Subgroups — #30651462

Order 16256.

382 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550603	C₂	1	1	yes	maximal, characteristic, complemented
2	#22550547	C₂	1	1	yes	maximal, characteristic
3	#22550242	C₂	1	1	yes	maximal, characteristic, complemented
4	#22550239	C₂	1	1	yes	maximal, characteristic, complemented
5	#22550221	C₂	1	1	yes	maximal, characteristic, complemented
6	#22550175	C₂	1	1	yes	maximal, characteristic, complemented
7	#22550155	C₂	1	1	yes	maximal, characteristic, complemented
8	#22550077	C₂	1	1	yes	maximal, characteristic, complemented
9	#22550075	C₂	1	1	yes	maximal, characteristic, complemented
10	#22550075	C₂	1	1	yes	maximal, characteristic, complemented
11	#22550005	C₂	1	1	yes	maximal, characteristic, complemented
12	#22549976	C₂	1	1	yes	maximal, characteristic, complemented
13	#22549975	C₂	1	1	yes	maximal, characteristic, complemented
14	#22549963	C₂	1	1	yes	maximal, characteristic, complemented
15	#478×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14523248	—	2	1	no	—
17	C₂×#14523237	C₂²	1	1	yes	characteristic
18	C₂×#14523237	C₂²	1	1	yes	characteristic
19	C₂×#14523236	C₂²	1	1	yes	characteristic
20	#16861208	—	2	1	no	—
21	#16861208	C₂²	1	1	yes	characteristic
22	#16861206	C₂²	1	1	yes	characteristic
23	#16861205	C₂²	1	1	yes	characteristic
24	#16861205	C₂²	1	1	yes	characteristic
25	#16861200	C₂²	1	1	yes	characteristic
26	#16861199	—	2	1	no	—
27	#16861199	—	2	1	no	—
28	#16861183	—	2	1	no	—
29	#16861182	—	2	1	no	—
30	#16861182	—	2	1	no	—
31	C₄×Dic₂₅₄	C₂²	1	1	yes	characteristic
32	C₄×Dic₂₅₄	C₂²	1	1	yes	characteristic
33	#16861174	—	2	1	no	—
34	#16861174	C₂²	1	1	yes	characteristic
35	#16861174	C₂²	1	1	yes	characteristic
36	#16861171	C₂²	1	1	yes	characteristic
37	#16861171	C₂²	1	1	yes	characteristic
38	#16861142	C₂²	1	1	yes	characteristic
39	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
40	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
41	#16861132	C₂²	1	1	yes	characteristic
42	#16861131	C₂²	1	1	yes	characteristic
43	#16861131	C₂²	1	1	yes	characteristic
44	#16861130	C₂²	1	1	yes	characteristic
45	#16861129	C₂²	1	1	yes	characteristic
46	#16861129	C₂²	1	1	yes	characteristic
47	#16861129	C₂²	1	1	yes	characteristic
48	#16861129	C₂²	1	1	yes	characteristic
49	#16861120	C₂²	1	1	yes	characteristic
50	#16861119	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
52	#125×C₁₂₇	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
54	#107×C₁₂₇	C₂²	1	1	yes	characteristic
55	#106×C₁₂₇	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
57	#102×C₁₂₇	C₂²	1	1	yes	characteristic
58	#102×C₁₂₇	C₂²	1	1	yes	characteristic
59	#14523248	—	4	4	no	—
60	C₂×Dic₂₅₄	—	2	1	no	—
61	C₂×Dic₂₅₄	D₄	1	1	yes	characteristic
62	C₂×Dic₂₅₄	C₂³	1	1	yes	characteristic
63	C₂×D₅₀₈	—	2	1	no	—
64	C₂×D₅₀₈	—	2	1	no	—
