Subgroups — #18085327

Order 4864.

686 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926918	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926892	C₂	1	1	yes	maximal, characteristic
3	#14564597	C₂	1	1	yes	maximal, characteristic, complemented
4	#14564441	C₂	1	1	yes	maximal, characteristic
5	#14564438	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₉×#500	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#64527	—	2	2	no	—
9	C₂×#10597320	C₂²	1	1	yes	characteristic
10	#11926918	—	2	2	no	—
11	#11926918	—	2	2	no	—
12	#11926892	—	2	2	no	—
13	#11926892	—	2	2	no	—
14	C₂×#10597292	—	2	2	no	—
15	C₂×#10597278	—	2	2	no	—
16	C₂×#10597266	—	2	2	no	—
17	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
19	#11926414	C₂²	1	1	yes	characteristic
20	#11926306	—	2	2	no	—
21	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
22	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
23	C₁₉×#500	—	2	2	no	—
24	C₁₉×#500	—	2	2	no	—
25	C₂×C₁₉×#107	—	2	2	no	—
26	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
27	C₂²×Q₈×C₁₉	—	2	1	no	—
28	C₂²×Q₈×C₁₉	—	2	1	no	—
29	C₂²×D₇₆	—	2	1	no	—
30	C₂²×D₇₆	—	2	1	no	—
31	C₂²×D₇₆	—	2	1	no	—
32	C₂²×C₈×C₁₉	D₄	1	2	yes	—
33	C₁₉×C₄⋊Q₈	—	2	1	no	—
34	C₁₉×C₄⋊Q₈	—	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	—	2	1	no	—
39	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
40	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
42	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
43	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
44	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
45	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
46	C₂×#64527	—	2	8	no	—
47	C₂×#64527	—	4	4	no	—
48	C₂×#64527	—	4	4	no	—
49	C₂×C₄⋊Dic₁₉	—	2	1	no	—
50	C₂×C₄⋊Dic₁₉	—	2	1	no	—
51	C₂×C₄⋊Dic₁₉	—	2	1	no	—
52	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
53	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
54	#10597320	—	2	1	no	—
55	#10597320	—	2	1	no	—
56	#10597320	—	2	1	no	—
57	#10597320	D₄	1	2	yes	—
58	C₁₉×#107	—	2	8	no	—
59	C₄×C₈×C₁₉	—	2	1	no	—
60	C₄×C₈×C₁₉	—	2	1	no	—
61	#10597292	—	4	4	no	—
62	#10597278	—	4	4	no	—
63	#10597266	—	2	8	no	—
64	C₄×C₁₉⋊C₈	—	2	1	no	—
65	C₄×C₁₉⋊C₈	—	2	1	no	—
66	C₄×C₁₉⋊C₈	—	2	1	no	—
67	C₄×C₁₉⋊C₈	—	2	1	no	—
68	C₂³×D₁₉	—	4	1	no	—
69	C₂×Q₈×C₁₉	—	2	2	no	—
70	C₂×Q₈×C₁₉	—	2	2	no	—
71	C₂×Q₈×C₁₉	—	4	1	no	—
72	C₂×Q₈×C₁₉	—	4	1	no	—
73	C₂×Q₈×C₁₉	—	4	2	no	—
74	C₂×Q₈×C₁₉	—	4	2	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	—	4	1	no	—
79	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×Dic₁₉	—	4	1	no	—
83	C₂×D₇₆	—	2	2	no	—
84	C₂×D₇₆	—	2	2	no	—
85	C₂×D₇₆	—	2	2	no	—
86	C₂×D₇₆	—	4	2	no	—
87	C₂×D₇₆	—	4	2	no	—
88	C₂×D₇₆	—	4	2	no	—
89	C₂×D₇₆	—	4	2	no	—
90	C₂×D₇₆	—	4	2	no	—
91	C₂×D₇₆	—	4	2	no	—
92	C₂×D₇₆	—	4	1	no	—
93	C₂×D₇₆	—	4	1	no	—
94	C₂×D₇₆	—	4	1	no	—
95	C₂×C₈×C₁₉	—	2	2	no	—
96	C₂×C₈×C₁₉	—	2	2	no	—
97	C₂×C₈×C₁₉	SD₁₆	1	4	yes	complemented
