Subgroups — #18081776

Order 4864.

586 conjugacy classes of subgroups.

The subgroup lattice has 376 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	#14564562	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11926294	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11926290	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11926287	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1988	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#64527	—	2	1	no	—
9	C₂²×#64527	—	2	1	no	—
10	C₂×#10597320	C₂²	1	1	yes	characteristic
11	#11926918	—	2	2	no	—
12	#11926918	C₄	1	4	yes	complemented
13	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×#10597266	—	2	2	no	—
16	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
17	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
18	#11926299	—	2	1	no	—
19	#11926294	—	2	2	no	—
20	#11926290	—	2	2	no	—
21	#11926287	—	2	2	no	—
22	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
23	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
25	C₁₉×#340	—	2	2	no	—
26	C₂²×Q₈×C₁₉	—	2	1	no	—
27	C₂²×D₇₆	—	2	1	no	—
28	C₂²×D₇₆	—	2	1	no	—
29	C₂²×D₇₆	—	2	1	no	—
30	C₂²×C₈×C₁₉	—	2	1	no	—
31	C₁₉×C₄⋊Q₈	—	2	1	no	—
32	C₁₉×C₄⋊Q₈	—	2	1	no	—
33	C₁₉×C₄⋊Q₈	—	2	1	no	—
34	C₁₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
36	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
37	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
38	C₂×C₄²×C₁₉	—	2	1	no	—
39	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
40	C₂×#64527	—	2	4	no	—
41	C₂×#64527	—	2	4	no	—
42	C₂×#64527	—	4	2	no	—
43	C₂×#64527	—	4	2	no	—
44	C₂×#64527	—	4	2	no	—
45	C₂×#64527	—	4	2	no	—
46	C₂²×C₁₉⋊C₈	—	2	1	no	—
47	C₂²×C₁₉⋊C₈	—	2	1	no	—
48	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
49	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
50	#10597320	—	2	1	no	—
51	#10597320	—	2	1	no	—
52	#10597320	—	2	1	no	—
53	#10597320	C₂×C₄	1	2	yes	—
54	C₁₉×C₄⋊C₈	—	2	1	no	—
55	C₁₉×C₄⋊C₈	—	2	1	no	—
56	C₁₉×#95	—	2	1	no	—
57	#10597266	—	4	4	no	—
58	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
59	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
60	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
61	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
62	C₄×C₁₉⋊C₈	—	4	2	no	—
63	C₄×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
64	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
65	C₂³×D₁₉	—	4	1	no	—
66	C₂×Q₈×C₁₉	—	2	2	no	—
67	C₂×Q₈×C₁₉	—	4	1	no	—
68	C₂×Q₈×C₁₉	—	4	1	no	—
69	C₂×Q₈×C₁₉	—	4	1	no	—
70	C₂×Q₈×C₁₉	—	4	1	no	—
71	C₂×Q₈×C₁₉	—	4	1	no	—
72	C₂²×C₄×C₁₉	—	2	2	no	—
73	C₂²×C₄×C₁₉	—	4	1	no	—
74	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
77	C₂×D₇₆	—	2	2	no	—
78	C₂×D₇₆	—	2	2	no	—
79	C₂×D₇₆	—	2	2	no	—
80	C₂×D₇₆	—	4	1	no	—
81	C₂×D₇₆	—	4	1	no	—
82	C₂×D₇₆	—	4	2	no	—
83	C₂×D₇₆	—	4	2	no	—
84	C₂×D₇₆	—	4	1	no	—
85	C₂×D₇₆	—	4	1	no	—
86	C₂×D₇₆	—	4	1	no	—
87	C₂×D₇₆	—	4	1	no	—
