Subgroups — #44408538

Order 29056.

548 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20300314	C₂	1	1	yes	maximal, characteristic
2	#28409500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₄×Dic₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#28409141	C₂	1	2	yes	maximal, complemented
5	#28409129	C₂	1	2	yes	maximal, complemented
6	#28409123	C₂	1	2	yes	maximal, complemented
7	#28409112	C₂	1	2	yes	maximal, complemented
8	#28409075	C₂	1	1	yes	maximal, characteristic, complemented
9	#28409034	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×Dic₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#418×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#16813884	C₂²	1	1	yes	characteristic
13	C₂³×Dic₂₂₇	—	2	1	no	—
14	#20300314	C₄	1	8	yes	complemented
15	C₂×#16813872	C₂²	1	2	yes	—
16	C₂×#16813869	C₂²	1	2	yes	—
17	C₂×#16813868	C₂²	1	1	yes	characteristic
18	#20300253	C₂²	1	1	yes	characteristic, complemented
19	#20300253	C₂²	1	1	yes	characteristic, complemented
20	D₄×Dic₂₂₇	—	2	1	no	—
21	D₄×Dic₂₂₇	—	2	1	no	—
22	D₄×Dic₂₂₇	C₂²	1	2	yes	—
23	D₄×Dic₂₂₇	C₂²	1	2	yes	—
24	#20300250	C₂²	1	1	yes	characteristic, complemented
25	#20300250	C₂²	1	1	yes	characteristic, complemented
26	(C₂²⋊D₂₂₇)⋊C₄	—	2	1	no	—
27	(C₂²⋊D₂₂₇)⋊C₄	—	2	1	no	—
28	#20300245	—	2	1	no	—
29	C₂×C₄⋊Dic₂₂₇	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊Dic₂₂₇	—	2	1	no	—
31	C₂×C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
32	C₄²×D₂₂₇	—	2	1	no	—
33	#20300221	C₂²	1	2	yes	complemented
34	#20300219	C₂²	1	2	yes	complemented
35	#20300213	C₂²	1	2	yes	complemented
36	#20300210	C₂²	1	2	yes	complemented
37	#20300198	C₂²	1	2	yes	complemented
38	C₈×Dic₂₂₇	C₂²	1	2	yes	complemented
39	C₂²×D₄×C₂₂₇	C₂²	1	1	yes	characteristic
40	C₂×M₁₆×C₂₂₇	C₂²	1	1	yes	characteristic
41	#117×C₂₂₇	C₂²	1	1	yes	characteristic
42	#102×C₂₂₇	C₂²	1	2	yes	—
43	#102×C₂₂₇	C₂²	1	2	yes	—
44	#16813884	—	2	2	no	—
45	#16813884	C₂×C₄	1	4	yes	complemented
46	C₂×Dic₄₅₄	C₂³	1	1	yes	characteristic
47	C₂×D₉₀₈	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊D₂₂₇	—	2	1	no	—
49	C₂²×Dic₂₂₇	—	2	2	no	—
50	C₂²×Dic₂₂₇	—	2	2	no	—
51	C₂²×Dic₂₂₇	—	2	1	no	—
