Subgroups — C₁₀₁×#10891 #38793042

Order 25856.

906 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#384×C₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₁₀₁×#2274	C₂	1	1	yes	maximal, characteristic
3	C₂×#343×C₁₀₁	C₂	1	1	yes	maximal, characteristic
4	C₁₀₁×#1736	C₂	1	2	yes	maximal
5	C₂×#339×C₁₀₁	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₁₀₁	C₂²	1	2	yes	—
7	C₂×#117×C₁₀₁	C₂²	1	2	yes	—
8	C₂²×C₄⋊C₄×C₁₀₁	—	2	2	no	—
9	C₂²×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
12	#384×C₁₀₁	—	2	2	no	—
13	#384×C₁₀₁	C₄	1	4	yes	complemented
14	C₄×C₄⋊C₄×C₁₀₁	—	2	4	no	—
15	C₄×C₂²⋊C₄×C₁₀₁	—	2	2	no	—
16	#343×C₁₀₁	—	2	2	no	—
17	#340×C₁₀₁	—	2	4	no	—
18	#339×C₁₀₁	C₄	1	8	yes	complemented
19	C₂³×C₄×C₁₀₁	—	2	1	no	—
20	C₂³×C₄×C₁₀₁	—	2	1	no	—
21	C₂³×C₄×C₁₀₁	D₄	1	2	yes	—
22	C₂³×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
23	C₂×M₁₆×C₁₀₁	—	2	2	no	—
24	C₂×M₁₆×C₁₀₁	—	2	2	no	—
25	C₂×M₁₆×C₁₀₁	—	2	2	no	—
26	C₂×M₁₆×C₁₀₁	—	2	2	no	—
27	C₂×M₁₆×C₁₀₁	—	2	2	no	—
28	C₂×M₁₆×C₁₀₁	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₁₀₁	—	2	2	no	—
30	#117×C₁₀₁	—	2	2	no	—
31	#117×C₁₀₁	—	2	2	no	—
32	#117×C₁₀₁	C₂×C₄	1	4	yes	—
33	#117×C₁₀₁	Q₈	1	4	yes	—
34	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
38	C₂×C₄⋊C₄×C₁₀₁	—	2	8	no	—
39	C₂×C₄⋊C₄×C₁₀₁	—	2	1	no	—
40	C₂×C₄⋊C₄×C₁₀₁	—	4	8	no	—
41	C₂×C₄⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
42	C₂×C₄⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₁₀₁	D₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₁₀₁	D₄	1	4	yes	—
45	C₂×C₂²⋊C₄×C₁₀₁	—	2	2	no	—
46	C₂×C₄²×C₁₀₁	—	2	1	no	—
47	C₂×C₄²×C₁₀₁	—	2	2	no	—
48	C₂×C₄²×C₁₀₁	—	2	2	no	—
49	C₂×C₄²×C₁₀₁	—	2	4	no	—
50	C₂×C₄²×C₁₀₁	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₁₀₁	C₂×C₄	1	2	yes	—
52	#95×C₁₀₁	—	2	4	no	—
53	C₂⁴×C₁₀₁	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₀₁	—	2	4	no	—
55	C₂²×C₄×C₁₀₁	—	2	4	no	—
56	C₂²×C₄×C₁₀₁	—	2	2	no	—
57	C₂²×C₄×C₁₀₁	—	2	4	no	—
58	C₂²×C₄×C₁₀₁	—	2	4	no	—
59	C₂²×C₄×C₁₀₁	—	2	2	no	—
60	C₂²×C₄×C₁₀₁	—	2	2	no	—
61	C₂²×C₄×C₁₀₁	—	2	2	no	—
62	C₂²×C₄×C₁₀₁	—	4	4	no	—
63	C₂²×C₄×C₁₀₁	—	4	4	no	—
64	C₂²×C₄×C₁₀₁	—	4	1	no	—
65	C₂²×C₄×C₁₀₁	—	4	1	no	—
66	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	4	yes	—
67	C₂²×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₁₀₁	#41	1	2	yes	—
73	C₂²×C₄×C₁₀₁	#41	1	2	yes	—
74	C₂²×C₄×C₁₀₁	#41	1	2	yes	—
75	M₁₆×C₁₀₁	—	2	4	no	—
76	M₁₆×C₁₀₁	—	2	4	no	—
77	M₁₆×C₁₀₁	—	4	2	no	—
78	M₁₆×C₁₀₁	—	4	2	no	—
79	C₂×C₈×C₁₀₁	—	2	4	no	—
80	C₂×C₈×C₁₀₁	—	4	2	no	—
