Subgroups — #18082410

Order 4864.

516 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14574194	C₂	1	1	yes	maximal, characteristic
2	#14564612	C₂	1	1	yes	maximal, characteristic, complemented
3	#14564463	C₂	1	1	yes	maximal, characteristic
4	#14562391	C₂	1	1	yes	maximal, characteristic, complemented
5	#14562091	C₂	1	1	yes	maximal, characteristic, complemented
6	#14562087	C₂	1	1	yes	maximal, characteristic
7	C₄²×C₈×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#11927101	C₂²	1	1	yes	characteristic
9	#11927094	C₂²	1	1	yes	characteristic
10	#11926942	C₄	1	1	yes	characteristic, complemented
11	#11926917	C₄	1	1	yes	characteristic, complemented
12	#11926895	C₄	1	1	yes	characteristic, complemented
13	#11926891	C₄	1	1	yes	characteristic, complemented
14	#11926773	—	2	2	no	—
15	C₄×#64516	—	2	2	no	—
16	#11926568	C₂²	1	1	yes	characteristic
17	#11926568	C₂²	1	1	yes	characteristic
18	C₂×#10597262	—	2	2	no	—
19	#11926304	—	2	2	no	—
20	#11926295	—	2	2	no	—
21	#11926276	—	2	2	no	—
22	#11926273	—	2	2	no	—
23	C₂×C₄×C₈×C₁₉	—	2	2	no	—
24	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
26	C₄³×C₁₉	C₂²	1	1	yes	characteristic
27	C₂×#64541	—	2	1	no	—
28	C₂²×C₈×C₁₉	—	2	1	no	—
29	C₂²×C₈×C₁₉	D₄	1	2	yes	—
30	C₂×C₄²×C₁₉	—	2	1	no	—
31	C₂×C₄²×C₁₉	—	2	1	no	—
32	C₂×C₄²×C₁₉	D₄	1	2	yes	—
33	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
34	#10597376	—	2	1	no	—
35	#10597376	—	2	1	no	—
36	#10597372	—	2	1	no	—
37	#10597370	—	2	1	no	—
38	#10597370	—	2	1	no	—
39	C₂×#64520	—	2	1	no	—
40	C₂×#64520	—	2	1	no	—
41	C₂×#64516	—	2	2	no	—
42	#10597321	—	2	1	no	—
43	#10597320	C₂×C₄	1	1	yes	characteristic
44	C₄×D₇₆	—	2	1	no	—
45	C₄×D₇₆	—	2	1	no	—
46	#10597316	—	2	1	no	—
47	#10597315	C₂×C₄	1	1	yes	characteristic
48	#10597315	C₂×C₄	1	1	yes	characteristic
49	#10597315	C₂×C₄	1	1	yes	characteristic
50	C₄×Dic₃₈	—	2	1	no	—
51	C₄×Dic₃₈	—	2	1	no	—
52	C₄×C₈×C₁₉	—	2	8	no	—
53	C₄×C₈×C₁₉	—	2	1	no	—
54	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₁₉	D₄	1	2	yes	—
57	C₄×C₈×C₁₉	D₄	1	2	yes	—
58	#10597278	—	2	2	no	—
59	#10597274	—	2	2	no	—
60	#10597273	—	2	2	no	—
61	#10597262	—	4	4	no	—
62	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
63	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
64	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
65	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
66	C₂²×C₄×C₁₉	—	2	2	no	—
67	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
70	C₂×C₂²⋊D₁₉	—	4	1	no	—
71	#64541	—	4	2	no	—
72	#64541	—	4	2	no	—
73	C₂×D₇₆	—	2	1	no	—
74	C₂×D₇₆	—	2	1	no	—
75	C₂×D₇₆	—	2	1	no	—
