Subgroups — C₁₀₁×#8114 #38790265

Order 25856.

942 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#407×C₁₀₁	C₂	1	2	yes	maximal
2	C₂×#381×C₁₀₁	C₂	1	1	yes	maximal, characteristic
3	C₁₀₁×#1306	C₂	1	4	yes	maximal, complemented
4	C₂⁴×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
5	C₂²×M₁₆×C₁₀₁	—	2	2	no	—
6	C₂²×C₂²⋊C₄×C₁₀₁	C₄	1	2	yes	—
7	#407×C₁₀₁	—	2	2	no	—
8	#407×C₁₀₁	—	2	2	no	—
9	#407×C₁₀₁	C₄	1	4	yes	complemented
10	#407×C₁₀₁	C₂²	1	4	yes	—
11	C₂×C₂²⋊C₈×C₁₀₁	—	2	2	no	—
12	C₂×C₂²⋊C₈×C₁₀₁	C₄	1	4	yes	complemented
13	#381×C₁₀₁	—	2	2	no	—
14	#381×C₁₀₁	C₄	1	2	yes	—
15	#381×C₁₀₁	C₂²	1	2	yes	—
16	C₂×#95×C₁₀₁	—	2	2	no	—
17	C₂⁵×C₁₀₁	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₁₀₁	—	2	2	no	—
19	C₂³×C₄×C₁₀₁	—	4	4	no	—
20	C₂³×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₁₀₁	D₄	1	2	yes	—
24	C₂³×C₄×C₁₀₁	D₄	1	2	yes	—
25	C₂³×C₄×C₁₀₁	D₄	1	2	yes	—
26	C₂³×C₄×C₁₀₁	Q₈	1	2	yes	—
27	C₂³×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₁₀₁	—	2	4	no	—
29	C₂×M₁₆×C₁₀₁	—	2	4	no	—
30	C₂×M₁₆×C₁₀₁	—	4	8	no	—
31	C₂²×C₈×C₁₀₁	—	2	4	no	—
32	C₂×C₂²⋊C₄×C₁₀₁	—	2	2	no	—
33	C₂×C₂²⋊C₄×C₁₀₁	—	4	4	no	—
34	C₂×C₂²⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	—
35	C₂²⋊C₈×C₁₀₁	—	2	4	no	—
36	C₂²⋊C₈×C₁₀₁	—	2	4	no	—
37	C₂²⋊C₈×C₁₀₁	—	2	4	no	—
38	C₂²⋊C₈×C₁₀₁	C₂×C₄	1	8	yes	complemented
39	#95×C₁₀₁	—	2	4	no	—
40	#95×C₁₀₁	—	2	4	no	—
41	C₂⁴×C₁₀₁	—	2	4	no	—
42	C₂⁴×C₁₀₁	—	4	4	no	—
43	C₂⁴×C₁₀₁	C₄²	1	2	yes	—
44	C₂⁴×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₂⁴×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₂⁴×C₁₀₁	C₄⋊C₄	1	2	yes	—
47	C₂⁴×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₀₁	—	2	8	no	—
49	C₂²×C₄×C₁₀₁	—	2	4	no	—
50	C₂²×C₄×C₁₀₁	—	4	4	no	—
51	C₂²×C₄×C₁₀₁	—	4	4	no	—
52	C₂²×C₄×C₁₀₁	—	4	2	no	—
53	C₂²×C₄×C₁₀₁	—	4	4	no	—
54	C₂²×C₄×C₁₀₁	—	4	4	no	—
55	C₂²×C₄×C₁₀₁	—	4	4	no	—
56	C₂²×C₄×C₁₀₁	—	4	2	no	—
57	C₂²×C₄×C₁₀₁	—	4	4	no	—
58	C₂²×C₄×C₁₀₁	—	4	8	no	—
59	C₂²×C₄×C₁₀₁	C₄²	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₁	C₄²	1	1	yes	characteristic
61	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₁₀₁	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₁₀₁	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₁₀₁	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₁₀₁	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₁₀₁	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
