Subgroups — #30648946

Order 16256.

372 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550488	C₂	1	1	yes	maximal, characteristic, complemented
2	#22550356	C₂	1	1	yes	maximal, characteristic
3	#22550212	C₂	1	1	yes	maximal, characteristic, complemented
4	#22550207	C₂	1	1	yes	maximal, characteristic, complemented
5	#22550184	C₂	1	1	yes	maximal, characteristic, complemented
6	#22550177	C₂	1	1	yes	maximal, characteristic, complemented
7	#22550087	C₂	1	1	yes	maximal, characteristic, complemented
8	#22550082	C₂	1	1	yes	maximal, characteristic, complemented
9	#22550040	C₂	1	1	yes	maximal, characteristic
10	#22550040	C₂	1	1	yes	maximal, characteristic
11	#22549984	C₂	1	1	yes	maximal, characteristic, complemented
12	#22549978	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×Dic₅₀₈	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#14523237	C₂	1	1	yes	maximal, characteristic, complemented
15	#433×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14523248	C₂²	1	1	yes	characteristic
17	C₂×Dic₅₀₈	C₂²	1	1	yes	characteristic
18	C₂×Dic₅₀₈	C₂²	1	1	yes	characteristic
19	C₂×#14523237	C₂²	1	1	yes	characteristic
20	C₂×#14523237	C₂²	1	1	yes	characteristic
21	#16861209	C₂²	1	1	yes	characteristic
22	#16861202	—	2	1	no	—
23	#16861202	—	2	1	no	—
24	#16861201	C₂²	1	1	yes	characteristic
25	#16861199	—	2	1	no	—
26	#16861199	—	2	1	no	—
27	#16861188	C₂²	1	1	yes	characteristic
28	#16861188	C₂²	1	1	yes	characteristic
29	C₄×Dic₂₅₄	C₂²	1	1	yes	characteristic
30	C₄×Dic₂₅₄	C₂²	1	1	yes	characteristic
31	C₄×Dic₂₅₄	C₂²	1	1	yes	characteristic
32	C₄×Dic₂₅₄	C₂²	1	1	yes	characteristic
33	C₄×D₅₀₈	C₂²	1	1	yes	characteristic
34	C₄×D₅₀₈	C₂²	1	1	yes	characteristic
35	(C₂²⋊D₁₂₇)⋊C₄	—	2	1	no	—
36	(C₂²⋊D₁₂₇)⋊C₄	—	2	1	no	—
37	#16861167	—	2	1	no	—
38	C₂×C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
39	C₄⋊C₄×D₁₂₇	—	2	1	no	—
40	#16861143	C₂²	1	1	yes	characteristic
41	#16861143	C₂²	1	1	yes	characteristic
42	#16861141	C₂²	1	1	yes	characteristic
43	#16861141	C₂²	1	1	yes	characteristic
44	#16861134	C₂²	1	1	yes	characteristic
45	#16861134	C₂²	1	1	yes	characteristic
46	#16861134	C₂²	1	1	yes	characteristic
47	#16861134	C₂²	1	1	yes	characteristic
48	#16861134	C₂²	1	1	yes	characteristic
49	#16861134	C₂²	1	1	yes	characteristic
50	#16861131	C₂²	1	1	yes	characteristic
51	#16861131	C₂²	1	1	yes	characteristic
52	#117×C₁₂₇	C₂²	1	1	yes	characteristic
53	C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
54	C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
55	C₂²⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
57	#97×C₁₂₇	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
59	#14523248	—	2	2	no	—
60	#14523248	—	2	2	no	—
