Subgroups — C₁₂₇×#20576 #44584249

Order 32512.

778 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#446×C₁₂₇	C₂	1	2	yes	maximal
2	C₁₂₇×#2295	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1771	C₂	1	2	yes	maximal
4	C₁₂₇×#1771	C₂	1	2	yes	maximal
5	C₁₂₇×#1770	C₂	1	2	yes	maximal
6	C₁₂₇×#1768	C₂	1	2	yes	maximal
7	C₁₂₇×#1767	C₂	1	1	yes	maximal, characteristic
8	C₁₂₇×#1746	C₂	1	2	yes	maximal
9	C₁₂₇×#1745	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
11	#446×C₁₂₇	C₄	1	16	yes	complemented
12	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
15	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
18	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
19	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
22	#391×C₁₂₇	C₄	1	2	yes	—
23	#389×C₁₂₇	C₄	1	2	yes	—
24	#387×C₁₂₇	C₄	1	4	yes	—
25	C₄×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
27	#376×C₁₂₇	C₂²	1	2	yes	—
28	C₄³×C₁₂₇	C₂²	1	1	yes	characteristic
29	#336×C₁₂₇	C₂²	1	1	yes	characteristic
30	#336×C₁₂₇	C₂²	1	1	yes	characteristic
31	#336×C₁₂₇	C₂²	1	2	yes	—
32	#336×C₁₂₇	C₂²	1	2	yes	—
33	#336×C₁₂₇	C₂²	1	2	yes	—
34	#336×C₁₂₇	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
39	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
40	C₄×Q₈×C₁₂₇	C₂×C₄	1	8	yes	—
41	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
42	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
52	C₄⋊C₈×C₁₂₇	C₂×C₄	1	8	yes	—
53	C₄⋊C₈×C₁₂₇	C₂×C₄	1	8	yes	—
54	C₄⋊C₈×C₁₂₇	C₂×C₄	1	8	yes	—
55	#97×C₁₂₇	—	4	2	no	—
56	#97×C₁₂₇	—	4	1	no	—
57	#97×C₁₂₇	—	4	1	no	—
58	#97×C₁₂₇	C₂×C₄	1	8	yes	—
59	#97×C₁₂₇	C₂×C₄	1	8	yes	—
60	C₄×C₈×C₁₂₇	—	4	2	no	—
61	C₄×C₈×C₁₂₇	C₂×C₄	1	8	yes	—
62	#95×C₁₂₇	C₂×C₄	1	4	yes	—
63	#95×C₁₂₇	C₂×C₄	1	4	yes	—
64	#95×C₁₂₇	C₂×C₄	1	2	yes	—
65	#95×C₁₂₇	C₂×C₄	1	4	yes	—
66	C₂×Q₈×C₁₂₇	—	4	2	no	—
67	C₂×Q₈×C₁₂₇	C₄²	1	4	yes	—
68	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
76	C₂×C₈×C₁₂₇	—	2	2	no	—
77	C₂×C₈×C₁₂₇	—	2	2	no	—
78	C₂×C₈×C₁₂₇	—	2	2	no	—
79	C₂×C₈×C₁₂₇	—	2	2	no	—
80	C₂×C₈×C₁₂₇	—	2	2	no	—
81	C₂×C₈×C₁₂₇	—	2	2	no	—
82	C₂×C₈×C₁₂₇	—	2	2	no	—
83	C₂×C₈×C₁₂₇	—	2	2	no	—
84	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
88	C₄⋊C₄×C₁₂₇	—	4	2	no	—
89	C₄⋊C₄×C₁₂₇	—	4	4	no	—
90	C₄⋊C₄×C₁₂₇	C₄²	1	8	yes	—
91	C₄⋊C₄×C₁₂₇	C₄²	1	4	yes	—
92	C₄²×C₁₂₇	—	2	2	no	—
93	C₄²×C₁₂₇	—	2	1	no	—
94	C₄²×C₁₂₇	—	2	2	no	—
95	C₄²×C₁₂₇	—	2	1	no	—
96	C₄²×C₁₂₇	—	2	2	no	—
97	C₄²×C₁₂₇	—	4	2	no	—
98	C₄²×C₁₂₇	—	4	2	no	—
99	C₄²×C₁₂₇	C₂²×C₄	1	4	yes	—
100	C₄²×C₁₂₇	C₂²×C₄	1	4	yes	—
101	C₄²×C₁₂₇	C₂²×C₄	1	4	yes	—
102	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
103	Q₈×C₁₂₇	—	4	4	no	—
104	C₂×C₄×C₁₂₇	—	2	2	no	—
105	C₂×C₄×C₁₂₇	—	2	2	no	—
106	C₂×C₄×C₁₂₇	—	2	4	no	—
107	C₂×C₄×C₁₂₇	—	2	4	no	—
108	C₂×C₄×C₁₂₇	—	2	2	no	—
109	C₂×C₄×C₁₂₇	—	2	4	no	—
110	C₂×C₄×C₁₂₇	—	2	2	no	—
111	C₂×C₄×C₁₂₇	—	2	2	no	—
112	C₂×C₄×C₁₂₇	—	2	2	no	—
113	C₂×C₄×C₁₂₇	—	2	2	no	—
114	C₂×C₄×C₁₂₇	—	2	2	no	—
115	C₂×C₄×C₁₂₇	—	2	4	no	—
116	C₂×C₄×C₁₂₇	—	2	4	no	—
117	C₂×C₄×C₁₂₇	C₂×C₄²	1	4	yes	—
118	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
119	C₂×C₄×C₁₂₇	C₂×C₄²	1	4	yes	—
120	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₁₂₇	C₂³×C₄	1	2	yes	—
122	C₂×C₄×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂×C₄×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
