Subgroups — C₂×#30632257 #44563678

Order 32128.

1512 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#16860809	C₂	1	1	yes	maximal, characteristic
2	C₂²×Q₈×D₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#30632257	C₂	1	16	yes	maximal, direct product factor
4	C₂×#22548715	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#22548713	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#22548702	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#22548697	C₂	1	4	yes	maximal, complemented
8	C₂×#22548692	C₂	1	4	yes	maximal, complemented
9	C₂×#22548672	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#124×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#16860809	—	2	8	no	—
12	C₂×#16860809	C₂²	1	8	yes	—
13	C₂²×Dic₅₀₂	C₂²	1	1	yes	characteristic
14	C₂²×Dic₅₀₂	C₂²	1	1	yes	characteristic
15	C₂²×Dic₅₀₂	C₂²	1	2	yes	—
16	C₂×Q₈×D₂₅₁	—	2	8	no	—
17	C₂×Q₈×D₂₅₁	C₂²	1	8	yes	complemented
18	C₂²×C₂²⋊D₂₅₁	C₂²	1	2	yes	—
19	C₂³×Dic₂₅₁	C₂²	1	2	yes	—
20	C₂²×C₄×D₂₅₁	C₂²	1	2	yes	—
21	C₂²×C₄×D₂₅₁	C₂²	1	1	yes	characteristic
22	#22548715	C₂²	1	8	yes	complemented
23	#22548713	C₂²	1	8	yes	complemented
24	#22548702	C₂²	1	8	yes	complemented
25	#22548697	C₂²	1	32	yes	complemented
26	#22548692	C₂²	1	32	yes	complemented
27	#22548672	C₂²	1	8	yes	complemented
28	C₂×C₄⋊Dic₂₅₁	C₂²	1	4	yes	—
29	C₂×Dic₂₅₁⋊C₄	C₂²	1	4	yes	—
30	C₂×Dic₂₅₁⋊C₄	C₂²	1	2	yes	—
31	C₂×C₂²⋊Dic₂₅₁	C₂²	1	4	yes	—
32	C₂×(C₂×D₂₅₁)⋊C₄	C₂²	1	2	yes	—
33	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
34	C₂²×Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
35	C₂²×D₄×C₂₅₁	C₂²	1	1	yes	characteristic
36	#124×C₂₅₁	C₂²	1	8	yes	complemented
37	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	4	yes	—
38	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
39	C₂³×D₂₅₁	C₂³	1	1	yes	characteristic
40	#16860809	—	2	32	no	—
41	C₂×Dic₅₀₂	—	2	8	no	—
42	C₂×Dic₅₀₂	—	2	4	no	—
43	C₂×Dic₅₀₂	—	2	4	no	—
44	C₂×Dic₅₀₂	C₂³	1	8	yes	—
45	C₂×Dic₅₀₂	C₂³	1	4	yes	—
46	C₂×Dic₅₀₂	C₂³	1	4	yes	—
47	Q₈×D₂₅₁	—	2	32	no	—
48	C₂×C₂²⋊D₂₅₁	—	2	8	no	—
49	C₂×C₂²⋊D₂₅₁	C₂³	1	8	yes	—
50	C₂²×Dic₂₅₁	—	2	8	no	—
51	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
52	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
53	C₂²×Dic₂₅₁	C₂³	1	8	yes	—
54	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
55	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
56	C₂×C₄×D₂₅₁	—	2	8	no	—
57	C₂×C₄×D₂₅₁	D₄	1	8	yes	—
58	C₂×C₄×D₂₅₁	C₂³	1	8	yes	—
59	C₂×C₄×D₂₅₁	C₂³	1	4	yes	—
60	C₄⋊Dic₂₅₁	C₂³	1	16	yes	—
61	Dic₂₅₁⋊C₄	C₂³	1	8	yes	—
62	Dic₂₅₁⋊C₄	C₂³	1	16	yes	—
63	C₂²⋊Dic₂₅₁	C₂³	1	16	yes	—
64	(C₂×D₂₅₁)⋊C₄	C₂³	1	8	yes	—
65	C₄×Dic₂₅₁	C₂³	1	4	yes	—
66	C₂⁴×C₂₅₁	C₂³	1	2	yes	—
67	C₂×Q₈×C₂₅₁	—	2	4	no	—
