Subgroups — #30648525

Order 16256.

440 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550432	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₄×D₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#22550177	C₂	1	1	yes	maximal, characteristic, complemented
4	#22550176	C₂	1	1	yes	maximal, characteristic, complemented
5	#22550173	C₂	1	1	yes	maximal, characteristic, complemented
6	#22550165	C₂	1	1	yes	maximal, characteristic, complemented
7	#22550164	C₂	1	1	yes	maximal, characteristic, complemented
8	#22550155	C₂	1	1	yes	maximal, characteristic, complemented
9	#22550154	C₂	1	1	yes	maximal, characteristic, complemented
10	#22549917	C₂	1	1	yes	maximal, characteristic, complemented
11	#22549915	C₂	1	1	yes	maximal, characteristic, complemented
12	#22549898	C₂	1	1	yes	maximal, characteristic, complemented
13	#22549879	C₂	1	1	yes	maximal, characteristic
14	#22549879	C₂	1	1	yes	maximal, characteristic
15	#478×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₁₂₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₂₇	—	2	1	no	—
18	C₂×#14523231	C₂²	1	1	yes	characteristic
19	C₂×#14523231	C₂²	1	1	yes	characteristic
20	#16861208	C₂²	1	1	yes	characteristic, complemented
21	#16861208	C₂²	1	1	yes	characteristic, complemented
22	#16861201	C₂²	1	1	yes	characteristic, complemented
23	#16861187	—	2	1	no	—
24	#16861187	—	2	1	no	—
25	#16861174	—	2	1	no	—
26	#16861174	—	2	1	no	—
27	C₄×D₅₀₈	C₂²	1	1	yes	characteristic
28	D₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
29	#16861171	C₂²	1	1	yes	characteristic
30	C₄×C₂²⋊D₁₂₇	—	2	1	no	—
31	(C₂²⋊D₁₂₇)⋊C₄	—	2	1	no	—
32	#16861167	C₂²	1	1	yes	characteristic
33	#16861167	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₁₂₇	C₂²	1	1	yes	characteristic
35	C₂²⋊C₄×D₁₂₇	—	2	1	no	—
36	C₄²×D₁₂₇	C₂²	1	1	yes	characteristic
37	#16861143	C₂²	1	1	yes	characteristic, complemented
38	#16861142	C₂²	1	1	yes	characteristic, complemented
39	#16861142	C₂²	1	1	yes	characteristic, complemented
40	#16861141	C₂²	1	1	yes	characteristic, complemented
41	#16861140	C₂²	1	1	yes	characteristic, complemented
42	#16861140	C₂²	1	1	yes	characteristic, complemented
43	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic, complemented
44	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
45	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
46	#16861131	C₂²	1	1	yes	characteristic
47	#16861131	C₂²	1	1	yes	characteristic
48	#16861130	C₂²	1	1	yes	characteristic
49	#16861130	C₂²	1	1	yes	characteristic
50	#16861129	C₂²	1	1	yes	characteristic
51	#16861129	C₂²	1	1	yes	characteristic
52	#125×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
53	C₄×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
54	#107×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
55	#106×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
56	C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
57	#102×C₁₂₇	C₂²	1	1	yes	characteristic
