Subgroups — C₁₀₁×#16110 #38798261

Order 25856.

1242 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#391×C₁₀₁	C₂	1	2	yes	maximal
2	C₁₀₁×#1446	C₂	1	1	yes	maximal, characteristic
3	C₁₀₁×#1445	C₂	1	2	yes	maximal
4	C₁₀₁×#1444	C₂	1	4	yes	maximal
5	C₁₀₁×#1442	C₂	1	2	yes	maximal
6	C₁₀₁×#1434	C₂	1	4	yes	maximal
7	C₂³×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
9	C₂²×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
10	C₂²×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
11	C₂²×C₄²×C₁₀₁	C₂²	1	2	yes	—
12	#391×C₁₀₁	—	2	8	no	—
13	C₂×#95×C₁₀₁	C₂²	1	4	yes	—
14	C₂×#95×C₁₀₁	C₂²	1	4	yes	—
15	C₂×#95×C₁₀₁	C₂²	1	4	yes	—
16	C₂×#95×C₁₀₁	C₂²	1	4	yes	—
17	C₂×#95×C₁₀₁	C₂²	1	2	yes	—
18	C₂×#95×C₁₀₁	C₂²	1	4	yes	—
19	C₂×#95×C₁₀₁	C₂²	1	1	yes	characteristic
20	C₂×#95×C₁₀₁	C₂²	1	1	yes	characteristic
21	C₂×#95×C₁₀₁	C₂²	1	2	yes	—
22	C₂²×Q₈×C₁₀₁	—	2	8	no	—
23	C₂²×Q₈×C₁₀₁	—	4	1	no	—
24	C₂²×Q₈×C₁₀₁	—	4	1	no	—
25	C₂²×Q₈×C₁₀₁	D₄	1	4	yes	—
26	C₂³×C₄×C₁₀₁	C₂³	1	4	yes	—
27	C₂³×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
28	C₂³×C₄×C₁₀₁	C₂³	1	4	yes	—
29	C₂³×C₄×C₁₀₁	C₂³	1	2	yes	—
30	C₂³×C₄×C₁₀₁	C₂³	1	2	yes	—
31	C₂³×C₄×C₁₀₁	C₂³	1	2	yes	—
32	C₂×C₄⋊C₄×C₁₀₁	—	2	4	no	—
33	C₂×C₄⋊C₄×C₁₀₁	—	2	4	no	—
34	C₂×C₄⋊C₄×C₁₀₁	—	4	2	no	—
35	C₂×C₄⋊C₄×C₁₀₁	—	4	2	no	—
36	C₂×C₄⋊C₄×C₁₀₁	—	4	2	no	—
37	C₂×C₄⋊C₄×C₁₀₁	—	4	2	no	—
38	C₂×C₄⋊C₄×C₁₀₁	—	4	4	no	—
39	C₂×C₄⋊C₄×C₁₀₁	—	4	2	no	—
40	C₂×C₄²×C₁₀₁	—	2	4	no	—
41	C₂×C₄²×C₁₀₁	—	2	8	no	—
42	#95×C₁₀₁	—	2	8	no	—
43	#95×C₁₀₁	—	2	8	no	—
44	#95×C₁₀₁	—	2	8	no	—
45	#95×C₁₀₁	—	2	8	no	—
46	#95×C₁₀₁	—	4	2	no	—
47	#95×C₁₀₁	—	4	4	no	—
48	#95×C₁₀₁	—	4	1	no	—
49	#95×C₁₀₁	—	4	1	no	—
50	#95×C₁₀₁	Q₈	1	8	yes	complemented
51	C₂⁴×C₁₀₁	C₂⁴	1	1	yes	characteristic
52	C₂×Q₈×C₁₀₁	—	2	16	no	—
53	C₂×Q₈×C₁₀₁	—	4	4	no	—
54	C₂×Q₈×C₁₀₁	—	4	8	no	—
55	C₂×Q₈×C₁₀₁	—	4	8	no	—
