Subgroups — #44558715

Order 32128.

644 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22548714	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#22548692	C₂	1	2	yes	maximal
3	C₂²×C₄×Dic₂₅₁	C₂	1	1	yes	maximal, characteristic
4	#30631591	C₂	1	2	yes	maximal, complemented
5	#30631586	C₂	1	1	yes	maximal, characteristic, complemented
6	#30631584	C₂	1	2	yes	maximal, complemented
7	#30631583	C₂	1	1	yes	maximal, characteristic, complemented
8	#30631556	C₂	1	1	yes	maximal, characteristic, complemented
9	#30631547	C₂	1	1	yes	maximal, characteristic
10	#30631545	C₂	1	2	yes	maximal
11	#380×C₂₅₁	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₅₀₂	C₂²	1	2	yes	—
13	C₂³×Dic₂₅₁	C₂²	1	1	yes	characteristic
14	C₂³×Dic₂₅₁	C₂²	1	1	yes	characteristic
15	#22548714	—	2	4	no	—
16	#22548692	—	2	8	no	—
17	C₂×C₄⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₂₅₁	C₂²	1	2	yes	—
20	C₂×Dic₂₅₁⋊C₄	C₂²	1	2	yes	—
21	C₂×Dic₂₅₁⋊C₄	C₂²	1	2	yes	—
22	C₂×Dic₂₅₁⋊C₄	C₂²	1	2	yes	—
23	C₂×Dic₂₅₁⋊C₄	C₂²	1	2	yes	—
24	C₂×C₂²⋊Dic₂₅₁	C₂²	1	2	yes	—
25	C₂×C₄×Dic₂₅₁	—	2	4	no	—
26	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
30	C₂₅₁⋊#95	C₂²	1	2	yes	—
31	C₂₅₁⋊#95	C₂²	1	2	yes	—
32	C₂₅₁⋊#95	C₂²	1	2	yes	—
33	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	2	yes	—
37	#95×C₂₅₁	C₂²	1	2	yes	—
38	C₂×Dic₅₀₂	—	2	4	no	—
39	C₂×Dic₅₀₂	—	2	4	no	—
40	C₂×Dic₅₀₂	—	2	4	no	—
41	C₂²×Dic₂₅₁	—	2	2	no	—
42	C₂²×Dic₂₅₁	—	2	2	no	—
43	C₂²×Dic₂₅₁	—	2	2	no	—
44	C₂²×Dic₂₅₁	—	2	2	no	—
45	C₂²×Dic₂₅₁	D₄	1	4	yes	—
46	C₂²×Dic₂₅₁	Q₈	1	4	yes	complemented
47	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
48	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
49	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
50	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
51	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
52	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
53	C₄⋊Dic₂₅₁	—	2	4	no	—
54	C₄⋊Dic₂₅₁	—	2	2	no	—
55	C₄⋊Dic₂₅₁	—	2	2	no	—
56	Dic₂₅₁⋊C₄	—	2	4	no	—
57	Dic₂₅₁⋊C₄	—	2	4	no	—
58	Dic₂₅₁⋊C₄	—	2	4	no	—
59	Dic₂₅₁⋊C₄	—	2	4	no	—
60	C₂²⋊Dic₂₅₁	—	2	4	no	—
61	C₄×Dic₂₅₁	—	2	2	no	—
62	C₄×Dic₂₅₁	—	2	2	no	—
63	C₄×Dic₂₅₁	—	2	2	no	—
64	C₄×Dic₂₅₁	D₄	1	4	yes	—
65	C₂⁴×C₂₅₁	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₂₅₁	—	2	2	no	—
67	C₂²×C₄×C₂₅₁	—	2	2	no	—
68	C₂²×C₄×C₂₅₁	Q₈	1	4	yes	complemented
69	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
70	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
