Subgroups — #30647833

Order 16256.

314 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550430	C₂	1	1	yes	maximal, characteristic
2	#22550007	C₂	1	1	yes	maximal, characteristic, complemented
3	#22550006	C₂	1	1	yes	maximal, characteristic, complemented
4	#22550005	C₂	1	1	yes	maximal, characteristic, complemented
5	#22550005	C₂	1	1	yes	maximal, characteristic, complemented
6	#22550004	C₂	1	1	yes	maximal, characteristic, complemented
7	#22550004	C₂	1	1	yes	maximal, characteristic, complemented
8	#22550003	C₂	1	1	yes	maximal, characteristic, complemented
9	#22549938	C₂	1	1	yes	maximal, characteristic
10	#22549936	C₂	1	1	yes	maximal, characteristic
11	#22549936	C₂	1	1	yes	maximal, characteristic
12	#22549935	C₂	1	1	yes	maximal, characteristic
13	#22549932	C₂	1	1	yes	maximal, characteristic
14	#22549931	C₂	1	1	yes	maximal, characteristic
15	#446×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
16	#16861208	C₄	1	1	yes	characteristic
17	#16861207	C₄	1	1	yes	characteristic
18	#16861199	C₄	1	1	yes	characteristic
19	#16861199	C₄	1	1	yes	characteristic
20	#16861189	C₄	1	1	yes	characteristic
21	#16861187	C₄	1	1	yes	characteristic
22	#16861187	C₄	1	1	yes	characteristic
23	#16861185	C₄	1	1	yes	characteristic
24	C₄×Dic₂₅₄	C₂²	1	1	yes	characteristic
25	C₄×Dic₂₅₄	C₂²	1	1	yes	characteristic
26	C₄×Dic₂₅₄	C₂²	1	1	yes	characteristic
27	#16861166	C₂²	1	1	yes	characteristic
28	#16861166	C₂²	1	1	yes	characteristic
29	C₄²×D₁₂₇	C₂²	1	1	yes	characteristic
30	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
31	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
32	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
33	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
34	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
35	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
36	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
37	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
38	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
39	(C₂×D₁₂₇)⋊C₈	C₂²	1	1	yes	characteristic
40	(C₂×D₁₂₇)⋊C₈	C₂²	1	1	yes	characteristic
41	(C₂×D₁₂₇)⋊C₈	C₂²	1	1	yes	characteristic
42	(C₂×D₁₂₇)⋊C₈	C₂²	1	1	yes	characteristic
43	#16861120	C₂²	1	1	yes	characteristic
44	#16861120	C₂²	1	1	yes	characteristic
45	#16861119	C₂²	1	1	yes	characteristic
46	#16861119	C₂²	1	1	yes	characteristic
47	#16861119	C₂²	1	1	yes	characteristic
48	#16861119	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	C₄×Q₈×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	C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
56	#97×C₁₂₇	C₂²	1	1	yes	characteristic
57	#97×C₁₂₇	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₅₄	C₂×C₄	1	1	yes	characteristic
60	C₂×Dic₂₅₄	C₂×C₄	1	1	yes	characteristic
61	C₂×Dic₂₅₄	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
65	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
66	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
67	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
68	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
69	C₄⋊Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
70	C₄⋊Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
71	C₄⋊Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
72	Dic₁₂₇⋊C₄	C₂×C₄	1	1	yes	characteristic
73	Dic₁₂₇⋊C₄	C₂×C₄	1	1	yes	characteristic
74	Dic₁₂₇⋊C₄	C₂×C₄	1	1	yes	characteristic
75	Dic₁₂₇⋊C₄	C₂×C₄	1	1	yes	characteristic
76	Dic₁₂₇⋊C₄	C₂×C₄	1	1	yes	characteristic
77	Dic₁₂₇⋊C₄	C₂×C₄	1	1	yes	characteristic
78	Dic₁₂₇⋊C₄	C₂×C₄	1	1	yes	characteristic
79	Dic₁₂₇⋊C₄	C₂×C₄	1	1	yes	characteristic
80	Dic₁₂₇⋊C₄	C₂×C₄	1	1	yes	characteristic
81	Dic₁₂₇⋊C₄	C₂×C₄	1	1	yes	characteristic
82	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
