Subgroups — C₁₀₁×#10940 #38793091

Order 25856.

898 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#101×C₁₀₁	C₂	1	2	yes	maximal
2	C₁₀₁×#2873	C₂	1	1	yes	maximal, characteristic
3	C₁₀₁×#2310	C₂	1	1	yes	maximal, characteristic
4	C₂×#344×C₁₀₁	C₂	1	2	yes	maximal
5	C₁₀₁×#1742	C₂	1	1	yes	maximal, characteristic
6	C₂³×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₁₀₁	C₂²	1	2	yes	—
8	C₂²×M₁₆×C₁₀₁	C₂²	1	2	yes	—
9	C₂×#117×C₁₀₁	C₂²	1	1	yes	characteristic
10	C₂×#117×C₁₀₁	C₂²	1	1	yes	characteristic
11	C₂×#101×C₁₀₁	—	2	4	no	—
12	C₂×#101×C₁₀₁	—	2	4	no	—
13	C₂×#101×C₁₀₁	C₄	1	8	yes	complemented
14	#405×C₁₀₁	—	2	4	no	—
15	C₄×M₁₆×C₁₀₁	—	2	2	no	—
16	#391×C₁₀₁	—	2	2	no	—
17	#391×C₁₀₁	—	2	2	no	—
18	#381×C₁₀₁	C₄	1	4	yes	—
19	#344×C₁₀₁	—	2	4	no	—
20	#343×C₁₀₁	—	2	2	no	—
21	C₂²×Q₈×C₁₀₁	—	2	2	no	—
22	C₂²×Q₈×C₁₀₁	—	2	2	no	—
23	C₂²×Q₈×C₁₀₁	—	2	1	no	—
24	C₂²×Q₈×C₁₀₁	—	2	1	no	—
25	C₂²×Q₈×C₁₀₁	—	2	2	no	—
26	C₂²×Q₈×C₁₀₁	—	2	2	no	—
27	C₂²×Q₈×C₁₀₁	—	2	2	no	—
28	C₂²×Q₈×C₁₀₁	C₂×C₄	1	4	yes	—
29	C₂³×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
31	C₂×M₁₆×C₁₀₁	—	2	2	no	—
32	C₂×M₁₆×C₁₀₁	—	2	2	no	—
33	C₂×M₁₆×C₁₀₁	—	2	2	no	—
34	C₂×M₁₆×C₁₀₁	—	2	2	no	—
35	C₂×M₁₆×C₁₀₁	—	2	2	no	—
36	C₂×M₁₆×C₁₀₁	—	2	2	no	—
37	C₂×M₁₆×C₁₀₁	—	2	2	no	—
38	C₂×M₁₆×C₁₀₁	—	2	2	no	—
39	C₂×M₁₆×C₁₀₁	C₂×C₄	1	4	yes	—
40	C₂×M₁₆×C₁₀₁	C₂×C₄	1	4	yes	—
41	C₂×M₁₆×C₁₀₁	D₄	1	4	yes	—
42	C₂×M₁₆×C₁₀₁	D₄	1	4	yes	—
43	C₂²×C₈×C₁₀₁	—	2	2	no	—
44	C₂²×C₈×C₁₀₁	—	2	2	no	—
45	#117×C₁₀₁	—	2	2	no	—
46	#117×C₁₀₁	—	2	2	no	—
47	#117×C₁₀₁	—	2	2	no	—
48	#117×C₁₀₁	—	2	2	no	—
49	C₂×C₄⋊C₄×C₁₀₁	—	2	1	no	—
50	C₂×C₄⋊C₄×C₁₀₁	D₄	1	2	yes	—
51	C₂×C₂²⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
52	C₂×C₂²⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₁₀₁	—	2	1	no	—
54	C₂×C₄²×C₁₀₁	D₄	1	2	yes	—
55	#101×C₁₀₁	—	2	16	no	—
56	#97×C₁₀₁	—	2	4	no	—
57	C₄×C₈×C₁₀₁	—	2	4	no	—
58	#95×C₁₀₁	—	2	8	no	—
59	C₂⁴×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×Q₈×C₁₀₁	—	2	4	no	—
61	C₂×Q₈×C₁₀₁	—	2	4	no	—
62	C₂×Q₈×C₁₀₁	—	2	4	no	—
63	C₂×Q₈×C₁₀₁	—	2	4	no	—
64	C₂×Q₈×C₁₀₁	—	2	4	no	—
65	C₂×Q₈×C₁₀₁	—	2	4	no	—
66	C₂×Q₈×C₁₀₁	—	4	1	no	—
