Subgroups — #18085134

Order 4864.

626 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926923	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11926825	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926821	C₂	1	1	yes	maximal, characteristic
4	#14564428	C₂	1	1	yes	maximal, characteristic
5	C₂×#11926291	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561469	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₉×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#64528	—	2	2	no	—
9	C₂×#10597315	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₇₆	C₂²	1	1	yes	characteristic
11	#11926923	—	2	2	no	—
12	#11926923	—	2	2	no	—
13	#11926825	—	2	2	no	—
14	#11926825	—	2	2	no	—
15	#11926821	—	2	2	no	—
16	#11926821	—	2	2	no	—
17	C₂×#10597275	—	2	1	no	—
18	C₂×#10597264	—	2	1	no	—
19	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
20	C₂×#10597278	—	2	1	no	—
21	C₂×#10597266	—	2	1	no	—
22	C₂×(C₂×D₁₉)⋊C₈	—	2	2	no	—
23	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
24	#11926291	—	2	2	no	—
25	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
26	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₁₉×#332	—	2	2	no	—
29	C₂²×Q₈×C₁₉	—	2	1	no	—
30	C₂²×D₇₆	D₄	1	1	yes	characteristic
31	C₂²×C₄×D₁₉	—	2	1	no	—
32	C₂²×Dic₃₈	—	2	1	no	—
33	C₂²×C₈×C₁₉	—	2	1	no	—
34	C₂²×C₈×C₁₉	—	2	1	no	—
35	C₁₉×C₄⋊Q₈	—	2	1	no	—
36	C₁₉×C₄⋊Q₈	—	2	1	no	—
37	C₁₉×C₄⋊Q₈	—	2	1	no	—
38	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
40	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
41	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
42	C₂×#64528	—	2	8	no	—
43	C₂×#64528	—	4	4	no	—
44	C₂×#64528	—	4	4	no	—
45	C₂×(C₂×D₁₉)⋊C₄	—	2	1	no	—
46	C₂×C₄⋊Dic₁₉	—	2	1	no	—
47	C₂×C₄⋊Dic₁₉	—	2	1	no	—
48	C₂×C₄⋊Dic₁₉	D₄	1	1	yes	characteristic
49	C₂²×C₁₉⋊C₈	—	2	1	no	—
50	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
51	C₄×D₇₆	—	2	1	no	—
52	C₄×D₇₆	—	2	1	no	—
53	C₄×D₇₆	—	2	1	no	—
54	C₄×D₇₆	—	2	1	no	—
55	#10597315	—	2	1	no	—
56	#10597315	—	2	1	no	—
57	#10597315	—	2	1	no	—
58	#10597315	D₄	1	2	yes	—
59	C₁₉×C₄⋊C₈	—	2	1	no	—
60	C₁₉×C₄⋊C₈	—	2	1	no	—
61	C₁₉×C₄⋊C₈	—	2	1	no	—
62	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
63	#10597278	—	2	4	no	—
64	(C₂×D₁₉)⋊C₈	—	4	4	no	—
65	#10597275	—	2	4	no	—
66	#10597266	—	2	4	no	—
67	#10597264	—	2	4	no	—
68	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
69	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
