Subgroups — #389×D₁₀₁ #27173579

Order 12928.

800 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₄×D₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₄×D₁₀₁	C₂	1	2	yes	maximal
3	C₂×C₄⋊C₄×D₁₀₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄²×D₁₀₁	C₂	1	1	yes	maximal, characteristic
5	#20265076	C₂	1	1	yes	maximal, characteristic, complemented
6	#20265075	C₂	1	1	yes	maximal, characteristic, complemented
7	#20265074	C₂	1	2	yes	maximal, complemented
8	#20265073	C₂	1	1	yes	maximal, characteristic, complemented
9	#20265072	C₂	1	2	yes	maximal, complemented
10	#95×D₁₀₁	C₂	1	2	yes	maximal
11	#389×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
13	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
14	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
20	C₂×#13108053	C₂²	1	2	yes	—
21	C₂×#13108053	C₂²	1	2	yes	—
22	C₂×#13108053	C₂²	1	2	yes	—
23	C₂×#13108053	C₂²	1	2	yes	—
24	C₄⋊C₄×D₁₀₁	—	2	8	no	—
25	C₄⋊C₄×D₁₀₁	—	2	4	no	—
26	C₄⋊C₄×D₁₀₁	C₄	1	8	yes	complemented
27	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
30	C₄²×D₁₀₁	—	2	4	no	—
31	#15692162	C₂²	1	2	yes	—
32	#15692162	C₂²	1	2	yes	—
33	#15692162	C₂²	1	2	yes	—
34	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
35	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
37	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
38	#95×C₁₀₁	C₂²	1	2	yes	—
39	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
40	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
41	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
42	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
43	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
44	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
45	C₂×C₄×D₁₀₁	—	2	2	no	—
46	C₂×C₄×D₁₀₁	—	2	2	no	—
47	C₂×C₄×D₁₀₁	—	2	4	no	—
48	C₂×C₄×D₁₀₁	—	2	4	no	—
49	C₂×C₄×D₁₀₁	—	2	2	no	—
50	C₂×C₄×D₁₀₁	—	2	2	no	—
51	C₂×C₄×D₁₀₁	—	2	4	no	—
52	C₂×C₄×D₁₀₁	—	2	4	no	—
53	C₂×C₄×D₁₀₁	—	2	4	no	—
54	C₂×C₄×D₁₀₁	C₂×C₄	1	8	yes	—
55	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
56	C₂×C₄×D₁₀₁	D₄	1	4	yes	—
57	C₂×C₄×D₁₀₁	D₄	1	4	yes	—
58	C₂×C₄×D₁₀₁	Q₈	1	4	yes	—
59	C₂×C₄×D₁₀₁	Q₈	1	4	yes	—
60	C₄⋊Dic₁₀₁	—	2	2	no	—
61	C₄⋊Dic₁₀₁	—	2	4	no	—
62	C₄⋊Dic₁₀₁	C₂×C₄	1	4	yes	complemented
63	#13108053	—	2	4	no	—
64	#13108053	—	2	4	no	—
65	#13108053	—	2	4	no	—
66	#13108053	C₂×C₄	1	8	yes	complemented
67	C₄×Dic₁₀₁	—	2	2	no	—
68	C₄×Dic₁₀₁	—	2	2	no	—
69	C₄×Dic₁₀₁	—	2	2	no	—
70	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
71	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
72	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
73	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
74	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₁₀₁	—	2	4	no	—
76	C₄⋊C₄×C₁₀₁	—	2	2	no	—
77	C₄⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
78	C₄²×C₁₀₁	—	2	2	no	—
79	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
80	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
81	C₂²×D₁₀₁	C₂×D₄	1	2	yes	—
82	C₂²×D₁₀₁	C₂×D₄	1	2	yes	—
83	C₂²×D₁₀₁	C₂×Q₈	1	2	yes	—
84	C₂²×D₁₀₁	C₂×Q₈	1	2	yes	—
85	C₂²×D₁₀₁	#41	1	2	yes	—
86	C₂²×D₁₀₁	#41	1	2	yes	—
87	C₂×C₄×C₁₀₁	—	2	1	no	—
88	C₂×C₄×C₁₀₁	—	2	1	no	—
89	C₂×C₄×C₁₀₁	—	2	2	no	—
90	C₂×C₄×C₁₀₁	—	2	2	no	—
91	C₂×C₄×C₁₀₁	—	2	1	no	—
92	C₂×C₄×C₁₀₁	—	2	1	no	—
93	C₂×C₄×C₁₀₁	—	2	2	no	—
94	C₂×C₄×C₁₀₁	—	2	2	no	—
95	C₂×C₄×C₁₀₁	—	2	2	no	—
