Subgroups — #44564445

Order 32128.

1104 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	#30632423	C₂	1	3	yes	maximal, complemented
2	Q₈×Dic₅₀₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#30632394	C₂	1	3	yes	maximal, complemented
4	#30632344	C₂	1	2	yes	maximal, complemented
5	#30632337	C₂	1	6	yes	maximal, complemented
6	#30632304	C₂	1	6	yes	maximal, complemented
7	#30632235	C₂	1	3	yes	maximal, complemented
8	#30632219	C₂	1	3	yes	maximal, complemented
9	#30632170	C₂	1	2	yes	maximal, complemented
10	C₄×Q₈×D₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
11	#550×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#16860809	C₂²	1	6	yes	—
13	C₂×Q₈×D₂₅₁	C₂²	1	1	yes	characteristic
14	#22548715	C₂²	1	3	yes	—
15	#22548714	C₂²	1	6	yes	—
16	#22548708	C₂²	1	6	yes	—
17	#22548706	C₂²	1	3	yes	—
18	#22548702	C₂²	1	3	yes	—
19	#22548702	C₂²	1	6	yes	—
20	#22548702	C₂²	1	6	yes	—
21	#22548697	C₂²	1	6	yes	—
22	#22548696	C₂²	1	12	yes	—
23	#22548695	C₂²	1	1	yes	characteristic
24	#22548694	C₂²	1	6	yes	—
25	#22548692	C₂²	1	12	yes	—
26	#22548686	C₂²	1	3	yes	—
27	#22548686	C₂²	1	6	yes	—
28	Q₈×Dic₂₅₁	C₂²	1	2	yes	—
29	Q₈×Dic₂₅₁	C₂²	1	1	yes	characteristic
30	C₄×Dic₅₀₂	C₂²	1	3	yes	—
31	C₄×Dic₅₀₂	C₂²	1	3	yes	—
32	#22548680	C₂²	1	3	yes	—
33	D₄×Dic₂₅₁	C₂²	1	6	yes	—
34	D₄×Dic₂₅₁	C₂²	1	6	yes	—
35	D₄×Dic₂₅₁	C₂²	1	6	yes	—
36	C₄×C₂²⋊D₂₅₁	C₂²	1	6	yes	—
37	#22548674	C₂²	1	6	yes	—
38	C₂×C₄⋊Dic₂₅₁	C₂²	1	6	yes	—
39	C₄⋊C₄×D₂₅₁	C₂²	1	3	yes	—
40	C₄²×D₂₅₁	C₂²	1	3	yes	—
41	C₂×#41×C₂₅₁	C₂²	1	2	yes	—
42	#127×C₂₅₁	C₂²	1	3	yes	—
43	#121×C₂₅₁	C₂²	1	6	yes	—
44	C₄×Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
45	C₄×D₄×C₂₅₁	C₂²	1	3	yes	—
46	#16860809	—	4	12	no	—
47	C₂×Dic₅₀₂	C₂³	1	3	yes	—
48	C₂×Dic₅₀₂	C₂³	1	1	yes	characteristic
49	C₂×Dic₅₀₂	C₂³	1	6	yes	—
50	Q₈×D₂₅₁	—	2	4	no	—
51	C₂×C₂²⋊D₂₅₁	C₂³	1	6	yes	—
52	C₂²×Dic₂₅₁	C₂³	1	12	yes	—
53	C₂×C₄×D₂₅₁	C₂³	1	3	yes	—
54	C₂×C₄×D₂₅₁	C₂³	1	3	yes	—
55	C₂×C₄×D₂₅₁	C₂³	1	1	yes	characteristic
56	C₄⋊Dic₂₅₁	C₂³	1	3	yes	—
57	C₄⋊Dic₂₅₁	C₂³	1	6	yes	—
58	C₄⋊Dic₂₅₁	C₂³	1	6	yes	—
59	C₄⋊Dic₂₅₁	C₂³	1	2	yes	—
