Subgroups — #44553284

Order 32128.

552 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₅₀₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#22548680	C₂	1	1	yes	maximal, characteristic
3	C₂×#22548680	C₂	1	1	yes	maximal, characteristic
4	#30631586	C₂	1	1	yes	maximal, characteristic
5	#30631584	C₂	1	2	yes	maximal
6	#30631582	C₂	1	2	yes	maximal
7	#30631552	C₂	1	1	yes	maximal, characteristic
8	#30631551	C₂	1	2	yes	maximal
9	C₄⋊C₄×Dic₂₅₁	C₂	1	2	yes	maximal
10	C₄×Dic₂₅₁⋊C₄	C₂	1	1	yes	maximal, characteristic
11	#389×C₂₅₁	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₅₀₂	C₂²	1	1	yes	characteristic
13	C₄×Dic₅₀₂	—	2	4	no	—
14	#22548680	—	2	4	no	—
15	#22548680	C₄	1	8	yes	complemented
16	C₂×C₄⋊Dic₂₅₁	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
18	C₂×Dic₂₅₁⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×Dic₂₅₁⋊C₄	C₂²	1	1	yes	characteristic
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×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₂₅₁	C₂²	1	2	yes	—
25	C₂×C₄×Dic₂₅₁	C₂²	1	2	yes	—
26	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×Dic₂₅₁	C₂²	1	2	yes	—
28	C₂₅₁⋊#95	C₂²	1	1	yes	characteristic
29	C₂₅₁⋊#95	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	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	2	yes	—
34	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
36	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
37	#95×C₂₅₁	C₂²	1	2	yes	—
38	C₂×Dic₅₀₂	C₂×C₄	1	4	yes	—
39	C₂×Dic₅₀₂	D₄	1	4	yes	—
40	C₂×Dic₅₀₂	Q₈	1	4	yes	—
41	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
42	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
43	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
44	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
45	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
46	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
47	C₄⋊Dic₂₅₁	—	2	4	no	—
48	C₄⋊Dic₂₅₁	—	2	2	no	—
49	Dic₂₅₁⋊C₄	—	2	4	no	—
50	Dic₂₅₁⋊C₄	—	2	4	no	—
51	Dic₂₅₁⋊C₄	—	2	2	no	—
52	Dic₂₅₁⋊C₄	—	2	2	no	—
53	Dic₂₅₁⋊C₄	C₂×C₄	1	8	yes	—
54	C₄×Dic₂₅₁	—	2	4	no	—
55	C₄×Dic₂₅₁	—	2	4	no	—
56	C₄×Dic₂₅₁	—	2	2	no	—
57	C₄×Dic₂₅₁	C₂×C₄	1	8	yes	—
58	C₄×Dic₂₅₁	C₂×C₄	1	4	yes	—
59	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
61	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
62	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₂₅₁	—	2	2	no	—
65	C₄⋊C₄×C₂₅₁	—	2	4	no	—
66	C₄⋊C₄×C₂₅₁	C₂×C₄	1	4	yes	—
67	C₄²×C₂₅₁	—	2	2	no	—
68	Dic₅₀₂	C₄⋊C₄	1	16	yes	complemented
69	C₂×Dic₂₅₁	—	2	1	no	—
70	C₂×Dic₂₅₁	—	2	1	no	—
71	C₂×Dic₂₅₁	—	2	1	no	—
72	C₂×Dic₂₅₁	—	2	1	no	—
73	C₂×Dic₂₅₁	—	2	1	no	—
74	C₂×Dic₂₅₁	—	2	1	no	—
75	C₂×Dic₂₅₁	—	2	1	no	—
76	C₂×Dic₂₅₁	—	2	2	no	—
77	C₂×Dic₂₅₁	—	2	1	no	—
78	C₂×Dic₂₅₁	—	2	2	no	—
79	C₂×Dic₂₅₁	—	2	1	no	—
80	C₂×Dic₂₅₁	—	2	2	no	—
81	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