88	C₂×D₇₆	—	4	1	no	—
89	C₂×D₇₆	—	8	2	no	—
90	C₂×C₈×C₁₉	—	4	1	no	—
91	C₂×C₈×C₁₉	—	4	2	no	—
92	C₄⋊C₄×C₁₉	—	2	2	no	—
93	C₄⋊C₄×C₁₉	—	2	2	no	—
94	C₄⋊C₄×C₁₉	—	2	2	no	—
95	C₄⋊C₄×C₁₉	—	4	1	no	—
96	C₄⋊C₄×C₁₉	—	4	1	no	—
97	C₄⋊C₄×C₁₉	SD₁₆	1	4	yes	complemented
98	C₄²×C₁₉	—	4	1	no	—
99	C₄²×C₁₉	—	4	1	no	—
100	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₁₉	#41	1	1	yes	characteristic
104	#64527	—	4	8	no	—
105	#64527	—	4	8	no	—
106	C₂×C₁₉⋊C₈	—	2	1	no	—
107	C₂×C₁₉⋊C₈	—	2	1	no	—
108	C₂×C₁₉⋊C₈	—	2	2	no	—
109	C₂×C₁₉⋊C₈	—	4	1	no	—
110	C₂×C₁₉⋊C₈	—	4	1	no	—
111	C₂×C₁₉⋊C₈	—	4	1	no	—
112	C₂×C₁₉⋊C₈	—	4	2	no	—
113	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	2	yes	—
114	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	2	yes	—
115	C₂×C₁₉⋊C₈	D₈	1	2	yes	—
116	C₂×C₁₉⋊C₈	SD₁₆	1	2	yes	—
117	#9732	—	19	1	no	maximal, Sylow
118	C₂³×C₁₉	#121	1	1	yes	characteristic
119	C₂²×D₁₉	—	4	2	no	—
120	C₂²×D₁₉	—	8	1	no	—
121	C₂²×D₁₉	—	8	1	no	—
122	C₂²×D₁₉	—	8	2	no	—
123	C₂²×D₁₉	—	8	2	no	—
124	Q₈×C₁₉	—	4	2	no	—
125	Q₈×C₁₉	—	4	2	no	—
126	Q₈×C₁₉	—	8	1	no	—
127	Q₈×C₁₉	—	8	1	no	—
128	C₂×C₄×C₁₉	—	2	2	no	—
129	C₂×C₄×C₁₉	—	2	4	no	—
130	C₂×C₄×C₁₉	—	4	2	no	—
131	C₂×C₄×C₁₉	—	4	2	no	—
132	C₂×C₄×C₁₉	—	4	2	no	—
133	C₂×C₄×C₁₉	—	8	1	no	—
134	C₂×C₄×C₁₉	—	8	1	no	—
135	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
139	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
140	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
141	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
143	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
144	C₂×C₄×C₁₉	#135	1	2	yes	—
145	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
147	D₇₆	—	4	2	no	—
148	D₇₆	—	4	2	no	—
149	D₇₆	—	4	4	no	—
150	D₇₆	—	8	2	no	—
151	D₇₆	—	8	2	no	—
152	D₇₆	—	8	1	no	—
153	D₇₆	—	8	1	no	—
154	D₇₆	—	8	1	no	—
155	D₇₆	—	8	1	no	—
156	C₈×C₁₉	—	8	1	no	—
157	C₁₉⋊C₈	—	2	2	no	—
158	C₁₉⋊C₈	—	4	2	no	—
159	C₁₉⋊C₈	—	8	1	no	—
160	C₁₉⋊C₈	#102	1	4	yes	complemented
161	C₂×#492	—	19	1	no	—
162	#1988	D₁₉	1	1	yes	characteristic, complemented
163	#1982	—	19	1	no	—
164	C₈×C₄⋊C₄	—	19	1	no	—
165	C₂×#334	—	19	1	no	—
166	C₂×#332	—	19	1	no	—
167	C₂×#331	—	19	1	no	—
168	C₂²×C₁₉	#390	1	1	yes	characteristic
169	C₂²×C₁₉	#459	1	1	yes	characteristic
170	C₂²×C₁₉	#460	1	1	yes	characteristic
171	C₂²×C₁₉	#465	1	2	yes	—
172	C₂²×C₁₉	#471	1	2	yes	—
173	C₂×D₁₉	—	8	2	no	—
174	C₂×D₁₉	—	8	2	no	—
175	C₂×D₁₉	—	8	2	no	—
176	C₂×D₁₉	—	16	2	no	—
177	C₂×D₁₉	—	16	2	no	—
178	C₄×C₁₉	—	2	1	no	—
179	C₄×C₁₉	—	2	1	no	—
180	C₄×C₁₉	—	4	4	no	—
181	C₄×C₁₉	—	8	2	no	—
182	C₄×C₁₉	C₂×#102	1	1	yes	characteristic
183	C₄×C₁₉	#418	1	1	yes	characteristic
184	C₄×C₁₉	C₄×SD₁₆	1	2	yes	—
185	C₄×C₁₉	#459	1	1	yes	characteristic
186	C₄×C₁₉	#460	1	1	yes	characteristic
187	C₄×C₁₉	#492	1	1	yes	characteristic
188	C₄×C₁₉	#495	1	1	yes	characteristic
189	C₂²×SD₁₆	—	38	1	no	—
190	C₂²×SD₁₆	—	38	1	no	—
191	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