52	C₂²×Dic₂₂₇	—	4	4	no	—
53	C₂²×Dic₂₂₇	D₄	1	1	yes	characteristic
54	C₂²×Dic₂₂₇	D₄	1	1	yes	characteristic
55	C₂×C₄×D₂₂₇	—	2	1	no	—
56	C₂×C₄×D₂₂₇	—	2	1	no	—
57	C₂×C₄×D₂₂₇	—	2	1	no	—
58	#16813872	C₂×C₄	1	8	yes	complemented
59	#16813869	C₂×C₄	1	8	yes	complemented
60	#16813868	C₂×C₄	1	4	yes	complemented
61	C₂×C₂₂₇⋊C₈	C₂³	1	2	yes	—
62	C₄⋊Dic₂₂₇	—	2	1	no	—
63	C₄⋊Dic₂₂₇	C₂³	1	2	yes	—
64	Dic₂₂₇⋊C₄	—	2	1	no	—
65	Dic₂₂₇⋊C₄	—	2	1	no	—
66	C₂²⋊Dic₂₂₇	—	2	2	no	—
67	C₂²⋊Dic₂₂₇	—	2	2	no	—
68	(C₂×D₂₂₇)⋊C₄	—	2	1	no	—
69	(C₂×D₂₂₇)⋊C₄	—	2	1	no	—
70	C₄×Dic₂₂₇	—	2	1	no	—
71	C₄×Dic₂₂₇	—	2	1	no	—
72	C₄×Dic₂₂₇	D₄	1	1	yes	characteristic
73	C₄×Dic₂₂₇	D₄	1	1	yes	characteristic
74	C₄×Dic₂₂₇	C₂³	1	1	yes	characteristic
75	C₄×Dic₂₂₇	C₂³	1	1	yes	characteristic
76	C₂⁴×C₂₂₇	—	2	1	no	—
77	C₂×D₄×C₂₂₇	—	2	1	no	—
78	C₂×D₄×C₂₂₇	—	2	2	no	—
79	C₂×D₄×C₂₂₇	C₂×C₄	1	4	yes	complemented
80	C₂×D₄×C₂₂₇	C₂³	1	2	yes	—
81	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
82	M₁₆×C₂₂₇	—	2	2	no	—
83	C₂×C₈×C₂₂₇	C₂³	1	2	yes	—
84	C₄⋊C₄×C₂₂₇	C₂³	1	1	yes	characteristic
85	C₄⋊C₄×C₂₂₇	C₂³	1	1	yes	characteristic
86	C₂²⋊C₄×C₂₂₇	—	2	1	no	—
87	C₄²×C₂₂₇	—	2	1	no	—
88	C₂²×D₂₂₇	—	2	1	no	—
89	Dic₄₅₄	—	2	1	no	—
90	Dic₄₅₄	C₂²×C₄	1	2	yes	complemented
91	D₉₀₈	—	2	1	no	—
92	D₉₀₈	C₂²×C₄	1	2	yes	complemented
93	C₂²⋊D₂₂₇	—	2	4	no	—
94	C₂×Dic₂₂₇	—	2	1	no	—
95	C₂×Dic₂₂₇	—	2	2	no	—
96	C₂×Dic₂₂₇	—	2	2	no	—
97	C₂×Dic₂₂₇	—	4	1	no	—
98	C₂×Dic₂₂₇	—	4	4	no	—
99	C₂×Dic₂₂₇	—	4	2	no	—
100	C₂×Dic₂₂₇	—	4	4	no	—
101	C₂×Dic₂₂₇	C₂×D₄	1	1	yes	characteristic
102	C₂×Dic₂₂₇	C₂×D₄	1	1	yes	characteristic
103	C₂×Dic₂₂₇	C₂×D₄	1	1	yes	characteristic
104	C₂×Dic₂₂₇	C₂×D₄	1	1	yes	characteristic
105	C₄×D₂₂₇	—	2	4	no	—
106	C₄×D₂₂₇	—	4	2	no	—
107	C₄×D₂₂₇	—	4	2	no	—
108	C₂₂₇⋊C₈	C₂²×C₄	1	4	yes	complemented
109	C₂³×C₂₂₇	—	2	4	no	—
110	C₂³×C₂₂₇	—	2	2	no	—
111	C₂³×C₂₂₇	—	4	2	no	—
112	C₂³×C₂₂₇	—	4	2	no	—
113	C₂³×C₂₂₇	C₂×D₄	1	1	yes	characteristic
114	D₄×C₂₂₇	—	2	4	no	—
115	D₄×C₂₂₇	—	2	2	no	—
116	D₄×C₂₂₇	C₂²×C₄	1	4	yes	—
117	C₂×C₄×C₂₂₇	—	2	1	no	—
118	C₂×C₄×C₂₂₇	—	2	1	no	—
119	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
120	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₂₂₇	#41	1	2	yes	—