70	C₄×C₁₉⋊C₈	—	2	1	no	—
71	C₄×C₁₉⋊C₈	—	2	1	no	—
72	C₂³×D₁₉	—	2	1	no	—
73	C₂×Q₈×C₁₉	—	2	2	no	—
74	C₂×Q₈×C₁₉	—	4	1	no	—
75	C₂×Q₈×C₁₉	—	4	2	no	—
76	C₂×Q₈×C₁₉	—	4	2	no	—
77	C₂²×C₄×C₁₉	—	4	1	no	—
78	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
81	C₂²×Dic₁₉	—	2	1	no	—
82	C₂²×Dic₁₉	—	4	1	no	—
83	C₂×D₇₆	—	2	1	no	—
84	C₂×D₇₆	—	2	1	no	—
85	C₂×D₇₆	—	2	1	no	—
86	C₂×D₇₆	—	2	1	no	—
87	C₂×D₇₆	—	2	1	no	—
88	C₂×D₇₆	Q₁₆	1	2	yes	complemented
89	C₂×C₄×D₁₉	—	4	1	no	—
90	C₂×C₄×D₁₉	—	4	1	no	—
91	C₂×C₄×D₁₉	—	4	2	no	—
92	C₂×C₄×D₁₉	—	4	2	no	—
93	C₂×Dic₃₈	—	2	2	no	—
94	C₂×Dic₃₈	—	4	2	no	—
95	C₂×Dic₃₈	—	4	2	no	—
96	C₂×Dic₃₈	—	4	1	no	—
97	C₂×C₈×C₁₉	—	2	2	no	—
98	C₂×C₈×C₁₉	—	4	1	no	—
99	C₂×C₈×C₁₉	—	4	1	no	—
100	C₂×C₈×C₁₉	—	4	1	no	—
101	C₂×C₈×C₁₉	—	4	2	no	—
102	C₄⋊C₄×C₁₉	—	2	2	no	—
103	C₄⋊C₄×C₁₉	—	2	2	no	—
104	C₄⋊C₄×C₁₉	—	4	1	no	—
105	C₄⋊C₄×C₁₉	—	4	1	no	—
106	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
110	#64528	—	4	16	no	—
111	(C₂×D₁₉)⋊C₄	—	4	1	no	—
112	(C₂×D₁₉)⋊C₄	—	4	1	no	—
113	C₄⋊Dic₁₉	—	2	1	no	—
114	C₄⋊Dic₁₉	—	2	1	no	—
115	C₄⋊Dic₁₉	—	2	2	no	—
116	C₄⋊Dic₁₉	—	2	2	no	—
117	C₄⋊Dic₁₉	—	4	1	no	—
118	C₄⋊Dic₁₉	—	4	1	no	—
119	C₂×C₁₉⋊C₈	—	2	2	no	—
120	C₂×C₁₉⋊C₈	—	2	2	no	—
121	C₂×C₁₉⋊C₈	—	2	2	no	—
122	C₂×C₁₉⋊C₈	—	4	1	no	—
123	C₂×C₁₉⋊C₈	—	4	1	no	—
124	C₂×C₁₉⋊C₈	D₈	1	4	yes	—
125	#10274	—	19	1	no	maximal, Sylow
126	C₂³×C₁₉	#121	1	1	yes	characteristic
127	C₂²×D₁₉	—	2	2	no	—
128	C₂²×D₁₉	—	4	1	no	—
129	C₂²×D₁₉	—	4	2	no	—
130	C₂²×D₁₉	—	4	1	no	—
131	C₂²×D₁₉	—	4	2	no	—
132	Q₈×C₁₉	—	4	4	no	—
133	Q₈×C₁₉	—	8	1	no	—
134	Q₈×C₁₉	—	8	1	no	—
135	C₂×C₄×C₁₉	—	2	2	no	—
136	C₂×C₄×C₁₉	—	4	2	no	—
137	C₂×C₄×C₁₉	—	8	2	no	—
138	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
139	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
140	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
141	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
142	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
143	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
144	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
145	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
146	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
147	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
148	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
149	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
150	C₂×Dic₁₉	—	4	2	no	—
