Subgroups — #18085099

Order 4864.

594 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926892	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926836	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×#333	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#64516	—	2	1	no	—
9	C₂×#10597320	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₃₈	C₂²	1	1	yes	characteristic
11	#11926892	—	2	2	no	—
12	#11926892	—	2	2	no	—
13	#11926836	—	2	2	no	—
14	#11926836	—	2	2	no	—
15	#11926745	—	2	2	no	—
16	#11926745	—	2	2	no	—
17	C₂×#10597264	—	2	1	no	—
18	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
19	C₂×#10597267	—	2	1	no	—
20	C₂×#10597278	—	2	1	no	—
21	C₂×#10597266	—	2	2	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×#333	—	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×C₄⋊Dic₁₉	D₄	1	1	yes	characteristic
44	C₂×Dic₁₉⋊C₄	—	2	1	no	—
45	C₂×C₄×Dic₁₉	—	2	1	no	—
46	C₂²×C₁₉⋊C₈	—	2	1	no	—
47	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
48	C₂×#64516	—	2	4	no	—
49	C₂×#64516	—	4	2	no	—
50	C₂×#64516	—	4	2	no	—
51	#10597320	—	2	1	no	—
52	#10597320	—	2	1	no	—
53	#10597320	—	2	1	no	—
54	#10597320	D₄	1	2	yes	—
55	C₄×Dic₃₈	—	2	1	no	—
56	C₄×Dic₃₈	—	2	1	no	—
57	C₄×Dic₃₈	—	2	1	no	—
58	C₄×Dic₃₈	—	2	1	no	—
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	#10597267	—	2	4	no	—
65	#10597266	—	2	8	no	—
66	#10597264	—	2	4	no	—
67	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
68	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
69	C₄×C₁₉⋊C₈	—	2	1	no	—
70	C₄×C₁₉⋊C₈	—	2	1	no	—
71	C₂³×D₁₉	—	4	1	no	—
72	C₂×Q₈×C₁₉	—	2	2	no	—
73	C₂×Q₈×C₁₉	—	4	1	no	—
74	C₂×Q₈×C₁₉	—	8	2	no	—
75	C₂²×C₄×C₁₉	—	4	1	no	—
76	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
79	C₂²×Dic₁₉	—	2	1	no	—
80	C₂²×Dic₁₉	—	2	1	no	—
81	C₂×D₇₆	—	2	2	no	—
82	C₂×D₇₆	—	2	2	no	—
83	C₂×D₇₆	—	2	2	no	—
84	C₂×D₇₆	—	4	1	no	—
85	C₂×D₇₆	—	4	2	no	—
86	C₂×D₇₆	—	4	1	no	—
87	C₂×D₇₆	—	4	1	no	—
88	C₂×D₇₆	—	4	1	no	—
89	C₂×D₇₆	—	4	2	no	—
90	C₂×D₇₆	—	4	1	no	—
91	C₂×D₇₆	—	4	1	no	—
92	C₂×D₇₆	—	4	1	no	—
93	C₂×D₇₆	—	8	2	no	—
94	C₂×Dic₃₈	—	2	1	no	—
95	C₂×Dic₃₈	—	2	1	no	—
96	C₂×Dic₃₈	—	2	1	no	—
97	C₂×Dic₃₈	—	2	1	no	—
98	C₂×Dic₃₈	—	2	1	no	—
99	C₂×Dic₃₈	SD₁₆	1	2	yes	—
100	C₂×C₈×C₁₉	—	2	2	no	—
101	C₂×C₈×C₁₉	—	4	1	no	—
102	C₂×C₈×C₁₉	—	4	1	no	—
103	C₂×C₈×C₁₉	—	4	1	no	—
104	C₂×C₈×C₁₉	—	4	2	no	—
105	C₄⋊C₄×C₁₉	—	2	2	no	—
106	C₄⋊C₄×C₁₉	—	2	2	no	—
107	C₄⋊C₄×C₁₉	—	4	1	no	—
108	C₄⋊C₄×C₁₉	—	4	1	no	—
109	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
113	C₄⋊Dic₁₉	—	2	1	no	—
114	C₄⋊Dic₁₉	—	2	1	no	—
115	Dic₁₉⋊C₄	—	4	1	no	—
116	Dic₁₉⋊C₄	—	4	1	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	—	2	2	no	—
