Subgroups — #30651445

Order 16256.

372 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550595	C₂	1	1	yes	maximal, characteristic, complemented
2	#22550541	C₂	1	1	yes	maximal, characteristic, complemented
3	#22550234	C₂	1	1	yes	maximal, characteristic, complemented
4	#22550225	C₂	1	1	yes	maximal, characteristic, complemented
5	#22550169	C₂	1	1	yes	maximal, characteristic, complemented
6	#22550168	C₂	1	1	yes	maximal, characteristic, complemented
7	#22550154	C₂	1	1	yes	maximal, characteristic, complemented
8	#22550090	C₂	1	1	yes	maximal, characteristic
9	#22550088	C₂	1	1	yes	maximal, characteristic
10	#22550080	C₂	1	1	yes	maximal, characteristic, complemented
11	#22550004	C₂	1	1	yes	maximal, characteristic, complemented
12	#22549979	C₂	1	1	yes	maximal, characteristic, complemented
13	#22549978	C₂	1	1	yes	maximal, characteristic, complemented
14	#22549956	C₂	1	1	yes	maximal, characteristic
15	#489×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₂₇	—	2	1	no	—
17	C₂×#14523237	C₂²	1	1	yes	characteristic
18	C₂×#14523235	C₂²	1	1	yes	characteristic
19	C₂×#14523233	C₂²	1	1	yes	characteristic
20	#16861208	C₂²	1	1	yes	characteristic
21	#16861208	C₂²	1	1	yes	characteristic
22	#16861207	C₂²	1	1	yes	characteristic
23	#16861202	—	2	1	no	—
24	#16861202	—	2	1	no	—
25	#16861196	C₂²	1	1	yes	characteristic
26	#16861187	—	2	1	no	—
27	#16861187	—	2	1	no	—
28	#16861186	—	2	1	no	—
29	#16861185	—	2	1	no	—
30	Q₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
31	C₄×Dic₂₅₄	C₂²	1	1	yes	characteristic
32	C₄×Dic₂₅₄	C₂²	1	1	yes	characteristic
33	C₄×D₅₀₈	C₂²	1	1	yes	characteristic
34	D₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
35	#16861171	C₂²	1	1	yes	characteristic
36	#16861171	C₂²	1	1	yes	characteristic
37	#16861165	—	2	1	no	—
38	#16861143	C₂²	1	1	yes	characteristic
39	#16861142	C₂²	1	1	yes	characteristic
40	#16861142	C₂²	1	1	yes	characteristic
41	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
42	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
43	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
44	#16861134	C₂²	1	1	yes	characteristic
45	#16861133	C₂²	1	1	yes	characteristic
46	#16861132	C₂²	1	1	yes	characteristic
47	#16861132	C₂²	1	1	yes	characteristic
48	#16861131	C₂²	1	1	yes	characteristic
49	#16861130	C₂²	1	1	yes	characteristic
50	#16861119	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
53	#124×C₁₂₇	C₂²	1	1	yes	characteristic
54	#103×C₁₂₇	C₂²	1	1	yes	characteristic
55	#103×C₁₂₇	C₂²	1	1	yes	characteristic
56	#102×C₁₂₇	C₂²	1	1	yes	characteristic
57	#102×C₁₂₇	C₂²	1	1	yes	characteristic
58	#97×C₁₂₇	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₅₄	—	2	1	no	—
60	C₂×Dic₂₅₄	—	2	1	no	—
61	C₂×Dic₂₅₄	D₄	1	1	yes	characteristic
62	C₂×Dic₂₅₄	C₂³	1	1	yes	characteristic
