Subgroups — C₂×#11927355 #14578395

Order 2432.

1460 conjugacy classes of subgroups.

The subgroup lattice has 242 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	1	yes	maximal, characteristic, complemented
2	#11927355	C₂	1	16	yes	maximal, direct product factor
3	C₂×#10597397	C₂	1	3	yes	maximal, complemented
4	C₂×#10597378	C₂	1	3	yes	maximal, complemented
5	C₂×#10597396	C₂	1	3	yes	maximal, complemented
6	C₂²×#64526	C₂	1	3	yes	maximal, complemented
7	C₂×#10597394	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₉×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₉×H(5,𝔽₂)	—	2	4	no	—
10	C₁₉×H(5,𝔽₂)	C₂²	1	8	yes	complemented
11	C₂×#41×C₁₉	—	2	1	no	—
12	C₂×#41×C₁₉	C₂²	1	3	yes	—
13	C₂×#41×C₁₉	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₁₉	—	2	3	no	—
15	C₂²×D₄×C₁₉	C₂²	1	3	yes	—
16	#10597436	—	2	4	no	—
17	#10597436	C₂²	1	8	yes	complemented
18	C₂×Q₈×D₁₉	—	2	1	no	—
19	C₂×#64543	—	2	3	no	—
20	C₂×#64541	C₂²	1	3	yes	complemented
21	C₂²×Dic₃₈	C₂²	1	3	yes	complemented
22	#10597397	C₂²	1	24	yes	complemented
23	#10597396	C₂²	1	24	yes	complemented
24	#10597394	C₂²	1	8	yes	complemented
25	C₂×#64528	C₂²	1	3	yes	complemented
26	C₂×#64527	C₂²	1	1	yes	characteristic, complemented
27	C₂×#64526	C₂²	1	3	yes	complemented
28	C₂×#64526	C₂²	1	6	yes	complemented
29	C₂×#64526	C₂²	1	24	yes	complemented
30	#10597378	C₂²	1	24	yes	complemented
31	C₂×#64525	C₂²	1	3	yes	complemented
32	C₂×#64520	C₂²	1	3	yes	complemented
33	C₂²×C₁₉⋊C₈	C₂²	1	3	yes	complemented
34	C₂⁴×C₁₉	—	2	3	no	—
35	#41×C₁₉	—	2	8	no	—
36	#41×C₁₉	—	2	6	no	—
37	#41×C₁₉	D₄	1	4	yes	—
38	#41×C₁₉	C₂³	1	12	yes	—
39	#41×C₁₉	C₂³	1	4	yes	—
40	C₂×Q₈×C₁₉	D₄	1	1	yes	characteristic
41	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
42	C₂×D₄×C₁₉	—	2	3	no	—
43	C₂×D₄×C₁₉	—	2	24	no	—
44	C₂×D₄×C₁₉	—	2	3	no	—
45	C₂×D₄×C₁₉	—	2	6	no	—
46	C₂×D₄×C₁₉	D₄	1	3	yes	—
47	C₂×D₄×C₁₉	C₂³	1	3	yes	—
48	C₂×D₄×C₁₉	C₂³	1	12	yes	—
49	C₂²×C₄×C₁₉	—	2	3	no	—
50	C₂²×C₄×C₁₉	C₂³	1	3	yes	—
51	C₂×C₂²⋊D₁₉	—	2	3	no	—
52	C₂²×Dic₁₉	—	2	3	no	—
53	Q₈×D₁₉	—	2	8	no	—
54	#64543	—	2	24	no	—
55	#64541	—	2	6	no	—
56	#64541	C₂³	1	12	yes	complemented
57	C₂×D₇₆	C₂³	1	1	yes	characteristic