82	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
83	C₂×Dic₂₅₁	C₂²×C₄	1	4	yes	—
84	C₂×Dic₂₅₁	C₂×D₄	1	4	yes	—
85	C₂×Dic₂₅₁	C₂×Q₈	1	4	yes	—
86	C₂×Dic₂₅₁	#41	1	2	yes	—
87	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
88	C₂×C₄×C₂₅₁	—	2	1	no	—
89	C₂×C₄×C₂₅₁	—	2	2	no	—
90	C₂×C₄×C₂₅₁	—	2	2	no	—
91	C₂×C₄×C₂₅₁	—	2	1	no	—
92	C₂×C₄×C₂₅₁	—	2	1	no	—
93	C₂×C₄×C₂₅₁	—	2	2	no	—
94	C₂×C₄×C₂₅₁	—	2	2	no	—
95	C₂×C₄×C₂₅₁	—	2	1	no	—
96	C₂×C₄×C₂₅₁	—	2	2	no	—
97	C₂×C₄×C₂₅₁	—	2	1	no	—
98	C₂×C₄×C₂₅₁	C₂²×C₄	1	4	yes	—
99	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
100	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
101	C₂×C₄×C₂₅₁	C₂×Q₈	1	2	yes	—
102	C₂×C₄×C₂₅₁	#41	1	2	yes	—
103	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
104	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
105	C₂²×C₂₅₁	C₂²×Q₈	1	1	yes	characteristic
106	C₂²×C₂₅₁	C₂×#41	1	1	yes	characteristic
107	C₂²×C₂₅₁	C₂×#41	1	1	yes	characteristic
108	C₂²×C₂₅₁	2₋¹⁺⁴	1	1	yes	characteristic
109	C₂²×C₂₅₁	2₋¹⁺⁴	1	1	yes	characteristic
110	C₄×C₂₅₁	—	2	4	no	—
111	C₄×C₂₅₁	—	2	2	no	—
112	C₄×C₂₅₁	—	4	2	no	—
113	C₄×C₂₅₁	—	4	1	no	—
114	C₄×C₂₅₁	C₂×C₄⋊C₄	1	4	yes	—
115	Dic₂₅₁	—	2	2	no	—
116	Dic₂₅₁	—	2	2	no	—
117	Dic₂₅₁	—	2	2	no	—
118	Dic₂₅₁	—	4	1	no	—
119	Dic₂₅₁	—	4	2	no	—
120	Dic₂₅₁	C₂×C₄⋊C₄	1	8	yes	—
121	C₂×C₂₅₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₂₅₁	C₄×#41	1	1	yes	characteristic
123	C₂×C₂₅₁	#519	1	1	yes	characteristic
124	C₂×C₂₅₁	#548	1	1	yes	characteristic
125	C₂×C₂₅₁	#548	1	1	yes	characteristic
126	C₂×C₂₅₁	#557	1	1	yes	characteristic
127	C₂×C₂₅₁	#557	1	1	yes	characteristic
128	C₂₅₁	#2351	1	1	yes	characteristic, complemented, Sylow
129	#2351	—	251	1	no	maximal, Sylow
130	C₂×C₄×Q₈	—	251	1	no	—
131	C₂×C₄×Q₈	—	251	1	no	—
132	C₂×C₄×Q₈	—	251	1	no	—
133	#389	—	251	2	no	—
134	#389	—	251	1	no	—
135	#389	—	251	2	no	—
136	#389	D₂₅₁	1	1	yes	characteristic
137	#382	—	251	1	no	—
138	#382	—	251	2	no	—
139	C₄×C₄⋊C₄	—	251	2	no	—
140	C₄×C₄⋊C₄	—	251	1	no	—
141	C₂²×Q₈	—	251	1	no	—
142	C₄×Q₈	—	502	4	no	—
143	C₄×Q₈	—	502	4	no	—
144	C₄×Q₈	—	251	8	no	—
145	C₂×C₄⋊C₄	—	251	2	no	—
146	C₂×C₄⋊C₄	—	251	2	no	—
147	C₂×C₄⋊C₄	—	251	1	no	—
148	C₂×C₄⋊C₄	—	251	1	no	—
149	C₂×C₄⋊C₄	—	251	1	no	—
150	C₂×C₄⋊C₄	—	251	2	no	—
151	C₂×C₄⋊C₄	—	251	2	no	—
152	C₂×C₄⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
153	C₂×C₄⋊C₄	C₂×D₂₅₁	1	2	yes	—
154	C₂×C₄⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
155	C₂×C₄²	—	251	2	no	—
156	C₂×C₄²	—	251	2	no	—
157	C₂×C₄²	—	251	1	no	—
158	C₂×C₄²	—	251	1	no	—
159	C₂×C₄²	—	251	2	no	—
160	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
161	#95	—	251	1	no	—
162	#95	—	251	2	no	—
163	#95	—	251	1	no	—
164	#95	—	251	2	no	—
165	#95	—	251	2	no	—
166	#95	C₂×D₂₅₁	1	2	yes	—
167	C₂×Q₈	—	251	4	no	—
168	C₂×Q₈	—	251	4	no	—
169	C₂×Q₈	—	251	4	no	—
