Subgroups — C₁₀₁×#21333 #38803484

Order 25856.

1338 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#543×C₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₁₀₁×#2315	C₂	1	1	yes	maximal, characteristic
3	C₁₀₁×#2273	C₂	1	1	yes	maximal, characteristic
4	C₁₀₁×#1996	C₂	1	8	yes	maximal, complemented
5	C₂×#342×C₁₀₁	C₂	1	4	yes	maximal
6	#543×C₁₀₁	—	2	4	no	—
7	#543×C₁₀₁	—	2	4	no	—
8	C₂×C₂²⋊Q₈×C₁₀₁	C₄	1	8	yes	complemented
9	C₂×C₂²⋊Q₈×C₁₀₁	C₂²	1	4	yes	—
10	C₂²×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
11	C₂²×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
12	C₂²×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
13	C₂²×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
14	#385×C₁₀₁	—	2	4	no	—
15	#380×C₁₀₁	C₂²	1	4	yes	—
16	C₄×C₂²⋊C₄×C₁₀₁	—	2	4	no	—
17	C₄×C₂²⋊C₄×C₁₀₁	C₂²	1	4	yes	—
18	#342×C₁₀₁	C₂²	1	16	yes	—
19	C₂⁵×C₁₀₁	C₂³	1	1	yes	characteristic
20	C₂²×Q₈×C₁₀₁	C₂×C₄	1	4	yes	complemented
21	C₂³×C₄×C₁₀₁	—	2	2	no	—
22	C₂³×C₄×C₁₀₁	—	2	2	no	—
23	C₂³×C₄×C₁₀₁	C₂×C₄	1	4	yes	—
24	C₂³×C₄×C₁₀₁	C₂³	1	2	yes	—
25	C₂²⋊Q₈×C₁₀₁	—	2	8	no	—
26	C₂²⋊Q₈×C₁₀₁	—	2	8	no	—
27	C₂²⋊Q₈×C₁₀₁	—	2	32	no	—
28	C₂×C₄⋊C₄×C₁₀₁	—	2	4	no	—
29	C₂×C₄⋊C₄×C₁₀₁	C₂×C₄	1	12	yes	complemented
30	C₂×C₂²⋊C₄×C₁₀₁	—	2	4	no	—
31	C₂×C₂²⋊C₄×C₁₀₁	—	2	4	no	—
32	C₂×C₂²⋊C₄×C₁₀₁	—	2	2	no	—
33	C₂×C₂²⋊C₄×C₁₀₁	—	2	4	no	—
34	C₂×C₂²⋊C₄×C₁₀₁	—	2	8	no	—
35	C₂×C₂²⋊C₄×C₁₀₁	—	2	2	no	—
36	C₂×C₂²⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	—
37	C₂×C₂²⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	—
38	C₂×C₂²⋊C₄×C₁₀₁	D₄	1	4	yes	—
39	C₂×C₂²⋊C₄×C₁₀₁	D₄	1	4	yes	—
40	C₂×C₂²⋊C₄×C₁₀₁	Q₈	1	8	yes	complemented
41	C₂×C₂²⋊C₄×C₁₀₁	C₂³	1	4	yes	—
42	C₂×C₂²⋊C₄×C₁₀₁	C₂³	1	4	yes	—
43	C₂×C₂²⋊C₄×C₁₀₁	C₂³	1	4	yes	—
44	C₂×C₄²×C₁₀₁	—	2	4	no	—
45	#95×C₁₀₁	—	2	4	no	—
46	#95×C₁₀₁	—	2	4	no	—
47	C₂⁴×C₁₀₁	—	8	2	no	—
48	C₂⁴×C₁₀₁	C₂²×C₄	1	2	yes	—
49	C₂⁴×C₁₀₁	C₂×D₄	1	2	yes	—
50	C₂⁴×C₁₀₁	C₂×D₄	1	2	yes	—
