Subgroups — #18082634

Order 4864.

468 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14571998	C₂	1	1	yes	maximal, characteristic
2	#14563488	C₂	1	1	yes	maximal, characteristic
3	#14561517	C₂	1	4	yes	maximal
4	C₄×C₁₉×#97	C₂	1	1	yes	maximal, characteristic
5	C₂×#10597316	C₄	1	2	yes	—
6	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
7	C₂×#10597260	C₂²	1	2	yes	—
8	#11926442	—	2	1	no	—
9	#11926442	—	2	1	no	—
10	#11926442	—	2	1	no	—
11	#11926442	—	2	1	no	—
12	#11926414	C₄	1	1	yes	characteristic
13	#11926414	C₄	1	1	yes	characteristic
14	#11926410	C₂²	1	2	yes	—
15	#11926299	—	2	1	no	—
16	C₂×C₁₉×#97	—	2	1	no	—
17	C₂×C₁₉×#97	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₉	—	2	1	no	—
19	C₄³×C₁₉	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₁₉	—	2	2	no	—
21	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₉	D₄	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	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊Dic₁₉	—	2	1	no	—
29	C₂×C₄⋊Dic₁₉	—	2	1	no	—
30	C₂×C₄⋊Dic₁₉	—	2	1	no	—
31	C₂×C₄⋊Dic₁₉	—	2	1	no	—
32	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
34	C₂×Dic₁₉⋊C₄	—	2	2	no	—
35	C₂×Dic₁₉⋊C₄	—	2	2	no	—
36	C₂²×C₁₉⋊C₈	—	2	2	no	—
37	#10597316	—	4	4	no	—
38	C₁₉×#97	—	2	4	no	—
39	C₁₉×#97	—	2	2	no	—
40	C₁₉×#97	Q₈	1	4	yes	—
41	C₄×C₈×C₁₉	—	2	4	no	—
42	C₁₉⋊#95	—	2	2	no	—
43	C₁₉⋊#95	—	2	2	no	—
44	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
45	#10597260	—	2	4	no	—
46	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₁₉	—	2	2	no	—
54	C₂²×Dic₁₉	—	2	2	no	—
55	C₂×C₈×C₁₉	—	2	4	no	—
56	C₂×C₈×C₁₉	—	2	4	no	—
57	C₂×C₈×C₁₉	—	2	4	no	—
58	C₄²×C₁₉	—	2	2	no	—
59	C₄²×C₁₉	—	2	2	no	—
60	C₄²×C₁₉	—	2	2	no	—
61	C₄²×C₁₉	—	2	2	no	—
62	C₄²×C₁₉	—	2	2	no	—
63	C₄²×C₁₉	—	4	1	no	—
64	C₄²×C₁₉	C₂×Q₈	1	2	yes	—
65	C₄²×C₁₉	#41	1	2	yes	—
66	C₄⋊Dic₁₉	—	4	2	no	—
67	C₄⋊Dic₁₉	—	4	2	no	—
68	C₄⋊Dic₁₉	—	4	2	no	—
69	C₄⋊Dic₁₉	—	4	2	no	—
70	C₄⋊Dic₁₉	—	4	2	no	—
71	C₄⋊Dic₁₉	—	4	2	no	—
72	Dic₁₉⋊C₄	—	4	4	no	—
73	Dic₁₉⋊C₄	—	4	4	no	—
74	C₂×C₁₉⋊C₈	—	2	4	no	—
75	C₂×C₁₉⋊C₈	—	4	2	no	—
76	C₂×C₁₉⋊C₈	—	4	2	no	—
77	#9891	—	19	1	no	maximal, Sylow
78	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₉	—	2	1	no	—
80	C₂×C₄×C₁₉	—	2	1	no	—
81	C₂×C₄×C₁₉	—	2	1	no	—
82	C₂×C₄×C₁₉	—	2	1	no	—
83	C₂×C₄×C₁₉	—	2	1	no	—
84	C₂×C₄×C₁₉	—	2	1	no	—
85	C₂×C₄×C₁₉	—	2	1	no	—
86	C₂×C₄×C₁₉	—	2	1	no	—
87	C₂×C₄×C₁₉	—	2	1	no	—
88	C₂×C₄×C₁₉	—	2	1	no	—
89	C₂×C₄×C₁₉	—	2	1	no	—
90	C₂×C₄×C₁₉	—	2	1	no	—
91	C₂×C₄×C₁₉	—	2	1	no	—
92	C₂×C₄×C₁₉	#100	1	2	yes	—
93	C₂×C₄×C₁₉	#101	1	2	yes	—
94	C₂×C₄×C₁₉	#104	1	2	yes	—
