Subgroups — C₂×#30632501 #44565129

Order 32128.

1408 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22548762	C₂	1	2	yes	maximal, complemented
2	#30632501	C₂	1	16	yes	maximal, direct product factor
3	C₂×#22548745	C₂	1	2	yes	maximal, complemented
4	C₂×#22548744	C₂	1	4	yes	maximal, complemented
5	C₂×#22548734	C₂	1	2	yes	maximal, complemented
6	C₂²×Dic₁₀₀₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×Q₁₆×D₂₅₁	C₂	1	2	yes	maximal, complemented
8	C₂×#22548721	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
10	#22548762	—	2	8	no	—
11	#22548762	C₂²	1	16	yes	complemented
12	C₂×#16860810	—	2	2	no	—
13	C₂×#16860810	C₂²	1	1	yes	characteristic
14	C₂×#16860809	—	2	4	no	—
15	C₂×#16860809	C₂²	1	4	yes	—
16	C₂²×Dic₅₀₂	—	2	2	no	—
17	C₂²×Dic₅₀₂	C₂²	1	2	yes	—
18	C₂×Q₈×D₂₅₁	C₂²	1	4	yes	—
19	#22548745	C₂²	1	16	yes	complemented
20	#22548744	C₂²	1	32	yes	complemented
21	#22548734	C₂²	1	16	yes	complemented
22	C₂×Dic₁₀₀₄	C₂²	1	2	yes	—
23	C₂×Dic₁₀₀₄	C₂²	1	2	yes	—
24	C₂×Dic₁₀₀₄	C₂²	1	8	yes	complemented
25	C₂×#16860800	C₂²	1	4	yes	—
26	Q₁₆×D₂₅₁	C₂²	1	16	yes	complemented
27	C₂×#16860798	C₂²	1	4	yes	—
28	#22548721	C₂²	1	8	yes	complemented
29	C₂×#16860794	C₂²	1	1	yes	characteristic
30	C₂×#16860793	C₂²	1	2	yes	—
31	C₂×C₈×D₂₅₁	C₂²	1	2	yes	—
32	C₂×#41×C₂₅₁	C₂²	1	2	yes	—
33	#135×C₂₅₁	C₂²	1	8	yes	complemented
34	C₂×Q₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
35	C₂×SD₁₆×C₂₅₁	C₂²	1	2	yes	—
36	C₂×D₈×C₂₅₁	C₂²	1	1	yes	characteristic
37	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
38	#16860810	—	2	16	no	—
39	#16860810	D₄	1	4	yes	—
40	#16860810	C₂³	1	4	yes	—
41	#16860809	—	2	32	no	—
42	#16860809	—	2	8	no	—
43	#16860809	C₂³	1	16	yes	—
44	C₂×Dic₅₀₂	—	2	16	no	—
45	C₂×Dic₅₀₂	—	2	4	no	—
46	C₂×Dic₅₀₂	—	2	2	no	—
47	C₂×Dic₅₀₂	—	2	4	no	—
48	C₂×Dic₅₀₂	D₄	1	1	yes	characteristic
49	C₂×Dic₅₀₂	C₂³	1	8	yes	—
50	C₂×Dic₅₀₂	C₂³	1	4	yes	—
51	Q₈×D₂₅₁	—	2	8	no	—
52	Q₈×D₂₅₁	C₂³	1	16	yes	—
53	C₂×D₁₀₀₄	D₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₂₅₁	—	2	4	no	—
55	C₂×C₂²⋊D₂₅₁	D₄	1	2	yes	—
56	C₂²×Dic₂₅₁	—	2	4	no	—
57	C₂²×Dic₂₅₁	—	2	2	no	—
58	C₂×C₄×D₂₅₁	—	2	4	no	—
59	C₂×C₄×D₂₅₁	C₂³	1	2	yes	—
60	Dic₁₀₀₄	C₂³	1	8	yes	—
61	Dic₁₀₀₄	C₂³	1	8	yes	—
62	#16860800	C₂³	1	16	yes	—
63	#16860798	C₂³	1	16	yes	—