81	C₂×C₈×C₁₀₁	—	4	2	no	—
82	C₄⋊C₄×C₁₀₁	—	2	4	no	—
83	C₄⋊C₄×C₁₀₁	—	2	4	no	—
84	C₄⋊C₄×C₁₀₁	—	2	4	no	—
85	C₄⋊C₄×C₁₀₁	—	2	4	no	—
86	C₄⋊C₄×C₁₀₁	—	4	16	no	—
87	C₄⋊C₄×C₁₀₁	—	4	2	no	—
88	C₄⋊C₄×C₁₀₁	C₄⋊C₄	1	8	yes	complemented
89	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
90	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
91	C₄²×C₁₀₁	—	2	2	no	—
92	C₄²×C₁₀₁	—	2	4	no	—
93	C₄²×C₁₀₁	—	2	4	no	—
94	C₄²×C₁₀₁	—	2	4	no	—
95	C₄²×C₁₀₁	—	8	4	no	—
96	C₄²×C₁₀₁	C₄²	1	4	yes	complemented
97	C₂³×C₁₀₁	—	2	2	no	—
98	C₂³×C₁₀₁	—	2	2	no	—
99	C₂³×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂³×C₁₀₁	C₄×D₄	1	2	yes	—
101	C₂³×C₁₀₁	#120	1	1	yes	characteristic
102	C₂³×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
103	C₂³×C₁₀₁	#123	1	1	yes	characteristic
104	C₂×C₄×C₁₀₁	—	2	4	no	—
105	C₂×C₄×C₁₀₁	—	2	16	no	—
106	C₂×C₄×C₁₀₁	—	2	4	no	—
107	C₂×C₄×C₁₀₁	—	2	4	no	—
108	C₂×C₄×C₁₀₁	—	2	8	no	—
109	C₂×C₄×C₁₀₁	—	2	4	no	—
110	C₂×C₄×C₁₀₁	—	4	4	no	—
111	C₂×C₄×C₁₀₁	—	4	4	no	—
112	C₂×C₄×C₁₀₁	—	4	4	no	—
113	C₂×C₄×C₁₀₁	—	4	4	no	—
114	C₂×C₄×C₁₀₁	—	4	4	no	—
115	C₂×C₄×C₁₀₁	—	4	2	no	—
116	C₂×C₄×C₁₀₁	—	8	2	no	—
117	C₂×C₄×C₁₀₁	—	8	2	no	—
118	C₂×C₄×C₁₀₁	—	8	2	no	—
119	C₂×C₄×C₁₀₁	C₂×C₄²	1	1	yes	characteristic
120	C₂×C₄×C₁₀₁	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₁₀₁	#117	1	2	yes	—
122	C₂×C₄×C₁₀₁	#117	1	1	yes	characteristic
123	C₂×C₄×C₁₀₁	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₁₀₁	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₁₀₁	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₁₀₁	C₄×Q₈	1	2	yes	—
127	C₂×C₄×C₁₀₁	C₄×Q₈	1	2	yes	—
128	C₂×C₄×C₁₀₁	#121	1	2	yes	—
129	C₂×C₄×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
130	C₂×C₄×C₁₀₁	#123	1	2	yes	—
131	C₂×C₄×C₁₀₁	#124	1	1	yes	characteristic
132	C₂×C₄×C₁₀₁	#125	1	2	yes	—
133	C₂×C₄×C₁₀₁	#126	1	2	yes	—
134	C₂×C₄×C₁₀₁	C₄⋊Q₈	1	1	yes	characteristic
135	C₈×C₁₀₁	—	4	4	no	—
136	C₂²×C₁₀₁	—	2	4	no	—
137	C₂²×C₁₀₁	—	2	4	no	—
138	C₂²×C₁₀₁	—	2	1	no	—
139	C₂²×C₁₀₁	—	2	1	no	—
140	C₂²×C₁₀₁	—	2	1	no	—
141	C₂²×C₁₀₁	—	2	1	no	—
142	C₂²×C₁₀₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
143	C₂²×C₁₀₁	#385	1	2	yes	—
144	C₂²×C₁₀₁	#386	1	1	yes	characteristic
145	C₂²×C₁₀₁	#387	1	2	yes	—
146	C₂²×C₁₀₁	#391	1	1	yes	characteristic
147	C₂²×C₁₀₁	#453	1	2	yes	—
148	C₂²×C₁₀₁	#456	1	2	yes	—
149	C₄×C₁₀₁	—	2	8	no	—
150	C₄×C₁₀₁	—	4	8	no	—
151	C₄×C₁₀₁	—	4	4	no	—
152	C₄×C₁₀₁	—	8	2	no	—
153	C₄×C₁₀₁	C₄×C₄⋊C₄	1	2	yes	—
154	C₄×C₁₀₁	#387	1	2	yes	—
155	C₄×C₁₀₁	#388	1	2	yes	—
156	C₄×C₁₀₁	#389	1	2	yes	—
