Subgroups — C₁₀₁×#8452 #38790603

Order 25856.

868 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#555×C₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₁₀₁×#1956	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₀₁×#1802	C₂	1	2	yes	maximal, complemented
4	C₂×#333×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₄⋊C₈×C₁₀₁	C₂	1	2	yes	maximal, complemented
6	#555×C₁₀₁	—	2	2	no	—
7	#555×C₁₀₁	—	2	4	no	—
8	#555×C₁₀₁	C₄	1	4	yes	complemented
9	C₂×C₄⋊Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
10	C₂×#124×C₁₀₁	—	2	2	no	—
11	C₂×#121×C₁₀₁	—	2	2	no	—
12	C₂×C₄×D₄×C₁₀₁	C₂²	1	2	yes	—
13	C₂×#117×C₁₀₁	—	2	2	no	—
14	C₂×C₄⋊C₈×C₁₀₁	C₂²	1	2	yes	—
15	C₂×#103×C₁₀₁	—	2	2	no	—
16	C₂×#102×C₁₀₁	—	2	2	no	—
17	C₂×C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
18	#384×C₁₀₁	C₂²	1	1	yes	characteristic
19	#340×C₁₀₁	—	2	2	no	—
20	#333×C₁₀₁	—	2	2	no	—
21	D₄⋊C₈×C₁₀₁	—	2	4	no	—
22	C₂²×Q₈×C₁₀₁	D₄	1	2	yes	—
23	C₂²×D₄×C₁₀₁	—	2	2	no	—
24	C₂²×D₄×C₁₀₁	D₄	1	2	yes	—
25	C₂³×C₄×C₁₀₁	—	2	2	no	—
26	C₂²×C₈×C₁₀₁	—	2	1	no	—
27	C₂²×C₈×C₁₀₁	—	2	2	no	—
28	C₄⋊Q₈×C₁₀₁	—	2	2	no	—
29	C₄⋊Q₈×C₁₀₁	C₂×C₄	1	2	yes	—
30	C₄⋊Q₈×C₁₀₁	D₄	1	2	yes	—
31	#124×C₁₀₁	—	2	8	no	—
32	#124×C₁₀₁	—	2	8	no	—
33	#121×C₁₀₁	—	2	8	no	—
34	#121×C₁₀₁	—	2	8	no	—
35	C₄×D₄×C₁₀₁	—	2	2	no	—
36	C₄×D₄×C₁₀₁	—	2	2	no	—
37	C₄×D₄×C₁₀₁	C₂×C₄	1	4	yes	—
38	C₄×D₄×C₁₀₁	D₄	1	4	yes	—
39	#117×C₁₀₁	—	2	8	no	—
40	#117×C₁₀₁	—	2	8	no	—
41	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₀₁	—	2	1	no	—
44	C₂×C₄⋊C₄×C₁₀₁	D₄	1	2	yes	—
45	C₂×C₂²⋊C₄×C₁₀₁	—	2	4	no	—
46	C₂×C₂²⋊C₄×C₁₀₁	—	2	2	no	—
47	C₂×C₄²×C₁₀₁	—	2	2	no	—
48	C₂×C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₁₀₁	—	2	2	no	—
50	C₄⋊C₈×C₁₀₁	—	2	2	no	—
51	#103×C₁₀₁	—	4	4	no	—
52	#102×C₁₀₁	—	4	4	no	—
53	C₄×C₈×C₁₀₁	—	2	2	no	—
54	C₂⁴×C₁₀₁	—	2	2	no	—
55	C₂×Q₈×C₁₀₁	—	2	2	no	—
56	C₂×Q₈×C₁₀₁	—	4	2	no	—
57	C₂×Q₈×C₁₀₁	—	4	2	no	—
58	C₂×Q₈×C₁₀₁	C₂²⋊C₄	1	4	yes	—
59	C₂×D₄×C₁₀₁	—	2	8	no	—
60	C₂×D₄×C₁₀₁	—	2	2	no	—
61	C₂×D₄×C₁₀₁	—	2	2	no	—
62	C₂×D₄×C₁₀₁	—	2	2	no	—
63	C₂×D₄×C₁₀₁	—	2	2	no	—
64	C₂×D₄×C₁₀₁	—	2	2	no	—
65	C₂×D₄×C₁₀₁	—	8	2	no	—
66	C₂×D₄×C₁₀₁	—	8	2	no	—
67	C₂×D₄×C₁₀₁	C₂²⋊C₄	1	4	yes	—
68	C₂²×C₄×C₁₀₁	—	2	2	no	—
69	C₂²×C₄×C₁₀₁	—	2	2	no	—
70	C₂²×C₄×C₁₀₁	—	2	4	no	—
71	C₂²×C₄×C₁₀₁	—	2	4	no	—
72	C₂²×C₄×C₁₀₁	—	4	2	no	—
73	C₂²×C₄×C₁₀₁	—	4	2	no	—
74	C₂²×C₄×C₁₀₁	—	4	2	no	—
75	C₂²×C₄×C₁₀₁	—	4	2	no	—
76	C₂²×C₄×C₁₀₁	—	4	1	no	—
77	C₂²×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₁₀₁	—	2	2	no	—
