Subgroups — #562×D₁₀₁ #27185254

Order 12928.

1152 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265925	C₂	1	1	yes	maximal, characteristic, complemented
2	#20265924	C₂	1	4	yes	maximal, complemented
3	#20265923	C₂	1	4	yes	maximal, complemented
4	#20265922	C₂	1	4	yes	maximal, complemented
5	#20265921	C₂	1	2	yes	maximal, complemented
6	#125×D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#124×D₁₀₁	C₂	1	2	yes	maximal, complemented
8	#123×D₁₀₁	C₂	1	4	yes	maximal, complemented
9	C₂²⋊Q₈×D₁₀₁	C₂	1	4	yes	maximal, complemented
10	#121×D₁₀₁	C₂	1	4	yes	maximal, complemented
11	#562×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₁₀₁	C₂²	1	2	yes	—
13	C₂×D₄×D₁₀₁	C₂²	1	4	yes	—
14	C₂×D₄×D₁₀₁	C₂²	1	2	yes	—
15	C₂²×C₄×D₁₀₁	C₂²	1	4	yes	—
16	#15692247	C₂²	1	1	yes	characteristic, complemented
17	#15692246	C₂²	1	2	yes	complemented
18	#15692245	C₂²	1	4	yes	complemented
19	#15692244	C₂²	1	2	yes	complemented
20	#15692243	C₂²	1	2	yes	complemented
21	#15692242	C₂²	1	2	yes	complemented
22	#15692241	C₂²	1	4	yes	complemented
23	#15692241	C₂²	1	4	yes	complemented
24	#15692240	C₂²	1	4	yes	complemented
25	#15692240	C₂²	1	4	yes	complemented
26	#15692239	C₂²	1	4	yes	complemented
27	#15692238	C₂²	1	4	yes	complemented
28	#15692237	C₂²	1	4	yes	complemented
29	#15692237	C₂²	1	4	yes	complemented
30	#15692236	C₂²	1	4	yes	complemented
31	#15692235	C₂²	1	4	yes	complemented
32	#15692234	C₂²	1	4	yes	complemented
33	#15692233	C₂²	1	4	yes	complemented
34	#15692233	C₂²	1	4	yes	complemented
35	#15692232	C₂²	1	4	yes	complemented
36	#15692232	C₂²	1	4	yes	complemented
37	#15692231	C₂²	1	4	yes	complemented
38	#15692230	C₂²	1	4	yes	complemented
39	#15692229	C₂²	1	4	yes	complemented
40	#15692228	C₂²	1	4	yes	complemented
41	#15692228	C₂²	1	4	yes	complemented
42	#15692227	C₂²	1	4	yes	complemented
43	#15692227	C₂²	1	4	yes	complemented
44	#15692226	C₂²	1	4	yes	complemented
45	C₄⋊C₄×D₁₀₁	C₂²	1	4	yes	—
46	C₄⋊C₄×D₁₀₁	C₂²	1	4	yes	—
47	C₄⋊C₄×D₁₀₁	C₂²	1	2	yes	—
48	C₂²⋊C₄×D₁₀₁	C₂²	1	4	yes	—
49	C₂²⋊C₄×D₁₀₁	C₂²	1	4	yes	—
50	C₂²⋊C₄×D₁₀₁	C₂²	1	4	yes	—
51	C₄²×D₁₀₁	C₂²	1	1	yes	characteristic
52	#125×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
53	#124×C₁₀₁	C₂²	1	2	yes	complemented
54	#123×C₁₀₁	C₂²	1	4	yes	complemented
55	C₂²⋊Q₈×C₁₀₁	C₂²	1	4	yes	complemented
56	#121×C₁₀₁	C₂²	1	4	yes	complemented
57	C₂³×D₁₀₁	C₂³	1	4	yes	—
58	C₂×Dic₂₀₂	C₂³	1	2	yes	—
59	C₂×Dic₂₀₂	C₂³	1	4	yes	—