56	C₂²×C₄×C₁₀₁	—	2	4	no	—
57	C₂²×C₄×C₁₀₁	—	2	2	no	—
58	C₂²×C₄×C₁₀₁	—	2	2	no	—
59	C₂²×C₄×C₁₀₁	—	2	4	no	—
60	C₂²×C₄×C₁₀₁	—	2	4	no	—
61	C₂²×C₄×C₁₀₁	—	2	4	no	—
62	C₂²×C₄×C₁₀₁	—	2	4	no	—
63	C₂²×C₄×C₁₀₁	—	2	4	no	—
64	C₂²×C₄×C₁₀₁	—	2	4	no	—
65	C₂²×C₄×C₁₀₁	—	2	4	no	—
66	C₂²×C₄×C₁₀₁	—	2	4	no	—
67	C₂²×C₄×C₁₀₁	—	2	2	no	—
68	C₂²×C₄×C₁₀₁	—	2	2	no	—
69	C₂²×C₄×C₁₀₁	—	2	2	no	—
70	C₂²×C₄×C₁₀₁	—	2	4	no	—
71	C₂²×C₄×C₁₀₁	—	2	1	no	—
72	C₂²×C₄×C₁₀₁	—	2	1	no	—
73	C₂²×C₄×C₁₀₁	—	2	2	no	—
74	C₂²×C₄×C₁₀₁	—	2	4	no	—
75	C₂²×C₄×C₁₀₁	—	2	4	no	—
76	C₂²×C₄×C₁₀₁	—	2	4	no	—
77	C₂²×C₄×C₁₀₁	—	2	4	no	—
78	C₂²×C₄×C₁₀₁	—	2	4	no	—
79	C₂²×C₄×C₁₀₁	—	2	4	no	—
80	C₂²×C₄×C₁₀₁	—	2	4	no	—
81	C₂²×C₄×C₁₀₁	—	2	4	no	—
82	C₂²×C₄×C₁₀₁	—	2	4	no	—
83	C₂²×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
85	C₂²×C₄×C₁₀₁	C₂×Q₈	1	8	yes	—
86	C₂²×C₄×C₁₀₁	C₂×Q₈	1	4	yes	—
87	C₂²×C₄×C₁₀₁	#41	1	4	yes	—
88	C₂²×C₄×C₁₀₁	#41	1	8	yes	—
89	C₄⋊C₄×C₁₀₁	—	4	8	no	—
90	C₄⋊C₄×C₁₀₁	—	4	8	no	—
91	C₄⋊C₄×C₁₀₁	—	4	8	no	—
92	C₄²×C₁₀₁	—	4	8	no	—
93	C₂³×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
94	C₂³×C₁₀₁	C₂²×Q₈	1	2	yes	—
95	C₂³×C₁₀₁	C₂×#41	1	4	yes	—
96	C₂³×C₁₀₁	C₂×#41	1	2	yes	—
97	C₂³×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
98	C₂³×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
99	C₂³×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
100	C₂³×C₁₀₁	2₋¹⁺⁴	1	2	yes	—
101	Q₈×C₁₀₁	—	4	16	no	—
102	C₂×C₄×C₁₀₁	—	2	4	no	—
103	C₂×C₄×C₁₀₁	—	2	4	no	—
104	C₂×C₄×C₁₀₁	—	2	8	no	—
105	C₂×C₄×C₁₀₁	—	2	8	no	—
106	C₂×C₄×C₁₀₁	—	2	4	no	—
107	C₂×C₄×C₁₀₁	—	2	8	no	—
108	C₂×C₄×C₁₀₁	—	2	2	no	—
109	C₂×C₄×C₁₀₁	—	2	4	no	—
110	C₂×C₄×C₁₀₁	—	2	4	no	—
111	C₂×C₄×C₁₀₁	—	2	8	no	—
112	C₂×C₄×C₁₀₁	—	2	8	no	—
113	C₂×C₄×C₁₀₁	—	2	8	no	—
114	C₂×C₄×C₁₀₁	—	2	8	no	—
115	C₂×C₄×C₁₀₁	—	4	4	no	—
116	C₂×C₄×C₁₀₁	—	4	4	no	—
117	C₂×C₄×C₁₀₁	—	4	4	no	—
118	C₂×C₄×C₁₀₁	—	4	4	no	—
