Subgroups — C₁₀₁×#11359 #38793510

Order 25856.

794 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₁×#2285	C₂	1	1	yes	maximal, characteristic
2	C₂×#339×C₁₀₁	C₂	1	4	yes	maximal
3	C₁₀₁×#1728	C₂	1	1	yes	maximal, characteristic
4	C₄×#95×C₁₀₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×M₁₆×C₁₀₁	C₂²	1	2	yes	—
6	C₂×#117×C₁₀₁	C₂²	1	2	yes	—
7	C₂²×C₄²×C₁₀₁	C₂²	1	2	yes	—
8	C₂²×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
9	#384×C₁₀₁	C₄	1	2	yes	complemented
10	#382×C₁₀₁	C₄	1	2	yes	complemented
11	#380×C₁₀₁	—	2	2	no	—
12	C₄×C₄⋊C₄×C₁₀₁	—	2	2	no	—
13	C₂×#95×C₁₀₁	—	2	2	no	—
14	#339×C₁₀₁	—	2	8	no	—
15	#336×C₁₀₁	—	2	2	no	—
16	C₂³×C₄×C₁₀₁	—	2	2	no	—
17	C₂³×C₄×C₁₀₁	D₄	1	2	yes	—
18	C₂³×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
19	C₂×M₁₆×C₁₀₁	—	2	2	no	—
20	C₂×M₁₆×C₁₀₁	—	2	4	no	—
21	C₂×M₁₆×C₁₀₁	—	2	4	no	—
22	C₂×M₁₆×C₁₀₁	D₄	1	4	yes	—
23	C₂²×C₈×C₁₀₁	—	2	2	no	—
24	#117×C₁₀₁	—	2	4	no	—
25	#117×C₁₀₁	—	2	4	no	—
26	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
27	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	—
29	C₂×C₂²⋊C₄×C₁₀₁	—	2	2	no	—
30	C₂×C₄²×C₁₀₁	—	2	4	no	—
31	C₂×C₄²×C₁₀₁	—	2	4	no	—
32	C₂×C₄²×C₁₀₁	—	2	4	no	—
33	C₂×C₄²×C₁₀₁	—	2	2	no	—
34	C₂×C₄²×C₁₀₁	—	2	2	no	—
35	C₂×C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₀₁	D₄	1	2	yes	—
38	C₂×C₄²×C₁₀₁	Q₈	1	4	yes	—
39	#95×C₁₀₁	—	2	8	no	—
40	#95×C₁₀₁	—	2	4	no	—
41	C₂⁴×C₁₀₁	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₀₁	—	2	8	no	—
43	C₂²×C₄×C₁₀₁	—	2	4	no	—
44	C₂²×C₄×C₁₀₁	—	2	4	no	—
45	C₂²×C₄×C₁₀₁	—	2	4	no	—
46	C₂²×C₄×C₁₀₁	—	2	4	no	—
47	C₂²×C₄×C₁₀₁	—	4	2	no	—
48	C₂²×C₄×C₁₀₁	—	4	4	no	—
49	C₂²×C₄×C₁₀₁	—	4	2	no	—
50	C₂²×C₄×C₁₀₁	—	4	2	no	—
51	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₁₀₁	#41	1	4	yes	—
58	C₂²×C₄×C₁₀₁	#41	1	2	yes	—
59	C₂²×C₄×C₁₀₁	#41	1	2	yes	—
60	M₁₆×C₁₀₁	—	2	8	no	—
61	M₁₆×C₁₀₁	—	4	4	no	—
62	C₂×C₈×C₁₀₁	—	2	4	no	—
63	C₂×C₈×C₁₀₁	—	4	2	no	—
64	C₂×C₈×C₁₀₁	—	4	2	no	—
65	C₄⋊C₄×C₁₀₁	—	2	8	no	—
66	C₄⋊C₄×C₁₀₁	—	4	4	no	—
67	C₄⋊C₄×C₁₀₁	—	4	4	no	—
68	C₂²⋊C₄×C₁₀₁	—	4	2	no	—
69	C₂²⋊C₄×C₁₀₁	—	4	2	no	—
70	C₄²×C₁₀₁	—	2	2	no	—
71	C₄²×C₁₀₁	—	2	2	no	—
