Subgroups — C₁₀₁×#11373 #38793524

Order 25856.

886 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#384×C₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#340×C₁₀₁	C₂	1	4	yes	maximal
3	C₂×#336×C₁₀₁	C₂	1	1	yes	maximal, characteristic
4	C₁₀₁×C₄⋊#95	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₁₀₁	C₂²	1	2	yes	—
6	C₂²×C₄⋊C₄×C₁₀₁	—	2	2	no	—
7	C₂²×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
8	C₂²×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
9	C₂²×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
10	#384×C₁₀₁	—	2	2	no	—
11	#384×C₁₀₁	C₄	1	4	yes	complemented
12	C₂×#95×C₁₀₁	—	2	2	no	—
13	#340×C₁₀₁	—	2	8	no	—
14	#336×C₁₀₁	—	2	2	no	—
15	C₂³×C₄×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	C₂²×C₈×C₁₀₁	—	2	2	no	—
20	C₂²×C₈×C₁₀₁	—	2	2	no	—
21	C₂²×C₈×C₁₀₁	—	2	4	no	—
22	C₂²×C₈×C₁₀₁	—	2	4	no	—
23	C₂²×C₈×C₁₀₁	D₄	1	4	yes	—
24	C₂×C₄⋊C₄×C₁₀₁	—	2	4	no	—
25	C₂×C₄⋊C₄×C₁₀₁	—	2	4	no	—
26	C₂×C₄⋊C₄×C₁₀₁	—	2	4	no	—
27	C₂×C₄⋊C₄×C₁₀₁	—	2	4	no	—
28	C₂×C₄⋊C₄×C₁₀₁	—	2	8	no	—
29	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
31	C₂×C₄⋊C₄×C₁₀₁	—	4	4	no	—
32	C₂×C₄⋊C₄×C₁₀₁	—	4	4	no	—
33	C₂×C₄⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	—
34	C₂×C₄⋊C₄×C₁₀₁	Q₈	1	4	yes	—
35	C₂×C₄²×C₁₀₁	—	2	4	no	—
36	C₂×C₄²×C₁₀₁	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₁₀₁	D₄	1	2	yes	—
38	#95×C₁₀₁	—	4	4	no	—
39	C₂⁴×C₁₀₁	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₀₁	—	2	4	no	—
41	C₂²×C₄×C₁₀₁	—	2	4	no	—
42	C₂²×C₄×C₁₀₁	—	2	4	no	—
43	C₂²×C₄×C₁₀₁	—	2	4	no	—
44	C₂²×C₄×C₁₀₁	—	4	2	no	—
45	C₂²×C₄×C₁₀₁	—	4	2	no	—
46	C₂²×C₄×C₁₀₁	—	4	2	no	—
47	C₂²×C₄×C₁₀₁	—	4	2	no	—
48	C₂²×C₄×C₁₀₁	—	4	4	no	—
49	C₂²×C₄×C₁₀₁	—	4	4	no	—
50	C₂²×C₄×C₁₀₁	—	4	4	no	—
51	C₂²×C₄×C₁₀₁	—	4	4	no	—
52	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	4	yes	—
53	C₂²×C₄×C₁₀₁	D₈	1	4	yes	—
54	C₂²×C₄×C₁₀₁	SD₁₆	1	4	yes	—
55	C₂²×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
59	C₂²×C₄×C₁₀₁	#41	1	4	yes	—
60	C₂²×C₄×C₁₀₁	#41	1	2	yes	—
61	C₂²×C₄×C₁₀₁	#41	1	2	yes	—
62	C₂×C₈×C₁₀₁	—	2	4	no	—
63	C₂×C₈×C₁₀₁	—	2	8	no	—
64	C₂×C₈×C₁₀₁	—	4	2	no	—
65	C₂×C₈×C₁₀₁	—	4	2	no	—
66	C₂×C₈×C₁₀₁	—	4	4	no	—
67	C₄⋊C₄×C₁₀₁	—	2	8	no	—
68	C₄⋊C₄×C₁₀₁	—	2	8	no	—
