Subgroups — C₁₀₁×#25861 #38808012

Order 25856.

1282 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#491×C₁₀₁	C₂	1	2	yes	maximal, complemented
3	C₁₀₁×#2851	C₂	1	1	yes	maximal, characteristic
4	C₁₀₁×#2043	C₂	1	4	yes	maximal, complemented
5	C₁₀₁×#1960	C₂	1	2	yes	maximal, complemented
6	C₁₀₁×#1787	C₂	1	4	yes	maximal
7	C₁₀₁×#1749	C₂	1	1	yes	maximal, characteristic
8	C₂³×Q₈×C₁₀₁	C₂²	1	2	yes	—
9	C₂×C₄⋊Q₈×C₁₀₁	—	2	8	no	—
10	C₂×C₄⋊Q₈×C₁₀₁	—	2	2	no	—
11	C₂×C₄⋊Q₈×C₁₀₁	C₂²	1	2	yes	—
12	C₂×C₄⋊Q₈×C₁₀₁	C₂²	1	2	yes	—
13	C₂×#125×C₁₀₁	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊C₄×C₁₀₁	—	2	2	no	—
15	C₂²×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
16	#491×C₁₀₁	—	2	8	no	—
17	C₂×#103×C₁₀₁	C₂²	1	8	yes	—
18	C₂×#103×C₁₀₁	C₂²	1	8	yes	—
19	C₂×C₂²⋊C₈×C₁₀₁	C₂²	1	2	yes	—
20	C₂×#97×C₁₀₁	C₂²	1	2	yes	—
21	#389×C₁₀₁	C₂²	1	2	yes	—
22	#387×C₁₀₁	—	2	2	no	—
23	#385×C₁₀₁	—	2	2	no	—
24	#382×C₁₀₁	C₂²	1	1	yes	characteristic
25	#380×C₁₀₁	C₂²	1	2	yes	—
26	#336×C₁₀₁	C₂²	1	2	yes	—
27	C₂²×Q₈×C₁₀₁	—	2	8	no	—
28	C₂²×Q₈×C₁₀₁	—	2	2	no	—
29	C₂²×Q₈×C₁₀₁	—	2	8	no	—
30	C₂²×Q₈×C₁₀₁	—	4	4	no	—
31	C₂²×Q₈×C₁₀₁	C₂³	1	4	yes	—
32	C₂³×C₄×C₁₀₁	—	2	2	no	—
33	C₂³×C₄×C₁₀₁	D₄	1	2	yes	—
34	C₂³×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₁₀₁	C₂³	1	4	yes	—
36	C₄⋊Q₈×C₁₀₁	—	2	16	no	—
37	C₄⋊Q₈×C₁₀₁	—	2	8	no	—
38	C₄⋊Q₈×C₁₀₁	—	2	4	no	—
39	C₄⋊Q₈×C₁₀₁	D₄	1	8	yes	—
40	#125×C₁₀₁	—	2	4	no	—
41	C₂×C₄⋊C₄×C₁₀₁	—	2	16	no	—
42	C₂×C₄⋊C₄×C₁₀₁	—	2	4	no	—
43	C₂×C₄⋊C₄×C₁₀₁	—	2	4	no	—
44	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
45	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₁₀₁	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₁₀₁	C₂³	1	2	yes	—
48	C₂×C₂²⋊C₄×C₁₀₁	—	2	2	no	—
49	C₂×C₄²×C₁₀₁	—	2	4	no	—
50	C₂×C₄²×C₁₀₁	D₄	1	2	yes	—
51	C₂×C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
53	#103×C₁₀₁	—	2	16	no	—
54	#103×C₁₀₁	—	4	8	no	—
55	C₂²⋊C₈×C₁₀₁	—	2	4	no	—
56	#97×C₁₀₁	—	2	4	no	—
57	#95×C₁₀₁	—	2	2	no	—
58	#95×C₁₀₁	—	2	2	no	—
59	#95×C₁₀₁	—	2	2	no	—
60	C₂⁴×C₁₀₁	C₂×D₄	1	1	yes	characteristic
61	C₂×Q₈×C₁₀₁	—	2	8	no	—
62	C₂×Q₈×C₁₀₁	—	2	4	no	—
63	C₂×Q₈×C₁₀₁	—	2	8	no	—
64	C₂×Q₈×C₁₀₁	—	2	8	no	—
65	C₂×Q₈×C₁₀₁	—	4	32	no	—
66	C₂×Q₈×C₁₀₁	—	4	4	no	—
67	C₂×Q₈×C₁₀₁	—	4	4	no	—
68	C₂×Q₈×C₁₀₁	C₂×D₄	1	8	yes	—
69	C₂²×C₄×C₁₀₁	—	2	4	no	—
70	C₂²×C₄×C₁₀₁	—	2	4	no	—
71	C₂²×C₄×C₁₀₁	—	2	4	no	—
