Subgroups — #18082339

Order 4864.

578 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926891	C₂	1	1	yes	maximal, characteristic
2	#14564596	C₂	1	1	yes	maximal, characteristic, complemented
3	#14564441	C₂	1	1	yes	maximal, characteristic
4	#14564441	C₂	1	1	yes	maximal, characteristic
5	#14564437	C₂	1	1	yes	maximal, characteristic, complemented
6	#14563795	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₉×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10597315	C₂²	1	1	yes	characteristic
9	C₂×#10597320	C₂²	1	1	yes	characteristic
10	#11926891	—	2	2	no	—
11	#11926891	C₄	1	4	yes	complemented
12	C₂×#10597275	—	2	2	no	—
13	C₂×#10597273	—	2	2	no	—
14	C₂×#10597278	—	2	2	no	—
15	C₂×#10597278	—	2	2	no	—
16	C₂×#10597278	—	2	2	no	—
17	#11926414	C₂²	1	1	yes	characteristic
18	#11926414	C₂²	1	1	yes	characteristic
19	#11926307	—	2	2	no	—
20	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
23	C₁₉×#322	—	2	2	no	—
24	C₂²×D₇₆	—	2	1	no	—
25	C₂²×D₇₆	—	2	1	no	—
26	C₂²×D₇₆	—	2	1	no	—
27	C₂²×Dic₃₈	—	2	1	no	—
28	C₂²×C₈×C₁₉	D₄	1	2	yes	—
29	C₂²×C₈×C₁₉	D₄	1	2	yes	—
30	C₂²×C₈×C₁₉	D₄	1	2	yes	—
31	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊Dic₁₉	—	2	1	no	—
33	C₂×C₄⋊Dic₁₉	—	2	1	no	—
34	C₂×C₄⋊Dic₁₉	—	2	1	no	—
35	C₂×C₄⋊Dic₁₉	—	2	1	no	—
36	C₂×C₄⋊Dic₁₉	—	2	1	no	—
37	C₂×C₄⋊Dic₁₉	—	2	1	no	—
38	C₂×C₄⋊Dic₁₉	—	2	1	no	—
39	C₂×C₄⋊Dic₁₉	—	2	1	no	—
40	#10597320	—	2	1	no	—
41	#10597320	—	2	1	no	—
42	#10597320	—	2	1	no	—
43	#10597320	C₂×C₄	1	2	yes	—
44	#10597315	—	2	1	no	—
45	#10597315	—	2	1	no	—
46	#10597315	—	2	1	no	—
47	#10597315	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₁₉	—	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	2	yes	—
53	C₄×C₈×C₁₉	D₄	1	2	yes	—
54	#10597278	—	2	8	no	—
55	#10597278	—	4	4	no	—
56	#10597278	—	4	4	no	—
57	#10597275	—	4	4	no	—
58	#10597273	—	4	4	no	—
59	C₂³×D₁₉	—	4	1	no	—
60	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×Dic₁₉	—	4	1	no	—
64	C₂²×Dic₁₉	—	4	1	no	—
65	C₂²×Dic₁₉	—	4	1	no	—
66	C₂×D₇₆	—	2	2	no	—
67	C₂×D₇₆	—	2	2	no	—
68	C₂×D₇₆	—	2	2	no	—
69	C₂×D₇₆	—	4	2	no	—
70	C₂×D₇₆	—	4	2	no	—
71	C₂×D₇₆	—	4	2	no	—
72	C₂×D₇₆	—	4	2	no	—
73	C₂×D₇₆	—	4	2	no	—
74	C₂×D₇₆	—	4	2	no	—
75	C₂×D₇₆	—	4	1	no	—
76	C₂×D₇₆	—	4	1	no	—
77	C₂×D₇₆	—	4	1	no	—
78	C₂×Dic₃₈	—	2	2	no	—
79	C₂×Dic₃₈	—	4	2	no	—
80	C₂×Dic₃₈	—	4	2	no	—
81	C₂×Dic₃₈	—	4	1	no	—
82	C₂×C₈×C₁₉	—	2	2	no	—
83	C₂×C₈×C₁₉	—	2	2	no	—
84	C₂×C₈×C₁₉	—	2	2	no	—
85	C₂×C₈×C₁₉	—	2	2	no	—
86	C₂×C₈×C₁₉	—	2	2	no	—
87	C₂×C₈×C₁₉	—	2	2	no	—
88	C₂×C₈×C₁₉	—	2	2	no	—
89	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
