Subgroups — #14567433

Order 2432.

512 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10597321	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₃₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926479	C₂	1	2	yes	maximal, complemented
4	#11926473	C₂	1	2	yes	maximal, complemented
5	#11926450	C₂	1	1	yes	maximal, characteristic, 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	#11926434	C₂	1	2	yes	maximal, complemented
10	#11926433	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×Dic₃₈	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
16	C₂×C₄²×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×(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
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	1	yes	characteristic
26	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
27	C₂×Dic₁₉⋊C₄	C₂²	1	2	yes	—
28	C₂×Dic₁₉⋊C₄	C₂²	1	2	yes	—
29	C₂×C₄×Dic₁₉	C₂²	1	2	yes	—
30	#10597321	—	2	4	no	—
31	C₄×Dic₃₈	—	2	4	no	—
32	C₁₉×#95	C₂²	1	1	yes	characteristic
33	C₁₉×#95	C₂²	1	1	yes	characteristic
34	C₁₉⋊#95	C₂²	1	2	yes	—
35	C₁₉⋊#95	C₂²	1	2	yes	—
36	C₁₉⋊#95	C₂²	1	2	yes	—
37	C₂³×D₁₉	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
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×Dic₁₉	C₂³	1	2	yes	—
46	C₂²×Dic₁₉	C₂³	1	2	yes	—
47	C₂²×Dic₁₉	C₂³	1	2	yes	—
48	C₂×D₇₆	—	2	2	no	—
49	C₂×D₇₆	—	2	2	no	—
50	C₂×D₇₆	—	2	2	no	—
51	C₂×Dic₃₈	—	2	2	no	—
52	C₂×Dic₃₈	—	2	2	no	—
53	C₂×Dic₃₈	D₄	1	4	yes	—
54	C₄⋊C₄×C₁₉	—	2	2	no	—
55	C₄⋊C₄×C₁₉	—	2	2	no	—
56	C₄²×C₁₉	—	2	2	no	—
57	C₄²×C₁₉	—	2	2	no	—
58	C₄²×C₁₉	—	2	2	no	—
59	(C₂×D₁₉)⋊C₄	—	2	4	no	—
60	(C₂×D₁₉)⋊C₄	—	2	4	no	—
61	(C₂×D₁₉)⋊C₄	—	2	4	no	—
62	(C₂×D₁₉)⋊C₄	—	2	4	no	—
63	(C₂×D₁₉)⋊C₄	—	4	2	no	—
64	(C₂×D₁₉)⋊C₄	—	4	2	no	—
65	C₄⋊Dic₁₉	—	2	2	no	—
66	C₄⋊Dic₁₉	—	2	2	no	—
67	Dic₁₉⋊C₄	—	2	4	no	—
68	Dic₁₉⋊C₄	—	4	2	no	—
69	C₄×Dic₁₉	—	2	4	no	—
70	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
71	C₂²×D₁₉	—	2	2	no	—
72	C₂²×D₁₉	—	2	2	no	—
73	C₂²×D₁₉	—	2	2	no	—
74	C₂²×D₁₉	—	2	2	no	—
75	C₂²×D₁₉	—	2	2	no	—
76	C₂²×D₁₉	—	2	2	no	—
77	C₂²×D₁₉	—	2	2	no	—
78	C₂×C₄×C₁₉	—	2	1	no	—
79	C₂×C₄×C₁₉	—	2	1	no	—
80	C₂×C₄×C₁₉	—	2	1	no	—
81	C₂×C₄×C₁₉	—	2	1	no	—
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	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₁₉	#41	1	2	yes	—
96	C₂×C₄×C₁₉	#41	1	2	yes	—
97	C₂×C₄×C₁₉	#41	1	2	yes	—
98	C₂×C₄×C₁₉	#41	1	2	yes	—
99	C₂×Dic₁₉	—	2	2	no	—
100	C₂×Dic₁₉	—	2	2	no	—
101	C₂×Dic₁₉	—	2	2	no	—
102	C₂×Dic₁₉	—	2	2	no	—
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	C₂×D₄	1	4	yes	—
108	D₇₆	—	4	4	no	—
109	Dic₃₈	—	2	8	no	—
110	#2448	—	19	1	no	maximal, Sylow
111	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