60	Q₈×D₁₀₁	—	4	4	no	—
61	C₂×D₄₀₄	C₂³	1	2	yes	—
62	C₂×D₄₀₄	C₂³	1	4	yes	—
63	C₂×C₂²⋊D₁₀₁	C₂³	1	4	yes	—
64	C₂×C₂²⋊D₁₀₁	C₂³	1	4	yes	—
65	C₂×C₂²⋊D₁₀₁	C₂³	1	4	yes	—
66	D₄×D₁₀₁	—	4	8	no	—
67	D₄×D₁₀₁	—	4	4	no	—
68	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
69	C₂×C₄×D₁₀₁	—	4	8	no	—
70	C₂×C₄×D₁₀₁	C₂³	1	4	yes	—
71	C₂×C₄×D₁₀₁	C₂³	1	4	yes	—
72	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
73	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
74	C₂×C₄×D₁₀₁	C₂³	1	4	yes	—
75	C₂×C₄×D₁₀₁	C₂³	1	4	yes	—
76	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
77	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
78	C₄⋊Dic₁₀₁	C₂³	1	2	yes	—
79	#13108053	C₂³	1	4	yes	—
80	#13108053	C₂³	1	4	yes	—
81	#13108053	C₂³	1	4	yes	—
82	#13108053	C₂³	1	4	yes	—
83	#13108053	C₂³	1	4	yes	—
84	C₂²⋊Dic₁₀₁	C₂³	1	4	yes	—
85	C₂²⋊Dic₁₀₁	C₂³	1	4	yes	—
86	C₂²⋊Dic₁₀₁	C₂³	1	4	yes	—
87	#13108051	C₂³	1	4	yes	—
88	#13108051	C₂³	1	4	yes	—
89	#13108051	C₂³	1	4	yes	—
90	#13108051	C₂³	1	4	yes	—
91	#13108051	C₂³	1	4	yes	—
92	#13108051	C₂³	1	4	yes	—
93	C₄×Dic₁₀₁	C₂³	1	1	yes	characteristic
94	C₄×Dic₁₀₁	C₂³	1	2	yes	—
95	C₂×Q₈×C₁₀₁	C₂³	1	2	yes	—
96	C₂×D₄×C₁₀₁	C₂³	1	4	yes	—
97	C₂×D₄×C₁₀₁	C₂³	1	2	yes	—
98	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
99	C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
100	C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
101	C₄⋊C₄×C₁₀₁	C₂³	1	2	yes	—
102	C₂²⋊C₄×C₁₀₁	C₂³	1	4	yes	—
103	C₂²⋊C₄×C₁₀₁	C₂³	1	4	yes	—
104	C₂²⋊C₄×C₁₀₁	C₂³	1	4	yes	—
105	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
106	C₂³×C₁₀₁	C₂⁴	1	4	yes	—
107	C₂²×D₁₀₁	—	2	8	no	—
108	C₂²×D₁₀₁	—	2	8	no	—
109	C₂²×D₁₀₁	—	2	8	no	—
110	C₂²×D₁₀₁	C₂⁴	1	4	yes	—
111	C₂²×D₁₀₁	C₂⁴	1	1	yes	characteristic
112	Q₈×C₁₀₁	—	4	2	no	—
113	D₄×C₁₀₁	—	4	2	no	—
114	D₄×C₁₀₁	—	4	4	no	—
115	C₂×C₄×C₁₀₁	—	4	4	no	—
116	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
117	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
118	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
119	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
120	C₂²⋊D₁₀₁	—	4	4	no	—
121	C₂²⋊D₁₀₁	—	4	4	no	—
122	C₂²⋊D₁₀₁	—	4	4	no	—
123	C₂×Dic₁₀₁	—	4	4	no	—
124	C₂×Dic₁₀₁	C₂⁴	1	1	yes	characteristic
125	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
126	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
127	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
