Subgroups — C₂×C₁₂₇×#1689 #44582805

Order 32512.

758 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#494×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#474×C₁₂₇	C₂	1	2	yes	maximal
3	C₂×#436×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1689	C₂	1	8	yes	maximal, direct product factor
5	C₂×#334×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#332×C₁₂₇	C₂	1	2	yes	maximal, complemented
7	C₂²×Q₁₆×C₁₂₇	—	2	1	no	—
8	C₂²×Q₁₆×C₁₂₇	—	2	1	no	—
9	C₂²×M₁₆×C₁₂₇	—	2	1	no	—
10	C₂×C₄⋊Q₈×C₁₂₇	C₂²	1	2	yes	—
11	C₂×C₄×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
12	#494×C₁₂₇	—	2	2	no	—
13	#494×C₁₂₇	C₂²	1	4	yes	complemented
14	#474×C₁₂₇	—	2	4	no	—
15	#474×C₁₂₇	C₂²	1	8	yes	—
16	#436×C₁₂₇	—	2	2	no	—
17	#436×C₁₂₇	C₂²	1	4	yes	—
18	C₂×#107×C₁₂₇	—	2	2	no	—
19	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₂×#102×C₁₂₇	—	2	2	no	—
22	C₂×C₂²⋊C₈×C₁₂₇	—	2	1	no	—
23	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
24	#334×C₁₂₇	C₂²	1	4	yes	complemented
25	#332×C₁₂₇	C₂²	1	8	yes	complemented
26	C₂²×Q₈×C₁₂₇	—	2	2	no	—
27	C₂²×D₄×C₁₂₇	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₂₇	—	2	1	no	—
29	C₂×Q₁₆×C₁₂₇	—	2	4	no	—
30	C₂×Q₁₆×C₁₂₇	—	2	4	no	—
31	C₂×Q₁₆×C₁₂₇	—	4	4	no	—
32	C₂×Q₁₆×C₁₂₇	—	4	4	no	—
33	C₂×M₁₆×C₁₂₇	—	2	4	no	—
34	C₂×M₁₆×C₁₂₇	—	4	4	no	—
35	C₂²×C₈×C₁₂₇	—	2	1	no	—
36	C₂²×C₈×C₁₂₇	—	2	2	no	—
37	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
39	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
40	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
41	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
42	C₄⋊Q₈×C₁₂₇	C₂³	1	4	yes	—
43	C₄×D₄×C₁₂₇	—	2	1	no	—
44	C₄×D₄×C₁₂₇	—	2	1	no	—
45	C₄×D₄×C₁₂₇	D₄	1	2	yes	—
46	C₄×D₄×C₁₂₇	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₁₂₇	D₄	1	1	yes	characteristic
50	C₂×C₂²⋊C₄×C₁₂₇	—	2	1	no	—
51	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
52	#107×C₁₂₇	—	2	8	no	—
53	C₄⋊C₈×C₁₂₇	—	2	1	no	—
54	C₄⋊C₈×C₁₂₇	—	2	2	no	—
55	C₄⋊C₈×C₁₂₇	C₂³	1	2	yes	—
56	C₄⋊C₈×C₁₂₇	C₂³	1	4	yes	—
57	#102×C₁₂₇	—	2	8	no	—
58	C₂²⋊C₈×C₁₂₇	—	2	4	no	—
59	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
60	C₄×C₈×C₁₂₇	C₂³	1	2	yes	—
61	C₂⁴×C₁₂₇	—	2	1	no	—
62	C₂×Q₈×C₁₂₇	—	2	4	no	—
63	C₂×Q₈×C₁₂₇	—	4	2	no	—
64	C₂×Q₈×C₁₂₇	—	4	4	no	—
65	C₂×Q₈×C₁₂₇	—	4	4	no	—
66	C₂×D₄×C₁₂₇	—	2	1	no	—
67	C₂×D₄×C₁₂₇	—	2	4	no	—
68	C₂×D₄×C₁₂₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₁₂₇	—	2	1	no	—
70	C₂²×C₄×C₁₂₇	—	2	2	no	—
71	C₂²×C₄×C₁₂₇	—	2	2	no	—
72	C₂²×C₄×C₁₂₇	—	4	2	no	—
73	C₂²×C₄×C₁₂₇	—	4	2	no	—
74	C₂²×C₄×C₁₂₇	—	4	2	no	—