65	C₂×D₅₀₈	C₂³	1	1	yes	characteristic
66	C₂×D₅₀₈	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₁₂₇	—	2	1	no	—
68	C₂×C₂²⋊D₁₂₇	—	2	1	no	—
69	C₂×C₂²⋊D₁₂₇	—	2	1	no	—
70	C₂×C₂²⋊D₁₂₇	—	2	1	no	—
71	C₂×C₄×D₁₂₇	—	2	1	no	—
72	C₂×C₄×D₁₂₇	—	2	1	no	—
73	#14523237	—	2	2	no	—
74	#14523237	—	2	2	no	—
75	#14523236	—	2	2	no	—
76	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
77	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₁₂₇	—	2	1	no	—
79	C₄⋊Dic₁₂₇	D₄	1	1	yes	characteristic
80	C₄⋊Dic₁₂₇	C₂³	1	1	yes	characteristic
81	Dic₁₂₇⋊C₄	—	2	1	no	—
82	Dic₁₂₇⋊C₄	—	2	1	no	—
83	Dic₁₂₇⋊C₄	—	2	1	no	—
84	Dic₁₂₇⋊C₄	—	2	1	no	—
85	Dic₁₂₇⋊C₄	—	2	1	no	—
86	Dic₁₂₇⋊C₄	D₄	1	1	yes	characteristic
87	Dic₁₂₇⋊C₄	D₄	1	1	yes	characteristic
88	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
89	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
90	C₄×Dic₁₂₇	—	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×D₄×C₁₂₇	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₁₂₇	—	2	1	no	—
97	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₁₂₇	—	2	1	no	—
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	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
104	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
105	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
106	C₂²×D₁₂₇	—	2	1	no	—
107	C₂²×D₁₂₇	—	2	1	no	—
108	Dic₂₅₄	—	2	1	no	—
109	Dic₂₅₄	—	2	2	no	—
110	Dic₂₅₄	—	4	2	no	—
111	Dic₂₅₄	#41	1	2	yes	complemented
112	D₅₀₈	—	2	1	no	—
113	D₅₀₈	—	2	1	no	—
114	D₅₀₈	—	2	1	no	—
115	D₅₀₈	—	2	1	no	—
116	D₅₀₈	—	4	2	no	—
117	D₅₀₈	—	8	1	no	—
118	C₂²⋊D₁₂₇	—	4	2	no	—
119	C₂²⋊D₁₂₇	—	4	2	no	—
120	C₂²⋊D₁₂₇	—	8	1	no	—
121	C₂²⋊D₁₂₇	—	8	1	no	—
122	C₂×Dic₁₂₇	—	2	1	no	—
123	C₂×Dic₁₂₇	—	2	1	no	—
124	C₂×Dic₁₂₇	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₁₂₇	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₁₂₇	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₁₂₇	C₂×D₄	1	1	yes	characteristic
128	C₄×D₁₂₇	—	4	2	no	—
129	C₄×D₁₂₇	—	4	2	no	—
130	C₁₂₇⋊C₈	—	2	1	no	—
131	C₁₂₇⋊C₈	—	2	1	no	—
132	C₂³×C₁₂₇	—	2	1	no	—
133	D₄×C₁₂₇	—	2	1	no	—
134	D₄×C₁₂₇	—	2	1	no	—
135	C₂×C₄×C₁₂₇	—	2	1	no	—
136	C₂×C₄×C₁₂₇	—	2	1	no	—
137	C₂×C₄×C₁₂₇	—	2	1	no	—
138	C₂×C₄×C₁₂₇	—	4	2	no	—
139	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
142	C₈×C₁₂₇	—	2	1	no	—
143	C₈×C₁₂₇	—	2	1	no	—
144	C₂²×C₁₂₇	—	4	1	no	—
145	C₂²×C₁₂₇	—	4	1	no	—
146	C₂²×C₁₂₇	—	4	1	no	—
147	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
148	C₂×D₁₂₇	—	4	1	no	—
149	C₂×D₁₂₇	—	4	1	no	—
150	C₂×D₁₂₇	—	4	1	no	—
151	C₂×D₁₂₇	—	4	1	no	—
152	C₂×D₁₂₇	—	4	1	no	—
153	C₂×D₁₂₇	—	4	1	no	—
154	C₄×C₁₂₇	—	2	1	no	—
155	C₄×C₁₂₇	—	2	1	no	—
156	C₄×C₁₂₇	—	4	1	no	—
157	C₄×C₁₂₇	—	4	1	no	—
158	C₄×C₁₂₇	—	4	1	no	—
159	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
160	C₄×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
