Subgroups — #44408641

Order 29056.

458 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28409830	C₂	1	1	yes	maximal, characteristic
2	#28409798	C₂	1	1	yes	maximal, characteristic, complemented
3	#28409152	C₂	1	4	yes	maximal, complemented
4	#28409147	C₂	1	4	yes	maximal, complemented
5	#28409080	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#20300215	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#20300214	C₂	1	1	yes	maximal, characteristic, complemented
8	M₁₆×Dic₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#568×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16813884	C₂²	1	1	yes	characteristic
11	#20300319	C₄	1	2	yes	complemented
12	#20300318	C₄	1	2	yes	complemented
13	#20300314	C₄	1	2	yes	complemented
14	#20300313	C₄	1	2	yes	complemented
15	#20300308	C₂²	1	4	yes	—
16	#20300296	—	2	2	no	—
17	#20300294	C₂²	1	4	yes	complemented
18	C₂×#16813868	C₂²	1	1	yes	characteristic
19	C₂×#16813868	C₂²	1	1	yes	characteristic
20	M₁₆×D₂₂₇	—	2	2	no	—
21	C₂×C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
22	#20300223	C₂²	1	4	yes	complemented
23	#20300215	C₂²	1	2	yes	complemented
24	#20300215	C₂²	1	2	yes	complemented
25	#20300214	C₂²	1	2	yes	complemented
26	#20300214	C₂²	1	2	yes	complemented
27	#20300198	C₂²	1	2	yes	complemented
28	C₈×Dic₂₂₇	C₂²	1	2	yes	complemented
29	C₂×#41×C₂₂₇	C₂²	1	1	yes	characteristic
30	#131×C₂₂₇	—	2	2	no	—
31	#131×C₂₂₇	C₂²	1	4	yes	complemented
32	C₂×M₁₆×C₂₂₇	—	2	2	no	—
33	C₂×M₁₆×C₂₂₇	C₂²	1	1	yes	characteristic
34	C₂×M₁₆×C₂₂₇	C₂²	1	1	yes	characteristic
35	#16813884	—	2	1	no	—
36	#16813884	—	2	1	no	—
37	#16813884	C₂×C₄	1	2	yes	—
38	#16813884	C₂³	1	2	yes	—
39	C₂×Dic₄₅₄	C₂×C₄	1	1	yes	characteristic
40	C₂×D₉₀₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₂²⋊D₂₂₇	—	2	1	no	—
42	C₂²×Dic₂₂₇	—	2	1	no	—
43	C₂×C₄×D₂₂₇	—	2	1	no	—
44	#16813874	C₂×C₄	1	4	yes	complemented
45	#16813872	C₂×C₄	1	4	yes	complemented
46	#16813871	C₂×C₄	1	4	yes	complemented
47	#16813869	C₂×C₄	1	4	yes	complemented
48	#16813868	—	2	1	no	—
49	#16813868	C₂×C₄	1	2	yes	—
50	#16813868	C₂×C₄	1	2	yes	—
51	#16813868	C₂³	1	2	yes	—
52	#16813867	—	2	4	no	—
53	C₂×C₂₂₇⋊C₈	—	2	1	no	—
54	C₂×C₂₂₇⋊C₈	C₂³	1	2	yes	—
55	C₈×D₂₂₇	—	2	4	no	—
56	C₄×Dic₂₂₇	—	2	1	no	—
57	C₄×Dic₂₂₇	C₂³	1	1	yes	characteristic
58	C₄×Dic₂₂₇	C₂³	1	1	yes	characteristic
59	#41×C₂₂₇	—	2	1	no	—
60	#41×C₂₂₇	—	2	1	no	—
61	#41×C₂₂₇	C₂×C₄	1	2	yes	—
62	#41×C₂₂₇	C₂³	1	2	yes	—
63	C₂×Q₈×C₂₂₇	C₂×C₄	1	1	yes	characteristic
64	C₂×D₄×C₂₂₇	—	2	1	no	—
65	C₂×D₄×C₂₂₇	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₂₇	—	2	1	no	—
67	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
68	M₁₆×C₂₂₇	—	2	1	no	—
69	M₁₆×C₂₂₇	—	2	4	no	—
70	M₁₆×C₂₂₇	D₄	1	2	yes	—
71	M₁₆×C₂₂₇	D₄	1	2	yes	—
72	M₁₆×C₂₂₇	C₂³	1	2	yes	—
