Subgroups — #30649032

Order 16256.

358 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550489	C₂	1	1	yes	maximal, characteristic
2	#22550352	C₂	1	1	yes	maximal, characteristic
3	#22550220	C₂	1	1	yes	maximal, characteristic
4	#22550219	C₂	1	1	yes	maximal, characteristic
5	#22550180	C₂	1	1	yes	maximal, characteristic, complemented
6	#22550087	C₂	1	1	yes	maximal, characteristic, complemented
7	#22550086	C₂	1	1	yes	maximal, characteristic, complemented
8	#22550033	C₂	1	1	yes	maximal, characteristic
9	#22550033	C₂	1	1	yes	maximal, characteristic
10	#22549982	C₂	1	1	yes	maximal, characteristic, complemented
11	#22549980	C₂	1	1	yes	maximal, characteristic, complemented
12	#22549970	C₂	1	1	yes	maximal, characteristic, complemented
13	#22549968	C₂	1	1	yes	maximal, characteristic, complemented
14	#22549936	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×Dic₅₀₈	C₂²	1	1	yes	characteristic
18	C₂×Dic₅₀₈	C₂²	1	1	yes	characteristic
19	C₂×#14523238	C₂²	1	1	yes	characteristic
20	C₂×#14523238	C₂²	1	1	yes	characteristic
21	#16861209	C₂²	1	1	yes	characteristic
22	#16861204	—	2	1	no	—
23	#16861204	—	2	1	no	—
24	#16861200	—	2	1	no	—
25	#16861200	—	2	1	no	—
26	#16861189	C₂²	1	1	yes	characteristic
27	#16861189	C₂²	1	1	yes	characteristic
28	Q₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
29	C₄×Dic₂₅₄	—	2	1	no	—
30	C₄×Dic₂₅₄	C₂²	1	1	yes	characteristic
31	#16861174	—	2	1	no	—
32	#16861174	C₂²	1	1	yes	characteristic
33	#16861174	C₂²	1	1	yes	characteristic
34	#16861174	C₂²	1	1	yes	characteristic
35	#16861167	—	2	1	no	—
36	#16861167	C₂²	1	1	yes	characteristic
37	#16861166	—	2	1	no	—
38	#16861166	C₂²	1	1	yes	characteristic
39	#16861142	C₂²	1	1	yes	characteristic
40	#16861140	C₂²	1	1	yes	characteristic
41	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
42	#16861134	C₂²	1	1	yes	characteristic
43	#16861134	C₂²	1	1	yes	characteristic
44	#16861132	C₂²	1	1	yes	characteristic
45	#16861132	C₂²	1	1	yes	characteristic
46	#16861132	C₂²	1	1	yes	characteristic
47	#16861132	C₂²	1	1	yes	characteristic
48	(C₂×D₁₂₇)⋊C₈	C₂²	1	1	yes	characteristic
49	(C₂×D₁₂₇)⋊C₈	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×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	C₂³	1	1	yes	characteristic
61	C₂×Dic₂₅₄	C₂³	1	1	yes	characteristic
62	C₂×Dic₂₅₄	C₂³	1	1	yes	characteristic
63	Q₈×D₁₂₇	—	2	2	no	—
64	Q₈×D₁₂₇	—	2	2	no	—
65	C₂×C₄×D₁₂₇	—	2	1	no	—
66	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
67	Dic₅₀₈	—	2	2	no	—
68	Dic₅₀₈	—	2	2	no	—
69	#14523238	—	2	2	no	—
70	#14523238	—	2	2	no	—
71	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₁₂₇	—	2	1	no	—
74	C₄⋊Dic₁₂₇	—	2	1	no	—
75	C₄⋊Dic₁₂₇	—	2	1	no	—
76	C₄⋊Dic₁₂₇	—	2	1	no	—
77	C₄⋊Dic₁₂₇	—	2	1	no	—
78	C₄⋊Dic₁₂₇	C₂³	1	1	yes	characteristic
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	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
86	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
87	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
88	(C₂×D₁₂₇)⋊C₄	D₄	1	1	yes	characteristic
89	(C₂×D₁₂₇)⋊C₄	D₄	1	1	yes	characteristic
90	C₄×Dic₁₂₇	—	2	1	no	—
91	C₄×Dic₁₂₇	—	2	1	no	—
