Subgroups — C₂²×#14583734 #26010489

Order 11008.

60366 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583734	C₂	1	96	yes	maximal, direct product factor
2	C₂²×#13063372	C₂	1	6	yes	maximal, complemented
3	C₂²×#13063370	C₂	1	9	yes	maximal, complemented
4	C₂³×#10621900	C₂	1	9	yes	maximal, complemented
5	C₂²×#41×D₄₃	C₂	1	6	yes	maximal, complemented
6	C₂²×H(5,𝔽₂)×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583734	C₂²	1	1024	yes	direct product factor
8	C₂×#13063372	C₂²	1	288	yes	complemented
9	C₂×#13063370	C₂²	1	432	yes	complemented
10	C₂²×#10621902	C₂²	1	2	yes	complemented
11	C₂²×#10621898	C₂²	1	18	yes	complemented
12	C₂²×#10621900	C₂²	1	18	yes	complemented
13	C₂²×#10621900	C₂²	1	36	yes	complemented
14	C₂²×#10621900	C₂²	1	432	yes	complemented
15	C₂×#41×D₄₃	C₂²	1	288	yes	complemented
16	C₂³×Dic₈₆	C₂²	1	9	yes	complemented
17	C₂²×Q₈×D₄₃	C₂²	1	6	yes	complemented
18	C₂³×C₂²⋊D₄₃	C₂²	1	18	yes	complemented
19	C₂²×D₄×D₄₃	C₂²	1	18	yes	complemented
20	C₂⁴×Dic₄₃	C₂²	1	6	yes	complemented
21	C₂³×C₄×D₄₃	C₂²	1	9	yes	complemented
22	C₂×H(5,𝔽₂)×C₄₃	C₂²	1	48	yes	complemented
23	C₂²×#41×C₄₃	C₂²	1	6	yes	complemented
24	C₂³×D₄×C₄₃	C₂²	1	9	yes	complemented
25	C₂⁴×D₄₃	C₂³	1	9	yes	—
26	#13063372	C₂³	1	1536	yes	complemented
27	#13063370	C₂³	1	2304	yes	complemented
28	C₂×#10621902	C₂³	1	48	yes	complemented
29	C₂×#10621898	C₂³	1	432	yes	complemented
30	C₂×#10621900	C₂³	1	432	yes	complemented
31	C₂×#10621900	C₂³	1	864	yes	complemented
32	C₂×#10621900	C₂³	1	2304	yes	complemented
33	#41×D₄₃	C₂³	1	1536	yes	complemented
34	C₂²×Dic₈₆	C₂³	1	18	yes	—
35	C₂²×Dic₈₆	C₂³	1	216	yes	complemented
36	C₂×Q₈×D₄₃	C₂³	1	144	yes	complemented
37	C₂²×D₁₇₂	C₂³	1	6	yes	—
38	C₂²×C₂²⋊D₄₃	C₂³	1	36	yes	—
39	C₂²×C₂²⋊D₄₃	C₂³	1	432	yes	complemented
40	C₂×D₄×D₄₃	C₂³	1	432	yes	complemented
41	C₂³×Dic₄₃	C₂³	1	9	yes	—
42	C₂³×Dic₄₃	C₂³	1	18	yes	—
43	C₂³×Dic₄₃	C₂³	1	144	yes	complemented
44	C₂²×C₄×D₄₃	C₂³	1	6	yes	—
45	C₂²×C₄×D₄₃	C₂³	1	18	yes	—
46	C₂²×C₄×D₄₃	C₂³	1	216	yes	complemented
47	C₂⁵×C₄₃	C₂³	1	6	yes	—
