Subgroups — #14574954

Order 2432.

398 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927089	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927075	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926847	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926846	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926838	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926832	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926778	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926773	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926771	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926758	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926757	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926743	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926737	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×Dic₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#466	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#64543	—	2	1	no	—
17	C₂×D₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₁₉	C₂²	1	1	yes	characteristic
19	C₁₉×#121	C₂²	1	1	yes	characteristic
20	C₁₉×#121	C₂²	1	1	yes	characteristic
21	#10597386	C₂²	1	1	yes	characteristic
22	#10597385	—	2	1	no	—
23	#10597382	C₂²	1	1	yes	characteristic
24	#10597381	—	2	1	no	—
25	D₄×Dic₁₉	—	2	1	no	—
26	D₄×Dic₁₉	C₂²	1	1	yes	characteristic
27	D₄×Dic₁₉	C₂²	1	1	yes	characteristic
28	C₂×#64526	C₂²	1	1	yes	characteristic
29	C₂×#64525	C₂²	1	1	yes	characteristic
30	#10597376	C₂²	1	1	yes	characteristic
31	#10597370	C₂²	1	1	yes	characteristic
32	C₂×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
33	C₂²×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
34	C₂×#64516	C₂²	1	1	yes	characteristic
35	#10597339	C₂²	1	1	yes	characteristic
36	#10597336	C₂²	1	1	yes	characteristic
37	#10597334	C₂²	1	1	yes	characteristic
38	#10597333	C₂²	1	1	yes	characteristic
39	#10597331	—	2	1	no	—
40	(C₂²⋊D₁₉)⋊C₄	—	2	1	no	—
41	#10597325	—	2	1	no	—
42	#10597323	—	2	1	no	—
43	C₁₉×#107	C₂²	1	1	yes	characteristic
44	C₁₉×#102	C₂²	1	1	yes	characteristic
45	C₁₉×#102	C₂²	1	1	yes	characteristic
46	#10597289	C₂²	1	1	yes	characteristic
47	#10597289	C₂²	1	1	yes	characteristic
48	#10597278	C₂²	1	1	yes	characteristic
49	#10597274	C₂²	1	1	yes	characteristic
50	#10597273	C₂²	1	1	yes	characteristic
51	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
52	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
53	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
54	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
55	#10597267	C₂²	1	1	yes	characteristic
56	#10597266	C₂²	1	1	yes	characteristic
57	#10597265	C₂²	1	1	yes	characteristic
58	#10597264	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₁₉	—	2	1	no	—
60	C₂×D₄×C₁₉	—	2	1	no	—
61	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₉	—	2	1	no	—
65	C₂×C₂²⋊D₁₉	—	2	1	no	—
