Subgroups — #41×D₄₀₄ #27184623

Order 12928.

1648 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×D₁₀₁	C₂	1	2	yes	maximal, complemented
2	#20265849	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×D₄₀₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265839	C₂	1	2	yes	maximal, complemented
5	#20265831	C₂	1	3	yes	maximal, complemented
6	#20265822	C₂	1	6	yes	maximal, complemented
7	D₄×D₄₀₄	C₂	1	3	yes	maximal, complemented
8	#20265727	C₂	1	3	yes	maximal, complemented
9	#20265689	C₂	1	6	yes	maximal, complemented
10	C₂×C₄×D₄₀₄	C₂	1	3	yes	maximal, complemented
11	C₄×#41×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13108075	C₂²	1	2	yes	complemented
13	C₂×#13108074	C₂²	1	6	yes	complemented
14	C₂×#13108073	C₂²	1	6	yes	complemented
15	#41×D₁₀₁	—	2	16	no	—
16	C₂×Q₈×D₁₀₁	C₂²	1	2	yes	complemented
17	C₂²×D₄₀₄	C₂²	1	3	yes	—
18	C₂×D₄×D₁₀₁	C₂²	1	6	yes	complemented
19	C₂²×C₄×D₁₀₁	C₂²	1	6	yes	complemented
20	#15692255	C₂²	1	3	yes	complemented
21	#15692252	C₂²	1	3	yes	complemented
22	#15692243	C₂²	1	6	yes	complemented
23	#15692239	C₂²	1	12	yes	complemented
24	#15692235	C₂²	1	6	yes	complemented
25	#15692235	C₂²	1	6	yes	complemented
26	#15692230	C₂²	1	6	yes	complemented
27	#15692230	C₂²	1	6	yes	complemented
28	#15692229	C₂²	1	6	yes	complemented
29	#15692229	C₂²	1	6	yes	complemented
30	#15692225	C₂²	1	12	yes	complemented
31	C₄×D₄₀₄	C₂²	1	3	yes	—
32	C₄×D₄₀₄	C₂²	1	1	yes	characteristic
33	C₄×D₄₀₄	C₂²	1	6	yes	complemented
34	C₄×D₄₀₄	C₂²	1	3	yes	complemented
35	C₄×D₄₀₄	C₂²	1	6	yes	complemented
36	C₄×D₄₀₄	C₂²	1	3	yes	complemented
37	C₄×D₄₀₄	C₂²	1	6	yes	complemented
38	C₂×C₄⋊Dic₁₀₁	C₂²	1	3	yes	complemented
39	C₂×#13108051	C₂²	1	6	yes	complemented
40	C₂×#41×C₁₀₁	C₂²	1	1	yes	characteristic
41	C₄×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
42	C₄×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
43	C₄×D₄×C₁₀₁	C₂²	1	3	yes	complemented
44	C₄×D₄×C₁₀₁	C₂²	1	3	yes	complemented
45	#117×C₁₀₁	C₂²	1	3	yes	complemented
46	C₂×C₄²×C₁₀₁	C₂²	1	3	yes	complemented
47	C₂³×D₁₀₁	C₂³	1	6	yes	—
48	#13108075	—	2	8	no	—
49	#13108074	—	2	24	no	—
50	#13108073	—	2	24	no	—
51	C₂×Dic₂₀₂	C₂³	1	6	yes	—
52	Q₈×D₁₀₁	—	2	8	no	—
53	C₂×D₄₀₄	—	2	12	no	—
54	C₂×D₄₀₄	C₂³	1	6	yes	—
55	C₂×D₄₀₄	C₂³	1	6	yes	—
56	C₂×D₄₀₄	C₂³	1	3	yes	—
