Subgroups — #459×D₁₀₁ #27180540

Order 12928.

648 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈×D₁₀₁	C₂	1	2	yes	maximal, complemented
2	#127×D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₄×D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265373	C₂	1	1	yes	maximal, characteristic, complemented
5	#20265372	C₂	1	1	yes	maximal, characteristic, complemented
6	#20265371	C₂	1	2	yes	maximal, complemented
7	#20265370	C₂	1	2	yes	maximal, complemented
8	#20265369	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄⋊C₈×D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	#102×D₁₀₁	C₂	1	2	yes	maximal, complemented
11	#459×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₁₀₁	—	2	1	no	—
13	C₂×D₄×D₁₀₁	—	2	1	no	—
14	C₂×D₄×D₁₀₁	C₂²	1	2	yes	—
15	C₂×D₄×D₁₀₁	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₁₀₁	—	2	1	no	—
17	C₂×D₈₀₈	C₂²	1	2	yes	complemented
18	C₂×#13108059	C₂²	1	2	yes	complemented
19	C₂×#13108059	C₂²	1	2	yes	complemented
20	D₈×D₁₀₁	—	2	8	no	—
21	C₂×C₈×D₁₀₁	C₂²	1	2	yes	—
22	#15692252	C₂²	1	1	yes	characteristic, complemented
23	#15692251	—	2	1	no	—
24	#15692251	—	2	1	no	—
25	#15692251	C₂²	1	2	yes	complemented
26	C₄×D₄₀₄	C₂²	1	1	yes	characteristic, complemented
27	D₄×Dic₁₀₁	C₂²	1	1	yes	characteristic, complemented
28	#15692217	C₂²	1	1	yes	characteristic, complemented
29	C₄×C₂²⋊D₁₀₁	—	2	1	no	—
30	#15692215	—	2	1	no	—
31	C₄⋊C₄×D₁₀₁	C₂²	1	1	yes	characteristic
32	C₂²⋊C₄×D₁₀₁	—	2	1	no	—
33	C₄²×D₁₀₁	C₂²	1	1	yes	characteristic
34	C₄⋊#11717579	C₂²	1	1	yes	characteristic, complemented
35	#15692184	C₂²	1	2	yes	complemented
36	#15692177	C₂²	1	2	yes	complemented
37	#15692176	C₂²	1	2	yes	complemented
38	#15692175	C₂²	1	2	yes	complemented
39	C₂×D₈×C₁₀₁	C₂²	1	2	yes	complemented
40	#127×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
41	C₄×D₄×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
42	C₄⋊C₈×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
43	#102×C₁₀₁	C₂²	1	2	yes	complemented
44	C₂³×D₁₀₁	—	2	1	no	—
45	C₂³×D₁₀₁	—	2	2	no	—
46	C₂×D₄₀₄	—	2	1	no	—
47	C₂×D₄₀₄	—	2	1	no	—
48	C₂×D₄₀₄	C₂³	1	2	yes	—
49	C₂×D₄₀₄	C₂³	1	1	yes	characteristic
50	C₂×C₂²⋊D₁₀₁	—	2	1	no	—
51	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
52	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
53	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
54	D₄×D₁₀₁	—	2	4	no	—
55	D₄×D₁₀₁	—	2	2	no	—