69	C₄⋊C₄×C₁₀₁	—	4	16	no	—
70	C₄⋊C₄×C₁₀₁	—	4	4	no	—
71	C₄⋊C₄×C₁₀₁	—	4	4	no	—
72	C₄²×C₁₀₁	—	2	4	no	—
73	C₄²×C₁₀₁	—	2	4	no	—
74	C₂³×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂³×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
76	C₂³×C₁₀₁	#123	1	2	yes	—
77	C₂³×C₁₀₁	#124	1	1	yes	characteristic
78	C₂³×C₁₀₁	#127	1	1	yes	characteristic
79	C₂³×C₁₀₁	C₂×D₈	1	2	yes	—
80	C₂³×C₁₀₁	C₂×SD₁₆	1	2	yes	—
81	C₂³×C₁₀₁	#137	1	4	yes	—
82	C₂×C₄×C₁₀₁	—	2	4	no	—
83	C₂×C₄×C₁₀₁	—	2	4	no	—
84	C₂×C₄×C₁₀₁	—	2	8	no	—
85	C₂×C₄×C₁₀₁	—	2	8	no	—
86	C₂×C₄×C₁₀₁	—	4	4	no	—
87	C₂×C₄×C₁₀₁	—	4	4	no	—
88	C₂×C₄×C₁₀₁	—	4	8	no	—
89	C₂×C₄×C₁₀₁	—	4	8	no	—
90	C₂×C₄×C₁₀₁	—	8	4	no	—
91	C₂×C₄×C₁₀₁	—	8	4	no	—
92	C₂×C₄×C₁₀₁	—	8	2	no	—
93	C₂×C₄×C₁₀₁	—	8	2	no	—
94	C₂×C₄×C₁₀₁	—	8	2	no	—
95	C₂×C₄×C₁₀₁	—	8	2	no	—
96	C₂×C₄×C₁₀₁	#102	1	8	yes	—
97	C₂×C₄×C₁₀₁	#117	1	2	yes	—
98	C₂×C₄×C₁₀₁	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₁₀₁	C₄×Q₈	1	2	yes	—
100	C₂×C₄×C₁₀₁	#121	1	2	yes	—
101	C₂×C₄×C₁₀₁	C₂²⋊Q₈	1	4	yes	—
102	C₂×C₄×C₁₀₁	#123	1	4	yes	—
103	C₂×C₄×C₁₀₁	#123	1	2	yes	—
104	C₂×C₄×C₁₀₁	#124	1	1	yes	characteristic
105	C₂×C₄×C₁₀₁	#125	1	4	yes	—
106	C₂×C₄×C₁₀₁	#126	1	4	yes	—
107	C₂×C₄×C₁₀₁	C₄⋊Q₈	1	1	yes	characteristic
108	C₈×C₁₀₁	—	4	4	no	—
109	C₂²×C₁₀₁	#381	1	1	yes	characteristic
110	C₂²×C₁₀₁	#390	1	1	yes	characteristic
111	C₂²×C₁₀₁	#391	1	1	yes	characteristic
112	C₂²×C₁₀₁	#399	1	2	yes	—
113	C₂²×C₁₀₁	#400	1	2	yes	—
114	C₂²×C₁₀₁	C₂×#102	1	2	yes	—
115	C₂²×C₁₀₁	#419	1	2	yes	—
116	C₂²×C₁₀₁	#474	1	4	yes	—
117	C₂²×C₁₀₁	#475	1	4	yes	—
118	C₂²×C₁₀₁	#480	1	4	yes	—
119	C₂²×C₁₀₁	#483	1	4	yes	—
120	C₂²×C₁₀₁	#486	1	2	yes	—
121	C₂²×C₁₀₁	#491	1	2	yes	—
122	C₂²×C₁₀₁	#492	1	1	yes	characteristic
123	C₂²×C₁₀₁	#493	1	1	yes	characteristic
124	C₂²×C₁₀₁	#497	1	2	yes	—
125	C₄×C₁₀₁	—	2	8	no	—
126	C₄×C₁₀₁	—	8	4	no	—
127	C₄×C₁₀₁	—	8	4	no	—
128	C₄×C₁₀₁	#387	1	4	yes	—
129	C₄×C₁₀₁	#397	1	2	yes	—
130	C₄×C₁₀₁	#398	1	2	yes	—
131	#11373	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂×C₁₀₁	#1445	1	1	yes	characteristic
133	C₂×C₁₀₁	#1823	1	2	yes	—
134	C₂×C₁₀₁	#1957	1	1	yes	characteristic
135	C₂×C₁₀₁	#1960	1	1	yes	characteristic
136	C₂×C₁₀₁	#1988	1	2	yes	—
137	C₂×C₁₀₁	#2093	1	2	yes	—
