Subgroups — #44559210

Order 32128.

470 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22548748	C₂	1	1	yes	maximal, characteristic
2	#30632167	C₂	1	1	yes	maximal, characteristic
3	#30632164	C₂	1	1	yes	maximal, characteristic, complemented
4	#30631795	C₂	1	2	yes	maximal, complemented
5	#30631789	C₂	1	2	yes	maximal, complemented
6	C₄×Dic₁₀₀₄	C₂	1	2	yes	maximal, complemented
7	C₄×#16860799	C₂	1	2	yes	maximal, complemented
8	#30631729	C₂	1	1	yes	maximal, characteristic, complemented
9	#30631684	C₂	1	1	yes	maximal, characteristic
10	C₂×#22548640	C₂	1	1	yes	maximal, characteristic, complemented
11	#405×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#16860810	C₂²	1	1	yes	characteristic
13	C₂²×Dic₅₀₂	C₂²	1	1	yes	characteristic
14	#22548748	C₄	1	8	yes	complemented
15	C₂×Dic₁₀₀₄	C₂²	1	2	yes	—
16	C₂×#16860799	C₂²	1	2	yes	—
17	C₄×Dic₅₀₂	C₂²	1	2	yes	—
18	C₄×Dic₅₀₂	C₂²	1	2	yes	—
19	C₄×Dic₅₀₂	C₂²	1	2	yes	—
20	#22548680	—	2	1	no	—
21	#22548680	—	2	1	no	—
22	C₄×D₁₀₀₄	C₂²	1	2	yes	—
23	(C₂²⋊D₂₅₁)⋊C₄	—	2	2	no	—
24	#22548674	C₂²	1	1	yes	characteristic
25	#22548673	—	2	1	no	—
26	#22548671	—	2	2	no	—
27	C₂×C₄⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×D₂₅₁	—	2	1	no	—
29	#22548649	C₂²	1	2	yes	—
30	#22548647	C₂²	1	2	yes	—
31	#22548640	C₂²	1	1	yes	characteristic
32	#22548640	C₂²	1	1	yes	characteristic
33	#22548640	C₂²	1	2	yes	—
34	#22548640	C₂²	1	2	yes	—
35	#22548637	C₂²	1	1	yes	characteristic
36	#22548637	C₂²	1	1	yes	characteristic
37	C₂×M₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
38	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
39	#117×C₂₅₁	C₂²	1	1	yes	characteristic
40	#97×C₂₅₁	C₂²	1	2	yes	—
41	C₄×C₈×C₂₅₁	C₂²	1	2	yes	—
42	#16860810	—	2	2	no	—
43	#16860810	C₂×C₄	1	4	yes	—
44	C₂×Dic₅₀₂	—	2	1	no	—
45	C₂×Dic₅₀₂	—	2	2	no	—
46	C₂×Dic₅₀₂	C₂×C₄	1	4	yes	complemented
47	C₂×Dic₅₀₂	C₂³	1	2	yes	—
48	C₂×Dic₅₀₂	C₂³	1	1	yes	characteristic
49	C₂×D₁₀₀₄	C₂³	1	1	yes	characteristic
50	C₂×C₂²⋊D₂₅₁	—	2	1	no	—
51	C₂²×Dic₂₅₁	—	2	1	no	—
52	C₂²×Dic₂₅₁	—	2	1	no	—
53	C₂×C₄×D₂₅₁	—	2	1	no	—
54	C₂×C₄×D₂₅₁	—	2	2	no	—
55	Dic₁₀₀₄	C₂×C₄	1	8	yes	complemented
56	#16860799	C₂×C₄	1	8	yes	complemented
57	C₄⋊Dic₂₅₁	—	2	1	no	—
58	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
59	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
60	C₄⋊Dic₂₅₁	C₂³	1	2	yes	—
61	Dic₂₅₁⋊C₄	—	2	2	no	—
62	Dic₂₅₁⋊C₄	—	2	2	no	—
63	Dic₂₅₁⋊C₄	—	2	2	no	—
64	C₂²⋊Dic₂₅₁	—	2	1	no	—
65	(C₂×D₂₅₁)⋊C₄	—	2	2	no	—
66	C₄×Dic₂₅₁	—	2	2	no	—
67	C₄×Dic₂₅₁	—	2	2	no	—
68	C₄×Dic₂₅₁	—	2	2	no	—
69	C₄×Dic₂₅₁	—	2	1	no	—
70	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
71	M₁₆×C₂₅₁	C₂×C₄	1	4	yes	complemented
72	C₂×C₈×C₂₅₁	—	2	2	no	—
73	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
76	C₄⋊C₄×C₂₅₁	D₄	1	1	yes	characteristic
77	C₄⋊C₄×C₂₅₁	D₄	1	1	yes	characteristic
78	C₂²⋊C₄×C₂₅₁	D₄	1	2	yes	—
79	C₄²×C₂₅₁	C₂³	1	2	yes	—
80	C₂²×D₂₅₁	—	2	1	no	—
81	Dic₅₀₂	—	2	4	no	—
82	Dic₅₀₂	—	2	2	no	—
