Subgroups — D₄×C₂²⋊Dic₁₀₁ #27173954

Order 12928.

1252 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Dic₁₀₁	C₂	1	2	yes	maximal, complemented
2	C₂²×C₂²⋊Dic₁₀₁	C₂	1	2	yes	maximal, complemented
3	#20265080	C₂	1	2	yes	maximal, complemented
4	#20265057	C₂	1	4	yes	maximal, complemented
5	#20265037	C₂	1	1	yes	maximal, characteristic, complemented
6	#20265033	C₂	1	2	yes	maximal, complemented
7	C₄×C₂²⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic
9	C₂³×Dic₁₀₁	C₂²	1	4	yes	complemented
10	D₄×Dic₁₀₁	—	2	8	no	—
11	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	complemented
12	C₂×C₂²⋊Dic₁₀₁	—	2	8	no	—
13	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
15	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊Dic₁₀₁	C₂²	1	4	yes	complemented
17	C₂×C₂²⋊Dic₁₀₁	C₂²	1	4	yes	complemented
18	C₂×C₂²⋊Dic₁₀₁	C₂²	1	4	yes	complemented
19	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	complemented
20	#15692162	C₂²	1	2	yes	complemented
21	#15692162	C₂²	1	2	yes	complemented
22	C₂⁵×C₁₀₁	C₂²	1	2	yes	—
23	C₂²×D₄×C₁₀₁	—	2	8	no	—
24	C₂²×D₄×C₁₀₁	C₄	1	8	yes	complemented
25	C₂²×D₄×C₁₀₁	C₂²	1	2	yes	—
26	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
27	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
28	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
29	C₂²×Dic₁₀₁	—	2	8	no	—
30	C₂²×Dic₁₀₁	—	2	8	no	—
31	C₂²×Dic₁₀₁	—	4	4	no	—
32	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
33	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
34	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
35	C₄⋊Dic₁₀₁	—	2	4	no	—
36	C₂²⋊Dic₁₀₁	—	2	4	no	—
37	C₂²⋊Dic₁₀₁	—	2	8	no	—
38	C₂²⋊Dic₁₀₁	—	2	2	no	—
39	C₂²⋊Dic₁₀₁	—	2	8	no	—
40	C₂²⋊Dic₁₀₁	—	4	4	no	—
41	C₂²⋊Dic₁₀₁	D₄	1	4	yes	direct product factor
42	C₄×Dic₁₀₁	—	2	4	no	—
43	C₂⁴×C₁₀₁	—	2	8	no	—
44	C₂⁴×C₁₀₁	—	2	4	no	—
45	C₂⁴×C₁₀₁	—	2	4	no	—
46	C₂⁴×C₁₀₁	—	4	4	no	—
47	C₂⁴×C₁₀₁	C₂×C₄	1	8	yes	complemented
48	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
49	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
50	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
51	C₂×D₄×C₁₀₁	—	2	8	no	—
52	C₂×D₄×C₁₀₁	—	2	32	no	—
53	C₂×D₄×C₁₀₁	—	2	4	no	—
54	C₂×D₄×C₁₀₁	C₂×C₄	1	4	yes	—
55	C₂×D₄×C₁₀₁	C₂×C₄	1	8	yes	complemented
56	C₂×D₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
57	C₂×D₄×C₁₀₁	D₄	1	8	yes	—
58	C₂²×C₄×C₁₀₁	—	2	4	no	—
59	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
60	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
62	C₂³×C₁₀₁	—	2	2	no	—
63	C₂³×C₁₀₁	—	2	2	no	—
64	C₂³×C₁₀₁	—	2	2	no	—
65	C₂³×C₁₀₁	—	2	2	no	—
66	C₂³×C₁₀₁	—	2	4	no	—