119	C₂×C₄×C₁₀₁	—	4	4	no	—
120	C₂×C₄×C₁₀₁	—	4	2	no	—
121	C₂×C₄×C₁₀₁	—	4	2	no	—
122	C₂×C₄×C₁₀₁	—	4	2	no	—
123	C₂×C₄×C₁₀₁	—	4	2	no	—
124	C₂×C₄×C₁₀₁	—	4	4	no	—
125	C₂×C₄×C₁₀₁	—	4	2	no	—
126	C₂×C₄×C₁₀₁	—	4	2	no	—
127	C₂×C₄×C₁₀₁	—	4	4	no	—
128	C₂×C₄×C₁₀₁	—	4	4	no	—
129	C₂×C₄×C₁₀₁	—	4	4	no	—
130	C₂×C₄×C₁₀₁	—	4	4	no	—
131	C₂×C₄×C₁₀₁	—	4	4	no	—
132	C₂×C₄×C₁₀₁	—	4	4	no	—
133	C₂×C₄×C₁₀₁	—	4	4	no	—
134	C₂×C₄×C₁₀₁	#123	1	8	yes	—
135	C₂²×C₁₀₁	C₂×#123	1	2	yes	—
136	C₂²×C₁₀₁	#533	1	4	yes	—
137	C₂²×C₁₀₁	#534	1	1	yes	characteristic
138	C₂²×C₁₀₁	#538	1	2	yes	—
139	C₂²×C₁₀₁	#542	1	4	yes	—
140	C₂²×C₁₀₁	#544	1	2	yes	—
141	C₂²×C₁₀₁	D₄×Q₈	1	2	yes	—
142	C₂²×C₁₀₁	#550	1	2	yes	—
143	C₂²×C₁₀₁	#551	1	2	yes	—
144	C₂²×C₁₀₁	#551	1	2	yes	—
145	C₂²×C₁₀₁	#552	1	4	yes	—
146	C₂²×C₁₀₁	#554	1	4	yes	—
147	C₂²×C₁₀₁	#555	1	1	yes	characteristic
148	C₂²×C₁₀₁	Q₈²	1	1	yes	characteristic
149	C₂²×C₁₀₁	H(3,𝔽₄)	1	1	yes	characteristic
150	C₂²×C₁₀₁	#563	1	1	yes	characteristic
151	C₄×C₁₀₁	—	2	8	no	—
152	C₄×C₁₀₁	—	4	8	no	—
153	C₄×C₁₀₁	—	4	2	no	—
154	C₄×C₁₀₁	—	4	4	no	—
155	C₄×C₁₀₁	—	4	4	no	—
156	C₄×C₁₀₁	—	8	4	no	—
157	#16110	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂×C₁₀₁	#2423	1	4	yes	—
159	C₂×C₁₀₁	#2483	1	1	yes	characteristic
160	C₂×C₁₀₁	#2586	1	2	yes	—
161	C₂×C₁₀₁	#2612	1	2	yes	—
162	C₂×C₁₀₁	#2733	1	1	yes	characteristic
163	C₂×C₁₀₁	#2734	1	1	yes	characteristic
164	C₂×C₁₀₁	#2763	1	2	yes	—
165	C₂×C₁₀₁	#2764	1	2	yes	—
166	C₂×#391	C₂×C₁₀₁	1	2	yes	—
167	#1446	C₂×C₁₀₁	1	1	yes	characteristic
168	#1445	C₂×C₁₀₁	1	2	yes	—
169	#1444	C₂×C₁₀₁	1	4	yes	—
170	#1442	C₂×C₁₀₁	1	2	yes	—
171	#1434	C₂×C₁₀₁	1	4	yes	—
172	C₁₀₁	#16110	1	1	yes	characteristic, direct product factor, Sylow
173	C₂³×Q₈	C₂²×C₁₀₁	1	1	yes	characteristic
174	C₂²×C₄⋊C₄	C₂²×C₁₀₁	1	2	yes	—
175	C₂²×C₄⋊C₄	C₂²×C₁₀₁	1	2	yes	—
176	C₂²×C₄⋊C₄	C₂²×C₁₀₁	1	2	yes	—
