Subgroups — #18075723

Order 4864.

408 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10597276	C₂	1	1	yes	maximal, characteristic, complemented
2	#14562391	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562074	C₂	1	2	yes	maximal
4	#14561526	C₂	1	2	yes	maximal, complemented
5	C₈×#41×C₁₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597276	—	2	1	no	—
7	#11926568	C₂²	1	2	yes	—
8	C₄×#64520	C₂²	1	2	yes	—
9	#11926309	—	2	1	no	—
10	#11926295	—	2	1	no	—
11	#11926295	—	2	1	no	—
12	Q₈⋊(C₁₉⋊C₈)	—	2	2	no	—
13	D₄⋊(C₁₉⋊C₈)	—	2	2	no	—
14	C₄×#41×C₁₉	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₁₉	C₄	1	1	yes	characteristic
16	C₈×Q₈×C₁₉	C₄	1	1	yes	characteristic
17	C₈×D₄×C₁₉	—	2	1	no	—
18	C₈×D₄×C₁₉	—	2	1	no	—
19	C₈×D₄×C₁₉	C₄	1	1	yes	characteristic
20	C₈×D₄×C₁₉	C₄	1	1	yes	characteristic
21	C₁₉×#431	—	2	1	no	—
22	C₁₉×#431	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₁₉	—	2	1	no	—
24	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
25	C₂×#41×C₁₉	Q₈	1	1	yes	characteristic
26	C₂²×C₈×C₁₉	—	2	1	no	—
27	C₂²×C₈×C₁₉	—	2	1	no	—
28	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
30	C₄×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₄×Q₈×C₁₉	D₄	1	1	yes	characteristic
32	C₄×D₄×C₁₉	—	2	1	no	—
33	C₄×D₄×C₁₉	—	2	1	no	—
34	C₄×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
35	C₄×D₄×C₁₉	D₄	1	1	yes	characteristic
36	C₁₉×#117	—	2	1	no	—
37	C₁₉×#117	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₁₉	—	2	1	no	—
39	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
40	C₂×#64520	—	2	2	no	—
41	C₂×#64520	—	2	2	no	—
42	C₁₉×C₄⋊C₈	—	2	1	no	—
43	C₁₉×C₄⋊C₈	—	2	1	no	—
44	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₁₉×C₂²⋊C₈	—	2	1	no	—
47	C₁₉×C₂²⋊C₈	—	2	1	no	—
48	C₁₉×C₂²⋊C₈	—	2	1	no	—
49	C₄×C₈×C₁₉	—	2	1	no	—
50	C₄×C₈×C₁₉	—	2	1	no	—
51	C₄×C₈×C₁₉	—	2	1	no	—
52	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
55	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
56	#10597276	—	2	4	no	—
57	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
58	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
59	#10597260	—	2	2	no	—
60	C₄×C₁₉⋊C₈	—	2	2	no	—
61	#41×C₁₉	—	2	2	no	—
62	#41×C₁₉	—	2	2	no	—
63	C₂×Q₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
64	C₂×D₄×C₁₉	—	2	1	no	—
65	C₂×D₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	—	2	1	no	—
67	C₂²×C₄×C₁₉	—	2	1	no	—
68	C₂²×C₄×C₁₉	—	2	1	no	—
69	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
72	C₂×C₈×C₁₉	—	2	2	no	—
73	C₂×C₈×C₁₉	—	2	1	no	—
74	C₂×C₈×C₁₉	—	2	1	no	—
75	C₂×C₈×C₁₉	—	4	1	no	—
76	C₂×C₈×C₁₉	—	4	2	no	—
77	C₂×C₈×C₁₉	—	4	2	no	—
78	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
82	C₄⋊C₄×C₁₉	—	2	1	no	—
83	C₄⋊C₄×C₁₉	—	2	1	no	—
84	C₄⋊C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
85	C₄⋊C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
86	C₂²⋊C₄×C₁₉	—	2	1	no	—
87	C₂²⋊C₄×C₁₉	—	2	1	no	—
88	C₂²⋊C₄×C₁₉	—	2	1	no	—
89	C₄²×C₁₉	—	2	1	no	—
90	C₄²×C₁₉	—	2	1	no	—
91	C₄²×C₁₉	—	2	1	no	—
92	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₁₉	#41	1	1	yes	characteristic
96	#64520	—	2	4	no	—
97	#64520	—	4	2	no	—
98	#64520	—	8	2	no	—
99	C₂×C₁₉⋊C₈	—	4	2	no	—
100	C₂×C₁₉⋊C₈	—	4	2	no	—
