Subgroups — C₁₀₁×#20324 #38802475

Order 25856.

570 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₁×#3385	C₂	1	2	yes	maximal, complemented
2	C₁₀₁×#3357	C₂	1	2	yes	maximal
3	C₁₀₁×#1713	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₁×#1696	C₂	1	2	yes	maximal, complemented
5	C₁₀₁×#1695	C₂	1	2	yes	maximal, complemented
6	C₁₀₁×#1634	C₂	1	1	yes	maximal, characteristic
7	C₁₀₁×#1633	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₀₁×#1632	C₂	1	2	yes	maximal, complemented
9	C₁₀₁×#1590	C₂	1	2	yes	maximal, complemented
10	Q₈²×C₁₀₁	C₂²	1	1	yes	characteristic
11	#555×C₁₀₁	C₂²	1	1	yes	characteristic
12	#554×C₁₀₁	C₂²	1	2	yes	—
13	#501×C₁₀₁	—	2	2	no	—
14	#500×C₁₀₁	C₂²	1	2	yes	—
15	#500×C₁₀₁	C₂²	1	2	yes	—
16	#494×C₁₀₁	C₂²	1	1	yes	characteristic
17	#494×C₁₀₁	C₂²	1	1	yes	characteristic
18	#485×C₁₀₁	—	2	2	no	—
19	#480×C₁₀₁	—	2	2	no	—
20	#479×C₁₀₁	—	2	2	no	—
21	#477×C₁₀₁	C₂²	1	2	yes	—
22	#477×C₁₀₁	C₂²	1	2	yes	—
23	#476×C₁₀₁	C₂²	1	2	yes	—
24	#476×C₁₀₁	C₂²	1	2	yes	—
25	#474×C₁₀₁	—	2	2	no	—
26	#468×C₁₀₁	—	2	2	no	—
27	#446×C₁₀₁	C₂²	1	2	yes	—
28	#440×C₁₀₁	—	2	2	no	—
29	C₄×SD₁₆×C₁₀₁	C₂²	1	2	yes	—
30	#436×C₁₀₁	C₂²	1	2	yes	—
31	#428×C₁₀₁	—	2	2	no	—
32	#334×C₁₀₁	C₂²	1	2	yes	—
33	#333×C₁₀₁	C₂²	1	2	yes	—
34	#333×C₁₀₁	C₂²	1	1	yes	characteristic
35	#333×C₁₀₁	C₂²	1	1	yes	characteristic
36	Q₈⋊C₈×C₁₀₁	C₂²	1	2	yes	—
37	D₄⋊C₈×C₁₀₁	C₂²	1	2	yes	—
38	C₈²×C₁₀₁	C₂²	1	1	yes	characteristic
39	C₂×Q₁₆×C₁₀₁	—	4	1	no	—
40	C₂×Q₁₆×C₁₀₁	—	4	1	no	—
41	C₂×SD₁₆×C₁₀₁	D₄	1	2	yes	—
42	C₂×M₁₆×C₁₀₁	—	2	2	no	—
43	C₄⋊Q₈×C₁₀₁	—	2	2	no	—
44	C₄⋊Q₈×C₁₀₁	—	4	1	no	—
45	C₄⋊Q₈×C₁₀₁	C₂³	1	1	yes	characteristic
46	C₄⋊Q₈×C₁₀₁	C₂³	1	2	yes	—
47	C₄⋊Q₈×C₁₀₁	C₂³	1	1	yes	characteristic
48	#125×C₁₀₁	—	2	2	no	—
49	#124×C₁₀₁	—	4	1	no	—
50	C₂²⋊Q₈×C₁₀₁	—	2	2	no	—
51	C₂²⋊Q₈×C₁₀₁	—	4	2	no	—
52	#121×C₁₀₁	—	2	2	no	—
53	C₄×Q₈×C₁₀₁	—	2	2	no	—
54	C₄×Q₈×C₁₀₁	C₂³	1	2	yes	—
55	C₄×D₄×C₁₀₁	—	2	2	no	—
56	C₄×D₄×C₁₀₁	C₂³	1	2	yes	—
