Subgroups — C₂×#20265650 #27184317

Order 12928.

1920 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	C₂³×C₂²⋊D₁₀₁	C₂	1	1	yes	maximal, characteristic
2	#20265650	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×C₂²⋊D₁₀₁	C₂	1	4	yes	maximal, complemented
4	C₂×#15692215	C₂	1	4	yes	maximal, complemented
5	C₂×#15692211	C₂	1	2	yes	maximal, complemented
6	C₂²×C₂²⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂²⋊C₄×D₁₀₁	C₂	1	2	yes	maximal, complemented
8	C₂²×C₂²⋊C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×D₁₀₁	C₂²	1	1	yes	characteristic
10	C₂²×C₂²⋊D₁₀₁	C₄	1	16	yes	complemented
11	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
12	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
13	C₂²×C₂²⋊D₁₀₁	C₂²	1	8	yes	—
14	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
15	C₂³×Dic₁₀₁	C₂²	1	2	yes	complemented
16	C₂²×C₄×D₁₀₁	C₂²	1	4	yes	complemented
17	C₄×C₂²⋊D₁₀₁	C₂²	1	32	yes	complemented
18	#15692215	C₂²	1	32	yes	complemented
19	#15692211	C₂²	1	16	yes	complemented
20	C₂×#13108053	C₂²	1	4	yes	complemented
21	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
22	C₂×C₂²⋊Dic₁₀₁	C₂²	1	8	yes	complemented
23	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	complemented
24	C₂×#13108051	C₂²	1	4	yes	complemented
25	C₂²⋊C₄×D₁₀₁	C₂²	1	16	yes	complemented
26	C₂×C₄×Dic₁₀₁	C₂²	1	4	yes	complemented
27	C₂⁵×C₁₀₁	C₂²	1	1	yes	characteristic
28	C₂³×C₄×C₁₀₁	C₂²	1	2	yes	complemented
29	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
30	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	8	yes	complemented
31	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	complemented
32	C₂³×D₁₀₁	—	4	4	no	—
33	C₂³×D₁₀₁	C₂×C₄	1	8	yes	complemented
34	C₂³×D₁₀₁	C₂³	1	2	yes	—
35	C₂³×D₁₀₁	C₂³	1	4	yes	—
36	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	16	yes	complemented
37	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	16	yes	complemented
38	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	64	yes	complemented
39	C₂×C₂²⋊D₁₀₁	C₂³	1	8	yes	—
40	C₂×C₂²⋊D₁₀₁	C₂³	1	8	yes	—
41	C₂²×Dic₁₀₁	—	4	4	no	—
42	C₂²×Dic₁₀₁	C₂×C₄	1	8	yes	complemented
43	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
44	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
45	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
46	C₂²×Dic₁₀₁	C₂³	1	8	yes	complemented
47	C₂×C₄×D₁₀₁	—	4	8	no	—
48	C₂×C₄×D₁₀₁	C₂³	1	16	yes	complemented
49	#13108053	C₂³	1	16	yes	complemented
50	C₂²⋊Dic₁₀₁	C₂³	1	8	yes	—
51	C₂²⋊Dic₁₀₁	C₂³	1	8	yes	complemented
52	#13108051	C₂³	1	16	yes	complemented
53	C₄×Dic₁₀₁	C₂³	1	16	yes	complemented
54	C₂⁴×C₁₀₁	—	4	4	no	—
55	C₂⁴×C₁₀₁	C₂×C₄	1	8	yes	complemented
