Subgroups — #18081434

Order 4864.

586 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₁₅₂	C₂	1	1	yes	maximal, characteristic
2	#14564428	C₂	1	2	yes	maximal, complemented
3	C₂×#11926271	C₂	1	2	yes	maximal, complemented
4	C₂×#11926266	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₉×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₅₂	C₄	1	1	yes	characteristic
7	C₂×C₄×D₇₆	C₂²	1	2	yes	—
8	C₄×D₁₅₂	—	2	2	no	—
9	C₄×D₁₅₂	—	2	2	no	—
10	C₂×#10597275	C₄	1	1	yes	characteristic
11	C₂×#10597278	C₄	1	2	yes	—
12	C₂×(C₂×D₁₉)⋊C₈	—	2	4	no	—
13	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
15	#11926271	—	2	4	no	—
16	#11926266	—	2	2	no	—
17	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
19	C₁₉×#335	—	2	2	no	—
20	C₂²×D₇₆	C₂×C₄	1	2	yes	—
21	C₂²×C₄×D₁₉	—	2	2	no	—
22	C₂²×C₈×C₁₉	—	2	2	no	—
23	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
26	C₂×(C₂×D₁₉)⋊C₄	—	2	2	no	—
27	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
28	C₂²×C₁₉⋊C₈	—	2	1	no	—
29	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
30	C₂×D₁₅₂	—	2	4	no	—
31	C₂×D₁₅₂	—	2	2	no	—
32	C₄×D₇₆	—	2	2	no	—
33	C₄×D₇₆	—	2	2	no	—
34	C₄×D₇₆	—	2	2	no	—
35	C₄×D₇₆	—	2	2	no	—
36	C₁₉×C₄⋊C₈	—	2	2	no	—
37	C₁₉×C₄⋊C₈	—	2	2	no	—
38	C₄×C₈×C₁₉	D₄	1	2	yes	—
39	C₄×C₈×C₁₉	D₄	1	2	yes	—
40	#10597278	—	2	4	no	—
41	(C₂×D₁₉)⋊C₈	—	4	8	no	—
42	#10597275	—	2	2	no	—
43	C₄×C₁₉⋊C₈	—	2	1	no	—
44	C₄×C₁₉⋊C₈	—	2	1	no	—
45	C₄×C₁₉⋊C₈	—	2	1	no	—
46	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
47	C₂³×D₁₉	—	2	2	no	—
48	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
51	C₂²×Dic₁₉	—	2	2	no	—
52	C₂×D₇₆	—	2	2	no	—
53	C₂×D₇₆	—	2	2	no	—
54	C₂×D₇₆	—	2	2	no	—
55	C₂×D₇₆	—	2	2	no	—
56	C₂×D₇₆	—	2	2	no	—
57	C₂×D₇₆	M₁₆	1	4	yes	—
58	C₂×C₄×D₁₉	—	4	2	no	—
59	C₂×C₄×D₁₉	—	4	2	no	—
60	C₂×C₄×D₁₉	—	4	4	no	—
61	C₂×C₄×D₁₉	—	4	4	no	—
62	C₂×C₈×C₁₉	—	2	2	no	—
63	C₂×C₈×C₁₉	—	2	2	no	—
64	C₂×C₈×C₁₉	—	4	2	no	—
65	C₂×C₈×C₁₉	—	4	4	no	—
66	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₁₉	#41	1	1	yes	characteristic
72	(C₂×D₁₉)⋊C₄	—	4	2	no	—
73	(C₂×D₁₉)⋊C₄	—	4	2	no	—
74	C₄⋊Dic₁₉	—	2	2	no	—
75	C₄⋊Dic₁₉	—	2	2	no	—
76	C₂×C₁₉⋊C₈	—	2	2	no	—
77	C₂×C₁₉⋊C₈	—	2	2	no	—
78	C₂×C₁₉⋊C₈	—	2	2	no	—
79	C₂×C₁₉⋊C₈	—	4	2	no	—
80	C₂×C₁₉⋊C₈	D₈	1	4	yes	complemented
81	D₁₅₂	—	2	8	no	—
82	#9663	—	19	1	no	maximal, Sylow
83	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
84	C₂²×D₁₉	—	2	4	no	—
85	C₂²×D₁₉	—	4	2	no	—
86	C₂²×D₁₉	—	4	2	no	—
87	C₂²×D₁₉	—	4	4	no	—
88	C₂²×D₁₉	—	4	4	no	—
89	C₂×C₄×C₁₉	—	2	2	no	—
90	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
93	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
94	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
95	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
96	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
