Subgroups — C₂×#16838210 #21438853

Order 7552.

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×#14273773	C₂	1	1	yes	maximal, characteristic, complemented
2	#16838210	C₂	1	16	yes	maximal, direct product factor
3	C₂×#14273758	C₂	1	3	yes	maximal, complemented
4	C₂×#14273752	C₂	1	3	yes	maximal, complemented
5	C₂×#14273747	C₂	1	3	yes	maximal, complemented
6	C₂²×#11745671	C₂	1	3	yes	maximal, complemented
7	C₂×#14273734	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×2₋¹⁺⁴×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#14273773	—	2	4	no	—
10	#14273773	C₂²	1	8	yes	complemented
11	C₂×#11745689	—	2	1	no	—
12	C₂×#11745685	C₂²	1	3	yes	—
13	C₂²×D₂₃₆	C₂²	1	3	yes	—
14	C₂×D₄×D₅₉	—	2	3	no	—
15	#14273758	C₂²	1	24	yes	complemented
16	#14273752	C₂²	1	24	yes	complemented
17	#14273747	C₂²	1	24	yes	complemented
18	C₂×#11745672	C₂²	1	3	yes	complemented
19	C₂×#11745670	C₂²	1	1	yes	characteristic, complemented
20	C₂×#11745671	C₂²	1	3	yes	complemented
21	C₂×#11745671	C₂²	1	6	yes	complemented
22	C₂×#11745671	C₂²	1	24	yes	complemented
23	C₂×#11745669	C₂²	1	3	yes	complemented
24	#14273734	C₂²	1	8	yes	complemented
25	C₂×#11745664	C₂²	1	3	yes	complemented
26	C₂²×C₅₉⋊C₈	C₂²	1	3	yes	complemented
27	2₋¹⁺⁴×C₅₉	—	2	4	no	—
28	2₋¹⁺⁴×C₅₉	C₂²	1	8	yes	complemented
29	C₂×#41×C₅₉	—	2	3	no	—
30	C₂×#41×C₅₉	C₂²	1	1	yes	characteristic
31	C₂×#41×C₅₉	C₂²	1	3	yes	complemented
32	C₂²×Q₈×C₅₉	—	2	1	no	—
33	C₂²×Q₈×C₅₉	C₂²	1	3	yes	complemented
34	C₂³×D₅₉	—	2	3	no	—
35	#41×C₅₉	—	2	24	no	—
36	#41×C₅₉	—	2	6	no	—
37	#41×C₅₉	D₄	1	4	yes	—
38	#41×C₅₉	C₂³	1	4	yes	—
39	#41×C₅₉	C₂³	1	12	yes	complemented
40	C₂×Q₈×C₅₉	—	2	8	no	—
41	C₂×Q₈×C₅₉	—	2	3	no	—
42	C₂×Q₈×C₅₉	—	2	6	no	—
43	C₂×Q₈×C₅₉	D₄	1	1	yes	characteristic
44	C₂×Q₈×C₅₉	C₂³	1	3	yes	—
45	C₂×Q₈×C₅₉	C₂³	1	12	yes	complemented
46	C₂×D₄×C₅₉	—	2	3	no	—
47	C₂×D₄×C₅₉	D₄	1	3	yes	—
48	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₅₉	—	2	3	no	—
50	C₂²×C₄×C₅₉	—	2	3	no	—
51	C₂²×C₄×C₅₉	C₂³	1	3	yes	—
52	C₂×C₂²⋊D₅₉	—	2	3	no	—
53	#11745689	—	2	8	no	—
54	D₄×D₅₉	—	2	24	no	—
55	#11745685	—	2	6	no	—
56	#11745685	C₂³	1	12	yes	—
57	C₂×D₂₃₆	—	2	3	no	—
58	C₂×D₂₃₆	—	2	6	no	—
59	C₂×D₂₃₆	C₂³	1	3	yes	—
60	C₂×D₂₃₆	C₂³	1	12	yes	—