56	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
57	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
58	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
59	C₂⁴×C₁₀₁	C₂³	1	4	yes	—
60	C₂²×C₄×C₁₀₁	—	4	4	no	—
61	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
62	C₂²×C₄×C₁₀₁	C₂³	1	8	yes	complemented
63	C₂²⋊C₄×C₁₀₁	C₂³	1	8	yes	—
64	C₂²⋊C₄×C₁₀₁	C₂³	1	8	yes	complemented
65	C₂³×C₁₀₁	—	2	4	no	—
66	C₂³×C₁₀₁	—	2	4	no	—
67	C₂³×C₁₀₁	—	2	4	no	—
68	C₂³×C₁₀₁	—	4	16	no	—
69	C₂³×C₁₀₁	—	4	8	no	—
70	C₂³×C₁₀₁	C₂²×C₄	1	4	yes	—
71	C₂³×C₁₀₁	C₂²×C₄	1	4	yes	—
72	C₂³×C₁₀₁	C₂²×C₄	1	4	yes	—
73	C₂³×C₁₀₁	C₂²×C₄	1	16	yes	complemented
74	C₂³×C₁₀₁	C₂⁴	1	2	yes	—
75	C₂³×C₁₀₁	C₂⁴	1	2	yes	—
76	C₂³×C₁₀₁	C₂⁴	1	2	yes	—
77	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
78	C₂²×D₁₀₁	—	2	8	no	—
79	C₂²×D₁₀₁	—	4	16	no	—
80	C₂²×D₁₀₁	—	4	8	no	—
81	C₂²×D₁₀₁	C₂²×C₄	1	8	yes	—
82	C₂²×D₁₀₁	C₂²×C₄	1	16	yes	complemented
83	C₂²×D₁₀₁	C₂⁴	1	4	yes	—
84	C₂×C₄×C₁₀₁	—	2	8	no	—
85	C₂×C₄×C₁₀₁	—	4	8	no	—
86	C₂×C₄×C₁₀₁	C₂⁴	1	8	yes	—
87	C₂²⋊D₁₀₁	C₂²×C₄	1	32	yes	complemented
88	C₂²⋊D₁₀₁	C₂²×C₄	1	32	yes	complemented
89	C₂×Dic₁₀₁	—	2	8	no	—
90	C₂×Dic₁₀₁	—	4	8	no	—
91	C₂×Dic₁₀₁	C₂²×C₄	1	8	yes	—
92	C₂×Dic₁₀₁	C₂²×C₄	1	16	yes	complemented
93	C₂×Dic₁₀₁	C₂⁴	1	8	yes	—
94	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
95	C₄×D₁₀₁	—	4	16	no	—
96	C₂²×C₁₀₁	—	2	8	no	—
97	C₂²×C₁₀₁	—	2	8	no	—
98	C₂²×C₁₀₁	—	2	8	no	—
99	C₂²×C₁₀₁	—	2	4	no	—
100	C₂²×C₁₀₁	—	2	4	no	—
101	C₂²×C₁₀₁	—	2	4	no	—
102	C₂²×C₁₀₁	—	4	16	no	—
103	C₂²×C₁₀₁	C₂³×C₄	1	4	yes	—
104	C₂²×C₁₀₁	C₂³×C₄	1	4	yes	—
105	C₂²×C₁₀₁	C₂³×C₄	1	4	yes	—
106	C₂²×C₁₀₁	C₂³×C₄	1	2	yes	—
107	C₂²×C₁₀₁	H(5,𝔽₂)	1	4	yes	—
108	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
109	C₂×D₁₀₁	—	2	16	no	—
110	C₂×D₁₀₁	—	2	8	no	—
111	C₂×D₁₀₁	—	4	16	no	—
112	C₂×D₁₀₁	C₂³×C₄	1	8	yes	—
113	C₄×C₁₀₁	—	2	8	no	—
114	Dic₁₀₁	—	2	8	no	—
115	Dic₁₀₁	C₂³×C₄	1	8	yes	—
116	C₂×C₁₀₁	—	2	4	no	—
117	C₂×C₁₀₁	—	2	4	no	—
118	C₂×C₁₀₁	—	2	4	no	—
119	C₂×C₁₀₁	#518	1	4	yes	complemented
120	C₂×C₁₀₁	C₂⁴×C₄	1	1	yes	characteristic
121	C₂×C₁₀₁	C₂×H(5,𝔽₂)	1	2	yes	—
122	D₁₀₁	—	2	8	no	—
123	C₂×#518	—	101	1	no	maximal, Sylow
124	C₁₀₁	C₂×#518	1	1	yes	characteristic, complemented, Sylow
125	C₂³×D₄	—	101	1	no	—
126	#518	—	101	16	no	—
127	C₂×C₄×D₄	—	101	4	no	—
128	C₂×C₄×D₄	—	101	4	no	—
129	C₂×#117	—	101	2	no	—
130	C₂²×C₂²⋊C₄	—	101	1	no	—
131	C₂²×C₂²⋊C₄	—	101	2	no	—
132	C₂²×C₂²⋊C₄	D₁₀₁	1	1	yes	characteristic, complemented
133	C₂⁵	—	101	1	no	—
134	C₂⁵	C₂×D₁₀₁	1	1	yes	characteristic
135	C₂²×D₄	—	101	16	no	—
136	C₂²×D₄	—	101	2	no	—
137	C₂²×D₄	—	101	2	no	—
138	C₂²×D₄	—	101	8	no	—
139	C₂³×C₄	—	101	1	no	—