97	C₂×C₄×C₁₉	#131	1	2	yes	—
98	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
99	C₂×C₄×C₁₉	#135	1	2	yes	—
100	C₂×Dic₁₉	—	4	2	no	—
101	C₂×Dic₁₉	—	4	4	no	—
102	D₇₆	—	2	4	no	—
103	D₇₆	—	2	4	no	—
104	D₇₆	—	4	2	no	—
105	D₇₆	—	4	2	no	—
106	C₄×D₁₉	—	8	4	no	—
107	C₄×D₁₉	—	8	2	no	—
108	C₄×D₁₉	—	8	2	no	—
109	C₈×C₁₉	—	2	2	no	—
110	C₈×C₁₉	—	2	2	no	—
111	C₈×C₁₉	—	8	2	no	—
112	C₁₉⋊C₈	—	2	4	no	—
113	C₁₉⋊C₈	—	4	2	no	—
114	C₂×C₄×D₈	—	19	1	no	—
115	#1954	—	19	2	no	—
116	C₂×#335	D₁₉	1	1	yes	characteristic, complemented
117	C₂×D₄⋊C₈	—	19	2	no	—
118	C₂×#322	—	19	1	no	—
119	C₂²×C₁₉	#390	1	1	yes	characteristic
120	C₂²×C₁₉	#435	1	2	yes	—
121	C₂²×C₁₉	C₄×D₈	1	2	yes	—
122	C₂²×C₁₉	#444	1	2	yes	—
123	C₂×D₁₉	—	4	8	no	—
124	C₂×D₁₉	—	4	4	no	—
125	C₂×D₁₉	—	8	4	no	—
126	C₂×D₁₉	—	8	2	no	—
127	C₂×D₁₉	—	8	2	no	—
128	C₄×C₁₉	—	2	1	no	—
129	C₄×C₁₉	—	2	1	no	—
130	C₄×C₁₉	#407	1	1	yes	characteristic
131	C₄×C₁₉	#408	1	1	yes	characteristic
132	C₄×C₁₉	#435	1	2	yes	—
133	C₄×C₁₉	#466	1	2	yes	—
134	C₄×C₁₉	#493	1	1	yes	characteristic
135	C₄×C₁₉	#495	1	1	yes	characteristic
136	Dic₁₉	—	8	2	no	—
137	C₂²×D₈	—	19	1	no	—
138	C₂×C₄×D₄	—	19	2	no	—
139	C₄×D₈	—	38	2	no	—
140	C₄×D₈	—	38	2	no	—
141	C₂×#107	—	19	1	no	—
142	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
143	C₂×#102	—	19	2	no	—
144	C₂×C₂²⋊C₈	—	38	4	no	—
145	C₂×C₄×C₈	—	19	1	no	—
146	C₂×C₄×C₈	—	19	1	no	—
147	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
148	#335	—	2	2	no	—
149	D₄⋊C₈	—	38	4	no	—
150	#322	—	38	2	no	—
151	C₂×C₁₉	#1581	1	4	yes	—
152	C₂×C₁₉	#1955	1	1	yes	characteristic
153	C₂×C₁₉	#1970	1	1	yes	characteristic
154	C₂×C₁₉	#1975	1	1	yes	characteristic
155	D₁₉	—	8	4	no	—
156	C₂²×D₄	—	19	2	no	—
157	C₂³×C₄	—	38	2	no	—
158	C₂×D₈	—	38	4	no	—
159	C₂×D₈	—	38	2	no	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	19	2	no	—
162	C₂²×C₈	—	38	1	no	—
163	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
164	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
165	C₄×D₄	—	38	2	no	—
166	C₄×D₄	—	38	2	no	—
167	C₄×D₄	—	38	2	no	—
168	C₄×D₄	—	38	2	no	—
169	C₂×C₄⋊C₄	—	19	2	no	—
170	C₂×C₂²⋊C₄	—	38	2	no	—
171	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
172	#107	—	38	2	no	—
173	C₄⋊C₈	—	2	2	no	—
174	C₄⋊C₈	—	2	2	no	—
175	#102	—	38	4	no	—
176	C₂²⋊C₈	—	76	8	no	—
177	C₄×C₈	—	19	2	no	—
178	C₄×C₈	—	38	1	no	—
179	C₄×C₈	—	38	1	no	—
180	C₄×C₈	—	38	1	no	—
181	C₄×C₈	D₇₆	1	2	yes	—
182	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
183	C₁₉	#9663	1	1	yes	characteristic, complemented, Sylow
184	C₂⁴	—	38	2	no	—
185	C₂×D₄	—	19	4	no	—
186	C₂×D₄	—	38	2	no	—
187	C₂×D₄	—	38	2	no	—
188	C₂×D₄	—	38	2	no	—
189	C₂×D₄	—	38	2	no	—
190	C₂×D₄	—	38	2	no	—
191	C₂²×C₄	—	38	2	no	—
192	C₂²×C₄	—	76	2	no	—
193	C₂²×C₄	—	76	2	no	—
194	C₂²×C₄	—	76	4	no	—
195	C₂²×C₄	—	76	4	no	—
196	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