112	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
113	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
114	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
115	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
116	C₂²×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
117	C₂²×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
118	C₂×D₁₉	—	4	2	no	—
119	C₂×D₁₉	—	4	2	no	—
120	C₂×D₁₉	—	4	2	no	—
121	C₂×D₁₉	—	4	2	no	—
122	C₂×D₁₉	—	4	2	no	—
123	C₂×D₁₉	—	4	2	no	—
124	C₂×D₁₉	—	4	2	no	—
125	C₄×C₁₉	—	2	2	no	—
126	C₄×C₁₉	—	2	2	no	—
127	C₄×C₁₉	—	4	1	no	—
128	C₄×C₁₉	—	4	1	no	—
129	C₄×C₁₉	—	4	1	no	—
130	C₄×C₁₉	—	4	1	no	—
131	C₄×C₁₉	#123	1	4	yes	—
132	Dic₁₉	—	2	4	no	—
133	Dic₁₉	—	4	2	no	—
134	Dic₁₉	—	4	2	no	—
135	C₂×#124	—	19	1	no	—
136	C₂×C₄×Q₈	—	19	1	no	—
137	#399	—	19	2	no	—
138	#397	—	19	2	no	—
139	#391	—	19	1	no	—
140	#390	—	19	1	no	—
141	#390	—	19	1	no	—
142	#389	—	19	1	no	—
143	#388	—	19	2	no	—
144	#388	—	19	2	no	—
145	C₄×C₄⋊C₄	D₁₉	1	1	yes	characteristic, complemented
146	C₂×C₁₉	C₂×#123	1	1	yes	characteristic
147	C₂×C₁₉	#529	1	1	yes	characteristic
148	C₂×C₁₉	#542	1	1	yes	characteristic
149	C₂×C₁₉	#548	1	1	yes	characteristic
150	C₂×C₁₉	#550	1	1	yes	characteristic
151	C₂×C₁₉	#556	1	1	yes	characteristic
152	C₂×C₁₉	#559	1	1	yes	characteristic
153	D₁₉	—	8	2	no	—
154	C₂²×Q₈	—	19	1	no	—
155	C₂²×D₄	—	19	1	no	—
156	#124	—	38	4	no	—
157	C₄×Q₈	—	38	4	no	—
158	C₂×C₄⋊C₄	—	19	1	no	—
159	C₂×C₄⋊C₄	—	19	2	no	—
160	C₂×C₄⋊C₄	—	19	1	no	—
161	C₂×C₄⋊C₄	—	19	2	no	—
162	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
163	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
164	C₂×C₂²⋊C₄	—	19	2	no	—
165	C₂×C₂²⋊C₄	—	19	2	no	—
166	C₂×C₂²⋊C₄	—	19	2	no	—
167	C₂×C₂²⋊C₄	—	19	2	no	—
168	C₂×C₂²⋊C₄	—	19	2	no	—
169	C₂×C₂²⋊C₄	—	19	2	no	—
170	C₂×C₄²	—	19	2	no	—
171	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
172	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
173	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
174	#95	—	19	2	no	—
175	#95	—	19	2	no	—
176	#95	—	19	2	no	—
177	#95	C₂×D₁₉	1	1	yes	characteristic
178	#95	C₂×D₁₉	1	1	yes	characteristic
179	C₁₉	#2448	1	1	yes	characteristic, complemented, Sylow
180	C₂⁴	—	19	2	no	—
181	C₂×Q₈	—	19	4	no	—
182	C₂×Q₈	—	38	2	no	—
183	C₂×Q₈	—	38	2	no	—
184	C₂×D₄	—	38	2	no	—
185	C₂×D₄	—	38	2	no	—
186	C₂×D₄	—	38	2	no	—
187	C₂²×C₄	—	19	2	no	—
188	C₂²×C₄	—	19	2	no	—
189	C₂²×C₄	—	19	2	no	—
190	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
191	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
192	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
193	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
