Subgroups — #533×D₁₂₇ #30652015

Order 16256.

1168 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550433	C₂	1	1	yes	maximal, characteristic, complemented
2	#22550432	C₂	1	2	yes	maximal, complemented
3	#22550431	C₂	1	2	yes	maximal, complemented
4	#22550430	C₂	1	4	yes	maximal, complemented
5	#22550429	C₂	1	2	yes	maximal, complemented
6	#22550428	C₂	1	4	yes	maximal, complemented
7	C₄⋊Q₈×D₁₂₇	C₂	1	2	yes	maximal, complemented
8	#125×D₁₂₇	C₂	1	2	yes	maximal, complemented
9	C₂²⋊Q₈×D₁₂₇	C₂	1	4	yes	maximal
10	C₄×Q₈×D₁₂₇	C₂	1	4	yes	maximal, complemented
11	#117×D₁₂₇	C₂	1	2	yes	maximal
12	C₂×C₄²×D₁₂₇	C₂	1	1	yes	maximal, characteristic
13	#533×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×Q₈×D₁₂₇	C₂²	1	4	yes	—
15	C₂²×C₄×D₁₂₇	C₂²	1	2	yes	—
16	C₂²×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
17	#16861209	C₂²	1	2	yes	complemented
18	#16861208	C₂²	1	4	yes	complemented
19	#16861208	C₂²	1	4	yes	complemented
20	#16861207	C₂²	1	4	yes	complemented
21	#16861201	C₂²	1	2	yes	complemented
22	#16861200	C₂²	1	4	yes	complemented
23	#16861199	C₂²	1	8	yes	complemented
24	#16861189	C₂²	1	4	yes	—
25	#16861188	C₂²	1	4	yes	—
26	#16861187	C₂²	1	8	yes	—
27	#16861186	C₂²	1	8	yes	—
28	#16861185	C₂²	1	4	yes	—
29	Q₈×Dic₁₂₇	C₂²	1	4	yes	complemented
30	C₄×Dic₂₅₄	C₂²	1	8	yes	complemented
31	C₄×Dic₂₅₄	C₂²	1	4	yes	complemented
32	#16861174	C₂²	1	4	yes	complemented
33	#16861174	C₂²	1	8	yes	complemented
34	#16861168	C₂²	1	2	yes	—
35	#16861167	C₂²	1	4	yes	—
36	#16861166	C₂²	1	4	yes	—
37	#16861165	C₂²	1	4	yes	—
38	C₄⋊C₄×D₁₂₇	C₂²	1	4	yes	—
39	C₄⋊C₄×D₁₂₇	C₂²	1	8	yes	—
40	C₄⋊C₄×D₁₂₇	C₂²	1	4	yes	—
41	C₂²⋊C₄×D₁₂₇	C₂²	1	4	yes	—
42	C₂×C₄×Dic₁₂₇	C₂²	1	2	yes	—
43	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
44	C₄²×D₁₂₇	C₂²	1	2	yes	—
45	C₄²×D₁₂₇	C₂²	1	1	yes	characteristic
46	C₄²×D₁₂₇	C₂²	1	1	yes	characteristic
47	C₄²×D₁₂₇	C₂²	1	2	yes	—
48	C₄²×D₁₂₇	C₂²	1	1	yes	characteristic
49	C₄²×D₁₂₇	C₂²	1	1	yes	characteristic
50	C₄²×D₁₂₇	C₂²	1	4	yes	—
51	C₄⋊Q₈×C₁₂₇	C₂²	1	2	yes	complemented
52	#125×C₁₂₇	C₂²	1	2	yes	complemented
53	C₂²⋊Q₈×C₁₂₇	C₂²	1	4	yes	—
54	C₄×Q₈×C₁₂₇	C₂²	1	4	yes	complemented
55	#117×C₁₂₇	C₂²	1	2	yes	—
56	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
57	C₂³×D₁₂₇	C₂³	1	1	yes	characteristic
58	C₂×Dic₂₅₄	C₂³	1	8	yes	—
59	C₂×Dic₂₅₄	C₂³	1	4	yes	—
