Subgroups — C₂×#30632502 #44565130

Order 32128.

1408 conjugacy classes of subgroups.

The subgroup lattice has 238 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	1	yes	maximal, characteristic, complemented
2	#30632502	C₂	1	16	yes	maximal, direct product factor
3	C₂×#22548746	C₂	1	3	yes	maximal, complemented
4	C₂×#22548745	C₂	1	3	yes	maximal, complemented
5	C₂×#22548735	C₂	1	3	yes	maximal, complemented
6	C₂²×#16860800	C₂	1	3	yes	maximal, complemented
7	C₂×#22548722	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×2₋¹⁺⁴×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#22548762	—	2	4	no	—
10	#22548762	C₂²	1	8	yes	complemented
11	C₂×#16860810	C₂²	1	3	yes	—
12	C₂×#16860809	—	2	3	no	—
13	C₂²×Dic₅₀₂	C₂²	1	3	yes	—
14	C₂×Q₈×D₂₅₁	—	2	1	no	—
15	#22548746	C₂²	1	24	yes	complemented
16	#22548745	C₂²	1	24	yes	complemented
17	#22548735	C₂²	1	24	yes	complemented
18	C₂×#16860800	C₂²	1	3	yes	—
19	C₂×#16860800	C₂²	1	6	yes	—
20	C₂×#16860800	C₂²	1	24	yes	complemented
21	C₂×#16860798	C₂²	1	3	yes	—
22	C₂×#16860797	C₂²	1	3	yes	—
23	C₂×#16860795	C₂²	1	1	yes	characteristic
24	#22548722	C₂²	1	8	yes	complemented
25	C₂×#16860794	C₂²	1	3	yes	—
26	C₂²×C₂₅₁⋊C₈	C₂²	1	3	yes	—
27	2₋¹⁺⁴×C₂₅₁	—	2	4	no	—
28	2₋¹⁺⁴×C₂₅₁	C₂²	1	8	yes	complemented
29	C₂×#41×C₂₅₁	—	2	3	no	—
30	C₂×#41×C₂₅₁	C₂²	1	3	yes	—
31	C₂×#41×C₂₅₁	C₂²	1	1	yes	characteristic
32	C₂²×Q₈×C₂₅₁	—	2	1	no	—
33	C₂²×Q₈×C₂₅₁	C₂²	1	3	yes	—
34	#16860810	—	2	6	no	—
35	#16860810	C₂³	1	12	yes	—
36	#16860809	—	2	24	no	—
37	C₂×Dic₅₀₂	—	2	3	no	—
38	C₂×Dic₅₀₂	—	2	6	no	—
39	C₂×Dic₅₀₂	C₂³	1	12	yes	—
40	C₂×Dic₅₀₂	C₂³	1	3	yes	—
41	Q₈×D₂₅₁	—	2	8	no	—
42	C₂×D₁₀₀₄	C₂³	1	1	yes	characteristic
43	C₂×C₂²⋊D₂₅₁	—	2	3	no	—
44	C₂²×Dic₂₅₁	—	2	3	no	—
45	C₂×C₄×D₂₅₁	—	2	3	no	—
46	#16860800	C₂³	1	24	yes	—
47	#16860800	C₂³	1	12	yes	—
48	#16860798	C₂³	1	12	yes	—
49	#16860797	C₂³	1	12	yes	—
50	#16860795	C₂³	1	4	yes	—
51	#16860794	C₂³	1	12	yes	—
52	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
53	C₂×C₂₅₁⋊C₈	C₂³	1	3	yes	—
54	C₂×C₂₅₁⋊C₈	C₂³	1	12	yes	—
