Subgroups — C₁₂₇×#13214 #44576887

Order 32512.

490 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#495×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#362×C₁₂₇	C₂	1	2	yes	maximal, complemented
3	C₂×#361×C₁₂₇	C₂	1	2	yes	maximal
4	C₄×M₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₂₇×#1848	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₂²×SD₁₆×C₁₂₇	—	2	1	no	—
8	C₂²×D₈×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₂×#124×C₁₂₇	—	2	1	no	—
10	C₂×M₃₂×C₁₂₇	C₂²	1	2	yes	—
11	#495×C₁₂₇	—	2	1	no	—
12	#495×C₁₂₇	—	2	1	no	—
13	#495×C₁₂₇	C₄	1	2	yes	—
14	#495×C₁₂₇	C₄	1	2	yes	—
15	C₂×#108×C₁₂₇	—	2	1	no	—
16	C₂×#108×C₁₂₇	C₂²	1	2	yes	—
17	#423×C₁₂₇	—	2	1	no	—
18	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
19	#362×C₁₂₇	—	2	4	no	—
20	#361×C₁₂₇	—	2	4	no	—
21	#346×C₁₂₇	—	2	1	no	—
22	C₄×C₁₆×C₁₂₇	—	2	1	no	—
23	C₂²×Q₈×C₁₂₇	—	2	1	no	—
24	C₂²×D₄×C₁₂₇	—	2	1	no	—
25	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
26	C₂×Q₁₆×C₁₂₇	—	2	1	no	—
27	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
28	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
29	C₂×SD₁₆×C₁₂₇	—	2	2	no	—
30	C₂×SD₁₆×C₁₂₇	—	2	2	no	—
31	C₂×SD₁₆×C₁₂₇	—	4	2	no	—
32	C₂×SD₁₆×C₁₂₇	—	4	2	no	—
33	C₂×D₈×C₁₂₇	—	2	2	no	—
34	C₂×D₈×C₁₂₇	—	2	1	no	—
35	C₂×D₈×C₁₂₇	—	2	2	no	—
36	C₂×D₈×C₁₂₇	C₂×C₄	1	2	yes	—
37	C₂×M₁₆×C₁₂₇	—	2	2	no	—
38	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
40	#124×C₁₂₇	—	2	2	no	—
41	#124×C₁₂₇	—	2	2	no	—
42	#124×C₁₂₇	—	4	2	no	—
43	C₂×C₂²⋊C₄×C₁₂₇	—	4	2	no	—
44	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
45	M₃₂×C₁₂₇	—	2	2	no	—
46	M₃₂×C₁₂₇	D₄	1	4	yes	—
47	C₂×C₁₆×C₁₂₇	—	2	2	no	—
48	#108×C₁₂₇	—	2	2	no	—
49	#108×C₁₂₇	—	2	2	no	—
50	#108×C₁₂₇	—	4	2	no	—
51	C₄⋊C₈×C₁₂₇	—	4	2	no	—
52	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
55	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
56	C₂⁴×C₁₂₇	—	4	1	no	—
57	C₂×Q₈×C₁₂₇	—	2	2	no	—
58	C₂×Q₈×C₁₂₇	—	4	1	no	—
59	C₂×Q₈×C₁₂₇	—	4	1	no	—
60	C₂×Q₈×C₁₂₇	—	4	1	no	—
61	C₂×Q₈×C₁₂₇	—	4	1	no	—
62	C₂×Q₈×C₁₂₇	—	4	1	no	—
63	C₂×D₄×C₁₂₇	—	2	2	no	—
64	C₂×D₄×C₁₂₇	—	4	1	no	—
65	C₂×D₄×C₁₂₇	—	4	1	no	—
66	C₂×D₄×C₁₂₇	—	4	1	no	—
67	C₂×D₄×C₁₂₇	—	4	1	no	—
68	C₂×D₄×C₁₂₇	—	4	1	no	—
69	C₂²×C₄×C₁₂₇	—	4	1	no	—
70	C₂²×C₄×C₁₂₇	SD₁₆	1	2	yes	—
71	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
72	Q₁₆×C₁₂₇	—	2	4	no	—
73	Q₁₆×C₁₂₇	—	4	2	no	—
74	SD₁₆×C₁₂₇	—	4	4	no	—
75	SD₁₆×C₁₂₇	—	4	4	no	—
76	D₈×C₁₂₇	—	2	4	no	—
77	D₈×C₁₂₇	—	4	2	no	—
78	M₁₆×C₁₂₇	—	4	2	no	—
79	M₁₆×C₁₂₇	—	4	2	no	—
80	C₂×C₈×C₁₂₇	—	4	2	no	—
81	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₂₇	D₈	1	2	yes	—
84	C₂×C₈×C₁₂₇	SD₁₆	1	2	yes	—
85	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₁₂₇	#41	1	2	yes	—
89	C₂²⋊C₄×C₁₂₇	—	4	4	no	—
90	C₂²⋊C₄×C₁₂₇	—	4	4	no	—
91	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
92	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
93	C₄²×C₁₂₇	D₈	1	2	yes	—
94	C₁₆×C₁₂₇	—	2	4	no	—
95	C₂³×C₁₂₇	—	4	2	no	—
96	C₂³×C₁₂₇	—	8	4	no	—
97	C₂³×C₁₂₇	—	8	1	no	—
98	C₂³×C₁₂₇	—	8	1	no	—
99	C₂³×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
100	Q₈×C₁₂₇	—	4	2	no	—
101	Q₈×C₁₂₇	—	4	2	no	—
102	Q₈×C₁₂₇	—	8	1	no	—
103	Q₈×C₁₂₇	—	8	1	no	—
104	D₄×C₁₂₇	—	4	2	no	—
