Subgroups — #30651380

Order 16256.

350 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550598	C₂	1	1	yes	maximal, characteristic, complemented
2	#22550585	C₂	1	1	yes	maximal, characteristic, complemented
3	#22550232	C₂	1	1	yes	maximal, characteristic, complemented
4	#22550231	C₂	1	1	yes	maximal, characteristic, complemented
5	#22550225	C₂	1	1	yes	maximal, characteristic, complemented
6	#22550186	C₂	1	1	yes	maximal, characteristic, complemented
7	#22550183	C₂	1	1	yes	maximal, characteristic, complemented
8	#22550165	C₂	1	1	yes	maximal, characteristic, complemented
9	#22550093	C₂	1	1	yes	maximal, characteristic
10	#22550091	C₂	1	1	yes	maximal, characteristic
11	#22550079	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈×Dic₂₅₄	C₂	1	1	yes	maximal, characteristic, complemented
13	#22549967	C₂	1	1	yes	maximal, characteristic, complemented
14	#22549956	C₂	1	1	yes	maximal, characteristic
15	C₄×SD₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14523236	C₂²	1	1	yes	characteristic
17	C₂×#14523233	C₂²	1	1	yes	characteristic
18	#16861208	C₂²	1	1	yes	characteristic
19	#16861207	C₂²	1	1	yes	characteristic
20	#16861204	—	2	1	no	—
21	#16861203	—	2	1	no	—
22	#16861201	C₂²	1	1	yes	characteristic
23	#16861200	C₂²	1	1	yes	characteristic
24	#16861198	C₂²	1	1	yes	characteristic
25	#16861197	C₂²	1	1	yes	characteristic
26	#16861188	—	2	1	no	—
27	#16861187	—	2	1	no	—
28	#16861186	—	2	1	no	—
29	#16861186	—	2	1	no	—
30	C₄×Dic₂₅₄	C₂²	1	1	yes	characteristic
31	#16861174	C₂²	1	1	yes	characteristic
32	D₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
33	D₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
34	#16861171	C₂²	1	1	yes	characteristic
35	#16861166	—	2	1	no	—
36	C₂×Dic₁₂₇⋊C₄	—	2	1	no	—
37	#16861142	C₂²	1	1	yes	characteristic
38	#16861140	C₂²	1	1	yes	characteristic
39	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
40	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
41	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
42	#16861134	C₂²	1	1	yes	characteristic
43	#16861134	C₂²	1	1	yes	characteristic
44	#16861133	C₂²	1	1	yes	characteristic
45	#16861133	C₂²	1	1	yes	characteristic
46	#16861132	C₂²	1	1	yes	characteristic
47	#16861131	C₂²	1	1	yes	characteristic
48	#16861131	C₂²	1	1	yes	characteristic
49	#16861129	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
55	#106×C₁₂₇	C₂²	1	1	yes	characteristic
56	#103×C₁₂₇	C₂²	1	1	yes	characteristic
57	#102×C₁₂₇	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₅₄	—	2	1	no	—
60	C₂×Dic₂₅₄	D₄	1	1	yes	characteristic
61	C₂×Dic₂₅₄	C₂³	1	1	yes	characteristic
62	C₂×D₅₀₈	C₂³	1	1	yes	characteristic
63	C₂²×Dic₁₂₇	—	2	1	no	—
64	C₂²×Dic₁₂₇	—	2	1	no	—
65	C₂×C₄×D₁₂₇	—	2	1	no	—
66	#14523236	—	2	2	no	—
67	#14523233	—	2	2	no	—
68	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₁₂₇	—	2	1	no	—
71	C₄⋊Dic₁₂₇	D₄	1	1	yes	characteristic
72	C₄⋊Dic₁₂₇	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₁₂₇	C₂³	1	1	yes	characteristic
74	Dic₁₂₇⋊C₄	—	2	1	no	—
75	Dic₁₂₇⋊C₄	—	2	1	no	—
76	Dic₁₂₇⋊C₄	—	2	1	no	—
77	Dic₁₂₇⋊C₄	—	2	1	no	—
78	Dic₁₂₇⋊C₄	—	2	1	no	—
79	Dic₁₂₇⋊C₄	—	2	1	no	—
80	Dic₁₂₇⋊C₄	—	2	1	no	—
81	Dic₁₂₇⋊C₄	D₄	1	1	yes	characteristic
82	Dic₁₂₇⋊C₄	D₄	1	1	yes	characteristic
83	C₂²⋊Dic₁₂₇	—	2	1	no	—
84	C₂²⋊Dic₁₂₇	—	2	1	no	—
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	—	2	1	no	—
