Subgroups — C₁₀₁×#18411 #38800562

Order 25856.

870 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₁×#3502	C₂	1	1	yes	maximal, characteristic
2	C₁₀₁×#3230	C₂	1	2	yes	maximal
3	C₁₀₁×#2568	C₂	1	1	yes	maximal, characteristic
4	C₁₀₁×#1617	C₂	1	8	yes	maximal
5	C₁₀₁×#1503	C₂	1	2	yes	maximal
6	C₁₀₁×#1461	C₂	1	1	yes	maximal, characteristic
7	Q₈²×C₁₀₁	—	2	2	no	—
8	#557×C₁₀₁	—	2	2	no	—
9	#557×C₁₀₁	—	2	4	no	—
10	#544×C₁₀₁	—	2	1	no	—
11	#544×C₁₀₁	C₂²	1	4	yes	—
12	#533×C₁₀₁	—	2	2	no	—
13	#533×C₁₀₁	C₂²	1	2	yes	—
14	C₂×C₄⋊Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
15	#519×C₁₀₁	—	2	1	no	—
16	C₂×C₄×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
17	#500×C₁₀₁	—	2	2	no	—
18	#487×C₁₀₁	—	2	2	no	—
19	#484×C₁₀₁	C₂²	1	8	yes	—
20	#477×C₁₀₁	—	2	4	no	—
21	#431×C₁₀₁	C₂²	1	2	yes	—
22	#400×C₁₀₁	C₂²	1	4	yes	—
23	#391×C₁₀₁	—	2	1	no	—
24	#391×C₁₀₁	—	2	2	no	—
25	#382×C₁₀₁	—	2	1	no	—
26	C₄×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
27	#328×C₁₀₁	C₂²	1	8	yes	—
28	#324×C₁₀₁	C₂²	1	4	yes	—
29	C₂²×Q₈×C₁₀₁	—	2	1	no	—
30	C₂²×Q₈×C₁₀₁	D₄	1	1	yes	characteristic
31	C₄⋊Q₈×C₁₀₁	—	2	2	no	—
32	C₄⋊Q₈×C₁₀₁	—	2	2	no	—
33	C₄⋊Q₈×C₁₀₁	—	2	4	no	—
34	C₄⋊Q₈×C₁₀₁	—	2	8	no	—
35	C₄⋊Q₈×C₁₀₁	—	2	2	no	—
36	C₄⋊Q₈×C₁₀₁	—	2	2	no	—
37	C₄⋊Q₈×C₁₀₁	D₄	1	2	yes	—
38	C₄⋊Q₈×C₁₀₁	D₄	1	2	yes	—
39	#125×C₁₀₁	—	2	4	no	—
40	#125×C₁₀₁	—	2	4	no	—
41	#125×C₁₀₁	—	2	8	no	—
42	#125×C₁₀₁	—	2	2	no	—
43	C₂²⋊Q₈×C₁₀₁	—	2	2	no	—
44	C₂²⋊Q₈×C₁₀₁	—	2	4	no	—
45	C₂²⋊Q₈×C₁₀₁	C₂³	1	4	yes	—
46	C₄×Q₈×C₁₀₁	—	2	4	no	—
47	C₄×Q₈×C₁₀₁	—	2	4	no	—
48	C₄×Q₈×C₁₀₁	—	2	4	no	—
49	C₄×Q₈×C₁₀₁	—	2	4	no	—
50	C₄×Q₈×C₁₀₁	—	2	2	no	—
51	C₄×Q₈×C₁₀₁	—	2	2	no	—
52	C₄×Q₈×C₁₀₁	—	2	2	no	—
53	C₄×Q₈×C₁₀₁	D₄	1	4	yes	—
54	#117×C₁₀₁	—	2	4	no	—
55	#117×C₁₀₁	—	2	2	no	—
56	C₂×C₄⋊C₄×C₁₀₁	—	2	1	no	—
57	C₂×C₄⋊C₄×C₁₀₁	—	2	1	no	—
58	C₂×C₄⋊C₄×C₁₀₁	D₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₁₀₁	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₁₀₁	C₂³	1	2	yes	—
61	C₂×C₄²×C₁₀₁	—	2	1	no	—
62	C₂×C₄²×C₁₀₁	D₄	1	2	yes	—
63	C₂×C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
64	#107×C₁₀₁	—	2	8	no	—
65	C₄⋊C₈×C₁₀₁	—	2	2	no	—
66	#103×C₁₀₁	—	2	8	no	—
67	C₂²⋊C₈×C₁₀₁	C₂³	1	4	yes	—
68	C₄×C₈×C₁₀₁	—	2	2	no	—
69	#95×C₁₀₁	—	2	4	no	—
70	#95×C₁₀₁	—	2	4	no	—
71	#95×C₁₀₁	C₂³	1	2	yes	—