161	Dic₁₂₇	—	2	2	no	—
162	Dic₁₂₇	—	2	1	no	—
163	Dic₁₂₇	—	2	1	no	—
164	Dic₁₂₇	—	2	1	no	—
165	Dic₁₂₇	—	4	1	no	—
166	Dic₁₂₇	AGL(1,ℤ/8)	1	2	yes	—
167	C₂×C₁₂₇	—	8	1	no	—
168	C₂×C₁₂₇	#548	1	1	yes	characteristic
169	C₂×C₁₂₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
170	C₂×C₁₂₇	#577	1	1	yes	characteristic
171	D₁₂₇	—	8	1	no	—
172	D₁₂₇	—	8	1	no	—
173	#3368	—	127	1	no	maximal, Sylow
174	C₁₂₇	#3368	1	1	yes	characteristic, complemented, Sylow
175	#557	—	127	1	no	—
176	#550	—	127	1	no	—
177	#490	—	127	1	no	—
178	#490	—	127	1	no	—
179	#486	—	127	1	no	—
180	#478	—	127	1	no	—
181	#478	D₁₂₇	1	1	yes	characteristic, complemented
182	#474	—	127	1	no	—
183	#460	—	127	1	no	—
184	#460	—	127	1	no	—
185	#460	—	127	1	no	—
186	#446	—	127	1	no	—
187	#440	—	127	1	no	—
188	#440	—	127	1	no	—
189	C₄×SD₁₆	—	127	1	no	—
190	C₂×#41	—	254	1	no	—
191	C₂×SD₁₆	—	127	1	no	—
192	C₂×SD₁₆	—	127	1	no	—
193	C₂×SD₁₆	—	127	1	no	—
194	C₄⋊Q₈	—	127	1	no	—
195	C₄⋊Q₈	—	254	1	no	—
196	#127	—	127	1	no	—
197	#127	—	127	1	no	—
198	#127	—	127	1	no	—
199	#125	—	127	1	no	—
200	#125	—	254	1	no	—
201	#125	—	254	1	no	—
202	#125	C₂×D₁₂₇	1	1	yes	characteristic
203	#121	—	254	1	no	—
204	#121	—	254	1	no	—
205	#121	—	254	1	no	—
206	C₄×Q₈	—	127	1	no	—
207	C₄×Q₈	—	127	1	no	—
208	C₄×Q₈	—	127	1	no	—
209	C₄×Q₈	—	127	1	no	—
210	C₄×Q₈	—	254	1	no	—
211	C₄×D₄	—	127	1	no	—
212	C₄×D₄	—	127	1	no	—
213	C₄×D₄	C₂×D₁₂₇	1	1	yes	characteristic
214	#107	—	127	1	no	—
215	#107	C₂×D₁₂₇	1	1	yes	characteristic
216	#106	C₂×D₁₂₇	1	1	yes	characteristic
217	C₄⋊C₈	—	127	1	no	—
218	C₄⋊C₈	—	127	1	no	—
219	C₄⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic
220	#103	—	127	1	no	—
221	#102	—	127	1	no	—
222	#102	—	127	1	no	—
223	#102	—	127	1	no	—
224	#102	—	127	1	no	—
225	#102	—	127	1	no	—
226	#102	—	127	1	no	—
227	#102	—	127	1	no	—
228	#102	C₂×D₁₂₇	1	1	yes	characteristic
229	#102	C₂×D₁₂₇	1	1	yes	characteristic
230	#97	—	127	1	no	—
231	#97	—	127	1	no	—
232	C₄×C₈	—	127	1	no	—
233	#41	—	508	4	no	—
234	C₂×Q₈	—	127	1	no	—
235	C₂×Q₈	—	127	1	no	—
236	C₂×Q₈	—	254	1	no	—
237	C₂×D₄	—	127	1	no	—
238	C₂×D₄	—	127	1	no	—
239	C₂×D₄	—	254	1	no	—
240	C₂×D₄	—	254	1	no	—
241	C₂×D₄	—	254	1	no	—
242	C₂×D₄	—	254	1	no	—
243	C₂×D₄	—	254	1	no	—
244	C₂×D₄	—	254	1	no	—
245	C₂×D₄	C₂²×D₁₂₇	1	1	yes	characteristic
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	254	1	no	—
248	C₂²×C₄	—	254	1	no	—
249	SD₁₆	—	254	2	no	—
250	SD₁₆	—	254	2	no	—
251	SD₁₆	—	254	2	no	—
252	C₂×C₈	—	127	1	no	—
253	C₂×C₈	—	127	1	no	—
254	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
255	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	127	1	no	—
259	C₄⋊C₄	—	127	1	no	—
260	C₄⋊C₄	—	127	1	no	—
261	C₄⋊C₄	—	127	1	no	—
262	C₄⋊C₄	—	254	1	no	—
263	C₄⋊C₄	—	254	1	no	—
264	C₄⋊C₄	—	254	1	no	—
265	C₄⋊C₄	—	254	1	no	—