128	D₄₀₄	—	4	2	no	—
129	D₄₀₄	—	4	4	no	—
130	C₄×D₁₀₁	—	2	8	no	—
131	C₄×D₁₀₁	—	2	8	no	—
132	C₄×D₁₀₁	—	2	4	no	—
133	C₄×D₁₀₁	—	2	2	no	—
134	C₄×D₁₀₁	—	4	4	no	—
135	C₄×D₁₀₁	—	4	4	no	—
136	Dic₂₀₂	—	4	2	no	—
137	Dic₂₀₂	—	4	4	no	—
138	C₂²×C₁₀₁	—	2	4	no	—
139	C₂²×C₁₀₁	—	2	4	no	—
140	C₂²×C₁₀₁	—	2	4	no	—
141	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
142	C₂×D₁₀₁	—	2	4	no	—
143	C₂×D₁₀₁	—	2	4	no	—
144	C₂×D₁₀₁	—	2	4	no	—
145	C₂×D₁₀₁	—	4	16	no	—
146	C₂×D₁₀₁	H(5,𝔽₂)	1	4	yes	—
147	C₂×D₁₀₁	2₋¹⁺⁴	1	2	yes	—
148	C₄×C₁₀₁	—	2	4	no	—
149	C₄×C₁₀₁	—	2	2	no	—
150	C₄×C₁₀₁	—	2	4	no	—
151	C₄×C₁₀₁	—	2	1	no	—
152	Dic₁₀₁	—	2	4	no	—
153	Dic₁₀₁	—	2	2	no	—
154	Dic₁₀₁	—	2	4	no	—
155	Dic₁₀₁	—	2	1	no	—
156	C₂×C₁₀₁	—	4	4	no	—
157	C₂×C₁₀₁	C₂×H(5,𝔽₂)	1	2	yes	—
158	C₂×C₁₀₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
159	D₁₀₁	—	4	4	no	—
160	D₁₀₁	#562	1	4	yes	direct product factor
161	C₂×#562	—	101	1	no	maximal, Sylow
162	C₁₀₁	C₂×#562	1	1	yes	characteristic, complemented, Sylow
163	#562	—	101	1	no	—
164	#562	—	101	2	no	—
165	#562	—	101	4	no	—
166	#562	—	101	4	no	—
167	#562	—	101	4	no	—
168	#562	D₁₀₁	1	1	yes	characteristic, direct product factor
169	C₂×#125	—	101	1	no	—
170	C₂×#124	—	101	2	no	—
171	C₂×#123	—	101	4	no	—
172	C₂×C₂²⋊Q₈	—	101	4	no	—
173	C₂×#121	—	101	4	no	—
174	C₂²×Q₈	—	101	2	no	—
175	C₂²×D₄	—	101	4	no	—
176	C₂²×D₄	—	101	2	no	—
177	C₂³×C₄	—	101	4	no	—
178	#125	—	101	4	no	—
179	#125	—	101	2	no	—
180	#125	—	101	1	no	—
181	#125	C₂×D₁₀₁	1	1	yes	characteristic, complemented
182	#124	—	101	2	no	—
183	#124	—	101	2	no	—
184	#124	—	101	4	no	—
185	#124	—	101	4	no	—
186	#124	—	101	2	no	—
187	#124	C₂×D₁₀₁	1	2	yes	complemented
188	#123	—	101	4	no	—
189	#123	—	101	4	no	—
190	#123	—	101	4	no	—
191	#123	—	101	4	no	—
192	#123	—	101	4	no	—
193	#123	—	101	4	no	—
194	#123	—	101	4	no	—
195	#123	C₂×D₁₀₁	1	4	yes	complemented
196	C₂²⋊Q₈	—	101	4	no	—
197	C₂²⋊Q₈	—	101	4	no	—
198	C₂²⋊Q₈	—	101	4	no	—
199	C₂²⋊Q₈	—	101	4	no	—
200	C₂²⋊Q₈	—	101	4	no	—
201	C₂²⋊Q₈	—	101	4	no	—
202	C₂²⋊Q₈	—	101	4	no	—
203	C₂²⋊Q₈	C₂×D₁₀₁	1	4	yes	complemented
204	#121	—	101	4	no	—
205	#121	—	101	4	no	—
206	#121	—	101	4	no	—
207	#121	—	101	4	no	—
208	#121	—	101	4	no	—
209	#121	—	101	4	no	—
210	#121	—	101	4	no	—
211	#121	C₂×D₁₀₁	1	4	yes	complemented
212	C₂×C₄⋊C₄	—	101	4	no	—
213	C₂×C₄⋊C₄	—	101	4	no	—
214	C₂×C₄⋊C₄	—	101	2	no	—
215	C₂×C₂²⋊C₄	—	101	4	no	—
216	C₂×C₂²⋊C₄	—	101	4	no	—
217	C₂×C₂²⋊C₄	—	101	4	no	—
