Subgroups — #18070663

Order 4864.

440 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	#14563778	C₂	1	1	yes	maximal, characteristic
2	#14563778	C₂	1	1	yes	maximal, characteristic
3	#14561458	C₂	1	4	yes	maximal
4	C₄×C₁₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
5	#11926299	—	2	1	no	—
6	#11926299	—	2	1	no	—
7	#11926299	—	2	1	no	—
8	#11926299	—	2	1	no	—
9	#11926299	C₂²	1	2	yes	—
10	#11926299	C₂²	1	2	yes	—
11	C₂×C₁₉×C₄⋊Q₈	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₁₉	—	2	1	no	—
13	C₁₉×#391	—	2	1	no	—
14	C₁₉×#389	—	2	1	no	—
15	C₁₉×#389	C₂²	1	2	yes	—
16	C₁₉×#384	C₄	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₁₉	—	2	1	no	—
18	C₄×C₄⋊C₄×C₁₉	C₄	1	2	yes	—
19	C₄³×C₁₉	C₂²	1	1	yes	characteristic
20	C₂²×Q₈×C₁₉	—	2	1	no	—
21	C₁₉×C₄⋊Q₈	—	2	2	no	—
22	C₁₉×C₄⋊Q₈	C₈	1	4	yes	complemented
23	C₄×Q₈×C₁₉	—	4	4	no	—
24	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
25	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
26	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
27	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
29	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₉	—	2	2	no	—
31	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
38	C₂²×C₁₉⋊C₈	—	2	2	no	—
39	C₂²×C₁₉⋊C₈	—	2	2	no	—
40	C₁₉×#95	—	2	2	no	—
41	C₁₉×#95	—	2	2	no	—
42	C₂×Q₈×C₁₉	—	2	4	no	—
43	C₂×Q₈×C₁₉	—	4	2	no	—
44	C₂×Q₈×C₁₉	—	4	2	no	—
45	C₂²×C₄×C₁₉	—	2	2	no	—
46	C₂²×C₄×C₁₉	—	2	2	no	—
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	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
54	C₄⋊C₄×C₁₉	—	2	4	no	—
55	C₄⋊C₄×C₁₉	—	2	2	no	—
56	C₄⋊C₄×C₁₉	—	4	2	no	—
57	C₄⋊C₄×C₁₉	—	4	2	no	—
58	C₄⋊C₄×C₁₉	—	4	2	no	—
59	C₄⋊C₄×C₁₉	—	4	4	no	—
60	C₄⋊C₄×C₁₉	C₂×C₈	1	4	yes	—
61	C₄²×C₁₉	—	2	2	no	—
62	C₄²×C₁₉	—	2	2	no	—
63	C₄²×C₁₉	—	2	2	no	—
64	C₄²×C₁₉	—	4	1	no	—
65	C₄²×C₁₉	—	4	1	no	—
66	C₄²×C₁₉	—	4	1	no	—
67	C₄²×C₁₉	—	4	4	no	—
68	C₄²×C₁₉	C₂×C₈	1	2	yes	—
69	C₄²×C₁₉	M₁₆	1	2	yes	—
70	C₂×C₁₉⋊C₈	—	4	2	no	—
71	C₂×C₁₉⋊C₈	—	4	2	no	—
72	C₂×C₁₉⋊C₈	—	4	2	no	—
73	C₂×C₁₉⋊C₈	—	4	2	no	—
74	C₂×C₁₉⋊C₈	—	4	2	no	—
75	C₂×C₁₉⋊C₈	—	4	2	no	—
76	#7604	—	19	1	no	maximal, Sylow
77	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	Q₈×C₁₉	—	4	8	no	—
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	4	no	—
92	C₂×C₄×C₁₉	—	2	1	no	—
93	C₂×C₄×C₁₉	—	2	1	no	—
94	C₂×C₄×C₁₉	—	2	1	no	—
95	C₂×C₄×C₁₉	—	4	2	no	—
96	C₂×C₄×C₁₉	—	4	2	no	—
97	C₂×C₄×C₁₉	—	4	2	no	—
98	C₂×C₄×C₁₉	—	4	2	no	—
99	C₂×C₄×C₁₉	—	4	2	no	—
100	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
101	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
102	C₂×C₄×C₁₉	#101	1	2	yes	—
103	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
105	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
106	C₂×C₄×C₁₉	#137	1	2	yes	—
107	C₁₉⋊C₈	—	8	2	no	—
108	C₁₉⋊C₈	—	8	2	no	—
109	C₄×C₄⋊Q₈	D₁₉	1	1	yes	characteristic
110	#1839	—	19	1	no	—
111	#1839	—	19	1	no	—
112	#1271	—	19	4	no	—
113	C₂²×C₁₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
114	C₂²×C₁₉	#410	1	1	yes	characteristic
115	C₂²×C₁₉	C₂×#101	1	1	yes	characteristic
116	C₂²×C₁₉	#418	1	1	yes	characteristic
117	C₂²×C₁₉	#419	1	1	yes	characteristic
118	C₂²×C₁₉	#421	1	1	yes	characteristic
119	C₂²×C₁₉	#421	1	1	yes	characteristic
120	C₄×C₁₉	—	2	2	no	—
121	C₄×C₁₉	—	2	1	no	—
122	C₄×C₁₉	—	2	1	no	—
123	C₄×C₁₉	—	2	2	no	—
124	C₄×C₁₉	—	2	2	no	—
125	C₄×C₁₉	—	2	2	no	—
126	C₄×C₁₉	—	2	2	no	—
127	C₄×C₁₉	—	4	1	no	—
128	C₄×C₁₉	—	4	4	no	—
129	C₄×C₁₉	—	8	2	no	—
130	C₂×C₄⋊Q₈	Dic₁₉	1	1	yes	characteristic
131	C₂×C₄×Q₈	—	2	1	no	—
132	#391	—	2	1	no	—
133	#389	—	2	1	no	—
134	#389	C₂×D₁₉	1	2	yes	—
135	#384	Dic₁₉	1	1	yes	characteristic
