Subgroups — C₁₀₁×#19061 #38801212

Order 25856.

942 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₁×#3548	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₀₁×#3214	C₂	1	2	yes	maximal, complemented
3	C₁₀₁×#3135	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₁×#1662	C₂	1	4	yes	maximal, complemented
5	C₁₀₁×#1656	C₂	1	4	yes	maximal, complemented
6	C₁₀₁×#1537	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₀₁×#1530	C₂	1	2	yes	maximal, complemented
8	#562×C₁₀₁	C₂²	1	4	yes	—
9	#554×C₁₀₁	—	2	2	no	—
10	#550×C₁₀₁	—	2	2	no	—
11	#547×C₁₀₁	—	2	2	no	—
12	#540×C₁₀₁	—	2	2	no	—
13	#537×C₁₀₁	—	2	4	no	—
14	#533×C₁₀₁	C₂²	1	1	yes	characteristic
15	#532×C₁₀₁	C₂²	1	1	yes	characteristic
16	#532×C₁₀₁	C₂²	1	2	yes	—
17	#496×C₁₀₁	C₂²	1	4	yes	—
18	#491×C₁₀₁	C₂²	1	2	yes	—
19	#490×C₁₀₁	C₂²	1	2	yes	—
20	#489×C₁₀₁	—	2	2	no	—
21	#489×C₁₀₁	C₂²	1	4	yes	—
22	#461×C₁₀₁	—	2	2	no	—
23	#459×C₁₀₁	—	2	2	no	—
24	#449×C₁₀₁	—	2	4	no	—
25	#432×C₁₀₁	C₂²	1	2	yes	—
26	#416×C₁₀₁	—	2	2	no	—
27	C₂×#97×C₁₀₁	C₂²	1	1	yes	characteristic
28	#330×C₁₀₁	C₂²	1	4	yes	—
29	#329×C₁₀₁	C₂²	1	8	yes	—
30	C₂×#41×C₁₀₁	—	2	4	no	—
31	C₂×#41×C₁₀₁	—	2	1	no	—
32	C₂×#41×C₁₀₁	D₄	1	4	yes	—
33	C₂×SD₁₆×C₁₀₁	—	2	4	no	—
34	C₂×D₈×C₁₀₁	—	2	4	no	—
35	C₂²×C₈×C₁₀₁	—	2	1	no	—
36	C₄⋊Q₈×C₁₀₁	D₄	1	2	yes	—
37	C₄⋊Q₈×C₁₀₁	D₄	1	2	yes	—
38	C₄⋊Q₈×C₁₀₁	C₂³	1	1	yes	characteristic
39	#127×C₁₀₁	—	2	2	no	—
40	#127×C₁₀₁	D₄	1	2	yes	—
41	#127×C₁₀₁	C₂³	1	1	yes	characteristic
42	#125×C₁₀₁	—	2	2	no	—
43	#125×C₁₀₁	C₂³	1	2	yes	—
44	#124×C₁₀₁	—	2	1	no	—
45	#124×C₁₀₁	C₂³	1	4	yes	—
46	#123×C₁₀₁	—	2	8	no	—
47	C₂²⋊Q₈×C₁₀₁	—	2	4	no	—
48	C₂²⋊Q₈×C₁₀₁	—	2	8	no	—
49	C₂²⋊Q₈×C₁₀₁	—	2	2	no	—
50	#121×C₁₀₁	—	2	8	no	—
51	#121×C₁₀₁	—	2	4	no	—
52	#121×C₁₀₁	—	2	8	no	—
53	#121×C₁₀₁	—	2	2	no	—
54	#121×C₁₀₁	—	2	2	no	—
55	#121×C₁₀₁	—	2	2	no	—
56	C₄×Q₈×C₁₀₁	—	2	2	no	—
57	C₄×D₄×C₁₀₁	—	2	4	no	—
58	C₄×D₄×C₁₀₁	—	2	2	no	—
59	C₄×D₄×C₁₀₁	—	2	2	no	—
60	C₄×D₄×C₁₀₁	—	2	2	no	—
61	#117×C₁₀₁	D₄	1	2	yes	—
62	C₂×C₄⋊C₄×C₁₀₁	—	2	4	no	—
63	C₂×C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
64	C₄⋊C₈×C₁₀₁	—	2	2	no	—
65	#103×C₁₀₁	—	2	4	no	—
66	#103×C₁₀₁	—	2	4	no	—
67	#102×C₁₀₁	—	2	4	no	—
68	#102×C₁₀₁	—	2	4	no	—
69	C₂²⋊C₈×C₁₀₁	—	2	2	no	—
70	#97×C₁₀₁	—	2	1	no	—
71	#97×C₁₀₁	C₂³	1	2	yes	—
72	#97×C₁₀₁	C₂³	1	4	yes	—
73	#41×C₁₀₁	—	4	4	no	—
74	#41×C₁₀₁	—	8	4	no	—
75	#41×C₁₀₁	—	8	8	no	—
76	C₂×Q₈×C₁₀₁	—	2	2	no	—
77	C₂×Q₈×C₁₀₁	—	2	1	no	—
78	C₂×Q₈×C₁₀₁	C₂×D₄	1	4	yes	—
79	C₂×D₄×C₁₀₁	—	2	8	no	—
