Subgroups — C₁₀₁×#9865 #38792016

Order 25856.

528 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊Q₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic
4	C₁₀₁×#1279	C₂	1	4	yes	maximal
5	C₂×C₄⋊Q₈×C₁₀₁	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₁₀₁	—	2	1	no	—
7	C₂×C₄⋊C₈×C₁₀₁	—	2	1	no	—
8	C₂×C₄⋊C₈×C₁₀₁	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₀₁	—	2	1	no	—
10	C₂×C₄×C₈×C₁₀₁	C₂²	1	2	yes	—
11	#391×C₁₀₁	—	2	1	no	—
12	#389×C₁₀₁	—	2	1	no	—
13	#389×C₁₀₁	C₄	1	2	yes	—
14	#384×C₁₀₁	C₄	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₁₀₁	—	2	1	no	—
16	C₄×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
17	C₄³×C₁₀₁	C₂²	1	1	yes	characteristic
18	#336×C₁₀₁	—	2	1	no	—
19	#336×C₁₀₁	—	2	1	no	—
20	C₂²×Q₈×C₁₀₁	—	2	1	no	—
21	C₂²×C₈×C₁₀₁	—	2	2	no	—
22	C₂²×C₈×C₁₀₁	—	2	2	no	—
23	C₄⋊Q₈×C₁₀₁	—	2	2	no	—
24	C₄⋊Q₈×C₁₀₁	C₈	1	4	yes	complemented
25	C₄×Q₈×C₁₀₁	—	4	2	no	—
26	C₄×Q₈×C₁₀₁	—	4	2	no	—
27	C₂×C₄⋊C₄×C₁₀₁	—	2	1	no	—
28	C₂×C₄⋊C₄×C₁₀₁	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₀₁	—	2	1	no	—
31	C₂×C₄⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₀₁	—	2	2	no	—
34	C₂×C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₀₁	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₀₁	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₀₁	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₀₁	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₁₀₁	—	2	4	no	—
42	C₄⋊C₈×C₁₀₁	—	2	2	no	—
43	C₄⋊C₈×C₁₀₁	Q₈	1	4	yes	complemented
44	C₄×C₈×C₁₀₁	—	2	4	no	—
45	C₄×C₈×C₁₀₁	—	4	2	no	—
46	#95×C₁₀₁	—	2	2	no	—
47	#95×C₁₀₁	—	2	2	no	—
48	C₂×Q₈×C₁₀₁	—	2	4	no	—
49	C₂×Q₈×C₁₀₁	—	4	2	no	—
50	C₂×Q₈×C₁₀₁	—	4	2	no	—
51	C₂²×C₄×C₁₀₁	—	2	2	no	—
52	C₂²×C₄×C₁₀₁	—	2	2	no	—
53	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₀₁	—	2	4	no	—
61	C₂×C₈×C₁₀₁	—	2	4	no	—
62	C₂×C₈×C₁₀₁	—	4	2	no	—
63	C₂×C₈×C₁₀₁	—	4	2	no	—
64	C₂×C₈×C₁₀₁	—	4	2	no	—
65	C₂×C₈×C₁₀₁	—	4	2	no	—
66	C₄⋊C₄×C₁₀₁	—	2	4	no	—
67	C₄⋊C₄×C₁₀₁	—	2	2	no	—
68	C₄⋊C₄×C₁₀₁	—	2	4	no	—
69	C₄⋊C₄×C₁₀₁	—	4	2	no	—
70	C₄⋊C₄×C₁₀₁	—	4	4	no	—
71	C₄⋊C₄×C₁₀₁	—	4	2	no	—
72	C₄⋊C₄×C₁₀₁	C₂×C₈	1	4	yes	—
73	C₄²×C₁₀₁	—	2	2	no	—
74	C₄²×C₁₀₁	—	2	2	no	—
75	C₄²×C₁₀₁	—	2	2	no	—
76	C₄²×C₁₀₁	—	4	1	no	—
77	C₄²×C₁₀₁	—	4	2	no	—
78	C₄²×C₁₀₁	—	4	2	no	—
79	C₄²×C₁₀₁	C₂×C₈	1	2	yes	—
80	C₄²×C₁₀₁	M₁₆	1	2	yes	—