60	Q₈×D₁₂₇	—	2	16	no	—
61	C₂²×Dic₁₂₇	C₂³	1	2	yes	—
62	C₂²×Dic₁₂₇	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₁₂₇	—	2	8	no	—
64	C₂×C₄×D₁₂₇	Q₈	1	8	yes	complemented
65	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
66	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
67	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
68	C₂×C₄×D₁₂₇	C₂³	1	8	yes	—
69	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
70	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
71	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
72	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
74	C₄⋊Dic₁₂₇	C₂³	1	4	yes	—
75	C₄⋊Dic₁₂₇	C₂³	1	8	yes	—
76	C₄⋊Dic₁₂₇	C₂³	1	4	yes	—
77	Dic₁₂₇⋊C₄	C₂³	1	8	yes	—
78	Dic₁₂₇⋊C₄	C₂³	1	8	yes	—
79	Dic₁₂₇⋊C₄	C₂³	1	8	yes	—
80	Dic₁₂₇⋊C₄	C₂³	1	8	yes	—
81	C₂²⋊Dic₁₂₇	C₂³	1	4	yes	—
82	(C₂×D₁₂₇)⋊C₄	C₂³	1	8	yes	—
83	C₄×Dic₁₂₇	C₂³	1	2	yes	—
84	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
85	C₄×Dic₁₂₇	C₂³	1	2	yes	—
86	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
87	C₄×Dic₁₂₇	C₂³	1	2	yes	—
88	C₄×Dic₁₂₇	C₂³	1	2	yes	—
89	C₄×Dic₁₂₇	C₂³	1	2	yes	—
90	C₄×Dic₁₂₇	C₂³	1	8	yes	—
91	C₄×Dic₁₂₇	C₂³	1	4	yes	—
92	C₂×Q₈×C₁₂₇	C₂³	1	4	yes	—
93	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
94	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
95	C₄⋊C₄×C₁₂₇	C₂³	1	8	yes	—
96	C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
97	C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
98	C₂²⋊C₄×C₁₂₇	C₂³	1	4	yes	—
99	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
100	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
101	C₄²×C₁₂₇	C₂³	1	2	yes	—
102	C₄²×C₁₂₇	C₂³	1	4	yes	—
103	C₂²×D₁₂₇	—	2	4	no	—
104	C₂²×D₁₂₇	C₂×Q₈	1	4	yes	—
105	C₂²×D₁₂₇	C₂⁴	1	1	yes	characteristic
106	C₂²×D₁₂₇	C₂⁴	1	1	yes	characteristic
107	Dic₂₅₄	—	2	8	no	—
108	Dic₂₅₄	—	2	16	no	—
109	C₂×Dic₁₂₇	—	2	4	no	—
110	C₂×Dic₁₂₇	C₂×Q₈	1	4	yes	complemented
111	C₂×Dic₁₂₇	C₂⁴	1	2	yes	—
112	C₂×Dic₁₂₇	C₂⁴	1	8	yes	—
113	C₂×Dic₁₂₇	C₂⁴	1	1	yes	characteristic
114	C₂×Dic₁₂₇	C₂⁴	1	1	yes	characteristic
115	C₂×Dic₁₂₇	C₂⁴	1	2	yes	—
116	C₄×D₁₂₇	—	2	4	no	—
117	C₄×D₁₂₇	—	2	16	no	—
118	C₄×D₁₂₇	—	2	4	no	—
119	C₄×D₁₂₇	—	2	4	no	—
120	C₄×D₁₂₇	C₂×Q₈	1	4	yes	—
121	C₄×D₁₂₇	C₂×Q₈	1	4	yes	—
122	C₄×D₁₂₇	C₂×Q₈	1	4	yes	complemented
123	C₄×D₁₂₇	C₂×Q₈	1	4	yes	complemented
124	C₄×D₁₂₇	#41	1	8	yes	—
