Subgroups — #14567485

Order 2432.

620 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10597315	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#10597342	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×D₇₆	C₂	1	1	yes	maximal, characteristic
4	#11926466	C₂	1	2	yes	maximal, complemented
5	#11926457	C₂	1	2	yes	maximal, complemented
6	#11926446	C₂	1	1	yes	maximal, characteristic
7	#11926446	C₂	1	1	yes	maximal, characteristic
8	#11926442	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926442	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926418	C₂	1	2	yes	maximal, complemented
11	C₄×C₄⋊C₄×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₇₆	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₁₉	C₂²	1	2	yes	—
14	C₂²×Dic₃₈	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
20	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
21	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
22	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
23	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
24	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
25	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
26	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
29	C₂×Dic₁₉⋊C₄	C₂²	1	2	yes	—
30	C₂×Dic₁₉⋊C₄	C₂²	1	2	yes	—
31	#10597342	—	2	8	no	—
32	C₄×D₇₆	—	2	4	no	—
33	#10597315	—	2	4	no	—
34	C₁₉×#95	C₂²	1	1	yes	characteristic
35	C₁₉×#95	C₂²	1	1	yes	characteristic
36	C₁₉⋊#95	C₂²	1	2	yes	—
37	C₁₉⋊#95	C₂²	1	2	yes	—
38	C₂³×D₁₉	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
46	C₂²×Dic₁₉	C₂³	1	2	yes	—
47	C₂²×Dic₁₉	C₂³	1	2	yes	—
48	C₂²×Dic₁₉	C₂³	1	2	yes	—
49	C₂×D₇₆	—	2	2	no	—
50	C₂×D₇₆	—	2	2	no	—
51	C₂×D₇₆	Q₈	1	4	yes	complemented
52	C₂×C₄×D₁₉	—	2	4	no	—
53	C₂×C₄×D₁₉	—	2	4	no	—
54	C₂×C₄×D₁₉	—	4	2	no	—
55	C₂×Dic₃₈	—	2	4	no	—
56	C₂×Dic₃₈	—	2	4	no	—
57	C₂×Dic₃₈	—	4	2	no	—
58	C₄⋊C₄×C₁₉	—	2	2	no	—
59	C₄⋊C₄×C₁₉	D₄	1	4	yes	complemented
60	C₄²×C₁₉	—	2	2	no	—
61	C₄²×C₁₉	—	2	2	no	—
62	C₄²×C₁₉	—	2	2	no	—
63	(C₂×D₁₉)⋊C₄	—	2	4	no	—
64	(C₂×D₁₉)⋊C₄	—	2	4	no	—
65	(C₂×D₁₉)⋊C₄	—	2	4	no	—
66	(C₂×D₁₉)⋊C₄	—	2	4	no	—
67	(C₂×D₁₉)⋊C₄	—	4	2	no	—
68	C₄⋊Dic₁₉	—	2	2	no	—
69	C₄⋊Dic₁₉	—	2	2	no	—
70	C₄⋊Dic₁₉	—	2	4	no	—
71	C₄⋊Dic₁₉	D₄	1	4	yes	—
72	Dic₁₉⋊C₄	—	2	4	no	—
73	Dic₁₉⋊C₄	—	2	4	no	—
74	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
75	C₂²×D₁₉	—	2	2	no	—
76	C₂²×D₁₉	—	2	2	no	—
77	C₂²×D₁₉	—	2	2	no	—
78	C₂²×D₁₉	—	2	2	no	—
79	C₂²×D₁₉	—	2	2	no	—
80	C₂²×D₁₉	—	2	2	no	—
81	C₂²×D₁₉	C₂×Q₈	1	4	yes	—
82	C₂×C₄×C₁₉	—	2	1	no	—
83	C₂×C₄×C₁₉	—	2	1	no	—
84	C₂×C₄×C₁₉	—	2	1	no	—
85	C₂×C₄×C₁₉	—	2	1	no	—
86	C₂×C₄×C₁₉	—	2	1	no	—
87	C₂×C₄×C₁₉	—	2	1	no	—
88	C₂×C₄×C₁₉	—	2	1	no	—
89	C₂×C₄×C₁₉	—	2	1	no	—
90	C₂×C₄×C₁₉	—	2	1	no	—
91	C₂×C₄×C₁₉	—	2	1	no	—
92	C₂×C₄×C₁₉	—	2	1	no	—
93	C₂×C₄×C₁₉	—	2	1	no	—
94	C₂×C₄×C₁₉	—	2	1	no	—
95	C₂×C₄×C₁₉	—	2	1	no	—
96	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
100	C₂×C₄×C₁₉	C₂×Q₈	1	2	yes	—
101	C₂×C₄×C₁₉	#41	1	2	yes	—
102	C₂×C₄×C₁₉	#41	1	2	yes	—
103	C₂×Dic₁₉	—	2	2	no	—
104	C₂×Dic₁₉	—	2	2	no	—
