Subgroups — #30647317

Order 16256.

492 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×D₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#16861167	C₂	1	1	yes	maximal, characteristic
3	#117×D₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#22549919	C₂	1	1	yes	maximal, characteristic, complemented
5	#22549918	C₂	1	1	yes	maximal, characteristic, complemented
6	#22549917	C₂	1	2	yes	maximal, complemented
7	#22549916	C₂	1	1	yes	maximal, characteristic, complemented
8	#22549915	C₂	1	2	yes	maximal, complemented
9	#22549913	C₂	1	2	yes	maximal, complemented
10	#22549911	C₂	1	2	yes	maximal, complemented
11	#428×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×#14523232	C₂²	1	1	yes	characteristic
14	C₂×#14523231	C₂²	1	2	yes	—
15	M₁₆×D₁₂₇	C₄	1	8	yes	complemented
16	C₂×C₈×D₁₂₇	C₂²	1	2	yes	—
17	#16861168	C₂²	1	1	yes	characteristic
18	#16861167	—	2	1	no	—
19	#16861167	—	2	1	no	—
20	#16861167	C₂²	1	1	yes	characteristic
21	#16861167	C₂²	1	1	yes	characteristic
22	#16861167	C₂²	1	2	yes	—
23	#16861167	C₂²	1	2	yes	—
24	#16861166	—	2	1	no	—
25	#16861165	—	2	1	no	—
26	C₂×C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×D₁₂₇	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×D₁₂₇	C₂²	1	1	yes	characteristic
29	C₂×(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
30	C₂²⋊C₄×D₁₂₇	—	2	1	no	—
31	C₂×C₄×Dic₁₂₇	—	2	1	no	—
32	C₄²×D₁₂₇	—	2	1	no	—
33	#16861143	C₂²	1	2	yes	complemented
34	#16861142	C₂²	1	2	yes	complemented
35	#16861142	C₂²	1	2	yes	complemented
36	#16861141	C₂²	1	2	yes	complemented
37	#16861140	C₂²	1	2	yes	complemented
38	#16861140	C₂²	1	2	yes	complemented
39	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
40	#117×C₁₂₇	C₂²	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
42	#107×C₁₂₇	C₂²	1	2	yes	complemented
43	#106×C₁₂₇	C₂²	1	2	yes	complemented
44	C₂³×D₁₂₇	D₄	1	1	yes	characteristic
45	C₂²×Dic₁₂₇	—	2	1	no	—
46	C₂²×Dic₁₂₇	D₄	1	1	yes	characteristic
47	C₂×C₄×D₁₂₇	—	2	1	no	—
48	C₂×C₄×D₁₂₇	—	2	1	no	—
49	C₂×C₄×D₁₂₇	C₂×C₄	1	4	yes	—
50	C₂×C₄×D₁₂₇	D₄	1	1	yes	characteristic
51	C₂×C₄×D₁₂₇	D₄	1	1	yes	characteristic
52	C₂×C₄×D₁₂₇	Q₈	1	4	yes	—
53	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
54	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
55	#14523232	C₂×C₄	1	4	yes	complemented
56	#14523231	C₂×C₄	1	8	yes	complemented
57	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
58	C₈×D₁₂₇	C₂×C₄	1	8	yes	complemented
59	C₄⋊Dic₁₂₇	—	2	1	no	—
60	C₄⋊Dic₁₂₇	C₂³	1	1	yes	characteristic
61	C₄⋊Dic₁₂₇	C₂³	1	2	yes	—
62	C₄⋊Dic₁₂₇	C₂³	1	1	yes	characteristic
63	Dic₁₂₇⋊C₄	—	2	1	no	—
64	Dic₁₂₇⋊C₄	—	2	1	no	—
65	C₂²⋊Dic₁₂₇	—	2	1	no	—
