Subgroups — C₁₀₁×#11613 #38793764

Order 25856.

378 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#41×C₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₁₀₁×#2162	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₁₆×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₁×Q₈⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	C₁₀₁×D₄⋊C₁₆	C₂	1	2	yes	maximal, complemented
6	C₄×#41×C₁₀₁	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₁₀₁	—	2	1	no	—
8	C₈×Q₈×C₁₀₁	C₄	1	1	yes	characteristic
9	C₈×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
10	C₈×D₄×C₁₀₁	—	2	1	no	—
11	C₈×D₄×C₁₀₁	—	2	1	no	—
12	C₈×D₄×C₁₀₁	C₄	1	1	yes	characteristic
13	C₈×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
14	#431×C₁₀₁	—	2	1	no	—
15	#431×C₁₀₁	C₄	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₀₁	—	2	1	no	—
17	C₂×C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
18	C₄⋊C₁₆×C₁₀₁	C₂²	1	2	yes	—
19	C₂²⋊C₁₆×C₁₀₁	—	2	1	no	—
20	C₄×C₁₆×C₁₀₁	—	2	1	no	—
21	C₄×C₁₆×C₁₀₁	—	2	1	no	—
22	C₄×C₁₆×C₁₀₁	C₂²	1	2	yes	—
23	C₂×#41×C₁₀₁	C₈	1	1	yes	characteristic
24	C₂²×C₈×C₁₀₁	—	2	1	no	—
25	C₂²×C₈×C₁₀₁	—	2	1	no	—
26	C₂²×C₈×C₁₀₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₀₁	D₄	1	1	yes	characteristic
28	C₄×Q₈×C₁₀₁	C₈	1	1	yes	characteristic
29	C₄×Q₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
30	C₄×D₄×C₁₀₁	—	2	1	no	—
31	C₄×D₄×C₁₀₁	—	2	1	no	—
32	C₄×D₄×C₁₀₁	C₈	1	1	yes	characteristic
33	C₄×D₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
34	#117×C₁₀₁	—	2	1	no	—
35	#117×C₁₀₁	C₈	1	1	yes	characteristic
36	C₂×C₄²×C₁₀₁	—	2	1	no	—
37	C₂×C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₁₆×C₁₀₁	—	2	4	no	—
39	C₂×C₁₆×C₁₀₁	—	2	2	no	—
40	C₂×C₁₆×C₁₀₁	—	2	2	no	—
41	C₄⋊C₈×C₁₀₁	—	2	1	no	—
42	C₄⋊C₈×C₁₀₁	—	2	1	no	—
43	C₄⋊C₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
44	C₄⋊C₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
45	C₂²⋊C₈×C₁₀₁	—	2	1	no	—
46	C₂²⋊C₈×C₁₀₁	—	2	1	no	—
47	C₂²⋊C₈×C₁₀₁	—	2	1	no	—
48	C₄×C₈×C₁₀₁	—	2	1	no	—
49	C₄×C₈×C₁₀₁	—	2	1	no	—
50	C₄×C₈×C₁₀₁	—	2	1	no	—
51	C₄×C₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₁₀₁	D₄	1	1	yes	characteristic
53	C₄×C₈×C₁₀₁	D₄	1	1	yes	characteristic
54	C₄×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
55	#41×C₁₀₁	—	2	2	no	—
56	#41×C₁₀₁	C₁₆	1	4	yes	complemented
57	C₂×Q₈×C₁₀₁	C₂×C₈	1	1	yes	characteristic
58	C₂×D₄×C₁₀₁	—	2	1	no	—
59	C₂×D₄×C₁₀₁	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₁	—	2	1	no	—
61	C₂²×C₄×C₁₀₁	—	2	1	no	—
62	C₂²×C₄×C₁₀₁	—	2	1	no	—
63	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₀₁	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₁₀₁	M₁₆	1	1	yes	characteristic
