Subgroups — C₂×#30632406 #44563910

Order 32128.

1112 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632406	C₂	1	16	yes	maximal, direct product factor
2	C₂×#22548713	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#22548708	C₂	1	2	yes	maximal, complemented
4	C₂×#22548708	C₂	1	2	yes	maximal, complemented
5	C₂×#22548702	C₂	1	1	yes	maximal, characteristic
6	C₂×#22548692	C₂	1	2	yes	maximal, complemented
7	C₂×Q₈×Dic₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×Dic₅₀₂	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×Dic₅₀₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×D₄×Dic₂₅₁	C₂	1	1	yes	maximal, characteristic
11	C₂×#22548671	C₂	1	2	yes	maximal, complemented
12	C₂×#124×C₂₅₁	C₂	1	1	yes	maximal, characteristic
13	C₂²×Dic₅₀₂	C₂²	1	1	yes	characteristic
14	C₂²×Dic₅₀₂	C₂²	1	1	yes	characteristic
15	C₂³×Dic₂₅₁	C₂²	1	2	yes	—
16	#22548713	C₂²	1	8	yes	complemented
17	#22548708	C₂²	1	16	yes	complemented
18	#22548708	C₂²	1	16	yes	complemented
19	#22548702	C₂²	1	8	yes	—
20	#22548692	C₂²	1	16	yes	complemented
21	Q₈×Dic₂₅₁	C₂²	1	8	yes	complemented
22	C₄×Dic₅₀₂	C₂²	1	8	yes	complemented
23	C₄×Dic₅₀₂	C₂²	1	8	yes	complemented
24	D₄×Dic₂₅₁	C₂²	1	8	yes	—
25	#22548671	C₂²	1	16	yes	complemented
26	C₂×C₄⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊Dic₂₅₁	C₂²	1	2	yes	—
28	C₂×C₄⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
29	C₂×Dic₂₅₁⋊C₄	C₂²	1	2	yes	—
30	C₂×Dic₂₅₁⋊C₄	C₂²	1	1	yes	characteristic
31	C₂×Dic₂₅₁⋊C₄	C₂²	1	2	yes	—
32	C₂×Dic₂₅₁⋊C₄	C₂²	1	2	yes	—
33	C₂×Dic₂₅₁⋊C₄	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊Dic₂₅₁	C₂²	1	2	yes	—
35	C₂×C₂²⋊Dic₂₅₁	C₂²	1	2	yes	—
36	C₂×C₂²⋊Dic₂₅₁	C₂²	1	2	yes	—
37	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
38	C₂×C₄×Dic₂₅₁	C₂²	1	2	yes	—
39	C₂×C₄×Dic₂₅₁	C₂²	1	2	yes	—
40	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
42	C₂²×D₄×C₂₅₁	C₂²	1	1	yes	characteristic
43	#124×C₂₅₁	C₂²	1	8	yes	—
44	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	2	yes	—
45	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	2	yes	—
46	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
47	C₂×Dic₅₀₂	—	2	4	no	—
48	C₂×Dic₅₀₂	—	2	4	no	—
49	C₂×Dic₅₀₂	C₂³	1	4	yes	—
50	C₂×Dic₅₀₂	C₂³	1	4	yes	—
51	C₂²×Dic₂₅₁	—	2	8	no	—
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	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
57	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
