Subgroups — C₁₀₁×#16254 #38798405

Order 25856.

2138 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊Q₈×C₁₀₁	C₂	1	2	yes	maximal
2	C₂×#395×C₁₀₁	C₂	1	4	yes	maximal, complemented
3	C₁₀₁×#1441	C₂	1	4	yes	maximal, complemented
4	C₁₀₁×#1439	C₂	1	4	yes	maximal
5	C₁₀₁×C₂²⋊#95	C₂	1	1	yes	maximal, characteristic
6	C₂³×Q₈×C₁₀₁	C₂²	1	2	yes	—
7	C₂⁴×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊Q₈×C₁₀₁	—	2	8	no	—
9	C₂×C₂²⋊Q₈×C₁₀₁	—	2	16	no	—
10	C₂²×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
11	C₂²×C₄⋊C₄×C₁₀₁	C₂²	1	4	yes	—
12	C₂²×C₄⋊C₄×C₁₀₁	C₂²	1	4	yes	—
13	C₂²×C₂²⋊C₄×C₁₀₁	C₂²	1	4	yes	—
14	C₂²×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
15	#395×C₁₀₁	—	2	16	no	—
16	C₂×#95×C₁₀₁	C₂²	1	4	yes	—
17	C₂×#95×C₁₀₁	C₂²	1	8	yes	—
18	C₂×#95×C₁₀₁	C₂²	1	4	yes	—
19	C₂⁵×C₁₀₁	C₂³	1	1	yes	characteristic
20	C₂²×Q₈×C₁₀₁	—	2	8	no	—
21	C₂²×Q₈×C₁₀₁	—	2	4	no	—
22	C₂²×Q₈×C₁₀₁	—	2	8	no	—
23	C₂²×Q₈×C₁₀₁	—	4	4	no	—
24	C₂³×C₄×C₁₀₁	—	2	8	no	—
25	C₂³×C₄×C₁₀₁	—	4	2	no	—
26	C₂³×C₄×C₁₀₁	D₄	1	4	yes	—
27	C₂³×C₄×C₁₀₁	C₂³	1	4	yes	—
28	C₂³×C₄×C₁₀₁	C₂³	1	4	yes	—
29	C₂³×C₄×C₁₀₁	C₂³	1	4	yes	—
30	C₂³×C₄×C₁₀₁	C₂³	1	2	yes	—
31	C₂²⋊Q₈×C₁₀₁	—	2	64	no	—
32	C₂×C₄⋊C₄×C₁₀₁	—	2	8	no	—
33	C₂×C₄⋊C₄×C₁₀₁	—	2	4	no	—
34	C₂×C₄⋊C₄×C₁₀₁	—	2	8	no	—
35	C₂×C₄⋊C₄×C₁₀₁	—	2	8	no	—
36	C₂×C₄⋊C₄×C₁₀₁	—	2	16	no	—
37	C₂×C₄⋊C₄×C₁₀₁	—	4	4	no	—
38	C₂×C₄⋊C₄×C₁₀₁	D₄	1	16	yes	—
39	C₂×C₂²⋊C₄×C₁₀₁	—	2	8	no	—
40	C₂×C₂²⋊C₄×C₁₀₁	—	2	8	no	—
41	C₂×C₂²⋊C₄×C₁₀₁	—	2	8	no	—
42	C₂×C₂²⋊C₄×C₁₀₁	—	4	2	no	—
43	C₂×C₂²⋊C₄×C₁₀₁	—	4	2	no	—
44	C₂×C₂²⋊C₄×C₁₀₁	Q₈	1	8	yes	complemented
45	#95×C₁₀₁	—	2	8	no	—
46	#95×C₁₀₁	—	2	16	no	—
47	#95×C₁₀₁	—	4	4	no	—
48	C₂⁴×C₁₀₁	—	2	4	no	—
49	C₂⁴×C₁₀₁	—	2	1	no	—
50	C₂⁴×C₁₀₁	—	2	2	no	—
51	C₂⁴×C₁₀₁	—	2	4	no	—
52	C₂⁴×C₁₀₁	—	2	1	no	—
53	C₂⁴×C₁₀₁	C₂×D₄	1	2	yes	—
54	C₂⁴×C₁₀₁	C₂×Q₈	1	4	yes	—
55	C₂⁴×C₁₀₁	C₂⁴	1	1	yes	characteristic
56	C₂×Q₈×C₁₀₁	—	2	16	no	—
57	C₂×Q₈×C₁₀₁	—	4	32	no	—
58	C₂×Q₈×C₁₀₁	—	4	8	no	—
