Subgroups — #30647644

Order 16256.

452 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×D₁₂₇	C₂	1	1	yes	maximal, characteristic
2	#22550641	C₂	1	2	yes	maximal, complemented
3	#22549999	C₂	1	1	yes	maximal, characteristic, complemented
4	#22549998	C₂	1	1	yes	maximal, characteristic, complemented
5	#22549997	C₂	1	1	yes	maximal, characteristic, complemented
6	#22549991	C₂	1	2	yes	maximal, complemented
7	#22549990	C₂	1	2	yes	maximal, complemented
8	#108×D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#22549891	C₂	1	2	yes	maximal, complemented
10	#22549833	C₂	1	1	yes	maximal, characteristic, complemented
11	#444×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₁₂₇	C₂²	1	2	yes	—
13	#16861232	C₄	1	1	yes	characteristic, complemented
14	#16861231	C₂²	1	1	yes	characteristic
15	#16861230	C₄	1	2	yes	complemented
16	#16861229	C₂²	1	2	yes	—
17	#16861228	C₄	1	1	yes	characteristic, complemented
18	Q₁₆×D₁₂₇	C₄	1	1	yes	characteristic, complemented
19	SD₁₆×D₁₂₇	C₄	1	2	yes	complemented
20	D₈×D₁₂₇	C₄	1	1	yes	characteristic, complemented
21	#16861217	—	2	2	no	—
22	#16861217	C₂²	1	2	yes	complemented
23	#16861216	C₂²	1	2	yes	complemented
24	#16861215	—	2	2	no	—
25	#16861215	C₂²	1	2	yes	complemented
26	C₂×#14523231	—	2	2	no	—
27	C₂×#14523231	C₂²	1	1	yes	characteristic
28	M₁₆×D₁₂₇	—	2	2	no	—
29	M₁₆×D₁₂₇	C₂²	1	2	yes	—
30	C₂×C₈×D₁₂₇	C₂²	1	1	yes	characteristic
31	#16861166	C₂²	1	1	yes	characteristic
32	#16861145	C₂²	1	1	yes	characteristic, complemented
33	#16861144	C₂²	1	2	yes	complemented
34	#16861137	C₂²	1	2	yes	complemented
35	#16861136	C₂²	1	2	yes	complemented
36	#16861135	C₂²	1	2	yes	complemented
37	#16861119	C₂²	1	1	yes	characteristic, complemented
38	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
39	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic, complemented
40	#135×C₁₂₇	C₂²	1	1	yes	characteristic
41	#131×C₁₂₇	C₂²	1	2	yes	complemented
42	#108×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
43	#104×C₁₂₇	C₂²	1	2	yes	complemented
44	#97×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
45	#14523249	C₂×C₄	1	2	yes	—
46	#14523248	—	2	2	no	—
47	#14523248	C₂³	1	2	yes	—
48	#14523247	—	2	2	no	—
49	#14523247	C₂×C₄	1	2	yes	—
50	Q₈×D₁₂₇	C₂×C₄	1	2	yes	—
51	D₄×D₁₂₇	—	2	2	no	—
52	D₄×D₁₂₇	C₂×C₄	1	2	yes	—
53	C₂×C₄×D₁₂₇	—	2	2	no	—
54	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
55	Dic₅₀₈	C₂×C₄	1	1	yes	characteristic
56	#14523238	C₂×C₄	1	2	yes	—
57	#14523237	C₂×C₄	1	2	yes	—
58	#14523236	C₂×C₄	1	2	yes	—
59	#14523235	C₂×C₄	1	2	yes	—
60	D₁₀₁₆	C₂×C₄	1	1	yes	characteristic
61	#14523233	C₂×C₄	1	2	yes	—
62	#14523232	—	2	2	no	—
63	#14523232	C₂³	1	2	yes	—
64	#14523231	—	2	2	no	—
65	#14523231	—	2	2	no	—
66	#14523231	—	2	2	no	—
67	#14523231	D₄	1	2	yes	—