80	C₂×D₄×C₁₀₁	—	2	4	no	—
81	C₂×D₄×C₁₀₁	—	2	2	no	—
82	C₂×D₄×C₁₀₁	—	2	2	no	—
83	C₂×D₄×C₁₀₁	—	2	1	no	—
84	C₂×D₄×C₁₀₁	—	2	2	no	—
85	C₂×D₄×C₁₀₁	—	4	1	no	—
86	C₂×D₄×C₁₀₁	C₂×D₄	1	4	yes	—
87	C₂²×C₄×C₁₀₁	—	2	8	no	—
88	C₂²×C₄×C₁₀₁	—	2	4	no	—
89	C₂²×C₄×C₁₀₁	—	4	1	no	—
90	C₂²×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
92	SD₁₆×C₁₀₁	—	8	4	no	—
93	D₈×C₁₀₁	—	8	4	no	—
94	C₂×C₈×C₁₀₁	—	2	2	no	—
95	C₂×C₈×C₁₀₁	—	2	4	no	—
96	C₂×C₈×C₁₀₁	—	4	2	no	—
97	C₄⋊C₄×C₁₀₁	—	2	8	no	—
98	C₄⋊C₄×C₁₀₁	—	2	2	no	—
99	C₄⋊C₄×C₁₀₁	—	2	2	no	—
100	C₄⋊C₄×C₁₀₁	—	2	2	no	—
101	C₄⋊C₄×C₁₀₁	—	2	4	no	—
102	C₄⋊C₄×C₁₀₁	—	2	2	no	—
103	C₄⋊C₄×C₁₀₁	C₂×D₄	1	4	yes	—
104	C₂²⋊C₄×C₁₀₁	—	2	8	no	—
105	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
106	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
107	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
108	C₂²⋊C₄×C₁₀₁	—	4	2	no	—
109	C₄²×C₁₀₁	—	2	2	no	—
110	C₄²×C₁₀₁	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₁₀₁	C₂×D₄	1	2	yes	—
112	C₄²×C₁₀₁	C₂⁴	1	1	yes	characteristic
113	C₂³×C₁₀₁	—	2	4	no	—
114	C₂³×C₁₀₁	—	4	1	no	—
115	C₂³×C₁₀₁	#120	1	1	yes	characteristic
116	Q₈×C₁₀₁	—	4	4	no	—
117	Q₈×C₁₀₁	—	4	4	no	—
118	Q₈×C₁₀₁	—	4	2	no	—
119	D₄×C₁₀₁	—	4	4	no	—
120	D₄×C₁₀₁	—	4	4	no	—
121	D₄×C₁₀₁	—	4	2	no	—
122	D₄×C₁₀₁	—	4	2	no	—
123	D₄×C₁₀₁	—	4	2	no	—
124	D₄×C₁₀₁	—	4	4	no	—
125	D₄×C₁₀₁	—	8	4	no	—
126	D₄×C₁₀₁	—	8	8	no	—
127	D₄×C₁₀₁	—	8	2	no	—
128	C₂×C₄×C₁₀₁	—	2	2	no	—
129	C₂×C₄×C₁₀₁	—	2	4	no	—
130	C₂×C₄×C₁₀₁	—	2	4	no	—
131	C₂×C₄×C₁₀₁	—	4	2	no	—
132	C₂×C₄×C₁₀₁	—	4	8	no	—
133	C₂×C₄×C₁₀₁	—	4	1	no	—
134	C₂×C₄×C₁₀₁	—	8	8	no	—
135	C₂×C₄×C₁₀₁	—	8	2	no	—
136	C₂×C₄×C₁₀₁	#120	1	2	yes	—
137	C₂×C₄×C₁₀₁	#120	1	1	yes	characteristic
138	C₂×C₄×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₀₁	C₂²×D₄	1	2	yes	—
140	C₈×C₁₀₁	—	4	2	no	—
141	C₈×C₁₀₁	—	4	4	no	—
142	C₂²×C₁₀₁	—	2	1	no	—
143	C₂²×C₁₀₁	—	2	2	no	—
144	C₂²×C₁₀₁	—	4	4	no	—
145	C₂²×C₁₀₁	—	4	4	no	—
146	C₂²×C₁₀₁	—	4	4	no	—
147	C₂²×C₁₀₁	—	8	1	no	—
148	C₂²×C₁₀₁	—	8	2	no	—
149	C₂²×C₁₀₁	C₂×#120	1	1	yes	characteristic
150	C₄×C₁₀₁	—	2	1	no	—
151	C₄×C₁₀₁	—	2	2	no	—
152	C₄×C₁₀₁	—	2	2	no	—
153	C₄×C₁₀₁	—	4	4	no	—
154	C₄×C₁₀₁	—	4	4	no	—
155	C₄×C₁₀₁	—	8	1	no	—
156	C₄×C₁₀₁	#454	1	2	yes	—
157	#19061	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂×C₁₀₁	—	4	1	no	—
159	C₂×C₁₀₁	—	8	4	no	—
160	C₂×C₁₀₁	—	16	1	no	—
161	C₂×C₁₀₁	C₂×#454	1	1	yes	characteristic
162	C₂×C₁₀₁	#3019	1	2	yes	—
163	#3548	C₂×C₁₀₁	1	1	yes	characteristic, complemented