57	C₂×D₄₀₄	C₂³	1	6	yes	—
58	C₂×D₄₀₄	C₂³	1	1	yes	characteristic
59	C₂×C₂²⋊D₁₀₁	C₂³	1	12	yes	—
60	D₄×D₁₀₁	—	2	24	no	—
61	C₂²×Dic₁₀₁	C₂³	1	6	yes	—
62	C₂×C₄×D₁₀₁	—	2	24	no	—
63	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
64	C₂×C₄×D₁₀₁	C₂³	1	6	yes	—
65	C₂×C₄×D₁₀₁	C₂³	1	6	yes	—
66	C₂×C₄×D₁₀₁	C₂³	1	6	yes	—
67	C₄⋊Dic₁₀₁	C₂³	1	1	yes	characteristic
68	C₄⋊Dic₁₀₁	C₂³	1	3	yes	—
69	C₄⋊Dic₁₀₁	C₂³	1	6	yes	—
70	C₄⋊Dic₁₀₁	C₂³	1	6	yes	—
71	#13108051	C₂³	1	2	yes	—
72	#13108051	C₂³	1	6	yes	—
73	#13108051	C₂³	1	6	yes	—
74	#13108051	C₂³	1	6	yes	—
75	#13108051	C₂³	1	12	yes	—
76	#41×C₁₀₁	D₄	1	8	yes	complemented
77	C₂×Q₈×C₁₀₁	C₂³	1	1	yes	characteristic
78	C₂×D₄×C₁₀₁	C₂³	1	3	yes	—
79	C₂²×C₄×C₁₀₁	C₂³	1	3	yes	—
80	C₂²×C₄×C₁₀₁	C₂³	1	3	yes	—
81	C₂²×C₄×C₁₀₁	C₂³	1	3	yes	—
82	C₄⋊C₄×C₁₀₁	C₂³	1	3	yes	—
83	C₄⋊C₄×C₁₀₁	C₂³	1	3	yes	—
84	C₂²⋊C₄×C₁₀₁	C₂³	1	6	yes	—
85	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
86	C₄²×C₁₀₁	C₂³	1	3	yes	—
87	C₄²×C₁₀₁	C₂³	1	3	yes	—
88	C₄²×C₁₀₁	C₂³	1	3	yes	—
89	C₂³×C₁₀₁	C₂⁴	1	3	yes	—
90	C₂²×D₁₀₁	—	2	12	no	—
91	C₂²×D₁₀₁	—	2	12	no	—
92	C₂²×D₁₀₁	—	4	6	no	—
93	C₂²×D₁₀₁	C₂⁴	1	6	yes	—
94	C₂²×D₁₀₁	C₂⁴	1	2	yes	—
95	Q₈×C₁₀₁	C₂×D₄	1	4	yes	complemented
96	D₄×C₁₀₁	C₂×D₄	1	12	yes	complemented
97	C₂×C₄×C₁₀₁	—	2	6	no	—
98	C₂×C₄×C₁₀₁	—	4	3	no	—
99	C₂×C₄×C₁₀₁	C₂×D₄	1	12	yes	complemented
100	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
101	C₂×C₄×C₁₀₁	C₂⁴	1	3	yes	—
102	C₂×C₄×C₁₀₁	C₂⁴	1	3	yes	—
103	C₂×C₄×C₁₀₁	C₂⁴	1	3	yes	—
104	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
105	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
106	C₂²⋊D₁₀₁	—	2	24	no	—
107	C₂×Dic₁₀₁	—	2	12	no	—
108	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
109	C₂×Dic₁₀₁	C₂⁴	1	6	yes	—
110	D₄₀₄	—	2	12	no	—
111	D₄₀₄	—	2	12	no	—
112	D₄₀₄	—	2	6	no	—
113	D₄₀₄	—	4	6	no	—
114	D₄₀₄	#41	1	4	yes	direct product factor
115	C₄×D₁₀₁	—	2	4	no	—
116	C₄×D₁₀₁	—	2	12	no	—
117	C₄×D₁₀₁	—	2	12	no	—
118	C₄×D₁₀₁	—	2	12	no	—
119	Dic₂₀₂	—	2	12	no	—
120	C₂²×C₁₀₁	—	2	3	no	—
121	C₂²×C₁₀₁	—	2	3	no	—
122	C₂²×C₁₀₁	C₂²×D₄	1	6	yes	—
123	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
124	C₂×D₁₀₁	—	2	2	no	—
125	C₂×D₁₀₁	—	2	2	no	—
126	C₂×D₁₀₁	—	2	6	no	—
127	C₂×D₁₀₁	—	2	6	no	—
128	C₂×D₁₀₁	—	2	6	no	—
129	C₂×D₁₀₁	—	4	24	no	—
