Subgroups — C₁₀₁×#7563 #38789714

Order 25856.

690 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₁×#2297	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊C₈×C₁₀₁	C₂	1	2	yes	maximal, complemented
3	C₁₀₁×#1272	C₂	1	2	yes	maximal, complemented
4	C₁₀₁×#1270	C₂	1	2	yes	maximal, complemented
5	C₂²×#41×C₁₀₁	C₄	1	1	yes	characteristic
6	C₂×#117×C₁₀₁	C₄	1	1	yes	characteristic
7	C₂²×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊C₈×C₁₀₁	—	2	2	no	—
9	C₂×C₄×C₈×C₁₀₁	—	2	2	no	—
10	#391×C₁₀₁	C₄	1	1	yes	characteristic
11	#391×C₁₀₁	C₂²	1	1	yes	characteristic
12	#390×C₁₀₁	C₄	1	1	yes	characteristic
13	#390×C₁₀₁	C₂²	1	1	yes	characteristic
14	#386×C₁₀₁	—	2	2	no	—
15	C₄×C₂²⋊C₄×C₁₀₁	—	2	2	no	—
16	#336×C₁₀₁	C₂²	1	4	yes	—
17	C₂×#41×C₁₀₁	—	2	2	no	—
18	C₂×#41×C₁₀₁	—	4	2	no	—
19	C₂×#41×C₁₀₁	—	4	2	no	—
20	C₂×#41×C₁₀₁	C₈	1	4	yes	complemented
21	C₂²×Q₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×D₄×C₁₀₁	—	2	1	no	—
23	C₂²×D₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₀₁	—	2	1	no	—
25	C₂³×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₀₁	D₄	1	2	yes	—
27	C₂²×C₈×C₁₀₁	—	2	4	no	—
28	#117×C₁₀₁	—	4	2	no	—
29	C₂×C₄⋊C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₂²⋊C₄×C₁₀₁	—	2	1	no	—
32	C₂×C₂²⋊C₄×C₁₀₁	—	2	2	no	—
33	C₂×C₂²⋊C₄×C₁₀₁	—	2	2	no	—
34	C₂×C₄²×C₁₀₁	—	2	1	no	—
35	C₂×C₄²×C₁₀₁	—	4	2	no	—
36	C₂×C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₀₁	D₄	1	2	yes	—
38	C₂×C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
39	C₂²⋊C₈×C₁₀₁	—	2	8	no	—
40	C₄×C₈×C₁₀₁	—	4	4	no	—
41	#95×C₁₀₁	—	2	2	no	—
42	#95×C₁₀₁	—	2	2	no	—
43	C₂⁴×C₁₀₁	—	2	1	no	—
44	C₂⁴×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
45	#41×C₁₀₁	—	4	8	no	—
46	#41×C₁₀₁	—	4	8	no	—
47	C₂×Q₈×C₁₀₁	—	2	1	no	—
48	C₂×Q₈×C₁₀₁	—	4	1	no	—
49	C₂×Q₈×C₁₀₁	—	4	1	no	—
50	C₂×Q₈×C₁₀₁	C₂×C₈	1	2	yes	—
51	C₂×D₄×C₁₀₁	—	2	4	no	—
52	C₂×D₄×C₁₀₁	—	2	1	no	—
53	C₂×D₄×C₁₀₁	—	4	1	no	—
54	C₂×D₄×C₁₀₁	—	4	1	no	—
55	C₂×D₄×C₁₀₁	—	4	2	no	—
56	C₂×D₄×C₁₀₁	—	4	2	no	—
57	C₂×D₄×C₁₀₁	C₂×C₈	1	2	yes	—
58	C₂²×C₄×C₁₀₁	—	2	2	no	—
59	C₂²×C₄×C₁₀₁	—	2	2	no	—
60	C₂²×C₄×C₁₀₁	—	2	1	no	—
61	C₂²×C₄×C₁₀₁	—	2	4	no	—
62	C₂²×C₄×C₁₀₁	—	2	4	no	—
63	C₂²×C₄×C₁₀₁	—	2	4	no	—
64	C₂²×C₄×C₁₀₁	—	2	1	no	—
65	C₂²×C₄×C₁₀₁	—	2	1	no	—
66	C₂²×C₄×C₁₀₁	—	4	2	no	—
67	C₂²×C₄×C₁₀₁	—	4	2	no	—
68	C₂²×C₄×C₁₀₁	—	4	2	no	—
69	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₀₁	C₂×C₈	1	2	yes	—
72	C₂²×C₄×C₁₀₁	M₁₆	1	2	yes	—