72	C₄²×C₁₀₁	—	2	4	no	—
73	C₄²×C₁₀₁	—	2	8	no	—
74	C₄²×C₁₀₁	—	4	4	no	—
75	C₄²×C₁₀₁	—	4	4	no	—
76	C₂³×C₁₀₁	—	2	2	no	—
77	C₂³×C₁₀₁	—	2	2	no	—
78	C₂³×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂³×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
80	C₂³×C₁₀₁	#123	1	2	yes	—
81	C₂³×C₁₀₁	#124	1	1	yes	characteristic
82	C₂³×C₁₀₁	#127	1	1	yes	characteristic
83	C₂×C₄×C₁₀₁	—	2	2	no	—
84	C₂×C₄×C₁₀₁	—	2	2	no	—
85	C₂×C₄×C₁₀₁	—	2	8	no	—
86	C₂×C₄×C₁₀₁	—	2	2	no	—
87	C₂×C₄×C₁₀₁	—	2	2	no	—
88	C₂×C₄×C₁₀₁	—	2	2	no	—
89	C₂×C₄×C₁₀₁	—	4	8	no	—
90	C₂×C₄×C₁₀₁	—	4	8	no	—
91	C₂×C₄×C₁₀₁	—	4	8	no	—
92	C₂×C₄×C₁₀₁	—	4	4	no	—
93	C₂×C₄×C₁₀₁	—	4	4	no	—
94	C₂×C₄×C₁₀₁	—	4	4	no	—
95	C₂×C₄×C₁₀₁	—	4	4	no	—
96	C₂×C₄×C₁₀₁	—	4	4	no	—
97	C₂×C₄×C₁₀₁	#104	1	4	yes	—
98	C₂×C₄×C₁₀₁	#117	1	2	yes	—
99	C₂×C₄×C₁₀₁	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₁₀₁	C₄×Q₈	1	2	yes	—
101	C₂×C₄×C₁₀₁	#121	1	2	yes	—
102	C₂×C₄×C₁₀₁	C₂²⋊Q₈	1	4	yes	—
103	C₂×C₄×C₁₀₁	#123	1	4	yes	—
104	C₂×C₄×C₁₀₁	#123	1	2	yes	—
105	C₂×C₄×C₁₀₁	#124	1	1	yes	characteristic
106	C₂×C₄×C₁₀₁	#125	1	4	yes	—
107	C₂×C₄×C₁₀₁	#126	1	4	yes	—
108	C₂×C₄×C₁₀₁	C₄⋊Q₈	1	1	yes	characteristic
109	C₈×C₁₀₁	—	4	4	no	—
110	C₂²×C₁₀₁	—	2	4	no	—
111	C₂²×C₁₀₁	—	2	2	no	—
112	C₂²×C₁₀₁	—	2	2	no	—
113	C₂²×C₁₀₁	—	2	2	no	—
114	C₂²×C₁₀₁	—	2	2	no	—
115	C₂²×C₁₀₁	#381	1	1	yes	characteristic
116	C₂²×C₁₀₁	#390	1	1	yes	characteristic
117	C₂²×C₁₀₁	#391	1	1	yes	characteristic
118	C₂²×C₁₀₁	#399	1	2	yes	—
119	C₂²×C₁₀₁	#400	1	2	yes	—
120	C₂²×C₁₀₁	C₂×#104	1	2	yes	—
121	C₂²×C₁₀₁	#421	1	2	yes	—
122	C₄×C₁₀₁	—	2	4	no	—
123	C₄×C₁₀₁	—	4	8	no	—
124	C₄×C₁₀₁	—	4	2	no	—
125	C₄×C₁₀₁	—	8	4	no	—
126	C₄×C₁₀₁	#387	1	4	yes	—
127	C₄×C₁₀₁	#397	1	2	yes	—
128	C₄×C₁₀₁	#398	1	2	yes	—
129	#11359	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂×C₁₀₁	—	2	2	no	—
131	C₂×C₁₀₁	—	2	2	no	—
132	C₂×C₁₀₁	#1445	1	1	yes	characteristic
133	C₂×C₁₀₁	#1826	1	2	yes	—
134	C₂×C₁₀₁	#1980	1	2	yes	—
135	C₂×C₁₀₁	#1994	1	1	yes	characteristic
136	C₂×C₁₀₁	#1995	1	1	yes	characteristic
137	#2285	C₂×C₁₀₁	1	1	yes	characteristic
138	C₂×#339	C₂×C₁₀₁	1	4	yes	—
139	#1728	C₂×C₁₀₁	1	1	yes	characteristic