123	C₂×C₁₉⋊C₈	—	4	1	no	—
124	C₂×C₁₉⋊C₈	—	4	1	no	—
125	#64516	—	4	8	no	—
126	#10269	—	19	1	no	maximal, Sylow
127	C₂³×C₁₉	#121	1	1	yes	characteristic
128	C₂²×D₁₉	—	4	2	no	—
129	C₂²×D₁₉	—	8	2	no	—
130	C₂²×D₁₉	—	8	1	no	—
131	C₂²×D₁₉	—	8	1	no	—
132	C₂²×D₁₉	—	8	2	no	—
133	Q₈×C₁₉	—	8	2	no	—
134	Q₈×C₁₉	—	8	2	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×SD₁₆	1	1	yes	characteristic
145	C₂×C₄×C₁₉	#135	1	2	yes	—
146	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
147	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
149	C₂×C₄×C₁₉	#137	1	2	yes	—
150	C₂×Dic₁₉	—	2	2	no	—
151	C₂×Dic₁₉	—	4	1	no	—
152	C₂×Dic₁₉	—	4	1	no	—
153	C₂×Dic₁₉	—	4	1	no	—
154	C₂×Dic₁₉	—	4	2	no	—
155	D₇₆	—	4	2	no	—
156	D₇₆	—	4	2	no	—
157	D₇₆	—	4	4	no	—
158	D₇₆	—	8	2	no	—
159	D₇₆	—	8	2	no	—
160	D₇₆	—	8	1	no	—
161	D₇₆	—	8	1	no	—
162	D₇₆	—	8	1	no	—
163	D₇₆	—	8	1	no	—
164	Dic₃₈	—	2	2	no	—
165	Dic₃₈	—	2	2	no	—
166	Dic₃₈	—	4	1	no	—
167	Dic₃₈	—	4	1	no	—
168	C₈×C₁₉	—	4	2	no	—
169	C₈×C₁₉	—	8	1	no	—
170	C₁₉⋊C₈	—	4	2	no	—
171	C₁₉⋊C₈	—	4	2	no	—
172	C₂×#486	—	19	1	no	—
173	C₂×#477	—	19	1	no	—
174	C₂×#460	—	19	1	no	—
175	#1984	—	19	1	no	—
176	C₂×#333	D₁₉	1	1	yes	characteristic, complemented
177	C₂×#331	—	19	1	no	—
178	#1277	—	19	1	no	—
179	C₂²×C₁₉	#392	1	1	yes	characteristic
180	C₂²×C₁₉	#464	1	2	yes	—
181	C₂²×C₁₉	#465	1	1	yes	characteristic
182	C₂²×C₁₉	#468	1	1	yes	characteristic
183	C₂²×C₁₉	#472	1	2	yes	—
184	C₂×D₁₉	—	8	2	no	—
185	C₂×D₁₉	—	8	2	no	—
186	C₂×D₁₉	—	8	2	no	—
187	C₂×D₁₉	—	16	2	no	—
188	C₂×D₁₉	—	16	2	no	—
189	C₄×C₁₉	—	2	1	no	—
190	C₄×C₁₉	—	2	1	no	—
191	C₄×C₁₉	—	8	2	no	—
192	C₄×C₁₉	#448	1	1	yes	characteristic
193	C₄×C₁₉	#449	1	1	yes	characteristic
194	C₄×C₁₉	#450	1	1	yes	characteristic
195	C₄×C₁₉	#452	1	1	yes	characteristic
196	C₄×C₁₉	#452	1	2	yes	—
197	C₄×C₁₉	#495	1	1	yes	characteristic
198	C₄×C₁₉	#497	1	1	yes	characteristic
199	Dic₁₉	—	4	2	no	—
200	Dic₁₉	—	8	1	no	—
201	C₂²×SD₁₆	—	38	1	no	—
202	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
203	C₂×#127	—	19	1	no	—
204	C₂×C₄×Q₈	—	19	1	no	—
205	#486	—	38	2	no	—
206	#486	—	38	2	no	—
207	#477	—	38	2	no	—
208	#477	—	38	2	no	—
209	#460	—	38	2	no	—
210	#460	—	38	2	no	—
211	C₂×#107	—	38	1	no	—
212	C₂×C₄⋊C₈	—	19	1	no	—
213	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
214	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
215	C₂×#103	—	38	1	no	—
216	C₂×#102	—	38	2	no	—
217	C₂×#102	—	38	1	no	—
218	C₂×C₄×C₈	—	19	1	no	—
219	#336	—	38	2	no	—
220	#333	—	2	2	no	—
221	#331	—	38	2	no	—
222	C₂×C₁₉	#1678	1	2	yes	—
223	C₂×C₁₉	#1698	1	2	yes	—
224	C₂×C₁₉	#2000	1	1	yes	characteristic
225	C₂×C₁₉	#2015	1	1	yes	characteristic
