Subgroups — C₂×#11927346 #14578385

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×#10597435	C₂	1	2	yes	maximal, complemented
2	#11927346	C₂	1	16	yes	maximal, direct product factor
3	C₂×#10597360	C₂	1	4	yes	maximal, complemented
4	C₂×#10597395	C₂	1	2	yes	maximal, complemented
5	C₂×#10597365	C₂	1	2	yes	maximal, complemented
6	C₂²×D₁₅₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₈×D₁₉	C₂	1	2	yes	maximal, complemented
8	C₂×#10597347	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₉×#135	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₁₉	C₂²	1	2	yes	complemented
11	#10597435	—	2	8	no	—
12	#10597435	C₂²	1	16	yes	complemented
13	C₂×#64545	C₂²	1	4	yes	complemented
14	C₂×D₄×D₁₉	—	2	4	no	—
15	C₂×D₄×D₁₉	C₂²	1	4	yes	complemented
16	C₂×#64541	—	2	2	no	—
17	C₂×#64541	C₂²	1	1	yes	characteristic
18	C₂²×D₇₆	—	2	2	no	—
19	C₂²×D₇₆	C₂²	1	2	yes	complemented
20	C₁₉×#135	C₂²	1	8	yes	complemented
21	C₂×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic, complemented
22	C₂×SD₁₆×C₁₉	C₂²	1	2	yes	complemented
23	C₂×D₈×C₁₉	C₂²	1	1	yes	characteristic, complemented
24	C₂²×C₈×C₁₉	C₂²	1	1	yes	characteristic, complemented
25	#10597395	C₂²	1	16	yes	complemented
26	C₂×#64527	C₂²	1	4	yes	complemented
27	C₂×#64525	C₂²	1	4	yes	complemented
28	C₂×#64520	C₂²	1	1	yes	characteristic, complemented
29	#10597365	C₂²	1	16	yes	complemented
30	#10597360	C₂²	1	32	yes	complemented
31	D₈×D₁₉	C₂²	1	16	yes	complemented
32	C₂×D₁₅₂	C₂²	1	2	yes	complemented
33	C₂×D₁₅₂	C₂²	1	2	yes	complemented
34	C₂×D₁₅₂	C₂²	1	8	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	C₂³×D₁₉	—	2	2	no	—
39	C₂³×D₁₉	—	2	4	no	—
40	#41×C₁₉	—	2	4	no	—
41	#41×C₁₉	C₂³	1	8	yes	complemented
42	C₂×Q₈×C₁₉	C₂³	1	2	yes	—
43	C₂×D₄×C₁₉	—	2	2	no	—
44	C₂×D₄×C₁₉	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₉	—	2	2	no	—
46	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
47	C₂×C₂²⋊D₁₉	—	2	4	no	—
48	C₂×C₂²⋊D₁₉	D₄	1	2	yes	—
49	#64545	—	2	8	no	—
50	#64545	C₂³	1	16	yes	complemented
51	D₄×D₁₉	—	2	32	no	—
52	D₄×D₁₉	—	2	8	no	—
53	D₄×D₁₉	C₂³	1	16	yes	complemented
54	#64541	—	2	16	no	—
55	#64541	D₄	1	4	yes	—
56	#64541	C₂³	1	4	yes	—
57	C₂×D₇₆	—	2	4	no	—
58	C₂×D₇₆	—	2	2	no	—
59	C₂×D₇₆	—	2	4	no	—
60	C₂×D₇₆	—	2	16	no	—
61	C₂×D₇₆	D₄	1	1	yes	characteristic
62	C₂×D₇₆	C₂³	1	4	yes	—
63	C₂×D₇₆	C₂³	1	8	yes	complemented
