Subgroups — C₂×#30632256 #44563676

Order 32128.

1896 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#16860810	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×D₁₀₀₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#30632256	C₂	1	16	yes	maximal, direct product factor
4	C₂×#22548712	C₂	1	2	yes	maximal, complemented
5	C₂×#22548703	C₂	1	2	yes	maximal, complemented
6	C₂×#22548690	C₂	1	4	yes	maximal, complemented
7	C₂×#22548685	C₂	1	4	yes	maximal, complemented
8	C₂×#117×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×D₂₅₁	C₂²	1	2	yes	—
10	C₂×#16860810	—	2	8	no	—
11	C₂×#16860810	C₂²	1	8	yes	complemented
12	C₂²×Dic₅₀₂	C₂²	1	1	yes	characteristic, complemented
13	C₂²×D₁₀₀₄	—	2	8	no	—
14	C₂²×D₁₀₀₄	C₂²	1	2	yes	—
15	C₂²×D₁₀₀₄	C₂²	1	2	yes	—
16	C₂²×D₁₀₀₄	C₂²	1	8	yes	complemented
17	C₂²×D₁₀₀₄	C₂²	1	1	yes	characteristic, complemented
18	C₂²×D₁₀₀₄	C₂²	1	8	yes	complemented
19	C₂²×C₂²⋊D₂₅₁	C₂²	1	2	yes	—
20	C₂²×C₄×D₂₅₁	C₂²	1	2	yes	complemented
21	#22548712	C₂²	1	16	yes	complemented
22	#22548703	C₂²	1	16	yes	complemented
23	#22548690	C₂²	1	32	yes	complemented
24	#22548685	C₂²	1	32	yes	complemented
25	C₂×(C₂×D₂₅₁)⋊C₄	C₂²	1	8	yes	complemented
26	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
27	#117×C₂₅₁	C₂²	1	8	yes	complemented
28	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	2	yes	complemented
29	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	2	yes	—
30	C₂×C₄²×C₂₅₁	C₂²	1	2	yes	complemented
31	C₂³×D₂₅₁	—	2	8	no	—
32	C₂³×D₂₅₁	—	4	8	no	—
33	C₂³×D₂₅₁	C₂³	1	2	yes	—
34	C₂³×D₂₅₁	C₂³	1	4	yes	—
35	C₂³×D₂₅₁	C₂³	1	8	yes	—
36	#16860810	—	2	32	no	—
37	C₂×Dic₅₀₂	—	2	4	no	—
38	C₂×Dic₅₀₂	C₂³	1	4	yes	complemented
39	C₂×D₁₀₀₄	—	2	8	no	—
40	C₂×D₁₀₀₄	—	2	8	no	—
41	C₂×D₁₀₀₄	—	2	4	no	—
42	C₂×D₁₀₀₄	—	2	32	no	—
43	C₂×D₁₀₀₄	—	4	16	no	—
44	C₂×D₁₀₀₄	C₂³	1	8	yes	—
45	C₂×D₁₀₀₄	C₂³	1	8	yes	—
46	C₂×D₁₀₀₄	C₂³	1	4	yes	complemented
47	C₂×D₁₀₀₄	C₂³	1	32	yes	complemented
48	C₂×C₂²⋊D₂₅₁	—	2	8	no	—
49	C₂×C₂²⋊D₂₅₁	C₂³	1	8	yes	—
50	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
51	C₂×C₄×D₂₅₁	—	2	8	no	—
52	C₂×C₄×D₂₅₁	C₂³	1	8	yes	complemented
53	(C₂×D₂₅₁)⋊C₄	C₂³	1	32	yes	complemented
54	C₂⁴×C₂₅₁	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₂₅₁	D₄	1	8	yes	complemented