58	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
59	C₄⋊Dic₂₅₁	C₂³	1	4	yes	—
60	C₄⋊Dic₂₅₁	C₂³	1	4	yes	—
61	C₄⋊Dic₂₅₁	C₂³	1	8	yes	—
62	Dic₂₅₁⋊C₄	C₂³	1	4	yes	—
63	Dic₂₅₁⋊C₄	C₂³	1	8	yes	—
64	Dic₂₅₁⋊C₄	C₂³	1	8	yes	—
65	Dic₂₅₁⋊C₄	C₂³	1	4	yes	—
66	Dic₂₅₁⋊C₄	C₂³	1	8	yes	—
67	C₂²⋊Dic₂₅₁	C₂³	1	8	yes	—
68	C₂²⋊Dic₂₅₁	C₂³	1	8	yes	—
69	C₂²⋊Dic₂₅₁	C₂³	1	8	yes	—
70	C₄×Dic₂₅₁	C₂³	1	8	yes	—
71	C₄×Dic₂₅₁	C₂³	1	8	yes	—
72	C₄×Dic₂₅₁	C₂³	1	4	yes	—
73	C₄×Dic₂₅₁	C₂³	1	4	yes	—
74	C₂⁴×C₂₅₁	C₂³	1	2	yes	—
75	C₂×Q₈×C₂₅₁	—	2	4	no	—
76	C₂×Q₈×C₂₅₁	C₂³	1	4	yes	—
77	C₂×D₄×C₂₅₁	—	2	4	no	—
78	C₂×D₄×C₂₅₁	C₂³	1	4	yes	—
79	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
80	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
81	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
82	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
83	C₂²⋊C₄×C₂₅₁	C₂³	1	8	yes	—
84	C₂²⋊C₄×C₂₅₁	C₂³	1	8	yes	—
85	C₄²×C₂₅₁	C₂³	1	4	yes	—
86	Dic₅₀₂	—	2	8	no	—
87	Dic₅₀₂	—	2	8	no	—
88	C₂×Dic₂₅₁	—	2	4	no	—
89	C₂×Dic₂₅₁	—	2	2	no	—
90	C₂×Dic₂₅₁	—	2	4	no	—
91	C₂×Dic₂₅₁	—	2	2	no	—
92	C₂×Dic₂₅₁	—	2	2	no	—
93	C₂×Dic₂₅₁	—	2	16	no	—
94	C₂×Dic₂₅₁	#41	1	4	yes	—
95	C₂×Dic₂₅₁	C₂⁴	1	2	yes	—
96	C₂×Dic₂₅₁	C₂⁴	1	4	yes	—
97	C₂×Dic₂₅₁	C₂⁴	1	2	yes	—
98	C₂×Dic₂₅₁	C₂⁴	1	4	yes	—
99	C₂×Dic₂₅₁	C₂⁴	1	2	yes	—
100	C₂×Dic₂₅₁	C₂⁴	1	2	yes	—
101	C₂³×C₂₅₁	—	2	4	no	—
102	C₂³×C₂₅₁	—	2	4	no	—
103	C₂³×C₂₅₁	—	2	4	no	—
104	C₂³×C₂₅₁	C₂⁴	1	4	yes	—
105	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
106	Q₈×C₂₅₁	—	2	8	no	—
107	D₄×C₂₅₁	—	2	8	no	—
108	C₂×C₄×C₂₅₁	—	2	4	no	—
109	C₂×C₄×C₂₅₁	—	2	2	no	—
110	C₂×C₄×C₂₅₁	—	2	2	no	—
111	C₂×C₄×C₂₅₁	#41	1	4	yes	—
112	C₂×C₄×C₂₅₁	C₂⁴	1	2	yes	—
113	C₂×C₄×C₂₅₁	C₂⁴	1	2	yes	—
114	C₂×C₄×C₂₅₁	C₂⁴	1	4	yes	—
115	C₂×C₄×C₂₅₁	C₂⁴	1	2	yes	—
116	C₂²×C₂₅₁	—	2	4	no	—
117	C₂²×C₂₅₁	—	2	4	no	—
118	C₂²×C₂₅₁	—	2	4	no	—
119	C₂²×C₂₅₁	—	4	8	no	—
120	C₂²×C₂₅₁	C₂×#41	1	2	yes	—
121	C₂²×C₂₅₁	C₂×#41	1	2	yes	—
122	C₂²×C₂₅₁	2₋¹⁺⁴	1	2	yes	—
123	C₂²×C₂₅₁	C₂⁵	1	1	yes	characteristic
124	C₄×C₂₅₁	—	2	4	no	—
125	C₄×C₂₅₁	—	2	2	no	—
126	C₄×C₂₅₁	—	2	2	no	—
127	C₄×C₂₅₁	C₂×#41	1	4	yes	—
128	Dic₂₅₁	—	2	4	no	—
129	Dic₂₅₁	—	2	2	no	—
130	Dic₂₅₁	—	2	4	no	—
131	Dic₂₅₁	—	2	2	no	—
132	Dic₂₅₁	—	2	2	no	—
133	Dic₂₅₁	C₂×#41	1	4	yes	—
134	C₂×C₂₅₁	—	4	4	no	—
135	C₂×C₂₅₁	#556	1	4	yes	complemented
136	C₂×C₂₅₁	C₂²×#41	1	1	yes	characteristic
137	C₂×C₂₅₁	C₂²×#41	1	1	yes	characteristic
138	C₂×C₂₅₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
