Subgroups — C₁₀₁×#13007 #38795158

Order 25856.

492 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#472×C₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₁₀₁×#2916	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#361×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#361×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₁×#2117	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₀₁×#1388	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₀₁×#1385	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×AGL(1,ℤ/8)×C₁₀₁	—	2	1	no	—
9	C₂²×D₈×C₁₀₁	C₂²	1	1	yes	characteristic
10	#568×C₁₀₁	—	2	1	no	—
11	C₂×#131×C₁₀₁	C₂²	1	1	yes	characteristic
12	C₂×M₃₂×C₁₀₁	C₂²	1	1	yes	characteristic
13	C₂×M₃₂×C₁₀₁	C₂²	1	1	yes	characteristic
14	#472×C₁₀₁	—	2	2	no	—
15	#472×C₁₀₁	C₄	1	4	yes	—
16	#430×C₁₀₁	—	2	1	no	—
17	#430×C₁₀₁	—	2	2	no	—
18	C₂×#108×C₁₀₁	—	2	2	no	—
19	C₂×#108×C₁₀₁	C₂²	1	1	yes	characteristic
20	C₂×#108×C₁₀₁	C₂²	1	1	yes	characteristic
21	C₂×#108×C₁₀₁	C₂²	1	1	yes	characteristic
22	C₂×#100×C₁₀₁	—	2	1	no	—
23	#361×C₁₀₁	—	2	2	no	—
24	#361×C₁₀₁	—	2	2	no	—
25	#350×C₁₀₁	—	2	2	no	—
26	C₂×#41×C₁₀₁	D₄	1	1	yes	characteristic
27	C₂²×D₄×C₁₀₁	—	2	1	no	—
28	AGL(1,ℤ/8)×C₁₀₁	—	2	4	no	—
29	AGL(1,ℤ/8)×C₁₀₁	—	4	2	no	—
30	C₂×SD₁₆×C₁₀₁	—	4	1	no	—
31	C₂×D₈×C₁₀₁	—	2	1	no	—
32	C₂×D₈×C₁₀₁	—	2	1	no	—
33	C₂×D₈×C₁₀₁	—	2	1	no	—
34	C₂×D₈×C₁₀₁	—	2	1	no	—
35	C₂×D₈×C₁₀₁	—	2	1	no	—
36	C₂×D₈×C₁₀₁	—	4	1	no	—
37	C₂×D₈×C₁₀₁	C₂×C₄	1	2	yes	—
38	#131×C₁₀₁	—	2	1	no	—
39	#131×C₁₀₁	—	2	1	no	—
40	#131×C₁₀₁	—	4	2	no	—
41	#131×C₁₀₁	—	4	2	no	—
42	#131×C₁₀₁	C₂×C₄	1	2	yes	—
43	#131×C₁₀₁	D₄	1	2	yes	—
44	C₂×M₁₆×C₁₀₁	—	2	1	no	—
45	C₂×M₁₆×C₁₀₁	—	2	4	no	—
46	C₂×M₁₆×C₁₀₁	—	2	1	no	—
47	C₂×M₁₆×C₁₀₁	—	2	1	no	—
48	C₂×M₁₆×C₁₀₁	—	2	1	no	—
49	C₂×M₁₆×C₁₀₁	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₁₀₁	—	2	1	no	—
51	C₂²×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
52	M₃₂×C₁₀₁	—	2	1	no	—
53	M₃₂×C₁₀₁	—	2	1	no	—
54	M₃₂×C₁₀₁	—	2	1	no	—
55	M₃₂×C₁₀₁	—	2	1	no	—
56	C₂×C₁₆×C₁₀₁	—	2	1	no	—
57	C₂×C₁₆×C₁₀₁	—	2	1	no	—
58	#108×C₁₀₁	—	2	4	no	—
59	#108×C₁₀₁	—	2	1	no	—
60	#108×C₁₀₁	—	2	1	no	—
61	#108×C₁₀₁	—	2	1	no	—
62	#108×C₁₀₁	C₂×C₄	1	2	yes	—
63	#108×C₁₀₁	D₄	1	2	yes	—
64	#108×C₁₀₁	D₄	1	2	yes	—