138	C₂×C₁₀₁	#2096	1	2	yes	—
139	C₂×C₁₀₁	#2099	1	4	yes	—
140	C₂×#384	C₂×C₁₀₁	1	1	yes	characteristic
141	C₂×#340	C₂×C₁₀₁	1	4	yes	—
142	C₂×#336	C₂×C₁₀₁	1	1	yes	characteristic
143	C₄⋊#95	C₂×C₁₀₁	1	1	yes	characteristic
144	C₁₀₁	#11373	1	1	yes	characteristic, direct product factor, Sylow
145	C₂³×C₈	C₂²×C₁₀₁	1	2	yes	—
146	C₂²×C₄⋊C₄	—	2	2	no	—
147	C₂²×C₄⋊C₄	C₂²×C₁₀₁	1	2	yes	—
148	C₂²×C₄⋊C₄	C₂²×C₁₀₁	1	2	yes	—
149	C₂²×C₄²	C₂²×C₁₀₁	1	1	yes	characteristic
150	#384	—	2	2	no	—
151	#384	C₄×C₁₀₁	1	4	yes	complemented
152	C₂×#95	—	2	2	no	—
153	#340	—	2	8	no	—
154	#336	—	2	2	no	—
155	C₂³×C₄	—	2	2	no	—
156	C₂³×C₄	—	2	2	no	—
157	C₂³×C₄	D₄×C₁₀₁	1	2	yes	—
158	C₂³×C₄	C₂³×C₁₀₁	1	1	yes	characteristic
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	2	4	no	—
162	C₂²×C₈	—	2	4	no	—
163	C₂²×C₈	D₄×C₁₀₁	1	4	yes	—
164	C₂×C₄⋊C₄	—	2	4	no	—
165	C₂×C₄⋊C₄	—	2	4	no	—
166	C₂×C₄⋊C₄	—	2	4	no	—
167	C₂×C₄⋊C₄	—	2	4	no	—
168	C₂×C₄⋊C₄	—	2	8	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	4	4	no	—
172	C₂×C₄⋊C₄	—	4	4	no	—
173	C₂×C₄⋊C₄	C₂×C₄×C₁₀₁	1	4	yes	—
174	C₂×C₄⋊C₄	Q₈×C₁₀₁	1	4	yes	—
175	C₂×C₄²	—	2	4	no	—
176	C₂×C₄²	C₂×C₄×C₁₀₁	1	2	yes	—
177	C₂×C₄²	D₄×C₁₀₁	1	2	yes	—
178	#95	—	4	4	no	—
179	C₂⁴	C₂×D₄×C₁₀₁	1	1	yes	characteristic
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	2	no	—
186	C₂²×C₄	—	4	2	no	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	—	4	4	no	—
189	C₂²×C₄	—	4	4	no	—
190	C₂²×C₄	—	4	4	no	—
191	C₂²×C₄	—	4	4	no	—
192	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	4	yes	—
193	C₂²×C₄	D₈×C₁₀₁	1	4	yes	—
194	C₂²×C₄	SD₁₆×C₁₀₁	1	4	yes	—
195	C₂²×C₄	C₂²×C₄×C₁₀₁	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₄×C₁₀₁	1	2	yes	—
197	C₂²×C₄	C₂×D₄×C₁₀₁	1	1	yes	characteristic
198	C₂²×C₄	C₂×Q₈×C₁₀₁	1	2	yes	—
199	C₂²×C₄	#41×C₁₀₁	1	4	yes	—
200	C₂²×C₄	#41×C₁₀₁	1	2	yes	—
201	C₂²×C₄	#41×C₁₀₁	1	2	yes	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	2	8	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	4	no	—
207	C₄⋊C₄	—	2	8	no	—
208	C₄⋊C₄	—	2	8	no	—
209	C₄⋊C₄	—	4	16	no	—
210	C₄⋊C₄	—	4	4	no	—
211	C₄⋊C₄	—	4	4	no	—
212	C₄²	—	2	4	no	—
213	C₄²	—	2	4	no	—
214	C₂³	C₂×C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
215	C₂³	C₂²⋊Q₈×C₁₀₁	1	2	yes	—