136	C₄×C₄⋊C₄	—	2	1	no	—
137	C₄×C₄⋊C₄	Dic₁₉	1	2	yes	—
138	C₄³	C₂×D₁₉	1	1	yes	characteristic
139	#336	—	19	2	no	—
140	#336	—	19	2	no	—
141	#336	—	38	1	no	—
142	#336	—	38	1	no	—
143	#336	—	38	1	no	—
144	#336	—	38	1	no	—
145	C₂×C₁₉	#1461	1	1	yes	characteristic
146	C₂×C₁₉	#1478	1	1	yes	characteristic
147	C₂×C₁₉	#1479	1	1	yes	characteristic
148	C₂×C₁₉	#1514	1	1	yes	characteristic
149	C₂×C₁₉	#1515	1	1	yes	characteristic
150	C₂×C₁₉	#1537	1	1	yes	characteristic
151	C₂×C₁₉	#1556	1	1	yes	characteristic
152	C₂²×Q₈	—	2	1	no	—
153	C₂²×C₈	—	38	2	no	—
154	C₂²×C₈	—	38	2	no	—
155	C₄⋊Q₈	—	2	2	no	—
156	C₄⋊Q₈	C₁₉⋊C₈	1	4	yes	complemented
157	C₄×Q₈	—	4	4	no	—
158	C₂×C₄⋊C₄	—	2	1	no	—
159	C₂×C₄⋊C₄	—	2	1	no	—
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
163	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
164	C₂×C₄²	—	2	2	no	—
165	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
166	C₂×C₄²	C₂×Dic₁₉	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	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
171	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
172	#95	—	2	2	no	—
173	#95	—	2	2	no	—
174	C₁₉	#7604	1	1	yes	characteristic, complemented, Sylow
175	C₂×Q₈	—	2	4	no	—
176	C₂×Q₈	—	4	2	no	—
177	C₂×Q₈	—	4	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
181	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
182	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
183	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
184	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
187	C₂×C₈	—	76	2	no	—
188	C₂×C₈	—	76	2	no	—
189	C₂×C₈	—	76	2	no	—
190	C₂×C₈	—	76	2	no	—
191	C₂×C₈	—	76	2	no	—
192	C₂×C₈	—	76	2	no	—
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	4	no	—
199	C₄⋊C₄	C₂×C₁₉⋊C₈	1	4	yes	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	4	1	no	—
204	C₄²	—	4	1	no	—
205	C₄²	—	4	1	no	—
206	C₄²	—	4	4	no	—
207	C₄²	C₂×C₁₉⋊C₈	1	2	yes	—
208	C₄²	#64520	1	2	yes	—
209	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
210	Q₈	—	4	8	no	—
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	—	2	1	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
233	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
234	C₂×C₄	#10597293	1	2	yes	—
235	C₂×C₄	C₂²×C₁₉⋊C₈	1	1	yes	characteristic
236	C₂×C₄	C₂×#64520	1	1	yes	characteristic
237	C₂×C₄	#10597378	1	2	yes	—
238	C₂×C₄	#10597388	1	2	yes	—
239	C₈	—	152	2	no	—
240	C₈	—	152	2	no	—
241	C₂²	C₂×C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
242	C₂²	#11926518	1	1	yes	characteristic
243	C₂²	C₂×#10597293	1	1	yes	characteristic
244	C₂²	#11926549	1	1	yes	characteristic
245	C₂²	#11926553	1	1	yes	characteristic
246	C₂²	#11926561	1	1	yes	characteristic
247	C₂²	#11926561	1	1	yes	characteristic
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	4	1	no	—
256	C₄	—	4	4	no	—
257	C₄	—	8	2	no	—
258	C₂	#14561959	1	1	yes	characteristic
259	C₂	#14562026	1	1	yes	characteristic
260	C₂	#14562030	1	1	yes	characteristic
261	C₂	#14562182	1	1	yes	characteristic
262	C₂	#14562186	1	1	yes	characteristic
263	C₂	#14562273	1	1	yes	characteristic
264	C₂	#14562352	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₉
commutator	C₂×C₄×C₁₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₄×C₁₉×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄×C₄⋊Q₈	D₁₉
hypocenter	C₁₉	#7604
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
22. C₁₉×C₄⋊Q₈	C₈	semidirect
156. C₄⋊Q₈	C₁₉⋊C₈	semidirect
174. C₁₉	#7604	semidirect

Maximal subgroups

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

subgroup	index
#14563778	2
#14561458	2
C₄×C₁₉×C₄⋊Q₈	2
#7604	19