55	#41×C₂₅₁	—	2	24	no	—
56	#41×C₂₅₁	—	2	6	no	—
57	#41×C₂₅₁	D₄	1	4	yes	—
58	#41×C₂₅₁	C₂³	1	12	yes	—
59	#41×C₂₅₁	C₂³	1	4	yes	—
60	C₂×Q₈×C₂₅₁	—	2	8	no	—
61	C₂×Q₈×C₂₅₁	—	2	3	no	—
62	C₂×Q₈×C₂₅₁	—	2	6	no	—
63	C₂×Q₈×C₂₅₁	D₄	1	1	yes	characteristic
64	C₂×Q₈×C₂₅₁	C₂³	1	12	yes	—
65	C₂×Q₈×C₂₅₁	C₂³	1	3	yes	—
66	C₂×D₄×C₂₅₁	—	2	3	no	—
67	C₂×D₄×C₂₅₁	D₄	1	3	yes	—
68	C₂×D₄×C₂₅₁	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₂₅₁	—	2	3	no	—
70	C₂²×C₄×C₂₅₁	—	2	3	no	—
71	C₂²×C₄×C₂₅₁	C₂³	1	3	yes	—
72	C₂²×D₂₅₁	—	2	1	no	—
73	Dic₅₀₂	—	2	12	no	—
74	Dic₅₀₂	—	2	3	no	—
75	Dic₅₀₂	C₂⁴	1	6	yes	—
76	D₁₀₀₄	—	2	1	no	—
77	D₁₀₀₄	C₂⁴	1	2	yes	—
78	C₂²⋊D₂₅₁	—	2	12	no	—
79	C₂×Dic₂₅₁	—	2	3	no	—
80	C₂×Dic₂₅₁	—	2	12	no	—
81	C₄×D₂₅₁	—	2	12	no	—
82	C₂₅₁⋊C₈	C₂⁴	1	2	yes	—
83	C₂₅₁⋊C₈	C₂⁴	1	6	yes	—
84	C₂³×C₂₅₁	—	2	1	no	—
85	C₂³×C₂₅₁	C₂×D₄	1	3	yes	—
86	Q₈×C₂₅₁	—	2	12	no	—
87	Q₈×C₂₅₁	—	2	3	no	—
88	Q₈×C₂₅₁	C₂×D₄	1	4	yes	—
89	Q₈×C₂₅₁	C₂⁴	1	6	yes	—
90	D₄×C₂₅₁	—	2	12	no	—
91	D₄×C₂₅₁	—	2	1	no	—
92	D₄×C₂₅₁	C₂×D₄	1	12	yes	—
93	D₄×C₂₅₁	C₂⁴	1	2	yes	—
94	C₂×C₄×C₂₅₁	—	2	3	no	—
95	C₂×C₄×C₂₅₁	—	2	12	no	—
96	C₂×C₄×C₂₅₁	—	2	12	no	—
97	C₂×C₄×C₂₅₁	C₂×D₄	1	3	yes	—
98	C₂×C₄×C₂₅₁	C₂×D₄	1	6	yes	—
99	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
100	C₂×C₄×C₂₅₁	C₂⁴	1	6	yes	—
101	C₂²×C₂₅₁	—	2	6	no	—
102	C₂²×C₂₅₁	—	2	2	no	—
103	C₂²×C₂₅₁	—	4	2	no	—
104	C₂²×C₂₅₁	C₂²×D₄	1	6	yes	—
105	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
106	C₂×D₂₅₁	—	2	2	no	—
107	C₂×D₂₅₁	—	4	2	no	—
108	C₄×C₂₅₁	—	2	6	no	—
109	C₄×C₂₅₁	C₂²×D₄	1	6	yes	—
110	C₄×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
111	C₄×C₂₅₁	C₂⁵	1	1	yes	characteristic
112	Dic₂₅₁	—	2	6	no	—
113	C₂×C₂₅₁	—	2	6	no	—
114	C₂×C₂₅₁	—	4	2	no	—
115	C₂×C₂₅₁	#578	1	2	yes	complemented
116	C₂×C₂₅₁	C₂³×D₄	1	1	yes	characteristic
117	D₂₅₁	—	4	2	no	—
118	C₂₅₁	C₂×#578	1	1	yes	characteristic, complemented, Sylow
119	C₂×#578	—	251	1	no	maximal, Sylow
120	C₂×2₋¹⁺⁴	—	251	1	no	—
121	C₂×2₋¹⁺⁴	D₂₅₁	1	1	yes	characteristic, complemented
122	#578	—	251	16	no	—
123	C₂×#137	—	251	3	no	—
124	C₂×#137	—	251	3	no	—