80	C₂×C₈×C₁₀₁	—	4	1	no	—
81	C₂×C₈×C₁₀₁	—	4	1	no	—
82	C₂×C₈×C₁₀₁	—	4	2	no	—
83	C₂×C₈×C₁₀₁	—	4	2	no	—
84	C₂×C₈×C₁₀₁	—	4	2	no	—
85	C₄⋊C₄×C₁₀₁	—	2	4	no	—
86	C₄⋊C₄×C₁₀₁	—	2	4	no	—
87	C₄⋊C₄×C₁₀₁	—	2	2	no	—
88	C₄⋊C₄×C₁₀₁	—	4	2	no	—
89	C₄⋊C₄×C₁₀₁	—	4	2	no	—
90	C₄⋊C₄×C₁₀₁	—	4	2	no	—
91	C₄⋊C₄×C₁₀₁	C₂²⋊C₄	1	4	yes	—
92	C₂²⋊C₄×C₁₀₁	—	2	16	no	—
93	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
94	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
95	C₄²×C₁₀₁	—	2	8	no	—
96	C₄²×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₁₀₁	C₂×D₄	1	2	yes	—
98	C₄²×C₁₀₁	C₂×D₄	1	1	yes	characteristic
99	C₂³×C₁₀₁	—	2	4	no	—
100	C₂³×C₁₀₁	—	4	2	no	—
101	C₂³×C₁₀₁	—	4	2	no	—
102	C₂³×C₁₀₁	—	4	2	no	—
103	C₂³×C₁₀₁	—	4	2	no	—
104	C₂³×C₁₀₁	—	4	2	no	—
105	C₂³×C₁₀₁	—	4	2	no	—
106	C₂³×C₁₀₁	#120	1	1	yes	characteristic
107	Q₈×C₁₀₁	—	4	4	no	—
108	Q₈×C₁₀₁	—	8	2	no	—
109	D₄×C₁₀₁	—	2	4	no	—
110	D₄×C₁₀₁	—	2	4	no	—
111	D₄×C₁₀₁	—	4	2	no	—
112	D₄×C₁₀₁	—	4	2	no	—
113	D₄×C₁₀₁	—	8	8	no	—
114	C₂×C₄×C₁₀₁	—	2	1	no	—
115	C₂×C₄×C₁₀₁	—	2	1	no	—
116	C₂×C₄×C₁₀₁	—	2	4	no	—
117	C₂×C₄×C₁₀₁	—	2	4	no	—
118	C₂×C₄×C₁₀₁	—	4	2	no	—
119	C₂×C₄×C₁₀₁	—	4	2	no	—
120	C₂×C₄×C₁₀₁	—	4	2	no	—
121	C₂×C₄×C₁₀₁	—	4	2	no	—
122	C₂×C₄×C₁₀₁	—	4	4	no	—
123	C₂×C₄×C₁₀₁	—	4	4	no	—
124	C₂×C₄×C₁₀₁	—	4	4	no	—
125	C₂×C₄×C₁₀₁	—	4	4	no	—
126	C₂×C₄×C₁₀₁	—	8	2	no	—
127	C₂×C₄×C₁₀₁	—	8	1	no	—
128	C₂×C₄×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
130	C₂×C₄×C₁₀₁	#120	1	1	yes	characteristic
131	C₂×C₄×C₁₀₁	#120	1	2	yes	—
132	C₂×C₄×C₁₀₁	#135	1	2	yes	—
133	C₂×C₄×C₁₀₁	#135	1	2	yes	—
134	C₂×C₄×C₁₀₁	AGL(1,ℤ/8)	1	2	yes	—
135	C₂×C₄×C₁₀₁	#137	1	2	yes	—
136	C₈×C₁₀₁	—	4	2	no	—
137	C₈×C₁₀₁	—	8	2	no	—
138	C₂²×C₁₀₁	—	4	4	no	—
139	C₂²×C₁₀₁	—	4	4	no	—
140	C₂²×C₁₀₁	—	4	4	no	—
141	C₂²×C₁₀₁	—	8	2	no	—
142	C₂²×C₁₀₁	—	8	2	no	—
143	C₂²×C₁₀₁	—	8	2	no	—
144	C₂²×C₁₀₁	—	8	2	no	—
145	C₂²×C₁₀₁	#379	1	1	yes	characteristic
146	C₂²×C₁₀₁	#449	1	2	yes	—
147	C₂²×C₁₀₁	#452	1	2	yes	—
148	C₂²×C₁₀₁	#453	1	1	yes	characteristic
149	C₂²×C₁₀₁	#455	1	1	yes	characteristic
150	C₄×C₁₀₁	—	2	1	no	—
151	C₄×C₁₀₁	—	2	1	no	—
152	C₄×C₁₀₁	—	4	4	no	—
153	C₄×C₁₀₁	—	4	4	no	—
154	C₄×C₁₀₁	—	16	1	no	—
155	C₄×C₁₀₁	#416	1	2	yes	—
156	C₄×C₁₀₁	#417	1	2	yes	—
157	C₄×C₁₀₁	#449	1	2	yes	—
158	C₄×C₁₀₁	#452	1	2	yes	—
159	#8452	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂×C₁₀₁	—	8	4	no	—
161	C₂×C₁₀₁	#1624	1	1	yes	characteristic
162	C₂×C₁₀₁	#1634	1	1	yes	characteristic