151	C₂×Dic₁₉	—	4	1	no	—
152	C₂×Dic₁₉	—	4	2	no	—
153	C₂×Dic₁₉	—	8	2	no	—
154	D₇₆	—	2	2	no	—
155	D₇₆	—	2	2	no	—
156	D₇₆	—	4	1	no	—
157	D₇₆	—	4	1	no	—
158	C₄×D₁₉	—	8	2	no	—
159	C₄×D₁₉	—	8	1	no	—
160	C₄×D₁₉	—	8	1	no	—
161	Dic₃₈	—	4	4	no	—
162	Dic₃₈	—	8	1	no	—
163	Dic₃₈	—	8	1	no	—
164	C₈×C₁₉	—	4	2	no	—
165	C₈×C₁₉	—	8	1	no	—
166	C₁₉⋊C₈	—	2	4	no	—
167	C₁₉⋊C₈	—	4	2	no	—
168	C₂×#494	—	19	1	no	—
169	C₂×#474	—	19	1	no	—
170	C₂×#474	—	19	1	no	—
171	#1954	—	19	1	no	—
172	C₂×#332	—	19	1	no	—
173	C₂×#332	D₁₉	1	1	yes	characteristic, complemented
174	#1277	—	19	1	no	—
175	C₂²×C₁₉	#392	1	1	yes	characteristic
176	C₂²×C₁₉	#459	1	2	yes	—
177	C₂²×C₁₉	#467	1	1	yes	characteristic
178	C₂²×C₁₉	#470	1	1	yes	characteristic
179	C₂²×C₁₉	#473	1	2	yes	—
180	C₂×D₁₉	—	4	4	no	—
181	C₂×D₁₉	—	4	2	no	—
182	C₂×D₁₉	—	8	2	no	—
183	C₂×D₁₉	—	8	1	no	—
184	C₂×D₁₉	—	8	1	no	—
185	C₄×C₁₉	—	2	1	no	—
186	C₄×C₁₉	—	2	1	no	—
187	C₄×C₁₉	—	8	2	no	—
188	C₄×C₁₉	#447	1	2	yes	—
189	C₄×C₁₉	#451	1	1	yes	characteristic
190	C₄×C₁₉	#451	1	1	yes	characteristic
191	C₄×C₁₉	#452	1	1	yes	characteristic
192	C₄×C₁₉	#452	1	1	yes	characteristic
193	C₄×C₁₉	#493	1	1	yes	characteristic
194	C₄×C₁₉	#496	1	1	yes	characteristic
195	Dic₁₉	—	8	2	no	—
196	Dic₁₉	—	8	1	no	—
197	C₂²×Q₁₆	—	38	2	no	—
198	C₂×C₄⋊Q₈	—	19	1	no	—
199	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
200	C₂×C₄×D₄	—	19	1	no	—
201	#494	—	38	2	no	—
202	#494	—	38	2	no	—
203	#474	—	38	2	no	—
204	#474	—	38	2	no	—
205	#474	—	38	2	no	—
206	#474	—	38	2	no	—
207	C₂×#107	—	38	1	no	—
208	C₂×#107	—	38	1	no	—
209	C₂×C₄⋊C₈	—	19	1	no	—
210	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
211	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
212	C₂×#102	—	38	1	no	—
213	C₂×#102	—	38	1	no	—
214	C₂×C₂²⋊C₈	—	38	2	no	—
215	C₂×C₄×C₈	—	19	1	no	—
216	#332	—	2	2	no	—
217	#332	—	38	2	no	—
218	C₂×C₁₉	#1669	1	2	yes	—
219	C₂×C₁₉	#1689	1	2	yes	—
220	C₂×C₁₉	#2001	1	1	yes	characteristic
221	C₂×C₁₉	#2011	1	1	yes	characteristic
222	C₂×C₁₉	#2020	1	1	yes	characteristic
223	D₁₉	—	8	2	no	—
224	C₂²×Q₈	—	2	1	no	—
225	C₂²×Q₈	—	38	1	no	—
226	C₂²×D₄	—	19	1	no	—
227	C₂³×C₄	—	38	1	no	—
228	C₂×Q₁₆	—	38	8	no	—
229	C₂×Q₁₆	—	76	4	no	—
230	C₂×Q₁₆	—	76	4	no	—
231	C₂²×C₈	—	2	1	no	—
232	C₂²×C₈	—	2	1	no	—
233	C₂²×C₈	—	19	2	no	—
234	C₂²×C₈	—	38	1	no	—
235	C₄⋊Q₈	—	2	1	no	—
236	C₄⋊Q₈	—	2	1	no	—
237	C₄⋊Q₈	—	2	1	no	—
238	C₄⋊Q₈	—	19	2	no	—
239	C₄⋊Q₈	—	38	1	no	—
