Subgroups — #14575322

Order 2432.

526 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10597350	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₁₅₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#11927190	C₂	1	2	yes	maximal, complemented
4	#11926928	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926785	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926780	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926760	C₂	1	2	yes	maximal, complemented
8	#11926697	C₂	1	2	yes	maximal, complemented
9	#11926691	C₂	1	2	yes	maximal, complemented
10	C₄×D₁₅₂	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#64541	C₂²	1	2	yes	complemented
13	C₂²×D₇₆	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₉	C₂²	1	1	yes	characteristic, complemented
16	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
17	#10597383	—	2	2	no	—
18	#10597376	C₂²	1	2	yes	complemented
19	#10597370	C₂²	1	2	yes	—
20	C₂×Dic₇₆	C₂²	1	1	yes	characteristic, complemented
21	#10597350	—	2	4	no	—
22	C₂×D₁₅₂	—	2	4	no	—
23	C₂×D₁₅₂	C₂²	1	1	yes	characteristic
24	C₂×D₁₅₂	C₂²	1	2	yes	—
25	C₂×D₁₅₂	C₂²	1	1	yes	characteristic
26	C₂×#64516	C₂²	1	2	yes	complemented
27	C₂×#64516	C₂²	1	2	yes	complemented
28	#10597340	—	2	2	no	—
29	#10597330	—	2	2	no	—
30	C₂²⋊C₄×D₁₉	—	2	2	no	—
31	#10597321	C₂²	1	2	yes	complemented
32	C₄×D₇₆	C₂²	1	2	yes	complemented
33	C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
34	C₁₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
35	C₁₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic, complemented
36	C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic, complemented
37	#10597278	C₂²	1	2	yes	complemented
38	#10597278	C₂²	1	2	yes	complemented
39	#10597275	C₂²	1	1	yes	characteristic, complemented
40	#10597273	C₂²	1	2	yes	complemented
41	C₂³×D₁₉	—	2	2	no	—
42	C₂×D₄×C₁₉	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
45	C₂×C₂²⋊D₁₉	—	2	2	no	—
46	C₂×C₂²⋊D₁₉	—	2	2	no	—
47	#64541	—	2	4	no	—
48	#64541	—	4	2	no	—
49	C₂×D₇₆	—	2	4	no	—
50	C₂×D₇₆	—	2	2	no	—
51	C₂×D₇₆	—	4	2	no	—
52	C₂×D₇₆	C₂³	1	2	yes	—
53	C₂×D₇₆	C₂³	1	2	yes	—
54	C₂×C₄×D₁₉	—	2	2	no	—
55	C₂×C₄×D₁₉	—	2	2	no	—
56	C₂×Dic₃₈	C₂³	1	2	yes	—
57	C₂×C₈×C₁₉	—	2	2	no	—
58	C₂×C₈×C₁₉	—	2	2	no	—
59	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
62	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
63	C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
64	C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
65	C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
66	C₄²×C₁₉	C₂³	1	1	yes	characteristic
67	C₂²⋊Dic₁₉	—	2	2	no	—
68	(C₂×D₁₉)⋊C₄	—	2	2	no	—
69	(C₂×D₁₉)⋊C₄	—	2	2	no	—
70	(C₂×D₁₉)⋊C₄	—	2	2	no	—
71	(C₂×D₁₉)⋊C₄	—	2	2	no	—
72	C₄⋊Dic₁₉	C₂³	1	2	yes	—
73	Dic₁₉⋊C₄	—	2	2	no	—
74	Dic₇₆	—	2	2	no	—
75	D₁₅₂	—	2	4	no	—
76	D₁₅₂	—	2	2	no	—
77	D₁₅₂	D₄	1	4	yes	complemented
78	#64516	—	2	4	no	—
79	#64516	—	4	2	no	—
80	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×D₁₉	—	2	2	no	—
83	C₂²×D₁₉	—	2	2	no	—
84	C₂²×D₁₉	—	4	4	no	—
85	C₂²×D₁₉	—	4	4	no	—
86	C₂²×D₁₉	—	8	2	no	—
87	D₄×C₁₉	—	2	2	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	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₁₉	—	4	4	no	—
98	C₂²⋊D₁₉	—	8	2	no	—
