Subgroups — C₁₂₇×#21733 #44585406

Order 32512.

654 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#3160	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#2961	C₂	1	2	yes	maximal
3	C₁₂₇×#1995	C₂	1	2	yes	maximal
4	C₁₂₇×#1946	C₂	1	2	yes	maximal
5	C₁₂₇×#1945	C₂	1	2	yes	maximal
6	C₁₂₇×#1848	C₂	1	1	yes	maximal, characteristic
7	C₁₂₇×#1773	C₂	1	2	yes	maximal
8	C₁₂₇×#1772	C₂	1	2	yes	maximal
9	C₁₂₇×#1726	C₂	1	1	yes	maximal, characteristic
10	#533×C₁₂₇	C₂²	1	2	yes	—
11	#500×C₁₂₇	C₄	1	2	yes	complemented
12	#499×C₁₂₇	C₄	1	2	yes	complemented
13	#499×C₁₂₇	C₄	1	2	yes	complemented
14	#498×C₁₂₇	C₄	1	2	yes	complemented
15	#446×C₁₂₇	—	2	8	no	—
16	C₂×#108×C₁₂₇	C₂²	1	2	yes	—
17	C₂×#108×C₁₂₇	C₂²	1	2	yes	—
18	#427×C₁₂₇	C₂²	1	2	yes	—
19	#427×C₁₂₇	C₂²	1	2	yes	—
20	#424×C₁₂₇	C₂²	1	4	yes	—
21	#423×C₁₂₇	C₂²	1	2	yes	—
22	#405×C₁₂₇	C₂²	1	4	yes	—
23	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
24	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
25	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
29	#376×C₁₂₇	C₂²	1	1	yes	characteristic
30	#339×C₁₂₇	C₂²	1	4	yes	—
31	#339×C₁₂₇	C₂²	1	4	yes	—
32	C₂×M₁₆×C₁₂₇	C₂³	1	4	yes	—
33	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
37	C₄⋊Q₈×C₁₂₇	C₂×C₄	1	4	yes	—
38	#125×C₁₂₇	C₂×C₄	1	4	yes	—
39	C₂²⋊Q₈×C₁₂₇	—	2	4	no	—
40	C₄×Q₈×C₁₂₇	—	2	4	no	—
41	#117×C₁₂₇	C₂³	1	4	yes	—
42	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
45	#108×C₁₂₇	—	2	4	no	—
46	#108×C₁₂₇	—	2	4	no	—
47	#107×C₁₂₇	C₂×C₄	1	4	yes	—
48	#107×C₁₂₇	C₂×C₄	1	4	yes	—
49	#106×C₁₂₇	C₂×C₄	1	4	yes	—
50	#106×C₁₂₇	C₂×C₄	1	4	yes	—
51	C₄⋊C₈×C₁₂₇	—	2	8	no	—
52	C₄⋊C₈×C₁₂₇	—	2	4	no	—
53	#97×C₁₂₇	—	2	4	no	—
54	#97×C₁₂₇	—	2	2	no	—
55	#97×C₁₂₇	D₄	1	4	yes	—
56	#97×C₁₂₇	Q₈	1	4	yes	complemented
57	C₄×C₈×C₁₂₇	—	2	4	no	—
58	C₄×C₈×C₁₂₇	—	2	2	no	—
59	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₁₂₇	Q₈	1	4	yes	complemented
64	#95×C₁₂₇	—	2	2	no	—
65	C₂×Q₈×C₁₂₇	—	2	4	no	—
66	C₂²×C₄×C₁₂₇	—	2	1	no	—
67	C₂²×C₄×C₁₂₇	—	2	1	no	—
68	C₂²×C₄×C₁₂₇	#41	1	2	yes	—
69	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
70	M₁₆×C₁₂₇	—	2	4	no	—
71	M₁₆×C₁₂₇	—	2	4	no	—
72	C₂×C₈×C₁₂₇	—	2	1	no	—
73	C₂×C₈×C₁₂₇	—	2	1	no	—
74	C₂×C₈×C₁₂₇	—	2	4	no	—
75	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₁₂₇	C₂×Q₈	1	2	yes	—
82	C₂×C₈×C₁₂₇	C₂×Q₈	1	2	yes	—
83	C₂×C₈×C₁₂₇	C₂×Q₈	1	2	yes	—
84	C₂×C₈×C₁₂₇	C₂×Q₈	1	2	yes	—
85	C₂×C₈×C₁₂₇	#41	1	2	yes	—
86	C₂×C₈×C₁₂₇	#41	1	2	yes	—
87	C₄⋊C₄×C₁₂₇	—	2	8	no	—
88	C₄⋊C₄×C₁₂₇	—	2	4	no	—
89	C₄⋊C₄×C₁₂₇	C₂²×C₄	1	8	yes	—
90	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
91	C₄²×C₁₂₇	—	2	4	no	—
92	C₄²×C₁₂₇	—	2	1	no	—
93	C₄²×C₁₂₇	—	2	1	no	—
94	C₄²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
97	C₄²×C₁₂₇	C₂×Q₈	1	2	yes	—
98	C₄²×C₁₂₇	C₂×Q₈	1	2	yes	—
99	C₂³×C₁₂₇	C₂×#41	1	1	yes	characteristic
100	Q₈×C₁₂₇	—	8	4	no	—
101	C₂×C₄×C₁₂₇	—	2	8	no	—
102	C₂×C₄×C₁₂₇	—	2	2	no	—
103	C₂×C₄×C₁₂₇	—	2	2	no	—
104	C₂×C₄×C₁₂₇	—	4	2	no	—
105	C₂×C₄×C₁₂₇	—	4	2	no	—
106	C₂×C₄×C₁₂₇	—	4	2	no	—
107	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
115	C₈×C₁₂₇	—	2	2	no	—
116	C₈×C₁₂₇	—	4	4	no	—
117	C₈×C₁₂₇	C₄×Q₈	1	2	yes	—
118	C₈×C₁₂₇	C₄×Q₈	1	2	yes	—
119	C₈×C₁₂₇	C₄×Q₈	1	2	yes	—
120	C₈×C₁₂₇	C₄×Q₈	1	2	yes	—
121	C₈×C₁₂₇	C₄⋊Q₈	1	4	yes	—