61	C₂×Dic₂₅₄	C₂³	1	1	yes	characteristic
62	C₂×Dic₂₅₄	C₂³	1	1	yes	characteristic
63	C₂×Dic₂₅₄	C₂³	1	1	yes	characteristic
64	C₂×D₅₀₈	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊D₁₂₇	—	2	1	no	—
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	Dic₅₀₈	—	2	2	no	—
71	Dic₅₀₈	—	2	2	no	—
72	#14523237	—	2	2	no	—
73	#14523237	—	2	2	no	—
74	C₄⋊Dic₁₂₇	—	2	1	no	—
75	C₄⋊Dic₁₂₇	—	2	1	no	—
76	C₄⋊Dic₁₂₇	—	2	1	no	—
77	C₄⋊Dic₁₂₇	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	Dic₁₂₇⋊C₄	—	2	1	no	—
87	Dic₁₂₇⋊C₄	—	2	1	no	—
88	Dic₁₂₇⋊C₄	—	2	1	no	—
89	C₂²⋊Dic₁₂₇	—	2	1	no	—
90	C₂²⋊Dic₁₂₇	—	2	1	no	—
91	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
92	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
93	C₄×Dic₁₂₇	—	2	1	no	—
94	C₄×Dic₁₂₇	—	2	1	no	—
95	C₄×Dic₁₂₇	—	2	1	no	—
96	C₄×Dic₁₂₇	—	2	1	no	—
97	C₂²×C₄×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	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₂₇	D₄	1	1	yes	characteristic
103	C₄⋊C₄×C₁₂₇	D₄	1	1	yes	characteristic
104	C₂²⋊C₄×C₁₂₇	D₄	1	1	yes	characteristic
105	C₂²⋊C₄×C₁₂₇	D₄	1	1	yes	characteristic
106	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
107	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
108	C₂²×D₁₂₇	—	2	1	no	—
109	Dic₂₅₄	—	2	1	no	—
110	Dic₂₅₄	—	2	1	no	—
111	Dic₂₅₄	—	2	1	no	—
112	Dic₂₅₄	—	2	1	no	—
113	Dic₂₅₄	—	2	1	no	—
114	Dic₂₅₄	#41	1	2	yes	—
115	D₅₀₈	—	2	1	no	—
116	D₅₀₈	#41	1	2	yes	—
117	C₂²⋊D₁₂₇	—	2	4	no	—
118	C₂×Dic₁₂₇	—	2	1	no	—
119	C₂×Dic₁₂₇	—	2	1	no	—
120	C₂×Dic₁₂₇	—	2	1	no	—
121	C₂×Dic₁₂₇	—	2	1	no	—
122	C₂×Dic₁₂₇	—	2	1	no	—
123	C₂×Dic₁₂₇	—	2	1	no	—
124	C₂×Dic₁₂₇	—	2	1	no	—
125	C₂×Dic₁₂₇	—	8	1	no	—
126	C₄×D₁₂₇	—	2	4	no	—
127	C₄×D₁₂₇	—	8	1	no	—
128	C₄×D₁₂₇	—	8	1	no	—
129	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₂₇	—	2	1	no	—
131	C₂×C₄×C₁₂₇	—	2	1	no	—
132	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
138	C₈×C₁₂₇	—	2	1	no	—
139	C₈×C₁₂₇	—	2	1	no	—
140	C₈×C₁₂₇	—	2	1	no	—
141	C₈×C₁₂₇	—	2	1	no	—
142	C₂²×C₁₂₇	—	2	1	no	—
143	C₂²×C₁₂₇	—	2	1	no	—
144	C₂²×C₁₂₇	—	2	1	no	—
145	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
146	C₂×D₁₂₇	—	2	2	no	—
147	C₂×D₁₂₇	—	4	1	no	—
148	C₂×D₁₂₇	—	4	1	no	—
149	C₄×C₁₂₇	—	2	1	no	—
150	C₄×C₁₂₇	—	2	1	no	—
151	C₄×C₁₂₇	—	2	1	no	—
152	C₄×C₁₂₇	—	2	1	no	—
153	C₄×C₁₂₇	—	2	1	no	—
154	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
155	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
156	Dic₁₂₇	—	2	2	no	—
157	Dic₁₂₇	—	4	1	no	—
158	Dic₁₂₇	—	4	1	no	—
159	Dic₁₂₇	—	4	1	no	—
160	Dic₁₂₇	—	4	1	no	—
161	Dic₁₂₇	—	4	1	no	—
162	Dic₁₂₇	—	4	1	no	—
163	C₂×C₁₂₇	—	4	1	no	—
164	C₂×C₁₂₇	#525	1	1	yes	characteristic
165	C₂×C₁₂₇	#577	1	1	yes	characteristic
166	C₂×C₁₂₇	#578	1	1	yes	characteristic
