Subgroups — #559×D₁₀₁ #27184716

Order 12928.

1240 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265913	C₂	1	1	yes	maximal, characteristic, complemented
2	#20265912	C₂	1	2	yes	maximal, complemented
3	#20265911	C₂	1	4	yes	maximal, complemented
4	#20265910	C₂	1	4	yes	maximal, complemented
5	#20265909	C₂	1	4	yes	maximal, complemented
6	#127×D₁₀₁	C₂	1	1	yes	maximal, characteristic
7	#124×D₁₀₁	C₂	1	4	yes	maximal, complemented
8	#123×D₁₀₁	C₂	1	4	yes	maximal, complemented
9	C₄×Q₈×D₁₀₁	C₂	1	2	yes	maximal, complemented
10	C₄×D₄×D₁₀₁	C₂	1	4	yes	maximal, complemented
11	#559×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₁₀₁	C₂²	1	2	yes	—
13	C₂×D₄×D₁₀₁	C₂²	1	4	yes	—
14	C₂×D₄×D₁₀₁	C₂²	1	2	yes	—
15	C₂²×C₄×D₁₀₁	C₂²	1	4	yes	—
16	#15692252	C₂²	1	1	yes	characteristic
17	#15692251	C₂²	1	4	yes	—
18	#15692251	C₂²	1	2	yes	—
19	#15692244	C₂²	1	4	yes	complemented
20	#15692243	C₂²	1	4	yes	complemented
21	#15692242	C₂²	1	4	yes	complemented
22	#15692241	C₂²	1	8	yes	complemented
23	#15692241	C₂²	1	8	yes	complemented
24	#15692240	C₂²	1	8	yes	complemented
25	#15692239	C₂²	1	4	yes	complemented
26	#15692238	C₂²	1	4	yes	complemented
27	#15692237	C₂²	1	8	yes	complemented
28	#15692236	C₂²	1	4	yes	complemented
29	Q₈×Dic₁₀₁	C₂²	1	2	yes	complemented
30	C₄×Dic₂₀₂	C₂²	1	4	yes	complemented
31	C₄×Dic₂₀₂	C₂²	1	2	yes	complemented
32	#15692220	C₂²	1	2	yes	complemented
33	#15692220	C₂²	1	4	yes	complemented
34	C₄×D₄₀₄	C₂²	1	4	yes	complemented
35	D₄×Dic₁₀₁	C₂²	1	4	yes	complemented
36	#15692217	C₂²	1	4	yes	complemented
37	C₄×C₂²⋊D₁₀₁	C₂²	1	8	yes	complemented
38	#15692215	C₂²	1	8	yes	complemented
39	C₄⋊C₄×D₁₀₁	C₂²	1	2	yes	—
40	C₄⋊C₄×D₁₀₁	C₂²	1	4	yes	—
41	C₂²⋊C₄×D₁₀₁	C₂²	1	8	yes	—
42	C₂²⋊C₄×D₁₀₁	C₂²	1	4	yes	—
43	C₄²×D₁₀₁	C₂²	1	4	yes	—
44	C₄²×D₁₀₁	C₂²	1	1	yes	characteristic
45	#127×C₁₀₁	C₂²	1	1	yes	characteristic
46	#124×C₁₀₁	C₂²	1	4	yes	complemented
47	#123×C₁₀₁	C₂²	1	4	yes	complemented
48	C₄×Q₈×C₁₀₁	C₂²	1	2	yes	complemented
49	C₄×D₄×C₁₀₁	C₂²	1	4	yes	complemented
50	C₂³×D₁₀₁	C₂³	1	4	yes	—
51	C₂×Dic₂₀₂	C₂³	1	4	yes	—
52	C₂×Dic₂₀₂	C₂³	1	2	yes	—
53	Q₈×D₁₀₁	—	2	8	no	—
54	C₂×D₄₀₄	C₂³	1	4	yes	—
55	C₂×D₄₀₄	C₂³	1	2	yes	—
56	C₂×C₂²⋊D₁₀₁	C₂³	1	8	yes	—
57	C₂×C₂²⋊D₁₀₁	C₂³	1	4	yes	—
58	D₄×D₁₀₁	—	2	16	no	—
59	D₄×D₁₀₁	—	4	4	no	—
60	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
61	C₂×C₄×D₁₀₁	—	4	8	no	—
62	C₂×C₄×D₁₀₁	C₂³	1	4	yes	—
63	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₁₀₁	C₂³	1	4	yes	—
65	C₂×C₄×D₁₀₁	C₂³	1	8	yes	—
66	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
67	C₄⋊Dic₁₀₁	C₂³	1	2	yes	—
68	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
69	#13108053	C₂³	1	4	yes	—
70	#13108053	C₂³	1	4	yes	—
71	#13108053	C₂³	1	4	yes	—
72	C₂²⋊Dic₁₀₁	C₂³	1	8	yes	—
73	C₂²⋊Dic₁₀₁	C₂³	1	4	yes	—
74	#13108051	C₂³	1	8	yes	—
75	#13108051	C₂³	1	8	yes	—
76	#13108051	C₂³	1	8	yes	—
77	C₄×Dic₁₀₁	C₂³	1	4	yes	—
78	C₄×Dic₁₀₁	C₂³	1	4	yes	—
79	C₄×Dic₁₀₁	C₂³	1	4	yes	—
80	C₄×Dic₁₀₁	C₂³	1	1	yes	characteristic
81	C₄×Dic₁₀₁	C₂³	1	2	yes	—