105	D₄×C₁₂₇	—	4	2	no	—
106	D₄×C₁₂₇	—	8	1	no	—
107	D₄×C₁₂₇	—	8	1	no	—
108	C₂×C₄×C₁₂₇	—	4	2	no	—
109	C₂×C₄×C₁₂₇	—	4	2	no	—
110	C₂×C₄×C₁₂₇	—	8	1	no	—
111	C₂×C₄×C₁₂₇	—	8	1	no	—
112	C₂×C₄×C₁₂₇	#102	1	2	yes	—
113	C₂×C₄×C₁₂₇	#102	1	2	yes	—
114	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₂₇	#124	1	1	yes	characteristic
116	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
117	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
120	C₈×C₁₂₇	—	8	2	no	—
121	C₈×C₁₂₇	#102	1	2	yes	—
122	C₈×C₁₂₇	#102	1	2	yes	—
123	C₈×C₁₂₇	C₄×D₄	1	2	yes	—
124	C₈×C₁₂₇	#121	1	2	yes	—
125	C₂²×C₁₂₇	—	2	2	no	—
126	C₂²×C₁₂₇	—	8	4	no	—
127	C₂²×C₁₂₇	—	8	2	no	—
128	C₂²×C₁₂₇	—	16	2	no	—
129	C₂²×C₁₂₇	—	16	2	no	—
130	C₂²×C₁₂₇	C₂×#102	1	1	yes	characteristic
131	C₂²×C₁₂₇	#486	1	1	yes	characteristic
132	C₂²×C₁₂₇	#492	1	1	yes	characteristic
133	C₄×C₁₂₇	—	2	2	no	—
134	C₄×C₁₂₇	—	2	2	no	—
135	C₄×C₁₂₇	—	8	2	no	—
136	C₄×C₁₂₇	#390	1	1	yes	characteristic
137	C₄×C₁₂₇	C₂×#102	1	1	yes	characteristic
138	C₄×C₁₂₇	#486	1	1	yes	characteristic
139	C₄×C₁₂₇	#493	1	1	yes	characteristic
140	#13214	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂×C₁₂₇	—	2	1	no	—
142	C₂×C₁₂₇	—	2	1	no	—
143	C₂×C₁₂₇	—	16	2	no	—
144	C₂×C₁₂₇	#1957	1	1	yes	characteristic
145	C₂×C₁₂₇	#2148	1	2	yes	—
146	C₂×#495	C₂×C₁₂₇	1	1	yes	characteristic
147	C₂×#362	C₂×C₁₂₇	1	2	yes	complemented
148	C₂×#361	C₂×C₁₂₇	1	2	yes	—
149	C₄×M₃₂	C₂×C₁₂₇	1	1	yes	characteristic, complemented
150	#1848	C₂×C₁₂₇	1	1	yes	characteristic, complemented
151	C₁₂₇	#13214	1	1	yes	characteristic, direct product factor, Sylow
152	C₂²×Q₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
153	C₂²×SD₁₆	—	2	1	no	—
154	C₂²×D₈	C₂²×C₁₂₇	1	1	yes	characteristic
155	C₂×#124	—	2	1	no	—
156	C₂×M₃₂	C₂²×C₁₂₇	1	2	yes	—
157	#495	—	2	1	no	—
158	#495	—	2	1	no	—
159	#495	C₄×C₁₂₇	1	2	yes	—
160	#495	C₄×C₁₂₇	1	2	yes	—
161	C₂×#108	—	2	1	no	—
162	C₂×#108	C₂²×C₁₂₇	1	2	yes	—
163	#423	—	2	1	no	—
164	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
165	#362	—	2	4	no	—
166	#361	—	2	4	no	—
167	#346	—	2	1	no	—
168	C₄×C₁₆	—	2	1	no	—
169	C₂²×Q₈	—	2	1	no	—
170	C₂²×D₄	—	2	1	no	—
171	C₂×Q₁₆	—	2	2	no	—
172	C₂×Q₁₆	—	2	1	no	—
173	C₂×Q₁₆	—	2	2	no	—
174	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
175	C₂×SD₁₆	—	2	2	no	—
176	C₂×SD₁₆	—	2	2	no	—
177	C₂×SD₁₆	—	4	2	no	—
178	C₂×SD₁₆	—	4	2	no	—
179	C₂×D₈	—	2	2	no	—
180	C₂×D₈	—	2	1	no	—
181	C₂×D₈	—	2	2	no	—
182	C₂×D₈	C₂×C₄×C₁₂₇	1	2	yes	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
185	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
186	#124	—	2	2	no	—
187	#124	—	2	2	no	—
188	#124	—	4	2	no	—
189	C₂×C₂²⋊C₄	—	4	2	no	—
190	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
191	M₃₂	—	2	2	no	—
192	M₃₂	D₄×C₁₂₇	1	4	yes	—
193	C₂×C₁₆	—	2	2	no	—
194	#108	—	2	2	no	—
195	#108	—	2	2	no	—
196	#108	—	4	2	no	—
197	C₄⋊C₈	—	4	2	no	—
198	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
199	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
200	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
201	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
202	C₂⁴	—	4	1	no	—
203	C₂×Q₈	—	2	2	no	—
204	C₂×Q₈	—	4	1	no	—
205	C₂×Q₈	—	4	1	no	—
206	C₂×Q₈	—	4	1	no	—
