Subgroups — #407×D₁₀₁ #27179565

Order 12928.

1008 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×D₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×D₁₀₁	C₂	1	2	yes	maximal, complemented
3	#20265141	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265140	C₂	1	2	yes	maximal, complemented
5	#20265139	C₂	1	4	yes	maximal, complemented
6	C₂²⋊C₈×D₁₀₁	C₂	1	4	yes	maximal, complemented
7	#407×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂⁴×D₁₀₁	C₄	1	1	yes	characteristic
9	C₂³×Dic₁₀₁	C₄	1	1	yes	characteristic
10	C₂²×C₄×D₁₀₁	—	2	8	no	—
11	C₂²×C₄×D₁₀₁	C₄	1	1	yes	characteristic
12	C₂²×C₄×D₁₀₁	C₄	1	1	yes	characteristic
13	C₂²×C₄×D₁₀₁	C₄	1	2	yes	—
14	C₂²×C₄×D₁₀₁	C₄	1	2	yes	—
15	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
16	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
17	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
18	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
19	C₂×#13108058	C₂²	1	2	yes	complemented
20	C₂×#13108057	C₂²	1	4	yes	complemented
21	M₁₆×D₁₀₁	—	2	8	no	—
22	C₂×C₈×D₁₀₁	C₂²	1	4	yes	complemented
23	C₂²⋊#11717579	C₂²	1	4	yes	complemented
24	#15692167	C₂²	1	8	yes	complemented
25	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
26	C₂×M₁₆×C₁₀₁	C₂²	1	2	yes	complemented
27	C₂²⋊C₈×C₁₀₁	C₂²	1	4	yes	complemented
28	C₂³×D₁₀₁	—	2	4	no	—
29	C₂³×D₁₀₁	—	2	8	no	—
30	C₂³×D₁₀₁	C₂×C₄	1	1	yes	characteristic
31	C₂³×D₁₀₁	C₂×C₄	1	2	yes	—
32	C₂³×D₁₀₁	C₂×C₄	1	1	yes	characteristic
33	C₂³×D₁₀₁	C₂×C₄	1	2	yes	—
34	C₂²×Dic₁₀₁	—	2	4	no	—
35	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
36	C₂²×Dic₁₀₁	C₂×C₄	1	2	yes	—
37	C₂²×Dic₁₀₁	C₂×C₄	1	2	yes	—
38	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×D₁₀₁	—	2	4	no	—
40	C₂×C₄×D₁₀₁	—	2	4	no	—
41	C₂×C₄×D₁₀₁	—	2	4	no	—
42	C₂×C₄×D₁₀₁	—	2	8	no	—
43	C₂×C₄×D₁₀₁	—	2	8	no	—
44	C₂×C₄×D₁₀₁	—	2	8	no	—
45	C₂×C₄×D₁₀₁	—	2	8	no	—
46	C₂×C₄×D₁₀₁	C₂×C₄	1	2	yes	—
47	C₂×C₄×D₁₀₁	C₂×C₄	1	2	yes	—
48	C₂×C₄×D₁₀₁	C₂×C₄	1	2	yes	—
49	C₂×C₄×D₁₀₁	C₂×C₄	1	2	yes	—
50	C₂×C₄×D₁₀₁	C₂×C₄	1	2	yes	—
51	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₁₀₁	D₄	1	8	yes	—
54	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
55	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
56	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
57	#13108058	—	2	4	no	—
58	#13108057	—	2	8	no	—
59	C₂×#11717579	C₂³	1	4	yes	complemented
60	C₈×D₁₀₁	—	2	8	no	—
61	C₂⁴×C₁₀₁	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₀₁	—	2	4	no	—
63	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
65	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
66	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
67	M₁₆×C₁₀₁	—	2	4	no	—
68	C₂×C₈×C₁₀₁	C₂³	1	4	yes	complemented
69	C₂³×C₁₀₁	—	2	2	no	—
70	C₂³×C₁₀₁	—	2	4	no	—
71	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
72	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
73	C₂²×D₁₀₁	—	2	32	no	—
74	C₂²×D₁₀₁	—	2	2	no	—
75	C₂²×D₁₀₁	—	2	4	no	—
76	C₂²×D₁₀₁	—	2	4	no	—
77	C₂²×D₁₀₁	—	2	4	no	—
78	C₂²×D₁₀₁	—	2	4	no	—
79	C₂²×D₁₀₁	—	2	4	no	—
80	C₂²×D₁₀₁	—	2	4	no	—
81	C₂²×D₁₀₁	—	2	4	no	—