92	C₄×Dic₁₂₇	—	2	1	no	—
93	C₄×Dic₁₂₇	—	2	1	no	—
94	C₄×Dic₁₂₇	—	2	1	no	—
95	C₄×Dic₁₂₇	—	2	1	no	—
96	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
97	C₂×Q₈×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	—	2	1	no	—
102	C₄⋊C₄×C₁₂₇	—	2	1	no	—
103	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
106	C₄²×C₁₂₇	—	2	1	no	—
107	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
108	C₂²×D₁₂₇	C₂×D₄	1	1	yes	characteristic
109	Dic₂₅₄	—	2	4	no	—
110	Dic₂₅₄	—	2	1	no	—
111	Dic₂₅₄	—	2	1	no	—
112	Dic₂₅₄	—	2	1	no	—
113	Dic₂₅₄	—	2	1	no	—
114	Dic₂₅₄	—	2	1	no	—
115	Dic₂₅₄	#41	1	2	yes	—
116	C₂×Dic₁₂₇	—	2	1	no	—
117	C₂×Dic₁₂₇	—	2	1	no	—
118	C₂×Dic₁₂₇	—	2	1	no	—
119	C₂×Dic₁₂₇	—	2	1	no	—
120	C₂×Dic₁₂₇	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₁₂₇	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₁₂₇	C₂×D₄	1	1	yes	characteristic
123	C₄×D₁₂₇	—	2	4	no	—
124	C₄×D₁₂₇	—	2	1	no	—
125	C₄×D₁₂₇	—	2	1	no	—
126	C₁₂₇⋊C₈	—	2	1	no	—
127	C₁₂₇⋊C₈	—	2	1	no	—
128	Q₈×C₁₂₇	—	2	1	no	—
129	Q₈×C₁₂₇	#41	1	2	yes	—
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	—	2	1	no	—
134	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
137	C₈×C₁₂₇	—	2	1	no	—
138	C₈×C₁₂₇	—	2	1	no	—
139	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
140	C₂×D₁₂₇	—	2	1	no	—
141	C₂×D₁₂₇	—	2	1	no	—
142	C₂×D₁₂₇	—	2	1	no	—
143	C₄×C₁₂₇	—	2	2	no	—
144	C₄×C₁₂₇	—	2	1	no	—
145	C₄×C₁₂₇	—	2	1	no	—
146	C₄×C₁₂₇	—	4	1	no	—
147	C₄×C₁₂₇	—	4	1	no	—
148	C₄×C₁₂₇	—	4	1	no	—
149	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
150	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
151	Dic₁₂₇	—	2	2	no	—
152	Dic₁₂₇	—	2	1	no	—
153	Dic₁₂₇	—	2	1	no	—
154	Dic₁₂₇	—	2	1	no	—
155	Dic₁₂₇	—	4	1	no	—
156	Dic₁₂₇	—	4	1	no	—
157	Dic₁₂₇	—	4	1	no	—
158	C₂×C₁₂₇	#525	1	1	yes	characteristic
159	C₂×C₁₂₇	#578	1	1	yes	characteristic
160	C₂×C₁₂₇	#578	1	1	yes	characteristic
161	D₁₂₇	—	4	1	no	—
162	#3127	—	127	1	no	maximal, Sylow
163	C₁₂₇	#3127	1	1	yes	characteristic, complemented, Sylow
164	#541	—	127	1	no	—
165	#519	—	127	1	no	—
166	#485	—	127	1	no	—
167	#485	—	127	1	no	—
168	#479	—	127	1	no	—
169	#479	D₁₂₇	1	1	yes	characteristic, complemented
170	#462	—	127	1	no	—
171	#462	—	127	1	no	—
172	#451	—	127	1	no	—
173	#451	—	127	1	no	—
174	#441	—	127	1	no	—
175	#441	—	127	1	no	—
176	C₄×Q₁₆	—	127	1	no	—
177	C₄×Q₁₆	—	127	1	no	—
178	#433	—	127	1	no	—
179	C₂²×Q₈	—	127	1	no	—
180	C₂×Q₁₆	—	127	1	no	—
181	C₂×Q₁₆	—	127	1	no	—
182	C₂×Q₁₆	—	127	1	no	—
183	C₂×Q₁₆	—	127	1	no	—
184	C₄⋊Q₈	—	127	1	no	—
185	#126	—	254	1	no	—
186	#126	—	254	1	no	—
187	#125	—	254	1	no	—
188	#125	—	254	1	no	—
189	#125	C₂×D₁₂₇	1	1	yes	characteristic
190	C₂²⋊Q₈	—	127	1	no	—
191	C₂²⋊Q₈	—	127	1	no	—
192	C₄×Q₈	—	127	1	no	—
193	C₄×Q₈	—	127	1	no	—
194	C₄×Q₈	—	127	1	no	—
195	C₄×Q₈	—	127	1	no	—
196	C₄×Q₈	—	127	1	no	—
197	C₄×Q₈	—	254	1	no	—
198	C₄×Q₈	—	254	1	no	—
199	C₄×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
200	#117	—	127	1	no	—
201	#117	—	127	1	no	—
202	#117	—	254	1	no	—
203	#117	—	254	1	no	—
204	#107	—	127	1	no	—
205	#107	C₂×D₁₂₇	1	1	yes	characteristic
206	#106	—	127	1	no	—
207	#106	C₂×D₁₂₇	1	1	yes	characteristic