218	C₂×C₄²	—	101	1	no	—
219	C₂⁴	—	101	4	no	—
220	C₂×Q₈	—	101	4	no	—
221	C₂×Q₈	—	101	2	no	—
222	C₂×Q₈	—	404	4	no	—
223	C₂×Q₈	C₂²×D₁₀₁	1	2	yes	—
224	C₂×D₄	—	101	4	no	—
225	C₂×D₄	—	101	4	no	—
226	C₂×D₄	—	101	4	no	—
227	C₂×D₄	—	101	4	no	—
228	C₂×D₄	—	101	2	no	—
229	C₂×D₄	—	404	8	no	—
230	C₂×D₄	—	404	4	no	—
231	C₂×D₄	C₂²×D₁₀₁	1	4	yes	—
232	C₂×D₄	C₂²×D₁₀₁	1	2	yes	—
233	C₂²×C₄	—	101	4	no	—
234	C₂²×C₄	—	101	4	no	—
235	C₂²×C₄	—	101	4	no	—
236	C₂²×C₄	—	101	1	no	—
237	C₂²×C₄	—	101	2	no	—
238	C₂²×C₄	—	101	4	no	—
239	C₂²×C₄	—	101	4	no	—
240	C₂²×C₄	—	404	8	no	—
241	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
242	C₄⋊C₄	—	101	4	no	—
243	C₄⋊C₄	—	101	4	no	—
244	C₄⋊C₄	—	101	4	no	—
245	C₄⋊C₄	—	101	4	no	—
246	C₄⋊C₄	—	101	4	no	—
247	C₄⋊C₄	—	101	4	no	—
248	C₄⋊C₄	—	101	4	no	—
249	C₄⋊C₄	—	101	2	no	—
250	C₄⋊C₄	C₂²×D₁₀₁	1	4	yes	—
251	C₄⋊C₄	C₂²×D₁₀₁	1	4	yes	—
252	C₄⋊C₄	C₂²×D₁₀₁	1	2	yes	—
253	C₂²⋊C₄	—	101	4	no	—
254	C₂²⋊C₄	—	101	4	no	—
255	C₂²⋊C₄	—	101	4	no	—
256	C₂²⋊C₄	—	101	4	no	—
257	C₂²⋊C₄	—	101	4	no	—
258	C₂²⋊C₄	—	101	4	no	—
259	C₂²⋊C₄	—	101	4	no	—
260	C₂²⋊C₄	—	101	4	no	—
261	C₂²⋊C₄	—	101	4	no	—
262	C₂²⋊C₄	C₂²×D₁₀₁	1	4	yes	—
263	C₂²⋊C₄	C₂²×D₁₀₁	1	4	yes	—
264	C₂²⋊C₄	C₂²×D₁₀₁	1	4	yes	—
265	C₄²	—	101	2	no	—
266	C₄²	—	101	1	no	—
267	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
268	C₂³	—	101	1	no	—
269	C₂³	—	101	4	no	—
270	C₂³	—	202	8	no	—
271	C₂³	—	202	8	no	—
272	C₂³	—	202	8	no	—
273	C₂³	C₂³×D₁₀₁	1	4	yes	—
274	Q₈	—	4	2	no	—
275	Q₈	—	404	4	no	—
276	Q₈	—	404	2	no	—
277	D₄	—	4	4	no	—
278	D₄	—	4	2	no	—
279	D₄	—	404	4	no	—
280	D₄	—	404	2	no	—
281	D₄	—	404	4	no	—
282	D₄	—	404	4	no	—
283	D₄	—	404	4	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	101	1	no	—
286	C₂×C₄	—	101	2	no	—
287	C₂×C₄	—	101	4	no	—
288	C₂×C₄	—	101	4	no	—
289	C₂×C₄	—	404	4	no	—
290	C₂×C₄	—	404	4	no	—
291	C₂×C₄	—	404	4	no	—
292	C₂×C₄	—	202	8	no	—
293	C₂×C₄	—	202	8	no	—
294	C₂×C₄	—	202	4	no	—
295	C₂×C₄	—	202	2	no	—
296	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
297	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
298	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
299	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	2	4	no	—
303	C₂²	—	101	4	no	—
304	C₂²	—	101	2	no	—
305	C₂²	—	404	16	no	—
306	C₂²	—	202	4	no	—
307	C₂²	—	202	4	no	—
308	C₂²	—	202	4	no	—
309	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
310	C₄	—	2	4	no	—