124	C₂×C₄×C₁₂₇	2₋¹⁺⁴	1	2	yes	—
125	C₈×C₁₂₇	—	2	4	no	—
126	C₈×C₁₂₇	—	2	4	no	—
127	C₈×C₁₂₇	—	2	4	no	—
128	C₈×C₁₂₇	—	2	4	no	—
129	C₂²×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
130	C₂²×C₁₂₇	#519	1	1	yes	characteristic
131	C₂²×C₁₂₇	#520	1	1	yes	characteristic
132	C₂²×C₁₂₇	#568	1	2	yes	—
133	C₂²×C₁₂₇	#568	1	2	yes	—
134	C₄×C₁₂₇	—	2	2	no	—
135	C₄×C₁₂₇	—	2	4	no	—
136	C₄×C₁₂₇	—	2	2	no	—
137	C₄×C₁₂₇	—	2	4	no	—
138	C₄×C₁₂₇	—	4	2	no	—
139	C₄×C₁₂₇	—	4	2	no	—
140	C₄×C₁₂₇	—	4	4	no	—
141	C₄×C₁₂₇	#519	1	2	yes	—
142	C₄×C₁₂₇	#520	1	2	yes	—
143	#20576	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂×C₁₂₇	#2276	1	1	yes	characteristic
145	C₂×C₁₂₇	#2873	1	1	yes	characteristic
146	C₂×C₁₂₇	#2875	1	1	yes	characteristic
147	C₂×C₁₂₇	#2987	1	2	yes	—
148	C₂×C₁₂₇	#2995	1	2	yes	—
149	C₂×#446	C₂×C₁₂₇	1	2	yes	—
150	#2295	C₂×C₁₂₇	1	1	yes	characteristic
151	#1771	C₂×C₁₂₇	1	2	yes	—
152	#1771	C₂×C₁₂₇	1	2	yes	—
153	#1770	C₂×C₁₂₇	1	2	yes	—
154	#1768	C₂×C₁₂₇	1	2	yes	—
155	#1767	C₂×C₁₂₇	1	1	yes	characteristic
156	#1746	C₂×C₁₂₇	1	2	yes	—
157	#1745	C₂×C₁₂₇	1	1	yes	characteristic
158	C₁₂₇	#20576	1	1	yes	characteristic, direct product factor, Sylow
159	C₂×C₄×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
160	#446	C₄×C₁₂₇	1	16	yes	complemented
161	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
162	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
163	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
164	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
165	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
166	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
167	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
168	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
169	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
170	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
171	#391	C₄×C₁₂₇	1	2	yes	—
172	#389	C₄×C₁₂₇	1	2	yes	—
173	#387	C₄×C₁₂₇	1	4	yes	—
174	C₄×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
175	C₄×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
176	#376	C₂²×C₁₂₇	1	2	yes	—
177	C₄³	C₂²×C₁₂₇	1	1	yes	characteristic
178	#336	C₂²×C₁₂₇	1	1	yes	characteristic
179	#336	C₂²×C₁₂₇	1	1	yes	characteristic
180	#336	C₂²×C₁₂₇	1	2	yes	—
181	#336	C₂²×C₁₂₇	1	2	yes	—
182	#336	C₂²×C₁₂₇	1	2	yes	—
183	#336	C₂²×C₁₂₇	1	2	yes	—
184	C₂²×Q₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
185	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
186	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
187	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
188	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
189	C₄×Q₈	C₂×C₄×C₁₂₇	1	8	yes	—
190	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
191	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
192	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
193	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
194	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
195	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
196	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
197	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
198	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