163	C₂×C₁₀₁	#1646	1	2	yes	—
164	C₂×C₁₀₁	#1788	1	2	yes	—
165	C₂×C₁₀₁	#1792	1	1	yes	characteristic
166	C₂×#555	C₂×C₁₀₁	1	1	yes	characteristic
167	#1956	C₂×C₁₀₁	1	1	yes	characteristic, complemented
168	#1802	C₂×C₁₀₁	1	2	yes	complemented
169	C₂×#333	C₂×C₁₀₁	1	1	yes	characteristic, complemented
170	C₂×D₄⋊C₈	C₂×C₁₀₁	1	2	yes	complemented
171	C₁₀₁	#8452	1	1	yes	characteristic, direct product factor, Sylow
172	#555	—	2	2	no	—
173	#555	—	2	4	no	—
174	#555	C₄×C₁₀₁	1	4	yes	complemented
175	C₂×C₄⋊Q₈	C₂²×C₁₀₁	1	1	yes	characteristic
176	C₂×#124	—	2	2	no	—
177	C₂×#121	—	2	2	no	—
178	C₂×C₄×D₄	C₂²×C₁₀₁	1	2	yes	—
179	C₂×#117	—	2	2	no	—
180	C₂×C₄⋊C₈	C₂²×C₁₀₁	1	2	yes	—
181	C₂×#103	—	2	2	no	—
182	C₂×#102	—	2	2	no	—
183	C₂×C₄×C₈	C₂²×C₁₀₁	1	1	yes	characteristic
184	#384	C₂²×C₁₀₁	1	1	yes	characteristic
185	#340	—	2	2	no	—
186	#333	—	2	2	no	—
187	D₄⋊C₈	—	2	4	no	—
188	C₂²×Q₈	D₄×C₁₀₁	1	2	yes	—
189	C₂²×D₄	—	2	2	no	—
190	C₂²×D₄	D₄×C₁₀₁	1	2	yes	—
191	C₂³×C₄	—	2	2	no	—
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	—	2	2	no	—
194	C₄⋊Q₈	—	2	2	no	—
195	C₄⋊Q₈	C₂×C₄×C₁₀₁	1	2	yes	—
196	C₄⋊Q₈	D₄×C₁₀₁	1	2	yes	—
197	#124	—	2	8	no	—
198	#124	—	2	8	no	—
199	#121	—	2	8	no	—
200	#121	—	2	8	no	—
201	C₄×D₄	—	2	2	no	—
202	C₄×D₄	—	2	2	no	—
203	C₄×D₄	C₂×C₄×C₁₀₁	1	4	yes	—
204	C₄×D₄	D₄×C₁₀₁	1	4	yes	—
205	#117	—	2	8	no	—
206	#117	—	2	8	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	1	no	—
210	C₂×C₄⋊C₄	D₄×C₁₀₁	1	2	yes	—
211	C₂×C₂²⋊C₄	—	2	4	no	—
212	C₂×C₂²⋊C₄	—	2	2	no	—
213	C₂×C₄²	—	2	2	no	—
214	C₂×C₄²	C₂³×C₁₀₁	1	1	yes	characteristic
215	C₄⋊C₈	—	2	2	no	—
216	C₄⋊C₈	—	2	2	no	—
217	#103	—	4	4	no	—
218	#102	—	4	4	no	—
219	C₄×C₈	—	2	2	no	—
220	C₂⁴	—	2	2	no	—
221	C₂×Q₈	—	2	2	no	—
222	C₂×Q₈	—	4	2	no	—
223	C₂×Q₈	—	4	2	no	—
224	C₂×Q₈	C₂²⋊C₄×C₁₀₁	1	4	yes	—
225	C₂×D₄	—	2	8	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	2	2	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	8	2	no	—
232	C₂×D₄	—	8	2	no	—
233	C₂×D₄	C₂²⋊C₄×C₁₀₁	1	4	yes	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	4	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	1	no	—
243	C₂²×C₄	C₂×D₄×C₁₀₁	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₁₀₁	1	2	yes	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	—	4	2	no	—
250	C₂×C₈	—	4	2	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	C₂²⋊C₄×C₁₀₁	1	4	yes	—
258	C₂²⋊C₄	—	2	16	no	—
259	C₂²⋊C₄	—	2	4	no	—
260	C₂²⋊C₄	—	2	4	no	—
261	C₄²	—	2	8	no	—