64	#16860794	C₂³	1	4	yes	—
65	#16860793	C₂³	1	8	yes	—
66	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
67	C₈×D₂₅₁	C₂³	1	8	yes	—
68	#41×C₂₅₁	—	2	4	no	—
69	#41×C₂₅₁	C₂³	1	8	yes	—
70	C₂×Q₈×C₂₅₁	C₂³	1	2	yes	—
71	C₂×D₄×C₂₅₁	—	2	2	no	—
72	C₂×D₄×C₂₅₁	C₂³	1	2	yes	—
73	C₂²×C₄×C₂₅₁	—	2	2	no	—
74	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
75	Q₁₆×C₂₅₁	C₂³	1	4	yes	—
76	SD₁₆×C₂₅₁	C₂³	1	8	yes	—
77	D₈×C₂₅₁	C₂³	1	4	yes	—
78	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
79	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
80	C₂×C₈×C₂₅₁	C₂³	1	4	yes	—
81	C₂²×D₂₅₁	C₂×D₄	1	2	yes	—
82	Dic₅₀₂	—	2	4	no	—
83	Dic₅₀₂	—	2	8	no	—
84	Dic₅₀₂	—	2	16	no	—
85	Dic₅₀₂	C₂×D₄	1	4	yes	—
86	Dic₅₀₂	C₂⁴	1	8	yes	—
87	D₁₀₀₄	C₂×D₄	1	4	yes	—
88	C₂²⋊D₂₅₁	—	2	16	no	—
89	C₂²⋊D₂₅₁	C₂×D₄	1	8	yes	—
90	C₂×Dic₂₅₁	—	2	4	no	—
91	C₂×Dic₂₅₁	—	2	8	no	—
92	C₂×Dic₂₅₁	—	2	16	no	—
93	C₂×Dic₂₅₁	C₂×D₄	1	2	yes	—
94	C₄×D₂₅₁	—	2	16	no	—
95	C₄×D₂₅₁	C₂×D₄	1	4	yes	—
96	C₄×D₂₅₁	C₂⁴	1	4	yes	—
97	C₂₅₁⋊C₈	C₂⁴	1	4	yes	—
98	C₂³×C₂₅₁	—	2	2	no	—
99	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
100	Q₈×C₂₅₁	—	2	2	no	—
101	Q₈×C₂₅₁	C₂⁴	1	4	yes	—
102	D₄×C₂₅₁	—	2	8	no	—
103	D₄×C₂₅₁	—	2	2	no	—
104	D₄×C₂₅₁	C₂⁴	1	4	yes	—
105	C₂×C₄×C₂₅₁	—	2	2	no	—
106	C₂×C₄×C₂₅₁	—	2	8	no	—
107	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
110	C₂×C₄×C₂₅₁	C₂⁴	1	2	yes	—
111	C₈×C₂₅₁	C₂⁴	1	2	yes	—
112	C₈×C₂₅₁	C₂⁴	1	2	yes	—
113	C₂²×C₂₅₁	—	2	2	no	—
114	C₂²×C₂₅₁	—	2	4	no	—
115	C₂²×C₂₅₁	—	4	4	no	—
116	C₂²×C₂₅₁	C₂²×D₄	1	2	yes	—
117	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
118	C₂×D₂₅₁	—	2	4	no	—
119	C₂×D₂₅₁	C₂²×D₄	1	4	yes	—
120	C₄×C₂₅₁	—	2	4	no	—
121	C₄×C₂₅₁	C₂²×D₄	1	2	yes	—
122	C₄×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
123	C₄×C₂₅₁	C₂⁵	1	1	yes	characteristic
124	Dic₂₅₁	—	2	8	no	—
125	Dic₂₅₁	C₂²×D₄	1	4	yes	—
126	C₂×C₂₅₁	—	2	2	no	—
127	C₂×C₂₅₁	—	4	4	no	—
128	C₂×C₂₅₁	#578	1	2	yes	complemented
129	C₂×C₂₅₁	C₂³×D₄	1	1	yes	characteristic
130	D₂₅₁	—	2	4	no	—
131	C₂₅₁	C₂×#578	1	1	yes	characteristic, complemented, Sylow
132	C₂×#578	—	251	1	no	maximal, Sylow
133	C₂×2₋¹⁺⁴	—	251	2	no	—
134	#578	—	251	16	no	—
135	C₂×#137	—	251	2	no	—
136	C₂×#137	—	251	4	no	—
137	C₂×#135	—	251	2	no	—
138	C₂×#135	D₂₅₁	1	1	yes	characteristic, complemented