130	C₂×D₁₀₁	C₂×#41	1	4	yes	complemented
131	C₄×C₁₀₁	—	2	3	no	—
132	C₄×C₁₀₁	—	2	3	no	—
133	C₄×C₁₀₁	—	2	1	no	—
134	C₄×C₁₀₁	C₂²×D₄	1	6	yes	—
135	C₄×C₁₀₁	C₂²×D₄	1	2	yes	—
136	C₄×C₁₀₁	C₂×#41	1	2	yes	complemented
137	Dic₁₀₁	—	2	6	no	—
138	Dic₁₀₁	—	2	2	no	—
139	C₂×C₁₀₁	—	2	6	no	—
140	C₂×C₁₀₁	C₂³×D₄	1	1	yes	characteristic
141	C₂×C₁₀₁	C₂²×#41	1	1	yes	characteristic
142	C₂×C₁₀₁	#585	1	1	yes	characteristic
143	D₁₀₁	—	2	4	no	—
144	D₁₀₁	—	4	6	no	—
145	D₄×#41	—	101	1	no	maximal, Sylow
146	C₁₀₁	D₄×#41	1	1	yes	characteristic, complemented, Sylow
147	C₂²×#41	—	101	2	no	—
148	#550	—	101	1	no	—
149	D₄×Q₈	—	101	1	no	—
150	#548	—	101	2	no	—
151	#547	—	101	3	no	—
152	#546	—	101	6	no	—
153	D₄²	—	101	3	no	—
154	#532	—	101	3	no	—
155	#525	—	101	6	no	—
156	C₄×#41	D₁₀₁	1	1	yes	characteristic, complemented
157	C₂×C₄×D₄	—	101	3	no	—
158	C₂×#41	—	101	2	no	—
159	C₂×#41	—	101	6	no	—
160	C₂×#41	—	101	6	no	—
161	C₂×#41	—	202	16	no	—
162	C₂×#41	C₂×D₁₀₁	1	1	yes	characteristic
163	C₂²×Q₈	—	101	2	no	—
164	C₂²×D₄	—	101	6	no	—
165	C₂²×D₄	—	101	3	no	—
166	C₂³×C₄	—	101	6	no	—
167	C₄⋊Q₈	—	101	3	no	—
168	#127	—	101	3	no	—
169	#124	—	101	6	no	—
170	#123	—	101	12	no	—
171	C₂²⋊Q₈	—	101	6	no	—
172	C₂²⋊Q₈	—	101	6	no	—
173	#121	—	101	6	no	—
174	#121	—	101	6	no	—
175	#121	—	101	6	no	—
176	#121	—	101	6	no	—
177	#120	—	101	12	no	—
178	C₄×Q₈	C₂×D₁₀₁	1	1	yes	characteristic, complemented
179	C₄×Q₈	C₂×D₁₀₁	1	1	yes	characteristic, complemented
180	C₄×D₄	—	101	1	no	—
181	C₄×D₄	—	101	6	no	—
182	C₄×D₄	—	101	3	no	—
183	C₄×D₄	—	101	3	no	—
184	C₄×D₄	—	101	6	no	—
185	C₄×D₄	—	101	3	no	—
186	C₄×D₄	—	101	6	no	—
187	C₄×D₄	C₂×D₁₀₁	1	3	yes	complemented
188	C₄×D₄	C₂×D₁₀₁	1	3	yes	complemented
189	#117	C₂×D₁₀₁	1	3	yes	complemented
190	C₂×C₄⋊C₄	—	101	3	no	—
191	C₂×C₂²⋊C₄	—	101	6	no	—
192	C₂×C₄²	C₂×D₁₀₁	1	3	yes	complemented
193	C₂⁴	—	101	6	no	—
194	#41	—	202	8	no	—
195	#41	—	202	24	no	—
196	#41	—	202	24	no	—
197	#41	D₄₀₄	1	8	yes	direct product factor
198	C₂×Q₈	—	101	6	no	—
199	C₂×Q₈	—	202	8	no	—
200	C₂×Q₈	C₂²×D₁₀₁	1	1	yes	characteristic
201	C₂×D₄	—	101	12	no	—
202	C₂×D₄	—	101	1	no	—
203	C₂×D₄	—	101	6	no	—
204	C₂×D₄	—	101	6	no	—
205	C₂×D₄	—	101	3	no	—
206	C₂×D₄	—	101	6	no	—
207	C₂×D₄	—	202	12	no	—
208	C₂×D₄	—	202	24	no	—
209	C₂×D₄	C₂²×D₁₀₁	1	3	yes	—
210	C₂²×C₄	—	101	6	no	—
