Subgroups — C₁₀₁×#23542 #38805693

Order 25856.

1362 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	C₂³×Q₁₆×C₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#424×C₁₀₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#419×C₁₀₁	C₂	1	2	yes	maximal
4	C₁₀₁×#2310	C₂	1	2	yes	maximal
5	C₁₀₁×#2056	C₂	1	8	yes	maximal, complemented
6	C₁₀₁×#1735	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₁₀₁	C₂²	1	2	yes	—
8	C₂²×Q₁₆×C₁₀₁	—	2	8	no	—
9	C₂²×Q₁₆×C₁₀₁	—	2	4	no	—
10	C₂²×Q₁₆×C₁₀₁	C₂²	1	4	yes	—
11	C₂²×M₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
13	C₂×#117×C₁₀₁	C₂²	1	2	yes	—
14	C₂×#117×C₁₀₁	C₂²	1	1	yes	characteristic
15	#424×C₁₀₁	—	2	4	no	—
16	C₂×C₄⋊C₈×C₁₀₁	C₂²	1	4	yes	—
17	#419×C₁₀₁	—	2	8	no	—
18	C₂×#103×C₁₀₁	C₂²	1	8	yes	—
19	#391×C₁₀₁	—	2	4	no	—
20	#391×C₁₀₁	C₂²	1	8	yes	—
21	#381×C₁₀₁	—	2	4	no	—
22	#340×C₁₀₁	C₂²	1	4	yes	—
23	C₂²×Q₈×C₁₀₁	—	2	2	no	—
24	C₂²×Q₈×C₁₀₁	—	2	4	no	—
25	C₂²×Q₈×C₁₀₁	—	2	4	no	—
26	C₂²×Q₈×C₁₀₁	—	2	4	no	—
27	C₂²×Q₈×C₁₀₁	—	2	4	no	—
28	C₂²×Q₈×C₁₀₁	—	2	4	no	—
29	C₂²×Q₈×C₁₀₁	D₄	1	8	yes	—
30	C₂²×Q₈×C₁₀₁	C₂³	1	4	yes	—
31	C₂³×C₄×C₁₀₁	—	2	2	no	—
32	C₂³×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
33	C₂×Q₁₆×C₁₀₁	—	2	32	no	—
34	C₂×Q₁₆×C₁₀₁	—	2	16	no	—
35	C₂×Q₁₆×C₁₀₁	—	2	8	no	—
36	C₂×M₁₆×C₁₀₁	—	2	4	no	—
37	C₂×M₁₆×C₁₀₁	—	2	2	no	—
38	C₂²×C₈×C₁₀₁	—	2	4	no	—
39	C₂²×C₈×C₁₀₁	D₄	1	4	yes	—
40	C₂²×C₈×C₁₀₁	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₀₁	C₂³	1	2	yes	—
42	#117×C₁₀₁	—	2	2	no	—
43	#117×C₁₀₁	—	2	2	no	—
44	#117×C₁₀₁	—	2	4	no	—
45	#117×C₁₀₁	—	2	4	no	—
46	C₂×C₄⋊C₄×C₁₀₁	D₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
48	C₂×C₂²⋊C₄×C₁₀₁	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₁₀₁	D₄	1	2	yes	—
50	C₂×C₄²×C₁₀₁	—	2	2	no	—
51	C₂×C₄²×C₁₀₁	C₂³	1	2	yes	—
52	C₄⋊C₈×C₁₀₁	—	2	8	no	—
53	#103×C₁₀₁	—	2	16	no	—
54	#95×C₁₀₁	—	2	16	no	—
55	C₂⁴×C₁₀₁	C₂×D₄	1	1	yes	characteristic
56	C₂×Q₈×C₁₀₁	—	2	8	no	—
57	C₂×Q₈×C₁₀₁	—	2	4	no	—
58	C₂×Q₈×C₁₀₁	—	2	16	no	—
59	C₂×Q₈×C₁₀₁	—	2	16	no	—
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	—	4	4	no	—
65	C₂×Q₈×C₁₀₁	—	4	4	no	—
66	C₂×Q₈×C₁₀₁	—	4	8	no	—
67	C₂×Q₈×C₁₀₁	—	4	8	no	—
68	C₂×Q₈×C₁₀₁	C₂×D₄	1	8	yes	—
69	C₂²×C₄×C₁₀₁	—	2	8	no	—