82	C₂×Q₈×C₁₀₁	C₂³	1	2	yes	—
83	C₂×D₄×C₁₀₁	C₂³	1	4	yes	—
84	C₂×D₄×C₁₀₁	C₂³	1	2	yes	—
85	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
86	C₄⋊C₄×C₁₀₁	C₂³	1	2	yes	—
87	C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
88	C₂²⋊C₄×C₁₀₁	C₂³	1	8	yes	—
89	C₂²⋊C₄×C₁₀₁	C₂³	1	4	yes	—
90	C₄²×C₁₀₁	C₂³	1	4	yes	—
91	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
92	C₂³×C₁₀₁	C₂⁴	1	4	yes	—
93	C₂²×D₁₀₁	—	2	16	no	—
94	C₂²×D₁₀₁	—	2	8	no	—
95	C₂²×D₁₀₁	C₂⁴	1	4	yes	—
96	C₂²×D₁₀₁	C₂⁴	1	1	yes	characteristic
97	Q₈×C₁₀₁	—	2	4	no	—
98	D₄×C₁₀₁	—	2	8	no	—
99	D₄×C₁₀₁	—	4	2	no	—
100	C₂×C₄×C₁₀₁	—	4	4	no	—
101	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
102	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
103	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
104	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
105	C₂²⋊D₁₀₁	—	2	16	no	—
106	C₂²⋊D₁₀₁	—	4	4	no	—
107	C₂×Dic₁₀₁	—	4	4	no	—
108	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
109	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
110	C₂×Dic₁₀₁	C₂⁴	1	1	yes	characteristic
111	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
112	D₄₀₄	—	2	8	no	—
113	D₄₀₄	—	4	2	no	—
114	C₄×D₁₀₁	—	2	8	no	—
115	C₄×D₁₀₁	—	2	8	no	—
116	C₄×D₁₀₁	—	2	2	no	—
117	C₄×D₁₀₁	—	4	8	no	—
118	C₄×D₁₀₁	#41	1	8	yes	—
119	Dic₂₀₂	—	2	4	no	—
120	Dic₂₀₂	—	2	8	no	—
121	C₂²×C₁₀₁	—	2	8	no	—
122	C₂²×C₁₀₁	—	2	4	no	—
123	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
124	C₂×D₁₀₁	—	2	8	no	—
125	C₂×D₁₀₁	—	2	4	no	—
126	C₂×D₁₀₁	—	4	16	no	—
127	C₂×D₁₀₁	C₂×#41	1	4	yes	—
128	C₂×D₁₀₁	H(5,𝔽₂)	1	2	yes	—
129	C₄×C₁₀₁	—	2	4	no	—
130	C₄×C₁₀₁	—	2	4	no	—
131	C₄×C₁₀₁	—	2	1	no	—
132	C₄×C₁₀₁	C₂×#41	1	4	yes	—
133	Dic₁₀₁	—	2	4	no	—
134	Dic₁₀₁	—	2	4	no	—
135	Dic₁₀₁	—	2	1	no	—
136	Dic₁₀₁	C₂×#41	1	4	yes	—
137	C₂×C₁₀₁	—	4	4	no	—
138	C₂×C₁₀₁	C₂²×#41	1	2	yes	—
139	C₂×C₁₀₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
140	D₁₀₁	—	4	4	no	—
141	D₁₀₁	#559	1	4	yes	direct product factor
142	C₂×#559	—	101	1	no	maximal, Sylow
143	C₁₀₁	C₂×#559	1	1	yes	characteristic, complemented, Sylow
144	#559	—	101	1	no	—
145	#559	—	101	4	no	—
146	#559	—	101	4	no	—
147	#559	—	101	2	no	—
148	#559	—	101	4	no	—
149	#559	D₁₀₁	1	1	yes	characteristic, direct product factor
150	C₂×#127	—	101	1	no	—
151	C₂×#124	—	101	4	no	—
152	C₂×#123	—	101	4	no	—
153	C₂×C₄×Q₈	—	101	2	no	—
154	C₂×C₄×D₄	—	101	4	no	—
155	C₂²×Q₈	—	101	2	no	—
156	C₂²×D₄	—	101	2	no	—
157	C₂²×D₄	—	101	4	no	—
158	C₂³×C₄	—	101	4	no	—
159	#127	—	101	4	no	—
160	#127	—	101	2	no	—
161	#127	—	101	1	no	—
162	#127	C₂×D₁₀₁	1	1	yes	characteristic
163	#124	—	101	4	no	—
164	#124	—	101	4	no	—
165	#124	—	101	8	no	—
166	#124	—	101	8	no	—
167	#124	—	101	4	no	—
168	#124	C₂×D₁₀₁	1	4	yes	complemented
169	#123	—	101	4	no	—
170	#123	—	101	4	no	—
171	#123	—	101	8	no	—
172	#123	—	101	8	no	—
173	#123	—	101	4	no	—
174	#123	C₂×D₁₀₁	1	4	yes	complemented
175	C₄×Q₈	—	101	2	no	—
176	C₄×Q₈	—	101	4	no	—
177	C₄×Q₈	—	101	2	no	—
178	C₄×Q₈	—	101	2	no	—
179	C₄×Q₈	—	101	4	no	—