122	C₂²×C₁₂₇	—	2	2	no	—
123	C₂²×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
124	C₂²×C₁₂₇	#533	1	1	yes	characteristic
125	C₂²×C₁₂₇	#533	1	1	yes	characteristic
126	C₄×C₁₂₇	—	2	2	no	—
127	C₄×C₁₂₇	—	2	2	no	—
128	C₄×C₁₂₇	—	4	2	no	—
129	C₄×C₁₂₇	—	4	2	no	—
130	C₄×C₁₂₇	—	4	8	no	—
131	C₄×C₁₂₇	C₂×C₄×Q₈	1	1	yes	characteristic
132	C₄×C₁₂₇	C₂×C₄×Q₈	1	1	yes	characteristic
133	C₄×C₁₂₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
134	C₄×C₁₂₇	#532	1	1	yes	characteristic
135	#21733	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂×C₁₂₇	—	2	1	no	—
137	C₂×C₁₂₇	—	2	1	no	—
138	C₂×C₁₂₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
139	C₂×C₁₂₇	#2957	1	2	yes	—
140	#3160	C₂×C₁₂₇	1	1	yes	characteristic
141	#2961	C₂×C₁₂₇	1	2	yes	—
142	#1995	C₂×C₁₂₇	1	2	yes	—
143	#1946	C₂×C₁₂₇	1	2	yes	—
144	#1945	C₂×C₁₂₇	1	2	yes	—
145	#1848	C₂×C₁₂₇	1	1	yes	characteristic
146	#1773	C₂×C₁₂₇	1	2	yes	—
147	#1772	C₂×C₁₂₇	1	2	yes	—
148	#1726	C₂×C₁₂₇	1	1	yes	characteristic
149	C₁₂₇	#21733	1	1	yes	characteristic, direct product factor, Sylow
150	#533	C₂²×C₁₂₇	1	2	yes	—
151	#500	C₄×C₁₂₇	1	2	yes	complemented
152	#499	C₄×C₁₂₇	1	2	yes	complemented
153	#499	C₄×C₁₂₇	1	2	yes	complemented
154	#498	C₄×C₁₂₇	1	2	yes	complemented
155	#446	—	2	8	no	—
156	C₂×#108	C₂²×C₁₂₇	1	2	yes	—
157	C₂×#108	C₂²×C₁₂₇	1	2	yes	—
158	#427	C₂²×C₁₂₇	1	2	yes	—
159	#427	C₂²×C₁₂₇	1	2	yes	—
160	#424	C₂²×C₁₂₇	1	4	yes	—
161	#423	C₂²×C₁₂₇	1	2	yes	—
162	#405	C₂²×C₁₂₇	1	4	yes	—
163	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
164	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
165	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
166	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
167	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
168	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
169	#376	C₂²×C₁₂₇	1	1	yes	characteristic
170	#339	C₂²×C₁₂₇	1	4	yes	—
171	#339	C₂²×C₁₂₇	1	4	yes	—
172	C₂×M₁₆	C₂³×C₁₂₇	1	4	yes	—
173	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
174	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
175	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
176	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
177	C₄⋊Q₈	C₂×C₄×C₁₂₇	1	4	yes	—
178	#125	C₂×C₄×C₁₂₇	1	4	yes	—
179	C₂²⋊Q₈	—	2	4	no	—
180	C₄×Q₈	—	2	4	no	—
181	#117	C₂³×C₁₂₇	1	4	yes	—
182	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
183	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
184	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
185	#108	—	2	4	no	—
186	#108	—	2	4	no	—
187	#107	C₂×C₄×C₁₂₇	1	4	yes	—
188	#107	C₂×C₄×C₁₂₇	1	4	yes	—
189	#106	C₂×C₄×C₁₂₇	1	4	yes	—
190	#106	C₂×C₄×C₁₂₇	1	4	yes	—
191	C₄⋊C₈	—	2	8	no	—
192	C₄⋊C₈	—	2	4	no	—
193	#97	—	2	4	no	—
194	#97	—	2	2	no	—
195	#97	D₄×C₁₂₇	1	4	yes	—
196	#97	Q₈×C₁₂₇	1	4	yes	complemented
197	C₄×C₈	—	2	4	no	—
198	C₄×C₈	—	2	2	no	—
199	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
200	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
201	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
202	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
203	C₄×C₈	Q₈×C₁₂₇	1	4	yes	complemented
204	#95	—	2	2	no	—
205	C₂×Q₈	—	2	4	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	#41×C₁₂₇	1	2	yes	—
