Subgroups — #30647358

Order 16256.

484 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×D₁₂₇	C₂	1	1	yes	maximal, characteristic
2	#22549932	C₂	1	2	yes	maximal, complemented
3	#22549930	C₂	1	2	yes	maximal, complemented
4	#22549929	C₂	1	1	yes	maximal, characteristic, complemented
5	#22549927	C₂	1	2	yes	maximal, complemented
6	#22549898	C₂	1	2	yes	maximal, complemented
7	#22549838	C₂	1	2	yes	maximal
8	C₄×#14523231	C₂	1	2	yes	maximal, complemented
9	#431×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₄×D₁₂₇	C₄	1	1	yes	characteristic
11	C₂²×C₄×D₁₂₇	C₄	1	1	yes	characteristic
12	C₂²×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×#14523231	C₂²	1	4	yes	—
14	C₂×C₄×Dic₁₂₇	C₄	1	1	yes	characteristic
15	C₂×C₄×Dic₁₂₇	C₄	1	1	yes	characteristic
16	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
17	C₄²×D₁₂₇	C₄	1	1	yes	characteristic
18	C₄²×D₁₂₇	C₄	1	1	yes	characteristic
19	C₄²×D₁₂₇	C₄	1	1	yes	characteristic
20	C₄²×D₁₂₇	C₄	1	1	yes	characteristic
21	C₄²×D₁₂₇	C₂²	1	1	yes	characteristic
22	C₄²×D₁₂₇	C₂²	1	1	yes	characteristic
23	C₄²×D₁₂₇	C₂²	1	1	yes	characteristic
24	C₄²×D₁₂₇	C₂²	1	1	yes	characteristic
25	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	complemented
26	Dic₁₂₇⋊C₈	C₂²	1	4	yes	complemented
27	C₂²⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
28	(C₂×D₁₂₇)⋊C₈	C₂²	1	4	yes	—
29	#16861119	C₂²	1	4	yes	complemented
30	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	complemented
31	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
32	C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	complemented
33	C₂²⋊C₈×C₁₂₇	C₂²	1	2	yes	—
34	C₄×C₈×C₁₂₇	C₂²	1	2	yes	complemented
35	C₂³×D₁₂₇	C₂×C₄	1	1	yes	characteristic
36	C₂²×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×D₁₂₇	—	2	4	no	—
40	C₂×C₄×D₁₂₇	—	2	4	no	—
41	C₂×C₄×D₁₂₇	—	2	4	no	—
42	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
54	#14523231	—	2	8	no	—
55	C₂×C₁₂₇⋊C₈	C₂³	1	4	yes	—
56	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
57	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
58	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
59	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
60	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
67	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
72	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
73	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
74	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
75	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
76	C₂²×D₁₂₇	—	2	2	no	—
77	C₂²×D₁₂₇	—	2	2	no	—
78	C₂²×D₁₂₇	—	2	2	no	—
79	C₂²×D₁₂₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×D₁₂₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₁₂₇	—	2	2	no	—
82	C₂×Dic₁₂₇	—	2	2	no	—
83	C₂×Dic₁₂₇	—	2	2	no	—
84	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
90	C₄×D₁₂₇	—	2	2	no	—
91	C₄×D₁₂₇	—	2	2	no	—
92	C₄×D₁₂₇	—	2	2	no	—
93	C₄×D₁₂₇	—	2	2	no	—
94	C₄×D₁₂₇	—	2	2	no	—
95	C₄×D₁₂₇	—	2	2	no	—
96	C₄×D₁₂₇	—	2	2	no	—
97	C₄×D₁₂₇	—	2	2	no	—
98	C₄×D₁₂₇	—	2	2	no	—
99	C₄×D₁₂₇	—	2	2	no	—
100	C₄×D₁₂₇	M₁₆	1	4	yes	complemented
101	C₄×D₁₂₇	#41	1	4	yes	—
102	C₁₂₇⋊C₈	—	2	4	no	—
103	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₂₇	—	2	2	no	—
105	C₂×C₄×C₁₂₇	—	2	2	no	—
106	C₂×C₄×C₁₂₇	M₁₆	1	4	yes	complemented
107	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
113	C₈×C₁₂₇	—	2	4	no	—
114	C₂²×C₁₂₇	—	2	1	no	—
115	C₂²×C₁₂₇	—	2	1	no	—
116	C₂²×C₁₂₇	C₂×M₁₆	1	2	yes	—
117	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
118	C₂×D₁₂₇	—	2	8	no	—
119	C₂×D₁₂₇	—	2	1	no	—
120	C₂×D₁₂₇	—	2	1	no	—
121	C₂×D₁₂₇	—	2	1	no	—
122	C₂×D₁₂₇	—	2	1	no	—
123	C₂×D₁₂₇	#117	1	2	yes	—
124	C₂×D₁₂₇	C₂×M₁₆	1	2	yes	—
125	C₄×C₁₂₇	—	2	1	no	—
126	C₄×C₁₂₇	—	2	1	no	—
127	C₄×C₁₂₇	C₂×M₁₆	1	2	yes	—
128	C₄×C₁₂₇	C₂×M₁₆	1	2	yes	—
129	C₄×C₁₂₇	C₂×M₁₆	1	2	yes	—
130	C₄×C₁₂₇	C₂×#41	1	2	yes	—
131	Dic₁₂₇	—	2	1	no	—
132	Dic₁₂₇	—	2	1	no	—
133	Dic₁₂₇	—	2	1	no	—
134	Dic₁₂₇	—	2	1	no	—
135	Dic₁₂₇	#117	1	2	yes	—
136	Dic₁₂₇	C₂×M₁₆	1	2	yes	—
137	C₂×C₁₂₇	—	2	2	no	—
138	C₂×C₁₂₇	C₂×#117	1	1	yes	characteristic
139	C₂×C₁₂₇	C₂²×M₁₆	1	1	yes	characteristic
140	C₂×C₁₂₇	C₂²×M₁₆	1	1	yes	characteristic
141	D₁₂₇	—	2	2	no	—
142	D₁₂₇	—	2	2	no	—
143	#2920	—	127	1	no	maximal, Sylow
