Subgroups — #18085092

Order 4864.

658 conjugacy classes of subgroups.

The subgroup lattice has 410 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×#11926826	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11926745	C₂	1	1	yes	maximal, characteristic
4	#14564570	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11926287	C₂	1	1	yes	maximal, characteristic
6	#14561469	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₉×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#64516	—	2	1	no	—
9	C₂²×#64527	—	2	2	no	—
10	C₂×#10597320	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₃₈	C₂²	1	1	yes	characteristic
12	#11926918	—	2	2	no	—
13	#11926918	—	2	2	no	—
14	#11926826	—	2	2	no	—
15	#11926826	—	2	2	no	—
16	#11926745	—	2	2	no	—
17	#11926745	—	2	2	no	—
18	C₂×#10597265	—	2	1	no	—
19	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
20	C₂×#10597267	—	2	1	no	—
21	C₂×#10597278	—	2	1	no	—
22	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
23	#11926298	—	2	2	no	—
24	#11926287	—	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	—	2	1	no	—
31	C₂²×D₇₆	—	2	1	no	—
32	C₂²×D₇₆	—	2	1	no	—
33	C₂²×Dic₃₈	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₉	—	2	1	no	—
35	C₂²×C₈×C₁₉	—	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	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
41	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
42	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
43	C₂×#64527	—	2	8	no	—
44	C₂×#64527	—	4	4	no	—
45	C₂×#64527	—	4	4	no	—
46	C₂×C₄⋊Dic₁₉	D₄	1	1	yes	characteristic
47	C₂×Dic₁₉⋊C₄	—	2	1	no	—
48	C₂×C₄×Dic₁₉	—	2	1	no	—
49	C₂²×C₁₉⋊C₈	—	2	1	no	—
50	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
51	C₂×#64516	—	2	4	no	—
52	C₂×#64516	—	4	2	no	—
53	C₂×#64516	—	4	2	no	—
54	#10597320	—	2	1	no	—
55	#10597320	—	2	1	no	—
56	#10597320	—	2	1	no	—
57	#10597320	D₄	1	2	yes	—
58	C₄×Dic₃₈	—	2	1	no	—
59	C₄×Dic₃₈	—	2	1	no	—
60	C₄×Dic₃₈	—	2	1	no	—
61	C₄×Dic₃₈	—	2	1	no	—
62	C₁₉×C₄⋊C₈	—	2	1	no	—
63	C₁₉×C₄⋊C₈	—	2	1	no	—
64	C₁₉×C₄⋊C₈	—	2	1	no	—
65	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
66	#10597278	—	2	4	no	—
67	#10597267	—	2	4	no	—
68	#10597265	—	2	4	no	—
69	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
70	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
71	C₄×C₁₉⋊C₈	—	2	1	no	—
72	C₄×C₁₉⋊C₈	—	2	1	no	—
73	C₂³×D₁₉	—	4	1	no	—
74	C₂×Q₈×C₁₉	—	2	2	no	—