51	C₂⁴×C₁₀₁	C₂×Q₈	1	4	yes	—
52	C₂⁴×C₁₀₁	#41	1	4	yes	—
53	C₂⁴×C₁₀₁	C₂⁴	1	1	yes	characteristic
54	C₂×Q₈×C₁₀₁	—	2	8	no	—
55	C₂×Q₈×C₁₀₁	—	8	4	no	—
56	C₂²×C₄×C₁₀₁	—	2	4	no	—
57	C₂²×C₄×C₁₀₁	—	2	8	no	—
58	C₂²×C₄×C₁₀₁	—	2	8	no	—
59	C₂²×C₄×C₁₀₁	—	2	4	no	—
60	C₂²×C₄×C₁₀₁	—	4	4	no	—
61	C₂²×C₄×C₁₀₁	—	8	2	no	—
62	C₂²×C₄×C₁₀₁	—	8	4	no	—
63	C₂²×C₄×C₁₀₁	—	8	4	no	—
64	C₂²×C₄×C₁₀₁	—	8	4	no	—
65	C₂²×C₄×C₁₀₁	C₂²×C₄	1	8	yes	—
66	C₂²×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
68	C₂²×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
69	C₄⋊C₄×C₁₀₁	—	2	24	no	—
70	C₄⋊C₄×C₁₀₁	—	8	4	no	—
71	C₂²⋊C₄×C₁₀₁	—	2	8	no	—
72	C₂²⋊C₄×C₁₀₁	—	2	8	no	—
73	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
74	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
75	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
76	C₂²⋊C₄×C₁₀₁	—	2	8	no	—
77	C₂²⋊C₄×C₁₀₁	—	4	4	no	—
78	C₂²⋊C₄×C₁₀₁	C₂²⋊C₄	1	16	yes	—
79	C₂²⋊C₄×C₁₀₁	C₂×Q₈	1	8	yes	—
80	C₄²×C₁₀₁	—	8	4	no	—
81	C₂³×C₁₀₁	—	4	4	no	—
82	C₂³×C₁₀₁	—	4	4	no	—
83	C₂³×C₁₀₁	—	4	4	no	—
84	C₂³×C₁₀₁	—	4	2	no	—
85	C₂³×C₁₀₁	—	8	4	no	—
86	C₂³×C₁₀₁	—	8	4	no	—
87	C₂³×C₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
88	C₂³×C₁₀₁	C₄×Q₈	1	8	yes	—
89	C₂³×C₁₀₁	C₂²⋊Q₈	1	4	yes	—
90	C₂³×C₁₀₁	C₂²⋊Q₈	1	4	yes	—
91	C₂³×C₁₀₁	#124	1	4	yes	—
92	C₂³×C₁₀₁	C₄⋊Q₈	1	4	yes	—
93	C₂³×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
94	C₂³×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
95	C₂³×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
96	C₂³×C₁₀₁	C₂²×Q₈	1	2	yes	—
97	C₂³×C₁₀₁	C₂×#41	1	2	yes	—
98	Q₈×C₁₀₁	—	8	8	no	—
99	C₂×C₄×C₁₀₁	—	2	8	no	—
100	C₂×C₄×C₁₀₁	—	2	8	no	—
101	C₂×C₄×C₁₀₁	—	4	4	no	—
102	C₂×C₄×C₁₀₁	—	4	4	no	—
103	C₂×C₄×C₁₀₁	—	4	4	no	—
104	C₂×C₄×C₁₀₁	—	4	8	no	—
105	C₂×C₄×C₁₀₁	—	4	8	no	—
106	C₂×C₄×C₁₀₁	—	4	8	no	—
107	C₂×C₄×C₁₀₁	—	4	4	no	—
108	C₂×C₄×C₁₀₁	—	8	8	no	—
109	C₂×C₄×C₁₀₁	—	8	8	no	—
110	C₂×C₄×C₁₀₁	—	8	8	no	—
