Subgroups — C₁₀₁×#8688 #38790839

Order 25856.

414 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×#2931	C₂	1	1	yes	maximal, characteristic
2	C₁₀₁×#1837	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#335×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₁×#1547	C₂	1	2	yes	maximal
5	C₁₀₁×#1292	C₂	1	2	yes	maximal, complemented
6	C₂×#131×C₁₀₁	—	2	1	no	—
7	#532×C₁₀₁	C₂²	1	1	yes	characteristic
8	#436×C₁₀₁	—	2	1	no	—
9	#436×C₁₀₁	C₄	1	2	yes	—
10	C₈×D₄×C₁₀₁	—	2	1	no	—
11	C₈×D₄×C₁₀₁	C₄	1	2	yes	—
12	#423×C₁₀₁	C₂²	1	2	yes	—
13	#423×C₁₀₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₁₀₁	C₂²	1	2	yes	—
15	#408×C₁₀₁	—	2	1	no	—
16	C₂×C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
17	#341×C₁₀₁	—	2	1	no	—
18	#341×C₁₀₁	—	2	1	no	—
19	#335×C₁₀₁	—	2	1	no	—
20	#335×C₁₀₁	—	2	1	no	—
21	#332×C₁₀₁	—	2	2	no	—
22	#331×C₁₀₁	—	2	2	no	—
23	C₂×#41×C₁₀₁	—	2	1	no	—
24	#131×C₁₀₁	—	4	4	no	—
25	C₂×M₁₆×C₁₀₁	—	2	1	no	—
26	C₂×M₁₆×C₁₀₁	—	2	2	no	—
27	C₂×M₁₆×C₁₀₁	—	2	2	no	—
28	C₂²×C₈×C₁₀₁	—	2	2	no	—
29	C₂²×C₈×C₁₀₁	—	2	2	no	—
30	C₂²×C₈×C₁₀₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₀₁	D₄	1	1	yes	characteristic
32	C₄⋊Q₈×C₁₀₁	D₄	1	1	yes	characteristic
33	#127×C₁₀₁	D₄	1	1	yes	characteristic
34	#124×C₁₀₁	—	2	1	no	—
35	#121×C₁₀₁	—	2	1	no	—
36	#121×C₁₀₁	Q₈	1	2	yes	—
37	C₄×D₄×C₁₀₁	—	2	1	no	—
38	C₄×D₄×C₁₀₁	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₁₀₁	—	2	1	no	—
41	C₄⋊C₈×C₁₀₁	—	2	2	no	—
42	C₄⋊C₈×C₁₀₁	—	2	2	no	—
43	C₄⋊C₈×C₁₀₁	—	2	2	no	—
44	C₄⋊C₈×C₁₀₁	—	2	2	no	—
45	C₄⋊C₈×C₁₀₁	C₂×C₄	1	2	yes	—
46	C₂²⋊C₈×C₁₀₁	—	2	2	no	—
47	C₂²⋊C₈×C₁₀₁	—	2	2	no	—
48	C₄×C₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₁₀₁	D₄	1	1	yes	characteristic
51	C₄×C₈×C₁₀₁	D₄	1	1	yes	characteristic
52	#41×C₁₀₁	—	4	2	no	—
53	#41×C₁₀₁	—	4	2	no	—
54	C₂×Q₈×C₁₀₁	—	2	1	no	—
55	C₂×D₄×C₁₀₁	—	2	2	no	—
56	C₂×D₄×C₁₀₁	—	2	1	no	—
57	C₂×D₄×C₁₀₁	—	2	1	no	—
58	C₂×D₄×C₁₀₁	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₁₀₁	—	2	2	no	—
60	C₂²×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₀₁	#41	1	1	yes	characteristic
63	M₁₆×C₁₀₁	—	4	4	no	—
64	M₁₆×C₁₀₁	—	4	2	no	—
65	M₁₆×C₁₀₁	—	4	2	no	—
66	M₁₆×C₁₀₁	—	4	2	no	—
67	C₂×C₈×C₁₀₁	—	2	2	no	—
68	C₂×C₈×C₁₀₁	—	2	2	no	—
69	C₂×C₈×C₁₀₁	—	2	2	no	—
70	C₂×C₈×C₁₀₁	—	4	4	no	—
71	C₂×C₈×C₁₀₁	—	4	2	no	—
72	C₂×C₈×C₁₀₁	—	4	2	no	—
73	C₂×C₈×C₁₀₁	—	8	2	no	—
74	C₂×C₈×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₄⋊C₄×C₁₀₁	—	2	1	no	—
79	C₄⋊C₄×C₁₀₁	C₄⋊C₄	1	2	yes	—
80	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
81	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
82	C₄²×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₁₀₁	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₁₀₁	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₁₀₁	#41	1	1	yes	characteristic
86	C₂³×C₁₀₁	—	2	2	no	—
87	C₂³×C₁₀₁	C₂²⋊Q₈	1	1	yes	characteristic
88	Q₈×C₁₀₁	—	4	2	no	—
89	D₄×C₁₀₁	—	4	2	no	—
90	D₄×C₁₀₁	—	4	4	no	—
91	D₄×C₁₀₁	—	4	2	no	—
92	D₄×C₁₀₁	—	4	1	no	—