57	#117×C₁₀₁	—	2	2	no	—
58	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
59	#107×C₁₀₁	—	2	2	no	—
60	#107×C₁₀₁	—	2	2	no	—
61	#107×C₁₀₁	—	2	2	no	—
62	#107×C₁₀₁	—	2	2	no	—
63	#107×C₁₀₁	—	2	2	no	—
64	#107×C₁₀₁	—	2	2	no	—
65	#106×C₁₀₁	—	2	2	no	—
66	#106×C₁₀₁	—	2	2	no	—
67	#106×C₁₀₁	D₄	1	2	yes	—
68	C₄⋊C₈×C₁₀₁	—	2	2	no	—
69	C₄⋊C₈×C₁₀₁	C₂³	1	2	yes	—
70	C₄⋊C₈×C₁₀₁	C₂³	1	2	yes	—
71	#103×C₁₀₁	—	2	2	no	—
72	#103×C₁₀₁	—	2	2	no	—
73	#103×C₁₀₁	D₄	1	2	yes	—
74	#102×C₁₀₁	—	2	2	no	—
75	#102×C₁₀₁	—	2	2	no	—
76	#102×C₁₀₁	D₄	1	2	yes	—
77	C₂²⋊C₈×C₁₀₁	—	2	2	no	—
78	#97×C₁₀₁	—	2	2	no	—
79	C₄×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₁₀₁	C₂³	1	2	yes	—
81	C₂×Q₈×C₁₀₁	—	2	2	no	—
82	C₂×Q₈×C₁₀₁	—	2	1	no	—
83	C₂×Q₈×C₁₀₁	—	2	1	no	—
84	C₂×Q₈×C₁₀₁	—	4	1	no	—
85	C₂×Q₈×C₁₀₁	C₂×D₄	1	2	yes	—
86	C₂×D₄×C₁₀₁	—	4	1	no	—
87	C₂×D₄×C₁₀₁	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₁₀₁	—	2	2	no	—
89	C₂²×C₄×C₁₀₁	—	4	2	no	—
90	Q₁₆×C₁₀₁	—	8	2	no	—
91	Q₁₆×C₁₀₁	—	8	2	no	—
92	SD₁₆×C₁₀₁	—	2	4	no	—
93	M₁₆×C₁₀₁	—	4	4	no	—
94	C₂×C₈×C₁₀₁	—	2	1	no	—
95	C₂×C₈×C₁₀₁	—	2	2	no	—
96	C₂×C₈×C₁₀₁	—	2	2	no	—
97	C₂×C₈×C₁₀₁	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₁₀₁	C₂×D₄	1	2	yes	—
99	C₄⋊C₄×C₁₀₁	—	2	2	no	—
100	C₄⋊C₄×C₁₀₁	—	2	2	no	—
101	C₄⋊C₄×C₁₀₁	—	2	2	no	—
102	C₄⋊C₄×C₁₀₁	—	2	2	no	—
103	C₄⋊C₄×C₁₀₁	—	2	2	no	—
104	C₄⋊C₄×C₁₀₁	—	4	2	no	—
105	C₄⋊C₄×C₁₀₁	—	4	2	no	—
106	C₄⋊C₄×C₁₀₁	—	4	2	no	—
107	C₄⋊C₄×C₁₀₁	C₂×D₄	1	2	yes	—
108	C₄⋊C₄×C₁₀₁	C₂×D₄	1	2	yes	—
109	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
110	C₂²⋊C₄×C₁₀₁	—	4	2	no	—
111	C₂²⋊C₄×C₁₀₁	—	4	2	no	—
112	C₄²×C₁₀₁	—	2	2	no	—
113	C₄²×C₁₀₁	—	4	1	no	—
114	C₄²×C₁₀₁	—	4	1	no	—
115	C₄²×C₁₀₁	C₂⁴	1	1	yes	characteristic
116	C₂³×C₁₀₁	—	2	2	no	—
117	Q₈×C₁₀₁	—	2	2	no	—
118	Q₈×C₁₀₁	—	2	2	no	—
119	Q₈×C₁₀₁	—	8	2	no	—
120	Q₈×C₁₀₁	—	8	1	no	—
121	Q₈×C₁₀₁	—	8	1	no	—
122	Q₈×C₁₀₁	—	16	1	no	—
123	D₄×C₁₀₁	—	2	2	no	—
124	D₄×C₁₀₁	—	2	2	no	—
125	D₄×C₁₀₁	—	16	1	no	—