140	C₂³×C₄	—	101	2	no	—
141	C₂³×C₄	—	101	4	no	—
142	C₂³×C₄	C₂×D₁₀₁	1	2	yes	complemented
143	C₄×D₄	—	101	32	no	—
144	C₄×D₄	—	101	32	no	—
145	#117	—	101	16	no	—
146	C₂×C₄⋊C₄	—	101	4	no	—
147	C₂×C₂²⋊C₄	—	101	2	no	—
148	C₂×C₂²⋊C₄	—	101	2	no	—
149	C₂×C₂²⋊C₄	—	101	8	no	—
150	C₂×C₂²⋊C₄	—	101	4	no	—
151	C₂×C₂²⋊C₄	—	101	16	no	—
152	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
153	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	8	yes	complemented
154	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	complemented
155	C₂×C₄²	—	101	4	no	—
156	C₂⁴	—	4	4	no	—
157	C₂⁴	—	101	8	no	—
158	C₂⁴	—	101	4	no	—
159	C₂⁴	—	101	2	no	—
160	C₂⁴	—	404	4	no	—
161	C₂⁴	C₄×D₁₀₁	1	8	yes	complemented
162	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
163	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
164	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
165	C₂⁴	C₂²×D₁₀₁	1	4	yes	—
166	C₂×D₄	—	101	16	no	—
167	C₂×D₄	—	101	16	no	—
168	C₂×D₄	—	101	64	no	—
169	C₂×D₄	—	101	8	no	—
170	C₂×D₄	—	101	8	no	—
171	C₂²×C₄	—	4	4	no	—
172	C₂²×C₄	—	101	8	no	—
173	C₂²×C₄	—	101	4	no	—
174	C₂²×C₄	—	101	2	no	—
175	C₂²×C₄	—	101	8	no	—
176	C₂²×C₄	—	101	4	no	—
177	C₂²×C₄	—	101	16	no	—
178	C₂²×C₄	—	404	8	no	—
179	C₂²×C₄	—	404	4	no	—
180	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
181	C₂²×C₄	C₂²×D₁₀₁	1	8	yes	complemented
182	C₄⋊C₄	—	101	16	no	—
183	C₂²⋊C₄	—	101	8	no	—
184	C₂²⋊C₄	—	101	8	no	—
185	C₂²⋊C₄	—	101	16	no	—
186	C₂²⋊C₄	C₂²×D₁₀₁	1	8	yes	—
187	C₂²⋊C₄	C₂²×D₁₀₁	1	8	yes	complemented
188	C₄²	—	101	16	no	—
189	C₂³	—	2	4	no	—
190	C₂³	—	2	4	no	—
191	C₂³	—	2	4	no	—
192	C₂³	—	4	16	no	—
193	C₂³	—	4	8	no	—
194	C₂³	—	101	16	no	—
195	C₂³	—	101	8	no	—
196	C₂³	—	101	4	no	—
197	C₂³	—	404	8	no	—
198	C₂³	—	404	16	no	—
199	C₂³	—	202	8	no	—
200	C₂³	C₂×C₄×D₁₀₁	1	4	yes	—
201	C₂³	C₂×C₄×D₁₀₁	1	4	yes	—
202	C₂³	C₂×C₄×D₁₀₁	1	4	yes	—
203	C₂³	C₂×C₄×D₁₀₁	1	16	yes	complemented
204	C₂³	C₂³×D₁₀₁	1	2	yes	—
205	C₂³	C₂³×D₁₀₁	1	2	yes	—
206	C₂³	C₂³×D₁₀₁	1	2	yes	—
207	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
208	D₄	—	101	32	no	—
209	D₄	—	101	32	no	—
210	C₂×C₄	—	2	8	no	—
211	C₂×C₄	—	4	8	no	—
212	C₂×C₄	—	101	8	no	—
213	C₂×C₄	—	101	16	no	—
214	C₂×C₄	—	101	4	no	—
215	C₂×C₄	—	101	8	no	—
216	C₂×C₄	—	404	16	no	—
217	C₂×C₄	—	404	8	no	—
218	C₂×C₄	—	202	8	no	—
219	C₂×C₄	C₂³×D₁₀₁	1	8	yes	—
220	C₂²	—	2	4	no	—
221	C₂²	—	2	4	no	—
222	C₂²	—	2	4	no	—
223	C₂²	—	2	8	no	—
224	C₂²	—	2	8	no	—
225	C₂²	—	2	8	no	—
226	C₂²	—	4	16	no	—
227	C₂²	—	101	8	no	—
228	C₂²	—	404	16	no	—
229	C₂²	—	202	8	no	—
230	C₂²	—	202	16	no	—
231	C₂²	C₂²×C₄×D₁₀₁	1	4	yes	—
232	C₂²	C₂²×C₄×D₁₀₁	1	4	yes	—
233	C₂²	C₂²×C₄×D₁₀₁	1	4	yes	—
