Subgroups — #30648533

Order 16256.

376 conjugacy classes of subgroups.

The subgroup lattice has 324 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
2	C₄×Q₈×D₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#22550184	C₂	1	1	yes	maximal, characteristic, complemented
4	#22550183	C₂	1	1	yes	maximal, characteristic, complemented
5	#22550180	C₂	1	1	yes	maximal, characteristic, complemented
6	#22550170	C₂	1	1	yes	maximal, characteristic, complemented
7	#22550169	C₂	1	1	yes	maximal, characteristic, complemented
8	#22550160	C₂	1	1	yes	maximal, characteristic, complemented
9	#22550159	C₂	1	1	yes	maximal, characteristic, complemented
10	#22549917	C₂	1	1	yes	maximal, characteristic
11	#22549915	C₂	1	1	yes	maximal, characteristic
12	#22549898	C₂	1	1	yes	maximal, characteristic
13	#22549883	C₂	1	1	yes	maximal, characteristic
14	#22549883	C₂	1	1	yes	maximal, characteristic
15	#479×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₂₇	C₂²	1	1	yes	characteristic
17	C₂×#14523231	C₂²	1	1	yes	characteristic
18	C₂×#14523231	C₂²	1	1	yes	characteristic
19	#16861208	C₂²	1	1	yes	characteristic
20	#16861208	C₂²	1	1	yes	characteristic
21	#16861201	C₂²	1	1	yes	characteristic
22	#16861187	—	2	1	no	—
23	#16861187	—	2	1	no	—
24	Q₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
25	C₄×Dic₂₅₄	—	2	1	no	—
26	C₄×Dic₂₅₄	C₂²	1	1	yes	characteristic
27	#16861174	—	2	1	no	—
28	#16861174	—	2	1	no	—
29	#16861174	—	2	1	no	—
30	#16861174	C₂²	1	1	yes	characteristic
31	#16861167	C₂²	1	1	yes	characteristic
32	#16861167	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₁₂₇	—	2	1	no	—
34	C₄⋊C₄×D₁₂₇	C₂²	1	1	yes	characteristic
35	C₄²×D₁₂₇	—	2	1	no	—
36	C₄²×D₁₂₇	C₂²	1	1	yes	characteristic
37	#16861143	C₂²	1	1	yes	characteristic
38	#16861142	C₂²	1	1	yes	characteristic
39	#16861142	C₂²	1	1	yes	characteristic
40	#16861141	C₂²	1	1	yes	characteristic
41	#16861140	C₂²	1	1	yes	characteristic
42	#16861140	C₂²	1	1	yes	characteristic
43	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
44	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
45	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
46	#16861134	C₂²	1	1	yes	characteristic
47	#16861134	C₂²	1	1	yes	characteristic
48	#16861133	C₂²	1	1	yes	characteristic
49	#16861133	C₂²	1	1	yes	characteristic
50	#16861132	C₂²	1	1	yes	characteristic
51	#16861132	C₂²	1	1	yes	characteristic
52	#125×C₁₂₇	C₂²	1	1	yes	characteristic
53	C₄×Q₈×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	#103×C₁₂₇	C₂²	1	1	yes	characteristic