67	C₂×Q₈×C₁₀₁	—	4	2	no	—
68	C₂×Q₈×C₁₀₁	—	4	2	no	—
69	C₂×Q₈×C₁₀₁	—	4	2	no	—
70	C₂×Q₈×C₁₀₁	—	4	1	no	—
71	C₂×Q₈×C₁₀₁	—	4	2	no	—
72	C₂×Q₈×C₁₀₁	—	4	2	no	—
73	C₂×Q₈×C₁₀₁	—	4	1	no	—
74	C₂×Q₈×C₁₀₁	—	4	2	no	—
75	C₂×Q₈×C₁₀₁	—	4	1	no	—
76	C₂²×C₄×C₁₀₁	—	2	2	no	—
77	C₂²×C₄×C₁₀₁	—	2	2	no	—
78	C₂²×C₄×C₁₀₁	—	2	4	no	—
79	C₂²×C₄×C₁₀₁	—	2	4	no	—
80	C₂²×C₄×C₁₀₁	—	2	2	no	—
81	C₂²×C₄×C₁₀₁	C₄²	1	4	yes	—
82	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	4	yes	—
83	C₂²×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₀₁	#41	1	2	yes	—
89	C₂²×C₄×C₁₀₁	#41	1	2	yes	—
90	C₂²×C₄×C₁₀₁	#41	1	2	yes	—
91	M₁₆×C₁₀₁	—	2	4	no	—
92	M₁₆×C₁₀₁	—	2	4	no	—
93	M₁₆×C₁₀₁	—	2	4	no	—
94	M₁₆×C₁₀₁	—	2	4	no	—
95	M₁₆×C₁₀₁	—	2	4	no	—
96	M₁₆×C₁₀₁	—	4	2	no	—
97	M₁₆×C₁₀₁	—	4	2	no	—
98	M₁₆×C₁₀₁	C₂²⋊C₄	1	8	yes	—
99	C₂×C₈×C₁₀₁	—	2	4	no	—
100	C₂×C₈×C₁₀₁	—	2	4	no	—
101	C₂×C₈×C₁₀₁	—	2	4	no	—
102	C₂×C₈×C₁₀₁	—	2	4	no	—
103	C₂×C₈×C₁₀₁	—	4	2	no	—
104	C₂×C₈×C₁₀₁	—	4	2	no	—
105	C₄⋊C₄×C₁₀₁	—	2	4	no	—
106	C₄⋊C₄×C₁₀₁	—	4	2	no	—
107	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
108	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
109	C₄²×C₁₀₁	—	2	2	no	—
110	C₄²×C₁₀₁	—	2	2	no	—
111	C₄²×C₁₀₁	—	4	1	no	—
112	C₄²×C₁₀₁	—	4	1	no	—
113	C₂³×C₁₀₁	—	4	2	no	—
114	C₂³×C₁₀₁	C₂×C₄²	1	1	yes	characteristic
115	C₂³×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂³×C₁₀₁	#117	1	1	yes	characteristic
117	C₂³×C₁₀₁	C₄×D₄	1	2	yes	—
118	C₂³×C₁₀₁	C₄×D₄	1	2	yes	—
119	Q₈×C₁₀₁	—	4	4	no	—
120	Q₈×C₁₀₁	—	4	4	no	—
121	Q₈×C₁₀₁	—	4	4	no	—
122	Q₈×C₁₀₁	—	4	4	no	—
123	C₂×C₄×C₁₀₁	—	2	8	no	—
124	C₂×C₄×C₁₀₁	—	2	4	no	—
125	C₂×C₄×C₁₀₁	—	2	4	no	—
126	C₂×C₄×C₁₀₁	—	4	2	no	—
127	C₂×C₄×C₁₀₁	—	4	4	no	—
128	C₂×C₄×C₁₀₁	—	4	2	no	—
129	C₂×C₄×C₁₀₁	—	4	4	no	—
130	C₂×C₄×C₁₀₁	—	4	4	no	—
131	C₂×C₄×C₁₀₁	—	4	2	no	—
132	C₂×C₄×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
133	C₂×C₄×C₁₀₁	#117	1	2	yes	—
134	C₂×C₄×C₁₀₁	C₄×D₄	1	2	yes	—
135	C₂×C₄×C₁₀₁	C₄×D₄	1	2	yes	—
136	C₂×C₄×C₁₀₁	C₄×D₄	1	2	yes	—
137	C₂×C₄×C₁₀₁	C₄×D₄	1	2	yes	—
138	C₂×C₄×C₁₀₁	#120	1	1	yes	characteristic
139	C₂×C₄×C₁₀₁	#121	1	2	yes	—
140	C₂×C₄×C₁₀₁	#121	1	2	yes	—
141	C₂×C₄×C₁₀₁	#121	1	2	yes	—