56	C₂²×C₄×C₂₅₁	C₂³	1	4	yes	—
57	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₂₅₁	C₂³	1	4	yes	—
60	C₄⋊C₄×C₂₅₁	C₂³	1	8	yes	complemented
61	C₂²⋊C₄×C₂₅₁	C₂³	1	8	yes	—
62	C₄²×C₂₅₁	C₂³	1	8	yes	complemented
63	C₂²×D₂₅₁	—	2	4	no	—
64	C₂²×D₂₅₁	—	2	16	no	—
65	C₂²×D₂₅₁	—	2	8	no	—
66	C₂²×D₂₅₁	—	2	8	no	—
67	C₂²×D₂₅₁	—	2	16	no	—
68	C₂²×D₂₅₁	—	4	16	no	—
69	C₂²×D₂₅₁	—	4	32	no	—
70	C₂²×D₂₅₁	C₂⁴	1	8	yes	—
71	C₂²×D₂₅₁	C₂⁴	1	4	yes	—
72	Dic₅₀₂	—	2	8	no	—
73	D₁₀₀₄	—	2	16	no	—
74	D₁₀₀₄	—	2	16	no	—
75	D₁₀₀₄	—	2	8	no	—
76	D₁₀₀₄	—	4	32	no	—
77	C₂²⋊D₂₅₁	—	2	16	no	—
78	C₂×Dic₂₅₁	—	2	4	no	—
79	C₂×Dic₂₅₁	C₂⁴	1	4	yes	—
80	C₄×D₂₅₁	—	2	16	no	—
81	C₂³×C₂₅₁	—	2	4	no	—
82	C₂³×C₂₅₁	C₂×D₄	1	4	yes	—
83	C₂³×C₂₅₁	C₂⁴	1	2	yes	—
84	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
85	C₂×C₄×C₂₅₁	—	2	8	no	—
86	C₂×C₄×C₂₅₁	C₂×D₄	1	4	yes	—
87	C₂×C₄×C₂₅₁	C₂×D₄	1	4	yes	—
88	C₂×C₄×C₂₅₁	C₂×D₄	1	16	yes	complemented
89	C₂×C₄×C₂₅₁	C₂⁴	1	8	yes	—
90	C₂×C₄×C₂₅₁	C₂⁴	1	2	yes	—
91	C₂×C₄×C₂₅₁	C₂⁴	1	2	yes	—
92	C₂²×C₂₅₁	—	2	8	no	—
93	C₂²×C₂₅₁	—	2	4	no	—
94	C₂²×C₂₅₁	C₂²×D₄	1	4	yes	—
95	C₂²×C₂₅₁	C₂²×D₄	1	2	yes	—
96	C₂²×C₂₅₁	H(5,𝔽₂)	1	4	yes	—
97	C₂²×C₂₅₁	C₂⁵	1	1	yes	characteristic
98	C₂×D₂₅₁	—	2	4	no	—
99	C₂×D₂₅₁	—	2	8	no	—
100	C₂×D₂₅₁	—	2	8	no	—
101	C₂×D₂₅₁	—	2	16	no	—
102	C₂×D₂₅₁	—	4	32	no	—
103	C₂×D₂₅₁	—	4	8	no	—
104	C₂×D₂₅₁	—	4	16	no	—
105	C₄×C₂₅₁	—	2	8	no	—
106	C₄×C₂₅₁	C₂²×D₄	1	4	yes	—
107	C₄×C₂₅₁	C₂²×D₄	1	4	yes	—
108	Dic₂₅₁	—	2	4	no	—
109	C₂×C₂₅₁	—	2	4	no	—
110	C₂×C₂₅₁	#535	1	4	yes	complemented
111	C₂×C₂₅₁	C₂³×D₄	1	1	yes	characteristic
112	C₂×C₂₅₁	C₂×H(5,𝔽₂)	1	2	yes	—
113	D₂₅₁	—	4	4	no	—
114	D₂₅₁	—	4	8	no	—
115	C₂₅₁	C₂×#535	1	1	yes	characteristic, complemented, Sylow
116	C₂×#535	—	251	1	no	maximal, Sylow
117	C₂²×#41	—	251	1	no	—
118	C₂³×D₄	—	251	1	no	—
119	#535	—	251	16	no	—
120	C₂×#127	—	251	2	no	—
121	C₂×#124	—	251	2	no	—
122	C₂×#121	—	251	4	no	—
123	C₂×#120	—	251	4	no	—
124	C₂×#117	D₂₅₁	1	1	yes	characteristic, complemented
125	C₂⁵	—	251	2	no	—
126	C₂×#41	—	502	8	no	—
127	C₂×#41	—	251	8	no	—
128	C₂²×Q₈	—	251	1	no	—
129	C₂²×D₄	—	502	8	no	—
130	C₂²×D₄	—	251	2	no	—
131	C₂²×D₄	—	251	8	no	—
132	C₂²×D₄	—	251	2	no	—
133	C₂²×D₄	—	251	2	no	—
134	C₂²×D₄	—	251	1	no	—
135	C₂²×D₄	—	251	8	no	—