234	C₂²	C₂²×C₄×D₁₀₁	1	2	yes	—
235	C₂²	#15692300	1	4	yes	—
236	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
237	C₄	—	2	8	no	—
238	C₄	—	101	8	no	—
239	C₄	—	202	8	no	—
240	C₂	—	2	4	no	—
241	C₂	—	2	4	no	—
242	C₂	—	2	4	no	—
243	C₂	—	202	8	no	—
244	C₂	#20265650	1	4	yes	direct product factor
245	C₂	C₂³×C₄×D₁₀₁	1	1	yes	characteristic
246	C₂	C₂×#15692300	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₁₀₁
commutator	C₂×C₁₀₁	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₀₁
Fitting	C₂²×C₂²⋊C₄×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂³×C₂²⋊D₁₀₁	C₂
cosocle	C₂²×C₁₀₁	C₂⁵
hypercenter	C₂²×C₂²⋊C₄	D₁₀₁
hypocenter	C₁₀₁	C₂×#518
LCS	G ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. #20265650	C₂	direct (N × H)
3. C₂×C₄×C₂²⋊D₁₀₁	C₂	semidirect
4. C₂×#15692215	C₂	semidirect
5. C₂×#15692211	C₂	semidirect
6. C₂²×C₂²⋊Dic₁₀₁	C₂	semidirect
7. C₂×C₂²⋊C₄×D₁₀₁	C₂	semidirect
8. C₂²×C₂²⋊C₄×C₁₀₁	C₂	semidirect
10. C₂²×C₂²⋊D₁₀₁	C₄	semidirect
15. C₂³×Dic₁₀₁	C₂²	semidirect
16. C₂²×C₄×D₁₀₁	C₂²	semidirect
17. C₄×C₂²⋊D₁₀₁	C₂²	semidirect
18. #15692215	C₂²	semidirect
19. #15692211	C₂²	semidirect
20. C₂×#13108053	C₂²	semidirect
22. C₂×C₂²⋊Dic₁₀₁	C₂²	semidirect
23. C₂×C₂²⋊Dic₁₀₁	C₂²	semidirect
24. C₂×#13108051	C₂²	semidirect
25. C₂²⋊C₄×D₁₀₁	C₂²	semidirect
26. C₂×C₄×Dic₁₀₁	C₂²	semidirect
28. C₂³×C₄×C₁₀₁	C₂²	semidirect
30. C₂×C₂²⋊C₄×C₁₀₁	C₂²	semidirect
31. C₂×C₂²⋊C₄×C₁₀₁	C₂²	semidirect
33. C₂³×D₁₀₁	C₂×C₄	semidirect
36. C₂×C₂²⋊D₁₀₁	C₂×C₄	semidirect
37. C₂×C₂²⋊D₁₀₁	C₂×C₄	semidirect
38. C₂×C₂²⋊D₁₀₁	C₂×C₄	semidirect
42. C₂²×Dic₁₀₁	C₂×C₄	semidirect
46. C₂²×Dic₁₀₁	C₂³	semidirect
48. C₂×C₄×D₁₀₁	C₂³	semidirect
49. #13108053	C₂³	semidirect
51. C₂²⋊Dic₁₀₁	C₂³	semidirect
52. #13108051	C₂³	semidirect
53. C₄×Dic₁₀₁	C₂³	semidirect
55. C₂⁴×C₁₀₁	C₂×C₄	semidirect
62. C₂²×C₄×C₁₀₁	C₂³	semidirect
64. C₂²⋊C₄×C₁₀₁	C₂³	semidirect
73. C₂³×C₁₀₁	C₂²×C₄	semidirect
82. C₂²×D₁₀₁	C₂²×C₄	semidirect
87. C₂²⋊D₁₀₁	C₂²×C₄	semidirect
88. C₂²⋊D₁₀₁	C₂²×C₄	semidirect
92. C₂×Dic₁₀₁	C₂²×C₄	semidirect
119. C₂×C₁₀₁	#518	semidirect
124. C₁₀₁	C₂×#518	semidirect
132. C₂²×C₂²⋊C₄	D₁₀₁	semidirect
142. C₂³×C₄	C₂×D₁₀₁	semidirect
153. C₂×C₂²⋊C₄	C₂×D₁₀₁	semidirect
154. C₂×C₂²⋊C₄	C₂×D₁₀₁	semidirect
161. C₂⁴	C₄×D₁₀₁	semidirect
181. C₂²×C₄	C₂²×D₁₀₁	semidirect
187. C₂²⋊C₄	C₂²×D₁₀₁	semidirect
203. C₂³	C₂×C₄×D₁₀₁	semidirect
244. C₂	#20265650	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₂²⋊D₁₀₁	2
#20265650	2
C₂×C₄×C₂²⋊D₁₀₁	2
C₂×#15692215	2
C₂×#15692211	2
C₂²×C₂²⋊Dic₁₀₁	2
C₂×C₂²⋊C₄×D₁₀₁	2
C₂²×C₂²⋊C₄×C₁₀₁	2
C₂×#518	101

Subgroups — C2×#20265650 #27184317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265650 #27184317