61	C₂×C₄×D₅₉	—	2	3	no	—
62	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
63	#11745672	C₂³	1	12	yes	complemented
64	#11745671	C₂³	1	12	yes	complemented
65	#11745671	C₂³	1	24	yes	complemented
66	#11745670	C₂³	1	4	yes	complemented
67	#11745669	C₂³	1	12	yes	complemented
68	#11745664	C₂³	1	12	yes	complemented
69	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₅₉⋊C₈	C₂³	1	3	yes	—
71	C₂×C₅₉⋊C₈	C₂³	1	12	yes	complemented
72	C₂³×C₅₉	—	2	1	no	—
73	C₂³×C₅₉	C₂×D₄	1	3	yes	—
74	C₂²×D₅₉	—	2	3	no	—
75	C₂²×D₅₉	—	2	12	no	—
76	C₂²×D₅₉	—	8	6	no	—
77	Q₈×C₅₉	—	2	3	no	—
78	Q₈×C₅₉	—	2	12	no	—
79	Q₈×C₅₉	C₂×D₄	1	4	yes	—
80	Q₈×C₅₉	C₂⁴	1	6	yes	—
81	D₄×C₅₉	—	2	12	no	—
82	D₄×C₅₉	—	2	1	no	—
83	D₄×C₅₉	C₂×D₄	1	12	yes	—
84	D₄×C₅₉	C₂⁴	1	2	yes	—
85	C₂×C₄×C₅₉	—	2	3	no	—
86	C₂×C₄×C₅₉	—	2	12	no	—
87	C₂×C₄×C₅₉	—	2	12	no	—
88	C₂×C₄×C₅₉	C₂×D₄	1	3	yes	—
89	C₂×C₄×C₅₉	C₂×D₄	1	6	yes	—
90	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
91	C₂×C₄×C₅₉	C₂⁴	1	6	yes	—
92	C₂²⋊D₅₉	—	2	12	no	—
93	C₂×Dic₅₉	—	2	1	no	—
94	D₂₃₆	—	2	3	no	—
95	D₂₃₆	—	2	12	no	—
96	D₂₃₆	C₂⁴	1	6	yes	—
97	C₄×D₅₉	—	2	12	no	—
98	Dic₁₁₈	—	2	1	no	—
99	Dic₁₁₈	C₂⁴	1	2	yes	—
100	C₅₉⋊C₈	C₂⁴	1	2	yes	—
101	C₅₉⋊C₈	C₂⁴	1	6	yes	—
102	C₂²×C₅₉	—	2	6	no	—
103	C₂²×C₅₉	—	2	2	no	—
104	C₂²×C₅₉	—	4	2	no	—
105	C₂²×C₅₉	C₂²×D₄	1	6	yes	—
106	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
107	C₂×D₅₉	—	2	6	no	—
108	C₂×D₅₉	—	4	6	no	—
109	C₂×D₅₉	—	8	12	no	—
110	C₄×C₅₉	—	2	6	no	—
111	C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
112	C₄×C₅₉	C₂²×D₄	1	6	yes	—
113	C₄×C₅₉	C₂⁵	1	1	yes	characteristic
114	Dic₅₉	—	2	2	no	—
115	C₂×#577	—	59	1	no	maximal, Sylow
116	C₂×C₅₉	—	2	6	no	—
117	C₂×C₅₉	—	4	2	no	—
118	C₂×C₅₉	#577	1	2	yes	complemented
119	C₂×C₅₉	C₂³×D₄	1	1	yes	characteristic
120	D₅₉	—	4	6	no	—
121	C₂×2₋¹⁺⁴	D₅₉	1	1	yes	characteristic, complemented
122	C₂×H(5,𝔽₂)	—	59	1	no	—
123	#577	—	59	16	no	—
124	C₂×#137	—	59	3	no	—
125	C₂×AGL(1,ℤ/8)	—	59	3	no	—
126	C₂×#135	—	59	3	no	—
127	C₂²×SD₁₆	—	59	3	no	—
128	C₂×#131	—	59	1	no	—
129	C₅₉	C₂×#577	1	1	yes	characteristic, complemented, Sylow
130	2₋¹⁺⁴	—	2	4	no	—
131	2₋¹⁺⁴	C₂×D₅₉	1	8	yes	complemented
132	H(5,𝔽₂)	—	59	8	no	—
133	H(5,𝔽₂)	—	118	4	no	—
