Subgroups — C₂×#30632426 #44564481

Order 32128.

1344 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632426	C₂	1	16	yes	maximal, direct product factor
2	C₂×#22548711	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#22548708	C₂	1	2	yes	maximal, complemented
4	C₂×#22548697	C₂	1	2	yes	maximal, complemented
5	C₂×#22548696	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#22548688	C₂	1	2	yes	maximal, complemented
7	C₂×#22548687	C₂	1	2	yes	maximal, complemented
8	C₂×#22548686	C₂	1	2	yes	maximal, complemented
9	C₂×#22548684	C₂	1	2	yes	maximal, complemented
10	C₂×#120×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×D₁₀₀₄	C₂²	1	1	yes	characteristic, complemented
12	C₂²×C₂²⋊D₂₅₁	C₂²	1	2	yes	—
13	C₂²×C₂²⋊D₂₅₁	C₂²	1	2	yes	—
14	C₂²×C₂²⋊D₂₅₁	C₂²	1	2	yes	—
15	C₂²×C₂²⋊D₂₅₁	C₂²	1	2	yes	complemented
16	C₂³×Dic₂₅₁	C₂²	1	1	yes	characteristic, complemented
17	C₂²×C₄×D₂₅₁	C₂²	1	2	yes	complemented
18	#22548711	C₂²	1	8	yes	complemented
19	#22548708	C₂²	1	16	yes	complemented
20	#22548697	C₂²	1	16	yes	complemented
21	#22548696	C₂²	1	8	yes	complemented
22	#22548688	C₂²	1	16	yes	complemented
23	#22548687	C₂²	1	16	yes	complemented
24	#22548686	C₂²	1	16	yes	complemented
25	#22548684	C₂²	1	16	yes	complemented
26	C₂×C₄⋊Dic₂₅₁	C₂²	1	2	yes	complemented
27	C₂×Dic₂₅₁⋊C₄	C₂²	1	2	yes	complemented
28	C₂×Dic₂₅₁⋊C₄	C₂²	1	2	yes	complemented
29	C₂×C₂²⋊Dic₂₅₁	C₂²	1	2	yes	—
30	C₂×C₂²⋊Dic₂₅₁	C₂²	1	2	yes	—
31	C₂×C₂²⋊Dic₂₅₁	C₂²	1	2	yes	—
32	C₂×C₂²⋊Dic₂₅₁	C₂²	1	1	yes	characteristic, complemented
33	C₂×(C₂×D₂₅₁)⋊C₄	C₂²	1	2	yes	complemented
34	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic, complemented
35	C₂⁵×C₂₅₁	C₂²	1	1	yes	characteristic
36	C₂²×D₄×C₂₅₁	C₂²	1	1	yes	characteristic
37	C₂²×D₄×C₂₅₁	C₂²	1	2	yes	—
38	#120×C₂₅₁	C₂²	1	8	yes	complemented
39	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	2	yes	—
40	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
41	C₂³×D₂₅₁	C₂³	1	2	yes	—
42	C₂×D₁₀₀₄	—	4	2	no	—
43	C₂×D₁₀₀₄	C₂³	1	4	yes	complemented
44	C₂×C₂²⋊D₂₅₁	—	4	4	no	—
45	C₂×C₂²⋊D₂₅₁	—	4	4	no	—
46	C₂×C₂²⋊D₂₅₁	—	4	4	no	—
47	C₂×C₂²⋊D₂₅₁	—	4	4	no	—
48	C₂×C₂²⋊D₂₅₁	C₂³	1	8	yes	—
49	C₂×C₂²⋊D₂₅₁	C₂³	1	8	yes	—
50	C₂×C₂²⋊D₂₅₁	C₂³	1	8	yes	—
51	C₂×C₂²⋊D₂₅₁	C₂³	1	8	yes	complemented
52	C₂²×Dic₂₅₁	—	4	2	no	—
53	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