60	C₄⋊Dic₂₅₁	C₂³	1	3	yes	—
61	Dic₂₅₁⋊C₄	C₂³	1	6	yes	—
62	Dic₂₅₁⋊C₄	C₂³	1	6	yes	—
63	Dic₂₅₁⋊C₄	C₂³	1	6	yes	—
64	C₂²⋊Dic₂₅₁	C₂³	1	6	yes	—
65	C₂²⋊Dic₂₅₁	C₂³	1	6	yes	—
66	C₂²⋊Dic₂₅₁	C₂³	1	6	yes	—
67	C₂²⋊Dic₂₅₁	C₂³	1	12	yes	—
68	(C₂×D₂₅₁)⋊C₄	C₂³	1	2	yes	—
69	C₄×Dic₂₅₁	C₂³	1	3	yes	—
70	C₄×Dic₂₅₁	C₂³	1	3	yes	—
71	C₄×Dic₂₅₁	C₂³	1	3	yes	—
72	C₄×Dic₂₅₁	C₂³	1	6	yes	—
73	#41×C₂₅₁	—	2	8	no	—
74	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
75	C₂×D₄×C₂₅₁	C₂³	1	3	yes	—
76	C₂×D₄×C₂₅₁	C₂³	1	6	yes	—
77	C₂×D₄×C₂₅₁	C₂³	1	6	yes	—
78	C₂²×C₄×C₂₅₁	C₂³	1	6	yes	—
79	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₂₅₁	C₂³	1	3	yes	—
81	C₂²⋊C₄×C₂₅₁	C₂³	1	6	yes	—
82	C₄²×C₂₅₁	C₂³	1	3	yes	—
83	C₂²×D₂₅₁	C₂⁴	1	1	yes	characteristic
84	Dic₅₀₂	—	2	6	no	—
85	Dic₅₀₂	—	2	2	no	—
86	Dic₅₀₂	—	4	6	no	—
87	C₂²⋊D₂₅₁	—	2	12	no	—
88	C₂×Dic₂₅₁	—	4	12	no	—
89	C₂×Dic₂₅₁	C₂⁴	1	1	yes	characteristic
90	C₂×Dic₂₅₁	C₂⁴	1	2	yes	—
91	C₂×Dic₂₅₁	C₂⁴	1	3	yes	—
92	C₂×Dic₂₅₁	C₂⁴	1	3	yes	—
93	C₂×Dic₂₅₁	C₂⁴	1	6	yes	—
94	C₄×D₂₅₁	—	2	6	no	—
95	C₄×D₂₅₁	—	2	6	no	—
96	C₄×D₂₅₁	—	2	2	no	—
97	C₂³×C₂₅₁	C₂⁴	1	6	yes	—
98	Q₈×C₂₅₁	#41	1	4	yes	complemented
99	D₄×C₂₅₁	—	2	12	no	—
100	D₄×C₂₅₁	—	2	6	no	—
101	D₄×C₂₅₁	—	4	6	no	—
102	C₂×C₄×C₂₅₁	—	2	12	no	—
103	C₂×C₄×C₂₅₁	C₂⁴	1	3	yes	—
104	C₂×C₄×C₂₅₁	C₂⁴	1	2	yes	—
105	C₂×C₄×C₂₅₁	C₂⁴	1	3	yes	—
106	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
107	C₂²×C₂₅₁	—	2	6	no	—
108	C₂²×C₂₅₁	—	2	6	no	—
109	C₂²×C₂₅₁	—	2	6	no	—
110	C₂²×C₂₅₁	C₂⁵	1	1	yes	characteristic
111	C₂×D₂₅₁	—	2	1	no	—
112	C₂×D₂₅₁	—	2	1	no	—
113	C₂×D₂₅₁	H(5,𝔽₂)	1	2	yes	—
114	C₄×C₂₅₁	—	2	2	no	—
115	C₄×C₂₅₁	—	2	3	no	—
116	C₄×C₂₅₁	C₂×#41	1	6	yes	—
117	C₄×C₂₅₁	2₋¹⁺⁴	1	2	yes	—
118	Dic₂₅₁	—	2	3	no	—
119	Dic₂₅₁	—	2	3	no	—
120	Dic₂₅₁	—	2	6	no	—
121	Dic₂₅₁	—	2	1	no	—
122	Dic₂₅₁	—	2	2	no	—
123	C₂×C₂₅₁	—	4	6	no	—
124	C₂×C₂₅₁	C₂²×#41	1	1	yes	characteristic
125	C₂×C₂₅₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
126	C₂×C₂₅₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
127	D₂₅₁	—	2	2	no	—
128	C₂₅₁	#3522	1	1	yes	characteristic, complemented, Sylow
129	#3522	—	251	1	no	maximal, Sylow