180	C₄×Q₈	C₂×D₁₀₁	1	2	yes	complemented
181	C₄×D₄	—	101	4	no	—
182	C₄×D₄	—	101	8	no	—
183	C₄×D₄	—	101	4	no	—
184	C₄×D₄	—	101	8	no	—
185	C₄×D₄	—	101	4	no	—
186	C₄×D₄	C₂×D₁₀₁	1	4	yes	complemented
187	C₂×C₄⋊C₄	—	101	2	no	—
188	C₂×C₄⋊C₄	—	101	4	no	—
189	C₂×C₂²⋊C₄	—	101	8	no	—
190	C₂×C₂²⋊C₄	—	101	4	no	—
191	C₂×C₄²	—	101	4	no	—
192	C₂×C₄²	—	101	1	no	—
193	C₂⁴	—	101	4	no	—
194	C₂×Q₈	—	101	4	no	—
195	C₂×Q₈	—	101	2	no	—
196	C₂×Q₈	—	202	8	no	—
197	C₂×Q₈	C₂²×D₁₀₁	1	2	yes	—
198	C₂×D₄	—	101	4	no	—
199	C₂×D₄	—	101	8	no	—
200	C₂×D₄	—	101	2	no	—
201	C₂×D₄	—	101	4	no	—
202	C₂×D₄	—	404	4	no	—
203	C₂×D₄	—	202	16	no	—
204	C₂×D₄	C₂²×D₁₀₁	1	4	yes	—
205	C₂×D₄	C₂²×D₁₀₁	1	2	yes	—
206	C₂²×C₄	—	101	2	no	—
207	C₂²×C₄	—	101	4	no	—
208	C₂²×C₄	—	101	8	no	—
209	C₂²×C₄	—	101	4	no	—
210	C₂²×C₄	—	101	4	no	—
211	C₂²×C₄	—	101	1	no	—
212	C₂²×C₄	—	404	8	no	—
213	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
214	C₄⋊C₄	—	101	4	no	—
215	C₄⋊C₄	—	101	4	no	—
216	C₄⋊C₄	—	101	4	no	—
217	C₄⋊C₄	—	101	2	no	—
218	C₄⋊C₄	—	101	4	no	—
219	C₄⋊C₄	C₂²×D₁₀₁	1	2	yes	—
220	C₄⋊C₄	C₂²×D₁₀₁	1	4	yes	—
221	C₂²⋊C₄	—	101	8	no	—
222	C₂²⋊C₄	—	101	4	no	—
223	C₂²⋊C₄	—	101	8	no	—
224	C₂²⋊C₄	—	101	8	no	—
225	C₂²⋊C₄	—	101	8	no	—
226	C₂²⋊C₄	C₂²×D₁₀₁	1	8	yes	—
227	C₂²⋊C₄	C₂²×D₁₀₁	1	4	yes	—
228	C₄²	—	101	4	no	—
229	C₄²	—	101	4	no	—
230	C₄²	—	101	4	no	—
231	C₄²	—	101	1	no	—
232	C₄²	—	101	2	no	—
233	C₄²	C₂²×D₁₀₁	1	4	yes	—
234	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
235	C₂³	—	101	1	no	—
236	C₂³	—	101	4	no	—
237	C₂³	—	202	8	no	—
238	C₂³	—	202	16	no	—
239	C₂³	C₂³×D₁₀₁	1	4	yes	—
240	Q₈	—	2	4	no	—
241	Q₈	—	202	8	no	—
242	Q₈	—	202	4	no	—
243	D₄	—	2	8	no	—
244	D₄	—	4	2	no	—
245	D₄	—	404	2	no	—
246	D₄	—	404	4	no	—
247	D₄	—	202	16	no	—
248	D₄	—	202	8	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	101	2	no	—
251	C₂×C₄	—	101	8	no	—
252	C₂×C₄	—	101	4	no	—
253	C₂×C₄	—	101	4	no	—
254	C₂×C₄	—	101	1	no	—
255	C₂×C₄	—	404	8	no	—
256	C₂×C₄	—	404	4	no	—
257	C₂×C₄	—	202	8	no	—
258	C₂×C₄	—	202	2	no	—
259	C₂×C₄	—	202	8	no	—
260	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
261	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
262	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
263	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
264	C₂²	—	2	8	no	—
265	C₂²	—	2	4	no	—
266	C₂²	—	101	4	no	—
267	C₂²	—	101	2	no	—
268	C₂²	—	404	16	no	—
269	C₂²	—	202	8	no	—
270	C₂²	—	202	4	no	—
271	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
272	C₄	—	2	4	no	—
273	C₄	—	2	4	no	—
274	C₄	—	2	1	no	—
275	C₄	—	101	4	no	—
276	C₄	—	202	4	no	—
277	C₄	—	202	4	no	—
278	C₄	—	202	1	no	—
279	C₄	#41×D₁₀₁	1	4	yes	—
280	C₂	—	4	4	no	—
281	C₂	—	101	4	no	—
282	C₂	—	404	4	no	—
283	C₂	C₂×#41×D₁₀₁	1	2	yes	—
284	C₂	H(5,𝔽₂)×D₁₀₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₁
commutator, socle, cosocle	C₂²×C₁₀₁	C₂⁵
Fitting	#559×C₁₀₁	C₂
prime radical	#127×D₁₀₁	C₂
hypercenter	#559	D₁₀₁
hypocenter	C₁₀₁	C₂×#559
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ #559

