Subgroups — C₁₀₁×#21295 #38803446

Order 25856.

722 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#135×C₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₁₀₁×#2923	C₂	1	2	yes	maximal, complemented
3	C₁₀₁×#1975	C₂	1	2	yes	maximal
4	C₁₀₁×#1946	C₂	1	2	yes	maximal, complemented
5	C₁₀₁×#1937	C₂	1	4	yes	maximal, complemented
6	C₄×#108×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₀₁×#1759	C₂	1	2	yes	maximal, complemented
8	C₁₀₁×#1726	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₁₀₁	C₂²	1	1	yes	characteristic
10	C₄×#41×C₁₀₁	C₂²	1	2	yes	—
11	C₄×Q₁₆×C₁₀₁	C₄	1	2	yes	complemented
12	C₄×SD₁₆×C₁₀₁	C₄	1	4	yes	complemented
13	C₄×D₈×C₁₀₁	C₄	1	2	yes	complemented
14	#433×C₁₀₁	—	2	4	no	—
15	#432×C₁₀₁	—	2	4	no	—
16	C₂×#108×C₁₀₁	C₂²	1	2	yes	—
17	#427×C₁₀₁	C₂²	1	1	yes	characteristic
18	#424×C₁₀₁	C₂²	1	4	yes	—
19	C₂×#104×C₁₀₁	C₂²	1	4	yes	—
20	#416×C₁₀₁	C₂²	1	2	yes	—
21	#408×C₁₀₁	C₂²	1	4	yes	—
22	C₄×M₁₆×C₁₀₁	C₂²	1	2	yes	—
23	C₂×#97×C₁₀₁	C₂²	1	2	yes	—
24	C₂×#97×C₁₀₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₁₀₁	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
27	#376×C₁₀₁	C₂²	1	1	yes	characteristic
28	#341×C₁₀₁	C₂²	1	2	yes	—
29	#339×C₁₀₁	C₂²	1	4	yes	—
30	C₂×#41×C₁₀₁	C₂³	1	2	yes	—
31	#135×C₁₀₁	—	2	4	no	—
32	C₂×Q₁₆×C₁₀₁	C₂×C₄	1	1	yes	characteristic
33	C₂×SD₁₆×C₁₀₁	C₂×C₄	1	2	yes	—
34	C₂×D₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
35	C₂×M₁₆×C₁₀₁	C₂³	1	4	yes	—
36	C₂²×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₁₀₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
39	C₄×Q₈×C₁₀₁	C₂×C₄	1	4	yes	—
40	C₄×D₄×C₁₀₁	—	2	4	no	—
41	C₄×D₄×C₁₀₁	C₂×C₄	1	4	yes	—
42	#117×C₁₀₁	—	2	2	no	—
43	#117×C₁₀₁	C₂³	1	2	yes	—
44	C₂×C₄²×C₁₀₁	—	2	2	no	—
45	C₂×C₄²×C₁₀₁	C₂³	1	2	yes	—
46	C₂×C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
47	#108×C₁₀₁	—	2	4	no	—
48	#107×C₁₀₁	C₂×C₄	1	2	yes	—
49	#106×C₁₀₁	C₂×C₄	1	2	yes	—
50	C₄⋊C₈×C₁₀₁	—	2	8	no	—
51	#104×C₁₀₁	—	2	8	no	—
52	#103×C₁₀₁	C₂×C₄	1	4	yes	—
53	#102×C₁₀₁	C₂×C₄	1	4	yes	—
54	C₂²⋊C₈×C₁₀₁	—	2	8	no	—
55	#97×C₁₀₁	—	2	4	no	—
56	#97×C₁₀₁	—	2	2	no	—
57	#97×C₁₀₁	D₄	1	2	yes	—
58	#97×C₁₀₁	D₄	1	2	yes	—
59	C₄×C₈×C₁₀₁	—	2	2	no	—
60	C₄×C₈×C₁₀₁	—	2	4	no	—
61	C₄×C₈×C₁₀₁	C₂×C₄	1	2	yes	—
62	C₄×C₈×C₁₀₁	C₂×C₄	1	2	yes	—
63	#95×C₁₀₁	—	2	2	no	—
64	#41×C₁₀₁	—	2	4	no	—
65	#41×C₁₀₁	—	2	4	no	—
66	C₂×Q₈×C₁₀₁	C₂²×C₄	1	2	yes	—
67	C₂×D₄×C₁₀₁	—	2	2	no	—
68	C₂×D₄×C₁₀₁	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₁₀₁	—	2	2	no	—
70	C₂²×C₄×C₁₀₁	—	2	2	no	—