59	C₂×Q₈×C₁₀₁	—	4	8	no	—
60	C₂²×C₄×C₁₀₁	—	2	4	no	—
61	C₂²×C₄×C₁₀₁	—	2	4	no	—
62	C₂²×C₄×C₁₀₁	—	2	8	no	—
63	C₂²×C₄×C₁₀₁	—	2	8	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	4	no	—
68	C₂²×C₄×C₁₀₁	—	2	4	no	—
69	C₂²×C₄×C₁₀₁	—	2	4	no	—
70	C₂²×C₄×C₁₀₁	—	2	4	no	—
71	C₂²×C₄×C₁₀₁	—	2	8	no	—
72	C₂²×C₄×C₁₀₁	—	2	4	no	—
73	C₂²×C₄×C₁₀₁	—	2	4	no	—
74	C₂²×C₄×C₁₀₁	—	2	4	no	—
75	C₂²×C₄×C₁₀₁	—	2	8	no	—
76	C₂²×C₄×C₁₀₁	—	2	16	no	—
77	C₂²×C₄×C₁₀₁	—	4	4	no	—
78	C₂²×C₄×C₁₀₁	—	4	16	no	—
79	C₂²×C₄×C₁₀₁	C₂×D₄	1	8	yes	—
80	C₂²×C₄×C₁₀₁	C₂×D₄	1	8	yes	—
81	C₂²×C₄×C₁₀₁	C₂×D₄	1	8	yes	—
82	C₂²×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
83	C₂²×C₄×C₁₀₁	C₂×Q₈	1	8	yes	—
84	C₄⋊C₄×C₁₀₁	—	2	16	no	—
85	C₄⋊C₄×C₁₀₁	—	2	32	no	—
86	C₄⋊C₄×C₁₀₁	—	4	16	no	—
87	C₂²⋊C₄×C₁₀₁	—	2	32	no	—
88	C₂²⋊C₄×C₁₀₁	—	4	8	no	—
89	C₂³×C₁₀₁	—	2	4	no	—
90	C₂³×C₁₀₁	—	2	2	no	—
91	C₂³×C₁₀₁	—	2	4	no	—
92	C₂³×C₁₀₁	—	2	2	no	—
93	C₂³×C₁₀₁	—	2	4	no	—
94	C₂³×C₁₀₁	—	2	4	no	—
95	C₂³×C₁₀₁	—	2	16	no	—
96	C₂³×C₁₀₁	—	4	4	no	—
97	C₂³×C₁₀₁	—	4	4	no	—
98	C₂³×C₁₀₁	—	4	4	no	—
99	C₂³×C₁₀₁	—	4	4	no	—
100	C₂³×C₁₀₁	C₄⋊Q₈	1	4	yes	—
101	C₂³×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
102	C₂³×C₁₀₁	C₂²×D₄	1	4	yes	—
103	C₂³×C₁₀₁	C₂²×Q₈	1	2	yes	—
104	C₂³×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
105	C₂³×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
106	C₂³×C₁₀₁	2₋¹⁺⁴	1	1	yes	characteristic
107	C₂³×C₁₀₁	2₋¹⁺⁴	1	4	yes	—
108	Q₈×C₁₀₁	—	4	32	no	—
109	C₂×C₄×C₁₀₁	—	2	4	no	—
110	C₂×C₄×C₁₀₁	—	2	4	no	—
111	C₂×C₄×C₁₀₁	—	2	16	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	—	2	8	no	—
116	C₂×C₄×C₁₀₁	—	2	8	no	—
117	C₂×C₄×C₁₀₁	—	2	16	no	—
118	C₂×C₄×C₁₀₁	—	2	16	no	—
119	C₂×C₄×C₁₀₁	—	4	16	no	—
120	C₂×C₄×C₁₀₁	—	4	4	no	—
121	C₂×C₄×C₁₀₁	—	4	4	no	—
122	C₂×C₄×C₁₀₁	—	4	4	no	—
123	C₂×C₄×C₁₀₁	—	4	4	no	—
124	C₂×C₄×C₁₀₁	—	4	4	no	—
125	C₂×C₄×C₁₀₁	—	4	4	no	—
126	C₂×C₄×C₁₀₁	—	4	8	no	—
127	C₂×C₄×C₁₀₁	—	4	4	no	—
128	C₂×C₄×C₁₀₁	—	4	8	no	—
