Subgroups — #30651382

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	#22550582	C₂	1	1	yes	maximal, characteristic, complemented
3	#22550233	C₂	1	1	yes	maximal, characteristic, complemented
4	#22550232	C₂	1	1	yes	maximal, characteristic, complemented
5	#22550225	C₂	1	1	yes	maximal, characteristic, complemented
6	#22550182	C₂	1	1	yes	maximal, characteristic, complemented
7	#22550165	C₂	1	1	yes	maximal, characteristic, complemented
8	#22550094	C₂	1	1	yes	maximal, characteristic
9	#22550092	C₂	1	1	yes	maximal, characteristic
10	#22550080	C₂	1	1	yes	maximal, characteristic, complemented
11	#22550000	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#14523235	C₂	1	1	yes	maximal, characteristic, complemented
13	#22549963	C₂	1	1	yes	maximal, characteristic, complemented
14	#22549958	C₂	1	1	yes	maximal, characteristic
15	#479×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14523237	C₂²	1	1	yes	characteristic
17	C₂×#14523235	C₂²	1	1	yes	characteristic
18	C₂×D₁₀₁₆	C₂²	1	1	yes	characteristic
19	#16861208	C₂²	1	1	yes	characteristic
20	#16861207	C₂²	1	1	yes	characteristic
21	#16861204	—	2	1	no	—
22	#16861203	—	2	1	no	—
23	#16861200	C₂²	1	1	yes	characteristic
24	#16861198	C₂²	1	1	yes	characteristic
25	#16861197	C₂²	1	1	yes	characteristic
26	#16861189	—	2	1	no	—
27	#16861187	—	2	1	no	—
28	#16861187	—	2	1	no	—
29	#16861187	—	2	1	no	—
30	C₄×Dic₂₅₄	C₂²	1	1	yes	characteristic
31	#16861174	C₂²	1	1	yes	characteristic
32	C₄×D₅₀₈	C₂²	1	1	yes	characteristic
33	#16861171	C₂²	1	1	yes	characteristic
34	#16861171	C₂²	1	1	yes	characteristic
35	#16861171	C₂²	1	1	yes	characteristic
36	#16861166	—	2	1	no	—
37	C₄⋊C₄×D₁₂₇	—	2	1	no	—
38	#16861140	C₂²	1	1	yes	characteristic
39	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
40	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
41	#16861134	C₂²	1	1	yes	characteristic
42	#16861133	C₂²	1	1	yes	characteristic
43	#16861132	C₂²	1	1	yes	characteristic
44	#16861132	C₂²	1	1	yes	characteristic
45	#16861131	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	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₂₇	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	D₄	1	1	yes	characteristic
61	C₂×Dic₂₅₄	C₂³	1	1	yes	characteristic
62	C₂×D₅₀₈	C₂³	1	1	yes	characteristic
63	C₂×D₅₀₈	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₁₂₇	—	2	1	no	—
65	C₂×C₄×D₁₂₇	—	2	1	no	—
66	C₂×C₄×D₁₂₇	—	2	1	no	—
67	C₂×C₄×D₁₂₇	—	2	1	no	—
68	#14523237	—	2	2	no	—
69	#14523235	—	2	2	no	—
70	D₁₀₁₆	—	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	D₄	1	1	yes	characteristic
75	C₄⋊Dic₁₂₇	C₂³	1	1	yes	characteristic
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	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
84	(C₂×D₁₂₇)⋊C₄	—	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	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
91	C₄×Dic₁₂₇	—	2	1	no	—
92	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
93	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₁₂₇	C₂³	1	1	yes	characteristic
95	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
96	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
97	C₄⋊C₄×C₁₂₇	—	2	1	no	—
98	C₄⋊C₄×C₁₂₇	—	2	1	no	—
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	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
103	C₄²×C₁₂₇	—	2	1	no	—
104	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
105	C₂²×D₁₂₇	—	2	1	no	—
106	C₂²×D₁₂₇	—	2	1	no	—
107	Dic₂₅₄	—	2	2	no	—
108	Dic₂₅₄	—	2	1	no	—
109	Dic₂₅₄	—	2	1	no	—
110	Dic₂₅₄	—	8	1	no	—
111	D₅₀₈	—	2	1	no	—
112	D₅₀₈	—	2	1	no	—
113	D₅₀₈	—	2	1	no	—
114	D₅₀₈	#41	1	2	yes	—
115	C₂×Dic₁₂₇	—	2	1	no	—
116	C₂×Dic₁₂₇	—	2	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	C₄×D₁₂₇	—	4	2	no	—