140	C₄×#95	C₂×C₁₀₁	1	1	yes	characteristic
141	C₁₀₁	#11359	1	1	yes	characteristic, direct product factor, Sylow
142	C₂²×M₁₆	C₂²×C₁₀₁	1	2	yes	—
143	C₂×#117	C₂²×C₁₀₁	1	2	yes	—
144	C₂²×C₄²	C₂²×C₁₀₁	1	2	yes	—
145	C₂²×C₄²	C₂²×C₁₀₁	1	1	yes	characteristic
146	#384	C₄×C₁₀₁	1	2	yes	complemented
147	#382	C₄×C₁₀₁	1	2	yes	complemented
148	#380	—	2	2	no	—
149	C₄×C₄⋊C₄	—	2	2	no	—
150	C₂×#95	—	2	2	no	—
151	#339	—	2	8	no	—
152	#336	—	2	2	no	—
153	C₂³×C₄	—	2	2	no	—
154	C₂³×C₄	D₄×C₁₀₁	1	2	yes	—
155	C₂³×C₄	C₂³×C₁₀₁	1	1	yes	characteristic
156	C₂×M₁₆	—	2	2	no	—
157	C₂×M₁₆	—	2	4	no	—
158	C₂×M₁₆	—	2	4	no	—
159	C₂×M₁₆	D₄×C₁₀₁	1	4	yes	—
160	C₂²×C₈	—	2	2	no	—
161	#117	—	2	4	no	—
162	#117	—	2	4	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	C₂×C₄×C₁₀₁	1	4	yes	—
166	C₂×C₂²⋊C₄	—	2	2	no	—
167	C₂×C₄²	—	2	4	no	—
168	C₂×C₄²	—	2	4	no	—
169	C₂×C₄²	—	2	4	no	—
170	C₂×C₄²	—	2	2	no	—
171	C₂×C₄²	—	2	2	no	—
172	C₂×C₄²	C₂×C₄×C₁₀₁	1	1	yes	characteristic
173	C₂×C₄²	C₂×C₄×C₁₀₁	1	1	yes	characteristic
174	C₂×C₄²	D₄×C₁₀₁	1	2	yes	—
175	C₂×C₄²	Q₈×C₁₀₁	1	4	yes	—
176	#95	—	2	8	no	—
177	#95	—	2	4	no	—
178	C₂⁴	C₂×D₄×C₁₀₁	1	1	yes	characteristic
179	C₂²×C₄	—	2	8	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	—	4	4	no	—
186	C₂²×C₄	—	4	2	no	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	2	yes	—
189	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	2	yes	—
190	C₂²×C₄	C₂²×C₄×C₁₀₁	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₄×C₁₀₁	1	2	yes	—
192	C₂²×C₄	C₂×D₄×C₁₀₁	1	1	yes	characteristic
193	C₂²×C₄	C₂×Q₈×C₁₀₁	1	2	yes	—
194	C₂²×C₄	#41×C₁₀₁	1	4	yes	—
195	C₂²×C₄	#41×C₁₀₁	1	2	yes	—
196	C₂²×C₄	#41×C₁₀₁	1	2	yes	—
197	M₁₆	—	2	8	no	—
198	M₁₆	—	4	4	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	4	2	no	—
202	C₄⋊C₄	—	2	8	no	—
203	C₄⋊C₄	—	4	4	no	—
204	C₄⋊C₄	—	4	4	no	—
205	C₂²⋊C₄	—	4	2	no	—
206	C₂²⋊C₄	—	4	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	4	no	—
210	C₄²	—	2	8	no	—
211	C₄²	—	4	4	no	—
212	C₄²	—	4	4	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	C₂×C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
216	C₂³	C₂²⋊Q₈×C₁₀₁	1	2	yes	—
217	C₂³	#123×C₁₀₁	1	2	yes	—