99	C₂×Dic₁₉	—	2	2	no	—
100	C₂×Dic₁₉	—	2	2	no	—
101	D₇₆	—	2	2	no	—
102	D₇₆	—	2	2	no	—
103	D₇₆	—	2	2	no	—
104	D₇₆	—	4	4	no	—
105	D₇₆	C₂×D₄	1	4	yes	—
106	C₄×D₁₉	—	4	4	no	—
107	C₄×D₁₉	—	8	2	no	—
108	Dic₃₈	—	2	2	no	—
109	Dic₃₈	—	2	2	no	—
110	C₈×C₁₉	—	2	1	no	—
111	C₈×C₁₉	—	2	1	no	—
112	C₈×C₁₉	—	2	1	no	—
113	C₈×C₁₉	C₂×D₄	1	2	yes	—
114	#3280	—	19	1	no	maximal, Sylow
115	C₂²×C₁₉	—	2	1	no	—
116	C₂²×C₁₉	—	2	1	no	—
117	C₂²×C₁₉	—	2	1	no	—
118	C₂²×C₁₉	—	2	1	no	—
119	C₂²×C₁₉	—	2	1	no	—
120	C₂²×C₁₉	#135	1	2	yes	complemented
121	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
122	C₂×D₁₉	—	2	4	no	—
123	C₂×D₁₉	—	4	2	no	—
124	C₂×D₁₉	—	4	2	no	—
125	C₂×D₁₉	—	4	2	no	—
126	C₂×D₁₉	—	4	2	no	—
127	C₂×D₁₉	—	4	2	no	—
128	C₂×D₁₉	—	8	8	no	—
129	C₄×C₁₉	—	2	1	no	—
130	C₄×C₁₉	—	2	1	no	—
131	C₄×C₁₉	—	2	1	no	—
132	C₄×C₁₉	—	2	1	no	—
133	C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
134	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
135	Dic₁₉	—	4	2	no	—
136	Dic₁₉	—	4	2	no	—
137	C₂×#135	—	19	1	no	—
138	C₂²×D₈	—	19	1	no	—
139	#546	—	19	2	no	—
140	#495	—	19	1	no	—
141	#467	—	19	1	no	—
142	#466	—	19	1	no	—
143	#463	—	19	2	no	—
144	#450	—	19	2	no	—
145	#449	—	19	2	no	—
146	C₄×D₈	—	19	1	no	—
147	C₈×D₄	D₁₉	1	1	yes	characteristic, complemented
148	C₂×C₁₉	—	2	2	no	—
149	C₂×C₁₉	—	4	1	no	—
150	C₂×C₁₉	D₄²	1	1	yes	characteristic
151	C₂×C₁₉	C₂×#135	1	1	yes	characteristic
152	C₂×C₁₉	#576	1	1	yes	characteristic
153	D₁₉	—	4	4	no	—
154	D₁₉	—	8	2	no	—
155	C₂×#41	—	19	2	no	—
156	C₂²×D₄	—	19	2	no	—
157	#135	—	38	4	no	—
158	C₂×Q₁₆	—	19	1	no	—
159	C₂×SD₁₆	—	19	2	no	—
160	C₂×SD₁₆	—	19	2	no	—
161	C₂×D₈	—	19	1	no	—
162	C₂×D₈	—	19	2	no	—
163	C₂×D₈	—	19	1	no	—
164	C₂×D₈	—	38	4	no	—
165	C₂²×C₈	C₂×D₁₉	1	1	yes	characteristic
166	C₂²×C₈	C₂×D₁₉	1	1	yes	characteristic, complemented
167	#124	—	19	2	no	—
168	#123	—	38	2	no	—
169	C₂²⋊Q₈	—	19	2	no	—
170	#121	—	19	2	no	—
171	#121	—	38	2	no	—
172	#120	—	38	2	no	—
173	C₄×D₄	—	19	2	no	—
174	C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
175	C₂×C₂²⋊C₄	—	38	2	no	—
176	#107	—	19	1	no	—
177	C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic, complemented
178	#103	—	19	2	no	—
179	#102	—	19	2	no	—
180	#102	—	19	2	no	—
181	C₂²⋊C₈	C₂×D₁₉	1	1	yes	characteristic
182	C₂²⋊C₈	C₂×D₁₉	1	1	yes	characteristic, complemented
183	C₄×C₈	C₂×D₁₉	1	1	yes	characteristic, complemented
184	C₁₉	#3280	1	1	yes	characteristic, complemented, Sylow
185	C₂⁴	—	38	2	no	—
186	#41	—	38	4	no	—
187	#41	—	76	2	no	—
188	C₂×Q₈	—	19	2	no	—
189	C₂×D₄	—	19	2	no	—
190	C₂×D₄	—	19	2	no	—
191	C₂×D₄	—	38	4	no	—
192	C₂×D₄	—	38	2	no	—
193	C₂×D₄	—	38	2	no	—
194	C₂×D₄	—	38	2	no	—
195	C₂×D₄	—	76	2	no	—
196	C₂×D₄	D₇₆	1	1	yes	characteristic
197	C₂²×C₄	—	38	2	no	—
198	C₂²×C₄	—	38	2	no	—
199	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
200	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
201	Q₁₆	—	38	2	no	—
202	SD₁₆	—	38	4	no	—
203	SD₁₆	—	76	2	no	—
204	D₈	—	19	4	no	—
205	D₈	—	38	4	no	—
206	D₈	—	38	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
210	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
211	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
212	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
