Subgroups — C₁₂₇×#21483 #44585156

Order 32512.

1328 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#3573	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₂₇×#3025	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#2929	C₂	1	3	yes	maximal, complemented
4	C₁₂₇×#1891	C₂	1	3	yes	maximal, complemented
5	C₁₂₇×#1868	C₂	1	3	yes	maximal, complemented
6	C₁₂₇×#1791	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₂₇×#1757	C₂	1	3	yes	maximal, complemented
8	#585×C₁₂₇	C₂²	1	1	yes	characteristic
9	#568×C₁₂₇	—	2	6	no	—
10	#568×C₁₂₇	C₂²	1	3	yes	—
11	C₂×#131×C₁₂₇	—	2	6	no	—
12	C₂×#131×C₁₂₇	C₂²	1	3	yes	—
13	C₂³×C₈×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
14	#525×C₁₂₇	C₂²	1	3	yes	—
15	#458×C₁₂₇	C₄	1	4	yes	—
16	B(4,𝔽₂)×C₁₂₇	C₄	1	4	yes	—
17	#435×C₁₂₇	—	2	6	no	—
18	C₈×D₄×C₁₂₇	—	2	6	no	—
19	#424×C₁₂₇	C₂²	1	3	yes	complemented
20	#413×C₁₂₇	C₂²	1	3	yes	—
21	#410×C₁₂₇	C₂²	1	3	yes	—
22	#408×C₁₂₇	C₂²	1	3	yes	—
23	#407×C₁₂₇	C₂²	1	3	yes	complemented
24	#405×C₁₂₇	C₂²	1	3	yes	complemented
25	#344×C₁₂₇	C₂²	1	6	yes	complemented
26	2₋¹⁺⁴×C₁₂₇	—	2	3	no	—
27	H(5,𝔽₂)×C₁₂₇	—	2	3	no	—
28	H(5,𝔽₂)×C₁₂₇	C₂×C₄	1	4	yes	—
29	C₂×#41×C₁₂₇	—	2	6	no	—
30	C₂×#41×C₁₂₇	C₂³	1	3	yes	—
31	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
32	#131×C₁₂₇	—	2	4	no	—
33	#131×C₁₂₇	—	2	12	no	—
34	#131×C₁₂₇	—	2	6	no	—
35	#131×C₁₂₇	—	2	12	no	—
36	#131×C₁₂₇	—	2	6	no	—
37	C₂×M₁₆×C₁₂₇	—	2	12	no	—
38	C₂×M₁₆×C₁₂₇	—	2	6	no	—
39	C₂×M₁₆×C₁₂₇	D₄	1	6	yes	complemented
40	C₂×M₁₆×C₁₂₇	C₂³	1	3	yes	—
41	C₂×M₁₆×C₁₂₇	C₂³	1	3	yes	—
42	C₂²×C₈×C₁₂₇	—	2	4	no	—
43	C₂²×C₈×C₁₂₇	—	2	3	no	—
44	C₂²×C₈×C₁₂₇	—	2	3	no	—
45	C₂²×C₈×C₁₂₇	D₄	1	6	yes	complemented
46	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
48	#123×C₁₂₇	C₂×C₄	1	6	yes	—
49	C₂²⋊Q₈×C₁₂₇	—	2	3	no	—
50	#121×C₁₂₇	—	2	3	no	—
51	#120×C₁₂₇	C₂×C₄	1	6	yes	—
52	C₄×D₄×C₁₂₇	—	2	6	no	—
53	#117×C₁₂₇	C₂³	1	3	yes	—
54	C₄⋊C₈×C₁₂₇	—	2	6	no	—
55	#101×C₁₂₇	—	2	3	no	—
56	#100×C₁₂₇	—	2	3	no	—