66	C₂×C₂²⋊D₁₉	—	2	1	no	—
67	C₂²×Dic₁₉	—	2	1	no	—
68	C₂²×Dic₁₉	—	2	1	no	—
69	C₂²×Dic₁₉	D₄	1	1	yes	characteristic
70	C₂²×Dic₁₉	D₄	1	1	yes	characteristic
71	#64543	—	4	4	no	—
72	C₂×D₇₆	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₁₉	—	2	1	no	—
74	C₂×Dic₃₈	—	2	1	no	—
75	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
76	D₈×C₁₉	—	4	1	no	—
77	C₂×C₈×C₁₉	—	4	1	no	—
78	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
79	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
82	C₂²⋊C₄×C₁₉	—	2	1	no	—
83	C₂²⋊C₄×C₁₉	—	2	1	no	—
84	C₂²⋊Dic₁₉	—	2	1	no	—
85	C₂²⋊Dic₁₉	—	2	1	no	—
86	C₂²⋊Dic₁₉	—	2	1	no	—
87	C₂²⋊Dic₁₉	—	2	1	no	—
88	C₂²⋊Dic₁₉	—	2	1	no	—
89	#64526	—	4	1	no	—
90	#64525	—	4	1	no	—
91	(C₂×D₁₉)⋊C₄	—	2	1	no	—
92	(C₂×D₁₉)⋊C₄	—	2	1	no	—
93	C₄⋊Dic₁₉	—	2	1	no	—
94	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
95	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
96	Dic₁₉⋊C₄	—	2	1	no	—
97	Dic₁₉⋊C₄	—	2	1	no	—
98	Dic₁₉⋊C₄	—	2	1	no	—
99	Dic₁₉⋊C₄	—	2	1	no	—
100	C₄×Dic₁₉	—	2	1	no	—
101	C₄×Dic₁₉	D₄	1	1	yes	characteristic
102	C₄×Dic₁₉	D₄	1	1	yes	characteristic
103	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
104	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
105	C₂×C₁₉⋊C₈	—	4	1	no	—
106	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
107	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
108	#64516	—	4	1	no	—
109	C₂³×C₁₉	—	2	1	no	—
110	C₂³×C₁₉	—	2	1	no	—
111	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
112	C₂²×D₁₉	—	2	1	no	—
113	D₄×C₁₉	—	2	1	no	—
114	D₄×C₁₉	—	2	1	no	—
115	D₄×C₁₉	—	2	1	no	—
116	D₄×C₁₉	—	2	1	no	—
117	D₄×C₁₉	—	4	2	no	—
118	D₄×C₁₉	—	4	2	no	—
119	C₂×C₄×C₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	—	2	1	no	—
123	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
125	C₂²⋊D₁₉	—	4	2	no	—
126	C₂²⋊D₁₉	—	4	2	no	—
127	C₂²⋊D₁₉	—	4	2	no	—
128	C₂×Dic₁₉	—	2	2	no	—
129	C₂×Dic₁₉	—	2	2	no	—
130	C₂×Dic₁₉	—	2	1	no	—
131	C₂×Dic₁₉	—	2	1	no	—
132	C₂×Dic₁₉	—	2	1	no	—
133	C₂×Dic₁₉	—	4	2	no	—
134	C₂×Dic₁₉	—	4	2	no	—
135	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
138	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
139	D₇₆	—	2	1	no	—
140	D₇₆	—	2	1	no	—
141	C₄×D₁₉	—	4	2	no	—
142	Dic₃₈	—	2	1	no	—
143	Dic₃₈	—	2	1	no	—
144	Dic₃₈	—	4	2	no	—
145	C₈×C₁₉	—	2	1	no	—
146	C₈×C₁₉	—	2	1	no	—
147	C₁₉⋊C₈	—	2	1	no	—
148	C₁₉⋊C₈	—	2	1	no	—
149	#3247	—	19	1	no	maximal, Sylow
150	C₂²×C₁₉	—	2	1	no	—
151	C₂²×C₁₉	—	2	1	no	—
152	C₂²×C₁₉	—	4	1	no	—
153	C₂²×C₁₉	—	4	1	no	—
154	C₂²×C₁₉	—	4	1	no	—
155	C₂²×C₁₉	—	4	1	no	—
156	C₂²×C₁₉	—	4	1	no	—
157	C₂²×C₁₉	—	4	1	no	—
158	C₂²×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
159	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
160	C₂×D₁₉	—	4	1	no	—
161	C₂×D₁₉	—	4	1	no	—
162	C₂×D₁₉	—	4	1	no	—
163	C₄×C₁₉	—	2	1	no	—
164	C₄×C₁₉	—	4	1	no	—
165	C₄×C₁₉	—	4	1	no	—
166	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
