Subgroups — C₁₀₁×#7609 #38789760

Order 25856.

676 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#391×C₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₁₀₁×#1817	C₂	1	1	yes	maximal, characteristic
3	C₁₀₁×#1728	C₂	1	1	yes	maximal, characteristic
4	C₁₀₁×#1272	C₂	1	4	yes	maximal, complemented
5	C₂³×Q₈×C₁₀₁	C₄	1	1	yes	characteristic
6	C₂²×M₁₆×C₁₀₁	—	2	1	no	—
7	C₂²×C₄⋊C₄×C₁₀₁	C₄	1	1	yes	characteristic
8	C₂²×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
9	C₂×C₂²⋊C₈×C₁₀₁	—	2	1	no	—
10	#391×C₁₀₁	—	2	1	no	—
11	#391×C₁₀₁	—	2	1	no	—
12	#391×C₁₀₁	C₄	1	2	yes	—
13	#391×C₁₀₁	C₂²	1	2	yes	—
14	C₂×#95×C₁₀₁	—	2	1	no	—
15	C₂×#95×C₁₀₁	—	2	1	no	—
16	#336×C₁₀₁	—	2	1	no	—
17	#336×C₁₀₁	—	2	1	no	—
18	#336×C₁₀₁	C₂²	1	2	yes	—
19	#336×C₁₀₁	C₂²	1	2	yes	—
20	C₂²×Q₈×C₁₀₁	—	2	4	no	—
21	C₂²×Q₈×C₁₀₁	—	2	1	no	—
22	C₂²×Q₈×C₁₀₁	—	4	2	no	—
23	C₂²×Q₈×C₁₀₁	—	4	2	no	—
24	C₂²×Q₈×C₁₀₁	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₁₀₁	—	2	1	no	—
26	C₂³×C₄×C₁₀₁	—	2	1	no	—
27	C₂³×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₀₁	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₁₀₁	D₄	1	1	yes	characteristic
30	C₂×M₁₆×C₁₀₁	—	2	4	no	—
31	C₂×M₁₆×C₁₀₁	—	4	2	no	—
32	C₂×M₁₆×C₁₀₁	—	4	2	no	—
33	C₂²×C₈×C₁₀₁	—	2	2	no	—
34	C₂²×C₈×C₁₀₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₁₀₁	—	2	1	no	—
36	C₂×C₄⋊C₄×C₁₀₁	—	4	2	no	—
37	C₂×C₄⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₁₀₁	—	4	2	no	—
39	C₂×C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₀₁	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₀₁	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
43	C₂²⋊C₈×C₁₀₁	—	4	2	no	—
44	#95×C₁₀₁	—	2	2	no	—
45	#95×C₁₀₁	—	2	2	no	—
46	#95×C₁₀₁	—	2	2	no	—
47	#95×C₁₀₁	—	2	2	no	—
48	C₂⁴×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×Q₈×C₁₀₁	—	2	4	no	—
50	C₂×Q₈×C₁₀₁	—	2	2	no	—
51	C₂×Q₈×C₁₀₁	—	4	16	no	—
52	C₂×Q₈×C₁₀₁	—	4	2	no	—
53	C₂×Q₈×C₁₀₁	—	4	2	no	—
54	C₂×Q₈×C₁₀₁	—	4	2	no	—
55	C₂×Q₈×C₁₀₁	—	4	2	no	—
56	C₂×Q₈×C₁₀₁	M₁₆	1	4	yes	complemented
57	C₂²×C₄×C₁₀₁	—	2	2	no	—
58	C₂²×C₄×C₁₀₁	—	2	2	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	—	2	4	no	—
63	C₂²×C₄×C₁₀₁	—	2	2	no	—
64	C₂²×C₄×C₁₀₁	—	2	2	no	—
65	C₂²×C₄×C₁₀₁	—	2	2	no	—
66	C₂²×C₄×C₁₀₁	—	2	2	no	—
67	C₂²×C₄×C₁₀₁	—	4	1	no	—
68	C₂²×C₄×C₁₀₁	—	4	2	no	—
69	C₂²×C₄×C₁₀₁	—	4	2	no	—
70	C₂²×C₄×C₁₀₁	—	4	2	no	—
71	C₂²×C₄×C₁₀₁	—	4	2	no	—
72	C₂²×C₄×C₁₀₁	—	4	2	no	—
73	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₀₁	SD₁₆	1	2	yes	—