54	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
55	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
56	C₂²×Dic₂₅₁	C₂³	1	4	yes	complemented
57	C₂×C₄×D₂₅₁	—	4	4	no	—
58	C₂×C₄×D₂₅₁	C₂³	1	8	yes	complemented
59	C₄⋊Dic₂₅₁	C₂³	1	8	yes	complemented
60	Dic₂₅₁⋊C₄	C₂³	1	8	yes	complemented
61	Dic₂₅₁⋊C₄	C₂³	1	8	yes	complemented
62	C₂²⋊Dic₂₅₁	C₂³	1	8	yes	—
63	C₂²⋊Dic₂₅₁	C₂³	1	8	yes	—
64	C₂²⋊Dic₂₅₁	C₂³	1	8	yes	—
65	C₂²⋊Dic₂₅₁	C₂³	1	4	yes	complemented
66	(C₂×D₂₅₁)⋊C₄	C₂³	1	8	yes	complemented
67	C₄×Dic₂₅₁	C₂³	1	4	yes	complemented
68	C₂⁴×C₂₅₁	—	4	4	no	—
69	C₂⁴×C₂₅₁	—	4	2	no	—
70	C₂⁴×C₂₅₁	C₂³	1	1	yes	characteristic
71	C₂⁴×C₂₅₁	C₂³	1	1	yes	characteristic
72	C₂⁴×C₂₅₁	C₂³	1	2	yes	—
73	C₂⁴×C₂₅₁	C₂³	1	4	yes	—
74	C₂×D₄×C₂₅₁	—	4	4	no	—
75	C₂×D₄×C₂₅₁	—	4	2	no	—
76	C₂×D₄×C₂₅₁	C₂³	1	8	yes	—
77	C₂×D₄×C₂₅₁	C₂³	1	4	yes	—
78	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
79	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
80	C₂²⋊C₄×C₂₅₁	C₂³	1	8	yes	—
81	C₂²⋊C₄×C₂₅₁	C₂³	1	4	yes	—
82	C₂²×D₂₅₁	—	2	4	no	—
83	C₂²×D₂₅₁	—	2	4	no	—
84	C₂²×D₂₅₁	—	2	4	no	—
85	C₂²×D₂₅₁	C₂⁴	1	4	yes	—
86	D₁₀₀₄	—	4	4	no	—
87	C₂²⋊D₂₅₁	—	4	8	no	—
88	C₂²⋊D₂₅₁	—	4	8	no	—
89	C₂²⋊D₂₅₁	—	4	8	no	—
90	C₂²⋊D₂₅₁	—	4	8	no	—
91	C₂×Dic₂₅₁	—	2	4	no	—
92	C₂×Dic₂₅₁	—	2	4	no	—
93	C₂×Dic₂₅₁	—	2	4	no	—
94	C₂×Dic₂₅₁	—	4	4	no	—
95	C₂×Dic₂₅₁	C₂⁴	1	4	yes	—
96	C₂×Dic₂₅₁	C₂⁴	1	4	yes	—
97	C₂×Dic₂₅₁	C₂⁴	1	4	yes	—
98	C₄×D₂₅₁	—	4	8	no	—
99	C₂³×C₂₅₁	—	2	4	no	—
100	C₂³×C₂₅₁	—	2	2	no	—
101	C₂³×C₂₅₁	—	2	2	no	—
102	C₂³×C₂₅₁	—	2	4	no	—
103	C₂³×C₂₅₁	—	2	4	no	—
104	C₂³×C₂₅₁	—	2	4	no	—
105	C₂³×C₂₅₁	—	2	4	no	—
106	C₂³×C₂₅₁	—	4	8	no	—
107	C₂³×C₂₅₁	—	4	4	no	—
108	C₂³×C₂₅₁	—	4	16	no	—
109	C₂³×C₂₅₁	C₂⁴	1	4	yes	—
110	C₂³×C₂₅₁	C₂⁴	1	2	yes	—
111	C₂³×C₂₅₁	C₂⁴	1	2	yes	—
112	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
113	D₄×C₂₅₁	—	4	8	no	—
114	D₄×C₂₅₁	—	4	4	no	—
115	C₂×C₄×C₂₅₁	—	2	4	no	—
116	C₂×C₄×C₂₅₁	—	2	2	no	—
117	C₂×C₄×C₂₅₁	C₂⁴	1	2	yes	—
118	C₂×C₄×C₂₅₁	C₂⁴	1	4	yes	—
119	C₂²×C₂₅₁	—	2	4	no	—
120	C₂²×C₂₅₁	—	2	2	no	—
121	C₂²×C₂₅₁	—	2	2	no	—
122	C₂²×C₂₅₁	—	2	4	no	—
123	C₂²×C₂₅₁	—	2	4	no	—
124	C₂²×C₂₅₁	—	2	4	no	—
125	C₂²×C₂₅₁	—	2	4	no	—
126	C₂²×C₂₅₁	—	4	16	no	—
127	C₂²×C₂₅₁	—	4	4	no	—
128	C₂²×C₂₅₁	—	4	8	no	—
129	C₂²×C₂₅₁	—	4	4	no	—
130	C₂²×C₂₅₁	H(5,𝔽₂)	1	4	yes	—