73	C₂×C₈×C₂₂₇	—	2	1	no	—
74	C₂×C₈×C₂₂₇	—	2	4	no	—
75	C₂×C₈×C₂₂₇	C₂³	1	2	yes	—
76	C₂²×D₂₂₇	—	2	1	no	—
77	Dic₄₅₄	—	2	1	no	—
78	Dic₄₅₄	C₂²×C₄	1	2	yes	—
79	D₉₀₈	—	2	1	no	—
80	D₉₀₈	C₂²×C₄	1	2	yes	—
81	C₂²⋊D₂₂₇	—	2	2	no	—
82	C₂²⋊D₂₂₇	—	2	2	no	—
83	C₂×Dic₂₂₇	—	2	1	no	—
84	C₂×Dic₂₂₇	—	2	1	no	—
85	C₂×Dic₂₂₇	—	2	1	no	—
86	C₂×Dic₂₂₇	—	4	2	no	—
87	C₄×D₂₂₇	—	2	2	no	—
88	C₄×D₂₂₇	—	2	2	no	—
89	C₂₂₇⋊C₈	—	2	2	no	—
90	C₂₂₇⋊C₈	C₂²×C₄	1	4	yes	—
91	C₂³×C₂₂₇	—	2	1	no	—
92	C₂³×C₂₂₇	#41	1	1	yes	characteristic
93	Q₈×C₂₂₇	—	2	1	no	—
94	Q₈×C₂₂₇	C₂²×C₄	1	2	yes	—
95	D₄×C₂₂₇	—	2	2	no	—
96	D₄×C₂₂₇	—	2	2	no	—
97	D₄×C₂₂₇	—	2	1	no	—
98	D₄×C₂₂₇	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₂₂₇	—	2	1	no	—
100	C₂×C₄×C₂₂₇	—	2	2	no	—
101	C₂×C₄×C₂₂₇	—	2	2	no	—
102	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₂₂₇	#41	1	1	yes	characteristic
107	C₂×C₄×C₂₂₇	C₂⁴	1	1	yes	characteristic
108	C₈×C₂₂₇	—	2	2	no	—
109	C₈×C₂₂₇	C₂×D₄	1	4	yes	—
110	C₂²×C₂₂₇	—	2	2	no	—
111	C₂²×C₂₂₇	—	4	1	no	—
112	C₂²×C₂₂₇	—	8	1	no	—
113	C₂²×C₂₂₇	C₄×D₄	1	1	yes	characteristic
114	C₂²×C₂₂₇	C₄×D₄	1	1	yes	characteristic
115	C₂²×C₂₂₇	C₂×#41	1	1	yes	characteristic
116	C₂×D₂₂₇	—	2	2	no	—
117	C₂×D₂₂₇	—	8	1	no	—
118	C₄×C₂₂₇	—	2	2	no	—
119	C₄×C₂₂₇	C₄×D₄	1	1	yes	characteristic
120	C₄×C₂₂₇	C₄×D₄	1	1	yes	characteristic
121	C₄×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
122	C₄×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
123	Dic₂₂₇	—	2	2	no	—
124	Dic₂₂₇	—	2	2	no	—
125	Dic₂₂₇	—	4	1	no	—
126	C₂×C₂₂₇	—	2	1	no	—
127	C₂×C₂₂₇	—	2	1	no	—
128	C₂×C₂₂₇	—	2	1	no	—
129	C₂×C₂₂₇	—	4	2	no	—
130	C₂×C₂₂₇	C₂×C₄×D₄	1	1	yes	characteristic
131	D₂₂₇	—	4	2	no	—
132	C₂₂₇	#2956	1	1	yes	characteristic, complemented, Sylow
133	#2956	—	227	1	no	maximal, Sylow
134	#575	—	227	1	no	—
135	#568	—	227	1	no	—
136	#568	D₂₂₇	1	1	yes	characteristic, complemented
137	#444	—	227	4	no	—
138	#443	—	227	4	no	—
139	#430	—	227	1	no	—
140	C₂×#104	—	227	1	no	—
141	C₂×#104	—	227	1	no	—
142	C₄×M₁₆	—	227	1	no	—
143	C₂×#41	—	227	1	no	—
144	C₂×#41	C₂×D₂₂₇	1	1	yes	characteristic
145	#137	—	227	2	no	—
146	#137	—	227	2	no	—
147	AGL(1,ℤ/8)	—	227	2	no	—
148	AGL(1,ℤ/8)	—	227	2	no	—
149	#135	—	227	4	no	—
150	#131	—	2	2	no	—
151	#131	—	454	2	no	—
152	#131	—	227	4	no	—
153	#131	C₂×D₂₂₇	1	4	yes	complemented
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	—	454	2	no	—
156	C₂×M₁₆	—	227	1	no	—
157	C₂×M₁₆	—	227	1	no	—
158	C₂×M₁₆	C₂×D₂₂₇	1	1	yes	characteristic
159	C₂×M₁₆	C₂×D₂₂₇	1	1	yes	characteristic
160	C₂×C₄²	—	227	1	no	—
161	#108	—	227	4	no	—
162	#104	—	227	2	no	—
163	#104	—	227	2	no	—
164	#104	—	227	2	no	—
165	#104	—	227	2	no	—
166	#97	—	227	2	no	—
167	C₄×C₈	—	227	2	no	—
168	#41	—	2	1	no	—
169	#41	—	2	1	no	—
170	#41	—	454	1	no	—
171	#41	—	454	1	no	—
172	#41	—	227	2	no	—
