Subgroups — C₂×#21437237 #29495201

Order 15104.

2126 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16838207	C₂	1	2	yes	maximal, complemented
2	#21437237	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16837840	C₂	1	6	yes	maximal, complemented
4	C₂×#16837834	C₂	1	3	yes	maximal, complemented
5	C₂²×D₉₄₄	C₂	1	3	yes	maximal, complemented
6	C₂×C₅₉×#504	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14273773	—	2	2	no	—
8	#16838207	—	2	8	no	—
9	#16838207	C₂²	1	16	yes	complemented
10	C₂×#14273755	—	2	6	no	—
11	C₂×#14273749	C₂²	1	6	yes	complemented
12	C₂²×D₄₇₂	C₂²	1	6	yes	complemented
13	#16837840	C₂²	1	48	yes	complemented
14	#16837834	C₂²	1	24	yes	complemented
15	C₂×Dic₄₇₂	C₂²	1	1	yes	characteristic, complemented
16	C₂×#14273672	C₂²	1	6	yes	complemented
17	C₂×D₉₄₄	C₂²	1	3	yes	complemented
18	C₂×D₉₄₄	C₂²	1	6	yes	complemented
19	C₂×D₉₄₄	C₂²	1	24	yes	complemented
20	C₂×#131×C₅₉	C₂²	1	1	yes	characteristic
21	C₅₉×#504	C₂²	1	8	yes	complemented
22	C₂×M₃₂×C₅₉	C₂²	1	3	yes	complemented
23	C₂²×C₁₆×C₅₉	C₂²	1	3	yes	complemented
24	#14273773	—	2	16	no	—
25	#14273773	—	4	8	no	—
26	C₂×#11745689	—	4	2	no	—
27	C₂×#11745685	—	2	6	no	—
28	C₂²×D₂₃₆	—	2	6	no	—
29	C₂×D₄×D₅₉	—	4	6	no	—
30	#14273755	—	2	48	no	—
31	#14273749	—	2	12	no	—
32	#14273749	C₂³	1	24	yes	complemented
33	C₂×Dic₂₃₆	C₂³	1	2	yes	—
34	C₂×#11745660	—	2	6	no	—
35	C₂×D₄₇₂	—	2	6	no	—
36	C₂×D₄₇₂	—	2	12	no	—
37	C₂×D₄₇₂	C₂³	1	6	yes	—
38	C₂×D₄₇₂	C₂³	1	24	yes	complemented
39	Dic₄₇₂	C₂³	1	4	yes	complemented
40	#14273672	C₂³	1	24	yes	complemented
41	D₉₄₄	C₂³	1	12	yes	complemented
42	D₉₄₄	C₂³	1	24	yes	complemented
43	C₂×#41×C₅₉	D₄	1	1	yes	characteristic
44	#131×C₅₉	D₄	1	4	yes	—
45	#131×C₅₉	C₂³	1	4	yes	—
46	C₂×M₁₆×C₅₉	D₄	1	3	yes	—
47	C₂²×C₈×C₅₉	C₂³	1	3	yes	—
48	M₃₂×C₅₉	C₂³	1	12	yes	complemented
49	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
50	C₂×C₁₆×C₅₉	C₂³	1	3	yes	—
51	C₂×C₁₆×C₅₉	C₂³	1	12	yes	complemented
52	C₂³×D₅₉	—	4	6	no	—
53	#41×C₅₉	D₈	1	4	yes	—
54	#41×C₅₉	C₂×D₄	1	4	yes	—
55	C₂×Q₈×C₅₉	D₈	1	1	yes	characteristic
56	C₂×D₄×C₅₉	D₈	1	3	yes	—
57	C₂²×C₄×C₅₉	C₂×D₄	1	3	yes	—
58	C₂×C₂²⋊D₅₉	—	4	6	no	—
