Subgroups — C₁₂₇×#20505 #44584178

Order 32512.

1054 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#441×C₁₂₇	C₂	1	2	yes	maximal
2	C₂×C₄×Q₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#2276	C₂	1	2	yes	maximal
4	C₁₂₇×#1776	C₂	1	2	yes	maximal
5	C₄×#106×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₁₂₇×#1763	C₂	1	2	yes	maximal
7	C₄×#103×C₁₂₇	C₂	1	4	yes	maximal
8	C₁₂₇×#1726	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₁₂₇	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₁₂₇	C₂²	1	4	yes	—
12	#441×C₁₂₇	C₄	1	16	yes	complemented
13	C₄×Q₁₆×C₁₂₇	C₄	1	8	yes	complemented
14	C₂×#107×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×#106×C₁₂₇	C₂²	1	2	yes	—
16	C₂×#103×C₁₂₇	C₂²	1	2	yes	—
17	C₂×#103×C₁₂₇	C₂²	1	4	yes	—
18	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₁₂₇	—	2	2	no	—
23	C₄×C₄⋊C₄×C₁₂₇	—	2	4	no	—
24	C₄×C₄⋊C₄×C₁₂₇	C₂²	1	4	yes	—
25	C₄×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
26	#376×C₁₂₇	—	2	2	no	—
27	#376×C₁₂₇	C₂²	1	1	yes	characteristic
28	#340×C₁₂₇	C₂²	1	2	yes	—
29	#340×C₁₂₇	C₂²	1	4	yes	—
30	C₂²×Q₈×C₁₂₇	C₂³	1	2	yes	—
31	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
32	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	8	yes	—
33	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
36	C₄×Q₈×C₁₂₇	—	2	8	no	—
37	C₄×Q₈×C₁₂₇	—	2	4	no	—
38	C₄×Q₈×C₁₂₇	C₂×C₄	1	8	yes	—
39	C₄×Q₈×C₁₂₇	C₂×C₄	1	16	yes	—
40	C₂×C₄⋊C₄×C₁₂₇	—	2	4	no	—
41	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
43	C₂×C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
44	C₂×C₄²×C₁₂₇	—	2	2	no	—
45	C₂×C₄²×C₁₂₇	—	2	4	no	—
46	C₂×C₄²×C₁₂₇	—	2	4	no	—
47	C₂×C₄²×C₁₂₇	—	2	2	no	—
48	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
50	#107×C₁₂₇	C₂×C₄	1	4	yes	—
51	#106×C₁₂₇	C₂×C₄	1	8	yes	—
52	#103×C₁₂₇	C₂×C₄	1	8	yes	—
53	#103×C₁₂₇	C₂×C₄	1	16	yes	—
54	#97×C₁₂₇	—	2	2	no	—
55	#97×C₁₂₇	C₂×C₄	1	8	yes	—
56	C₄×C₈×C₁₂₇	—	2	4	no	—
57	C₄×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
58	#95×C₁₂₇	—	2	2	no	—
59	#95×C₁₂₇	—	2	4	no	—
60	#95×C₁₂₇	D₄	1	2	yes	—
61	#95×C₁₂₇	D₄	1	2	yes	—
62	C₂×Q₈×C₁₂₇	—	2	4	no	—
63	C₂×Q₈×C₁₂₇	—	2	2	no	—
64	C₂×Q₈×C₁₂₇	C₂²×C₄	1	4	yes	—
65	C₂×Q₈×C₁₂₇	C₂²×C₄	1	8	yes	—
66	C₂²×C₄×C₁₂₇	—	2	2	no	—
67	C₂²×C₄×C₁₂₇	—	2	2	no	—
68	C₂²×C₄×C₁₂₇	—	2	4	no	—
69	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
71	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
73	Q₁₆×C₁₂₇	C₄²	1	16	yes	complemented
74	C₂×C₈×C₁₂₇	—	2	2	no	—
75	C₂×C₈×C₁₂₇	—	2	2	no	—
76	C₂×C₈×C₁₂₇	—	2	2	no	—
77	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
79	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
81	C₄⋊C₄×C₁₂₇	—	2	16	no	—
82	C₄⋊C₄×C₁₂₇	—	2	8	no	—
83	C₄⋊C₄×C₁₂₇	—	2	4	no	—
84	C₄⋊C₄×C₁₂₇	—	2	2	no	—
85	C₄⋊C₄×C₁₂₇	C₂²×C₄	1	8	yes	—
86	C₄⋊C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
87	C₄²×C₁₂₇	—	2	8	no	—
88	C₄²×C₁₂₇	—	2	16	no	—
89	C₄²×C₁₂₇	—	8	4	no	—
90	C₄²×C₁₂₇	—	8	2	no	—
91	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
92	C₄²×C₁₂₇	C₂²×C₄	1	4	yes	—
93	C₄²×C₁₂₇	#41	1	4	yes	—
94	C₄²×C₁₂₇	#41	1	2	yes	—
95	C₂³×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
96	Q₈×C₁₂₇	—	2	4	no	—
97	Q₈×C₁₂₇	—	2	8	no	—