75	C₂×Q₈×C₁₉	—	4	1	no	—
76	C₂×Q₈×C₁₉	—	4	2	no	—
77	C₂×Q₈×C₁₉	—	4	2	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	#41	1	1	yes	characteristic
82	C₂²×Dic₁₉	—	2	1	no	—
83	C₂²×Dic₁₉	—	2	1	no	—
84	C₂×D₇₆	—	2	2	no	—
85	C₂×D₇₆	—	2	2	no	—
86	C₂×D₇₆	—	2	2	no	—
87	C₂×D₇₆	—	4	2	no	—
88	C₂×D₇₆	—	4	1	no	—
89	C₂×D₇₆	—	4	1	no	—
90	C₂×D₇₆	—	4	2	no	—
91	C₂×D₇₆	—	4	1	no	—
92	C₂×D₇₆	—	4	1	no	—
93	C₂×D₇₆	—	4	1	no	—
94	C₂×D₇₆	—	4	1	no	—
95	C₂×D₇₆	—	4	1	no	—
96	C₂×D₇₆	—	8	2	no	—
97	C₂×Dic₃₈	—	2	1	no	—
98	C₂×Dic₃₈	—	2	1	no	—
99	C₂×Dic₃₈	—	2	1	no	—
100	C₂×Dic₃₈	—	2	1	no	—
101	C₂×Dic₃₈	—	2	1	no	—
102	C₂×Dic₃₈	SD₁₆	1	2	yes	complemented
103	C₂×C₈×C₁₉	—	2	2	no	—
104	C₂×C₈×C₁₉	—	4	1	no	—
105	C₂×C₈×C₁₉	—	4	1	no	—
106	C₂×C₈×C₁₉	—	4	1	no	—
107	C₂×C₈×C₁₉	—	4	2	no	—
108	C₄⋊C₄×C₁₉	—	2	2	no	—
109	C₄⋊C₄×C₁₉	—	2	2	no	—
110	C₄⋊C₄×C₁₉	—	4	1	no	—
111	C₄⋊C₄×C₁₉	—	4	1	no	—
112	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
116	#64527	—	4	16	no	—
117	C₄⋊Dic₁₉	—	2	1	no	—
118	C₄⋊Dic₁₉	—	2	1	no	—
119	Dic₁₉⋊C₄	—	4	1	no	—
120	Dic₁₉⋊C₄	—	4	1	no	—
121	C₄×Dic₁₉	—	4	1	no	—
122	C₄×Dic₁₉	—	4	1	no	—
123	C₂×C₁₉⋊C₈	—	2	2	no	—
124	C₂×C₁₉⋊C₈	—	2	2	no	—
125	C₂×C₁₉⋊C₈	—	2	2	no	—
126	C₂×C₁₉⋊C₈	—	4	1	no	—
127	C₂×C₁₉⋊C₈	—	4	1	no	—
128	C₂×C₁₉⋊C₈	SD₁₆	1	4	yes	complemented
129	#64516	—	4	8	no	—
130	#10268	—	19	1	no	maximal, Sylow
131	C₂³×C₁₉	#121	1	1	yes	characteristic
132	C₂²×D₁₉	—	4	2	no	—
133	C₂²×D₁₉	—	8	2	no	—
134	C₂²×D₁₉	—	8	1	no	—
135	C₂²×D₁₉	—	8	1	no	—
136	C₂²×D₁₉	—	8	2	no	—
137	Q₈×C₁₉	—	4	4	no	—
138	Q₈×C₁₉	—	8	1	no	—
139	Q₈×C₁₉	—	8	1	no	—
140	C₂×C₄×C₁₉	—	2	2	no	—
141	C₂×C₄×C₁₉	—	4	2	no	—
142	C₂×C₄×C₁₉	—	8	2	no	—
143	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
144	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
145	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
146	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
147	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
148	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
149	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
150	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
151	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
152	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
153	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
154	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
155	C₂×Dic₁₉	—	2	2	no	—
156	C₂×Dic₁₉	—	4	1	no	—
157	C₂×Dic₁₉	—	4	1	no	—