101	#8670	—	19	1	no	maximal, Sylow
102	C₂³×C₁₉	—	2	1	no	—
103	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
104	Q₈×C₁₉	—	2	1	no	—
105	Q₈×C₁₉	#108	1	2	yes	complemented
106	D₄×C₁₉	—	2	4	no	—
107	D₄×C₁₉	—	2	1	no	—
108	D₄×C₁₉	#108	1	2	yes	complemented
109	C₂×C₄×C₁₉	—	2	4	no	—
110	C₂×C₄×C₁₉	—	2	1	no	—
111	C₂×C₄×C₁₉	—	2	1	no	—
112	C₂×C₄×C₁₉	—	2	1	no	—
113	C₂×C₄×C₁₉	—	2	1	no	—
114	C₂×C₄×C₁₉	—	2	1	no	—
115	C₂×C₄×C₁₉	—	2	2	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	2	1	no	—
118	C₂×C₄×C₁₉	—	4	1	no	—
119	C₂×C₄×C₁₉	—	4	1	no	—
120	C₂×C₄×C₁₉	—	4	2	no	—
121	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
124	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
125	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
127	C₈×C₁₉	—	2	1	no	—
128	C₈×C₁₉	—	2	1	no	—
129	C₈×C₁₉	—	2	1	no	—
130	C₈×C₁₉	—	4	1	no	—
131	C₈×C₁₉	—	4	1	no	—
132	C₈×C₁₉	#104	1	2	yes	—
133	C₁₉⋊C₈	—	4	4	no	—
134	C₁₉⋊C₈	—	8	2	no	—
135	C₈×#41	D₁₉	1	1	yes	characteristic
136	C₄×#108	—	19	1	no	—
137	#1566	—	19	1	no	—
138	#1489	—	19	2	no	—
139	#1299	—	19	2	no	—
140	C₂²×C₁₉	—	2	1	no	—
141	C₂²×C₁₉	—	2	1	no	—
142	C₂²×C₁₉	—	2	2	no	—
143	C₂²×C₁₉	—	2	1	no	—
144	C₂²×C₁₉	—	4	1	no	—
145	C₂²×C₁₉	—	4	1	no	—
146	C₂²×C₁₉	#380	1	1	yes	characteristic
147	C₄×C₁₉	—	2	1	no	—
148	C₄×C₁₉	—	2	1	no	—
149	C₄×C₁₉	—	2	2	no	—
150	C₄×C₁₉	—	2	1	no	—
151	C₄×C₁₉	—	4	1	no	—
152	C₄×C₁₉	—	4	1	no	—
153	C₄×C₁₉	C₂×#104	1	1	yes	characteristic
154	C₄×C₁₉	#421	1	1	yes	characteristic
155	C₄×C₁₉	C₂×#108	1	1	yes	characteristic
156	C₄×C₁₉	C₂×#108	1	1	yes	characteristic
157	C₄×C₁₉	#471	1	2	yes	—
158	C₄×#41	C₂×D₁₉	1	1	yes	characteristic
159	C₈×Q₈	Dic₁₉	1	1	yes	characteristic
160	C₈×Q₈	Dic₁₉	1	1	yes	characteristic
161	C₈×D₄	—	2	1	no	—
162	C₈×D₄	—	2	1	no	—
163	C₈×D₄	Dic₁₉	1	1	yes	characteristic
164	C₈×D₄	Dic₁₉	1	1	yes	characteristic
165	#431	—	2	1	no	—
166	#431	C₂×D₁₉	1	1	yes	characteristic
167	C₂×#108	—	38	1	no	—
168	#423	—	19	2	no	—
169	C₄×M₁₆	—	19	2	no	—
170	C₂×C₄×C₈	—	2	1	no	—
171	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
172	#341	—	38	1	no	—
173	#334	—	38	1	no	—
174	#334	—	38	1	no	—
175	Q₈⋊C₈	—	38	2	no	—
176	D₄⋊C₈	—	38	2	no	—
177	C₂×C₁₉	—	4	1	no	—
178	C₂×C₁₉	—	4	2	no	—
179	C₂×C₁₉	#1806	1	1	yes	characteristic
180	C₂×C₁₉	#1810	1	1	yes	characteristic
181	C₂×C₁₉	#1815	1	1	yes	characteristic
182	C₂×#41	Dic₃₈	1	1	yes	characteristic
183	C₂×M₁₆	—	38	2	no	—
184	C₂×M₁₆	—	38	2	no	—
185	C₂²×C₈	—	2	1	no	—
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
188	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
189	C₄×Q₈	D₇₆	1	1	yes	characteristic
190	C₄×Q₈	C₂×Dic₁₉	1	1	yes	characteristic
191	C₄×D₄	—	2	1	no	—
192	C₄×D₄	—	2	1	no	—
193	C₄×D₄	D₇₆	1	1	yes	characteristic
194	C₄×D₄	C₂×Dic₁₉	1	1	yes	characteristic
195	#117	—	2	1	no	—
196	#117	Dic₃₈	1	1	yes	characteristic
197	C₂×C₄²	—	2	1	no	—
198	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
199	#108	—	38	4	no	—
200	C₄⋊C₈	—	2	1	no	—
201	C₄⋊C₈	—	2	1	no	—
202	C₄⋊C₈	—	38	2	no	—
203	C₄⋊C₈	—	38	2	no	—
204	C₄⋊C₈	C₂×Dic₁₉	1	1	yes	characteristic
205	C₄⋊C₈	C₂×Dic₁₉	1	1	yes	characteristic
206	C₂²⋊C₈	—	2	1	no	—
207	C₂²⋊C₈	—	2	1	no	—
208	C₂²⋊C₈	—	2	1	no	—
209	#97	—	38	2	no	—
210	C₄×C₈	—	2	1	no	—