71	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
72	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₂₅₁	—	2	2	no	—
74	C₄⋊C₄×C₂₅₁	—	2	2	no	—
75	C₂²⋊C₄×C₂₅₁	—	2	4	no	—
76	Dic₅₀₂	—	4	8	no	—
77	C₂×Dic₂₅₁	—	2	1	no	—
78	C₂×Dic₂₅₁	—	2	1	no	—
79	C₂×Dic₂₅₁	—	2	1	no	—
80	C₂×Dic₂₅₁	—	2	1	no	—
81	C₂×Dic₂₅₁	—	2	8	no	—
82	C₂×Dic₂₅₁	—	2	8	no	—
83	C₂×Dic₂₅₁	—	2	2	no	—
84	C₂×Dic₂₅₁	—	2	2	no	—
85	C₂×Dic₂₅₁	—	2	2	no	—
86	C₂×Dic₂₅₁	—	2	2	no	—
87	C₂×Dic₂₅₁	—	2	2	no	—
88	C₂×Dic₂₅₁	—	2	2	no	—
89	C₂×Dic₂₅₁	C₂×D₄	1	2	yes	—
90	C₂×Dic₂₅₁	C₂×D₄	1	2	yes	—
91	C₂×Dic₂₅₁	C₂×D₄	1	2	yes	—
92	C₂×Dic₂₅₁	C₂×D₄	1	2	yes	—
93	C₂×Dic₂₅₁	C₂×Q₈	1	2	yes	—
94	C₂×Dic₂₅₁	C₂×Q₈	1	2	yes	—
95	C₂×Dic₂₅₁	#41	1	2	yes	—
96	C₂×Dic₂₅₁	#41	1	2	yes	—
97	C₂³×C₂₅₁	—	2	1	no	—
98	C₂³×C₂₅₁	—	2	1	no	—
99	C₂³×C₂₅₁	—	2	1	no	—
100	C₂³×C₂₅₁	—	2	1	no	—
101	C₂³×C₂₅₁	C₂×D₄	1	2	yes	—
102	C₂³×C₂₅₁	C₂×Q₈	1	2	yes	—
103	C₂³×C₂₅₁	C₂×Q₈	1	2	yes	—
104	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
105	C₂×C₄×C₂₅₁	—	2	1	no	—
106	C₂×C₄×C₂₅₁	—	2	1	no	—
107	C₂×C₄×C₂₅₁	—	2	8	no	—
108	C₂×C₄×C₂₅₁	—	2	2	no	—
109	C₂×C₄×C₂₅₁	—	2	2	no	—
110	C₂×C₄×C₂₅₁	—	2	2	no	—
111	C₂×C₄×C₂₅₁	—	2	2	no	—
112	C₂×C₄×C₂₅₁	—	2	2	no	—
113	C₂×C₄×C₂₅₁	—	2	2	no	—
114	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
115	C₂×C₄×C₂₅₁	C₂×Q₈	1	2	yes	—
116	C₂×C₄×C₂₅₁	C₂×Q₈	1	2	yes	—
117	C₂×C₄×C₂₅₁	#41	1	2	yes	—
118	C₂²×C₂₅₁	—	2	2	no	—
119	C₂²×C₂₅₁	—	2	2	no	—
120	C₂²×C₂₅₁	—	2	2	no	—
121	C₂²×C₂₅₁	—	2	2	no	—
122	C₂²×C₂₅₁	—	2	2	no	—
123	C₂²×C₂₅₁	—	2	2	no	—
124	C₂²×C₂₅₁	C₄⋊Q₈	1	4	yes	complemented
125	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
126	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
127	C₂²×C₂₅₁	C₂²×Q₈	1	1	yes	characteristic
128	C₂²×C₂₅₁	C₂²×Q₈	1	1	yes	characteristic
129	C₂²×C₂₅₁	C₂×#41	1	1	yes	characteristic
130	C₂²×C₂₅₁	C₂×#41	1	1	yes	characteristic
131	C₂²×C₂₅₁	C₂×#41	1	1	yes	characteristic
132	C₄×C₂₅₁	—	2	2	no	—
133	C₄×C₂₅₁	—	4	2	no	—
134	C₄×C₂₅₁	—	4	2	no	—
135	C₄×C₂₅₁	C₂²⋊Q₈	1	4	yes	complemented
136	Dic₂₅₁	—	2	2	no	—
137	Dic₂₅₁	—	2	2	no	—
138	Dic₂₅₁	—	4	2	no	—
139	Dic₂₅₁	—	4	2	no	—
140	Dic₂₅₁	#121	1	4	yes	—
141	Dic₂₅₁	C₂²⋊Q₈	1	4	yes	complemented
142	C₂×C₂₅₁	—	2	4	no	—
143	C₂×C₂₅₁	C₂×#121	1	1	yes	characteristic
144	C₂×C₂₅₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
145	C₂×C₂₅₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
146	C₂×C₂₅₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
147	C₂×C₂₅₁	#532	1	1	yes	characteristic