142	C₂×C₄×C₁₀₁	#123	1	2	yes	—
143	C₂×C₄×C₁₀₁	#123	1	1	yes	characteristic
144	C₂×C₄×C₁₀₁	#124	1	1	yes	characteristic
145	C₂×C₄×C₁₀₁	#124	1	2	yes	—
146	C₂×C₄×C₁₀₁	#126	1	2	yes	—
147	C₂×C₄×C₁₀₁	#127	1	1	yes	characteristic
148	C₈×C₁₀₁	—	2	8	no	—
149	C₈×C₁₀₁	—	4	4	no	—
150	C₂²×C₁₀₁	—	2	2	no	—
151	C₂²×C₁₀₁	—	2	4	no	—
152	C₂²×C₁₀₁	—	4	1	no	—
153	C₂²×C₁₀₁	—	4	1	no	—
154	C₂²×C₁₀₁	—	4	2	no	—
155	C₂²×C₁₀₁	C₄×C₂²⋊C₄	1	2	yes	—
156	C₂²×C₁₀₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
157	C₂²×C₁₀₁	#386	1	1	yes	characteristic
158	C₂²×C₁₀₁	#388	1	2	yes	—
159	C₂²×C₁₀₁	#390	1	1	yes	characteristic
160	C₄×C₁₀₁	—	4	4	no	—
161	C₄×C₁₀₁	—	4	4	no	—
162	C₄×C₁₀₁	—	8	2	no	—
163	C₄×C₁₀₁	#386	1	2	yes	—
164	C₄×C₁₀₁	#388	1	2	yes	—
165	C₄×C₁₀₁	#388	1	2	yes	—
166	C₄×C₁₀₁	#390	1	2	yes	—
167	#10940	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₂×C₁₀₁	—	2	1	no	—
169	C₂×C₁₀₁	—	2	1	no	—
170	C₂×C₁₀₁	—	2	2	no	—
171	C₂×C₁₀₁	—	2	2	no	—
172	C₂×C₁₀₁	#1437	1	1	yes	characteristic
173	C₂×C₁₀₁	#1915	1	2	yes	—
174	C₂²×#101	C₂×C₁₀₁	1	2	yes	—
175	#2873	C₂×C₁₀₁	1	1	yes	characteristic
176	#2310	C₂×C₁₀₁	1	1	yes	characteristic
177	C₂×#344	C₂×C₁₀₁	1	2	yes	—
178	#1742	C₂×C₁₀₁	1	1	yes	characteristic
179	C₁₀₁	#10940	1	1	yes	characteristic, direct product factor, Sylow
180	C₂³×Q₈	C₂²×C₁₀₁	1	1	yes	characteristic
181	C₂²×M₁₆	C₂²×C₁₀₁	1	2	yes	—
182	C₂²×M₁₆	C₂²×C₁₀₁	1	2	yes	—
183	C₂×#117	C₂²×C₁₀₁	1	1	yes	characteristic
184	C₂×#117	C₂²×C₁₀₁	1	1	yes	characteristic
185	C₂×#101	—	2	4	no	—
186	C₂×#101	—	2	4	no	—
187	C₂×#101	C₄×C₁₀₁	1	8	yes	complemented
188	#405	—	2	4	no	—
189	C₄×M₁₆	—	2	2	no	—
190	#391	—	2	2	no	—
191	#391	—	2	2	no	—
192	#381	C₄×C₁₀₁	1	4	yes	—
193	#344	—	2	4	no	—
194	#343	—	2	2	no	—
195	C₂²×Q₈	—	2	1	no	—
196	C₂²×Q₈	—	2	1	no	—
197	C₂²×Q₈	—	2	2	no	—
198	C₂²×Q₈	—	2	2	no	—
199	C₂²×Q₈	—	2	2	no	—
200	C₂²×Q₈	—	2	2	no	—
201	C₂²×Q₈	—	2	2	no	—
202	C₂²×Q₈	C₂×C₄×C₁₀₁	1	4	yes	—
203	C₂³×C₄	C₂×C₄×C₁₀₁	1	2	yes	—
204	C₂³×C₄	C₂³×C₁₀₁	1	1	yes	characteristic
205	C₂×M₁₆	—	2	2	no	—
206	C₂×M₁₆	—	2	2	no	—
207	C₂×M₁₆	—	2	2	no	—
208	C₂×M₁₆	—	2	2	no	—
209	C₂×M₁₆	—	2	2	no	—
210	C₂×M₁₆	—	2	2	no	—