170	C₂²×C₄	—	251	2	no	—
171	C₂²×C₄	—	251	1	no	—
172	C₂²×C₄	—	251	1	no	—
173	C₂²×C₄	—	251	1	no	—
174	C₂²×C₄	—	251	1	no	—
175	C₂²×C₄	—	251	2	no	—
176	C₂²×C₄	C₂²×D₂₅₁	1	2	yes	—
177	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
178	C₂²×C₄	C₂²×D₂₅₁	1	2	yes	—
179	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
180	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	2	4	no	—
183	C₄⋊C₄	—	502	2	no	—
184	C₄⋊C₄	—	502	4	no	—
185	C₄⋊C₄	—	502	4	no	—
186	C₄⋊C₄	—	502	4	no	—
187	C₄⋊C₄	—	502	2	no	—
188	C₄⋊C₄	—	502	2	no	—
189	C₄⋊C₄	—	251	8	no	—
190	C₄⋊C₄	C₄×D₂₅₁	1	4	yes	—
191	C₄²	—	2	2	no	—
192	C₄²	—	502	4	no	—
193	C₄²	—	502	4	no	—
194	C₄²	—	502	2	no	—
195	C₄²	—	251	4	no	—
196	C₄²	—	251	8	no	—
197	C₂³	C₂³×D₂₅₁	1	1	yes	characteristic
198	Q₈	—	251	16	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	502	1	no	—
210	C₂×C₄	—	502	2	no	—
211	C₂×C₄	—	502	2	no	—
212	C₂×C₄	—	502	1	no	—
213	C₂×C₄	—	502	1	no	—
214	C₂×C₄	—	502	2	no	—
215	C₂×C₄	—	502	1	no	—
216	C₂×C₄	—	502	1	no	—
217	C₂×C₄	—	502	1	no	—
218	C₂×C₄	—	502	1	no	—
219	C₂×C₄	—	502	1	no	—
220	C₂×C₄	—	502	1	no	—
221	C₂×C₄	—	251	2	no	—
222	C₂×C₄	—	251	2	no	—
223	C₂×C₄	—	251	2	no	—
224	C₂×C₄	—	251	4	no	—
225	C₂×C₄	—	251	4	no	—
226	C₂×C₄	—	251	4	no	—
227	C₂×C₄	C₂×C₄×D₂₅₁	1	4	yes	—
228	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
229	C₂×C₄	D₄×D₂₅₁	1	2	yes	—
230	C₂×C₄	Q₈×D₂₅₁	1	2	yes	—
231	C₂×C₄	#16860809	1	2	yes	—
232	C₂²	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
233	C₂²	C₂×D₄×D₂₅₁	1	1	yes	characteristic
234	C₂²	C₂×Q₈×D₂₅₁	1	1	yes	characteristic
235	C₂²	#41×D₂₅₁	1	1	yes	characteristic
236	C₂²	C₂×#16860809	1	1	yes	characteristic
237	C₂²	#22548762	1	1	yes	characteristic
238	C₂²	#22548763	1	1	yes	characteristic
239	C₄	—	2	2	no	—
240	C₄	—	2	4	no	—
241	C₄	—	4	2	no	—
242	C₄	—	4	1	no	—
243	C₄	—	1004	1	no	—
244	C₄	—	1004	2	no	—
245	C₄	—	502	2	no	—
246	C₄	—	502	2	no	—
247	C₄	—	502	2	no	—
248	C₄	—	251	8	no	—
249	C₄	C₄⋊C₄×D₂₅₁	1	4	yes	—
250	C₂	C₂×C₄⋊C₄×D₂₅₁	1	1	yes	characteristic
251	C₂	C₄×#16860809	1	1	yes	characteristic
252	C₂	#30632163	1	1	yes	characteristic
253	C₂	#30632343	1	1	yes	characteristic
254	C₂	#30632347	1	1	yes	characteristic
255	C₂	#30632412	1	1	yes	characteristic
256	C₂	#30632414	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₅₁
commutator	C₂²×C₂₅₁	C₂³×C₄
Fitting	#389×C₂₅₁	C₂
socle, prime radical, cosocle	C₂³×C₂₅₁	C₂⁴
hypercenter	#389	D₂₅₁
hypocenter	C₂₅₁	#2351
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ #389

Complements

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

N	H ≅ G/N	type
15. #22548680	C₄	semidirect
68. Dic₅₀₂	C₄⋊C₄	semidirect
128. C₂₅₁	#2351	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₅₀₂	2
C₂×#22548680	2
#30631586	2
#30631584	2
#30631582	2
#30631552	2
#30631551	2
C₄⋊C₄×Dic₂₅₁	2
C₄×Dic₂₅₁⋊C₄	2
#389×C₂₅₁	2
#2351	251