144	C₁₂₇	#2920	1	1	yes	characteristic, complemented, Sylow
145	C₂²×C₄²	—	127	1	no	—
146	#432	—	127	2	no	—
147	#432	—	127	2	no	—
148	#431	—	127	1	no	—
149	#431	—	127	2	no	—
150	#431	D₁₂₇	1	1	yes	characteristic, complemented
151	#423	—	127	2	no	—
152	#407	—	127	2	no	—
153	C₄×M₁₆	—	127	2	no	—
154	C₂³×C₄	—	127	1	no	—
155	C₂³×C₄	—	127	1	no	—
156	C₂³×C₄	—	127	1	no	—
157	C₂×M₁₆	—	127	4	no	—
158	C₂×C₄²	—	127	1	no	—
159	C₂×C₄²	—	127	1	no	—
160	C₂×C₄²	—	127	1	no	—
161	C₂×C₄²	—	127	1	no	—
162	C₂×C₄²	—	127	1	no	—
163	C₂×C₄²	—	127	1	no	—
164	C₂×C₄²	—	127	1	no	—
165	C₂×C₄²	—	127	1	no	—
166	C₂×C₄²	—	127	1	no	—
167	C₂×C₄²	—	127	1	no	—
168	C₂×C₄²	—	127	1	no	—
169	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
170	C₄⋊C₈	—	127	4	no	—
171	C₄⋊C₈	—	127	2	no	—
172	C₄⋊C₈	C₂×D₁₂₇	1	2	yes	complemented
173	C₂²⋊C₈	—	127	2	no	—
174	C₂²⋊C₈	—	127	4	no	—
175	C₂²⋊C₈	C₂×D₁₂₇	1	2	yes	—
176	#97	—	127	4	no	—
177	C₄×C₈	—	127	2	no	—
178	C₄×C₈	C₂×D₁₂₇	1	2	yes	complemented
179	C₂⁴	—	127	1	no	—
180	C₂²×C₄	—	127	1	no	—
181	C₂²×C₄	—	127	1	no	—
182	C₂²×C₄	—	127	1	no	—
183	C₂²×C₄	—	127	1	no	—
184	C₂²×C₄	—	127	1	no	—
185	C₂²×C₄	—	127	1	no	—
186	C₂²×C₄	—	127	1	no	—
187	C₂²×C₄	—	127	1	no	—
188	C₂²×C₄	—	127	1	no	—
189	C₂²×C₄	—	127	1	no	—
190	C₂²×C₄	—	127	1	no	—
191	C₂²×C₄	—	127	1	no	—
192	C₂²×C₄	—	127	1	no	—
193	C₂²×C₄	—	127	1	no	—
194	C₂²×C₄	—	127	1	no	—
195	C₂²×C₄	—	254	4	no	—
196	C₂²×C₄	—	254	4	no	—
197	C₂²×C₄	—	254	4	no	—
198	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
199	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
200	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
201	M₁₆	—	254	8	no	—
202	C₂×C₈	—	127	4	no	—
203	C₂×C₈	C₂²×D₁₂₇	1	4	yes	—
204	C₄²	—	127	1	no	—
205	C₄²	—	127	1	no	—
206	C₄²	—	127	1	no	—
207	C₄²	—	127	1	no	—
208	C₄²	—	127	1	no	—
209	C₄²	—	127	1	no	—
210	C₄²	—	127	1	no	—
211	C₄²	—	127	1	no	—
212	C₄²	—	127	1	no	—
213	C₄²	—	127	1	no	—
214	C₄²	—	127	1	no	—
215	C₄²	—	127	1	no	—
216	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
217	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
218	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
219	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
220	C₂³	—	127	1	no	—
221	C₂³	—	127	1	no	—
222	C₂³	—	254	2	no	—
223	C₂³	—	254	2	no	—
224	C₂³	—	254	2	no	—
225	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	127	1	no	—
229	C₂×C₄	—	127	1	no	—
230	C₂×C₄	—	127	4	no	—
231	C₂×C₄	—	127	4	no	—
232	C₂×C₄	—	127	1	no	—
233	C₂×C₄	—	127	1	no	—
234	C₂×C₄	—	127	1	no	—
235	C₂×C₄	—	127	1	no	—
236	C₂×C₄	—	254	2	no	—
237	C₂×C₄	—	254	2	no	—
238	C₂×C₄	—	254	2	no	—
239	C₂×C₄	—	254	2	no	—
240	C₂×C₄	—	254	2	no	—
241	C₂×C₄	—	254	2	no	—
242	C₂×C₄	—	254	2	no	—
243	C₂×C₄	—	254	2	no	—
244	C₂×C₄	—	254	2	no	—
245	C₂×C₄	—	254	2	no	—
246	C₂×C₄	—	254	2	no	—
247	C₂×C₄	—	254	2	no	—
248	C₂×C₄	—	254	2	no	—
249	C₂×C₄	#14523231	1	4	yes	complemented
250	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
251	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
252	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
253	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
254	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
255	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
256	C₈	—	2	4	no	—
257	C₈	—	254	4	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	127	2	no	—
261	C₂²	—	127	2	no	—
262	C₂²	—	254	8	no	—
263	C₂²	—	254	1	no	—
264	C₂²	—	254	1	no	—
265	C₂²	—	254	1	no	—
266	C₂²	—	254	1	no	—
267	C₂²	C₂×#14523231	1	2	yes	—
268	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	127	2	no	—
272	C₄	—	127	2	no	—
273	C₄	—	254	1	no	—
274	C₄	—	254	1	no	—
275	C₄	—	254	1	no	—
276	C₄	—	254	1	no	—
277	C₄	M₁₆×D₁₂₇	1	2	yes	—
278	C₄	C₂×#14523231	1	2	yes	—
279	C₄	C₂×#14523231	1	2	yes	—
280	C₄	#41×D₁₂₇	1	2	yes	—
281	C₂	—	2	2	no	—
282	C₂	—	254	2	no	—
283	C₂	—	254	2	no	—
284	C₂	#117×D₁₂₇	1	1	yes	characteristic
285	C₂	C₂×M₁₆×D₁₂₇	1	1	yes	characteristic
286	C₂	C₂²×#14523231	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂²×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#431×C₁₂₇	C₂
prime radical	C₂³×D₁₂₇	C₂×C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#431	D₁₂₇
hypocenter	C₁₂₇	#2920
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄² ◁ #431

Complements

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

N	H ≅ G/N	type
2. #22549932	C₂	semidirect
3. #22549930	C₂	semidirect
4. #22549929	C₂	semidirect
5. #22549927	C₂	semidirect
6. #22549898	C₂	semidirect
8. C₄×#14523231	C₂	semidirect
9. #431×C₁₂₇	C₂	semidirect
25. C₄⋊(C₁₂₇⋊C₈)	C₂²	semidirect
26. Dic₁₂₇⋊C₈	C₂²	semidirect
29. #16861119	C₂²	semidirect
30. C₄×C₁₂₇⋊C₈	C₂²	semidirect
32. C₄⋊C₈×C₁₂₇	C₂²	semidirect
34. C₄×C₈×C₁₂₇	C₂²	semidirect
100. C₄×D₁₂₇	M₁₆	semidirect
106. C₂×C₄×C₁₂₇	M₁₆	semidirect
144. C₁₂₇	#2920	semidirect
150. #431	D₁₂₇	semidirect
172. C₄⋊C₈	C₂×D₁₂₇	semidirect
178. C₄×C₈	C₂×D₁₂₇	semidirect
249. C₂×C₄	#14523231	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄²×D₁₂₇	2
#22549932	2
#22549930	2
#22549929	2
#22549927	2
#22549898	2
#22549838	2
C₄×#14523231	2
#431×C₁₂₇	2
#2920	127