Subgroups — #18072007

Order 4864.

670 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	C₂×#11926601	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#14561469	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₉×#432	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₁₉⋊C₈	C₄	1	2	yes	—
6	#11926601	—	2	2	no	—
7	#11926601	C₄	1	4	yes	—
8	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
10	C₂×#10597260	—	2	1	no	—
11	C₂×C₄×C₁₉⋊C₈	—	2	2	no	—
12	C₂×C₄×C₁₉⋊C₈	—	2	2	no	—
13	C₂×C₄×C₁₉⋊C₈	C₄	1	2	yes	—
14	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
15	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
16	C₁₉×#432	—	2	2	no	—
17	C₁₉×#432	C₄	1	4	yes	—
18	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
19	C₂×C₁₉×C₂²⋊C₈	—	2	1	no	—
20	C₂×C₁₉×#97	—	2	1	no	—
21	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₉	Q₈	1	1	yes	characteristic
24	C₂²×C₈×C₁₉	—	2	2	no	—
25	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₉	D₄	1	2	yes	—
30	C₂×C₄²×C₁₉	D₄	1	2	yes	—
31	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
33	C₂²×C₁₉⋊C₈	—	2	4	no	—
34	C₂²×C₁₉⋊C₈	—	2	8	no	—
35	C₂²×C₁₉⋊C₈	—	2	2	no	—
36	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
37	C₁₉×C₄⋊C₈	—	2	2	no	—
38	C₁₉×C₄⋊C₈	C₂×C₄	1	4	yes	—
39	C₁₉×C₂²⋊C₈	—	2	4	no	—
40	C₁₉×#97	—	2	4	no	—
41	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
42	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
43	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	4	yes	—
44	#10597260	—	2	4	no	—
45	C₄×C₁₉⋊C₈	—	2	4	no	—
46	C₄×C₁₉⋊C₈	—	2	2	no	—
47	C₄×C₁₉⋊C₈	—	2	2	no	—
48	C₂⁴×C₁₉	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	—	2	4	no	—
50	C₂²×C₄×C₁₉	—	2	4	no	—
51	C₂²×C₄×C₁₉	—	2	4	no	—
52	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
53	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₈×C₁₉	—	2	4	no	—
65	C₂×C₈×C₁₉	—	4	4	no	—
66	C₄²×C₁₉	C₄²	1	2	yes	—
67	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
69	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
71	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
72	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₁₉⋊C₈	—	2	16	no	—
77	C₂×C₁₉⋊C₈	—	2	4	no	—
78	C₂×C₁₉⋊C₈	—	2	4	no	—
79	C₂×C₁₉⋊C₈	—	2	4	no	—
80	C₂×C₁₉⋊C₈	—	4	4	no	—
81	C₂×C₁₉⋊C₈	M₁₆	1	8	yes	—
82	#7875	—	19	1	no	maximal, Sylow
83	C₂³×C₁₉	—	2	2	no	—
84	C₂³×C₁₉	—	2	2	no	—
85	C₂³×C₁₉	#95	1	2	yes	—
86	C₂³×C₁₉	#108	1	4	yes	—
87	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₉	—	2	4	no	—
89	C₂×C₄×C₁₉	—	2	4	no	—
90	C₂×C₄×C₁₉	—	2	2	no	—
91	C₂×C₄×C₁₉	—	2	2	no	—
92	C₂×C₄×C₁₉	—	2	2	no	—
93	C₂×C₄×C₁₉	—	2	2	no	—
94	C₂×C₄×C₁₉	—	2	4	no	—
95	C₂×C₄×C₁₉	—	2	4	no	—
96	C₂×C₄×C₁₉	—	2	4	no	—
97	C₂×C₄×C₁₉	—	2	4	no	—
98	C₂×C₄×C₁₉	#95	1	2	yes	—
99	C₂×C₄×C₁₉	#95	1	2	yes	—
100	C₂×C₄×C₁₉	#95	1	2	yes	—
101	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₂×M₁₆	1	4	yes	—
108	C₂×C₄×C₁₉	#135	1	4	yes	—
109	C₈×C₁₉	—	4	4	no	—
110	C₁₉⋊C₈	—	2	8	no	—
111	C₁₉⋊C₈	—	4	4	no	—
112	C₂×#432	—	19	1	no	—
113	C₂×#432	D₁₉	1	1	yes	characteristic
114	C₂²×C₄×C₈	—	19	1	no	—
115	#1277	—	19	4	no	—
116	C₂²×C₁₉	—	2	8	no	—
117	C₂²×C₁₉	—	2	2	no	—
118	C₂²×C₁₉	—	2	2	no	—
119	C₂²×C₁₉	#341	1	4	yes	—
120	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
121	C₂²×C₁₉	#423	1	2	yes	—
122	C₂²×C₁₉	#427	1	2	yes	—
123	C₂²×C₁₉	C₂×#108	1	2	yes	—
124	C₄×C₁₉	—	2	2	no	—
125	C₄×C₁₉	—	2	2	no	—
126	C₄×C₁₉	—	2	2	no	—
127	C₄×C₁₉	—	2	1	no	—
128	C₄×C₁₉	—	2	1	no	—
129	C₄×C₁₉	C₄×M₁₆	1	2	yes	—
130	C₄×C₁₉	#407	1	2	yes	—
131	C₄×C₁₉	#416	1	2	yes	—
132	C₄×C₁₉	#416	1	2	yes	—
133	C₂³×C₈	—	19	2	no	—
134	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
135	#432	—	2	2	no	—
136	#432	—	19	4	no	—
137	#432	—	38	2	no	—
138	#432	Dic₁₉	1	4	yes	—
139	C₂×C₄⋊C₈	—	19	2	no	—
140	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
141	C₂×C₂²⋊C₈	—	2	1	no	—
142	C₂×C₂²⋊C₈	—	38	1	no	—
143	C₂×#97	—	2	1	no	—
144	C₂×#97	—	38	1	no	—
145	C₂×C₄×C₈	—	19	2	no	—
146	C₂×C₄×C₈	—	19	2	no	—
147	C₂×C₄×C₈	—	38	2	no	—