65	#100×C₁₀₁	—	2	4	no	—
66	C₂⁴×C₁₀₁	—	4	1	no	—
67	#41×C₁₀₁	—	2	1	no	—
68	#41×C₁₀₁	—	2	1	no	—
69	#41×C₁₀₁	C₂²⋊C₄	1	2	yes	—
70	#41×C₁₀₁	D₈	1	2	yes	—
71	C₂×Q₈×C₁₀₁	SD₁₆	1	1	yes	characteristic
72	C₂×D₄×C₁₀₁	—	2	1	no	—
73	C₂×D₄×C₁₀₁	—	2	2	no	—
74	C₂×D₄×C₁₀₁	—	4	1	no	—
75	C₂×D₄×C₁₀₁	—	4	1	no	—
76	C₂×D₄×C₁₀₁	—	4	1	no	—
77	C₂×D₄×C₁₀₁	—	4	1	no	—
78	C₂×D₄×C₁₀₁	—	4	1	no	—
79	C₂×D₄×C₁₀₁	SD₁₆	1	1	yes	characteristic
80	C₂²×C₄×C₁₀₁	—	2	1	no	—
81	C₂²×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
82	SD₁₆×C₁₀₁	—	4	4	no	—
83	D₈×C₁₀₁	—	2	2	no	—
84	D₈×C₁₀₁	—	2	2	no	—
85	D₈×C₁₀₁	—	4	4	no	—
86	D₈×C₁₀₁	—	4	1	no	—
87	D₈×C₁₀₁	—	4	1	no	—
88	M₁₆×C₁₀₁	—	2	1	no	—
89	M₁₆×C₁₀₁	—	2	2	no	—
90	M₁₆×C₁₀₁	—	2	2	no	—
91	M₁₆×C₁₀₁	—	2	2	no	—
92	M₁₆×C₁₀₁	—	2	2	no	—
93	M₁₆×C₁₀₁	—	2	2	no	—
94	M₁₆×C₁₀₁	—	4	4	no	—
95	M₁₆×C₁₀₁	—	4	1	no	—
96	M₁₆×C₁₀₁	—	4	1	no	—
97	M₁₆×C₁₀₁	—	4	1	no	—
98	M₁₆×C₁₀₁	C₂²⋊C₄	1	2	yes	—
99	C₂×C₈×C₁₀₁	—	2	1	no	—
100	C₂×C₈×C₁₀₁	—	2	2	no	—
101	C₂×C₈×C₁₀₁	—	2	2	no	—
102	C₂×C₈×C₁₀₁	—	4	4	no	—
103	C₂×C₈×C₁₀₁	—	4	1	no	—
104	C₂×C₈×C₁₀₁	—	4	1	no	—
105	C₂×C₈×C₁₀₁	—	4	1	no	—
106	C₂×C₈×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₁₀₁	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₁₀₁	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₁₀₁	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₁₀₁	#41	1	1	yes	characteristic
111	C₂×C₈×C₁₀₁	#41	1	1	yes	characteristic
112	C₁₆×C₁₀₁	—	4	1	no	—
113	C₁₆×C₁₀₁	—	4	1	no	—
114	C₂³×C₁₀₁	—	2	1	no	—
115	C₂³×C₁₀₁	—	4	2	no	—
116	C₂³×C₁₀₁	—	8	1	no	—
117	C₂³×C₁₀₁	—	8	1	no	—
118	C₂³×C₁₀₁	—	16	2	no	—
119	C₂³×C₁₀₁	#137	1	1	yes	characteristic
120	Q₈×C₁₀₁	—	2	1	no	—
121	Q₈×C₁₀₁	#102	1	2	yes	—
122	D₄×C₁₀₁	—	2	2	no	—
123	D₄×C₁₀₁	—	2	2	no	—
124	D₄×C₁₀₁	—	2	1	no	—
125	D₄×C₁₀₁	—	4	2	no	—
126	D₄×C₁₀₁	—	4	2	no	—
127	D₄×C₁₀₁	—	8	1	no	—
128	D₄×C₁₀₁	—	8	1	no	—
129	D₄×C₁₀₁	#102	1	2	yes	—
130	C₂×C₄×C₁₀₁	—	2	1	no	—
131	C₂×C₄×C₁₀₁	—	2	2	no	—
132	C₂×C₄×C₁₀₁	—	2	2	no	—
133	C₂×C₄×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₀₁	#120	1	1	yes	characteristic
135	C₂×C₄×C₁₀₁	#123	1	1	yes	characteristic
136	C₂×C₄×C₁₀₁	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₁₀₁	C₂×SD₁₆	1	1	yes	characteristic
138	C₂×C₄×C₁₀₁	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₈×C₁₀₁	—	2	2	no	—