72	C₂²×C₄×C₁₀₁	—	2	2	no	—
73	C₂²×C₄×C₁₀₁	—	2	2	no	—
74	C₂²×C₄×C₁₀₁	—	2	8	no	—
75	C₂²×C₄×C₁₀₁	—	2	4	no	—
76	C₂²×C₄×C₁₀₁	—	2	2	no	—
77	C₂²×C₄×C₁₀₁	—	2	2	no	—
78	C₂²×C₄×C₁₀₁	—	4	4	no	—
79	C₂²×C₄×C₁₀₁	—	4	4	no	—
80	C₂²×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₁₀₁	#41	1	4	yes	—
84	C₂²×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
85	C₂×C₈×C₁₀₁	—	2	4	no	—
86	C₂×C₈×C₁₀₁	—	2	4	no	—
87	C₂×C₈×C₁₀₁	—	2	4	no	—
88	C₄⋊C₄×C₁₀₁	—	2	16	no	—
89	C₄⋊C₄×C₁₀₁	—	2	16	no	—
90	C₄⋊C₄×C₁₀₁	—	2	4	no	—
91	C₄⋊C₄×C₁₀₁	—	2	4	no	—
92	C₄⋊C₄×C₁₀₁	—	4	4	no	—
93	C₄⋊C₄×C₁₀₁	—	4	4	no	—
94	C₂²⋊C₄×C₁₀₁	—	4	4	no	—
95	C₄²×C₁₀₁	—	2	4	no	—
96	C₄²×C₁₀₁	—	2	2	no	—
97	C₄²×C₁₀₁	C₂×D₄	1	4	yes	—
98	C₂³×C₁₀₁	—	2	1	no	—
99	C₂³×C₁₀₁	—	2	1	no	—
100	C₂³×C₁₀₁	—	2	2	no	—
101	C₂³×C₁₀₁	#124	1	2	yes	—
102	C₂³×C₁₀₁	#137	1	4	yes	—
103	C₂³×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
104	Q₈×C₁₀₁	—	2	16	no	—
105	Q₈×C₁₀₁	—	4	16	no	—
106	Q₈×C₁₀₁	—	4	8	no	—
107	C₂×C₄×C₁₀₁	—	2	16	no	—
108	C₂×C₄×C₁₀₁	—	2	2	no	—
109	C₂×C₄×C₁₀₁	—	2	1	no	—
110	C₂×C₄×C₁₀₁	—	2	1	no	—
111	C₂×C₄×C₁₀₁	—	2	2	no	—
112	C₂×C₄×C₁₀₁	—	2	2	no	—
113	C₂×C₄×C₁₀₁	—	2	2	no	—
114	C₂×C₄×C₁₀₁	—	2	8	no	—
115	C₂×C₄×C₁₀₁	—	2	8	no	—
116	C₂×C₄×C₁₀₁	—	4	16	no	—
117	C₂×C₄×C₁₀₁	—	4	4	no	—
118	C₂×C₄×C₁₀₁	—	4	4	no	—
119	C₂×C₄×C₁₀₁	—	4	2	no	—
120	C₂×C₄×C₁₀₁	—	4	2	no	—
121	C₂×C₄×C₁₀₁	—	4	2	no	—
122	C₂×C₄×C₁₀₁	—	4	2	no	—
123	C₂×C₄×C₁₀₁	—	4	2	no	—
124	C₂×C₄×C₁₀₁	—	4	2	no	—
125	C₂×C₄×C₁₀₁	#120	1	4	yes	—
126	C₂×C₄×C₁₀₁	#124	1	2	yes	—
127	C₂×C₄×C₁₀₁	#137	1	4	yes	—
128	C₂×C₄×C₁₀₁	C₂²×D₄	1	2	yes	—
129	C₂×C₄×C₁₀₁	C₂×#41	1	2	yes	—
130	C₂×C₄×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
131	C₈×C₁₀₁	—	4	4	no	—
132	C₂²×C₁₀₁	—	2	2	no	—
133	C₂²×C₁₀₁	—	2	2	no	—
134	C₂²×C₁₀₁	—	2	4	no	—
135	C₂²×C₁₀₁	—	2	4	no	—
136	C₂²×C₁₀₁	#491	1	4	yes	complemented
137	C₂²×C₁₀₁	C₂×#120	1	1	yes	characteristic
138	C₂²×C₁₀₁	C₂×#124	1	1	yes	characteristic
139	C₂²×C₁₀₁	#535	1	1	yes	characteristic
140	C₂²×C₁₀₁	C₂×#137	1	2	yes	—
141	C₂²×C₁₀₁	C₂×#137	1	2	yes	—
142	C₄×C₁₀₁	—	2	2	no	—
143	C₄×C₁₀₁	—	2	4	no	—
144	C₄×C₁₀₁	—	2	4	no	—
145	C₄×C₁₀₁	—	4	8	no	—
146	C₄×C₁₀₁	—	8	2	no	—
147	C₄×C₁₀₁	—	8	2	no	—
148	C₄×C₁₀₁	#546	1	4	yes	—
149	#25861	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂×C₁₀₁	—	2	4	no	—
151	C₂×C₁₀₁	#2435	1	1	yes	characteristic