157	#10891	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂×C₁₀₁	—	2	1	no	—
159	C₂×C₁₀₁	—	2	2	no	—
160	C₂×C₁₀₁	—	2	1	no	—
161	C₂×C₁₀₁	#1436	1	1	yes	characteristic
162	C₂×C₁₀₁	#1859	1	4	yes	—
163	C₂×C₁₀₁	#1886	1	1	yes	characteristic
164	C₂×C₁₀₁	#1887	1	1	yes	characteristic
165	C₂×#384	C₂×C₁₀₁	1	1	yes	characteristic
166	#2274	C₂×C₁₀₁	1	1	yes	characteristic
167	C₂×#343	C₂×C₁₀₁	1	1	yes	characteristic
168	#1736	C₂×C₁₀₁	1	2	yes	—
169	C₂×#339	C₂×C₁₀₁	1	2	yes	—
170	C₁₀₁	#10891	1	1	yes	characteristic, direct product factor, Sylow
171	C₂²×M₁₆	C₂²×C₁₀₁	1	2	yes	—
172	C₂×#117	C₂²×C₁₀₁	1	2	yes	—
173	C₂²×C₄⋊C₄	—	2	2	no	—
174	C₂²×C₄⋊C₄	C₂²×C₁₀₁	1	1	yes	characteristic
175	C₂²×C₄⋊C₄	C₂²×C₁₀₁	1	1	yes	characteristic
176	C₂²×C₄²	C₂²×C₁₀₁	1	1	yes	characteristic
177	#384	—	2	2	no	—
178	#384	C₄×C₁₀₁	1	4	yes	complemented
179	C₄×C₄⋊C₄	—	2	4	no	—
180	C₄×C₂²⋊C₄	—	2	2	no	—
181	#343	—	2	2	no	—
182	#340	—	2	4	no	—
183	#339	C₄×C₁₀₁	1	8	yes	complemented
184	C₂³×C₄	—	2	1	no	—
185	C₂³×C₄	—	2	1	no	—
186	C₂³×C₄	D₄×C₁₀₁	1	2	yes	—
187	C₂³×C₄	C₂³×C₁₀₁	1	1	yes	characteristic
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	C₂×C₄×C₁₀₁	1	4	yes	—
194	C₂²×C₈	—	2	2	no	—
195	#117	—	2	2	no	—
196	#117	—	2	2	no	—
197	#117	C₂×C₄×C₁₀₁	1	4	yes	—
198	#117	Q₈×C₁₀₁	1	4	yes	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	8	no	—
204	C₂×C₄⋊C₄	—	2	1	no	—
205	C₂×C₄⋊C₄	—	4	8	no	—
206	C₂×C₄⋊C₄	C₂×C₄×C₁₀₁	1	2	yes	—
207	C₂×C₄⋊C₄	C₂×C₄×C₁₀₁	1	2	yes	—
208	C₂×C₄⋊C₄	D₄×C₁₀₁	1	2	yes	—
209	C₂×C₄⋊C₄	D₄×C₁₀₁	1	4	yes	—
210	C₂×C₂²⋊C₄	—	2	2	no	—
211	C₂×C₄²	—	2	2	no	—
212	C₂×C₄²	—	2	1	no	—
213	C₂×C₄²	—	2	2	no	—
214	C₂×C₄²	—	2	4	no	—
215	C₂×C₄²	C₂×C₄×C₁₀₁	1	4	yes	—
216	C₂×C₄²	C₂×C₄×C₁₀₁	1	2	yes	—
217	#95	—	2	4	no	—
218	C₂⁴	C₂×D₄×C₁₀₁	1	1	yes	characteristic
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	4	4	no	—
228	C₂²×C₄	—	4	4	no	—
229	C₂²×C₄	—	4	1	no	—
230	C₂²×C₄	—	4	1	no	—
231	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	4	yes	—
232	C₂²×C₄	C₂²×C₄×C₁₀₁	1	2	yes	—
233	C₂²×C₄	C₂²×C₄×C₁₀₁	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₄×C₁₀₁	1	2	yes	—
235	C₂²×C₄	C₂×D₄×C₁₀₁	1	1	yes	characteristic
236	C₂²×C₄	C₂×Q₈×C₁₀₁	1	2	yes	—
237	C₂²×C₄	#41×C₁₀₁	1	2	yes	—
238	C₂²×C₄	#41×C₁₀₁	1	2	yes	—
239	C₂²×C₄	#41×C₁₀₁	1	2	yes	—
240	M₁₆	—	2	4	no	—
241	M₁₆	—	2	4	no	—
242	M₁₆	—	4	2	no	—
243	M₁₆	—	4	2	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	4	2	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	4	16	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	C₄⋊C₄×C₁₀₁	1	8	yes	complemented