82	C₂²×D₁₀₁	C₂²⋊C₄	1	4	yes	—
83	C₂²×D₁₀₁	M₁₆	1	8	yes	complemented
84	C₂²×D₁₀₁	C₂²×C₄	1	1	yes	characteristic
85	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
86	C₂²×D₁₀₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₀₁	—	2	2	no	—
88	C₂×C₄×C₁₀₁	—	2	4	no	—
89	C₂×C₄×C₁₀₁	—	2	4	no	—
90	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
93	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
94	C₂×Dic₁₀₁	—	2	2	no	—
95	C₂×Dic₁₀₁	—	2	4	no	—
96	C₂×Dic₁₀₁	—	2	4	no	—
97	C₂×Dic₁₀₁	C₂²⋊C₄	1	4	yes	—
98	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
99	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
100	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
101	C₄×D₁₀₁	—	2	4	no	—
102	C₄×D₁₀₁	—	2	4	no	—
103	C₄×D₁₀₁	—	2	4	no	—
104	C₄×D₁₀₁	—	2	4	no	—
105	C₄×D₁₀₁	—	2	4	no	—
106	C₄×D₁₀₁	—	2	4	no	—
107	C₄×D₁₀₁	C₂²⋊C₄	1	4	yes	—
108	C₄×D₁₀₁	C₂²⋊C₄	1	4	yes	—
109	C₄×D₁₀₁	C₂×D₄	1	4	yes	—
110	C₄×D₁₀₁	C₂×D₄	1	4	yes	—
111	C₈×C₁₀₁	—	2	4	no	—
112	#11717579	—	2	4	no	—
113	C₂²×C₁₀₁	—	2	8	no	—
114	C₂²×C₁₀₁	—	2	2	no	—
115	C₂²×C₁₀₁	—	2	2	no	—
116	C₂²×C₁₀₁	—	2	2	no	—
117	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
118	C₂²×C₁₀₁	C₂×M₁₆	1	4	yes	complemented
119	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
120	C₂×D₁₀₁	—	2	16	no	—
121	C₂×D₁₀₁	—	2	8	no	—
122	C₂×D₁₀₁	—	2	2	no	—
123	C₂×D₁₀₁	—	2	2	no	—
124	C₂×D₁₀₁	—	2	2	no	—
125	C₂×D₁₀₁	—	2	8	no	—
126	C₂×D₁₀₁	—	2	16	no	—
127	C₂×D₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
128	C₂×D₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
129	C₂×D₁₀₁	C₂×M₁₆	1	4	yes	—
130	C₂×D₁₀₁	C₂×M₁₆	1	4	yes	complemented
131	C₄×C₁₀₁	—	2	2	no	—
132	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
133	C₄×C₁₀₁	C₂²×D₄	1	2	yes	—
134	Dic₁₀₁	—	2	2	no	—
135	Dic₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
136	Dic₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
137	C₂×C₁₀₁	—	2	2	no	—
138	C₂×C₁₀₁	—	2	4	no	—
139	C₂×C₁₀₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₁₀₁	C₂²×M₁₆	1	2	yes	—
141	D₁₀₁	—	2	2	no	—
142	D₁₀₁	—	2	4	no	—
143	D₁₀₁	#407	1	4	yes	direct product factor
144	C₂×#407	—	101	1	no	maximal, Sylow
145	C₁₀₁	C₂×#407	1	1	yes	characteristic, complemented, Sylow
146	C₂⁴×C₄	—	101	1	no	—
147	C₂²×M₁₆	—	101	2	no	—
148	#407	—	101	1	no	—
149	#407	—	101	2	no	—
150	#407	—	101	4	no	—
151	#407	D₁₀₁	1	1	yes	characteristic, direct product factor
152	C₂×C₂²⋊C₈	—	101	4	no	—
153	C₂⁵	—	101	1	no	—
154	C₂³×C₄	—	101	1	no	—
155	C₂³×C₄	—	101	1	no	—
156	C₂³×C₄	—	101	1	no	—
157	C₂³×C₄	—	101	2	no	—
158	C₂³×C₄	—	101	2	no	—
159	C₂³×C₄	—	101	1	no	—
160	C₂³×C₄	—	101	2	no	—
161	C₂³×C₄	—	101	2	no	—
162	C₂³×C₄	—	101	1	no	—
163	C₂³×C₄	—	202	8	no	—
164	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
165	C₂×M₁₆	—	101	2	no	—
166	C₂×M₁₆	—	101	4	no	—
167	C₂×M₁₆	—	202	8	no	—
168	C₂×M₁₆	C₂×D₁₀₁	1	2	yes	complemented
169	C₂²×C₈	—	101	4	no	—
170	C₂²⋊C₈	—	101	4	no	—
171	C₂²⋊C₈	—	101	8	no	—
172	C₂²⋊C₈	C₂×D₁₀₁	1	4	yes	complemented
173	C₂⁴	—	101	1	no	—
174	C₂⁴	—	101	2	no	—
175	C₂⁴	—	101	2	no	—
176	C₂⁴	—	101	1	no	—
177	C₂⁴	—	202	4	no	—
178	C₂⁴	—	202	8	no	—