148	C₂×C₂₅₁	#533	1	1	yes	characteristic
149	C₂×C₂₅₁	#533	1	1	yes	characteristic
150	C₂₅₁	#2857	1	1	yes	characteristic, complemented, Sylow
151	#2857	—	251	1	no	maximal, Sylow
152	C₂×C₄⋊Q₈	—	251	1	no	—
153	C₂×C₂²⋊Q₈	—	251	2	no	—
154	C₂²×C₄²	—	251	1	no	—
155	#391	—	251	2	no	—
156	#389	—	251	1	no	—
157	#389	—	251	1	no	—
158	#389	—	251	2	no	—
159	#384	—	251	1	no	—
160	#380	—	251	1	no	—
161	#380	—	251	2	no	—
162	#380	D₂₅₁	1	1	yes	characteristic
163	C₂²×Q₈	—	251	2	no	—
164	C₂³×C₄	—	251	1	no	—
165	C₂³×C₄	—	251	1	no	—
166	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
167	C₄⋊Q₈	—	502	4	no	—
168	C₂²⋊Q₈	—	502	8	no	—
169	C₂×C₄⋊C₄	—	251	2	no	—
170	C₂×C₄⋊C₄	—	251	2	no	—
171	C₂×C₄⋊C₄	—	251	2	no	—
172	C₂×C₄⋊C₄	—	251	2	no	—
173	C₂×C₄⋊C₄	—	251	1	no	—
174	C₂×C₄⋊C₄	—	251	2	no	—
175	C₂×C₄⋊C₄	—	251	1	no	—
176	C₂×C₄⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
177	C₂×C₄⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
178	C₂×C₂²⋊C₄	—	251	2	no	—
179	C₂×C₂²⋊C₄	C₂×D₂₅₁	1	2	yes	—
180	C₂×C₄²	—	502	4	no	—
181	C₂×C₄²	—	251	1	no	—
182	C₂×C₄²	—	251	1	no	—
183	C₂×C₄²	—	251	1	no	—
184	C₂×C₄²	—	251	1	no	—
185	#95	—	251	2	no	—
186	#95	—	251	2	no	—
187	#95	—	251	2	no	—
188	#95	C₂×D₂₅₁	1	2	yes	—
189	C₂⁴	C₂²×D₂₅₁	1	1	yes	characteristic
190	C₂×Q₈	—	502	4	no	—
191	C₂×Q₈	—	502	4	no	—
192	C₂×Q₈	—	502	4	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	502	2	no	—
196	C₂²×C₄	—	502	2	no	—
197	C₂²×C₄	—	502	2	no	—
198	C₂²×C₄	—	502	2	no	—
199	C₂²×C₄	—	251	2	no	—
200	C₂²×C₄	—	251	2	no	—
201	C₂²×C₄	—	251	4	no	—
202	C₂²×C₄	—	251	4	no	—
203	C₂²×C₄	—	251	1	no	—
204	C₂²×C₄	—	251	1	no	—
205	C₂²×C₄	—	251	1	no	—
206	C₂²×C₄	—	251	1	no	—
207	C₂²×C₄	Dic₅₀₂	1	4	yes	complemented
208	C₂²×C₄	C₂²×D₂₅₁	1	2	yes	—
209	C₂²×C₄	C₂²×D₂₅₁	1	2	yes	—
210	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
211	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	502	4	no	—
215	C₄⋊C₄	—	502	4	no	—
216	C₄⋊C₄	—	502	4	no	—
217	C₄⋊C₄	—	502	4	no	—
218	C₄⋊C₄	—	502	4	no	—
219	C₄⋊C₄	—	502	2	no	—
220	C₄⋊C₄	—	502	2	no	—
221	C₂²⋊C₄	—	2	4	no	—
222	C₂²⋊C₄	—	502	4	no	—
223	C₄²	—	502	2	no	—
224	C₄²	—	502	2	no	—
225	C₄²	—	502	2	no	—
226	C₄²	—	251	4	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	2	1	no	—
229	C₂³	—	2	1	no	—
230	C₂³	—	2	1	no	—
231	C₂³	D₄×D₂₅₁	1	2	yes	—
232	C₂³	Q₈×D₂₅₁	1	2	yes	—
233	C₂³	C₂×Dic₅₀₂	1	2	yes	—
234	C₂³	C₂³×D₂₅₁	1	1	yes	characteristic
235	Q₈	—	1004	8	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	8	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	502	2	no	—
