Subgroups — C₂×#88251⋊C₃ #72276197

Order 2496.

1796 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄×C₁₃⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×(C₂²⋊D₁₃)⋊C₃	C₂	1	4	yes	maximal, complemented
3	#88251⋊C₃	C₂	1	16	yes	maximal, direct product factor
4	C₂×#6129⋊C₃	C₂	1	4	yes	maximal, complemented
5	C₂×D₄×D₁₃⋊C₃	C₂	1	4	yes	maximal, complemented
6	C₂×#6127⋊C₃	C₂	1	2	yes	maximal, complemented
7	C₂×#88251	C₃	1	1	yes	maximal, characteristic, complemented, Hall
8	C₂⁴×C₁₃⋊C₃	C₂²	1	2	yes	complemented
9	C₂³×D₁₃⋊C₃	C₂²	1	4	yes	complemented
10	C₂×D₄×C₁₃⋊C₃	C₂²	1	4	yes	complemented
11	C₂×D₄×C₁₃⋊C₃	C₂²	1	8	yes	complemented
12	C₂²×C₄×C₁₃⋊C₃	C₂²	1	1	yes	characteristic, complemented
13	C₂×(C₂²⋊D₁₃)⋊C₃	C₂²	1	4	yes	complemented
14	C₂×(C₂²⋊D₁₃)⋊C₃	C₂²	1	8	yes	complemented
15	C₂×(C₂²⋊D₁₃)⋊C₃	C₂²	1	32	yes	complemented
16	C₂²×Dic₁₃⋊C₃	C₂²	1	4	yes	complemented
17	#6129⋊C₃	C₂²	1	32	yes	complemented
18	D₄×D₁₃⋊C₃	C₂²	1	32	yes	complemented
19	#6127⋊C₃	C₂²	1	16	yes	complemented
20	C₂×C₄⋊(D₁₃⋊C₃)	C₂²	1	2	yes	complemented
21	C₂×C₄×D₁₃⋊C₃	C₂²	1	4	yes	complemented
22	C₂×(C₁₃⋊C₃)⋊Q₈	C₂²	1	2	yes	complemented
23	C₂²×D₄×C₁₃	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂²×C₂²⋊D₁₃	C₂×C₃	1	4	yes	complemented
25	#88251	C₂×C₃	1	16	yes	complemented
26	C₂×#6129	C₂×C₃	1	4	yes	complemented
27	C₂×D₄×D₁₃	C₂×C₃	1	4	yes	complemented
28	C₂×#6127	C₂×C₃	1	2	yes	complemented
29	C₂³×C₁₃⋊C₃	—	4	4	no	—
30	C₂³×C₁₃⋊C₃	C₂³	1	4	yes	—
31	C₂³×C₁₃⋊C₃	C₂³	1	1	yes	characteristic
32	C₂³×C₁₃⋊C₃	C₂³	1	8	yes	complemented
33	C₂²×D₁₃⋊C₃	—	4	8	no	—
34	C₂²×D₁₃⋊C₃	C₂³	1	4	yes	—
35	C₂²×D₁₃⋊C₃	C₂³	1	16	yes	complemented
36	D₄×C₁₃⋊C₃	C₂³	1	16	yes	complemented
37	C₂×C₄×C₁₃⋊C₃	C₂³	1	2	yes	—
38	C₂×C₄×C₁₃⋊C₃	C₂³	1	4	yes	complemented
39	(C₂²⋊D₁₃)⋊C₃	C₂³	1	16	yes	complemented
40	(C₂²⋊D₁₃)⋊C₃	C₂³	1	32	yes	complemented
41	C₂×Dic₁₃⋊C₃	C₂³	1	4	yes	—
42	C₂×Dic₁₃⋊C₃	C₂³	1	16	yes	complemented
43	C₄⋊(D₁₃⋊C₃)	C₂³	1	8	yes	complemented
44	C₄×D₁₃⋊C₃	C₂³	1	16	yes	complemented
45	(C₁₃⋊C₃)⋊Q₈	C₂³	1	8	yes	complemented