57	#99×C₁₂₇	C₂×C₄	1	12	yes	—
58	C₂²⋊C₈×C₁₂₇	—	2	6	no	—
59	C₂²⋊C₈×C₁₂₇	—	2	6	no	—
60	#97×C₁₂₇	—	2	3	no	—
61	C₄×C₈×C₁₂₇	—	2	3	no	—
62	C₂⁴×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
63	#41×C₁₂₇	—	2	4	no	—
64	#41×C₁₂₇	—	2	12	no	—
65	#41×C₁₂₇	—	2	12	no	—
66	#41×C₁₂₇	—	2	6	no	—
67	#41×C₁₂₇	—	2	6	no	—
68	C₂×Q₈×C₁₂₇	—	2	6	no	—
69	C₂×Q₈×C₁₂₇	#41	1	3	yes	—
70	C₂×D₄×C₁₂₇	—	2	6	no	—
71	C₂×D₄×C₁₂₇	—	2	12	no	—
72	C₂×D₄×C₁₂₇	—	2	6	no	—
73	C₂×D₄×C₁₂₇	C₂²×C₄	1	6	yes	—
74	C₂×D₄×C₁₂₇	#41	1	3	yes	—
75	C₂²×C₄×C₁₂₇	—	2	4	no	—
76	C₂²×C₄×C₁₂₇	—	2	3	no	—
77	C₂²×C₄×C₁₂₇	—	2	3	no	—
78	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₂₇	C₂×D₄	1	6	yes	—
80	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
81	M₁₆×C₁₂₇	—	2	12	no	—
82	M₁₆×C₁₂₇	—	2	6	no	—
83	M₁₆×C₁₂₇	—	2	12	no	—
84	M₁₆×C₁₂₇	—	2	6	no	—
85	C₂×C₈×C₁₂₇	—	2	12	no	—
86	C₂×C₈×C₁₂₇	—	2	6	no	—
87	C₂×C₈×C₁₂₇	—	2	6	no	—
88	C₂×C₈×C₁₂₇	—	2	3	no	—
89	C₂×C₈×C₁₂₇	—	2	3	no	—
90	C₂×C₈×C₁₂₇	—	4	3	no	—
91	C₂×C₈×C₁₂₇	—	4	1	no	—
92	C₂×C₈×C₁₂₇	C₂×D₄	1	6	yes	—
93	C₂×C₈×C₁₂₇	C₂×D₄	1	3	yes	—
94	C₂×C₈×C₁₂₇	C₂×D₄	1	3	yes	—
95	C₄⋊C₄×C₁₂₇	—	2	6	no	—
96	C₄⋊C₄×C₁₂₇	—	2	3	no	—
97	C₂²⋊C₄×C₁₂₇	—	2	6	no	—
98	C₂²⋊C₄×C₁₂₇	—	2	6	no	—
99	C₂²⋊C₄×C₁₂₇	C₂²×C₄	1	6	yes	—
100	C₄²×C₁₂₇	—	2	3	no	—
101	C₂³×C₁₂₇	—	2	4	no	—
102	C₂³×C₁₂₇	—	2	3	no	—
103	C₂³×C₁₂₇	—	4	2	no	—
104	C₂³×C₁₂₇	C₄×D₄	1	6	yes	—
105	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
106	Q₈×C₁₂₇	—	2	4	no	—
107	Q₈×C₁₂₇	—	2	6	no	—
108	D₄×C₁₂₇	—	2	12	no	—
109	D₄×C₁₂₇	—	2	12	no	—
110	D₄×C₁₂₇	—	2	6	no	—
111	D₄×C₁₂₇	—	4	12	no	—
112	C₂×C₄×C₁₂₇	—	2	12	no	—
113	C₂×C₄×C₁₂₇	—	2	6	no	—
114	C₂×C₄×C₁₂₇	—	2	6	no	—
115	C₂×C₄×C₁₂₇	—	2	6	no	—
116	C₂×C₄×C₁₂₇	—	2	3	no	—
117	C₂×C₄×C₁₂₇	—	4	1	no	—
118	C₂×C₄×C₁₂₇	—	4	3	no	—
119	C₂×C₄×C₁₂₇	C₄×D₄	1	6	yes	—
120	C₂×C₄×C₁₂₇	C₂²×D₄	1	3	yes	—
121	C₂×C₄×C₁₂₇	C₂×#41	1	3	yes	—
122	C₈×C₁₂₇	—	2	6	no	—
123	C₈×C₁₂₇	—	4	1	no	—
124	C₈×C₁₂₇	—	4	3	no	—
125	C₈×C₁₂₇	#120	1	3	yes	—
126	C₈×C₁₂₇	#120	1	1	yes	characteristic