83	Dic₅₀₂	—	2	1	no	—
84	Dic₅₀₂	C₂²×C₄	1	4	yes	—
85	Dic₅₀₂	C₂²×C₄	1	2	yes	—
86	D₁₀₀₄	—	2	1	no	—
87	D₁₀₀₄	C₂²×C₄	1	2	yes	—
88	C₂²⋊D₂₅₁	—	2	4	no	—
89	C₂×Dic₂₅₁	—	2	4	no	—
90	C₂×Dic₂₅₁	—	2	1	no	—
91	C₂×Dic₂₅₁	—	2	2	no	—
92	C₂×Dic₂₅₁	—	2	1	no	—
93	C₂×Dic₂₅₁	—	2	1	no	—
94	C₂×Dic₂₅₁	—	2	2	no	—
95	C₂×Dic₂₅₁	—	8	1	no	—
96	C₄×D₂₅₁	—	2	4	no	—
97	C₄×D₂₅₁	—	8	2	no	—
98	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
100	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
101	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₂₅₁	#41	1	2	yes	—
104	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
105	C₈×C₂₅₁	—	2	1	no	—
106	C₈×C₂₅₁	—	2	1	no	—
107	C₈×C₂₅₁	C₂²×C₄	1	4	yes	—
108	C₂²×C₂₅₁	—	2	1	no	—
109	C₂²×C₂₅₁	—	2	1	no	—
110	C₂²×C₂₅₁	C₄×D₄	1	2	yes	—
111	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
112	C₂×D₂₅₁	—	2	2	no	—
113	C₂×D₂₅₁	—	4	1	no	—
114	C₂×D₂₅₁	—	4	1	no	—
115	C₄×C₂₅₁	—	2	2	no	—
116	C₄×C₂₅₁	—	2	2	no	—
117	C₄×C₂₅₁	C₄×D₄	1	2	yes	—
118	C₄×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
119	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
120	Dic₂₅₁	—	2	2	no	—
121	Dic₂₅₁	—	2	4	no	—
122	Dic₂₅₁	—	4	2	no	—
123	Dic₂₅₁	—	4	1	no	—
124	Dic₂₅₁	—	4	1	no	—
125	C₂×C₂₅₁	—	2	2	no	—
126	C₂×C₂₅₁	C₂×C₄×D₄	1	1	yes	characteristic
127	C₂×C₂₅₁	#577	1	1	yes	characteristic
128	C₂×C₂₅₁	#578	1	1	yes	characteristic
129	D₂₅₁	—	4	2	no	—
130	C₂₅₁	#2947	1	1	yes	characteristic, complemented, Sylow
131	#2947	—	251	1	no	maximal, Sylow
132	C₂×#137	—	251	1	no	—
133	#520	—	251	1	no	—
134	#519	—	251	1	no	—
135	#441	—	251	2	no	—
136	#440	—	251	2	no	—
137	C₄×Q₁₆	—	251	2	no	—
138	C₄×SD₁₆	—	251	2	no	—
139	#428	—	251	1	no	—
140	#416	—	251	1	no	—
141	C₂×#103	—	251	1	no	—
142	#405	D₂₅₁	1	1	yes	characteristic, complemented
143	C₂×#41	—	251	1	no	—
144	C₂²×Q₈	—	251	1	no	—
145	#137	—	251	8	no	—
146	C₂×Q₁₆	—	251	2	no	—
147	C₂×SD₁₆	—	251	2	no	—
148	C₂×M₁₆	C₂×D₂₅₁	1	1	yes	characteristic
149	C₂²×C₈	C₂×D₂₅₁	1	1	yes	characteristic
150	C₄×Q₈	—	502	1	no	—
151	C₄×Q₈	—	502	1	no	—
152	C₄×Q₈	—	251	2	no	—
153	C₄×Q₈	—	251	2	no	—
154	C₄×Q₈	—	251	2	no	—
155	C₄×D₄	—	502	2	no	—
156	C₄×D₄	—	251	2	no	—
157	#117	—	502	1	no	—
158	#117	—	502	2	no	—
159	#117	—	251	1	no	—
160	#117	C₂×D₂₅₁	1	1	yes	characteristic
161	C₂×C₄⋊C₄	—	502	1	no	—
162	C₂×C₄⋊C₄	—	251	1	no	—
163	#107	—	251	2	no	—
164	#106	—	251	2	no	—
165	#103	—	251	2	no	—
166	#103	—	251	2	no	—
167	#103	—	251	1	no	—
168	#103	—	251	1	no	—
169	#102	—	251	1	no	—
170	#102	—	251	1	no	—
171	#97	C₂×D₂₅₁	1	2	yes	—
172	C₄×C₈	C₂×D₂₅₁	1	2	yes	—
173	#41	—	502	2	no	—
174	#41	—	251	4	no	—
175	C₂×Q₈	—	502	1	no	—
176	C₂×Q₈	—	502	2	no	—
177	C₂×Q₈	—	251	4	no	—
178	C₂×Q₈	—	251	2	no	—
179	C₂×Q₈	—	251	1	no	—
180	C₂×D₄	—	502	1	no	—
181	C₂×D₄	—	251	1	no	—
182	C₂²×C₄	—	502	1	no	—
183	C₂²×C₄	—	502	2	no	—
184	C₂²×C₄	—	502	1	no	—
185	C₂²×C₄	—	502	1	no	—
186	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
187	Q₁₆	—	251	8	no	—
188	SD₁₆	—	251	8	no	—