105	C₂×Dic₁₉	—	2	2	no	—
106	C₂×Dic₁₉	—	2	2	no	—
107	C₂×Dic₁₉	—	2	2	no	—
108	C₂×Dic₁₉	—	2	2	no	—
109	C₂×Dic₁₉	—	2	2	no	—
110	C₂×Dic₁₉	—	2	2	no	—
111	C₂×Dic₁₉	C₂×D₄	1	4	yes	—
112	D₇₆	—	2	8	no	—
113	C₄×D₁₉	—	4	8	no	—
114	Dic₃₈	—	4	8	no	—
115	#2452	—	19	1	no	maximal, Sylow
116	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
117	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
118	C₂²×C₁₉	C₂²×Q₈	1	1	yes	characteristic
119	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
120	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
121	C₂²×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
122	C₂²×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
123	C₂×D₁₉	—	2	4	no	—
124	C₂×D₁₉	—	2	4	no	—
125	C₂×D₁₉	—	2	4	no	—
126	C₂×D₁₉	—	4	2	no	—
127	C₂×D₁₉	—	4	2	no	—
128	C₂×D₁₉	—	4	2	no	—
129	C₂×D₁₉	—	4	2	no	—
130	C₄×C₁₉	—	2	2	no	—
131	C₄×C₁₉	—	2	2	no	—
132	C₄×C₁₉	—	2	2	no	—
133	C₄×C₁₉	—	2	2	no	—
134	C₄×C₁₉	—	4	1	no	—
135	C₄×C₁₉	—	4	1	no	—
136	C₄×C₁₉	C₂²⋊Q₈	1	4	yes	complemented
137	Dic₁₉	—	2	4	no	—
138	Dic₁₉	—	4	2	no	—
139	Dic₁₉	—	4	2	no	—
140	C₂×C₄⋊Q₈	—	19	1	no	—
141	C₂×C₂²⋊Q₈	—	19	2	no	—
142	C₂×C₄×D₄	—	19	1	no	—
143	#396	—	19	2	no	—
144	#393	—	19	2	no	—
145	#390	—	19	1	no	—
146	#390	—	19	1	no	—
147	#389	—	19	1	no	—
148	#389	—	19	1	no	—
149	#385	—	19	2	no	—
150	C₄×C₄⋊C₄	D₁₉	1	1	yes	characteristic, complemented
151	C₂×C₁₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
152	C₂×C₁₉	#532	1	1	yes	characteristic
153	C₂×C₁₉	#542	1	1	yes	characteristic
154	C₂×C₁₉	D₄²	1	1	yes	characteristic
155	C₂×C₁₉	#548	1	1	yes	characteristic
156	C₂×C₁₉	D₄×Q₈	1	1	yes	characteristic
157	C₂×C₁₉	#554	1	1	yes	characteristic
158	D₁₉	—	4	4	no	—
159	C₂²×Q₈	—	19	2	no	—
160	C₂²×D₄	—	19	1	no	—
161	C₂³×C₄	—	19	2	no	—
162	C₄⋊Q₈	—	38	4	no	—
163	C₂²⋊Q₈	—	38	8	no	—
164	C₄×D₄	—	38	4	no	—
165	C₂×C₄⋊C₄	—	19	1	no	—
166	C₂×C₄⋊C₄	—	19	2	no	—
167	C₂×C₄⋊C₄	—	19	2	no	—
168	C₂×C₄⋊C₄	—	19	2	no	—
169	C₂×C₄⋊C₄	—	19	1	no	—
170	C₂×C₄⋊C₄	—	19	1	no	—
171	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
172	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
173	C₂×C₂²⋊C₄	—	19	2	no	—
174	C₂×C₂²⋊C₄	—	19	2	no	—
175	C₂×C₂²⋊C₄	—	19	2	no	—
176	C₂×C₂²⋊C₄	—	19	2	no	—
177	C₂×C₂²⋊C₄	—	19	2	no	—
178	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
179	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
180	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
181	#95	—	19	2	no	—
182	#95	—	19	2	no	—
183	#95	C₂×D₁₉	1	1	yes	characteristic
184	#95	C₂×D₁₉	1	1	yes	characteristic
185	C₁₉	#2452	1	1	yes	characteristic, complemented, Sylow
186	C₂⁴	—	19	2	no	—
187	C₂×Q₈	—	38	4	no	—
188	C₂×Q₈	—	38	4	no	—
189	C₂×Q₈	—	76	2	no	—
190	C₂×D₄	—	19	4	no	—
191	C₂×D₄	—	38	2	no	—
192	C₂×D₄	—	38	2	no	—
193	C₂²×C₄	—	19	2	no	—
194	C₂²×C₄	—	19	2	no	—
195	C₂²×C₄	—	19	2	no	—
196	C₂²×C₄	—	38	4	no	—
197	C₂²×C₄	—	38	4	no	—
198	C₂²×C₄	—	76	2	no	—
199	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
200	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