130	#559	—	251	3	no	—
131	Q₈²	—	251	1	no	—
132	#554	—	251	3	no	—
133	#550	D₂₅₁	1	1	yes	characteristic, complemented
134	#548	—	251	2	no	—
135	#547	—	251	6	no	—
136	#542	—	251	6	no	—
137	#532	—	251	3	no	—
138	#529	—	251	3	no	—
139	#520	—	251	2	no	—
140	C₂×C₄×Q₈	—	251	1	no	—
141	C₂×#41	—	251	6	no	—
142	C₂×#41	C₂×D₂₅₁	1	2	yes	—
143	C₂²×Q₈	—	251	1	no	—
144	C₄⋊Q₈	—	251	6	no	—
145	C₄⋊Q₈	—	251	3	no	—
146	#127	C₂×D₂₅₁	1	3	yes	—
147	#126	—	251	6	no	—
148	#125	—	251	3	no	—
149	#124	—	251	3	no	—
150	#124	—	251	6	no	—
151	#124	—	251	6	no	—
152	#123	—	251	12	no	—
153	#123	—	251	6	no	—
154	C₂²⋊Q₈	—	251	6	no	—
155	C₂²⋊Q₈	—	251	1	no	—
156	C₂²⋊Q₈	—	251	12	no	—
157	#121	—	251	3	no	—
158	#121	—	251	6	no	—
159	#121	C₂×D₂₅₁	1	6	yes	—
160	C₄×Q₈	—	251	1	no	—
161	C₄×Q₈	—	251	2	no	—
162	C₄×Q₈	—	251	3	no	—
163	C₄×Q₈	—	251	3	no	—
164	C₄×Q₈	—	251	3	no	—
165	C₄×Q₈	C₂×D₂₅₁	1	1	yes	characteristic
166	C₄×D₄	—	251	6	no	—
167	C₄×D₄	—	251	6	no	—
168	C₄×D₄	—	251	6	no	—
169	C₄×D₄	—	251	6	no	—
170	C₄×D₄	C₂×D₂₅₁	1	3	yes	—
171	#117	—	251	6	no	—
172	C₂×C₄⋊C₄	—	251	6	no	—
173	C₂×C₄⋊C₄	—	251	3	no	—
174	C₂×C₄²	—	251	3	no	—
175	#41	—	2	8	no	—
176	#41	—	1004	12	no	—
177	C₂×Q₈	—	502	4	no	—
178	C₂×Q₈	—	251	6	no	—
179	C₂×Q₈	—	251	1	no	—
180	C₂×Q₈	—	251	3	no	—
181	C₂×Q₈	C₂²×D₂₅₁	1	1	yes	characteristic
182	C₂×D₄	—	251	6	no	—
183	C₂×D₄	C₂²×D₂₅₁	1	3	yes	—
184	C₂×D₄	C₂²×D₂₅₁	1	6	yes	—
185	C₂×D₄	C₂²×D₂₅₁	1	6	yes	—
186	C₂²×C₄	—	251	12	no	—
187	C₂²×C₄	—	251	1	no	—
188	C₂²×C₄	—	251	3	no	—
189	C₂²×C₄	—	251	3	no	—
190	C₂²×C₄	C₂²×D₂₅₁	1	6	yes	—
191	C₄⋊C₄	—	251	3	no	—
192	C₄⋊C₄	—	251	6	no	—
193	C₄⋊C₄	—	251	6	no	—
194	C₄⋊C₄	—	251	6	no	—
195	C₄⋊C₄	—	251	2	no	—
196	C₄⋊C₄	—	251	6	no	—
197	C₄⋊C₄	—	251	6	no	—
198	C₄⋊C₄	—	251	3	no	—
199	C₄⋊C₄	C₂²×D₂₅₁	1	3	yes	—
200	C₄⋊C₄	C₂²×D₂₅₁	1	1	yes	characteristic
201	C₂²⋊C₄	—	251	6	no	—
202	C₂²⋊C₄	—	251	6	no	—
203	C₂²⋊C₄	—	251	12	no	—
204	C₂²⋊C₄	—	251	6	no	—
205	C₂²⋊C₄	—	251	2	no	—
206	C₂²⋊C₄	C₂²×D₂₅₁	1	6	yes	—
207	C₄²	—	251	3	no	—
208	C₄²	—	251	3	no	—
209	C₄²	—	251	3	no	—
210	C₄²	—	251	6	no	—
211	C₄²	C₂²×D₂₅₁	1	3	yes	—