111	C₂×C₄×C₁₀₁	C₂×C₂²⋊C₄	1	8	yes	—
112	C₂²×C₁₀₁	—	2	2	no	—
113	C₂²×C₁₀₁	—	2	2	no	—
114	C₂²×C₁₀₁	—	2	2	no	—
115	C₂²×C₁₀₁	—	2	8	no	—
116	C₂²×C₁₀₁	—	4	4	no	—
117	C₂²×C₁₀₁	—	4	4	no	—
118	C₂²×C₁₀₁	—	4	8	no	—
119	C₂²×C₁₀₁	—	4	4	no	—
120	C₂²×C₁₀₁	—	4	4	no	—
121	C₂²×C₁₀₁	—	4	2	no	—
122	C₂²×C₁₀₁	—	4	2	no	—
123	C₂²×C₁₀₁	#391	1	8	yes	—
124	C₂²×C₁₀₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
125	C₂²×C₁₀₁	C₂×C₄×Q₈	1	2	yes	—
126	C₂²×C₁₀₁	C₂×C₂²⋊Q₈	1	2	yes	—
127	C₂²×C₁₀₁	C₂×#124	1	1	yes	characteristic
128	C₂²×C₁₀₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
129	C₄×C₁₀₁	—	2	8	no	—
130	C₄×C₁₀₁	—	4	8	no	—
131	C₄×C₁₀₁	—	4	8	no	—
132	C₄×C₁₀₁	—	8	4	no	—
133	#21333	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂×C₁₀₁	—	2	2	no	—
135	C₂×C₁₀₁	—	2	1	no	—
136	C₂×C₁₀₁	—	2	2	no	—
137	C₂×C₁₀₁	—	2	1	no	—
138	C₂×C₁₀₁	—	2	8	no	—
139	C₂×C₁₀₁	C₂×#391	1	1	yes	characteristic
140	C₂×C₁₀₁	#2883	1	2	yes	—
141	C₂×#543	C₂×C₁₀₁	1	1	yes	characteristic
142	#2315	C₂×C₁₀₁	1	1	yes	characteristic
143	#2273	C₂×C₁₀₁	1	1	yes	characteristic
144	#1996	C₂×C₁₀₁	1	8	yes	complemented
145	C₂×#342	C₂×C₁₀₁	1	4	yes	—
146	C₁₀₁	#21333	1	1	yes	characteristic, direct product factor, Sylow
147	#543	—	2	4	no	—
148	#543	—	2	4	no	—
149	C₂×C₂²⋊Q₈	C₄×C₁₀₁	1	8	yes	complemented
150	C₂×C₂²⋊Q₈	C₂²×C₁₀₁	1	4	yes	—
151	C₂²×C₂²⋊C₄	C₂²×C₁₀₁	1	2	yes	—
152	C₂²×C₂²⋊C₄	C₂²×C₁₀₁	1	2	yes	—
153	C₂²×C₂²⋊C₄	C₂²×C₁₀₁	1	2	yes	—
154	C₂²×C₂²⋊C₄	C₂²×C₁₀₁	1	1	yes	characteristic
155	#385	—	2	4	no	—
156	#380	C₂²×C₁₀₁	1	4	yes	—
157	C₄×C₂²⋊C₄	—	2	4	no	—
158	C₄×C₂²⋊C₄	C₂²×C₁₀₁	1	4	yes	—
159	#342	C₂²×C₁₀₁	1	16	yes	—
160	C₂⁵	C₂³×C₁₀₁	1	1	yes	characteristic
161	C₂²×Q₈	C₂×C₄×C₁₀₁	1	4	yes	complemented
162	C₂³×C₄	—	2	2	no	—
163	C₂³×C₄	—	2	2	no	—
164	C₂³×C₄	C₂×C₄×C₁₀₁	1	4	yes	—
165	C₂³×C₄	C₂³×C₁₀₁	1	2	yes	—
166	C₂²⋊Q₈	—	2	8	no	—
167	C₂²⋊Q₈	—	2	8	no	—
168	C₂²⋊Q₈	—	2	32	no	—
169	C₂×C₄⋊C₄	—	2	4	no	—
170	C₂×C₄⋊C₄	C₂×C₄×C₁₀₁	1	12	yes	complemented