266	C₄⋊C₄	—	254	1	no	—
267	C₄⋊C₄	—	254	1	no	—
268	C₄⋊C₄	C₂²×D₁₂₇	1	1	yes	characteristic
269	C₄⋊C₄	C₂²×D₁₂₇	1	1	yes	characteristic
270	C₄⋊C₄	C₂²×D₁₂₇	1	1	yes	characteristic
271	C₂²⋊C₄	—	2	1	no	—
272	C₂²⋊C₄	—	254	1	no	—
273	C₂²⋊C₄	—	254	1	no	—
274	C₄²	—	127	1	no	—
275	C₄²	—	127	1	no	—
276	C₄²	—	254	1	no	—
277	C₄²	—	254	1	no	—
278	C₄²	—	254	1	no	—
279	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
280	C₂³	—	2	1	no	—
281	C₂³	—	254	1	no	—
282	C₂³	—	254	1	no	—
283	Q₈	—	127	2	no	—
284	Q₈	—	508	2	no	—
285	Q₈	—	254	2	no	—
286	Q₈	—	254	1	no	—
287	D₄	—	2	1	no	—
288	D₄	—	2	1	no	—
289	D₄	—	1016	1	no	—
290	D₄	—	1016	1	no	—
291	D₄	—	1016	1	no	—
292	D₄	—	508	2	no	—
293	D₄	—	508	2	no	—
294	D₄	—	508	2	no	—
295	D₄	—	254	1	no	—
296	D₄	—	254	1	no	—
297	D₄	—	254	1	no	—
298	D₄	—	254	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	127	1	no	—
304	C₂×C₄	—	127	1	no	—
305	C₂×C₄	—	127	1	no	—
306	C₂×C₄	—	127	1	no	—
307	C₂×C₄	—	508	2	no	—
308	C₂×C₄	—	508	2	no	—
309	C₂×C₄	—	254	1	no	—
310	C₂×C₄	—	254	1	no	—
311	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
312	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
313	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	254	1	no	—
317	C₈	—	254	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	508	1	no	—
322	C₂²	—	508	1	no	—
323	C₂²	—	508	1	no	—
324	C₂²	—	508	1	no	—
325	C₂²	—	508	1	no	—
326	C₂²	—	508	1	no	—
327	C₂²	C₂×D₄×D₁₂₇	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	127	2	no	—
334	C₄	—	508	1	no	—
335	C₄	—	254	2	no	—
336	C₄	—	254	1	no	—
337	C₄	—	254	1	no	—
338	C₄	—	254	1	no	—
339	C₄	#41×D₁₂₇	1	1	yes	characteristic
340	C₄	#16861257	1	1	yes	characteristic
341	C₂	—	8	1	no	—
342	C₂	—	1016	1	no	—
343	C₂	—	1016	1	no	—
344	C₂	#22550536	1	1	yes	characteristic
345	C₂	AGL(1,ℤ/8)×D₁₂₇	1	1	yes	characteristic
346	C₂	#22550683	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₂₇
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#478×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	#22550547	C₂
hypercenter	#478	D₁₂₇
hypocenter	C₁₂₇	#3368
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #22550603	C₂	semidirect
3. #22550242	C₂	semidirect
4. #22550239	C₂	semidirect
5. #22550221	C₂	semidirect
6. #22550175	C₂	semidirect
7. #22550155	C₂	semidirect
8. #22550077	C₂	semidirect
9. #22550075	C₂	semidirect
10. #22550075	C₂	semidirect
11. #22550005	C₂	semidirect
12. #22549976	C₂	semidirect
13. #22549975	C₂	semidirect
14. #22549963	C₂	semidirect
15. #478×C₁₂₇	C₂	semidirect
111. Dic₂₅₄	#41	semidirect
174. C₁₂₇	#3368	semidirect
181. #478	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550603	2
#22550547	2
#22550242	2
#22550239	2
#22550221	2
#22550175	2
#22550155	2
#22550077	2
#22550075	2
#22550005	2
#22549976	2
#22549975	2
#22549963	2
#478×C₁₂₇	2
#3368	127