89	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
90	C₄×Dic₁₂₇	—	2	1	no	—
91	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
92	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
93	C₂×Q₈×C₁₂₇	C₂³	1	1	yes	characteristic
94	C₂×D₄×C₁₂₇	C₂³	1	1	yes	characteristic
95	C₂²×C₄×C₁₂₇	—	2	1	no	—
96	SD₁₆×C₁₂₇	—	2	2	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	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
102	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
103	C₄²×C₁₂₇	—	2	1	no	—
104	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
105	C₂²×D₁₂₇	—	2	1	no	—
106	Dic₂₅₄	—	2	2	no	—
107	Dic₂₅₄	—	2	1	no	—
108	Dic₂₅₄	—	2	1	no	—
109	Dic₂₅₄	—	8	1	no	—
110	D₅₀₈	—	2	1	no	—
111	D₅₀₈	—	2	1	no	—
112	C₂×Dic₁₂₇	—	2	1	no	—
113	C₂×Dic₁₂₇	—	2	1	no	—
114	C₂×Dic₁₂₇	—	2	1	no	—
115	C₂×Dic₁₂₇	—	4	2	no	—
116	C₂×Dic₁₂₇	—	8	1	no	—
117	C₂×Dic₁₂₇	C₂×D₄	1	1	yes	characteristic
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	C₄×D₁₂₇	—	4	2	no	—
122	C₁₂₇⋊C₈	—	2	1	no	—
123	C₁₂₇⋊C₈	—	2	1	no	—
124	C₂³×C₁₂₇	—	2	1	no	—
125	Q₈×C₁₂₇	—	2	1	no	—
126	Q₈×C₁₂₇	—	2	1	no	—
127	D₄×C₁₂₇	—	2	1	no	—
128	D₄×C₁₂₇	#41	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	—	8	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	—	2	2	no	—
139	C₂²×C₁₂₇	—	4	1	no	—
140	C₂²×C₁₂₇	—	4	1	no	—
141	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
142	C₂×D₁₂₇	—	4	1	no	—
143	C₂×D₁₂₇	—	4	1	no	—
144	C₂×D₁₂₇	—	4	1	no	—
145	C₄×C₁₂₇	—	2	1	no	—
146	C₄×C₁₂₇	—	2	1	no	—
147	C₄×C₁₂₇	—	4	1	no	—
148	C₄×C₁₂₇	—	4	1	no	—
149	C₄×C₁₂₇	—	4	1	no	—
150	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
151	C₄×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
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	Dic₁₂₇	#135	1	2	yes	—
159	C₂×C₁₂₇	—	4	2	no	—
160	C₂×C₁₂₇	#548	1	1	yes	characteristic
161	C₂×C₁₂₇	C₂×#135	1	1	yes	characteristic
162	C₂×C₁₂₇	#577	1	1	yes	characteristic
163	D₁₂₇	—	8	1	no	—
164	#3360	—	127	1	no	maximal, Sylow
165	C₁₂₇	#3360	1	1	yes	characteristic, complemented, Sylow
166	#556	—	127	1	no	—
167	#554	—	127	1	no	—
168	#488	—	127	1	no	—
169	#488	—	127	1	no	—
170	#487	—	127	1	no	—
171	#479	—	127	1	no	—
172	#479	—	127	1	no	—
173	#476	—	127	1	no	—
174	#463	—	127	1	no	—
175	#463	—	127	1	no	—
176	#461	—	127	1	no	—
177	C₈×Q₈	—	127	1	no	—
178	C₄×SD₁₆	—	127	1	no	—
179	C₄×SD₁₆	D₁₂₇	1	1	yes	characteristic, complemented
180	C₄×D₈	—	127	1	no	—
181	C₂×SD₁₆	—	127	1	no	—
182	C₂×SD₁₆	C₂×D₁₂₇	1	1	yes	characteristic
183	C₂×D₈	—	127	1	no	—
184	C₄⋊Q₈	—	127	1	no	—
185	C₄⋊Q₈	—	127	1	no	—
186	#126	—	254	1	no	—
187	#126	—	254	1	no	—
188	#125	—	127	1	no	—
189	#125	—	127	1	no	—
190	#124	—	127	1	no	—
191	#124	—	127	1	no	—
192	C₂²⋊Q₈	—	254	1	no	—
193	C₂²⋊Q₈	—	254	1	no	—
194	C₂²⋊Q₈	—	254	1	no	—
195	C₂²⋊Q₈	—	254	1	no	—
196	C₄×Q₈	—	127	1	no	—
197	C₄×Q₈	—	127	1	no	—
198	C₄×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
199	C₄×D₄	—	127	1	no	—
200	C₄×D₄	—	127	1	no	—
201	C₄×D₄	—	127	1	no	—
202	C₄×D₄	C₂×D₁₂₇	1	1	yes	characteristic
203	#117	—	254	1	no	—
204	C₂×C₄⋊C₄	—	254	1	no	—
205	#107	—	127	1	no	—
206	#106	—	127	1	no	—
207	#106	C₂×D₁₂₇	1	1	yes	characteristic
208	C₄⋊C₈	—	127	1	no	—
209	C₄⋊C₈	—	127	1	no	—
210	C₄⋊C₈	—	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	#102	—	127	1	no	—
218	#102	—	127	1	no	—
219	#102	—	127	1	no	—
220	#102	C₂×D₁₂₇	1	1	yes	characteristic