48	H(5,𝔽₂)×C₄₃	C₂³	1	256	yes	complemented
49	C₂×#41×C₄₃	C₂³	1	144	yes	complemented
50	C₂²×Q₈×C₄₃	C₂³	1	2	yes	—
51	C₂²×D₄×C₄₃	C₂³	1	18	yes	—
52	C₂²×D₄×C₄₃	C₂³	1	216	yes	complemented
53	C₂³×C₄×C₄₃	C₂³	1	9	yes	—
54	C₂⁴×C₄₃	—	4	24	no	—
55	C₂⁴×C₄₃	C₂⁴	1	9	yes	—
56	C₂⁴×C₄₃	C₂⁴	1	72	yes	—
57	C₂³×D₄₃	—	4	36	no	—
58	C₂³×D₄₃	C₂⁴	1	6	yes	—
59	C₂³×D₄₃	C₂⁴	1	108	yes	—
60	#41×C₄₃	C₂⁴	1	384	yes	complemented
61	C₂×Q₈×C₄₃	C₂⁴	1	24	yes	—
62	C₂×D₄×C₄₃	C₂⁴	1	216	yes	—
63	C₂×D₄×C₄₃	C₂⁴	1	576	yes	complemented
64	C₂²×C₄×C₄₃	C₂⁴	1	6	yes	—
65	C₂²×C₄×C₄₃	C₂⁴	1	108	yes	—
66	C₂×C₂²⋊D₄₃	C₂⁴	1	432	yes	—
67	C₂×C₂²⋊D₄₃	C₂⁴	1	1152	yes	complemented
68	C₂²×Dic₄₃	C₂⁴	1	1	yes	characteristic
69	C₂²×Dic₄₃	C₂⁴	1	9	yes	—
70	C₂²×Dic₄₃	C₂⁴	1	108	yes	—
71	C₂²×Dic₄₃	C₂⁴	1	216	yes	—
72	C₂²×Dic₄₃	C₂⁴	1	384	yes	complemented
73	#10621902	C₂⁴	1	128	yes	complemented
74	Q₈×D₄₃	C₂⁴	1	384	yes	complemented
75	#10621900	C₂⁴	1	1152	yes	complemented
76	#10621900	C₂⁴	1	2304	yes	complemented
77	D₄×D₄₃	C₂⁴	1	1152	yes	complemented
78	#10621898	C₂⁴	1	1152	yes	complemented
79	C₂×D₁₇₂	C₂⁴	1	72	yes	—
80	C₂×C₄×D₄₃	C₂⁴	1	72	yes	—
81	C₂×C₄×D₄₃	C₂⁴	1	216	yes	—
82	C₂×C₄×D₄₃	C₂⁴	1	576	yes	complemented
83	C₂×Dic₈₆	C₂⁴	1	216	yes	—
84	C₂×Dic₈₆	C₂⁴	1	576	yes	complemented
85	C₂³×C₄₃	—	2	36	no	—
86	C₂³×C₄₃	—	4	144	no	—
87	C₂³×C₄₃	C₂⁵	1	54	yes	—
88	C₂³×C₄₃	C₂⁵	1	96	yes	—
89	C₂³×C₄₃	C₂⁵	1	1	yes	characteristic
90	C₂²×D₄₃	—	2	24	no	—
91	C₂²×D₄₃	—	4	216	no	—
92	C₂²×D₄₃	C₂⁵	1	36	yes	—
93	C₂²×D₄₃	C₂⁵	1	144	yes	—
94	Q₈×C₄₃	C₂⁵	1	32	yes	—
95	D₄×C₄₃	C₂⁵	1	288	yes	—
96	C₂×C₄×C₄₃	C₂⁵	1	36	yes	—
97	C₂×C₄×C₄₃	C₂⁵	1	144	yes	—
98	C₂²⋊D₄₃	C₂⁵	1	576	yes	—
99	C₂×Dic₄₃	C₂⁵	1	54	yes	—
100	C₂×Dic₄₃	C₂⁵	1	288	yes	—
101	C₂×Dic₄₃	C₂⁵	1	6	yes	—
102	C₂×Dic₄₃	C₂⁵	1	144	yes	—
103	D₁₇₂	C₂⁵	1	96	yes	—
104	C₄×D₄₃	C₂⁵	1	96	yes	—
105	C₄×D₄₃	C₂⁵	1	288	yes	—
106	Dic₈₆	C₂⁵	1	288	yes	—
107	C₂²×#585	—	43	1	no	maximal, Sylow
108	C₂²×C₄₃	—	2	108	no	—
109	C₂²×C₄₃	—	4	96	no	—
110	C₂²×C₄₃	#585	1	4	yes	complemented
111	C₂²×C₄₃	C₂⁶	1	36	yes	—
112	C₂²×C₄₃	C₂⁶	1	3	yes	—
113	C₂×D₄₃	—	2	72	no	—
114	C₂×D₄₃	—	4	144	no	—