126	C₂×C₄×C₁₀₁	—	2	2	no	—
127	C₂×C₄×C₁₀₁	—	2	2	no	—
128	C₂×C₄×C₁₀₁	—	2	2	no	—
129	C₂×C₄×C₁₀₁	—	4	2	no	—
130	C₂×C₄×C₁₀₁	—	4	2	no	—
131	C₂×C₄×C₁₀₁	—	4	1	no	—
132	C₂×C₄×C₁₀₁	—	4	1	no	—
133	C₂×C₄×C₁₀₁	—	4	2	no	—
134	C₂×C₄×C₁₀₁	—	16	2	no	—
135	C₂×C₄×C₁₀₁	C₂²×D₄	1	2	yes	—
136	C₂×C₄×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
137	C₈×C₁₀₁	—	2	2	no	—
138	C₈×C₁₀₁	—	2	2	no	—
139	C₈×C₁₀₁	—	4	2	no	—
140	C₈×C₁₀₁	—	4	2	no	—
141	C₈×C₁₀₁	#137	1	4	yes	—
142	C₂²×C₁₀₁	—	4	2	no	—
143	C₂²×C₁₀₁	—	4	2	no	—
144	C₂²×C₁₀₁	—	4	2	no	—
145	C₂²×C₁₀₁	D₄²	1	1	yes	characteristic
146	C₄×C₁₀₁	—	2	1	no	—
147	C₄×C₁₀₁	—	4	2	no	—
148	C₄×C₁₀₁	—	4	2	no	—
149	C₄×C₁₀₁	—	4	2	no	—
150	C₄×C₁₀₁	—	8	2	no	—
151	C₄×C₁₀₁	—	8	1	no	—
152	C₄×C₁₀₁	—	8	1	no	—
153	C₄×C₁₀₁	C₂×#137	1	2	yes	—
154	C₄×C₁₀₁	#578	1	2	yes	—
155	#20324	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂×C₁₀₁	—	8	2	no	—
157	C₂×C₁₀₁	#3276	1	2	yes	—
158	C₂×C₁₀₁	#3292	1	1	yes	characteristic
159	#3385	C₂×C₁₀₁	1	2	yes	complemented
160	#3357	C₂×C₁₀₁	1	2	yes	—
161	#1713	C₂×C₁₀₁	1	1	yes	characteristic, complemented
162	#1696	C₂×C₁₀₁	1	2	yes	complemented
163	#1695	C₂×C₁₀₁	1	2	yes	complemented
164	#1634	C₂×C₁₀₁	1	1	yes	characteristic
165	#1633	C₂×C₁₀₁	1	1	yes	characteristic, complemented
166	#1632	C₂×C₁₀₁	1	2	yes	complemented
167	#1590	C₂×C₁₀₁	1	2	yes	complemented
168	C₁₀₁	#20324	1	1	yes	characteristic, direct product factor, Sylow
169	Q₈²	C₂²×C₁₀₁	1	1	yes	characteristic
170	#555	C₂²×C₁₀₁	1	1	yes	characteristic
171	#554	C₂²×C₁₀₁	1	2	yes	—
172	#501	—	2	2	no	—
173	#500	C₂²×C₁₀₁	1	2	yes	—
174	#500	C₂²×C₁₀₁	1	2	yes	—
175	#494	C₂²×C₁₀₁	1	1	yes	characteristic
176	#494	C₂²×C₁₀₁	1	1	yes	characteristic
177	#485	—	2	2	no	—
178	#480	—	2	2	no	—
179	#479	—	2	2	no	—
180	#477	C₂²×C₁₀₁	1	2	yes	—
181	#477	C₂²×C₁₀₁	1	2	yes	—
182	#476	C₂²×C₁₀₁	1	2	yes	—
183	#476	C₂²×C₁₀₁	1	2	yes	—
184	#474	—	2	2	no	—
185	#468	—	2	2	no	—
186	#446	C₂²×C₁₀₁	1	2	yes	—