96	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
97	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
98	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
100	C₂×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
101	C₂×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
102	C₂×Dic₁₀₁	—	2	1	no	—
103	C₂×Dic₁₀₁	—	2	1	no	—
104	C₂×Dic₁₀₁	—	2	2	no	—
105	C₂×Dic₁₀₁	—	2	2	no	—
106	C₂×Dic₁₀₁	—	2	1	no	—
107	C₂×Dic₁₀₁	—	2	1	no	—
108	C₂×Dic₁₀₁	—	2	2	no	—
109	C₂×Dic₁₀₁	—	2	2	no	—
110	C₂×Dic₁₀₁	—	2	2	no	—
111	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
112	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
113	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
114	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
115	C₂×Dic₁₀₁	C₂×Q₈	1	2	yes	—
116	C₂×Dic₁₀₁	C₂×Q₈	1	2	yes	—
117	C₄×D₁₀₁	—	2	16	no	—
118	C₄×D₁₀₁	—	2	8	no	—
119	C₄×D₁₀₁	—	2	8	no	—
120	C₄×D₁₀₁	—	4	8	no	—
121	C₄×D₁₀₁	C₄⋊C₄	1	16	yes	complemented
122	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
123	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
124	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
125	C₂²×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
126	C₂²×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
127	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
128	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
129	C₂×D₁₀₁	C₂×C₄⋊C₄	1	4	yes	—
130	C₂×D₁₀₁	C₄×D₄	1	4	yes	—
131	C₂×D₁₀₁	C₄×Q₈	1	4	yes	—
132	C₂×D₁₀₁	#121	1	4	yes	—
133	C₂×D₁₀₁	C₂²⋊Q₈	1	4	yes	—
134	C₂×D₁₀₁	#125	1	4	yes	—
135	C₂×D₁₀₁	C₄⋊Q₈	1	4	yes	—
136	C₄×C₁₀₁	—	2	4	no	—
137	C₄×C₁₀₁	—	2	2	no	—
138	C₄×C₁₀₁	—	2	2	no	—
139	C₄×C₁₀₁	—	4	2	no	—
140	C₄×C₁₀₁	C₂×C₄⋊C₄	1	4	yes	complemented
141	Dic₁₀₁	—	2	4	no	—
142	Dic₁₀₁	—	2	2	no	—
143	Dic₁₀₁	—	2	2	no	—
144	Dic₁₀₁	—	4	2	no	—
145	Dic₁₀₁	C₂×C₄⋊C₄	1	4	yes	complemented
146	C₂×C₁₀₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
148	C₂×C₁₀₁	C₂×C₄×Q₈	1	1	yes	characteristic
149	C₂×C₁₀₁	C₂×#121	1	1	yes	characteristic
150	C₂×C₁₀₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
151	C₂×C₁₀₁	C₂×#125	1	1	yes	characteristic
152	C₂×C₁₀₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
153	D₁₀₁	#389	1	8	yes	direct product factor
154	C₂×#389	—	101	1	no	maximal, Sylow
155	C₁₀₁	C₂×#389	1	1	yes	characteristic, complemented, Sylow
156	C₂²×C₄⋊C₄	—	101	1	no	—
157	C₂²×C₄⋊C₄	—	101	1	no	—
158	C₂²×C₄⋊C₄	—	101	2	no	—
159	C₂²×C₄²	—	101	1	no	—
160	#389	—	101	2	no	—
161	#389	—	101	1	no	—
162	#389	—	101	1	no	—
163	#389	—	101	1	no	—
164	#389	—	101	2	no	—
165	#389	D₁₀₁	1	1	yes	characteristic, direct product factor
166	C₂×#95	—	101	2	no	—
167	C₂³×C₄	—	101	1	no	—
168	C₂³×C₄	—	101	2	no	—
169	C₂³×C₄	—	101	1	no	—
170	C₂³×C₄	—	101	1	no	—
171	C₂³×C₄	—	101	2	no	—
172	C₂×C₄⋊C₄	—	101	2	no	—
173	C₂×C₄⋊C₄	—	101	1	no	—
174	C₂×C₄⋊C₄	—	101	8	no	—
175	C₂×C₄⋊C₄	—	101	2	no	—
176	C₂×C₄⋊C₄	—	101	2	no	—
177	C₂×C₄⋊C₄	—	101	2	no	—
178	C₂×C₄⋊C₄	—	101	2	no	—
179	C₂×C₄⋊C₄	—	101	1	no	—
180	C₂×C₄⋊C₄	—	202	4	no	—
181	C₂×C₄⋊C₄	—	202	8	no	—
182	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
183	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
184	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
185	C₂×C₄²	—	101	1	no	—
186	C₂×C₄²	—	101	1	no	—
187	C₂×C₄²	—	101	1	no	—
188	C₂×C₄²	—	202	4	no	—
189	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