123	C₄×D₁₂₇	—	8	1	no	—
124	C₄×D₁₂₇	—	8	1	no	—
125	C₁₂₇⋊C₈	—	2	1	no	—
126	C₁₂₇⋊C₈	—	2	1	no	—
127	Q₈×C₁₂₇	—	2	1	no	—
128	Q₈×C₁₂₇	—	2	1	no	—
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	2	no	—
140	C₂×D₁₂₇	—	4	1	no	—
141	C₂×D₁₂₇	—	4	1	no	—
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	—	4	1	no	—
151	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
152	C₄×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
153	Dic₁₂₇	—	2	1	no	—
154	Dic₁₂₇	—	2	1	no	—
155	Dic₁₂₇	—	2	1	no	—
156	Dic₁₂₇	—	4	1	no	—
157	Dic₁₂₇	—	4	1	no	—
158	Dic₁₂₇	#135	1	2	yes	—
159	C₂×C₁₂₇	#548	1	1	yes	characteristic
160	C₂×C₁₂₇	C₂×#135	1	1	yes	characteristic
161	C₂×C₁₂₇	#577	1	1	yes	characteristic
162	D₁₂₇	—	4	2	no	—
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	D₁₂₇	1	1	yes	characteristic, complemented
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	—	127	1	no	—
180	C₄×D₈	—	127	1	no	—
181	C₂×SD₁₆	—	127	1	no	—
182	C₂×SD₁₆	—	127	1	no	—
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	C₂×D₁₂₇	1	1	yes	characteristic
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	—	127	1	no	—
203	#117	—	254	1	no	—
204	C₂×C₄⋊C₄	—	254	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	—	127	1	no	—
210	C₄⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic
211	#103	—	127	1	no	—
212	#103	—	127	1	no	—
213	#103	—	127	1	no	—
214	#103	—	127	1	no	—
215	#103	C₂×D₁₂₇	1	1	yes	characteristic
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	—	127	1	no	—
221	C₄×C₈	—	127	1	no	—
222	C₄×C₈	—	127	1	no	—
223	C₄×C₈	—	127	1	no	—
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	—	127	1	no	—
230	C₂²×C₄	—	254	1	no	—
231	C₂²×C₄	—	254	1	no	—
232	C₂²×C₄	—	254	1	no	—
233	C₂²×C₄	—	254	1	no	—
234	SD₁₆	—	254	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	—	2	1	no	—
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	127	1	no	—
245	C₄⋊C₄	—	127	1	no	—
246	C₄⋊C₄	—	127	1	no	—
247	C₄⋊C₄	—	127	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	—	254	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	—	254	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	—	127	2	no	—
279	D₄	—	254	1	no	—
280	D₄	—	254	1	no	—
281	D₄	—	254	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	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	—	1016	1	no	—
292	C₂×C₄	—	508	2	no	—
293	C₂×C₄	—	508	2	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	—	508	1	no	—
304	C₂²	—	508	1	no	—
305	C₂²	—	508	1	no	—
306	C₂²	—	508	1	no	—
307	C₂²	—	508	1	no	—
308	C₂²	—	254	2	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	—	4	1	no	—
316	C₄	—	127	2	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	#41×D₁₂₇	1	1	yes	characteristic
323	C₄	#16861257	1	1	yes	characteristic
324	C₂	—	1016	1	no	—
325	C₂	—	508	2	no	—
326	C₂	#22550536	1	1	yes	characteristic
327	C₂	#135×D₁₂₇	1	1	yes	characteristic
328	C₂	#22550685	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	C₂×D₁₀₁₆	C₂²
hypercenter	#479	D₁₂₇
hypocenter	C₁₂₇	#3360
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
1. #22550598	C₂	semidirect
2. #22550582	C₂	semidirect
3. #22550233	C₂	semidirect
4. #22550232	C₂	semidirect
5. #22550225	C₂	semidirect
6. #22550182	C₂	semidirect
7. #22550165	C₂	semidirect
10. #22550080	C₂	semidirect
11. #22550000	C₂	semidirect
12. C₄×#14523235	C₂	semidirect
13. #22549963	C₂	semidirect
15. #479×C₁₂₇	C₂	semidirect
165. C₁₂₇	#3360	semidirect
172. #479	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
#22550582	2
#22550233	2
#22550232	2
#22550225	2
#22550182	2
#22550165	2
#22550094	2
#22550092	2
#22550080	2
#22550000	2
C₄×#14523235	2
#22549963	2
#22549958	2
#479×C₁₂₇	2
#3360	127