75	M₁₆×C₁₀₁	—	4	8	no	—
76	M₁₆×C₁₀₁	—	4	8	no	—
77	C₂×C₈×C₁₀₁	—	2	8	no	—
78	C₂×C₈×C₁₀₁	—	4	8	no	—
79	C₂²⋊C₄×C₁₀₁	—	4	4	no	—
80	C₂²⋊C₄×C₁₀₁	—	4	4	no	—
81	C₂³×C₁₀₁	—	2	4	no	—
82	C₂³×C₁₀₁	—	2	2	no	—
83	C₂³×C₁₀₁	—	2	4	no	—
84	C₂³×C₁₀₁	—	2	4	no	—
85	C₂³×C₁₀₁	—	2	4	no	—
86	C₂³×C₁₀₁	—	2	4	no	—
87	C₂³×C₁₀₁	—	2	2	no	—
88	C₂³×C₁₀₁	—	4	2	no	—
89	C₂³×C₁₀₁	—	4	2	no	—
90	C₂³×C₁₀₁	—	4	2	no	—
91	C₂³×C₁₀₁	—	4	2	no	—
92	C₂³×C₁₀₁	—	4	16	no	—
93	C₂³×C₁₀₁	#95	1	2	yes	—
94	C₂³×C₁₀₁	#95	1	2	yes	—
95	C₂³×C₁₀₁	#99	1	4	yes	—
96	C₂³×C₁₀₁	C₂×C₄²	1	1	yes	characteristic
97	C₂³×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂³×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₀₁	—	2	4	no	—
100	C₂×C₄×C₁₀₁	—	2	4	no	—
101	C₂×C₄×C₁₀₁	—	2	4	no	—
102	C₂×C₄×C₁₀₁	—	2	4	no	—
103	C₂×C₄×C₁₀₁	—	2	4	no	—
104	C₂×C₄×C₁₀₁	—	2	4	no	—
105	C₂×C₄×C₁₀₁	—	4	1	no	—
106	C₂×C₄×C₁₀₁	—	4	1	no	—
107	C₂×C₄×C₁₀₁	—	4	2	no	—
108	C₂×C₄×C₁₀₁	—	4	2	no	—
109	C₂×C₄×C₁₀₁	—	4	2	no	—
110	C₂×C₄×C₁₀₁	—	4	2	no	—
111	C₂×C₄×C₁₀₁	—	4	2	no	—
112	C₂×C₄×C₁₀₁	—	4	2	no	—
113	C₂×C₄×C₁₀₁	—	4	1	no	—
114	C₂×C₄×C₁₀₁	—	4	1	no	—
115	C₂×C₄×C₁₀₁	—	4	8	no	—
116	C₂×C₄×C₁₀₁	—	4	4	no	—
117	C₂×C₄×C₁₀₁	—	8	8	no	—
118	C₂×C₄×C₁₀₁	#95	1	2	yes	—
119	C₂×C₄×C₁₀₁	#95	1	2	yes	—
120	C₂×C₄×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
121	C₂×C₄×C₁₀₁	C₂×C₄⋊C₄	1	2	yes	—
122	C₈×C₁₀₁	—	4	8	no	—
123	C₂²×C₁₀₁	—	2	8	no	—
124	C₂²×C₁₀₁	—	2	2	no	—
125	C₂²×C₁₀₁	—	2	2	no	—
126	C₂²×C₁₀₁	—	2	2	no	—
127	C₂²×C₁₀₁	—	2	2	no	—
128	C₂²×C₁₀₁	—	2	2	no	—
129	C₂²×C₁₀₁	—	2	1	no	—
130	C₂²×C₁₀₁	—	2	1	no	—
131	C₂²×C₁₀₁	—	2	2	no	—
132	C₂²×C₁₀₁	—	2	2	no	—
133	C₂²×C₁₀₁	—	4	4	no	—
134	C₂²×C₁₀₁	—	4	8	no	—
135	C₂²×C₁₀₁	—	4	8	no	—
136	C₂²×C₁₀₁	—	4	4	no	—
137	C₂²×C₁₀₁	#342	1	4	yes	—
138	C₂²×C₁₀₁	#344	1	4	yes	—
139	C₂²×C₁₀₁	C₂×#95	1	1	yes	characteristic
140	C₂²×C₁₀₁	C₂×#99	1	2	yes	—
141	C₄×C₁₀₁	—	2	2	no	—
142	C₄×C₁₀₁	—	2	2	no	—
143	C₄×C₁₀₁	—	2	2	no	—
144	C₄×C₁₀₁	—	2	2	no	—
145	C₄×C₁₀₁	—	8	4	no	—
146	#8114	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
147	C₂×C₁₀₁	—	2	2	no	—
148	C₂×C₁₀₁	—	2	4	no	—
149	C₂×C₁₀₁	—	2	4	no	—
150	C₂×C₁₀₁	—	4	1	no	—
151	C₂×C₁₀₁	—	4	1	no	—
152	C₂×C₁₀₁	C₂×#342	1	1	yes	characteristic
153	C₂×C₁₀₁	C₂×#344	1	2	yes	—
154	C₂×#407	C₂×C₁₀₁	1	2	yes	—
155	C₂×#381	C₂×C₁₀₁	1	1	yes	characteristic