58	#102×C₁₂₇	C₂²	1	1	yes	characteristic
59	C₂³×D₁₂₇	—	2	1	no	—
60	C₂×Dic₂₅₄	—	2	1	no	—
61	C₂×Dic₂₅₄	—	2	1	no	—
62	C₂×D₅₀₈	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₂₇	—	2	1	no	—
64	D₄×D₁₂₇	—	2	2	no	—
65	D₄×D₁₂₇	Q₈	1	4	yes	complemented
66	C₂²×Dic₁₂₇	—	2	1	no	—
67	C₂×C₄×D₁₂₇	—	2	1	no	—
68	C₂×C₄×D₁₂₇	—	2	1	no	—
69	C₂×C₄×D₁₂₇	—	2	1	no	—
70	C₂×C₄×D₁₂₇	—	4	4	no	—
71	C₂×C₄×D₁₂₇	D₄	1	1	yes	characteristic
72	C₂×C₄×D₁₂₇	D₄	1	1	yes	characteristic
73	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
74	#14523231	—	2	2	no	—
75	#14523231	—	2	2	no	—
76	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
77	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₁₂₇	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₁₂₇	C₂³	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	C₂²⋊Dic₁₂₇	—	2	1	no	—
87	(C₂×D₁₂₇)⋊C₄	—	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	D₄	1	1	yes	characteristic
94	C₄×Dic₁₂₇	D₄	1	1	yes	characteristic
95	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
96	C₂×D₄×C₁₂₇	C₂³	1	1	yes	characteristic
97	C₂²×C₄×C₁₂₇	—	2	1	no	—
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	—	2	1	no	—
102	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
105	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
106	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
107	C₂²×D₁₂₇	—	2	1	no	—
108	C₂²×D₁₂₇	—	2	4	no	—
109	C₂²×D₁₂₇	—	4	2	no	—
110	C₂²×D₁₂₇	—	4	2	no	—
111	C₂²×D₁₂₇	C₂×D₄	1	1	yes	characteristic
112	Dic₂₅₄	—	4	2	no	—
113	Dic₂₅₄	—	4	2	no	—
114	D₅₀₈	—	2	1	no	—
115	D₅₀₈	C₂×Q₈	1	2	yes	—
116	C₂²⋊D₁₂₇	—	2	4	no	—
117	C₂×Dic₁₂₇	—	2	1	no	—
118	C₂×Dic₁₂₇	—	2	1	no	—
119	C₂×Dic₁₂₇	—	2	1	no	—
120	C₂×Dic₁₂₇	—	4	2	no	—
121	C₂×Dic₁₂₇	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₁₂₇	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₁₂₇	C₂×D₄	1	1	yes	characteristic
124	C₄×D₁₂₇	—	2	2	no	—
125	C₄×D₁₂₇	—	4	1	no	—
126	C₄×D₁₂₇	—	4	2	no	—
127	C₄×D₁₂₇	—	4	2	no	—
128	C₄×D₁₂₇	—	4	2	no	—
129	C₄×D₁₂₇	C₂×Q₈	1	2	yes	—
130	C₄×D₁₂₇	#41	1	2	yes	—
131	C₁₂₇⋊C₈	—	2	1	no	—
132	C₁₂₇⋊C₈	—	2	1	no	—
133	C₂³×C₁₂₇	—	2	1	no	—
134	D₄×C₁₂₇	—	2	1	no	—
135	D₄×C₁₂₇	C₂×Q₈	1	2	yes	—
136	C₂×C₄×C₁₂₇	—	2	1	no	—
137	C₂×C₄×C₁₂₇	—	2	1	no	—
138	C₂×C₄×C₁₂₇	—	2	1	no	—
139	C₂×C₄×C₁₂₇	—	4	2	no	—
140	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
143	C₈×C₁₂₇	—	2	1	no	—
144	C₈×C₁₂₇	—	2	1	no	—
145	C₂²×C₁₂₇	—	2	2	no	—
146	C₂²×C₁₂₇	—	4	1	no	—
147	C₂²×C₁₂₇	—	4	1	no	—
148	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
149	C₂×D₁₂₇	—	2	1	no	—
150	C₂×D₁₂₇	—	2	1	no	—
151	C₂×D₁₂₇	—	2	2	no	—
152	C₂×D₁₂₇	—	4	8	no	—
153	C₂×D₁₂₇	—	4	1	no	—
154	C₂×D₁₂₇	—	4	1	no	—
155	C₂×D₁₂₇	C₂²⋊Q₈	1	2	yes	—
156	C₄×C₁₂₇	—	2	1	no	—
157	C₄×C₁₂₇	—	2	1	no	—
158	C₄×C₁₂₇	—	4	1	no	—
159	C₄×C₁₂₇	—	4	1	no	—
160	C₄×C₁₂₇	—	4	1	no	—
161	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
162	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
163	Dic₁₂₇	—	2	1	no	—