58	#103×C₁₂₇	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₅₄	—	2	1	no	—
60	C₂×Dic₂₅₄	—	2	1	no	—
61	C₂×Dic₂₅₄	—	2	1	no	—
62	C₂×Dic₂₅₄	C₂³	1	1	yes	characteristic
63	Q₈×D₁₂₇	—	2	2	no	—
64	Q₈×D₁₂₇	Q₈	1	4	yes	complemented
65	C₂×C₄×D₁₂₇	—	2	1	no	—
66	C₂×C₄×D₁₂₇	—	2	1	no	—
67	C₂×C₄×D₁₂₇	D₄	1	1	yes	characteristic
68	C₂×C₄×D₁₂₇	D₄	1	1	yes	characteristic
69	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
70	#14523231	—	2	2	no	—
71	#14523231	—	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	C₂³	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	—	2	1	no	—
84	Dic₁₂₇⋊C₄	—	2	1	no	—
85	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
86	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
87	C₄×Dic₁₂₇	—	2	1	no	—
88	C₄×Dic₁₂₇	—	2	1	no	—
89	C₄×Dic₁₂₇	—	2	1	no	—
90	C₄×Dic₁₂₇	—	2	1	no	—
91	C₄×Dic₁₂₇	—	2	1	no	—
92	C₄×Dic₁₂₇	D₄	1	1	yes	characteristic
93	C₄×Dic₁₂₇	D₄	1	1	yes	characteristic
94	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₁₂₇	C₂³	1	1	yes	characteristic
96	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₁₂₇	—	2	1	no	—
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	C₄²×C₁₂₇	—	2	1	no	—
105	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
106	C₂²×D₁₂₇	C₂×D₄	1	1	yes	characteristic
107	Dic₂₅₄	—	2	1	no	—
108	Dic₂₅₄	—	2	4	no	—
109	Dic₂₅₄	—	4	2	no	—
110	Dic₂₅₄	—	4	2	no	—
111	Dic₂₅₄	C₂×Q₈	1	2	yes	—
112	C₂×Dic₁₂₇	—	2	1	no	—
113	C₂×Dic₁₂₇	—	2	1	no	—
114	C₂×Dic₁₂₇	—	2	1	no	—
115	C₂×Dic₁₂₇	—	2	1	no	—
116	C₂×Dic₁₂₇	C₂×D₄	1	1	yes	characteristic
117	C₂×Dic₁₂₇	C₂×D₄	1	1	yes	characteristic
118	C₂×Dic₁₂₇	C₂×D₄	1	1	yes	characteristic
119	C₄×D₁₂₇	—	2	4	no	—
120	C₄×D₁₂₇	—	2	2	no	—
121	C₄×D₁₂₇	—	4	1	no	—
122	C₄×D₁₂₇	—	4	2	no	—
123	C₄×D₁₂₇	C₂×Q₈	1	2	yes	—
124	C₄×D₁₂₇	#41	1	2	yes	—
125	C₁₂₇⋊C₈	—	2	1	no	—
126	C₁₂₇⋊C₈	—	2	1	no	—
127	Q₈×C₁₂₇	—	2	1	no	—
128	Q₈×C₁₂₇	C₂×Q₈	1	2	yes	—
129	C₂×C₄×C₁₂₇	—	2	1	no	—
130	C₂×C₄×C₁₂₇	—	2	1	no	—
131	C₂×C₄×C₁₂₇	—	2	1	no	—
132	C₂×C₄×C₁₂₇	—	2	1	no	—
133	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
136	C₈×C₁₂₇	—	2	1	no	—
137	C₈×C₁₂₇	—	2	1	no	—
138	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×D₁₂₇	—	2	1	no	—
140	C₂×D₁₂₇	—	2	1	no	—
141	C₂×D₁₂₇	C₂²⋊Q₈	1	2	yes	—
142	C₄×C₁₂₇	—	2	2	no	—
143	C₄×C₁₂₇	—	2	1	no	—
144	C₄×C₁₂₇	—	2	1	no	—
145	C₄×C₁₂₇	—	4	1	no	—
146	C₄×C₁₂₇	—	4	1	no	—
147	C₄×C₁₂₇	—	4	1	no	—
148	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
149	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
