Subgroups — #18075618

Order 4864.

438 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563763	C₂	1	1	yes	maximal, characteristic
2	#14563640	C₂	1	2	yes	maximal
3	C₂×#11926297	C₂	1	1	yes	maximal, characteristic
4	#14561519	C₂	1	2	yes	maximal
5	C₂×C₈×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×#10597292	—	2	2	no	—
8	#11926414	C₂²	1	2	yes	—
9	#11926307	—	2	1	no	—
10	#11926307	—	2	1	no	—
11	#11926306	—	2	2	no	—
12	#11926297	—	2	2	no	—
13	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
14	C₈×Q₈×C₁₉	—	2	2	no	—
15	C₈×Q₈×C₁₉	C₄	1	4	yes	complemented
16	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
17	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₉	—	2	1	no	—
19	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
20	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₉	—	2	1	no	—
22	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
24	C₄×Q₈×C₁₉	—	2	1	no	—
25	C₄×Q₈×C₁₉	—	2	1	no	—
26	C₄×Q₈×C₁₉	C₂×C₄	1	2	yes	—
27	C₄×Q₈×C₁₉	Q₈	1	2	yes	—
28	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₉	—	2	1	no	—
31	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊Dic₁₉	—	2	2	no	—
33	C₂×C₄⋊Dic₁₉	—	2	2	no	—
34	C₂×C₄⋊Dic₁₉	—	2	2	no	—
35	C₂²×C₁₉⋊C₈	—	2	2	no	—
36	C₁₉×C₄⋊C₈	—	2	4	no	—
37	C₁₉×C₄⋊C₈	—	2	1	no	—
38	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₁₉	—	2	4	no	—
40	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₁₉	D₄	1	2	yes	—
42	#10597292	—	4	4	no	—
43	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
44	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
45	C₂×Q₈×C₁₉	—	2	1	no	—
46	C₂×Q₈×C₁₉	—	2	1	no	—
47	C₂×Q₈×C₁₉	—	2	1	no	—
48	C₂×Q₈×C₁₉	C₄⋊C₄	1	2	yes	—
49	C₂×Q₈×C₁₉	D₈	1	2	yes	—
50	C₂×Q₈×C₁₉	Q₁₆	1	2	yes	—
51	C₂²×C₄×C₁₉	—	2	1	no	—
52	C₂²×C₄×C₁₉	—	2	1	no	—
53	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
56	C₂²×Dic₁₉	—	4	2	no	—
57	C₂×C₈×C₁₉	—	2	2	no	—
58	C₂×C₈×C₁₉	—	2	2	no	—
59	C₂×C₈×C₁₉	—	2	2	no	—
60	C₂×C₈×C₁₉	—	4	2	no	—
61	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
63	C₄⋊C₄×C₁₉	—	2	2	no	—
64	C₄⋊C₄×C₁₉	—	2	1	no	—
65	C₄⋊C₄×C₁₉	—	2	2	no	—
66	C₄⋊C₄×C₁₉	C₄⋊C₄	1	2	yes	—
67	C₄²×C₁₉	—	2	2	no	—
68	C₄²×C₁₉	—	2	2	no	—
69	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
72	C₄²×C₁₉	#41	1	1	yes	characteristic
73	C₄⋊Dic₁₉	—	4	2	no	—
74	C₄⋊Dic₁₉	—	4	2	no	—
75	C₄⋊Dic₁₉	—	4	2	no	—
76	C₄⋊Dic₁₉	—	4	2	no	—
77	C₄⋊Dic₁₉	—	8	2	no	—
78	C₂×C₁₉⋊C₈	—	4	2	no	—
79	C₂×C₁₉⋊C₈	—	4	2	no	—
80	C₂×C₁₉⋊C₈	—	4	2	no	—
81	#8649	—	19	1	no	maximal, Sylow
82	C₂³×C₁₉	#121	1	1	yes	characteristic
83	Q₈×C₁₉	—	2	2	no	—
84	Q₈×C₁₉	—	2	2	no	—
85	Q₈×C₁₉	—	2	2	no	—
86	Q₈×C₁₉	#107	1	4	yes	—
87	C₂×C₄×C₁₉	—	2	2	no	—
88	C₂×C₄×C₁₉	—	2	2	no	—
89	C₂×C₄×C₁₉	—	2	2	no	—
90	C₂×C₄×C₁₉	—	2	2	no	—
91	C₂×C₄×C₁₉	—	2	2	no	—
92	C₂×C₄×C₁₉	—	4	2	no	—
93	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