189	M₁₆	C₄×D₂₅₁	1	4	yes	complemented
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
192	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
193	C₂×C₈	C₂²×D₂₅₁	1	2	yes	—
194	C₄⋊C₄	—	502	1	no	—
195	C₄⋊C₄	—	502	2	no	—
196	C₄⋊C₄	—	502	2	no	—
197	C₄⋊C₄	—	502	2	no	—
198	C₄⋊C₄	—	251	1	no	—
199	C₄⋊C₄	—	251	1	no	—
200	C₄⋊C₄	—	251	2	no	—
201	C₄⋊C₄	D₁₀₀₄	1	1	yes	characteristic
202	C₄⋊C₄	D₁₀₀₄	1	1	yes	characteristic
203	C₂²⋊C₄	—	502	1	no	—
204	C₂²⋊C₄	—	502	2	no	—
205	C₂²⋊C₄	D₁₀₀₄	1	2	yes	—
206	C₄²	—	502	2	no	—
207	C₄²	—	502	2	no	—
208	C₄²	—	502	2	no	—
209	C₄²	—	502	1	no	—
210	C₄²	C₂²×D₂₅₁	1	2	yes	—
211	C₂³	—	502	1	no	—
212	C₂³	C₂×D₁₀₀₄	1	1	yes	characteristic
213	Q₈	—	502	2	no	—
214	Q₈	—	502	1	no	—
215	Q₈	—	502	4	no	—
216	Q₈	—	251	4	no	—
217	Q₈	—	251	2	no	—
218	D₄	—	502	4	no	—
219	D₄	—	502	1	no	—
220	D₄	—	251	2	no	—
221	C₂×C₄	—	2008	2	no	—
222	C₂×C₄	—	2008	1	no	—
223	C₂×C₄	—	502	1	no	—
224	C₂×C₄	—	502	1	no	—
225	C₂×C₄	—	502	2	no	—
226	C₂×C₄	—	502	1	no	—
227	C₂×C₄	—	502	2	no	—
228	C₂×C₄	—	502	4	no	—
229	C₂×C₄	—	502	4	no	—
230	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
231	C₂×C₄	C₂×D₁₀₀₄	1	2	yes	—
232	C₂×C₄	C₂×D₁₀₀₄	1	2	yes	—
233	C₂×C₄	C₂×D₁₀₀₄	1	1	yes	characteristic
234	C₂×C₄	#16860810	1	2	yes	—
235	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	C₂×C₄×D₂₅₁	1	4	yes	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	1004	1	no	—
242	C₂²	—	1004	1	no	—
243	C₂²	—	502	2	no	—
244	C₂²	C₄×D₁₀₀₄	1	2	yes	—
245	C₂²	C₂²×D₁₀₀₄	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	1004	2	no	—
249	C₄	—	1004	1	no	—
250	C₄	—	1004	1	no	—
251	C₄	—	502	2	no	—
252	C₄	—	502	4	no	—
253	C₄	C₄×D₁₀₀₄	1	2	yes	—
254	C₄	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
255	C₄	C₂×#16860810	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	1004	2	no	—
258	C₂	C₂×C₄×D₁₀₀₄	1	1	yes	characteristic
259	C₂	#30632499	1	1	yes	characteristic
260	C₂	#30632504	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₀₀₄
commutator	C₄×C₂₅₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	#405×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
prime radical	C₂×#16860810	C₂²
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	#405	D₂₅₁
hypocenter	C₂₅₁	#2947
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
3. #30632164	C₂	semidirect
4. #30631795	C₂	semidirect
5. #30631789	C₂	semidirect
6. C₄×Dic₁₀₀₄	C₂	semidirect
7. C₄×#16860799	C₂	semidirect
8. #30631729	C₂	semidirect
10. C₂×#22548640	C₂	semidirect
11. #405×C₂₅₁	C₂	semidirect
14. #22548748	C₄	semidirect
46. C₂×Dic₅₀₂	C₂×C₄	semidirect
55. Dic₁₀₀₄	C₂×C₄	semidirect
56. #16860799	C₂×C₄	semidirect
71. M₁₆×C₂₅₁	C₂×C₄	semidirect
130. C₂₅₁	#2947	semidirect
142. #405	D₂₅₁	semidirect
189. M₁₆	C₄×D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22548748	2
#30632167	2
#30632164	2
#30631795	2
#30631789	2
C₄×Dic₁₀₀₄	2
C₄×#16860799	2
#30631729	2
#30631684	2
C₂×#22548640	2
#405×C₂₅₁	2
#2947	251