177	C₂²×C₄²	C₂²×C₁₀₁	1	2	yes	—
178	#391	—	2	8	no	—
179	C₂×#95	C₂²×C₁₀₁	1	4	yes	—
180	C₂×#95	C₂²×C₁₀₁	1	4	yes	—
181	C₂×#95	C₂²×C₁₀₁	1	4	yes	—
182	C₂×#95	C₂²×C₁₀₁	1	4	yes	—
183	C₂×#95	C₂²×C₁₀₁	1	2	yes	—
184	C₂×#95	C₂²×C₁₀₁	1	4	yes	—
185	C₂×#95	C₂²×C₁₀₁	1	1	yes	characteristic
186	C₂×#95	C₂²×C₁₀₁	1	1	yes	characteristic
187	C₂×#95	C₂²×C₁₀₁	1	2	yes	—
188	C₂²×Q₈	—	2	8	no	—
189	C₂²×Q₈	—	4	1	no	—
190	C₂²×Q₈	—	4	1	no	—
191	C₂²×Q₈	D₄×C₁₀₁	1	4	yes	—
192	C₂³×C₄	C₂³×C₁₀₁	1	4	yes	—
193	C₂³×C₄	C₂³×C₁₀₁	1	1	yes	characteristic
194	C₂³×C₄	C₂³×C₁₀₁	1	2	yes	—
195	C₂³×C₄	C₂³×C₁₀₁	1	4	yes	—
196	C₂³×C₄	C₂³×C₁₀₁	1	2	yes	—
197	C₂³×C₄	C₂³×C₁₀₁	1	2	yes	—
198	C₂×C₄⋊C₄	—	2	4	no	—
199	C₂×C₄⋊C₄	—	2	4	no	—
200	C₂×C₄⋊C₄	—	4	2	no	—
201	C₂×C₄⋊C₄	—	4	2	no	—
202	C₂×C₄⋊C₄	—	4	2	no	—
203	C₂×C₄⋊C₄	—	4	2	no	—
204	C₂×C₄⋊C₄	—	4	4	no	—
205	C₂×C₄⋊C₄	—	4	2	no	—
206	C₂×C₄²	—	2	4	no	—
207	C₂×C₄²	—	2	8	no	—
208	#95	—	2	8	no	—
209	#95	—	2	8	no	—
210	#95	—	2	8	no	—
211	#95	—	2	8	no	—
212	#95	—	4	2	no	—
213	#95	—	4	4	no	—
214	#95	—	4	1	no	—
215	#95	—	4	1	no	—
216	#95	Q₈×C₁₀₁	1	8	yes	complemented
217	C₂⁴	C₂⁴×C₁₀₁	1	1	yes	characteristic
218	C₂×Q₈	—	2	16	no	—
219	C₂×Q₈	—	4	4	no	—
220	C₂×Q₈	—	4	8	no	—
221	C₂×Q₈	—	4	8	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	2	4	no	—
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	—	2	4	no	—
249	C₂²×C₄	C₂×D₄×C₁₀₁	1	2	yes	—
250	C₂²×C₄	C₂×D₄×C₁₀₁	1	4	yes	—
251	C₂²×C₄	C₂×Q₈×C₁₀₁	1	8	yes	—
252	C₂²×C₄	C₂×Q₈×C₁₀₁	1	4	yes	—
253	C₂²×C₄	#41×C₁₀₁	1	4	yes	—
254	C₂²×C₄	#41×C₁₀₁	1	8	yes	—
255	C₄⋊C₄	—	4	8	no	—
256	C₄⋊C₄	—	4	8	no	—
257	C₄⋊C₄	—	4	8	no	—
258	C₄²	—	4	8	no	—
259	C₂³	C₂²×D₄×C₁₀₁	1	1	yes	characteristic
260	C₂³	C₂²×Q₈×C₁₀₁	1	2	yes	—
261	C₂³	C₂×#41×C₁₀₁	1	4	yes	—
262	C₂³	C₂×#41×C₁₀₁	1	2	yes	—
263	C₂³	H(5,𝔽₂)×C₁₀₁	1	1	yes	characteristic
264	C₂³	H(5,𝔽₂)×C₁₀₁	1	1	yes	characteristic