150	Dic₁₂₇	—	2	2	no	—
151	Dic₁₂₇	—	2	1	no	—
152	Dic₁₂₇	—	4	1	no	—
153	Dic₁₂₇	—	4	1	no	—
154	Dic₁₂₇	—	4	1	no	—
155	Dic₁₂₇	C₂²⋊Q₈	1	2	yes	—
156	Dic₁₂₇	#137	1	2	yes	—
157	C₂×C₁₂₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
158	C₂×C₁₂₇	C₂×#137	1	1	yes	characteristic
159	C₂×C₁₂₇	#575	1	1	yes	characteristic
160	D₁₂₇	—	2	2	no	—
161	#3078	—	127	1	no	maximal, Sylow
162	C₁₂₇	#3078	1	1	yes	characteristic, complemented, Sylow
163	#533	—	127	1	no	—
164	C₂×C₄×Q₈	—	127	1	no	—
165	#479	—	127	1	no	—
166	#479	—	127	1	no	—
167	#479	—	127	1	no	—
168	#479	D₁₂₇	1	1	yes	characteristic, complemented
169	#477	—	127	1	no	—
170	#477	—	127	1	no	—
171	#475	—	127	1	no	—
172	#475	—	127	1	no	—
173	#428	—	127	1	no	—
174	#428	—	127	1	no	—
175	#423	—	127	1	no	—
176	#419	—	127	1	no	—
177	#419	—	127	1	no	—
178	C₂²×Q₈	—	127	1	no	—
179	C₂×M₁₆	—	127	1	no	—
180	C₂×M₁₆	—	127	1	no	—
181	C₄⋊Q₈	—	127	1	no	—
182	C₄⋊Q₈	—	127	1	no	—
183	#125	—	127	1	no	—
184	#125	C₂×D₁₂₇	1	1	yes	characteristic
185	C₂²⋊Q₈	—	254	1	no	—
186	C₂²⋊Q₈	—	254	1	no	—
187	C₄×Q₈	—	127	1	no	—
188	C₄×Q₈	—	127	1	no	—
189	C₄×Q₈	—	127	1	no	—
190	C₄×Q₈	—	254	1	no	—
191	C₄×Q₈	—	254	1	no	—
192	C₄×Q₈	—	254	1	no	—
193	C₄×Q₈	—	254	1	no	—
194	C₄×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
195	#117	—	127	1	no	—
196	#117	—	127	1	no	—
197	C₂×C₄⋊C₄	—	127	1	no	—
198	C₂×C₄⋊C₄	—	254	1	no	—
199	C₂×C₄²	—	127	1	no	—
200	C₂×C₄²	—	254	1	no	—
201	#107	—	127	1	no	—
202	#107	—	127	1	no	—
203	#107	—	127	1	no	—
204	#107	C₂×D₁₂₇	1	1	yes	characteristic
205	#106	—	127	1	no	—
206	#106	—	127	1	no	—
207	#106	—	127	1	no	—
208	#106	C₂×D₁₂₇	1	1	yes	characteristic
209	C₄⋊C₈	—	127	1	no	—
210	C₄⋊C₈	—	127	1	no	—
211	C₄⋊C₈	—	127	1	no	—
212	C₄⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic
213	#103	—	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	—	127	1	no	—
219	#103	C₂×D₁₂₇	1	1	yes	characteristic
220	#103	C₂×D₁₂₇	1	1	yes	characteristic
221	C₂×Q₈	—	127	4	no	—
222	C₂×Q₈	—	127	1	no	—
223	C₂×Q₈	—	254	1	no	—
224	C₂×Q₈	—	254	1	no	—
225	C₂×Q₈	—	254	1	no	—
226	C₂×Q₈	—	254	2	no	—
227	C₂×Q₈	C₂²×D₁₂₇	1	1	yes	characteristic
228	C₂²×C₄	—	127	1	no	—
229	C₂²×C₄	—	127	1	no	—
230	C₂²×C₄	—	127	1	no	—
231	C₂²×C₄	—	254	1	no	—
232	C₂²×C₄	—	254	1	no	—
233	M₁₆	—	254	2	no	—
234	M₁₆	—	254	2	no	—
235	C₂×C₈	—	127	1	no	—
236	C₂×C₈	—	127	1	no	—
237	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	127	1	no	—
243	C₄⋊C₄	—	127	1	no	—
244	C₄⋊C₄	—	127	1	no	—
245	C₄⋊C₄	—	254	1	no	—
246	C₄⋊C₄	—	254	1	no	—
247	C₄⋊C₄	—	254	1	no	—