207	C₂×Q₈	—	4	1	no	—
208	C₂×Q₈	—	4	1	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	4	1	no	—
211	C₂×D₄	—	4	1	no	—
212	C₂×D₄	—	4	1	no	—
213	C₂×D₄	—	4	1	no	—
214	C₂×D₄	—	4	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	SD₁₆×C₁₂₇	1	2	yes	—
217	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
218	Q₁₆	—	2	4	no	—
219	Q₁₆	—	4	2	no	—
220	SD₁₆	—	4	4	no	—
221	SD₁₆	—	4	4	no	—
222	D₈	—	2	4	no	—
223	D₈	—	4	2	no	—
224	M₁₆	—	4	2	no	—
225	M₁₆	—	4	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
228	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
229	C₂×C₈	D₈×C₁₂₇	1	2	yes	—
230	C₂×C₈	SD₁₆×C₁₂₇	1	2	yes	—
231	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
232	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
233	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
234	C₂×C₈	#41×C₁₂₇	1	2	yes	—
235	C₂²⋊C₄	—	4	4	no	—
236	C₂²⋊C₄	—	4	4	no	—
237	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
238	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
239	C₄²	D₈×C₁₂₇	1	2	yes	—
240	C₁₆	—	2	4	no	—
241	C₂³	—	4	2	no	—
242	C₂³	—	8	4	no	—
243	C₂³	—	8	1	no	—
244	C₂³	—	8	1	no	—
245	C₂³	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
246	Q₈	—	4	2	no	—
247	Q₈	—	4	2	no	—
248	Q₈	—	8	1	no	—
249	Q₈	—	8	1	no	—
250	D₄	—	4	2	no	—
251	D₄	—	4	2	no	—
252	D₄	—	8	1	no	—
253	D₄	—	8	1	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	8	1	no	—
257	C₂×C₄	—	8	1	no	—
258	C₂×C₄	#102×C₁₂₇	1	2	yes	—
259	C₂×C₄	#102×C₁₂₇	1	2	yes	—
260	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
261	C₂×C₄	#124×C₁₂₇	1	1	yes	characteristic
262	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
263	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
264	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
265	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
266	C₈	—	8	2	no	—
267	C₈	#102×C₁₂₇	1	2	yes	—
268	C₈	#102×C₁₂₇	1	2	yes	—
269	C₈	C₄×D₄×C₁₂₇	1	2	yes	—
270	C₈	#121×C₁₂₇	1	2	yes	—
271	C₂²	—	2	2	no	—
272	C₂²	—	8	4	no	—
273	C₂²	—	8	2	no	—
274	C₂²	—	16	2	no	—
275	C₂²	—	16	2	no	—
276	C₂²	C₂×#102×C₁₂₇	1	1	yes	characteristic
277	C₂²	#486×C₁₂₇	1	1	yes	characteristic
278	C₂²	#492×C₁₂₇	1	1	yes	characteristic
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	8	2	no	—
282	C₄	#390×C₁₂₇	1	1	yes	characteristic
283	C₄	C₂×#102×C₁₂₇	1	1	yes	characteristic
284	C₄	#486×C₁₂₇	1	1	yes	characteristic
285	C₄	#493×C₁₂₇	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	—	16	2	no	—
289	C₂	C₁₂₇×#1957	1	1	yes	characteristic
290	C₂	C₁₂₇×#2148	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₂₇	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
prime radical	C₂²×D₈×C₁₂₇	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#362×C₁₂₇	C₂	semidirect
4. C₄×M₃₂×C₁₂₇	C₂	semidirect
5. C₁₂₇×#1848	C₂	semidirect
140. #13214	C₁₂₇	direct (N × H)
147. C₂×#362	C₂×C₁₂₇	semidirect
149. C₄×M₃₂	C₂×C₁₂₇	semidirect
150. #1848	C₂×C₁₂₇	semidirect
151. C₁₂₇	#13214	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#495×C₁₂₇	2
C₂×#362×C₁₂₇	2
C₂×#361×C₁₂₇	2
C₄×M₃₂×C₁₂₇	2
C₁₂₇×#1848	2
#13214	127

Subgroups — C127×#13214 #44576887

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#13214 #44576887