134	C₂×#41	—	2	3	no	—
135	C₂×#41	—	59	3	no	—
136	C₂×#41	—	118	1	no	—
137	C₂×#41	C₂×D₅₉	1	1	yes	characteristic
138	C₂×#41	C₂×D₅₉	1	3	yes	complemented
139	C₂²×Q₈	—	2	1	no	—
140	C₂²×Q₈	C₂×D₅₉	1	3	yes	complemented
141	C₂²×D₄	—	59	3	no	—
142	C₂²×D₄	—	118	3	no	—
143	#137	—	59	24	no	—
144	AGL(1,ℤ/8)	—	59	24	no	—
145	#135	—	59	24	no	—
146	C₂×Q₁₆	—	59	3	no	—
147	C₂×SD₁₆	—	59	3	no	—
148	C₂×SD₁₆	—	59	6	no	—
149	C₂×SD₁₆	—	59	24	no	—
150	C₂×SD₁₆	—	59	1	no	—
151	C₂×D₈	—	59	3	no	—
152	#131	—	59	8	no	—
153	C₂×M₁₆	—	59	3	no	—
154	C₂²×C₈	—	59	3	no	—
155	C₂⁴	—	118	3	no	—
156	#41	—	2	24	no	—
157	#41	—	2	6	no	—
158	#41	—	59	12	no	—
159	#41	—	118	8	no	—
160	#41	—	118	6	no	—
161	#41	C₂²⋊D₅₉	1	4	yes	—
162	#41	C₂²×D₅₉	1	4	yes	—
163	#41	C₂²×D₅₉	1	12	yes	complemented
164	C₂×Q₈	—	2	8	no	—
165	C₂×Q₈	—	2	3	no	—
166	C₂×Q₈	—	2	6	no	—
167	C₂×Q₈	—	59	1	no	—
168	C₂×Q₈	C₂²⋊D₅₉	1	1	yes	characteristic
169	C₂×Q₈	C₂²×D₅₉	1	3	yes	—
170	C₂×Q₈	C₂²×D₅₉	1	12	yes	complemented
171	C₂×D₄	—	2	3	no	—
172	C₂×D₄	—	59	3	no	—
173	C₂×D₄	—	59	12	no	—
174	C₂×D₄	—	118	3	no	—
175	C₂×D₄	—	118	24	no	—
176	C₂×D₄	—	118	3	no	—
177	C₂×D₄	—	118	6	no	—
178	C₂×D₄	C₂²⋊D₅₉	1	3	yes	—
179	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
180	C₂²×C₄	—	2	3	no	—
181	C₂²×C₄	—	2	3	no	—
182	C₂²×C₄	—	118	3	no	—
183	C₂²×C₄	C₂²×D₅₉	1	3	yes	—
184	Q₁₆	—	59	12	no	—
185	SD₁₆	—	59	12	no	—
186	SD₁₆	—	59	24	no	—
187	SD₁₆	—	59	4	no	—
188	D₈	—	59	12	no	—
189	M₁₆	—	59	12	no	—
190	C₂×C₈	—	59	1	no	—
191	C₂×C₈	—	59	3	no	—
192	C₂×C₈	—	59	12	no	—
193	C₂³	—	2	1	no	—
194	C₂³	—	118	12	no	—
195	C₂³	—	118	3	no	—
196	C₂³	—	472	6	no	—
197	C₂³	C₂×C₂²⋊D₅₉	1	3	yes	—
198	Q₈	—	2	3	no	—
199	Q₈	—	2	12	no	—
200	Q₈	—	59	2	no	—
201	Q₈	—	118	1	no	—
202	Q₈	C₂×C₂²⋊D₅₉	1	4	yes	—
203	Q₈	C₂³×D₅₉	1	6	yes	—
204	D₄	—	2	12	no	—
205	D₄	—	2	1	no	—
206	D₄	—	59	6	no	—
207	D₄	—	118	12	no	—
208	D₄	—	118	12	no	—
209	D₄	—	118	3	no	—
210	D₄	C₂×C₂²⋊D₅₉	1	12	yes	—
211	D₄	C₂³×D₅₉	1	2	yes	—
212	C₂×C₄	—	2	12	no	—
213	C₂×C₄	—	2	3	no	—
214	C₂×C₄	—	2	12	no	—
215	C₂×C₄	—	118	1	no	—
216	C₂×C₄	—	118	12	no	—
217	C₂×C₄	C₂×C₂²⋊D₅₉	1	6	yes	—
218	C₂×C₄	C₂×C₂²⋊D₅₉	1	3	yes	—
219	C₂×C₄	C₂³×D₅₉	1	6	yes	—
220	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