164	Dic₁₂₇	—	4	1	no	—
165	Dic₁₂₇	—	4	1	no	—
166	Dic₁₂₇	—	4	1	no	—
167	Dic₁₂₇	C₂²⋊Q₈	1	2	yes	—
168	Dic₁₂₇	AGL(1,ℤ/8)	1	2	yes	—
169	C₂×C₁₂₇	—	4	2	no	—
170	C₂×C₁₂₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
171	C₂×C₁₂₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
172	C₂×C₁₂₇	#575	1	1	yes	characteristic
173	D₁₂₇	—	2	2	no	—
174	D₁₂₇	—	4	2	no	—
175	#3077	—	127	1	no	maximal, Sylow
176	C₁₂₇	#3077	1	1	yes	characteristic, complemented, Sylow
177	#533	—	127	1	no	—
178	C₂×C₄×D₄	—	127	1	no	—
179	#478	—	127	1	no	—
180	#478	—	127	1	no	—
181	#478	—	127	1	no	—
182	#478	D₁₂₇	1	1	yes	characteristic, complemented
183	#476	—	127	1	no	—
184	#476	—	127	1	no	—
185	#474	—	127	1	no	—
186	#474	—	127	1	no	—
187	#428	—	127	1	no	—
188	#428	—	127	1	no	—
189	#423	—	127	1	no	—
190	#418	—	127	1	no	—
191	#418	—	127	1	no	—
192	C₂²×D₄	—	127	1	no	—
193	C₂³×C₄	—	254	1	no	—
194	C₂×M₁₆	—	127	1	no	—
195	C₂×M₁₆	—	127	1	no	—
196	C₄⋊Q₈	—	127	1	no	—
197	C₄⋊Q₈	—	127	1	no	—
198	#125	—	127	1	no	—
199	#125	C₂×D₁₂₇	1	1	yes	characteristic, complemented
200	C₂²⋊Q₈	—	254	1	no	—
201	C₂²⋊Q₈	—	254	1	no	—
202	C₄×Q₈	—	254	1	no	—
203	C₄×Q₈	—	254	1	no	—
204	C₄×D₄	—	127	1	no	—
205	C₄×D₄	—	127	1	no	—
206	C₄×D₄	—	127	1	no	—
207	C₄×D₄	—	254	1	no	—
208	C₄×D₄	—	254	1	no	—
209	C₄×D₄	C₂×D₁₂₇	1	1	yes	characteristic
210	#117	—	127	1	no	—
211	#117	—	127	1	no	—
212	C₂×C₄⋊C₄	—	127	1	no	—
213	C₂×C₂²⋊C₄	—	254	1	no	—
214	C₂×C₄²	—	127	1	no	—
215	#107	—	127	1	no	—
216	#107	—	127	1	no	—
217	#107	—	127	1	no	—
218	#107	C₂×D₁₂₇	1	1	yes	characteristic, complemented
219	#106	—	127	1	no	—
220	#106	—	127	1	no	—
221	#106	—	127	1	no	—
222	#106	C₂×D₁₂₇	1	1	yes	characteristic, complemented
223	C₄⋊C₈	—	127	1	no	—
224	C₄⋊C₈	—	127	1	no	—
225	C₄⋊C₈	—	127	1	no	—
226	C₄⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic, complemented
227	#102	—	127	1	no	—
228	#102	—	127	1	no	—
229	#102	—	127	1	no	—
230	#102	—	127	1	no	—
231	#102	—	127	1	no	—
232	#102	—	127	1	no	—
233	#102	C₂×D₁₂₇	1	1	yes	characteristic
234	#102	C₂×D₁₂₇	1	1	yes	characteristic
235	C₂⁴	—	254	1	no	—
236	C₂×Q₈	—	254	1	no	—
237	C₂×Q₈	—	254	1	no	—
238	C₂×D₄	—	127	4	no	—
239	C₂×D₄	—	127	1	no	—
240	C₂×D₄	—	254	2	no	—
241	C₂×D₄	—	254	1	no	—
242	C₂×D₄	C₂²×D₁₂₇	1	1	yes	characteristic
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	127	1	no	—
245	C₂²×C₄	—	127	1	no	—
246	C₂²×C₄	—	127	1	no	—
247	C₂²×C₄	—	508	4	no	—
248	C₂²×C₄	—	254	1	no	—
249	C₂²×C₄	—	254	1	no	—
250	C₂²×C₄	—	254	1	no	—
251	C₂²×C₄	—	254	1	no	—
252	M₁₆	—	254	2	no	—
253	M₁₆	—	254	2	no	—
254	C₂×C₈	—	127	1	no	—
255	C₂×C₈	—	127	1	no	—
256	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
257	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	127	1	no	—
261	C₄⋊C₄	—	127	1	no	—
262	C₄⋊C₄	—	127	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	C₂²⋊C₄	—	254	1	no	—
275	C₂²⋊C₄	—	254	1	no	—
276	C₂²⋊C₄	—	254	1	no	—
277	C₄²	—	127	1	no	—
278	C₄²	—	127	1	no	—