152	C₂×C₁₀₁	#3004	1	2	yes	—
153	C₂×C₁₀₁	C₂×#491	1	1	yes	characteristic
154	C₂×C₁₀₁	#3139	1	1	yes	characteristic
155	C₂×C₁₀₁	#3218	1	2	yes	—
156	C₂²×C₄⋊Q₈	C₂×C₁₀₁	1	1	yes	characteristic
157	C₂×#491	C₂×C₁₀₁	1	2	yes	complemented
158	#2851	C₂×C₁₀₁	1	1	yes	characteristic
159	#2043	C₂×C₁₀₁	1	4	yes	complemented
160	#1960	C₂×C₁₀₁	1	2	yes	complemented
161	#1787	C₂×C₁₀₁	1	4	yes	—
162	#1749	C₂×C₁₀₁	1	1	yes	characteristic
163	C₁₀₁	#25861	1	1	yes	characteristic, direct product factor, Sylow
164	C₂³×Q₈	C₂²×C₁₀₁	1	2	yes	—
165	C₂×C₄⋊Q₈	—	2	8	no	—
166	C₂×C₄⋊Q₈	—	2	2	no	—
167	C₂×C₄⋊Q₈	C₂²×C₁₀₁	1	2	yes	—
168	C₂×C₄⋊Q₈	C₂²×C₁₀₁	1	2	yes	—
169	C₂×#125	C₂²×C₁₀₁	1	1	yes	characteristic
170	C₂²×C₄⋊C₄	—	2	2	no	—
171	C₂²×C₄²	C₂²×C₁₀₁	1	1	yes	characteristic
172	#491	—	2	8	no	—
173	C₂×#103	C₂²×C₁₀₁	1	8	yes	—
174	C₂×#103	C₂²×C₁₀₁	1	8	yes	—
175	C₂×C₂²⋊C₈	C₂²×C₁₀₁	1	2	yes	—
176	C₂×#97	C₂²×C₁₀₁	1	2	yes	—
177	#389	C₂²×C₁₀₁	1	2	yes	—
178	#387	—	2	2	no	—
179	#385	—	2	2	no	—
180	#382	C₂²×C₁₀₁	1	1	yes	characteristic
181	#380	C₂²×C₁₀₁	1	2	yes	—
182	#336	C₂²×C₁₀₁	1	2	yes	—
183	C₂²×Q₈	—	2	8	no	—
184	C₂²×Q₈	—	2	2	no	—
185	C₂²×Q₈	—	2	8	no	—
186	C₂²×Q₈	—	4	4	no	—
187	C₂²×Q₈	C₂³×C₁₀₁	1	4	yes	—
188	C₂³×C₄	—	2	2	no	—
189	C₂³×C₄	D₄×C₁₀₁	1	2	yes	—
190	C₂³×C₄	C₂³×C₁₀₁	1	1	yes	characteristic
191	C₂²×C₈	C₂³×C₁₀₁	1	4	yes	—
192	C₄⋊Q₈	—	2	16	no	—
193	C₄⋊Q₈	—	2	8	no	—
194	C₄⋊Q₈	—	2	4	no	—
195	C₄⋊Q₈	D₄×C₁₀₁	1	8	yes	—
196	#125	—	2	4	no	—
197	C₂×C₄⋊C₄	—	2	16	no	—
198	C₂×C₄⋊C₄	—	2	4	no	—
199	C₂×C₄⋊C₄	—	2	4	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	C₂³×C₁₀₁	1	2	yes	—
203	C₂×C₄⋊C₄	C₂³×C₁₀₁	1	2	yes	—
204	C₂×C₂²⋊C₄	—	2	2	no	—
205	C₂×C₄²	—	2	4	no	—
206	C₂×C₄²	D₄×C₁₀₁	1	2	yes	—
207	C₂×C₄²	C₂³×C₁₀₁	1	1	yes	characteristic
208	C₂×C₄²	C₂³×C₁₀₁	1	1	yes	characteristic
209	#103	—	2	16	no	—
210	#103	—	4	8	no	—
211	C₂²⋊C₈	—	2	4	no	—
212	#97	—	2	4	no	—
213	#95	—	2	2	no	—
214	#95	—	2	2	no	—
215	#95	—	2	2	no	—
216	C₂⁴	C₂×D₄×C₁₀₁	1	1	yes	characteristic
217	C₂×Q₈	—	2	8	no	—
218	C₂×Q₈	—	2	4	no	—
219	C₂×Q₈	—	2	8	no	—
220	C₂×Q₈	—	2	8	no	—
221	C₂×Q₈	—	4	32	no	—
222	C₂×Q₈	—	4	4	no	—
223	C₂×Q₈	—	4	4	no	—
224	C₂×Q₈	C₂×D₄×C₁₀₁	1	8	yes	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	8	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	4	4	no	—
235	C₂²×C₄	—	4	4	no	—
236	C₂²×C₄	C₂×D₄×C₁₀₁	1	2	yes	—