187	#440	—	2	2	no	—
188	C₄×SD₁₆	C₂²×C₁₀₁	1	2	yes	—
189	#436	C₂²×C₁₀₁	1	2	yes	—
190	#428	—	2	2	no	—
191	#334	C₂²×C₁₀₁	1	2	yes	—
192	#333	C₂²×C₁₀₁	1	2	yes	—
193	#333	C₂²×C₁₀₁	1	1	yes	characteristic
194	#333	C₂²×C₁₀₁	1	1	yes	characteristic
195	Q₈⋊C₈	C₂²×C₁₀₁	1	2	yes	—
196	D₄⋊C₈	C₂²×C₁₀₁	1	2	yes	—
197	C₈²	C₂²×C₁₀₁	1	1	yes	characteristic
198	C₂×Q₁₆	—	4	1	no	—
199	C₂×Q₁₆	—	4	1	no	—
200	C₂×SD₁₆	D₄×C₁₀₁	1	2	yes	—
201	C₂×M₁₆	—	2	2	no	—
202	C₄⋊Q₈	—	2	2	no	—
203	C₄⋊Q₈	—	4	1	no	—
204	C₄⋊Q₈	C₂³×C₁₀₁	1	2	yes	—
205	C₄⋊Q₈	C₂³×C₁₀₁	1	1	yes	characteristic
206	C₄⋊Q₈	C₂³×C₁₀₁	1	1	yes	characteristic
207	#125	—	2	2	no	—
208	#124	—	4	1	no	—
209	C₂²⋊Q₈	—	2	2	no	—
210	C₂²⋊Q₈	—	4	2	no	—
211	#121	—	2	2	no	—
212	C₄×Q₈	—	2	2	no	—
213	C₄×Q₈	C₂³×C₁₀₁	1	2	yes	—
214	C₄×D₄	—	2	2	no	—
215	C₄×D₄	C₂³×C₁₀₁	1	2	yes	—
216	#117	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	#107	—	2	2	no	—
219	#107	—	2	2	no	—
220	#107	—	2	2	no	—
221	#107	—	2	2	no	—
222	#107	—	2	2	no	—
223	#107	—	2	2	no	—
224	#106	—	2	2	no	—
225	#106	—	2	2	no	—
226	#106	D₄×C₁₀₁	1	2	yes	—
227	C₄⋊C₈	—	2	2	no	—
228	C₄⋊C₈	C₂³×C₁₀₁	1	2	yes	—
229	C₄⋊C₈	C₂³×C₁₀₁	1	2	yes	—
230	#103	—	2	2	no	—
231	#103	—	2	2	no	—
232	#103	D₄×C₁₀₁	1	2	yes	—
233	#102	—	2	2	no	—
234	#102	—	2	2	no	—
235	#102	D₄×C₁₀₁	1	2	yes	—
236	C₂²⋊C₈	—	2	2	no	—
237	#97	—	2	2	no	—
238	C₄×C₈	C₂³×C₁₀₁	1	1	yes	characteristic
239	C₄×C₈	C₂³×C₁₀₁	1	2	yes	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	1	no	—
242	C₂×Q₈	—	2	1	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×Q₈	C₂×D₄×C₁₀₁	1	2	yes	—
245	C₂×D₄	—	4	1	no	—
246	C₂×D₄	C₂×D₄×C₁₀₁	1	2	yes	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	4	2	no	—
249	Q₁₆	—	8	2	no	—
250	Q₁₆	—	8	2	no	—
251	SD₁₆	—	2	4	no	—
252	M₁₆	—	4	4	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	C₂×D₄×C₁₀₁	1	2	yes	—
257	C₂×C₈	C₂×D₄×C₁₀₁	1	2	yes	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	C₂×D₄×C₁₀₁	1	2	yes	—