72	C₂×Q₈×C₁₀₁	—	2	4	no	—
73	C₂×Q₈×C₁₀₁	—	2	2	no	—
74	C₂×Q₈×C₁₀₁	—	2	2	no	—
75	C₂×Q₈×C₁₀₁	—	2	2	no	—
76	C₂×Q₈×C₁₀₁	—	4	4	no	—
77	C₂×Q₈×C₁₀₁	—	4	1	no	—
78	C₂×Q₈×C₁₀₁	—	4	1	no	—
79	C₂×Q₈×C₁₀₁	Q₁₆	1	4	yes	—
80	C₂×Q₈×C₁₀₁	C₂×D₄	1	4	yes	—
81	C₂²×C₄×C₁₀₁	—	2	2	no	—
82	C₂²×C₄×C₁₀₁	—	2	2	no	—
83	C₂²×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
88	C₂×C₈×C₁₀₁	—	2	4	no	—
89	C₄⋊C₄×C₁₀₁	—	2	4	no	—
90	C₄⋊C₄×C₁₀₁	—	2	4	no	—
91	C₄⋊C₄×C₁₀₁	—	2	4	no	—
92	C₄⋊C₄×C₁₀₁	—	2	8	no	—
93	C₄⋊C₄×C₁₀₁	—	2	2	no	—
94	C₄⋊C₄×C₁₀₁	—	2	2	no	—
95	C₄⋊C₄×C₁₀₁	—	2	2	no	—
96	C₄⋊C₄×C₁₀₁	—	2	2	no	—
97	C₄⋊C₄×C₁₀₁	—	2	4	no	—
98	C₄⋊C₄×C₁₀₁	—	2	4	no	—
99	C₄⋊C₄×C₁₀₁	—	2	4	no	—
100	C₄⋊C₄×C₁₀₁	—	2	4	no	—
101	C₄⋊C₄×C₁₀₁	—	4	2	no	—
102	C₄⋊C₄×C₁₀₁	—	4	2	no	—
103	C₄⋊C₄×C₁₀₁	C₂×D₄	1	4	yes	—
104	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
105	C₄²×C₁₀₁	—	2	8	no	—
106	C₄²×C₁₀₁	—	2	2	no	—
107	C₄²×C₁₀₁	—	2	2	no	—
108	C₄²×C₁₀₁	—	2	4	no	—
109	C₄²×C₁₀₁	—	4	2	no	—
110	C₄²×C₁₀₁	C₂×D₄	1	2	yes	—
111	C₄²×C₁₀₁	C₂×D₄	1	2	yes	—
112	C₂³×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
113	Q₈×C₁₀₁	—	2	8	no	—
114	Q₈×C₁₀₁	—	4	4	no	—
115	Q₈×C₁₀₁	—	4	4	no	—
116	Q₈×C₁₀₁	—	8	2	no	—
117	Q₈×C₁₀₁	—	8	4	no	—
118	Q₈×C₁₀₁	—	8	2	no	—
119	C₂×C₄×C₁₀₁	—	2	2	no	—
120	C₂×C₄×C₁₀₁	—	2	2	no	—
121	C₂×C₄×C₁₀₁	—	2	2	no	—
122	C₂×C₄×C₁₀₁	—	2	4	no	—
123	C₂×C₄×C₁₀₁	—	2	4	no	—
124	C₂×C₄×C₁₀₁	—	2	2	no	—
125	C₂×C₄×C₁₀₁	—	4	1	no	—
126	C₂×C₄×C₁₀₁	—	4	2	no	—
127	C₂×C₄×C₁₀₁	—	4	2	no	—
128	C₂×C₄×C₁₀₁	—	4	2	no	—
129	C₂×C₄×C₁₀₁	—	4	4	no	—
130	C₂×C₄×C₁₀₁	—	4	2	no	—
131	C₂×C₄×C₁₀₁	—	4	2	no	—
132	C₂×C₄×C₁₀₁	#120	1	1	yes	characteristic
133	C₂×C₄×C₁₀₁	#120	1	2	yes	—
134	C₂×C₄×C₁₀₁	#120	1	1	yes	characteristic
135	C₂×C₄×C₁₀₁	C₂×Q₁₆	1	4	yes	—
136	C₂×C₄×C₁₀₁	C₂×Q₁₆	1	2	yes	—
137	C₂×C₄×C₁₀₁	#137	1	2	yes	—
138	C₂×C₄×C₁₀₁	C₂²×D₄	1	2	yes	—
139	C₈×C₁₀₁	—	4	4	no	—
140	C₂²×C₁₀₁	—	2	1	no	—
141	C₂²×C₁₀₁	—	2	1	no	—
142	C₂²×C₁₀₁	C₂×#120	1	1	yes	characteristic
143	C₂²×C₁₀₁	C₂²×Q₁₆	1	1	yes	characteristic
144	C₂²×C₁₀₁	C₂×#137	1	1	yes	characteristic
145	C₄×C₁₀₁	—	2	4	no	—
146	C₄×C₁₀₁	—	2	1	no	—
147	C₄×C₁₀₁	—	2	2	no	—
148	C₄×C₁₀₁	—	4	4	no	—
149	C₄×C₁₀₁	—	4	2	no	—
150	C₄×C₁₀₁	—	4	4	no	—
151	C₄×C₁₀₁	—	8	2	no	—
152	C₄×C₁₀₁	—	8	2	no	—
153	C₄×C₁₀₁	#451	1	4	yes	—
154	C₄×C₁₀₁	#454	1	2	yes	—