129	C₂×C₄×C₁₀₁	—	4	4	no	—
130	C₂×C₄×C₁₀₁	#120	1	8	yes	—
131	C₂²×C₁₀₁	—	2	4	no	—
132	C₂²×C₁₀₁	—	2	8	no	—
133	C₂²×C₁₀₁	—	4	2	no	—
134	C₂²×C₁₀₁	—	4	4	no	—
135	C₂²×C₁₀₁	—	4	2	no	—
136	C₂²×C₁₀₁	—	4	8	no	—
137	C₂²×C₁₀₁	—	4	8	no	—
138	C₂²×C₁₀₁	C₂×#120	1	2	yes	—
139	C₂²×C₁₀₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
140	C₂²×C₁₀₁	#534	1	1	yes	characteristic
141	C₂²×C₁₀₁	#536	1	2	yes	—
142	C₂²×C₁₀₁	#536	1	4	yes	—
143	C₂²×C₁₀₁	#537	1	1	yes	characteristic
144	C₂²×C₁₀₁	#537	1	4	yes	—
145	C₂²×C₁₀₁	#543	1	2	yes	—
146	C₂²×C₁₀₁	#544	1	2	yes	—
147	C₂²×C₁₀₁	D₄²	1	4	yes	—
148	C₂²×C₁₀₁	D₄×Q₈	1	8	yes	—
149	C₂²×C₁₀₁	#563	1	4	yes	—
150	C₄×C₁₀₁	—	2	8	no	—
151	C₄×C₁₀₁	—	4	8	no	—
152	C₄×C₁₀₁	—	4	8	no	—
153	C₄×C₁₀₁	—	4	8	no	—
154	#16254	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂×C₁₀₁	—	4	4	no	—
156	C₂×C₁₀₁	#2403	1	1	yes	characteristic
157	C₂×C₁₀₁	#2434	1	2	yes	—
158	C₂×C₁₀₁	#2616	1	2	yes	—
159	C₂×C₁₀₁	#2659	1	1	yes	characteristic
160	C₂×C₁₀₁	#2677	1	4	yes	—
161	C₂×C₁₀₁	#2683	1	4	yes	—
162	C₂×C₁₀₁	#2817	1	1	yes	characteristic
163	C₂²×C₂²⋊Q₈	C₂×C₁₀₁	1	2	yes	—
164	C₂×#395	C₂×C₁₀₁	1	4	yes	complemented
165	#1441	C₂×C₁₀₁	1	4	yes	complemented
166	#1439	C₂×C₁₀₁	1	4	yes	—
167	C₂²⋊#95	C₂×C₁₀₁	1	1	yes	characteristic
168	C₁₀₁	#16254	1	1	yes	characteristic, direct product factor, Sylow
169	C₂³×Q₈	C₂²×C₁₀₁	1	2	yes	—
170	C₂⁴×C₄	C₂²×C₁₀₁	1	1	yes	characteristic
171	C₂×C₂²⋊Q₈	—	2	8	no	—
172	C₂×C₂²⋊Q₈	—	2	16	no	—
173	C₂²×C₄⋊C₄	C₂²×C₁₀₁	1	2	yes	—
174	C₂²×C₄⋊C₄	C₂²×C₁₀₁	1	4	yes	—
175	C₂²×C₄⋊C₄	C₂²×C₁₀₁	1	4	yes	—
176	C₂²×C₂²⋊C₄	C₂²×C₁₀₁	1	4	yes	—
177	C₂²×C₂²⋊C₄	C₂²×C₁₀₁	1	2	yes	—
178	#395	—	2	16	no	—
179	C₂×#95	C₂²×C₁₀₁	1	4	yes	—
180	C₂×#95	C₂²×C₁₀₁	1	8	yes	—
181	C₂×#95	C₂²×C₁₀₁	1	4	yes	—
182	C₂⁵	C₂³×C₁₀₁	1	1	yes	characteristic
183	C₂²×Q₈	—	2	8	no	—
184	C₂²×Q₈	—	2	4	no	—
185	C₂²×Q₈	—	2	8	no	—
186	C₂²×Q₈	—	4	4	no	—
187	C₂³×C₄	—	2	8	no	—
188	C₂³×C₄	—	4	2	no	—
189	C₂³×C₄	D₄×C₁₀₁	1	4	yes	—