127	C₂²×C₁₂₇	—	2	6	no	—
128	C₂²×C₁₂₇	—	2	6	no	—
129	C₂²×C₁₂₇	—	4	1	no	—
130	C₂²×C₁₂₇	—	4	6	no	—
131	C₂²×C₁₂₇	—	4	3	no	—
132	C₂²×C₁₂₇	—	4	1	no	—
133	C₂²×C₁₂₇	—	8	4	no	—
134	C₂²×C₁₂₇	C₂×C₄×D₄	1	3	yes	—
135	C₄×C₁₂₇	—	2	6	no	—
136	C₄×C₁₂₇	—	4	6	no	—
137	C₄×C₁₂₇	—	4	1	no	—
138	C₄×C₁₂₇	C₂×#120	1	1	yes	characteristic
139	C₄×C₁₂₇	#525	1	3	yes	—
140	#21483	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂×C₁₂₇	—	2	3	no	—
142	C₂×C₁₂₇	—	4	2	no	—
143	C₂×C₁₂₇	—	4	6	no	—
144	C₂×C₁₂₇	C₄×#120	1	1	yes	characteristic
145	#3573	C₂×C₁₂₇	1	1	yes	characteristic, complemented
146	#3025	C₂×C₁₂₇	1	1	yes	characteristic
147	#2929	C₂×C₁₂₇	1	3	yes	complemented
148	#1891	C₂×C₁₂₇	1	3	yes	complemented
149	#1868	C₂×C₁₂₇	1	3	yes	complemented
150	#1791	C₂×C₁₂₇	1	1	yes	characteristic, complemented
151	#1757	C₂×C₁₂₇	1	3	yes	complemented
152	C₁₂₇	#21483	1	1	yes	characteristic, direct product factor, Sylow
153	#585	C₂²×C₁₂₇	1	1	yes	characteristic
154	#568	—	2	6	no	—
155	#568	C₂²×C₁₂₇	1	3	yes	—
156	C₂×#131	—	2	6	no	—
157	C₂×#131	C₂²×C₁₂₇	1	3	yes	—
158	C₂³×C₈	C₂²×C₁₂₇	1	1	yes	characteristic, complemented
159	#525	C₂²×C₁₂₇	1	3	yes	—
160	#458	C₄×C₁₂₇	1	4	yes	—
161	B(4,𝔽₂)	C₄×C₁₂₇	1	4	yes	—
162	#435	—	2	6	no	—
163	C₈×D₄	—	2	6	no	—
164	#424	C₂²×C₁₂₇	1	3	yes	complemented
165	#413	C₂²×C₁₂₇	1	3	yes	—
166	#410	C₂²×C₁₂₇	1	3	yes	—
167	#408	C₂²×C₁₂₇	1	3	yes	—
168	#407	C₂²×C₁₂₇	1	3	yes	complemented
169	#405	C₂²×C₁₂₇	1	3	yes	complemented
170	#344	C₂²×C₁₂₇	1	6	yes	complemented
171	2₋¹⁺⁴	—	2	3	no	—
172	H(5,𝔽₂)	—	2	3	no	—
173	H(5,𝔽₂)	C₂×C₄×C₁₂₇	1	4	yes	—
174	C₂×#41	—	2	6	no	—
175	C₂×#41	C₂³×C₁₂₇	1	3	yes	—
176	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
177	#131	—	2	4	no	—
178	#131	—	2	12	no	—
179	#131	—	2	6	no	—
180	#131	—	2	12	no	—
181	#131	—	2	6	no	—
182	C₂×M₁₆	—	2	12	no	—
183	C₂×M₁₆	—	2	6	no	—
184	C₂×M₁₆	D₄×C₁₂₇	1	6	yes	complemented
185	C₂×M₁₆	C₂³×C₁₂₇	1	3	yes	—
186	C₂×M₁₆	C₂³×C₁₂₇	1	3	yes	—
187	C₂²×C₈	—	2	4	no	—
188	C₂²×C₈	—	2	3	no	—
