Subgroups — C₂×#11927345 #14578384

Order 2432.

1512 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10597430	C₂	1	2	yes	maximal, complemented
2	#11927345	C₂	1	16	yes	maximal, direct product factor
3	C₂×#10597378	C₂	1	2	yes	maximal, complemented
4	C₂×#10597357	C₂	1	4	yes	maximal, complemented
5	C₂×#10597350	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10597358	C₂	1	2	yes	maximal, complemented
7	C₂×D₈×D₁₉	C₂	1	2	yes	maximal, complemented
8	C₂×#10597347	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₄×C₁₉	C₂²	1	2	yes	complemented
11	C₂²×C₂²⋊D₁₉	—	2	4	no	—
12	#10597430	—	2	8	no	—
13	#10597430	C₂²	1	16	yes	complemented
14	C₂×#64543	—	2	2	no	—
15	C₂×#64543	C₂²	1	4	yes	complemented
16	C₂×D₄×D₁₉	—	2	2	no	—
17	C₂×D₄×D₁₉	C₂²	1	4	yes	complemented
18	C₂×#64541	C₂²	1	1	yes	characteristic
19	C₂×#64541	C₂²	1	2	yes	complemented
20	C₂×D₈×C₁₉	C₂²	1	2	yes	complemented
21	C₂×D₈×C₁₉	C₂²	1	2	yes	complemented
22	C₂×D₈×C₁₉	C₂²	1	8	yes	complemented
23	C₂²×C₈×C₁₉	C₂²	1	1	yes	characteristic, complemented
24	C₂×#64526	C₂²	1	4	yes	complemented
25	#10597378	C₂²	1	16	yes	complemented
26	C₂×#64525	C₂²	1	4	yes	complemented
27	C₂×#64520	C₂²	1	1	yes	characteristic, complemented
28	#10597358	C₂²	1	16	yes	complemented
29	#10597357	C₂²	1	32	yes	complemented
30	D₈×D₁₉	C₂²	1	16	yes	complemented
31	C₂×Dic₇₆	C₂²	1	1	yes	characteristic, complemented
32	#10597350	C₂²	1	8	yes	complemented
33	C₂×D₁₅₂	C₂²	1	1	yes	characteristic, complemented
34	C₂×#64516	C₂²	1	2	yes	complemented
35	#10597347	C₂²	1	8	yes	complemented
36	C₂×#64515	C₂²	1	2	yes	complemented
37	C₂×C₈×D₁₉	C₂²	1	2	yes	complemented
38	C₂⁴×C₁₉	—	2	2	no	—
39	C₂³×D₁₉	—	2	4	no	—
40	C₂×D₄×C₁₉	—	2	2	no	—
41	C₂×D₄×C₁₉	—	2	4	no	—
42	C₂×D₄×C₁₉	C₂³	1	4	yes	—
43	C₂×D₄×C₁₉	C₂³	1	8	yes	complemented
44	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
45	C₂×C₂²⋊D₁₉	—	2	2	no	—
46	C₂×C₂²⋊D₁₉	—	2	4	no	—
47	C₂×C₂²⋊D₁₉	—	2	32	no	—
48	C₂×C₂²⋊D₁₉	—	2	4	no	—
49	C₂×C₂²⋊D₁₉	D₄	1	2	yes	—
50	C₂²×Dic₁₉	—	2	4	no	—
51	#64543	—	2	8	no	—
52	#64543	—	2	16	no	—
53	#64543	C₂³	1	16	yes	complemented
54	D₄×D₁₉	—	2	16	no	—
55	D₄×D₁₉	—	2	8	no	—
56	D₄×D₁₉	C₂³	1	16	yes	complemented
57	#64541	—	2	4	no	—
58	#64541	D₄	1	4	yes	—
59	#64541	C₂³	1	4	yes	—
60	#64541	C₂³	1	8	yes	complemented
61	C₂×D₇₆	D₄	1	1	yes	characteristic
62	C₂×D₇₆	C₂³	1	2	yes	—
63	C₂×C₄×D₁₉	—	2	2	no	—
64	C₂×C₄×D₁₉	C₂³	1	2	yes	—
65	C₂×Dic₃₈	D₄	1	1	yes	characteristic
66	C₂×Dic₃₈	C₂³	1	2	yes	—
67	D₈×C₁₉	C₂³	1	8	yes	complemented
68	D₈×C₁₉	C₂³	1	8	yes	complemented
69	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₉	C₂³	1	4	yes	complemented