96	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
97	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
101	C₂×C₄×C₁₉	#135	1	2	yes	—
102	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
103	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
104	C₂×C₄×C₁₉	#137	1	2	yes	—
105	C₂×Dic₁₉	—	8	2	no	—
106	C₂×Dic₁₉	—	8	2	no	—
107	C₂×Dic₁₉	—	8	2	no	—
108	C₈×C₁₉	—	2	2	no	—
109	C₈×C₁₉	—	2	2	no	—
110	C₈×C₁₉	—	4	2	no	—
111	C₁₉⋊C₈	—	8	2	no	—
112	C₂×C₈×Q₈	D₁₉	1	1	yes	characteristic
113	#1834	—	19	1	no	—
114	#1803	—	19	2	no	—
115	C₂×#335	—	19	1	no	—
116	#1296	—	19	2	no	—
117	C₂²×C₁₉	#380	1	1	yes	characteristic
118	C₂²×C₁₉	#464	1	2	yes	—
119	C₂²×C₁₉	#466	1	1	yes	characteristic
120	C₂²×C₁₉	#467	1	1	yes	characteristic
121	C₂²×C₁₉	#471	1	2	yes	—
122	C₄×C₁₉	—	2	4	no	—
123	C₄×C₁₉	—	2	1	no	—
124	C₄×C₁₉	—	2	1	no	—
125	C₄×C₁₉	—	4	2	no	—
126	C₄×C₁₉	#416	1	1	yes	characteristic
127	C₄×C₁₉	#419	1	1	yes	characteristic
128	C₄×C₁₉	C₂×#107	1	1	yes	characteristic
129	C₄×C₁₉	#427	1	1	yes	characteristic
130	C₄×C₁₉	#466	1	1	yes	characteristic
131	C₄×C₁₉	#467	1	1	yes	characteristic
132	C₄×C₁₉	#479	1	2	yes	—
133	Dic₁₉	—	16	2	no	—
134	C₂×C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
135	C₈×Q₈	—	2	2	no	—
136	C₈×Q₈	Dic₁₉	1	4	yes	complemented
137	C₂×C₄⋊C₈	—	2	1	no	—
138	C₂×C₄⋊C₈	—	19	2	no	—
139	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
140	C₂×#103	—	38	2	no	—
141	C₂×C₄×C₈	—	2	1	no	—
142	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
143	#384	—	19	2	no	—
144	#340	—	38	1	no	—
145	#340	—	38	1	no	—
146	#340	—	38	2	no	—
147	#335	—	38	2	no	—
148	C₂×C₁₉	#1682	1	2	yes	—
149	C₂×C₁₉	#1684	1	2	yes	—
150	C₂×C₁₉	#1805	1	1	yes	characteristic
151	C₂×C₁₉	#1808	1	1	yes	characteristic
152	C₂×C₁₉	#1815	1	1	yes	characteristic
153	C₂²×Q₈	D₇₆	1	1	yes	characteristic
154	C₂²×C₈	—	2	1	no	—
155	C₂²×C₈	—	38	2	no	—
156	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
157	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
158	C₄×Q₈	—	2	1	no	—
159	C₄×Q₈	—	2	1	no	—
160	C₄×Q₈	Dic₃₈	1	2	yes	—
161	C₄×Q₈	C₂×Dic₁₉	1	2	yes	—
162	C₂×C₄⋊C₄	—	2	1	no	—
163	C₂×C₄⋊C₄	—	38	2	no	—
164	C₂×C₄⋊C₄	—	38	2	no	—
165	C₂×C₄⋊C₄	—	38	2	no	—
166	C₂×C₄⋊C₄	D₇₆	1	1	yes	characteristic
167	C₂×C₄²	—	2	1	no	—
168	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
169	C₄⋊C₈	—	2	4	no	—
170	C₄⋊C₈	—	2	1	no	—
171	C₄⋊C₈	—	38	2	no	—
172	C₄⋊C₈	—	38	2	no	—
173	C₄⋊C₈	C₂×Dic₁₉	1	2	yes	—
174	#103	—	76	4	no	—
175	C₄×C₈	—	2	4	no	—
176	C₄×C₈	C₂×Dic₁₉	1	2	yes	—
177	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
178	C₁₉	#8649	1	1	yes	characteristic, complemented, Sylow
179	C₂×Q₈	—	2	1	no	—
180	C₂×Q₈	—	2	1	no	—
181	C₂×Q₈	—	2	1	no	—
182	C₂×Q₈	D₁₅₂	1	2	yes	—
183	C₂×Q₈	Dic₇₆	1	2	yes	—
184	C₂×Q₈	C₄⋊Dic₁₉	1	2	yes	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	76	2	no	—