64	C₂×C₄×D₁₉	—	2	4	no	—
65	C₂×C₄×D₁₉	C₂³	1	2	yes	—
66	C₂×Dic₃₈	D₄	1	1	yes	characteristic
67	Q₁₆×C₁₉	C₂³	1	4	yes	complemented
68	SD₁₆×C₁₉	C₂³	1	8	yes	complemented
69	D₈×C₁₉	C₂³	1	4	yes	complemented
70	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₁₉	C₂³	1	4	yes	complemented
73	#64527	C₂³	1	16	yes	complemented
74	#64525	C₂³	1	16	yes	complemented
75	#64520	C₂³	1	4	yes	complemented
76	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
77	D₁₅₂	C₂³	1	8	yes	complemented
78	D₁₅₂	C₂³	1	8	yes	complemented
79	#64515	C₂³	1	8	yes	complemented
80	C₈×D₁₉	C₂³	1	8	yes	complemented
81	C₂³×C₁₉	—	2	2	no	—
82	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
83	C₂²×D₁₉	—	2	4	no	—
84	C₂²×D₁₉	—	2	8	no	—
85	C₂²×D₁₉	—	2	16	no	—
86	C₂²×D₁₉	—	8	8	no	—
87	C₂²×D₁₉	—	8	4	no	—
88	C₂²×D₁₉	C₂×D₄	1	2	yes	—
89	Q₈×C₁₉	—	2	2	no	—
90	Q₈×C₁₉	C₂⁴	1	4	yes	—
91	D₄×C₁₉	—	2	8	no	—
92	D₄×C₁₉	—	2	2	no	—
93	D₄×C₁₉	C₂⁴	1	4	yes	—
94	C₂×C₄×C₁₉	—	2	2	no	—
95	C₂×C₄×C₁₉	—	2	8	no	—
96	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
99	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
100	C₂²⋊D₁₉	—	2	16	no	—
101	C₂²⋊D₁₉	C₂×D₄	1	8	yes	—
102	C₂×Dic₁₉	C₂×D₄	1	2	yes	—
103	D₇₆	—	2	16	no	—
104	D₇₆	—	2	8	no	—
105	D₇₆	—	2	4	no	—
106	D₇₆	C₂×D₄	1	4	yes	—
107	D₇₆	C₂⁴	1	8	yes	—
108	C₄×D₁₉	—	2	16	no	—
109	C₄×D₁₉	C₂×D₄	1	4	yes	—
110	C₄×D₁₉	C₂⁴	1	4	yes	—
111	Dic₃₈	C₂×D₄	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	4	no	—
118	C₂²×C₁₉	—	4	4	no	—
119	C₂²×C₁₉	C₂²×D₄	1	2	yes	—
120	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
121	C₂×D₁₉	—	2	4	no	—
122	C₂×D₁₉	—	2	8	no	—
123	C₂×D₁₉	—	4	8	no	—
124	C₂×D₁₉	—	8	16	no	—
125	C₂×D₁₉	—	8	8	no	—
126	C₂×D₁₉	C₂²×D₄	1	4	yes	—
127	C₄×C₁₉	—	2	4	no	—
128	C₄×C₁₉	C₂²×D₄	1	2	yes	—
129	C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
130	C₄×C₁₉	C₂⁵	1	1	yes	characteristic
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	D₁₉	1	1	yes	characteristic, complemented
138	C₂²×D₈	—	19	2	no	—
139	C₂²×D₈	—	19	1	no	—
140	C₂×#131	—	19	1	no	—
141	C₂×C₁₉	—	2	2	no	—
142	C₂×C₁₉	—	4	4	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	8	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	—	38	2	no	—
152	C₂×#41	C₂×D₁₉	1	2	yes	complemented
153	C₂²×D₄	—	19	4	no	—
154	C₂²×D₄	—	19	2	no	—
155	C₂²×D₄	—	38	2	no	—
156	C₂²×D₄	—	38	4	no	—
157	AGL(1,ℤ/8)	—	19	16	no	—
158	AGL(1,ℤ/8)	—	19	32	no	—