77	C₂²×C₄×C₁₀₁	Q₁₆	1	2	yes	—
78	C₂²×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
81	M₁₆×C₁₀₁	—	4	8	no	—
82	C₂×C₈×C₁₀₁	—	2	4	no	—
83	C₂×C₈×C₁₀₁	—	4	2	no	—
84	C₂×C₈×C₁₀₁	—	4	2	no	—
85	C₂×C₈×C₁₀₁	—	4	2	no	—
86	C₂×C₈×C₁₀₁	—	4	2	no	—
87	C₂×C₈×C₁₀₁	—	4	2	no	—
88	C₄⋊C₄×C₁₀₁	—	4	2	no	—
89	C₄⋊C₄×C₁₀₁	—	4	2	no	—
90	C₄²×C₁₀₁	—	4	1	no	—
91	C₄²×C₁₀₁	—	4	1	no	—
92	C₄²×C₁₀₁	—	4	1	no	—
93	C₄²×C₁₀₁	—	4	1	no	—
94	C₂³×C₁₀₁	—	2	1	no	—
95	C₂³×C₁₀₁	—	2	1	no	—
96	C₂³×C₁₀₁	—	2	1	no	—
97	C₂³×C₁₀₁	—	2	1	no	—
98	C₂³×C₁₀₁	#99	1	2	yes	—
99	C₂³×C₁₀₁	#103	1	2	yes	—
100	C₂³×C₁₀₁	#104	1	2	yes	—
101	C₂³×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	Q₈×C₁₀₁	—	4	4	no	—
103	Q₈×C₁₀₁	—	4	8	no	—
104	Q₈×C₁₀₁	—	4	4	no	—
105	C₂×C₄×C₁₀₁	—	2	8	no	—
106	C₂×C₄×C₁₀₁	—	2	4	no	—
107	C₂×C₄×C₁₀₁	—	2	4	no	—
108	C₂×C₄×C₁₀₁	—	2	1	no	—
109	C₂×C₄×C₁₀₁	—	2	1	no	—
110	C₂×C₄×C₁₀₁	—	2	1	no	—
111	C₂×C₄×C₁₀₁	—	2	1	no	—
112	C₂×C₄×C₁₀₁	—	2	1	no	—
113	C₂×C₄×C₁₀₁	—	2	1	no	—
114	C₂×C₄×C₁₀₁	—	2	1	no	—
115	C₂×C₄×C₁₀₁	—	2	1	no	—
116	C₂×C₄×C₁₀₁	—	2	1	no	—
117	C₂×C₄×C₁₀₁	—	2	1	no	—
118	C₂×C₄×C₁₀₁	—	2	4	no	—
119	C₂×C₄×C₁₀₁	—	2	1	no	—
120	C₂×C₄×C₁₀₁	—	2	1	no	—
121	C₂×C₄×C₁₀₁	—	2	1	no	—
122	C₂×C₄×C₁₀₁	—	4	4	no	—
123	C₂×C₄×C₁₀₁	—	4	8	no	—
124	C₂×C₄×C₁₀₁	—	4	2	no	—
125	C₂×C₄×C₁₀₁	—	4	2	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	—	8	4	no	—
130	C₂×C₄×C₁₀₁	#100	1	2	yes	—
131	C₂×C₄×C₁₀₁	#103	1	2	yes	—
132	C₂×C₄×C₁₀₁	C₂×M₁₆	1	2	yes	—
133	C₂×C₄×C₁₀₁	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₁₀₁	C₂×Q₁₆	1	1	yes	characteristic
135	C₂×C₄×C₁₀₁	#137	1	2	yes	—
136	C₈×C₁₀₁	—	4	4	no	—
137	C₈×C₁₀₁	—	8	2	no	—
138	C₂²×C₁₀₁	—	2	2	no	—
139	C₂²×C₁₀₁	—	2	2	no	—
140	C₂²×C₁₀₁	—	2	2	no	—
141	C₂²×C₁₀₁	—	2	2	no	—
142	C₂²×C₁₀₁	—	2	2	no	—
143	C₂²×C₁₀₁	—	2	2	no	—
144	C₂²×C₁₀₁	#328	1	4	yes	—
145	C₂²×C₁₀₁	#407	1	1	yes	characteristic
146	C₂²×C₁₀₁	C₂×#99	1	1	yes	characteristic
147	C₂²×C₁₀₁	C₂×#100	1	1	yes	characteristic
148	C₂²×C₁₀₁	C₂×#103	1	1	yes	characteristic
149	C₂²×C₁₀₁	#419	1	1	yes	characteristic
150	C₂²×C₁₀₁	C₂×#104	1	1	yes	characteristic
151	C₂²×C₁₀₁	#421	1	1	yes	characteristic
152	C₄×C₁₀₁	—	2	2	no	—
153	C₄×C₁₀₁	—	2	2	no	—
154	C₄×C₁₀₁	—	2	1	no	—
155	C₄×C₁₀₁	—	2	2	no	—
156	C₄×C₁₀₁	—	2	1	no	—
157	C₄×C₁₀₁	—	2	2	no	—
158	C₄×C₁₀₁	—	4	4	no	—
159	C₄×C₁₀₁	—	4	1	no	—
160	C₄×C₁₀₁	—	8	2	no	—
161	#7609	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₂×C₁₀₁	—	2	4	no	—
163	C₂×C₁₀₁	#1462	1	1	yes	characteristic