70	C₂²×C₄×C₁₀₁	—	2	4	no	—
71	C₂²×C₄×C₁₀₁	—	2	4	no	—
72	C₂²×C₄×C₁₀₁	—	4	4	no	—
73	C₂²×C₄×C₁₀₁	—	4	4	no	—
74	C₂²×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
76	C₂²×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₀₁	#41	1	2	yes	—
79	C₂²×C₄×C₁₀₁	#41	1	4	yes	—
80	C₂²×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
81	Q₁₆×C₁₀₁	—	2	32	no	—
82	M₁₆×C₁₀₁	—	4	4	no	—
83	C₂×C₈×C₁₀₁	—	2	8	no	—
84	C₂×C₈×C₁₀₁	—	2	4	no	—
85	C₂×C₈×C₁₀₁	—	2	4	no	—
86	C₂×C₈×C₁₀₁	—	2	2	no	—
87	C₂×C₈×C₁₀₁	C₂×D₄	1	4	yes	—
88	C₄⋊C₄×C₁₀₁	—	2	4	no	—
89	C₄⋊C₄×C₁₀₁	—	2	4	no	—
90	C₄⋊C₄×C₁₀₁	—	2	4	no	—
91	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
92	C₂²⋊C₄×C₁₀₁	—	4	4	no	—
93	C₄²×C₁₀₁	—	2	2	no	—
94	C₄²×C₁₀₁	—	2	2	no	—
95	C₄²×C₁₀₁	—	4	4	no	—
96	C₂³×C₁₀₁	—	2	4	no	—
97	C₂³×C₁₀₁	#121	1	2	yes	—
98	C₂³×C₁₀₁	#121	1	4	yes	—
99	C₂³×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
100	Q₈×C₁₀₁	—	2	8	no	—
101	Q₈×C₁₀₁	—	2	8	no	—
102	Q₈×C₁₀₁	—	4	4	no	—
103	Q₈×C₁₀₁	—	4	4	no	—
104	Q₈×C₁₀₁	—	4	16	no	—
105	C₂×C₄×C₁₀₁	—	2	4	no	—
106	C₂×C₄×C₁₀₁	—	2	8	no	—
107	C₂×C₄×C₁₀₁	—	2	8	no	—
108	C₂×C₄×C₁₀₁	—	4	4	no	—
109	C₂×C₄×C₁₀₁	—	4	4	no	—
110	C₂×C₄×C₁₀₁	—	4	16	no	—
111	C₂×C₄×C₁₀₁	—	4	4	no	—
112	C₂×C₄×C₁₀₁	—	4	8	no	—
113	C₂×C₄×C₁₀₁	#120	1	4	yes	—
114	C₂×C₄×C₁₀₁	#121	1	4	yes	—
115	C₂×C₄×C₁₀₁	#121	1	2	yes	—
116	C₂×C₄×C₁₀₁	#123	1	8	yes	—
117	C₂×C₄×C₁₀₁	#124	1	4	yes	—
118	C₂×C₄×C₁₀₁	C₂²×D₄	1	2	yes	—
119	C₂×C₄×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₀₁	C₂×#41	1	1	yes	characteristic
121	C₂×C₄×C₁₀₁	C₂×#41	1	2	yes	—
122	C₈×C₁₀₁	—	2	4	no	—
123	C₈×C₁₀₁	—	4	2	no	—
124	C₂²×C₁₀₁	—	2	8	no	—
125	C₂²×C₁₀₁	—	2	4	no	—
126	C₂²×C₁₀₁	#394	1	4	yes	—
127	C₂²×C₁₀₁	C₂×#121	1	2	yes	—
128	C₂²×C₁₀₁	C₂×#121	1	1	yes	characteristic
129	C₂²×C₁₀₁	#578	1	4	yes	—
130	C₄×C₁₀₁	—	4	8	no	—
131	C₄×C₁₀₁	—	4	4	no	—
132	C₄×C₁₀₁	—	8	4	no	—
133	C₄×C₁₀₁	#394	1	4	yes	—
134	C₄×C₁₀₁	C₂×#120	1	1	yes	characteristic
135	C₄×C₁₀₁	C₂×#123	1	2	yes	—
136	C₄×C₁₀₁	C₂×#124	1	1	yes	characteristic
137	#23542	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₂×C₁₀₁	—	2	4	no	—
139	C₂×C₁₀₁	C₂×#394	1	1	yes	characteristic
140	C₂×C₁₀₁	#3044	1	4	yes	—
141	C₂×C₁₀₁	#3059	1	2	yes	—