72	#64526	C₂³	1	16	yes	complemented
73	#64525	C₂³	1	16	yes	complemented
74	#64520	C₂³	1	4	yes	complemented
75	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
76	Dic₇₆	C₂³	1	4	yes	complemented
77	D₁₅₂	C₂³	1	4	yes	complemented
78	#64516	C₂³	1	8	yes	complemented
79	#64515	C₂³	1	8	yes	complemented
80	C₈×D₁₉	C₂³	1	8	yes	complemented
81	C₂³×C₁₉	—	2	4	no	—
82	C₂³×C₁₉	—	2	8	no	—
83	C₂³×C₁₉	—	8	4	no	—
84	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
85	C₂²×D₁₉	—	2	2	no	—
86	C₂²×D₁₉	—	2	16	no	—
87	C₂²×D₁₉	—	8	8	no	—
88	C₂²×D₁₉	C₂×D₄	1	2	yes	—
89	D₄×C₁₉	—	2	4	no	—
90	D₄×C₁₉	—	2	8	no	—
91	D₄×C₁₉	C₂⁴	1	8	yes	—
92	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
95	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
96	C₂²⋊D₁₉	—	2	8	no	—
97	C₂²⋊D₁₉	—	2	16	no	—
98	C₂²⋊D₁₉	—	2	16	no	—
99	C₂²⋊D₁₉	C₂×D₄	1	8	yes	—
100	C₂×Dic₁₉	—	2	2	no	—
101	C₂×Dic₁₉	—	2	16	no	—
102	C₂×Dic₁₉	C₂×D₄	1	2	yes	—
103	D₇₆	—	2	2	no	—
104	D₇₆	C₂×D₄	1	4	yes	—
105	D₇₆	C₂⁴	1	4	yes	—
106	C₄×D₁₉	—	2	8	no	—
107	C₄×D₁₉	C₂×D₄	1	4	yes	—
108	C₄×D₁₉	C₂⁴	1	4	yes	—
109	Dic₃₈	—	2	2	no	—
110	Dic₃₈	C₂×D₄	1	4	yes	—
111	Dic₃₈	C₂⁴	1	4	yes	—
112	C₈×C₁₉	C₂⁴	1	2	yes	—
113	C₈×C₁₉	C₂⁴	1	2	yes	—
114	C₁₉⋊C₈	C₂⁴	1	4	yes	—
115	C₂×#576	—	19	1	no	maximal, Sylow
116	C₂²×C₁₉	—	2	2	no	—
117	C₂²×C₁₉	—	2	8	no	—
118	C₂²×C₁₉	—	4	8	no	—
119	C₂²×C₁₉	—	8	8	no	—
120	C₂²×C₁₉	C₂²×D₄	1	2	yes	—
121	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
122	C₂×D₁₉	—	2	4	no	—
123	C₂×D₁₉	—	2	4	no	—
124	C₂×D₁₉	—	4	4	no	—
125	C₂×D₁₉	—	8	16	no	—
126	C₂×D₁₉	C₂²×D₄	1	4	yes	—
127	C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
128	C₄×C₁₉	C₂²×D₄	1	2	yes	—
129	C₄×C₁₉	C₂⁵	1	1	yes	characteristic
130	Dic₁₉	—	2	4	no	—
131	Dic₁₉	C₂²×D₄	1	4	yes	—
132	C₂×H(5,𝔽₂)	—	19	2	no	—
133	#576	—	19	16	no	—
134	C₂×AGL(1,ℤ/8)	—	19	2	no	—
135	C₂×AGL(1,ℤ/8)	—	19	4	no	—
136	C₂×#135	—	19	2	no	—
137	C₂×#135	—	19	1	no	—
138	C₂²×D₈	—	19	2	no	—
139	C₂²×D₈	D₁₉	1	1	yes	characteristic, complemented
140	C₂×#131	—	19	1	no	—
141	C₂×C₁₉	—	2	2	no	—
142	C₂×C₁₉	—	4	8	no	—
143	C₂×C₁₉	#576	1	2	yes	complemented
144	C₂×C₁₉	C₂³×D₄	1	1	yes	characteristic
145	D₁₉	—	2	4	no	—
146	D₁₉	—	4	4	no	—
147	H(5,𝔽₂)	—	19	16	no	—
148	H(5,𝔽₂)	—	38	8	no	—
149	C₂×#41	—	19	1	no	—
150	C₂×#41	—	19	4	no	—
151	C₂×#41	—	19	2	no	—
152	C₂×#41	—	38	2	no	—
153	C₂²×D₄	—	19	4	no	—
154	C₂²×D₄	—	38	2	no	—
155	C₂²×D₄	—	38	4	no	—
156	C₂²×D₄	C₂×D₁₉	1	2	yes	complemented
157	AGL(1,ℤ/8)	—	19	16	no	—
158	AGL(1,ℤ/8)	—	19	32	no	—
159	#135	—	19	16	no	—
160	#135	—	19	8	no	—
161	C₂×Q₁₆	—	19	1	no	—
162	C₂×SD₁₆	—	19	4	no	—
163	C₂×SD₁₆	—	19	2	no	—
