Subgroups — C₂×#30632499 #44565126

Order 32128.

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×#22548762	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#22548761	C₂	1	1	yes	maximal, characteristic, complemented
3	#30632499	C₂	1	16	yes	maximal, direct product factor
4	C₂×#22548748	C₂	1	3	yes	maximal, complemented
5	C₂×#22548743	C₂	1	3	yes	maximal, complemented
6	C₂×#22548737	C₂	1	3	yes	maximal, complemented
7	C₂²×#16860799	C₂	1	3	yes	maximal, complemented
8	C₂×#131×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#22548762	—	2	4	no	—
10	#22548762	C₂²	1	8	yes	complemented
11	#22548761	—	2	4	no	—
12	#22548761	C₂²	1	8	yes	complemented
13	C₂×#16860811	—	2	1	no	—
14	C₂×#16860810	C₂²	1	3	yes	—
15	C₂×#16860810	C₂²	1	3	yes	complemented
16	C₂×#16860809	—	2	3	no	—
17	C₂²×Dic₅₀₂	C₂²	1	3	yes	complemented
18	C₂×Q₈×D₂₅₁	—	2	1	no	—
19	C₂²×D₁₀₀₄	C₂²	1	3	yes	—
20	C₂×D₄×D₂₅₁	—	2	3	no	—
21	#22548748	C₂²	1	24	yes	complemented
22	#22548743	C₂²	1	24	yes	complemented
23	#22548737	C₂²	1	24	yes	complemented
24	C₂×Dic₁₀₀₄	C₂²	1	3	yes	complemented
25	C₂×#16860799	C₂²	1	3	yes	complemented
26	C₂×#16860799	C₂²	1	6	yes	complemented
27	C₂×#16860799	C₂²	1	1	yes	characteristic, complemented
28	C₂×#16860799	C₂²	1	24	yes	complemented
29	C₂×D₂₀₀₈	C₂²	1	3	yes	complemented
30	C₂×#41×C₂₅₁	C₂²	1	1	yes	characteristic
31	#131×C₂₅₁	C₂²	1	8	yes	complemented
32	C₂×M₁₆×C₂₅₁	C₂²	1	3	yes	complemented
33	C₂²×C₈×C₂₅₁	C₂²	1	3	yes	complemented
34	C₂³×D₂₅₁	—	2	3	no	—
35	#16860811	—	2	8	no	—
36	#16860810	—	2	6	no	—
37	#16860810	—	2	6	no	—
38	#16860810	C₂³	1	12	yes	—
39	#16860810	C₂³	1	12	yes	complemented
40	#16860809	—	2	24	no	—
41	C₂×Dic₅₀₂	—	2	3	no	—
42	C₂×Dic₅₀₂	—	2	6	no	—
43	C₂×Dic₅₀₂	C₂³	1	3	yes	—
44	C₂×Dic₅₀₂	C₂³	1	1	yes	characteristic
45	C₂×Dic₅₀₂	C₂³	1	12	yes	complemented
46	Q₈×D₂₅₁	—	2	8	no	—
47	C₂×D₁₀₀₄	—	2	3	no	—
48	C₂×D₁₀₀₄	—	2	6	no	—
49	C₂×D₁₀₀₄	C₂³	1	12	yes	—
50	C₂×D₁₀₀₄	C₂³	1	3	yes	—
51	C₂×D₁₀₀₄	C₂³	1	1	yes	characteristic
52	C₂×C₂²⋊D₂₅₁	—	2	3	no	—
53	C₂×C₂²⋊D₂₅₁	—	2	3	no	—
54	D₄×D₂₅₁	—	2	24	no	—
55	C₂²×Dic₂₅₁	—	2	3	no	—
56	C₂×C₄×D₂₅₁	—	2	3	no	—
57	C₂×C₄×D₂₅₁	—	2	3	no	—
58	Dic₁₀₀₄	C₂³	1	12	yes	complemented
59	#16860799	C₂³	1	4	yes	complemented
60	#16860799	C₂³	1	12	yes	complemented
