Subgroups — C₂×#11927358 #14578399

Order 2432.

1408 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10597436	C₂	1	2	yes	maximal, complemented
2	#11927358	C₂	1	16	yes	maximal, direct product factor
3	C₂×#10597397	C₂	1	2	yes	maximal, complemented
4	C₂×#10597361	C₂	1	4	yes	maximal, complemented
5	C₂×#10597358	C₂	1	2	yes	maximal, complemented
6	C₂²×Dic₇₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×Q₁₆×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	—
11	#10597436	—	2	8	no	—
12	#10597436	C₂²	1	16	yes	complemented
13	C₂×Q₈×D₁₉	C₂²	1	4	yes	—
14	C₂×#64543	—	2	4	no	—
15	C₂×#64543	C₂²	1	4	yes	—
16	C₂×#64541	—	2	2	no	—
17	C₂×#64541	C₂²	1	1	yes	characteristic
18	C₂²×Dic₃₈	—	2	2	no	—
19	C₂²×Dic₃₈	C₂²	1	2	yes	—
20	C₁₉×#135	C₂²	1	8	yes	complemented
21	C₂×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
22	C₂×SD₁₆×C₁₉	C₂²	1	2	yes	—
23	C₂×D₈×C₁₉	C₂²	1	1	yes	characteristic
24	C₂²×C₈×C₁₉	C₂²	1	1	yes	characteristic
25	#10597397	C₂²	1	16	yes	complemented
26	C₂×#64528	C₂²	1	4	yes	—
27	C₂×#64526	C₂²	1	4	yes	—
28	C₂×#64520	C₂²	1	1	yes	characteristic
29	Q₁₆×D₁₉	C₂²	1	16	yes	complemented
30	#10597361	C₂²	1	32	yes	complemented
31	#10597358	C₂²	1	16	yes	complemented
32	C₂×Dic₇₆	C₂²	1	2	yes	—
33	C₂×Dic₇₆	C₂²	1	2	yes	—
34	C₂×Dic₇₆	C₂²	1	8	yes	complemented
35	#10597347	C₂²	1	8	yes	complemented
36	C₂×#64515	C₂²	1	2	yes	—
37	C₂×C₈×D₁₉	C₂²	1	2	yes	—
38	#41×C₁₉	—	2	4	no	—
39	#41×C₁₉	C₂³	1	8	yes	—
40	C₂×Q₈×C₁₉	C₂³	1	2	yes	—
41	C₂×D₄×C₁₉	—	2	2	no	—
42	C₂×D₄×C₁₉	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₉	—	2	2	no	—
44	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
45	C₂×C₂²⋊D₁₉	—	2	4	no	—
46	C₂×C₂²⋊D₁₉	D₄	1	2	yes	—
47	C₂²×Dic₁₉	—	2	2	no	—
48	C₂²×Dic₁₉	—	2	4	no	—
49	Q₈×D₁₉	—	2	8	no	—
50	Q₈×D₁₉	C₂³	1	16	yes	—
51	#64543	—	2	32	no	—
52	#64543	—	2	8	no	—
53	#64543	C₂³	1	16	yes	—
54	#64541	—	2	16	no	—
55	#64541	D₄	1	4	yes	—
56	#64541	C₂³	1	4	yes	—
57	C₂×D₇₆	D₄	1	1	yes	characteristic
58	C₂×C₄×D₁₉	—	2	4	no	—
59	C₂×C₄×D₁₉	C₂³	1	2	yes	—
60	C₂×Dic₃₈	—	2	2	no	—
61	C₂×Dic₃₈	—	2	4	no	—
62	C₂×Dic₃₈	—	2	4	no	—
63	C₂×Dic₃₈	—	2	16	no	—
64	C₂×Dic₃₈	D₄	1	1	yes	characteristic
65	C₂×Dic₃₈	C₂³	1	4	yes	—
66	C₂×Dic₃₈	C₂³	1	8	yes	—
67	Q₁₆×C₁₉	C₂³	1	4	yes	—
68	SD₁₆×C₁₉	C₂³	1	8	yes	—
69	D₈×C₁₉	C₂³	1	4	yes	—
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	—
73	#64528	C₂³	1	16	yes	—
74	#64526	C₂³	1	16	yes	—
75	#64520	C₂³	1	4	yes	—
76	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
77	Dic₇₆	C₂³	1	8	yes	—
78	Dic₇₆	C₂³	1	8	yes	—
79	#64515	C₂³	1	8	yes	—
80	C₈×D₁₉	C₂³	1	8	yes	—
81	C₂³×C₁₉	—	2	2	no	—
82	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
83	C₂²×D₁₉	C₂×D₄	1	2	yes	—
84	Q₈×C₁₉	—	2	2	no	—
85	Q₈×C₁₉	C₂⁴	1	4	yes	—
86	D₄×C₁₉	—	2	8	no	—
87	D₄×C₁₉	—	2	2	no	—
88	D₄×C₁₉	C₂⁴	1	4	yes	—
89	C₂×C₄×C₁₉	—	2	2	no	—
90	C₂×C₄×C₁₉	—	2	8	no	—