164	C₂×D₈	—	19	4	no	—
165	C₂×D₈	—	19	16	no	—
166	C₂×D₈	—	19	1	no	—
167	C₂×D₈	C₂×D₁₉	1	2	yes	complemented
168	C₂×D₈	C₂×D₁₉	1	2	yes	complemented
169	C₂×D₈	C₂×D₁₉	1	8	yes	complemented
170	#131	—	19	8	no	—
171	C₂×M₁₆	—	19	1	no	—
172	C₂×M₁₆	—	19	2	no	—
173	C₂²×C₈	—	19	2	no	—
174	C₂²×C₈	C₂×D₁₉	1	1	yes	characteristic, complemented
175	C₁₉	C₂×#576	1	1	yes	characteristic, complemented, Sylow
176	C₂⁴	—	2	2	no	—
177	C₂⁴	—	38	4	no	—
178	#41	—	19	4	no	—
179	#41	—	19	4	no	—
180	#41	—	19	16	no	—
181	#41	—	19	8	no	—
182	#41	—	38	8	no	—
183	#41	—	38	16	no	—
184	#41	—	38	4	no	—
185	C₂×Q₈	—	19	1	no	—
186	C₂×Q₈	—	19	2	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	2	4	no	—
189	C₂×D₄	—	19	2	no	—
190	C₂×D₄	—	19	16	no	—
191	C₂×D₄	—	19	1	no	—
192	C₂×D₄	—	19	2	no	—
193	C₂×D₄	—	38	8	no	—
194	C₂×D₄	—	38	4	no	—
195	C₂×D₄	—	38	16	no	—
196	C₂×D₄	—	38	32	no	—
197	C₂×D₄	—	38	4	no	—
198	C₂×D₄	—	38	2	no	—
199	C₂×D₄	C₂²×D₁₉	1	4	yes	—
200	C₂×D₄	C₂²×D₁₉	1	8	yes	complemented
201	C₂²×C₄	—	19	2	no	—
202	C₂²×C₄	—	38	4	no	—
203	C₂²×C₄	—	38	2	no	—
204	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
205	Q₁₆	—	19	4	no	—
206	SD₁₆	—	19	16	no	—
207	SD₁₆	—	19	8	no	—
208	D₈	—	19	16	no	—
209	D₈	—	19	4	no	—
210	D₈	C₂²×D₁₉	1	8	yes	complemented
211	D₈	C₂²×D₁₉	1	8	yes	complemented
212	M₁₆	—	19	4	no	—
213	M₁₆	—	19	8	no	—
214	C₂×C₈	—	19	2	no	—
215	C₂×C₈	—	19	8	no	—
216	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
217	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
218	C₂×C₈	C₂²×D₁₉	1	4	yes	complemented
219	C₂³	—	2	4	no	—
220	C₂³	—	2	8	no	—
221	C₂³	—	8	4	no	—
222	C₂³	—	19	2	no	—
223	C₂³	—	38	16	no	—
224	C₂³	—	38	2	no	—
225	C₂³	—	152	8	no	—
226	C₂³	D₄×D₁₉	1	1	yes	characteristic
227	Q₈	—	19	4	no	—
228	Q₈	—	19	4	no	—
229	Q₈	—	38	2	no	—
230	D₄	—	2	4	no	—
231	D₄	—	2	8	no	—
232	D₄	—	19	8	no	—
233	D₄	—	19	4	no	—
234	D₄	—	19	4	no	—
235	D₄	—	38	16	no	—
236	D₄	—	38	16	no	—
237	D₄	—	38	8	no	—
238	D₄	—	38	2	no	—
239	D₄	C₂³×D₁₉	1	8	yes	—
240	C₂×C₄	—	19	2	no	—
241	C₂×C₄	—	19	4	no	—
242	C₂×C₄	—	19	4	no	—
243	C₂×C₄	—	38	16	no	—
244	C₂×C₄	—	38	2	no	—
245	C₂×C₄	—	38	8	no	—
246	C₂×C₄	D₄×D₁₉	1	2	yes	—
247	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
248	C₂×C₄	C₂³×D₁₉	1	2	yes	—
249	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
250	C₈	—	19	4	no	—
251	C₈	C₂³×D₁₉	1	2	yes	—
252	C₈	C₂³×D₁₉	1	2	yes	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	8	no	—
255	C₂²	—	4	8	no	—
256	C₂²	—	8	8	no	—
257	C₂²	—	19	4	no	—
258	C₂²	—	38	4	no	—
259	C₂²	—	38	4	no	—
260	C₂²	—	76	4	no	—
261	C₂²	—	152	16	no	—
262	C₂²	C₂×D₄×D₁₉	1	2	yes	—