66	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
67	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
68	(C₂×D₁₂₇)⋊C₄	—	2	2	no	—
69	(C₂×D₁₂₇)⋊C₄	—	2	2	no	—
70	C₄×Dic₁₂₇	—	2	1	no	—
71	C₄×Dic₁₂₇	—	2	1	no	—
72	C₄×Dic₁₂₇	—	2	2	no	—
73	C₄×Dic₁₂₇	—	2	2	no	—
74	C₄×Dic₁₂₇	—	2	1	no	—
75	C₂²×C₄×C₁₂₇	—	2	1	no	—
76	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
77	M₁₆×C₁₂₇	C₂×C₄	1	4	yes	complemented
78	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
79	C₄⋊C₄×C₁₂₇	—	2	1	no	—
80	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
82	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
83	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
84	C₄²×C₁₂₇	—	2	1	no	—
85	C₂²×D₁₂₇	—	4	1	no	—
86	C₂²×D₁₂₇	—	4	1	no	—
87	C₂²×D₁₂₇	C₄⋊C₄	1	4	yes	—
88	C₂²×D₁₂₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×D₁₂₇	C₂×D₄	1	1	yes	characteristic
90	C₂×Dic₁₂₇	—	2	2	no	—
91	C₂×Dic₁₂₇	—	2	1	no	—
92	C₂×Dic₁₂₇	—	2	1	no	—
93	C₂×Dic₁₂₇	—	4	2	no	—
94	C₂×Dic₁₂₇	C₄⋊C₄	1	4	yes	—
95	C₂×Dic₁₂₇	C₂×D₄	1	1	yes	characteristic
96	C₂×Dic₁₂₇	C₂×D₄	1	1	yes	characteristic
97	C₄×D₁₂₇	—	4	2	no	—
98	C₄×D₁₂₇	—	4	2	no	—
99	C₄×D₁₂₇	C₄⋊C₄	1	4	yes	—
100	C₄×D₁₂₇	C₄⋊C₄	1	4	yes	—
101	C₄×D₁₂₇	C₂²×C₄	1	2	yes	—
102	C₄×D₁₂₇	C₂²×C₄	1	2	yes	—
103	C₄×D₁₂₇	C₂×Q₈	1	2	yes	—
104	C₄×D₁₂₇	C₂×Q₈	1	2	yes	—
105	C₁₂₇⋊C₈	C₂²×C₄	1	4	yes	—
106	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₂₇	—	2	1	no	—
108	C₂×C₄×C₁₂₇	—	2	1	no	—
109	C₂×C₄×C₁₂₇	—	2	2	no	—
110	C₂×C₄×C₁₂₇	—	4	2	no	—
111	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
112	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
114	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
115	C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
116	C₂²×C₁₂₇	—	2	1	no	—
117	C₂²×C₁₂₇	—	2	1	no	—
118	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
119	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×D₁₂₇	—	2	1	no	—
121	C₂×D₁₂₇	—	2	1	no	—
122	C₂×D₁₂₇	—	2	1	no	—
123	C₂×D₁₂₇	—	2	1	no	—
124	C₂×D₁₂₇	—	4	4	no	—
125	C₂×D₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×D₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
127	C₄×C₁₂₇	—	4	2	no	—
128	C₄×C₁₂₇	—	4	1	no	—
129	C₄×C₁₂₇	—	4	1	no	—
130	C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
131	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
132	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
133	Dic₁₂₇	—	4	1	no	—
134	Dic₁₂₇	—	4	1	no	—
135	Dic₁₂₇	—	4	2	no	—
136	Dic₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
137	Dic₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
138	C₂×C₁₂₇	—	2	2	no	—
139	C₂×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₁₂₇	#575	1	1	yes	characteristic
