Subgroups — C₁₂₇×#13060 #44576733

Order 32512.

526 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#467×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#358×C₁₂₇	C₂	1	2	yes	maximal, complemented
4	C₁₂₇×#1808	C₂	1	1	yes	maximal, characteristic
5	C₁₂₇×#1380	C₂	1	2	yes	maximal, complemented
6	C₂²×Q₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊Q₈×C₁₂₇	—	2	1	no	—
9	C₂²×C₁₆×C₁₂₇	—	2	2	no	—
10	C₂²×C₁₆×C₁₂₇	—	2	4	no	—
11	C₂²×C₁₆×C₁₂₇	C₂²	1	2	yes	—
12	#467×C₁₂₇	—	2	2	no	—
13	#467×C₁₂₇	C₄	1	4	yes	complemented
14	C₂×#107×C₁₂₇	C₂²	1	2	yes	—
15	C₂×#107×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×#103×C₁₂₇	—	2	1	no	—
17	#380×C₁₂₇	—	2	1	no	—
18	#358×C₁₂₇	—	2	4	no	—
19	#340×C₁₂₇	—	2	1	no	—
20	C₂²×Q₈×C₁₂₇	—	2	1	no	—
21	C₂³×C₄×C₁₂₇	D₄	1	1	yes	characteristic
22	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
23	C₂×Q₁₆×C₁₂₇	—	2	1	no	—
24	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
25	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₂₇	—	2	2	no	—
27	C₂²×C₈×C₁₂₇	—	2	2	no	—
28	C₂²×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₂₇	D₄	1	2	yes	—
31	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
32	C₂²⋊Q₈×C₁₂₇	—	2	4	no	—
33	C₂²⋊Q₈×C₁₂₇	—	4	2	no	—
34	C₂×C₄⋊C₄×C₁₂₇	—	2	1	no	—
35	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₁₂₇	—	4	1	no	—
37	C₂×C₂²⋊C₄×C₁₂₇	—	4	1	no	—
38	C₂×C₂²⋊C₄×C₁₂₇	—	4	1	no	—
39	C₂×C₁₆×C₁₂₇	—	2	8	no	—
40	C₂×C₁₆×C₁₂₇	—	2	2	no	—
41	C₂×C₁₆×C₁₂₇	—	2	2	no	—
42	C₂×C₁₆×C₁₂₇	D₄	1	4	yes	—
43	#107×C₁₂₇	—	2	1	no	—
44	#107×C₁₂₇	—	2	2	no	—
45	#107×C₁₂₇	—	2	2	no	—
46	#107×C₁₂₇	C₂×C₄	1	2	yes	—
47	#103×C₁₂₇	—	2	4	no	—
48	#95×C₁₂₇	—	4	1	no	—
49	C₂⁴×C₁₂₇	D₈	1	1	yes	characteristic
50	C₂×Q₈×C₁₂₇	—	2	2	no	—
51	C₂×Q₈×C₁₂₇	—	4	1	no	—
52	C₂×Q₈×C₁₂₇	—	4	2	no	—
53	C₂×Q₈×C₁₂₇	—	4	2	no	—
54	C₂²×C₄×C₁₂₇	—	2	2	no	—
55	C₂²×C₄×C₁₂₇	—	2	2	no	—
56	C₂²×C₄×C₁₂₇	—	4	1	no	—
57	C₂²×C₄×C₁₂₇	—	4	1	no	—
58	C₂²×C₄×C₁₂₇	—	4	2	no	—
59	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₂₇	SD₁₆	1	2	yes	—
62	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
63	Q₁₆×C₁₂₇	—	2	4	no	—
64	Q₁₆×C₁₂₇	—	4	1	no	—
65	Q₁₆×C₁₂₇	—	4	1	no	—
66	C₂×C₈×C₁₂₇	—	2	2	no	—
67	C₂×C₈×C₁₂₇	—	2	4	no	—
68	C₂×C₈×C₁₂₇	—	2	2	no	—
69	C₂×C₈×C₁₂₇	—	2	2	no	—
70	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	2	yes	—
71	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₁₂₇	#41	1	2	yes	—
75	C₄⋊C₄×C₁₂₇	—	2	2	no	—
76	C₄⋊C₄×C₁₂₇	—	4	4	no	—
77	C₄⋊C₄×C₁₂₇	—	4	1	no	—
78	C₄⋊C₄×C₁₂₇	—	4	2	no	—
79	C₄⋊C₄×C₁₂₇	—	4	2	no	—
80	C₂²⋊C₄×C₁₂₇	—	4	4	no	—
81	C₂²⋊C₄×C₁₂₇	—	8	2	no	—
82	C₁₆×C₁₂₇	—	2	4	no	—
83	C₂³×C₁₂₇	—	2	2	no	—
84	C₂³×C₁₂₇	—	2	2	no	—
85	C₂³×C₁₂₇	#102	1	2	yes	—
86	C₂³×C₁₂₇	SD₃₂	1	2	yes	—
87	C₂³×C₁₂₇	Q₃₂	1	2	yes	—
88	C₂³×C₁₂₇	C₂×D₈	1	1	yes	characteristic
89	Q₈×C₁₂₇	—	4	2	no	—
90	Q₈×C₁₂₇	—	4	2	no	—
91	Q₈×C₁₂₇	—	8	2	no	—
92	C₂×C₄×C₁₂₇	—	2	2	no	—
93	C₂×C₄×C₁₂₇	—	2	2	no	—
94	C₂×C₄×C₁₂₇	—	2	2	no	—
95	C₂×C₄×C₁₂₇	—	2	4	no	—
96	C₂×C₄×C₁₂₇	—	4	2	no	—