98	C₄⋊C₄×C₁₉	—	2	2	no	—
99	C₄⋊C₄×C₁₉	—	2	2	no	—
100	C₄⋊C₄×C₁₉	—	2	2	no	—
101	C₄⋊C₄×C₁₉	—	2	2	no	—
102	C₄⋊C₄×C₁₉	—	4	1	no	—
103	C₄⋊C₄×C₁₉	—	4	1	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	#41	1	1	yes	characteristic
110	#64527	—	4	16	no	—
111	C₄⋊Dic₁₉	—	4	2	no	—
112	C₄⋊Dic₁₉	—	4	2	no	—
113	C₄⋊Dic₁₉	—	4	2	no	—
114	C₂×C₁₉⋊C₈	—	2	2	no	—
115	C₂×C₁₉⋊C₈	—	2	2	no	—
116	C₂×C₁₉⋊C₈	—	2	2	no	—
117	C₂×C₁₉⋊C₈	—	2	2	no	—
118	C₂×C₁₉⋊C₈	—	2	2	no	—
119	C₂×C₁₉⋊C₈	D₈	1	4	yes	—
120	#10308	—	19	1	no	maximal, Sylow
121	C₂³×C₁₉	#127	1	1	yes	characteristic
122	C₂²×D₁₉	—	4	2	no	—
123	C₂²×D₁₉	—	8	1	no	—
124	C₂²×D₁₉	—	8	1	no	—
125	C₂²×D₁₉	—	8	1	no	—
126	C₂²×D₁₉	—	8	1	no	—
127	C₂²×D₁₉	—	8	1	no	—
128	C₂²×D₁₉	—	8	1	no	—
129	Q₈×C₁₉	—	4	4	no	—
130	Q₈×C₁₉	—	4	4	no	—
131	Q₈×C₁₉	—	8	2	no	—
132	Q₈×C₁₉	—	8	2	no	—
133	C₂×C₄×C₁₉	—	4	2	no	—
134	C₂×C₄×C₁₉	—	4	2	no	—
135	C₂×C₄×C₁₉	—	8	1	no	—
136	C₂×C₄×C₁₉	—	8	1	no	—
137	C₂×C₄×C₁₉	—	8	1	no	—
138	C₂×C₄×C₁₉	—	8	1	no	—
139	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
140	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
141	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
142	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
143	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
144	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
145	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
146	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
147	C₂×C₄×C₁₉	#135	1	2	yes	—
148	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
149	C₂×C₄×C₁₉	#137	1	2	yes	—
150	C₂×C₄×C₁₉	#137	1	2	yes	—
151	C₂×Dic₁₉	—	8	1	no	—
152	C₂×Dic₁₉	—	8	1	no	—
153	C₂×Dic₁₉	—	8	1	no	—
154	D₇₆	—	4	4	no	—
155	D₇₆	—	4	4	no	—
156	D₇₆	—	4	4	no	—
157	D₇₆	—	8	2	no	—
158	D₇₆	—	8	2	no	—
159	D₇₆	—	8	2	no	—
160	C₈×C₁₉	—	2	4	no	—
161	C₁₉⋊C₈	—	2	4	no	—
162	C₁₉⋊C₈	—	4	2	no	—
163	C₂×#500	D₁₉	1	1	yes	characteristic, complemented
164	C₂×#492	—	19	1	no	—
165	C₂×#486	—	19	1	no	—
166	#1990	—	19	1	no	—
167	#1957	—	19	1	no	—
168	#1956	—	19	1	no	—
169	#1273	—	19	1	no	—
170	C₂²×C₁₉	#392	1	1	yes	characteristic
171	C₂²×C₁₉	#492	1	1	yes	characteristic
172	C₂²×C₁₉	#493	1	1	yes	characteristic
173	C₂²×C₁₉	#495	1	1	yes	characteristic
174	C₂²×C₁₉	#496	1	1	yes	characteristic
175	C₂²×C₁₉	#497	1	1	yes	characteristic
176	C₂²×C₁₉	#497	1	1	yes	characteristic
177	C₂×D₁₉	—	8	2	no	—
178	C₂×D₁₉	—	8	2	no	—
179	C₂×D₁₉	—	8	2	no	—
180	C₂×D₁₉	—	16	1	no	—
181	C₂×D₁₉	—	16	1	no	—
182	C₂×D₁₉	—	16	1	no	—
183	C₂×D₁₉	—	16	1	no	—
184	C₄×C₁₉	—	8	2	no	—
185	C₄×C₁₉	—	8	2	no	—
186	C₄×C₁₉	#447	1	2	yes	—
187	C₄×C₁₉	#448	1	2	yes	—
188	C₄×C₁₉	#452	1	2	yes	—
189	C₄×C₁₉	#459	1	2	yes	—
190	C₄×C₁₉	#461	1	2	yes	—
191	C₄×C₁₉	#464	1	2	yes	—
192	Dic₁₉	—	16	1	no	—
193	C₂²×SD₁₆	—	38	2	no	—
194	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
195	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
196	C₂×#127	—	19	1	no	—
197	#500	—	2	2	no	—
