Subgroups — C₁₀₁×#21155 #38803306

Order 25856.

982 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#427×C₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×#106×C₁₀₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#405×C₁₀₁	C₂	1	1	yes	maximal, characteristic
4	C₁₀₁×#2317	C₂	1	2	yes	maximal
5	C₁₀₁×#1943	C₂	1	4	yes	maximal, complemented
6	C₁₀₁×#1941	C₂	1	4	yes	maximal, complemented
7	C₁₀₁×#1736	C₂	1	2	yes	maximal
8	C₂²×M₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
10	C₂×#117×C₁₀₁	C₂²	1	2	yes	—
11	C₂×#117×C₁₀₁	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
13	#427×C₁₀₁	C₄	1	8	yes	complemented
14	C₂×#107×C₁₀₁	C₂²	1	2	yes	—
15	C₂×#106×C₁₀₁	—	2	4	no	—
16	C₂×#106×C₁₀₁	C₂²	1	2	yes	—
17	C₂×#106×C₁₀₁	C₂²	1	2	yes	—
18	C₂×#106×C₁₀₁	C₂²	1	2	yes	—
19	#405×C₁₀₁	—	2	4	no	—
20	C₂×#97×C₁₀₁	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₀₁	C₂²	1	2	yes	—
22	#389×C₁₀₁	C₂²	1	8	yes	—
23	#385×C₁₀₁	—	2	4	no	—
24	#384×C₁₀₁	—	2	2	no	—
25	#382×C₁₀₁	—	2	2	no	—
26	#340×C₁₀₁	C₂²	1	8	yes	—
27	C₂³×C₄×C₁₀₁	—	2	2	no	—
28	C₂³×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
29	C₂×M₁₆×C₁₀₁	—	2	2	no	—
30	C₂×M₁₆×C₁₀₁	—	2	2	no	—
31	C₂×M₁₆×C₁₀₁	Q₈	1	4	yes	—
32	C₂²×C₈×C₁₀₁	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₁₀₁	D₄	1	4	yes	—
34	C₂²×C₈×C₁₀₁	Q₈	1	4	yes	—
35	C₂²×C₈×C₁₀₁	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
38	#117×C₁₀₁	—	2	2	no	—
39	#117×C₁₀₁	—	2	4	no	—
40	#117×C₁₀₁	—	2	2	no	—
41	#117×C₁₀₁	C₂×C₄	1	8	yes	—
42	C₂×C₄⋊C₄×C₁₀₁	—	2	4	no	—
43	C₂×C₄⋊C₄×C₁₀₁	—	2	4	no	—
44	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
45	C₂×C₄⋊C₄×C₁₀₁	—	2	8	no	—
46	C₂×C₄⋊C₄×C₁₀₁	D₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
48	C₂×C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
49	C₂×C₂²⋊C₄×C₁₀₁	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₁₀₁	D₄	1	2	yes	—
51	C₂×C₄²×C₁₀₁	—	2	2	no	—
52	C₂×C₄²×C₁₀₁	C₂³	1	2	yes	—
53	#107×C₁₀₁	C₂×C₄	1	8	yes	complemented
54	#106×C₁₀₁	—	2	4	no	—
55	#106×C₁₀₁	—	2	4	no	—
56	#106×C₁₀₁	C₂×C₄	1	8	yes	complemented
57	#97×C₁₀₁	—	2	4	no	—
58	C₄×C₈×C₁₀₁	—	2	4	no	—
59	#95×C₁₀₁	—	2	8	no	—
60	C₂⁴×C₁₀₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₀₁	—	2	4	no	—
62	C₂²×C₄×C₁₀₁	—	2	4	no	—
63	C₂²×C₄×C₁₀₁	—	4	4	no	—
64	C₂²×C₄×C₁₀₁	—	8	4	no	—
65	C₂²×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
67	C₂²×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₁₀₁	#41	1	2	yes	—
72	C₂²×C₄×C₁₀₁	#41	1	2	yes	—
73	C₂²×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
74	M₁₆×C₁₀₁	—	2	8	no	—
75	C₂×C₈×C₁₀₁	—	2	2	no	—
76	C₂×C₈×C₁₀₁	—	2	2	no	—
