Subgroups — C₁₂₇×#20513 #44584186

Order 32512.

998 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#444×C₁₂₇	C₂	1	2	yes	maximal
2	C₄×#135×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#2875	C₂	1	2	yes	maximal, complemented
4	C₄×#108×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₂₇×#1773	C₂	1	2	yes	maximal, complemented
6	C₁₂₇×#1764	C₂	1	4	yes	maximal, complemented
7	C₁₂₇×#1759	C₂	1	2	yes	maximal, complemented
8	C₁₂₇×#1726	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂×#131×C₁₂₇	C₂²	1	4	yes	—
11	C₄×#41×C₁₂₇	C₂²	1	2	yes	—
12	#444×C₁₂₇	C₄	1	16	yes	complemented
13	C₄×Q₁₆×C₁₂₇	C₄	1	2	yes	complemented
14	C₄×SD₁₆×C₁₂₇	C₄	1	4	yes	complemented
15	C₄×D₈×C₁₂₇	C₄	1	2	yes	complemented
16	C₂×#108×C₁₂₇	C₂²	1	2	yes	—
17	#427×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂×#104×C₁₂₇	C₂²	1	4	yes	—
19	#416×C₁₂₇	C₂²	1	2	yes	—
20	#405×C₁₂₇	—	2	4	no	—
21	#405×C₁₂₇	C₂²	1	4	yes	—
22	C₄×M₁₆×C₁₂₇	—	2	2	no	—
23	C₄×M₁₆×C₁₂₇	C₂²	1	2	yes	—
24	C₂×#97×C₁₂₇	—	2	2	no	—
25	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
26	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
29	#376×C₁₂₇	C₂²	1	1	yes	characteristic
30	#341×C₁₂₇	C₂²	1	2	yes	—
31	#339×C₁₂₇	C₂²	1	4	yes	—
32	C₂×#41×C₁₂₇	C₂³	1	2	yes	—
33	#135×C₁₂₇	C₂×C₄	1	8	yes	—
34	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
35	C₂×SD₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
36	C₂×D₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
37	#131×C₁₂₇	—	2	8	no	—
38	#131×C₁₂₇	C₂×C₄	1	16	yes	complemented
39	C₂×M₁₆×C₁₂₇	—	2	4	no	—
40	C₂×M₁₆×C₁₂₇	C₂³	1	4	yes	—
41	C₂²×C₈×C₁₂₇	—	2	4	no	—
42	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
45	C₄×Q₈×C₁₂₇	C₂×C₄	1	4	yes	—
46	C₄×D₄×C₁₂₇	—	2	4	no	—
47	C₄×D₄×C₁₂₇	C₂×C₄	1	4	yes	—
48	#117×C₁₂₇	—	2	2	no	—
49	#117×C₁₂₇	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₂₇	—	2	2	no	—
51	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
52	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
53	#108×C₁₂₇	C₂×C₄	1	8	yes	complemented
54	#107×C₁₂₇	C₂×C₄	1	2	yes	—
55	#106×C₁₂₇	C₂×C₄	1	2	yes	—
56	#104×C₁₂₇	C₂×C₄	1	16	yes	complemented
57	#103×C₁₂₇	C₂×C₄	1	4	yes	—
58	#102×C₁₂₇	C₂×C₄	1	4	yes	—
59	#97×C₁₂₇	—	2	4	no	—
60	#97×C₁₂₇	—	2	2	no	—
61	#97×C₁₂₇	C₂×C₄	1	8	yes	complemented
62	#97×C₁₂₇	D₄	1	2	yes	—
63	#97×C₁₂₇	D₄	1	2	yes	—
64	C₄×C₈×C₁₂₇	—	2	4	no	—
65	C₄×C₈×C₁₂₇	—	2	4	no	—
66	C₄×C₈×C₁₂₇	—	2	2	no	—
67	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
69	#95×C₁₂₇	—	2	2	no	—
70	#41×C₁₂₇	—	2	4	no	—
71	#41×C₁₂₇	C₂²×C₄	1	8	yes	—
72	C₂×Q₈×C₁₂₇	C₂²×C₄	1	2	yes	—
73	C₂×D₄×C₁₂₇	—	2	2	no	—
74	C₂×D₄×C₁₂₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₁₂₇	—	2	2	no	—
76	C₂²×C₄×C₁₂₇	—	2	2	no	—
77	C₂²×C₄×C₁₂₇	—	2	4	no	—
78	C₂²×C₄×C₁₂₇	#41	1	2	yes	—
79	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
80	Q₁₆×C₁₂₇	C₄²	1	4	yes	—
81	SD₁₆×C₁₂₇	C₄²	1	8	yes	—
82	D₈×C₁₂₇	C₄²	1	4	yes	—