63	Q₈×D₁₂₇	—	4	4	no	—
64	C₂×D₅₀₈	C₂³	1	1	yes	characteristic
65	C₂²×Dic₁₂₇	—	2	1	no	—
66	C₂×C₄×D₁₂₇	—	2	1	no	—
67	C₂×C₄×D₁₂₇	—	2	1	no	—
68	C₂×C₄×D₁₂₇	—	2	1	no	—
69	#14523237	—	2	2	no	—
70	#14523235	—	2	2	no	—
71	#14523233	—	2	2	no	—
72	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₂₇	—	2	1	no	—
75	C₄⋊Dic₁₂₇	D₄	1	1	yes	characteristic
76	C₄⋊Dic₁₂₇	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₂₇	C₂³	1	1	yes	characteristic
78	Dic₁₂₇⋊C₄	—	2	1	no	—
79	Dic₁₂₇⋊C₄	—	2	1	no	—
80	Dic₁₂₇⋊C₄	—	2	1	no	—
81	Dic₁₂₇⋊C₄	—	2	1	no	—
82	Dic₁₂₇⋊C₄	—	2	1	no	—
83	Dic₁₂₇⋊C₄	D₄	1	1	yes	characteristic
84	Dic₁₂₇⋊C₄	D₄	1	1	yes	characteristic
85	C₂²⋊Dic₁₂₇	—	2	1	no	—
86	C₂²⋊Dic₁₂₇	—	2	1	no	—
87	C₂²⋊Dic₁₂₇	—	2	1	no	—
88	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
89	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
90	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
91	C₄×Dic₁₂₇	—	2	1	no	—
92	C₄×Dic₁₂₇	—	2	1	no	—
93	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
94	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₁₂₇	—	2	1	no	—
96	C₂×Q₈×C₁₂₇	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₁₂₇	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₁₂₇	—	2	1	no	—
101	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
103	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
104	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
105	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
106	C₂²×D₁₂₇	—	2	1	no	—
107	Dic₂₅₄	—	2	2	no	—
108	Dic₂₅₄	—	2	1	no	—
109	Dic₂₅₄	—	2	1	no	—
110	Dic₂₅₄	—	4	2	no	—
111	Dic₂₅₄	—	4	2	no	—
112	D₅₀₈	—	2	1	no	—
113	D₅₀₈	#41	1	2	yes	—
114	C₂×Dic₁₂₇	—	2	1	no	—
115	C₂×Dic₁₂₇	—	2	1	no	—
116	C₂×Dic₁₂₇	—	2	1	no	—
117	C₂×Dic₁₂₇	—	4	2	no	—
118	C₂×Dic₁₂₇	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₁₂₇	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₁₂₇	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₁₂₇	C₂×D₄	1	1	yes	characteristic
122	C₄×D₁₂₇	—	4	2	no	—
123	C₄×D₁₂₇	—	4	2	no	—
124	C₄×D₁₂₇	—	4	2	no	—
125	C₁₂₇⋊C₈	—	2	1	no	—
126	C₁₂₇⋊C₈	—	2	1	no	—
127	C₂³×C₁₂₇	—	2	1	no	—
128	Q₈×C₁₂₇	—	2	1	no	—
129	Q₈×C₁₂₇	—	2	1	no	—
130	Q₈×C₁₂₇	—	4	2	no	—
131	D₄×C₁₂₇	—	2	1	no	—
132	D₄×C₁₂₇	—	2	1	no	—
133	C₂×C₄×C₁₂₇	—	2	1	no	—
134	C₂×C₄×C₁₂₇	—	2	1	no	—
135	C₂×C₄×C₁₂₇	—	2	1	no	—
136	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
139	C₈×C₁₂₇	—	2	1	no	—
140	C₈×C₁₂₇	—	2	1	no	—
141	C₂²×C₁₂₇	—	4	1	no	—
142	C₂²×C₁₂₇	—	4	1	no	—
143	C₂²×C₁₂₇	—	4	1	no	—
144	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
145	C₂×D₁₂₇	—	2	2	no	—
146	C₂×D₁₂₇	—	4	1	no	—
147	C₂×D₁₂₇	—	4	1	no	—
148	C₄×C₁₂₇	—	2	1	no	—
149	C₄×C₁₂₇	—	2	1	no	—
150	C₄×C₁₂₇	—	4	1	no	—
151	C₄×C₁₂₇	—	4	1	no	—
152	C₄×C₁₂₇	—	4	1	no	—
153	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