199	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
200	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
201	C₄⋊C₈	C₂×C₄×C₁₂₇	1	8	yes	—
202	C₄⋊C₈	C₂×C₄×C₁₂₇	1	8	yes	—
203	C₄⋊C₈	C₂×C₄×C₁₂₇	1	8	yes	—
204	#97	—	4	2	no	—
205	#97	—	4	1	no	—
206	#97	—	4	1	no	—
207	#97	C₂×C₄×C₁₂₇	1	8	yes	—
208	#97	C₂×C₄×C₁₂₇	1	8	yes	—
209	C₄×C₈	—	4	2	no	—
210	C₄×C₈	C₂×C₄×C₁₂₇	1	8	yes	—
211	#95	C₂×C₄×C₁₂₇	1	4	yes	—
212	#95	C₂×C₄×C₁₂₇	1	4	yes	—
213	#95	C₂×C₄×C₁₂₇	1	2	yes	—
214	#95	C₂×C₄×C₁₂₇	1	4	yes	—
215	C₂×Q₈	—	4	2	no	—
216	C₂×Q₈	C₄²×C₁₂₇	1	4	yes	—
217	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
218	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
219	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
220	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
221	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
222	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
223	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
224	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
234	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
235	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
236	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	—	4	4	no	—
239	C₄⋊C₄	C₄²×C₁₂₇	1	8	yes	—
240	C₄⋊C₄	C₄²×C₁₂₇	1	4	yes	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	1	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	4	2	no	—
247	C₄²	—	4	2	no	—
248	C₄²	C₂²×C₄×C₁₂₇	1	4	yes	—
249	C₄²	C₂²×C₄×C₁₂₇	1	4	yes	—
250	C₄²	C₂²×C₄×C₁₂₇	1	4	yes	—
251	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
252	Q₈	—	4	4	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	C₂×C₄²×C₁₂₇	1	4	yes	—
267	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
268	C₂×C₄	C₂×C₄²×C₁₂₇	1	4	yes	—
269	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
270	C₂×C₄	C₂³×C₄×C₁₂₇	1	2	yes	—
271	C₂×C₄	H(5,𝔽₂)×C₁₂₇	1	1	yes	characteristic
272	C₂×C₄	2₋¹⁺⁴×C₁₂₇	1	1	yes	characteristic
273	C₂×C₄	2₋¹⁺⁴×C₁₂₇	1	2	yes	—
274	C₈	—	2	4	no	—
275	C₈	—	2	4	no	—
276	C₈	—	2	4	no	—
277	C₈	—	2	4	no	—
278	C₂²	C₂²×C₄²×C₁₂₇	1	1	yes	characteristic
279	C₂²	#519×C₁₂₇	1	1	yes	characteristic
280	C₂²	#520×C₁₂₇	1	1	yes	characteristic
281	C₂²	#568×C₁₂₇	1	2	yes	—
282	C₂²	#568×C₁₂₇	1	2	yes	—
283	C₄	—	2	2	no	—
284	C₄	—	2	4	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	4	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	2	no	—
289	C₄	—	4	4	no	—
290	C₄	#519×C₁₂₇	1	2	yes	—
291	C₄	#520×C₁₂₇	1	2	yes	—
292	C₂	C₁₂₇×#2276	1	1	yes	characteristic
293	C₂	C₁₂₇×#2873	1	1	yes	characteristic
294	C₂	C₁₂₇×#2875	1	1	yes	characteristic
295	C₂	C₁₂₇×#2987	1	2	yes	—
296	C₂	C₁₂₇×#2995	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₂₇	C₂⁴
commutator	C₂²	C₂²×C₄²×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle, prime radical	C₂³×C₁₂₇	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
11. #446×C₁₂₇	C₄	semidirect
143. #20576	C₁₂₇	direct (N × H)
158. C₁₂₇	#20576	direct (N × H)
160. #446	C₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#446×C₁₂₇	2
C₁₂₇×#2295	2
C₁₂₇×#1771	2
C₁₂₇×#1770	2
C₁₂₇×#1768	2
C₁₂₇×#1767	2
C₁₂₇×#1746	2
C₁₂₇×#1745	2
#20576	127

Subgroups — C127×#20576 #44584249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#20576 #44584249