59	#11745689	—	4	16	no	—
60	D₄×D₅₉	—	4	48	no	—
61	#11745685	—	2	24	no	—
62	#11745685	—	4	12	no	—
63	C₂×D₂₃₆	—	2	6	no	—
64	C₂×D₂₃₆	—	2	24	no	—
65	C₂×D₂₃₆	—	4	6	no	—
66	C₂×D₂₃₆	—	4	12	no	—
67	C₂×C₄×D₅₉	—	4	6	no	—
68	C₂×Dic₁₁₈	—	2	2	no	—
69	M₁₆×C₅₉	C₂×D₄	1	12	yes	—
70	C₂×C₈×C₅₉	C₂×D₄	1	3	yes	—
71	C₂×C₈×C₅₉	C₂×D₄	1	6	yes	—
72	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
73	C₂×C₈×C₅₉	C₂⁴	1	6	yes	—
74	Dic₂₃₆	—	2	2	no	—
75	Dic₂₃₆	C₂⁴	1	4	yes	—
76	D₄₇₂	—	2	6	no	—
77	D₄₇₂	—	2	24	no	—
78	D₄₇₂	C₂⁴	1	12	yes	—
79	#11745660	—	2	24	no	—
80	C₁₆×C₅₉	C₂⁴	1	2	yes	—
81	C₁₆×C₅₉	C₂⁴	1	6	yes	—
82	C₂³×C₅₉	C₂×D₈	1	3	yes	—
83	C₂²×D₅₉	—	4	6	no	—
84	C₂²×D₅₉	—	4	24	no	—
85	C₂²×D₅₉	—	16	12	no	—
86	Q₈×C₅₉	C₂×D₈	1	4	yes	—
87	D₄×C₅₉	C₂×D₈	1	12	yes	—
88	C₂×C₄×C₅₉	C₂×D₈	1	3	yes	—
89	C₂×C₄×C₅₉	C₂×D₈	1	6	yes	—
90	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
91	C₂×C₄×C₅₉	C₂²×D₄	1	6	yes	—
92	C₂²⋊D₅₉	—	4	24	no	—
93	C₂×Dic₅₉	—	4	2	no	—
94	D₂₃₆	—	2	12	no	—
95	D₂₃₆	—	4	6	no	—
96	D₂₃₆	—	4	24	no	—
97	C₄×D₅₉	—	4	24	no	—
98	Dic₁₁₈	—	2	4	no	—
99	Dic₁₁₈	—	4	2	no	—
100	C₈×C₅₉	C₂²×D₄	1	6	yes	—
101	C₈×C₅₉	C₂²×D₄	1	1	yes	characteristic
102	C₈×C₅₉	C₂⁵	1	1	yes	characteristic
103	C₂×#3415	—	59	1	no	maximal, Sylow
104	C₂²×C₅₉	—	2	6	no	—
105	C₂²×C₅₉	C₂²×D₈	1	6	yes	—
106	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
107	C₂×D₅₉	—	4	12	no	—
108	C₂×D₅₉	—	8	12	no	—
109	C₂×D₅₉	—	16	24	no	—
110	C₄×C₅₉	C₂²×D₈	1	1	yes	characteristic
111	C₄×C₅₉	C₂²×D₈	1	6	yes	—
112	C₄×C₅₉	C₂³×D₄	1	1	yes	characteristic
113	Dic₅₉	—	4	4	no	—
114	C₂×#576	—	59	2	no	—
115	#3415	—	59	16	no	—
116	C₂×#509	—	59	6	no	—
117	C₂×#508	—	59	3	no	—
118	C₂²×D₁₆	—	59	3	no	—
119	C₂×#504	D₅₉	1	1	yes	characteristic, complemented
120	C₂×C₅₉	—	2	6	no	—
121	C₂×C₅₉	#3415	1	2	yes	complemented
122	C₂×C₅₉	C₂³×D₈	1	1	yes	characteristic
123	D₅₉	—	8	12	no	—
124	C₂×H(5,𝔽₂)	—	118	2	no	—
125	#576	—	59	16	no	—
126	#576	—	118	8	no	—
127	C₂×AGL(1,ℤ/8)	—	118	6	no	—
128	C₂×#135	—	59	6	no	—
129	C₂²×D₈	—	59	6	no	—
130	C₂×#131	C₂×D₅₉	1	1	yes	characteristic
131	#509	—	59	48	no	—
132	#508	—	59	24	no	—
133	C₂×Q₃₂	—	59	1	no	—
134	C₂×SD₃₂	—	59	6	no	—
135	C₂×D₁₆	—	59	3	no	—