91	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
94	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
95	C₂²⋊D₁₉	—	2	16	no	—
96	C₂²⋊D₁₉	C₂×D₄	1	8	yes	—
97	C₂×Dic₁₉	—	2	4	no	—
98	C₂×Dic₁₉	—	2	8	no	—
99	C₂×Dic₁₉	—	2	16	no	—
100	C₂×Dic₁₉	C₂×D₄	1	2	yes	—
101	D₇₆	C₂×D₄	1	4	yes	—
102	C₄×D₁₉	—	2	16	no	—
103	C₄×D₁₉	C₂×D₄	1	4	yes	—
104	C₄×D₁₉	C₂⁴	1	4	yes	—
105	Dic₃₈	—	2	8	no	—
106	Dic₃₈	—	2	4	no	—
107	Dic₃₈	—	2	16	no	—
108	Dic₃₈	C₂×D₄	1	4	yes	—
109	Dic₃₈	C₂⁴	1	8	yes	—
110	C₈×C₁₉	C₂⁴	1	2	yes	—
111	C₈×C₁₉	C₂⁴	1	2	yes	—
112	C₁₉⋊C₈	C₂⁴	1	4	yes	—
113	C₂×#578	—	19	1	no	maximal, Sylow
114	C₂²×C₁₉	—	2	2	no	—
115	C₂²×C₁₉	—	2	4	no	—
116	C₂²×C₁₉	—	4	4	no	—
117	C₂²×C₁₉	C₂²×D₄	1	2	yes	—
118	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
119	C₂×D₁₉	—	2	4	no	—
120	C₂×D₁₉	C₂²×D₄	1	4	yes	—
121	C₄×C₁₉	—	2	4	no	—
122	C₄×C₁₉	C₂²×D₄	1	2	yes	—
123	C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
124	C₄×C₁₉	C₂⁵	1	1	yes	characteristic
125	Dic₁₉	—	2	8	no	—
126	Dic₁₉	C₂²×D₄	1	4	yes	—
127	C₂×2₋¹⁺⁴	—	19	2	no	—
128	#578	—	19	16	no	—
129	C₂×#137	—	19	2	no	—
130	C₂×#137	—	19	4	no	—
131	C₂×#135	—	19	2	no	—
132	C₂×#135	D₁₉	1	1	yes	characteristic, complemented
133	C₂²×Q₁₆	—	19	2	no	—
134	C₂²×Q₁₆	—	19	1	no	—
135	C₂×#131	—	19	1	no	—
136	C₂×C₁₉	—	2	2	no	—
137	C₂×C₁₉	—	4	4	no	—
138	C₂×C₁₉	#578	1	2	yes	complemented
139	C₂×C₁₉	C₂³×D₄	1	1	yes	characteristic
140	D₁₉	—	2	4	no	—
141	2₋¹⁺⁴	—	19	16	no	—
142	2₋¹⁺⁴	—	38	8	no	—
143	C₂×#41	—	19	1	no	—
144	C₂×#41	—	19	4	no	—
145	C₂×#41	—	38	2	no	—
146	C₂×#41	—	38	4	no	—
147	C₂×#41	C₂×D₁₉	1	2	yes	—
148	C₂²×Q₈	—	19	4	no	—
149	C₂²×Q₈	—	19	2	no	—
150	C₂²×Q₈	—	38	2	no	—
151	#137	—	19	16	no	—
152	#137	—	19	32	no	—
153	#135	—	19	16	no	—
154	#135	C₂×D₁₉	1	8	yes	complemented
155	C₂×Q₁₆	—	19	4	no	—
156	C₂×Q₁₆	—	19	16	no	—
157	C₂×Q₁₆	—	19	2	no	—
158	C₂×Q₁₆	—	19	2	no	—
159	C₂×Q₁₆	—	19	8	no	—
160	C₂×Q₁₆	C₂×D₁₉	1	1	yes	characteristic
161	C₂×SD₁₆	—	19	4	no	—
162	C₂×SD₁₆	C₂×D₁₉	1	2	yes	—
163	C₂×D₈	C₂×D₁₉	1	1	yes	characteristic
164	#131	—	19	8	no	—
165	C₂×M₁₆	—	19	1	no	—
166	C₂×M₁₆	—	19	2	no	—
167	C₂²×C₈	—	19	2	no	—
168	C₂²×C₈	C₂×D₁₉	1	1	yes	characteristic
169	C₁₉	C₂×#578	1	1	yes	characteristic, complemented, Sylow
170	#41	—	2	4	no	—
171	#41	—	19	4	no	—
172	#41	—	19	4	no	—
173	#41	—	19	16	no	—
174	#41	—	38	16	no	—
175	#41	—	38	8	no	—
176	#41	—	38	32	no	—
177	#41	C₂²×D₁₉	1	8	yes	—
178	C₂×Q₈	—	19	1	no	—
179	C₂×Q₈	—	19	16	no	—
180	C₂×Q₈	—	19	4	no	—
181	C₂×Q₈	—	19	8	no	—
182	C₂×Q₈	—	38	4	no	—
183	C₂×Q₈	—	38	16	no	—
184	C₂×Q₈	—	38	8	no	—
185	C₂×Q₈	—	38	2	no	—
186	C₂×Q₈	—	38	4	no	—
187	C₂×Q₈	C₂²×D₁₉	1	2	yes	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	19	2	no	—
190	C₂×D₄	—	19	1	no	—
191	C₂×D₄	—	38	4	no	—
192	C₂×D₄	C₂²×D₁₉	1	2	yes	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	19	2	no	—
195	C₂²×C₄	—	38	4	no	—