73	C₂²×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₁₀₁	—	2	8	no	—
76	C₂×C₈×C₁₀₁	—	4	4	no	—
77	C₂×C₈×C₁₀₁	—	4	4	no	—
78	C₄⋊C₄×C₁₀₁	—	4	1	no	—
79	C₄⋊C₄×C₁₀₁	—	4	1	no	—
80	C₂²⋊C₄×C₁₀₁	—	4	2	no	—
81	C₂²⋊C₄×C₁₀₁	—	4	4	no	—
82	C₂²⋊C₄×C₁₀₁	—	4	4	no	—
83	C₄²×C₁₀₁	—	2	4	no	—
84	C₄²×C₁₀₁	—	4	1	no	—
85	C₄²×C₁₀₁	—	4	1	no	—
86	C₄²×C₁₀₁	—	4	2	no	—
87	C₂³×C₁₀₁	—	2	1	no	—
88	C₂³×C₁₀₁	—	2	1	no	—
89	C₂³×C₁₀₁	—	2	2	no	—
90	C₂³×C₁₀₁	—	2	2	no	—
91	C₂³×C₁₀₁	—	2	2	no	—
92	C₂³×C₁₀₁	—	4	1	no	—
93	C₂³×C₁₀₁	—	4	1	no	—
94	C₂³×C₁₀₁	—	4	2	no	—
95	C₂³×C₁₀₁	—	4	2	no	—
96	C₂³×C₁₀₁	C₂²⋊C₈	1	2	yes	—
97	C₂³×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	Q₈×C₁₀₁	—	4	2	no	—
99	Q₈×C₁₀₁	—	4	2	no	—
100	D₄×C₁₀₁	—	4	2	no	—
101	D₄×C₁₀₁	—	4	2	no	—
102	D₄×C₁₀₁	—	4	8	no	—
103	C₂×C₄×C₁₀₁	—	2	2	no	—
104	C₂×C₄×C₁₀₁	—	2	2	no	—
105	C₂×C₄×C₁₀₁	—	2	1	no	—
106	C₂×C₄×C₁₀₁	—	2	1	no	—
107	C₂×C₄×C₁₀₁	—	2	2	no	—
108	C₂×C₄×C₁₀₁	—	2	2	no	—
109	C₂×C₄×C₁₀₁	—	2	4	no	—
110	C₂×C₄×C₁₀₁	—	2	4	no	—
111	C₂×C₄×C₁₀₁	—	2	2	no	—
112	C₂×C₄×C₁₀₁	—	4	8	no	—
113	C₂×C₄×C₁₀₁	—	4	1	no	—
114	C₂×C₄×C₁₀₁	—	4	1	no	—
115	C₂×C₄×C₁₀₁	—	4	8	no	—
116	C₂×C₄×C₁₀₁	—	4	1	no	—
117	C₂×C₄×C₁₀₁	—	4	1	no	—
118	C₂×C₄×C₁₀₁	—	4	2	no	—
119	C₂×C₄×C₁₀₁	—	4	2	no	—
120	C₂×C₄×C₁₀₁	C₂²⋊C₈	1	2	yes	—
121	C₂×C₄×C₁₀₁	#99	1	2	yes	—
122	C₂×C₄×C₁₀₁	#100	1	2	yes	—
123	C₂×C₄×C₁₀₁	#104	1	4	yes	—
124	C₂×C₄×C₁₀₁	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₁₀₁	C₂×M₁₆	1	1	yes	characteristic
126	C₂×C₄×C₁₀₁	#135	1	4	yes	—
127	C₈×C₁₀₁	—	4	8	no	—
128	C₂²×C₁₀₁	—	2	2	no	—
129	C₂²×C₁₀₁	—	2	4	no	—
130	C₂²×C₁₀₁	—	4	1	no	—
131	C₂²×C₁₀₁	—	4	2	no	—
132	C₂²×C₁₀₁	—	4	1	no	—
133	C₂²×C₁₀₁	—	4	2	no	—
134	C₂²×C₁₀₁	—	4	4	no	—
135	C₂²×C₁₀₁	—	8	1	no	—
136	C₂²×C₁₀₁	—	8	2	no	—
137	C₂²×C₁₀₁	—	8	1	no	—
138	C₂²×C₁₀₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
139	C₂²×C₁₀₁	C₂×#99	1	1	yes	characteristic
140	C₂²×C₁₀₁	C₂×#100	1	1	yes	characteristic
141	C₂²×C₁₀₁	#416	1	2	yes	—
142	C₂²×C₁₀₁	C₂×#104	1	2	yes	—
143	C₄×C₁₀₁	—	2	1	no	—
144	C₄×C₁₀₁	—	2	1	no	—
145	C₄×C₁₀₁	—	2	4	no	—
146	C₄×C₁₀₁	—	2	4	no	—
147	C₄×C₁₀₁	—	4	2	no	—
148	C₄×C₁₀₁	—	8	1	no	—
149	C₄×C₁₀₁	—	8	1	no	—
150	C₄×C₁₀₁	#323	1	4	yes	—
151	#7563	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂×C₁₀₁	—	4	2	no	—
153	C₂×C₁₀₁	—	8	2	no	—
154	C₂×C₁₀₁	C₂×#323	1	1	yes	characteristic
155	C₂×C₁₀₁	#1476	1	2	yes	—
156	C₂×C₁₀₁	#1501	1	2	yes	—
157	C₂×C₁₀₁	#1525	1	1	yes	characteristic