190	C₂³×C₄	C₂³×C₁₀₁	1	4	yes	—
191	C₂³×C₄	C₂³×C₁₀₁	1	4	yes	—
192	C₂³×C₄	C₂³×C₁₀₁	1	4	yes	—
193	C₂³×C₄	C₂³×C₁₀₁	1	2	yes	—
194	C₂²⋊Q₈	—	2	64	no	—
195	C₂×C₄⋊C₄	—	2	8	no	—
196	C₂×C₄⋊C₄	—	2	4	no	—
197	C₂×C₄⋊C₄	—	2	8	no	—
198	C₂×C₄⋊C₄	—	2	8	no	—
199	C₂×C₄⋊C₄	—	2	16	no	—
200	C₂×C₄⋊C₄	—	4	4	no	—
201	C₂×C₄⋊C₄	D₄×C₁₀₁	1	16	yes	—
202	C₂×C₂²⋊C₄	—	2	8	no	—
203	C₂×C₂²⋊C₄	—	2	8	no	—
204	C₂×C₂²⋊C₄	—	2	8	no	—
205	C₂×C₂²⋊C₄	—	4	2	no	—
206	C₂×C₂²⋊C₄	—	4	2	no	—
207	C₂×C₂²⋊C₄	Q₈×C₁₀₁	1	8	yes	complemented
208	#95	—	2	8	no	—
209	#95	—	2	16	no	—
210	#95	—	4	4	no	—
211	C₂⁴	—	2	4	no	—
212	C₂⁴	—	2	1	no	—
213	C₂⁴	—	2	2	no	—
214	C₂⁴	—	2	4	no	—
215	C₂⁴	—	2	1	no	—
216	C₂⁴	C₂×D₄×C₁₀₁	1	2	yes	—
217	C₂⁴	C₂×Q₈×C₁₀₁	1	4	yes	—
218	C₂⁴	C₂⁴×C₁₀₁	1	1	yes	characteristic
219	C₂×Q₈	—	2	16	no	—
220	C₂×Q₈	—	4	32	no	—
221	C₂×Q₈	—	4	8	no	—
222	C₂×Q₈	—	4	8	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	8	no	—
226	C₂²×C₄	—	2	8	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	4	no	—
234	C₂²×C₄	—	2	8	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	2	8	no	—
239	C₂²×C₄	—	2	16	no	—
240	C₂²×C₄	—	4	4	no	—
241	C₂²×C₄	—	4	16	no	—
242	C₂²×C₄	C₂×D₄×C₁₀₁	1	8	yes	—
243	C₂²×C₄	C₂×D₄×C₁₀₁	1	8	yes	—
244	C₂²×C₄	C₂×D₄×C₁₀₁	1	8	yes	—
245	C₂²×C₄	C₂×D₄×C₁₀₁	1	4	yes	—
246	C₂²×C₄	C₂×Q₈×C₁₀₁	1	8	yes	—
247	C₄⋊C₄	—	2	16	no	—
248	C₄⋊C₄	—	2	32	no	—
249	C₄⋊C₄	—	4	16	no	—
250	C₂²⋊C₄	—	2	32	no	—
251	C₂²⋊C₄	—	4	8	no	—
252	C₂³	—	2	4	no	—
253	C₂³	—	2	2	no	—
254	C₂³	—	2	4	no	—
255	C₂³	—	2	2	no	—
256	C₂³	—	2	4	no	—
257	C₂³	—	2	4	no	—
258	C₂³	—	2	16	no	—
259	C₂³	—	4	4	no	—
260	C₂³	—	4	4	no	—
261	C₂³	—	4	4	no	—
262	C₂³	—	4	4	no	—
263	C₂³	C₄⋊Q₈×C₁₀₁	1	4	yes	—
264	C₂³	C₂²×D₄×C₁₀₁	1	1	yes	characteristic
265	C₂³	C₂²×D₄×C₁₀₁	1	4	yes	—
266	C₂³	C₂²×Q₈×C₁₀₁	1	2	yes	—
267	C₂³	H(5,𝔽₂)×C₁₀₁	1	2	yes	—
268	C₂³	H(5,𝔽₂)×C₁₀₁	1	1	yes	characteristic
269	C₂³	2₋¹⁺⁴×C₁₀₁	1	1	yes	characteristic
270	C₂³	2₋¹⁺⁴×C₁₀₁	1	4	yes	—