136	C₂³×C₄	—	251	2	no	—
137	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
138	#127	—	251	16	no	—
139	#124	—	251	16	no	—
140	#121	—	251	32	no	—
141	#120	—	251	32	no	—
142	#117	C₂×D₂₅₁	1	8	yes	complemented
143	C₂×C₄⋊C₄	C₂×D₂₅₁	1	2	yes	complemented
144	C₂×C₂²⋊C₄	—	251	8	no	—
145	C₂×C₂²⋊C₄	C₂×D₂₅₁	1	2	yes	—
146	C₂×C₄²	C₂×D₂₅₁	1	2	yes	complemented
147	C₂⁴	—	1004	8	no	—
148	C₂⁴	—	502	8	no	—
149	C₂⁴	—	251	8	no	—
150	C₂⁴	—	251	4	no	—
151	C₂⁴	—	251	2	no	—
152	C₂⁴	C₂²×D₂₅₁	1	1	yes	characteristic
153	#41	—	502	32	no	—
154	C₂×Q₈	—	502	4	no	—
155	C₂×Q₈	—	251	4	no	—
156	C₂×D₄	—	1004	16	no	—
157	C₂×D₄	—	502	8	no	—
158	C₂×D₄	—	502	8	no	—
159	C₂×D₄	—	502	8	no	—
160	C₂×D₄	—	502	4	no	—
161	C₂×D₄	—	502	32	no	—
162	C₂×D₄	—	251	8	no	—
163	C₂×D₄	—	251	32	no	—
164	C₂×D₄	—	251	8	no	—
165	C₂×D₄	—	251	8	no	—
166	C₂×D₄	—	251	4	no	—
167	C₂²×C₄	—	502	8	no	—
168	C₂²×C₄	—	251	2	no	—
169	C₂²×C₄	—	251	8	no	—
170	C₂²×C₄	D₁₀₀₄	1	8	yes	complemented
171	C₂²×C₄	C₂²×D₂₅₁	1	4	yes	—
172	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
173	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
174	C₂²×C₄	C₂²×D₂₅₁	1	4	yes	—
175	C₄⋊C₄	C₂²×D₂₅₁	1	8	yes	complemented
176	C₂²⋊C₄	—	251	32	no	—
177	C₂²⋊C₄	C₂²×D₂₅₁	1	8	yes	—
178	C₄²	C₂²×D₂₅₁	1	8	yes	complemented
179	C₂³	—	2	4	no	—
180	C₂³	—	1004	32	no	—
181	C₂³	—	1004	16	no	—
182	C₂³	—	502	4	no	—
183	C₂³	—	502	16	no	—
184	C₂³	—	502	8	no	—
185	C₂³	—	502	8	no	—
186	C₂³	—	502	16	no	—
187	C₂³	—	251	8	no	—
188	C₂³	—	251	4	no	—
189	C₂³	C₂×D₁₀₀₄	1	4	yes	—
190	C₂³	C₂³×D₂₅₁	1	2	yes	—
191	C₂³	C₂³×D₂₅₁	1	1	yes	characteristic
192	Q₈	—	502	8	no	—
193	D₄	—	1004	32	no	—
194	D₄	—	502	16	no	—
195	D₄	—	502	16	no	—
196	D₄	—	502	16	no	—
197	D₄	—	502	8	no	—
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	—	502	4	no	—
200	C₂×C₄	—	502	16	no	—
201	C₂×C₄	—	251	4	no	—
202	C₂×C₄	C₂×D₁₀₀₄	1	4	yes	—
203	C₂×C₄	C₂×D₁₀₀₄	1	4	yes	—
204	C₂×C₄	C₂×D₁₀₀₄	1	16	yes	complemented
205	C₂×C₄	C₂³×D₂₅₁	1	8	yes	—
206	C₂×C₄	C₂³×D₂₅₁	1	2	yes	—
207	C₂×C₄	C₂³×D₂₅₁	1	2	yes	—
208	C₂²	—	2	8	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	1004	32	no	—
211	C₂²	—	1004	16	no	—
212	C₂²	—	1004	8	no	—
213	C₂²	—	502	4	no	—
214	C₂²	—	502	8	no	—
215	C₂²	—	502	8	no	—