198	#500	—	2	2	no	—
199	#492	—	38	2	no	—
200	#492	—	38	2	no	—
201	#486	—	38	2	no	—
202	#486	—	38	2	no	—
203	C₂×#107	—	2	2	no	—
204	C₂×#103	—	38	2	no	—
205	C₂×#102	—	38	2	no	—
206	C₂×#102	—	38	2	no	—
207	C₂×C₄×C₈	—	19	1	no	—
208	C₂×C₄×C₈	—	19	1	no	—
209	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
210	#384	—	19	1	no	—
211	#340	—	38	2	no	—
212	C₂×C₁₉	#1706	1	1	yes	characteristic
213	C₂×C₁₉	#1714	1	1	yes	characteristic
214	C₂×C₁₉	#1719	1	1	yes	characteristic
215	C₂×C₁₉	#1722	1	1	yes	characteristic
216	C₂×C₁₉	#2011	1	1	yes	characteristic
217	C₂×C₁₉	#2013	1	1	yes	characteristic
218	C₂×C₁₉	#2015	1	1	yes	characteristic
219	D₁₉	—	16	2	no	—
220	C₂²×Q₈	—	2	1	no	—
221	C₂²×Q₈	—	2	1	no	—
222	C₂²×D₄	—	38	1	no	—
223	C₂²×D₄	—	38	1	no	—
224	C₂²×D₄	—	38	1	no	—
225	C₂×SD₁₆	—	38	8	no	—
226	C₂×SD₁₆	—	76	4	no	—
227	C₂×SD₁₆	—	76	4	no	—
228	C₂²×C₈	—	19	2	no	—
229	C₂²×C₈	—	19	2	no	—
230	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
231	C₄⋊Q₈	—	2	1	no	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	2	1	no	—
234	C₄⋊Q₈	—	2	1	no	—
235	C₄⋊Q₈	—	2	1	no	—
236	C₄⋊Q₈	—	2	1	no	—
237	C₄⋊Q₈	C₂²⋊D₁₉	1	2	yes	—
238	C₄⋊Q₈	C₂²⋊D₁₉	1	2	yes	—
239	#127	—	19	2	no	—
240	#127	—	38	1	no	—
241	#127	—	38	1	no	—
242	#127	—	38	1	no	—
243	C₂×C₄⋊C₄	—	2	1	no	—
244	C₂×C₄⋊C₄	—	2	1	no	—
245	C₂×C₄⋊C₄	—	2	1	no	—
246	C₂×C₄⋊C₄	—	2	1	no	—
247	C₂×C₄⋊C₄	—	38	1	no	—
248	C₂×C₄⋊C₄	—	38	1	no	—
249	C₂×C₄⋊C₄	—	38	1	no	—
250	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
251	#107	—	2	8	no	—
252	#103	—	76	4	no	—
253	#102	—	38	8	no	—
254	#102	—	76	4	no	—
255	C₄×C₈	—	2	1	no	—
256	C₄×C₈	—	2	1	no	—
257	C₄×C₈	—	38	1	no	—
258	C₄×C₈	—	38	1	no	—
259	C₄×C₈	—	38	1	no	—
260	C₄×C₈	—	38	1	no	—
261	C₁₉	#10308	1	1	yes	characteristic, complemented, Sylow
262	C₂⁴	—	76	1	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	—	4	1	no	—
266	C₂×Q₈	—	4	1	no	—
267	C₂×Q₈	—	4	2	no	—
268	C₂×Q₈	—	4	2	no	—
269	C₂×Q₈	—	4	2	no	—
270	C₂×Q₈	—	4	2	no	—
271	C₂×D₄	—	38	2	no	—
272	C₂×D₄	—	38	2	no	—
273	C₂×D₄	—	38	2	no	—
274	C₂×D₄	—	76	1	no	—
275	C₂×D₄	—	76	1	no	—
276	C₂×D₄	—	76	2	no	—
277	C₂×D₄	—	76	2	no	—
278	C₂×D₄	—	76	1	no	—
279	C₂×D₄	—	76	2	no	—
280	C₂×D₄	—	76	2	no	—
281	C₂×D₄	—	76	2	no	—
282	C₂×D₄	—	76	2	no	—
283	C₂²×C₄	—	4	1	no	—
284	C₂²×C₄	—	4	1	no	—
285	C₂²×C₄	—	76	1	no	—
286	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
287	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
288	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
289	SD₁₆	—	76	16	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	19	4	no	—
293	C₂×C₈	—	38	2	no	—
294	C₂×C₈	—	38	2	no	—
295	C₂×C₈	—	38	2	no	—
296	C₂×C₈	—	38	2	no	—
297	C₂×C₈	—	38	2	no	—
298	C₂×C₈	#64527	1	4	yes	complemented
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	2	2	no	—
303	C₄⋊C₄	—	4	1	no	—
304	C₄⋊C₄	—	4	1	no	—
305	C₄⋊C₄	—	4	1	no	—
306	C₄⋊C₄	—	4	1	no	—
307	C₄⋊C₄	—	76	2	no	—
308	C₄⋊C₄	—	76	2	no	—
309	C₄⋊C₄	—	76	2	no	—