136	C₂×D₁₆	—	59	6	no	—
137	C₂×D₁₆	—	59	24	no	—
138	#504	C₂×D₅₉	1	8	yes	complemented
139	C₂×M₃₂	C₂×D₅₉	1	3	yes	complemented
140	C₂²×C₁₆	C₂×D₅₉	1	3	yes	complemented
141	C₅₉	C₂×#3415	1	1	yes	characteristic, complemented, Sylow
142	H(5,𝔽₂)	—	118	16	no	—
143	H(5,𝔽₂)	—	236	8	no	—
144	C₂×#41	—	118	6	no	—
145	C₂×#41	—	236	2	no	—
146	C₂×#41	D₂₃₆	1	1	yes	characteristic
147	C₂²×D₄	—	118	6	no	—
148	C₂²×D₄	—	236	6	no	—
149	AGL(1,ℤ/8)	—	118	48	no	—
150	#135	—	59	24	no	—
151	#135	—	118	12	no	—
152	C₂×Q₁₆	—	59	2	no	—
153	C₂×SD₁₆	—	118	6	no	—
154	C₂×D₈	—	59	6	no	—
155	C₂×D₈	—	59	24	no	—
156	C₂×D₈	—	118	6	no	—
157	C₂×D₈	—	118	12	no	—
158	#131	D₂₃₆	1	4	yes	—
159	#131	C₂²×D₅₉	1	4	yes	—
160	C₂×M₁₆	D₂₃₆	1	3	yes	—
161	C₂²×C₈	C₂²×D₅₉	1	3	yes	—
162	Q₃₂	—	59	4	no	—
163	SD₃₂	—	59	24	no	—
164	D₁₆	—	59	12	no	—
165	D₁₆	—	59	24	no	—
166	M₃₂	C₂²×D₅₉	1	12	yes	complemented
167	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
168	C₂×C₁₆	C₂²×D₅₉	1	3	yes	—
169	C₂×C₁₆	C₂²×D₅₉	1	12	yes	complemented
170	C₂⁴	—	236	6	no	—
171	#41	—	118	24	no	—
172	#41	—	236	16	no	—
173	#41	—	236	12	no	—
174	#41	D₄₇₂	1	4	yes	—
175	#41	C₂×D₂₃₆	1	4	yes	—
176	C₂×Q₈	—	118	2	no	—
177	C₂×Q₈	D₄₇₂	1	1	yes	characteristic
178	C₂×D₄	—	118	6	no	—
179	C₂×D₄	—	118	24	no	—
180	C₂×D₄	—	236	6	no	—
181	C₂×D₄	—	236	48	no	—
182	C₂×D₄	—	236	6	no	—
183	C₂×D₄	—	236	12	no	—
184	C₂×D₄	D₄₇₂	1	3	yes	—
185	C₂²×C₄	—	236	6	no	—
186	C₂²×C₄	C₂×D₂₃₆	1	3	yes	—
187	Q₁₆	—	59	4	no	—
188	Q₁₆	—	118	2	no	—
189	SD₁₆	—	118	24	no	—
190	D₈	—	59	12	no	—
191	D₈	—	118	6	no	—
192	D₈	—	118	24	no	—
193	M₁₆	C₂×D₂₃₆	1	12	yes	—
194	C₂×C₈	C₂×D₂₃₆	1	3	yes	—
195	C₂×C₈	C₂×D₂₃₆	1	6	yes	—
196	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
197	C₂×C₈	C₂³×D₅₉	1	6	yes	—
198	C₁₆	C₂³×D₅₉	1	2	yes	—
199	C₁₆	C₂³×D₅₉	1	6	yes	—
200	C₂³	—	944	12	no	—
201	C₂³	—	236	24	no	—
202	C₂³	—	236	6	no	—
203	C₂³	C₂×D₄₇₂	1	3	yes	—
204	Q₈	—	118	4	no	—
205	Q₈	—	236	2	no	—
206	Q₈	C₂×D₄₇₂	1	4	yes	—
207	D₄	—	118	12	no	—
208	D₄	—	236	24	no	—
209	D₄	—	236	24	no	—
210	D₄	—	236	6	no	—
211	D₄	C₂×D₄₇₂	1	12	yes	—
212	C₂×C₄	—	236	2	no	—
213	C₂×C₄	—	236	24	no	—
214	C₂×C₄	C₂×D₄₇₂	1	6	yes	—
215	C₂×C₄	C₂×D₄₇₂	1	3	yes	—