97	C₂×C₄×C₁₂₇	—	4	2	no	—
98	C₂×C₄×C₁₂₇	—	8	2	no	—
99	C₂×C₄×C₁₂₇	—	8	2	no	—
100	C₂×C₄×C₁₂₇	—	8	2	no	—
101	C₂×C₄×C₁₂₇	—	8	2	no	—
102	C₂×C₄×C₁₂₇	—	8	2	no	—
103	C₂×C₄×C₁₂₇	#102	1	2	yes	—
104	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₂₇	#121	1	2	yes	—
106	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₁₂₇	#135	1	2	yes	—
108	C₈×C₁₂₇	—	2	2	no	—
109	C₈×C₁₂₇	C₄×D₄	1	2	yes	—
110	C₈×C₁₂₇	#120	1	1	yes	characteristic
111	C₈×C₁₂₇	#123	1	1	yes	characteristic
112	C₂²×C₁₂₇	—	2	2	no	—
113	C₂²×C₁₂₇	—	2	2	no	—
114	C₂²×C₁₂₇	—	2	4	no	—
115	C₂²×C₁₂₇	—	2	4	no	—
116	C₂²×C₁₂₇	#358	1	4	yes	complemented
117	C₂²×C₁₂₇	C₂×#102	1	1	yes	characteristic
118	C₂²×C₁₂₇	#466	1	2	yes	—
119	C₂²×C₁₂₇	C₂×SD₃₂	1	1	yes	characteristic
120	C₂²×C₁₂₇	C₂×Q₃₂	1	1	yes	characteristic
121	C₂²×C₁₂₇	#508	1	2	yes	—
122	C₄×C₁₂₇	—	2	2	no	—
123	C₄×C₁₂₇	—	8	2	no	—
124	C₄×C₁₂₇	—	8	2	no	—
125	C₄×C₁₂₇	—	16	2	no	—
126	C₄×C₁₂₇	#386	1	1	yes	characteristic
127	C₄×C₁₂₇	#416	1	1	yes	characteristic
128	C₄×C₁₂₇	#465	1	2	yes	—
129	#13060	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂×C₁₂₇	—	2	4	no	—
131	C₂×C₁₂₇	#1893	1	1	yes	characteristic
132	C₂×C₁₂₇	C₂×#358	1	1	yes	characteristic
133	C₂×C₁₂₇	#2138	1	1	yes	characteristic
134	C₂×C₁₂₇	#2216	1	2	yes	—
135	C₂×C₁₂₇	#2217	1	2	yes	—
136	C₂³×C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
137	C₂×#467	C₂×C₁₂₇	1	1	yes	characteristic
138	C₂×#358	C₂×C₁₂₇	1	2	yes	complemented
139	#1808	C₂×C₁₂₇	1	1	yes	characteristic
140	#1380	C₂×C₁₂₇	1	2	yes	complemented
141	C₁₂₇	#13060	1	1	yes	characteristic, direct product factor, Sylow
142	C₂²×Q₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
143	C₂³×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
144	C₂×C₂²⋊Q₈	—	2	1	no	—
145	C₂²×C₁₆	—	2	2	no	—
146	C₂²×C₁₆	—	2	4	no	—
147	C₂²×C₁₆	C₂²×C₁₂₇	1	2	yes	—
148	#467	—	2	2	no	—
149	#467	C₄×C₁₂₇	1	4	yes	complemented
150	C₂×#107	C₂²×C₁₂₇	1	2	yes	—
151	C₂×#107	C₂²×C₁₂₇	1	1	yes	characteristic
152	C₂×#103	—	2	1	no	—
153	#380	—	2	1	no	—
154	#358	—	2	4	no	—
155	#340	—	2	1	no	—
156	C₂²×Q₈	—	2	1	no	—
157	C₂³×C₄	D₄×C₁₂₇	1	1	yes	characteristic
158	C₂×Q₁₆	—	2	2	no	—
159	C₂×Q₁₆	—	2	1	no	—
160	C₂×Q₁₆	—	2	2	no	—
161	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
165	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
166	C₂²×C₈	D₄×C₁₂₇	1	2	yes	—
167	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
168	C₂²⋊Q₈	—	2	4	no	—
169	C₂²⋊Q₈	—	4	2	no	—
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	4	1	no	—
173	C₂×C₂²⋊C₄	—	4	1	no	—
174	C₂×C₂²⋊C₄	—	4	1	no	—
175	C₂×C₁₆	—	2	8	no	—
176	C₂×C₁₆	—	2	2	no	—
177	C₂×C₁₆	—	2	2	no	—
178	C₂×C₁₆	D₄×C₁₂₇	1	4	yes	—
179	#107	—	2	1	no	—
180	#107	—	2	2	no	—
181	#107	—	2	2	no	—
182	#107	C₂×C₄×C₁₂₇	1	2	yes	—
183	#103	—	2	4	no	—
184	#95	—	4	1	no	—
185	C₂⁴	D₈×C₁₂₇	1	1	yes	characteristic
186	C₂×Q₈	—	2	2	no	—
187	C₂×Q₈	—	4	1	no	—
188	C₂×Q₈	—	4	2	no	—
189	C₂×Q₈	—	4	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	4	1	no	—