75	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₂₇	#41	1	1	yes	characteristic
78	Q₁₆×C₁₂₇	—	4	8	no	—
79	Q₁₆×C₁₂₇	—	4	8	no	—
80	M₁₆×C₁₂₇	—	4	8	no	—
81	C₂×C₈×C₁₂₇	—	2	2	no	—
82	C₂×C₈×C₁₂₇	—	2	2	no	—
83	C₂×C₈×C₁₂₇	—	2	4	no	—
84	C₂×C₈×C₁₂₇	—	4	2	no	—
85	C₂×C₈×C₁₂₇	—	4	4	no	—
86	C₂×C₈×C₁₂₇	D₈	1	4	yes	—
87	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
89	C₄⋊C₄×C₁₂₇	—	2	1	no	—
90	C₄⋊C₄×C₁₂₇	—	2	4	no	—
91	C₄⋊C₄×C₁₂₇	—	2	4	no	—
92	C₄⋊C₄×C₁₂₇	—	4	2	no	—
93	C₄⋊C₄×C₁₂₇	—	4	2	no	—
94	C₄⋊C₄×C₁₂₇	C₂×D₄	1	2	yes	—
95	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
96	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
97	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₁₂₇	#41	1	1	yes	characteristic
100	C₄²×C₁₂₇	C₂⁴	1	1	yes	characteristic
101	C₂³×C₁₂₇	—	2	2	no	—
102	C₂³×C₁₂₇	—	4	2	no	—
103	C₂³×C₁₂₇	—	4	2	no	—
104	C₂³×C₁₂₇	—	4	2	no	—
105	C₂³×C₁₂₇	#121	1	1	yes	characteristic
106	Q₈×C₁₂₇	—	4	4	no	—
107	Q₈×C₁₂₇	—	4	4	no	—
108	Q₈×C₁₂₇	—	8	4	no	—
109	D₄×C₁₂₇	—	2	4	no	—
110	D₄×C₁₂₇	—	4	2	no	—
111	C₂×C₄×C₁₂₇	—	2	2	no	—
112	C₂×C₄×C₁₂₇	—	2	2	no	—
113	C₂×C₄×C₁₂₇	—	4	2	no	—
114	C₂×C₄×C₁₂₇	—	4	2	no	—
115	C₂×C₄×C₁₂₇	—	4	2	no	—
116	C₂×C₄×C₁₂₇	—	4	4	no	—
117	C₂×C₄×C₁₂₇	—	4	4	no	—
118	C₂×C₄×C₁₂₇	—	8	4	no	—
119	C₂×C₄×C₁₂₇	#121	1	1	yes	characteristic
120	C₂×C₄×C₁₂₇	#121	1	1	yes	characteristic
121	C₂×C₄×C₁₂₇	#121	1	1	yes	characteristic
122	C₂×C₄×C₁₂₇	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₁₂₇	AGL(1,ℤ/8)	1	2	yes	—
124	C₂×C₄×C₁₂₇	#137	1	4	yes	—
125	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
128	C₈×C₁₂₇	—	2	2	no	—
129	C₈×C₁₂₇	—	4	2	no	—
130	C₈×C₁₂₇	—	4	4	no	—
131	C₈×C₁₂₇	C₂×D₈	1	4	yes	—
132	C₂²×C₁₂₇	—	4	2	no	—
133	C₂²×C₁₂₇	—	4	2	no	—
134	C₂²×C₁₂₇	—	4	2	no	—
135	C₂²×C₁₂₇	—	8	4	no	—
136	C₂²×C₁₂₇	#459	1	2	yes	—
137	C₂²×C₁₂₇	#470	1	2	yes	—
138	C₂²×C₁₂₇	#473	1	2	yes	—
139	C₂²×C₁₂₇	C₂×#121	1	1	yes	characteristic
140	C₄×C₁₂₇	—	2	1	no	—
141	C₄×C₁₂₇	—	2	1	no	—
142	C₄×C₁₂₇	—	4	2	no	—
143	C₄×C₁₂₇	—	8	4	no	—
144	C₄×C₁₂₇	#459	1	2	yes	—
145	C₄×C₁₂₇	#470	1	2	yes	—
146	C₄×C₁₂₇	C₂²×D₈	1	1	yes	characteristic
147	C₄×C₁₂₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₄×C₁₂₇	C₂×#137	1	2	yes	—
149	C₂×#1689	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂×C₁₂₇	—	8	2	no	—
151	C₂×C₁₂₇	#1689	1	4	yes	direct product factor
152	C₂×C₁₂₇	C₂×#459	1	1	yes	characteristic
153	C₂×C₁₂₇	C₂×#470	1	1	yes	characteristic
154	C₂×C₁₂₇	C₂×#473	1	1	yes	characteristic
155	C₂×#494	C₂×C₁₂₇	1	1	yes	characteristic, complemented
156	C₂×#474	C₂×C₁₂₇	1	2	yes	—
157	C₂×#436	C₂×C₁₂₇	1	1	yes	characteristic