265	C₂³	H(5,𝔽₂)×C₁₀₁	1	2	yes	—
266	C₂³	2₋¹⁺⁴×C₁₀₁	1	2	yes	—
267	Q₈	—	4	16	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	8	no	—
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	2	8	no	—
279	C₂×C₄	—	2	8	no	—
280	C₂×C₄	—	2	8	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	#123×C₁₀₁	1	8	yes	—
301	C₂²	C₂×#123×C₁₀₁	1	2	yes	—
302	C₂²	#533×C₁₀₁	1	4	yes	—
303	C₂²	#534×C₁₀₁	1	1	yes	characteristic
304	C₂²	#538×C₁₀₁	1	2	yes	—
305	C₂²	#542×C₁₀₁	1	4	yes	—
306	C₂²	#544×C₁₀₁	1	2	yes	—
307	C₂²	D₄×Q₈×C₁₀₁	1	2	yes	—
308	C₂²	#550×C₁₀₁	1	2	yes	—
309	C₂²	#551×C₁₀₁	1	2	yes	—
310	C₂²	#551×C₁₀₁	1	2	yes	—
311	C₂²	#552×C₁₀₁	1	4	yes	—
312	C₂²	#554×C₁₀₁	1	4	yes	—
313	C₂²	#555×C₁₀₁	1	1	yes	characteristic
314	C₂²	Q₈²×C₁₀₁	1	1	yes	characteristic
315	C₂²	H(3,𝔽₄)×C₁₀₁	1	1	yes	characteristic
316	C₂²	#563×C₁₀₁	1	1	yes	characteristic
317	C₄	—	2	8	no	—
318	C₄	—	4	8	no	—
319	C₄	—	4	2	no	—
320	C₄	—	4	4	no	—
321	C₄	—	4	4	no	—
322	C₄	—	8	4	no	—
323	C₂	C₁₀₁×#2423	1	4	yes	—
324	C₂	C₁₀₁×#2483	1	1	yes	characteristic
325	C₂	C₁₀₁×#2586	1	2	yes	—
326	C₂	C₁₀₁×#2612	1	2	yes	—
327	C₂	C₁₀₁×#2733	1	1	yes	characteristic
328	C₂	C₁₀₁×#2734	1	1	yes	characteristic
329	C₂	C₁₀₁×#2763	1	2	yes	—
330	C₂	C₁₀₁×#2764	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₀₁	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₁₀₁
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₀₁ ◁ G

Complements

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

N	H ≅ G/N	type
50. #95×C₁₀₁	Q₈	semidirect
157. #16110	C₁₀₁	direct (N × H)
172. C₁₀₁	#16110	direct (N × H)
216. #95	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×#391×C₁₀₁	2
C₁₀₁×#1446	2
C₁₀₁×#1445	2
C₁₀₁×#1444	2
C₁₀₁×#1442	2
C₁₀₁×#1434	2
#16110	101

Subgroups — C101×#16110 #38798261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#16110 #38798261