131	C₂²×C₂₅₁	H(5,𝔽₂)	1	2	yes	—
132	C₂²×C₂₅₁	C₂⁵	1	1	yes	characteristic
133	C₂×D₂₅₁	—	2	4	no	—
134	C₂×D₂₅₁	—	2	4	no	—
135	C₂×D₂₅₁	—	2	4	no	—
136	C₂×D₂₅₁	—	4	8	no	—
137	C₄×C₂₅₁	—	2	4	no	—
138	C₄×C₂₅₁	—	2	2	no	—
139	Dic₂₅₁	—	2	4	no	—
140	Dic₂₅₁	—	2	4	no	—
141	Dic₂₅₁	—	2	4	no	—
142	C₂×C₂₅₁	—	4	2	no	—
143	C₂×C₂₅₁	—	4	4	no	—
144	C₂×C₂₅₁	—	4	2	no	—
145	C₂×C₂₅₁	#560	1	4	yes	complemented
146	C₂×C₂₅₁	C₂×H(5,𝔽₂)	1	2	yes	—
147	C₂×C₂₅₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
148	D₂₅₁	—	4	4	no	—
149	C₂₅₁	C₂×#560	1	1	yes	characteristic, complemented, Sylow
150	C₂×#560	—	251	1	no	maximal, Sylow
151	#560	—	251	16	no	—
152	C₂×#127	—	251	1	no	—
153	C₂×#126	—	251	2	no	—
154	C₂×#123	—	251	1	no	—
155	C₂×#123	—	251	2	no	—
156	C₂×#121	—	251	2	no	—
157	C₂×#121	—	251	2	no	—
158	C₂×#121	—	251	2	no	—
159	C₂×#120	—	251	2	no	—
160	C₂×#120	D₂₅₁	1	1	yes	characteristic, complemented
161	C₂⁵	C₂×D₂₅₁	1	1	yes	characteristic
162	C₂²×D₄	—	251	2	no	—
163	C₂²×D₄	—	251	2	no	—
164	C₂²×D₄	—	251	2	no	—
165	C₂²×D₄	—	251	2	no	—
166	C₂²×D₄	—	251	1	no	—
167	C₂²×D₄	C₂×D₂₅₁	1	2	yes	—
168	C₂²×D₄	C₂×D₂₅₁	1	1	yes	characteristic
169	C₂³×C₄	—	251	1	no	—
170	C₂³×C₄	—	251	2	no	—
171	#127	—	251	8	no	—
172	#126	—	251	16	no	—
173	#123	—	251	8	no	—
174	#123	—	251	16	no	—
175	#121	—	251	16	no	—
176	#121	—	251	16	no	—
177	#121	—	251	16	no	—
178	#120	—	251	16	no	—
179	#120	C₂×D₂₅₁	1	8	yes	complemented
180	C₂×C₄⋊C₄	—	251	2	no	—
181	C₂×C₄⋊C₄	—	251	2	no	—
182	C₂×C₄⋊C₄	—	251	2	no	—
183	C₂×C₂²⋊C₄	—	251	2	no	—
184	C₂×C₂²⋊C₄	—	251	2	no	—
185	C₂×C₂²⋊C₄	—	251	2	no	—
186	C₂×C₂²⋊C₄	—	251	1	no	—
187	C₂×C₂²⋊C₄	—	251	2	no	—
188	C₂×C₂²⋊C₄	C₂×D₂₅₁	1	2	yes	—
189	C₂×C₂²⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
190	C₂×C₄²	—	251	1	no	—
191	C₂⁴	—	4	4	no	—
192	C₂⁴	—	4	2	no	—
193	C₂⁴	—	251	2	no	—
194	C₂⁴	C₂²×D₂₅₁	1	2	yes	—
195	C₂⁴	C₂²×D₂₅₁	1	1	yes	characteristic
196	C₂⁴	C₂²×D₂₅₁	1	1	yes	characteristic
197	C₂⁴	C₂²×D₂₅₁	1	4	yes	—
198	C₂×D₄	—	4	4	no	—
199	C₂×D₄	—	4	2	no	—
200	C₂×D₄	—	1004	2	no	—
201	C₂×D₄	—	1004	4	no	—
202	C₂×D₄	—	1004	4	no	—
203	C₂×D₄	—	1004	4	no	—
204	C₂×D₄	—	1004	4	no	—
205	C₂×D₄	—	251	8	no	—
206	C₂×D₄	—	251	8	no	—
207	C₂×D₄	—	251	8	no	—
208	C₂×D₄	—	251	8	no	—
209	C₂×D₄	—	251	4	no	—
210	C₂×D₄	C₂²×D₂₅₁	1	8	yes	—
211	C₂×D₄	C₂²×D₂₅₁	1	4	yes	—
212	C₂²×C₄	—	1004	4	no	—