164	#3214	C₂×C₁₀₁	1	2	yes	complemented
165	#3135	C₂×C₁₀₁	1	1	yes	characteristic, complemented
166	#1662	C₂×C₁₀₁	1	4	yes	complemented
167	#1656	C₂×C₁₀₁	1	4	yes	complemented
168	#1537	C₂×C₁₀₁	1	1	yes	characteristic, complemented
169	#1530	C₂×C₁₀₁	1	2	yes	complemented
170	C₁₀₁	#19061	1	1	yes	characteristic, direct product factor, Sylow
171	#562	C₂²×C₁₀₁	1	4	yes	—
172	#554	—	2	2	no	—
173	#550	—	2	2	no	—
174	#547	—	2	2	no	—
175	#540	—	2	2	no	—
176	#537	—	2	4	no	—
177	#533	C₂²×C₁₀₁	1	1	yes	characteristic
178	#532	C₂²×C₁₀₁	1	1	yes	characteristic
179	#532	C₂²×C₁₀₁	1	2	yes	—
180	#496	C₂²×C₁₀₁	1	4	yes	—
181	#491	C₂²×C₁₀₁	1	2	yes	—
182	#490	C₂²×C₁₀₁	1	2	yes	—
183	#489	—	2	2	no	—
184	#489	C₂²×C₁₀₁	1	4	yes	—
185	#461	—	2	2	no	—
186	#459	—	2	2	no	—
187	#449	—	2	4	no	—
188	#432	C₂²×C₁₀₁	1	2	yes	—
189	#416	—	2	2	no	—
190	C₂×#97	C₂²×C₁₀₁	1	1	yes	characteristic
191	#330	C₂²×C₁₀₁	1	4	yes	—
192	#329	C₂²×C₁₀₁	1	8	yes	—
193	C₂×#41	—	2	4	no	—
194	C₂×#41	—	2	1	no	—
195	C₂×#41	D₄×C₁₀₁	1	4	yes	—
196	C₂×SD₁₆	—	2	4	no	—
197	C₂×D₈	—	2	4	no	—
198	C₂²×C₈	—	2	1	no	—
199	C₄⋊Q₈	D₄×C₁₀₁	1	2	yes	—
200	C₄⋊Q₈	D₄×C₁₀₁	1	2	yes	—
201	C₄⋊Q₈	C₂³×C₁₀₁	1	1	yes	characteristic
202	#127	—	2	2	no	—
203	#127	D₄×C₁₀₁	1	2	yes	—
204	#127	C₂³×C₁₀₁	1	1	yes	characteristic
205	#125	—	2	2	no	—
206	#125	C₂³×C₁₀₁	1	2	yes	—
207	#124	—	2	1	no	—
208	#124	C₂³×C₁₀₁	1	4	yes	—
209	#123	—	2	8	no	—
210	C₂²⋊Q₈	—	2	4	no	—
211	C₂²⋊Q₈	—	2	8	no	—
212	C₂²⋊Q₈	—	2	2	no	—
213	#121	—	2	8	no	—
214	#121	—	2	4	no	—
215	#121	—	2	8	no	—
216	#121	—	2	2	no	—
217	#121	—	2	2	no	—
218	#121	—	2	2	no	—
219	C₄×Q₈	—	2	2	no	—
220	C₄×D₄	—	2	4	no	—
221	C₄×D₄	—	2	2	no	—
222	C₄×D₄	—	2	2	no	—
223	C₄×D₄	—	2	2	no	—
224	#117	D₄×C₁₀₁	1	2	yes	—
225	C₂×C₄⋊C₄	—	2	4	no	—
226	C₂×C₄²	C₂³×C₁₀₁	1	1	yes	characteristic
227	C₄⋊C₈	—	2	2	no	—
228	#103	—	2	4	no	—
229	#103	—	2	4	no	—
230	#102	—	2	4	no	—
231	#102	—	2	4	no	—
232	C₂²⋊C₈	—	2	2	no	—
233	#97	—	2	1	no	—
234	#97	C₂³×C₁₀₁	1	2	yes	—
235	#97	C₂³×C₁₀₁	1	4	yes	—
236	#41	—	4	4	no	—
237	#41	—	8	4	no	—
238	#41	—	8	8	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	2	1	no	—
241	C₂×Q₈	C₂×D₄×C₁₀₁	1	4	yes	—
242	C₂×D₄	—	2	8	no	—
243	C₂×D₄	—	2	4	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂×D₄	C₂×D₄×C₁₀₁	1	4	yes	—
250	C₂²×C₄	—	2	8	no	—
251	C₂²×C₄	—	2	4	no	—
252	C₂²×C₄	—	4	1	no	—
253	C₂²×C₄	C₂×D₄×C₁₀₁	1	1	yes	characteristic
254	C₂²×C₄	C₂×D₄×C₁₀₁	1	2	yes	—
255	SD₁₆	—	8	4	no	—