125	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
126	Q₈×C₁₂₇	—	2	8	no	—
127	C₂×C₄×C₁₂₇	—	2	4	no	—
128	C₂×C₄×C₁₂₇	C₂×Q₈	1	4	yes	complemented
129	C₂×C₄×C₁₂₇	C₂⁴	1	2	yes	—
130	C₂×C₄×C₁₂₇	C₂⁴	1	8	yes	—
131	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
132	C₂×C₄×C₁₂₇	C₂⁴	1	2	yes	—
133	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
134	C₂²×C₁₂₇	—	2	2	no	—
135	C₂²×C₁₂₇	C₂²×Q₈	1	2	yes	—
136	C₂²×C₁₂₇	C₂⁵	1	1	yes	characteristic
137	C₂×D₁₂₇	—	2	2	no	—
138	C₂×D₁₂₇	—	2	8	no	—
139	C₂×D₁₂₇	C₂²×Q₈	1	2	yes	—
140	C₂×D₁₂₇	C₂²×Q₈	1	2	yes	—
141	C₂×D₁₂₇	C₂×#41	1	4	yes	—
142	C₄×C₁₂₇	—	2	2	no	—
143	C₄×C₁₂₇	—	2	8	no	—
144	C₄×C₁₂₇	C₂²×Q₈	1	2	yes	—
145	C₄×C₁₂₇	C₂²×Q₈	1	2	yes	—
146	C₄×C₁₂₇	C₂×#41	1	4	yes	—
147	Dic₁₂₇	—	2	2	no	—
148	Dic₁₂₇	—	2	8	no	—
149	Dic₁₂₇	C₂²×Q₈	1	2	yes	—
150	Dic₁₂₇	C₂²×Q₈	1	2	yes	—
151	Dic₁₂₇	C₂×#41	1	4	yes	—
152	C₂×C₁₂₇	—	2	2	no	—
153	C₂×C₁₂₇	C₂³×Q₈	1	1	yes	characteristic
154	C₂×C₁₂₇	C₂²×#41	1	2	yes	—
155	D₁₂₇	—	2	2	no	—
156	D₁₂₇	#533	1	4	yes	direct product factor
157	C₂×#533	—	127	1	no	maximal, Sylow
158	C₁₂₇	C₂×#533	1	1	yes	characteristic, complemented, Sylow
159	#533	—	127	2	no	—
160	#533	—	127	2	no	—
161	#533	—	127	4	no	—
162	#533	—	127	4	no	—
163	#533	—	127	2	no	—
164	#533	—	127	1	no	—
165	#533	D₁₂₇	1	1	yes	characteristic, direct product factor
166	C₂×C₄⋊Q₈	—	127	2	no	—
167	C₂×#125	—	127	2	no	—
168	C₂×C₂²⋊Q₈	—	127	4	no	—
169	C₂×C₄×Q₈	—	127	4	no	—
170	C₂×#117	—	127	2	no	—
171	C₂²×C₄²	—	127	1	no	—
172	C₂²×Q₈	—	127	4	no	—
173	C₂³×C₄	—	127	1	no	—
174	C₂³×C₄	—	127	2	no	—
175	C₄⋊Q₈	—	127	4	no	—
176	C₄⋊Q₈	—	127	4	no	—
177	C₄⋊Q₈	—	127	4	no	—
178	C₄⋊Q₈	—	127	2	no	—
179	C₄⋊Q₈	C₂×D₁₂₇	1	2	yes	complemented
180	#125	—	127	8	no	—
181	#125	—	127	4	no	—
182	#125	—	127	2	no	—
183	#125	C₂×D₁₂₇	1	2	yes	complemented
184	C₂²⋊Q₈	—	127	4	no	—
185	C₂²⋊Q₈	—	127	4	no	—
186	C₂²⋊Q₈	—	127	8	no	—
187	C₂²⋊Q₈	—	127	4	no	—
188	C₂²⋊Q₈	—	127	8	no	—
189	C₂²⋊Q₈	C₂×D₁₂₇	1	4	yes	—
190	C₄×Q₈	—	127	4	no	—
191	C₄×Q₈	—	127	4	no	—
192	C₄×Q₈	—	127	8	no	—
193	C₄×Q₈	—	127	8	no	—
194	C₄×Q₈	—	127	4	no	—
195	C₄×Q₈	C₂×D₁₂₇	1	4	yes	complemented
196	#117	—	127	2	no	—
197	#117	—	127	4	no	—
198	#117	—	127	4	no	—
199	#117	—	127	4	no	—
200	#117	C₂×D₁₂₇	1	2	yes	—
201	C₂×C₄⋊C₄	—	127	4	no	—
202	C₂×C₄⋊C₄	—	127	8	no	—
203	C₂×C₄⋊C₄	—	127	4	no	—
204	C₂×C₂²⋊C₄	—	127	4	no	—