77	C₂×C₈×C₁₀₁	C₄⋊C₄	1	16	yes	complemented
78	C₂×C₈×C₁₀₁	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₁₀₁	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₁₀₁	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₁₀₁	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₁₀₁	C₂×Q₈	1	4	yes	—
83	C₂×C₈×C₁₀₁	C₂×Q₈	1	2	yes	—
84	C₂×C₈×C₁₀₁	C₂×Q₈	1	2	yes	—
85	C₄⋊C₄×C₁₀₁	—	2	4	no	—
86	C₄⋊C₄×C₁₀₁	—	2	4	no	—
87	C₄⋊C₄×C₁₀₁	—	2	4	no	—
88	C₄⋊C₄×C₁₀₁	—	2	4	no	—
89	C₄⋊C₄×C₁₀₁	—	4	4	no	—
90	C₄⋊C₄×C₁₀₁	—	8	8	no	—
91	C₄⋊C₄×C₁₀₁	C₂²×C₄	1	8	yes	—
92	C₂²⋊C₄×C₁₀₁	—	2	8	no	—
93	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
94	C₄²×C₁₀₁	—	2	8	no	—
95	C₄²×C₁₀₁	—	2	2	no	—
96	C₄²×C₁₀₁	—	2	2	no	—
97	C₂³×C₁₀₁	—	2	4	no	—
98	C₂³×C₁₀₁	C₄×D₄	1	2	yes	—
99	C₂³×C₁₀₁	#121	1	2	yes	—
100	C₂³×C₁₀₁	C₄⋊Q₈	1	2	yes	—
101	C₂³×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₀₁	—	2	8	no	—
103	C₂×C₄×C₁₀₁	—	2	4	no	—
104	C₂×C₄×C₁₀₁	—	2	8	no	—
105	C₂×C₄×C₁₀₁	—	4	4	no	—
106	C₂×C₄×C₁₀₁	—	4	4	no	—
107	C₂×C₄×C₁₀₁	—	4	4	no	—
108	C₂×C₄×C₁₀₁	—	4	8	no	—
109	C₂×C₄×C₁₀₁	—	8	8	no	—
110	C₂×C₄×C₁₀₁	C₂×C₄⋊C₄	1	4	yes	—
111	C₂×C₄×C₁₀₁	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₁₀₁	C₄×Q₈	1	4	yes	—
113	C₂×C₄×C₁₀₁	#121	1	2	yes	—
114	C₂×C₄×C₁₀₁	C₂²⋊Q₈	1	4	yes	—
115	C₂×C₄×C₁₀₁	#125	1	4	yes	—
116	C₂×C₄×C₁₀₁	C₄⋊Q₈	1	2	yes	—
117	C₂×C₄×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₁₀₁	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₁₀₁	C₂×#41	1	1	yes	characteristic
123	C₈×C₁₀₁	—	2	4	no	—
124	C₈×C₁₀₁	C₂×C₄⋊C₄	1	4	yes	—
125	C₈×C₁₀₁	C₂×C₄⋊C₄	1	4	yes	—
126	C₂²×C₁₀₁	—	2	4	no	—
127	C₂²×C₁₀₁	—	2	4	no	—
128	C₂²×C₁₀₁	—	2	4	no	—
129	C₂²×C₁₀₁	#389	1	4	yes	—
130	C₂²×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
131	C₂²×C₁₀₁	C₂×#121	1	1	yes	characteristic
132	C₂²×C₁₀₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
133	C₂²×C₁₀₁	#577	1	4	yes	—
134	C₄×C₁₀₁	—	4	8	no	—
135	C₄×C₁₀₁	—	4	4	no	—
136	C₄×C₁₀₁	—	8	4	no	—
137	C₄×C₁₀₁	#389	1	4	yes	—
138	C₄×C₁₀₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
139	C₄×C₁₀₁	C₂×C₄×Q₈	1	1	yes	characteristic
140	C₄×C₁₀₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
141	C₄×C₁₀₁	C₂×#125	1	1	yes	characteristic
142	#21155	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂×C₁₀₁	—	2	4	no	—
144	C₂×C₁₀₁	C₂×#389	1	1	yes	characteristic
145	C₂×C₁₀₁	#2952	1	2	yes	—
146	C₂×C₁₀₁	#3057	1	2	yes	—
147	C₂×C₁₀₁	#3163	1	2	yes	—
148	C₂×#427	C₂×C₁₀₁	1	1	yes	characteristic
149	C₂²×#106	C₂×C₁₀₁	1	1	yes	characteristic
150	C₂×#405	C₂×C₁₀₁	1	1	yes	characteristic
151	#2317	C₂×C₁₀₁	1	2	yes	—