142	C₂³×Q₁₆	C₂×C₁₀₁	1	1	yes	characteristic
143	C₂×#424	C₂×C₁₀₁	1	1	yes	characteristic
144	C₂×#419	C₂×C₁₀₁	1	2	yes	—
145	#2310	C₂×C₁₀₁	1	2	yes	—
146	#2056	C₂×C₁₀₁	1	8	yes	complemented
147	#1735	C₂×C₁₀₁	1	1	yes	characteristic
148	C₁₀₁	#23542	1	1	yes	characteristic, direct product factor, Sylow
149	C₂³×Q₈	C₂²×C₁₀₁	1	2	yes	—
150	C₂²×Q₁₆	—	2	8	no	—
151	C₂²×Q₁₆	—	2	4	no	—
152	C₂²×Q₁₆	C₂²×C₁₀₁	1	4	yes	—
153	C₂²×M₁₆	C₂²×C₁₀₁	1	1	yes	characteristic
154	C₂³×C₈	C₂²×C₁₀₁	1	1	yes	characteristic
155	C₂×#117	C₂²×C₁₀₁	1	2	yes	—
156	C₂×#117	C₂²×C₁₀₁	1	1	yes	characteristic
157	#424	—	2	4	no	—
158	C₂×C₄⋊C₈	C₂²×C₁₀₁	1	4	yes	—
159	#419	—	2	8	no	—
160	C₂×#103	C₂²×C₁₀₁	1	8	yes	—
161	#391	—	2	4	no	—
162	#391	C₂²×C₁₀₁	1	8	yes	—
163	#381	—	2	4	no	—
164	#340	C₂²×C₁₀₁	1	4	yes	—
165	C₂²×Q₈	—	2	2	no	—
166	C₂²×Q₈	—	2	4	no	—
167	C₂²×Q₈	—	2	4	no	—
168	C₂²×Q₈	—	2	4	no	—
169	C₂²×Q₈	—	2	4	no	—
170	C₂²×Q₈	—	2	4	no	—
171	C₂²×Q₈	D₄×C₁₀₁	1	8	yes	—
172	C₂²×Q₈	C₂³×C₁₀₁	1	4	yes	—
173	C₂³×C₄	—	2	2	no	—
174	C₂³×C₄	C₂³×C₁₀₁	1	1	yes	characteristic
175	C₂×Q₁₆	—	2	32	no	—
176	C₂×Q₁₆	—	2	16	no	—
177	C₂×Q₁₆	—	2	8	no	—
178	C₂×M₁₆	—	2	4	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂²×C₈	—	2	4	no	—
181	C₂²×C₈	D₄×C₁₀₁	1	4	yes	—
182	C₂²×C₈	C₂³×C₁₀₁	1	2	yes	—
183	C₂²×C₈	C₂³×C₁₀₁	1	2	yes	—
184	#117	—	2	2	no	—
185	#117	—	2	2	no	—
186	#117	—	2	4	no	—
187	#117	—	2	4	no	—
188	C₂×C₄⋊C₄	D₄×C₁₀₁	1	2	yes	—
189	C₂×C₄⋊C₄	C₂³×C₁₀₁	1	4	yes	—
190	C₂×C₂²⋊C₄	—	2	2	no	—
191	C₂×C₂²⋊C₄	D₄×C₁₀₁	1	2	yes	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	C₂³×C₁₀₁	1	2	yes	—
194	C₄⋊C₈	—	2	8	no	—
195	#103	—	2	16	no	—
196	#95	—	2	16	no	—
197	C₂⁴	C₂×D₄×C₁₀₁	1	1	yes	characteristic
198	C₂×Q₈	—	2	8	no	—
199	C₂×Q₈	—	2	4	no	—
200	C₂×Q₈	—	2	16	no	—
201	C₂×Q₈	—	2	16	no	—
202	C₂×Q₈	—	2	4	no	—
203	C₂×Q₈	—	2	4	no	—
204	C₂×Q₈	—	2	4	no	—
205	C₂×Q₈	—	2	4	no	—
206	C₂×Q₈	—	4	4	no	—
207	C₂×Q₈	—	4	4	no	—
208	C₂×Q₈	—	4	8	no	—
209	C₂×Q₈	—	4	8	no	—
210	C₂×Q₈	C₂×D₄×C₁₀₁	1	8	yes	—
211	C₂²×C₄	—	2	8	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	4	4	no	—
215	C₂²×C₄	—	4	4	no	—
216	C₂²×C₄	C₂×D₄×C₁₀₁	1	4	yes	—
217	C₂²×C₄	C₂×D₄×C₁₀₁	1	4	yes	—
218	C₂²×C₄	C₂×D₄×C₁₀₁	1	2	yes	—
219	C₂²×C₄	C₂×D₄×C₁₀₁	1	1	yes	characteristic