83	M₁₆×C₁₂₇	—	2	4	no	—
84	M₁₆×C₁₂₇	—	2	16	no	—
85	M₁₆×C₁₂₇	C₂²×C₄	1	8	yes	—
86	C₂×C₈×C₁₂₇	—	2	4	no	—
87	C₂×C₈×C₁₂₇	—	2	2	no	—
88	C₂×C₈×C₁₂₇	—	2	16	no	—
89	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
93	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
94	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
95	C₂×C₈×C₁₂₇	#41	1	4	yes	—
96	C₄⋊C₄×C₁₂₇	—	2	4	no	—
97	C₄⋊C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
98	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
99	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
100	C₄²×C₁₂₇	—	2	4	no	—
101	C₄²×C₁₂₇	—	2	2	no	—
102	C₄²×C₁₂₇	—	2	2	no	—
103	C₄²×C₁₂₇	C₂²×C₄	1	4	yes	—
104	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
105	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
107	C₂³×C₁₂₇	—	2	2	no	—
108	C₂³×C₁₂₇	C₂×#41	1	1	yes	characteristic
109	Q₈×C₁₂₇	—	2	2	no	—
110	Q₈×C₁₂₇	C₂×C₄²	1	4	yes	—
111	D₄×C₁₂₇	—	2	8	no	—
112	D₄×C₁₂₇	—	2	2	no	—
113	D₄×C₁₂₇	C₂×C₄²	1	4	yes	—
114	C₂×C₄×C₁₂₇	—	2	2	no	—
115	C₂×C₄×C₁₂₇	—	2	8	no	—
116	C₂×C₄×C₁₂₇	—	2	4	no	—
117	C₂×C₄×C₁₂₇	—	2	4	no	—
118	C₂×C₄×C₁₂₇	—	4	2	no	—
119	C₂×C₄×C₁₂₇	—	4	4	no	—
120	C₂×C₄×C₁₂₇	—	8	4	no	—
121	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₁₂₇	C₂³×C₄	1	2	yes	—
124	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₂₇	C₂×#41	1	2	yes	—
127	C₈×C₁₂₇	—	2	8	no	—
128	C₈×C₁₂₇	—	2	2	no	—
129	C₈×C₁₂₇	C₂×C₄²	1	2	yes	—
130	C₈×C₁₂₇	C₂×C₄²	1	2	yes	—
131	C₈×C₁₂₇	C₄×D₄	1	8	yes	—
132	C₂²×C₁₂₇	—	2	2	no	—
133	C₂²×C₁₂₇	—	2	4	no	—
134	C₂²×C₁₂₇	—	4	4	no	—
135	C₂²×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂²×C₁₂₇	C₄×#41	1	2	yes	—
137	C₄×C₁₂₇	—	2	2	no	—
138	C₄×C₁₂₇	—	2	2	no	—
139	C₄×C₁₂₇	—	2	4	no	—
140	C₄×C₁₂₇	—	4	4	no	—
141	C₄×C₁₂₇	—	4	4	no	—
142	C₄×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
143	C₄×C₁₂₇	C₂×C₄×D₄	1	2	yes	—
144	C₄×C₁₂₇	C₄×#41	1	1	yes	characteristic
145	#20513	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂×C₁₂₇	—	2	2	no	—
147	C₂×C₁₂₇	—	4	4	no	—
148	C₂×C₁₂₇	C₄²×D₄	1	1	yes	characteristic
149	C₂×C₁₂₇	#2957	1	2	yes	—
150	C₂×#444	C₂×C₁₂₇	1	2	yes	—
151	C₄×#135	C₂×C₁₂₇	1	1	yes	characteristic
152	#2875	C₂×C₁₂₇	1	2	yes	complemented
153	C₄×#108	C₂×C₁₂₇	1	1	yes	characteristic, complemented
154	#1773	C₂×C₁₂₇	1	2	yes	complemented
155	#1764	C₂×C₁₂₇	1	4	yes	complemented
156	#1759	C₂×C₁₂₇	1	2	yes	complemented
157	#1726	C₂×C₁₂₇	1	1	yes	characteristic, complemented
158	C₁₂₇	#20513	1	1	yes	characteristic, direct product factor, Sylow
159	C₂×#135	C₂²×C₁₂₇	1	1	yes	characteristic
160	C₂×#131	C₂²×C₁₂₇	1	4	yes	—
161	C₄×#41	C₂²×C₁₂₇	1	2	yes	—
162	#444	C₄×C₁₂₇	1	16	yes	complemented
163	C₄×Q₁₆	C₄×C₁₂₇	1	2	yes	complemented
164	C₄×SD₁₆	C₄×C₁₂₇	1	4	yes	complemented
165	C₄×D₈	C₄×C₁₂₇	1	2	yes	complemented
166	C₂×#108	C₂²×C₁₂₇	1	2	yes	—
167	#427	C₂²×C₁₂₇	1	1	yes	characteristic
168	C₂×#104	C₂²×C₁₂₇	1	4	yes	—
169	#416	C₂²×C₁₂₇	1	2	yes	—