122	C₂×C₄×C₂₂₇	C₂⁴	1	1	yes	characteristic
123	C₈×C₂₂₇	—	2	2	no	—
124	C₂²×C₂₂₇	—	2	1	no	—
125	C₂²×C₂₂₇	—	2	1	no	—
126	C₂²×C₂₂₇	—	2	4	no	—
127	C₂²×C₂₂₇	—	4	8	no	—
128	C₂²×C₂₂₇	—	4	2	no	—
129	C₂²×C₂₂₇	—	4	2	no	—
130	C₂²×C₂₂₇	C₄×D₄	1	2	yes	—
131	C₂²×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
132	C₂×D₂₂₇	—	2	2	no	—
133	C₂×D₂₂₇	—	4	1	no	—
134	C₂×D₂₂₇	—	4	1	no	—
135	C₄×C₂₂₇	—	4	1	no	—
136	C₄×C₂₂₇	—	4	1	no	—
137	C₄×C₂₂₇	C₄×D₄	1	2	yes	—
138	C₄×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
139	C₄×C₂₂₇	C₂×#41	1	1	yes	characteristic
140	Dic₂₂₇	—	2	2	no	—
141	Dic₂₂₇	—	2	1	no	—
142	Dic₂₂₇	—	2	1	no	—
143	Dic₂₂₇	—	2	1	no	—
144	Dic₂₂₇	—	4	2	no	—
145	Dic₂₂₇	AGL(1,ℤ/8)	1	2	yes	complemented
146	C₂×C₂₂₇	—	2	2	no	—
147	C₂×C₂₂₇	—	4	4	no	—
148	C₂×C₂₂₇	C₂×C₄×D₄	1	1	yes	characteristic
149	C₂×C₂₂₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₂×C₂₂₇	#575	1	1	yes	characteristic
151	D₂₂₇	—	4	2	no	—
152	C₂₂₇	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
153	C₄×AGL(1,ℤ/8)	—	227	1	no	maximal, Sylow
154	C₂×AGL(1,ℤ/8)	—	227	1	no	—
155	C₄×#41	—	227	1	no	—
156	C₂×C₄×D₄	—	227	1	no	—
157	#442	—	227	2	no	—
158	#440	—	227	2	no	—
159	C₄×SD₁₆	—	227	2	no	—
160	C₄×D₈	—	227	2	no	—
161	#428	—	227	1	no	—
162	#418	D₂₂₇	1	1	yes	characteristic, complemented
163	#417	—	227	1	no	—
164	C₄×M₁₆	—	227	1	no	—
165	C₂×#41	—	227	1	no	—
166	C₂²×D₄	C₂×D₂₂₇	1	1	yes	characteristic
167	C₂³×C₄	—	454	1	no	—
168	AGL(1,ℤ/8)	—	227	8	no	—
169	C₂×SD₁₆	—	227	2	no	—
170	C₂×D₈	—	227	2	no	—
171	C₂×M₁₆	—	227	1	no	—
172	C₂×M₁₆	C₂×D₂₂₇	1	1	yes	characteristic
173	C₄×Q₈	—	227	1	no	—
174	C₄×Q₈	—	227	1	no	—
175	C₄×D₄	—	454	1	no	—
176	C₄×D₄	—	454	1	no	—
177	C₄×D₄	—	454	1	no	—
178	C₄×D₄	—	454	1	no	—
179	C₄×D₄	—	227	2	no	—
180	C₄×D₄	—	227	2	no	—
181	C₄×D₄	—	227	1	no	—
182	C₄×D₄	—	227	1	no	—
183	#117	—	454	1	no	—
184	#117	C₂×D₂₂₇	1	1	yes	characteristic
185	C₂×C₄⋊C₄	—	227	1	no	—
186	C₂×C₂²⋊C₄	—	454	1	no	—
187	C₂×C₄²	—	454	1	no	—
188	C₂×C₄²	—	227	1	no	—
189	#107	—	227	2	no	—
190	#106	—	227	2	no	—
191	#103	—	227	2	no	—
192	#102	—	227	2	no	—
193	#102	C₂×D₂₂₇	1	2	yes	—
194	#102	C₂×D₂₂₇	1	2	yes	—
195	#97	—	227	2	no	—