156	#1306	C₂×C₁₀₁	1	4	yes	complemented
157	C₁₀₁	#8114	1	1	yes	characteristic, direct product factor, Sylow
158	C₂⁴×C₄	C₂²×C₁₀₁	1	1	yes	characteristic
159	C₂²×M₁₆	—	2	2	no	—
160	C₂²×C₂²⋊C₄	C₄×C₁₀₁	1	2	yes	—
161	#407	—	2	2	no	—
162	#407	—	2	2	no	—
163	#407	C₄×C₁₀₁	1	4	yes	complemented
164	#407	C₂²×C₁₀₁	1	4	yes	—
165	C₂×C₂²⋊C₈	—	2	2	no	—
166	C₂×C₂²⋊C₈	C₄×C₁₀₁	1	4	yes	complemented
167	#381	—	2	2	no	—
168	#381	C₄×C₁₀₁	1	2	yes	—
169	#381	C₂²×C₁₀₁	1	2	yes	—
170	C₂×#95	—	2	2	no	—
171	C₂⁵	C₂×C₄×C₁₀₁	1	1	yes	characteristic
172	C₂³×C₄	—	2	2	no	—
173	C₂³×C₄	—	4	4	no	—
174	C₂³×C₄	C₂×C₄×C₁₀₁	1	1	yes	characteristic
175	C₂³×C₄	C₂×C₄×C₁₀₁	1	2	yes	—
176	C₂³×C₄	C₂×C₄×C₁₀₁	1	2	yes	—
177	C₂³×C₄	D₄×C₁₀₁	1	2	yes	—
178	C₂³×C₄	D₄×C₁₀₁	1	2	yes	—
179	C₂³×C₄	D₄×C₁₀₁	1	2	yes	—
180	C₂³×C₄	Q₈×C₁₀₁	1	2	yes	—
181	C₂³×C₄	C₂³×C₁₀₁	1	1	yes	characteristic
182	C₂×M₁₆	—	2	4	no	—
183	C₂×M₁₆	—	2	4	no	—
184	C₂×M₁₆	—	4	8	no	—
185	C₂²×C₈	—	2	4	no	—
186	C₂×C₂²⋊C₄	—	2	2	no	—
187	C₂×C₂²⋊C₄	—	4	4	no	—
188	C₂×C₂²⋊C₄	C₂×C₄×C₁₀₁	1	4	yes	—
189	C₂²⋊C₈	—	2	4	no	—
190	C₂²⋊C₈	—	2	4	no	—
191	C₂²⋊C₈	—	2	4	no	—
192	C₂²⋊C₈	C₂×C₄×C₁₀₁	1	8	yes	complemented
193	#95	—	2	4	no	—
194	#95	—	2	4	no	—
195	C₂⁴	—	2	4	no	—
196	C₂⁴	—	4	4	no	—
197	C₂⁴	C₄²×C₁₀₁	1	2	yes	—
198	C₂⁴	C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
199	C₂⁴	C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
200	C₂⁴	C₄⋊C₄×C₁₀₁	1	2	yes	—
201	C₂⁴	C₂²×C₄×C₁₀₁	1	1	yes	characteristic
202	C₂²×C₄	—	2	8	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	4	2	no	—
205	C₂²×C₄	—	4	4	no	—
206	C₂²×C₄	—	4	4	no	—
207	C₂²×C₄	—	4	4	no	—
208	C₂²×C₄	—	4	4	no	—
209	C₂²×C₄	—	4	4	no	—
210	C₂²×C₄	—	4	2	no	—
211	C₂²×C₄	—	4	4	no	—
212	C₂²×C₄	—	4	8	no	—
213	C₂²×C₄	C₄²×C₁₀₁	1	1	yes	characteristic
214	C₂²×C₄	C₄²×C₁₀₁	1	1	yes	characteristic
215	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	2	yes	—
216	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	2	yes	—
217	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	2	yes	—
218	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	2	yes	—
219	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	2	yes	—
220	C₂²×C₄	C₄⋊C₄×C₁₀₁	1	2	yes	—
221	C₂²×C₄	C₄⋊C₄×C₁₀₁	1	2	yes	—
222	C₂²×C₄	C₄⋊C₄×C₁₀₁	1	2	yes	—
223	C₂²×C₄	C₄⋊C₄×C₁₀₁	1	2	yes	—
224	C₂²×C₄	C₄⋊C₄×C₁₀₁	1	2	yes	—
225	C₂²×C₄	C₂²×C₄×C₁₀₁	1	2	yes	—