139	C₂₅₁	C₂×#556	1	1	yes	characteristic, complemented, Sylow
140	C₂×#556	—	251	1	no	maximal, Sylow
141	#556	—	251	16	no	—
142	C₂×C₄⋊Q₈	—	251	1	no	—
143	C₂×#126	—	251	2	no	—
144	C₂×#126	—	251	2	no	—
145	C₂×#124	—	251	1	no	—
146	C₂×#124	D₂₅₁	1	1	yes	characteristic
147	C₂×C₂²⋊Q₈	—	251	2	no	—
148	C₂×C₄×Q₈	—	251	1	no	—
149	C₂×C₄×Q₈	—	251	1	no	—
150	C₂×C₄×Q₈	—	251	1	no	—
151	C₂×C₄×D₄	—	251	1	no	—
152	C₂×#117	—	251	2	no	—
153	C₂²×Q₈	—	251	1	no	—
154	C₂²×Q₈	—	251	1	no	—
155	C₂²×Q₈	C₂×D₂₅₁	1	1	yes	characteristic
156	C₂²×D₄	C₂×D₂₅₁	1	1	yes	characteristic
157	C₂³×C₄	—	251	2	no	—
158	C₄⋊Q₈	—	251	8	no	—
159	#126	—	251	16	no	—
160	#126	—	251	16	no	—
161	#124	—	251	8	no	—
162	#124	C₂×D₂₅₁	1	8	yes	—
163	C₂²⋊Q₈	—	251	16	no	—
164	C₄×Q₈	—	251	8	no	—
165	C₄×Q₈	—	251	8	no	—
166	C₄×Q₈	—	251	8	no	—
167	C₄×D₄	—	251	8	no	—
168	#117	—	251	16	no	—
169	C₂×C₄⋊C₄	—	251	1	no	—
170	C₂×C₄⋊C₄	—	251	2	no	—
171	C₂×C₄⋊C₄	—	251	1	no	—
172	C₂×C₄⋊C₄	—	251	2	no	—
173	C₂×C₄⋊C₄	—	251	2	no	—
174	C₂×C₄⋊C₄	—	251	1	no	—
175	C₂×C₄⋊C₄	—	251	1	no	—
176	C₂×C₄⋊C₄	—	251	2	no	—
177	C₂×C₂²⋊C₄	—	251	2	no	—
178	C₂×C₂²⋊C₄	—	251	2	no	—
179	C₂×C₂²⋊C₄	—	251	2	no	—
180	C₂×C₂²⋊C₄	C₂×D₂₅₁	1	2	yes	—
181	C₂×C₂²⋊C₄	C₂×D₂₅₁	1	2	yes	—
182	C₂×C₄²	—	251	2	no	—
183	C₂×C₄²	—	251	2	no	—
184	C₂×C₄²	—	251	1	no	—
185	C₂×C₄²	—	251	1	no	—
186	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
187	C₂⁴	C₂²×D₂₅₁	1	2	yes	—
188	C₂×Q₈	—	2	4	no	—
189	C₂×Q₈	—	502	4	no	—
190	C₂×Q₈	—	502	4	no	—
191	C₂×Q₈	—	251	4	no	—
192	C₂×Q₈	—	251	4	no	—
193	C₂×Q₈	C₂²×D₂₅₁	1	4	yes	—
194	C₂×D₄	—	2	4	no	—
195	C₂×D₄	C₂²×D₂₅₁	1	4	yes	—
196	C₂²×C₄	—	502	8	no	—
197	C₂²×C₄	—	251	8	no	—
198	C₂²×C₄	—	251	1	no	—
199	C₂²×C₄	—	251	2	no	—
200	C₂²×C₄	—	251	1	no	—
201	C₂²×C₄	—	251	1	no	—
202	C₂²×C₄	—	251	2	no	—
203	C₂²×C₄	—	251	1	no	—
204	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
205	C₂²×C₄	C₂²×D₂₅₁	1	2	yes	—
206	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
207	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
208	C₄⋊C₄	—	251	4	no	—
209	C₄⋊C₄	—	251	8	no	—
210	C₄⋊C₄	—	251	8	no	—
211	C₄⋊C₄	—	251	4	no	—
212	C₄⋊C₄	—	251	8	no	—
213	C₄⋊C₄	—	251	4	no	—
214	C₄⋊C₄	—	251	8	no	—
215	C₄⋊C₄	—	251	4	no	—
216	C₂²⋊C₄	—	251	8	no	—
217	C₂²⋊C₄	—	251	8	no	—
218	C₂²⋊C₄	—	251	8	no	—
219	C₂²⋊C₄	C₂²×D₂₅₁	1	8	yes	—
220	C₂²⋊C₄	C₂²×D₂₅₁	1	8	yes	—
221	C₄²	—	251	8	no	—
222	C₄²	—	251	8	no	—
223	C₄²	—	251	4	no	—
224	C₄²	—	251	4	no	—
225	C₄²	C₂²×D₂₅₁	1	4	yes	—