211	C₂×M₁₆	—	2	2	no	—
212	C₂×M₁₆	—	2	2	no	—
213	C₂×M₁₆	C₂×C₄×C₁₀₁	1	4	yes	—
214	C₂×M₁₆	C₂×C₄×C₁₀₁	1	4	yes	—
215	C₂×M₁₆	D₄×C₁₀₁	1	4	yes	—
216	C₂×M₁₆	D₄×C₁₀₁	1	4	yes	—
217	C₂²×C₈	—	2	2	no	—
218	C₂²×C₈	—	2	2	no	—
219	#117	—	2	2	no	—
220	#117	—	2	2	no	—
221	#117	—	2	2	no	—
222	#117	—	2	2	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	D₄×C₁₀₁	1	2	yes	—
225	C₂×C₂²⋊C₄	C₂×C₄×C₁₀₁	1	2	yes	—
226	C₂×C₂²⋊C₄	C₂×C₄×C₁₀₁	1	2	yes	—
227	C₂×C₄²	—	2	1	no	—
228	C₂×C₄²	D₄×C₁₀₁	1	2	yes	—
229	#101	—	2	16	no	—
230	#97	—	2	4	no	—
231	C₄×C₈	—	2	4	no	—
232	#95	—	2	8	no	—
233	C₂⁴	C₂²×C₄×C₁₀₁	1	1	yes	characteristic
234	C₂×Q₈	—	2	4	no	—
235	C₂×Q₈	—	2	4	no	—
236	C₂×Q₈	—	2	4	no	—
237	C₂×Q₈	—	2	4	no	—
238	C₂×Q₈	—	2	4	no	—
239	C₂×Q₈	—	2	4	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×Q₈	—	4	2	no	—
242	C₂×Q₈	—	4	2	no	—
243	C₂×Q₈	—	4	2	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂×Q₈	—	4	2	no	—
246	C₂×Q₈	—	4	2	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	4	2	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	4	no	—
253	C₂²×C₄	—	2	4	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	C₄²×C₁₀₁	1	4	yes	—
256	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	4	yes	—
257	C₂²×C₄	C₂²×C₄×C₁₀₁	1	2	yes	—
258	C₂²×C₄	C₂×D₄×C₁₀₁	1	2	yes	—
259	C₂²×C₄	C₂×D₄×C₁₀₁	1	2	yes	—
260	C₂²×C₄	C₂×D₄×C₁₀₁	1	1	yes	characteristic
261	C₂²×C₄	C₂×D₄×C₁₀₁	1	1	yes	characteristic
262	C₂²×C₄	#41×C₁₀₁	1	2	yes	—
263	C₂²×C₄	#41×C₁₀₁	1	2	yes	—
264	C₂²×C₄	#41×C₁₀₁	1	2	yes	—
265	M₁₆	—	2	4	no	—
266	M₁₆	—	2	4	no	—
267	M₁₆	—	2	4	no	—
268	M₁₆	—	2	4	no	—
269	M₁₆	—	2	4	no	—
270	M₁₆	—	4	2	no	—
271	M₁₆	—	4	2	no	—
272	M₁₆	C₂²⋊C₄×C₁₀₁	1	8	yes	—
273	C₂×C₈	—	2	4	no	—
274	C₂×C₈	—	2	4	no	—
275	C₂×C₈	—	2	4	no	—
276	C₂×C₈	—	2	4	no	—
277	C₂×C₈	—	4	2	no	—
278	C₂×C₈	—	4	2	no	—
279	C₄⋊C₄	—	2	4	no	—
280	C₄⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	2	4	no	—
282	C₂²⋊C₄	—	2	4	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	4	1	no	—
286	C₄²	—	4	1	no	—
287	C₂³	—	4	2	no	—
288	C₂³	C₂×C₄²×C₁₀₁	1	1	yes	characteristic
289	C₂³	C₂×C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