140	C₈×C₁₀₁	—	4	2	no	—
141	C₈×C₁₀₁	—	4	2	no	—
142	C₈×C₁₀₁	—	4	2	no	—
143	C₈×C₁₀₁	C₄×D₄	1	1	yes	characteristic
144	C₈×C₁₀₁	C₄×D₄	1	1	yes	characteristic
145	C₈×C₁₀₁	#121	1	1	yes	characteristic
146	C₈×C₁₀₁	#121	1	1	yes	characteristic
147	C₂²×C₁₀₁	—	2	2	no	—
148	C₂²×C₁₀₁	—	4	1	no	—
149	C₂²×C₁₀₁	—	8	1	no	—
150	C₂²×C₁₀₁	—	8	2	no	—
151	C₂²×C₁₀₁	—	16	2	no	—
152	C₂²×C₁₀₁	—	16	2	no	—
153	C₂²×C₁₀₁	—	16	2	no	—
154	C₂²×C₁₀₁	#417	1	1	yes	characteristic
155	C₂²×C₁₀₁	#448	1	1	yes	characteristic
156	C₂²×C₁₀₁	#480	1	1	yes	characteristic
157	C₄×C₁₀₁	—	2	2	no	—
158	C₄×C₁₀₁	#386	1	1	yes	characteristic
159	C₄×C₁₀₁	C₂×#102	1	1	yes	characteristic
160	C₄×C₁₀₁	#447	1	1	yes	characteristic
161	C₄×C₁₀₁	#481	1	1	yes	characteristic
162	#13007	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂×C₁₀₁	—	2	1	no	—
164	C₂×C₁₀₁	—	2	1	no	—
165	C₂×C₁₀₁	—	2	1	no	—
166	C₂×C₁₀₁	—	4	2	no	—
167	C₂×C₁₀₁	—	16	2	no	—
168	C₂×C₁₀₁	#1874	1	1	yes	characteristic
169	C₂×#472	C₂×C₁₀₁	1	1	yes	characteristic
170	#2916	C₂×C₁₀₁	1	1	yes	characteristic, complemented
171	C₂×#361	C₂×C₁₀₁	1	1	yes	characteristic, complemented
172	C₂×#361	C₂×C₁₀₁	1	1	yes	characteristic, complemented
173	#2117	C₂×C₁₀₁	1	1	yes	characteristic, complemented
174	#1388	C₂×C₁₀₁	1	1	yes	characteristic, complemented
175	#1385	C₂×C₁₀₁	1	1	yes	characteristic, complemented
176	C₁₀₁	#13007	1	1	yes	characteristic, direct product factor, Sylow
177	C₂×AGL(1,ℤ/8)	—	2	1	no	—
178	C₂²×D₈	C₂²×C₁₀₁	1	1	yes	characteristic
179	#568	—	2	1	no	—
180	C₂×#131	C₂²×C₁₀₁	1	1	yes	characteristic
181	C₂×M₃₂	C₂²×C₁₀₁	1	1	yes	characteristic
182	C₂×M₃₂	C₂²×C₁₀₁	1	1	yes	characteristic
183	#472	—	2	2	no	—
184	#472	C₄×C₁₀₁	1	4	yes	—
185	#430	—	2	1	no	—
186	#430	—	2	2	no	—
187	C₂×#108	—	2	2	no	—
188	C₂×#108	C₂²×C₁₀₁	1	1	yes	characteristic
189	C₂×#108	C₂²×C₁₀₁	1	1	yes	characteristic
190	C₂×#108	C₂²×C₁₀₁	1	1	yes	characteristic
191	C₂×#100	—	2	1	no	—
192	#361	—	2	2	no	—
193	#361	—	2	2	no	—
194	#350	—	2	2	no	—
195	C₂×#41	D₄×C₁₀₁	1	1	yes	characteristic
196	C₂²×D₄	—	2	1	no	—
197	AGL(1,ℤ/8)	—	2	4	no	—
198	AGL(1,ℤ/8)	—	4	2	no	—
199	C₂×SD₁₆	—	4	1	no	—
200	C₂×D₈	—	2	1	no	—
201	C₂×D₈	—	2	1	no	—
202	C₂×D₈	—	2	1	no	—
203	C₂×D₈	—	2	1	no	—
204	C₂×D₈	—	2	1	no	—