148	C₂×C₄×C₈	—	38	2	no	—
149	C₂×C₁₉	—	2	4	no	—
150	C₂×C₁₉	#1568	1	4	yes	—
151	C₂×C₁₉	#1728	1	1	yes	characteristic
152	C₂×C₁₉	#1735	1	1	yes	characteristic
153	C₂×C₁₉	C₂×#341	1	1	yes	characteristic
154	C₂³×C₄	Dic₃₈	1	1	yes	characteristic
155	C₂³×C₄	C₄×D₁₉	1	1	yes	characteristic
156	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	19	4	no	—
159	C₂²×C₈	—	38	8	no	—
160	C₂²×C₈	—	38	4	no	—
161	C₂²×C₈	—	38	2	no	—
162	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
163	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
164	C₂×C₄²	C₄×D₁₉	1	2	yes	—
165	C₂×C₄²	D₇₆	1	2	yes	—
166	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
167	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
168	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
169	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	19	4	no	—
172	C₄⋊C₈	—	38	2	no	—
173	C₄⋊C₈	C₂×Dic₁₉	1	4	yes	—
174	C₂²⋊C₈	—	2	4	no	—
175	C₂²⋊C₈	—	38	4	no	—
176	#97	—	2	4	no	—
177	#97	—	38	4	no	—
178	C₄×C₈	—	38	4	no	—
179	C₄×C₈	—	38	2	no	—
180	C₄×C₈	—	38	2	no	—
181	C₁₉	#7875	1	1	yes	characteristic, complemented, Sylow
182	C₂⁴	Dic₁₉⋊C₄	1	1	yes	characteristic
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
187	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
188	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
189	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
190	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
191	C₂²×C₄	C₄⋊Dic₁₉	1	2	yes	—
192	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
193	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
194	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
195	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	4	4	no	—
200	C₂×C₈	—	19	8	no	—
201	C₂×C₈	—	38	16	no	—
202	C₂×C₈	—	38	4	no	—
203	C₂×C₈	—	38	4	no	—
204	C₂×C₈	—	38	4	no	—
205	C₂×C₈	—	76	4	no	—
206	C₄²	C₄×Dic₁₉	1	2	yes	—
207	C₄²	Dic₁₉⋊C₄	1	2	yes	—
208	C₄²	C₄⋊Dic₁₉	1	2	yes	—
209	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
210	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
211	C₄²	C₂²⋊Dic₁₉	1	2	yes	—
212	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
213	C₄²	C₂×D₇₆	1	1	yes	characteristic
214	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
215	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
216	C₂³	—	2	2	no	—
217	C₂³	—	2	2	no	—
218	C₂³	#10597279	1	4	yes	—
219	C₂³	C₁₉⋊#95	1	2	yes	—
220	C₂³	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	C₁₉⋊#95	1	2	yes	—
232	C₂×C₄	C₁₉⋊#95	1	2	yes	—
233	C₂×C₄	C₁₉⋊#95	1	2	yes	—
234	C₂×C₄	M₁₆×D₁₉	1	4	yes	—
235	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
236	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
237	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
238	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
239	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
240	C₂×C₄	#10597396	1	4	yes	—
241	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
242	C₈	—	4	4	no	—
243	C₈	—	38	8	no	—
244	C₈	—	76	4	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	8	no	—
248	C₂²	#11926308	1	4	yes	—
249	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
250	C₂²	#11926567	1	2	yes	—
251	C₂²	#11926580	1	2	yes	—
252	C₂²	C₂×#10597279	1	2	yes	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	M₁₆×Dic₁₉	1	2	yes	—
259	C₄	#11926507	1	2	yes	—
260	C₄	#11926538	1	2	yes	—
261	C₄	#11926540	1	2	yes	—
262	C₂	—	2	4	no	—
263	C₂	#14562396	1	4	yes	—
264	C₂	#14563328	1	1	yes	characteristic
265	C₂	#14563351	1	1	yes	characteristic
266	C₂	C₂×#11926308	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#432	C₂
socle	C₂³×C₁₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₉	C₄⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#432	D₁₉
hypocenter	C₁₉	#7875
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
3. #14561469	C₂	semidirect
181. C₁₉	#7875	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926601	2
C₂²×C₄×C₁₉⋊C₈	2
#14561469	2
C₂×C₁₉×#432	2
#7875	19