216	C₂³	#123×C₁₀₁	1	2	yes	—
217	C₂³	#124×C₁₀₁	1	1	yes	characteristic
218	C₂³	#127×C₁₀₁	1	1	yes	characteristic
219	C₂³	C₂×D₈×C₁₀₁	1	2	yes	—
220	C₂³	C₂×SD₁₆×C₁₀₁	1	2	yes	—
221	C₂³	#137×C₁₀₁	1	4	yes	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	8	no	—
228	C₂×C₄	—	4	8	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	8	4	no	—
231	C₂×C₄	—	8	4	no	—
232	C₂×C₄	—	8	2	no	—
233	C₂×C₄	—	8	2	no	—
234	C₂×C₄	—	8	2	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	#102×C₁₀₁	1	8	yes	—
237	C₂×C₄	#117×C₁₀₁	1	2	yes	—
238	C₂×C₄	C₄×D₄×C₁₀₁	1	2	yes	—
239	C₂×C₄	C₄×Q₈×C₁₀₁	1	2	yes	—
240	C₂×C₄	#121×C₁₀₁	1	2	yes	—
241	C₂×C₄	C₂²⋊Q₈×C₁₀₁	1	4	yes	—
242	C₂×C₄	#123×C₁₀₁	1	2	yes	—
243	C₂×C₄	#123×C₁₀₁	1	4	yes	—
244	C₂×C₄	#124×C₁₀₁	1	1	yes	characteristic
245	C₂×C₄	#125×C₁₀₁	1	4	yes	—
246	C₂×C₄	#126×C₁₀₁	1	4	yes	—
247	C₂×C₄	C₄⋊Q₈×C₁₀₁	1	1	yes	characteristic
248	C₈	—	4	4	no	—
249	C₂²	#381×C₁₀₁	1	1	yes	characteristic
250	C₂²	#390×C₁₀₁	1	1	yes	characteristic
251	C₂²	#391×C₁₀₁	1	1	yes	characteristic
252	C₂²	#399×C₁₀₁	1	2	yes	—
253	C₂²	#400×C₁₀₁	1	2	yes	—
254	C₂²	C₂×#102×C₁₀₁	1	2	yes	—
255	C₂²	#419×C₁₀₁	1	2	yes	—
256	C₂²	#474×C₁₀₁	1	4	yes	—
257	C₂²	#475×C₁₀₁	1	4	yes	—
258	C₂²	#480×C₁₀₁	1	4	yes	—
259	C₂²	#483×C₁₀₁	1	4	yes	—
260	C₂²	#486×C₁₀₁	1	2	yes	—
261	C₂²	#491×C₁₀₁	1	2	yes	—
262	C₂²	#492×C₁₀₁	1	1	yes	characteristic
263	C₂²	#493×C₁₀₁	1	1	yes	characteristic
264	C₂²	#497×C₁₀₁	1	2	yes	—
265	C₄	—	2	8	no	—
266	C₄	—	8	4	no	—
267	C₄	—	8	4	no	—
268	C₄	#387×C₁₀₁	1	4	yes	—
269	C₄	#397×C₁₀₁	1	2	yes	—
270	C₄	#398×C₁₀₁	1	2	yes	—
271	C₂	C₁₀₁×#1445	1	1	yes	characteristic
272	C₂	C₁₀₁×#1823	1	2	yes	—
273	C₂	C₁₀₁×#1957	1	1	yes	characteristic
274	C₂	C₁₀₁×#1960	1	1	yes	characteristic
275	C₂	C₁₀₁×#1988	1	2	yes	—
276	C₂	C₁₀₁×#2093	1	2	yes	—
277	C₂	C₁₀₁×#2096	1	2	yes	—
278	C₂	C₁₀₁×#2099	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
11. #384×C₁₀₁	C₄	semidirect
131. #11373	C₁₀₁	direct (N × H)
144. C₁₀₁	#11373	direct (N × H)
151. #384	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×#384×C₁₀₁	2
C₂×#340×C₁₀₁	2
C₂×#336×C₁₀₁	2
C₁₀₁×C₄⋊#95	2
#11373	101

Subgroups — C101×#11373 #38793524

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#11373 #38793524