213	C₄⋊C₄	—	19	2	no	—
214	C₄⋊C₄	—	38	2	no	—
215	C₄⋊C₄	D₇₆	1	1	yes	characteristic
216	C₂²⋊C₄	—	38	2	no	—
217	C₂²⋊C₄	—	38	2	no	—
218	C₂²⋊C₄	—	38	2	no	—
219	C₂²⋊C₄	—	38	2	no	—
220	C₂²⋊C₄	—	38	2	no	—
221	C₂²⋊C₄	D₇₆	1	1	yes	characteristic
222	C₂²⋊C₄	D₇₆	1	1	yes	characteristic
223	C₄²	C₂²×D₁₉	1	1	yes	characteristic
224	C₂³	—	38	2	no	—
225	C₂³	—	38	2	no	—
226	C₂³	—	76	4	no	—
227	C₂³	—	76	4	no	—
228	C₂³	—	152	2	no	—
229	C₂³	C₂×D₇₆	1	1	yes	characteristic
230	C₂³	C₂×D₇₆	1	1	yes	characteristic
231	Q₈	—	38	2	no	—
232	Q₈	—	38	2	no	—
233	D₄	—	2	2	no	—
234	D₄	—	19	4	no	—
235	D₄	—	38	2	no	—
236	D₄	—	38	2	no	—
237	D₄	—	38	2	no	—
238	D₄	—	76	4	no	—
239	D₄	—	76	4	no	—
240	D₄	—	152	2	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	—	38	2	no	—
246	C₂×C₄	—	38	2	no	—
247	C₂×C₄	—	76	4	no	—
248	C₂×C₄	—	152	2	no	—
249	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
250	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
251	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
252	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
253	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	D₄×D₁₉	1	2	yes	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	38	4	no	—
264	C₂²	—	76	2	no	—
265	C₂²	—	76	2	no	—
266	C₂²	—	76	2	no	—
267	C₂²	—	76	2	no	—
268	C₂²	—	76	2	no	—
269	C₂²	—	152	8	no	—
270	C₂²	#10597350	1	2	yes	complemented
271	C₂²	C₂²×D₇₆	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	76	2	no	—
277	C₄	—	76	2	no	—
278	C₄	C₂×D₄×D₁₉	1	1	yes	characteristic
279	C₄	#10597435	1	1	yes	characteristic
280	C₂	—	2	2	no	—
281	C₂	—	4	1	no	—
282	C₂	—	76	4	no	—
283	C₂	—	152	2	no	—
284	C₂	D₄×D₇₆	1	1	yes	characteristic
285	C₂	C₂×#10597350	1	1	yes	characteristic
286	C₂	#11927349	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₆
commutator, cosocle	C₂×C₄×C₁₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉
Fitting	C₈×D₄×C₁₉	C₂
socle	C₂²×C₁₉	C₂²×D₄
hypercenter	C₈×D₄	D₁₉
hypocenter	C₁₉	#3280
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#10597350	C₂	semidirect
2. C₂²×D₁₅₂	C₂	semidirect
3. #11927190	C₂	semidirect
4. #11926928	C₂	semidirect
5. #11926785	C₂	semidirect
6. #11926780	C₂	semidirect
7. #11926760	C₂	semidirect
8. #11926697	C₂	semidirect
9. #11926691	C₂	semidirect
10. C₄×D₁₅₂	C₂	semidirect
11. C₈×D₄×C₁₉	C₂	semidirect
12. C₂×#64541	C₂²	semidirect
15. C₂²×C₈×C₁₉	C₂²	semidirect
18. #10597376	C₂²	semidirect
20. C₂×Dic₇₆	C₂²	semidirect
26. C₂×#64516	C₂²	semidirect
27. C₂×#64516	C₂²	semidirect
31. #10597321	C₂²	semidirect
32. C₄×D₇₆	C₂²	semidirect
33. C₁₉×C₄⋊C₈	C₂²	semidirect
35. C₁₉×C₂²⋊C₈	C₂²	semidirect
36. C₄×C₈×C₁₉	C₂²	semidirect
37. #10597278	C₂²	semidirect
38. #10597278	C₂²	semidirect
39. #10597275	C₂²	semidirect
40. #10597273	C₂²	semidirect
77. D₁₅₂	D₄	semidirect
120. C₂²×C₁₉	#135	semidirect
147. C₈×D₄	D₁₉	semidirect
166. C₂²×C₈	C₂×D₁₉	semidirect
177. C₄⋊C₈	C₂×D₁₉	semidirect
182. C₂²⋊C₈	C₂×D₁₉	semidirect
183. C₄×C₈	C₂×D₁₉	semidirect
184. C₁₉	#3280	semidirect
270. C₂²	#10597350	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10597350	2
C₂²×D₁₅₂	2
#11927190	2
#11926928	2
#11926785	2
#11926780	2
#11926760	2
#11926697	2
#11926691	2
C₄×D₁₅₂	2
C₈×D₄×C₁₉	2
#3280	19