221	C₄×C₈	—	127	1	no	—
222	C₄×C₈	—	127	1	no	—
223	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
224	C₂×Q₈	—	127	1	no	—
225	C₂×Q₈	—	127	1	no	—
226	C₂×Q₈	—	254	1	no	—
227	C₂×Q₈	C₂²×D₁₂₇	1	1	yes	characteristic
228	C₂×D₄	—	127	1	no	—
229	C₂×D₄	C₂²×D₁₂₇	1	1	yes	characteristic
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	254	1	no	—
232	C₂²×C₄	—	254	1	no	—
233	C₂²×C₄	—	254	1	no	—
234	SD₁₆	—	2	2	no	—
235	SD₁₆	—	254	2	no	—
236	D₈	—	254	2	no	—
237	C₂×C₈	—	127	1	no	—
238	C₂×C₈	—	127	1	no	—
239	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
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	—	127	1	no	—
246	C₄⋊C₄	—	127	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	—	254	1	no	—
255	C₄⋊C₄	C₂²×D₁₂₇	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₁₂₇	1	1	yes	characteristic
257	C₂²⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	254	1	no	—
259	C₂²⋊C₄	—	254	1	no	—
260	C₂²⋊C₄	—	254	1	no	—
261	C₂²⋊C₄	—	254	1	no	—
262	C₂²⋊C₄	—	254	1	no	—
263	C₂²⋊C₄	—	254	1	no	—
264	C₂²⋊C₄	—	254	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	127	1	no	—
267	C₄²	—	127	1	no	—
268	C₄²	—	254	1	no	—
269	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
270	C₂³	—	2	1	no	—
271	C₂³	—	254	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	1016	1	no	—
275	Q₈	—	254	2	no	—
276	Q₈	—	254	1	no	—
277	Q₈	—	254	1	no	—
278	D₄	—	2	1	no	—
279	D₄	—	254	1	no	—
280	D₄	—	254	1	no	—
281	D₄	#14523248	1	2	yes	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	8	1	no	—
286	C₂×C₄	—	127	1	no	—
287	C₂×C₄	—	127	1	no	—
288	C₂×C₄	—	127	1	no	—
289	C₂×C₄	—	127	1	no	—
290	C₂×C₄	—	1016	1	no	—
291	C₂×C₄	—	508	2	no	—
292	C₂×C₄	—	508	2	no	—
293	C₂×C₄	—	254	1	no	—
294	C₂×C₄	—	254	1	no	—
295	C₂×C₄	—	254	1	no	—
296	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
297	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	254	1	no	—
302	C₈	—	254	1	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	508	1	no	—
307	C₂²	—	508	1	no	—
308	C₂²	—	508	1	no	—
309	C₂²	C₂×D₄×D₁₂₇	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	127	2	no	—
316	C₄	—	508	1	no	—
317	C₄	—	508	1	no	—
318	C₄	—	508	1	no	—
319	C₄	—	254	1	no	—
320	C₄	—	254	1	no	—
321	C₄	—	254	1	no	—
322	C₄	C₂×#14523248	1	1	yes	characteristic
323	C₄	#16861256	1	1	yes	characteristic
324	C₂	—	4	2	no	—
325	C₂	—	1016	1	no	—
326	C₂	#22550535	1	1	yes	characteristic
327	C₂	#135×D₁₂₇	1	1	yes	characteristic
328	C₂	#22550682	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₂₇
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₄×SD₁₆×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	C₂×#14523233	C₂²
hypercenter	C₄×SD₁₆	D₁₂₇
hypocenter	C₁₂₇	#3360
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #22550598	C₂	semidirect
2. #22550585	C₂	semidirect
3. #22550232	C₂	semidirect
4. #22550231	C₂	semidirect
5. #22550225	C₂	semidirect
6. #22550186	C₂	semidirect
7. #22550183	C₂	semidirect
8. #22550165	C₂	semidirect
11. #22550079	C₂	semidirect
12. C₈×Dic₂₅₄	C₂	semidirect
13. #22549967	C₂	semidirect
15. C₄×SD₁₆×C₁₂₇	C₂	semidirect
165. C₁₂₇	#3360	semidirect
179. C₄×SD₁₆	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550598	2
#22550585	2
#22550232	2
#22550231	2
#22550225	2
#22550186	2
#22550183	2
#22550165	2
#22550093	2
#22550091	2
#22550079	2
C₈×Dic₂₅₄	2
#22549967	2
#22549956	2
C₄×SD₁₆×C₁₂₇	2
#3360	127