201	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
202	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
203	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
204	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
205	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	19	4	no	—
208	C₄⋊C₄	—	38	4	no	—
209	C₄⋊C₄	—	38	4	no	—
210	C₄⋊C₄	—	38	4	no	—
211	C₄⋊C₄	—	38	2	no	—
212	C₄⋊C₄	—	38	2	no	—
213	C₄⋊C₄	D₇₆	1	4	yes	complemented
214	C₂²⋊C₄	—	38	4	no	—
215	C₂²⋊C₄	—	38	4	no	—
216	C₂²⋊C₄	—	38	4	no	—
217	C₂²⋊C₄	—	38	4	no	—
218	C₂²⋊C₄	—	76	2	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₂³	—	19	4	no	—
223	C₂³	—	38	2	no	—
224	C₂³	—	38	2	no	—
225	C₂³	—	38	2	no	—
226	C₂³	—	38	2	no	—
227	C₂³	—	38	2	no	—
228	C₂³	—	38	2	no	—
229	C₂³	C₂³×D₁₉	1	1	yes	characteristic
230	Q₈	—	76	8	no	—
231	D₄	—	38	8	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	19	4	no	—
247	C₂×C₄	—	38	2	no	—
248	C₂×C₄	—	38	2	no	—
249	C₂×C₄	—	38	2	no	—
250	C₂×C₄	—	38	2	no	—
251	C₂×C₄	—	38	2	no	—
252	C₂×C₄	—	38	2	no	—
253	C₂×C₄	—	38	2	no	—
254	C₂×C₄	—	38	2	no	—
255	C₂×C₄	—	76	8	no	—
256	C₂×C₄	C₂×D₇₆	1	2	yes	—
257	C₂×C₄	C₂×D₇₆	1	2	yes	—
258	C₂×C₄	C₂×D₇₆	1	2	yes	—
259	C₂×C₄	#64541	1	2	yes	—
260	C₂×C₄	#64541	1	2	yes	—
261	C₂×C₄	D₄×D₁₉	1	2	yes	—
262	C₂×C₄	Q₈×D₁₉	1	2	yes	—
263	C₂²	—	38	4	no	—
264	C₂²	—	38	4	no	—
265	C₂²	—	38	4	no	—
266	C₂²	—	76	2	no	—
267	C₂²	—	76	2	no	—
268	C₂²	—	76	2	no	—
269	C₂²	—	76	2	no	—
270	C₂²	C₂²×D₇₆	1	1	yes	characteristic
271	C₂²	C₂×#64541	1	1	yes	characteristic
272	C₂²	C₂×#64541	1	1	yes	characteristic
273	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
274	C₂²	C₂×Q₈×D₁₉	1	1	yes	characteristic
275	C₂²	#10597435	1	1	yes	characteristic
276	C₂²	#10597436	1	1	yes	characteristic
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	4	1	no	—
282	C₄	—	4	1	no	—
283	C₄	—	38	4	no	—
284	C₄	—	76	2	no	—
285	C₄	—	76	2	no	—
286	C₄	#10597342	1	4	yes	complemented
287	C₂	—	76	4	no	—
288	C₂	C₂×#10597342	1	1	yes	characteristic
289	C₂	#11927094	1	1	yes	characteristic
290	C₂	#11927165	1	1	yes	characteristic
291	C₂	D₄×D₇₆	1	1	yes	characteristic
292	C₂	#11927203	1	1	yes	characteristic
293	C₂	Q₈×D₇₆	1	1	yes	characteristic
294	C₂	#11927254	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉
commutator, socle, cosocle	C₂³×C₁₉	C₂⁴
Fitting	C₄×C₄⋊C₄×C₁₉	C₂
prime radical	C₂²×D₇₆	C₂²
hypercenter	C₄×C₄⋊C₄	D₁₉
hypocenter	C₁₉	#2452
LCS	G ▷ C₂³×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#10597315	C₂	semidirect
2. C₂×#10597342	C₂	semidirect
4. #11926466	C₂	semidirect
5. #11926457	C₂	semidirect
8. #11926442	C₂	semidirect
9. #11926442	C₂	semidirect
10. #11926418	C₂	semidirect
11. C₄×C₄⋊C₄×C₁₉	C₂	semidirect
51. C₂×D₇₆	Q₈	semidirect
59. C₄⋊C₄×C₁₉	D₄	semidirect
136. C₄×C₁₉	C₂²⋊Q₈	semidirect
150. C₄×C₄⋊C₄	D₁₉	semidirect
185. C₁₉	#2452	semidirect
213. C₄⋊C₄	D₇₆	semidirect
286. C₄	#10597342	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10597315	2
C₂×#10597342	2
C₂×C₄×D₇₆	2
#11926466	2
#11926457	2
#11926446	2
#11926442	2
#11926418	2
C₄×C₄⋊C₄×C₁₉	2
#2452	19