71	C₂²×C₄×C₁₀₁	—	2	4	no	—
72	C₂²×C₄×C₁₀₁	#41	1	2	yes	—
73	C₂²×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
74	Q₁₆×C₁₀₁	—	2	2	no	—
75	SD₁₆×C₁₀₁	—	2	4	no	—
76	D₈×C₁₀₁	—	2	2	no	—
77	M₁₆×C₁₀₁	—	2	4	no	—
78	M₁₆×C₁₀₁	—	2	4	no	—
79	C₂×C₈×C₁₀₁	—	2	2	no	—
80	C₂×C₈×C₁₀₁	—	2	2	no	—
81	C₂×C₈×C₁₀₁	—	2	2	no	—
82	C₂×C₈×C₁₀₁	—	2	4	no	—
83	C₂×C₈×C₁₀₁	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₀₁	C₂×D₄	1	4	yes	—
87	C₂×C₈×C₁₀₁	#41	1	4	yes	—
88	C₄⋊C₄×C₁₀₁	—	2	4	no	—
89	C₄⋊C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
90	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
91	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
92	C₄²×C₁₀₁	—	2	4	no	—
93	C₄²×C₁₀₁	—	2	2	no	—
94	C₄²×C₁₀₁	—	2	2	no	—
95	C₄²×C₁₀₁	C₂²×C₄	1	2	yes	—
96	C₄²×C₁₀₁	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₁₀₁	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₁₀₁	#41	1	4	yes	—
99	C₂³×C₁₀₁	—	2	2	no	—
100	C₂³×C₁₀₁	C₂×#41	1	1	yes	characteristic
101	Q₈×C₁₀₁	—	2	2	no	—
102	Q₈×C₁₀₁	—	2	2	no	—
103	D₄×C₁₀₁	—	2	2	no	—
104	D₄×C₁₀₁	—	2	2	no	—
105	D₄×C₁₀₁	—	4	4	no	—
106	C₂×C₄×C₁₀₁	—	2	4	no	—
107	C₂×C₄×C₁₀₁	—	2	2	no	—
108	C₂×C₄×C₁₀₁	—	2	4	no	—
109	C₂×C₄×C₁₀₁	—	4	2	no	—
110	C₂×C₄×C₁₀₁	—	4	4	no	—
111	C₂×C₄×C₁₀₁	—	4	4	no	—
112	C₂×C₄×C₁₀₁	—	8	4	no	—
113	C₂×C₄×C₁₀₁	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₁₀₁	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₀₁	C₂×#41	1	2	yes	—
118	C₂×C₄×C₁₀₁	C₂×#41	1	2	yes	—
119	C₈×C₁₀₁	—	2	2	no	—
120	C₈×C₁₀₁	—	2	4	no	—
121	C₈×C₁₀₁	—	4	4	no	—
122	C₈×C₁₀₁	#117	1	2	yes	—
123	C₈×C₁₀₁	#117	1	2	yes	—
124	C₂²×C₁₀₁	—	2	2	no	—
125	C₂²×C₁₀₁	—	4	4	no	—
126	C₂²×C₁₀₁	—	4	2	no	—
127	C₂²×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
128	C₂²×C₁₀₁	#529	1	2	yes	—
129	C₄×C₁₀₁	—	2	2	no	—
130	C₄×C₁₀₁	—	2	2	no	—
131	C₄×C₁₀₁	—	4	4	no	—
132	C₄×C₁₀₁	—	4	4	no	—
133	C₄×C₁₀₁	—	4	2	no	—
134	C₄×C₁₀₁	C₂×#117	1	1	yes	characteristic
135	C₄×C₁₀₁	C₄×#41	1	1	yes	characteristic
136	C₄×C₁₀₁	#532	1	2	yes	—
137	#21295	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₂×C₁₀₁	—	2	2	no	—
139	C₂×C₁₀₁	—	8	2	no	—
140	C₂×C₁₀₁	#2293	1	1	yes	characteristic
141	C₂×C₁₀₁	#2957	1	2	yes	—
142	C₄×#135	C₂×C₁₀₁	1	1	yes	characteristic
143	#2923	C₂×C₁₀₁	1	2	yes	complemented
144	#1975	C₂×C₁₀₁	1	2	yes	—
145	#1946	C₂×C₁₀₁	1	2	yes	complemented
146	#1937	C₂×C₁₀₁	1	4	yes	complemented
147	C₄×#108	C₂×C₁₀₁	1	1	yes	characteristic, complemented
148	#1759	C₂×C₁₀₁	1	2	yes	complemented
149	#1726	C₂×C₁₀₁	1	1	yes	characteristic, complemented