81	C₄²×C₁₀₁	D₈	1	2	yes	—
82	C₄²×C₁₀₁	SD₁₆	1	2	yes	—
83	C₄²×C₁₀₁	C₂×Q₈	1	2	yes	—
84	C₄²×C₁₀₁	#41	1	2	yes	—
85	C₂³×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	Q₈×C₁₀₁	—	4	8	no	—
87	C₂×C₄×C₁₀₁	—	2	1	no	—
88	C₂×C₄×C₁₀₁	—	2	1	no	—
89	C₂×C₄×C₁₀₁	—	2	1	no	—
90	C₂×C₄×C₁₀₁	—	2	1	no	—
91	C₂×C₄×C₁₀₁	—	2	1	no	—
92	C₂×C₄×C₁₀₁	—	2	1	no	—
93	C₂×C₄×C₁₀₁	—	2	4	no	—
94	C₂×C₄×C₁₀₁	—	2	1	no	—
95	C₂×C₄×C₁₀₁	—	2	1	no	—
96	C₂×C₄×C₁₀₁	—	2	1	no	—
97	C₂×C₄×C₁₀₁	—	4	2	no	—
98	C₂×C₄×C₁₀₁	—	4	2	no	—
99	C₂×C₄×C₁₀₁	—	4	2	no	—
100	C₂×C₄×C₁₀₁	—	4	2	no	—
101	C₂×C₄×C₁₀₁	—	4	2	no	—
102	C₂×C₄×C₁₀₁	C₂²⋊C₈	1	2	yes	—
103	C₂×C₄×C₁₀₁	C₂²⋊C₈	1	2	yes	—
104	C₂×C₄×C₁₀₁	#102	1	2	yes	—
105	C₂×C₄×C₁₀₁	#102	1	2	yes	—
106	C₂×C₄×C₁₀₁	#104	1	2	yes	—
107	C₂×C₄×C₁₀₁	C₄×Q₈	1	2	yes	—
108	C₂×C₄×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₁₀₁	#124	1	1	yes	characteristic
110	C₂×C₄×C₁₀₁	C₄⋊Q₈	1	1	yes	characteristic
111	C₂×C₄×C₁₀₁	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₁₀₁	C₂×M₁₆	1	1	yes	characteristic
113	C₂×C₄×C₁₀₁	#131	1	2	yes	—
114	C₂×C₄×C₁₀₁	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₁₀₁	C₂×SD₁₆	1	1	yes	characteristic
116	C₂×C₄×C₁₀₁	#137	1	2	yes	—
117	C₈×C₁₀₁	—	4	4	no	—
118	C₈×C₁₀₁	—	4	4	no	—
119	C₂²×C₁₀₁	#391	1	1	yes	characteristic
120	C₂²×C₁₀₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
121	C₂²×C₁₀₁	#408	1	1	yes	characteristic
122	C₂²×C₁₀₁	C₂×#102	1	1	yes	characteristic
123	C₂²×C₁₀₁	#419	1	1	yes	characteristic
124	C₂²×C₁₀₁	C₂×#104	1	1	yes	characteristic
125	C₂²×C₁₀₁	#421	1	1	yes	characteristic
126	C₄×C₁₀₁	—	2	1	no	—
127	C₄×C₁₀₁	—	2	1	no	—
128	C₄×C₁₀₁	—	2	2	no	—
129	C₄×C₁₀₁	—	2	2	no	—
130	C₄×C₁₀₁	—	4	4	no	—
131	C₄×C₁₀₁	—	4	1	no	—
132	C₄×C₁₀₁	—	8	2	no	—
133	C₄×C₁₀₁	D₄⋊C₈	1	4	yes	—
134	C₄×C₁₀₁	C₈×Q₈	1	2	yes	—
135	C₄×C₁₀₁	#446	1	2	yes	—
136	C₄×C₁₀₁	#474	1	2	yes	—
137	C₄×C₁₀₁	#475	1	2	yes	—
138	#9865	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂×C₁₀₁	C₂×D₄⋊C₈	1	1	yes	characteristic
140	C₂×C₁₀₁	#1479	1	1	yes	characteristic
141	C₂×C₁₀₁	#1480	1	1	yes	characteristic
142	C₂×C₁₀₁	#1485	1	1	yes	characteristic
143	C₂×C₁₀₁	#1984	1	1	yes	characteristic
144	C₂×C₁₀₁	#1986	1	1	yes	characteristic
145	C₂×C₁₀₁	#1988	1	1	yes	characteristic
146	C₄×C₄⋊Q₈	C₂×C₁₀₁	1	1	yes	characteristic
147	C₄×C₄⋊C₈	C₂×C₁₀₁	1	1	yes	characteristic
148	C₄×C₄⋊C₈	C₂×C₁₀₁	1	1	yes	characteristic