246	C₂×C₄	—	502	2	no	—
247	C₂×C₄	—	502	2	no	—
248	C₂×C₄	—	502	2	no	—
249	C₂×C₄	—	502	2	no	—
250	C₂×C₄	—	502	2	no	—
251	C₂×C₄	—	502	1	no	—
252	C₂×C₄	—	502	1	no	—
253	C₂×C₄	—	502	1	no	—
254	C₂×C₄	—	502	1	no	—
255	C₂×C₄	—	502	8	no	—
256	C₂×C₄	—	502	8	no	—
257	C₂×C₄	—	251	2	no	—
258	C₂×C₄	—	251	2	no	—
259	C₂×C₄	—	251	2	no	—
260	C₂×C₄	—	251	2	no	—
261	C₂×C₄	—	251	2	no	—
262	C₂×C₄	—	251	2	no	—
263	C₂×C₄	—	251	2	no	—
264	C₂×C₄	—	251	2	no	—
265	C₂×C₄	D₄×D₂₅₁	1	2	yes	—
266	C₂×C₄	C₂×Dic₅₀₂	1	2	yes	—
267	C₂×C₄	C₂×Dic₅₀₂	1	2	yes	—
268	C₂×C₄	#16860809	1	2	yes	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	#22548714	1	4	yes	complemented
276	C₂²	C₂×D₄×D₂₅₁	1	1	yes	characteristic
277	C₂²	C₂×D₄×D₂₅₁	1	1	yes	characteristic
278	C₂²	C₂×Q₈×D₂₅₁	1	1	yes	characteristic
279	C₂²	C₂²×Dic₅₀₂	1	1	yes	characteristic
280	C₂²	#41×D₂₅₁	1	1	yes	characteristic
281	C₂²	#41×D₂₅₁	1	1	yes	characteristic
282	C₂²	C₂×#16860809	1	1	yes	characteristic
283	C₄	—	2	2	no	—
284	C₄	—	4	2	no	—
285	C₄	—	4	2	no	—
286	C₄	—	1004	2	no	—
287	C₄	—	1004	2	no	—
288	C₄	—	502	2	no	—
289	C₄	—	502	2	no	—
290	C₄	—	251	4	no	—
291	C₄	—	251	4	no	—
292	C₄	#22548692	1	4	yes	complemented
293	C₂	—	2	4	no	—
294	C₂	#121×D₂₅₁	1	1	yes	characteristic
295	C₂	C₂²⋊Q₈×D₂₅₁	1	1	yes	characteristic
296	C₂	C₂×#22548692	1	1	yes	characteristic
297	C₂	C₂×#22548714	1	1	yes	characteristic
298	C₂	#30632232	1	1	yes	characteristic
299	C₂	#30632239	1	1	yes	characteristic
300	C₂	#30632240	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₅₁
commutator, socle, cosocle	C₂³×C₂₅₁	C₂⁴
Fitting	#380×C₂₅₁	C₂
prime radical	C₂⁴×C₂₅₁	C₂³
hypercenter	#380	D₂₅₁
hypocenter	C₂₅₁	#2857
LCS	G ▷ C₂³×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ #380

Complements

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

N	H ≅ G/N	type
1. C₂×#22548714	C₂	semidirect
4. #30631591	C₂	semidirect
5. #30631586	C₂	semidirect
6. #30631584	C₂	semidirect
7. #30631583	C₂	semidirect
8. #30631556	C₂	semidirect
46. C₂²×Dic₂₅₁	Q₈	semidirect
68. C₂²×C₄×C₂₅₁	Q₈	semidirect
124. C₂²×C₂₅₁	C₄⋊Q₈	semidirect
135. C₄×C₂₅₁	C₂²⋊Q₈	semidirect
141. Dic₂₅₁	C₂²⋊Q₈	semidirect
150. C₂₅₁	#2857	semidirect
207. C₂²×C₄	Dic₅₀₂	semidirect
275. C₂²	#22548714	semidirect
292. C₄	#22548692	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22548714	2
C₂×#22548692	2
C₂²×C₄×Dic₂₅₁	2
#30631591	2
#30631586	2
#30631584	2
#30631583	2
#30631556	2
#30631547	2
#30631545	2
#380×C₂₅₁	2
#2857	251