Subgroups — C₁₂₇×#21840 #44585513

Order 32512.

1238 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#432×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#2283	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1993	C₂	1	4	yes	maximal, complemented
5	C₁₂₇×#1987	C₂	1	4	yes	maximal, complemented
6	C₁₂₇×#1823	C₂	1	4	yes	maximal
7	C₂³×Q₈×C₁₂₇	—	2	1	no	—
8	C₂×C₄⋊Q₈×C₁₂₇	—	2	1	no	—
9	C₂×C₄⋊Q₈×C₁₂₇	—	2	1	no	—
10	C₂×C₄⋊Q₈×C₁₂₇	—	2	4	no	—
11	C₂×C₄⋊Q₈×C₁₂₇	C₄	1	8	yes	complemented
12	C₂×C₄⋊Q₈×C₁₂₇	C₂²	1	2	yes	—
13	C₂×C₄⋊Q₈×C₁₂₇	C₂²	1	2	yes	—
14	C₂²×C₄⋊C₄×C₁₂₇	—	2	1	no	—
15	C₂²×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
17	#432×C₁₂₇	—	2	4	no	—
18	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₈×C₁₂₇	C₂²	1	2	yes	—
20	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
21	#383×C₁₂₇	—	2	1	no	—
22	#382×C₁₂₇	C₂²	1	2	yes	—
23	#381×C₁₂₇	—	2	1	no	—
24	#380×C₁₂₇	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
26	C₄×C₂²⋊C₄×C₁₂₇	—	2	1	no	—
27	#376×C₁₂₇	—	2	1	no	—
28	#340×C₁₂₇	C₂²	1	16	yes	—
29	C₂²×Q₈×C₁₂₇	—	2	8	no	—
30	C₂²×Q₈×C₁₂₇	—	2	2	no	—
31	C₂²×Q₈×C₁₂₇	—	2	2	no	—
32	C₂²×Q₈×C₁₂₇	—	4	8	no	—
33	C₂³×C₄×C₁₂₇	—	2	2	no	—
34	C₂³×C₄×C₁₂₇	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₁₂₇	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₁₂₇	C₂³	1	4	yes	—
38	C₄⋊Q₈×C₁₂₇	—	2	8	no	—
39	C₄⋊Q₈×C₁₂₇	—	2	8	no	—
40	C₄⋊Q₈×C₁₂₇	—	2	4	no	—
41	C₄⋊Q₈×C₁₂₇	—	2	4	no	—
42	C₄⋊Q₈×C₁₂₇	C₂×C₄	1	8	yes	—
43	C₄⋊Q₈×C₁₂₇	C₂×C₄	1	8	yes	—
44	C₂×C₄⋊C₄×C₁₂₇	—	2	8	no	—
45	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₁₂₇	—	2	4	no	—
49	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	8	yes	—
50	C₂×C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
51	C₂×C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
52	C₂×C₂²⋊C₄×C₁₂₇	—	2	2	no	—
53	C₂×C₄²×C₁₂₇	—	2	2	no	—
54	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
55	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₁₂₇	D₄	1	4	yes	—
58	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₁₂₇	—	2	4	no	—
61	C₂²⋊C₈×C₁₂₇	—	2	4	no	—
62	#97×C₁₂₇	—	2	4	no	—
63	#95×C₁₂₇	—	2	2	no	—
64	#95×C₁₂₇	—	2	2	no	—
65	#95×C₁₂₇	—	2	2	no	—
66	#95×C₁₂₇	—	2	2	no	—
67	C₂⁴×C₁₂₇	C₂×D₄	1	1	yes	characteristic
68	C₂×Q₈×C₁₂₇	—	2	8	no	—
69	C₂×Q₈×C₁₂₇	—	2	8	no	—
70	C₂×Q₈×C₁₂₇	—	4	8	no	—
71	C₂×Q₈×C₁₂₇	—	4	8	no	—
72	C₂×Q₈×C₁₂₇	—	4	32	no	—
73	C₂²×C₄×C₁₂₇	—	2	4	no	—
74	C₂²×C₄×C₁₂₇	—	2	4	no	—
75	C₂²×C₄×C₁₂₇	—	2	4	no	—
76	C₂²×C₄×C₁₂₇	—	2	8	no	—
77	C₂²×C₄×C₁₂₇	—	2	4	no	—
78	C₂²×C₄×C₁₂₇	—	2	4	no	—
79	C₂²×C₄×C₁₂₇	—	4	8	no	—
80	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