226	C₂³	—	2	4	no	—
227	C₂³	—	2	4	no	—
228	C₂³	—	2	4	no	—
229	C₂³	C₂³×D₂₅₁	1	4	yes	—
230	C₂³	C₂³×D₂₅₁	1	1	yes	characteristic
231	Q₈	—	2	8	no	—
232	Q₈	—	502	8	no	—
233	Q₈	—	502	8	no	—
234	D₄	—	2	8	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	502	4	no	—
239	C₂×C₄	—	502	2	no	—
240	C₂×C₄	—	502	4	no	—
241	C₂×C₄	—	502	2	no	—
242	C₂×C₄	—	502	2	no	—
243	C₂×C₄	—	502	16	no	—
244	C₂×C₄	—	251	2	no	—
245	C₂×C₄	—	251	4	no	—
246	C₂×C₄	—	251	2	no	—
247	C₂×C₄	—	251	2	no	—
248	C₂×C₄	—	251	4	no	—
249	C₂×C₄	—	251	4	no	—
250	C₂×C₄	—	251	2	no	—
251	C₂×C₄	#16860809	1	4	yes	—
252	C₂×C₄	C₂³×D₂₅₁	1	2	yes	—
253	C₂×C₄	C₂³×D₂₅₁	1	4	yes	—
254	C₂×C₄	C₂³×D₂₅₁	1	2	yes	—
255	C₂×C₄	C₂³×D₂₅₁	1	2	yes	—
256	C₂²	—	2	4	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	2	4	no	—
259	C₂²	—	4	8	no	—
260	C₂²	#41×D₂₅₁	1	2	yes	—
261	C₂²	C₂×#16860809	1	2	yes	—
262	C₂²	#22548762	1	2	yes	—
263	C₂²	C₂⁴×D₂₅₁	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	4	no	—
266	C₄	—	2	2	no	—
267	C₄	—	502	4	no	—
268	C₄	—	502	2	no	—
269	C₄	—	502	4	no	—
270	C₄	—	502	2	no	—
271	C₄	—	502	2	no	—
272	C₄	—	251	4	no	—
273	C₄	C₂×#16860809	1	4	yes	—
274	C₂	—	4	4	no	—
275	C₂	#30632406	1	4	yes	direct product factor
276	C₂	C₂×#41×D₂₅₁	1	1	yes	characteristic
277	C₂	C₂²×#16860809	1	1	yes	characteristic
278	C₂	C₂×#22548762	1	1	yes	characteristic

	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×D₄×C₂₅₁	C₂²
hypercenter	C₂×#124	D₂₅₁
hypocenter	C₂₅₁	C₂×#556
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
1. #30632406	C₂	direct (N × H)
2. C₂×#22548713	C₂	semidirect
3. C₂×#22548708	C₂	semidirect
4. C₂×#22548708	C₂	semidirect
6. C₂×#22548692	C₂	semidirect
7. C₂×Q₈×Dic₂₅₁	C₂	semidirect
8. C₂×C₄×Dic₅₀₂	C₂	semidirect
9. C₂×C₄×Dic₅₀₂	C₂	semidirect
11. C₂×#22548671	C₂	semidirect
16. #22548713	C₂²	semidirect
17. #22548708	C₂²	semidirect
18. #22548708	C₂²	semidirect
20. #22548692	C₂²	semidirect
21. Q₈×Dic₂₅₁	C₂²	semidirect
22. C₄×Dic₅₀₂	C₂²	semidirect
23. C₄×Dic₅₀₂	C₂²	semidirect
25. #22548671	C₂²	semidirect
135. C₂×C₂₅₁	#556	semidirect
139. C₂₅₁	C₂×#556	semidirect
275. C₂	#30632406	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#30632406	2
C₂×#22548713	2
C₂×#22548708	2
C₂×#22548702	2
C₂×#22548692	2
C₂×Q₈×Dic₂₅₁	2
C₂×C₄×Dic₅₀₂	2
C₂×D₄×Dic₂₅₁	2
C₂×#22548671	2
C₂×#124×C₂₅₁	2
C₂×#556	251

Subgroups — C2×#30632406 #44563910

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#30632406 #44563910