149	#1279	C₂×C₁₀₁	1	4	yes	—
150	C₁₀₁	#9865	1	1	yes	characteristic, direct product factor, Sylow
151	C₂×C₄⋊Q₈	C₄×C₁₀₁	1	1	yes	characteristic
152	C₂×C₄×Q₈	—	2	1	no	—
153	C₂×C₄⋊C₈	—	2	1	no	—
154	C₂×C₄⋊C₈	C₂²×C₁₀₁	1	2	yes	—
155	C₂×C₄×C₈	—	2	1	no	—
156	C₂×C₄×C₈	C₂²×C₁₀₁	1	2	yes	—
157	#391	—	2	1	no	—
158	#389	—	2	1	no	—
159	#389	C₄×C₁₀₁	1	2	yes	—
160	#384	C₄×C₁₀₁	1	1	yes	characteristic
161	C₄×C₄⋊C₄	—	2	1	no	—
162	C₄×C₄⋊C₄	C₂²×C₁₀₁	1	2	yes	—
163	C₄³	C₂²×C₁₀₁	1	1	yes	characteristic
164	#336	—	2	1	no	—
165	#336	—	2	1	no	—
166	C₂²×Q₈	—	2	1	no	—
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	—	2	2	no	—
169	C₄⋊Q₈	—	2	2	no	—
170	C₄⋊Q₈	C₈×C₁₀₁	1	4	yes	complemented
171	C₄×Q₈	—	4	2	no	—
172	C₄×Q₈	—	4	2	no	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₄⋊C₄	—	2	1	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	—	2	1	no	—
177	C₂×C₄⋊C₄	C₂×C₄×C₁₀₁	1	2	yes	—
178	C₂×C₄⋊C₄	C₂×C₄×C₁₀₁	1	2	yes	—
179	C₂×C₄²	—	2	2	no	—
180	C₂×C₄²	C₂×C₄×C₁₀₁	1	1	yes	characteristic
181	C₂×C₄²	C₂×C₄×C₁₀₁	1	1	yes	characteristic
182	C₂×C₄²	D₄×C₁₀₁	1	1	yes	characteristic
183	C₂×C₄²	D₄×C₁₀₁	1	1	yes	characteristic
184	C₂×C₄²	D₄×C₁₀₁	1	1	yes	characteristic
185	C₂×C₄²	D₄×C₁₀₁	1	1	yes	characteristic
186	C₂×C₄²	C₂³×C₁₀₁	1	1	yes	characteristic
187	C₄⋊C₈	—	2	4	no	—
188	C₄⋊C₈	—	2	2	no	—
189	C₄⋊C₈	Q₈×C₁₀₁	1	4	yes	complemented
190	C₄×C₈	—	2	4	no	—
191	C₄×C₈	—	4	2	no	—
192	#95	—	2	2	no	—
193	#95	—	2	2	no	—
194	C₂×Q₈	—	2	4	no	—
195	C₂×Q₈	—	4	2	no	—
196	C₂×Q₈	—	4	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
200	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
201	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
202	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
203	C₂²×C₄	C₂²×C₄×C₁₀₁	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₄×C₁₀₁	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₄×C₁₀₁	1	1	yes	characteristic
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	—	4	2	no	—
211	C₂×C₈	—	4	2	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	4	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄⋊C₄	C₂×C₈×C₁₀₁	1	4	yes	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	4	1	no	—
223	C₄²	—	4	2	no	—
224	C₄²	—	4	2	no	—
225	C₄²	C₂×C₈×C₁₀₁	1	2	yes	—
226	C₄²	M₁₆×C₁₀₁	1	2	yes	—
227	C₄²	D₈×C₁₀₁	1	2	yes	—
228	C₄²	SD₁₆×C₁₀₁	1	2	yes	—