67	C₂³×C₁₀₁	—	2	16	no	—
68	C₂³×C₁₀₁	—	2	16	no	—
69	C₂³×C₁₀₁	—	2	4	no	—
70	C₂³×C₁₀₁	—	2	2	no	—
71	C₂³×C₁₀₁	—	2	2	no	—
72	C₂³×C₁₀₁	—	2	4	no	—
73	C₂³×C₁₀₁	—	4	8	no	—
74	C₂³×C₁₀₁	—	4	16	no	—
75	C₂³×C₁₀₁	—	4	16	no	—
76	C₂³×C₁₀₁	C₂²×C₄	1	4	yes	—
77	C₂³×C₁₀₁	C₂²×C₄	1	4	yes	—
78	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
79	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
80	C₂³×C₁₀₁	C₂×D₄	1	8	yes	—
81	C₂³×C₁₀₁	#41	1	2	yes	—
82	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
83	D₄×C₁₀₁	—	2	16	no	—
84	D₄×C₁₀₁	—	2	8	no	—
85	D₄×C₁₀₁	C₂²⋊C₄	1	16	yes	complemented
86	C₂×C₄×C₁₀₁	—	2	2	no	—
87	C₂×C₄×C₁₀₁	—	2	8	no	—
88	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
91	C₂×Dic₁₀₁	—	2	2	no	—
92	C₂×Dic₁₀₁	—	2	2	no	—
93	C₂×Dic₁₀₁	—	2	2	no	—
94	C₂×Dic₁₀₁	—	2	4	no	—
95	C₂×Dic₁₀₁	—	2	4	no	—
96	C₂×Dic₁₀₁	—	2	2	no	—
97	C₂×Dic₁₀₁	—	2	2	no	—
98	C₂×Dic₁₀₁	—	2	4	no	—
99	C₂×Dic₁₀₁	—	4	16	no	—
100	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	complemented
101	C₂²×C₁₀₁	—	2	2	no	—
102	C₂²×C₁₀₁	—	2	4	no	—
103	C₂²×C₁₀₁	—	2	2	no	—
104	C₂²×C₁₀₁	—	2	4	no	—
105	C₂²×C₁₀₁	—	2	8	no	—
106	C₂²×C₁₀₁	—	2	4	no	—
107	C₂²×C₁₀₁	—	2	4	no	—
108	C₂²×C₁₀₁	—	2	8	no	—
109	C₂²×C₁₀₁	—	2	4	no	—
110	C₂²×C₁₀₁	—	2	4	no	—
111	C₂²×C₁₀₁	—	2	4	no	—
112	C₂²×C₁₀₁	—	4	32	no	—
113	C₂²×C₁₀₁	—	4	4	no	—
114	C₂²×C₁₀₁	—	4	4	no	—
115	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	8	yes	complemented
116	C₂²×C₁₀₁	C₄×D₄	1	4	yes	complemented
117	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
118	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
119	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
120	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
121	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
122	C₄×C₁₀₁	—	2	2	no	—
123	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	4	yes	complemented
124	Dic₁₀₁	—	2	4	no	—
125	Dic₁₀₁	—	4	4	no	—
126	Dic₁₀₁	—	4	2	no	—
127	C₂×C₁₀₁	—	2	4	no	—
128	C₂×C₁₀₁	—	2	8	no	—
129	C₂×C₁₀₁	—	4	4	no	—
130	C₂×C₁₀₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂×C₁₀₁	#518	1	1	yes	characteristic
133	C₂×C₁₀₁	D₄²	1	2	yes	—
134	C₂×C₁₀₁	#546	1	2	yes	—
135	D₄×C₂²⋊C₄	—	101	1	no	maximal, Sylow
136	C₁₀₁	D₄×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
137	C₂³×D₄	D₁₀₁	1	1	yes	characteristic
138	C₂×C₄×D₄	—	101	2	no	—
139	C₂²×C₂²⋊C₄	—	101	2	no	—
140	#390	—	101	2	no	—
141	#386	—	101	4	no	—
142	#380	—	101	1	no	—
143	#379	—	101	2	no	—
144	C₄×C₂²⋊C₄	—	101	1	no	—
145	C₂⁵	C₂×D₁₀₁	1	2	yes	—
146	C₂²×D₄	—	2	8	no	—
147	C₂²×D₄	Dic₁₀₁	1	8	yes	complemented
148	C₂²×D₄	C₂×D₁₀₁	1	2	yes	—