179	C₂⁴	C₄×D₁₀₁	1	1	yes	characteristic
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	101	1	no	—
182	C₂²×C₄	—	101	1	no	—
183	C₂²×C₄	—	101	1	no	—
184	C₂²×C₄	—	101	8	no	—
185	C₂²×C₄	—	101	1	no	—
186	C₂²×C₄	—	101	1	no	—
187	C₂²×C₄	—	101	2	no	—
188	C₂²×C₄	—	101	2	no	—
189	C₂²×C₄	—	101	1	no	—
190	C₂²×C₄	—	101	2	no	—
191	C₂²×C₄	—	101	2	no	—
192	C₂²×C₄	—	101	2	no	—
193	C₂²×C₄	—	101	2	no	—
194	C₂²×C₄	—	101	2	no	—
195	C₂²×C₄	—	101	2	no	—
196	C₂²×C₄	—	202	4	no	—
197	C₂²×C₄	—	202	8	no	—
198	C₂²×C₄	—	202	8	no	—
199	C₂²×C₄	—	202	8	no	—
200	C₂²×C₄	—	202	8	no	—
201	C₂²×C₄	—	202	4	no	—
202	C₂²×C₄	—	202	4	no	—
203	C₂²×C₄	—	202	4	no	—
204	C₂²×C₄	C₄×D₁₀₁	1	1	yes	characteristic
205	C₂²×C₄	C₄×D₁₀₁	1	2	yes	—
206	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
207	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
208	M₁₆	—	2	4	no	—
209	M₁₆	—	202	8	no	—
210	M₁₆	—	202	4	no	—
211	C₂×C₈	—	101	4	no	—
212	C₂×C₈	—	202	8	no	—
213	C₂×C₈	C₂²×D₁₀₁	1	4	yes	complemented
214	C₂³	—	2	2	no	—
215	C₂³	—	2	4	no	—
216	C₂³	—	101	1	no	—
217	C₂³	—	101	1	no	—
218	C₂³	—	101	2	no	—
219	C₂³	—	101	4	no	—
220	C₂³	—	101	8	no	—
221	C₂³	—	202	4	no	—
222	C₂³	—	202	4	no	—
223	C₂³	—	202	4	no	—
224	C₂³	—	202	4	no	—
225	C₂³	—	202	4	no	—
226	C₂³	—	202	4	no	—
227	C₂³	—	202	32	no	—
228	C₂³	—	202	2	no	—
229	C₂³	—	202	4	no	—
230	C₂³	C₂×C₄×D₁₀₁	1	1	yes	characteristic
231	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	101	1	no	—
236	C₂×C₄	—	101	1	no	—
237	C₂×C₄	—	101	4	no	—
238	C₂×C₄	—	101	4	no	—
239	C₂×C₄	—	101	4	no	—
240	C₂×C₄	—	101	4	no	—
241	C₂×C₄	—	101	4	no	—
242	C₂×C₄	—	101	2	no	—
243	C₂×C₄	—	202	4	no	—
244	C₂×C₄	—	202	4	no	—
245	C₂×C₄	—	202	4	no	—
246	C₂×C₄	—	202	4	no	—
247	C₂×C₄	—	202	4	no	—
248	C₂×C₄	—	202	4	no	—
249	C₂×C₄	—	202	2	no	—
250	C₂×C₄	—	202	4	no	—
251	C₂×C₄	—	202	4	no	—
252	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
253	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
254	C₂×C₄	D₄×D₁₀₁	1	4	yes	—
255	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
256	C₈	—	2	4	no	—
257	C₈	—	202	4	no	—
258	C₂²	—	2	8	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	101	2	no	—
263	C₂²	—	101	4	no	—
264	C₂²	—	101	2	no	—
265	C₂²	—	101	4	no	—
266	C₂²	—	202	8	no	—
267	C₂²	—	202	16	no	—
268	C₂²	—	202	8	no	—
269	C₂²	—	202	16	no	—
270	C₂²	—	202	2	no	—
271	C₂²	—	202	2	no	—
272	C₂²	—	202	2	no	—
273	C₂²	C₂²⋊C₄×D₁₀₁	1	2	yes	—
274	C₂²	M₁₆×D₁₀₁	1	4	yes	complemented
275	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
276	C₄	—	2	2	no	—
277	C₄	—	101	2	no	—
278	C₄	—	101	2	no	—
279	C₄	—	202	2	no	—
280	C₄	C₂²⋊C₄×D₁₀₁	1	2	yes	—
281	C₄	C₂×D₄×D₁₀₁	1	2	yes	—
282	C₂	—	2	2	no	—
283	C₂	—	2	4	no	—
284	C₂	—	101	4	no	—
285	C₂	—	202	2	no	—
286	C₂	—	202	4	no	—
287	C₂	C₂×C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
288	C₂	C₂×M₁₆×D₁₀₁	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₁
commutator, socle	C₂²×C₁₀₁	C₂³×C₄
Fitting	#407×C₁₀₁	C₂
prime radical	C₂⁴×D₁₀₁	C₄
cosocle	C₂×C₄×C₁₀₁	C₂⁴
hypercenter	#407	D₁₀₁
hypocenter	C₁₀₁	C₂×#407
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ #407