56	D₄×D₁₀₁	—	2	4	no	—
57	D₄×D₁₀₁	—	4	4	no	—
58	D₄×D₁₀₁	—	4	4	no	—
59	D₄×D₁₀₁	D₄	1	4	yes	complemented
60	C₂²×Dic₁₀₁	—	2	1	no	—
61	C₂×C₄×D₁₀₁	—	2	1	no	—
62	C₂×C₄×D₁₀₁	—	2	1	no	—
63	C₂×C₄×D₁₀₁	—	2	1	no	—
64	C₂×C₄×D₁₀₁	—	4	4	no	—
65	C₂×C₄×D₁₀₁	D₄	1	1	yes	characteristic
66	C₂×C₄×D₁₀₁	D₄	1	1	yes	characteristic
67	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
68	D₈₀₈	—	2	4	no	—
69	#13108059	—	2	4	no	—
70	#13108059	—	2	4	no	—
71	C₂×#11717579	C₂³	1	2	yes	—
72	C₈×D₁₀₁	—	2	4	no	—
73	C₄⋊Dic₁₀₁	C₂³	1	1	yes	characteristic
74	#13108053	—	2	1	no	—
75	C₂²⋊Dic₁₀₁	—	2	1	no	—
76	#13108051	—	2	1	no	—
77	#13108051	—	2	1	no	—
78	C₄×Dic₁₀₁	D₄	1	1	yes	characteristic
79	C₄×Dic₁₀₁	D₄	1	1	yes	characteristic
80	C₄×Dic₁₀₁	C₂³	1	1	yes	characteristic
81	C₂×D₄×C₁₀₁	—	2	1	no	—
82	C₂×D₄×C₁₀₁	—	2	1	no	—
83	C₂×D₄×C₁₀₁	C₂³	1	2	yes	—
84	C₂×D₄×C₁₀₁	C₂³	1	1	yes	characteristic
85	C₂²×C₄×C₁₀₁	—	2	1	no	—
86	D₈×C₁₀₁	—	2	4	no	—
87	C₂×C₈×C₁₀₁	C₂³	1	2	yes	—
88	C₄⋊C₄×C₁₀₁	C₂³	1	1	yes	characteristic
89	C₂²⋊C₄×C₁₀₁	—	2	1	no	—
90	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
91	C₂³×C₁₀₁	—	2	1	no	—
92	C₂³×C₁₀₁	—	2	2	no	—
93	C₂²×D₁₀₁	—	2	1	no	—
94	C₂²×D₁₀₁	—	2	4	no	—
95	C₂²×D₁₀₁	—	2	2	no	—
96	C₂²×D₁₀₁	—	4	2	no	—
97	C₂²×D₁₀₁	—	4	2	no	—
98	C₂²×D₁₀₁	—	4	4	no	—
99	C₂²×D₁₀₁	—	4	4	no	—
100	C₂²×D₁₀₁	—	4	4	no	—
101	C₂²×D₁₀₁	C₂×D₄	1	1	yes	characteristic
102	D₄×C₁₀₁	—	2	2	no	—
103	D₄×C₁₀₁	—	2	1	no	—
104	D₄×C₁₀₁	—	2	2	no	—
105	D₄×C₁₀₁	—	4	2	no	—
106	D₄×C₁₀₁	—	4	2	no	—
107	D₄×C₁₀₁	C₂×D₄	1	2	yes	complemented
108	C₂×C₄×C₁₀₁	—	2	1	no	—
109	C₂×C₄×C₁₀₁	—	4	2	no	—
110	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
113	C₂²⋊D₁₀₁	—	2	4	no	—
114	C₂²⋊D₁₀₁	—	4	4	no	—
115	C₂²⋊D₁₀₁	—	4	4	no	—
116	C₂²⋊D₁₀₁	—	4	4	no	—
117	C₂×Dic₁₀₁	—	2	1	no	—
118	C₂×Dic₁₀₁	—	4	2	no	—
119	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
122	D₄₀₄	—	2	2	no	—
123	D₄₀₄	—	2	1	no	—
124	D₄₀₄	—	2	2	no	—
125	D₄₀₄	—	4	2	no	—
126	D₄₀₄	—	4	2	no	—
127	D₄₀₄	C₂×D₄	1	2	yes	complemented
128	C₄×D₁₀₁	—	2	2	no	—
129	C₄×D₁₀₁	—	4	2	no	—
130	C₄×D₁₀₁	—	4	2	no	—
131	C₄×D₁₀₁	—	4	2	no	—
132	C₄×D₁₀₁	D₈	1	4	yes	complemented
133	C₄×D₁₀₁	C₂×D₄	1	2	yes	—
134	C₄×D₁₀₁	#41	1	2	yes	—
135	C₈×C₁₀₁	—	2	2	no	—
136	#11717579	—	2	2	no	—
137	C₂²×C₁₀₁	—	2	2	no	—
138	C₂²×C₁₀₁	—	4	1	no	—
139	C₂²×C₁₀₁	—	4	1	no	—
140	C₂²×C₁₀₁	—	4	2	no	—
141	C₂²×C₁₀₁	—	4	2	no	—
142	C₂²×C₁₀₁	—	4	2	no	—
143	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