205	C₂×D₈	—	4	1	no	—
206	C₂×D₈	C₂×C₄×C₁₀₁	1	2	yes	—
207	#131	—	2	1	no	—
208	#131	—	2	1	no	—
209	#131	—	4	2	no	—
210	#131	—	4	2	no	—
211	#131	C₂×C₄×C₁₀₁	1	2	yes	—
212	#131	D₄×C₁₀₁	1	2	yes	—
213	C₂×M₁₆	—	2	1	no	—
214	C₂×M₁₆	—	2	4	no	—
215	C₂×M₁₆	—	2	1	no	—
216	C₂×M₁₆	—	2	1	no	—
217	C₂×M₁₆	—	2	1	no	—
218	C₂×M₁₆	D₄×C₁₀₁	1	1	yes	characteristic
219	C₂²×C₈	—	2	1	no	—
220	C₂²×C₈	C₂³×C₁₀₁	1	1	yes	characteristic
221	M₃₂	—	2	1	no	—
222	M₃₂	—	2	1	no	—
223	M₃₂	—	2	1	no	—
224	M₃₂	—	2	1	no	—
225	C₂×C₁₆	—	2	1	no	—
226	C₂×C₁₆	—	2	1	no	—
227	#108	—	2	4	no	—
228	#108	—	2	1	no	—
229	#108	—	2	1	no	—
230	#108	—	2	1	no	—
231	#108	C₂×C₄×C₁₀₁	1	2	yes	—
232	#108	D₄×C₁₀₁	1	2	yes	—
233	#108	D₄×C₁₀₁	1	2	yes	—
234	#100	—	2	4	no	—
235	C₂⁴	—	4	1	no	—
236	#41	—	2	1	no	—
237	#41	—	2	1	no	—
238	#41	C₂²⋊C₄×C₁₀₁	1	2	yes	—
239	#41	D₈×C₁₀₁	1	2	yes	—
240	C₂×Q₈	SD₁₆×C₁₀₁	1	1	yes	characteristic
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	4	1	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂×D₄	—	4	1	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂×D₄	—	4	1	no	—
248	C₂×D₄	SD₁₆×C₁₀₁	1	1	yes	characteristic
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	C₂×D₄×C₁₀₁	1	1	yes	characteristic
251	SD₁₆	—	4	4	no	—
252	D₈	—	2	2	no	—
253	D₈	—	2	2	no	—
254	D₈	—	4	4	no	—
255	D₈	—	4	1	no	—
256	D₈	—	4	1	no	—
257	M₁₆	—	2	1	no	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	2	2	no	—
260	M₁₆	—	2	2	no	—
261	M₁₆	—	2	2	no	—
262	M₁₆	—	2	2	no	—
263	M₁₆	—	4	4	no	—
264	M₁₆	—	4	1	no	—
265	M₁₆	—	4	1	no	—
266	M₁₆	—	4	1	no	—
267	M₁₆	C₂²⋊C₄×C₁₀₁	1	2	yes	—
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	4	4	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	4	1	no	—
275	C₂×C₈	C₂²×C₄×C₁₀₁	1	1	yes	characteristic
276	C₂×C₈	C₂×D₄×C₁₀₁	1	1	yes	characteristic
277	C₂×C₈	C₂×D₄×C₁₀₁	1	1	yes	characteristic
278	C₂×C₈	C₂×D₄×C₁₀₁	1	1	yes	characteristic
279	C₂×C₈	#41×C₁₀₁	1	1	yes	characteristic
280	C₂×C₈	#41×C₁₀₁	1	1	yes	characteristic
281	C₁₆	—	4	1	no	—
282	C₁₆	—	4	1	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	4	2	no	—
285	C₂³	—	8	1	no	—
286	C₂³	—	8	1	no	—
287	C₂³	—	16	2	no	—
288	C₂³	#137×C₁₀₁	1	1	yes	characteristic
289	Q₈	—	2	1	no	—
290	Q₈	#102×C₁₀₁	1	2	yes	—
291	D₄	—	2	2	no	—