115	C₂×D₄₃	C₂⁶	1	24	yes	—
116	C₄×C₄₃	C₂⁶	1	24	yes	—
117	Dic₄₃	C₂⁶	1	36	yes	—
118	Dic₄₃	C₂⁶	1	4	yes	—
119	C₂×#585	—	43	96	no	—
120	C₂²×2₋¹⁺⁴	—	43	6	no	—
121	C₂²×H(5,𝔽₂)	—	43	9	no	—
122	C₂²×H(5,𝔽₂)	D₄₃	1	1	yes	characteristic, complemented
123	C₂³×#41	—	43	6	no	—
124	C₂³×#41	—	43	9	no	—
125	C₂×C₄₃	—	2	36	no	—
126	C₂×C₄₃	C₂×#585	1	6	yes	complemented
127	C₂×C₄₃	C₂⁷	1	1	yes	characteristic
128	D₄₃	—	2	24	no	—
129	#585	—	43	1024	no	—
130	C₂×2₋¹⁺⁴	—	43	288	no	—
131	C₂×H(5,𝔽₂)	—	43	432	no	—
132	C₂×H(5,𝔽₂)	C₂×D₄₃	1	48	yes	complemented
133	C₂²×#41	—	43	288	no	—
134	C₂²×#41	—	43	2	no	—
135	C₂²×#41	—	43	18	no	—
136	C₂²×#41	—	43	36	no	—
137	C₂²×#41	—	43	432	no	—
138	C₂²×#41	—	43	18	no	—
139	C₂²×#41	C₂×D₄₃	1	6	yes	complemented
140	C₂³×Q₈	—	43	6	no	—
141	C₂³×Q₈	—	43	9	no	—
142	C₂³×D₄	—	43	18	no	—
143	C₂³×D₄	—	43	18	no	—
144	C₂³×D₄	C₂×D₄₃	1	9	yes	complemented
145	C₂⁴×C₄	—	43	6	no	—
146	C₂⁴×C₄	—	43	9	no	—
147	C₄₃	C₂²×#585	1	1	yes	characteristic, complemented, Sylow
148	C₂⁵	—	43	9	no	—
149	C₂⁵	C₂²×D₄₃	1	6	yes	—
150	2₋¹⁺⁴	—	43	1536	no	—
151	H(5,𝔽₂)	—	43	2304	no	—
152	H(5,𝔽₂)	C₂²×D₄₃	1	256	yes	complemented
153	C₂×#41	—	43	1536	no	—
154	C₂×#41	—	43	48	no	—
155	C₂×#41	—	43	432	no	—
156	C₂×#41	—	43	864	no	—
157	C₂×#41	—	43	2304	no	—
158	C₂×#41	—	43	432	no	—
159	C₂×#41	C₂²×D₄₃	1	144	yes	complemented
160	C₂²×Q₈	—	43	144	no	—
161	C₂²×Q₈	—	43	18	no	—
162	C₂²×Q₈	—	43	216	no	—
163	C₂²×Q₈	C₂²×D₄₃	1	2	yes	—
164	C₂²×D₄	—	43	36	no	—
165	C₂²×D₄	—	43	432	no	—
166	C₂²×D₄	—	43	432	no	—
167	C₂²×D₄	—	43	6	no	—
168	C₂²×D₄	C₂²×D₄₃	1	18	yes	—
169	C₂²×D₄	C₂²×D₄₃	1	216	yes	complemented
170	C₂³×C₄	—	43	9	no	—
171	C₂³×C₄	—	43	18	no	—
172	C₂³×C₄	—	43	144	no	—
173	C₂³×C₄	—	43	6	no	—
174	C₂³×C₄	—	43	18	no	—
175	C₂³×C₄	—	43	216	no	—
176	C₂³×C₄	C₂²×D₄₃	1	9	yes	—
177	C₂⁴	—	4	24	no	—
178	C₂⁴	—	43	6	no	—
179	C₂⁴	—	43	108	no	—
180	C₂⁴	—	172	36	no	—
181	C₂⁴	C₂³×D₄₃	1	9	yes	—
182	C₂⁴	C₂³×D₄₃	1	72	yes	—
183	#41	—	43	128	no	—
184	#41	—	43	1152	no	—
185	#41	—	43	2304	no	—
186	#41	—	43	1152	no	—
187	#41	C₂³×D₄₃	1	384	yes	complemented