196	C₄×C₈	—	227	2	no	—
197	C₂⁴	—	2	1	no	—
198	#41	—	454	2	no	—
199	#41	—	227	4	no	—
200	C₂×Q₈	—	227	1	no	—
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	454	1	no	—
204	C₂×D₄	—	227	1	no	—
205	C₂×D₄	C₄×D₂₂₇	1	4	yes	complemented
206	C₂×D₄	C₂²×D₂₂₇	1	2	yes	—
207	C₂²×C₄	—	908	4	no	—
208	C₂²×C₄	—	454	1	no	—
209	C₂²×C₄	—	454	1	no	—
210	C₂²×C₄	—	454	1	no	—
211	C₂²×C₄	—	454	2	no	—
212	C₂²×C₄	—	454	2	no	—
213	C₂²×C₄	—	454	1	no	—
214	C₂²×C₄	—	227	1	no	—
215	C₂²×C₄	—	227	1	no	—
216	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
217	SD₁₆	—	227	8	no	—
218	D₈	—	227	8	no	—
219	M₁₆	—	2	2	no	—
220	M₁₆	—	227	4	no	—
221	C₂×C₈	—	227	2	no	—
222	C₂×C₈	C₂²×D₂₂₇	1	2	yes	—
223	C₄⋊C₄	—	454	1	no	—
224	C₄⋊C₄	—	454	1	no	—
225	C₄⋊C₄	—	454	1	no	—
226	C₄⋊C₄	—	227	2	no	—
227	C₄⋊C₄	C₂²×D₂₂₇	1	1	yes	characteristic
228	C₄⋊C₄	C₂²×D₂₂₇	1	1	yes	characteristic
229	C₂²⋊C₄	—	2	1	no	—
230	C₂²⋊C₄	—	454	1	no	—
231	C₂²⋊C₄	—	454	1	no	—
232	C₂²⋊C₄	—	454	2	no	—
233	C₂²⋊C₄	—	454	2	no	—
234	C₄²	—	2	1	no	—
235	C₄²	—	454	1	no	—
236	C₄²	—	454	1	no	—
237	C₄²	—	227	1	no	—
238	C₄²	—	227	1	no	—
239	C₄²	—	227	1	no	—
240	C₄²	—	227	1	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	4	no	—
243	C₂³	—	4	2	no	—
244	C₂³	—	4	2	no	—
245	C₂³	—	454	1	no	—
246	C₂³	D₄×D₂₂₇	1	1	yes	characteristic
247	Q₈	—	454	1	no	—
248	Q₈	—	227	2	no	—
249	D₄	—	2	2	no	—
250	D₄	—	2	4	no	—
251	D₄	—	454	4	no	—
252	D₄	—	454	1	no	—
253	D₄	—	227	2	no	—
254	D₄	C₂×C₄×D₂₂₇	1	4	yes	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	908	2	no	—
258	C₂×C₄	—	908	2	no	—
259	C₂×C₄	—	908	2	no	—
260	C₂×C₄	—	908	4	no	—
261	C₂×C₄	—	908	4	no	—
262	C₂×C₄	—	908	1	no	—
263	C₂×C₄	—	454	2	no	—
264	C₂×C₄	—	454	1	no	—
265	C₂×C₄	—	454	2	no	—
266	C₂×C₄	—	454	4	no	—
267	C₂×C₄	—	227	1	no	—
268	C₂×C₄	—	227	1	no	—
269	C₂×C₄	—	227	1	no	—
270	C₂×C₄	—	227	1	no	—
271	C₂×C₄	C₂×C₄×D₂₂₇	1	2	yes	—
272	C₂×C₄	D₄×D₂₂₇	1	1	yes	characteristic
273	C₂×C₄	#16813883	1	2	yes	—
274	C₂×C₄	C₂³×D₂₂₇	1	1	yes	characteristic
275	C₈	—	2	2	no	—
276	C₈	—	227	4	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	4	8	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	908	1	no	—
