Subgroups — C₄×#22548734 #44559132

Order 32128.

500 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22548734	C₂	1	1	yes	maximal, characteristic
2	C₄×#16860809	C₂	1	2	yes	maximal, complemented
3	C₄×Dic₁₀₀₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#30631781	C₂	1	1	yes	maximal, characteristic, complemented
5	#30631778	C₂	1	2	yes	maximal, complemented
6	C₄×#16860798	C₂	1	2	yes	maximal, complemented
7	D₈×Dic₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#30631724	C₂	1	1	yes	maximal, characteristic, complemented
9	#30631680	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₈×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#16860809	C₂²	1	2	yes	—
13	#22548734	C₄	1	8	yes	direct product factor
14	C₂×Dic₁₀₀₄	C₂²	1	1	yes	characteristic
15	C₂×#16860798	C₂²	1	2	yes	—
16	C₂×C₈×D₂₅₁	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₂₅₁	C₂²	1	1	yes	characteristic
18	C₄×Dic₅₀₂	C₂²	1	2	yes	—
19	#22548680	C₂²	1	2	yes	—
20	D₄×Dic₂₅₁	C₂²	1	2	yes	—
21	C₄×C₂²⋊D₂₅₁	—	2	2	no	—
22	(C₂²⋊D₂₅₁)⋊C₄	—	2	2	no	—
23	#22548673	C₂²	1	2	yes	—
24	#22548671	—	2	2	no	—
25	C₂×C₄×Dic₂₅₁	—	2	2	no	—
26	C₄²×D₂₅₁	C₂²	1	1	yes	characteristic
27	#22548649	C₂²	1	1	yes	characteristic
28	#22548646	C₂²	1	2	yes	—
29	#22548640	C₂²	1	2	yes	—
30	#22548638	C₂²	1	2	yes	—
31	#22548636	C₂²	1	2	yes	—
32	C₄×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
33	C₈×Dic₂₅₁	C₂²	1	1	yes	characteristic
34	C₈×Dic₂₅₁	C₂²	1	1	yes	characteristic
35	C₂×D₈×C₂₅₁	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₂₅₁	C₂²	1	2	yes	—
37	#107×C₂₅₁	C₂²	1	1	yes	characteristic
38	#102×C₂₅₁	C₂²	1	2	yes	—
39	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
40	#16860809	—	2	4	no	—
41	#16860809	C₂×C₄	1	8	yes	complemented
42	C₂×Dic₅₀₂	C₂³	1	2	yes	—
43	C₂×C₂²⋊D₂₅₁	—	2	2	no	—
44	C₂²×Dic₂₅₁	—	2	2	no	—
45	C₂²×Dic₂₅₁	—	2	2	no	—
46	C₂×C₄×D₂₅₁	D₄	1	1	yes	characteristic
47	C₂×C₄×D₂₅₁	D₄	1	1	yes	characteristic
48	C₂×C₄×D₂₅₁	C₂³	1	1	yes	characteristic
49	Dic₁₀₀₄	C₂×C₄	1	4	yes	complemented
50	#16860798	C₂×C₄	1	8	yes	complemented
51	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
52	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
53	C₈×D₂₅₁	—	2	2	no	—
54	C₈×D₂₅₁	C₂×C₄	1	4	yes	complemented
55	C₄⋊Dic₂₅₁	C₂³	1	2	yes	—
56	Dic₂₅₁⋊C₄	—	2	2	no	—
57	Dic₂₅₁⋊C₄	—	2	2	no	—
58	C₂²⋊Dic₂₅₁	—	2	2	no	—
59	(C₂×D₂₅₁)⋊C₄	—	2	2	no	—
60	C₄×Dic₂₅₁	—	2	2	no	—
61	C₄×Dic₂₅₁	—	2	2	no	—
62	C₄×Dic₂₅₁	—	2	2	no	—
63	C₄×Dic₂₅₁	D₄	1	1	yes	characteristic
64	C₄×Dic₂₅₁	D₄	1	1	yes	characteristic
65	C₄×Dic₂₅₁	C₂³	1	1	yes	characteristic
66	C₂×D₄×C₂₅₁	C₂³	1	2	yes	—
67	C₂²×C₄×C₂₅₁	—	2	2	no	—
68	D₈×C₂₅₁	C₂×C₄	1	4	yes	complemented
69	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₂₅₁	C₂³	1	2	yes	—
72	C₂²⋊C₄×C₂₅₁	—	2	2	no	—
73	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
74	C₂²×D₂₅₁	C₂×D₄	1	1	yes	characteristic
75	Dic₅₀₂	—	2	2	no	—
76	Dic₅₀₂	C₂²×C₄	1	4	yes	—
77	C₂²⋊D₂₅₁	—	2	8	no	—
78	C₂×Dic₂₅₁	—	2	2	no	—
79	C₂×Dic₂₅₁	—	2	8	no	—
80	C₂×Dic₂₅₁	—	2	2	no	—