196	C₂²×C₄	—	38	4	no	—
197	C₂²×C₄	—	38	2	no	—
198	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
199	Q₁₆	—	19	16	no	—
200	Q₁₆	—	19	8	no	—
201	Q₁₆	—	19	8	no	—
202	Q₁₆	C₂²×D₁₉	1	4	yes	—
203	SD₁₆	—	19	16	no	—
204	SD₁₆	C₂²×D₁₉	1	8	yes	—
205	D₈	C₂²×D₁₉	1	4	yes	—
206	M₁₆	—	19	4	no	—
207	M₁₆	—	19	8	no	—
208	C₂×C₈	—	19	2	no	—
209	C₂×C₈	—	19	8	no	—
210	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
211	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
212	C₂×C₈	C₂²×D₁₉	1	4	yes	—
213	C₂³	—	2	2	no	—
214	C₂³	—	19	2	no	—
215	C₂³	D₄×D₁₉	1	1	yes	characteristic
216	Q₈	—	2	2	no	—
217	Q₈	—	19	4	no	—
218	Q₈	—	19	8	no	—
219	Q₈	—	38	16	no	—
220	Q₈	—	38	8	no	—
221	Q₈	—	38	4	no	—
222	Q₈	C₂³×D₁₉	1	4	yes	—
223	D₄	—	2	8	no	—
224	D₄	—	2	2	no	—
225	D₄	—	19	8	no	—
226	D₄	—	19	4	no	—
227	D₄	—	38	16	no	—
228	D₄	C₂³×D₁₉	1	4	yes	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	—	19	2	no	—
232	C₂×C₄	—	19	4	no	—
233	C₂×C₄	—	19	4	no	—
234	C₂×C₄	—	38	16	no	—
235	C₂×C₄	—	38	16	no	—
236	C₂×C₄	—	38	8	no	—
237	C₂×C₄	—	38	4	no	—
238	C₂×C₄	D₄×D₁₉	1	2	yes	—
239	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
240	C₂×C₄	C₂³×D₁₉	1	2	yes	—
241	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
242	C₈	—	19	4	no	—
243	C₈	C₂³×D₁₉	1	2	yes	—
244	C₈	C₂³×D₁₉	1	2	yes	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	4	no	—
247	C₂²	—	4	4	no	—
248	C₂²	—	19	4	no	—
249	C₂²	—	38	4	no	—
250	C₂²	C₂×D₄×D₁₉	1	2	yes	—
251	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
252	C₄	—	2	4	no	—
253	C₄	—	19	4	no	—
254	C₄	—	38	8	no	—
255	C₄	C₂×D₄×D₁₉	1	2	yes	—
256	C₄	C₂×D₄×D₁₉	1	1	yes	characteristic
257	C₄	C₂⁴×D₁₉	1	1	yes	characteristic
258	C₂	—	2	2	no	—
259	C₂	—	4	4	no	—
260	C₂	—	38	4	no	—
261	C₂	#11927358	1	2	yes	direct product factor
262	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×#578
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×#10597436	C₂	semidirect
2. #11927358	C₂	direct (N × H)
3. C₂×#10597397	C₂	semidirect
4. C₂×#10597361	C₂	semidirect
5. C₂×#10597358	C₂	semidirect
6. C₂²×Dic₇₆	C₂	semidirect
7. C₂×Q₁₆×D₁₉	C₂	semidirect
8. C₂×#10597347	C₂	semidirect
9. C₂×C₁₉×#135	C₂	semidirect
12. #10597436	C₂²	semidirect
20. C₁₉×#135	C₂²	semidirect
25. #10597397	C₂²	semidirect
29. Q₁₆×D₁₉	C₂²	semidirect
30. #10597361	C₂²	semidirect
31. #10597358	C₂²	semidirect
34. C₂×Dic₇₆	C₂²	semidirect
35. #10597347	C₂²	semidirect
132. C₂×#135	D₁₉	semidirect
138. C₂×C₁₉	#578	semidirect
154. #135	C₂×D₁₉	semidirect
169. C₁₉	C₂×#578	semidirect
261. C₂	#11927358	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×#10597436	2
#11927358	2
C₂×#10597397	2
C₂×#10597361	2
C₂×#10597358	2
C₂²×Dic₇₆	2
C₂×Q₁₆×D₁₉	2
C₂×#10597347	2
C₂×C₁₉×#135	2
C₂×#578	19

Subgroups — C2×#11927358 #14578399

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11927358 #14578399