167	D₁₂₇	—	4	2	no	—
168	#3120	—	127	1	no	maximal, Sylow
169	C₁₂₇	#3120	1	1	yes	characteristic, complemented, Sylow
170	#541	—	127	1	no	—
171	#520	—	127	1	no	—
172	#484	—	127	1	no	—
173	#483	—	127	1	no	—
174	#479	—	127	1	no	—
175	#478	—	127	1	no	—
176	#462	—	127	1	no	—
177	#461	—	127	1	no	—
178	#452	—	127	1	no	—
179	#452	—	127	1	no	—
180	#441	—	127	1	no	—
181	#440	—	127	1	no	—
182	C₄×Q₁₆	—	127	1	no	—
183	C₄×SD₁₆	—	127	1	no	—
184	#433	D₁₂₇	1	1	yes	characteristic, complemented
185	C₂×#41	—	127	1	no	—
186	C₂×Q₁₆	—	127	1	no	—
187	C₂×Q₁₆	—	127	1	no	—
188	C₂×SD₁₆	—	127	1	no	—
189	C₂×SD₁₆	—	127	1	no	—
190	C₄⋊Q₈	—	127	1	no	—
191	#126	—	254	1	no	—
192	#126	—	254	1	no	—
193	#125	—	127	1	no	—
194	#125	—	254	1	no	—
195	#125	—	254	1	no	—
196	C₂²⋊Q₈	—	127	1	no	—
197	C₂²⋊Q₈	—	127	1	no	—
198	C₄×Q₈	—	127	1	no	—
199	C₄×Q₈	—	127	1	no	—
200	C₄×Q₈	—	127	1	no	—
201	C₄×Q₈	—	127	1	no	—
202	C₄×D₄	—	127	1	no	—
203	C₄×D₄	—	127	1	no	—
204	C₄×D₄	—	254	1	no	—
205	C₄×D₄	—	254	1	no	—
206	#117	—	254	1	no	—
207	#117	C₂×D₁₂₇	1	1	yes	characteristic
208	C₂×C₄⋊C₄	—	127	1	no	—
209	C₂×C₄⋊C₄	—	254	1	no	—
210	#107	—	127	1	no	—
211	#107	—	127	1	no	—
212	#106	—	127	1	no	—
213	#106	—	127	1	no	—
214	C₄⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic
215	C₄⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic
216	#103	—	127	1	no	—
217	#103	—	127	1	no	—
218	#103	—	127	1	no	—
219	#103	—	127	1	no	—
220	#103	—	127	1	no	—
221	#103	—	127	1	no	—
222	#102	—	127	1	no	—
223	#102	—	127	1	no	—
224	C₂²⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic
225	C₂²⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic
226	#97	C₂×D₁₂₇	1	1	yes	characteristic
227	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
228	#41	—	254	2	no	—
229	#41	—	254	2	no	—
230	C₂×Q₈	—	127	1	no	—
231	C₂×Q₈	—	127	1	no	—
232	C₂×Q₈	—	127	1	no	—
233	C₂×D₄	—	127	1	no	—
234	C₂×D₄	—	254	1	no	—
235	C₂²×C₄	—	254	1	no	—
236	C₂²×C₄	—	254	1	no	—
237	C₂²×C₄	—	254	1	no	—
238	C₂²×C₄	—	254	1	no	—
239	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
240	Q₁₆	—	254	2	no	—
241	Q₁₆	—	254	2	no	—
242	SD₁₆	—	254	2	no	—
243	SD₁₆	—	254	2	no	—
244	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
245	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
246	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
248	C₄⋊C₄	—	127	1	no	—
249	C₄⋊C₄	—	127	1	no	—
250	C₄⋊C₄	—	127	1	no	—
251	C₄⋊C₄	—	127	1	no	—
252	C₄⋊C₄	—	254	1	no	—
253	C₄⋊C₄	—	254	1	no	—
254	C₄⋊C₄	—	254	1	no	—
255	C₄⋊C₄	—	254	1	no	—
256	C₄⋊C₄	—	254	1	no	—
257	C₄⋊C₄	—	254	1	no	—
258	C₄⋊C₄	—	254	1	no	—
259	C₄⋊C₄	—	254	1	no	—
260	C₄⋊C₄	—	254	1	no	—
261	C₄⋊C₄	—	254	1	no	—
262	C₄⋊C₄	—	254	1	no	—
263	C₄⋊C₄	D₅₀₈	1	1	yes	characteristic