158	#1689	C₂×C₁₂₇	1	8	yes	direct product factor
159	C₂×#334	C₂×C₁₂₇	1	1	yes	characteristic, complemented
160	C₂×#332	C₂×C₁₂₇	1	2	yes	complemented
161	C₁₂₇	C₂×#1689	1	1	yes	characteristic, direct product factor, Sylow
162	C₂²×Q₁₆	—	2	1	no	—
163	C₂²×Q₁₆	—	2	1	no	—
164	C₂²×M₁₆	—	2	1	no	—
165	C₂×C₄⋊Q₈	C₂²×C₁₂₇	1	2	yes	—
166	C₂×C₄×D₄	C₂²×C₁₂₇	1	1	yes	characteristic
167	#494	—	2	2	no	—
168	#494	C₂²×C₁₂₇	1	4	yes	complemented
169	#474	—	2	4	no	—
170	#474	C₂²×C₁₂₇	1	8	yes	—
171	#436	—	2	2	no	—
172	#436	C₂²×C₁₂₇	1	4	yes	—
173	C₂×#107	—	2	2	no	—
174	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	1	yes	characteristic
175	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
176	C₂×#102	—	2	2	no	—
177	C₂×C₂²⋊C₈	—	2	1	no	—
178	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
179	#334	C₂²×C₁₂₇	1	4	yes	complemented
180	#332	C₂²×C₁₂₇	1	8	yes	complemented
181	C₂²×Q₈	—	2	2	no	—
182	C₂²×D₄	D₄×C₁₂₇	1	1	yes	characteristic
183	C₂³×C₄	—	2	1	no	—
184	C₂×Q₁₆	—	2	4	no	—
185	C₂×Q₁₆	—	2	4	no	—
186	C₂×Q₁₆	—	4	4	no	—
187	C₂×Q₁₆	—	4	4	no	—
188	C₂×M₁₆	—	2	4	no	—
189	C₂×M₁₆	—	4	4	no	—
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
193	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
194	C₄⋊Q₈	—	2	2	no	—
195	C₄⋊Q₈	—	2	2	no	—
196	C₄⋊Q₈	—	2	2	no	—
197	C₄⋊Q₈	C₂³×C₁₂₇	1	4	yes	—
198	C₄×D₄	—	2	1	no	—
199	C₄×D₄	—	2	1	no	—
200	C₄×D₄	D₄×C₁₂₇	1	2	yes	—
201	C₄×D₄	C₂³×C₁₂₇	1	2	yes	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	D₄×C₁₂₇	1	1	yes	characteristic
205	C₂×C₂²⋊C₄	—	2	1	no	—
206	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
207	#107	—	2	8	no	—
208	C₄⋊C₈	—	2	1	no	—
209	C₄⋊C₈	—	2	2	no	—
210	C₄⋊C₈	C₂³×C₁₂₇	1	2	yes	—
211	C₄⋊C₈	C₂³×C₁₂₇	1	4	yes	—
212	#102	—	2	8	no	—
213	C₂²⋊C₈	—	2	4	no	—
214	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
215	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
216	C₂⁴	—	2	1	no	—
217	C₂×Q₈	—	2	4	no	—
218	C₂×Q₈	—	4	2	no	—
219	C₂×Q₈	—	4	4	no	—
220	C₂×Q₈	—	4	4	no	—
221	C₂×D₄	—	2	1	no	—
222	C₂×D₄	—	2	4	no	—
223	C₂×D₄	C₂×D₄×C₁₂₇	1	2	yes	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	4	2	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
232	C₂²×C₄	#41×C₁₂₇	1	1	yes	characteristic
233	Q₁₆	—	4	8	no	—
234	Q₁₆	—	4	8	no	—
235	M₁₆	—	4	8	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	4	no	—
241	C₂×C₈	D₈×C₁₂₇	1	4	yes	—
242	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
243	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	C₂×D₄×C₁₂₇	1	2	yes	—
250	C₂²⋊C₄	—	2	2	no	—
251	C₂²⋊C₄	—	2	2	no	—