284	C₂²	—	908	1	no	—
285	C₂²	—	454	2	no	—
286	C₂²	(C₂²⋊D₂₂₇)⋊C₄	1	2	yes	—
287	C₂²	C₂×D₄×D₂₂₇	1	1	yes	characteristic
288	C₄	—	4	1	no	—
289	C₄	—	4	1	no	—
290	C₄	—	908	2	no	—
291	C₄	—	454	2	no	—
292	C₄	—	454	1	no	—
293	C₄	—	454	1	no	—
294	C₄	—	454	1	no	—
295	C₄	—	227	2	no	—
296	C₄	(C₂²⋊D₂₂₇)⋊C₄	1	2	yes	—
297	C₄	C₂²×C₄×D₂₂₇	1	1	yes	characteristic
298	C₄	C₂×#16813883	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	4	4	no	—
301	C₂	—	908	2	no	—
302	C₂	C₂×(C₂²⋊D₂₂₇)⋊C₄	1	1	yes	characteristic
303	C₂	AGL(1,ℤ/8)×D₂₂₇	1	1	yes	characteristic
304	C₂	#28409828	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₂₇
commutator	C₄×C₂₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₂₇
Fitting	#418×C₂₂₇	C₂
socle	C₂²×C₂₂₇	C₂²×D₄
prime radical	C₂×#20300314	C₂
cosocle	C₂×C₄×C₂₂₇	C₂⁴
hypercenter	#418	D₂₂₇
hypocenter	C₂₂₇	C₄×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #418

Complements

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

N	H ≅ G/N	type
2. #28409500	C₂	semidirect
3. C₂×D₄×Dic₂₂₇	C₂	semidirect
4. #28409141	C₂	semidirect
5. #28409129	C₂	semidirect
6. #28409123	C₂	semidirect
7. #28409112	C₂	semidirect
8. #28409075	C₂	semidirect
9. #28409034	C₂	semidirect
10. M₁₆×Dic₂₂₇	C₂	semidirect
11. #418×C₂₂₇	C₂	semidirect
14. #20300314	C₄	semidirect
18. #20300253	C₂²	semidirect
19. #20300253	C₂²	semidirect
24. #20300250	C₂²	semidirect
25. #20300250	C₂²	semidirect
33. #20300221	C₂²	semidirect
34. #20300219	C₂²	semidirect
35. #20300213	C₂²	semidirect
36. #20300210	C₂²	semidirect
37. #20300198	C₂²	semidirect
38. C₈×Dic₂₂₇	C₂²	semidirect
45. #16813884	C₂×C₄	semidirect
58. #16813872	C₂×C₄	semidirect
59. #16813869	C₂×C₄	semidirect
60. #16813868	C₂×C₄	semidirect
79. C₂×D₄×C₂₂₇	C₂×C₄	semidirect
90. Dic₄₅₄	C₂²×C₄	semidirect
92. D₉₀₈	C₂²×C₄	semidirect
108. C₂₂₇⋊C₈	C₂²×C₄	semidirect
145. Dic₂₂₇	AGL(1,ℤ/8)	semidirect
152. C₂₂₇	C₄×AGL(1,ℤ/8)	semidirect
162. #418	D₂₂₇	semidirect
205. C₂×D₄	C₄×D₂₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20300314	2
#28409500	2
C₂×D₄×Dic₂₂₇	2
#28409141	2
#28409129	2
#28409123	2
#28409112	2
#28409075	2
#28409034	2
M₁₆×Dic₂₂₇	2
#418×C₂₂₇	2
C₄×AGL(1,ℤ/8)	227