144	C₂×D₁₀₁	—	2	2	no	—
145	C₂×D₁₀₁	—	4	1	no	—
146	C₂×D₁₀₁	—	4	1	no	—
147	C₂×D₁₀₁	—	4	8	no	—
148	C₂×D₁₀₁	—	4	2	no	—
149	C₂×D₁₀₁	—	4	2	no	—
150	C₂×D₁₀₁	—	4	2	no	—
151	C₂×D₁₀₁	—	8	8	no	—
152	C₂×D₁₀₁	#121	1	2	yes	—
153	C₂×D₁₀₁	C₂×D₈	1	2	yes	—
154	C₂×D₁₀₁	AGL(1,ℤ/8)	1	2	yes	—
155	C₄×C₁₀₁	—	2	1	no	—
156	C₄×C₁₀₁	—	4	1	no	—
157	C₄×C₁₀₁	C₂×D₈	1	2	yes	complemented
158	C₄×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
159	C₄×C₁₀₁	C₂×#41	1	1	yes	characteristic
160	Dic₁₀₁	—	2	1	no	—
161	Dic₁₀₁	—	4	1	no	—
162	Dic₁₀₁	#121	1	2	yes	—
163	Dic₁₀₁	C₂×D₈	1	2	yes	complemented
164	C₂×C₁₀₁	—	4	2	no	—
165	C₂×C₁₀₁	—	8	2	no	—
166	C₂×C₁₀₁	C₂×#121	1	1	yes	characteristic
167	C₂×C₁₀₁	C₂²×D₈	1	1	yes	characteristic
168	C₂×C₁₀₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
169	D₁₀₁	—	4	2	no	—
170	D₁₀₁	—	8	2	no	—
171	D₁₀₁	#459	1	4	yes	direct product factor
172	C₂×#459	—	101	1	no	maximal, Sylow
173	C₁₀₁	C₂×#459	1	1	yes	characteristic, complemented, Sylow
174	C₂²×D₈	—	101	2	no	—
175	C₂×#127	—	101	1	no	—
176	C₂×C₄×D₄	—	101	1	no	—
177	#459	—	101	2	no	—
178	#459	—	101	1	no	—
179	#459	—	101	1	no	—
180	#459	—	101	1	no	—
181	#459	—	101	2	no	—
182	#459	D₁₀₁	1	1	yes	characteristic, direct product factor
183	C₂×C₄⋊C₈	—	101	1	no	—
184	C₂×#102	—	101	2	no	—
185	C₂²×D₄	—	101	1	no	—
186	C₂²×D₄	—	101	2	no	—
187	C₂²×D₄	—	202	1	no	—
188	C₂²×D₄	—	202	1	no	—
189	C₂³×C₄	—	202	1	no	—
190	C₂×D₈	—	101	2	no	—
191	C₂×D₈	—	101	2	no	—
192	C₂×D₈	—	101	2	no	—
193	C₂×D₈	—	202	8	no	—
194	C₂×D₈	C₂×D₁₀₁	1	2	yes	complemented
195	C₂²×C₈	—	101	2	no	—
196	#127	—	101	2	no	—
197	#127	—	101	1	no	—
198	#127	—	202	1	no	—
199	#127	—	202	1	no	—
200	#127	C₂×D₁₀₁	1	1	yes	characteristic, complemented
201	C₄×D₄	—	101	1	no	—
202	C₄×D₄	—	101	1	no	—
203	C₄×D₄	—	101	1	no	—
204	C₄×D₄	—	202	1	no	—
205	C₄×D₄	—	202	1	no	—
206	C₄×D₄	C₂×D₁₀₁	1	1	yes	characteristic, complemented
207	C₂×C₄⋊C₄	—	101	1	no	—
208	C₂×C₂²⋊C₄	—	202	1	no	—
209	C₂×C₄²	—	101	1	no	—
210	C₄⋊C₈	—	101	2	no	—
211	C₄⋊C₈	—	101	1	no	—
212	C₄⋊C₈	C₂×D₁₀₁	1	1	yes	characteristic, complemented
213	#102	—	101	2	no	—
214	#102	—	101	2	no	—
215	#102	—	101	2	no	—
216	#102	C₂×D₁₀₁	1	2	yes	complemented
217	C₂⁴	—	202	2	no	—
218	C₂⁴	—	202	1	no	—
219	C₂×D₄	—	2	1	no	—
220	C₂×D₄	—	2	1	no	—
221	C₂×D₄	—	101	4	no	—
222	C₂×D₄	—	101	1	no	—
223	C₂×D₄	—	101	2	no	—
224	C₂×D₄	—	404	4	no	—
225	C₂×D₄	—	404	4	no	—
226	C₂×D₄	—	202	4	no	—