205	C₂×C₄²	—	127	2	no	—
206	C₂×C₄²	—	127	1	no	—
207	C₂×C₄²	—	127	4	no	—
208	C₂×C₄²	—	127	2	no	—
209	C₂×C₄²	—	127	1	no	—
210	C₂×C₄²	—	127	1	no	—
211	C₂×C₄²	—	127	1	no	—
212	C₂×C₄²	—	127	1	no	—
213	C₂×C₄²	—	127	2	no	—
214	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
215	C₂⁴	—	127	1	no	—
216	C₂×Q₈	—	127	8	no	—
217	C₂×Q₈	—	127	4	no	—
218	C₂×Q₈	—	254	16	no	—
219	C₂×Q₈	C₂²×D₁₂₇	1	4	yes	—
220	C₂²×C₄	—	127	1	no	—
221	C₂²×C₄	—	127	8	no	—
222	C₂²×C₄	—	127	1	no	—
223	C₂²×C₄	—	127	2	no	—
224	C₂²×C₄	—	127	1	no	—
225	C₂²×C₄	—	127	1	no	—
226	C₂²×C₄	—	127	1	no	—
227	C₂²×C₄	—	127	2	no	—
228	C₂²×C₄	—	127	2	no	—
229	C₂²×C₄	—	127	2	no	—
230	C₂²×C₄	—	127	2	no	—
231	C₂²×C₄	—	127	8	no	—
232	C₂²×C₄	—	254	8	no	—
233	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
234	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
235	C₄⋊C₄	—	127	8	no	—
236	C₄⋊C₄	—	127	8	no	—
237	C₄⋊C₄	—	127	8	no	—
238	C₄⋊C₄	—	127	8	no	—
239	C₄⋊C₄	—	127	4	no	—
240	C₄⋊C₄	—	127	8	no	—
241	C₄⋊C₄	—	127	4	no	—
242	C₄⋊C₄	C₂²×D₁₂₇	1	4	yes	—
243	C₄⋊C₄	C₂²×D₁₂₇	1	8	yes	—
244	C₄⋊C₄	C₂²×D₁₂₇	1	4	yes	—
245	C₂²⋊C₄	—	127	4	no	—
246	C₂²⋊C₄	—	127	8	no	—
247	C₂²⋊C₄	C₂²×D₁₂₇	1	4	yes	—
248	C₄²	—	127	2	no	—
249	C₄²	—	127	2	no	—
250	C₄²	—	127	2	no	—
251	C₄²	—	127	1	no	—
252	C₄²	—	127	2	no	—
253	C₄²	—	127	1	no	—
254	C₄²	—	127	8	no	—
255	C₄²	—	127	2	no	—
256	C₄²	—	127	4	no	—
257	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
258	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
259	C₄²	C₂²×D₁₂₇	1	2	yes	—
260	C₄²	C₂²×D₁₂₇	1	4	yes	—
261	C₂³	—	127	1	no	—
262	C₂³	—	127	4	no	—
263	C₂³	—	127	1	no	—
264	C₂³	—	254	4	no	—
265	C₂³	C₂³×D₁₂₇	1	1	yes	characteristic
266	Q₈	—	2	8	no	—
267	Q₈	—	254	16	no	—
268	Q₈	—	254	8	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	127	4	no	—
271	C₂×C₄	—	127	1	no	—
272	C₂×C₄	—	127	1	no	—
273	C₂×C₄	—	127	2	no	—
274	C₂×C₄	—	127	4	no	—
275	C₂×C₄	—	127	4	no	—
276	C₂×C₄	—	127	4	no	—
277	C₂×C₄	—	127	4	no	—
278	C₂×C₄	—	127	8	no	—
279	C₂×C₄	—	127	2	no	—
280	C₂×C₄	—	127	8	no	—
281	C₂×C₄	—	254	4	no	—
282	C₂×C₄	—	254	4	no	—
283	C₂×C₄	—	254	16	no	—
284	C₂×C₄	—	254	4	no	—
285	C₂×C₄	—	254	4	no	—
286	C₂×C₄	Q₈×D₁₂₇	1	4	yes	complemented
287	C₂×C₄	C₂³×D₁₂₇	1	2	yes	—
288	C₂×C₄	C₂³×D₁₂₇	1	8	yes	—
289	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
290	C₂×C₄	C₂³×D₁₂₇	1	2	yes	—