76	C₂×C₄×D₁₉	—	4	1	no	—
77	C₂×Dic₃₈	—	2	1	no	—
78	C₂×Dic₃₈	—	2	1	no	—
79	C₂×Dic₃₈	—	2	1	no	—
80	C₂×C₈×C₁₉	—	2	4	no	—
81	C₂×C₈×C₁₉	—	2	2	no	—
82	C₂×C₈×C₁₉	—	2	4	no	—
83	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
84	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
85	C₄²×C₁₉	—	2	4	no	—
86	C₄²×C₁₉	—	2	4	no	—
87	C₄²×C₁₉	—	2	2	no	—
88	C₄²×C₁₉	SD₁₆	1	4	yes	complemented
89	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₁₉	#41	1	1	yes	characteristic
93	C₂²⋊Dic₁₉	—	4	1	no	—
94	(C₂×D₁₉)⋊C₄	—	4	1	no	—
95	(C₂×D₁₉)⋊C₄	—	4	1	no	—
96	C₄⋊Dic₁₉	—	2	1	no	—
97	C₄⋊Dic₁₉	—	2	1	no	—
98	C₄⋊Dic₁₉	—	2	1	no	—
99	C₄⋊Dic₁₉	—	2	1	no	—
100	C₄⋊Dic₁₉	—	2	1	no	—
101	C₄⋊Dic₁₉	—	2	1	no	—
102	Dic₁₉⋊C₄	—	4	1	no	—
103	Dic₁₉⋊C₄	—	4	1	no	—
104	C₄×Dic₁₉	—	4	1	no	—
105	#64520	—	4	2	no	—
106	#64520	—	4	2	no	—
107	C₂×C₁₉⋊C₈	—	4	1	no	—
108	C₂×C₁₉⋊C₈	—	4	1	no	—
109	#64516	—	4	4	no	—
110	#9843	—	19	1	no	maximal, Sylow
111	C₂³×C₁₉	#121	1	1	yes	characteristic
112	C₂²×D₁₉	—	4	1	no	—
113	C₂×C₄×C₁₉	—	2	4	no	—
114	C₂×C₄×C₁₉	—	2	4	no	—
115	C₂×C₄×C₁₉	—	2	4	no	—
116	C₂×C₄×C₁₉	—	2	2	no	—
117	C₂×C₄×C₁₉	—	2	2	no	—
118	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
122	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
123	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
125	C₂×C₄×C₁₉	#135	1	2	yes	—
126	C₂²⋊D₁₉	—	8	2	no	—
127	C₂×Dic₁₉	—	4	1	no	—
128	C₂×Dic₁₉	—	4	1	no	—
129	C₂×Dic₁₉	—	4	1	no	—
130	D₇₆	—	4	2	no	—
131	D₇₆	—	4	2	no	—
132	D₇₆	—	4	2	no	—
133	C₄×D₁₉	—	8	1	no	—
134	C₄×D₁₉	—	8	1	no	—
135	Dic₃₈	—	4	2	no	—
136	Dic₃₈	—	4	2	no	—
137	Dic₃₈	—	4	2	no	—
138	C₈×C₁₉	—	2	4	no	—
139	C₈×C₁₉	—	2	2	no	—
140	C₈×C₁₉	#104	1	4	yes	complemented
141	C₁₉⋊C₈	—	8	1	no	—
142	C₁₉⋊C₈	—	8	1	no	—
143	#3167	—	19	1	no	—
144	#1994	—	19	1	no	—
145	#1962	—	19	1	no	—
146	C₄²×C₈	D₁₉	1	1	yes	characteristic, complemented
147	#1566	—	19	1	no	—
148	#1493	—	19	1	no	—
149	#1492	—	19	1	no	—
150	C₂²×C₁₉	—	2	1	no	—
151	C₂²×C₁₉	—	2	1	no	—
152	C₂²×C₁₉	#390	1	1	yes	characteristic
153	C₂²×C₁₉	#465	1	2	yes	—
154	C₂×D₁₉	—	8	1	no	—
155	C₂×D₁₉	—	8	1	no	—
156	C₂×D₁₉	—	8	1	no	—
157	C₄×C₁₉	—	2	8	no	—
158	C₄×C₁₉	—	2	1	no	—
159	C₄×C₁₉	—	2	1	no	—
160	C₄×C₁₉	C₂×#104	1	2	yes	—
161	C₄×C₁₉	C₄×SD₁₆	1	2	yes	—
162	C₄×C₁₉	#443	1	2	yes	—
163	C₄×C₁₉	#492	1	1	yes	characteristic
164	C₄×C₁₉	#495	1	1	yes	characteristic
165	Dic₁₉	—	8	1	no	—
166	Dic₁₉	—	8	1	no	—
167	Dic₁₉	—	8	1	no	—
168	#533	—	19	1	no	—
169	#532	—	19	1	no	—
170	#498	—	19	1	no	—
171	#492	—	19	1	no	—
172	#487	—	19	1	no	—
173	#486	—	19	1	no	—
174	#465	—	38	2	no	—
175	C₄×SD₁₆	—	38	2	no	—