154	C₄×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
155	Dic₁₂₇	—	2	1	no	—
156	Dic₁₂₇	—	2	1	no	—
157	Dic₁₂₇	—	2	1	no	—
158	Dic₁₂₇	—	4	1	no	—
159	Dic₁₂₇	—	4	1	no	—
160	Dic₁₂₇	—	4	1	no	—
161	Dic₁₂₇	AGL(1,ℤ/8)	1	2	yes	—
162	C₂×C₁₂₇	—	8	1	no	—
163	C₂×C₁₂₇	#548	1	1	yes	characteristic
164	C₂×C₁₂₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
165	C₂×C₁₂₇	#578	1	1	yes	characteristic
166	D₁₂₇	—	4	2	no	—
167	#3367	—	127	1	no	maximal, Sylow
168	C₁₂₇	#3367	1	1	yes	characteristic, complemented, Sylow
169	#556	—	127	1	no	—
170	D₄×Q₈	—	127	1	no	—
171	#489	—	127	1	no	—
172	#489	D₁₂₇	1	1	yes	characteristic, complemented
173	#487	—	127	1	no	—
174	#477	—	127	1	no	—
175	#477	—	127	1	no	—
176	#474	—	127	1	no	—
177	#463	—	127	1	no	—
178	#463	—	127	1	no	—
179	#461	—	127	1	no	—
180	#446	—	127	1	no	—
181	#440	—	127	1	no	—
182	#440	—	127	1	no	—
183	C₄×D₈	—	127	1	no	—
184	C₂²×Q₈	—	254	1	no	—
185	C₂×SD₁₆	—	127	1	no	—
186	C₂×SD₁₆	—	127	1	no	—
187	C₂×D₈	—	127	1	no	—
188	C₄⋊Q₈	—	127	1	no	—
189	C₄⋊Q₈	—	127	1	no	—
190	C₄⋊Q₈	—	127	1	no	—
191	C₄⋊Q₈	C₂×D₁₂₇	1	1	yes	characteristic
192	#126	—	254	1	no	—
193	#126	—	254	1	no	—
194	#124	—	127	1	no	—
195	#124	C₂×D₁₂₇	1	1	yes	characteristic
196	C₂²⋊Q₈	—	254	1	no	—
197	C₂²⋊Q₈	—	254	1	no	—
198	C₂²⋊Q₈	—	254	1	no	—
199	C₂²⋊Q₈	—	254	1	no	—
200	C₄×Q₈	—	127	1	no	—
201	C₄×Q₈	—	127	1	no	—
202	C₄×Q₈	—	127	1	no	—
203	C₄×D₄	—	127	1	no	—
204	C₄×D₄	—	127	1	no	—
205	C₄×D₄	—	127	1	no	—
206	C₄×D₄	—	127	1	no	—
207	#117	—	254	1	no	—
208	#107	—	127	1	no	—
209	#107	—	127	1	no	—
210	#107	—	127	1	no	—
211	C₄⋊C₈	—	127	1	no	—
212	C₄⋊C₈	—	127	1	no	—
213	C₄⋊C₈	—	127	1	no	—
214	#103	—	127	1	no	—
215	#103	—	127	1	no	—
216	#103	—	127	1	no	—
217	#103	—	127	1	no	—
218	#103	C₂×D₁₂₇	1	1	yes	characteristic
219	#103	C₂×D₁₂₇	1	1	yes	characteristic
220	#102	—	127	1	no	—
221	#102	—	127	1	no	—
222	#102	C₂×D₁₂₇	1	1	yes	characteristic
223	#102	C₂×D₁₂₇	1	1	yes	characteristic
224	#97	—	127	1	no	—
225	#97	C₂×D₁₂₇	1	1	yes	characteristic
226	C₄×C₈	—	127	1	no	—
227	C₂×Q₈	—	2	1	no	—
228	C₂×Q₈	—	127	1	no	—
229	C₂×Q₈	—	127	1	no	—
230	C₂×Q₈	—	508	4	no	—
231	C₂×Q₈	—	254	1	no	—
232	C₂×Q₈	—	254	1	no	—
233	C₂×Q₈	C₂²×D₁₂₇	1	1	yes	characteristic
234	C₂×D₄	—	127	1	no	—
235	C₂×D₄	C₂²×D₁₂₇	1	1	yes	characteristic
236	C₂²×C₄	—	254	1	no	—
237	C₂²×C₄	—	254	1	no	—
238	C₂²×C₄	—	254	1	no	—
239	C₂²×C₄	—	254	1	no	—
240	SD₁₆	—	254	2	no	—
241	SD₁₆	—	254	2	no	—
242	D₈	—	254	2	no	—
243	C₂×C₈	—	127	1	no	—
244	C₂×C₈	—	127	1	no	—
245	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
246	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	127	1	no	—
249	C₄⋊C₄	—	127	1	no	—
250	C₄⋊C₄	—	127	1	no	—
251	C₄⋊C₄	—	127	1	no	—
252	C₄⋊C₄	—	127	1	no	—
253	C₄⋊C₄	—	254	1	no	—
254	C₄⋊C₄	—	254	1	no	—
255	C₄⋊C₄	—	254	1	no	—
256	C₄⋊C₄	—	254	1	no	—