66	C₂×C₈×C₁₀₁	—	2	2	no	—
67	C₂×C₈×C₁₀₁	—	2	2	no	—
68	C₂×C₈×C₁₀₁	—	2	1	no	—
69	C₂×C₈×C₁₀₁	—	2	1	no	—
70	C₂×C₈×C₁₀₁	—	4	2	no	—
71	C₂×C₈×C₁₀₁	—	4	2	no	—
72	C₂×C₈×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₀₁	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₁₀₁	C₂×D₄	1	1	yes	characteristic
76	C₄⋊C₄×C₁₀₁	—	2	1	no	—
77	C₄⋊C₄×C₁₀₁	—	2	1	no	—
78	C₄⋊C₄×C₁₀₁	C₂×C₈	1	1	yes	characteristic
79	C₄⋊C₄×C₁₀₁	C₂×C₈	1	1	yes	characteristic
80	C₂²⋊C₄×C₁₀₁	—	2	1	no	—
81	C₂²⋊C₄×C₁₀₁	—	2	1	no	—
82	C₂²⋊C₄×C₁₀₁	—	2	1	no	—
83	C₄²×C₁₀₁	—	2	1	no	—
84	C₄²×C₁₀₁	—	2	1	no	—
85	C₄²×C₁₀₁	—	2	1	no	—
86	C₄²×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
87	C₄²×C₁₀₁	C₂×C₈	1	1	yes	characteristic
88	C₄²×C₁₀₁	M₁₆	1	1	yes	characteristic
89	C₄²×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
90	C₁₆×C₁₀₁	—	2	4	no	—
91	C₁₆×C₁₀₁	—	4	2	no	—
92	C₂³×C₁₀₁	—	2	1	no	—
93	C₂³×C₁₀₁	C₂²⋊C₈	1	1	yes	characteristic
94	Q₈×C₁₀₁	—	2	1	no	—
95	Q₈×C₁₀₁	C₂×C₁₆	1	2	yes	complemented
96	D₄×C₁₀₁	—	2	4	no	—
97	D₄×C₁₀₁	—	2	1	no	—
98	D₄×C₁₀₁	C₂×C₁₆	1	2	yes	complemented
99	C₂×C₄×C₁₀₁	—	2	1	no	—
100	C₂×C₄×C₁₀₁	—	2	4	no	—
101	C₂×C₄×C₁₀₁	—	2	2	no	—
102	C₂×C₄×C₁₀₁	—	2	1	no	—
103	C₂×C₄×C₁₀₁	—	2	1	no	—
104	C₂×C₄×C₁₀₁	—	2	1	no	—
105	C₂×C₄×C₁₀₁	—	2	1	no	—
106	C₂×C₄×C₁₀₁	—	4	2	no	—
107	C₂×C₄×C₁₀₁	—	4	1	no	—
108	C₂×C₄×C₁₀₁	—	4	1	no	—
109	C₂×C₄×C₁₀₁	C₂²⋊C₈	1	1	yes	characteristic
110	C₂×C₄×C₁₀₁	C₂²⋊C₈	1	1	yes	characteristic
111	C₂×C₄×C₁₀₁	C₂²⋊C₈	1	1	yes	characteristic
112	C₂×C₄×C₁₀₁	C₂×C₁₆	1	2	yes	—
113	C₂×C₄×C₁₀₁	M₃₂	1	2	yes	—
114	C₂×C₄×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₀₁	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₁₀₁	C₂×M₁₆	1	1	yes	characteristic
117	C₈×C₁₀₁	—	2	1	no	—
118	C₈×C₁₀₁	—	2	1	no	—
119	C₈×C₁₀₁	—	4	1	no	—
120	C₈×C₁₀₁	—	4	1	no	—
121	C₈×C₁₀₁	#104	1	2	yes	—
122	C₈×C₁₀₁	#135	1	2	yes	—
123	C₂²×C₁₀₁	—	2	1	no	—
124	C₂²×C₁₀₁	—	2	1	no	—
125	C₂²×C₁₀₁	—	2	2	no	—
126	C₂²×C₁₀₁	—	4	1	no	—
127	C₂²×C₁₀₁	—	4	1	no	—
128	C₂²×C₁₀₁	C₂²⋊C₁₆	1	2	yes	—
129	C₂²×C₁₀₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
130	C₄×C₁₀₁	—	2	2	no	—
131	C₄×C₁₀₁	—	2	1	no	—
132	C₄×C₁₀₁	—	2	1	no	—
133	C₄×C₁₀₁	—	4	1	no	—
134	C₄×C₁₀₁	—	4	1	no	—
135	C₄×C₁₀₁	C₂²⋊C₁₆	1	2	yes	—
136	C₄×C₁₀₁	#350	1	2	yes	—
137	C₄×C₁₀₁	#416	1	1	yes	characteristic
138	C₄×C₁₀₁	C₂×#104	1	1	yes	characteristic
139	C₄×C₁₀₁	C₂²×C₁₆	1	1	yes	characteristic
140	C₄×C₁₀₁	C₂×M₃₂	1	1	yes	characteristic
141	#11613	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂×C₁₀₁	—	2	2	no	—
143	C₂×C₁₀₁	—	4	2	no	—
144	C₂×C₁₀₁	#1476	1	1	yes	characteristic
145	C₂×C₁₀₁	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
146	C₂×C₁₀₁	C₂×#350	1	1	yes	characteristic