211	C₂²×C₄	—	101	6	no	—
212	C₂²×C₄	—	101	6	no	—
213	C₂²×C₄	—	101	2	no	—
214	C₂²×C₄	—	101	6	no	—
215	C₂²×C₄	—	202	24	no	—
216	C₂²×C₄	C₂²×D₁₀₁	1	3	yes	—
217	C₂²×C₄	C₂²×D₁₀₁	1	3	yes	—
218	C₂²×C₄	C₂²×D₁₀₁	1	3	yes	—
219	C₄⋊C₄	—	101	1	no	—
220	C₄⋊C₄	—	101	3	no	—
221	C₄⋊C₄	—	101	6	no	—
222	C₄⋊C₄	—	101	6	no	—
223	C₄⋊C₄	C₂²×D₁₀₁	1	3	yes	—
224	C₄⋊C₄	C₂²×D₁₀₁	1	3	yes	—
225	C₂²⋊C₄	—	101	2	no	—
226	C₂²⋊C₄	—	101	6	no	—
227	C₂²⋊C₄	—	101	6	no	—
228	C₂²⋊C₄	—	101	6	no	—
229	C₂²⋊C₄	—	101	12	no	—
230	C₂²⋊C₄	C₂²×D₁₀₁	1	6	yes	—
231	C₄²	C₂²×D₁₀₁	1	3	yes	—
232	C₄²	C₂²×D₁₀₁	1	3	yes	—
233	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
234	C₄²	C₂²×D₁₀₁	1	3	yes	—
235	C₂³	—	101	2	no	—
236	C₂³	—	101	6	no	—
237	C₂³	—	404	6	no	—
238	C₂³	—	202	12	no	—
239	C₂³	—	202	12	no	—
240	C₂³	C₂³×D₁₀₁	1	3	yes	—
241	Q₈	—	202	12	no	—
242	Q₈	C₂×D₄₀₄	1	4	yes	complemented
243	D₄	—	101	4	no	—
244	D₄	—	404	6	no	—
245	D₄	—	202	12	no	—
246	D₄	—	202	24	no	—
247	D₄	—	202	12	no	—
248	D₄	—	202	6	no	—
249	D₄	C₂×D₄₀₄	1	12	yes	complemented
250	C₂×C₄	—	2	6	no	—
251	C₂×C₄	—	4	3	no	—
252	C₂×C₄	—	101	2	no	—
253	C₂×C₄	—	101	6	no	—
254	C₂×C₄	—	202	12	no	—
255	C₂×C₄	—	202	4	no	—
256	C₂×C₄	—	202	12	no	—
257	C₂×C₄	—	202	12	no	—
258	C₂×C₄	—	202	12	no	—
259	C₂×C₄	C₂×D₄₀₄	1	12	yes	complemented
260	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
261	C₂×C₄	C₂³×D₁₀₁	1	3	yes	—
262	C₂×C₄	C₂³×D₁₀₁	1	3	yes	—
263	C₂×C₄	C₂³×D₁₀₁	1	3	yes	—
264	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
265	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
266	C₂²	—	2	3	no	—
267	C₂²	—	2	3	no	—
268	C₂²	—	101	4	no	—
269	C₂²	—	404	24	no	—
270	C₂²	—	202	6	no	—
271	C₂²	—	202	6	no	—
272	C₂²	—	202	6	no	—
273	C₂²	—	202	2	no	—
274	C₂²	—	202	2	no	—
275	C₂²	C₂²×D₄₀₄	1	6	yes	—
276	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
277	C₄	—	2	3	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	3	no	—
280	C₄	—	202	2	no	—
281	C₄	—	202	6	no	—
282	C₄	C₂²×D₄₀₄	1	2	yes	—
283	C₄	C₂²×D₄₀₄	1	6	yes	—
284	C₄	#41×D₁₀₁	1	2	yes	complemented
285	C₂	—	2	6	no	—
286	C₂	—	404	6	no	—
287	C₂	—	202	4	no	—
288	C₂	C₂³×D₄₀₄	1	1	yes	characteristic
289	C₂	C₂×#41×D₁₀₁	1	1	yes	characteristic
290	C₂	#20266013	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₀₁