227	C₂×D₄	—	202	4	no	—
228	C₂×D₄	—	202	2	no	—
229	C₂×D₄	—	202	2	no	—
230	C₂×D₄	—	202	2	no	—
231	C₂×D₄	—	202	2	no	—
232	C₂×D₄	—	202	1	no	—
233	C₂×D₄	—	202	1	no	—
234	C₂×D₄	—	202	1	no	—
235	C₂×D₄	C₂²×D₁₀₁	1	2	yes	—
236	C₂×D₄	C₂²×D₁₀₁	1	1	yes	characteristic
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	101	1	no	—
239	C₂²×C₄	—	101	1	no	—
240	C₂²×C₄	—	101	1	no	—
241	C₂²×C₄	—	404	4	no	—
242	C₂²×C₄	—	202	1	no	—
243	C₂²×C₄	—	202	1	no	—
244	C₂²×C₄	—	202	1	no	—
245	C₂²×C₄	—	202	1	no	—
246	D₈	—	2	4	no	—
247	D₈	—	202	4	no	—
248	D₈	—	202	4	no	—
249	D₈	—	202	4	no	—
250	C₂×C₈	—	101	2	no	—
251	C₂×C₈	—	202	4	no	—
252	C₂×C₈	C₂²×D₁₀₁	1	2	yes	—
253	C₄⋊C₄	—	101	1	no	—
254	C₄⋊C₄	—	202	1	no	—
255	C₄⋊C₄	C₂²×D₁₀₁	1	1	yes	characteristic
256	C₂²⋊C₄	—	2	1	no	—
257	C₂²⋊C₄	—	202	1	no	—
258	C₂²⋊C₄	—	202	1	no	—
259	C₂²⋊C₄	—	202	1	no	—
260	C₄²	—	101	1	no	—
261	C₄²	—	101	1	no	—
262	C₄²	—	101	1	no	—
263	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
264	C₂³	—	2	1	no	—
265	C₂³	—	2	2	no	—
266	C₂³	—	101	1	no	—
267	C₂³	—	404	4	no	—
268	C₂³	—	404	4	no	—
269	C₂³	—	404	4	no	—
270	C₂³	—	404	2	no	—
271	C₂³	—	404	2	no	—
272	C₂³	—	202	4	no	—
273	C₂³	—	202	2	no	—
274	C₂³	—	202	1	no	—
275	D₄	—	2	2	no	—
276	D₄	—	2	1	no	—
277	D₄	—	2	2	no	—
278	D₄	—	4	2	no	—
279	D₄	—	4	2	no	—
280	D₄	—	101	2	no	—
281	D₄	—	404	2	no	—
282	D₄	—	404	4	no	—
283	D₄	—	404	4	no	—
284	D₄	—	404	4	no	—
285	D₄	—	404	2	no	—
286	D₄	—	202	4	no	—
287	D₄	—	202	2	no	—
288	D₄	—	202	2	no	—
289	D₄	—	202	1	no	—
290	D₄	D₄×D₁₀₁	1	2	yes	complemented
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	101	1	no	—
294	C₂×C₄	—	101	1	no	—
295	C₂×C₄	—	101	2	no	—
296	C₂×C₄	—	101	2	no	—
297	C₂×C₄	—	101	4	no	—
298	C₂×C₄	—	101	1	no	—
299	C₂×C₄	—	404	2	no	—
300	C₂×C₄	—	404	2	no	—
301	C₂×C₄	—	404	2	no	—
302	C₂×C₄	—	404	2	no	—
303	C₂×C₄	—	202	2	no	—
304	C₂×C₄	—	202	1	no	—
305	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
306	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	—	202	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	2	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	808	8	no	—
317	C₂²	—	101	2	no	—
318	C₂²	—	101	2	no	—
319	C₂²	—	101	2	no	—
320	C₂²	—	404	8	no	—
321	C₂²	—	404	2	no	—
322	C₂²	—	404	2	no	—
323	C₂²	—	404	2	no	—
324	C₂²	—	404	1	no	—
325	C₂²	—	404	1	no	—
326	C₂²	—	202	2	no	—
327	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	101	2	no	—