158	C₂×Dic₁₉	—	4	1	no	—
159	C₂×Dic₁₉	—	4	2	no	—
160	D₇₆	—	4	2	no	—
161	D₇₆	—	4	4	no	—
162	D₇₆	—	4	2	no	—
163	D₇₆	—	8	2	no	—
164	D₇₆	—	8	2	no	—
165	D₇₆	—	8	1	no	—
166	D₇₆	—	8	1	no	—
167	D₇₆	—	8	1	no	—
168	D₇₆	—	8	1	no	—
169	Dic₃₈	—	2	2	no	—
170	Dic₃₈	—	2	2	no	—
171	Dic₃₈	—	4	1	no	—
172	Dic₃₈	—	4	1	no	—
173	C₈×C₁₉	—	4	2	no	—
174	C₈×C₁₉	—	8	1	no	—
175	C₁₉⋊C₈	—	2	4	no	—
176	C₁₉⋊C₈	—	4	2	no	—
177	C₂×#492	—	19	1	no	—
178	C₂×#475	—	19	1	no	—
179	C₂×#460	—	19	1	no	—
180	#1984	—	19	1	no	—
181	C₂×#332	D₁₉	1	1	yes	characteristic, complemented
182	C₂×#331	—	19	1	no	—
183	#1277	—	19	1	no	—
184	C₂²×C₁₉	#392	1	1	yes	characteristic
185	C₂²×C₁₉	#460	1	2	yes	—
186	C₂²×C₁₉	#465	1	1	yes	characteristic
187	C₂²×C₁₉	#468	1	1	yes	characteristic
188	C₂²×C₁₉	#471	1	2	yes	—
189	C₂×D₁₉	—	8	2	no	—
190	C₂×D₁₉	—	8	2	no	—
191	C₂×D₁₉	—	8	2	no	—
192	C₂×D₁₉	—	16	2	no	—
193	C₂×D₁₉	—	16	2	no	—
194	C₄×C₁₉	—	2	1	no	—
195	C₄×C₁₉	—	2	1	no	—
196	C₄×C₁₉	—	8	2	no	—
197	C₄×C₁₉	#448	1	1	yes	characteristic
198	C₄×C₁₉	#449	1	1	yes	characteristic
199	C₄×C₁₉	#450	1	1	yes	characteristic
200	C₄×C₁₉	#450	1	2	yes	—
201	C₄×C₁₉	#452	1	1	yes	characteristic
202	C₄×C₁₉	#492	1	1	yes	characteristic
203	C₄×C₁₉	#496	1	1	yes	characteristic
204	Dic₁₉	—	4	2	no	—
205	Dic₁₉	—	8	1	no	—
206	C₂²×SD₁₆	—	38	1	no	—
207	C₂²×SD₁₆	—	38	2	no	—
208	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
209	C₂×#127	—	19	1	no	—
210	C₂×C₄×Q₈	—	19	1	no	—
211	#492	—	38	2	no	—
212	#492	—	38	2	no	—
213	#475	—	38	2	no	—
214	#475	—	38	2	no	—
215	#460	—	38	2	no	—
216	#460	—	38	2	no	—
217	C₂×#106	—	38	1	no	—
218	C₂×C₄⋊C₈	—	19	1	no	—
219	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
220	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
221	C₂×#103	—	38	1	no	—
222	C₂×#102	—	38	1	no	—
223	C₂×C₄×C₈	—	19	1	no	—
224	#336	—	38	2	no	—
225	#332	—	2	2	no	—
226	#331	—	38	2	no	—
227	C₂×C₁₉	#1666	1	2	yes	—
228	C₂×C₁₉	#1686	1	2	yes	—
229	C₂×C₁₉	#2000	1	1	yes	characteristic
230	C₂×C₁₉	#2014	1	1	yes	characteristic
231	C₂×C₁₉	#2019	1	1	yes	characteristic
232	D₁₉	—	16	2	no	—
233	C₂²×Q₈	—	2	1	no	—
234	C₂²×Q₈	—	19	1	no	—
235	C₂²×D₄	—	38	1	no	—
236	C₂²×D₄	—	38	1	no	—
237	C₂²×D₄	—	38	1	no	—
238	C₂×SD₁₆	—	38	4	no	—
239	C₂×SD₁₆	—	38	8	no	—
240	C₂×SD₁₆	—	76	2	no	—
241	C₂×SD₁₆	—	76	2	no	—
242	C₂×SD₁₆	—	76	4	no	—
243	C₂×SD₁₆	—	76	4	no	—
244	C₂²×C₈	—	2	1	no	—
245	C₂²×C₈	—	2	1	no	—
246	C₂²×C₈	—	19	2	no	—
247	C₂²×C₈	—	38	1	no	—