141	C₂×C₁₂₇	#575	1	1	yes	characteristic
142	D₁₂₇	—	2	2	no	—
143	D₁₂₇	—	2	2	no	—
144	#2911	—	127	1	no	maximal, Sylow
145	C₁₂₇	#2911	1	1	yes	characteristic, complemented, Sylow
146	C₂²×M₁₆	—	127	1	no	—
147	C₂×#117	—	127	1	no	—
148	C₂×#117	—	127	1	no	—
149	#428	—	127	1	no	—
150	#428	—	127	1	no	—
151	#428	—	127	1	no	—
152	#428	—	127	2	no	—
153	#428	—	127	2	no	—
154	#428	D₁₂₇	1	1	yes	characteristic, complemented
155	#427	—	127	2	no	—
156	#427	—	127	2	no	—
157	C₂³×C₄	—	127	1	no	—
158	C₂×M₁₆	—	127	1	no	—
159	C₂×M₁₆	—	127	8	no	—
160	C₂×M₁₆	—	127	2	no	—
161	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
162	C₂²×C₈	—	127	2	no	—
163	#117	—	127	1	no	—
164	#117	—	127	1	no	—
165	#117	—	127	2	no	—
166	#117	—	127	1	no	—
167	#117	—	127	2	no	—
168	#117	—	254	1	no	—
169	#117	—	254	1	no	—
170	#117	—	254	1	no	—
171	#117	—	254	1	no	—
172	#117	C₂×D₁₂₇	1	1	yes	characteristic
173	C₂×C₄⋊C₄	—	127	1	no	—
174	C₂×C₄⋊C₄	—	127	1	no	—
175	C₂×C₄⋊C₄	—	127	1	no	—
176	C₂×C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
177	C₂×C₂²⋊C₄	—	254	1	no	—
178	C₂×C₂²⋊C₄	—	254	1	no	—
179	C₂×C₄²	—	254	1	no	—
180	C₂×C₄²	—	254	1	no	—
181	#107	—	127	2	no	—
182	#107	—	127	2	no	—
183	#107	—	127	2	no	—
184	#107	C₂×D₁₂₇	1	2	yes	complemented
185	#106	—	127	2	no	—
186	#106	—	127	2	no	—
187	#106	—	127	2	no	—
188	#106	C₂×D₁₂₇	1	2	yes	complemented
189	C₂⁴	—	127	1	no	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	127	1	no	—
192	C₂²×C₄	—	127	4	no	—
193	C₂²×C₄	—	127	1	no	—
194	C₂²×C₄	—	127	1	no	—
195	C₂²×C₄	—	127	4	no	—
196	C₂²×C₄	—	127	1	no	—
197	C₂²×C₄	—	127	1	no	—
198	C₂²×C₄	—	254	1	no	—
199	C₂²×C₄	—	254	1	no	—
200	C₂²×C₄	—	254	1	no	—
201	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
202	M₁₆	—	127	4	no	—
203	M₁₆	—	127	8	no	—
204	M₁₆	C₄×D₁₂₇	1	4	yes	complemented
205	C₂×C₈	—	127	2	no	—
206	C₂×C₈	—	127	8	no	—
207	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	127	1	no	—
210	C₄⋊C₄	—	127	2	no	—
211	C₄⋊C₄	—	127	1	no	—
212	C₄⋊C₄	—	254	1	no	—
213	C₄⋊C₄	—	254	1	no	—
214	C₄⋊C₄	—	254	1	no	—
215	C₄⋊C₄	C₂²×D₁₂₇	1	1	yes	characteristic
216	C₄⋊C₄	C₂²×D₁₂₇	1	2	yes	—
217	C₄⋊C₄	C₂²×D₁₂₇	1	1	yes	characteristic
218	C₂²⋊C₄	—	2	1	no	—
219	C₂²⋊C₄	—	254	1	no	—
220	C₂²⋊C₄	—	254	1	no	—
221	C₂²⋊C₄	—	254	2	no	—
222	C₂²⋊C₄	—	254	2	no	—
223	C₂²⋊C₄	—	254	1	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	254	1	no	—
226	C₄²	—	254	1	no	—
227	C₄²	—	254	2	no	—
228	C₄²	—	254	2	no	—
229	C₄²	—	254	1	no	—
230	C₂³	—	127	4	no	—
231	C₂³	—	127	1	no	—
232	C₂³	—	127	1	no	—
233	C₂³	—	508	1	no	—
234	C₂³	—	508	1	no	—
235	C₂³	D₄×D₁₂₇	1	1	yes	characteristic