279	C₄²	—	127	1	no	—
280	C₄²	—	254	1	no	—
281	C₄²	—	254	1	no	—
282	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
283	C₂³	—	2	1	no	—
284	C₂³	—	127	1	no	—
285	C₂³	—	508	2	no	—
286	C₂³	—	508	2	no	—
287	C₂³	—	254	4	no	—
288	C₂³	—	254	1	no	—
289	Q₈	—	508	2	no	—
290	Q₈	—	508	2	no	—
291	D₄	—	2	1	no	—
292	D₄	—	127	2	no	—
293	D₄	—	254	4	no	—
294	D₄	—	254	1	no	—
295	D₄	Q₈×D₁₂₇	1	2	yes	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	127	1	no	—
301	C₂×C₄	—	127	1	no	—
302	C₂×C₄	—	127	2	no	—
303	C₂×C₄	—	127	2	no	—
304	C₂×C₄	—	127	1	no	—
305	C₂×C₄	—	508	2	no	—
306	C₂×C₄	—	508	1	no	—
307	C₂×C₄	—	508	2	no	—
308	C₂×C₄	—	508	2	no	—
309	C₂×C₄	—	508	2	no	—
310	C₂×C₄	—	254	2	no	—
311	C₂×C₄	—	254	1	no	—
312	C₂×C₄	—	254	1	no	—
313	C₂×C₄	—	254	1	no	—
314	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
315	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	254	1	no	—
320	C₈	—	254	1	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	127	2	no	—
325	C₂²	—	508	8	no	—
326	C₂²	—	508	1	no	—
327	C₂²	—	508	1	no	—
328	C₂²	—	254	2	no	—
329	C₂²	—	254	1	no	—
330	C₂²	—	254	1	no	—
331	C₂²	C₂×D₄×D₁₂₇	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	127	2	no	—
338	C₄	—	127	2	no	—
339	C₄	—	508	1	no	—
340	C₄	—	508	1	no	—
341	C₄	—	508	1	no	—
342	C₄	—	254	1	no	—
343	C₄	C₂×Q₈×D₁₂₇	1	1	yes	characteristic
344	C₄	#41×D₁₂₇	1	1	yes	characteristic
345	C₂	—	4	2	no	—
346	C₂	—	508	2	no	—
347	C₂	—	254	2	no	—
348	C₂	C₂²⋊Q₈×D₁₂₇	1	1	yes	characteristic
349	C₂	AGL(1,ℤ/8)×D₁₂₇	1	1	yes	characteristic
350	C₂	#22550673	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	C₂×D₄×D₁₂₇	C₂²
hypercenter	#478	D₁₂₇
hypocenter	C₁₂₇	#3077
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. #22550432	C₂	semidirect
3. #22550177	C₂	semidirect
4. #22550176	C₂	semidirect
5. #22550173	C₂	semidirect
6. #22550165	C₂	semidirect
7. #22550164	C₂	semidirect
8. #22550155	C₂	semidirect
9. #22550154	C₂	semidirect
10. #22549917	C₂	semidirect
11. #22549915	C₂	semidirect
12. #22549898	C₂	semidirect
15. #478×C₁₂₇	C₂	semidirect
20. #16861208	C₂²	semidirect
21. #16861208	C₂²	semidirect
22. #16861201	C₂²	semidirect
37. #16861143	C₂²	semidirect
38. #16861142	C₂²	semidirect
39. #16861142	C₂²	semidirect
40. #16861141	C₂²	semidirect
41. #16861140	C₂²	semidirect
42. #16861140	C₂²	semidirect
43. C₄⋊(C₁₂₇⋊C₈)	C₂²	semidirect
44. Dic₁₂₇⋊C₈	C₂²	semidirect
45. Dic₁₂₇⋊C₈	C₂²	semidirect
52. #125×C₁₂₇	C₂²	semidirect
54. #107×C₁₂₇	C₂²	semidirect
55. #106×C₁₂₇	C₂²	semidirect
56. C₄⋊C₈×C₁₂₇	C₂²	semidirect
65. D₄×D₁₂₇	Q₈	semidirect
176. C₁₂₇	#3077	semidirect
182. #478	D₁₂₇	semidirect
199. #125	C₂×D₁₂₇	semidirect
218. #107	C₂×D₁₂₇	semidirect
222. #106	C₂×D₁₂₇	semidirect
226. C₄⋊C₈	C₂×D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550432	2
C₄×D₄×D₁₂₇	2
#22550177	2
#22550176	2
#22550173	2
#22550165	2
#22550164	2
#22550155	2
#22550154	2
#22549917	2
#22549915	2
#22549898	2
#22549879	2
#478×C₁₂₇	2
#3077	127