220	C₂²×C₄	#41×C₁₀₁	1	2	yes	—
221	C₂²×C₄	#41×C₁₀₁	1	4	yes	—
222	C₂²×C₄	C₂⁴×C₁₀₁	1	1	yes	characteristic
223	Q₁₆	—	2	32	no	—
224	M₁₆	—	4	4	no	—
225	C₂×C₈	—	2	8	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	C₂×D₄×C₁₀₁	1	4	yes	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₂²⋊C₄	—	4	4	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	4	4	no	—
238	C₂³	—	2	4	no	—
239	C₂³	#121×C₁₀₁	1	2	yes	—
240	C₂³	#121×C₁₀₁	1	4	yes	—
241	C₂³	C₂²×D₄×C₁₀₁	1	1	yes	characteristic
242	Q₈	—	2	8	no	—
243	Q₈	—	2	8	no	—
244	Q₈	—	4	4	no	—
245	Q₈	—	4	4	no	—
246	Q₈	—	4	16	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	8	no	—
249	C₂×C₄	—	2	8	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	16	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	8	no	—
255	C₂×C₄	#120×C₁₀₁	1	4	yes	—
256	C₂×C₄	#121×C₁₀₁	1	4	yes	—
257	C₂×C₄	#121×C₁₀₁	1	2	yes	—
258	C₂×C₄	#123×C₁₀₁	1	8	yes	—
259	C₂×C₄	#124×C₁₀₁	1	4	yes	—
260	C₂×C₄	C₂²×D₄×C₁₀₁	1	2	yes	—
261	C₂×C₄	C₂²×D₄×C₁₀₁	1	1	yes	characteristic
262	C₂×C₄	C₂×#41×C₁₀₁	1	1	yes	characteristic
263	C₂×C₄	C₂×#41×C₁₀₁	1	2	yes	—
264	C₈	—	2	4	no	—
265	C₈	—	4	2	no	—
266	C₂²	—	2	8	no	—
267	C₂²	—	2	4	no	—
268	C₂²	#394×C₁₀₁	1	4	yes	—
269	C₂²	C₂×#121×C₁₀₁	1	2	yes	—
270	C₂²	C₂×#121×C₁₀₁	1	1	yes	characteristic
271	C₂²	#578×C₁₀₁	1	4	yes	—
272	C₄	—	4	4	no	—
273	C₄	—	4	8	no	—
274	C₄	—	8	4	no	—
275	C₄	#394×C₁₀₁	1	4	yes	—
276	C₄	C₂×#120×C₁₀₁	1	1	yes	characteristic
277	C₄	C₂×#123×C₁₀₁	1	2	yes	—
278	C₄	C₂×#124×C₁₀₁	1	1	yes	characteristic
279	C₂	—	2	4	no	—
280	C₂	C₂×#394×C₁₀₁	1	1	yes	characteristic
281	C₂	C₁₀₁×#3044	1	4	yes	—
282	C₂	C₁₀₁×#3059	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₀₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₁
prime radical	C₂⁴×C₁₀₁	C₂×D₄
LCS	G ▷ C₂²×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×#2056	C₂	semidirect
137. #23542	C₁₀₁	direct (N × H)
146. #2056	C₂×C₁₀₁	semidirect
148. C₁₀₁	#23542	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Q₁₆×C₁₀₁	2
C₂×#424×C₁₀₁	2
C₂×#419×C₁₀₁	2
C₁₀₁×#2310	2
C₁₀₁×#2056	2
C₁₀₁×#1735	2
#23542	101

Subgroups — C101×#23542 #38805693

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#23542 #38805693