226	C₂²×C₄	C₂×D₄×C₁₀₁	1	2	yes	—
227	C₂²×C₄	C₂×D₄×C₁₀₁	1	1	yes	characteristic
228	C₂²×C₄	C₂×Q₈×C₁₀₁	1	1	yes	characteristic
229	M₁₆	—	4	8	no	—
230	M₁₆	—	4	8	no	—
231	C₂×C₈	—	2	8	no	—
232	C₂×C₈	—	4	8	no	—
233	C₂²⋊C₄	—	4	4	no	—
234	C₂²⋊C₄	—	4	4	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	2	4	no	—
237	C₂³	—	2	2	no	—
238	C₂³	—	2	4	no	—
239	C₂³	—	2	4	no	—
240	C₂³	—	2	4	no	—
241	C₂³	—	2	4	no	—
242	C₂³	—	4	2	no	—
243	C₂³	—	4	2	no	—
244	C₂³	—	4	2	no	—
245	C₂³	—	4	2	no	—
246	C₂³	—	4	16	no	—
247	C₂³	#95×C₁₀₁	1	2	yes	—
248	C₂³	#95×C₁₀₁	1	2	yes	—
249	C₂³	#99×C₁₀₁	1	4	yes	—
250	C₂³	C₂×C₄²×C₁₀₁	1	1	yes	characteristic
251	C₂³	C₂×C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
252	C₂³	C₂×C₄⋊C₄×C₁₀₁	1	1	yes	characteristic
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	8	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	8	8	no	—
272	C₂×C₄	#95×C₁₀₁	1	2	yes	—
273	C₂×C₄	#95×C₁₀₁	1	2	yes	—
274	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₁	1	2	yes	—
275	C₂×C₄	C₂×C₄⋊C₄×C₁₀₁	1	2	yes	—
276	C₈	—	4	8	no	—
277	C₂²	—	2	8	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	4	no	—
288	C₂²	—	4	8	no	—
289	C₂²	—	4	8	no	—
290	C₂²	—	4	4	no	—
291	C₂²	#342×C₁₀₁	1	4	yes	—
292	C₂²	#344×C₁₀₁	1	4	yes	—
293	C₂²	C₂×#95×C₁₀₁	1	1	yes	characteristic
294	C₂²	C₂×#99×C₁₀₁	1	2	yes	—
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	8	4	no	—
300	C₂	—	2	2	no	—
301	C₂	—	2	4	no	—
302	C₂	—	2	4	no	—
303	C₂	—	4	1	no	—
304	C₂	—	4	1	no	—
305	C₂	C₂×#342×C₁₀₁	1	1	yes	characteristic
306	C₂	C₂×#344×C₁₀₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₀₁	C₂×#95
commutator	C₂³	C₂×C₄²×C₁₀₁
Frattini, cosocle	C₂³×C₄	C₂³×C₁₀₁
prime radical	C₂⁵×C₁₀₁	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₁ ◁ C₂⁴×C₄×C₁₀₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₀₁×#1306	C₂	semidirect
9. #407×C₁₀₁	C₄	semidirect
12. C₂×C₂²⋊C₈×C₁₀₁	C₄	semidirect
38. C₂²⋊C₈×C₁₀₁	C₂×C₄	semidirect
146. #8114	C₁₀₁	direct (N × H)
156. #1306	C₂×C₁₀₁	semidirect
157. C₁₀₁	#8114	direct (N × H)
163. #407	C₄×C₁₀₁	semidirect
166. C₂×C₂²⋊C₈	C₄×C₁₀₁	semidirect
192. C₂²⋊C₈	C₂×C₄×C₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#407×C₁₀₁	2
C₂×#381×C₁₀₁	2
C₁₀₁×#1306	2
#8114	101

Subgroups — C101×#8114 #38790265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#8114 #38790265