95	C₂×C₄×C₁₉	#104	1	2	yes	—
96	C₂×C₄×C₁₉	#104	1	2	yes	—
97	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
98	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
99	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
100	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
101	C₂×Dic₁₉	—	4	2	no	—
102	C₂×Dic₁₉	—	4	2	no	—
103	C₂×Dic₁₉	—	4	2	no	—
104	C₂×Dic₁₉	—	4	2	no	—
105	C₂×Dic₁₉	—	4	2	no	—
106	C₂×Dic₁₉	—	4	2	no	—
107	C₈×C₁₉	—	2	8	no	—
108	C₁₉⋊C₈	—	4	4	no	—
109	#2859	—	19	1	no	—
110	#1767	—	19	1	no	—
111	C₄×#97	D₁₉	1	1	yes	characteristic
112	#1295	—	19	4	no	—
113	C₂²×C₁₉	#391	1	1	yes	characteristic
114	C₂²×C₁₉	C₂×#100	1	1	yes	characteristic
115	C₂²×C₁₉	C₂×#101	1	1	yes	characteristic
116	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
117	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
118	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
119	C₂²×C₁₉	#421	1	1	yes	characteristic
120	C₄×C₁₉	—	2	1	no	—
121	C₄×C₁₉	—	2	1	no	—
122	C₄×C₁₉	—	2	2	no	—
123	C₄×C₁₉	—	2	2	no	—
124	C₄×C₁₉	—	2	2	no	—
125	C₄×C₁₉	—	2	2	no	—
126	C₄×C₁₉	—	4	1	no	—
127	C₄×C₁₉	#330	1	4	yes	—
128	Dic₁₉	—	8	2	no	—
129	Dic₁₉	—	8	2	no	—
130	C₂×#125	—	19	2	no	—
131	C₂×C₄⋊C₈	—	38	1	no	—
132	C₂×#97	—	2	1	no	—
133	C₂×#97	—	19	2	no	—
134	C₂×#97	C₂×D₁₉	1	2	yes	—
135	C₂×C₄×C₈	—	2	1	no	—
136	#389	—	38	1	no	—
137	#389	—	38	1	no	—
138	#389	—	38	1	no	—
139	#389	—	38	1	no	—
140	#384	—	19	1	no	—
141	#384	—	19	1	no	—
142	#382	—	19	2	no	—
143	C₄³	C₂×D₁₉	1	1	yes	characteristic
144	#336	—	38	1	no	—
145	C₂×C₁₉	C₂×#330	1	1	yes	characteristic
146	C₂×C₁₉	#1525	1	1	yes	characteristic
147	C₂×C₁₉	#1528	1	1	yes	characteristic
148	C₂×C₁₉	#1530	1	1	yes	characteristic
149	C₂×C₁₉	#1986	1	1	yes	characteristic
150	C₂×C₁₉	#1994	1	1	yes	characteristic
151	C₂×C₁₉	#1997	1	1	yes	characteristic
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	—	38	2	no	—
154	#125	—	76	4	no	—
155	C₂×C₄⋊C₄	—	19	2	no	—
156	C₂×C₄⋊C₄	—	19	2	no	—
157	C₂×C₄⋊C₄	—	38	2	no	—
158	C₂×C₄⋊C₄	—	38	2	no	—
159	C₂×C₄⋊C₄	—	38	1	no	—
160	C₂×C₄⋊C₄	—	38	1	no	—
161	C₂×C₄⋊C₄	—	38	1	no	—
162	C₂×C₄⋊C₄	—	38	1	no	—
163	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
164	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
165	C₂×C₄²	D₇₆	1	1	yes	characteristic
166	C₂×C₄²	D₇₆	1	1	yes	characteristic
167	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
168	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
169	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
170	C₄⋊C₈	—	38	4	no	—
171	#97	—	2	4	no	—
172	#97	—	2	2	no	—