139	C₂²×Q₁₆	—	251	2	no	—
140	C₂²×Q₁₆	—	251	1	no	—
141	C₂×#131	—	251	1	no	—
142	2₋¹⁺⁴	—	502	8	no	—
143	2₋¹⁺⁴	—	251	16	no	—
144	C₂×#41	—	502	2	no	—
145	C₂×#41	—	502	4	no	—
146	C₂×#41	—	251	4	no	—
147	C₂×#41	—	251	1	no	—
148	C₂×#41	C₂×D₂₅₁	1	2	yes	—
149	C₂²×Q₈	—	502	2	no	—
150	C₂²×Q₈	—	251	4	no	—
151	C₂²×Q₈	—	251	2	no	—
152	#137	—	251	16	no	—
153	#137	—	251	32	no	—
154	#135	—	251	16	no	—
155	#135	C₂×D₂₅₁	1	8	yes	complemented
156	C₂×Q₁₆	—	251	4	no	—
157	C₂×Q₁₆	—	251	16	no	—
158	C₂×Q₁₆	—	251	2	no	—
159	C₂×Q₁₆	—	251	2	no	—
160	C₂×Q₁₆	—	251	8	no	—
161	C₂×Q₁₆	C₂×D₂₅₁	1	1	yes	characteristic
162	C₂×SD₁₆	—	251	4	no	—
163	C₂×SD₁₆	C₂×D₂₅₁	1	2	yes	—
164	C₂×D₈	C₂×D₂₅₁	1	1	yes	characteristic
165	#131	—	251	8	no	—
166	C₂×M₁₆	—	251	1	no	—
167	C₂×M₁₆	—	251	2	no	—
168	C₂²×C₈	—	251	2	no	—
169	C₂²×C₈	C₂×D₂₅₁	1	1	yes	characteristic
170	#41	—	2	4	no	—
171	#41	—	502	16	no	—
172	#41	—	502	32	no	—
173	#41	—	502	8	no	—
174	#41	—	251	16	no	—
175	#41	—	251	4	no	—
176	#41	—	251	4	no	—
177	#41	C₂²×D₂₅₁	1	8	yes	—
178	C₂×Q₈	—	502	16	no	—
179	C₂×Q₈	—	502	4	no	—
180	C₂×Q₈	—	502	8	no	—
181	C₂×Q₈	—	502	2	no	—
182	C₂×Q₈	—	502	4	no	—
183	C₂×Q₈	—	251	16	no	—
184	C₂×Q₈	—	251	4	no	—
185	C₂×Q₈	—	251	8	no	—
186	C₂×Q₈	—	251	1	no	—
187	C₂×Q₈	C₂²×D₂₅₁	1	2	yes	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	502	4	no	—
190	C₂×D₄	—	251	2	no	—
191	C₂×D₄	—	251	1	no	—
192	C₂×D₄	C₂²×D₂₅₁	1	2	yes	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	502	4	no	—
195	C₂²×C₄	—	502	2	no	—
196	C₂²×C₄	—	502	4	no	—
197	C₂²×C₄	—	251	2	no	—
198	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
199	Q₁₆	—	251	16	no	—
200	Q₁₆	—	251	8	no	—
201	Q₁₆	—	251	8	no	—
202	Q₁₆	C₂²×D₂₅₁	1	4	yes	—
203	SD₁₆	—	251	16	no	—
204	SD₁₆	C₂²×D₂₅₁	1	8	yes	—
205	D₈	C₂²×D₂₅₁	1	4	yes	—
206	M₁₆	—	251	4	no	—
207	M₁₆	—	251	8	no	—
208	C₂×C₈	—	251	2	no	—
209	C₂×C₈	—	251	8	no	—
210	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
211	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
212	C₂×C₈	C₂²×D₂₅₁	1	4	yes	—
213	C₂³	—	2	2	no	—
214	C₂³	—	251	2	no	—
215	C₂³	D₄×D₂₅₁	1	1	yes	characteristic
216	Q₈	—	2	2	no	—
217	Q₈	—	502	16	no	—
218	Q₈	—	502	8	no	—
219	Q₈	—	502	4	no	—
220	Q₈	—	251	8	no	—
221	Q₈	—	251	4	no	—
222	Q₈	C₂³×D₂₅₁	1	4	yes	—
223	D₄	—	2	8	no	—
224	D₄	—	2	2	no	—