192	C₂×#127	—	19	1	no	—
193	#492	—	19	4	no	—
194	#492	—	38	2	no	—
195	C₂×C₄⋊C₈	—	19	1	no	—
196	C₂×C₄⋊C₈	—	19	1	no	—
197	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
198	C₂×#102	—	38	2	no	—
199	C₂×C₄×C₈	—	19	1	no	—
200	C₂×C₄×C₈	—	38	1	no	—
201	C₄×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
202	#340	—	2	2	no	—
203	#336	—	38	1	no	—
204	#334	—	38	2	no	—
205	#332	—	38	2	no	—
206	#331	—	38	2	no	—
207	C₂×C₁₉	#1665	1	2	yes	—
208	C₂×C₁₉	#1666	1	2	yes	—
209	C₂×C₁₉	#1968	1	1	yes	characteristic
210	C₂×C₁₉	#1977	1	1	yes	characteristic
211	C₂×C₁₉	#1982	1	1	yes	characteristic
212	D₁₉	—	16	2	no	—
213	C₂²×Q₈	—	2	1	no	—
214	C₂²×D₄	—	38	1	no	—
215	C₂²×D₄	—	38	1	no	—
216	C₂²×D₄	—	38	1	no	—
217	C₂×SD₁₆	—	38	4	no	—
218	C₂×SD₁₆	—	38	4	no	—
219	C₂×SD₁₆	—	76	2	no	—
220	C₂×SD₁₆	—	76	2	no	—
221	C₂×SD₁₆	—	76	2	no	—
222	C₂×SD₁₆	—	76	2	no	—
223	C₂²×C₈	—	2	1	no	—
224	C₂²×C₈	—	19	1	no	—
225	C₂²×C₈	—	19	1	no	—
226	C₂²×C₈	—	38	1	no	—
227	C₂²×C₈	—	38	1	no	—
228	C₄⋊Q₈	—	2	1	no	—
229	C₄⋊Q₈	—	2	1	no	—
230	C₄⋊Q₈	—	2	1	no	—
231	C₄⋊Q₈	C₄×D₁₉	1	2	yes	—
232	#127	—	19	2	no	—
233	#127	—	38	1	no	—
234	#127	—	38	1	no	—
235	#127	—	38	1	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	—	2	1	no	—
238	C₂×C₄⋊C₄	C₂²⋊D₁₉	1	2	yes	—
239	C₂×C₄²	—	2	1	no	—
240	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
241	C₄⋊C₈	—	2	1	no	—
242	C₄⋊C₈	—	2	1	no	—
243	C₄⋊C₈	—	19	2	no	—
244	C₄⋊C₈	—	38	1	no	—
245	C₄⋊C₈	—	38	1	no	—
246	C₄⋊C₈	—	38	1	no	—
247	#102	—	76	4	no	—
248	C₄×C₈	—	19	2	no	—
249	C₄×C₈	—	19	2	no	—
250	C₄×C₈	—	76	2	no	—
251	#95	—	2	1	no	—
252	C₁₉	#9732	1	1	yes	characteristic, complemented, Sylow
253	C₂⁴	—	76	1	no	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	4	1	no	—
256	C₂×Q₈	—	4	1	no	—
257	C₂×Q₈	—	4	1	no	—
258	C₂×Q₈	—	4	1	no	—
259	C₂×Q₈	—	4	1	no	—
260	C₂×D₄	—	38	2	no	—
261	C₂×D₄	—	38	2	no	—
262	C₂×D₄	—	38	2	no	—
263	C₂×D₄	—	76	1	no	—
264	C₂×D₄	—	76	2	no	—
265	C₂×D₄	—	76	2	no	—
266	C₂×D₄	—	76	1	no	—
267	C₂×D₄	—	76	1	no	—
268	C₂×D₄	—	76	1	no	—
269	C₂×D₄	—	76	1	no	—
270	C₂×D₄	—	76	1	no	—
271	C₂×D₄	—	76	1	no	—
272	C₂×D₄	—	152	2	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	4	1	no	—
275	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
276	C₂²×C₄	#64543	1	1	yes	characteristic
277	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
278	SD₁₆	—	76	8	no	—
279	SD₁₆	—	76	8	no	—
280	C₂×C₈	—	4	1	no	—
281	C₂×C₈	—	4	2	no	—
282	C₂×C₈	—	19	2	no	—
283	C₂×C₈	—	19	2	no	—
284	C₂×C₈	—	19	2	no	—
285	C₂×C₈	—	19	2	no	—
286	C₂×C₈	—	38	1	no	—
287	C₂×C₈	—	38	1	no	—
288	C₂×C₈	—	38	2	no	—
289	C₂×C₈	—	76	1	no	—
290	C₂×C₈	—	76	1	no	—
291	C₂×C₈	—	76	1	no	—
292	C₂×C₈	—	76	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₄⋊C₄	—	4	1	no	—
298	C₄⋊C₄	#64527	1	4	yes	complemented
299	C₄²	—	4	1	no	—
300	C₄²	—	4	1	no	—
301	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
302	C₄²	#64541	1	1	yes	characteristic