173	#97	—	38	4	no	—
174	#97	Dic₃₈	1	4	yes	—
175	C₄×C₈	—	2	4	no	—
176	#95	—	38	2	no	—
177	#95	—	38	2	no	—
178	C₁₉	#9891	1	1	yes	characteristic, complemented, Sylow
179	C₂²×C₄	—	38	2	no	—
180	C₂²×C₄	—	38	2	no	—
181	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
182	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
183	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
184	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
187	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	38	4	no	—
192	C₂×C₈	—	76	2	no	—
193	C₂×C₈	—	76	2	no	—
194	C₄⋊C₄	—	76	2	no	—
195	C₄⋊C₄	—	76	2	no	—
196	C₄⋊C₄	—	76	4	no	—
197	C₄⋊C₄	—	76	2	no	—
198	C₄⋊C₄	—	76	2	no	—
199	C₄⋊C₄	—	76	2	no	—
200	C₄⋊C₄	—	76	4	no	—
201	C₄⋊C₄	—	76	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	4	1	no	—
208	C₄²	C₂×Dic₃₈	1	2	yes	—
209	C₄²	#64541	1	2	yes	—
210	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	76	2	no	—
225	C₂×C₄	—	76	2	no	—
226	C₂×C₄	—	76	2	no	—
227	C₂×C₄	—	76	2	no	—
228	C₂×C₄	—	76	2	no	—
229	C₂×C₄	—	76	2	no	—
230	C₂×C₄	#10597262	1	2	yes	—
231	C₂×C₄	#10597262	1	2	yes	—
232	C₂×C₄	#10597280	1	2	yes	—
233	C₂×C₄	#10597281	1	2	yes	—
234	C₂×C₄	#10597282	1	2	yes	—
235	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
236	C₂×C₄	#10597315	1	1	yes	characteristic
237	C₂×C₄	#10597321	1	1	yes	characteristic
238	C₂×C₄	#10597370	1	2	yes	—
239	C₈	—	2	8	no	—
240	C₈	—	76	4	no	—
241	C₂²	#11926450	1	1	yes	characteristic
242	C₂²	C₂×#10597280	1	1	yes	characteristic
243	C₂²	C₂×#10597281	1	1	yes	characteristic
244	C₂²	C₂×#10597282	1	1	yes	characteristic
245	C₂²	C₂×#10597262	1	1	yes	characteristic
246	C₂²	C₂×#10597262	1	1	yes	characteristic
247	C₂²	#11926560	1	1	yes	characteristic
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	4	1	no	—
255	C₄	—	152	2	no	—
256	C₄	—	152	2	no	—
257	C₄	#11926285	1	4	yes	—
258	C₂	C₂×#11926285	1	1	yes	characteristic
259	C₂	#14562226	1	1	yes	characteristic
260	C₂	#14562236	1	1	yes	characteristic
261	C₂	#14562245	1	1	yes	characteristic
262	C₂	#14564579	1	1	yes	characteristic
263	C₂	#14564612	1	1	yes	characteristic
264	C₂	#14564624	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₁₉)⋊C₄
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₄×C₁₉×#97	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄×#97	D₁₉
hypocenter	C₁₉	#9891
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
178. C₁₉	#9891	semidirect

Maximal subgroups

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

subgroup	index
#14571998	2
#14563488	2
#14561517	2
C₄×C₁₉×#97	2
#9891	19