213	C₂²×C₄	—	1004	2	no	—
214	C₂²×C₄	—	251	4	no	—
215	C₂²×C₄	—	251	8	no	—
216	C₂²×C₄	—	251	2	no	—
217	C₂²×C₄	—	251	2	no	—
218	C₂²×C₄	—	251	2	no	—
219	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
220	C₂²×C₄	C₂²×D₂₅₁	1	2	yes	—
221	C₄⋊C₄	—	251	8	no	—
222	C₄⋊C₄	—	251	8	no	—
223	C₄⋊C₄	—	251	8	no	—
224	C₂²⋊C₄	—	251	8	no	—
225	C₂²⋊C₄	—	251	8	no	—
226	C₂²⋊C₄	—	251	8	no	—
227	C₂²⋊C₄	—	251	4	no	—
228	C₂²⋊C₄	—	251	8	no	—
229	C₂²⋊C₄	C₂²×D₂₅₁	1	8	yes	—
230	C₂²⋊C₄	C₂²×D₂₅₁	1	4	yes	—
231	C₄²	—	251	4	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	4	no	—
234	C₂³	—	2	4	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	2	4	no	—
237	C₂³	—	2	4	no	—
238	C₂³	—	2	4	no	—
239	C₂³	—	4	8	no	—
240	C₂³	—	4	4	no	—
241	C₂³	—	4	16	no	—
242	C₂³	—	502	4	no	—
243	C₂³	—	502	4	no	—
244	C₂³	—	502	4	no	—
245	C₂³	—	251	4	no	—
246	C₂³	C₂³×D₂₅₁	1	4	yes	—
247	C₂³	C₂³×D₂₅₁	1	2	yes	—
248	C₂³	C₂³×D₂₅₁	1	2	yes	—
249	C₂³	C₂³×D₂₅₁	1	1	yes	characteristic
250	D₄	—	4	8	no	—
251	D₄	—	4	4	no	—
252	D₄	—	1004	4	no	—
253	D₄	—	1004	8	no	—
254	D₄	—	1004	8	no	—
255	D₄	—	1004	8	no	—
256	D₄	—	1004	8	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	1004	8	no	—
260	C₂×C₄	—	1004	4	no	—
261	C₂×C₄	—	502	4	no	—
262	C₂×C₄	—	502	4	no	—
263	C₂×C₄	—	502	4	no	—
264	C₂×C₄	—	251	4	no	—
265	C₂×C₄	—	251	4	no	—
266	C₂×C₄	—	251	4	no	—
267	C₂×C₄	C₂³×D₂₅₁	1	2	yes	—
268	C₂×C₄	C₂³×D₂₅₁	1	4	yes	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	4	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	4	no	—
276	C₂²	—	4	16	no	—
277	C₂²	—	4	8	no	—
278	C₂²	—	4	4	no	—
279	C₂²	—	4	4	no	—
280	C₂²	—	1004	8	no	—
281	C₂²	—	502	4	no	—
282	C₂²	—	502	4	no	—
283	C₂²	—	502	4	no	—
284	C₂²	#22548760	1	4	yes	—
285	C₂²	#22548760	1	2	yes	—
286	C₂²	C₂⁴×D₂₅₁	1	1	yes	characteristic
287	C₄	—	2	4	no	—
288	C₄	—	2	2	no	—
289	C₄	—	502	4	no	—
290	C₄	—	502	4	no	—
291	C₄	—	502	4	no	—
292	C₂	—	4	4	no	—
293	C₂	—	4	2	no	—
294	C₂	—	4	2	no	—
295	C₂	—	1004	4	no	—
296	C₂	#30632426	1	4	yes	direct product factor
297	C₂	C₂×#22548760	1	2	yes	—
298	C₂	C₂×#22548760	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₂₅₁
commutator, cosocle	C₂²×C₂₅₁	C₂⁵
Frattini	C₂²	C₂⁴×D₂₅₁
Fitting	C₂×#120×C₂₅₁	C₂
socle	C₂³×C₂₅₁	C₂⁴
hypercenter	C₂×#120	D₂₅₁
hypocenter	C₂₅₁	C₂×#560
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ C₂×#120