262	C₄²	C₂²×C₄×C₁₀₁	1	1	yes	characteristic
263	C₄²	C₂×D₄×C₁₀₁	1	2	yes	—
264	C₄²	C₂×D₄×C₁₀₁	1	1	yes	characteristic
265	C₂³	—	2	4	no	—
266	C₂³	—	4	2	no	—
267	C₂³	—	4	2	no	—
268	C₂³	—	4	2	no	—
269	C₂³	—	4	2	no	—
270	C₂³	—	4	2	no	—
271	C₂³	—	4	2	no	—
272	C₂³	#120×C₁₀₁	1	1	yes	characteristic
273	Q₈	—	4	4	no	—
274	Q₈	—	8	2	no	—
275	D₄	—	2	4	no	—
276	D₄	—	2	4	no	—
277	D₄	—	4	2	no	—
278	D₄	—	4	2	no	—
279	D₄	—	8	8	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	8	2	no	—
293	C₂×C₄	—	8	1	no	—
294	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
295	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₁	1	2	yes	—
296	C₂×C₄	#120×C₁₀₁	1	1	yes	characteristic
297	C₂×C₄	#120×C₁₀₁	1	2	yes	—
298	C₂×C₄	#135×C₁₀₁	1	2	yes	—
299	C₂×C₄	#135×C₁₀₁	1	2	yes	—
300	C₂×C₄	AGL(1,ℤ/8)×C₁₀₁	1	2	yes	—
301	C₂×C₄	#137×C₁₀₁	1	2	yes	—
302	C₈	—	4	2	no	—
303	C₈	—	8	2	no	—
304	C₂²	—	4	4	no	—
305	C₂²	—	4	4	no	—
306	C₂²	—	4	4	no	—
307	C₂²	—	8	2	no	—
308	C₂²	—	8	2	no	—
309	C₂²	—	8	2	no	—
310	C₂²	—	8	2	no	—
311	C₂²	#379×C₁₀₁	1	1	yes	characteristic
312	C₂²	#449×C₁₀₁	1	2	yes	—
313	C₂²	#452×C₁₀₁	1	2	yes	—
314	C₂²	#453×C₁₀₁	1	1	yes	characteristic
315	C₂²	#455×C₁₀₁	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	4	no	—
319	C₄	—	4	4	no	—
320	C₄	—	16	1	no	—
321	C₄	#416×C₁₀₁	1	2	yes	—
322	C₄	#417×C₁₀₁	1	2	yes	—
323	C₄	#449×C₁₀₁	1	2	yes	—
324	C₄	#452×C₁₀₁	1	2	yes	—
325	C₂	—	8	4	no	—
326	C₂	C₁₀₁×#1624	1	1	yes	characteristic
327	C₂	C₁₀₁×#1634	1	1	yes	characteristic
328	C₂	C₁₀₁×#1646	1	2	yes	—
329	C₂	C₁₀₁×#1788	1	2	yes	—
330	C₂	C₁₀₁×#1792	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₀₁	#120
commutator	C₄²	C₂²×C₄×C₁₀₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₁
prime radical	C₂×#555×C₁₀₁	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₁ ◁ C₂×C₄²×C₁₀₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₀₁×#1956	C₂	semidirect
3. C₁₀₁×#1802	C₂	semidirect
4. C₂×#333×C₁₀₁	C₂	semidirect
5. C₂×D₄⋊C₈×C₁₀₁	C₂	semidirect
8. #555×C₁₀₁	C₄	semidirect
159. #8452	C₁₀₁	direct (N × H)
167. #1956	C₂×C₁₀₁	semidirect
168. #1802	C₂×C₁₀₁	semidirect
169. C₂×#333	C₂×C₁₀₁	semidirect
170. C₂×D₄⋊C₈	C₂×C₁₀₁	semidirect
171. C₁₀₁	#8452	direct (N × H)
174. #555	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×#555×C₁₀₁	2
C₁₀₁×#1956	2
C₁₀₁×#1802	2
C₂×#333×C₁₀₁	2
C₂×D₄⋊C₈×C₁₀₁	2
#8452	101

Subgroups — C101×#8452 #38790603

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#8452 #38790603