194	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
195	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
196	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	38	4	no	—
200	C₄⋊C₄	—	38	2	no	—
201	C₄⋊C₄	—	38	2	no	—
202	C₄⋊C₄	—	76	2	no	—
203	C₂²⋊C₄	—	38	4	no	—
204	C₂²⋊C₄	—	38	4	no	—
205	C₂²⋊C₄	—	38	4	no	—
206	C₂²⋊C₄	—	38	4	no	—
207	C₂²⋊C₄	—	76	2	no	—
208	C₂²⋊C₄	—	76	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	38	4	no	—
213	C₂³	—	38	2	no	—
214	C₂³	—	38	2	no	—
215	C₂³	—	38	2	no	—
216	C₂³	—	38	2	no	—
217	C₂³	—	38	2	no	—
218	C₂³	—	38	2	no	—
219	C₂³	—	38	2	no	—
220	C₂³	C₂³×D₁₉	1	1	yes	characteristic
221	Q₈	—	38	8	no	—
222	D₄	—	76	4	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	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	—	19	4	no	—
240	C₂×C₄	—	38	2	no	—
241	C₂×C₄	—	38	2	no	—
242	C₂×C₄	—	38	2	no	—
243	C₂×C₄	—	38	2	no	—
244	C₂×C₄	—	38	2	no	—
245	C₂×C₄	—	38	2	no	—
246	C₂×C₄	—	38	2	no	—
247	C₂×C₄	—	38	2	no	—
248	C₂×C₄	#64541	1	2	yes	—
249	C₂×C₄	#64541	1	2	yes	—
250	C₂×C₄	#64541	1	2	yes	—
251	C₂×C₄	D₄×D₁₉	1	2	yes	—
252	C₂×C₄	#64545	1	2	yes	—
253	C₂²	—	76	2	no	—
254	C₂²	—	76	2	no	—
255	C₂²	—	76	2	no	—
256	C₂²	—	76	2	no	—
257	C₂²	—	76	2	no	—
258	C₂²	—	76	2	no	—
259	C₂²	—	76	2	no	—
260	C₂²	C₂×#64541	1	1	yes	characteristic
261	C₂²	C₂×#64541	1	1	yes	characteristic
262	C₂²	C₂×#64541	1	1	yes	characteristic
263	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
264	C₂²	C₂×#64545	1	1	yes	characteristic
265	C₂²	#10597435	1	1	yes	characteristic
266	C₂²	#10597436	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	4	1	no	—
270	C₄	—	4	1	no	—
271	C₄	—	4	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	38	4	no	—
274	C₄	—	76	2	no	—
275	C₄	—	76	2	no	—
276	C₄	#10597340	1	4	yes	—
277	C₂	—	152	2	no	—
278	C₂	C₂×#10597340	1	1	yes	characteristic
279	C₂	#11927078	1	1	yes	characteristic
280	C₂	#11927165	1	1	yes	characteristic
281	C₂	#11927203	1	1	yes	characteristic
282	C₂	#11927215	1	1	yes	characteristic
283	C₂	#11927267	1	1	yes	characteristic
284	C₂	#11927279	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	#2448
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
2. C₂×C₄×Dic₃₈	C₂	semidirect
3. #11926479	C₂	semidirect
4. #11926473	C₂	semidirect
5. #11926450	C₂	semidirect
8. #11926442	C₂	semidirect
9. #11926434	C₂	semidirect
10. #11926433	C₂	semidirect
11. C₄×C₄⋊C₄×C₁₉	C₂	semidirect
145. C₄×C₄⋊C₄	D₁₉	semidirect
179. C₁₉	#2448	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10597321	2
C₂×C₄×Dic₃₈	2
#11926479	2
#11926473	2
#11926450	2
#11926446	2
#11926442	2
#11926434	2
#11926433	2
C₄×C₄⋊C₄×C₁₉	2
#2448	19