46	C₂⁴×C₁₃	C₂²×C₃	1	2	yes	complemented
47	C₂³×D₁₃	C₂²×C₃	1	4	yes	complemented
48	C₂×D₄×C₁₃	C₂²×C₃	1	4	yes	complemented
49	C₂×D₄×C₁₃	C₂²×C₃	1	8	yes	complemented
50	C₂²×C₄×C₁₃	C₂²×C₃	1	1	yes	characteristic, complemented
51	C₂×C₂²⋊D₁₃	C₂²×C₃	1	4	yes	complemented
52	C₂×C₂²⋊D₁₃	C₂²×C₃	1	8	yes	complemented
53	C₂×C₂²⋊D₁₃	C₂²×C₃	1	32	yes	complemented
54	C₂²×Dic₁₃	C₂²×C₃	1	4	yes	complemented
55	#6129	C₂²×C₃	1	32	yes	complemented
56	D₄×D₁₃	C₂²×C₃	1	32	yes	complemented
57	#6127	C₂²×C₃	1	16	yes	complemented
58	C₂×D₅₂	C₂²×C₃	1	2	yes	complemented
59	C₂×C₄×D₁₃	C₂²×C₃	1	4	yes	complemented
60	C₂×Dic₂₆	C₂²×C₃	1	2	yes	complemented
61	C₂×C₃×H(5,𝔽₂)	—	13	1	no	maximal, Hall
62	C₂²×C₁₃⋊C₃	—	2	8	no	—
63	C₂²×C₁₃⋊C₃	—	2	2	no	—
64	C₂²×C₁₃⋊C₃	—	4	8	no	—
65	C₂²×C₁₃⋊C₃	C₂⁴	1	8	yes	—
66	C₂²×C₁₃⋊C₃	C₂⁴	1	2	yes	—
67	C₂²×C₁₃⋊C₃	C₂⁴	1	1	yes	characteristic
68	C₂×D₁₃⋊C₃	—	2	8	no	—
69	C₂×D₁₃⋊C₃	—	4	16	no	—
70	C₂×D₁₃⋊C₃	C₂⁴	1	8	yes	—
71	C₄×C₁₃⋊C₃	C₂⁴	1	4	yes	—
72	Dic₁₃⋊C₃	C₂⁴	1	8	yes	—
73	C₂³×C₁₃	—	4	4	no	—
74	C₂³×C₁₃	C₂³×C₃	1	4	yes	—
75	C₂³×C₁₃	C₂³×C₃	1	1	yes	characteristic
76	C₂³×C₁₃	C₂³×C₃	1	8	yes	complemented
77	C₂²×D₁₃	—	4	8	no	—
78	C₂²×D₁₃	C₂³×C₃	1	4	yes	—
79	C₂²×D₁₃	C₂³×C₃	1	16	yes	complemented
80	D₄×C₁₃	C₂³×C₃	1	16	yes	complemented
81	C₂×C₄×C₁₃	C₂³×C₃	1	2	yes	—
82	C₂×C₄×C₁₃	C₂³×C₃	1	4	yes	complemented
83	C₂²⋊D₁₃	C₂³×C₃	1	16	yes	complemented
84	C₂²⋊D₁₃	C₂³×C₃	1	32	yes	complemented
85	C₂×Dic₁₃	C₂³×C₃	1	4	yes	—
86	C₂×Dic₁₃	C₂³×C₃	1	16	yes	complemented
87	D₅₂	C₂³×C₃	1	8	yes	complemented
88	C₄×D₁₃	C₂³×C₃	1	16	yes	complemented
89	Dic₂₆	C₂³×C₃	1	8	yes	complemented
90	C₃×H(5,𝔽₂)	—	13	16	no	—
91	C₂×C₃×#41	—	13	2	no	—
92	C₂×C₃×#41	—	13	4	no	—
93	C₂²×C₃×D₄	—	13	4	no	—
94	C₂²×C₃×D₄	—	13	4	no	—
95	C₂²×C₃×D₄	—	13	1	no	—
96	C₂×C₁₃⋊C₃	—	2	8	no	—
97	C₂×C₁₃⋊C₃	—	2	2	no	—
98	C₂×C₁₃⋊C₃	H(5,𝔽₂)	1	2	yes	complemented
99	C₂×C₁₃⋊C₃	C₂⁵	1	1	yes	characteristic
100	D₁₃⋊C₃	—	2	8	no	—
101	C₂×H(5,𝔽₂)	—	13	1	no	Sylow
102	C₂²×C₁₃	—	2	8	no	—
103	C₂²×C₁₃	—	2	2	no	—
104	C₂²×C₁₃	—	4	8	no	—
105	C₂²×C₁₃	C₂⁴×C₃	1	8	yes	—
