Subgroups — C₁₂₇×#21705 #44585378

Order 32512.

1046 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#3147	C₂	1	1	yes	maximal, characteristic
2	C₂×#443×C₁₂₇	C₂	1	4	yes	maximal, complemented
3	C₁₂₇×#2931	C₂	1	2	yes	maximal, complemented
4	C₁₂₇×#1974	C₂	1	4	yes	maximal, complemented
5	C₁₂₇×#1962	C₂	1	2	yes	maximal, complemented
6	C₁₂₇×#1848	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄²×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₁₂₇	C₂²	1	4	yes	—
9	C₂×#131×C₁₂₇	C₂²	1	4	yes	—
10	#532×C₁₂₇	C₂²	1	2	yes	—
11	#495×C₁₂₇	C₄	1	4	yes	complemented
12	#494×C₁₂₇	C₄	1	1	yes	characteristic, complemented
13	#493×C₁₂₇	C₄	1	1	yes	characteristic, complemented
14	#492×C₁₂₇	C₄	1	2	yes	complemented
15	#443×C₁₂₇	—	2	16	no	—
16	#436×C₁₂₇	—	2	4	no	—
17	C₈×D₄×C₁₂₇	—	2	4	no	—
18	C₂×#108×C₁₂₇	C₂²	1	4	yes	—
19	#423×C₁₂₇	C₂²	1	2	yes	—
20	C₂×#104×C₁₂₇	C₂²	1	8	yes	—
21	#408×C₁₂₇	C₂²	1	4	yes	—
22	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
26	C₄³×C₁₂₇	C₂²	1	1	yes	characteristic
27	C₂×#41×C₁₂₇	C₂³	1	4	yes	—
28	#135×C₁₂₇	—	2	8	no	—
29	#135×C₁₂₇	—	2	8	no	—
30	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
31	C₂×SD₁₆×C₁₂₇	C₂×C₄	1	8	yes	—
32	C₂×D₈×C₁₂₇	C₂×C₄	1	4	yes	—
33	#131×C₁₂₇	—	2	16	no	—
34	C₂×M₁₆×C₁₂₇	—	2	4	no	—
35	C₂×M₁₆×C₁₂₇	C₂³	1	4	yes	—
36	C₂²×C₈×C₁₂₇	—	2	4	no	—
37	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
39	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₁₂₇	C₂×C₄	1	2	yes	—
41	#127×C₁₂₇	C₂×C₄	1	2	yes	—
42	#124×C₁₂₇	C₂×C₄	1	4	yes	—
43	#121×C₁₂₇	—	2	4	no	—
44	C₄×D₄×C₁₂₇	—	2	4	no	—
45	C₂×C₄²×C₁₂₇	—	2	2	no	—
46	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
47	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
48	#108×C₁₂₇	—	2	8	no	—
49	C₄⋊C₈×C₁₂₇	—	2	4	no	—
50	#104×C₁₂₇	—	2	16	no	—
51	C₂²⋊C₈×C₁₂₇	—	2	8	no	—
52	C₄×C₈×C₁₂₇	—	2	4	no	—
53	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
56	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
57	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
58	C₄×C₈×C₁₂₇	D₄	1	8	yes	complemented
59	#41×C₁₂₇	—	2	8	no	—
60	#41×C₁₂₇	—	2	8	no	—
61	C₂×Q₈×C₁₂₇	C₂²×C₄	1	4	yes	—
62	C₂×D₄×C₁₂₇	—	2	4	no	—
63	C₂×D₄×C₁₂₇	—	2	4	no	—
64	C₂×D₄×C₁₂₇	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₁₂₇	—	2	2	no	—
66	C₂²×C₄×C₁₂₇	—	2	4	no	—
67	C₂²×C₄×C₁₂₇	#41	1	2	yes	—
68	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
69	Q₁₆×C₁₂₇	—	2	8	no	—
70	SD₁₆×C₁₂₇	—	2	16	no	—
71	D₈×C₁₂₇	—	2	8	no	—
72	M₁₆×C₁₂₇	—	2	8	no	—
73	M₁₆×C₁₂₇	—	4	8	no	—
74	C₂×C₈×C₁₂₇	—	2	2	no	—
75	C₂×C₈×C₁₂₇	—	2	4	no	—
76	C₂×C₈×C₁₂₇	—	4	8	no	—
77	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
81	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
83	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₂₇	#41	1	4	yes	—
85	C₄⋊C₄×C₁₂₇	—	2	4	no	—
86	C₂²⋊C₄×C₁₂₇	—	2	8	no	—
87	C₄²×C₁₂₇	—	2	8	no	—
88	C₄²×C₁₂₇	—	2	2	no	—
89	C₄²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₁₂₇	C₂×D₄	1	4	yes	—
92	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
93	C₂³×C₁₂₇	—	2	4	no	—
94	C₂³×C₁₂₇	C₂×#41	1	1	yes	characteristic
95	Q₈×C₁₂₇	—	2	8	no	—
96	D₄×C₁₂₇	—	2	8	no	—
97	D₄×C₁₂₇	—	4	8	no	—
98	D₄×C₁₂₇	—	4	8	no	—
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	4	no	—
103	C₂×C₄×C₁₂₇	—	4	4	no	—
104	C₂×C₄×C₁₂₇	—	4	4	no	—