159	#135	—	19	16	no	—
160	#135	C₂×D₁₉	1	8	yes	complemented
161	C₂×Q₁₆	C₂×D₁₉	1	1	yes	characteristic, complemented
162	C₂×SD₁₆	—	19	4	no	—
163	C₂×SD₁₆	C₂×D₁₉	1	2	yes	complemented
164	C₂×D₈	—	19	4	no	—
165	C₂×D₈	—	19	16	no	—
166	C₂×D₈	—	19	2	no	—
167	C₂×D₈	—	19	2	no	—
168	C₂×D₈	—	19	8	no	—
169	C₂×D₈	C₂×D₁₉	1	1	yes	characteristic, 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	—	38	4	no	—
177	C₂⁴	—	38	2	no	—
178	#41	—	2	4	no	—
179	#41	—	19	4	no	—
180	#41	—	19	4	no	—
181	#41	—	19	16	no	—
182	#41	—	38	16	no	—
183	#41	—	38	8	no	—
184	#41	C₂²×D₁₉	1	8	yes	complemented
185	C₂×Q₈	—	19	1	no	—
186	C₂×Q₈	C₂²×D₁₉	1	2	yes	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	19	2	no	—
189	C₂×D₄	—	19	16	no	—
190	C₂×D₄	—	19	1	no	—
191	C₂×D₄	—	19	4	no	—
192	C₂×D₄	—	19	8	no	—
193	C₂×D₄	—	38	16	no	—
194	C₂×D₄	—	38	4	no	—
195	C₂×D₄	—	38	8	no	—
196	C₂×D₄	—	38	4	no	—
197	C₂×D₄	—	38	32	no	—
198	C₂×D₄	—	38	2	no	—
199	C₂×D₄	—	38	4	no	—
200	C₂×D₄	C₂²×D₁₉	1	2	yes	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	19	2	no	—
203	C₂²×C₄	—	38	4	no	—
204	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
205	Q₁₆	C₂²×D₁₉	1	4	yes	complemented
206	SD₁₆	—	19	16	no	—
207	SD₁₆	C₂²×D₁₉	1	8	yes	complemented
208	D₈	—	19	16	no	—
209	D₈	—	19	8	no	—
210	D₈	—	19	8	no	—
211	D₈	C₂²×D₁₉	1	4	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	2	no	—
220	C₂³	—	19	2	no	—
221	C₂³	—	38	16	no	—
222	C₂³	—	38	4	no	—
223	C₂³	—	38	8	no	—
224	C₂³	—	152	4	no	—
225	C₂³	—	152	8	no	—
226	C₂³	D₄×D₁₉	1	1	yes	characteristic
227	Q₈	—	2	2	no	—
228	Q₈	—	19	4	no	—
229	Q₈	C₂³×D₁₉	1	4	yes	—
230	D₄	—	2	8	no	—
231	D₄	—	2	2	no	—
232	D₄	—	19	8	no	—
233	D₄	—	19	4	no	—
234	D₄	—	19	8	no	—
235	D₄	—	38	16	no	—
236	D₄	—	38	16	no	—
237	D₄	—	38	8	no	—
238	D₄	—	38	4	no	—
239	D₄	C₂³×D₁₉	1	4	yes	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	19	2	no	—
243	C₂×C₄	—	19	4	no	—
244	C₂×C₄	—	19	4	no	—
245	C₂×C₄	—	38	16	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	4	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	19	4	no	—
257	C₂²	—	38	4	no	—
258	C₂²	—	38	8	no	—
259	C₂²	—	76	8	no	—
260	C₂²	—	152	8	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	—	2	4	no	—
265	C₄	—	19	4	no	—
266	C₄	C₂×D₄×D₁₉	1	2	yes	—
267	C₄	C₂×D₄×D₁₉	1	1	yes	characteristic
268	C₄	C₂⁴×D₁₉	1	1	yes	characteristic
269	C₂	—	2	2	no	—