303	C₄²	D₄×D₁₉	1	1	yes	characteristic
304	C₄²	D₄×D₁₉	1	1	yes	characteristic
305	C₂³	—	76	2	no	—
306	C₂³	—	152	2	no	—
307	C₂³	—	152	1	no	—
308	C₂³	—	152	1	no	—
309	C₂³	—	152	2	no	—
310	C₂³	#10597385	1	1	yes	characteristic
311	Q₈	—	4	2	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	8	1	no	—
314	Q₈	—	8	1	no	—
315	D₄	—	76	2	no	—
316	D₄	—	76	2	no	—
317	D₄	—	76	4	no	—
318	D₄	—	152	2	no	—
319	D₄	—	152	2	no	—
320	D₄	—	152	1	no	—
321	D₄	—	152	1	no	—
322	D₄	—	152	1	no	—
323	D₄	—	152	1	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	4	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	8	1	no	—
330	C₂×C₄	—	8	1	no	—
331	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
332	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
333	C₂×C₄	#10597330	1	1	yes	characteristic
334	C₂×C₄	#10597331	1	1	yes	characteristic
335	C₂×C₄	D₈×D₁₉	1	1	yes	characteristic
336	C₂×C₄	#10597357	1	1	yes	characteristic
337	C₂×C₄	SD₁₆×D₁₉	1	1	yes	characteristic
338	C₂×C₄	#10597360	1	1	yes	characteristic
339	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
340	C₂×C₄	#10597386	1	1	yes	characteristic
341	C₂×C₄	C₂×#64527	1	2	yes	—
342	C₂×C₄	#10597396	1	2	yes	—
343	C₈	—	8	1	no	—
344	C₈	—	19	4	no	—
345	C₈	—	38	2	no	—
346	C₈	—	76	2	no	—
347	C₈	—	152	1	no	—
348	C₂²	—	304	2	no	—
349	C₂²	—	304	2	no	—
350	C₂²	—	152	2	no	—
351	C₂²	—	152	2	no	—
352	C₂²	—	152	2	no	—
353	C₂²	#11926444	1	1	yes	characteristic
354	C₂²	#11926737	1	1	yes	characteristic
355	C₂²	#11926742	1	1	yes	characteristic
356	C₂²	#11926775	1	2	yes	—
357	C₂²	#11926809	1	2	yes	—
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	4	4	no	—
361	C₄	—	8	2	no	—
362	C₄	#102×D₁₉	1	1	yes	characteristic
363	C₄	#11926547	1	1	yes	characteristic
364	C₄	C₄×#64527	1	2	yes	—
365	C₄	#11926738	1	1	yes	characteristic
366	C₄	#11926743	1	1	yes	characteristic
367	C₄	#11926918	1	1	yes	characteristic
368	C₄	#11926926	1	1	yes	characteristic
369	C₂	—	304	2	no	—
370	C₂	#14562978	1	2	yes	—
371	C₂	#14562985	1	2	yes	—
372	C₂	#14564492	1	1	yes	characteristic
373	C₂	#14564537	1	1	yes	characteristic
374	C₂	#14564560	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597385
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1988	C₂
socle	C₂³×C₁₉	#121
prime radical	C₂×#10597320	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1988	D₁₉
hypocenter	C₁₉	#9732
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1988

Complements

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

N	H ≅ G/N	type
3. C₄⋊C₄×C₁₉⋊C₈	C₂	semidirect
4. C₂×#11926294	C₂	semidirect
5. C₂×#11926290	C₂	semidirect
7. C₁₉×#1988	C₂	semidirect
12. #11926918	C₄	semidirect
97. C₄⋊C₄×C₁₉	SD₁₆	semidirect
160. C₁₉⋊C₈	#102	semidirect
162. #1988	D₁₉	semidirect
252. C₁₉	#9732	semidirect
298. C₄⋊C₄	#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
#14564562	2
C₄⋊C₄×C₁₉⋊C₈	2
C₂×#11926294	2
C₂×#11926290	2
C₂×#11926287	2
C₁₉×#1988	2
#9732	19