Subgroups — C₁₂₇×#17334 #44581007

Order 32512.

806 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#2920	C₂	1	2	yes	maximal
2	C₂×C₄×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1455	C₂	1	4	yes	maximal, complemented
4	C₁₂₇×#1453	C₂	1	4	yes	maximal, complemented
5	C₁₂₇×#1452	C₂	1	2	yes	maximal, complemented
6	C₁₂₇×#1450	C₂	1	2	yes	maximal, complemented
7	C₂²×M₁₆×C₁₂₇	C₄	1	2	yes	—
8	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
9	#432×C₁₂₇	C₄	1	4	yes	—
10	#432×C₁₂₇	C₄	1	4	yes	—
11	#431×C₁₂₇	C₂²	1	4	yes	—
12	#431×C₁₂₇	C₂²	1	4	yes	—
13	#423×C₁₂₇	C₄	1	4	yes	—
14	#407×C₁₂₇	C₄	1	4	yes	—
15	C₄×M₁₆×C₁₂₇	C₄	1	2	yes	—
16	C₄×M₁₆×C₁₂₇	C₄	1	2	yes	—
17	C₄×M₁₆×C₁₂₇	C₂²	1	2	yes	—
18	C₄×M₁₆×C₁₂₇	C₂²	1	2	yes	—
19	C₄×M₁₆×C₁₂₇	C₂²	1	4	yes	—
20	C₂×#97×C₁₂₇	C₄	1	2	yes	—
21	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
23	#322×C₁₂₇	C₂²	1	4	yes	complemented
24	#322×C₁₂₇	C₂²	1	8	yes	complemented
25	C₈²×C₁₂₇	C₂²	1	4	yes	complemented
26	C₂³×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
28	C₂×M₁₆×C₁₂₇	—	2	8	no	—
29	C₂×M₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
30	C₂×M₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
31	C₂×M₁₆×C₁₂₇	C₂×C₄	1	8	yes	—
32	C₂²×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₁₂₇	C₂×C₄	1	8	yes	—
44	C₄⋊C₈×C₁₂₇	C₂×C₄	1	8	yes	—
45	C₂²⋊C₈×C₁₂₇	C₂×C₄	1	8	yes	—
46	C₂²⋊C₈×C₁₂₇	C₂×C₄	1	8	yes	—
47	#97×C₁₂₇	C₂×C₄	1	4	yes	—
48	#97×C₁₂₇	C₂×C₄	1	4	yes	—
49	#97×C₁₂₇	C₂×C₄	1	8	yes	—
50	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₁₂₇	C₂³	1	2	yes	—
53	C₄×C₈×C₁₂₇	C₂³	1	8	yes	—
54	C₄×C₈×C₁₂₇	C₂³	1	2	yes	—
55	C₂⁴×C₁₂₇	C₄²	1	1	yes	characteristic
56	C₂²×C₄×C₁₂₇	—	2	4	no	—
57	C₂²×C₄×C₁₂₇	—	4	4	no	—
58	C₂²×C₄×C₁₂₇	C₄²	1	1	yes	characteristic
59	C₂²×C₄×C₁₂₇	C₄²	1	1	yes	characteristic
60	C₂²×C₄×C₁₂₇	C₄²	1	1	yes	characteristic
61	C₂²×C₄×C₁₂₇	C₄²	1	2	yes	—
62	C₂²×C₄×C₁₂₇	C₄²	1	2	yes	—
63	C₂²×C₄×C₁₂₇	C₄²	1	2	yes	—
64	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
70	M₁₆×C₁₂₇	—	2	8	no	—
71	M₁₆×C₁₂₇	—	2	8	no	—
72	M₁₆×C₁₂₇	—	4	8	no	—
73	C₂×C₈×C₁₂₇	—	2	4	no	—
74	C₂×C₈×C₁₂₇	M₁₆	1	8	yes	complemented
75	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
76	C₂×C₈×C₁₂₇	C₂²×C₄	1	16	yes	—
77	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
78	C₄²×C₁₂₇	C₄²	1	2	yes	—
79	C₄²×C₁₂₇	C₄²	1	2	yes	—
80	C₄²×C₁₂₇	C₄²	1	2	yes	—
81	C₄²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
85	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
86	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
87	C₄²×C₁₂₇	C₂⁴	1	1	yes	characteristic
88	C₂³×C₁₂₇	—	2	2	no	—
89	C₂³×C₁₂₇	—	4	2	no	—
90	C₂³×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
91	C₂³×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
92	C₂³×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
93	C₂×C₄×C₁₂₇	—	2	2	no	—
94	C₂×C₄×C₁₂₇	—	2	2	no	—
95	C₂×C₄×C₁₂₇	—	2	2	no	—
96	C₂×C₄×C₁₂₇	—	2	4	no	—
97	C₂×C₄×C₁₂₇	—	2	4	no	—
98	C₂×C₄×C₁₂₇	—	2	4	no	—
99	C₂×C₄×C₁₂₇	—	2	2	no	—
100	C₂×C₄×C₁₂₇	—	2	2	no	—
101	C₂×C₄×C₁₂₇	—	4	2	no	—
102	C₂×C₄×C₁₂₇	—	4	2	no	—
103	C₂×C₄×C₁₂₇	—	4	2	no	—
104	C₂×C₄×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₄×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
106	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
107	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
108	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