167	C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
168	Dic₁₉	—	2	1	no	—
169	Dic₁₉	—	2	1	no	—
170	Dic₁₉	—	2	1	no	—
171	Dic₁₉	—	4	1	no	—
172	Dic₁₉	—	4	1	no	—
173	Dic₁₉	—	4	1	no	—
174	Dic₁₉	#135	1	2	yes	—
175	#532	—	19	1	no	—
176	#529	—	19	1	no	—
177	#478	—	19	1	no	—
178	#478	—	19	1	no	—
179	#477	—	19	1	no	—
180	#476	—	19	1	no	—
181	#466	—	19	1	no	—
182	#466	D₁₉	1	1	yes	characteristic, complemented
183	#465	—	19	1	no	—
184	#465	—	19	1	no	—
185	#463	—	19	1	no	—
186	#463	—	19	1	no	—
187	#460	—	19	1	no	—
188	#459	—	19	1	no	—
189	C₂×C₄⋊C₈	—	19	1	no	—
190	C₂×C₁₉	—	2	2	no	—
191	C₂×C₁₉	—	8	1	no	—
192	C₂×C₁₉	—	8	1	no	—
193	C₂×C₁₉	#537	1	1	yes	characteristic
194	C₂×C₁₉	C₂×#135	1	1	yes	characteristic
195	C₂×C₁₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
196	D₁₉	—	8	1	no	—
197	C₂×#41	—	38	1	no	—
198	C₂×SD₁₆	—	19	1	no	—
199	C₂×SD₁₆	—	19	1	no	—
200	C₂×D₈	—	19	1	no	—
201	C₂×D₈	C₂×D₁₉	1	1	yes	characteristic
202	C₂²×C₈	—	19	1	no	—
203	C₂²×C₈	C₂×D₁₉	1	1	yes	characteristic
204	C₄⋊Q₈	—	19	1	no	—
205	#127	—	19	1	no	—
206	#126	—	38	1	no	—
207	#125	—	19	1	no	—
208	#124	—	19	1	no	—
209	#124	—	38	1	no	—
210	#123	—	38	1	no	—
211	C₂²⋊Q₈	—	19	1	no	—
212	#121	—	19	1	no	—
213	#121	—	38	1	no	—
214	#121	C₂×D₁₉	1	1	yes	characteristic
215	#121	C₂×D₁₉	1	1	yes	characteristic
216	C₄×Q₈	—	19	1	no	—
217	C₄×D₄	—	19	1	no	—
218	C₄×D₄	—	19	1	no	—
219	C₄×D₄	—	19	1	no	—
220	C₄×D₄	—	38	1	no	—
221	C₄×D₄	—	38	1	no	—
222	#117	—	38	1	no	—
223	C₂×C₄²	—	19	1	no	—
224	#107	—	19	1	no	—
225	#107	C₂×D₁₉	1	1	yes	characteristic
226	#106	—	19	1	no	—
227	#106	—	19	1	no	—
228	C₄⋊C₈	—	19	1	no	—
229	C₄⋊C₈	—	19	1	no	—
230	C₄⋊C₈	—	19	1	no	—
231	C₄⋊C₈	—	19	1	no	—
232	#103	—	19	1	no	—
233	#103	—	19	1	no	—
234	#102	—	19	1	no	—
235	#102	—	19	1	no	—
236	#102	—	19	1	no	—
237	#102	—	19	1	no	—
238	#102	C₂×D₁₉	1	1	yes	characteristic
239	#102	C₂×D₁₉	1	1	yes	characteristic
240	C₁₉	#3247	1	1	yes	characteristic, complemented, Sylow
241	#41	—	76	4	no	—
242	C₂×Q₈	—	19	1	no	—
243	C₂×Q₈	—	38	1	no	—
244	C₂×D₄	—	2	1	no	—
245	C₂×D₄	—	2	1	no	—
246	C₂×D₄	—	19	1	no	—
247	C₂×D₄	—	38	1	no	—
248	C₂×D₄	—	38	1	no	—
249	C₂×D₄	—	38	1	no	—
250	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
251	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
252	C₂²×C₄	—	19	1	no	—
253	C₂²×C₄	—	19	1	no	—
254	C₂²×C₄	—	38	1	no	—
255	C₂²×C₄	—	38	1	no	—
256	C₂²×C₄	—	38	1	no	—
257	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
258	SD₁₆	—	76	1	no	—
259	SD₁₆	—	76	1	no	—
260	D₈	—	4	1	no	—
261	D₈	—	76	1	no	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	—	19	1	no	—
264	C₂×C₈	—	19	1	no	—
265	C₂×C₈	—	76	1	no	—
266	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
267	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
268	C₄⋊C₄	—	19	1	no	—
269	C₄⋊C₄	—	19	1	no	—
270	C₄⋊C₄	—	38	1	no	—
271	C₄⋊C₄	—	38	1	no	—
272	C₄⋊C₄	—	38	1	no	—
273	C₄⋊C₄	—	38	1	no	—
274	C₄⋊C₄	—	38	1	no	—