171	C₂×C₂²⋊C₄	—	2	4	no	—
172	C₂×C₂²⋊C₄	—	2	4	no	—
173	C₂×C₂²⋊C₄	—	2	2	no	—
174	C₂×C₂²⋊C₄	—	2	4	no	—
175	C₂×C₂²⋊C₄	—	2	8	no	—
176	C₂×C₂²⋊C₄	—	2	2	no	—
177	C₂×C₂²⋊C₄	C₂×C₄×C₁₀₁	1	4	yes	—
178	C₂×C₂²⋊C₄	C₂×C₄×C₁₀₁	1	4	yes	—
179	C₂×C₂²⋊C₄	D₄×C₁₀₁	1	4	yes	—
180	C₂×C₂²⋊C₄	D₄×C₁₀₁	1	4	yes	—
181	C₂×C₂²⋊C₄	Q₈×C₁₀₁	1	8	yes	complemented
182	C₂×C₂²⋊C₄	C₂³×C₁₀₁	1	4	yes	—
183	C₂×C₂²⋊C₄	C₂³×C₁₀₁	1	4	yes	—
184	C₂×C₂²⋊C₄	C₂³×C₁₀₁	1	4	yes	—
185	C₂×C₄²	—	2	4	no	—
186	#95	—	2	4	no	—
187	#95	—	2	4	no	—
188	C₂⁴	—	8	2	no	—
189	C₂⁴	C₂²×C₄×C₁₀₁	1	2	yes	—
190	C₂⁴	C₂×D₄×C₁₀₁	1	2	yes	—
191	C₂⁴	C₂×D₄×C₁₀₁	1	2	yes	—
192	C₂⁴	C₂×Q₈×C₁₀₁	1	4	yes	—
193	C₂⁴	#41×C₁₀₁	1	4	yes	—
194	C₂⁴	C₂⁴×C₁₀₁	1	1	yes	characteristic
195	C₂×Q₈	—	2	8	no	—
196	C₂×Q₈	—	8	4	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	2	8	no	—
199	C₂²×C₄	—	2	8	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	4	4	no	—
202	C₂²×C₄	—	8	2	no	—
203	C₂²×C₄	—	8	4	no	—
204	C₂²×C₄	—	8	4	no	—
205	C₂²×C₄	—	8	4	no	—
206	C₂²×C₄	C₂²×C₄×C₁₀₁	1	8	yes	—
207	C₂²×C₄	C₂²×C₄×C₁₀₁	1	4	yes	—
208	C₂²×C₄	C₂×D₄×C₁₀₁	1	4	yes	—
209	C₂²×C₄	C₂×D₄×C₁₀₁	1	4	yes	—
210	C₄⋊C₄	—	2	24	no	—
211	C₄⋊C₄	—	8	4	no	—
212	C₂²⋊C₄	—	2	8	no	—
213	C₂²⋊C₄	—	2	8	no	—
214	C₂²⋊C₄	—	2	4	no	—
215	C₂²⋊C₄	—	2	4	no	—
216	C₂²⋊C₄	—	2	4	no	—
217	C₂²⋊C₄	—	2	8	no	—
218	C₂²⋊C₄	—	4	4	no	—
219	C₂²⋊C₄	C₂²⋊C₄×C₁₀₁	1	16	yes	—
220	C₂²⋊C₄	C₂×Q₈×C₁₀₁	1	8	yes	—
221	C₄²	—	8	4	no	—
222	C₂³	—	4	4	no	—
223	C₂³	—	4	4	no	—
224	C₂³	—	4	4	no	—
225	C₂³	—	4	2	no	—
226	C₂³	—	8	4	no	—
227	C₂³	—	8	4	no	—
228	C₂³	C₂×C₂²⋊C₄×C₁₀₁	1	4	yes	—
229	C₂³	C₄×Q₈×C₁₀₁	1	8	yes	—
230	C₂³	C₂²⋊Q₈×C₁₀₁	1	4	yes	—
231	C₂³	C₂²⋊Q₈×C₁₀₁	1	4	yes	—
232	C₂³	#124×C₁₀₁	1	4	yes	—
233	C₂³	C₄⋊Q₈×C₁₀₁	1	4	yes	—
234	C₂³	C₂³×C₄×C₁₀₁	1	1	yes	characteristic
235	C₂³	C₂²×D₄×C₁₀₁	1	1	yes	characteristic
236	C₂³	C₂²×D₄×C₁₀₁	1	1	yes	characteristic
237	C₂³	C₂²×Q₈×C₁₀₁	1	2	yes	—
238	C₂³	C₂×#41×C₁₀₁	1	2	yes	—
239	Q₈	—	8	8	no	—