164	C₂×C₁₀₁	#1487	1	1	yes	characteristic
165	C₂×C₁₀₁	#1491	1	1	yes	characteristic
166	C₂×C₁₀₁	C₂×#328	1	1	yes	characteristic
167	C₂×C₁₀₁	#1511	1	1	yes	characteristic
168	C₂×C₁₀₁	#1527	1	1	yes	characteristic
169	C₂×C₁₀₁	#1548	1	1	yes	characteristic
170	C₂×#391	C₂×C₁₀₁	1	1	yes	characteristic
171	#1817	C₂×C₁₀₁	1	1	yes	characteristic
172	#1728	C₂×C₁₀₁	1	1	yes	characteristic
173	#1272	C₂×C₁₀₁	1	4	yes	complemented
174	C₁₀₁	#7609	1	1	yes	characteristic, direct product factor, Sylow
175	C₂³×Q₈	C₄×C₁₀₁	1	1	yes	characteristic
176	C₂²×M₁₆	—	2	1	no	—
177	C₂²×C₄⋊C₄	C₄×C₁₀₁	1	1	yes	characteristic
178	C₂²×C₄²	C₂²×C₁₀₁	1	1	yes	characteristic
179	C₂×C₂²⋊C₈	—	2	1	no	—
180	#391	—	2	1	no	—
181	#391	—	2	1	no	—
182	#391	C₄×C₁₀₁	1	2	yes	—
183	#391	C₂²×C₁₀₁	1	2	yes	—
184	C₂×#95	—	2	1	no	—
185	C₂×#95	—	2	1	no	—
186	#336	—	2	1	no	—
187	#336	—	2	1	no	—
188	#336	C₂²×C₁₀₁	1	2	yes	—
189	#336	C₂²×C₁₀₁	1	2	yes	—
190	C₂²×Q₈	—	2	4	no	—
191	C₂²×Q₈	—	2	1	no	—
192	C₂²×Q₈	—	4	2	no	—
193	C₂²×Q₈	—	4	2	no	—
194	C₂²×Q₈	C₂×C₄×C₁₀₁	1	2	yes	—
195	C₂³×C₄	—	2	1	no	—
196	C₂³×C₄	—	2	1	no	—
197	C₂³×C₄	C₂×C₄×C₁₀₁	1	1	yes	characteristic
198	C₂³×C₄	D₄×C₁₀₁	1	1	yes	characteristic
199	C₂³×C₄	D₄×C₁₀₁	1	1	yes	characteristic
200	C₂×M₁₆	—	2	4	no	—
201	C₂×M₁₆	—	4	2	no	—
202	C₂×M₁₆	—	4	2	no	—
203	C₂²×C₈	—	2	2	no	—
204	C₂²×C₈	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	4	2	no	—
207	C₂×C₄⋊C₄	C₂×C₄×C₁₀₁	1	2	yes	—
208	C₂×C₄²	—	4	2	no	—
209	C₂×C₄²	C₂×C₄×C₁₀₁	1	1	yes	characteristic
210	C₂×C₄²	D₄×C₁₀₁	1	1	yes	characteristic
211	C₂×C₄²	D₄×C₁₀₁	1	1	yes	characteristic
212	C₂×C₄²	C₂³×C₁₀₁	1	1	yes	characteristic
213	C₂²⋊C₈	—	4	2	no	—
214	#95	—	2	2	no	—
215	#95	—	2	2	no	—
216	#95	—	2	2	no	—
217	#95	—	2	2	no	—
218	C₂⁴	C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
219	C₂×Q₈	—	2	4	no	—
220	C₂×Q₈	—	2	2	no	—
221	C₂×Q₈	—	4	16	no	—
222	C₂×Q₈	—	4	2	no	—
223	C₂×Q₈	—	4	2	no	—
224	C₂×Q₈	—	4	2	no	—
225	C₂×Q₈	—	4	2	no	—
226	C₂×Q₈	M₁₆×C₁₀₁	1	4	yes	complemented
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	—	4	2	no	—
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	—	4	2	no	—
243	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
244	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
245	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
246	C₂²×C₄	SD₁₆×C₁₀₁	1	2	yes	—
247	C₂²×C₄	Q₁₆×C₁₀₁	1	2	yes	—
248	C₂²×C₄	C₂²×C₄×C₁₀₁	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₄×C₁₀₁	1	1	yes	characteristic