106	C₂²×C₁₃	C₂⁴×C₃	1	2	yes	—
107	C₂²×C₁₃	C₂⁴×C₃	1	1	yes	characteristic
108	C₂×D₁₃	—	2	8	no	—
109	C₂×D₁₃	—	4	16	no	—
110	C₂×D₁₃	C₂⁴×C₃	1	8	yes	—
111	C₄×C₁₃	C₂⁴×C₃	1	4	yes	—
112	Dic₁₃	C₂⁴×C₃	1	8	yes	—
113	C₂⁴×C₃	—	13	4	no	—
114	C₂⁴×C₃	—	13	2	no	—
115	C₃×#41	—	13	16	no	—
116	C₃×#41	—	13	32	no	—
117	C₂×C₃×Q₈	—	13	2	no	—
118	C₂×C₃×D₄	—	13	2	no	—
119	C₂×C₃×D₄	—	13	32	no	—
120	C₂×C₃×D₄	—	13	4	no	—
121	C₂×C₃×D₄	—	13	8	no	—
122	C₂×C₃×D₄	—	13	32	no	—
123	C₂×C₃×D₄	—	13	4	no	—
124	C₂×C₃×D₄	—	13	8	no	—
125	C₂²×C₃×C₄	—	13	4	no	—
126	C₂²×C₃×C₄	—	13	4	no	—
127	C₂²×C₃×C₄	—	13	1	no	—
128	C₁₃⋊C₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Hall
129	H(5,𝔽₂)	—	13	16	no	—
130	C₂×#41	—	13	4	no	—
131	C₂×#41	—	13	2	no	—
132	C₂²×D₄	—	13	4	no	—
133	C₂²×D₄	—	13	4	no	—
134	C₂²×D₄	D₁₃⋊C₃	1	1	yes	characteristic, complemented
135	C₂×C₁₃	—	2	8	no	—
136	C₂×C₁₃	—	2	2	no	—
137	C₂×C₁₃	C₃×H(5,𝔽₂)	1	2	yes	complemented
138	C₂×C₁₃	C₂⁵×C₃	1	1	yes	characteristic
139	D₁₃	—	2	8	no	—
140	C₂³×C₃	—	13	4	no	—
141	C₂³×C₃	—	13	16	no	—
142	C₂³×C₃	—	13	4	no	—
143	C₂³×C₃	—	13	1	no	—
144	C₂³×C₃	—	13	8	no	—
145	C₂³×C₃	—	52	4	no	—
146	C₂³×C₃	—	52	8	no	—
147	C₃×Q₈	—	13	8	no	—
148	C₃×D₄	—	13	8	no	—
149	C₃×D₄	—	13	16	no	—
150	C₃×D₄	—	13	32	no	—
151	C₃×D₄	—	13	16	no	—
152	C₂×C₃×C₄	—	13	16	no	—
153	C₂×C₃×C₄	—	13	4	no	—
154	C₂×C₃×C₄	—	13	16	no	—
155	C₂×C₃×C₄	—	13	2	no	—
156	C₂×C₃×C₄	—	13	4	no	—
157	C₂⁴	—	13	4	no	—
158	C₂⁴	C₂×D₁₃⋊C₃	1	2	yes	complemented
159	#41	—	13	32	no	—
160	#41	—	13	16	no	—
161	C₂×Q₈	—	13	2	no	—
162	C₂×D₄	—	13	8	no	—
163	C₂×D₄	—	13	4	no	—
164	C₂×D₄	—	13	32	no	—
165	C₂×D₄	—	13	32	no	—
166	C₂×D₄	—	13	2	no	—
167	C₂×D₄	C₂×D₁₃⋊C₃	1	4	yes	complemented
168	C₂×D₄	C₂×D₁₃⋊C₃	1	8	yes	complemented
169	C₂²×C₄	—	13	4	no	—
170	C₂²×C₄	—	13	4	no	—
171	C₂²×C₄	C₂×D₁₃⋊C₃	1	1	yes	characteristic, complemented
172	C₁₃	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
173	C₂²×C₃	—	13	8	no	—
174	C₂²×C₃	—	13	8	no	—
175	C₂²×C₃	—	13	2	no	—
176	C₂²×C₃	—	13	1	no	—
177	C₂²×C₃	—	26	2	no	—
178	C₂²×C₃	—	26	8	no	—