189	C₂²×C₈	—	2	3	no	—
190	C₂²×C₈	D₄×C₁₂₇	1	6	yes	complemented
191	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
192	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
193	#123	C₂×C₄×C₁₂₇	1	6	yes	—
194	C₂²⋊Q₈	—	2	3	no	—
195	#121	—	2	3	no	—
196	#120	C₂×C₄×C₁₂₇	1	6	yes	—
197	C₄×D₄	—	2	6	no	—
198	#117	C₂³×C₁₂₇	1	3	yes	—
199	C₄⋊C₈	—	2	6	no	—
200	#101	—	2	3	no	—
201	#100	—	2	3	no	—
202	#99	C₂×C₄×C₁₂₇	1	12	yes	—
203	C₂²⋊C₈	—	2	6	no	—
204	C₂²⋊C₈	—	2	6	no	—
205	#97	—	2	3	no	—
206	C₄×C₈	—	2	3	no	—
207	C₂⁴	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
208	#41	—	2	4	no	—
209	#41	—	2	12	no	—
210	#41	—	2	12	no	—
211	#41	—	2	6	no	—
212	#41	—	2	6	no	—
213	C₂×Q₈	—	2	6	no	—
214	C₂×Q₈	#41×C₁₂₇	1	3	yes	—
215	C₂×D₄	—	2	6	no	—
216	C₂×D₄	—	2	12	no	—
217	C₂×D₄	—	2	6	no	—
218	C₂×D₄	C₂²×C₄×C₁₂₇	1	6	yes	—
219	C₂×D₄	#41×C₁₂₇	1	3	yes	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	3	no	—
222	C₂²×C₄	—	2	3	no	—
223	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₁₂₇	1	6	yes	—
225	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
226	M₁₆	—	2	12	no	—
227	M₁₆	—	2	6	no	—
228	M₁₆	—	2	12	no	—
229	M₁₆	—	2	6	no	—
230	C₂×C₈	—	2	12	no	—
231	C₂×C₈	—	2	6	no	—
232	C₂×C₈	—	2	6	no	—
233	C₂×C₈	—	2	3	no	—
234	C₂×C₈	—	2	3	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	4	3	no	—
237	C₂×C₈	C₂×D₄×C₁₂₇	1	6	yes	—
238	C₂×C₈	C₂×D₄×C₁₂₇	1	3	yes	—
239	C₂×C₈	C₂×D₄×C₁₂₇	1	3	yes	—
240	C₄⋊C₄	—	2	6	no	—
241	C₄⋊C₄	—	2	3	no	—
242	C₂²⋊C₄	—	2	6	no	—
243	C₂²⋊C₄	—	2	6	no	—
244	C₂²⋊C₄	C₂²×C₄×C₁₂₇	1	6	yes	—
245	C₄²	—	2	3	no	—
246	C₂³	—	2	4	no	—
247	C₂³	—	2	3	no	—
248	C₂³	—	4	2	no	—
249	C₂³	C₄×D₄×C₁₂₇	1	6	yes	—
250	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
251	Q₈	—	2	4	no	—
252	Q₈	—	2	6	no	—
253	D₄	—	2	12	no	—
254	D₄	—	2	12	no	—
255	D₄	—	2	6	no	—
256	D₄	—	4	12	no	—
257	C₂×C₄	—	2	12	no	—
258	C₂×C₄	—	2	6	no	—
259	C₂×C₄	—	2	6	no	—
260	C₂×C₄	—	2	6	no	—
261	C₂×C₄	—	2	3	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	3	no	—
