Subgroups — #30648516

Order 16256.

388 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550429	C₂	1	1	yes	maximal, characteristic, complemented
2	#22550344	C₂	1	1	yes	maximal, characteristic
3	#22550186	C₂	1	1	yes	maximal, characteristic, complemented
4	#22550182	C₂	1	1	yes	maximal, characteristic, complemented
5	#22550179	C₂	1	1	yes	maximal, characteristic, complemented
6	#22550175	C₂	1	1	yes	maximal, characteristic, complemented
7	#22550170	C₂	1	1	yes	maximal, characteristic, complemented
8	#22550165	C₂	1	1	yes	maximal, characteristic, complemented
9	#22550160	C₂	1	1	yes	maximal, characteristic, complemented
10	#22550155	C₂	1	1	yes	maximal, characteristic, complemented
11	#22549912	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
13	#22549873	C₂	1	1	yes	maximal, characteristic
14	#22549870	C₂	1	1	yes	maximal, characteristic
15	#427×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14523248	C₂²	1	1	yes	characteristic
17	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
18	#16861208	C₂²	1	1	yes	characteristic
19	#16861208	C₂²	1	1	yes	characteristic
20	#16861200	C₂²	1	1	yes	characteristic
21	#16861200	C₂²	1	1	yes	characteristic
22	#16861186	—	2	1	no	—
23	#16861186	—	2	1	no	—
24	C₄×Dic₂₅₄	—	2	1	no	—
25	C₄×Dic₂₅₄	—	2	1	no	—
26	#16861174	C₂²	1	1	yes	characteristic
27	#16861174	C₂²	1	1	yes	characteristic
28	#16861171	C₂²	1	1	yes	characteristic
29	#16861171	C₂²	1	1	yes	characteristic
30	(C₂²⋊D₁₂₇)⋊C₄	—	2	1	no	—
31	(C₂²⋊D₁₂₇)⋊C₄	—	2	1	no	—
32	#16861168	C₂²	1	1	yes	characteristic
33	#16861166	—	2	1	no	—
34	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
35	C₄²×D₁₂₇	—	2	1	no	—
36	#16861142	C₂²	1	1	yes	characteristic
37	#16861142	C₂²	1	1	yes	characteristic
38	#16861140	C₂²	1	1	yes	characteristic
39	#16861140	C₂²	1	1	yes	characteristic
40	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
41	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
42	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
43	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
44	#16861134	C₂²	1	1	yes	characteristic
45	#16861134	C₂²	1	1	yes	characteristic
46	#16861132	C₂²	1	1	yes	characteristic
47	#16861132	C₂²	1	1	yes	characteristic
48	#16861131	C₂²	1	1	yes	characteristic
49	#16861131	C₂²	1	1	yes	characteristic
50	#16861129	C₂²	1	1	yes	characteristic
51	#16861129	C₂²	1	1	yes	characteristic
52	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
53	#117×C₁₂₇	C₂²	1	1	yes	characteristic
54	#117×C₁₂₇	C₂²	1	1	yes	characteristic
55	#107×C₁₂₇	C₂²	1	1	yes	characteristic
56	#107×C₁₂₇	C₂²	1	1	yes	characteristic
57	#106×C₁₂₇	C₂²	1	1	yes	characteristic
58	#106×C₁₂₇	C₂²	1	1	yes	characteristic
59	#14523248	—	2	2	no	—
60	#14523248	Q₈	1	4	yes	complemented
61	C₂×Dic₂₅₄	—	2	1	no	—
62	C₂×Dic₂₅₄	—	2	1	no	—
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	D₄	1	1	yes	characteristic
67	C₂²×Dic₁₂₇	D₄	1	1	yes	characteristic
68	C₂×C₄×D₁₂₇	—	2	1	no	—
69	C₂×C₄×D₁₂₇	—	2	1	no	—
70	C₂×C₄×D₁₂₇	—	2	1	no	—
71	C₂×C₁₂₇⋊C₈	—	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	—	2	1	no	—
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	C₂²⋊Dic₁₂₇	—	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	D₄	1	1	yes	characteristic
91	C₄×Dic₁₂₇	D₄	1	1	yes	characteristic
92	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