270	C₂	—	4	4	no	—
271	C₂	—	38	4	no	—
272	C₂	—	76	8	no	—
273	C₂	#11927346	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	C₂×C₁₉×#135	C₂
socle	C₂²×C₁₉	C₂²×D₄
hypercenter	C₂×#135	D₁₉
hypocenter	C₁₉	C₂×#576
LCS	G ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#10597435	C₂	semidirect
2. #11927346	C₂	direct (N × H)
3. C₂×#10597360	C₂	semidirect
4. C₂×#10597395	C₂	semidirect
5. C₂×#10597365	C₂	semidirect
6. C₂²×D₁₅₂	C₂	semidirect
7. C₂×D₈×D₁₉	C₂	semidirect
8. C₂×#10597347	C₂	semidirect
9. C₂×C₁₉×#135	C₂	semidirect
10. C₂×#41×C₁₉	C₂²	semidirect
12. #10597435	C₂²	semidirect
13. C₂×#64545	C₂²	semidirect
15. C₂×D₄×D₁₉	C₂²	semidirect
19. C₂²×D₇₆	C₂²	semidirect
20. C₁₉×#135	C₂²	semidirect
21. C₂×Q₁₆×C₁₉	C₂²	semidirect
22. C₂×SD₁₆×C₁₉	C₂²	semidirect
23. C₂×D₈×C₁₉	C₂²	semidirect
24. C₂²×C₈×C₁₉	C₂²	semidirect
25. #10597395	C₂²	semidirect
26. C₂×#64527	C₂²	semidirect
27. C₂×#64525	C₂²	semidirect
28. C₂×#64520	C₂²	semidirect
29. #10597365	C₂²	semidirect
30. #10597360	C₂²	semidirect
31. D₈×D₁₉	C₂²	semidirect
32. C₂×D₁₅₂	C₂²	semidirect
33. C₂×D₁₅₂	C₂²	semidirect
34. C₂×D₁₅₂	C₂²	semidirect
35. #10597347	C₂²	semidirect
36. C₂×#64515	C₂²	semidirect
37. C₂×C₈×D₁₉	C₂²	semidirect
41. #41×C₁₉	C₂³	semidirect
50. #64545	C₂³	semidirect
53. D₄×D₁₉	C₂³	semidirect
63. C₂×D₇₆	C₂³	semidirect
67. Q₁₆×C₁₉	C₂³	semidirect
68. SD₁₆×C₁₉	C₂³	semidirect
69. D₈×C₁₉	C₂³	semidirect
72. C₂×C₈×C₁₉	C₂³	semidirect
73. #64527	C₂³	semidirect
74. #64525	C₂³	semidirect
75. #64520	C₂³	semidirect
77. D₁₅₂	C₂³	semidirect
78. D₁₅₂	C₂³	semidirect
79. #64515	C₂³	semidirect
80. C₈×D₁₉	C₂³	semidirect
137. C₂×#135	D₁₉	semidirect
143. C₂×C₁₉	#576	semidirect
152. C₂×#41	C₂×D₁₉	semidirect
160. #135	C₂×D₁₉	semidirect
161. C₂×Q₁₆	C₂×D₁₉	semidirect
163. C₂×SD₁₆	C₂×D₁₉	semidirect
169. C₂×D₈	C₂×D₁₉	semidirect
174. C₂²×C₈	C₂×D₁₉	semidirect
175. C₁₉	C₂×#576	semidirect
184. #41	C₂²×D₁₉	semidirect
205. Q₁₆	C₂²×D₁₉	semidirect
207. SD₁₆	C₂²×D₁₉	semidirect
211. D₈	C₂²×D₁₉	semidirect
218. C₂×C₈	C₂²×D₁₉	semidirect
273. C₂	#11927346	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×#10597435	2
#11927346	2
C₂×#10597360	2
C₂×#10597395	2
C₂×#10597365	2
C₂²×D₁₅₂	2
C₂×D₈×D₁₉	2
C₂×#10597347	2
C₂×C₁₉×#135	2
C₂×#576	19

Subgroups — C2×#11927346 #14578385

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11927346 #14578385