248	C₄⋊Q₈	—	2	1	no	—
249	C₄⋊Q₈	—	2	1	no	—
250	C₄⋊Q₈	—	2	1	no	—
251	C₄⋊Q₈	D₇₆	1	2	yes	—
252	#127	—	19	2	no	—
253	#127	—	38	1	no	—
254	#127	—	38	1	no	—
255	#127	—	38	1	no	—
256	C₄×Q₈	—	38	1	no	—
257	C₄×Q₈	—	38	1	no	—
258	C₄×Q₈	—	38	1	no	—
259	C₄×Q₈	—	38	1	no	—
260	C₂×C₄⋊C₄	—	2	1	no	—
261	C₂×C₄⋊C₄	—	2	1	no	—
262	C₂×C₄⋊C₄	—	19	1	no	—
263	C₂×C₄⋊C₄	—	38	1	no	—
264	C₂×C₄²	—	38	1	no	—
265	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
266	#106	—	38	4	no	—
267	C₄⋊C₈	—	2	1	no	—
268	C₄⋊C₈	—	2	1	no	—
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	—	19	2	no	—
271	C₄⋊C₈	—	38	1	no	—
272	C₄⋊C₈	C₂²⋊D₁₉	1	2	yes	—
273	#103	—	38	4	no	—
274	#102	—	38	4	no	—
275	C₄×C₈	—	38	1	no	—
276	C₄×C₈	—	38	1	no	—
277	C₁₉	#10268	1	1	yes	characteristic, complemented, Sylow
278	C₂⁴	—	76	1	no	—
279	C₂×Q₈	—	2	2	no	—
280	C₂×Q₈	—	4	1	no	—
281	C₂×Q₈	—	4	2	no	—
282	C₂×Q₈	—	4	2	no	—
283	C₂×Q₈	—	19	2	no	—
284	C₂×Q₈	—	38	1	no	—
285	C₂×Q₈	—	38	1	no	—
286	C₂×Q₈	—	38	1	no	—
287	C₂×Q₈	—	38	1	no	—
288	C₂×Q₈	—	38	1	no	—
289	C₂×D₄	—	38	2	no	—
290	C₂×D₄	—	38	2	no	—
291	C₂×D₄	—	38	2	no	—
292	C₂×D₄	—	76	1	no	—
293	C₂×D₄	—	76	1	no	—
294	C₂×D₄	—	76	1	no	—
295	C₂×D₄	—	76	2	no	—
296	C₂×D₄	—	76	2	no	—
297	C₂×D₄	—	76	1	no	—
298	C₂×D₄	—	76	1	no	—
299	C₂×D₄	—	76	1	no	—
300	C₂×D₄	—	76	1	no	—
301	C₂×D₄	—	152	2	no	—
302	C₂²×C₄	—	4	1	no	—
303	C₂²×C₄	—	38	1	no	—
304	C₂²×C₄	—	38	1	no	—
305	C₂²×C₄	#64541	1	1	yes	characteristic
306	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
307	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
308	SD₁₆	—	76	8	no	—
309	SD₁₆	—	76	16	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	4	1	no	—
312	C₂×C₈	—	4	1	no	—
313	C₂×C₈	—	4	1	no	—
314	C₂×C₈	—	4	2	no	—
315	C₂×C₈	—	19	4	no	—
316	C₂×C₈	—	38	2	no	—
317	C₂×C₈	—	38	2	no	—
318	C₂×C₈	—	38	2	no	—
319	C₂×C₈	—	76	1	no	—
320	C₂×C₈	—	76	1	no	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	—	4	1	no	—
324	C₄⋊C₄	—	4	1	no	—
325	C₄⋊C₄	—	38	1	no	—
326	C₄⋊C₄	—	38	1	no	—
327	C₄⋊C₄	—	76	1	no	—
328	C₄⋊C₄	—	76	1	no	—
329	C₄²	—	76	1	no	—
330	C₄²	—	76	1	no	—
331	C₄²	C₂×D₇₆	1	1	yes	characteristic
332	C₄²	D₄×D₁₉	1	1	yes	characteristic
333	C₄²	D₄×D₁₉	1	1	yes	characteristic
334	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
335	C₂³	—	76	2	no	—
336	C₂³	—	152	1	no	—
337	C₂³	—	152	2	no	—
338	C₂³	—	152	2	no	—
339	C₂³	—	152	1	no	—
340	C₂³	#10597330	1	1	yes	characteristic
341	Q₈	—	4	4	no	—