170	#405	—	2	4	no	—
171	#405	C₂²×C₁₂₇	1	4	yes	—
172	C₄×M₁₆	—	2	2	no	—
173	C₄×M₁₆	C₂²×C₁₂₇	1	2	yes	—
174	C₂×#97	—	2	2	no	—
175	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
176	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
177	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
178	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
179	#376	C₂²×C₁₂₇	1	1	yes	characteristic
180	#341	C₂²×C₁₂₇	1	2	yes	—
181	#339	C₂²×C₁₂₇	1	4	yes	—
182	C₂×#41	C₂³×C₁₂₇	1	2	yes	—
183	#135	C₂×C₄×C₁₂₇	1	8	yes	—
184	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
185	C₂×SD₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
186	C₂×D₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
187	#131	—	2	8	no	—
188	#131	C₂×C₄×C₁₂₇	1	16	yes	complemented
189	C₂×M₁₆	—	2	4	no	—
190	C₂×M₁₆	C₂³×C₁₂₇	1	4	yes	—
191	C₂²×C₈	—	2	4	no	—
192	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
193	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
194	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
195	C₄×Q₈	C₂×C₄×C₁₂₇	1	4	yes	—
196	C₄×D₄	—	2	4	no	—
197	C₄×D₄	C₂×C₄×C₁₂₇	1	4	yes	—
198	#117	—	2	2	no	—
199	#117	C₂³×C₁₂₇	1	2	yes	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
202	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
203	#108	C₂×C₄×C₁₂₇	1	8	yes	complemented
204	#107	C₂×C₄×C₁₂₇	1	2	yes	—
205	#106	C₂×C₄×C₁₂₇	1	2	yes	—
206	#104	C₂×C₄×C₁₂₇	1	16	yes	complemented
207	#103	C₂×C₄×C₁₂₇	1	4	yes	—
208	#102	C₂×C₄×C₁₂₇	1	4	yes	—
209	#97	—	2	4	no	—
210	#97	—	2	2	no	—
211	#97	C₂×C₄×C₁₂₇	1	8	yes	complemented
212	#97	D₄×C₁₂₇	1	2	yes	—
213	#97	D₄×C₁₂₇	1	2	yes	—
214	C₄×C₈	—	2	4	no	—
215	C₄×C₈	—	2	4	no	—
216	C₄×C₈	—	2	2	no	—
217	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
218	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
219	#95	—	2	2	no	—
220	#41	—	2	4	no	—
221	#41	C₂²×C₄×C₁₂₇	1	8	yes	—
222	C₂×Q₈	C₂²×C₄×C₁₂₇	1	2	yes	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	C₂²×C₄×C₁₂₇	1	2	yes	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	#41×C₁₂₇	1	2	yes	—
229	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
230	Q₁₆	C₄²×C₁₂₇	1	4	yes	—
231	SD₁₆	C₄²×C₁₂₇	1	8	yes	—
232	D₈	C₄²×C₁₂₇	1	4	yes	—
233	M₁₆	—	2	4	no	—
234	M₁₆	—	2	16	no	—
235	M₁₆	C₂²×C₄×C₁₂₇	1	8	yes	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	16	no	—
239	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
240	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
241	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
242	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
243	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
244	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
245	C₂×C₈	#41×C₁₂₇	1	4	yes	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
248	C₂²⋊C₄	—	2	2	no	—
249	C₂²⋊C₄	—	2	4	no	—
250	C₄²	—	2	4	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	C₂²×C₄×C₁₂₇	1	4	yes	—