226	C₂×C₁₉	#2018	1	1	yes	characteristic
227	D₁₉	—	16	2	no	—
228	C₂²×Q₈	—	2	1	no	—
229	C₂²×Q₈	—	19	1	no	—
230	C₂²×D₄	—	38	1	no	—
231	C₂²×D₄	—	38	1	no	—
232	C₂²×D₄	—	38	1	no	—
233	C₂×SD₁₆	—	38	4	no	—
234	C₂×SD₁₆	—	76	2	no	—
235	C₂×SD₁₆	—	76	2	no	—
236	C₂²×C₈	—	2	1	no	—
237	C₂²×C₈	—	2	1	no	—
238	C₂²×C₈	—	19	2	no	—
239	C₂²×C₈	—	38	1	no	—
240	C₄⋊Q₈	—	2	1	no	—
241	C₄⋊Q₈	—	2	1	no	—
242	C₄⋊Q₈	—	2	1	no	—
243	C₄⋊Q₈	D₇₆	1	2	yes	—
244	#127	—	19	2	no	—
245	#127	—	38	1	no	—
246	#127	—	38	1	no	—
247	#127	—	38	1	no	—
248	C₄×Q₈	—	38	1	no	—
249	C₄×Q₈	—	38	1	no	—
250	C₄×Q₈	—	38	1	no	—
251	C₄×Q₈	—	38	1	no	—
252	C₂×C₄⋊C₄	—	2	1	no	—
253	C₂×C₄⋊C₄	—	2	1	no	—
254	C₂×C₄⋊C₄	—	19	1	no	—
255	C₂×C₄⋊C₄	—	38	1	no	—
256	C₂×C₄²	—	38	1	no	—
257	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
258	#107	—	38	4	no	—
259	C₄⋊C₈	—	2	1	no	—
260	C₄⋊C₈	—	2	1	no	—
261	C₄⋊C₈	—	2	1	no	—
262	C₄⋊C₈	—	19	2	no	—
263	C₄⋊C₈	—	38	1	no	—
264	C₄⋊C₈	C₂²⋊D₁₉	1	2	yes	—
265	#103	—	38	4	no	—
266	#102	—	38	8	no	—
267	#102	—	38	4	no	—
268	C₄×C₈	—	38	1	no	—
269	C₄×C₈	—	38	1	no	—
270	C₁₉	#10269	1	1	yes	characteristic, complemented, Sylow
271	C₂⁴	—	76	1	no	—
272	C₂×Q₈	—	2	2	no	—
273	C₂×Q₈	—	4	1	no	—
274	C₂×Q₈	—	8	2	no	—
275	C₂×Q₈	—	19	2	no	—
276	C₂×Q₈	—	38	1	no	—
277	C₂×Q₈	—	38	1	no	—
278	C₂×Q₈	—	38	1	no	—
279	C₂×Q₈	—	38	1	no	—
280	C₂×Q₈	—	38	1	no	—
281	C₂×D₄	—	38	2	no	—
282	C₂×D₄	—	38	2	no	—
283	C₂×D₄	—	38	2	no	—
284	C₂×D₄	—	76	1	no	—
285	C₂×D₄	—	76	1	no	—
286	C₂×D₄	—	76	1	no	—
287	C₂×D₄	—	76	1	no	—
288	C₂×D₄	—	76	1	no	—
289	C₂×D₄	—	76	2	no	—
290	C₂×D₄	—	76	2	no	—
291	C₂×D₄	—	76	1	no	—
292	C₂×D₄	—	76	1	no	—
293	C₂×D₄	—	152	2	no	—
294	C₂²×C₄	—	4	1	no	—
295	C₂²×C₄	—	38	1	no	—
296	C₂²×C₄	—	38	1	no	—
297	C₂²×C₄	#64541	1	1	yes	characteristic
298	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
299	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
300	SD₁₆	—	76	8	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	4	1	no	—
303	C₂×C₈	—	4	1	no	—
304	C₂×C₈	—	4	1	no	—
305	C₂×C₈	—	4	2	no	—
306	C₂×C₈	—	38	2	no	—
307	C₂×C₈	—	38	2	no	—
308	C₂×C₈	—	38	2	no	—
309	C₂×C₈	—	38	2	no	—
310	C₂×C₈	—	76	1	no	—
311	C₂×C₈	—	76	1	no	—
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	4	1	no	—
315	C₄⋊C₄	—	4	1	no	—
316	C₄⋊C₄	—	38	1	no	—
317	C₄⋊C₄	—	38	1	no	—
318	C₄⋊C₄	—	76	1	no	—
319	C₄⋊C₄	—	76	1	no	—
320	C₄²	—	76	1	no	—
321	C₄²	—	76	1	no	—
322	C₄²	C₂×D₇₆	1	1	yes	characteristic
323	C₄²	D₄×D₁₉	1	1	yes	characteristic
324	C₄²	D₄×D₁₉	1	1	yes	characteristic
325	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
326	C₂³	—	76	2	no	—
327	C₂³	—	152	1	no	—
328	C₂³	—	152	2	no	—
329	C₂³	—	152	2	no	—
330	C₂³	—	152	1	no	—
