Subgroups — C₁₂₇×#13197 #44576870

Order 32512.

424 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#443×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#364×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₂₇×#2117	C₂	1	2	yes	maximal, complemented
4	C₁₂₇×#2115	C₂	1	2	yes	maximal, complemented
5	C₄×M₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#135×C₁₂₇	C₄	1	1	yes	characteristic
7	C₂×#131×C₁₂₇	C₂²	1	2	yes	—
8	C₂×M₃₂×C₁₂₇	C₂²	1	2	yes	—
9	C₂×M₃₂×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂×M₃₂×C₁₂₇	C₂²	1	1	yes	characteristic
11	#443×C₁₂₇	—	2	1	no	—
12	#443×C₁₂₇	—	2	1	no	—
13	#443×C₁₂₇	—	2	1	no	—
14	#443×C₁₂₇	—	2	1	no	—
15	C₂×#108×C₁₂₇	C₄	1	1	yes	characteristic
16	C₂×#104×C₁₂₇	C₄	1	2	yes	—
17	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
18	#364×C₁₂₇	—	2	1	no	—
19	#364×C₁₂₇	—	2	1	no	—
20	#350×C₁₂₇	—	2	4	no	—
21	#349×C₁₂₇	—	2	4	no	—
22	#346×C₁₂₇	—	2	1	no	—
23	C₄×C₁₆×C₁₂₇	—	2	1	no	—
24	C₂×#41×C₁₂₇	C₂×C₄	1	2	yes	—
25	#135×C₁₂₇	—	2	1	no	—
26	#135×C₁₂₇	—	2	1	no	—
27	#135×C₁₂₇	—	2	1	no	—
28	#135×C₁₂₇	—	2	1	no	—
29	C₂×Q₁₆×C₁₂₇	C₈	1	1	yes	characteristic
30	C₂×SD₁₆×C₁₂₇	C₈	1	2	yes	—
31	C₂×D₈×C₁₂₇	C₈	1	1	yes	characteristic
32	#131×C₁₂₇	—	2	2	no	—
33	#131×C₁₂₇	—	2	2	no	—
34	#131×C₁₂₇	—	2	2	no	—
35	#131×C₁₂₇	—	2	2	no	—
36	C₂×M₁₆×C₁₂₇	—	2	2	no	—
37	C₂×M₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₁₂₇	—	2	2	no	—
39	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
42	M₃₂×C₁₂₇	—	2	1	no	—
43	M₃₂×C₁₂₇	—	2	1	no	—
44	M₃₂×C₁₂₇	—	2	2	no	—
45	M₃₂×C₁₂₇	—	2	2	no	—
46	M₃₂×C₁₂₇	D₄	1	2	yes	—
47	M₃₂×C₁₂₇	D₄	1	2	yes	—
48	C₂×C₁₆×C₁₂₇	—	2	1	no	—
49	C₂×C₁₆×C₁₂₇	—	2	1	no	—
50	C₂×C₁₆×C₁₂₇	—	2	2	no	—
51	#108×C₁₂₇	—	2	1	no	—
52	#108×C₁₂₇	—	2	1	no	—
53	#104×C₁₂₇	—	2	2	no	—
54	#104×C₁₂₇	—	2	2	no	—
55	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
56	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
57	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
58	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
59	#41×C₁₂₇	—	2	2	no	—
60	#41×C₁₂₇	—	2	2	no	—
61	#41×C₁₂₇	—	2	2	no	—
62	#41×C₁₂₇	—	2	2	no	—
63	C₂×Q₈×C₁₂₇	C₂×C₈	1	2	yes	—
64	C₂×D₄×C₁₂₇	—	2	2	no	—
65	C₂×D₄×C₁₂₇	C₂×C₈	1	2	yes	—
66	C₂²×C₄×C₁₂₇	—	2	1	no	—
67	C₂²×C₄×C₁₂₇	—	2	2	no	—
68	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
69	Q₁₆×C₁₂₇	—	2	1	no	—
70	Q₁₆×C₁₂₇	—	2	1	no	—
71	SD₁₆×C₁₂₇	—	2	2	no	—
72	SD₁₆×C₁₂₇	—	2	2	no	—
73	D₈×C₁₂₇	—	2	1	no	—
74	D₈×C₁₂₇	—	2	1	no	—
75	M₁₆×C₁₂₇	—	2	2	no	—
76	M₁₆×C₁₂₇	—	2	2	no	—
77	M₁₆×C₁₂₇	—	4	2	no	—
78	M₁₆×C₁₂₇	—	4	2	no	—
79	C₂×C₈×C₁₂₇	—	2	2	no	—
80	C₂×C₈×C₁₂₇	—	4	2	no	—
81	C₂×C₈×C₁₂₇	—	4	2	no	—
82	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
85	C₂×C₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
86	C₂×C₈×C₁₂₇	M₁₆	1	1	yes	characteristic
87	C₂×C₈×C₁₂₇	M₁₆	1	1	yes	characteristic
88	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