211	C₄×C₈	—	2	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	38	2	no	—
214	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
215	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
216	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
217	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
218	C₁₉	#8670	1	1	yes	characteristic, complemented, Sylow
219	#41	—	2	2	no	—
220	#41	—	2	2	no	—
221	C₂×Q₈	C₄⋊Dic₁₉	1	1	yes	characteristic
222	C₂×D₄	—	2	1	no	—
223	C₂×D₄	C₄⋊Dic₁₉	1	1	yes	characteristic
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
228	C₂²×C₄	#64541	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
230	M₁₆	—	38	4	no	—
231	M₁₆	—	76	2	no	—
232	M₁₆	—	152	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	76	2	no	—
240	C₂×C₈	—	76	2	no	—
241	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
242	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
243	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
244	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
248	C₄⋊C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
249	C₂²⋊C₄	—	2	1	no	—
250	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	2	1	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₄²	C₂×D₇₆	1	1	yes	characteristic
256	C₄²	#64541	1	1	yes	characteristic
257	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
258	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
259	C₂³	—	2	1	no	—
260	C₂³	#10597370	1	1	yes	characteristic
261	Q₈	—	2	1	no	—
262	Q₈	#10597276	1	2	yes	complemented
263	D₄	—	2	4	no	—
264	D₄	—	2	1	no	—
265	D₄	#10597276	1	2	yes	complemented
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	#10597370	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
280	C₂×C₄	#10597372	1	1	yes	characteristic
281	C₂×C₄	#10597376	1	1	yes	characteristic
282	C₂×C₄	#10597376	1	1	yes	characteristic
283	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	—	4	1	no	—
288	C₈	—	4	1	no	—
289	C₈	—	76	4	no	—
290	C₈	—	152	2	no	—
291	C₈	#10597294	1	2	yes	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	1	no	—
298	C₂²	#11926404	1	1	yes	characteristic
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	2	no	—
302	C₄	—	2	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	C₂×#10597294	1	1	yes	characteristic
306	C₄	#11926561	1	1	yes	characteristic
307	C₄	C₂×#10597276	1	1	yes	characteristic
308	C₄	C₂×#10597276	1	1	yes	characteristic
309	C₄	#11926807	1	2	yes	—
310	C₂	—	4	1	no	—
311	C₂	—	4	2	no	—
312	C₂	#14563655	1	1	yes	characteristic
313	C₂	#14563673	1	1	yes	characteristic
314	C₂	#14563694	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597376
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₈×#41×C₁₉	C₂
socle	C₂²×C₁₉	#380
prime radical	C₂×#41×C₁₉	Q₈
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₈×#41	D₁₉
hypocenter	C₁₉	#8670
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
1. C₄×#10597276	C₂	semidirect
2. #14562391	C₂	semidirect
4. #14561526	C₂	semidirect
105. Q₈×C₁₉	#108	semidirect
108. D₄×C₁₉	#108	semidirect
218. C₁₉	#8670	semidirect
262. Q₈	#10597276	semidirect
265. D₄	#10597276	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10597276	2
#14562391	2
#14562074	2
#14561526	2
C₈×#41×C₁₉	2
#8670	19