109	C₂×C₄×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₁₂₇	C₂×M₁₆	1	4	yes	—
111	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₂₇	C₂³×C₄	1	2	yes	—
113	C₈×C₁₂₇	—	2	4	no	—
114	C₈×C₁₂₇	—	2	16	no	—
115	C₈×C₁₂₇	C₂×M₁₆	1	8	yes	—
116	C₂²×C₁₂₇	—	2	2	no	—
117	C₂²×C₁₂₇	—	2	1	no	—
118	C₂²×C₁₂₇	—	2	1	no	—
119	C₂²×C₁₂₇	—	2	2	no	—
120	C₂²×C₁₂₇	—	2	2	no	—
121	C₂²×C₁₂₇	—	4	4	no	—
122	C₂²×C₁₂₇	C₄×M₁₆	1	2	yes	—
123	C₂²×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
124	C₄×C₁₂₇	—	2	1	no	—
125	C₄×C₁₂₇	—	2	1	no	—
126	C₄×C₁₂₇	—	2	2	no	—
127	C₄×C₁₂₇	—	2	2	no	—
128	C₄×C₁₂₇	—	2	2	no	—
129	C₄×C₁₂₇	C₄×M₁₆	1	2	yes	—
130	C₄×C₁₂₇	C₂²×M₁₆	1	2	yes	—
131	C₄×C₁₂₇	#568	1	4	yes	—
132	#17334	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂×C₁₂₇	—	2	2	no	—
134	C₂×C₁₂₇	—	4	1	no	—
135	C₂×C₁₂₇	—	4	1	no	—
136	C₂×C₁₂₇	C₂×C₄×M₁₆	1	1	yes	characteristic
137	C₂×C₁₂₇	#2873	1	2	yes	—
138	#2920	C₂×C₁₂₇	1	2	yes	—
139	C₂×C₄×M₁₆	C₂×C₁₂₇	1	1	yes	characteristic
140	#1455	C₂×C₁₂₇	1	4	yes	complemented
141	#1453	C₂×C₁₂₇	1	4	yes	complemented
142	#1452	C₂×C₁₂₇	1	2	yes	complemented
143	#1450	C₂×C₁₂₇	1	2	yes	complemented
144	C₁₂₇	#17334	1	1	yes	characteristic, direct product factor, Sylow
145	C₂²×M₁₆	C₄×C₁₂₇	1	2	yes	—
146	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
147	#432	C₄×C₁₂₇	1	4	yes	—
148	#432	C₄×C₁₂₇	1	4	yes	—
149	#431	C₂²×C₁₂₇	1	4	yes	—
150	#431	C₂²×C₁₂₇	1	4	yes	—
151	#423	C₄×C₁₂₇	1	4	yes	—
152	#407	C₄×C₁₂₇	1	4	yes	—
153	C₄×M₁₆	C₄×C₁₂₇	1	2	yes	—
154	C₄×M₁₆	C₄×C₁₂₇	1	2	yes	—
155	C₄×M₁₆	C₂²×C₁₂₇	1	2	yes	—
156	C₄×M₁₆	C₂²×C₁₂₇	1	2	yes	—
157	C₄×M₁₆	C₂²×C₁₂₇	1	4	yes	—
158	C₂×#97	C₄×C₁₂₇	1	2	yes	—
159	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
160	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
161	#322	C₂²×C₁₂₇	1	4	yes	complemented
162	#322	C₂²×C₁₂₇	1	8	yes	complemented
163	C₈²	C₂²×C₁₂₇	1	4	yes	complemented
164	C₂³×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
165	C₂³×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
166	C₂×M₁₆	—	2	8	no	—
167	C₂×M₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
168	C₂×M₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
169	C₂×M₁₆	C₂×C₄×C₁₂₇	1	8	yes	—
170	C₂²×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
171	C₂²×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
172	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
173	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
174	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
175	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
176	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
177	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
178	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
179	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
180	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
181	C₄⋊C₈	C₂×C₄×C₁₂₇	1	8	yes	—
182	C₄⋊C₈	C₂×C₄×C₁₂₇	1	8	yes	—
183	C₂²⋊C₈	C₂×C₄×C₁₂₇	1	8	yes	—
184	C₂²⋊C₈	C₂×C₄×C₁₂₇	1	8	yes	—
185	#97	C₂×C₄×C₁₂₇	1	4	yes	—
186	#97	C₂×C₄×C₁₂₇	1	4	yes	—
187	#97	C₂×C₄×C₁₂₇	1	8	yes	—
188	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
189	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
190	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
191	C₄×C₈	C₂³×C₁₂₇	1	8	yes	—
192	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
193	C₂⁴	C₄²×C₁₂₇	1	1	yes	characteristic