155	#18411	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂×C₁₀₁	—	2	1	no	—
157	C₂×C₁₀₁	—	2	1	no	—
158	C₂×C₁₀₁	C₂×#451	1	1	yes	characteristic
159	C₂×C₁₀₁	C₂×#454	1	1	yes	characteristic
160	C₂×C₁₀₁	#3028	1	1	yes	characteristic
161	#3502	C₂×C₁₀₁	1	1	yes	characteristic
162	#3230	C₂×C₁₀₁	1	2	yes	—
163	#2568	C₂×C₁₀₁	1	1	yes	characteristic
164	#1617	C₂×C₁₀₁	1	8	yes	—
165	#1503	C₂×C₁₀₁	1	2	yes	—
166	#1461	C₂×C₁₀₁	1	1	yes	characteristic
167	C₁₀₁	#18411	1	1	yes	characteristic, direct product factor, Sylow
168	Q₈²	—	2	2	no	—
169	#557	—	2	2	no	—
170	#557	—	2	4	no	—
171	#544	—	2	1	no	—
172	#544	C₂²×C₁₀₁	1	4	yes	—
173	#533	—	2	2	no	—
174	#533	C₂²×C₁₀₁	1	2	yes	—
175	C₂×C₄⋊Q₈	C₂²×C₁₀₁	1	1	yes	characteristic
176	#519	—	2	1	no	—
177	C₂×C₄×Q₈	C₂²×C₁₀₁	1	1	yes	characteristic
178	#500	—	2	2	no	—
179	#487	—	2	2	no	—
180	#484	C₂²×C₁₀₁	1	8	yes	—
181	#477	—	2	4	no	—
182	#431	C₂²×C₁₀₁	1	2	yes	—
183	#400	C₂²×C₁₀₁	1	4	yes	—
184	#391	—	2	1	no	—
185	#391	—	2	2	no	—
186	#382	—	2	1	no	—
187	C₄×C₄⋊C₄	C₂²×C₁₀₁	1	1	yes	characteristic
188	#328	C₂²×C₁₀₁	1	8	yes	—
189	#324	C₂²×C₁₀₁	1	4	yes	—
190	C₂²×Q₈	—	2	1	no	—
191	C₂²×Q₈	D₄×C₁₀₁	1	1	yes	characteristic
192	C₄⋊Q₈	—	2	2	no	—
193	C₄⋊Q₈	—	2	2	no	—
194	C₄⋊Q₈	—	2	4	no	—
195	C₄⋊Q₈	—	2	8	no	—
196	C₄⋊Q₈	—	2	2	no	—
197	C₄⋊Q₈	—	2	2	no	—
198	C₄⋊Q₈	D₄×C₁₀₁	1	2	yes	—
199	C₄⋊Q₈	D₄×C₁₀₁	1	2	yes	—
200	#125	—	2	4	no	—
201	#125	—	2	4	no	—
202	#125	—	2	8	no	—
203	#125	—	2	2	no	—
204	C₂²⋊Q₈	—	2	2	no	—
205	C₂²⋊Q₈	—	2	4	no	—
206	C₂²⋊Q₈	C₂³×C₁₀₁	1	4	yes	—
207	C₄×Q₈	—	2	4	no	—
208	C₄×Q₈	—	2	4	no	—
209	C₄×Q₈	—	2	4	no	—
210	C₄×Q₈	—	2	4	no	—
211	C₄×Q₈	—	2	2	no	—
212	C₄×Q₈	—	2	2	no	—
213	C₄×Q₈	—	2	2	no	—
214	C₄×Q₈	D₄×C₁₀₁	1	4	yes	—
215	#117	—	2	4	no	—
216	#117	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	D₄×C₁₀₁	1	1	yes	characteristic
220	C₂×C₄⋊C₄	C₂³×C₁₀₁	1	2	yes	—
221	C₂×C₄⋊C₄	C₂³×C₁₀₁	1	2	yes	—
222	C₂×C₄²	—	2	1	no	—
223	C₂×C₄²	D₄×C₁₀₁	1	2	yes	—
224	C₂×C₄²	C₂³×C₁₀₁	1	1	yes	characteristic
225	#107	—	2	8	no	—
226	C₄⋊C₈	—	2	2	no	—
227	#103	—	2	8	no	—
228	C₂²⋊C₈	C₂³×C₁₀₁	1	4	yes	—
229	C₄×C₈	—	2	2	no	—
230	#95	—	2	4	no	—
231	#95	—	2	4	no	—
232	#95	C₂³×C₁₀₁	1	2	yes	—
233	C₂×Q₈	—	2	4	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	4	4	no	—