Complements

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

N	H ≅ G/N	type
1. #20265913	C₂	semidirect
2. #20265912	C₂	semidirect
3. #20265911	C₂	semidirect
4. #20265910	C₂	semidirect
5. #20265909	C₂	semidirect
7. #124×D₁₀₁	C₂	semidirect
8. #123×D₁₀₁	C₂	semidirect
9. C₄×Q₈×D₁₀₁	C₂	semidirect
10. C₄×D₄×D₁₀₁	C₂	semidirect
11. #559×C₁₀₁	C₂	semidirect
19. #15692244	C₂²	semidirect
20. #15692243	C₂²	semidirect
21. #15692242	C₂²	semidirect
22. #15692241	C₂²	semidirect
23. #15692241	C₂²	semidirect
24. #15692240	C₂²	semidirect
25. #15692239	C₂²	semidirect
26. #15692238	C₂²	semidirect
27. #15692237	C₂²	semidirect
28. #15692236	C₂²	semidirect
29. Q₈×Dic₁₀₁	C₂²	semidirect
30. C₄×Dic₂₀₂	C₂²	semidirect
31. C₄×Dic₂₀₂	C₂²	semidirect
32. #15692220	C₂²	semidirect
33. #15692220	C₂²	semidirect
34. C₄×D₄₀₄	C₂²	semidirect
35. D₄×Dic₁₀₁	C₂²	semidirect
36. #15692217	C₂²	semidirect
37. C₄×C₂²⋊D₁₀₁	C₂²	semidirect
38. #15692215	C₂²	semidirect
46. #124×C₁₀₁	C₂²	semidirect
47. #123×C₁₀₁	C₂²	semidirect
48. C₄×Q₈×C₁₀₁	C₂²	semidirect
49. C₄×D₄×C₁₀₁	C₂²	semidirect
141. D₁₀₁	#559	direct (N × H)
143. C₁₀₁	C₂×#559	semidirect
149. #559	D₁₀₁	direct (N × H)
168. #124	C₂×D₁₀₁	semidirect
174. #123	C₂×D₁₀₁	semidirect
180. C₄×Q₈	C₂×D₁₀₁	semidirect
186. C₄×D₄	C₂×D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
#20265913	2
#20265912	2
#20265911	2
#20265910	2
#20265909	2
#127×D₁₀₁	2
#124×D₁₀₁	2
#123×D₁₀₁	2
C₄×Q₈×D₁₀₁	2
C₄×D₄×D₁₀₁	2
#559×C₁₀₁	2
C₂×#559	101

Subgroups — #559×D101 #27184716

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #559×D₁₀₁ #27184716