173	#41	—	227	2	no	—
174	#41	C₄×D₂₂₇	1	2	yes	—
175	#41	C₂²×D₂₂₇	1	2	yes	—
176	C₂×Q₈	—	227	1	no	—
177	C₂×Q₈	C₄×D₂₂₇	1	1	yes	characteristic
178	C₂×D₄	—	2	1	no	—
179	C₂×D₄	—	454	1	no	—
180	C₂×D₄	—	227	1	no	—
181	C₂×D₄	C₄×D₂₂₇	1	1	yes	characteristic
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	454	1	no	—
184	C₂²×C₄	—	454	1	no	—
185	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
186	Q₁₆	—	227	4	no	—
187	SD₁₆	—	227	4	no	—
188	SD₁₆	—	227	4	no	—
189	D₈	—	227	4	no	—
190	M₁₆	—	2	1	no	—
191	M₁₆	—	2	4	no	—
192	M₁₆	—	454	1	no	—
193	M₁₆	—	454	4	no	—
194	M₁₆	—	227	2	no	—
195	M₁₆	—	227	2	no	—
196	M₁₆	—	227	2	no	—
197	M₁₆	C₂²⋊D₂₂₇	1	2	yes	—
198	M₁₆	C₂²⋊D₂₂₇	1	2	yes	—
199	M₁₆	C₂²×D₂₂₇	1	2	yes	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	454	1	no	—
203	C₂×C₈	—	454	4	no	—
204	C₂×C₈	—	227	2	no	—
205	C₂×C₈	C₂²×D₂₂₇	1	2	yes	—
206	C₄²	—	454	1	no	—
207	C₄²	—	227	1	no	—
208	C₄²	—	227	1	no	—
209	C₂³	—	2	1	no	—
210	C₂³	—	454	1	no	—
211	C₂³	#16813884	1	1	yes	characteristic
212	Q₈	—	2	1	no	—
213	Q₈	—	454	1	no	—
214	Q₈	—	227	2	no	—
215	Q₈	C₂×C₄×D₂₂₇	1	2	yes	—
216	D₄	—	2	2	no	—
217	D₄	—	2	2	no	—
218	D₄	—	2	1	no	—
219	D₄	—	454	2	no	—
220	D₄	—	454	2	no	—
221	D₄	—	454	1	no	—
222	D₄	—	227	2	no	—
223	D₄	C₂×C₄×D₂₂₇	1	2	yes	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	908	2	no	—
228	C₂×C₄	—	454	1	no	—
229	C₂×C₄	—	454	1	no	—
230	C₂×C₄	—	454	1	no	—
231	C₂×C₄	—	454	2	no	—
232	C₂×C₄	—	454	2	no	—
233	C₂×C₄	C₂×C₄×D₂₂₇	1	1	yes	characteristic
234	C₂×C₄	C₂×C₄×D₂₂₇	1	1	yes	characteristic
235	C₂×C₄	C₂×C₂²⋊D₂₂₇	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊D₂₂₇	1	1	yes	characteristic
237	C₂×C₄	#16813884	1	1	yes	characteristic
238	C₂×C₄	C₂³×D₂₂₇	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	—	454	2	no	—
241	C₈	—	227	4	no	—
242	C₈	C₂×C₂²⋊D₂₂₇	1	4	yes	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	8	1	no	—
246	C₂²	—	1816	1	no	—
247	C₂²	—	454	2	no	—
248	C₂²	C₄×C₂²⋊D₂₂₇	1	1	yes	characteristic
249	C₂²	C₄×C₂²⋊D₂₂₇	1	1	yes	characteristic
250	C₂²	C₂×#16813884	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	908	1	no	—
253	C₄	—	454	2	no	—
254	C₄	—	454	2	no	—
255	C₄	C₄×C₂²⋊D₂₂₇	1	1	yes	characteristic
256	C₄	C₄×C₂²⋊D₂₂₇	1	1	yes	characteristic
257	C₄	C₂²×C₄×D₂₂₇	1	1	yes	characteristic
258	C₄	C₂²×C₂²⋊D₂₂₇	1	1	yes	characteristic
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	4	2	no	—
263	C₂	—	908	2	no	—
264	C₂	C₂×C₄×C₂²⋊D₂₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₂²⋊D₂₂₇
commutator	C₄×C₂₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₂₇
Fitting	#568×C₂₂₇	C₂
socle	C₂×C₂₂₇	C₂×C₄×D₄
prime radical	#28409830	C₂
cosocle	C₂×C₄×C₂₂₇	C₂⁴
hypercenter	#568	D₂₂₇
hypocenter	C₂₂₇	#2956
LCS	G ▷ C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #568