68	C₂×Q₈×C₂₅₁	C₂³	1	4	yes	—
69	C₂×D₄×C₂₅₁	—	2	4	no	—
70	C₂×D₄×C₂₅₁	C₂³	1	4	yes	—
71	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
72	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
73	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
74	C₂²⋊C₄×C₂₅₁	C₂³	1	16	yes	—
75	C₄²×C₂₅₁	C₂³	1	4	yes	—
76	C₂²×D₂₅₁	—	2	4	no	—
77	C₂²×D₂₅₁	C₂×D₄	1	4	yes	—
78	C₂²×D₂₅₁	C₂⁴	1	2	yes	—
79	Dic₅₀₂	—	2	8	no	—
80	Dic₅₀₂	—	2	8	no	—
81	Dic₅₀₂	—	2	16	no	—
82	C₂²⋊D₂₅₁	—	2	16	no	—
83	C₂×Dic₂₅₁	—	2	4	no	—
84	C₂×Dic₂₅₁	—	2	4	no	—
85	C₂×Dic₂₅₁	—	2	4	no	—
86	C₂×Dic₂₅₁	—	2	16	no	—
87	C₂×Dic₂₅₁	C₂×D₄	1	4	yes	—
88	C₂×Dic₂₅₁	C₂⁴	1	4	yes	—
89	C₂×Dic₂₅₁	C₂⁴	1	4	yes	—
90	C₂×Dic₂₅₁	C₂⁴	1	4	yes	—
91	C₂×Dic₂₅₁	C₂⁴	1	2	yes	—
92	C₄×D₂₅₁	—	2	16	no	—
93	C₄×D₂₅₁	C₂×D₄	1	16	yes	—
94	C₂³×C₂₅₁	—	2	4	no	—
95	C₂³×C₂₅₁	—	2	8	no	—
96	C₂³×C₂₅₁	C₂⁴	1	4	yes	—
97	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
98	Q₈×C₂₅₁	—	2	8	no	—
99	D₄×C₂₅₁	—	2	8	no	—
100	C₂×C₄×C₂₅₁	—	2	4	no	—
101	C₂×C₄×C₂₅₁	—	2	4	no	—
102	C₂×C₄×C₂₅₁	C₂×D₄	1	4	yes	—
103	C₂×C₄×C₂₅₁	C₂⁴	1	4	yes	—
104	C₂×C₄×C₂₅₁	C₂⁴	1	4	yes	—
105	C₂×C₄×C₂₅₁	C₂⁴	1	2	yes	—
106	C₂²×C₂₅₁	—	2	4	no	—
107	C₂²×C₂₅₁	—	2	8	no	—
108	C₂²×C₂₅₁	—	4	8	no	—
109	C₂²×C₂₅₁	C₂²×D₄	1	2	yes	—
110	C₂²×C₂₅₁	2₋¹⁺⁴	1	4	yes	—
111	C₂²×C₂₅₁	C₂⁵	1	1	yes	characteristic
112	C₂×D₂₅₁	—	2	8	no	—
113	C₂×D₂₅₁	—	2	4	no	—
114	C₂×D₂₅₁	C₂²×D₄	1	4	yes	—
115	C₄×C₂₅₁	—	2	4	no	—
116	C₄×C₂₅₁	—	2	4	no	—
117	C₄×C₂₅₁	C₂²×D₄	1	4	yes	—
118	Dic₂₅₁	—	2	4	no	—
119	Dic₂₅₁	—	2	4	no	—
120	Dic₂₅₁	—	2	4	no	—
121	Dic₂₅₁	C₂²×D₄	1	4	yes	—
122	C₂×C₂₅₁	—	4	4	no	—
123	C₂×C₂₅₁	#536	1	4	yes	complemented
124	C₂×C₂₅₁	C₂³×D₄	1	1	yes	characteristic
125	C₂×C₂₅₁	C₂×2₋¹⁺⁴	1	2	yes	—
126	D₂₅₁	—	2	4	no	—
127	C₂₅₁	C₂×#536	1	1	yes	characteristic, complemented, Sylow
128	C₂×#536	—	251	1	no	maximal, Sylow
129	C₂²×#41	—	251	1	no	—
130	C₂³×Q₈	—	251	1	no	—
131	#536	—	251	16	no	—
132	C₂×C₄⋊Q₈	—	251	1	no	—
133	C₂×C₄⋊Q₈	—	251	1	no	—
134	C₂×#124	—	251	1	no	—
135	C₂×#124	D₂₅₁	1	1	yes	characteristic, complemented
136	C₂×#123	—	251	4	no	—
137	C₂×C₂²⋊Q₈	—	251	4	no	—
138	C₂×#117	—	251	1	no	—
139	C₂×#41	—	502	8	no	—
140	C₂×#41	—	251	8	no	—
141	C₂²×Q₈	—	502	8	no	—
142	C₂²×Q₈	—	251	8	no	—
143	C₂²×Q₈	—	251	2	no	—
144	C₂²×Q₈	—	251	1	no	—
145	C₂²×Q₈	—	251	1	no	—
146	C₂²×Q₈	C₂×D₂₅₁	1	1	yes	characteristic
147	C₂²×D₄	—	251	2	no	—
148	C₂²×D₄	C₂×D₂₅₁	1	1	yes	characteristic
149	C₂³×C₄	—	251	2	no	—
150	C₂³×C₄	—	251	2	no	—
151	C₂³×C₄	—	251	1	no	—
152	C₄⋊Q₈	—	251	8	no	—
153	C₄⋊Q₈	—	251	8	no	—
154	#124	—	251	8	no	—
155	#124	C₂×D₂₅₁	1	8	yes	complemented
156	#123	—	251	32	no	—
157	C₂²⋊Q₈	—	251	32	no	—
158	#117	—	251	8	no	—
159	C₂×C₄⋊C₄	—	251	2	no	—
160	C₂×C₄⋊C₄	—	251	4	no	—