292	D₄	—	2	2	no	—
293	D₄	—	2	1	no	—
294	D₄	—	4	2	no	—
295	D₄	—	4	2	no	—
296	D₄	—	8	1	no	—
297	D₄	—	8	1	no	—
298	D₄	#102×C₁₀₁	1	2	yes	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
303	C₂×C₄	#120×C₁₀₁	1	1	yes	characteristic
304	C₂×C₄	#123×C₁₀₁	1	1	yes	characteristic
305	C₂×C₄	C₂×D₈×C₁₀₁	1	1	yes	characteristic
306	C₂×C₄	C₂×SD₁₆×C₁₀₁	1	1	yes	characteristic
307	C₂×C₄	AGL(1,ℤ/8)×C₁₀₁	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	—	4	2	no	—
310	C₈	—	4	2	no	—
311	C₈	—	4	2	no	—
312	C₈	C₄×D₄×C₁₀₁	1	1	yes	characteristic
313	C₈	C₄×D₄×C₁₀₁	1	1	yes	characteristic
314	C₈	#121×C₁₀₁	1	1	yes	characteristic
315	C₈	#121×C₁₀₁	1	1	yes	characteristic
316	C₂²	—	2	2	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	8	1	no	—
319	C₂²	—	8	2	no	—
320	C₂²	—	16	2	no	—
321	C₂²	—	16	2	no	—
322	C₂²	—	16	2	no	—
323	C₂²	#417×C₁₀₁	1	1	yes	characteristic
324	C₂²	#448×C₁₀₁	1	1	yes	characteristic
325	C₂²	#480×C₁₀₁	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	#386×C₁₀₁	1	1	yes	characteristic
328	C₄	C₂×#102×C₁₀₁	1	1	yes	characteristic
329	C₄	#447×C₁₀₁	1	1	yes	characteristic
330	C₄	#481×C₁₀₁	1	1	yes	characteristic
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	2	1	no	—
334	C₂	—	4	2	no	—
335	C₂	—	16	2	no	—
336	C₂	C₁₀₁×#1874	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₀₁	#1874
commutator	C₂×C₈	C₂²×C₄×C₁₀₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₁
prime radical	C₂×#472×C₁₀₁	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₁ ◁ C₂²×C₄×C₁₀₁ ◁ C₂×#131×C₁₀₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₀₁×#2916	C₂	semidirect
3. C₂×#361×C₁₀₁	C₂	semidirect
4. C₂×#361×C₁₀₁	C₂	semidirect
5. C₁₀₁×#2117	C₂	semidirect
6. C₁₀₁×#1388	C₂	semidirect
7. C₁₀₁×#1385	C₂	semidirect
162. #13007	C₁₀₁	direct (N × H)
170. #2916	C₂×C₁₀₁	semidirect
171. C₂×#361	C₂×C₁₀₁	semidirect
172. C₂×#361	C₂×C₁₀₁	semidirect
173. #2117	C₂×C₁₀₁	semidirect
174. #1388	C₂×C₁₀₁	semidirect
175. #1385	C₂×C₁₀₁	semidirect
176. C₁₀₁	#13007	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#472×C₁₀₁	2
C₁₀₁×#2916	2
C₂×#361×C₁₀₁	2
C₁₀₁×#2117	2
C₁₀₁×#1388	2
C₁₀₁×#1385	2
#13007	101

Subgroups — C101×#13007 #38795158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#13007 #38795158