188	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
189	C₂²×C₄	#64541	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	76	2	no	—
196	C₂×C₈	—	76	2	no	—
197	C₂×C₈	—	76	2	no	—
198	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
199	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
200	C₄⋊C₄	—	2	1	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	76	2	no	—
204	C₄⋊C₄	—	76	2	no	—
205	C₄⋊C₄	—	76	2	no	—
206	C₄⋊C₄	—	76	2	no	—
207	C₄⋊C₄	—	152	2	no	—
208	C₄⋊C₄	C₄⋊Dic₁₉	1	2	yes	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
212	C₄²	#64541	1	1	yes	characteristic
213	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
214	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
215	C₂³	#10597376	1	1	yes	characteristic
216	Q₈	—	2	2	no	—
217	Q₈	—	2	2	no	—
218	Q₈	—	2	2	no	—
219	Q₈	#10597275	1	4	yes	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	152	2	no	—
227	C₂×C₄	—	152	2	no	—
228	C₂×C₄	—	152	2	no	—
229	C₂×C₄	C₂×D₁₅₂	1	1	yes	characteristic
230	C₂×C₄	#10597350	1	1	yes	characteristic
231	C₂×C₄	#10597350	1	1	yes	characteristic
232	C₂×C₄	C₂×Dic₇₆	1	1	yes	characteristic
233	C₂×C₄	#10597370	1	1	yes	characteristic
234	C₂×C₄	#10597370	1	1	yes	characteristic
235	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
236	C₂×C₄	#10597372	1	1	yes	characteristic
237	C₂×C₄	#10597376	1	1	yes	characteristic
238	C₂×C₄	#10597388	1	2	yes	—
239	C₂×C₄	#10597396	1	2	yes	—
240	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
241	C₈	—	2	2	no	—
242	C₈	—	2	2	no	—
243	C₈	—	4	2	no	—
244	C₈	—	152	2	no	—
245	C₂²	#11926404	1	1	yes	characteristic
246	C₂²	#11926766	1	2	yes	—
247	C₂²	#11926780	1	1	yes	characteristic
248	C₂²	#11926785	1	1	yes	characteristic
249	C₂²	#11926807	1	2	yes	—
250	C₄	—	2	4	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	4	2	no	—
254	C₄	—	304	2	no	—
255	C₄	#11926540	1	1	yes	characteristic
256	C₄	#11926553	1	1	yes	characteristic
257	C₄	C₂×#10597275	1	1	yes	characteristic
258	C₄	#11926582	1	1	yes	characteristic
259	C₄	#11926780	1	1	yes	characteristic
260	C₄	#11926785	1	1	yes	characteristic
261	C₄	#11926851	1	2	yes	—
262	C₂	#14563079	1	2	yes	—
263	C₂	#14563093	1	2	yes	—
264	C₂	#14563650	1	1	yes	characteristic
265	C₂	#14563664	1	1	yes	characteristic
266	C₂	#14563694	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597376
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₈×Q₈×C₁₉	C₂
socle, prime radical	C₂³×C₁₉	#121
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×C₈×Q₈	D₁₉
hypocenter	C₁₉	#8649
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
15. C₈×Q₈×C₁₉	C₄	semidirect
136. C₈×Q₈	Dic₁₉	semidirect
178. C₁₉	#8649	semidirect

Maximal subgroups

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

subgroup	index
#14563763	2
#14563640	2
C₂×#11926297	2
#14561519	2
C₂×C₈×Q₈×C₁₉	2
#8649	19