310	C₄²	D₄×D₁₉	1	1	yes	characteristic
311	C₄²	#64543	1	1	yes	characteristic
312	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
313	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
314	C₂³	—	76	2	no	—
315	C₂³	—	152	1	no	—
316	C₂³	—	152	1	no	—
317	C₂³	—	152	1	no	—
318	C₂³	—	152	1	no	—
319	C₂³	—	152	1	no	—
320	C₂³	—	152	1	no	—
321	C₂³	#10597386	1	1	yes	characteristic
322	Q₈	—	4	4	no	—
323	Q₈	—	4	4	no	—
324	Q₈	—	8	2	no	—
325	Q₈	—	8	2	no	—
326	D₄	—	76	4	no	—
327	D₄	—	76	4	no	—
328	D₄	—	76	4	no	—
329	D₄	—	152	2	no	—
330	D₄	—	152	2	no	—
331	D₄	—	152	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	8	1	no	—
335	C₂×C₄	—	8	1	no	—
336	C₂×C₄	—	8	1	no	—
337	C₂×C₄	—	8	1	no	—
338	C₂×C₄	—	152	1	no	—
339	C₂×C₄	—	152	1	no	—
340	C₂×C₄	—	152	1	no	—
341	C₂×C₄	D₈×D₁₉	1	2	yes	—
342	C₂×C₄	#10597357	1	2	yes	—
343	C₂×C₄	#10597364	1	2	yes	—
344	C₂×C₄	#10597365	1	2	yes	—
345	C₂×C₄	#10597383	1	1	yes	characteristic
346	C₂×C₄	#10597383	1	1	yes	characteristic
347	C₂×C₄	#10597384	1	1	yes	characteristic
348	C₂×C₄	#10597385	1	1	yes	characteristic
349	C₂×C₄	#10597385	1	1	yes	characteristic
350	C₂×C₄	C₂×#64527	1	2	yes	—
351	C₂×C₄	#10597388	1	2	yes	—
352	C₂×C₄	#10597399	1	1	yes	characteristic
353	C₈	—	2	4	no	—
354	C₈	—	38	4	no	—
355	C₈	—	76	2	no	—
356	C₂²	—	304	1	no	—
357	C₂²	—	304	1	no	—
358	C₂²	—	304	1	no	—
359	C₂²	—	304	1	no	—
360	C₂²	—	152	2	no	—
361	C₂²	—	152	2	no	—
362	C₂²	—	152	2	no	—
363	C₂²	#11926452	1	1	yes	characteristic
364	C₂²	#11926918	1	1	yes	characteristic
365	C₂²	#11926919	1	1	yes	characteristic
366	C₂²	#11926929	1	1	yes	characteristic
367	C₂²	#11926932	1	1	yes	characteristic
368	C₂²	#11926936	1	1	yes	characteristic
369	C₂²	#11926937	1	1	yes	characteristic
370	C₄	—	8	2	no	—
371	C₄	—	8	2	no	—
372	C₄	—	304	1	no	—
373	C₄	#11926676	1	2	yes	—
374	C₄	#11926683	1	2	yes	—
375	C₄	#11926705	1	2	yes	—
376	C₄	#11926737	1	2	yes	—
377	C₄	#11926746	1	2	yes	—
378	C₄	#11926768	1	2	yes	—
379	C₂	—	304	2	no	—
380	C₂	#14563228	1	1	yes	characteristic
381	C₂	#14563258	1	1	yes	characteristic
382	C₂	#14563291	1	1	yes	characteristic
383	C₂	#14563305	1	1	yes	characteristic
384	C₂	#14564704	1	1	yes	characteristic
385	C₂	#14564718	1	1	yes	characteristic
386	C₂	#14564726	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597386
commutator, cosocle	C₂×C₄²×C₁₉	C₂³
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#500	C₂
socle	C₂³×C₁₉	#127
prime radical	C₂×#10597320	C₂²
hypercenter	C₂×#500	D₁₉
hypocenter	C₁₉	#10308
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
3. #14564597	C₂	semidirect
5. #14564438	C₂	semidirect
6. C₁₉⋊#1273	C₂	semidirect
7. C₂×C₁₉×#500	C₂	semidirect
97. C₂×C₈×C₁₉	SD₁₆	semidirect
163. C₂×#500	D₁₉	semidirect
261. C₁₉	#10308	semidirect
298. C₂×C₈	#64527	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926918	2
C₂×#11926892	2
#14564597	2
#14564441	2
#14564438	2
C₁₉⋊#1273	2
C₂×C₁₉×#500	2
#10308	19