254	C₂²⋊C₄	—	2	4	no	—
255	C₂²⋊C₄	—	2	4	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	4	no	—
258	C₄²	—	2	4	no	—
259	C₄²	—	2	4	no	—
260	C₄²	—	8	4	no	—
261	C₄²	C₄²×C₁₀₁	1	4	yes	complemented
262	C₂³	—	2	2	no	—
263	C₂³	—	2	2	no	—
264	C₂³	C₂×C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
265	C₂³	C₄×D₄×C₁₀₁	1	2	yes	—
266	C₂³	#120×C₁₀₁	1	1	yes	characteristic
267	C₂³	C₂²⋊Q₈×C₁₀₁	1	2	yes	—
268	C₂³	#123×C₁₀₁	1	1	yes	characteristic
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	16	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	8	2	no	—
282	C₂×C₄	—	8	2	no	—
283	C₂×C₄	—	8	2	no	—
284	C₂×C₄	C₂×C₄²×C₁₀₁	1	1	yes	characteristic
285	C₂×C₄	C₂×C₄⋊C₄×C₁₀₁	1	2	yes	—
286	C₂×C₄	#117×C₁₀₁	1	2	yes	—
287	C₂×C₄	#117×C₁₀₁	1	1	yes	characteristic
288	C₂×C₄	C₄×D₄×C₁₀₁	1	2	yes	—
289	C₂×C₄	C₄×D₄×C₁₀₁	1	2	yes	—
290	C₂×C₄	C₄×D₄×C₁₀₁	1	2	yes	—
291	C₂×C₄	C₄×Q₈×C₁₀₁	1	2	yes	—
292	C₂×C₄	C₄×Q₈×C₁₀₁	1	2	yes	—
293	C₂×C₄	#121×C₁₀₁	1	2	yes	—
294	C₂×C₄	C₂²⋊Q₈×C₁₀₁	1	2	yes	—
295	C₂×C₄	#123×C₁₀₁	1	2	yes	—
296	C₂×C₄	#124×C₁₀₁	1	1	yes	characteristic
297	C₂×C₄	#125×C₁₀₁	1	2	yes	—
298	C₂×C₄	#126×C₁₀₁	1	2	yes	—
299	C₂×C₄	C₄⋊Q₈×C₁₀₁	1	1	yes	characteristic
300	C₈	—	4	4	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	2	4	no	—
307	C₂²	C₄×C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
308	C₂²	#385×C₁₀₁	1	2	yes	—
309	C₂²	#386×C₁₀₁	1	1	yes	characteristic
310	C₂²	#387×C₁₀₁	1	2	yes	—
311	C₂²	#391×C₁₀₁	1	1	yes	characteristic
312	C₂²	#453×C₁₀₁	1	2	yes	—
313	C₂²	#456×C₁₀₁	1	2	yes	—
314	C₄	—	2	8	no	—
315	C₄	—	4	4	no	—
316	C₄	—	4	8	no	—
317	C₄	—	8	2	no	—
318	C₄	C₄×C₄⋊C₄×C₁₀₁	1	2	yes	—
319	C₄	#387×C₁₀₁	1	2	yes	—
320	C₄	#388×C₁₀₁	1	2	yes	—
321	C₄	#389×C₁₀₁	1	2	yes	—
322	C₂	—	2	1	no	—
323	C₂	—	2	2	no	—
324	C₂	—	2	1	no	—
325	C₂	C₁₀₁×#1436	1	1	yes	characteristic
326	C₂	C₁₀₁×#1859	1	4	yes	—
327	C₂	C₁₀₁×#1886	1	1	yes	characteristic
328	C₂	C₁₀₁×#1887	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
13. #384×C₁₀₁	C₄	semidirect
18. #339×C₁₀₁	C₄	semidirect
88. C₄⋊C₄×C₁₀₁	C₄⋊C₄	semidirect
96. C₄²×C₁₀₁	C₄²	semidirect
157. #10891	C₁₀₁	direct (N × H)
170. C₁₀₁	#10891	direct (N × H)
178. #384	C₄×C₁₀₁	semidirect
183. #339	C₄×C₁₀₁	semidirect
253. C₄⋊C₄	C₄⋊C₄×C₁₀₁	semidirect
261. C₄²	C₄²×C₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#384×C₁₀₁	2
C₁₀₁×#2274	2
C₂×#343×C₁₀₁	2
C₁₀₁×#1736	2
C₂×#339×C₁₀₁	2
#10891	101

Subgroups — C101×#10891 #38793042

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#10891 #38793042