149	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
150	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
151	C₂³×C₄	—	101	4	no	—
152	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
153	C₄×D₄	—	202	8	no	—
154	C₂×C₄⋊C₄	—	101	2	no	—
155	C₂×C₂²⋊C₄	—	101	1	no	—
156	C₂×C₂²⋊C₄	—	101	4	no	—
157	C₂×C₂²⋊C₄	—	101	2	no	—
158	C₂×C₂²⋊C₄	—	101	4	no	—
159	C₂×C₂²⋊C₄	—	101	4	no	—
160	C₂×C₂²⋊C₄	—	101	1	no	—
161	C₂×C₂²⋊C₄	—	202	8	no	—
162	C₂×C₄²	—	101	2	no	—
163	#95	—	101	2	no	—
164	#95	—	101	2	no	—
165	C₂⁴	—	2	4	no	—
166	C₂⁴	—	2	8	no	—
167	C₂⁴	—	2	4	no	—
168	C₂⁴	—	4	4	no	—
169	C₂⁴	C₂×Dic₁₀₁	1	8	yes	complemented
170	C₂⁴	C₂²×D₁₀₁	1	2	yes	—
171	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
172	C₂⁴	C₂²×D₁₀₁	1	2	yes	—
173	C₂×D₄	—	2	8	no	—
174	C₂×D₄	—	2	32	no	—
175	C₂×D₄	—	2	4	no	—
176	C₂×D₄	C₂×Dic₁₀₁	1	4	yes	—
177	C₂×D₄	C₂×Dic₁₀₁	1	8	yes	complemented
178	C₂×D₄	C₂×Dic₁₀₁	1	4	yes	complemented
179	C₂×D₄	C₂²⋊D₁₀₁	1	8	yes	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	101	2	no	—
182	C₂²×C₄	—	101	2	no	—
183	C₂²×C₄	—	101	4	no	—
184	C₂²×C₄	—	404	4	no	—
185	C₂²×C₄	—	202	8	no	—
186	C₂²×C₄	—	202	8	no	—
187	C₂²×C₄	C₂×Dic₁₀₁	1	4	yes	complemented
188	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
189	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
190	C₄⋊C₄	—	202	4	no	—
191	C₂²⋊C₄	—	101	4	no	—
192	C₂²⋊C₄	—	404	4	no	—
193	C₂²⋊C₄	—	202	8	no	—
194	C₂²⋊C₄	—	202	4	no	—
195	C₂²⋊C₄	—	202	8	no	—
196	C₂²⋊C₄	—	202	2	no	—
197	C₄²	—	202	4	no	—
198	C₂³	—	2	2	no	—
199	C₂³	—	2	2	no	—
200	C₂³	—	2	2	no	—
201	C₂³	—	2	4	no	—
202	C₂³	—	2	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	4	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	2	4	no	—
207	C₂³	—	2	16	no	—
208	C₂³	—	2	16	no	—
209	C₂³	—	4	8	no	—
210	C₂³	—	4	16	no	—
211	C₂³	—	4	16	no	—
212	C₂³	C₂²×Dic₁₀₁	1	4	yes	—
213	C₂³	C₂²×Dic₁₀₁	1	4	yes	—
214	C₂³	C₂²×Dic₁₀₁	1	2	yes	—
215	C₂³	D₄×D₁₀₁	1	2	yes	—
216	C₂³	C₂×C₂²⋊D₁₀₁	1	8	yes	—
217	C₂³	#13108073	1	2	yes	—
218	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
219	D₄	—	2	16	no	—
220	D₄	—	2	8	no	—
221	D₄	C₂²⋊Dic₁₀₁	1	16	yes	direct product factor
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	101	4	no	—
225	C₂×C₄	—	404	16	no	—
226	C₂×C₄	—	202	4	no	—
227	C₂×C₄	—	202	2	no	—
228	C₂×C₄	—	202	4	no	—
229	C₂×C₄	—	202	2	no	—
230	C₂×C₄	—	202	2	no	—
231	C₂×C₄	—	202	4	no	—
232	C₂×C₄	—	202	2	no	—
233	C₂×C₄	—	202	2	no	—
234	C₂×C₄	C₂²×Dic₁₀₁	1	2	yes	—
235	C₂×C₄	C₂²×Dic₁₀₁	1	2	yes	—
236	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	4	yes	—
237	C₂²	—	2	8	no	—
238	C₂²	—	2	4	no	—
239	C₂²	—	2	4	no	—