209	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
210	M₁₆	—	2	4	no	—
211	M₁₆	—	2	4	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
216	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
217	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
218	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
219	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
220	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
221	C₂×C₈	C₂×Q₈×C₁₂₇	1	2	yes	—
222	C₂×C₈	C₂×Q₈×C₁₂₇	1	2	yes	—
223	C₂×C₈	C₂×Q₈×C₁₂₇	1	2	yes	—
224	C₂×C₈	C₂×Q₈×C₁₂₇	1	2	yes	—
225	C₂×C₈	#41×C₁₂₇	1	2	yes	—
226	C₂×C₈	#41×C₁₂₇	1	2	yes	—
227	C₄⋊C₄	—	2	8	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	C₂²×C₄×C₁₂₇	1	8	yes	—
230	C₂²⋊C₄	—	2	4	no	—
231	C₄²	—	2	4	no	—
232	C₄²	—	2	1	no	—
233	C₄²	—	2	1	no	—
234	C₄²	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
235	C₄²	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
236	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
237	C₄²	C₂×Q₈×C₁₂₇	1	2	yes	—
238	C₄²	C₂×Q₈×C₁₂₇	1	2	yes	—
239	C₂³	C₂×#41×C₁₂₇	1	1	yes	characteristic
240	Q₈	—	8	4	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
248	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
249	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
250	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
251	C₂×C₄	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
252	C₂×C₄	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
253	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
254	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	—	4	4	no	—
257	C₈	C₄×Q₈×C₁₂₇	1	2	yes	—
258	C₈	C₄×Q₈×C₁₂₇	1	2	yes	—
259	C₈	C₄×Q₈×C₁₂₇	1	2	yes	—
260	C₈	C₄×Q₈×C₁₂₇	1	2	yes	—
261	C₈	C₄⋊Q₈×C₁₂₇	1	4	yes	—
262	C₂²	—	2	2	no	—
263	C₂²	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
264	C₂²	#533×C₁₂₇	1	1	yes	characteristic
265	C₂²	#533×C₁₂₇	1	1	yes	characteristic
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	4	2	no	—
269	C₄	—	4	2	no	—
270	C₄	—	4	8	no	—
271	C₄	C₂×C₄×Q₈×C₁₂₇	1	1	yes	characteristic
272	C₄	C₂×C₄×Q₈×C₁₂₇	1	1	yes	characteristic
273	C₄	C₂×C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
274	C₄	#532×C₁₂₇	1	1	yes	characteristic
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	C₄×C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
278	C₂	C₁₂₇×#2957	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle	C₂²×C₁₂₇	C₂×C₄×D₄
prime radical	C₂³×C₁₂₇	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂×#97×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
11. #500×C₁₂₇	C₄	semidirect
12. #499×C₁₂₇	C₄	semidirect
13. #499×C₁₂₇	C₄	semidirect
14. #498×C₁₂₇	C₄	semidirect
56. #97×C₁₂₇	Q₈	semidirect
63. C₄×C₈×C₁₂₇	Q₈	semidirect
135. #21733	C₁₂₇	direct (N × H)
149. C₁₂₇	#21733	direct (N × H)
151. #500	C₄×C₁₂₇	semidirect
152. #499	C₄×C₁₂₇	semidirect
153. #499	C₄×C₁₂₇	semidirect
154. #498	C₄×C₁₂₇	semidirect
196. #97	Q₈×C₁₂₇	semidirect
203. C₄×C₈	Q₈×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×#3160	2
C₁₂₇×#2961	2
C₁₂₇×#1995	2
C₁₂₇×#1946	2
C₁₂₇×#1945	2
C₁₂₇×#1848	2
C₁₂₇×#1773	2
C₁₂₇×#1772	2
C₁₂₇×#1726	2
#21733	127

Subgroups — C127×#21733 #44585406

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#21733 #44585406