208	C₄⋊C₈	—	127	1	no	—
209	C₄⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic
210	#103	—	127	1	no	—
211	#103	—	127	1	no	—
212	#103	—	127	1	no	—
213	#103	—	127	1	no	—
214	#103	—	127	1	no	—
215	#103	—	127	1	no	—
216	#103	C₂×D₁₂₇	1	1	yes	characteristic
217	#103	C₂×D₁₂₇	1	1	yes	characteristic
218	C₂²⋊C₈	—	127	1	no	—
219	C₂²⋊C₈	—	127	1	no	—
220	#97	—	127	1	no	—
221	C₄×C₈	—	127	1	no	—
222	C₂×Q₈	—	127	1	no	—
223	C₂×Q₈	—	127	1	no	—
224	C₂×Q₈	—	127	1	no	—
225	C₂×Q₈	—	254	1	no	—
226	C₂×Q₈	—	254	2	no	—
227	C₂×Q₈	—	254	2	no	—
228	C₂×Q₈	C₂²×D₁₂₇	1	1	yes	characteristic
229	C₂²×C₄	—	127	1	no	—
230	C₂²×C₄	—	254	1	no	—
231	Q₁₆	—	254	2	no	—
232	Q₁₆	—	254	2	no	—
233	Q₁₆	—	254	2	no	—
234	Q₁₆	—	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	—	254	1	no	—
254	C₄⋊C₄	C₂²×D₁₂₇	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₁₂₇	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₁₂₇	1	1	yes	characteristic
257	C₂²⋊C₄	—	127	1	no	—
258	C₂²⋊C₄	—	127	1	no	—
259	C₂²⋊C₄	—	254	1	no	—
260	C₂²⋊C₄	—	254	1	no	—
261	C₂²⋊C₄	—	254	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	127	1	no	—
264	C₄²	—	254	1	no	—
265	C₄²	—	254	1	no	—
266	C₄²	—	254	1	no	—
267	C₄²	—	254	1	no	—
268	C₄²	—	254	1	no	—
269	C₄²	—	254	1	no	—
270	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
271	C₂³	—	127	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	127	2	no	—
274	Q₈	—	254	4	no	—
275	Q₈	—	254	1	no	—
276	Q₈	—	254	1	no	—
277	Q₈	—	254	1	no	—
278	Q₈	—	254	1	no	—
279	Q₈	—	254	1	no	—
280	Q₈	#14523249	1	2	yes	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	127	1	no	—
286	C₂×C₄	—	127	1	no	—
287	C₂×C₄	—	127	1	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	—	254	1	no	—
294	C₂×C₄	—	254	1	no	—
295	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
296	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
297	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	254	1	no	—
301	C₈	—	254	1	no	—
302	C₂²	—	254	1	no	—
303	C₂²	—	254	1	no	—
304	C₂²	—	254	1	no	—
305	C₂²	C₂×D₄×D₁₂₇	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	4	1	no	—
312	C₄	—	508	1	no	—
313	C₄	—	508	1	no	—
314	C₄	—	508	1	no	—
315	C₄	—	254	1	no	—
316	C₄	—	254	2	no	—
317	C₄	—	254	1	no	—
318	C₄	—	254	1	no	—
319	C₄	#41×D₁₂₇	1	1	yes	characteristic
320	C₄	C₂×#14523249	1	1	yes	characteristic
321	C₂	—	508	1	no	—
322	C₂	#22550389	1	1	yes	characteristic
323	C₂	#22550690	1	1	yes	characteristic
324	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	#479×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	C₂²×D₁₂₇	C₂×D₄
hypercenter	#479	D₁₂₇
hypocenter	C₁₂₇	#3127
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
5. #22550180	C₂	semidirect
6. #22550087	C₂	semidirect
7. #22550086	C₂	semidirect
10. #22549982	C₂	semidirect
11. #22549980	C₂	semidirect
12. #22549970	C₂	semidirect
13. #22549968	C₂	semidirect
15. #479×C₁₂₇	C₂	semidirect
163. C₁₂₇	#3127	semidirect
169. #479	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550489	2
#22550352	2
#22550220	2
#22550219	2
#22550180	2
#22550087	2
#22550086	2
#22550033	2
#22549982	2
#22549980	2
#22549970	2
#22549968	2
#22549936	2
#479×C₁₂₇	2
#3127	127