229	C₄²	C₂×Q₈×C₁₀₁	1	2	yes	—
230	C₄²	#41×C₁₀₁	1	2	yes	—
231	C₂³	C₂×C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
232	Q₈	—	4	8	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	C₂²⋊C₈×C₁₀₁	1	2	yes	—
249	C₂×C₄	C₂²⋊C₈×C₁₀₁	1	2	yes	—
250	C₂×C₄	#102×C₁₀₁	1	2	yes	—
251	C₂×C₄	#102×C₁₀₁	1	2	yes	—
252	C₂×C₄	#104×C₁₀₁	1	2	yes	—
253	C₂×C₄	C₄×Q₈×C₁₀₁	1	2	yes	—
254	C₂×C₄	C₂²⋊Q₈×C₁₀₁	1	2	yes	—
255	C₂×C₄	#124×C₁₀₁	1	1	yes	characteristic
256	C₂×C₄	C₄⋊Q₈×C₁₀₁	1	1	yes	characteristic
257	C₂×C₄	C₂²×C₈×C₁₀₁	1	1	yes	characteristic
258	C₂×C₄	C₂×M₁₆×C₁₀₁	1	1	yes	characteristic
259	C₂×C₄	#131×C₁₀₁	1	2	yes	—
260	C₂×C₄	C₂×D₈×C₁₀₁	1	1	yes	characteristic
261	C₂×C₄	C₂×SD₁₆×C₁₀₁	1	1	yes	characteristic
262	C₂×C₄	#137×C₁₀₁	1	2	yes	—
263	C₈	—	4	4	no	—
264	C₈	—	4	4	no	—
265	C₂²	#391×C₁₀₁	1	1	yes	characteristic
266	C₂²	C₂×C₂²⋊C₈×C₁₀₁	1	1	yes	characteristic
267	C₂²	#408×C₁₀₁	1	1	yes	characteristic
268	C₂²	C₂×#102×C₁₀₁	1	1	yes	characteristic
269	C₂²	#419×C₁₀₁	1	1	yes	characteristic
270	C₂²	C₂×#104×C₁₀₁	1	1	yes	characteristic
271	C₂²	#421×C₁₀₁	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	4	4	no	—
277	C₄	—	4	1	no	—
278	C₄	—	8	2	no	—
279	C₄	D₄⋊C₈×C₁₀₁	1	4	yes	—
280	C₄	C₈×Q₈×C₁₀₁	1	2	yes	—
281	C₄	#446×C₁₀₁	1	2	yes	—
282	C₄	#474×C₁₀₁	1	2	yes	—
283	C₄	#475×C₁₀₁	1	2	yes	—
284	C₂	C₂×D₄⋊C₈×C₁₀₁	1	1	yes	characteristic
285	C₂	C₁₀₁×#1479	1	1	yes	characteristic
286	C₂	C₁₀₁×#1480	1	1	yes	characteristic
287	C₂	C₁₀₁×#1485	1	1	yes	characteristic
288	C₂	C₁₀₁×#1984	1	1	yes	characteristic
289	C₂	C₁₀₁×#1986	1	1	yes	characteristic
290	C₂	C₁₀₁×#1988	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₀₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₀₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₁
socle, prime radical	C₂³×C₁₀₁	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₀₁ ◁ C₄³×C₁₀₁ ◁ G

Complements

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

N	H ≅ G/N	type
24. C₄⋊Q₈×C₁₀₁	C₈	semidirect
43. C₄⋊C₈×C₁₀₁	Q₈	semidirect
138. #9865	C₁₀₁	direct (N × H)
150. C₁₀₁	#9865	direct (N × H)
170. C₄⋊Q₈	C₈×C₁₀₁	semidirect
189. C₄⋊C₈	Q₈×C₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊Q₈×C₁₀₁	2
C₄×C₄⋊C₈×C₁₀₁	2
C₁₀₁×#1279	2
#9865	101

Subgroups — C101×#9865 #38792016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#9865 #38792016