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	4	4	no	—
196	C₂²×C₄	C₄²×C₁₂₇	1	1	yes	characteristic
197	C₂²×C₄	C₄²×C₁₂₇	1	1	yes	characteristic
198	C₂²×C₄	C₄²×C₁₂₇	1	1	yes	characteristic
199	C₂²×C₄	C₄²×C₁₂₇	1	2	yes	—
200	C₂²×C₄	C₄²×C₁₂₇	1	2	yes	—
201	C₂²×C₄	C₄²×C₁₂₇	1	2	yes	—
202	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
203	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
204	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
205	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
206	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
207	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
208	M₁₆	—	2	8	no	—
209	M₁₆	—	2	8	no	—
210	M₁₆	—	4	8	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	M₁₆×C₁₂₇	1	8	yes	complemented
213	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
214	C₂×C₈	C₂²×C₄×C₁₂₇	1	16	yes	—
215	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
216	C₄²	C₄²×C₁₂₇	1	2	yes	—
217	C₄²	C₄²×C₁₂₇	1	2	yes	—
218	C₄²	C₄²×C₁₂₇	1	2	yes	—
219	C₄²	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
220	C₄²	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
221	C₄²	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
222	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
223	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
224	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
225	C₄²	C₂⁴×C₁₂₇	1	1	yes	characteristic
226	C₂³	—	2	2	no	—
227	C₂³	—	4	2	no	—
228	C₂³	C₂×C₄²×C₁₂₇	1	1	yes	characteristic
229	C₂³	C₂×C₄²×C₁₂₇	1	1	yes	characteristic
230	C₂³	C₂×C₄²×C₁₂₇	1	1	yes	characteristic
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	C₂×C₄²×C₁₂₇	1	1	yes	characteristic
243	C₂×C₄	C₂×C₄²×C₁₂₇	1	1	yes	characteristic
244	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
245	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
246	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
247	C₂×C₄	C₂×C₄²×C₁₂₇	1	1	yes	characteristic
248	C₂×C₄	C₂×M₁₆×C₁₂₇	1	4	yes	—
249	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
250	C₂×C₄	C₂³×C₄×C₁₂₇	1	2	yes	—
251	C₈	—	2	4	no	—
252	C₈	—	2	16	no	—
253	C₈	C₂×M₁₆×C₁₂₇	1	8	yes	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	4	4	no	—
260	C₂²	C₄×M₁₆×C₁₂₇	1	2	yes	—
261	C₂²	C₂²×C₄²×C₁₂₇	1	1	yes	characteristic
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	C₄×M₁₆×C₁₂₇	1	2	yes	—
268	C₄	C₂²×M₁₆×C₁₂₇	1	2	yes	—
269	C₄	#568×C₁₂₇	1	4	yes	—
270	C₂	—	2	2	no	—
271	C₂	—	4	1	no	—
272	C₂	—	4	1	no	—
273	C₂	C₂×C₄×M₁₆×C₁₂₇	1	1	yes	characteristic
274	C₂	C₁₂₇×#2873	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₁₂₇×#1455	C₂	semidirect
4. C₁₂₇×#1453	C₂	semidirect
5. C₁₂₇×#1452	C₂	semidirect
6. C₁₂₇×#1450	C₂	semidirect
23. #322×C₁₂₇	C₂²	semidirect
24. #322×C₁₂₇	C₂²	semidirect
25. C₈²×C₁₂₇	C₂²	semidirect
74. C₂×C₈×C₁₂₇	M₁₆	semidirect
132. #17334	C₁₂₇	direct (N × H)
140. #1455	C₂×C₁₂₇	semidirect
141. #1453	C₂×C₁₂₇	semidirect
142. #1452	C₂×C₁₂₇	semidirect
143. #1450	C₂×C₁₂₇	semidirect
144. C₁₂₇	#17334	direct (N × H)
161. #322	C₂²×C₁₂₇	semidirect
162. #322	C₂²×C₁₂₇	semidirect
163. C₈²	C₂²×C₁₂₇	semidirect
212. C₂×C₈	M₁₆×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×#2920	2
C₂×C₄×M₁₆×C₁₂₇	2
C₁₂₇×#1455	2
C₁₂₇×#1453	2
C₁₂₇×#1452	2
C₁₂₇×#1450	2
#17334	127

Subgroups — C127×#17334 #44581007

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#17334 #44581007