176	#423	—	19	1	no	—
177	#423	—	19	1	no	—
178	C₂×#104	—	38	2	no	—
179	C₂×C₄×C₈	—	2	2	no	—
180	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
181	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
182	C₄³	C₂×D₁₉	1	1	yes	characteristic
183	#339	—	38	2	no	—
184	#334	—	38	2	no	—
185	Q₈⋊C₈	—	38	2	no	—
186	D₄⋊C₈	—	38	2	no	—
187	C₂×C₁₉	—	2	2	no	—
188	C₂×C₁₉	#1962	1	1	yes	characteristic
189	C₂×C₁₉	#1968	1	1	yes	characteristic
190	C₂×C₁₉	#1974	1	1	yes	characteristic
191	D₁₉	—	16	1	no	—
192	C₂×#41	—	38	1	no	—
193	C₂×SD₁₆	—	38	2	no	—
194	C₂×M₁₆	—	38	1	no	—
195	C₂×M₁₆	—	38	1	no	—
196	C₂²×C₈	—	2	1	no	—
197	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
198	C₄⋊Q₈	—	19	1	no	—
199	C₄⋊Q₈	—	19	1	no	—
200	C₄⋊Q₈	—	19	1	no	—
201	#127	—	19	1	no	—
202	#125	—	38	1	no	—
203	#124	—	38	1	no	—
204	C₂²⋊Q₈	—	38	1	no	—
205	C₂²⋊Q₈	—	38	1	no	—
206	#121	—	38	1	no	—
207	#121	—	38	1	no	—
208	C₄×Q₈	—	38	1	no	—
209	C₄×Q₈	—	38	1	no	—
210	C₄×D₄	—	38	1	no	—
211	C₄×D₄	—	38	1	no	—
212	#117	—	38	1	no	—
213	C₂×C₄²	—	2	1	no	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	D₇₆	1	2	yes	—
216	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
217	#106	—	38	2	no	—
218	C₄⋊C₈	—	38	1	no	—
219	C₄⋊C₈	—	38	1	no	—
220	C₄⋊C₈	—	38	1	no	—
221	C₄⋊C₈	—	38	1	no	—
222	#104	—	76	4	no	—
223	#103	—	38	2	no	—
224	#102	—	38	2	no	—
225	C₄×C₈	—	2	8	no	—
226	C₄×C₈	—	2	1	no	—
227	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
228	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
229	C₄×C₈	D₇₆	1	2	yes	—
230	C₄×C₈	D₇₆	1	2	yes	—
231	C₁₉	#9843	1	1	yes	characteristic, complemented, Sylow
232	#41	—	76	2	no	—
233	#41	—	76	2	no	—
234	C₂×Q₈	—	38	1	no	—
235	C₂×Q₈	—	38	1	no	—
236	C₂×Q₈	—	38	1	no	—
237	C₂×D₄	—	38	1	no	—
238	C₂×D₄	—	38	1	no	—
239	C₂×D₄	—	38	1	no	—
240	C₂×D₄	—	76	1	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	76	1	no	—
243	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
244	C₂²×C₄	#64541	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
246	SD₁₆	—	76	4	no	—
247	M₁₆	—	76	2	no	—
248	M₁₆	—	76	2	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	76	1	no	—
253	C₂×C₈	—	76	1	no	—
254	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
255	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
256	C₄⋊C₄	—	38	1	no	—
257	C₄⋊C₄	—	38	1	no	—
258	C₄⋊C₄	—	38	1	no	—
259	C₄⋊C₄	—	38	1	no	—
260	C₄⋊C₄	—	38	1	no	—
261	C₄⋊C₄	—	38	1	no	—
262	C₄⋊C₄	—	76	1	no	—
263	C₄⋊C₄	—	76	1	no	—
264	C₂²⋊C₄	—	76	1	no	—
265	C₂²⋊C₄	—	76	1	no	—
266	C₂²⋊C₄	—	76	1	no	—
267	C₄²	—	2	4	no	—
268	C₄²	—	2	4	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	76	1	no	—