238	C₂×Q₈	—	4	1	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	Q₁₆×C₁₀₁	1	4	yes	—
241	C₂×Q₈	C₂×D₄×C₁₀₁	1	4	yes	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	C₂×D₄×C₁₀₁	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₁₀₁	1	2	yes	—
246	C₂²×C₄	C₂×D₄×C₁₀₁	1	2	yes	—
247	C₂²×C₄	C₂×D₄×C₁₀₁	1	1	yes	characteristic
248	C₂²×C₄	C₂⁴×C₁₀₁	1	1	yes	characteristic
249	C₂×C₈	—	2	4	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	8	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	C₂×D₄×C₁₀₁	1	4	yes	—
265	C₂²⋊C₄	—	2	4	no	—
266	C₄²	—	2	8	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	4	no	—
270	C₄²	—	4	2	no	—
271	C₄²	C₂×D₄×C₁₀₁	1	2	yes	—
272	C₄²	C₂×D₄×C₁₀₁	1	2	yes	—
273	C₂³	C₂²×D₄×C₁₀₁	1	1	yes	characteristic
274	Q₈	—	2	8	no	—
275	Q₈	—	4	4	no	—
276	Q₈	—	4	4	no	—
277	Q₈	—	8	2	no	—
278	Q₈	—	8	4	no	—
279	Q₈	—	8	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	#120×C₁₀₁	1	1	yes	characteristic
294	C₂×C₄	#120×C₁₀₁	1	2	yes	—
295	C₂×C₄	#120×C₁₀₁	1	1	yes	characteristic
296	C₂×C₄	C₂×Q₁₆×C₁₀₁	1	4	yes	—
297	C₂×C₄	C₂×Q₁₆×C₁₀₁	1	2	yes	—
298	C₂×C₄	#137×C₁₀₁	1	2	yes	—
299	C₂×C₄	C₂²×D₄×C₁₀₁	1	2	yes	—
300	C₈	—	4	4	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	C₂×#120×C₁₀₁	1	1	yes	characteristic
304	C₂²	C₂²×Q₁₆×C₁₀₁	1	1	yes	characteristic
305	C₂²	C₂×#137×C₁₀₁	1	1	yes	characteristic
306	C₄	—	2	4	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	2	no	—
309	C₄	—	4	4	no	—
310	C₄	—	4	2	no	—
311	C₄	—	4	4	no	—
312	C₄	—	8	2	no	—
313	C₄	—	8	2	no	—
314	C₄	#451×C₁₀₁	1	4	yes	—
315	C₄	#454×C₁₀₁	1	2	yes	—
316	C₂	—	2	1	no	—
317	C₂	—	2	1	no	—
318	C₂	C₂×#451×C₁₀₁	1	1	yes	characteristic
319	C₂	C₂×#454×C₁₀₁	1	1	yes	characteristic
320	C₂	C₁₀₁×#3028	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₁	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₁
prime radical	C₂³×C₁₀₁	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₁ ◁ C₂×C₄²×C₁₀₁ ◁ G

Complements

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

N	H ≅ G/N	type
155. #18411	C₁₀₁	direct (N × H)
167. C₁₀₁	#18411	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₀₁×#3502	2
C₁₀₁×#3230	2
C₁₀₁×#2568	2
C₁₀₁×#1617	2
C₁₀₁×#1503	2
C₁₀₁×#1461	2
#18411	101

Subgroups — C101×#18411 #38800562

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#18411 #38800562