190	#95	—	101	2	no	—
191	#95	—	101	2	no	—
192	#95	—	101	2	no	—
193	#95	C₂×D₁₀₁	1	2	yes	—
194	C₂⁴	—	101	1	no	—
195	C₂²×C₄	—	101	1	no	—
196	C₂²×C₄	—	101	4	no	—
197	C₂²×C₄	—	101	4	no	—
198	C₂²×C₄	—	101	4	no	—
199	C₂²×C₄	—	101	1	no	—
200	C₂²×C₄	—	101	2	no	—
201	C₂²×C₄	—	101	1	no	—
202	C₂²×C₄	—	101	8	no	—
203	C₂²×C₄	—	101	4	no	—
204	C₂²×C₄	—	101	4	no	—
205	C₂²×C₄	—	101	2	no	—
206	C₂²×C₄	—	202	4	no	—
207	C₂²×C₄	—	202	4	no	—
208	C₂²×C₄	—	202	4	no	—
209	C₂²×C₄	—	202	2	no	—
210	C₂²×C₄	—	202	2	no	—
211	C₂²×C₄	—	202	4	no	—
212	C₂²×C₄	—	202	4	no	—
213	C₂²×C₄	—	202	2	no	—
214	C₂²×C₄	—	202	2	no	—
215	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
216	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
217	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
218	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
219	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	101	8	no	—
223	C₄⋊C₄	—	101	4	no	—
224	C₄⋊C₄	—	202	4	no	—
225	C₄⋊C₄	—	202	4	no	—
226	C₄⋊C₄	—	202	4	no	—
227	C₄⋊C₄	—	202	4	no	—
228	C₄⋊C₄	—	202	2	no	—
229	C₄⋊C₄	C₄×D₁₀₁	1	4	yes	complemented
230	C₄²	—	2	2	no	—
231	C₄²	—	202	2	no	—
232	C₄²	—	202	2	no	—
233	C₄²	—	202	2	no	—
234	C₂³	—	101	2	no	—
235	C₂³	—	101	2	no	—
236	C₂³	—	101	2	no	—
237	C₂³	—	101	2	no	—
238	C₂³	—	101	2	no	—
239	C₂³	—	101	2	no	—
240	C₂³	—	101	2	no	—
241	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	101	2	no	—
252	C₂×C₄	—	101	2	no	—
253	C₂×C₄	—	101	2	no	—
254	C₂×C₄	—	101	16	no	—
255	C₂×C₄	—	101	4	no	—
256	C₂×C₄	—	101	2	no	—
257	C₂×C₄	—	101	2	no	—
258	C₂×C₄	—	404	8	no	—
259	C₂×C₄	—	202	16	no	—
260	C₂×C₄	—	202	8	no	—
261	C₂×C₄	—	202	8	no	—
262	C₂×C₄	—	202	2	no	—
263	C₂×C₄	—	202	2	no	—
264	C₂×C₄	—	202	2	no	—
265	C₂×C₄	—	202	1	no	—
266	C₂×C₄	—	202	1	no	—
267	C₂×C₄	—	202	2	no	—
268	C₂×C₄	—	202	2	no	—
269	C₂×C₄	—	202	1	no	—
270	C₂×C₄	—	202	1	no	—
271	C₂×C₄	C₂×C₄×D₁₀₁	1	4	yes	—
272	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
273	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
274	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
275	C₂×C₄	Q₈×D₁₀₁	1	2	yes	—
276	C₂×C₄	Q₈×D₁₀₁	1	2	yes	—
277	C₂²	—	101	4	no	—
278	C₂²	—	101	4	no	—
279	C₂²	—	101	4	no	—
280	C₂²	—	101	4	no	—
281	C₂²	—	101	4	no	—
282	C₂²	—	101	4	no	—
283	C₂²	—	101	4	no	—
284	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
285	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
286	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
287	C₂²	C₂×Q₈×D₁₀₁	1	1	yes	characteristic
288	C₂²	C₂×Q₈×D₁₀₁	1	1	yes	characteristic
289	C₂²	#41×D₁₀₁	1	1	yes	characteristic
290	C₂²	#41×D₁₀₁	1	1	yes	characteristic
291	C₄	—	2	4	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	4	2	no	—
295	C₄	—	101	4	no	—
296	C₄	—	404	2	no	—
297	C₄	—	202	4	no	—
298	C₄	—	202	2	no	—
299	C₄	—	202	2	no	—
300	C₄	C₄⋊C₄×D₁₀₁	1	4	yes	complemented
301	C₂	—	101	8	no	—
302	C₂	C₂×C₄⋊C₄×D₁₀₁	1	1	yes	characteristic
303	C₂	C₄×D₄×D₁₀₁	1	1	yes	characteristic
304	C₂	C₄×Q₈×D₁₀₁	1	1	yes	characteristic
305	C₂	#121×D₁₀₁	1	1	yes	characteristic
306	C₂	C₂²⋊Q₈×D₁₀₁	1	1	yes	characteristic
307	C₂	#125×D₁₀₁	1	1	yes	characteristic
308	C₂	C₄⋊Q₈×D₁₀₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	#389×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂³×D₁₀₁	C₂³
hypercenter	#389	D₁₀₁
hypocenter	C₁₀₁	C₂×#389
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ #389