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	2	yes	—
196	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
197	C₂²×C₄	SD₁₆×C₁₂₇	1	2	yes	—
198	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
199	Q₁₆	—	2	4	no	—
200	Q₁₆	—	4	1	no	—
201	Q₁₆	—	4	1	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	2	yes	—
207	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
208	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
209	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
210	C₂×C₈	#41×C₁₂₇	1	2	yes	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	4	4	no	—
213	C₄⋊C₄	—	4	1	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₂²⋊C₄	—	4	4	no	—
217	C₂²⋊C₄	—	8	2	no	—
218	C₁₆	—	2	4	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	#102×C₁₂₇	1	2	yes	—
222	C₂³	SD₃₂×C₁₂₇	1	2	yes	—
223	C₂³	Q₃₂×C₁₂₇	1	2	yes	—
224	C₂³	C₂×D₈×C₁₂₇	1	1	yes	characteristic
225	Q₈	—	4	2	no	—
226	Q₈	—	4	2	no	—
227	Q₈	—	8	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	8	2	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	8	2	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	#102×C₁₂₇	1	2	yes	—
240	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
241	C₂×C₄	#121×C₁₂₇	1	2	yes	—
242	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
243	C₂×C₄	#135×C₁₂₇	1	2	yes	—
244	C₈	—	2	2	no	—
245	C₈	C₄×D₄×C₁₂₇	1	2	yes	—
246	C₈	#120×C₁₂₇	1	1	yes	characteristic
247	C₈	#123×C₁₂₇	1	1	yes	characteristic
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	4	no	—
252	C₂²	#358×C₁₂₇	1	4	yes	complemented
253	C₂²	C₂×#102×C₁₂₇	1	1	yes	characteristic
254	C₂²	#466×C₁₂₇	1	2	yes	—
255	C₂²	C₂×SD₃₂×C₁₂₇	1	1	yes	characteristic
256	C₂²	C₂×Q₃₂×C₁₂₇	1	1	yes	characteristic
257	C₂²	#508×C₁₂₇	1	2	yes	—
258	C₄	—	2	2	no	—
259	C₄	—	8	2	no	—
260	C₄	—	8	2	no	—
261	C₄	—	16	2	no	—
262	C₄	#386×C₁₂₇	1	1	yes	characteristic
263	C₄	#416×C₁₂₇	1	1	yes	characteristic
264	C₄	#465×C₁₂₇	1	2	yes	—
265	C₂	—	2	4	no	—
266	C₂	C₁₂₇×#1893	1	1	yes	characteristic
267	C₂	C₂×#358×C₁₂₇	1	1	yes	characteristic
268	C₂	C₁₂₇×#2138	1	1	yes	characteristic
269	C₂	C₁₂₇×#2216	1	2	yes	—
270	C₂	C₁₂₇×#2217	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₂₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
prime radical	C₂⁴×C₁₂₇	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ C₂³×C₄×C₁₂₇ ◁ C₂³×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#358×C₁₂₇	C₂	semidirect
5. C₁₂₇×#1380	C₂	semidirect
13. #467×C₁₂₇	C₄	semidirect
116. C₂²×C₁₂₇	#358	semidirect
129. #13060	C₁₂₇	direct (N × H)
138. C₂×#358	C₂×C₁₂₇	semidirect
140. #1380	C₂×C₁₂₇	semidirect
141. C₁₂₇	#13060	direct (N × H)
149. #467	C₄×C₁₂₇	semidirect
252. C₂²	#358×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₁₆×C₁₂₇	2
C₂×#467×C₁₂₇	2
C₂×#358×C₁₂₇	2
C₁₂₇×#1808	2
C₁₂₇×#1380	2
#13060	127

Subgroups — C127×#13060 #44576733

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#13060 #44576733