267	C₄⋊C₄	C₂×D₄×C₁₀₁	1	2	yes	—
268	C₂²⋊C₄	—	2	2	no	—
269	C₂²⋊C₄	—	4	2	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	4	1	no	—
273	C₄²	—	4	1	no	—
274	C₄²	C₂⁴×C₁₀₁	1	1	yes	characteristic
275	C₂³	—	2	2	no	—
276	Q₈	—	2	2	no	—
277	Q₈	—	2	2	no	—
278	Q₈	—	8	2	no	—
279	Q₈	—	8	1	no	—
280	Q₈	—	8	1	no	—
281	Q₈	—	16	1	no	—
282	D₄	—	2	2	no	—
283	D₄	—	2	2	no	—
284	D₄	—	16	1	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	16	2	no	—
294	C₂×C₄	C₂²×D₄×C₁₀₁	1	2	yes	—
295	C₂×C₄	H(5,𝔽₂)×C₁₀₁	1	1	yes	characteristic
296	C₈	—	2	2	no	—
297	C₈	—	2	2	no	—
298	C₈	—	4	2	no	—
299	C₈	—	4	2	no	—
300	C₈	#137×C₁₀₁	1	4	yes	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	—	4	2	no	—
304	C₂²	D₄²×C₁₀₁	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	4	2	no	—
307	C₄	—	4	2	no	—
308	C₄	—	4	2	no	—
309	C₄	—	8	2	no	—
310	C₄	—	8	1	no	—
311	C₄	—	8	1	no	—
312	C₄	C₂×#137×C₁₀₁	1	2	yes	—
313	C₄	#578×C₁₀₁	1	2	yes	—
314	C₂	—	8	2	no	—
315	C₂	C₁₀₁×#3276	1	2	yes	—
316	C₂	C₁₀₁×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₀₁
prime radical	#555×C₁₀₁	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₁ ◁ C₄²×C₁₀₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₀₁×#3385	C₂	semidirect
3. C₁₀₁×#1713	C₂	semidirect
4. C₁₀₁×#1696	C₂	semidirect
5. C₁₀₁×#1695	C₂	semidirect
7. C₁₀₁×#1633	C₂	semidirect
8. C₁₀₁×#1632	C₂	semidirect
9. C₁₀₁×#1590	C₂	semidirect
155. #20324	C₁₀₁	direct (N × H)
159. #3385	C₂×C₁₀₁	semidirect
161. #1713	C₂×C₁₀₁	semidirect
162. #1696	C₂×C₁₀₁	semidirect
163. #1695	C₂×C₁₀₁	semidirect
165. #1633	C₂×C₁₀₁	semidirect
166. #1632	C₂×C₁₀₁	semidirect
167. #1590	C₂×C₁₀₁	semidirect
168. C₁₀₁	#20324	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×#3385	2
C₁₀₁×#3357	2
C₁₀₁×#1713	2
C₁₀₁×#1696	2
C₁₀₁×#1695	2
C₁₀₁×#1634	2
C₁₀₁×#1633	2
C₁₀₁×#1632	2
C₁₀₁×#1590	2
#20324	101

Subgroups — C101×#20324 #38802475

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#20324 #38802475