257	C₄⋊C₄	—	254	1	no	—
258	C₄⋊C₄	—	254	1	no	—
259	C₄⋊C₄	C₂²×D₁₂₇	1	1	yes	characteristic
260	C₄⋊C₄	C₂²×D₁₂₇	1	1	yes	characteristic
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	254	1	no	—
264	C₂²⋊C₄	—	254	1	no	—
265	C₂²⋊C₄	—	254	1	no	—
266	C₂²⋊C₄	—	254	1	no	—
267	C₂²⋊C₄	—	254	1	no	—
268	C₂²⋊C₄	—	254	1	no	—
269	C₄²	—	127	1	no	—
270	C₄²	—	127	1	no	—
271	C₄²	—	254	1	no	—
272	C₄²	—	254	1	no	—
273	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
274	C₂³	—	2	1	no	—
275	C₂³	—	254	1	no	—
276	Q₈	—	2	1	no	—
277	Q₈	—	2	1	no	—
278	Q₈	—	4	2	no	—
279	Q₈	—	508	2	no	—
280	Q₈	—	508	2	no	—
281	Q₈	—	254	2	no	—
282	Q₈	—	254	1	no	—
283	Q₈	—	254	1	no	—
284	D₄	—	2	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	127	2	no	—
287	D₄	—	254	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	127	1	no	—
292	C₂×C₄	—	127	1	no	—
293	C₂×C₄	—	127	1	no	—
294	C₂×C₄	—	127	1	no	—
295	C₂×C₄	—	508	2	no	—
296	C₂×C₄	—	508	2	no	—
297	C₂×C₄	—	508	2	no	—
298	C₂×C₄	—	508	2	no	—
299	C₂×C₄	—	254	1	no	—
300	C₂×C₄	—	254	1	no	—
301	C₂×C₄	—	254	1	no	—
302	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
303	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
304	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	254	1	no	—
308	C₈	—	254	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	508	1	no	—
313	C₂²	—	508	1	no	—
314	C₂²	—	254	2	no	—
315	C₂²	C₂×D₄×D₁₂₇	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	127	2	no	—
322	C₄	—	508	1	no	—
323	C₄	—	508	1	no	—
324	C₄	—	508	1	no	—
325	C₄	—	254	1	no	—
326	C₄	—	254	1	no	—
327	C₄	—	254	1	no	—
328	C₄	#41×D₁₂₇	1	1	yes	characteristic
329	C₄	#16861256	1	1	yes	characteristic
330	C₂	—	8	1	no	—
331	C₂	—	508	2	no	—
332	C₂	#22550534	1	1	yes	characteristic
333	C₂	AGL(1,ℤ/8)×D₁₂₇	1	1	yes	characteristic
334	C₂	#22550692	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₂₇
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#489×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	C₂×#14523233	C₂²
hypercenter	#489	D₁₂₇
hypocenter	C₁₂₇	#3367
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #489

Complements

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

N	H ≅ G/N	type
1. #22550595	C₂	semidirect
2. #22550541	C₂	semidirect
3. #22550234	C₂	semidirect
4. #22550225	C₂	semidirect
5. #22550169	C₂	semidirect
6. #22550168	C₂	semidirect
7. #22550154	C₂	semidirect
10. #22550080	C₂	semidirect
11. #22550004	C₂	semidirect
12. #22549979	C₂	semidirect
13. #22549978	C₂	semidirect
15. #489×C₁₂₇	C₂	semidirect
168. C₁₂₇	#3367	semidirect
172. #489	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550595	2
#22550541	2
#22550234	2
#22550225	2
#22550169	2
#22550168	2
#22550154	2
#22550090	2
#22550088	2
#22550080	2
#22550004	2
#22549979	2
#22549978	2
#22549956	2
#489×C₁₂₇	2
#3367	127