105	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
109	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₂₇	C₂×#41	1	2	yes	—
111	C₈×C₁₂₇	—	2	2	no	—
112	C₈×C₁₂₇	—	4	4	no	—
113	C₈×C₁₂₇	C₄×D₄	1	4	yes	—
114	C₈×C₁₂₇	C₄×D₄	1	4	yes	—
115	C₈×C₁₂₇	#127	1	2	yes	—
116	C₈×C₁₂₇	#127	1	2	yes	—
117	C₂²×C₁₂₇	—	2	2	no	—
118	C₂²×C₁₂₇	—	4	4	no	—
119	C₂²×C₁₂₇	—	4	8	no	—
120	C₂²×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
121	C₂²×C₁₂₇	#532	1	2	yes	—
122	C₄×C₁₂₇	—	2	8	no	—
123	C₄×C₁₂₇	—	2	2	no	—
124	C₄×C₁₂₇	—	4	4	no	—
125	C₄×C₁₂₇	#443	1	4	yes	complemented
126	C₄×C₁₂₇	C₂×C₄×D₄	1	2	yes	—
127	C₄×C₁₂₇	C₂×#127	1	1	yes	characteristic
128	C₄×C₁₂₇	#532	1	1	yes	characteristic
129	#21705	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂×C₁₂₇	—	2	2	no	—
131	C₂×C₁₂₇	—	8	4	no	—
132	C₂×C₁₂₇	C₄×#127	1	1	yes	characteristic
133	C₂×C₁₂₇	C₂×#443	1	2	yes	—
134	#3147	C₂×C₁₂₇	1	1	yes	characteristic
135	C₂×#443	C₂×C₁₂₇	1	4	yes	complemented
136	#2931	C₂×C₁₂₇	1	2	yes	complemented
137	#1974	C₂×C₁₂₇	1	4	yes	complemented
138	#1962	C₂×C₁₂₇	1	2	yes	complemented
139	#1848	C₂×C₁₂₇	1	1	yes	characteristic, complemented
140	C₄²×C₈	C₂×C₁₂₇	1	1	yes	characteristic, complemented
141	C₁₂₇	#21705	1	1	yes	characteristic, direct product factor, Sylow
142	C₂×#135	C₂²×C₁₂₇	1	4	yes	—
143	C₂×#131	C₂²×C₁₂₇	1	4	yes	—
144	#532	C₂²×C₁₂₇	1	2	yes	—
145	#495	C₄×C₁₂₇	1	4	yes	complemented
146	#494	C₄×C₁₂₇	1	1	yes	characteristic, complemented
147	#493	C₄×C₁₂₇	1	1	yes	characteristic, complemented
148	#492	C₄×C₁₂₇	1	2	yes	complemented
149	#443	—	2	16	no	—
150	#436	—	2	4	no	—
151	C₈×D₄	—	2	4	no	—
152	C₂×#108	C₂²×C₁₂₇	1	4	yes	—
153	#423	C₂²×C₁₂₇	1	2	yes	—
154	C₂×#104	C₂²×C₁₂₇	1	8	yes	—
155	#408	C₂²×C₁₂₇	1	4	yes	—
156	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
157	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
158	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
160	C₄³	C₂²×C₁₂₇	1	1	yes	characteristic
161	C₂×#41	C₂³×C₁₂₇	1	4	yes	—
162	#135	—	2	8	no	—
163	#135	—	2	8	no	—
164	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
165	C₂×SD₁₆	C₂×C₄×C₁₂₇	1	8	yes	—
166	C₂×D₈	C₂×C₄×C₁₂₇	1	4	yes	—
167	#131	—	2	16	no	—
168	C₂×M₁₆	—	2	4	no	—
169	C₂×M₁₆	C₂³×C₁₂₇	1	4	yes	—
170	C₂²×C₈	—	2	4	no	—
171	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
172	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
173	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
174	C₄⋊Q₈	C₂×C₄×C₁₂₇	1	2	yes	—
175	#127	C₂×C₄×C₁₂₇	1	2	yes	—
176	#124	C₂×C₄×C₁₂₇	1	4	yes	—
177	#121	—	2	4	no	—
178	C₄×D₄	—	2	4	no	—
179	C₂×C₄²	—	2	2	no	—
180	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
181	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
182	#108	—	2	8	no	—
183	C₄⋊C₈	—	2	4	no	—
184	#104	—	2	16	no	—
185	C₂²⋊C₈	—	2	8	no	—
186	C₄×C₈	—	2	4	no	—
187	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
188	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
189	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
190	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
191	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
192	C₄×C₈	D₄×C₁₂₇	1	8	yes	complemented
193	#41	—	2	8	no	—
194	#41	—	2	8	no	—
195	C₂×Q₈	C₂²×C₄×C₁₂₇	1	4	yes	—
196	C₂×D₄	—	2	4	no	—
197	C₂×D₄	—	2	4	no	—