68	#14523231	D₄	1	2	yes	—
69	C₂×C₁₂₇⋊C₈	—	2	2	no	—
70	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
72	C₈×D₁₂₇	—	2	2	no	—
73	C₈×D₁₂₇	—	2	2	no	—
74	C₈×D₁₂₇	—	2	2	no	—
75	C₈×D₁₂₇	C₂×C₄	1	1	yes	characteristic
76	C₈×D₁₂₇	C₂×C₄	1	1	yes	characteristic
77	C₈×D₁₂₇	C₂×C₄	1	1	yes	characteristic
78	C₈×D₁₂₇	C₂×C₄	1	1	yes	characteristic
79	Dic₁₂₇⋊C₄	—	2	1	no	—
80	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
81	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
82	#41×C₁₂₇	C₂³	1	2	yes	—
83	Q₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
84	SD₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
85	D₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
86	M₁₆×C₁₂₇	—	2	2	no	—
87	M₁₆×C₁₂₇	C₂³	1	2	yes	—
88	C₂×C₈×C₁₂₇	—	2	2	no	—
89	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
90	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
91	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
92	C₂²×D₁₂₇	—	2	2	no	—
93	C₂²×D₁₂₇	#41	1	1	yes	characteristic
94	Dic₂₅₄	—	2	2	no	—
95	Dic₂₅₄	C₂²×C₄	1	2	yes	—
96	D₅₀₈	—	2	2	no	—
97	D₅₀₈	C₂²×C₄	1	2	yes	—
98	C₂²⋊D₁₂₇	—	2	2	no	—
99	C₂²⋊D₁₂₇	—	2	2	no	—
100	C₂²⋊D₁₂₇	—	2	2	no	—
101	C₂×Dic₁₂₇	—	2	2	no	—
102	C₂×Dic₁₂₇	—	2	1	no	—
103	C₂×Dic₁₂₇	#41	1	1	yes	characteristic
104	C₄×D₁₂₇	—	2	2	no	—
105	C₄×D₁₂₇	—	2	2	no	—
106	C₄×D₁₂₇	—	2	2	no	—
107	C₄×D₁₂₇	C₂²×C₄	1	1	yes	characteristic
108	C₄×D₁₂₇	C₂²×C₄	1	1	yes	characteristic
109	C₄×D₁₂₇	C₂×D₄	1	1	yes	characteristic
110	C₄×D₁₂₇	C₂×D₄	1	1	yes	characteristic
111	C₁₂₇⋊C₈	—	2	2	no	—
112	C₁₂₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
113	C₁₂₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
114	C₁₂₇⋊C₈	C₂×D₄	1	2	yes	—
115	Q₈×C₁₂₇	C₂²×C₄	1	2	yes	—
116	D₄×C₁₂₇	—	2	2	no	—
117	D₄×C₁₂₇	C₂²×C₄	1	2	yes	—
118	C₂×C₄×C₁₂₇	—	2	2	no	—
119	C₂×C₄×C₁₂₇	—	2	1	no	—
120	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
121	C₈×C₁₂₇	—	2	2	no	—
122	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
123	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
124	C₈×C₁₂₇	C₂×D₄	1	2	yes	—
125	C₂²×C₁₂₇	—	2	2	no	—
126	C₂²×C₁₂₇	C₂×#41	1	1	yes	characteristic
127	C₂×D₁₂₇	—	2	2	no	—
128	C₂×D₁₂₇	—	4	1	no	—
129	C₂×D₁₂₇	—	8	2	no	—
130	C₂×D₁₂₇	C₄×D₄	1	1	yes	characteristic
131	C₂×D₁₂₇	C₄×D₄	1	1	yes	characteristic
132	C₄×C₁₂₇	—	2	2	no	—
133	C₄×C₁₂₇	—	4	1	no	—
134	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
135	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
136	Dic₁₂₇	—	2	2	no	—
137	Dic₁₂₇	—	4	1	no	—
138	Dic₁₂₇	C₄×D₄	1	1	yes	characteristic
139	Dic₁₂₇	C₄×D₄	1	1	yes	characteristic
140	C₂×C₁₂₇	—	2	1	no	—
141	C₂×C₁₂₇	—	4	2	no	—
142	C₂×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
143	D₁₂₇	—	2	1	no	—
144	D₁₂₇	—	2	1	no	—
145	D₁₂₇	—	4	2	no	—