58	C₂×C₄×D₁₉	—	2	3	no	—
59	C₂×Dic₃₈	—	2	3	no	—
60	C₂×Dic₃₈	—	2	6	no	—
61	C₂×Dic₃₈	C₂³	1	3	yes	—
62	C₂×Dic₃₈	C₂³	1	12	yes	complemented
63	#64528	C₂³	1	12	yes	complemented
64	#64527	C₂³	1	4	yes	complemented
65	#64526	C₂³	1	24	yes	complemented
66	#64526	C₂³	1	12	yes	complemented
67	#64525	C₂³	1	12	yes	complemented
68	#64520	C₂³	1	12	yes	complemented
69	C₂×C₁₉⋊C₈	C₂³	1	3	yes	—
70	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₁₉⋊C₈	C₂³	1	12	yes	complemented
72	C₂³×C₁₉	—	2	3	no	—
73	C₂³×C₁₉	—	2	12	no	—
74	C₂³×C₁₉	—	8	6	no	—
75	C₂³×C₁₉	C₂×D₄	1	3	yes	—
76	C₂²×D₁₉	—	2	1	no	—
77	Q₈×C₁₉	—	2	1	no	—
78	Q₈×C₁₉	C₂×D₄	1	4	yes	—
79	Q₈×C₁₉	C₂⁴	1	2	yes	—
80	D₄×C₁₉	—	2	12	no	—
81	D₄×C₁₉	—	2	3	no	—
82	D₄×C₁₉	—	2	12	no	—
83	D₄×C₁₉	C₂×D₄	1	12	yes	—
84	D₄×C₁₉	C₂⁴	1	6	yes	—
85	C₂×C₄×C₁₉	—	2	1	no	—
86	C₂×C₄×C₁₉	—	2	12	no	—
87	C₂×C₄×C₁₉	C₂×D₄	1	3	yes	—
88	C₂×C₄×C₁₉	C₂×D₄	1	6	yes	—
89	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
90	C₂×C₄×C₁₉	C₂⁴	1	6	yes	—
91	C₂²⋊D₁₉	—	2	12	no	—
92	C₂×Dic₁₉	—	2	3	no	—
93	C₂×Dic₁₉	—	2	12	no	—
94	D₇₆	—	2	1	no	—
95	D₇₆	C₂⁴	1	2	yes	—
96	C₄×D₁₉	—	2	12	no	—
97	Dic₃₈	—	2	3	no	—
98	Dic₃₈	—	2	12	no	—
99	Dic₃₈	C₂⁴	1	6	yes	—
100	C₁₉⋊C₈	C₂⁴	1	2	yes	—
101	C₁₉⋊C₈	C₂⁴	1	6	yes	—
102	C₂×#577	—	19	1	no	maximal, Sylow
103	C₂²×C₁₉	—	2	6	no	—
104	C₂²×C₁₉	—	2	6	no	—
105	C₂²×C₁₉	—	4	6	no	—
106	C₂²×C₁₉	—	8	12	no	—
107	C₂²×C₁₉	C₂²×D₄	1	6	yes	—
108	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
109	C₂×D₁₉	—	2	2	no	—
110	C₂×D₁₉	—	4	2	no	—
111	C₄×C₁₉	—	2	2	no	—
112	C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
113	C₄×C₁₉	C₂²×D₄	1	6	yes	—
114	C₄×C₁₉	C₂⁵	1	1	yes	characteristic
115	Dic₁₉	—	2	6	no	—
116	C₂×2₋¹⁺⁴	—	19	1	no	—
117	C₂×H(5,𝔽₂)	D₁₉	1	1	yes	characteristic, complemented
118	#577	—	19	16	no	—
119	C₂×#137	—	19	3	no	—
120	C₂×AGL(1,ℤ/8)	—	19	3	no	—
121	C₂×#135	—	19	3	no	—
122	C₂²×SD₁₆	—	19	3	no	—
123	C₂×#131	—	19	1	no	—
124	C₂×C₁₉	—	2	6	no	—
125	C₂×C₁₉	—	4	6	no	—
126	C₂×C₁₉	#577	1	2	yes	complemented
127	C₂×C₁₉	C₂³×D₄	1	1	yes	characteristic
128	D₁₉	—	4	2	no	—
129	2₋¹⁺⁴	—	19	8	no	—
130	2₋¹⁺⁴	—	38	4	no	—
131	H(5,𝔽₂)	—	2	4	no	—
132	H(5,𝔽₂)	C₂×D₁₉	1	8	yes	complemented
133	C₂×#41	—	2	1	no	—