161	C₂×C₄⋊C₄	—	251	4	no	—
162	C₂×C₂²⋊C₄	—	251	4	no	—
163	C₂×C₂²⋊C₄	—	251	2	no	—
164	C₂×C₂²⋊C₄	C₂×D₂₅₁	1	4	yes	—
165	C₂×C₄²	—	251	1	no	—
166	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
167	C₂⁴	—	251	1	no	—
168	C₂⁴	C₂²×D₂₅₁	1	2	yes	—
169	#41	—	502	32	no	—
170	C₂×Q₈	—	2	4	no	—
171	C₂×Q₈	—	502	8	no	—
172	C₂×Q₈	—	502	32	no	—
173	C₂×Q₈	—	502	4	no	—
174	C₂×Q₈	—	502	4	no	—
175	C₂×Q₈	—	251	8	no	—
176	C₂×Q₈	—	251	4	no	—
177	C₂×Q₈	—	251	4	no	—
178	C₂×Q₈	C₂²×D₂₅₁	1	4	yes	—
179	C₂×D₄	—	2	4	no	—
180	C₂×D₄	—	502	8	no	—
181	C₂×D₄	—	251	8	no	—
182	C₂×D₄	C₂²×D₂₅₁	1	4	yes	—
183	C₂²×C₄	—	502	8	no	—
184	C₂²×C₄	—	502	8	no	—
185	C₂²×C₄	—	251	8	no	—
186	C₂²×C₄	—	251	2	no	—
187	C₂²×C₄	—	251	2	no	—
188	C₂²×C₄	—	251	2	no	—
189	C₂²×C₄	—	251	8	no	—
190	C₂²×C₄	—	251	1	no	—
191	C₂²×C₄	—	251	8	no	—
192	C₂²×C₄	—	251	4	no	—
193	C₂²×C₄	C₂²×D₂₅₁	1	2	yes	—
194	C₂²×C₄	C₂²×D₂₅₁	1	2	yes	—
195	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
196	C₄⋊C₄	—	251	8	no	—
197	C₄⋊C₄	—	251	16	no	—
198	C₄⋊C₄	—	251	16	no	—
199	C₂²⋊C₄	—	251	16	no	—
200	C₂²⋊C₄	—	251	8	no	—
201	C₂²⋊C₄	C₂²×D₂₅₁	1	16	yes	—
202	C₄²	—	251	4	no	—
203	C₄²	C₂²×D₂₅₁	1	4	yes	—
204	C₂³	—	2	4	no	—
205	C₂³	—	2	8	no	—
206	C₂³	—	502	4	no	—
207	C₂³	—	251	4	no	—
208	C₂³	—	251	2	no	—
209	C₂³	C₂³×D₂₅₁	1	4	yes	—
210	C₂³	C₂³×D₂₅₁	1	1	yes	characteristic
211	Q₈	—	2	8	no	—
212	Q₈	—	502	16	no	—
213	Q₈	—	502	8	no	—
214	Q₈	—	502	8	no	—
215	D₄	—	2	8	no	—
216	D₄	—	502	16	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	502	16	no	—
220	C₂×C₄	—	502	4	no	—
221	C₂×C₄	—	502	4	no	—
222	C₂×C₄	—	502	4	no	—
223	C₂×C₄	—	502	16	no	—
224	C₂×C₄	—	251	4	no	—
225	C₂×C₄	—	251	4	no	—
226	C₂×C₄	—	251	4	no	—
227	C₂×C₄	—	251	4	no	—
228	C₂×C₄	—	251	2	no	—
229	C₂×C₄	—	251	16	no	—
230	C₂×C₄	D₄×D₂₅₁	1	4	yes	—
231	C₂×C₄	C₂³×D₂₅₁	1	4	yes	—
232	C₂×C₄	C₂³×D₂₅₁	1	4	yes	—
233	C₂×C₄	C₂³×D₂₅₁	1	2	yes	—
234	C₂²	—	2	4	no	—
235	C₂²	—	2	8	no	—
236	C₂²	—	4	8	no	—
237	C₂²	—	502	8	no	—
238	C₂²	—	502	4	no	—
239	C₂²	—	251	4	no	—
240	C₂²	C₂×D₄×D₂₅₁	1	2	yes	—
241	C₂²	#22548762	1	4	yes	—
242	C₂²	C₂⁴×D₂₅₁	1	1	yes	characteristic
243	C₄	—	2	4	no	—
244	C₄	—	2	4	no	—
245	C₄	—	502	4	no	—
246	C₄	—	502	4	no	—
247	C₄	—	502	4	no	—
248	C₄	—	251	4	no	—
249	C₄	C₂×D₄×D₂₅₁	1	4	yes	—
250	C₂	—	4	4	no	—
251	C₂	—	502	4	no	—
252	C₂	#30632257	1	4	yes	direct product factor
253	C₂	C₂²×D₄×D₂₅₁	1	1	yes	characteristic
254	C₂	C₂×#22548762	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₂₅₁
commutator, cosocle	C₂²×C₂₅₁	C₂⁵
Frattini	C₂²	C₂⁴×D₂₅₁
Fitting	C₂×#124×C₂₅₁	C₂
socle	C₂³×C₂₅₁	C₂⁴
prime radical	C₂²×#16860809	C₂
hypercenter	C₂×#124	D₂₅₁
hypocenter	C₂₅₁	C₂×#536
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ C₂×#124