83	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
84	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
85	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
86	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
87	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
88	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
89	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
90	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
91	C₂×Q₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
93	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
94	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
95	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
96	C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
97	C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
98	C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
99	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
100	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
101	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
102	C₂²×D₁₂₇	C₂²×C₄	1	1	yes	characteristic
103	Dic₂₅₄	—	4	1	no	—
104	Dic₂₅₄	—	4	1	no	—
105	Dic₂₅₄	—	4	1	no	—
106	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
107	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
108	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
110	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
113	C₄×D₁₂₇	—	2	1	no	—
114	C₄×D₁₂₇	—	2	1	no	—
115	C₄×D₁₂₇	—	4	1	no	—
116	C₄×D₁₂₇	—	4	1	no	—
117	C₁₂₇⋊C₈	—	2	1	no	—
118	C₁₂₇⋊C₈	—	2	1	no	—
119	C₁₂₇⋊C₈	—	2	1	no	—
120	C₁₂₇⋊C₈	—	2	1	no	—
121	Q₈×C₁₂₇	—	4	1	no	—
122	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
129	C₈×C₁₂₇	—	2	1	no	—
130	C₈×C₁₂₇	—	2	1	no	—
131	C₈×C₁₂₇	—	2	1	no	—
132	C₈×C₁₂₇	—	2	1	no	—
133	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
134	C₂×D₁₂₇	—	2	1	no	—
135	C₂×D₁₂₇	—	2	1	no	—
136	C₂×D₁₂₇	—	2	1	no	—
137	C₄×C₁₂₇	—	2	1	no	—
138	C₄×C₁₂₇	—	2	1	no	—
139	C₄×C₁₂₇	—	2	1	no	—
140	C₄×C₁₂₇	—	2	1	no	—
141	C₄×C₁₂₇	—	2	1	no	—
142	C₄×C₁₂₇	—	2	1	no	—
143	C₄×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
144	C₄×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
145	Dic₁₂₇	—	2	1	no	—
146	Dic₁₂₇	—	2	1	no	—
147	Dic₁₂₇	—	2	1	no	—
148	Dic₁₂₇	—	2	1	no	—
149	Dic₁₂₇	—	2	1	no	—
150	Dic₁₂₇	—	2	1	no	—
151	Dic₁₂₇	—	2	1	no	—
152	C₂×C₁₂₇	#519	1	1	yes	characteristic
153	C₂×C₁₂₇	#568	1	1	yes	characteristic
154	C₂×C₁₂₇	#568	1	1	yes	characteristic
155	D₁₂₇	—	4	1	no	—
156	#2987	—	127	1	no	maximal, Sylow
157	C₁₂₇	#2987	1	1	yes	characteristic, complemented, Sylow
158	#533	—	127	1	no	—
159	#446	—	127	1	no	—
160	#446	—	127	1	no	—
161	#446	—	127	1	no	—
162	#446	—	127	1	no	—
163	#446	—	127	1	no	—
164	#446	—	127	1	no	—
165	#446	—	127	1	no	—
166	#446	D₁₂₇	1	1	yes	characteristic, complemented
167	#433	—	127	1	no	—
168	#433	—	127	1	no	—
169	#433	—	127	1	no	—
170	#433	—	127	1	no	—
171	#432	—	127	1	no	—
172	#432	—	127	1	no	—
173	C₄⋊Q₈	—	127	1	no	—
174	C₄⋊Q₈	—	127	1	no	—
175	#125	—	127	1	no	—
176	#125	—	127	1	no	—
177	C₂²⋊Q₈	—	127	1	no	—
178	C₂²⋊Q₈	—	127	1	no	—
179	C₂²⋊Q₈	—	127	1	no	—
180	C₂²⋊Q₈	—	127	1	no	—
181	C₄×Q₈	—	127	1	no	—
182	C₄×Q₈	—	127	1	no	—
183	C₄×Q₈	—	127	1	no	—
184	C₄×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
185	#117	—	127	1	no	—
186	#117	—	127	1	no	—
187	C₂×C₄²	—	127	1	no	—
188	C₄⋊C₈	—	127	1	no	—
189	C₄⋊C₈	—	127	1	no	—
190	C₄⋊C₈	—	127	1	no	—
191	C₄⋊C₈	—	127	1	no	—
192	C₄⋊C₈	—	127	1	no	—
193	C₄⋊C₈	—	127	1	no	—
194	C₄⋊C₈	—	127	1	no	—
195	C₄⋊C₈	—	127	1	no	—
196	C₄⋊C₈	—	127	1	no	—
197	C₄⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic
198	C₄⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic
199	C₄⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic
200	C₂²⋊C₈	—	127	1	no	—
201	C₂²⋊C₈	—	127	1	no	—
202	C₂²⋊C₈	—	127	1	no	—
203	C₂²⋊C₈	—	127	1	no	—
204	#97	—	127	1	no	—
205	#97	—	127	1	no	—
206	#97	—	127	1	no	—
207	#97	—	127	1	no	—
208	#97	—	127	1	no	—
209	#97	—	127	1	no	—
210	#97	C₂×D₁₂₇	1	1	yes	characteristic
211	#97	C₂×D₁₂₇	1	1	yes	characteristic
212	C₄×C₈	—	127	1	no	—
213	C₄×C₈	—	127	1	no	—
214	C₄×C₈	—	127	1	no	—
215	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
216	C₂×Q₈	—	127	1	no	—
217	C₂×Q₈	—	127	1	no	—
218	C₂×Q₈	—	127	1	no	—
219	C₂×Q₈	C₄×D₁₂₇	1	1	yes	characteristic
220	C₂²×C₄	—	127	1	no	—
221	C₂²×C₄	—	127	1	no	—
222	C₂²×C₄	—	127	1	no	—
223	C₂×C₈	—	127	1	no	—
224	C₂×C₈	—	127	1	no	—
225	C₂×C₈	—	127	1	no	—
226	C₂×C₈	—	127	1	no	—
227	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
228	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
229	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
230	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
231	C₄⋊C₄	—	127	1	no	—
232	C₄⋊C₄	—	127	1	no	—
233	C₄⋊C₄	—	127	1	no	—
234	C₄⋊C₄	—	127	1	no	—
235	C₄⋊C₄	—	127	1	no	—
236	C₄⋊C₄	—	127	1	no	—
237	C₄⋊C₄	—	127	1	no	—
238	C₄⋊C₄	—	127	1	no	—
239	C₄⋊C₄	—	127	1	no	—
240	C₄⋊C₄	—	127	1	no	—
241	C₄⋊C₄	—	127	1	no	—
242	C₄⋊C₄	—	127	1	no	—
243	C₄⋊C₄	—	127	1	no	—
244	C₄⋊C₄	C₄×D₁₂₇	1	1	yes	characteristic
245	C₄⋊C₄	C₄×D₁₂₇	1	1	yes	characteristic
246	C₄⋊C₄	C₄×D₁₂₇	1	1	yes	characteristic
247	C₂²⋊C₄	—	127	1	no	—
248	C₂²⋊C₄	—	127	1	no	—
249	C₂²⋊C₄	—	127	1	no	—
250	C₂²⋊C₄	—	127	1	no	—
251	C₄²	—	127	1	no	—
252	C₄²	—	127	1	no	—
253	C₄²	—	127	1	no	—
254	C₄²	—	127	1	no	—
255	C₄²	—	127	1	no	—
256	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
257	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
258	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
259	C₂³	—	127	1	no	—
260	Q₈	—	4	1	no	—
261	Q₈	—	508	1	no	—
262	Q₈	—	508	1	no	—
263	Q₈	—	508	1	no	—
264	C₂×C₄	—	127	1	no	—
265	C₂×C₄	—	127	1	no	—
266	C₂×C₄	—	127	1	no	—
267	C₂×C₄	—	127	1	no	—
268	C₂×C₄	—	127	1	no	—
269	C₂×C₄	—	127	1	no	—
270	C₂×C₄	—	127	1	no	—
271	C₂×C₄	—	508	1	no	—
272	C₂×C₄	—	508	1	no	—
273	C₂×C₄	—	254	1	no	—
274	C₂×C₄	—	254	1	no	—
275	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
276	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
277	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
278	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
280	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
281	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	254	1	no	—
287	C₈	—	254	1	no	—
288	C₈	—	254	1	no	—
289	C₈	—	254	1	no	—
290	C₂²	—	254	1	no	—
291	C₂²	—	254	1	no	—
292	C₂²	—	254	1	no	—
293	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	254	1	no	—
301	C₄	—	254	1	no	—
302	C₄	—	254	1	no	—
303	C₄	—	254	1	no	—
304	C₄	—	254	1	no	—
305	C₄	—	254	1	no	—
306	C₄	—	254	1	no	—
307	C₄	#16861256	1	1	yes	characteristic
308	C₄	#16861257	1	1	yes	characteristic
309	C₂	—	508	1	no	—
310	C₂	#22550352	1	1	yes	characteristic
311	C₂	#22550641	1	1	yes	characteristic
312	C₂	#22550642	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Fitting	#446×C₁₂₇	C₂
prime radical	C₂²×D₁₂₇	C₂²×C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#446	D₁₂₇
hypocenter	C₁₂₇	#2987
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #22550007	C₂	semidirect
3. #22550006	C₂	semidirect
4. #22550005	C₂	semidirect
5. #22550005	C₂	semidirect
6. #22550004	C₂	semidirect
7. #22550004	C₂	semidirect
8. #22550003	C₂	semidirect
15. #446×C₁₂₇	C₂	semidirect
157. C₁₂₇	#2987	semidirect
166. #446	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550430	2
#22550007	2
#22550006	2
#22550005	2
#22550004	2
#22550003	2
#22549938	2
#22549936	2
#22549935	2
#22549932	2
#22549931	2
#446×C₁₂₇	2
#2987	127