179	C₂²×C₃	—	26	8	no	—
180	C₂²×C₃	—	52	8	no	—
181	C₂²×C₃	—	52	16	no	—
182	C₃×C₄	—	13	8	no	—
183	C₃×C₄	—	13	4	no	—
184	C₂³	—	4	4	no	—
185	C₂³	—	13	4	no	—
186	C₂³	—	13	16	no	—
187	C₂³	—	52	8	no	—
188	C₂³	C₂²×D₁₃⋊C₃	1	4	yes	—
189	C₂³	C₂²×D₁₃⋊C₃	1	1	yes	characteristic
190	C₂³	C₂²×D₁₃⋊C₃	1	8	yes	complemented
191	Q₈	—	13	8	no	—
192	D₄	—	13	32	no	—
193	D₄	—	13	16	no	—
194	D₄	—	13	8	no	—
195	D₄	C₂²×D₁₃⋊C₃	1	16	yes	complemented
196	C₂×C₄	—	13	16	no	—
197	C₂×C₄	—	13	4	no	—
198	C₂×C₄	—	13	16	no	—
199	C₂×C₄	C₂²×D₁₃⋊C₃	1	2	yes	—
200	C₂×C₄	C₂²×D₁₃⋊C₃	1	4	yes	complemented
201	C₂×C₃	—	13	2	no	—
202	C₂×C₃	—	13	1	no	—
203	C₂×C₃	—	26	2	no	—
204	C₂×C₃	—	26	8	no	—
205	C₂×C₃	—	26	8	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	8	no	—
208	C₂²	—	4	8	no	—
209	C₂²	—	13	8	no	—
210	C₂²	—	26	8	no	—
211	C₂²	—	52	16	no	—
212	C₂²	C₂³×D₁₃⋊C₃	1	8	yes	—
213	C₂²	C₂³×D₁₃⋊C₃	1	2	yes	—
214	C₂²	C₂³×D₁₃⋊C₃	1	1	yes	characteristic
215	C₄	—	13	8	no	—
216	C₄	C₂³×D₁₃⋊C₃	1	4	yes	—
217	C₃	—	13	1	no	Sylow
218	C₂	—	2	2	no	—
219	C₂	—	2	8	no	—
220	C₂	—	26	8	no	—
221	C₂	#88251⋊C₃	1	2	yes	direct product factor
222	C₂	C₂⁴×D₁₃⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×D₁₃⋊C₃
commutator, cosocle	C₂×C₁₃	C₂⁵×C₃
Frattini	C₂	C₂⁴×D₁₃⋊C₃
Fitting	C₂²×D₄×C₁₃	C₂×C₃
socle	C₂²×C₁₃	C₂⁴×C₃
hypercenter	C₂²×D₄	D₁₃⋊C₃
hypocenter	C₁₃	C₂×C₃×H(5,𝔽₂)
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×D₄×C₁₃⋊C₃	C₂	semidirect
2. C₂²×(C₂²⋊D₁₃)⋊C₃	C₂	semidirect
3. #88251⋊C₃	C₂	direct (N × H)
4. C₂×#6129⋊C₃	C₂	semidirect
5. C₂×D₄×D₁₃⋊C₃	C₂	semidirect
6. C₂×#6127⋊C₃	C₂	semidirect
7. C₂×#88251	C₃	semidirect
8. C₂⁴×C₁₃⋊C₃	C₂²	semidirect
9. C₂³×D₁₃⋊C₃	C₂²	semidirect
10. C₂×D₄×C₁₃⋊C₃	C₂²	semidirect
11. C₂×D₄×C₁₃⋊C₃	C₂²	semidirect
12. C₂²×C₄×C₁₃⋊C₃	C₂²	semidirect
13. C₂×(C₂²⋊D₁₃)⋊C₃	C₂²	semidirect
14. C₂×(C₂²⋊D₁₃)⋊C₃	C₂²	semidirect
15. C₂×(C₂²⋊D₁₃)⋊C₃	C₂²	semidirect
16. C₂²×Dic₁₃⋊C₃	C₂²	semidirect
17. #6129⋊C₃	C₂²	semidirect
18. D₄×D₁₃⋊C₃	C₂²	semidirect
19. #6127⋊C₃	C₂²	semidirect
20. C₂×C₄⋊(D₁₃⋊C₃)	C₂²	semidirect