216	C₂×C₄	C₂²×D₂₃₆	1	6	yes	—
217	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
218	C₈	C₂²×D₂₃₆	1	1	yes	characteristic
219	C₈	C₂²×D₂₃₆	1	6	yes	—
220	C₈	C₂⁴×D₅₉	1	1	yes	characteristic
221	C₂²	—	2	6	no	—
222	C₂²	—	944	24	no	—
223	C₂²	—	472	12	no	—
224	C₂²	—	236	12	no	—
225	C₂²	C₂²×D₄₇₂	1	6	yes	—
226	C₂²	C₂²×D₄₇₂	1	1	yes	characteristic
227	C₄	—	236	4	no	—
228	C₄	C₂²×D₄₇₂	1	6	yes	—
229	C₄	C₂²×D₄₇₂	1	1	yes	characteristic
230	C₄	C₂³×D₂₃₆	1	1	yes	characteristic
231	C₂	—	2	6	no	—
232	C₂	—	472	12	no	—
233	C₂	#21437237	1	2	yes	direct product factor
234	C₂	C₂³×D₄₇₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₇₂
commutator, cosocle	C₈×C₅₉	C₂⁵
Frattini	C₈	C₂⁴×D₅₉
Fitting	C₂×C₅₉×#504	C₂
socle	C₂²×C₅₉	C₂²×D₈
hypercenter	C₂×#504	D₅₉
hypocenter	C₅₉	C₂×#3415
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂×#16838207	C₂	semidirect
2. #21437237	C₂	direct (N × H)
3. C₂×#16837840	C₂	semidirect
4. C₂×#16837834	C₂	semidirect
5. C₂²×D₉₄₄	C₂	semidirect
6. C₂×C₅₉×#504	C₂	semidirect
9. #16838207	C₂²	semidirect
11. C₂×#14273749	C₂²	semidirect
12. C₂²×D₄₇₂	C₂²	semidirect
13. #16837840	C₂²	semidirect
14. #16837834	C₂²	semidirect
15. C₂×Dic₄₇₂	C₂²	semidirect
16. C₂×#14273672	C₂²	semidirect
17. C₂×D₉₄₄	C₂²	semidirect
18. C₂×D₉₄₄	C₂²	semidirect
19. C₂×D₉₄₄	C₂²	semidirect
21. C₅₉×#504	C₂²	semidirect
22. C₂×M₃₂×C₅₉	C₂²	semidirect
23. C₂²×C₁₆×C₅₉	C₂²	semidirect
32. #14273749	C₂³	semidirect
38. C₂×D₄₇₂	C₂³	semidirect
39. Dic₄₇₂	C₂³	semidirect
40. #14273672	C₂³	semidirect
41. D₉₄₄	C₂³	semidirect
42. D₉₄₄	C₂³	semidirect
48. M₃₂×C₅₉	C₂³	semidirect
51. C₂×C₁₆×C₅₉	C₂³	semidirect
119. C₂×#504	D₅₉	semidirect
121. C₂×C₅₉	#3415	semidirect
138. #504	C₂×D₅₉	semidirect
139. C₂×M₃₂	C₂×D₅₉	semidirect
140. C₂²×C₁₆	C₂×D₅₉	semidirect
141. C₅₉	C₂×#3415	semidirect
166. M₃₂	C₂²×D₅₉	semidirect
169. C₂×C₁₆	C₂²×D₅₉	semidirect
233. C₂	#21437237	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×#16838207	2
#21437237	2
C₂×#16837840	2
C₂×#16837834	2
C₂²×D₉₄₄	2
C₂×C₅₉×#504	2
C₂×#3415	59

Subgroups — C2×#21437237 #29495201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21437237 #29495201