81	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
82	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
90	C₂×C₈×C₁₂₇	—	2	4	no	—
91	C₂×C₈×C₁₂₇	—	2	8	no	—
92	C₄⋊C₄×C₁₂₇	—	2	8	no	—
93	C₄⋊C₄×C₁₂₇	—	2	8	no	—
94	C₄⋊C₄×C₁₂₇	—	2	8	no	—
95	C₄⋊C₄×C₁₂₇	—	2	4	no	—
96	C₄⋊C₄×C₁₂₇	—	2	8	no	—
97	C₄⋊C₄×C₁₂₇	C₂²×C₄	1	8	yes	—
98	C₂²⋊C₄×C₁₂₇	—	4	4	no	—
99	C₄²×C₁₂₇	—	4	4	no	—
100	C₄²×C₁₂₇	C₂²⋊C₄	1	4	yes	—
101	C₄²×C₁₂₇	C₂²⋊C₄	1	4	yes	—
102	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
103	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
104	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
105	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
106	C₂³×C₁₂₇	—	2	2	no	—
107	C₂³×C₁₂₇	—	2	2	no	—
108	C₂³×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
109	C₂³×C₁₂₇	AGL(1,ℤ/8)	1	4	yes	—
110	C₂³×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
111	Q₈×C₁₂₇	—	4	16	no	—
112	Q₈×C₁₂₇	—	4	16	no	—
113	C₂×C₄×C₁₂₇	—	2	2	no	—
114	C₂×C₄×C₁₂₇	—	2	2	no	—
115	C₂×C₄×C₁₂₇	—	2	2	no	—
116	C₂×C₄×C₁₂₇	—	2	2	no	—
117	C₂×C₄×C₁₂₇	—	2	8	no	—
118	C₂×C₄×C₁₂₇	—	2	4	no	—
119	C₂×C₄×C₁₂₇	—	2	4	no	—
120	C₂×C₄×C₁₂₇	—	2	8	no	—
121	C₂×C₄×C₁₂₇	—	2	8	no	—
122	C₂×C₄×C₁₂₇	—	4	8	no	—
123	C₂×C₄×C₁₂₇	—	4	16	no	—
124	C₂×C₄×C₁₂₇	—	4	4	no	—
125	C₂×C₄×C₁₂₇	—	4	4	no	—
126	C₂×C₄×C₁₂₇	—	4	4	no	—
127	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
128	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
129	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
130	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
131	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
132	C₂×C₄×C₁₂₇	#137	1	4	yes	—
133	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₂₇	2₋¹⁺⁴	1	4	yes	—
136	C₈×C₁₂₇	—	4	4	no	—
137	C₂²×C₁₂₇	—	2	2	no	—
138	C₂²×C₁₂₇	—	2	2	no	—
139	C₂²×C₁₂₇	—	2	8	no	—
140	C₂²×C₁₂₇	#418	1	4	yes	—
141	C₂²×C₁₂₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₁₂₇	#536	1	2	yes	—
143	C₂²×C₁₂₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
144	C₂²×C₁₂₇	C₂×#137	1	2	yes	—
145	C₄×C₁₂₇	—	2	2	no	—
146	C₄×C₁₂₇	—	2	2	no	—
147	C₄×C₁₂₇	—	2	2	no	—
148	C₄×C₁₂₇	—	2	2	no	—
149	C₄×C₁₂₇	—	4	8	no	—
150	C₄×C₁₂₇	—	8	4	no	—
151	C₄×C₁₂₇	#419	1	4	yes	—
152	C₄×C₁₂₇	#519	1	2	yes	—
153	C₄×C₁₂₇	#536	1	2	yes	—
154	#21840	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂×C₁₂₇	—	2	4	no	—
156	C₂×C₁₂₇	#2310	1	1	yes	characteristic
157	C₂×C₁₂₇	C₂×#418	1	1	yes	characteristic
158	C₂×C₁₂₇	C₂×#419	1	1	yes	characteristic
159	C₂×C₁₂₇	#3238	1	4	yes	—
160	C₂²×C₄⋊Q₈	C₂×C₁₂₇	1	1	yes	characteristic
161	C₂×#432	C₂×C₁₂₇	1	1	yes	characteristic
162	#2283	C₂×C₁₂₇	1	1	yes	characteristic
163	#1993	C₂×C₁₂₇	1	4	yes	complemented