98	Q₈×C₁₂₇	C₂×C₄²	1	8	yes	—
99	C₂×C₄×C₁₂₇	—	2	8	no	—
100	C₂×C₄×C₁₂₇	—	2	4	no	—
101	C₂×C₄×C₁₂₇	—	2	4	no	—
102	C₂×C₄×C₁₂₇	—	2	2	no	—
103	C₂×C₄×C₁₂₇	—	2	4	no	—
104	C₂×C₄×C₁₂₇	—	2	2	no	—
105	C₂×C₄×C₁₂₇	—	2	8	no	—
106	C₂×C₄×C₁₂₇	—	2	4	no	—
107	C₂×C₄×C₁₂₇	—	4	4	no	—
108	C₂×C₄×C₁₂₇	—	4	2	no	—
109	C₂×C₄×C₁₂₇	—	4	2	no	—
110	C₂×C₄×C₁₂₇	—	4	4	no	—
111	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₁₂₇	C₄×D₄	1	8	yes	—
113	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₁₂₇	C₂³×C₄	1	2	yes	—
115	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₂₇	C₂×#41	1	2	yes	—
117	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
118	C₈×C₁₂₇	—	2	2	no	—
119	C₈×C₁₂₇	—	2	4	no	—
120	C₈×C₁₂₇	C₂×C₄²	1	4	yes	—
121	C₂²×C₁₂₇	C₂×C₄×D₄	1	2	yes	—
122	C₂²×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
123	C₂²×C₁₂₇	#577	1	2	yes	—
124	C₂²×C₁₂₇	#578	1	2	yes	—
125	C₄×C₁₂₇	—	2	8	no	—
126	C₄×C₁₂₇	—	2	2	no	—
127	C₄×C₁₂₇	—	2	2	no	—
128	C₄×C₁₂₇	—	2	8	no	—
129	C₄×C₁₂₇	—	4	8	no	—
130	C₄×C₁₂₇	—	4	4	no	—
131	C₄×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
132	C₄×C₁₂₇	C₄×#41	1	1	yes	characteristic
133	C₄×C₁₂₇	C₄×#41	1	2	yes	—
134	#20505	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂×C₁₂₇	C₄²×D₄	1	1	yes	characteristic
136	C₂×C₁₂₇	#2947	1	4	yes	—
137	C₂×C₁₂₇	#2950	1	1	yes	characteristic
138	C₂×C₁₂₇	#2952	1	1	yes	characteristic
139	C₂×#441	C₂×C₁₂₇	1	2	yes	—
140	C₂×C₄×Q₁₆	C₂×C₁₂₇	1	1	yes	characteristic
141	#2276	C₂×C₁₂₇	1	2	yes	—
142	#1776	C₂×C₁₂₇	1	2	yes	—
143	C₄×#106	C₂×C₁₂₇	1	1	yes	characteristic
144	#1763	C₂×C₁₂₇	1	2	yes	—
145	C₄×#103	C₂×C₁₂₇	1	4	yes	—
146	#1726	C₂×C₁₂₇	1	1	yes	characteristic
147	C₁₂₇	#20505	1	1	yes	characteristic, direct product factor, Sylow
148	C₂²×Q₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
149	C₂×C₄×Q₈	C₂²×C₁₂₇	1	2	yes	—
150	C₂×C₄×Q₈	C₂²×C₁₂₇	1	4	yes	—
151	#441	C₄×C₁₂₇	1	16	yes	complemented
152	C₄×Q₁₆	C₄×C₁₂₇	1	8	yes	complemented
153	C₂×#107	C₂²×C₁₂₇	1	1	yes	characteristic
154	C₂×#106	C₂²×C₁₂₇	1	2	yes	—
155	C₂×#103	C₂²×C₁₂₇	1	2	yes	—
156	C₂×#103	C₂²×C₁₂₇	1	4	yes	—
157	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
158	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
159	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
160	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
161	C₄×C₄⋊C₄	—	2	2	no	—
162	C₄×C₄⋊C₄	—	2	4	no	—
163	C₄×C₄⋊C₄	C₂²×C₁₂₇	1	4	yes	—
164	C₄×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
165	#376	—	2	2	no	—
166	#376	C₂²×C₁₂₇	1	1	yes	characteristic
167	#340	C₂²×C₁₂₇	1	2	yes	—
168	#340	C₂²×C₁₂₇	1	4	yes	—
169	C₂²×Q₈	C₂³×C₁₂₇	1	2	yes	—
170	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
171	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	8	yes	—
172	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
173	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
174	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
175	C₄×Q₈	—	2	8	no	—
176	C₄×Q₈	—	2	4	no	—
177	C₄×Q₈	C₂×C₄×C₁₂₇	1	8	yes	—
178	C₄×Q₈	C₂×C₄×C₁₂₇	1	16	yes	—
179	C₂×C₄⋊C₄	—	2	4	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	C₂³×C₁₂₇	1	4	yes	—
182	C₂×C₄⋊C₄	C₂³×C₁₂₇	1	2	yes	—
183	C₂×C₄²	—	2	2	no	—
184	C₂×C₄²	—	2	4	no	—