240	C₄⋊Q₈	—	38	1	no	—
241	C₄⋊Q₈	—	38	1	no	—
242	C₄⋊Q₈	D₇₆	1	2	yes	—
243	C₄×D₄	—	38	1	no	—
244	C₄×D₄	—	38	1	no	—
245	C₄×D₄	—	38	1	no	—
246	C₄×D₄	—	38	1	no	—
247	C₂×C₄⋊C₄	—	2	1	no	—
248	C₂×C₄⋊C₄	—	2	1	no	—
249	C₂×C₄⋊C₄	—	19	1	no	—
250	C₂×C₄⋊C₄	—	38	1	no	—
251	C₂×C₄⋊C₄	—	38	1	no	—
252	C₂×C₂²⋊C₄	—	38	1	no	—
253	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
254	#107	—	38	4	no	—
255	#107	—	38	4	no	—
256	C₄⋊C₈	—	2	1	no	—
257	C₄⋊C₈	—	2	1	no	—
258	C₄⋊C₈	—	2	1	no	—
259	C₄⋊C₈	—	19	2	no	—
260	C₄⋊C₈	—	38	1	no	—
261	C₄⋊C₈	C₂²⋊D₁₉	1	2	yes	—
262	#102	—	38	4	no	—
263	#102	—	38	4	no	—
264	C₂²⋊C₈	—	76	4	no	—
265	C₄×C₈	—	38	1	no	—
266	C₄×C₈	—	38	1	no	—
267	C₁₉	#10274	1	1	yes	characteristic, complemented, Sylow
268	C₂⁴	—	38	1	no	—
269	C₂×Q₈	—	2	2	no	—
270	C₂×Q₈	—	4	1	no	—
271	C₂×Q₈	—	4	2	no	—
272	C₂×Q₈	—	4	2	no	—
273	C₂×Q₈	—	38	2	no	—
274	C₂×Q₈	—	76	1	no	—
275	C₂×Q₈	—	76	2	no	—
276	C₂×Q₈	—	76	2	no	—
277	C₂×D₄	—	19	2	no	—
278	C₂×D₄	—	38	1	no	—
279	C₂×D₄	—	38	1	no	—
280	C₂×D₄	—	38	1	no	—
281	C₂×D₄	—	38	1	no	—
282	C₂×D₄	—	38	1	no	—
283	C₂²×C₄	—	4	1	no	—
284	C₂²×C₄	—	38	1	no	—
285	C₂²×C₄	—	76	1	no	—
286	C₂²×C₄	—	76	1	no	—
287	C₂²×C₄	—	76	1	no	—
288	C₂²×C₄	—	76	2	no	—
289	C₂²×C₄	—	76	2	no	—
290	C₂²×C₄	#64541	1	1	yes	characteristic
291	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
292	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
293	Q₁₆	—	76	16	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	4	1	no	—
296	C₂×C₈	—	4	1	no	—
297	C₂×C₈	—	4	1	no	—
298	C₂×C₈	—	4	2	no	—
299	C₂×C₈	—	19	4	no	—
300	C₂×C₈	—	38	2	no	—
301	C₂×C₈	—	38	2	no	—
302	C₂×C₈	—	38	2	no	—
303	C₂×C₈	—	76	1	no	—
304	C₂×C₈	—	76	1	no	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	4	1	no	—
308	C₄⋊C₄	—	4	1	no	—
309	C₄⋊C₄	—	38	2	no	—
310	C₄⋊C₄	—	38	1	no	—
311	C₄⋊C₄	—	38	1	no	—
312	C₄⋊C₄	—	38	2	no	—
313	C₄⋊C₄	—	76	1	no	—
314	C₄⋊C₄	—	76	1	no	—
315	C₂²⋊C₄	—	76	1	no	—
316	C₂²⋊C₄	—	76	1	no	—
317	C₄²	C₂×D₇₆	1	1	yes	characteristic
318	C₄²	D₄×D₁₉	1	1	yes	characteristic
319	C₄²	D₄×D₁₉	1	1	yes	characteristic
320	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
321	C₂³	—	38	2	no	—
322	C₂³	—	76	1	no	—
323	C₂³	—	76	2	no	—
324	C₂³	—	76	1	no	—
325	C₂³	—	76	2	no	—
326	C₂³	#10597330	1	1	yes	characteristic
327	Q₈	—	4	4	no	—
328	Q₈	—	8	1	no	—
329	Q₈	—	8	1	no	—
330	Q₈	—	76	4	no	—
331	Q₈	—	152	1	no	—
332	Q₈	—	152	1	no	—
333	D₄	—	38	2	no	—
334	D₄	—	38	2	no	—