147	C₈×#41	C₂×C₁₀₁	1	1	yes	characteristic
148	#2162	C₂×C₁₀₁	1	1	yes	characteristic, complemented
149	C₂×C₄×C₁₆	C₂×C₁₀₁	1	1	yes	characteristic, complemented
150	Q₈⋊C₁₆	C₂×C₁₀₁	1	2	yes	complemented
151	D₄⋊C₁₆	C₂×C₁₀₁	1	2	yes	complemented
152	C₁₀₁	#11613	1	1	yes	characteristic, direct product factor, Sylow
153	C₄×#41	C₄×C₁₀₁	1	1	yes	characteristic
154	C₂²×C₁₆	—	2	1	no	—
155	C₈×Q₈	C₄×C₁₀₁	1	1	yes	characteristic
156	C₈×Q₈	C₂²×C₁₀₁	1	1	yes	characteristic
157	C₈×D₄	—	2	1	no	—
158	C₈×D₄	—	2	1	no	—
159	C₈×D₄	C₄×C₁₀₁	1	1	yes	characteristic
160	C₈×D₄	C₂²×C₁₀₁	1	1	yes	characteristic
161	#431	—	2	1	no	—
162	#431	C₄×C₁₀₁	1	1	yes	characteristic
163	C₂×C₄×C₈	—	2	1	no	—
164	C₂×C₄×C₈	C₂²×C₁₀₁	1	1	yes	characteristic
165	C₄⋊C₁₆	C₂²×C₁₀₁	1	2	yes	—
166	C₂²⋊C₁₆	—	2	1	no	—
167	C₄×C₁₆	—	2	1	no	—
168	C₄×C₁₆	—	2	1	no	—
169	C₄×C₁₆	C₂²×C₁₀₁	1	2	yes	—
170	C₂×#41	C₈×C₁₀₁	1	1	yes	characteristic
171	C₂²×C₈	—	2	1	no	—
172	C₂²×C₈	—	2	1	no	—
173	C₂²×C₈	D₄×C₁₀₁	1	1	yes	characteristic
174	C₂²×C₈	D₄×C₁₀₁	1	1	yes	characteristic
175	C₄×Q₈	C₈×C₁₀₁	1	1	yes	characteristic
176	C₄×Q₈	C₂×C₄×C₁₀₁	1	1	yes	characteristic
177	C₄×D₄	—	2	1	no	—
178	C₄×D₄	—	2	1	no	—
179	C₄×D₄	C₈×C₁₀₁	1	1	yes	characteristic
180	C₄×D₄	C₂×C₄×C₁₀₁	1	1	yes	characteristic
181	#117	—	2	1	no	—
182	#117	C₈×C₁₀₁	1	1	yes	characteristic
183	C₂×C₄²	—	2	1	no	—
184	C₂×C₄²	C₂×C₄×C₁₀₁	1	1	yes	characteristic
185	C₂×C₁₆	—	2	4	no	—
186	C₂×C₁₆	—	2	2	no	—
187	C₂×C₁₆	—	2	2	no	—
188	C₄⋊C₈	—	2	1	no	—
189	C₄⋊C₈	—	2	1	no	—
190	C₄⋊C₈	C₂×C₄×C₁₀₁	1	1	yes	characteristic
191	C₄⋊C₈	C₂×C₄×C₁₀₁	1	1	yes	characteristic
192	C₂²⋊C₈	—	2	1	no	—
193	C₂²⋊C₈	—	2	1	no	—
194	C₂²⋊C₈	—	2	1	no	—
195	C₄×C₈	—	2	1	no	—
196	C₄×C₈	—	2	1	no	—
197	C₄×C₈	—	2	1	no	—
198	C₄×C₈	C₂×C₄×C₁₀₁	1	1	yes	characteristic
199	C₄×C₈	D₄×C₁₀₁	1	1	yes	characteristic
200	C₄×C₈	D₄×C₁₀₁	1	1	yes	characteristic
201	C₄×C₈	C₂³×C₁₀₁	1	1	yes	characteristic
202	#41	—	2	2	no	—
203	#41	C₁₆×C₁₀₁	1	4	yes	complemented
204	C₂×Q₈	C₂×C₈×C₁₀₁	1	1	yes	characteristic
205	C₂×D₄	—	2	1	no	—
206	C₂×D₄	C₂×C₈×C₁₀₁	1	1	yes	characteristic
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₈×C₁₀₁	1	1	yes	characteristic
212	C₂²×C₄	M₁₆×C₁₀₁	1	1	yes	characteristic
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
220	C₂×C₈	C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
221	C₂×C₈	C₂×D₄×C₁₀₁	1	1	yes	characteristic
222	C₂×C₈	C₂×D₄×C₁₀₁	1	1	yes	characteristic
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	2	1	no	—
225	C₄⋊C₄	C₂×C₈×C₁₀₁	1	1	yes	characteristic
226	C₄⋊C₄	C₂×C₈×C₁₀₁	1	1	yes	characteristic
227	C₂²⋊C₄	—	2	1	no	—
228	C₂²⋊C₄	—	2	1	no	—
229	C₂²⋊C₄	—	2	1	no	—
230	C₄²	—	2	1	no	—
231	C₄²	—	2	1	no	—
232	C₄²	—	2	1	no	—