331	C₂³	#10597330	1	1	yes	characteristic
332	Q₈	—	8	2	no	—
333	Q₈	—	8	2	no	—
334	Q₈	—	38	2	no	—
335	Q₈	—	38	2	no	—
336	Q₈	—	76	1	no	—
337	Q₈	—	76	1	no	—
338	D₄	—	76	2	no	—
339	D₄	—	76	2	no	—
340	D₄	—	76	4	no	—
341	D₄	—	152	2	no	—
342	D₄	—	152	2	no	—
343	D₄	—	152	1	no	—
344	D₄	—	152	1	no	—
345	D₄	—	152	1	no	—
346	D₄	—	152	1	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	8	2	no	—
350	C₂×C₄	—	38	2	no	—
351	C₂×C₄	—	76	1	no	—
352	C₂×C₄	—	76	1	no	—
353	C₂×C₄	—	76	2	no	—
354	C₂×C₄	—	76	1	no	—
355	C₂×C₄	#10597328	1	1	yes	characteristic
356	C₂×C₄	#10597328	1	1	yes	characteristic
357	C₂×C₄	#10597330	1	1	yes	characteristic
358	C₂×C₄	SD₁₆×D₁₉	1	1	yes	characteristic
359	C₂×C₄	#10597360	1	1	yes	characteristic
360	C₂×C₄	#10597362	1	2	yes	—
361	C₂×C₄	#10597364	1	2	yes	—
362	C₂×C₄	#10597364	1	1	yes	characteristic
363	C₂×C₄	#10597365	1	1	yes	characteristic
364	C₂×C₄	#10597376	1	1	yes	characteristic
365	C₂×C₄	#10597383	1	1	yes	characteristic
366	C₂×C₄	#10597386	1	1	yes	characteristic
367	C₈	—	4	2	no	—
368	C₈	—	8	1	no	—
369	C₈	—	76	2	no	—
370	C₈	—	76	2	no	—
371	C₂²	—	304	2	no	—
372	C₂²	—	304	2	no	—
373	C₂²	—	152	2	no	—
374	C₂²	—	152	2	no	—
375	C₂²	—	152	2	no	—
376	C₂²	#11926453	1	1	yes	characteristic
377	C₂²	#11926769	1	2	yes	—
378	C₂²	#11926776	1	1	yes	characteristic
379	C₂²	#11926793	1	1	yes	characteristic
380	C₂²	#11926813	1	2	yes	—
381	C₄	—	2	1	no	—
382	C₄	—	2	1	no	—
383	C₄	—	8	2	no	—
384	C₄	—	76	2	no	—
385	C₄	—	152	1	no	—
386	C₄	#11926684	1	1	yes	characteristic
387	C₄	#11926692	1	1	yes	characteristic
388	C₄	#11926693	1	1	yes	characteristic
389	C₄	#11926705	1	1	yes	characteristic
390	C₄	#11926709	1	2	yes	—
391	C₄	#11926925	1	1	yes	characteristic
392	C₄	#11926937	1	1	yes	characteristic
393	C₂	—	304	2	no	—
394	C₂	#14563054	1	2	yes	—
395	C₂	#14563178	1	2	yes	—
396	C₂	#14564637	1	1	yes	characteristic
397	C₂	#14564730	1	1	yes	characteristic
398	C₂	#14564743	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597330
commutator, cosocle	C₂×C₄²×C₁₉	C₂³
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#333	C₂
socle	C₂³×C₁₉	#121
prime radical	C₂×#10597320	C₂²
hypercenter	C₂×#333	D₁₉
hypocenter	C₁₉	#10269
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#333

Complements

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

N	H ≅ G/N	type
2. C₂×#11926836	C₂	semidirect
4. #14564570	C₂	semidirect
6. #14561469	C₂	semidirect
7. C₂×C₁₉×#333	C₂	semidirect
176. C₂×#333	D₁₉	semidirect
270. C₁₉	#10269	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926892	2
C₂×#11926836	2
C₂×#11926745	2
#14564570	2
C₂×#11926287	2
#14561469	2
C₂×C₁₉×#333	2
#10269	19