21. C₂×C₄×D₁₃⋊C₃	C₂²	semidirect
22. C₂×(C₁₃⋊C₃)⋊Q₈	C₂²	semidirect
23. C₂²×D₄×C₁₃	C₂×C₃	semidirect
24. C₂²×C₂²⋊D₁₃	C₂×C₃	semidirect
25. #88251	C₂×C₃	semidirect
26. C₂×#6129	C₂×C₃	semidirect
27. C₂×D₄×D₁₃	C₂×C₃	semidirect
28. C₂×#6127	C₂×C₃	semidirect
32. C₂³×C₁₃⋊C₃	C₂³	semidirect
35. C₂²×D₁₃⋊C₃	C₂³	semidirect
36. D₄×C₁₃⋊C₃	C₂³	semidirect
38. C₂×C₄×C₁₃⋊C₃	C₂³	semidirect
39. (C₂²⋊D₁₃)⋊C₃	C₂³	semidirect
40. (C₂²⋊D₁₃)⋊C₃	C₂³	semidirect
42. C₂×Dic₁₃⋊C₃	C₂³	semidirect
43. C₄⋊(D₁₃⋊C₃)	C₂³	semidirect
44. C₄×D₁₃⋊C₃	C₂³	semidirect
45. (C₁₃⋊C₃)⋊Q₈	C₂³	semidirect
46. C₂⁴×C₁₃	C₂²×C₃	semidirect
47. C₂³×D₁₃	C₂²×C₃	semidirect
48. C₂×D₄×C₁₃	C₂²×C₃	semidirect
49. C₂×D₄×C₁₃	C₂²×C₃	semidirect
50. C₂²×C₄×C₁₃	C₂²×C₃	semidirect
51. C₂×C₂²⋊D₁₃	C₂²×C₃	semidirect
52. C₂×C₂²⋊D₁₃	C₂²×C₃	semidirect
53. C₂×C₂²⋊D₁₃	C₂²×C₃	semidirect
54. C₂²×Dic₁₃	C₂²×C₃	semidirect
55. #6129	C₂²×C₃	semidirect
56. D₄×D₁₃	C₂²×C₃	semidirect
57. #6127	C₂²×C₃	semidirect
58. C₂×D₅₂	C₂²×C₃	semidirect
59. C₂×C₄×D₁₃	C₂²×C₃	semidirect
60. C₂×Dic₂₆	C₂²×C₃	semidirect
76. C₂³×C₁₃	C₂³×C₃	semidirect
79. C₂²×D₁₃	C₂³×C₃	semidirect
80. D₄×C₁₃	C₂³×C₃	semidirect
82. C₂×C₄×C₁₃	C₂³×C₃	semidirect
83. C₂²⋊D₁₃	C₂³×C₃	semidirect
84. C₂²⋊D₁₃	C₂³×C₃	semidirect
86. C₂×Dic₁₃	C₂³×C₃	semidirect
87. D₅₂	C₂³×C₃	semidirect
88. C₄×D₁₃	C₂³×C₃	semidirect
89. Dic₂₆	C₂³×C₃	semidirect
98. C₂×C₁₃⋊C₃	H(5,𝔽₂)	semidirect
128. C₁₃⋊C₃	C₂×H(5,𝔽₂)	semidirect
134. C₂²×D₄	D₁₃⋊C₃	semidirect
137. C₂×C₁₃	C₃×H(5,𝔽₂)	semidirect
158. C₂⁴	C₂×D₁₃⋊C₃	semidirect
167. C₂×D₄	C₂×D₁₃⋊C₃	semidirect
168. C₂×D₄	C₂×D₁₃⋊C₃	semidirect
171. C₂²×C₄	C₂×D₁₃⋊C₃	semidirect
172. C₁₃	C₂×C₃×H(5,𝔽₂)	semidirect
190. C₂³	C₂²×D₁₃⋊C₃	semidirect
195. D₄	C₂²×D₁₃⋊C₃	semidirect
200. C₂×C₄	C₂²×D₁₃⋊C₃	semidirect
221. C₂	#88251⋊C₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₄×C₁₃⋊C₃	2
C₂²×(C₂²⋊D₁₃)⋊C₃	2
#88251⋊C₃	2
C₂×#6129⋊C₃	2
C₂×D₄×D₁₃⋊C₃	2
C₂×#6127⋊C₃	2
C₂×#88251	3
C₂×C₃×H(5,𝔽₂)	13

Subgroups — C2×#88251⋊C3 #72276197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#88251⋊C₃ #72276197