152	#1943	C₂×C₁₀₁	1	4	yes	complemented
153	#1941	C₂×C₁₀₁	1	4	yes	complemented
154	#1736	C₂×C₁₀₁	1	2	yes	—
155	C₁₀₁	#21155	1	1	yes	characteristic, direct product factor, Sylow
156	C₂²×M₁₆	C₂²×C₁₀₁	1	1	yes	characteristic
157	C₂³×C₈	C₂²×C₁₀₁	1	1	yes	characteristic
158	C₂×#117	C₂²×C₁₀₁	1	2	yes	—
159	C₂×#117	C₂²×C₁₀₁	1	1	yes	characteristic
160	C₂²×C₄⋊C₄	C₂²×C₁₀₁	1	2	yes	—
161	#427	C₄×C₁₀₁	1	8	yes	complemented
162	C₂×#107	C₂²×C₁₀₁	1	2	yes	—
163	C₂×#106	—	2	4	no	—
164	C₂×#106	C₂²×C₁₀₁	1	2	yes	—
165	C₂×#106	C₂²×C₁₀₁	1	2	yes	—
166	C₂×#106	C₂²×C₁₀₁	1	2	yes	—
167	#405	—	2	4	no	—
168	C₂×#97	C₂²×C₁₀₁	1	2	yes	—
169	C₂×C₄×C₈	C₂²×C₁₀₁	1	2	yes	—
170	#389	C₂²×C₁₀₁	1	8	yes	—
171	#385	—	2	4	no	—
172	#384	—	2	2	no	—
173	#382	—	2	2	no	—
174	#340	C₂²×C₁₀₁	1	8	yes	—
175	C₂³×C₄	—	2	2	no	—
176	C₂³×C₄	C₂³×C₁₀₁	1	1	yes	characteristic
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	Q₈×C₁₀₁	1	4	yes	—
180	C₂²×C₈	C₂×C₄×C₁₀₁	1	4	yes	—
181	C₂²×C₈	D₄×C₁₀₁	1	4	yes	—
182	C₂²×C₈	Q₈×C₁₀₁	1	4	yes	—
183	C₂²×C₈	C₂³×C₁₀₁	1	2	yes	—
184	C₂²×C₈	C₂³×C₁₀₁	1	1	yes	characteristic
185	C₂²×C₈	C₂³×C₁₀₁	1	1	yes	characteristic
186	#117	—	2	2	no	—
187	#117	—	2	4	no	—
188	#117	—	2	2	no	—
189	#117	C₂×C₄×C₁₀₁	1	8	yes	—
190	C₂×C₄⋊C₄	—	2	4	no	—
191	C₂×C₄⋊C₄	—	2	4	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	8	no	—
194	C₂×C₄⋊C₄	D₄×C₁₀₁	1	2	yes	—
195	C₂×C₄⋊C₄	C₂³×C₁₀₁	1	4	yes	—
196	C₂×C₄⋊C₄	C₂³×C₁₀₁	1	4	yes	—
197	C₂×C₂²⋊C₄	—	2	2	no	—
198	C₂×C₂²⋊C₄	D₄×C₁₀₁	1	2	yes	—
199	C₂×C₄²	—	2	2	no	—
200	C₂×C₄²	C₂³×C₁₀₁	1	2	yes	—
201	#107	C₂×C₄×C₁₀₁	1	8	yes	complemented
202	#106	—	2	4	no	—
203	#106	—	2	4	no	—
204	#106	C₂×C₄×C₁₀₁	1	8	yes	complemented
205	#97	—	2	4	no	—
206	C₄×C₈	—	2	4	no	—
207	#95	—	2	8	no	—
208	C₂⁴	C₂×D₄×C₁₀₁	1	1	yes	characteristic
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	4	4	no	—
212	C₂²×C₄	—	8	4	no	—
213	C₂²×C₄	C₂²×C₄×C₁₀₁	1	2	yes	—
214	C₂²×C₄	C₂×D₄×C₁₀₁	1	4	yes	—
215	C₂²×C₄	C₂×D₄×C₁₀₁	1	2	yes	—
216	C₂²×C₄	C₂×D₄×C₁₀₁	1	1	yes	characteristic
217	C₂²×C₄	C₂×Q₈×C₁₀₁	1	2	yes	—
218	C₂²×C₄	C₂×Q₈×C₁₀₁	1	2	yes	—
219	C₂²×C₄	#41×C₁₀₁	1	2	yes	—
220	C₂²×C₄	#41×C₁₀₁	1	2	yes	—
221	C₂²×C₄	C₂⁴×C₁₀₁	1	1	yes	characteristic
222	M₁₆	—	2	8	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	C₄⋊C₄×C₁₀₁	1	16	yes	complemented
226	C₂×C₈	C₂²×C₄×C₁₀₁	1	2	yes	—
227	C₂×C₈	C₂²×C₄×C₁₀₁	1	2	yes	—
228	C₂×C₈	C₂×D₄×C₁₀₁	1	2	yes	—
229	C₂×C₈	C₂×D₄×C₁₀₁	1	2	yes	—
230	C₂×C₈	C₂×Q₈×C₁₀₁	1	4	yes	—
231	C₂×C₈	C₂×Q₈×C₁₀₁	1	2	yes	—
232	C₂×C₈	C₂×Q₈×C₁₀₁	1	2	yes	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	4	4	no	—
238	C₄⋊C₄	—	8	8	no	—
239	C₄⋊C₄	C₂²×C₄×C₁₀₁	1	8	yes	—
