂×#344	1	2	yes	—
134	C₂³×C₄	C₄×D₁₉	1	2	yes	—
135	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
136	C₂×M₁₆	—	2	4	no	—
137	C₂×M₁₆	—	2	4	no	—
138	C₂×M₁₆	C₁₉⋊C₈	1	8	yes	complemented
139	C₂²×C₈	—	38	4	no	—
140	C₂²×C₈	C₂×Dic₁₉	1	4	yes	—
141	C₂×C₄²	Dic₃₈	1	2	yes	—
142	C₂×C₄²	C₄×D₁₉	1	2	yes	—
143	C₂×C₄²	D₇₆	1	2	yes	—
144	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
145	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
146	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
147	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
148	C₄⋊C₈	—	38	4	no	—
149	C₄⋊C₈	—	38	4	no	—
150	C₂²⋊C₈	—	19	8	no	—
151	C₂²⋊C₈	—	38	4	no	—
152	#97	—	2	4	no	—
153	C₄×C₈	—	2	2	no	—
154	C₄×C₈	—	2	2	no	—
155	C₄×C₈	—	38	4	no	—
156	C₄×C₈	—	38	4	no	—
157	C₁₉	#7918	1	1	yes	characteristic, complemented, Sylow
158	C₂⁴	C₄×Dic₁₉	1	1	yes	characteristic
159	C₂²×C₄	—	4	4	no	—
160	C₂²×C₄	C₂×C₁₉⋊C₈	1	4	yes	—
161	C₂²×C₄	#64520	1	4	yes	—
162	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
163	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
164	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
165	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
166	C₂²×C₄	C₄⋊Dic₁₉	1	2	yes	—
167	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
168	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
169	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
170	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
171	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
172	M₁₆	—	2	16	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	38	8	no	—
176	C₂×C₈	—	76	8	no	—
177	C₂×C₈	C₂×C₁₉⋊C₈	1	8	yes	complemented
178	C₄²	Dic₁₉⋊C₄	1	2	yes	—
179	C₄²	Dic₁₉⋊C₄	1	2	yes	—
180	C₄²	C₄⋊Dic₁₉	1	2	yes	—
181	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
182	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
183	C₄²	C₂²⋊Dic₁₉	1	1	yes	characteristic
184	C₄²	C₂²⋊Dic₁₉	1	1	yes	characteristic
185	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
186	C₄²	C₂×D₇₆	1	1	yes	characteristic
187	C₄²	C₂×C₂²⋊D₁₉	1	2	yes	—
188	C₂³	—	4	2	no	—
189	C₂³	C₄×C₁₉⋊C₈	1	2	yes	—
190	C₂³	#10597260	1	2	yes	—
191	C₂³	C₁₉⋊#95	1	2	yes	—
192	C₂³	C₂×C₄×Dic₁₉	1	1	yes	characteristic
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	C₄×C₁₉⋊C₈	1	2	yes	—
200	C₂×C₄	#10597260	1	2	yes	—
201	C₂×C₄	C₄⋊(C₁₉⋊C₈)	1	8	yes	—
202	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	8	yes	—
203	C₂×C₄	C₁₉⋊#95	1	2	yes	—
204	C₂×C₄	C₁₉⋊#95	1	2	yes	—
205	C₂×C₄	C₁₉⋊#95	1	2	yes	—
206	C₂×C₄	C₂²×C₁₉⋊C₈	1	2	yes	—
207	C₂×C₄	C₂×#64520	1	2	yes	—
208	C₂×C₄	C₂×Dic₁₉⋊C₄	1	2	yes	—
209	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
210	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	2	yes	—
211	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
212	C₈	—	2	8	no	—
213	C₈	—	76	8	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	4	2	no	—
218	C₂²	—	4	2	no	—
219	C₂²	#11926299	1	4	yes	—
220	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
221	C₂²	C₂×C₄×C₁₉⋊C₈	1	1	yes	characteristic
222	C₂²	C₂×#10597260	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	#11926299	1	4	yes	—
227	C₄	#11926315	1	4	yes	—
228	C₄	C₂×C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
229	C₄	C₂×C₄⋊(C₁₉⋊C₈)	1	2	yes	—
230	C₂	—	2	4	no	—
231	C₂	—	2	1	no	—
232	C₂	—	2	1	no	—
233	C₂	C₂×#11926299	1	1	yes	characteristic
234	C₂	C₂×#11926315	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂²⋊Dic₁₉
commutator	C₂²×C₁₉	C₂×C₄×C₈
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	#7918
LCS	G ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₄² ◁ 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
21. C₂×M₁₆×C₁₉	C₈	semidirect
57. C₂×C₈×C₁₉	C₂×C₈	semidirect
138. C₂×M₁₆	C₁₉⋊C₈	semidirect
157. C₁₉	#7918	semidirect
177. C₂×C₈	C₂×C₁₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926597	2
#14561480	2
C₂×C₄×M₁₆×C₁₉	2
#7918	19