221	C₈	—	59	2	no	—
222	C₈	—	59	6	no	—
223	C₂²	—	2	6	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	—	118	6	no	—
227	C₂²	—	472	12	no	—
228	C₂²	—	236	6	no	—
229	C₂²	C₂²×C₂²⋊D₅₉	1	6	yes	—
230	C₂²	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
231	C₄	—	2	6	no	—
232	C₄	—	118	2	no	—
233	C₄	C₂²×C₂²⋊D₅₉	1	6	yes	—
234	C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
235	C₄	C₂⁴×D₅₉	1	1	yes	characteristic
236	C₂	—	2	6	no	—
237	C₂	—	4	2	no	—
238	C₂	—	236	6	no	—
239	C₂	#16838210	1	2	yes	direct product factor
240	C₂	C₂³×C₂²⋊D₅₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₅₉
commutator	C₄×C₅₉	C₂⁵
Frattini	C₄	C₂⁴×D₅₉
Fitting	C₂×2₋¹⁺⁴×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	C₂×2₋¹⁺⁴	D₅₉
hypocenter	C₅₉	C₂×#577
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. C₂×#14273773	C₂	semidirect
2. #16838210	C₂	direct (N × H)
3. C₂×#14273758	C₂	semidirect
4. C₂×#14273752	C₂	semidirect
5. C₂×#14273747	C₂	semidirect
6. C₂²×#11745671	C₂	semidirect
7. C₂×#14273734	C₂	semidirect
8. C₂×2₋¹⁺⁴×C₅₉	C₂	semidirect
10. #14273773	C₂²	semidirect
15. #14273758	C₂²	semidirect
16. #14273752	C₂²	semidirect
17. #14273747	C₂²	semidirect
18. C₂×#11745672	C₂²	semidirect
19. C₂×#11745670	C₂²	semidirect
20. C₂×#11745671	C₂²	semidirect
21. C₂×#11745671	C₂²	semidirect
22. C₂×#11745671	C₂²	semidirect
23. C₂×#11745669	C₂²	semidirect
24. #14273734	C₂²	semidirect
25. C₂×#11745664	C₂²	semidirect
26. C₂²×C₅₉⋊C₈	C₂²	semidirect
28. 2₋¹⁺⁴×C₅₉	C₂²	semidirect
31. C₂×#41×C₅₉	C₂²	semidirect
33. C₂²×Q₈×C₅₉	C₂²	semidirect
39. #41×C₅₉	C₂³	semidirect
45. C₂×Q₈×C₅₉	C₂³	semidirect
63. #11745672	C₂³	semidirect
64. #11745671	C₂³	semidirect
65. #11745671	C₂³	semidirect
66. #11745670	C₂³	semidirect
67. #11745669	C₂³	semidirect
68. #11745664	C₂³	semidirect
71. C₂×C₅₉⋊C₈	C₂³	semidirect
118. C₂×C₅₉	#577	semidirect
121. C₂×2₋¹⁺⁴	D₅₉	semidirect
129. C₅₉	C₂×#577	semidirect
131. 2₋¹⁺⁴	C₂×D₅₉	semidirect
138. C₂×#41	C₂×D₅₉	semidirect
140. C₂²×Q₈	C₂×D₅₉	semidirect
163. #41	C₂²×D₅₉	semidirect
170. C₂×Q₈	C₂²×D₅₉	semidirect
239. C₂	#16838210	direct (N × H)

Subgroups — C2×#16838210 #21438853

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16838210 #21438853