225	D₄	—	502	16	no	—
226	D₄	—	251	8	no	—
227	D₄	—	251	4	no	—
228	D₄	C₂³×D₂₅₁	1	4	yes	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	—	502	16	no	—
232	C₂×C₄	—	502	8	no	—
233	C₂×C₄	—	502	4	no	—
234	C₂×C₄	—	502	16	no	—
235	C₂×C₄	—	251	2	no	—
236	C₂×C₄	—	251	4	no	—
237	C₂×C₄	—	251	4	no	—
238	C₂×C₄	D₄×D₂₅₁	1	2	yes	—
239	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
240	C₂×C₄	C₂³×D₂₅₁	1	2	yes	—
241	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
242	C₈	—	251	4	no	—
243	C₈	C₂³×D₂₅₁	1	2	yes	—
244	C₈	C₂³×D₂₅₁	1	2	yes	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	4	no	—
247	C₂²	—	4	4	no	—
248	C₂²	—	502	4	no	—
249	C₂²	—	251	4	no	—
250	C₂²	C₂×D₄×D₂₅₁	1	2	yes	—
251	C₂²	C₂×D₄×D₂₅₁	1	1	yes	characteristic
252	C₄	—	2	4	no	—
253	C₄	—	502	8	no	—
254	C₄	—	251	4	no	—
255	C₄	C₂×D₄×D₂₅₁	1	2	yes	—
256	C₄	C₂×D₄×D₂₅₁	1	1	yes	characteristic
257	C₄	C₂⁴×D₂₅₁	1	1	yes	characteristic
258	C₂	—	2	2	no	—
259	C₂	—	4	4	no	—
260	C₂	—	502	4	no	—
261	C₂	#30632501	1	2	yes	direct product factor
262	C₂	C₂²×D₄×D₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₅₁
commutator, cosocle	C₄×C₂₅₁	C₂⁵
Frattini	C₄	C₂⁴×D₂₅₁
Fitting	C₂×#135×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
hypercenter	C₂×#135	D₂₅₁
hypocenter	C₂₅₁	C₂×#578
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#22548762	C₂	semidirect
2. #30632501	C₂	direct (N × H)
3. C₂×#22548745	C₂	semidirect
4. C₂×#22548744	C₂	semidirect
5. C₂×#22548734	C₂	semidirect
6. C₂²×Dic₁₀₀₄	C₂	semidirect
7. C₂×Q₁₆×D₂₅₁	C₂	semidirect
8. C₂×#22548721	C₂	semidirect
9. C₂×#135×C₂₅₁	C₂	semidirect
11. #22548762	C₂²	semidirect
19. #22548745	C₂²	semidirect
20. #22548744	C₂²	semidirect
21. #22548734	C₂²	semidirect
24. C₂×Dic₁₀₀₄	C₂²	semidirect
26. Q₁₆×D₂₅₁	C₂²	semidirect
28. #22548721	C₂²	semidirect
33. #135×C₂₅₁	C₂²	semidirect
128. C₂×C₂₅₁	#578	semidirect
131. C₂₅₁	C₂×#578	semidirect
138. C₂×#135	D₂₅₁	semidirect
155. #135	C₂×D₂₅₁	semidirect
261. C₂	#30632501	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×#22548762	2
#30632501	2
C₂×#22548745	2
C₂×#22548744	2
C₂×#22548734	2
C₂²×Dic₁₀₀₄	2
C₂×Q₁₆×D₂₅₁	2
C₂×#22548721	2
C₂×#135×C₂₅₁	2
C₂×#578	251

Subgroups — C2×#30632501 #44565129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#30632501 #44565129