90	C₂×C₈×C₁₉	D₈	1	4	yes	—
91	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₁₉	#41	1	1	yes	characteristic
94	C₄²×C₁₉	#41	1	1	yes	characteristic
95	C₄⋊Dic₁₉	—	2	2	no	—
96	C₄⋊Dic₁₉	—	2	2	no	—
97	C₄⋊Dic₁₉	—	4	1	no	—
98	C₄⋊Dic₁₉	—	4	1	no	—
99	C₄⋊Dic₁₉	—	4	2	no	—
100	C₄⋊Dic₁₉	—	4	2	no	—
101	C₄⋊Dic₁₉	—	4	2	no	—
102	C₄⋊Dic₁₉	—	4	2	no	—
103	C₄⋊Dic₁₉	—	4	2	no	—
104	C₄⋊Dic₁₉	—	4	2	no	—
105	#9830	—	19	1	no	maximal, Sylow
106	C₂³×C₁₉	#127	1	1	yes	characteristic
107	C₂²×D₁₉	—	4	2	no	—
108	C₂²×D₁₉	—	8	1	no	—
109	C₂²×D₁₉	—	8	1	no	—
110	C₂²×D₁₉	—	8	1	no	—
111	C₂²×D₁₉	—	8	1	no	—
112	C₂²×D₁₉	—	8	1	no	—
113	C₂²×D₁₉	—	8	1	no	—
114	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
118	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
119	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₁₉	#135	1	2	yes	—
122	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
123	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
124	C₂×C₄×C₁₉	#137	1	2	yes	—
125	C₂×C₄×C₁₉	#137	1	2	yes	—
126	C₂×Dic₁₉	—	4	2	no	—
127	C₂×Dic₁₉	—	8	1	no	—
128	C₂×Dic₁₉	—	8	1	no	—
129	C₂×Dic₁₉	—	8	1	no	—
130	C₂×Dic₁₉	—	8	1	no	—
131	C₂×Dic₁₉	—	8	1	no	—
132	C₂×Dic₁₉	—	8	1	no	—
133	C₂×Dic₁₉	—	8	1	no	—
134	C₂×Dic₁₉	—	8	1	no	—
135	D₇₆	—	4	4	no	—
136	D₇₆	—	4	4	no	—
137	D₇₆	—	4	4	no	—
138	D₇₆	—	8	2	no	—
139	D₇₆	—	8	2	no	—
140	D₇₆	—	8	2	no	—
141	Dic₃₈	—	4	4	no	—
142	Dic₃₈	—	8	2	no	—
143	C₈×C₁₉	—	2	4	no	—
144	C₈×C₁₉	—	2	4	no	—
145	C₈×C₁₉	—	4	2	no	—
146	C₂×#486	—	19	1	no	—
147	#1990	—	19	1	no	—
148	#1957	—	19	1	no	—
149	#1957	—	19	1	no	—
150	#1956	—	19	1	no	—
151	#1845	—	19	1	no	—
152	C₂×#322	D₁₉	1	1	yes	characteristic, complemented
153	C₂²×C₁₉	#390	1	1	yes	characteristic
154	C₂²×C₁₉	#493	1	1	yes	characteristic
155	C₂²×C₁₉	#495	1	1	yes	characteristic
156	C₂²×C₁₉	#496	1	1	yes	characteristic
157	C₂²×C₁₉	#496	1	1	yes	characteristic
158	C₂²×C₁₉	#497	1	1	yes	characteristic
159	C₂²×C₁₉	#497	1	1	yes	characteristic
160	C₂×D₁₉	—	8	2	no	—
161	C₂×D₁₉	—	8	2	no	—
162	C₂×D₁₉	—	8	2	no	—
163	C₂×D₁₉	—	16	1	no	—
164	C₂×D₁₉	—	16	1	no	—
165	C₂×D₁₉	—	16	1	no	—
166	C₂×D₁₉	—	16	1	no	—
167	C₄×C₁₉	#417	1	2	yes	—
168	C₄×C₁₉	C₄×D₈	1	2	yes	—
169	C₄×C₁₉	#440	1	2	yes	—
170	C₄×C₁₉	#466	1	2	yes	—
171	C₄×C₁₉	#468	1	2	yes	—
172	C₄×C₁₉	#489	1	2	yes	—
173	Dic₁₉	—	8	2	no	—
174	Dic₁₉	—	16	1	no	—
175	Dic₁₉	—	16	1	no	—
176	C₂×C₄⋊Q₈	—	19	1	no	—
177	C₂×#127	—	19	1	no	—
178	#486	—	19	4	no	—
179	#486	—	38	2	no	—
180	C₂×#107	—	38	2	no	—
181	C₂×#103	—	38	2	no	—
182	C₂×#102	—	38	2	no	—
183	C₂×#102	—	38	2	no	—
184	C₂×#102	—	38	2	no	—
185	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