Complements

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

N	H ≅ G/N	type
1. #30632426	C₂	direct (N × H)
2. C₂×#22548711	C₂	semidirect
3. C₂×#22548708	C₂	semidirect
4. C₂×#22548697	C₂	semidirect
5. C₂×#22548696	C₂	semidirect
6. C₂×#22548688	C₂	semidirect
7. C₂×#22548687	C₂	semidirect
8. C₂×#22548686	C₂	semidirect
9. C₂×#22548684	C₂	semidirect
10. C₂×#120×C₂₅₁	C₂	semidirect
11. C₂²×D₁₀₀₄	C₂²	semidirect
15. C₂²×C₂²⋊D₂₅₁	C₂²	semidirect
16. C₂³×Dic₂₅₁	C₂²	semidirect
17. C₂²×C₄×D₂₅₁	C₂²	semidirect
18. #22548711	C₂²	semidirect
19. #22548708	C₂²	semidirect
20. #22548697	C₂²	semidirect
21. #22548696	C₂²	semidirect
22. #22548688	C₂²	semidirect
23. #22548687	C₂²	semidirect
24. #22548686	C₂²	semidirect
25. #22548684	C₂²	semidirect
26. C₂×C₄⋊Dic₂₅₁	C₂²	semidirect
27. C₂×Dic₂₅₁⋊C₄	C₂²	semidirect
28. C₂×Dic₂₅₁⋊C₄	C₂²	semidirect
32. C₂×C₂²⋊Dic₂₅₁	C₂²	semidirect
33. C₂×(C₂×D₂₅₁)⋊C₄	C₂²	semidirect
34. C₂×C₄×Dic₂₅₁	C₂²	semidirect
38. #120×C₂₅₁	C₂²	semidirect
43. C₂×D₁₀₀₄	C₂³	semidirect
51. C₂×C₂²⋊D₂₅₁	C₂³	semidirect
56. C₂²×Dic₂₅₁	C₂³	semidirect
58. C₂×C₄×D₂₅₁	C₂³	semidirect
59. C₄⋊Dic₂₅₁	C₂³	semidirect
60. Dic₂₅₁⋊C₄	C₂³	semidirect
61. Dic₂₅₁⋊C₄	C₂³	semidirect
65. C₂²⋊Dic₂₅₁	C₂³	semidirect
66. (C₂×D₂₅₁)⋊C₄	C₂³	semidirect
67. C₄×Dic₂₅₁	C₂³	semidirect
145. C₂×C₂₅₁	#560	semidirect
149. C₂₅₁	C₂×#560	semidirect
160. C₂×#120	D₂₅₁	semidirect
179. #120	C₂×D₂₅₁	semidirect
296. C₂	#30632426	direct (N × H)

Maximal subgroups

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

subgroup	index
#30632426	2
C₂×#22548711	2
C₂×#22548708	2
C₂×#22548697	2
C₂×#22548696	2
C₂×#22548688	2
C₂×#22548687	2
C₂×#22548686	2
C₂×#22548684	2
C₂×#120×C₂₅₁	2
C₂×#560	251

Subgroups — C2×#30632426 #44564481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#30632426 #44564481