146	#2957	—	127	1	no	maximal, Sylow
147	C₁₂₇	#2957	1	1	yes	characteristic, complemented, Sylow
148	C₂×#135	—	127	1	no	—
149	#568	—	127	2	no	—
150	#444	—	127	1	no	—
151	#444	—	127	1	no	—
152	#444	—	127	1	no	—
153	#444	D₁₂₇	1	1	yes	characteristic, complemented
154	#443	—	127	2	no	—
155	#443	—	127	2	no	—
156	C₂×#108	—	127	1	no	—
157	#421	—	127	2	no	—
158	#405	—	127	1	no	—
159	C₂×#41	—	127	2	no	—
160	#135	—	127	2	no	—
161	#135	—	127	1	no	—
162	#135	—	127	2	no	—
163	#135	—	127	1	no	—
164	#135	—	127	1	no	—
165	#135	C₂×D₁₂₇	1	1	yes	characteristic
166	C₂×Q₁₆	—	127	1	no	—
167	C₂×SD₁₆	—	127	2	no	—
168	C₂×D₈	—	127	1	no	—
169	#131	—	127	2	no	—
170	#131	—	127	2	no	—
171	#131	—	127	2	no	—
172	#131	—	254	2	no	—
173	#131	—	254	2	no	—
174	#131	C₂×D₁₂₇	1	2	yes	complemented
175	C₂×M₁₆	—	127	1	no	—
176	C₂×M₁₆	—	127	2	no	—
177	C₂×M₁₆	—	254	2	no	—
178	C₂×M₁₆	—	254	2	no	—
179	C₂²×C₈	—	127	1	no	—
180	#117	—	127	1	no	—
181	#108	—	127	2	no	—
182	#108	—	127	1	no	—
183	#108	C₂×D₁₂₇	1	1	yes	characteristic, complemented
184	#104	—	127	2	no	—
185	#104	—	127	2	no	—
186	#104	—	127	2	no	—
187	#104	C₂×D₁₂₇	1	2	yes	complemented
188	#97	—	127	1	no	—
189	#97	C₂×D₁₂₇	1	1	yes	characteristic, complemented
190	C₄×C₈	—	127	1	no	—
191	C₄×C₈	—	127	1	no	—
192	#41	—	127	2	no	—
193	#41	—	127	2	no	—
194	#41	—	127	2	no	—
195	#41	—	254	2	no	—
196	#41	—	254	2	no	—
197	#41	C₂²×D₁₂₇	1	2	yes	—
198	C₂×Q₈	—	127	2	no	—
199	C₂×D₄	—	127	2	no	—
200	C₂×D₄	—	254	2	no	—
201	C₂²×C₄	—	127	1	no	—
202	C₂²×C₄	—	254	2	no	—
203	Q₁₆	—	127	2	no	—
204	Q₁₆	—	127	1	no	—
205	Q₁₆	C₄×D₁₂₇	1	1	yes	characteristic
206	SD₁₆	—	127	2	no	—
207	SD₁₆	—	127	2	no	—
208	SD₁₆	—	127	2	no	—
209	SD₁₆	C₄×D₁₂₇	1	2	yes	—
210	D₈	—	127	2	no	—
211	D₈	—	127	1	no	—
212	D₈	C₄×D₁₂₇	1	1	yes	characteristic
213	M₁₆	—	2	2	no	—
214	M₁₆	—	127	2	no	—
215	M₁₆	—	127	2	no	—
216	M₁₆	—	127	2	no	—
217	M₁₆	—	254	2	no	—
218	M₁₆	—	254	2	no	—
219	M₁₆	—	254	2	no	—
220	M₁₆	—	254	2	no	—
221	M₁₆	C₂²×D₁₂₇	1	2	yes	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	127	1	no	—
224	C₂×C₈	—	127	1	no	—
225	C₂×C₈	—	127	1	no	—
226	C₂×C₈	—	127	1	no	—
227	C₂×C₈	—	127	1	no	—
228	C₂×C₈	—	127	1	no	—
229	C₂×C₈	—	254	2	no	—
230	C₂×C₈	—	254	2	no	—
231	C₂×C₈	—	254	2	no	—
232	C₂×C₈	—	254	2	no	—
233	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
234	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
235	C₄⋊C₄	—	254	1	no	—
236	C₂²⋊C₄	—	254	1	no	—
237	C₄²	—	127	1	no	—
238	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
239	C₂³	—	127	1	no	—
240	C₂³	—	254	2	no	—
241	Q₈	—	127	2	no	—
242	Q₈	—	254	2	no	—
243	Q₈	C₂×C₄×D₁₂₇	1	2	yes	—
244	D₄	—	2	2	no	—
245	D₄	—	127	2	no	—
246	D₄	—	254	2	no	—
247	D₄	—	254	2	no	—
248	D₄	—	254	2	no	—
249	D₄	—	254	2	no	—