290	C₂³	#117×C₁₀₁	1	1	yes	characteristic
291	C₂³	C₄×D₄×C₁₀₁	1	2	yes	—
292	C₂³	C₄×D₄×C₁₀₁	1	2	yes	—
293	Q₈	—	4	4	no	—
294	Q₈	—	4	4	no	—
295	Q₈	—	4	4	no	—
296	Q₈	—	4	4	no	—
297	C₂×C₄	—	2	8	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	4	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₁	1	2	yes	—
307	C₂×C₄	#117×C₁₀₁	1	2	yes	—
308	C₂×C₄	C₄×D₄×C₁₀₁	1	2	yes	—
309	C₂×C₄	C₄×D₄×C₁₀₁	1	2	yes	—
310	C₂×C₄	C₄×D₄×C₁₀₁	1	2	yes	—
311	C₂×C₄	C₄×D₄×C₁₀₁	1	2	yes	—
312	C₂×C₄	#120×C₁₀₁	1	1	yes	characteristic
313	C₂×C₄	#121×C₁₀₁	1	2	yes	—
314	C₂×C₄	#121×C₁₀₁	1	2	yes	—
315	C₂×C₄	#121×C₁₀₁	1	2	yes	—
316	C₂×C₄	#123×C₁₀₁	1	2	yes	—
317	C₂×C₄	#123×C₁₀₁	1	1	yes	characteristic
318	C₂×C₄	#124×C₁₀₁	1	1	yes	characteristic
319	C₂×C₄	#124×C₁₀₁	1	2	yes	—
320	C₂×C₄	#126×C₁₀₁	1	2	yes	—
321	C₂×C₄	#127×C₁₀₁	1	1	yes	characteristic
322	C₈	—	2	8	no	—
323	C₈	—	4	4	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	2	4	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	2	no	—
329	C₂²	C₄×C₂²⋊C₄×C₁₀₁	1	2	yes	—
330	C₂²	C₄×C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
331	C₂²	#386×C₁₀₁	1	1	yes	characteristic
332	C₂²	#388×C₁₀₁	1	2	yes	—
333	C₂²	#390×C₁₀₁	1	1	yes	characteristic
334	C₄	—	4	4	no	—
335	C₄	—	4	4	no	—
336	C₄	—	8	2	no	—
337	C₄	#386×C₁₀₁	1	2	yes	—
338	C₄	#388×C₁₀₁	1	2	yes	—
339	C₄	#388×C₁₀₁	1	2	yes	—
340	C₄	#390×C₁₀₁	1	2	yes	—
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	2	2	no	—
344	C₂	—	2	2	no	—
345	C₂	C₁₀₁×#1437	1	1	yes	characteristic
346	C₂	C₁₀₁×#1915	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₀₁	#386
commutator	C₂³	C₂×C₄²×C₁₀₁
Frattini, cosocle	C₂³×C₄	C₂³×C₁₀₁
prime radical	C₂⁴×C₁₀₁	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₁ ◁ C₂³×C₄×C₁₀₁ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₂×#101×C₁₀₁	C₄	semidirect
167. #10940	C₁₀₁	direct (N × H)
179. C₁₀₁	#10940	direct (N × H)
187. C₂×#101	C₄×C₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#101×C₁₀₁	2
C₁₀₁×#2873	2
C₁₀₁×#2310	2
C₂×#344×C₁₀₁	2
C₁₀₁×#1742	2
#10940	101

Subgroups — C101×#10940 #38793091

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#10940 #38793091