218	C₂³	#124×C₁₀₁	1	1	yes	characteristic
219	C₂³	#127×C₁₀₁	1	1	yes	characteristic
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	8	no	—
227	C₂×C₄	—	4	8	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	4	8	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	#104×C₁₀₁	1	4	yes	—
235	C₂×C₄	#117×C₁₀₁	1	2	yes	—
236	C₂×C₄	C₄×D₄×C₁₀₁	1	2	yes	—
237	C₂×C₄	C₄×Q₈×C₁₀₁	1	2	yes	—
238	C₂×C₄	#121×C₁₀₁	1	2	yes	—
239	C₂×C₄	C₂²⋊Q₈×C₁₀₁	1	4	yes	—
240	C₂×C₄	#123×C₁₀₁	1	2	yes	—
241	C₂×C₄	#123×C₁₀₁	1	4	yes	—
242	C₂×C₄	#124×C₁₀₁	1	1	yes	characteristic
243	C₂×C₄	#125×C₁₀₁	1	4	yes	—
244	C₂×C₄	#126×C₁₀₁	1	4	yes	—
245	C₂×C₄	C₄⋊Q₈×C₁₀₁	1	1	yes	characteristic
246	C₈	—	4	4	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	4	no	—
252	C₂²	#381×C₁₀₁	1	1	yes	characteristic
253	C₂²	#390×C₁₀₁	1	1	yes	characteristic
254	C₂²	#391×C₁₀₁	1	1	yes	characteristic
255	C₂²	#399×C₁₀₁	1	2	yes	—
256	C₂²	#400×C₁₀₁	1	2	yes	—
257	C₂²	C₂×#104×C₁₀₁	1	2	yes	—
258	C₂²	#421×C₁₀₁	1	2	yes	—
259	C₄	—	2	4	no	—
260	C₄	—	4	8	no	—
261	C₄	—	4	2	no	—
262	C₄	—	8	4	no	—
263	C₄	#387×C₁₀₁	1	4	yes	—
264	C₄	#397×C₁₀₁	1	2	yes	—
265	C₄	#398×C₁₀₁	1	2	yes	—
266	C₂	—	2	2	no	—
267	C₂	—	2	2	no	—
268	C₂	C₁₀₁×#1445	1	1	yes	characteristic
269	C₂	C₁₀₁×#1826	1	2	yes	—
270	C₂	C₁₀₁×#1980	1	2	yes	—
271	C₂	C₁₀₁×#1994	1	1	yes	characteristic
272	C₂	C₁₀₁×#1995	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₀₁	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₁₀₁
Frattini, cosocle	C₂³×C₄	C₂³×C₁₀₁
socle	C₂³×C₁₀₁	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₀₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₀₁ ◁ C₂²×C₄²×C₁₀₁ ◁ G

Complements

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

N	H ≅ G/N	type
9. #384×C₁₀₁	C₄	semidirect
10. #382×C₁₀₁	C₄	semidirect
129. #11359	C₁₀₁	direct (N × H)
141. C₁₀₁	#11359	direct (N × H)
146. #384	C₄×C₁₀₁	semidirect
147. #382	C₄×C₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₁₀₁×#2285	2
C₂×#339×C₁₀₁	2
C₁₀₁×#1728	2
C₄×#95×C₁₀₁	2
#11359	101

Subgroups — C101×#11359 #38793510

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#11359 #38793510