263	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
264	C₄	—	19	4	no	—
265	C₄	—	38	4	no	—
266	C₄	C₂×D₄×D₁₉	1	1	yes	characteristic
267	C₄	C₂×D₄×D₁₉	1	2	yes	—
268	C₄	C₂⁴×D₁₉	1	1	yes	characteristic
269	C₂	—	2	2	no	—
270	C₂	—	4	8	no	—
271	C₂	—	38	4	no	—
272	C₂	—	76	4	no	—
273	C₂	#11927345	1	2	yes	direct product factor
274	C₂	C₂²×D₄×D₁₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₉
commutator, cosocle	C₄×C₁₉	C₂⁵
Frattini	C₄	C₂⁴×D₁₉
Fitting	C₂²×D₈×C₁₉	C₂
socle	C₂²×C₁₉	C₂²×D₄
hypercenter	C₂²×D₈	D₁₉
hypocenter	C₁₉	C₂×#576
LCS	G ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×#10597430	C₂	semidirect
2. #11927345	C₂	direct (N × H)
3. C₂×#10597378	C₂	semidirect
4. C₂×#10597357	C₂	semidirect
5. C₂×#10597350	C₂	semidirect
6. C₂×#10597358	C₂	semidirect
7. C₂×D₈×D₁₉	C₂	semidirect
8. C₂×#10597347	C₂	semidirect
9. C₂²×D₈×C₁₉	C₂	semidirect
10. C₂²×D₄×C₁₉	C₂²	semidirect
13. #10597430	C₂²	semidirect
15. C₂×#64543	C₂²	semidirect
17. C₂×D₄×D₁₉	C₂²	semidirect
19. C₂×#64541	C₂²	semidirect
20. C₂×D₈×C₁₉	C₂²	semidirect
21. C₂×D₈×C₁₉	C₂²	semidirect
22. C₂×D₈×C₁₉	C₂²	semidirect
23. C₂²×C₈×C₁₉	C₂²	semidirect
24. C₂×#64526	C₂²	semidirect
25. #10597378	C₂²	semidirect
26. C₂×#64525	C₂²	semidirect
27. C₂×#64520	C₂²	semidirect
28. #10597358	C₂²	semidirect
29. #10597357	C₂²	semidirect
30. D₈×D₁₉	C₂²	semidirect
31. C₂×Dic₇₆	C₂²	semidirect
32. #10597350	C₂²	semidirect
33. C₂×D₁₅₂	C₂²	semidirect
34. C₂×#64516	C₂²	semidirect
35. #10597347	C₂²	semidirect
36. C₂×#64515	C₂²	semidirect
37. C₂×C₈×D₁₉	C₂²	semidirect
43. C₂×D₄×C₁₉	C₂³	semidirect
53. #64543	C₂³	semidirect
56. D₄×D₁₉	C₂³	semidirect
60. #64541	C₂³	semidirect
67. D₈×C₁₉	C₂³	semidirect
68. D₈×C₁₉	C₂³	semidirect
71. C₂×C₈×C₁₉	C₂³	semidirect
72. #64526	C₂³	semidirect
73. #64525	C₂³	semidirect
74. #64520	C₂³	semidirect
76. Dic₇₆	C₂³	semidirect
77. D₁₅₂	C₂³	semidirect
78. #64516	C₂³	semidirect
79. #64515	C₂³	semidirect
80. C₈×D₁₉	C₂³	semidirect
139. C₂²×D₈	D₁₉	semidirect
143. C₂×C₁₉	#576	semidirect
156. C₂²×D₄	C₂×D₁₉	semidirect
167. C₂×D₈	C₂×D₁₉	semidirect
168. C₂×D₈	C₂×D₁₉	semidirect
169. C₂×D₈	C₂×D₁₉	semidirect
174. C₂²×C₈	C₂×D₁₉	semidirect
175. C₁₉	C₂×#576	semidirect
200. C₂×D₄	C₂²×D₁₉	semidirect
210. D₈	C₂²×D₁₉	semidirect
211. D₈	C₂²×D₁₉	semidirect
218. C₂×C₈	C₂²×D₁₉	semidirect
273. C₂	#11927345	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#10597430	2
#11927345	2
C₂×#10597378	2
C₂×#10597357	2
C₂×#10597350	2
C₂×#10597358	2
C₂×D₈×D₁₉	2
C₂×#10597347	2
C₂²×D₈×C₁₉	2
C₂×#576	19

Subgroups — C2×#11927345 #14578384

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11927345 #14578384