256	D₈	—	8	4	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	4	2	no	—
260	C₄⋊C₄	—	2	8	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	C₂×D₄×C₁₀₁	1	4	yes	—
267	C₂²⋊C₄	—	2	8	no	—
268	C₂²⋊C₄	—	2	2	no	—
269	C₂²⋊C₄	—	2	4	no	—
270	C₂²⋊C₄	—	2	4	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	C₂×D₄×C₁₀₁	1	1	yes	characteristic
274	C₄²	C₂×D₄×C₁₀₁	1	2	yes	—
275	C₄²	C₂⁴×C₁₀₁	1	1	yes	characteristic
276	C₂³	—	2	4	no	—
277	C₂³	—	4	1	no	—
278	C₂³	#120×C₁₀₁	1	1	yes	characteristic
279	Q₈	—	4	4	no	—
280	Q₈	—	4	4	no	—
281	Q₈	—	4	2	no	—
282	D₄	—	4	4	no	—
283	D₄	—	4	4	no	—
284	D₄	—	4	2	no	—
285	D₄	—	4	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	4	no	—
288	D₄	—	8	4	no	—
289	D₄	—	8	8	no	—
290	D₄	—	8	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	8	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	8	8	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	#120×C₁₀₁	1	2	yes	—
300	C₂×C₄	#120×C₁₀₁	1	1	yes	characteristic
301	C₂×C₄	C₂²×D₄×C₁₀₁	1	1	yes	characteristic
302	C₂×C₄	C₂²×D₄×C₁₀₁	1	2	yes	—
303	C₈	—	4	2	no	—
304	C₈	—	4	4	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	4	4	no	—
308	C₂²	—	4	4	no	—
309	C₂²	—	4	4	no	—
310	C₂²	—	8	1	no	—
311	C₂²	—	8	2	no	—
312	C₂²	C₂×#120×C₁₀₁	1	1	yes	characteristic
313	C₄	—	2	1	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	4	4	no	—
317	C₄	—	4	4	no	—
318	C₄	—	8	1	no	—
319	C₄	#454×C₁₀₁	1	2	yes	—
320	C₂	—	4	1	no	—
321	C₂	—	8	4	no	—
322	C₂	—	16	1	no	—
323	C₂	C₂×#454×C₁₀₁	1	1	yes	characteristic
324	C₂	C₁₀₁×#3019	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₀₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₀₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₁ ◁ C₂×C₄²×C₁₀₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₀₁×#3548	C₂	semidirect
2. C₁₀₁×#3214	C₂	semidirect
3. C₁₀₁×#3135	C₂	semidirect
4. C₁₀₁×#1662	C₂	semidirect
5. C₁₀₁×#1656	C₂	semidirect
6. C₁₀₁×#1537	C₂	semidirect
7. C₁₀₁×#1530	C₂	semidirect
157. #19061	C₁₀₁	direct (N × H)
163. #3548	C₂×C₁₀₁	semidirect
164. #3214	C₂×C₁₀₁	semidirect
165. #3135	C₂×C₁₀₁	semidirect
166. #1662	C₂×C₁₀₁	semidirect
167. #1656	C₂×C₁₀₁	semidirect
168. #1537	C₂×C₁₀₁	semidirect
169. #1530	C₂×C₁₀₁	semidirect
170. C₁₀₁	#19061	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₀₁×#3548	2
C₁₀₁×#3214	2
C₁₀₁×#3135	2
C₁₀₁×#1662	2
C₁₀₁×#1656	2
C₁₀₁×#1537	2
C₁₀₁×#1530	2
#19061	101

Subgroups — C101×#19061 #38801212

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#19061 #38801212