197	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
198	C₂²×C₄	#64545	1	1	yes	characteristic
199	D₈	—	38	8	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	—	4	4	no	—
204	C₂×C₈	—	19	4	no	—
205	C₂×C₈	—	38	2	no	—
206	C₂×C₈	—	38	2	no	—
207	C₂×C₈	—	38	2	no	—
208	C₂×C₈	—	76	2	no	—
209	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
210	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
211	C₄⋊C₄	—	38	2	no	—
212	C₄⋊C₄	—	38	2	no	—
213	C₂²⋊C₄	—	76	2	no	—
214	C₂²⋊C₄	—	76	2	no	—
215	C₄²	C₂×D₇₆	1	1	yes	characteristic
216	C₄²	D₄×D₁₉	1	1	yes	characteristic
217	C₄²	#64545	1	1	yes	characteristic
218	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
219	C₂³	—	38	4	no	—
220	C₂³	—	76	2	no	—
221	C₂³	—	76	4	no	—
222	C₂³	—	76	4	no	—
223	C₂³	—	76	2	no	—
224	C₂³	#10597339	1	1	yes	characteristic
225	D₄	—	38	4	no	—
226	D₄	—	38	4	no	—
227	D₄	—	76	2	no	—
228	D₄	—	76	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	76	4	no	—
231	C₂×C₄	—	76	2	no	—
232	C₂×C₄	—	152	2	no	—
233	C₂×C₄	—	152	2	no	—
234	C₂×C₄	—	152	4	no	—
235	C₂×C₄	#10597339	1	1	yes	characteristic
236	C₂×C₄	#10597341	1	1	yes	characteristic
237	C₂×C₄	#10597341	1	1	yes	characteristic
238	C₂×C₄	#10597347	1	2	yes	—
239	C₂×C₄	M₁₆×D₁₉	1	2	yes	—
240	C₂×C₄	D₈×D₁₉	1	2	yes	—
241	C₂×C₄	#10597365	1	2	yes	—
242	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
243	C₂×C₄	#10597386	1	1	yes	characteristic
244	C₂×C₄	#10597393	1	1	yes	characteristic
245	C₈	—	2	2	no	—
246	C₈	—	2	2	no	—
247	C₈	—	8	2	no	—
248	C₈	—	38	4	no	—
249	C₈	—	76	2	no	—
250	C₂²	—	76	4	no	—
251	C₂²	—	76	8	no	—
252	C₂²	—	152	2	no	—
253	C₂²	—	152	4	no	—
254	C₂²	—	152	2	no	—
255	C₂²	#11926445	1	1	yes	characteristic
256	C₂²	#11926614	1	2	yes	—
257	C₂²	#11926626	1	2	yes	—
258	C₂²	#11926665	1	2	yes	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	152	2	no	—
262	C₄	#11926507	1	1	yes	characteristic
263	C₄	#11926510	1	1	yes	characteristic
264	C₄	#11926614	1	2	yes	—
265	C₄	#11926781	1	2	yes	—
266	C₄	#11926919	1	1	yes	characteristic
267	C₄	#11926929	1	1	yes	characteristic
268	C₂	—	152	4	no	—
269	C₂	#14562463	1	4	yes	—
270	C₂	#14564432	1	1	yes	characteristic
271	C₂	#14564499	1	1	yes	characteristic
272	C₂	#14564526	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#335	C₂
socle	C₂³×C₁₉	C₄×D₄
prime radical	C₂²×D₁₅₂	C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#335	D₁₉
hypocenter	C₁₉	#9663
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
2. #14564428	C₂	semidirect
3. C₂×#11926271	C₂	semidirect
4. C₂×#11926266	C₂	semidirect
5. C₂×C₁₉×#335	C₂	semidirect
80. C₂×C₁₉⋊C₈	D₈	semidirect
116. C₂×#335	D₁₉	semidirect
183. C₁₉	#9663	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₁₅₂	2
#14564428	2
C₂×#11926271	2
C₂×#11926266	2
C₂×C₁₉×#335	2
#9663	19