134	C₂×#41	—	19	3	no	—
135	C₂×#41	—	38	3	no	—
136	C₂×#41	C₂×D₁₉	1	3	yes	—
137	C₂×#41	C₂×D₁₉	1	1	yes	characteristic
138	C₂²×Q₈	—	19	3	no	—
139	C₂²×Q₈	—	38	1	no	—
140	C₂²×D₄	—	2	3	no	—
141	C₂²×D₄	C₂×D₁₉	1	3	yes	—
142	#137	—	19	24	no	—
143	AGL(1,ℤ/8)	—	19	24	no	—
144	#135	—	19	24	no	—
145	C₂×Q₁₆	—	19	3	no	—
146	C₂×SD₁₆	—	19	1	no	—
147	C₂×SD₁₆	—	19	3	no	—
148	C₂×SD₁₆	—	19	6	no	—
149	C₂×SD₁₆	—	19	24	no	—
150	C₂×D₈	—	19	3	no	—
151	#131	—	19	8	no	—
152	C₂×M₁₆	—	19	3	no	—
153	C₂²×C₈	—	19	3	no	—
154	C₁₉	C₂×#577	1	1	yes	characteristic, complemented, Sylow
155	C₂⁴	—	2	3	no	—
156	#41	—	2	8	no	—
157	#41	—	2	6	no	—
158	#41	—	19	12	no	—
159	#41	—	38	24	no	—
160	#41	—	38	6	no	—
161	#41	C₂²⋊D₁₉	1	4	yes	—
162	#41	C₂²×D₁₉	1	12	yes	—
163	#41	C₂²×D₁₉	1	4	yes	—
164	C₂×Q₈	—	19	3	no	—
165	C₂×Q₈	—	19	12	no	—
166	C₂×Q₈	—	38	8	no	—
167	C₂×Q₈	—	38	3	no	—
168	C₂×Q₈	—	38	6	no	—
169	C₂×Q₈	C₂²⋊D₁₉	1	1	yes	characteristic
170	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
171	C₂×D₄	—	2	3	no	—
172	C₂×D₄	—	2	24	no	—
173	C₂×D₄	—	2	3	no	—
174	C₂×D₄	—	2	6	no	—
175	C₂×D₄	—	19	1	no	—
176	C₂×D₄	—	38	3	no	—
177	C₂×D₄	C₂²⋊D₁₉	1	3	yes	—
178	C₂×D₄	C₂²×D₁₉	1	3	yes	—
179	C₂×D₄	C₂²×D₁₉	1	12	yes	—
180	C₂²×C₄	—	2	3	no	—
181	C₂²×C₄	—	38	3	no	—
182	C₂²×C₄	—	38	3	no	—
183	C₂²×C₄	C₂²×D₁₉	1	3	yes	—
184	Q₁₆	—	19	12	no	—
185	SD₁₆	—	19	4	no	—
186	SD₁₆	—	19	12	no	—
187	SD₁₆	—	19	24	no	—
188	D₈	—	19	12	no	—
189	M₁₆	—	19	12	no	—
190	C₂×C₈	—	19	1	no	—
191	C₂×C₈	—	19	3	no	—
192	C₂×C₈	—	19	12	no	—
193	C₂³	—	2	3	no	—
194	C₂³	—	2	12	no	—
195	C₂³	—	8	6	no	—
196	C₂³	—	38	1	no	—
197	C₂³	C₂×C₂²⋊D₁₉	1	3	yes	—
198	Q₈	—	2	1	no	—
199	Q₈	—	19	6	no	—
200	Q₈	—	38	12	no	—
201	Q₈	—	38	3	no	—
202	Q₈	C₂×C₂²⋊D₁₉	1	4	yes	—
203	Q₈	C₂³×D₁₉	1	2	yes	—
204	D₄	—	2	12	no	—
205	D₄	—	2	12	no	—
206	D₄	—	2	3	no	—
207	D₄	—	19	2	no	—
208	D₄	—	38	12	no	—
209	D₄	—	38	1	no	—
210	D₄	C₂×C₂²⋊D₁₉	1	12	yes	—
211	D₄	C₂³×D₁₉	1	6	yes	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	12	no	—
214	C₂×C₄	—	38	12	no	—
215	C₂×C₄	—	38	3	no	—
216	C₂×C₄	—	38	12	no	—
217	C₂×C₄	C₂×C₂²⋊D₁₉	1	6	yes	—
218	C₂×C₄	C₂×C₂²⋊D₁₉	1	3	yes	—
219	C₂×C₄	C₂³×D₁₉	1	6	yes	—
220	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