commutator, socle, cosocle	C₂²×C₁₀₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₁
Fitting	C₄×#41×C₁₀₁	C₂
hypercenter	C₄×#41	D₁₀₁
hypocenter	C₁₀₁	D₄×#41
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#41×D₁₀₁	C₂	semidirect
2. #20265849	C₂	semidirect
3. Q₈×D₄₀₄	C₂	semidirect
4. #20265839	C₂	semidirect
5. #20265831	C₂	semidirect
6. #20265822	C₂	semidirect
7. D₄×D₄₀₄	C₂	semidirect
8. #20265727	C₂	semidirect
9. #20265689	C₂	semidirect
10. C₂×C₄×D₄₀₄	C₂	semidirect
11. C₄×#41×C₁₀₁	C₂	semidirect
12. C₂×#13108075	C₂²	semidirect
13. C₂×#13108074	C₂²	semidirect
14. C₂×#13108073	C₂²	semidirect
16. C₂×Q₈×D₁₀₁	C₂²	semidirect
18. C₂×D₄×D₁₀₁	C₂²	semidirect
19. C₂²×C₄×D₁₀₁	C₂²	semidirect
20. #15692255	C₂²	semidirect
21. #15692252	C₂²	semidirect
22. #15692243	C₂²	semidirect
23. #15692239	C₂²	semidirect
24. #15692235	C₂²	semidirect
25. #15692235	C₂²	semidirect
26. #15692230	C₂²	semidirect
27. #15692230	C₂²	semidirect
28. #15692229	C₂²	semidirect
29. #15692229	C₂²	semidirect
30. #15692225	C₂²	semidirect
33. C₄×D₄₀₄	C₂²	semidirect
34. C₄×D₄₀₄	C₂²	semidirect
35. C₄×D₄₀₄	C₂²	semidirect
36. C₄×D₄₀₄	C₂²	semidirect
37. C₄×D₄₀₄	C₂²	semidirect
38. C₂×C₄⋊Dic₁₀₁	C₂²	semidirect
39. C₂×#13108051	C₂²	semidirect
41. C₄×Q₈×C₁₀₁	C₂²	semidirect
42. C₄×Q₈×C₁₀₁	C₂²	semidirect
43. C₄×D₄×C₁₀₁	C₂²	semidirect
44. C₄×D₄×C₁₀₁	C₂²	semidirect
45. #117×C₁₀₁	C₂²	semidirect
46. C₂×C₄²×C₁₀₁	C₂²	semidirect
76. #41×C₁₀₁	D₄	semidirect
95. Q₈×C₁₀₁	C₂×D₄	semidirect
96. D₄×C₁₀₁	C₂×D₄	semidirect
99. C₂×C₄×C₁₀₁	C₂×D₄	semidirect
114. D₄₀₄	#41	direct (N × H)
130. C₂×D₁₀₁	C₂×#41	semidirect
136. C₄×C₁₀₁	C₂×#41	semidirect
146. C₁₀₁	D₄×#41	semidirect
156. C₄×#41	D₁₀₁	semidirect
178. C₄×Q₈	C₂×D₁₀₁	semidirect
179. C₄×Q₈	C₂×D₁₀₁	semidirect
187. C₄×D₄	C₂×D₁₀₁	semidirect
188. C₄×D₄	C₂×D₁₀₁	semidirect
189. #117	C₂×D₁₀₁	semidirect
192. C₂×C₄²	C₂×D₁₀₁	semidirect
197. #41	D₄₀₄	direct (N × H)
242. Q₈	C₂×D₄₀₄	semidirect
249. D₄	C₂×D₄₀₄	semidirect
259. C₂×C₄	C₂×D₄₀₄	semidirect
284. C₄	#41×D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41×D₁₀₁	2
#20265849	2
Q₈×D₄₀₄	2
#20265839	2
#20265831	2
#20265822	2
D₄×D₄₀₄	2
#20265727	2
#20265689	2
C₂×C₄×D₄₀₄	2
C₄×#41×C₁₀₁	2
D₄×#41	101

Subgroups — #41×D404 #27184623

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #41×D₄₀₄ #27184623