331	C₄	—	101	2	no	—
332	C₄	—	404	1	no	—
333	C₄	—	202	1	no	—
334	C₄	D₈×D₁₀₁	1	2	yes	complemented
335	C₄	C₂×D₄×D₁₀₁	1	1	yes	characteristic
336	C₄	#41×D₁₀₁	1	1	yes	characteristic
337	C₂	—	4	2	no	—
338	C₂	—	8	2	no	—
339	C₂	—	808	2	no	—
340	C₂	—	101	4	no	—
341	C₂	—	404	2	no	—
342	C₂	#121×D₁₀₁	1	1	yes	characteristic
343	C₂	C₂×D₈×D₁₀₁	1	1	yes	characteristic
344	C₂	AGL(1,ℤ/8)×D₁₀₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₁
commutator, cosocle	C₂×C₄×C₁₀₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting	#459×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂²×D₄
hypercenter	#459	D₁₀₁
hypocenter	C₁₀₁	C₂×#459
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #459

Complements

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

N	H ≅ G/N	type
1. C₂×D₈×D₁₀₁	C₂	semidirect
2. #127×D₁₀₁	C₂	semidirect
3. C₄×D₄×D₁₀₁	C₂	semidirect
4. #20265373	C₂	semidirect
5. #20265372	C₂	semidirect
6. #20265371	C₂	semidirect
7. #20265370	C₂	semidirect
8. #20265369	C₂	semidirect
9. C₄⋊C₈×D₁₀₁	C₂	semidirect
10. #102×D₁₀₁	C₂	semidirect
11. #459×C₁₀₁	C₂	semidirect
17. C₂×D₈₀₈	C₂²	semidirect
18. C₂×#13108059	C₂²	semidirect
19. C₂×#13108059	C₂²	semidirect
22. #15692252	C₂²	semidirect
25. #15692251	C₂²	semidirect
26. C₄×D₄₀₄	C₂²	semidirect
27. D₄×Dic₁₀₁	C₂²	semidirect
28. #15692217	C₂²	semidirect
34. C₄⋊#11717579	C₂²	semidirect
35. #15692184	C₂²	semidirect
36. #15692177	C₂²	semidirect
37. #15692176	C₂²	semidirect
38. #15692175	C₂²	semidirect
39. C₂×D₈×C₁₀₁	C₂²	semidirect
40. #127×C₁₀₁	C₂²	semidirect
41. C₄×D₄×C₁₀₁	C₂²	semidirect
42. C₄⋊C₈×C₁₀₁	C₂²	semidirect
43. #102×C₁₀₁	C₂²	semidirect
59. D₄×D₁₀₁	D₄	semidirect
107. D₄×C₁₀₁	C₂×D₄	semidirect
127. D₄₀₄	C₂×D₄	semidirect
132. C₄×D₁₀₁	D₈	semidirect
157. C₄×C₁₀₁	C₂×D₈	semidirect
163. Dic₁₀₁	C₂×D₈	semidirect
171. D₁₀₁	#459	direct (N × H)
173. C₁₀₁	C₂×#459	semidirect
182. #459	D₁₀₁	direct (N × H)
194. C₂×D₈	C₂×D₁₀₁	semidirect
200. #127	C₂×D₁₀₁	semidirect
206. C₄×D₄	C₂×D₁₀₁	semidirect
212. C₄⋊C₈	C₂×D₁₀₁	semidirect
216. #102	C₂×D₁₀₁	semidirect
290. D₄	D₄×D₁₀₁	semidirect
334. C₄	D₈×D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₈×D₁₀₁	2
#127×D₁₀₁	2
C₄×D₄×D₁₀₁	2
#20265373	2
#20265372	2
#20265371	2
#20265370	2
#20265369	2
C₄⋊C₈×D₁₀₁	2
#102×D₁₀₁	2
#459×C₁₀₁	2
C₂×#459	101

Subgroups — #459×D101 #27180540

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #459×D₁₀₁ #27180540