254	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
255	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
256	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
257	C₂³	—	2	2	no	—
258	C₂³	C₂×#41×C₁₂₇	1	1	yes	characteristic
259	Q₈	—	2	2	no	—
260	Q₈	C₂×C₄²×C₁₂₇	1	4	yes	—
261	D₄	—	2	8	no	—
262	D₄	—	2	2	no	—
263	D₄	C₂×C₄²×C₁₂₇	1	4	yes	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	8	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	8	4	no	—
271	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
272	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
273	C₂×C₄	C₂³×C₄×C₁₂₇	1	2	yes	—
274	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
275	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
276	C₂×C₄	C₂×#41×C₁₂₇	1	2	yes	—
277	C₈	—	2	8	no	—
278	C₈	—	2	2	no	—
279	C₈	C₂×C₄²×C₁₂₇	1	2	yes	—
280	C₈	C₂×C₄²×C₁₂₇	1	2	yes	—
281	C₈	C₄×D₄×C₁₂₇	1	8	yes	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	4	4	no	—
285	C₂²	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
286	C₂²	C₄×#41×C₁₂₇	1	2	yes	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	2	4	no	—
290	C₄	—	4	4	no	—
291	C₄	—	4	4	no	—
292	C₄	C₂²×C₄²×C₁₂₇	1	1	yes	characteristic
293	C₄	C₂×C₄×D₄×C₁₂₇	1	2	yes	—
294	C₄	C₄×#41×C₁₂₇	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	—	4	4	no	—
297	C₂	C₄²×D₄×C₁₂₇	1	1	yes	characteristic
298	C₂	C₁₂₇×#2957	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂²×D₄
commutator	C₄	C₂²×C₄²×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle	C₂²×C₁₂₇	C₂×C₄×D₄
prime radical	C₂×#135×C₁₂₇	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂×#97×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₂₇×#2875	C₂	semidirect
4. C₄×#108×C₁₂₇	C₂	semidirect
5. C₁₂₇×#1773	C₂	semidirect
6. C₁₂₇×#1764	C₂	semidirect
7. C₁₂₇×#1759	C₂	semidirect
8. C₁₂₇×#1726	C₂	semidirect
12. #444×C₁₂₇	C₄	semidirect
13. C₄×Q₁₆×C₁₂₇	C₄	semidirect
14. C₄×SD₁₆×C₁₂₇	C₄	semidirect
15. C₄×D₈×C₁₂₇	C₄	semidirect
38. #131×C₁₂₇	C₂×C₄	semidirect
53. #108×C₁₂₇	C₂×C₄	semidirect
56. #104×C₁₂₇	C₂×C₄	semidirect
61. #97×C₁₂₇	C₂×C₄	semidirect
145. #20513	C₁₂₇	direct (N × H)
152. #2875	C₂×C₁₂₇	semidirect
153. C₄×#108	C₂×C₁₂₇	semidirect
154. #1773	C₂×C₁₂₇	semidirect
155. #1764	C₂×C₁₂₇	semidirect
156. #1759	C₂×C₁₂₇	semidirect
157. #1726	C₂×C₁₂₇	semidirect
158. C₁₂₇	#20513	direct (N × H)
162. #444	C₄×C₁₂₇	semidirect
163. C₄×Q₁₆	C₄×C₁₂₇	semidirect
164. C₄×SD₁₆	C₄×C₁₂₇	semidirect
165. C₄×D₈	C₄×C₁₂₇	semidirect
188. #131	C₂×C₄×C₁₂₇	semidirect
203. #108	C₂×C₄×C₁₂₇	semidirect
206. #104	C₂×C₄×C₁₂₇	semidirect
211. #97	C₂×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×#444×C₁₂₇	2
C₄×#135×C₁₂₇	2
C₁₂₇×#2875	2
C₄×#108×C₁₂₇	2
C₁₂₇×#1773	2
C₁₂₇×#1764	2
C₁₂₇×#1759	2
C₁₂₇×#1726	2
#20513	127

Subgroups — C127×#20513 #44584186

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#20513 #44584186