158	C₂×C₁₀₁	#1543	1	1	yes	characteristic
159	#2297	C₂×C₁₀₁	1	1	yes	characteristic
160	C₄×C₂²⋊C₈	C₂×C₁₀₁	1	2	yes	complemented
161	#1272	C₂×C₁₀₁	1	2	yes	complemented
162	#1270	C₂×C₁₀₁	1	2	yes	complemented
163	C₁₀₁	#7563	1	1	yes	characteristic, direct product factor, Sylow
164	C₂²×#41	C₄×C₁₀₁	1	1	yes	characteristic
165	C₂×#117	C₄×C₁₀₁	1	1	yes	characteristic
166	C₂²×C₄²	C₂²×C₁₀₁	1	1	yes	characteristic
167	C₂×C₂²⋊C₈	—	2	2	no	—
168	C₂×C₄×C₈	—	2	2	no	—
169	#391	C₄×C₁₀₁	1	1	yes	characteristic
170	#391	C₂²×C₁₀₁	1	1	yes	characteristic
171	#390	C₄×C₁₀₁	1	1	yes	characteristic
172	#390	C₂²×C₁₀₁	1	1	yes	characteristic
173	#386	—	2	2	no	—
174	C₄×C₂²⋊C₄	—	2	2	no	—
175	#336	C₂²×C₁₀₁	1	4	yes	—
176	C₂×#41	—	2	2	no	—
177	C₂×#41	—	4	2	no	—
178	C₂×#41	—	4	2	no	—
179	C₂×#41	C₈×C₁₀₁	1	4	yes	complemented
180	C₂²×Q₈	C₂×C₄×C₁₀₁	1	1	yes	characteristic
181	C₂²×D₄	—	2	1	no	—
182	C₂²×D₄	C₂×C₄×C₁₀₁	1	1	yes	characteristic
183	C₂³×C₄	—	2	1	no	—
184	C₂³×C₄	C₂×C₄×C₁₀₁	1	1	yes	characteristic
185	C₂³×C₄	D₄×C₁₀₁	1	2	yes	—
186	C₂²×C₈	—	2	4	no	—
187	#117	—	4	2	no	—
188	C₂×C₄⋊C₄	C₂×C₄×C₁₀₁	1	1	yes	characteristic
189	C₂×C₄⋊C₄	C₂×C₄×C₁₀₁	1	1	yes	characteristic
190	C₂×C₂²⋊C₄	—	2	1	no	—
191	C₂×C₂²⋊C₄	—	2	2	no	—
192	C₂×C₂²⋊C₄	—	2	2	no	—
193	C₂×C₄²	—	2	1	no	—
194	C₂×C₄²	—	4	2	no	—
195	C₂×C₄²	C₂×C₄×C₁₀₁	1	1	yes	characteristic
196	C₂×C₄²	D₄×C₁₀₁	1	2	yes	—
197	C₂×C₄²	C₂³×C₁₀₁	1	1	yes	characteristic
198	C₂²⋊C₈	—	2	8	no	—
199	C₄×C₈	—	4	4	no	—
200	#95	—	2	2	no	—
201	#95	—	2	2	no	—
202	C₂⁴	—	2	1	no	—
203	C₂⁴	C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
204	#41	—	4	8	no	—
205	#41	—	4	8	no	—
206	C₂×Q₈	—	2	1	no	—
207	C₂×Q₈	—	4	1	no	—
208	C₂×Q₈	—	4	1	no	—
209	C₂×Q₈	C₂×C₈×C₁₀₁	1	2	yes	—
210	C₂×D₄	—	2	4	no	—
211	C₂×D₄	—	2	1	no	—
212	C₂×D₄	—	4	1	no	—
213	C₂×D₄	—	4	1	no	—
214	C₂×D₄	—	4	2	no	—
215	C₂×D₄	—	4	2	no	—
216	C₂×D₄	C₂×C₈×C₁₀₁	1	2	yes	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	—	4	2	no	—
227	C₂²×C₄	—	4	2	no	—
228	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	2	yes	—
229	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₈×C₁₀₁	1	2	yes	—
231	C₂²×C₄	M₁₆×C₁₀₁	1	2	yes	—
232	C₂²×C₄	C₂²×C₄×C₁₀₁	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₄×C₁₀₁	1	2	yes	—
234	C₂×C₈	—	2	8	no	—
235	C₂×C₈	—	4	4	no	—
236	C₂×C₈	—	4	4	no	—
237	C₄⋊C₄	—	4	1	no	—
238	C₄⋊C₄	—	4	1	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	4	no	—
241	C₂²⋊C₄	—	4	4	no	—