150	C₁₀₁	#21295	1	1	yes	characteristic, direct product factor, Sylow
151	C₂×#135	C₂²×C₁₀₁	1	1	yes	characteristic
152	C₄×#41	C₂²×C₁₀₁	1	2	yes	—
153	C₄×Q₁₆	C₄×C₁₀₁	1	2	yes	complemented
154	C₄×SD₁₆	C₄×C₁₀₁	1	4	yes	complemented
155	C₄×D₈	C₄×C₁₀₁	1	2	yes	complemented
156	#433	—	2	4	no	—
157	#432	—	2	4	no	—
158	C₂×#108	C₂²×C₁₀₁	1	2	yes	—
159	#427	C₂²×C₁₀₁	1	1	yes	characteristic
160	#424	C₂²×C₁₀₁	1	4	yes	—
161	C₂×#104	C₂²×C₁₀₁	1	4	yes	—
162	#416	C₂²×C₁₀₁	1	2	yes	—
163	#408	C₂²×C₁₀₁	1	4	yes	—
164	C₄×M₁₆	C₂²×C₁₀₁	1	2	yes	—
165	C₂×#97	C₂²×C₁₀₁	1	2	yes	—
166	C₂×#97	C₂²×C₁₀₁	1	1	yes	characteristic
167	C₂×C₄×C₈	C₂²×C₁₀₁	1	2	yes	—
168	C₂×C₄×C₈	C₂²×C₁₀₁	1	1	yes	characteristic
169	#376	C₂²×C₁₀₁	1	1	yes	characteristic
170	#341	C₂²×C₁₀₁	1	2	yes	—
171	#339	C₂²×C₁₀₁	1	4	yes	—
172	C₂×#41	C₂³×C₁₀₁	1	2	yes	—
173	#135	—	2	4	no	—
174	C₂×Q₁₆	C₂×C₄×C₁₀₁	1	1	yes	characteristic
175	C₂×SD₁₆	C₂×C₄×C₁₀₁	1	2	yes	—
176	C₂×D₈	C₂×C₄×C₁₀₁	1	1	yes	characteristic
177	C₂×M₁₆	C₂³×C₁₀₁	1	4	yes	—
178	C₂²×C₈	C₂³×C₁₀₁	1	1	yes	characteristic
179	C₂²×C₈	C₂³×C₁₀₁	1	2	yes	—
180	C₂²×C₈	C₂³×C₁₀₁	1	1	yes	characteristic
181	C₄×Q₈	C₂×C₄×C₁₀₁	1	4	yes	—
182	C₄×D₄	—	2	4	no	—
183	C₄×D₄	C₂×C₄×C₁₀₁	1	4	yes	—
184	#117	—	2	2	no	—
185	#117	C₂³×C₁₀₁	1	2	yes	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	C₂³×C₁₀₁	1	2	yes	—
188	C₂×C₄²	C₂³×C₁₀₁	1	1	yes	characteristic
189	#108	—	2	4	no	—
190	#107	C₂×C₄×C₁₀₁	1	2	yes	—
191	#106	C₂×C₄×C₁₀₁	1	2	yes	—
192	C₄⋊C₈	—	2	8	no	—
193	#104	—	2	8	no	—
194	#103	C₂×C₄×C₁₀₁	1	4	yes	—
195	#102	C₂×C₄×C₁₀₁	1	4	yes	—
196	C₂²⋊C₈	—	2	8	no	—
197	#97	—	2	4	no	—
198	#97	—	2	2	no	—
199	#97	D₄×C₁₀₁	1	2	yes	—
200	#97	D₄×C₁₀₁	1	2	yes	—
201	C₄×C₈	—	2	2	no	—
202	C₄×C₈	—	2	4	no	—
203	C₄×C₈	C₂×C₄×C₁₀₁	1	2	yes	—
204	C₄×C₈	C₂×C₄×C₁₀₁	1	2	yes	—
205	#95	—	2	2	no	—
206	#41	—	2	4	no	—
207	#41	—	2	4	no	—
208	C₂×Q₈	C₂²×C₄×C₁₀₁	1	2	yes	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	C₂²×C₄×C₁₀₁	1	2	yes	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	#41×C₁₀₁	1	2	yes	—
215	C₂²×C₄	C₂⁴×C₁₀₁	1	1	yes	characteristic
216	Q₁₆	—	2	2	no	—
217	SD₁₆	—	2	4	no	—
218	D₈	—	2	2	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	2	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	C₂²×C₄×C₁₀₁	1	2	yes	—
226	C₂×C₈	C₂²×C₄×C₁₀₁	1	1	yes	characteristic
227	C₂×C₈	C₂²×C₄×C₁₀₁	1	1	yes	characteristic
228	C₂×C₈	C₂×D₄×C₁₀₁	1	4	yes	—
229	C₂×C₈	#41×C₁₀₁	1	4	yes	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	C₂²×C₄×C₁₀₁	1	4	yes	—