81	C₂×Dic₂₅₁	—	4	4	no	—
82	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
83	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
84	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
85	C₄×D₂₅₁	—	2	2	no	—
86	C₄×D₂₅₁	—	2	2	no	—
87	C₄×D₂₅₁	C₂²×C₄	1	2	yes	—
88	C₄×D₂₅₁	#41	1	2	yes	—
89	C₂₅₁⋊C₈	—	2	1	no	—
90	C₂₅₁⋊C₈	C₂²×C₄	1	2	yes	—
91	C₂³×C₂₅₁	—	2	2	no	—
92	D₄×C₂₅₁	—	2	2	no	—
93	D₄×C₂₅₁	C₂²×C₄	1	4	yes	—
94	C₂×C₄×C₂₅₁	—	2	2	no	—
95	C₂×C₄×C₂₅₁	—	4	4	no	—
96	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
99	C₈×C₂₅₁	—	2	1	no	—
100	C₈×C₂₅₁	C₂²×C₄	1	2	yes	—
101	C₂²×C₂₅₁	—	2	4	no	—
102	C₂²×C₂₅₁	—	4	2	no	—
103	C₂²×C₂₅₁	—	4	2	no	—
104	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
105	C₂×D₂₅₁	—	2	1	no	—
106	C₂×D₂₅₁	—	2	1	no	—
107	C₂×D₂₅₁	C₄×D₄	1	2	yes	—
108	C₄×C₂₅₁	—	2	1	no	—
109	C₄×C₂₅₁	—	4	2	no	—
110	C₄×C₂₅₁	#135	1	2	yes	complemented
111	C₄×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
112	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
113	Dic₂₅₁	—	2	4	no	—
114	Dic₂₅₁	—	2	1	no	—
115	Dic₂₅₁	—	4	2	no	—
116	Dic₂₅₁	C₄×D₄	1	2	yes	—
117	Dic₂₅₁	#135	1	2	yes	complemented
118	C₂×C₂₅₁	—	4	4	no	—
119	C₂×C₂₅₁	C₂×C₄×D₄	1	1	yes	characteristic
120	C₂×C₂₅₁	C₂×#135	1	1	yes	characteristic
121	C₂×C₂₅₁	C₂×#135	1	1	yes	characteristic
122	D₂₅₁	—	2	2	no	—
123	C₂₅₁	C₄×#135	1	1	yes	characteristic, complemented, Sylow
124	C₄×#135	—	251	1	no	maximal, Sylow
125	C₂×#135	—	251	1	no	—
126	C₄×#41	—	251	2	no	—
127	C₄×Q₁₆	—	251	1	no	—
128	C₄×Q₁₆	—	251	1	no	—
129	C₄×SD₁₆	—	251	2	no	—
130	C₄×SD₁₆	—	251	2	no	—
131	C₄×D₈	—	251	1	no	—
132	C₄×D₈	D₂₅₁	1	1	yes	characteristic, complemented
133	#427	—	251	1	no	—
134	#416	—	251	2	no	—
135	C₂×C₄×C₈	—	251	1	no	—
136	C₂×#41	—	251	2	no	—
137	#135	—	251	8	no	—
138	C₂×Q₁₆	—	251	1	no	—
139	C₂×SD₁₆	—	251	2	no	—
140	C₂×D₈	C₂×D₂₅₁	1	1	yes	characteristic
141	C₂²×C₈	—	251	1	no	—
142	C₂²×C₈	—	251	1	no	—
143	C₄×Q₈	—	251	2	no	—
144	C₄×Q₈	—	251	2	no	—
145	C₄×D₄	—	502	2	no	—
146	C₄×D₄	—	502	2	no	—
147	C₄×D₄	—	251	2	no	—
148	C₄×D₄	C₂×D₂₅₁	1	2	yes	—
149	#117	—	502	2	no	—
150	#117	—	251	2	no	—
151	C₂×C₄²	—	502	2	no	—
152	C₂×C₄²	—	251	1	no	—
153	#107	—	251	1	no	—
154	#107	C₂×D₂₅₁	1	1	yes	characteristic
155	#106	—	251	2	no	—
156	#103	—	251	2	no	—
157	#103	—	251	2	no	—
158	#102	—	251	2	no	—
159	#102	C₂×D₂₅₁	1	2	yes	—
160	C₄×C₈	—	251	1	no	—
161	C₄×C₈	—	251	1	no	—
162	C₄×C₈	—	251	1	no	—
163	C₄×C₈	C₂×D₂₅₁	1	1	yes	characteristic
164	#41	—	502	4	no	—
165	#41	—	251	8	no	—
166	C₂×Q₈	—	251	2	no	—
167	C₂×D₄	—	502	2	no	—
168	C₂×D₄	C₂²×D₂₅₁	1	2	yes	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	502	2	no	—
171	C₂²×C₄	—	502	2	no	—
172	C₂²×C₄	—	251	1	no	—
173	C₂²×C₄	—	251	1	no	—
174	C₂²×C₄	—	251	1	no	—
175	Q₁₆	—	251	4	no	—
176	SD₁₆	—	251	8	no	—
177	D₈	C₄×D₂₅₁	1	4	yes	complemented
178	C₂×C₈	—	502	2	no	—
179	C₂×C₈	—	251	1	no	—
180	C₂×C₈	—	251	1	no	—
181	C₂×C₈	—	251	4	no	—
182	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
183	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