93	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
95	C₂×C₈×C₁₂₇	—	2	2	no	—
96	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
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	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
104	C₄²×C₁₂₇	—	2	1	no	—
105	C₄²×C₁₂₇	—	2	1	no	—
106	C₂²×D₁₂₇	—	2	1	no	—
107	Dic₂₅₄	—	2	1	no	—
108	Dic₂₅₄	—	4	2	no	—
109	Dic₂₅₄	—	4	2	no	—
110	Dic₂₅₄	C₂×Q₈	1	2	yes	—
111	D₅₀₈	—	2	1	no	—
112	D₅₀₈	C₂×Q₈	1	2	yes	—
113	C₂²⋊D₁₂₇	—	2	4	no	—
114	C₂×Dic₁₂₇	—	2	1	no	—
115	C₂×Dic₁₂₇	—	2	2	no	—
116	C₂×Dic₁₂₇	—	2	1	no	—
117	C₂×Dic₁₂₇	—	2	1	no	—
118	C₂×Dic₁₂₇	—	4	1	no	—
119	C₂×Dic₁₂₇	C₂×D₄	1	1	yes	characteristic
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	—	4	2	no	—
125	C₄×D₁₂₇	—	4	2	no	—
126	C₁₂₇⋊C₈	—	2	1	no	—
127	C₁₂₇⋊C₈	—	2	1	no	—
128	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
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×Q₈	1	2	yes	—
135	C₂×C₄×C₁₂₇	#41	1	2	yes	—
136	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
137	C₈×C₁₂₇	—	2	1	no	—
138	C₈×C₁₂₇	—	2	1	no	—
139	C₂²×C₁₂₇	—	2	1	no	—
140	C₂²×C₁₂₇	—	2	1	no	—
141	C₂²×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
142	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
143	C₂×D₁₂₇	—	2	2	no	—
144	C₂×D₁₂₇	—	4	1	no	—
145	C₂×D₁₂₇	—	4	1	no	—
146	C₄×C₁₂₇	—	4	1	no	—
147	C₄×C₁₂₇	—	4	1	no	—
148	C₄×C₁₂₇	—	4	1	no	—
149	C₄×C₁₂₇	—	4	1	no	—
150	C₄×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
151	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
152	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
153	Dic₁₂₇	—	2	2	no	—
154	Dic₁₂₇	—	2	1	no	—
155	Dic₁₂₇	—	2	1	no	—
156	Dic₁₂₇	—	2	1	no	—
157	Dic₁₂₇	—	4	1	no	—
158	Dic₁₂₇	—	4	1	no	—
159	Dic₁₂₇	#135	1	2	yes	—
160	C₂×C₁₂₇	—	2	2	no	—
161	C₂×C₁₂₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
162	C₂×C₁₂₇	C₂×#135	1	1	yes	characteristic
163	C₂×C₁₂₇	#575	1	1	yes	characteristic
164	D₁₂₇	—	4	2	no	—
165	#3076	—	127	1	no	maximal, Sylow
166	C₁₂₇	#3076	1	1	yes	characteristic, complemented, Sylow
167	#533	—	127	1	no	—
168	C₄×#41	—	127	1	no	—
169	#479	—	127	1	no	—
170	#479	—	127	1	no	—
171	#478	—	127	1	no	—
172	#478	—	127	1	no	—
173	#477	—	127	1	no	—
174	#476	—	127	1	no	—
175	#475	—	127	1	no	—
176	#474	—	127	1	no	—
177	#427	—	127	1	no	—
178	#427	D₁₂₇	1	1	yes	characteristic, complemented
179	C₂×C₄⋊C₈	—	127	1	no	—
180	#416	—	127	1	no	—
181	#416	—	127	1	no	—
182	C₂×#41	—	127	1	no	—
183	C₂²×C₈	—	127	1	no	—
184	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
185	C₄⋊Q₈	—	127	1	no	—
186	C₄⋊Q₈	—	127	1	no	—
187	#125	—	127	1	no	—
188	#125	—	127	1	no	—
189	C₂²⋊Q₈	—	254	1	no	—
190	C₂²⋊Q₈	—	254	1	no	—
191	C₄×Q₈	—	127	1	no	—
192	C₄×Q₈	—	127	1	no	—
193	C₄×Q₈	—	254	1	no	—
194	C₄×Q₈	—	254	1	no	—
195	C₄×D₄	—	127	1	no	—
196	C₄×D₄	—	127	1	no	—
197	C₄×D₄	—	254	1	no	—
198	C₄×D₄	—	254	1	no	—
199	#117	—	127	1	no	—
200	#117	—	254	1	no	—
201	#117	C₂×D₁₂₇	1	1	yes	characteristic
202	#117	C₂×D₁₂₇	1	1	yes	characteristic
203	C₂×C₄²	—	127	1	no	—
204	C₂×C₄²	—	254	1	no	—
205	#107	—	127	1	no	—
206	#107	—	127	1	no	—
207	#107	C₂×D₁₂₇	1	1	yes	characteristic
208	#107	C₂×D₁₂₇	1	1	yes	characteristic
209	#106	—	127	1	no	—
210	#106	—	127	1	no	—
211	#106	C₂×D₁₂₇	1	1	yes	characteristic
212	#106	C₂×D₁₂₇	1	1	yes	characteristic
213	C₄⋊C₈	—	127	1	no	—
214	C₄⋊C₈	—	127	1	no	—
215	C₄⋊C₈	—	127	1	no	—