311	C₄	—	2	4	no	—
312	C₄	—	2	2	no	—
313	C₄	—	2	1	no	—
314	C₄	—	202	4	no	—
315	C₄	—	202	4	no	—
316	C₄	—	202	2	no	—
317	C₄	—	202	1	no	—
318	C₂	—	4	4	no	—
319	C₂	—	101	4	no	—
320	C₂	—	404	4	no	—
321	C₂	H(5,𝔽₂)×D₁₀₁	1	2	yes	—
322	C₂	2₋¹⁺⁴×D₁₀₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₁
commutator, socle, cosocle	C₂²×C₁₀₁	C₂⁵
Fitting	#562×C₁₀₁	C₂
hypercenter	#562	D₁₀₁
hypocenter	C₁₀₁	C₂×#562
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ #562

Complements

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

N	H ≅ G/N	type
1. #20265925	C₂	semidirect
2. #20265924	C₂	semidirect
3. #20265923	C₂	semidirect
4. #20265922	C₂	semidirect
5. #20265921	C₂	semidirect
6. #125×D₁₀₁	C₂	semidirect
7. #124×D₁₀₁	C₂	semidirect
8. #123×D₁₀₁	C₂	semidirect
9. C₂²⋊Q₈×D₁₀₁	C₂	semidirect
10. #121×D₁₀₁	C₂	semidirect
11. #562×C₁₀₁	C₂	semidirect
16. #15692247	C₂²	semidirect
17. #15692246	C₂²	semidirect
18. #15692245	C₂²	semidirect
19. #15692244	C₂²	semidirect
20. #15692243	C₂²	semidirect
21. #15692242	C₂²	semidirect
22. #15692241	C₂²	semidirect
23. #15692241	C₂²	semidirect
24. #15692240	C₂²	semidirect
25. #15692240	C₂²	semidirect
26. #15692239	C₂²	semidirect
27. #15692238	C₂²	semidirect
28. #15692237	C₂²	semidirect
29. #15692237	C₂²	semidirect
30. #15692236	C₂²	semidirect
31. #15692235	C₂²	semidirect
32. #15692234	C₂²	semidirect
33. #15692233	C₂²	semidirect
34. #15692233	C₂²	semidirect
35. #15692232	C₂²	semidirect
36. #15692232	C₂²	semidirect
37. #15692231	C₂²	semidirect
38. #15692230	C₂²	semidirect
39. #15692229	C₂²	semidirect
40. #15692228	C₂²	semidirect
41. #15692228	C₂²	semidirect
42. #15692227	C₂²	semidirect
43. #15692227	C₂²	semidirect
44. #15692226	C₂²	semidirect
52. #125×C₁₀₁	C₂²	semidirect
53. #124×C₁₀₁	C₂²	semidirect
54. #123×C₁₀₁	C₂²	semidirect
55. C₂²⋊Q₈×C₁₀₁	C₂²	semidirect
56. #121×C₁₀₁	C₂²	semidirect
160. D₁₀₁	#562	direct (N × H)
162. C₁₀₁	C₂×#562	semidirect
168. #562	D₁₀₁	direct (N × H)
181. #125	C₂×D₁₀₁	semidirect
187. #124	C₂×D₁₀₁	semidirect
195. #123	C₂×D₁₀₁	semidirect
203. C₂²⋊Q₈	C₂×D₁₀₁	semidirect
211. #121	C₂×D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
#20265925	2
#20265924	2
#20265923	2
#20265922	2
#20265921	2
#125×D₁₀₁	2
#124×D₁₀₁	2
#123×D₁₀₁	2
C₂²⋊Q₈×D₁₀₁	2
#121×D₁₀₁	2
#562×C₁₀₁	2
C₂×#562	101

Subgroups — #562×D101 #27185254

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #562×D₁₀₁ #27185254