61	#16860799	C₂³	1	24	yes	complemented
62	D₂₀₀₈	C₂³	1	12	yes	complemented
63	#41×C₂₅₁	D₄	1	4	yes	—
64	#41×C₂₅₁	C₂³	1	4	yes	—
65	C₂×Q₈×C₂₅₁	D₄	1	1	yes	characteristic
66	C₂×D₄×C₂₅₁	D₄	1	3	yes	—
67	C₂²×C₄×C₂₅₁	C₂³	1	3	yes	—
68	M₁₆×C₂₅₁	C₂³	1	12	yes	complemented
69	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₂₅₁	C₂³	1	3	yes	—
71	C₂×C₈×C₂₅₁	C₂³	1	12	yes	complemented
72	C₂²×D₂₅₁	—	2	1	no	—
73	C₂²×D₂₅₁	—	2	3	no	—
74	C₂²×D₂₅₁	—	2	12	no	—
75	C₂²×D₂₅₁	—	8	6	no	—
76	Dic₅₀₂	—	2	12	no	—
77	Dic₅₀₂	—	2	3	no	—
78	Dic₅₀₂	—	2	1	no	—
79	Dic₅₀₂	C₂⁴	1	6	yes	—
80	Dic₅₀₂	C₂⁴	1	2	yes	—
81	D₁₀₀₄	—	2	12	no	—
82	D₁₀₀₄	—	2	3	no	—
83	D₁₀₀₄	—	2	1	no	—
84	D₁₀₀₄	C₂⁴	1	2	yes	—
85	D₁₀₀₄	C₂⁴	1	6	yes	—
86	C₂²⋊D₂₅₁	—	2	12	no	—
87	C₂²⋊D₂₅₁	—	2	12	no	—
88	C₂×Dic₂₅₁	—	2	1	no	—
89	C₂×Dic₂₅₁	—	2	3	no	—
90	C₂×Dic₂₅₁	—	2	12	no	—
91	C₄×D₂₅₁	—	2	12	no	—
92	C₄×D₂₅₁	—	2	12	no	—
93	C₂³×C₂₅₁	C₂×D₄	1	3	yes	—
94	Q₈×C₂₅₁	C₂×D₄	1	4	yes	—
95	D₄×C₂₅₁	C₂×D₄	1	12	yes	—
96	C₂×C₄×C₂₅₁	C₂×D₄	1	3	yes	—
97	C₂×C₄×C₂₅₁	C₂×D₄	1	6	yes	—
98	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
99	C₂×C₄×C₂₅₁	C₂⁴	1	6	yes	—
100	C₈×C₂₅₁	C₂⁴	1	2	yes	—
101	C₈×C₂₅₁	C₂⁴	1	6	yes	—
102	C₂²×C₂₅₁	—	2	6	no	—
103	C₂²×C₂₅₁	C₂²×D₄	1	6	yes	—
104	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
105	C₂×D₂₅₁	—	2	2	no	—
106	C₂×D₂₅₁	—	2	6	no	—
107	C₂×D₂₅₁	—	4	2	no	—
108	C₂×D₂₅₁	—	4	6	no	—
109	C₂×D₂₅₁	—	8	12	no	—
110	C₄×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
111	C₄×C₂₅₁	C₂²×D₄	1	6	yes	—
112	C₄×C₂₅₁	C₂⁵	1	1	yes	characteristic
113	Dic₂₅₁	—	2	2	no	—
114	Dic₂₅₁	—	2	6	no	—
115	C₂×C₂₅₁	—	2	6	no	—
116	C₂×C₂₅₁	#577	1	2	yes	complemented
117	C₂×C₂₅₁	C₂³×D₄	1	1	yes	characteristic
118	D₂₅₁	—	4	2	no	—
119	D₂₅₁	—	4	6	no	—
120	C₂₅₁	C₂×#577	1	1	yes	characteristic, complemented, Sylow
121	C₂×#577	—	251	1	no	maximal, Sylow
122	C₂×2₋¹⁺⁴	—	251	1	no	—
123	C₂×H(5,𝔽₂)	—	251	1	no	—
124	#577	—	251	16	no	—
125	C₂×#137	—	251	3	no	—
126	C₂×AGL(1,ℤ/8)	—	251	3	no	—
127	C₂×#135	—	251	3	no	—
128	C₂²×SD₁₆	—	251	3	no	—
129	C₂×#131	D₂₅₁	1	1	yes	characteristic, complemented
130	2₋¹⁺⁴	—	502	4	no	—
131	2₋¹⁺⁴	—	251	8	no	—
132	H(5,𝔽₂)	—	502	4	no	—
133	H(5,𝔽₂)	—	251	8	no	—
134	C₂×#41	—	502	1	no	—
135	C₂×#41	—	502	3	no	—
136	C₂×#41	—	251	3	no	—
137	C₂×#41	—	251	3	no	—
138	C₂×#41	C₂×D₂₅₁	1	1	yes	characteristic