184	C₄⋊C₄	—	502	2	no	—
185	C₄⋊C₄	—	502	2	no	—
186	C₄⋊C₄	—	251	2	no	—
187	C₄⋊C₄	C₂²×D₂₅₁	1	2	yes	—
188	C₂²⋊C₄	—	2	2	no	—
189	C₂²⋊C₄	—	502	2	no	—
190	C₂²⋊C₄	—	502	2	no	—
191	C₄²	—	502	2	no	—
192	C₄²	—	502	2	no	—
193	C₄²	—	502	2	no	—
194	C₄²	—	251	1	no	—
195	C₄²	—	251	1	no	—
196	C₄²	—	251	1	no	—
197	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
198	C₂³	—	2	2	no	—
199	C₂³	—	251	1	no	—
200	Q₈	—	502	2	no	—
201	Q₈	—	251	4	no	—
202	D₄	—	2	2	no	—
203	D₄	—	502	8	no	—
204	D₄	C₂×C₄×D₂₅₁	1	4	yes	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	1004	4	no	—
208	C₂×C₄	—	502	2	no	—
209	C₂×C₄	—	502	2	no	—
210	C₂×C₄	—	502	2	no	—
211	C₂×C₄	—	502	8	no	—
212	C₂×C₄	—	502	2	no	—
213	C₂×C₄	—	251	1	no	—
214	C₂×C₄	—	251	1	no	—
215	C₂×C₄	—	251	1	no	—
216	C₂×C₄	—	251	2	no	—
217	C₂×C₄	—	251	2	no	—
218	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
219	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
220	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
221	C₈	—	2	1	no	—
222	C₈	—	502	1	no	—
223	C₈	—	251	2	no	—
224	C₈	C₂×C₄×D₂₅₁	1	2	yes	—
225	C₂²	—	2	4	no	—
226	C₂²	—	4	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	502	1	no	—
229	C₂²	—	502	1	no	—
230	C₂²	—	251	2	no	—
231	C₂²	C₂×D₄×D₂₅₁	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	4	2	no	—
234	C₄	—	1004	2	no	—
235	C₄	—	502	1	no	—
236	C₄	—	502	4	no	—
237	C₄	—	251	2	no	—
238	C₄	—	251	2	no	—
239	C₄	#22548734	1	2	yes	direct product factor
240	C₄	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
241	C₄	#41×D₂₅₁	1	1	yes	characteristic
242	C₂	—	4	4	no	—
243	C₂	—	502	2	no	—
244	C₂	C₄×D₄×D₂₅₁	1	1	yes	characteristic
245	C₂	#135×D₂₅₁	1	1	yes	characteristic
246	C₂	C₂×#22548734	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₂₅₁
commutator	C₄×C₂₅₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	C₄×D₈×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
prime radical	C₂×#22548734	C₂
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	C₄×D₈	D₂₅₁
hypocenter	C₂₅₁	C₄×#135
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
2. C₄×#16860809	C₂	semidirect
3. C₄×Dic₁₀₀₄	C₂	semidirect
4. #30631781	C₂	semidirect
5. #30631778	C₂	semidirect
6. C₄×#16860798	C₂	semidirect
7. D₈×Dic₂₅₁	C₂	semidirect
8. #30631724	C₂	semidirect
9. #30631680	C₂	semidirect
10. C₄×C₈×D₂₅₁	C₂	semidirect
11. C₄×D₈×C₂₅₁	C₂	semidirect
13. #22548734	C₄	direct (N × H)
41. #16860809	C₂×C₄	semidirect
49. Dic₁₀₀₄	C₂×C₄	semidirect
50. #16860798	C₂×C₄	semidirect
54. C₈×D₂₅₁	C₂×C₄	semidirect
68. D₈×C₂₅₁	C₂×C₄	semidirect
110. C₄×C₂₅₁	#135	semidirect
117. Dic₂₅₁	#135	semidirect
123. C₂₅₁	C₄×#135	semidirect
132. C₄×D₈	D₂₅₁	semidirect
177. D₈	C₄×D₂₅₁	semidirect
239. C₄	#22548734	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#22548734	2
C₄×#16860809	2
C₄×Dic₁₀₀₄	2
#30631781	2
#30631778	2
C₄×#16860798	2
D₈×Dic₂₅₁	2
#30631724	2
#30631680	2
C₄×C₈×D₂₅₁	2
C₄×D₈×C₂₅₁	2
C₄×#135	251

Subgroups — C4×#22548734 #44559132

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#22548734 #44559132