Complements

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

N	H ≅ G/N	type
2. C₂×M₁₆×D₁₀₁	C₂	semidirect
3. #20265141	C₂	semidirect
4. #20265140	C₂	semidirect
5. #20265139	C₂	semidirect
6. C₂²⋊C₈×D₁₀₁	C₂	semidirect
7. #407×C₁₀₁	C₂	semidirect
19. C₂×#13108058	C₂²	semidirect
20. C₂×#13108057	C₂²	semidirect
22. C₂×C₈×D₁₀₁	C₂²	semidirect
23. C₂²⋊#11717579	C₂²	semidirect
24. #15692167	C₂²	semidirect
26. C₂×M₁₆×C₁₀₁	C₂²	semidirect
27. C₂²⋊C₈×C₁₀₁	C₂²	semidirect
59. C₂×#11717579	C₂³	semidirect
68. C₂×C₈×C₁₀₁	C₂³	semidirect
83. C₂²×D₁₀₁	M₁₆	semidirect
118. C₂²×C₁₀₁	C₂×M₁₆	semidirect
130. C₂×D₁₀₁	C₂×M₁₆	semidirect
143. D₁₀₁	#407	direct (N × H)
145. C₁₀₁	C₂×#407	semidirect
151. #407	D₁₀₁	direct (N × H)
168. C₂×M₁₆	C₂×D₁₀₁	semidirect
172. C₂²⋊C₈	C₂×D₁₀₁	semidirect
213. C₂×C₈	C₂²×D₁₀₁	semidirect
274. C₂²	M₁₆×D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₄×D₁₀₁	2
C₂×M₁₆×D₁₀₁	2
#20265141	2
#20265140	2
#20265139	2
C₂²⋊C₈×D₁₀₁	2
#407×C₁₀₁	2
C₂×#407	101

Subgroups — #407×D101 #27179565

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #407×D₁₀₁ #27179565