248	C₄⋊C₄	—	254	1	no	—
249	C₄⋊C₄	—	254	1	no	—
250	C₄⋊C₄	—	254	1	no	—
251	C₄⋊C₄	—	254	1	no	—
252	C₄⋊C₄	—	254	1	no	—
253	C₄⋊C₄	C₂²×D₁₂₇	1	1	yes	characteristic
254	C₄⋊C₄	C₂²×D₁₂₇	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₁₂₇	1	1	yes	characteristic
256	C₂²⋊C₄	—	254	1	no	—
257	C₂²⋊C₄	—	254	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	127	1	no	—
260	C₄²	—	127	1	no	—
261	C₄²	—	127	1	no	—
262	C₄²	—	254	1	no	—
263	C₄²	—	254	1	no	—
264	C₄²	—	254	1	no	—
265	C₄²	—	254	1	no	—
266	C₄²	—	254	1	no	—
267	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
268	C₂³	—	127	1	no	—
269	Q₈	—	2	1	no	—
270	Q₈	—	127	2	no	—
271	Q₈	—	508	2	no	—
272	Q₈	—	508	2	no	—
273	Q₈	—	254	4	no	—
274	Q₈	—	254	1	no	—
275	Q₈	Q₈×D₁₂₇	1	2	yes	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	127	1	no	—
281	C₂×C₄	—	127	1	no	—
282	C₂×C₄	—	127	2	no	—
283	C₂×C₄	—	127	2	no	—
284	C₂×C₄	—	127	1	no	—
285	C₂×C₄	—	508	1	no	—
286	C₂×C₄	—	508	2	no	—
287	C₂×C₄	—	254	2	no	—
288	C₂×C₄	—	254	4	no	—
289	C₂×C₄	—	254	1	no	—
290	C₂×C₄	—	254	1	no	—
291	C₂×C₄	—	254	1	no	—
292	C₂×C₄	—	254	1	no	—
293	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
294	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
295	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	254	1	no	—
299	C₈	—	254	1	no	—
300	C₂²	—	127	2	no	—
301	C₂²	—	254	1	no	—
302	C₂²	—	254	1	no	—
303	C₂²	C₂×D₄×D₁₂₇	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	127	2	no	—
311	C₄	—	127	2	no	—
312	C₄	—	508	1	no	—
313	C₄	—	508	1	no	—
314	C₄	—	508	1	no	—
315	C₄	—	254	2	no	—
316	C₄	—	254	1	no	—
317	C₄	C₂×Q₈×D₁₂₇	1	1	yes	characteristic
318	C₄	#41×D₁₂₇	1	1	yes	characteristic
319	C₂	—	254	2	no	—
320	C₂	C₂²⋊Q₈×D₁₂₇	1	1	yes	characteristic
321	C₂	#137×D₁₂₇	1	1	yes	characteristic
322	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	#479×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	C₂²×D₁₂₇	C₂×D₄
hypercenter	#479	D₁₂₇
hypocenter	C₁₂₇	#3078
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
3. #22550184	C₂	semidirect
4. #22550183	C₂	semidirect
5. #22550180	C₂	semidirect
6. #22550170	C₂	semidirect
7. #22550169	C₂	semidirect
8. #22550160	C₂	semidirect
9. #22550159	C₂	semidirect
15. #479×C₁₂₇	C₂	semidirect
64. Q₈×D₁₂₇	Q₈	semidirect
162. C₁₂₇	#3078	semidirect
168. #479	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×Q₈×D₁₂₇	2
#22550184	2
#22550183	2
#22550180	2
#22550170	2
#22550169	2
#22550160	2
#22550159	2
#22549917	2
#22549915	2
#22549898	2
#22549883	2
#479×C₁₂₇	2
#3078	127