164	#1987	C₂×C₁₂₇	1	4	yes	complemented
165	#1823	C₂×C₁₂₇	1	4	yes	—
166	C₁₂₇	#21840	1	1	yes	characteristic, direct product factor, Sylow
167	C₂³×Q₈	—	2	1	no	—
168	C₂×C₄⋊Q₈	—	2	1	no	—
169	C₂×C₄⋊Q₈	—	2	1	no	—
170	C₂×C₄⋊Q₈	—	2	4	no	—
171	C₂×C₄⋊Q₈	C₄×C₁₂₇	1	8	yes	complemented
172	C₂×C₄⋊Q₈	C₂²×C₁₂₇	1	2	yes	—
173	C₂×C₄⋊Q₈	C₂²×C₁₂₇	1	2	yes	—
174	C₂²×C₄⋊C₄	—	2	1	no	—
175	C₂²×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
176	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
177	#432	—	2	4	no	—
178	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
179	C₂×C₂²⋊C₈	C₂²×C₁₂₇	1	2	yes	—
180	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
181	#383	—	2	1	no	—
182	#382	C₂²×C₁₂₇	1	2	yes	—
183	#381	—	2	1	no	—
184	#380	C₂²×C₁₂₇	1	2	yes	—
185	C₄×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
186	C₄×C₂²⋊C₄	—	2	1	no	—
187	#376	—	2	1	no	—
188	#340	C₂²×C₁₂₇	1	16	yes	—
189	C₂²×Q₈	—	2	8	no	—
190	C₂²×Q₈	—	2	2	no	—
191	C₂²×Q₈	—	2	2	no	—
192	C₂²×Q₈	—	4	8	no	—
193	C₂³×C₄	—	2	2	no	—
194	C₂³×C₄	D₄×C₁₂₇	1	1	yes	characteristic
195	C₂³×C₄	D₄×C₁₂₇	1	1	yes	characteristic
196	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
197	C₂²×C₈	C₂³×C₁₂₇	1	4	yes	—
198	C₄⋊Q₈	—	2	8	no	—
199	C₄⋊Q₈	—	2	8	no	—
200	C₄⋊Q₈	—	2	4	no	—
201	C₄⋊Q₈	—	2	4	no	—
202	C₄⋊Q₈	C₂×C₄×C₁₂₇	1	8	yes	—
203	C₄⋊Q₈	C₂×C₄×C₁₂₇	1	8	yes	—
204	C₂×C₄⋊C₄	—	2	8	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	4	no	—
209	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	8	yes	—
210	C₂×C₄⋊C₄	C₂³×C₁₂₇	1	4	yes	—
211	C₂×C₄⋊C₄	C₂³×C₁₂₇	1	4	yes	—
212	C₂×C₂²⋊C₄	—	2	2	no	—
213	C₂×C₄²	—	2	2	no	—
214	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
215	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
216	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
217	C₂×C₄²	D₄×C₁₂₇	1	4	yes	—
218	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
219	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
220	C₄⋊C₈	—	2	4	no	—
221	C₂²⋊C₈	—	2	4	no	—
222	#97	—	2	4	no	—
223	#95	—	2	2	no	—
224	#95	—	2	2	no	—
225	#95	—	2	2	no	—
226	#95	—	2	2	no	—
227	C₂⁴	C₂×D₄×C₁₂₇	1	1	yes	characteristic
228	C₂×Q₈	—	2	8	no	—
229	C₂×Q₈	—	2	8	no	—
230	C₂×Q₈	—	4	8	no	—
231	C₂×Q₈	—	4	8	no	—
232	C₂×Q₈	—	4	32	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	8	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	4	8	no	—
240	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	4	yes	—
241	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	4	yes	—
242	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
243	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