264	C₄⋊C₄	D₅₀₈	1	1	yes	characteristic
265	C₂²⋊C₄	—	254	1	no	—
266	C₂²⋊C₄	—	254	1	no	—
267	C₂²⋊C₄	—	254	1	no	—
268	C₂²⋊C₄	—	254	1	no	—
269	C₂²⋊C₄	D₅₀₈	1	1	yes	characteristic
270	C₂²⋊C₄	D₅₀₈	1	1	yes	characteristic
271	C₄²	—	254	1	no	—
272	C₄²	—	254	1	no	—
273	C₄²	—	254	1	no	—
274	C₄²	—	254	1	no	—
275	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
276	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
277	C₂³	—	254	1	no	—
278	C₂³	C₂×D₅₀₈	1	1	yes	characteristic
279	Q₈	—	127	2	no	—
280	Q₈	—	254	1	no	—
281	Q₈	—	254	1	no	—
282	Q₈	—	254	1	no	—
283	Q₈	—	254	1	no	—
284	Q₈	—	254	1	no	—
285	D₄	—	127	2	no	—
286	D₄	—	254	4	no	—
287	D₄	—	254	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	1016	1	no	—
291	C₂×C₄	—	1016	1	no	—
292	C₂×C₄	—	1016	1	no	—
293	C₂×C₄	—	254	4	no	—
294	C₂×C₄	—	254	1	no	—
295	C₂×C₄	—	254	1	no	—
296	C₂×C₄	—	254	1	no	—
297	C₂×C₄	—	254	1	no	—
298	C₂×C₄	—	254	1	no	—
299	C₂×C₄	—	254	1	no	—
300	C₂×C₄	—	254	1	no	—
301	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
302	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
303	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
304	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
305	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	508	1	no	—
315	C₂²	—	508	1	no	—
316	C₂²	—	254	2	no	—
317	C₂²	C₂²×D₅₀₈	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	508	1	no	—
324	C₄	—	508	1	no	—
325	C₄	—	508	1	no	—
326	C₄	—	508	1	no	—
327	C₄	—	508	1	no	—
328	C₄	—	508	1	no	—
329	C₄	—	254	2	no	—
330	C₄	#41×D₁₂₇	1	1	yes	characteristic
331	C₄	#41×D₁₂₇	1	1	yes	characteristic
332	C₂	—	4	1	no	—
333	C₂	—	508	2	no	—
334	C₂	#22550388	1	1	yes	characteristic
335	C₂	#22550688	1	1	yes	characteristic
336	C₂	#22550693	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₀₈
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#433×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	C₂×#14523248	C₂²
hypercenter	#433	D₁₂₇
hypocenter	C₁₂₇	#3120
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #433

Complements

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

N	H ≅ G/N	type
1. #22550488	C₂	semidirect
3. #22550212	C₂	semidirect
4. #22550207	C₂	semidirect
5. #22550184	C₂	semidirect
6. #22550177	C₂	semidirect
7. #22550087	C₂	semidirect
8. #22550082	C₂	semidirect
11. #22549984	C₂	semidirect
12. #22549978	C₂	semidirect
13. C₄×Dic₅₀₈	C₂	semidirect
14. C₄×#14523237	C₂	semidirect
15. #433×C₁₂₇	C₂	semidirect
169. C₁₂₇	#3120	semidirect
184. #433	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550488	2
#22550356	2
#22550212	2
#22550207	2
#22550184	2
#22550177	2
#22550087	2
#22550082	2
#22550040	2
#22549984	2
#22549978	2
C₄×Dic₅₀₈	2
C₄×#14523237	2
#433×C₁₂₇	2
#3120	127