240	C₂×C₄	—	2	8	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	8	no	—
246	C₂×C₄	—	4	8	no	—
247	C₂×C₄	—	4	8	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	8	8	no	—
250	C₂×C₄	—	8	8	no	—
251	C₂×C₄	—	8	8	no	—
252	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₁	1	8	yes	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	8	no	—
257	C₂²	—	4	4	no	—
258	C₂²	—	4	4	no	—
259	C₂²	—	4	8	no	—
260	C₂²	—	4	4	no	—
261	C₂²	—	4	4	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	4	2	no	—
264	C₂²	#391×C₁₀₁	1	8	yes	—
265	C₂²	C₂²×C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
266	C₂²	C₂×C₄×Q₈×C₁₀₁	1	2	yes	—
267	C₂²	C₂×C₂²⋊Q₈×C₁₀₁	1	2	yes	—
268	C₂²	C₂×#124×C₁₀₁	1	1	yes	characteristic
269	C₂²	C₂×C₄⋊Q₈×C₁₀₁	1	1	yes	characteristic
270	C₄	—	2	8	no	—
271	C₄	—	4	8	no	—
272	C₄	—	4	8	no	—
273	C₄	—	8	4	no	—
274	C₂	—	2	2	no	—
275	C₂	—	2	1	no	—
276	C₂	—	2	2	no	—
277	C₂	—	2	1	no	—
278	C₂	—	2	8	no	—
279	C₂	C₂×#391×C₁₀₁	1	1	yes	characteristic
280	C₂	C₁₀₁×#2883	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₀₁	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₁₀₁
Frattini, cosocle	C₂⁴	C₂⁴×C₁₀₁
prime radical	C₂⁵×C₁₀₁	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₁ ◁ C₂²×C₂²⋊C₄×C₁₀₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₀₁×#1996	C₂	semidirect
8. C₂×C₂²⋊Q₈×C₁₀₁	C₄	semidirect
20. C₂²×Q₈×C₁₀₁	C₂×C₄	semidirect
29. C₂×C₄⋊C₄×C₁₀₁	C₂×C₄	semidirect
40. C₂×C₂²⋊C₄×C₁₀₁	Q₈	semidirect
133. #21333	C₁₀₁	direct (N × H)
144. #1996	C₂×C₁₀₁	semidirect
146. C₁₀₁	#21333	direct (N × H)
149. C₂×C₂²⋊Q₈	C₄×C₁₀₁	semidirect
161. C₂²×Q₈	C₂×C₄×C₁₀₁	semidirect
170. C₂×C₄⋊C₄	C₂×C₄×C₁₀₁	semidirect
181. C₂×C₂²⋊C₄	Q₈×C₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#543×C₁₀₁	2
C₁₀₁×#2315	2
C₁₀₁×#2273	2
C₁₀₁×#1996	2
C₂×#342×C₁₀₁	2
#21333	101

Subgroups — C101×#21333 #38803484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#21333 #38803484