248	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
249	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	2	8	no	—
252	C₄⋊C₄	—	2	8	no	—
253	C₄⋊C₄	—	2	8	no	—
254	C₄⋊C₄	—	2	8	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	8	no	—
257	C₄⋊C₄	C₂²×C₄×C₁₂₇	1	8	yes	—
258	C₂²⋊C₄	—	4	4	no	—
259	C₄²	—	4	4	no	—
260	C₄²	C₂²⋊C₄×C₁₂₇	1	4	yes	—
261	C₄²	C₂²⋊C₄×C₁₂₇	1	4	yes	—
262	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
263	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
264	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
265	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
269	C₂³	AGL(1,ℤ/8)×C₁₂₇	1	4	yes	—
270	C₂³	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
271	Q₈	—	4	16	no	—
272	Q₈	—	4	16	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	8	no	—
281	C₂×C₄	—	2	8	no	—
282	C₂×C₄	—	4	8	no	—
283	C₂×C₄	—	4	16	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
288	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
289	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
290	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
291	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
292	C₂×C₄	#137×C₁₂₇	1	4	yes	—
293	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
295	C₂×C₄	2₋¹⁺⁴×C₁₂₇	1	4	yes	—
296	C₈	—	4	4	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	8	no	—
299	C₂²	—	2	2	no	—
300	C₂²	#418×C₁₂₇	1	4	yes	—
301	C₂²	C₂²×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
302	C₂²	#536×C₁₂₇	1	2	yes	—
303	C₂²	C₂×AGL(1,ℤ/8)×C₁₂₇	1	2	yes	—
304	C₂²	C₂×#137×C₁₂₇	1	2	yes	—
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	4	8	no	—
310	C₄	—	8	4	no	—
311	C₄	#419×C₁₂₇	1	4	yes	—
312	C₄	#519×C₁₂₇	1	2	yes	—
313	C₄	#536×C₁₂₇	1	2	yes	—
314	C₂	—	2	4	no	—
315	C₂	C₁₂₇×#2310	1	1	yes	characteristic
316	C₂	C₂×#418×C₁₂₇	1	1	yes	characteristic
317	C₂	C₂×#419×C₁₂₇	1	1	yes	characteristic
318	C₂	C₁₂₇×#3238	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₂₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
prime radical	C₂⁴×C₁₂₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ C₂²×C₄²×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₂₇×#1993	C₂	semidirect
5. C₁₂₇×#1987	C₂	semidirect
11. C₂×C₄⋊Q₈×C₁₂₇	C₄	semidirect
154. #21840	C₁₂₇	direct (N × H)
163. #1993	C₂×C₁₂₇	semidirect
164. #1987	C₂×C₁₂₇	semidirect
166. C₁₂₇	#21840	direct (N × H)
171. C₂×C₄⋊Q₈	C₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Q₈×C₁₂₇	2
C₂×#432×C₁₂₇	2
C₁₂₇×#2283	2
C₁₂₇×#1993	2
C₁₂₇×#1987	2
C₁₂₇×#1823	2
#21840	127

Subgroups — C127×#21840 #44585513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#21840 #44585513