188	C₂×Q₈	—	43	384	no	—
189	C₂×Q₈	—	43	216	no	—
190	C₂×Q₈	—	43	576	no	—
191	C₂×Q₈	C₂³×D₄₃	1	24	yes	—
192	C₂×D₄	—	43	432	no	—
193	C₂×D₄	—	43	1152	no	—
194	C₂×D₄	—	43	1152	no	—
195	C₂×D₄	—	43	72	no	—
196	C₂×D₄	C₂³×D₄₃	1	216	yes	—
197	C₂×D₄	C₂³×D₄₃	1	576	yes	complemented
198	C₂²×C₄	—	43	108	no	—
199	C₂²×C₄	—	43	216	no	—
200	C₂²×C₄	—	43	384	no	—
201	C₂²×C₄	—	43	1	no	—
202	C₂²×C₄	—	43	9	no	—
203	C₂²×C₄	—	43	72	no	—
204	C₂²×C₄	—	43	216	no	—
205	C₂²×C₄	—	43	576	no	—
206	C₂²×C₄	C₂³×D₄₃	1	108	yes	—
207	C₂²×C₄	C₂³×D₄₃	1	6	yes	—
208	C₂³	—	2	36	no	—
209	C₂³	—	4	144	no	—
210	C₂³	—	43	36	no	—
211	C₂³	—	43	144	no	—
212	C₂³	—	86	24	no	—
213	C₂³	—	172	216	no	—
214	C₂³	C₂⁴×D₄₃	1	54	yes	—
215	C₂³	C₂⁴×D₄₃	1	96	yes	—
216	C₂³	C₂⁴×D₄₃	1	1	yes	characteristic
217	Q₈	—	43	288	no	—
218	Q₈	C₂⁴×D₄₃	1	32	yes	—
219	D₄	—	43	576	no	—
220	D₄	—	43	96	no	—
221	D₄	C₂⁴×D₄₃	1	288	yes	—
222	C₂×C₄	—	43	6	no	—
223	C₂×C₄	—	43	54	no	—
224	C₂×C₄	—	43	144	no	—
225	C₂×C₄	—	43	288	no	—
226	C₂×C₄	—	43	96	no	—
227	C₂×C₄	—	43	288	no	—
228	C₂×C₄	C₂⁴×D₄₃	1	36	yes	—
229	C₂×C₄	C₂⁴×D₄₃	1	144	yes	—
230	C₂²	—	2	108	no	—
231	C₂²	—	4	96	no	—
232	C₂²	—	43	24	no	—
233	C₂²	—	86	72	no	—
234	C₂²	—	172	144	no	—
235	C₂²	#14583734	1	4	yes	direct product factor
236	C₂²	C₂⁵×D₄₃	1	36	yes	—
237	C₂²	C₂⁵×D₄₃	1	3	yes	—
238	C₄	—	43	4	no	—
239	C₄	—	43	36	no	—
240	C₄	C₂⁵×D₄₃	1	24	yes	—
241	C₂	—	2	36	no	—
242	C₂	—	86	24	no	—
243	C₂	C₂×#14583734	1	6	yes	direct product factor
244	C₂	C₂⁶×D₄₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₄₃
commutator, cosocle	C₂×C₄₃	C₂⁷
Frattini	C₂	C₂⁶×D₄₃
Fitting	C₂²×H(5,𝔽₂)×C₄₃	C₂
socle	C₂³×C₄₃	C₂⁵
hypercenter	C₂²×H(5,𝔽₂)	D₄₃
hypocenter	C₄₃	C₂²×#585
LCS	G ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₂×#14583734	C₂	direct (N × H)
2. C₂²×#13063372	C₂	semidirect
3. C₂²×#13063370	C₂	semidirect
4. C₂³×#10621900	C₂	semidirect
5. C₂²×#41×D₄₃	C₂	semidirect
6. C₂²×H(5,𝔽₂)×C₄₃	C₂	semidirect
7. #14583734	C₂²	direct (N × H)
8. C₂×#13063372	C₂²	semidirect
9. C₂×#13063370	C₂²	semidirect