242	C₄²	—	2	4	no	—
243	C₄²	—	4	1	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	4	2	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	—	2	2	no	—
251	C₂³	—	4	1	no	—
252	C₂³	—	4	1	no	—
253	C₂³	—	4	2	no	—
254	C₂³	—	4	2	no	—
255	C₂³	C₂²⋊C₈×C₁₀₁	1	2	yes	—
256	C₂³	C₂×C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
257	Q₈	—	4	2	no	—
258	Q₈	—	4	2	no	—
259	D₄	—	4	2	no	—
260	D₄	—	4	2	no	—
261	D₄	—	4	8	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	4	8	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	8	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	C₂²⋊C₈×C₁₀₁	1	2	yes	—
280	C₂×C₄	#99×C₁₀₁	1	2	yes	—
281	C₂×C₄	#100×C₁₀₁	1	2	yes	—
282	C₂×C₄	#104×C₁₀₁	1	4	yes	—
283	C₂×C₄	C₂²×C₈×C₁₀₁	1	1	yes	characteristic
284	C₂×C₄	C₂×M₁₆×C₁₀₁	1	1	yes	characteristic
285	C₂×C₄	#135×C₁₀₁	1	4	yes	—
286	C₈	—	4	8	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	4	2	no	—
293	C₂²	—	4	4	no	—
294	C₂²	—	8	2	no	—
295	C₂²	—	8	1	no	—
296	C₂²	—	8	1	no	—
297	C₂²	C₂×C₂²⋊C₈×C₁₀₁	1	1	yes	characteristic
298	C₂²	C₂×#99×C₁₀₁	1	1	yes	characteristic
299	C₂²	C₂×#100×C₁₀₁	1	1	yes	characteristic
300	C₂²	#416×C₁₀₁	1	2	yes	—
301	C₂²	C₂×#104×C₁₀₁	1	2	yes	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	4	no	—
305	C₄	—	2	4	no	—
306	C₄	—	4	2	no	—
307	C₄	—	8	1	no	—
308	C₄	—	8	1	no	—
309	C₄	#323×C₁₀₁	1	4	yes	—
310	C₂	—	4	2	no	—
311	C₂	—	8	2	no	—
312	C₂	C₂×#323×C₁₀₁	1	1	yes	characteristic
313	C₂	C₁₀₁×#1476	1	2	yes	—
314	C₂	C₁₀₁×#1501	1	2	yes	—
315	C₂	C₁₀₁×#1525	1	1	yes	characteristic
316	C₂	C₁₀₁×#1543	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₀₁	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₁₀₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₁
socle	C₂³×C₁₀₁	C₂×C₂²⋊C₄
prime radical	C₂²×#41×C₁₀₁	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₀₁ ◁ C₂²×C₄²×C₁₀₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₂²⋊C₈×C₁₀₁	C₂	semidirect
3. C₁₀₁×#1272	C₂	semidirect
4. C₁₀₁×#1270	C₂	semidirect
20. C₂×#41×C₁₀₁	C₈	semidirect
151. #7563	C₁₀₁	direct (N × H)
160. C₄×C₂²⋊C₈	C₂×C₁₀₁	semidirect
161. #1272	C₂×C₁₀₁	semidirect
162. #1270	C₂×C₁₀₁	semidirect
163. C₁₀₁	#7563	direct (N × H)
179. C₂×#41	C₈×C₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₁₀₁×#2297	2
C₄×C₂²⋊C₈×C₁₀₁	2
C₁₀₁×#1272	2
C₁₀₁×#1270	2
#7563	101

Subgroups — C101×#7563 #38789714

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#7563 #38789714