221	C₈	—	19	2	no	—
222	C₈	—	19	6	no	—
223	C₂²	—	2	6	no	—
224	C₂²	—	2	6	no	—
225	C₂²	—	4	6	no	—
226	C₂²	—	8	12	no	—
227	C₂²	—	38	2	no	—
228	C₂²	—	76	2	no	—
229	C₂²	C₂²×C₂²⋊D₁₉	1	6	yes	—
230	C₂²	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	38	6	no	—
233	C₄	C₂²×C₂²⋊D₁₉	1	6	yes	—
234	C₄	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
235	C₄	C₂⁴×D₁₉	1	1	yes	characteristic
236	C₂	—	2	6	no	—
237	C₂	—	4	6	no	—
238	C₂	—	76	2	no	—
239	C₂	#11927355	1	2	yes	direct product factor
240	C₂	C₂³×C₂²⋊D₁₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₉
commutator, cosocle	C₄×C₁₉	C₂⁵
Frattini	C₄	C₂⁴×D₁₉
Fitting	C₂×C₁₉×H(5,𝔽₂)	C₂
socle	C₂²×C₁₉	C₂²×D₄
hypercenter	C₂×H(5,𝔽₂)	D₁₉
hypocenter	C₁₉	C₂×#577
LCS	G ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₂×#10597436	C₂	semidirect
2. #11927355	C₂	direct (N × H)
3. C₂×#10597397	C₂	semidirect
4. C₂×#10597378	C₂	semidirect
5. C₂×#10597396	C₂	semidirect
6. C₂²×#64526	C₂	semidirect
7. C₂×#10597394	C₂	semidirect
8. C₂×C₁₉×H(5,𝔽₂)	C₂	semidirect
10. C₁₉×H(5,𝔽₂)	C₂²	semidirect
17. #10597436	C₂²	semidirect
20. C₂×#64541	C₂²	semidirect
21. C₂²×Dic₃₈	C₂²	semidirect
22. #10597397	C₂²	semidirect
23. #10597396	C₂²	semidirect
24. #10597394	C₂²	semidirect
25. C₂×#64528	C₂²	semidirect
26. C₂×#64527	C₂²	semidirect
27. C₂×#64526	C₂²	semidirect
28. C₂×#64526	C₂²	semidirect
29. C₂×#64526	C₂²	semidirect
30. #10597378	C₂²	semidirect
31. C₂×#64525	C₂²	semidirect
32. C₂×#64520	C₂²	semidirect
33. C₂²×C₁₉⋊C₈	C₂²	semidirect
56. #64541	C₂³	semidirect
62. C₂×Dic₃₈	C₂³	semidirect
63. #64528	C₂³	semidirect
64. #64527	C₂³	semidirect
65. #64526	C₂³	semidirect
66. #64526	C₂³	semidirect
67. #64525	C₂³	semidirect
68. #64520	C₂³	semidirect
71. C₂×C₁₉⋊C₈	C₂³	semidirect
117. C₂×H(5,𝔽₂)	D₁₉	semidirect
126. C₂×C₁₉	#577	semidirect
132. H(5,𝔽₂)	C₂×D₁₉	semidirect
154. C₁₉	C₂×#577	semidirect
239. C₂	#11927355	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
#11927355	2
C₂×#10597397	2
C₂×#10597378	2
C₂×#10597396	2
C₂²×#64526	2
C₂×#10597394	2
C₂×C₁₉×H(5,𝔽₂)	2
C₂×#577	19

Subgroups — C2×#11927355 #14578395

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11927355 #14578395