291	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
292	C₂²	—	2	2	no	—
293	C₂²	—	127	2	no	—
294	C₂²	—	127	4	no	—
295	C₂²	—	127	2	no	—
296	C₂²	—	254	8	no	—
297	C₂²	—	254	2	no	—
298	C₂²	C₂×Q₈×D₁₂₇	1	2	yes	—
299	C₂²	C₂⁴×D₁₂₇	1	1	yes	characteristic
300	C₄	—	2	2	no	—
301	C₄	—	2	8	no	—
302	C₄	—	127	2	no	—
303	C₄	—	127	2	no	—
304	C₄	—	127	4	no	—
305	C₄	—	254	2	no	—
306	C₄	—	254	8	no	—
307	C₄	C₂×Q₈×D₁₂₇	1	2	yes	—
308	C₄	C₂×Q₈×D₁₂₇	1	2	yes	—
309	C₄	#41×D₁₂₇	1	4	yes	—
310	C₂	—	2	2	no	—
311	C₂	—	127	4	no	—
312	C₂	—	254	2	no	—
313	C₂	C₂²×Q₈×D₁₂₇	1	1	yes	characteristic
314	C₂	C₂×#41×D₁₂₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₂₇
commutator, socle, cosocle	C₂²×C₁₂₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₂₇
Fitting	#533×C₁₂₇	C₂
prime radical	C₂³×D₁₂₇	C₂³
hypercenter	#533	D₁₂₇
hypocenter	C₁₂₇	C₂×#533
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
1. #22550433	C₂	semidirect
2. #22550432	C₂	semidirect
3. #22550431	C₂	semidirect
4. #22550430	C₂	semidirect
5. #22550429	C₂	semidirect
6. #22550428	C₂	semidirect
7. C₄⋊Q₈×D₁₂₇	C₂	semidirect
8. #125×D₁₂₇	C₂	semidirect
10. C₄×Q₈×D₁₂₇	C₂	semidirect
13. #533×C₁₂₇	C₂	semidirect
17. #16861209	C₂²	semidirect
18. #16861208	C₂²	semidirect
19. #16861208	C₂²	semidirect
20. #16861207	C₂²	semidirect
21. #16861201	C₂²	semidirect
22. #16861200	C₂²	semidirect
23. #16861199	C₂²	semidirect
29. Q₈×Dic₁₂₇	C₂²	semidirect
30. C₄×Dic₂₅₄	C₂²	semidirect
31. C₄×Dic₂₅₄	C₂²	semidirect
32. #16861174	C₂²	semidirect
33. #16861174	C₂²	semidirect
51. C₄⋊Q₈×C₁₂₇	C₂²	semidirect
52. #125×C₁₂₇	C₂²	semidirect
54. C₄×Q₈×C₁₂₇	C₂²	semidirect
64. C₂×C₄×D₁₂₇	Q₈	semidirect
110. C₂×Dic₁₂₇	C₂×Q₈	semidirect
122. C₄×D₁₂₇	C₂×Q₈	semidirect
123. C₄×D₁₂₇	C₂×Q₈	semidirect
128. C₂×C₄×C₁₂₇	C₂×Q₈	semidirect
156. D₁₂₇	#533	direct (N × H)
158. C₁₂₇	C₂×#533	semidirect
165. #533	D₁₂₇	direct (N × H)
179. C₄⋊Q₈	C₂×D₁₂₇	semidirect
183. #125	C₂×D₁₂₇	semidirect
195. C₄×Q₈	C₂×D₁₂₇	semidirect
286. C₂×C₄	Q₈×D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550433	2
#22550432	2
#22550431	2
#22550430	2
#22550429	2
#22550428	2
C₄⋊Q₈×D₁₂₇	2
#125×D₁₂₇	2
C₂²⋊Q₈×D₁₂₇	2
C₄×Q₈×D₁₂₇	2
#117×D₁₂₇	2
C₂×C₄²×D₁₂₇	2
#533×C₁₂₇	2
C₂×#533	127

Subgroups — #533×D127 #30652015

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #533×D₁₂₇ #30652015