185	C₂×C₄²	—	2	4	no	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
188	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
189	#107	C₂×C₄×C₁₂₇	1	4	yes	—
190	#106	C₂×C₄×C₁₂₇	1	8	yes	—
191	#103	C₂×C₄×C₁₂₇	1	8	yes	—
192	#103	C₂×C₄×C₁₂₇	1	16	yes	—
193	#97	—	2	2	no	—
194	#97	C₂×C₄×C₁₂₇	1	8	yes	—
195	C₄×C₈	—	2	4	no	—
196	C₄×C₈	C₂×C₄×C₁₂₇	1	4	yes	—
197	#95	—	2	2	no	—
198	#95	—	2	4	no	—
199	#95	D₄×C₁₂₇	1	2	yes	—
200	#95	D₄×C₁₂₇	1	2	yes	—
201	C₂×Q₈	—	2	4	no	—
202	C₂×Q₈	—	2	2	no	—
203	C₂×Q₈	C₂²×C₄×C₁₂₇	1	4	yes	—
204	C₂×Q₈	C₂²×C₄×C₁₂₇	1	8	yes	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₄×C₁₂₇	1	4	yes	—
210	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
211	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
212	Q₁₆	C₄²×C₁₂₇	1	16	yes	complemented
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
217	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
218	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
219	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
220	C₄⋊C₄	—	2	16	no	—
221	C₄⋊C₄	—	2	8	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	C₂²×C₄×C₁₂₇	1	8	yes	—
225	C₄⋊C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
226	C₄²	—	2	8	no	—
227	C₄²	—	2	16	no	—
228	C₄²	—	8	4	no	—
229	C₄²	—	8	2	no	—
230	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
231	C₄²	C₂²×C₄×C₁₂₇	1	4	yes	—
232	C₄²	#41×C₁₂₇	1	4	yes	—
233	C₄²	#41×C₁₂₇	1	2	yes	—
234	C₂³	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
235	Q₈	—	2	4	no	—
236	Q₈	—	2	8	no	—
237	Q₈	C₂×C₄²×C₁₂₇	1	8	yes	—
238	C₂×C₄	—	2	8	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
251	C₂×C₄	C₄×D₄×C₁₂₇	1	8	yes	—
252	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
253	C₂×C₄	C₂³×C₄×C₁₂₇	1	2	yes	—
254	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
255	C₂×C₄	C₂×#41×C₁₂₇	1	2	yes	—
256	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	2	4	no	—
259	C₈	C₂×C₄²×C₁₂₇	1	4	yes	—
260	C₂²	C₂×C₄×D₄×C₁₂₇	1	2	yes	—
261	C₂²	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
262	C₂²	#577×C₁₂₇	1	2	yes	—
263	C₂²	#578×C₁₂₇	1	2	yes	—
264	C₄	—	2	8	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	8	no	—
268	C₄	—	4	8	no	—
269	C₄	—	4	4	no	—
270	C₄	C₂²×C₄²×C₁₂₇	1	1	yes	characteristic
271	C₄	C₄×#41×C₁₂₇	1	1	yes	characteristic
272	C₄	C₄×#41×C₁₂₇	1	2	yes	—
273	C₂	C₄²×D₄×C₁₂₇	1	1	yes	characteristic
274	C₂	C₁₂₇×#2947	1	4	yes	—
275	C₂	C₁₂₇×#2950	1	1	yes	characteristic
276	C₂	C₁₂₇×#2952	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₂₇	C₂²×D₄
commutator	C₄	C₂²×C₄²×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ #376×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
12. #441×C₁₂₇	C₄	semidirect
13. C₄×Q₁₆×C₁₂₇	C₄	semidirect
73. Q₁₆×C₁₂₇	C₄²	semidirect
134. #20505	C₁₂₇	direct (N × H)
147. C₁₂₇	#20505	direct (N × H)
151. #441	C₄×C₁₂₇	semidirect
152. C₄×Q₁₆	C₄×C₁₂₇	semidirect
212. Q₁₆	C₄²×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#441×C₁₂₇	2
C₂×C₄×Q₁₆×C₁₂₇	2
C₁₂₇×#2276	2
C₁₂₇×#1776	2
C₄×#106×C₁₂₇	2
C₁₂₇×#1763	2
C₄×#103×C₁₂₇	2
C₁₂₇×#1726	2
#20505	127

Subgroups — C127×#20505 #44584178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#20505 #44584178