233	C₄²	C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
234	C₄²	C₂×C₈×C₁₀₁	1	1	yes	characteristic
235	C₄²	M₁₆×C₁₀₁	1	1	yes	characteristic
236	C₄²	C₂²×C₄×C₁₀₁	1	1	yes	characteristic
237	C₁₆	—	2	4	no	—
238	C₁₆	—	4	2	no	—
239	C₂³	—	2	1	no	—
240	C₂³	C₂²⋊C₈×C₁₀₁	1	1	yes	characteristic
241	Q₈	—	2	1	no	—
242	Q₈	C₂×C₁₆×C₁₀₁	1	2	yes	complemented
243	D₄	—	2	4	no	—
244	D₄	—	2	1	no	—
245	D₄	C₂×C₁₆×C₁₀₁	1	2	yes	complemented
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	C₂²⋊C₈×C₁₀₁	1	1	yes	characteristic
257	C₂×C₄	C₂²⋊C₈×C₁₀₁	1	1	yes	characteristic
258	C₂×C₄	C₂²⋊C₈×C₁₀₁	1	1	yes	characteristic
259	C₂×C₄	C₂×C₁₆×C₁₀₁	1	2	yes	—
260	C₂×C₄	M₃₂×C₁₀₁	1	2	yes	—
261	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
262	C₂×C₄	C₂²×C₈×C₁₀₁	1	1	yes	characteristic
263	C₂×C₄	C₂×M₁₆×C₁₀₁	1	1	yes	characteristic
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	4	1	no	—
267	C₈	—	4	1	no	—
268	C₈	#104×C₁₀₁	1	2	yes	—
269	C₈	#135×C₁₀₁	1	2	yes	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	4	1	no	—
274	C₂²	—	4	1	no	—
275	C₂²	C₂²⋊C₁₆×C₁₀₁	1	2	yes	—
276	C₂²	C₂×C₂²⋊C₈×C₁₀₁	1	1	yes	characteristic
277	C₄	—	2	1	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	1	no	—
280	C₄	—	4	1	no	—
281	C₄	—	4	1	no	—
282	C₄	C₂²⋊C₁₆×C₁₀₁	1	2	yes	—
283	C₄	#350×C₁₀₁	1	2	yes	—
284	C₄	#416×C₁₀₁	1	1	yes	characteristic
285	C₄	C₂×#104×C₁₀₁	1	1	yes	characteristic
286	C₄	C₂²×C₁₆×C₁₀₁	1	1	yes	characteristic
287	C₄	C₂×M₃₂×C₁₀₁	1	1	yes	characteristic
288	C₂	—	2	2	no	—
289	C₂	—	4	2	no	—
290	C₂	C₁₀₁×#1476	1	1	yes	characteristic
291	C₂	C₂×C₂²⋊C₁₆×C₁₀₁	1	1	yes	characteristic
292	C₂	C₂×#350×C₁₀₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈×C₁₀₁	D₄
commutator	C₄	C₂²×C₁₆×C₁₀₁
Frattini, cosocle	C₄×C₈	C₂³×C₁₀₁
socle	C₂²×C₁₀₁	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₁₀₁	C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈×C₁₀₁ ◁ C₂×C₄×C₈×C₁₀₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₀₁×#2162	C₂	semidirect
3. C₂×C₄×C₁₆×C₁₀₁	C₂	semidirect
4. C₁₀₁×Q₈⋊C₁₆	C₂	semidirect
5. C₁₀₁×D₄⋊C₁₆	C₂	semidirect
56. #41×C₁₀₁	C₁₆	semidirect
95. Q₈×C₁₀₁	C₂×C₁₆	semidirect
98. D₄×C₁₀₁	C₂×C₁₆	semidirect
141. #11613	C₁₀₁	direct (N × H)
148. #2162	C₂×C₁₀₁	semidirect
149. C₂×C₄×C₁₆	C₂×C₁₀₁	semidirect
150. Q₈⋊C₁₆	C₂×C₁₀₁	semidirect
151. D₄⋊C₁₆	C₂×C₁₀₁	semidirect
152. C₁₀₁	#11613	direct (N × H)
203. #41	C₁₆×C₁₀₁	semidirect
242. Q₈	C₂×C₁₆×C₁₀₁	semidirect
245. D₄	C₂×C₁₆×C₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₈×#41×C₁₀₁	2
C₁₀₁×#2162	2
C₂×C₄×C₁₆×C₁₀₁	2
C₁₀₁×Q₈⋊C₁₆	2
C₁₀₁×D₄⋊C₁₆	2
#11613	101

Subgroups — C101×#11613 #38793764

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#11613 #38793764