139	C₂²×Q₈	—	502	1	no	—
140	C₂²×Q₈	—	251	3	no	—
141	C₂²×D₄	—	502	3	no	—
142	C₂²×D₄	—	251	3	no	—
143	#137	—	251	24	no	—
144	AGL(1,ℤ/8)	—	251	24	no	—
145	#135	—	251	24	no	—
146	C₂×Q₁₆	—	251	3	no	—
147	C₂×SD₁₆	—	251	3	no	—
148	C₂×SD₁₆	—	251	6	no	—
149	C₂×SD₁₆	—	251	1	no	—
150	C₂×SD₁₆	—	251	24	no	—
151	C₂×D₈	—	251	3	no	—
152	#131	C₂×D₂₅₁	1	8	yes	complemented
153	C₂×M₁₆	C₂×D₂₅₁	1	3	yes	complemented
154	C₂²×C₈	C₂×D₂₅₁	1	3	yes	complemented
155	C₂⁴	—	502	3	no	—
156	#41	—	502	8	no	—
157	#41	—	502	24	no	—
158	#41	—	502	6	no	—
159	#41	—	502	6	no	—
160	#41	—	251	12	no	—
161	#41	—	251	12	no	—
162	#41	D₁₀₀₄	1	4	yes	—
163	#41	C₂²×D₂₅₁	1	4	yes	—
164	C₂×Q₈	—	502	8	no	—
165	C₂×Q₈	—	502	3	no	—
166	C₂×Q₈	—	502	6	no	—
167	C₂×Q₈	—	251	3	no	—
168	C₂×Q₈	—	251	12	no	—
169	C₂×Q₈	—	251	1	no	—
170	C₂×Q₈	D₁₀₀₄	1	1	yes	characteristic
171	C₂×D₄	—	502	3	no	—
172	C₂×D₄	—	502	3	no	—
173	C₂×D₄	—	502	24	no	—
174	C₂×D₄	—	502	3	no	—
175	C₂×D₄	—	502	6	no	—
176	C₂×D₄	—	251	1	no	—
177	C₂×D₄	—	251	3	no	—
178	C₂×D₄	—	251	12	no	—
179	C₂×D₄	D₁₀₀₄	1	3	yes	—
180	C₂²×C₄	—	502	3	no	—
181	C₂²×C₄	—	502	3	no	—
182	C₂²×C₄	—	502	3	no	—
183	C₂²×C₄	C₂²×D₂₅₁	1	3	yes	—
184	Q₁₆	—	251	12	no	—
185	SD₁₆	—	251	12	no	—
186	SD₁₆	—	251	24	no	—
187	SD₁₆	—	251	4	no	—
188	D₈	—	251	12	no	—
189	M₁₆	C₂²×D₂₅₁	1	12	yes	complemented
190	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
191	C₂×C₈	C₂²×D₂₅₁	1	3	yes	—
192	C₂×C₈	C₂²×D₂₅₁	1	12	yes	complemented
193	C₂³	—	2008	6	no	—
194	C₂³	—	502	12	no	—
195	C₂³	—	502	1	no	—
196	C₂³	—	502	3	no	—
197	C₂³	C₂×D₁₀₀₄	1	3	yes	—
198	Q₈	—	502	12	no	—
199	Q₈	—	502	3	no	—
200	Q₈	—	502	1	no	—
201	Q₈	—	251	6	no	—
202	Q₈	—	251	2	no	—
203	Q₈	C₂×D₁₀₀₄	1	4	yes	—
204	D₄	—	502	12	no	—
205	D₄	—	502	12	no	—
206	D₄	—	502	12	no	—
207	D₄	—	502	3	no	—
208	D₄	—	502	1	no	—
209	D₄	—	251	2	no	—
210	D₄	—	251	6	no	—
211	D₄	C₂×D₁₀₀₄	1	12	yes	—
212	C₂×C₄	—	502	12	no	—
213	C₂×C₄	—	502	1	no	—
214	C₂×C₄	—	502	3	no	—
215	C₂×C₄	—	502	12	no	—
216	C₂×C₄	—	502	12	no	—
217	C₂×C₄	C₂×D₁₀₀₄	1	6	yes	—
218	C₂×C₄	C₂×D₁₀₀₄	1	3	yes	—
219	C₂×C₄	C₂³×D₂₅₁	1	6	yes	—
220	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
221	C₈	C₂³×D₂₅₁	1	2	yes	—
222	C₈	C₂³×D₂₅₁	1	6	yes	—
223	C₂²	—	2	6	no	—
224	C₂²	—	2008	12	no	—
225	C₂²	—	1004	6	no	—