240	C₂²⋊C₄	—	2	8	no	—
241	C₂²⋊C₄	—	2	4	no	—
242	C₄²	—	2	8	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₂³	—	2	4	no	—
246	C₂³	C₄×D₄×C₁₀₁	1	2	yes	—
247	C₂³	#121×C₁₀₁	1	2	yes	—
248	C₂³	C₄⋊Q₈×C₁₀₁	1	2	yes	—
249	C₂³	C₂²×D₄×C₁₀₁	1	1	yes	characteristic
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	8	no	—
257	C₂×C₄	—	8	8	no	—
258	C₂×C₄	C₂×C₄⋊C₄×C₁₀₁	1	4	yes	—
259	C₂×C₄	C₄×D₄×C₁₀₁	1	2	yes	—
260	C₂×C₄	C₄×Q₈×C₁₀₁	1	4	yes	—
261	C₂×C₄	#121×C₁₀₁	1	2	yes	—
262	C₂×C₄	C₂²⋊Q₈×C₁₀₁	1	4	yes	—
263	C₂×C₄	#125×C₁₀₁	1	4	yes	—
264	C₂×C₄	C₄⋊Q₈×C₁₀₁	1	2	yes	—
265	C₂×C₄	C₂³×C₄×C₁₀₁	1	1	yes	characteristic
266	C₂×C₄	C₂²×D₄×C₁₀₁	1	1	yes	characteristic
267	C₂×C₄	C₂²×Q₈×C₁₀₁	1	1	yes	characteristic
268	C₂×C₄	C₂²×Q₈×C₁₀₁	1	1	yes	characteristic
269	C₂×C₄	C₂×#41×C₁₀₁	1	1	yes	characteristic
270	C₂×C₄	C₂×#41×C₁₀₁	1	1	yes	characteristic
271	C₈	—	2	4	no	—
272	C₈	C₂×C₄⋊C₄×C₁₀₁	1	4	yes	—
273	C₈	C₂×C₄⋊C₄×C₁₀₁	1	4	yes	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	4	no	—
276	C₂²	—	2	4	no	—
277	C₂²	#389×C₁₀₁	1	4	yes	—
278	C₂²	C₂×C₄×D₄×C₁₀₁	1	1	yes	characteristic
279	C₂²	C₂×#121×C₁₀₁	1	1	yes	characteristic
280	C₂²	C₂×C₄⋊Q₈×C₁₀₁	1	1	yes	characteristic
281	C₂²	#577×C₁₀₁	1	4	yes	—
282	C₄	—	4	8	no	—
283	C₄	—	4	4	no	—
284	C₄	—	8	4	no	—
285	C₄	#389×C₁₀₁	1	4	yes	—
286	C₄	C₂²×C₄⋊C₄×C₁₀₁	1	1	yes	characteristic
287	C₄	C₂×C₄×Q₈×C₁₀₁	1	1	yes	characteristic
288	C₄	C₂×C₂²⋊Q₈×C₁₀₁	1	1	yes	characteristic
289	C₄	C₂×#125×C₁₀₁	1	1	yes	characteristic
290	C₂	—	2	4	no	—
291	C₂	C₂×#389×C₁₀₁	1	1	yes	characteristic
292	C₂	C₁₀₁×#2952	1	2	yes	—
293	C₂	C₁₀₁×#3057	1	2	yes	—
294	C₂	C₁₀₁×#3163	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₀₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₀₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₁
prime radical	C₂⁴×C₁₀₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₁ ◁ C₂×#117×C₁₀₁ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₁₀₁×#1943	C₂	semidirect
6. C₁₀₁×#1941	C₂	semidirect
13. #427×C₁₀₁	C₄	semidirect
53. #107×C₁₀₁	C₂×C₄	semidirect
56. #106×C₁₀₁	C₂×C₄	semidirect
77. C₂×C₈×C₁₀₁	C₄⋊C₄	semidirect
142. #21155	C₁₀₁	direct (N × H)
152. #1943	C₂×C₁₀₁	semidirect
153. #1941	C₂×C₁₀₁	semidirect
155. C₁₀₁	#21155	direct (N × H)
161. #427	C₄×C₁₀₁	semidirect
201. #107	C₂×C₄×C₁₀₁	semidirect
204. #106	C₂×C₄×C₁₀₁	semidirect
225. C₂×C₈	C₄⋊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×#427×C₁₀₁	2
C₂²×#106×C₁₀₁	2
C₂×#405×C₁₀₁	2
C₁₀₁×#2317	2
C₁₀₁×#1943	2
C₁₀₁×#1941	2
C₁₀₁×#1736	2
#21155	101

Subgroups — C101×#21155 #38803306

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#21155 #38803306