271	Q₈	—	4	32	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	16	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	2	8	no	—
279	C₂×C₄	—	2	8	no	—
280	C₂×C₄	—	2	16	no	—
281	C₂×C₄	—	2	16	no	—
282	C₂×C₄	—	4	16	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	8	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	8	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	#120×C₁₀₁	1	8	yes	—
294	C₂²	—	2	4	no	—
295	C₂²	—	2	8	no	—
296	C₂²	—	4	2	no	—
297	C₂²	—	4	8	no	—
298	C₂²	—	4	8	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	4	4	no	—
301	C₂²	C₂×#120×C₁₀₁	1	2	yes	—
302	C₂²	C₂×C₄⋊Q₈×C₁₀₁	1	1	yes	characteristic
303	C₂²	#534×C₁₀₁	1	1	yes	characteristic
304	C₂²	#536×C₁₀₁	1	4	yes	—
305	C₂²	#536×C₁₀₁	1	2	yes	—
306	C₂²	#537×C₁₀₁	1	1	yes	characteristic
307	C₂²	#537×C₁₀₁	1	4	yes	—
308	C₂²	#543×C₁₀₁	1	2	yes	—
309	C₂²	#544×C₁₀₁	1	2	yes	—
310	C₂²	D₄²×C₁₀₁	1	4	yes	—
311	C₂²	D₄×Q₈×C₁₀₁	1	8	yes	—
312	C₂²	#563×C₁₀₁	1	4	yes	—
313	C₄	—	2	8	no	—
314	C₄	—	4	8	no	—
315	C₄	—	4	8	no	—
316	C₄	—	4	8	no	—
317	C₂	—	4	4	no	—
318	C₂	C₁₀₁×#2403	1	1	yes	characteristic
319	C₂	C₁₀₁×#2434	1	2	yes	—
320	C₂	C₁₀₁×#2616	1	2	yes	—
321	C₂	C₁₀₁×#2659	1	1	yes	characteristic
322	C₂	C₁₀₁×#2677	1	4	yes	—
323	C₂	C₁₀₁×#2683	1	4	yes	—
324	C₂	C₁₀₁×#2817	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₁₀₁	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₁₀₁
prime radical	C₂⁵×C₁₀₁	C₂³
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
2. C₂×#395×C₁₀₁	C₂	semidirect
3. C₁₀₁×#1441	C₂	semidirect
44. C₂×C₂²⋊C₄×C₁₀₁	Q₈	semidirect
154. #16254	C₁₀₁	direct (N × H)
164. C₂×#395	C₂×C₁₀₁	semidirect
165. #1441	C₂×C₁₀₁	semidirect
168. C₁₀₁	#16254	direct (N × H)
207. 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
C₂²×C₂²⋊Q₈×C₁₀₁	2
C₂×#395×C₁₀₁	2
C₁₀₁×#1441	2
C₁₀₁×#1439	2
C₁₀₁×C₂²⋊#95	2
#16254	101

Subgroups — C101×#16254 #38798405

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#16254 #38798405