Complements

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

N	H ≅ G/N	type
5. #20265076	C₂	semidirect
6. #20265075	C₂	semidirect
7. #20265074	C₂	semidirect
8. #20265073	C₂	semidirect
9. #20265072	C₂	semidirect
11. #389×C₁₀₁	C₂	semidirect
26. C₄⋊C₄×D₁₀₁	C₄	semidirect
62. C₄⋊Dic₁₀₁	C₂×C₄	semidirect
66. #13108053	C₂×C₄	semidirect
77. C₄⋊C₄×C₁₀₁	C₂×C₄	semidirect
121. C₄×D₁₀₁	C₄⋊C₄	semidirect
140. C₄×C₁₀₁	C₂×C₄⋊C₄	semidirect
145. Dic₁₀₁	C₂×C₄⋊C₄	semidirect
153. D₁₀₁	#389	direct (N × H)
155. C₁₀₁	C₂×#389	semidirect
165. #389	D₁₀₁	direct (N × H)
229. C₄⋊C₄	C₄×D₁₀₁	semidirect
300. C₄	C₄⋊C₄×D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₄×D₁₀₁	2
C₂×C₄²×D₁₀₁	2
#20265076	2
#20265075	2
#20265074	2
#20265073	2
#20265072	2
#95×D₁₀₁	2
#389×C₁₀₁	2
C₂×#389	101

Subgroups — #389×D101 #27173579

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #389×D₁₀₁ #27173579