342	Q₈	—	8	1	no	—
343	Q₈	—	8	1	no	—
344	Q₈	—	38	2	no	—
345	Q₈	—	38	2	no	—
346	Q₈	—	76	1	no	—
347	Q₈	—	76	1	no	—
348	D₄	—	76	2	no	—
349	D₄	—	76	4	no	—
350	D₄	—	76	2	no	—
351	D₄	—	152	2	no	—
352	D₄	—	152	2	no	—
353	D₄	—	152	1	no	—
354	D₄	—	152	1	no	—
355	D₄	—	152	1	no	—
356	D₄	—	152	1	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	8	2	no	—
360	C₂×C₄	—	38	2	no	—
361	C₂×C₄	—	76	1	no	—
362	C₂×C₄	—	76	1	no	—
363	C₂×C₄	—	76	2	no	—
364	C₂×C₄	—	76	1	no	—
365	C₂×C₄	#10597328	1	1	yes	characteristic
366	C₂×C₄	#10597328	1	1	yes	characteristic
367	C₂×C₄	#10597330	1	1	yes	characteristic
368	C₂×C₄	#10597357	1	2	yes	—
369	C₂×C₄	SD₁₆×D₁₉	1	2	yes	—
370	C₂×C₄	SD₁₆×D₁₉	1	1	yes	characteristic
371	C₂×C₄	#10597360	1	1	yes	characteristic
372	C₂×C₄	#10597364	1	1	yes	characteristic
373	C₂×C₄	#10597365	1	1	yes	characteristic
374	C₂×C₄	#10597376	1	1	yes	characteristic
375	C₂×C₄	#10597383	1	1	yes	characteristic
376	C₂×C₄	#10597386	1	1	yes	characteristic
377	C₈	—	4	2	no	—
378	C₈	—	8	1	no	—
379	C₈	—	38	4	no	—
380	C₈	—	76	2	no	—
381	C₂²	—	304	2	no	—
382	C₂²	—	304	2	no	—
383	C₂²	—	152	2	no	—
384	C₂²	—	152	2	no	—
385	C₂²	—	152	2	no	—
386	C₂²	#11926453	1	1	yes	characteristic
387	C₂²	#11926743	1	2	yes	—
388	C₂²	#11926776	1	1	yes	characteristic
389	C₂²	#11926793	1	1	yes	characteristic
390	C₂²	#11926810	1	2	yes	—
391	C₄	—	2	1	no	—
392	C₄	—	2	1	no	—
393	C₄	—	8	2	no	—
394	C₄	—	76	2	no	—
395	C₄	—	152	1	no	—
396	C₄	#11926684	1	1	yes	characteristic
397	C₄	#11926692	1	1	yes	characteristic
398	C₄	#11926693	1	1	yes	characteristic
399	C₄	#11926696	1	2	yes	—
400	C₄	#11926705	1	1	yes	characteristic
401	C₄	#11926915	1	1	yes	characteristic
402	C₄	#11926932	1	1	yes	characteristic
403	C₂	—	304	2	no	—
404	C₂	#14562986	1	2	yes	—
405	C₂	#14563110	1	2	yes	—
406	C₂	#14564637	1	1	yes	characteristic
407	C₂	#14564724	1	1	yes	characteristic
408	C₂	#14564748	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	C₂×#10597320	C₂²
hypercenter	C₂×#332	D₁₉
hypocenter	C₁₉	#10268
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
2. C₂×#11926826	C₂	semidirect
4. #14564570	C₂	semidirect
6. #14561469	C₂	semidirect
7. C₂×C₁₉×#332	C₂	semidirect
102. C₂×Dic₃₈	SD₁₆	semidirect
128. C₂×C₁₉⋊C₈	SD₁₆	semidirect
181. C₂×#332	D₁₉	semidirect
277. C₁₉	#10268	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×#11926826	2
C₂×#11926745	2
#14564570	2
C₂×#11926287	2
#14561469	2
C₂×C₁₉×#332	2
#10268	19