226	C₂²	—	1004	2	no	—
227	C₂²	—	502	2	no	—
228	C₂²	—	502	6	no	—
229	C₂²	C₂²×D₁₀₀₄	1	6	yes	—
230	C₂²	C₂²×D₁₀₀₄	1	1	yes	characteristic
231	C₄	—	502	2	no	—
232	C₄	—	502	6	no	—
233	C₄	C₂²×D₁₀₀₄	1	6	yes	—
234	C₄	C₂²×D₁₀₀₄	1	1	yes	characteristic
235	C₄	C₂⁴×D₂₅₁	1	1	yes	characteristic
236	C₂	—	2	6	no	—
237	C₂	—	1004	6	no	—
238	C₂	—	1004	2	no	—
239	C₂	#30632499	1	2	yes	direct product factor
240	C₂	C₂³×D₁₀₀₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₀₀₄
commutator, cosocle	C₄×C₂₅₁	C₂⁵
Frattini	C₄	C₂⁴×D₂₅₁
Fitting	C₂×#131×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
hypercenter	C₂×#131	D₂₅₁
hypocenter	C₂₅₁	C₂×#577
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. C₂×#22548762	C₂	semidirect
2. C₂×#22548761	C₂	semidirect
3. #30632499	C₂	direct (N × H)
4. C₂×#22548748	C₂	semidirect
5. C₂×#22548743	C₂	semidirect
6. C₂×#22548737	C₂	semidirect
7. C₂²×#16860799	C₂	semidirect
8. C₂×#131×C₂₅₁	C₂	semidirect
10. #22548762	C₂²	semidirect
12. #22548761	C₂²	semidirect
15. C₂×#16860810	C₂²	semidirect
17. C₂²×Dic₅₀₂	C₂²	semidirect
21. #22548748	C₂²	semidirect
22. #22548743	C₂²	semidirect
23. #22548737	C₂²	semidirect
24. C₂×Dic₁₀₀₄	C₂²	semidirect
25. C₂×#16860799	C₂²	semidirect
26. C₂×#16860799	C₂²	semidirect
27. C₂×#16860799	C₂²	semidirect
28. C₂×#16860799	C₂²	semidirect
29. C₂×D₂₀₀₈	C₂²	semidirect
31. #131×C₂₅₁	C₂²	semidirect
32. C₂×M₁₆×C₂₅₁	C₂²	semidirect
33. C₂²×C₈×C₂₅₁	C₂²	semidirect
39. #16860810	C₂³	semidirect
45. C₂×Dic₅₀₂	C₂³	semidirect
58. Dic₁₀₀₄	C₂³	semidirect
59. #16860799	C₂³	semidirect
60. #16860799	C₂³	semidirect
61. #16860799	C₂³	semidirect
62. D₂₀₀₈	C₂³	semidirect
68. M₁₆×C₂₅₁	C₂³	semidirect
71. C₂×C₈×C₂₅₁	C₂³	semidirect
116. C₂×C₂₅₁	#577	semidirect
120. C₂₅₁	C₂×#577	semidirect
129. C₂×#131	D₂₅₁	semidirect
152. #131	C₂×D₂₅₁	semidirect
153. C₂×M₁₆	C₂×D₂₅₁	semidirect
154. C₂²×C₈	C₂×D₂₅₁	semidirect
189. M₁₆	C₂²×D₂₅₁	semidirect
192. C₂×C₈	C₂²×D₂₅₁	semidirect
239. C₂	#30632499	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×#22548762	2
C₂×#22548761	2
#30632499	2
C₂×#22548748	2
C₂×#22548743	2
C₂×#22548737	2
C₂²×#16860799	2
C₂×#131×C₂₅₁	2
C₂×#577	251

Subgroups — C2×#30632499 #44565126

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#30632499 #44565126