93	D₄×C₁₀₁	—	4	1	no	—
94	C₂×C₄×C₁₀₁	—	2	1	no	—
95	C₂×C₄×C₁₀₁	—	2	1	no	—
96	C₂×C₄×C₁₀₁	—	2	2	no	—
97	C₂×C₄×C₁₀₁	—	4	1	no	—
98	C₂×C₄×C₁₀₁	—	4	2	no	—
99	C₂×C₄×C₁₀₁	—	4	2	no	—
100	C₂×C₄×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₀₁	#117	1	1	yes	characteristic
103	C₂×C₄×C₁₀₁	#121	1	1	yes	characteristic
104	C₂×C₄×C₁₀₁	#121	1	1	yes	characteristic
105	C₂×C₄×C₁₀₁	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₁₀₁	#135	1	2	yes	—
107	C₂×C₄×C₁₀₁	AGL(1,ℤ/8)	1	2	yes	—
108	C₈×C₁₀₁	—	2	1	no	—
109	C₈×C₁₀₁	—	2	1	no	—
110	C₈×C₁₀₁	—	2	1	no	—
111	C₈×C₁₀₁	—	2	1	no	—
112	C₈×C₁₀₁	—	4	2	no	—
113	C₈×C₁₀₁	—	8	2	no	—
114	C₈×C₁₀₁	—	8	2	no	—
115	C₂²×C₁₀₁	—	2	1	no	—
116	C₂²×C₁₀₁	—	2	1	no	—
117	C₂²×C₁₀₁	—	4	2	no	—
118	C₂²×C₁₀₁	—	4	2	no	—
119	C₂²×C₁₀₁	—	4	2	no	—
120	C₂²×C₁₀₁	#380	1	1	yes	characteristic
121	C₂²×C₁₀₁	#478	1	2	yes	—
122	C₄×C₁₀₁	—	2	1	no	—
123	C₄×C₁₀₁	—	2	1	no	—
124	C₄×C₁₀₁	—	2	1	no	—
125	C₄×C₁₀₁	—	4	2	no	—
126	C₄×C₁₀₁	#416	1	1	yes	characteristic
127	C₄×C₁₀₁	#418	1	1	yes	characteristic
128	C₄×C₁₀₁	#463	1	2	yes	—
129	C₄×C₁₀₁	#472	1	1	yes	characteristic
130	C₄×C₁₀₁	#473	1	1	yes	characteristic
131	#8688	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂×C₁₀₁	—	2	2	no	—
133	C₂×C₁₀₁	—	8	2	no	—
134	C₂×C₁₀₁	#1804	1	1	yes	characteristic
135	C₂×C₁₀₁	#1813	1	1	yes	characteristic
136	C₂×C₁₀₁	#1816	1	1	yes	characteristic
137	#2931	C₂×C₁₀₁	1	1	yes	characteristic
138	#1837	C₂×C₁₀₁	1	1	yes	characteristic, complemented
139	C₂×#335	C₂×C₁₀₁	1	1	yes	characteristic, complemented
140	#1547	C₂×C₁₀₁	1	2	yes	—
141	#1292	C₂×C₁₀₁	1	2	yes	complemented
142	C₁₀₁	#8688	1	1	yes	characteristic, direct product factor, Sylow
143	C₂×#131	—	2	1	no	—
144	#532	C₂²×C₁₀₁	1	1	yes	characteristic
145	#436	—	2	1	no	—
146	#436	C₄×C₁₀₁	1	2	yes	—
147	C₈×D₄	—	2	1	no	—
148	C₈×D₄	C₄×C₁₀₁	1	2	yes	—
149	#423	C₂²×C₁₀₁	1	2	yes	—
150	#423	C₂²×C₁₀₁	1	1	yes	characteristic
151	C₂×C₄⋊C₈	C₂²×C₁₀₁	1	2	yes	—
152	#408	—	2	1	no	—
153	C₂×C₄×C₈	C₂²×C₁₀₁	1	1	yes	characteristic
154	#341	—	2	1	no	—
155	#341	—	2	1	no	—
156	#335	—	2	1	no	—
157	#335	—	2	1	no	—
158	#332	—	2	2	no	—
159	#331	—	2	2	no	—
160	C₂×#41	—	2	1	no	—
161	#131	—	4	4	no	—
162	C₂×M₁₆	—	2	1	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	D₄×C₁₀₁	1	1	yes	characteristic
168	C₂²×C₈	D₄×C₁₀₁	1	1	yes	characteristic
169	C₄⋊Q₈	D₄×C₁₀₁	1	1	yes	characteristic
170	#127	D₄×C₁₀₁	1	1	yes	characteristic
171	#124	—	2	1	no	—
172	#121	—	2	1	no	—
173	#121	Q₈×C₁₀₁	1	2	yes	—
174	C₄×D₄	—	2	1	no	—
175	C₄×D₄	C₂×C₄×C₁₀₁	1	2	yes	—
176	C₂×C₄²	C₂³×C₁₀₁	1	1	yes	characteristic
177	C₄⋊C₈	—	2	1	no	—
178	C₄⋊C₈	—	2	2	no	—
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	—	2	2	no	—
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	C₂×C₄×C₁₀₁	1	2	yes	—
183	C₂²⋊C₈	—	2	2	no	—
184	C₂²⋊C₈	—	2	2	no	—
185	C₄×C₈	C₂×C₄×C₁₀₁	1	1	yes	characteristic
186	C₄×C₈	C₂×C₄×C₁₀₁	1	1	yes	characteristic
187	C₄×C₈	D₄×C₁₀₁	1	1	yes	characteristic
188	C₄×C₈	D₄×C₁₀₁	1	1	yes	characteristic
189	#41	—	4	2	no	—