212	C₂³	—	251	1	no	—
213	C₂³	C₂³×D₂₅₁	1	6	yes	—
214	Q₈	—	1004	6	no	—
215	Q₈	—	502	6	no	—
216	Q₈	—	502	2	no	—
217	Q₈	#16860809	1	4	yes	complemented
218	D₄	—	2	12	no	—
219	D₄	—	2	6	no	—
220	D₄	—	4	6	no	—
221	D₄	—	502	12	no	—
222	C₂×C₄	—	2	12	no	—
223	C₂×C₄	—	1004	12	no	—
224	C₂×C₄	—	502	6	no	—
225	C₂×C₄	—	502	6	no	—
226	C₂×C₄	—	502	2	no	—
227	C₂×C₄	—	251	1	no	—
228	C₂×C₄	—	251	2	no	—
229	C₂×C₄	—	251	3	no	—
230	C₂×C₄	—	251	3	no	—
231	C₂×C₄	—	251	6	no	—
232	C₂×C₄	C₂³×D₂₅₁	1	3	yes	—
233	C₂×C₄	C₂³×D₂₅₁	1	2	yes	—
234	C₂×C₄	C₂³×D₂₅₁	1	3	yes	—
235	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
236	C₂²	—	2	6	no	—
237	C₂²	—	2	6	no	—
238	C₂²	—	2	6	no	—
239	C₂²	—	502	1	no	—
240	C₂²	—	502	1	no	—
241	C₂²	—	251	2	no	—
242	C₂²	C₂⁴×D₂₅₁	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	2	3	no	—
245	C₄	—	502	3	no	—
246	C₄	—	502	3	no	—
247	C₄	—	502	6	no	—
248	C₄	—	502	1	no	—
249	C₄	—	502	2	no	—
250	C₄	C₂×#16860809	1	6	yes	—
251	C₄	#22548762	1	2	yes	—
252	C₂	—	4	6	no	—
253	C₂	—	502	2	no	—
254	C₂	C₂²×#16860809	1	1	yes	characteristic
255	C₂	H(5,𝔽₂)×D₂₅₁	1	1	yes	characteristic
256	C₂	C₂×#22548762	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₂₅₁
commutator, socle, cosocle	C₂²×C₂₅₁	C₂⁵
Fitting	#550×C₂₅₁	C₂
hypercenter	#550	D₂₅₁
hypocenter	C₂₅₁	#3522
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ #550

Complements

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

N	H ≅ G/N	type
1. #30632423	C₂	semidirect
2. Q₈×Dic₅₀₂	C₂	semidirect
3. #30632394	C₂	semidirect
4. #30632344	C₂	semidirect
5. #30632337	C₂	semidirect
6. #30632304	C₂	semidirect
7. #30632235	C₂	semidirect
8. #30632219	C₂	semidirect
9. #30632170	C₂	semidirect
10. C₄×Q₈×D₂₅₁	C₂	semidirect
11. #550×C₂₅₁	C₂	semidirect
98. Q₈×C₂₅₁	#41	semidirect
128. C₂₅₁	#3522	semidirect
133. #550	D₂₅₁	semidirect
217. Q₈	#16860809	semidirect

Maximal subgroups

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

subgroup	index
#30632423	2
Q₈×Dic₅₀₂	2
#30632394	2
#30632344	2
#30632337	2
#30632304	2
#30632235	2
#30632219	2
#30632170	2
C₄×Q₈×D₂₅₁	2
#550×C₂₅₁	2
#3522	251