186	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
187	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
188	#384	—	19	1	no	—
189	#384	—	19	1	no	—
190	#340	—	38	2	no	—
191	#322	—	2	2	no	—
192	C₂×C₁₉	#1719	1	1	yes	characteristic
193	C₂×C₁₉	#1720	1	1	yes	characteristic
194	C₂×C₁₉	#1721	1	1	yes	characteristic
195	C₂×C₁₉	#1722	1	1	yes	characteristic
196	C₂×C₁₉	#1959	1	1	yes	characteristic
197	C₂×C₁₉	#1970	1	1	yes	characteristic
198	C₂×C₁₉	#1973	1	1	yes	characteristic
199	D₁₉	—	16	2	no	—
200	C₂²×Q₈	—	38	1	no	—
201	C₂²×D₄	—	38	1	no	—
202	C₂²×D₄	—	38	1	no	—
203	C₂²×D₄	—	38	1	no	—
204	C₂²×C₈	D₇₆	1	2	yes	—
205	C₂²×C₈	D₇₆	1	2	yes	—
206	C₂²×C₈	D₇₆	1	2	yes	—
207	C₄⋊Q₈	—	19	2	no	—
208	C₄⋊Q₈	—	38	1	no	—
209	C₄⋊Q₈	—	38	1	no	—
210	C₄⋊Q₈	—	38	1	no	—
211	#127	—	19	2	no	—
212	#127	—	38	1	no	—
213	#127	—	38	1	no	—
214	#127	—	38	1	no	—
215	C₂×C₄⋊C₄	—	38	1	no	—
216	C₂×C₄⋊C₄	—	38	1	no	—
217	C₂×C₄⋊C₄	—	38	1	no	—
218	C₂×C₄⋊C₄	—	38	1	no	—
219	C₂×C₄⋊C₄	—	38	1	no	—
220	C₂×C₄⋊C₄	—	38	1	no	—
221	C₂×C₄⋊C₄	—	38	1	no	—
222	C₂×C₄⋊C₄	—	38	1	no	—
223	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
224	#107	—	76	4	no	—
225	#103	—	76	4	no	—
226	#102	—	38	8	no	—
227	#102	—	76	4	no	—
228	#102	—	76	4	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	—	2	1	no	—
232	C₄×C₈	—	2	1	no	—
233	C₄×C₈	C₄×D₁₉	1	2	yes	—
234	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
235	C₁₉	#9830	1	1	yes	characteristic, complemented, Sylow
236	C₂⁴	—	76	1	no	—
237	C₂×Q₈	—	38	2	no	—
238	C₂×Q₈	—	76	1	no	—
239	C₂×Q₈	—	76	2	no	—
240	C₂×Q₈	—	76	2	no	—
241	C₂×D₄	—	38	2	no	—
242	C₂×D₄	—	38	2	no	—
243	C₂×D₄	—	38	2	no	—
244	C₂×D₄	—	76	1	no	—
245	C₂×D₄	—	76	1	no	—
246	C₂×D₄	—	76	1	no	—
247	C₂×D₄	—	76	2	no	—
248	C₂×D₄	—	76	2	no	—
249	C₂×D₄	—	76	2	no	—
250	C₂×D₄	—	76	2	no	—
251	C₂×D₄	—	76	2	no	—
252	C₂×D₄	—	76	2	no	—
253	C₂²×C₄	—	76	1	no	—
254	C₂²×C₄	—	76	1	no	—
255	C₂²×C₄	—	76	1	no	—
256	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	D₁₅₂	1	4	yes	—
267	C₂×C₈	(C₂×D₁₉)⋊C₄	1	4	yes	—
268	C₄⋊C₄	—	38	2	no	—
269	C₄⋊C₄	—	38	2	no	—
270	C₄⋊C₄	—	76	1	no	—
271	C₄⋊C₄	—	76	1	no	—
272	C₄⋊C₄	—	76	2	no	—
273	C₄⋊C₄	—	76	2	no	—
274	C₄⋊C₄	—	76	2	no	—
275	C₄⋊C₄	—	76	2	no	—
276	C₄⋊C₄	—	76	2	no	—
277	C₄⋊C₄	—	76	2	no	—
278	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
279	C₄²	#64541	1	1	yes	characteristic
280	C₄²	#64541	1	1	yes	characteristic
281	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
282	C₂³	—	76	2	no	—
283	C₂³	—	152	1	no	—
284	C₂³	—	152	1	no	—
285	C₂³	—	152	1	no	—
286	C₂³	—	152	1	no	—