190	#41	—	4	2	no	—
191	C₂×Q₈	—	2	1	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	2	1	no	—
194	C₂×D₄	—	2	1	no	—
195	C₂×D₄	C₄⋊C₄×C₁₀₁	1	2	yes	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	C₂×D₄×C₁₀₁	1	1	yes	characteristic
198	C₂²×C₄	C₂×Q₈×C₁₀₁	1	1	yes	characteristic
199	C₂²×C₄	#41×C₁₀₁	1	1	yes	characteristic
200	M₁₆	—	4	4	no	—
201	M₁₆	—	4	2	no	—
202	M₁₆	—	4	2	no	—
203	M₁₆	—	4	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	4	4	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	—	8	2	no	—
211	C₂×C₈	C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
212	C₂×C₈	C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
213	C₂×C₈	C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
214	C₂×C₈	C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
215	C₄⋊C₄	—	2	1	no	—
216	C₄⋊C₄	C₄⋊C₄×C₁₀₁	1	2	yes	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	—	2	2	no	—
219	C₄²	C₂²×C₄×C₁₀₁	1	1	yes	characteristic
220	C₄²	C₂×D₄×C₁₀₁	1	1	yes	characteristic
221	C₄²	C₂×D₄×C₁₀₁	1	1	yes	characteristic
222	C₄²	#41×C₁₀₁	1	1	yes	characteristic
223	C₂³	—	2	2	no	—
224	C₂³	C₂²⋊Q₈×C₁₀₁	1	1	yes	characteristic
225	Q₈	—	4	2	no	—
226	D₄	—	4	2	no	—
227	D₄	—	4	4	no	—
228	D₄	—	4	2	no	—
229	D₄	—	4	1	no	—
230	D₄	—	4	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄⋊C₄×C₁₀₁	1	1	yes	characteristic
239	C₂×C₄	#117×C₁₀₁	1	1	yes	characteristic
240	C₂×C₄	#121×C₁₀₁	1	1	yes	characteristic
241	C₂×C₄	#121×C₁₀₁	1	1	yes	characteristic
242	C₂×C₄	C₂²⋊Q₈×C₁₀₁	1	1	yes	characteristic
243	C₂×C₄	#135×C₁₀₁	1	2	yes	—
244	C₂×C₄	AGL(1,ℤ/8)×C₁₀₁	1	2	yes	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	—	2	1	no	—
249	C₈	—	4	2	no	—
250	C₈	—	8	2	no	—
251	C₈	—	8	2	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	#380×C₁₀₁	1	1	yes	characteristic
258	C₂²	#478×C₁₀₁	1	2	yes	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	4	2	no	—
263	C₄	#416×C₁₀₁	1	1	yes	characteristic
264	C₄	#418×C₁₀₁	1	1	yes	characteristic
265	C₄	#463×C₁₀₁	1	2	yes	—
266	C₄	#472×C₁₀₁	1	1	yes	characteristic
267	C₄	#473×C₁₀₁	1	1	yes	characteristic
268	C₂	—	2	2	no	—
269	C₂	—	8	2	no	—
270	C₂	C₁₀₁×#1804	1	1	yes	characteristic
271	C₂	C₁₀₁×#1813	1	1	yes	characteristic
272	C₂	C₁₀₁×#1816	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₁	#380
commutator	C₄²	C₂²×C₄×C₁₀₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₁
prime radical	#532×C₁₀₁	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₁ ◁ C₂×C₄²×C₁₀₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₀₁×#1837	C₂	semidirect
3. C₂×#335×C₁₀₁	C₂	semidirect
5. C₁₀₁×#1292	C₂	semidirect
131. #8688	C₁₀₁	direct (N × H)
138. #1837	C₂×C₁₀₁	semidirect
139. C₂×#335	C₂×C₁₀₁	semidirect
141. #1292	C₂×C₁₀₁	semidirect
142. C₁₀₁	#8688	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₀₁×#2931	2
C₁₀₁×#1837	2
C₂×#335×C₁₀₁	2
C₁₀₁×#1547	2
C₁₀₁×#1292	2
#8688	101

Subgroups — C101×#8688 #38790839

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#8688 #38790839