264	C₂×C₄	C₄×D₄×C₁₂₇	1	6	yes	—
265	C₂×C₄	C₂²×D₄×C₁₂₇	1	3	yes	—
266	C₂×C₄	C₂×#41×C₁₂₇	1	3	yes	—
267	C₈	—	2	6	no	—
268	C₈	—	4	1	no	—
269	C₈	—	4	3	no	—
270	C₈	#120×C₁₂₇	1	1	yes	characteristic
271	C₈	#120×C₁₂₇	1	3	yes	—
272	C₂²	—	2	6	no	—
273	C₂²	—	2	6	no	—
274	C₂²	—	4	1	no	—
275	C₂²	—	4	6	no	—
276	C₂²	—	4	3	no	—
277	C₂²	—	4	1	no	—
278	C₂²	—	8	4	no	—
279	C₂²	C₂×C₄×D₄×C₁₂₇	1	3	yes	—
280	C₄	—	2	6	no	—
281	C₄	—	4	6	no	—
282	C₄	—	4	1	no	—
283	C₄	C₂×#120×C₁₂₇	1	1	yes	characteristic
284	C₄	#525×C₁₂₇	1	3	yes	—
285	C₂	—	2	3	no	—
286	C₂	—	4	2	no	—
287	C₂	—	4	6	no	—
288	C₂	C₄×#120×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₂₇	#120
commutator	C₂³	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle	C₂×C₁₂₇	C₄×#120
prime radical	C₁₂₇×#3025	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₂₇ ◁ C₂²×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₂₇×#3573	C₂	semidirect
3. C₁₂₇×#2929	C₂	semidirect
4. C₁₂₇×#1891	C₂	semidirect
5. C₁₂₇×#1868	C₂	semidirect
6. C₁₂₇×#1791	C₂	semidirect
7. C₁₂₇×#1757	C₂	semidirect
13. C₂³×C₈×C₁₂₇	C₂²	semidirect
19. #424×C₁₂₇	C₂²	semidirect
23. #407×C₁₂₇	C₂²	semidirect
24. #405×C₁₂₇	C₂²	semidirect
25. #344×C₁₂₇	C₂²	semidirect
39. C₂×M₁₆×C₁₂₇	D₄	semidirect
45. C₂²×C₈×C₁₂₇	D₄	semidirect
140. #21483	C₁₂₇	direct (N × H)
145. #3573	C₂×C₁₂₇	semidirect
147. #2929	C₂×C₁₂₇	semidirect
148. #1891	C₂×C₁₂₇	semidirect
149. #1868	C₂×C₁₂₇	semidirect
150. #1791	C₂×C₁₂₇	semidirect
151. #1757	C₂×C₁₂₇	semidirect
152. C₁₂₇	#21483	direct (N × H)
158. C₂³×C₈	C₂²×C₁₂₇	semidirect
164. #424	C₂²×C₁₂₇	semidirect
168. #407	C₂²×C₁₂₇	semidirect
169. #405	C₂²×C₁₂₇	semidirect
170. #344	C₂²×C₁₂₇	semidirect
184. C₂×M₁₆	D₄×C₁₂₇	semidirect
190. C₂²×C₈	D₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×#3573	2
C₁₂₇×#3025	2
C₁₂₇×#2929	2
C₁₂₇×#1891	2
C₁₂₇×#1868	2
C₁₂₇×#1791	2
C₁₂₇×#1757	2
#21483	127

Subgroups — C127×#21483 #44585156

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#21483 #44585156