232	C₂²⋊C₄	—	2	2	no	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₄²	—	2	4	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	2	no	—
237	C₄²	C₂²×C₄×C₁₀₁	1	2	yes	—
238	C₄²	C₂×D₄×C₁₀₁	1	1	yes	characteristic
239	C₄²	C₂×D₄×C₁₀₁	1	1	yes	characteristic
240	C₄²	#41×C₁₀₁	1	4	yes	—
241	C₂³	—	2	2	no	—
242	C₂³	C₂×#41×C₁₀₁	1	1	yes	characteristic
243	Q₈	—	2	2	no	—
244	Q₈	—	2	2	no	—
245	D₄	—	2	2	no	—
246	D₄	—	2	2	no	—
247	D₄	—	4	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	8	4	no	—
255	C₂×C₄	C₄×D₄×C₁₀₁	1	2	yes	—
256	C₂×C₄	C₄×D₄×C₁₀₁	1	2	yes	—
257	C₂×C₄	C₂³×C₄×C₁₀₁	1	1	yes	characteristic
258	C₂×C₄	C₂²×D₄×C₁₀₁	1	1	yes	characteristic
259	C₂×C₄	C₂×#41×C₁₀₁	1	2	yes	—
260	C₂×C₄	C₂×#41×C₁₀₁	1	2	yes	—
261	C₈	—	2	2	no	—
262	C₈	—	2	4	no	—
263	C₈	—	4	4	no	—
264	C₈	#117×C₁₀₁	1	2	yes	—
265	C₈	#117×C₁₀₁	1	2	yes	—
266	C₂²	—	2	2	no	—
267	C₂²	—	4	4	no	—
268	C₂²	—	4	2	no	—
269	C₂²	C₂×C₄×D₄×C₁₀₁	1	1	yes	characteristic
270	C₂²	#529×C₁₀₁	1	2	yes	—
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	4	4	no	—
274	C₄	—	4	4	no	—
275	C₄	—	4	2	no	—
276	C₄	C₂×#117×C₁₀₁	1	1	yes	characteristic
277	C₄	C₄×#41×C₁₀₁	1	1	yes	characteristic
278	C₄	#532×C₁₀₁	1	2	yes	—
279	C₂	—	2	2	no	—
280	C₂	—	8	2	no	—
281	C₂	C₁₀₁×#2293	1	1	yes	characteristic
282	C₂	C₁₀₁×#2957	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₀₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₀₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₁
socle	C₂²×C₁₀₁	C₂×C₄×D₄
prime radical	C₂×#135×C₁₀₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₁ ◁ C₂×#97×C₁₀₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₀₁×#2923	C₂	semidirect
4. C₁₀₁×#1946	C₂	semidirect
5. C₁₀₁×#1937	C₂	semidirect
6. C₄×#108×C₁₀₁	C₂	semidirect
7. C₁₀₁×#1759	C₂	semidirect
8. C₁₀₁×#1726	C₂	semidirect
11. C₄×Q₁₆×C₁₀₁	C₄	semidirect
12. C₄×SD₁₆×C₁₀₁	C₄	semidirect
13. C₄×D₈×C₁₀₁	C₄	semidirect
137. #21295	C₁₀₁	direct (N × H)
143. #2923	C₂×C₁₀₁	semidirect
145. #1946	C₂×C₁₀₁	semidirect
146. #1937	C₂×C₁₀₁	semidirect
147. C₄×#108	C₂×C₁₀₁	semidirect
148. #1759	C₂×C₁₀₁	semidirect
149. #1726	C₂×C₁₀₁	semidirect
150. C₁₀₁	#21295	direct (N × H)
153. C₄×Q₁₆	C₄×C₁₀₁	semidirect
154. C₄×SD₁₆	C₄×C₁₀₁	semidirect
155. C₄×D₈	C₄×C₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×#135×C₁₀₁	2
C₁₀₁×#2923	2
C₁₀₁×#1975	2
C₁₀₁×#1946	2
C₁₀₁×#1937	2
C₄×#108×C₁₀₁	2
C₁₀₁×#1759	2
C₁₀₁×#1726	2
#21295	101

Subgroups — C101×#21295 #38803446

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#21295 #38803446