287	C₂³	—	152	1	no	—
288	C₂³	—	152	1	no	—
289	C₂³	#10597320	1	1	yes	characteristic
290	Q₈	—	76	4	no	—
291	Q₈	—	152	2	no	—
292	D₄	—	76	4	no	—
293	D₄	—	76	4	no	—
294	D₄	—	76	4	no	—
295	D₄	—	152	2	no	—
296	D₄	—	152	2	no	—
297	D₄	—	152	2	no	—
298	C₂×C₄	—	76	2	no	—
299	C₂×C₄	—	152	1	no	—
300	C₂×C₄	—	152	1	no	—
301	C₂×C₄	—	152	1	no	—
302	C₂×C₄	—	152	1	no	—
303	C₂×C₄	—	152	1	no	—
304	C₂×C₄	—	152	1	no	—
305	C₂×C₄	—	152	1	no	—
306	C₂×C₄	—	152	1	no	—
307	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
308	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
309	C₂×C₄	#10597321	1	1	yes	characteristic
310	C₂×C₄	C₂×D₁₅₂	1	2	yes	—
311	C₂×C₄	#10597350	1	2	yes	—
312	C₂×C₄	#10597354	1	2	yes	—
313	C₂×C₄	#10597354	1	2	yes	—
314	C₂×C₄	#10597355	1	2	yes	—
315	C₂×C₄	#10597355	1	2	yes	—
316	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
317	C₂×C₄	#10597376	1	1	yes	characteristic
318	C₂×C₄	#10597376	1	1	yes	characteristic
319	C₈	—	2	4	no	—
320	C₈	—	2	4	no	—
321	C₈	—	4	2	no	—
322	C₂²	—	304	1	no	—
323	C₂²	—	304	1	no	—
324	C₂²	—	304	1	no	—
325	C₂²	—	304	1	no	—
326	C₂²	—	152	2	no	—
327	C₂²	—	152	2	no	—
328	C₂²	—	152	2	no	—
329	C₂²	#11926446	1	1	yes	characteristic
330	C₂²	#11926921	1	1	yes	characteristic
331	C₂²	#11926928	1	1	yes	characteristic
332	C₂²	#11926933	1	1	yes	characteristic
333	C₂²	#11926933	1	1	yes	characteristic
334	C₂²	#11926938	1	1	yes	characteristic
335	C₂²	#11926938	1	1	yes	characteristic
336	C₄	—	304	1	no	—
337	C₄	—	304	1	no	—
338	C₄	—	152	2	no	—
339	C₄	#11926546	1	2	yes	—
340	C₄	C₄×D₁₅₂	1	2	yes	—
341	C₄	#11926646	1	2	yes	—
342	C₄	#11926780	1	2	yes	—
343	C₄	#11926790	1	2	yes	—
344	C₄	#11926904	1	2	yes	—
345	C₂	—	304	2	no	—
346	C₂	#14563288	1	1	yes	characteristic
347	C₂	#14563295	1	1	yes	characteristic
348	C₂	#14563301	1	1	yes	characteristic
349	C₂	#14563306	1	1	yes	characteristic
350	C₂	#14564451	1	1	yes	characteristic
351	C₂	#14564501	1	1	yes	characteristic
352	C₂	#14564516	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597320
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#322	C₂
socle	C₂³×C₁₉	#127
prime radical	C₂×#10597320	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#322	D₁₉
hypocenter	C₁₉	#9830
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. #14564596	C₂	semidirect
5. #14564437	C₂	semidirect
6. #14563795	C₂	semidirect
7. C₂×C₁₉×#322	C₂	semidirect
11. #11926891	C₄	semidirect
152. C₂×#322	D₁₉	semidirect
235. C₁₉	#9830	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926891	2
#14564596	2
#14564441	2
#14564437	2
#14563795	2
C₂×C₁₉×#322	2
#9830	19