271	C₄²	#64516	1	4	yes	complemented
272	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
273	C₄²	C₂×D₇₆	1	1	yes	characteristic
274	C₄²	C₂×D₇₆	1	1	yes	characteristic
275	C₄²	#64541	1	1	yes	characteristic
276	C₂³	—	76	1	no	—
277	C₂³	#10597376	1	1	yes	characteristic
278	Q₈	—	76	2	no	—
279	Q₈	—	76	2	no	—
280	Q₈	—	76	2	no	—
281	D₄	—	76	2	no	—
282	D₄	—	76	2	no	—
283	D₄	—	76	2	no	—
284	D₄	—	152	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	76	1	no	—
291	C₂×C₄	—	76	1	no	—
292	C₂×C₄	—	76	1	no	—
293	C₂×C₄	—	152	1	no	—
294	C₂×C₄	—	152	1	no	—
295	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
296	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
297	C₂×C₄	#10597320	1	1	yes	characteristic
298	C₂×C₄	#10597321	1	1	yes	characteristic
299	C₂×C₄	C₂×#64516	1	2	yes	—
300	C₂×C₄	#10597350	1	2	yes	—
301	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
302	C₂×C₄	#10597376	1	1	yes	characteristic
303	C₈	—	2	4	no	—
304	C₈	—	2	2	no	—
305	C₈	—	152	1	no	—
306	C₈	—	152	1	no	—
307	C₈	#10597262	1	4	yes	complemented
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	152	1	no	—
311	C₂²	—	152	1	no	—
312	C₂²	—	152	1	no	—
313	C₂²	#11926446	1	1	yes	characteristic
314	C₂²	#11926773	1	2	yes	—
315	C₄	—	2	8	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	152	1	no	—
319	C₄	—	152	1	no	—
320	C₄	—	152	1	no	—
321	C₄	C₂×#10597262	1	2	yes	—
322	C₄	C₄×#64516	1	2	yes	—
323	C₄	#11926662	1	2	yes	—
324	C₄	#11926917	1	1	yes	characteristic
325	C₄	#11926928	1	1	yes	characteristic
326	C₂	—	2	2	no	—
327	C₂	—	304	1	no	—
328	C₂	#14564463	1	1	yes	characteristic
329	C₂	#14564489	1	1	yes	characteristic
330	C₂	#14564523	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597320
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₄²×C₈×C₁₉	C₂
socle	C₂²×C₁₉	#390
prime radical	#11927094	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄²×C₈	D₁₉
hypocenter	C₁₉	#9843
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
2. #14564612	C₂	semidirect
4. #14562391	C₂	semidirect
5. #14562091	C₂	semidirect
7. C₄²×C₈×C₁₉	C₂	semidirect
10. #11926942	C₄	semidirect
11. #11926917	C₄	semidirect
12. #11926895	C₄	semidirect
13. #11926891	C₄	semidirect
88. C₄²×C₁₉	SD₁₆	semidirect
140. C₈×C₁₉	#104	semidirect
146. C₄²×C₈	D₁₉	semidirect
231. C₁₉	#9843	semidirect
271. C₄²	#64516	semidirect
307. C₈	#10597262	semidirect

Maximal subgroups

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

subgroup	index
#14574194	2
#14564612	2
#14564463	2
#14562391	2
#14562091	2
#14562087	2
C₄²×C₈×C₁₉	2
#9843	19