250	D₄	C₂×C₄×D₁₂₇	1	2	yes	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	127	1	no	—
254	C₂×C₄	—	127	1	no	—
255	C₂×C₄	—	127	1	no	—
256	C₂×C₄	—	127	1	no	—
257	C₂×C₄	—	127	1	no	—
258	C₂×C₄	—	254	2	no	—
259	C₂×C₄	—	254	2	no	—
260	C₂×C₄	—	254	2	no	—
261	C₂×C₄	—	254	1	no	—
262	C₂×C₄	—	254	2	no	—
263	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	127	1	no	—
266	C₈	—	127	1	no	—
267	C₈	—	127	2	no	—
268	C₈	—	254	2	no	—
269	C₈	C₂×C₄×D₁₂₇	1	1	yes	characteristic
270	C₈	C₂×C₄×D₁₂₇	1	1	yes	characteristic
271	C₈	D₄×D₁₂₇	1	2	yes	—
272	C₂²	—	2	2	no	—
273	C₂²	—	127	1	no	—
274	C₂²	—	127	1	no	—
275	C₂²	—	1016	2	no	—
276	C₂²	—	508	1	no	—
277	C₂²	—	254	2	no	—
278	C₂²	#41×D₁₂₇	1	1	yes	characteristic
279	C₄	—	2	2	no	—
280	C₄	—	4	1	no	—
281	C₄	—	127	1	no	—
282	C₄	—	127	1	no	—
283	C₄	—	508	1	no	—
284	C₄	—	254	2	no	—
285	C₄	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
286	C₄	C₂×D₄×D₁₂₇	1	1	yes	characteristic
287	C₂	—	2	1	no	—
288	C₂	—	4	2	no	—
289	C₂	—	508	2	no	—
290	C₂	—	254	1	no	—
291	C₂	—	254	1	no	—
292	C₂	C₄×D₄×D₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₁₂₇
commutator	C₄×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#444×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂×C₄×D₄
prime radical	#135×D₁₂₇	C₂
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#444	D₁₂₇
hypocenter	C₁₂₇	#2957
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #444

Complements

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

N	H ≅ G/N	type
2. #22550641	C₂	semidirect
3. #22549999	C₂	semidirect
4. #22549998	C₂	semidirect
5. #22549997	C₂	semidirect
6. #22549991	C₂	semidirect
7. #22549990	C₂	semidirect
8. #108×D₁₂₇	C₂	semidirect
9. #22549891	C₂	semidirect
10. #22549833	C₂	semidirect
11. #444×C₁₂₇	C₂	semidirect
13. #16861232	C₄	semidirect
15. #16861230	C₄	semidirect
17. #16861228	C₄	semidirect
18. Q₁₆×D₁₂₇	C₄	semidirect
19. SD₁₆×D₁₂₇	C₄	semidirect
20. D₈×D₁₂₇	C₄	semidirect
22. #16861217	C₂²	semidirect
23. #16861216	C₂²	semidirect
25. #16861215	C₂²	semidirect
32. #16861145	C₂²	semidirect
33. #16861144	C₂²	semidirect
34. #16861137	C₂²	semidirect
35. #16861136	C₂²	semidirect
36. #16861135	C₂²	semidirect
37. #16861119	C₂²	semidirect
38. C₄×C₁₂₇⋊C₈	C₂²	semidirect
39. C₈×Dic₁₂₇	C₂²	semidirect
41. #131×C₁₂₇	C₂²	semidirect
42. #108×C₁₂₇	C₂²	semidirect
43. #104×C₁₂₇	C₂²	semidirect
44. #97×C₁₂₇	C₂²	semidirect
147. C₁₂₇	#2957	semidirect
153. #444	D₁₂₇	semidirect
174. #131	C₂×D₁₂₇	semidirect
183. #108	C₂×D₁₂₇	semidirect
187. #104	C₂×D₁₂₇	semidirect
189. #97	C₂×D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#135×D₁₂₇	2
#22550641	2
#22549999	2
#22549998	2
#22549997	2
#22549991	2
#22549990	2
#108×D₁₂₇	2
#22549891	2
#22549833	2
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