198	C₂×D₄	C₂²×C₄×C₁₂₇	1	4	yes	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	#41×C₁₂₇	1	2	yes	—
202	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
203	Q₁₆	—	2	8	no	—
204	SD₁₆	—	2	16	no	—
205	D₈	—	2	8	no	—
206	M₁₆	—	2	8	no	—
207	M₁₆	—	4	8	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	4	8	no	—
211	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
212	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
213	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
214	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
215	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
216	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
217	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
218	C₂×C₈	#41×C₁₂₇	1	4	yes	—
219	C₄⋊C₄	—	2	4	no	—
220	C₂²⋊C₄	—	2	8	no	—
221	C₄²	—	2	8	no	—
222	C₄²	—	2	2	no	—
223	C₄²	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
224	C₄²	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
225	C₄²	C₂×D₄×C₁₂₇	1	4	yes	—
226	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
227	C₂³	—	2	4	no	—
228	C₂³	C₂×#41×C₁₂₇	1	1	yes	characteristic
229	Q₈	—	2	8	no	—
230	D₄	—	2	8	no	—
231	D₄	—	4	8	no	—
232	D₄	—	4	8	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
240	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
241	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
242	C₂×C₄	C₂²×D₄×C₁₂₇	1	2	yes	—
243	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
244	C₂×C₄	C₂×#41×C₁₂₇	1	2	yes	—
245	C₈	—	2	2	no	—
246	C₈	—	4	4	no	—
247	C₈	C₄×D₄×C₁₂₇	1	4	yes	—
248	C₈	C₄×D₄×C₁₂₇	1	4	yes	—
249	C₈	#127×C₁₂₇	1	2	yes	—
250	C₈	#127×C₁₂₇	1	2	yes	—
251	C₂²	—	2	2	no	—
252	C₂²	—	4	8	no	—
253	C₂²	—	4	4	no	—
254	C₂²	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
255	C₂²	#532×C₁₂₇	1	2	yes	—
256	C₄	—	2	8	no	—
257	C₄	—	2	2	no	—
258	C₄	—	4	4	no	—
259	C₄	#443×C₁₂₇	1	4	yes	complemented
260	C₄	C₂×C₄×D₄×C₁₂₇	1	2	yes	—
261	C₄	C₂×#127×C₁₂₇	1	1	yes	characteristic
262	C₄	#532×C₁₂₇	1	1	yes	characteristic
263	C₂	—	2	2	no	—
264	C₂	—	8	4	no	—
265	C₂	C₄×#127×C₁₂₇	1	1	yes	characteristic
266	C₂	C₂×#443×C₁₂₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₁₂₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle	C₂²×C₁₂₇	C₂×C₄×D₄
prime radical	C₁₂₇×#3147	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#443×C₁₂₇	C₂	semidirect
3. C₁₂₇×#2931	C₂	semidirect
4. C₁₂₇×#1974	C₂	semidirect
5. C₁₂₇×#1962	C₂	semidirect
6. C₁₂₇×#1848	C₂	semidirect
7. C₄²×C₈×C₁₂₇	C₂	semidirect
11. #495×C₁₂₇	C₄	semidirect
12. #494×C₁₂₇	C₄	semidirect
13. #493×C₁₂₇	C₄	semidirect
14. #492×C₁₂₇	C₄	semidirect
58. C₄×C₈×C₁₂₇	D₄	semidirect
125. C₄×C₁₂₇	#443	semidirect
129. #21705	C₁₂₇	direct (N × H)
135. C₂×#443	C₂×C₁₂₇	semidirect
136. #2931	C₂×C₁₂₇	semidirect
137. #1974	C₂×C₁₂₇	semidirect
138. #1962	C₂×C₁₂₇	semidirect
139. #1848	C₂×C₁₂₇	semidirect
140. C₄²×C₈	C₂×C₁₂₇	semidirect
141. C₁₂₇	#21705	direct (N × H)
145. #495	C₄×C₁₂₇	semidirect
146. #494	C₄×C₁₂₇	semidirect
147. #493	C₄×C₁₂₇	semidirect
148. #492	C₄×C₁₂₇	semidirect
192. C₄×C₈	D₄×C₁₂₇	semidirect
259. C₄	#443×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×#3147	2
C₂×#443×C₁₂₇	2
C₁₂₇×#2931	2
C₁₂₇×#1974	2
C₁₂₇×#1962	2
C₁₂₇×#1848	2
C₄²×C₈×C₁₂₇	2
#21705	127

Subgroups — C127×#21705 #44585378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#21705 #44585378