240	C₂²	—	2	8	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	4	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	4	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	2	4	no	—
247	C₂²	—	2	4	no	—
248	C₂²	—	4	32	no	—
249	C₂²	—	4	4	no	—
250	C₂²	—	4	4	no	—
251	C₂²	C₂×C₂²⋊Dic₁₀₁	1	8	yes	complemented
252	C₂²	D₄×Dic₁₀₁	1	4	yes	complemented
253	C₂²	C₂³×Dic₁₀₁	1	1	yes	characteristic
254	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
255	C₂²	C₂²×C₂²⋊D₁₀₁	1	2	yes	—
256	C₂²	C₂×#13108073	1	1	yes	characteristic
257	C₂²	#15692300	1	2	yes	—
258	C₄	—	2	2	no	—
259	C₄	—	404	4	no	—
260	C₄	—	404	2	no	—
261	C₄	—	202	4	no	—
262	C₄	C₂×C₂²⋊Dic₁₀₁	1	4	yes	complemented
263	C₂	—	2	4	no	—
264	C₂	—	2	8	no	—
265	C₂	—	4	4	no	—
266	C₂	C₂²×C₂²⋊Dic₁₀₁	1	1	yes	characteristic
267	C₂	C₂×D₄×Dic₁₀₁	1	1	yes	characteristic
268	C₂	#20265652	1	1	yes	characteristic
269	C₂	D₄×C₂²⋊D₁₀₁	1	2	yes	—
270	C₂	#20265821	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting, prime radical	C₂³×D₄×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
hypercenter	C₂³×D₄	D₁₀₁
hypocenter	C₁₀₁	D₄×C₂²⋊C₄
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂³×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×Dic₁₀₁	C₂	semidirect
2. C₂²×C₂²⋊Dic₁₀₁	C₂	semidirect
3. #20265080	C₂	semidirect
4. #20265057	C₂	semidirect
5. #20265037	C₂	semidirect
6. #20265033	C₂	semidirect
7. C₄×C₂²⋊Dic₁₀₁	C₂	semidirect
9. C₂³×Dic₁₀₁	C₂²	semidirect
11. C₂×C₄⋊Dic₁₀₁	C₂²	semidirect
16. C₂×C₂²⋊Dic₁₀₁	C₂²	semidirect
17. C₂×C₂²⋊Dic₁₀₁	C₂²	semidirect
18. C₂×C₂²⋊Dic₁₀₁	C₂²	semidirect
19. C₂×C₄×Dic₁₀₁	C₂²	semidirect
20. #15692162	C₂²	semidirect
21. #15692162	C₂²	semidirect
24. C₂²×D₄×C₁₀₁	C₄	semidirect
41. C₂²⋊Dic₁₀₁	D₄	direct (N × H)
47. C₂⁴×C₁₀₁	C₂×C₄	semidirect
55. C₂×D₄×C₁₀₁	C₂×C₄	semidirect
56. C₂×D₄×C₁₀₁	C₂×C₄	semidirect
59. C₂²×C₄×C₁₀₁	C₂×C₄	semidirect
85. D₄×C₁₀₁	C₂²⋊C₄	semidirect
100. C₂×Dic₁₀₁	C₂×D₄	semidirect
115. C₂²×C₁₀₁	C₂×C₂²⋊C₄	semidirect
116. C₂²×C₁₀₁	C₄×D₄	semidirect
123. C₄×C₁₀₁	C₂×C₂²⋊C₄	semidirect
136. C₁₀₁	D₄×C₂²⋊C₄	semidirect
147. C₂²×D₄	Dic₁₀₁	semidirect
169. C₂⁴	C₂×Dic₁₀₁	semidirect
177. C₂×D₄	C₂×Dic₁₀₁	semidirect
178. C₂×D₄	C₂×Dic₁₀₁	semidirect
187. C₂²×C₄	C₂×Dic₁₀₁	semidirect
221. D₄	C₂²⋊Dic₁₀₁	direct (N × H)
251. C₂²	C₂×C₂²⋊Dic₁₀₁	semidirect
252. C₂²	D₄×Dic₁₀₁	semidirect
262. C₄	C₂×C₂²⋊Dic₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×Dic₁₀₁	2
C₂²×C₂²⋊Dic₁₀₁	2
#20265080	2
#20265057	2
#20265037	2
#20265033	2
C₄×C₂²⋊Dic₁₀₁	2
C₂³×D₄×C₁₀₁	2
D₄×C₂²⋊C₄	101

Subgroups — D4×C22⋊Dic101 #27173954

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×C₂²⋊Dic₁₀₁ #27173954