250	C₂²×C₄	C₂×D₄×C₁₀₁	1	1	yes	characteristic
251	M₁₆	—	4	8	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	—	4	2	no	—
255	C₂×C₈	—	4	2	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄²	—	4	1	no	—
261	C₄²	—	4	1	no	—
262	C₄²	—	4	1	no	—
263	C₄²	—	4	1	no	—
264	C₂³	—	2	1	no	—
265	C₂³	—	2	1	no	—
266	C₂³	—	2	1	no	—
267	C₂³	—	2	1	no	—
268	C₂³	#99×C₁₀₁	1	2	yes	—
269	C₂³	#103×C₁₀₁	1	2	yes	—
270	C₂³	#104×C₁₀₁	1	2	yes	—
271	C₂³	C₂×C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
272	Q₈	—	4	4	no	—
273	Q₈	—	4	8	no	—
274	Q₈	—	4	4	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	8	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	8	4	no	—
300	C₂×C₄	#100×C₁₀₁	1	2	yes	—
301	C₂×C₄	#103×C₁₀₁	1	2	yes	—
302	C₂×C₄	C₂×M₁₆×C₁₀₁	1	2	yes	—
303	C₂×C₄	C₂×SD₁₆×C₁₀₁	1	1	yes	characteristic
304	C₂×C₄	C₂×Q₁₆×C₁₀₁	1	1	yes	characteristic
305	C₂×C₄	#137×C₁₀₁	1	2	yes	—
306	C₈	—	4	4	no	—
307	C₈	—	8	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	#328×C₁₀₁	1	4	yes	—
315	C₂²	#407×C₁₀₁	1	1	yes	characteristic
316	C₂²	C₂×#99×C₁₀₁	1	1	yes	characteristic
317	C₂²	C₂×#100×C₁₀₁	1	1	yes	characteristic
318	C₂²	C₂×#103×C₁₀₁	1	1	yes	characteristic
319	C₂²	#419×C₁₀₁	1	1	yes	characteristic
320	C₂²	C₂×#104×C₁₀₁	1	1	yes	characteristic
321	C₂²	#421×C₁₀₁	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	2	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	2	no	—
328	C₄	—	4	4	no	—
329	C₄	—	4	1	no	—
330	C₄	—	8	2	no	—
331	C₂	—	2	4	no	—
332	C₂	C₁₀₁×#1462	1	1	yes	characteristic
333	C₂	C₁₀₁×#1487	1	1	yes	characteristic
334	C₂	C₁₀₁×#1491	1	1	yes	characteristic
335	C₂	C₂×#328×C₁₀₁	1	1	yes	characteristic
336	C₂	C₁₀₁×#1511	1	1	yes	characteristic
337	C₂	C₁₀₁×#1527	1	1	yes	characteristic
338	C₂	C₁₀₁×#1548	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₀₁	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₀₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₁
prime radical	C₂⁴×C₁₀₁	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ 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
4. C₁₀₁×#1272	C₂	semidirect
56. C₂×Q₈×C₁₀₁	M₁₆	semidirect
161. #7609	C₁₀₁	direct (N × H)
173. #1272	C₂×C₁₀₁	semidirect
174. C₁₀₁	#7609	direct (N × H)
226. C₂×Q₈	M₁₆×C₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#391×C₁₀₁	2
C₁₀₁×#1817	2
C₁₀₁×#1728	2
C₁₀₁×#1272	2
#7609	101

Subgroups — C101×#7609 #38789760

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#7609 #38789760