Complements

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

N	H ≅ G/N	type
2. #28409798	C₂	semidirect
3. #28409152	C₂	semidirect
4. #28409147	C₂	semidirect
5. #28409080	C₂	semidirect
6. C₂×#20300215	C₂	semidirect
7. C₂×#20300214	C₂	semidirect
8. M₁₆×Dic₂₂₇	C₂	semidirect
9. #568×C₂₂₇	C₂	semidirect
11. #20300319	C₄	semidirect
12. #20300318	C₄	semidirect
13. #20300314	C₄	semidirect
14. #20300313	C₄	semidirect
17. #20300294	C₂²	semidirect
22. #20300223	C₂²	semidirect
23. #20300215	C₂²	semidirect
24. #20300215	C₂²	semidirect
25. #20300214	C₂²	semidirect
26. #20300214	C₂²	semidirect
27. #20300198	C₂²	semidirect
28. C₈×Dic₂₂₇	C₂²	semidirect
31. #131×C₂₂₇	C₂²	semidirect
44. #16813874	C₂×C₄	semidirect
45. #16813872	C₂×C₄	semidirect
46. #16813871	C₂×C₄	semidirect
47. #16813869	C₂×C₄	semidirect
132. C₂₂₇	#2956	semidirect
136. #568	D₂₂₇	semidirect
153. #131	C₂×D₂₂₇	semidirect

Maximal subgroups

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

subgroup	index
#28409830	2
#28409798	2
#28409152	2
#28409147	2
#28409080	2
C₂×#20300215	2
C₂×#20300214	2
M₁₆×Dic₂₂₇	2
#568×C₂₂₇	2
#2956	227