125	C₂×#135	—	251	3	no	—
126	C₂²×Q₁₆	—	251	3	no	—
127	C₂×#131	—	251	1	no	—
128	2₋¹⁺⁴	—	2	4	no	—
129	2₋¹⁺⁴	—	502	4	no	—
130	2₋¹⁺⁴	—	251	8	no	—
131	2₋¹⁺⁴	C₂×D₂₅₁	1	8	yes	complemented
132	C₂×#41	—	2	3	no	—
133	C₂×#41	—	502	3	no	—
134	C₂×#41	—	251	3	no	—
135	C₂×#41	C₂×D₂₅₁	1	3	yes	—
136	C₂×#41	C₂×D₂₅₁	1	1	yes	characteristic
137	C₂²×Q₈	—	2	1	no	—
138	C₂²×Q₈	—	502	1	no	—
139	C₂²×Q₈	—	251	3	no	—
140	C₂²×Q₈	C₂×D₂₅₁	1	3	yes	—
141	#137	—	251	24	no	—
142	#137	—	251	24	no	—
143	#135	—	251	24	no	—
144	C₂×Q₁₆	—	251	3	no	—
145	C₂×Q₁₆	—	251	6	no	—
146	C₂×Q₁₆	—	251	24	no	—
147	C₂×SD₁₆	—	251	3	no	—
148	C₂×SD₁₆	—	251	3	no	—
149	C₂×D₈	—	251	1	no	—
150	#131	—	251	8	no	—
151	C₂×M₁₆	—	251	3	no	—
152	C₂²×C₈	—	251	3	no	—
153	#41	—	2	24	no	—
154	#41	—	2	6	no	—
155	#41	—	502	24	no	—
156	#41	—	502	6	no	—
157	#41	—	251	12	no	—
158	#41	C₂²⋊D₂₅₁	1	4	yes	—
159	#41	C₂²×D₂₅₁	1	12	yes	—
160	#41	C₂²×D₂₅₁	1	4	yes	—
161	C₂×Q₈	—	2	8	no	—
162	C₂×Q₈	—	2	3	no	—
163	C₂×Q₈	—	2	6	no	—
164	C₂×Q₈	—	502	8	no	—
165	C₂×Q₈	—	502	3	no	—
166	C₂×Q₈	—	502	6	no	—
167	C₂×Q₈	—	251	3	no	—
168	C₂×Q₈	—	251	12	no	—
169	C₂×Q₈	C₂²⋊D₂₅₁	1	1	yes	characteristic
170	C₂×Q₈	C₂²×D₂₅₁	1	3	yes	—
171	C₂×Q₈	C₂²×D₂₅₁	1	12	yes	—
172	C₂×D₄	—	2	3	no	—
173	C₂×D₄	—	502	3	no	—
174	C₂×D₄	—	251	1	no	—
175	C₂×D₄	C₂²⋊D₂₅₁	1	3	yes	—
176	C₂×D₄	C₂²×D₂₅₁	1	1	yes	characteristic
177	C₂²×C₄	—	2	3	no	—
178	C₂²×C₄	—	2	3	no	—
179	C₂²×C₄	—	502	3	no	—
180	C₂²×C₄	—	502	3	no	—
181	C₂²×C₄	C₂²×D₂₅₁	1	3	yes	—
182	Q₁₆	—	251	12	no	—
183	Q₁₆	—	251	24	no	—
184	SD₁₆	—	251	12	no	—
185	SD₁₆	—	251	12	no	—
186	D₈	—	251	4	no	—
187	M₁₆	—	251	12	no	—
188	C₂×C₈	—	251	1	no	—
189	C₂×C₈	—	251	3	no	—
190	C₂×C₈	—	251	12	no	—
191	C₂³	—	2	1	no	—
192	C₂³	—	502	1	no	—
193	C₂³	C₂×C₂²⋊D₂₅₁	1	3	yes	—
194	Q₈	—	2	12	no	—
195	Q₈	—	2	3	no	—
196	Q₈	—	502	12	no	—
197	Q₈	—	502	3	no	—
198	Q₈	—	251	6	no	—
199	Q₈	C₂×C₂²⋊D₂₅₁	1	4	yes	—
200	Q₈	C₂³×D₂₅₁	1	6	yes	—
201	D₄	—	2	12	no	—
202	D₄	—	2	1	no	—
203	D₄	—	502	12	no	—
204	D₄	—	502	1	no	—
205	D₄	—	251	2	no	—
206	D₄	C₂×C₂²⋊D₂₅₁	1	12	yes	—
207	D₄	C₂³×D₂₅₁	1	2	yes	—