237	C₂²×C₄	C₂×D₄×C₁₀₁	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₁₀₁	1	2	yes	—
239	C₂²×C₄	#41×C₁₀₁	1	4	yes	—
240	C₂²×C₄	C₂⁴×C₁₀₁	1	1	yes	characteristic
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	4	no	—
244	C₄⋊C₄	—	2	16	no	—
245	C₄⋊C₄	—	2	16	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	4	4	no	—
249	C₄⋊C₄	—	4	4	no	—
250	C₂²⋊C₄	—	4	4	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	2	2	no	—
253	C₄²	C₂×D₄×C₁₀₁	1	4	yes	—
254	C₂³	—	2	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	—	2	2	no	—
257	C₂³	#124×C₁₀₁	1	2	yes	—
258	C₂³	#137×C₁₀₁	1	4	yes	—
259	C₂³	C₂²×D₄×C₁₀₁	1	1	yes	characteristic
260	Q₈	—	2	16	no	—
261	Q₈	—	4	16	no	—
262	Q₈	—	4	8	no	—
263	C₂×C₄	—	2	16	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	8	no	—
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	—	4	16	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	#120×C₁₀₁	1	4	yes	—
282	C₂×C₄	#124×C₁₀₁	1	2	yes	—
283	C₂×C₄	#137×C₁₀₁	1	4	yes	—
284	C₂×C₄	C₂²×D₄×C₁₀₁	1	2	yes	—
285	C₂×C₄	C₂×#41×C₁₀₁	1	2	yes	—
286	C₂×C₄	H(5,𝔽₂)×C₁₀₁	1	2	yes	—
287	C₈	—	4	4	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	2	2	no	—
292	C₂²	#491×C₁₀₁	1	4	yes	complemented
293	C₂²	C₂×#120×C₁₀₁	1	1	yes	characteristic
294	C₂²	C₂×#124×C₁₀₁	1	1	yes	characteristic
295	C₂²	#535×C₁₀₁	1	1	yes	characteristic
296	C₂²	C₂×#137×C₁₀₁	1	2	yes	—
297	C₂²	C₂×#137×C₁₀₁	1	2	yes	—
298	C₄	—	2	2	no	—
299	C₄	—	2	4	no	—
300	C₄	—	2	4	no	—
301	C₄	—	4	8	no	—
302	C₄	—	8	2	no	—
303	C₄	—	8	2	no	—
304	C₄	#546×C₁₀₁	1	4	yes	—
305	C₂	—	2	4	no	—
306	C₂	C₁₀₁×#2435	1	1	yes	characteristic
307	C₂	C₁₀₁×#3004	1	2	yes	—
308	C₂	C₂×#491×C₁₀₁	1	1	yes	characteristic
309	C₂	C₁₀₁×#3139	1	1	yes	characteristic
310	C₂	C₁₀₁×#3218	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×#491×C₁₀₁	C₂	semidirect
4. C₁₀₁×#2043	C₂	semidirect
5. C₁₀₁×#1960	C₂	semidirect
136. C₂²×C₁₀₁	#491	semidirect
149. #25861	C₁₀₁	direct (N × H)
157. C₂×#491	C₂×C₁₀₁	semidirect
159. #2043	C₂×C₁₀₁	semidirect
160. #1960	C₂×C₁₀₁	semidirect
163. C₁₀₁	#25861	direct (N × H)
292. C₂²	#491×C₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Q₈×C₁₀₁	2
C₂×#491×C₁₀₁	2
C₁₀₁×#2851	2
C₁₀₁×#2043	2
C₁₀₁×#1960	2
C₁₀₁×#1787	2
C₁₀₁×#1749	2
#25861	101

Subgroups — C101×#25861 #38808012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#25861 #38808012