275	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	38	1	no	—
280	C₂²⋊C₄	—	38	1	no	—
281	C₂²⋊C₄	—	38	1	no	—
282	C₂²⋊C₄	—	38	1	no	—
283	C₂²⋊C₄	—	38	1	no	—
284	C₂²⋊C₄	—	38	1	no	—
285	C₂²⋊C₄	—	38	1	no	—
286	C₄²	—	19	1	no	—
287	C₄²	—	19	1	no	—
288	C₄²	—	19	1	no	—
289	C₄²	—	19	1	no	—
290	C₄²	—	38	1	no	—
291	C₂³	—	2	1	no	—
292	C₂³	—	2	1	no	—
293	C₂³	—	38	1	no	—
294	C₂³	D₄×D₁₉	1	1	yes	characteristic
295	Q₈	—	38	1	no	—
296	Q₈	—	38	1	no	—
297	Q₈	—	76	2	no	—
298	D₄	—	2	1	no	—
299	D₄	—	2	1	no	—
300	D₄	—	2	1	no	—
301	D₄	—	2	1	no	—
302	D₄	—	4	2	no	—
303	D₄	—	4	2	no	—
304	D₄	—	38	1	no	—
305	D₄	—	38	1	no	—
306	D₄	—	76	2	no	—
307	D₄	—	76	2	no	—
308	D₄	—	76	2	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	19	1	no	—
314	C₂×C₄	—	19	1	no	—
315	C₂×C₄	—	19	1	no	—
316	C₂×C₄	—	19	1	no	—
317	C₂×C₄	—	38	2	no	—
318	C₂×C₄	—	38	2	no	—
319	C₂×C₄	—	38	1	no	—
320	C₂×C₄	—	38	1	no	—
321	C₂×C₄	—	38	1	no	—
322	C₂×C₄	—	76	2	no	—
323	C₂×C₄	—	76	2	no	—
324	C₂×C₄	—	76	2	no	—
325	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
326	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
327	C₈	—	2	1	no	—
328	C₈	—	2	1	no	—
329	C₈	—	38	1	no	—
330	C₈	—	38	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	4	1	no	—
339	C₂²	—	76	1	no	—
340	C₂²	—	76	1	no	—
341	C₂²	—	76	1	no	—
342	C₂²	#10597357	1	2	yes	—
343	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	19	2	no	—
348	C₄	—	38	1	no	—
349	C₄	—	38	1	no	—
350	C₄	—	38	1	no	—
351	C₄	—	76	1	no	—
352	C₄	—	76	1	no	—
353	C₄	—	76	1	no	—
354	C₄	#10597430	1	1	yes	characteristic
355	C₄	#10597436	1	1	yes	characteristic
356	C₂	—	2	2	no	—
357	C₂	—	8	1	no	—
358	C₂	—	8	1	no	—
359	C₂	—	152	1	no	—
360	C₂	#11927121	1	1	yes	characteristic
361	C₂	#135×D₁₉	1	1	yes	characteristic
362	C₂	C₂×#10597357	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₉
commutator, cosocle	C₂×C₄×C₁₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉
Fitting	C₁₉×#466	C₂
socle	C₂²×C₁₉	C₂²×D₄
hypercenter	#466	D₁₉
hypocenter	C₁₉	#3247
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #466

Complements

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

N	H ≅ G/N	type
1. #11927089	C₂	semidirect
2. #11927075	C₂	semidirect
3. #11926847	C₂	semidirect
4. #11926846	C₂	semidirect
5. #11926838	C₂	semidirect
6. #11926832	C₂	semidirect
7. #11926778	C₂	semidirect
8. #11926773	C₂	semidirect
9. #11926771	C₂	semidirect
10. #11926758	C₂	semidirect
11. #11926757	C₂	semidirect
12. #11926743	C₂	semidirect
13. #11926737	C₂	semidirect
14. C₂×Dic₁₉⋊C₈	C₂	semidirect
15. C₁₉×#466	C₂	semidirect
182. #466	D₁₉	semidirect
240. C₁₉	#3247	semidirect

Maximal subgroups

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

subgroup	index
#11927089	2
#11927075	2
#11926847	2
#11926846	2
#11926838	2
#11926832	2
#11926778	2
#11926773	2
#11926771	2
#11926758	2
#11926757	2
#11926743	2
#11926737	2
C₂×Dic₁₉⋊C₈	2
C₁₉×#466	2
#3247	19