208	C₂×C₄	—	2	12	no	—
209	C₂×C₄	—	2	3	no	—
210	C₂×C₄	—	2	12	no	—
211	C₂×C₄	—	502	12	no	—
212	C₂×C₄	—	502	3	no	—
213	C₂×C₄	—	502	12	no	—
214	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	6	yes	—
215	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	3	yes	—
216	C₂×C₄	C₂³×D₂₅₁	1	6	yes	—
217	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
218	C₈	—	251	2	no	—
219	C₈	—	251	6	no	—
220	C₂²	—	2	6	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	—	1004	2	no	—
224	C₂²	—	502	2	no	—
225	C₂²	C₂²×C₂²⋊D₂₅₁	1	6	yes	—
226	C₂²	C₂²×C₂²⋊D₂₅₁	1	1	yes	characteristic
227	C₄	—	2	6	no	—
228	C₄	—	502	6	no	—
229	C₄	C₂²×C₂²⋊D₂₅₁	1	6	yes	—
230	C₄	C₂²×C₂²⋊D₂₅₁	1	1	yes	characteristic
231	C₄	C₂⁴×D₂₅₁	1	1	yes	characteristic
232	C₂	—	2	6	no	—
233	C₂	—	4	2	no	—
234	C₂	—	1004	2	no	—
235	C₂	#30632502	1	2	yes	direct product factor
236	C₂	C₂³×C₂²⋊D₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₂₅₁
commutator, cosocle	C₄×C₂₅₁	C₂⁵
Frattini	C₄	C₂⁴×D₂₅₁
Fitting	C₂×2₋¹⁺⁴×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
hypercenter	C₂×2₋¹⁺⁴	D₂₅₁
hypocenter	C₂₅₁	C₂×#578
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. C₂×#22548762	C₂	semidirect
2. #30632502	C₂	direct (N × H)
3. C₂×#22548746	C₂	semidirect
4. C₂×#22548745	C₂	semidirect
5. C₂×#22548735	C₂	semidirect
6. C₂²×#16860800	C₂	semidirect
7. C₂×#22548722	C₂	semidirect
8. C₂×2₋¹⁺⁴×C₂₅₁	C₂	semidirect
10. #22548762	C₂²	semidirect
15. #22548746	C₂²	semidirect
16. #22548745	C₂²	semidirect
17. #22548735	C₂²	semidirect
20. C₂×#16860800	C₂²	semidirect
24. #22548722	C₂²	semidirect
28. 2₋¹⁺⁴×C₂₅₁	C₂²	semidirect
115. C₂×C₂₅₁	#578	semidirect
118. C₂₅₁	C₂×#578	semidirect
121. C₂×2₋¹⁺⁴	D₂₅₁	semidirect
131. 2₋¹⁺⁴	C₂×D₂₅₁	semidirect
235. C₂	#30632502	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
#30632502	2
C₂×#22548746	2
C₂×#22548745	2
C₂×#22548735	2
C₂²×#16860800	2
C₂×#22548722	2
C₂×2₋¹⁺⁴×C₂₅₁	2
C₂×#578	251

Subgroups — C2×#30632502 #44565130

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#30632502 #44565130