252	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
253	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
254	C₄²	#41×C₁₂₇	1	1	yes	characteristic
255	C₄²	C₂⁴×C₁₂₇	1	1	yes	characteristic
256	C₂³	—	2	2	no	—
257	C₂³	—	4	2	no	—
258	C₂³	—	4	2	no	—
259	C₂³	—	4	2	no	—
260	C₂³	#121×C₁₂₇	1	1	yes	characteristic
261	Q₈	—	4	4	no	—
262	Q₈	—	4	4	no	—
263	Q₈	—	8	4	no	—
264	D₄	—	2	4	no	—
265	D₄	—	4	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	8	4	no	—
274	C₂×C₄	#121×C₁₂₇	1	1	yes	characteristic
275	C₂×C₄	#121×C₁₂₇	1	1	yes	characteristic
276	C₂×C₄	#121×C₁₂₇	1	1	yes	characteristic
277	C₂×C₄	C₂×D₈×C₁₂₇	1	2	yes	—
278	C₂×C₄	AGL(1,ℤ/8)×C₁₂₇	1	2	yes	—
279	C₂×C₄	#137×C₁₂₇	1	4	yes	—
280	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
281	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
282	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	4	2	no	—
285	C₈	—	4	4	no	—
286	C₈	C₂×D₈×C₁₂₇	1	4	yes	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	8	4	no	—
291	C₂²	#459×C₁₂₇	1	2	yes	—
292	C₂²	#470×C₁₂₇	1	2	yes	—
293	C₂²	#473×C₁₂₇	1	2	yes	—
294	C₂²	C₂×#121×C₁₂₇	1	1	yes	characteristic
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	4	2	no	—
298	C₄	—	8	4	no	—
299	C₄	#459×C₁₂₇	1	2	yes	—
300	C₄	#470×C₁₂₇	1	2	yes	—
301	C₄	C₂²×D₈×C₁₂₇	1	1	yes	characteristic
302	C₄	C₂×AGL(1,ℤ/8)×C₁₂₇	1	1	yes	characteristic
303	C₄	C₂×#137×C₁₂₇	1	2	yes	—
304	C₂	—	8	2	no	—
305	C₂	C₁₂₇×#1689	1	4	yes	direct product factor
306	C₂	C₂×#459×C₁₂₇	1	1	yes	characteristic
307	C₂	C₂×#470×C₁₂₇	1	1	yes	characteristic
308	C₂	C₂×#473×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₂₇	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₂₇
prime radical	C₂²×D₄×C₁₂₇	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#494×C₁₂₇	C₂	semidirect
4. C₁₂₇×#1689	C₂	direct (N × H)
5. C₂×#334×C₁₂₇	C₂	semidirect
6. C₂×#332×C₁₂₇	C₂	semidirect
13. #494×C₁₂₇	C₂²	semidirect
24. #334×C₁₂₇	C₂²	semidirect
25. #332×C₁₂₇	C₂²	semidirect
149. C₂×#1689	C₁₂₇	direct (N × H)
151. C₂×C₁₂₇	#1689	direct (N × H)
155. C₂×#494	C₂×C₁₂₇	semidirect
158. #1689	C₂×C₁₂₇	direct (N × H)
159. C₂×#334	C₂×C₁₂₇	semidirect
160. C₂×#332	C₂×C₁₂₇	semidirect
161. C₁₂₇	C₂×#1689	direct (N × H)
168. #494	C₂²×C₁₂₇	semidirect
179. #334	C₂²×C₁₂₇	semidirect
180. #332	C₂²×C₁₂₇	semidirect
305. C₂	C₁₂₇×#1689	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#494×C₁₂₇	2
C₂×#474×C₁₂₇	2
C₂×#436×C₁₂₇	2
C₁₂₇×#1689	2
C₂×#334×C₁₂₇	2
C₂×#332×C₁₂₇	2
C₂×#1689	127

Subgroups — C2×C127×#1689 #44582805

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₂₇×#1689 #44582805