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	127	1	no	—
241	C₂×C₄	—	127	1	no	—
242	C₂×C₄	—	127	4	no	—
243	C₂×C₄	—	127	2	no	—
244	C₂×C₄	—	127	2	no	—
245	C₂×C₄	—	127	2	no	—
246	C₂×C₄	—	127	2	no	—
247	C₂×C₄	—	127	4	no	—
248	C₂×C₄	—	127	4	no	—
249	C₂×C₄	—	508	2	no	—
250	C₂×C₄	—	508	2	no	—
251	C₂×C₄	—	508	2	no	—
252	C₂×C₄	—	254	1	no	—
253	C₂×C₄	—	254	2	no	—
254	C₂×C₄	—	254	1	no	—
255	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
256	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
257	C₂×C₄	Q₈×D₁₂₇	1	2	yes	—
258	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
259	C₈	—	127	4	no	—
260	C₈	C₂×C₄×D₁₂₇	1	4	yes	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	127	2	no	—
264	C₂²	—	127	2	no	—
265	C₂²	—	508	4	no	—
266	C₂²	—	254	1	no	—
267	C₂²	—	254	1	no	—
268	C₂²	—	254	1	no	—
269	C₂²	—	254	1	no	—
270	C₂²	C₄⋊C₄×D₁₂₇	1	2	yes	—
271	C₂²	C₂×D₄×D₁₂₇	1	1	yes	characteristic
272	C₄	—	4	2	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	1	no	—
275	C₄	—	127	2	no	—
276	C₄	—	127	2	no	—
277	C₄	—	508	1	no	—
278	C₄	—	508	1	no	—
279	C₄	—	508	2	no	—
280	C₄	C₄⋊C₄×D₁₂₇	1	2	yes	—
281	C₄	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
282	C₄	C₂×Q₈×D₁₂₇	1	1	yes	characteristic
283	C₂	—	2	2	no	—
284	C₂	—	254	2	no	—
285	C₂	—	254	2	no	—
286	C₂	C₂×C₄⋊C₄×D₁₂₇	1	1	yes	characteristic
287	C₂	#22550672	1	1	yes	characteristic
288	C₂	#22550675	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₂₇
commutator	C₄×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#428×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	C₂³×D₁₂₇	D₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#428	D₁₂₇
hypocenter	C₁₂₇	#2911
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
4. #22549919	C₂	semidirect
5. #22549918	C₂	semidirect
6. #22549917	C₂	semidirect
7. #22549916	C₂	semidirect
8. #22549915	C₂	semidirect
9. #22549913	C₂	semidirect
10. #22549911	C₂	semidirect
11. #428×C₁₂₇	C₂	semidirect
15. M₁₆×D₁₂₇	C₄	semidirect
33. #16861143	C₂²	semidirect
34. #16861142	C₂²	semidirect
35. #16861142	C₂²	semidirect
36. #16861141	C₂²	semidirect
37. #16861140	C₂²	semidirect
38. #16861140	C₂²	semidirect
42. #107×C₁₂₇	C₂²	semidirect
43. #106×C₁₂₇	C₂²	semidirect
55. #14523232	C₂×C₄	semidirect
56. #14523231	C₂×C₄	semidirect
58. C₈×D₁₂₇	C₂×C₄	semidirect
77. M₁₆×C₁₂₇	C₂×C₄	semidirect
145. C₁₂₇	#2911	semidirect
154. #428	D₁₂₇	semidirect
184. #107	C₂×D₁₂₇	semidirect
188. #106	C₂×D₁₂₇	semidirect
204. M₁₆	C₄×D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×D₁₂₇	2
C₂×#16861167	2
#117×D₁₂₇	2
#22549919	2
#22549918	2
#22549917	2
#22549916	2
#22549915	2
#22549913	2
#22549911	2
#428×C₁₂₇	2
#2911	127