216	C₂²	—	502	16	no	—
217	C₂²	C₂²×D₁₀₀₄	1	4	yes	—
218	C₂²	C₂²×D₁₀₀₄	1	2	yes	—
219	C₂²	#22548761	1	4	yes	—
220	C₂²	C₂⁴×D₂₅₁	1	1	yes	characteristic
221	C₄	—	2	8	no	—
222	C₄	—	502	4	no	—
223	C₄	C₂²×D₁₀₀₄	1	4	yes	—
224	C₄	C₂²×D₁₀₀₄	1	4	yes	—
225	C₂	—	2	4	no	—
226	C₂	—	1004	8	no	—
227	C₂	—	1004	4	no	—
228	C₂	#30632256	1	4	yes	direct product factor
229	C₂	C₂³×D₁₀₀₄	1	1	yes	characteristic
230	C₂	C₂×#22548761	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₂₅₁
commutator, cosocle	C₂²×C₂₅₁	C₂⁵
Frattini	C₂²	C₂⁴×D₂₅₁
Fitting	C₂×#117×C₂₅₁	C₂
socle	C₂³×C₂₅₁	C₂⁴
hypercenter	C₂×#117	D₂₅₁
hypocenter	C₂₅₁	C₂×#535
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. C₂²×#16860810	C₂	semidirect
2. C₂³×D₁₀₀₄	C₂	semidirect
3. #30632256	C₂	direct (N × H)
4. C₂×#22548712	C₂	semidirect
5. C₂×#22548703	C₂	semidirect
6. C₂×#22548690	C₂	semidirect
7. C₂×#22548685	C₂	semidirect
8. C₂×#117×C₂₅₁	C₂	semidirect
11. C₂×#16860810	C₂²	semidirect
12. C₂²×Dic₅₀₂	C₂²	semidirect
16. C₂²×D₁₀₀₄	C₂²	semidirect
17. C₂²×D₁₀₀₄	C₂²	semidirect
18. C₂²×D₁₀₀₄	C₂²	semidirect
20. C₂²×C₄×D₂₅₁	C₂²	semidirect
21. #22548712	C₂²	semidirect
22. #22548703	C₂²	semidirect
23. #22548690	C₂²	semidirect
24. #22548685	C₂²	semidirect
25. C₂×(C₂×D₂₅₁)⋊C₄	C₂²	semidirect
27. #117×C₂₅₁	C₂²	semidirect
28. C₂×C₄⋊C₄×C₂₅₁	C₂²	semidirect
30. C₂×C₄²×C₂₅₁	C₂²	semidirect
38. C₂×Dic₅₀₂	C₂³	semidirect
46. C₂×D₁₀₀₄	C₂³	semidirect
47. C₂×D₁₀₀₄	C₂³	semidirect
52. C₂×C₄×D₂₅₁	C₂³	semidirect
53. (C₂×D₂₅₁)⋊C₄	C₂³	semidirect
55. C₂²×C₄×C₂₅₁	D₄	semidirect
60. C₄⋊C₄×C₂₅₁	C₂³	semidirect
62. C₄²×C₂₅₁	C₂³	semidirect
88. C₂×C₄×C₂₅₁	C₂×D₄	semidirect
110. C₂×C₂₅₁	#535	semidirect
115. C₂₅₁	C₂×#535	semidirect
124. C₂×#117	D₂₅₁	semidirect
142. #117	C₂×D₂₅₁	semidirect
143. C₂×C₄⋊C₄	C₂×D₂₅₁	semidirect
146. C₂×C₄²	C₂×D₂₅₁	semidirect
170. C₂²×C₄	D₁₀₀₄	semidirect
175. C₄⋊C₄	C₂²×D₂₅₁	semidirect
178. C₄²	C₂²×D₂₅₁	semidirect
204. C₂×C₄	C₂×D₁₀₀₄	semidirect
228. C₂	#30632256	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#16860810	2
C₂³×D₁₀₀₄	2
#30632256	2
C₂×#22548712	2
C₂×#22548703	2
C₂×#22548690	2
C₂×#22548685	2
C₂×#117×C₂₅₁	2
C₂×#535	251

Subgroups — C2×#30632256 #44563676

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#30632256 #44563676