216	C₄⋊C₈	—	127	1	no	—
217	#103	—	127	1	no	—
218	#103	—	127	1	no	—
219	#103	—	127	1	no	—
220	#103	—	127	1	no	—
221	#102	—	127	1	no	—
222	#102	—	127	1	no	—
223	#102	—	127	1	no	—
224	#102	—	127	1	no	—
225	#41	—	127	4	no	—
226	#41	—	254	2	no	—
227	C₂×Q₈	—	127	1	no	—
228	C₂×Q₈	—	254	1	no	—
229	C₂×Q₈	—	254	1	no	—
230	C₂×D₄	—	127	1	no	—
231	C₂×D₄	—	254	1	no	—
232	C₂²×C₄	—	127	1	no	—
233	C₂²×C₄	—	127	1	no	—
234	C₂²×C₄	—	254	1	no	—
235	C₂²×C₄	—	254	1	no	—
236	C₂²×C₄	—	254	1	no	—
237	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	127	1	no	—
240	C₂×C₈	—	127	1	no	—
241	C₂×C₈	—	254	2	no	—
242	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
244	C₄⋊C₄	—	127	1	no	—
245	C₄⋊C₄	—	127	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	C₄⋊C₄	C₂²×D₁₂₇	1	1	yes	characteristic
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	254	1	no	—
261	C₂²⋊C₄	—	254	1	no	—
262	C₂²⋊C₄	—	254	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	127	1	no	—
266	C₄²	—	127	1	no	—
267	C₄²	—	127	1	no	—
268	C₄²	—	127	1	no	—
269	C₄²	—	254	1	no	—
270	C₄²	—	254	1	no	—
271	C₄²	—	254	1	no	—
272	C₂³	—	254	1	no	—
273	C₂³	D₄×D₁₂₇	1	1	yes	characteristic
274	Q₈	—	127	2	no	—
275	Q₈	—	508	2	no	—
276	Q₈	—	508	2	no	—
277	Q₈	—	254	1	no	—
278	D₄	—	127	2	no	—
279	D₄	—	254	4	no	—
280	D₄	—	254	1	no	—
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	—	127	1	no	—
289	C₂×C₄	—	508	2	no	—
290	C₂×C₄	—	508	2	no	—
291	C₂×C₄	—	508	1	no	—
292	C₂×C₄	—	254	2	no	—
293	C₂×C₄	—	254	1	no	—
294	C₂×C₄	—	254	1	no	—
295	C₂×C₄	—	254	1	no	—
296	C₂×C₄	—	254	4	no	—
297	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
298	C₂×C₄	Q₈×D₁₂₇	1	2	yes	—
299	C₂×C₄	#14523248	1	2	yes	—
300	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	254	1	no	—
304	C₈	—	254	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	508	1	no	—
308	C₂²	—	508	1	no	—
309	C₂²	—	254	2	no	—
310	C₂²	#16861187	1	2	yes	—
311	C₂²	C₂×D₄×D₁₂₇	1	1	yes	characteristic
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	2	no	—
320	C₄	—	254	1	no	—
321	C₄	—	254	1	no	—
322	C₄	—	254	1	no	—
323	C₄	#16861187	1	2	yes	—
324	C₄	C₂×Q₈×D₁₂₇	1	1	yes	characteristic
325	C₄	C₂×#14523248	1	1	yes	characteristic
326	C₂	—	2	2	no	—
327	C₂	—	508	2	no	—
328	C₂	C₂×#16861187	1	1	yes	characteristic
329	C₂	#135×D₁₂₇	1	1	yes	characteristic
330	C₂	#22550673	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₂₇
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#427×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	C₂×#14523248	C₂²
hypercenter	#427	D₁₂₇
hypocenter	C₁₂₇	#3076
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

Complements

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

N	H ≅ G/N	type
1. #22550429	C₂	semidirect
3. #22550186	C₂	semidirect
4. #22550182	C₂	semidirect
5. #22550179	C₂	semidirect
6. #22550175	C₂	semidirect
7. #22550170	C₂	semidirect
8. #22550165	C₂	semidirect
9. #22550160	C₂	semidirect
10. #22550155	C₂	semidirect
11. #22549912	C₂	semidirect
12. C₂×Dic₁₂₇⋊C₈	C₂	semidirect
15. #427×C₁₂₇	C₂	semidirect
60. #14523248	Q₈	semidirect
166. C₁₂₇	#3076	semidirect
178. #427	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550429	2
#22550344	2
#22550186	2
#22550182	2
#22550179	2
#22550175	2
#22550170	2
#22550165	2
#22550160	2
#22550155	2
#22549912	2
C₂×Dic₁₂₇⋊C₈	2
#22549873	2
#22549870	2
#427×C₁₂₇	2
#3076	127