10. C₂²×#10621902	C₂²	semidirect
11. C₂²×#10621898	C₂²	semidirect
12. C₂²×#10621900	C₂²	semidirect
13. C₂²×#10621900	C₂²	semidirect
14. C₂²×#10621900	C₂²	semidirect
15. C₂×#41×D₄₃	C₂²	semidirect
16. C₂³×Dic₈₆	C₂²	semidirect
17. C₂²×Q₈×D₄₃	C₂²	semidirect
18. C₂³×C₂²⋊D₄₃	C₂²	semidirect
19. C₂²×D₄×D₄₃	C₂²	semidirect
20. C₂⁴×Dic₄₃	C₂²	semidirect
21. C₂³×C₄×D₄₃	C₂²	semidirect
22. C₂×H(5,𝔽₂)×C₄₃	C₂²	semidirect
23. C₂²×#41×C₄₃	C₂²	semidirect
24. C₂³×D₄×C₄₃	C₂²	semidirect
26. #13063372	C₂³	semidirect
27. #13063370	C₂³	semidirect
28. C₂×#10621902	C₂³	semidirect
29. C₂×#10621898	C₂³	semidirect
30. C₂×#10621900	C₂³	semidirect
31. C₂×#10621900	C₂³	semidirect
32. C₂×#10621900	C₂³	semidirect
33. #41×D₄₃	C₂³	semidirect
35. C₂²×Dic₈₆	C₂³	semidirect
36. C₂×Q₈×D₄₃	C₂³	semidirect
39. C₂²×C₂²⋊D₄₃	C₂³	semidirect
40. C₂×D₄×D₄₃	C₂³	semidirect
43. C₂³×Dic₄₃	C₂³	semidirect
46. C₂²×C₄×D₄₃	C₂³	semidirect
48. H(5,𝔽₂)×C₄₃	C₂³	semidirect
49. C₂×#41×C₄₃	C₂³	semidirect
52. C₂²×D₄×C₄₃	C₂³	semidirect
60. #41×C₄₃	C₂⁴	semidirect
63. C₂×D₄×C₄₃	C₂⁴	semidirect
67. C₂×C₂²⋊D₄₃	C₂⁴	semidirect
72. C₂²×Dic₄₃	C₂⁴	semidirect
73. #10621902	C₂⁴	semidirect
74. Q₈×D₄₃	C₂⁴	semidirect
75. #10621900	C₂⁴	semidirect
76. #10621900	C₂⁴	semidirect
77. D₄×D₄₃	C₂⁴	semidirect
78. #10621898	C₂⁴	semidirect
82. C₂×C₄×D₄₃	C₂⁴	semidirect
84. C₂×Dic₈₆	C₂⁴	semidirect
110. C₂²×C₄₃	#585	semidirect
122. C₂²×H(5,𝔽₂)	D₄₃	semidirect
126. C₂×C₄₃	C₂×#585	semidirect
132. C₂×H(5,𝔽₂)	C₂×D₄₃	semidirect
139. C₂²×#41	C₂×D₄₃	semidirect
144. C₂³×D₄	C₂×D₄₃	semidirect
147. C₄₃	C₂²×#585	semidirect
152. H(5,𝔽₂)	C₂²×D₄₃	semidirect
159. C₂×#41	C₂²×D₄₃	semidirect
169. C₂²×D₄	C₂²×D₄₃	semidirect
187. #41	C₂³×D₄₃	semidirect
197. C₂×D₄	C₂³×D₄₃	semidirect
235. C₂²	#14583734	direct (N × H)
243. C₂	C₂×#14583734	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×#14583734	2
C₂²×#13063372	2
C₂²×#13063370	2
C₂³×#10621900	2
C₂²×#41×D₄₃	2
C₂²×H(5,𝔽₂)×C₄₃	2
C₂²×#585	43

Subgroups — C22×#14583734 #26010489

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#14583734 #26010489