Complements

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

N	H ≅ G/N	type
2. C₂²×Q₈×D₂₅₁	C₂	semidirect
3. #30632257	C₂	direct (N × H)
4. C₂×#22548715	C₂	semidirect
5. C₂×#22548713	C₂	semidirect
6. C₂×#22548702	C₂	semidirect
7. C₂×#22548697	C₂	semidirect
8. C₂×#22548692	C₂	semidirect
9. C₂×#22548672	C₂	semidirect
10. C₂×#124×C₂₅₁	C₂	semidirect
17. C₂×Q₈×D₂₅₁	C₂²	semidirect
22. #22548715	C₂²	semidirect
23. #22548713	C₂²	semidirect
24. #22548702	C₂²	semidirect
25. #22548697	C₂²	semidirect
26. #22548692	C₂²	semidirect
27. #22548672	C₂²	semidirect
36. #124×C₂₅₁	C₂²	semidirect
123. C₂×C₂₅₁	#536	semidirect
127. C₂₅₁	C₂×#536	semidirect
135. C₂×#124	D₂₅₁	semidirect
155. #124	C₂×D₂₅₁	semidirect
252. C₂	#30632257	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×#16860809	2
C₂²×Q₈×D₂₅₁	2
#30632257	2
C₂×#22548715	2
C₂×#22548713	2
C₂×#22548702	2
C₂×#22548697	2
C₂×#22548692	2
C₂×#22548672	2
C₂×#124×C₂₅₁	2
C₂×#536	251

Subgroups — C2×#30632257 #44563678

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#30632257 #44563678