335	D₄	—	76	1	no	—
336	D₄	—	76	1	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	8	2	no	—
340	C₂×C₄	—	76	1	no	—
341	C₂×C₄	—	76	2	no	—
342	C₂×C₄	—	76	2	no	—
343	C₂×C₄	—	152	2	no	—
344	C₂×C₄	—	152	1	no	—
345	C₂×C₄	—	152	1	no	—
346	C₂×C₄	—	152	2	no	—
347	C₂×C₄	#10597328	1	1	yes	characteristic
348	C₂×C₄	#10597328	1	1	yes	characteristic
349	C₂×C₄	#10597330	1	1	yes	characteristic
350	C₂×C₄	D₈×D₁₉	1	2	yes	—
351	C₂×C₄	#10597360	1	2	yes	—
352	C₂×C₄	#10597361	1	1	yes	characteristic
353	C₂×C₄	#10597362	1	1	yes	characteristic
354	C₂×C₄	Q₁₆×D₁₉	1	1	yes	characteristic
355	C₂×C₄	#10597364	1	1	yes	characteristic
356	C₂×C₄	#10597376	1	1	yes	characteristic
357	C₂×C₄	#10597383	1	1	yes	characteristic
358	C₂×C₄	#10597386	1	1	yes	characteristic
359	C₈	—	4	2	no	—
360	C₈	—	8	1	no	—
361	C₈	—	38	4	no	—
362	C₈	—	76	2	no	—
363	C₂²	—	76	2	no	—
364	C₂²	—	76	4	no	—
365	C₂²	—	152	2	no	—
366	C₂²	—	152	1	no	—
367	C₂²	—	152	1	no	—
368	C₂²	#11926453	1	1	yes	characteristic
369	C₂²	#11926738	1	2	yes	—
370	C₂²	#11926784	1	1	yes	characteristic
371	C₂²	#11926801	1	1	yes	characteristic
372	C₂²	#11926818	1	2	yes	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	8	2	no	—
376	C₄	—	152	1	no	—
377	C₄	—	152	2	no	—
378	C₄	#11926679	1	2	yes	—
379	C₄	#11926698	1	1	yes	characteristic
380	C₄	#11926701	1	1	yes	characteristic
381	C₄	#11926703	1	1	yes	characteristic
382	C₄	#11926709	1	1	yes	characteristic
383	C₄	#11926919	1	1	yes	characteristic
384	C₄	#11926930	1	1	yes	characteristic
385	C₂	—	152	2	no	—
386	C₂	#14562999	1	2	yes	—
387	C₂	#14563123	1	2	yes	—
388	C₂	#14564644	1	1	yes	characteristic
389	C₂	#14564707	1	1	yes	characteristic
390	C₂	#14564755	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597330
commutator, cosocle	C₂×C₄²×C₁₉	C₂³
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#332	C₂
socle	C₂³×C₁₉	#121
prime radical	C₂²×D₇₆	D₄
hypercenter	C₂×#332	D₁₉
hypocenter	C₁₉	#10274
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#332

Complements

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

N	H ≅ G/N	type
1. C₂×#11926923	C₂	semidirect
5. C₂×#11926291	C₂	semidirect
6. #14561469	C₂	semidirect
7. C₂×C₁₉×#332	C₂	semidirect
88. C₂×D₇₆	Q₁₆	semidirect
173. C₂×#332	D₁₉	semidirect
267. C₁₉	#10274	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926923	2
C₂×#11926825	2
C₂×#11926821	2
#14564428	2
C₂×#11926291	2
#14561469	2
C₂×C₁₉×#332	2
#10274	19