93	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
94	C₄²×C₁₂₇	—	2	1	no	—
95	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
96	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
97	C₁₆×C₁₂₇	—	2	2	no	—
98	C₁₆×C₁₂₇	—	2	2	no	—
99	C₁₆×C₁₂₇	—	4	2	no	—
100	C₂³×C₁₂₇	—	2	2	no	—
101	C₂³×C₁₂₇	#131	1	1	yes	characteristic
102	Q₈×C₁₂₇	—	2	2	no	—
103	Q₈×C₁₂₇	—	2	2	no	—
104	D₄×C₁₂₇	—	2	2	no	—
105	D₄×C₁₂₇	—	2	2	no	—
106	D₄×C₁₂₇	—	4	2	no	—
107	D₄×C₁₂₇	—	4	2	no	—
108	C₂×C₄×C₁₂₇	—	2	1	no	—
109	C₂×C₄×C₁₂₇	—	2	1	no	—
110	C₂×C₄×C₁₂₇	—	2	2	no	—
111	C₂×C₄×C₁₂₇	—	4	2	no	—
112	C₂×C₄×C₁₂₇	—	4	2	no	—
113	C₂×C₄×C₁₂₇	—	4	2	no	—
114	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
119	C₂×C₄×C₁₂₇	#131	1	1	yes	characteristic
120	C₈×C₁₂₇	—	4	2	no	—
121	C₈×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
122	C₈×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
123	C₈×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
124	C₈×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
125	C₈×C₁₂₇	#121	1	1	yes	characteristic
126	C₈×C₁₂₇	#121	1	1	yes	characteristic
127	C₈×C₁₂₇	#124	1	1	yes	characteristic
128	C₈×C₁₂₇	#127	1	1	yes	characteristic
129	C₂²×C₁₂₇	—	4	1	no	—
130	C₂²×C₁₂₇	—	4	2	no	—
131	C₂²×C₁₂₇	—	8	2	no	—
132	C₂²×C₁₂₇	#408	1	1	yes	characteristic
133	C₂²×C₁₂₇	C₈×D₄	1	1	yes	characteristic
134	C₂²×C₁₂₇	#436	1	1	yes	characteristic
135	C₄×C₁₂₇	—	2	2	no	—
136	C₄×C₁₂₇	—	4	1	no	—
137	C₄×C₁₂₇	—	4	2	no	—
138	C₄×C₁₂₇	#390	1	1	yes	characteristic
139	C₄×C₁₂₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
140	C₄×C₁₂₇	C₈×D₄	1	1	yes	characteristic
141	C₄×C₁₂₇	#436	1	1	yes	characteristic
142	#13197	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂×C₁₂₇	—	2	1	no	—
144	C₂×C₁₂₇	—	2	1	no	—
145	C₂×C₁₂₇	—	2	1	no	—
146	C₂×C₁₂₇	—	8	2	no	—
147	C₂×C₁₂₇	#1954	1	1	yes	characteristic
148	C₂×#443	C₂×C₁₂₇	1	1	yes	characteristic
149	C₂×#364	C₂×C₁₂₇	1	1	yes	characteristic, complemented
150	#2117	C₂×C₁₂₇	1	2	yes	complemented
151	#2115	C₂×C₁₂₇	1	2	yes	complemented
152	C₄×M₃₂	C₂×C₁₂₇	1	1	yes	characteristic, complemented
153	C₁₂₇	#13197	1	1	yes	characteristic, direct product factor, Sylow
154	C₂×#135	C₄×C₁₂₇	1	1	yes	characteristic
155	C₂×#131	C₂²×C₁₂₇	1	2	yes	—
156	C₂×M₃₂	C₂²×C₁₂₇	1	2	yes	—
157	C₂×M₃₂	C₂²×C₁₂₇	1	1	yes	characteristic
158	C₂×M₃₂	C₂²×C₁₂₇	1	1	yes	characteristic
159	#443	—	2	1	no	—
160	#443	—	2	1	no	—
161	#443	—	2	1	no	—
162	#443	—	2	1	no	—
163	C₂×#108	C₄×C₁₂₇	1	1	yes	characteristic
164	C₂×#104	C₄×C₁₂₇	1	2	yes	—
165	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
166	#364	—	2	1	no	—
167	#364	—	2	1	no	—
168	#350	—	2	4	no	—
169	#349	—	2	4	no	—
170	#346	—	2	1	no	—
171	C₄×C₁₆	—	2	1	no	—
172	C₂×#41	C₂×C₄×C₁₂₇	1	2	yes	—
173	#135	—	2	1	no	—
174	#135	—	2	1	no	—
175	#135	—	2	1	no	—
176	#135	—	2	1	no	—
177	C₂×Q₁₆	C₈×C₁₂₇	1	1	yes	characteristic
178	C₂×SD₁₆	C₈×C₁₂₇	1	2	yes	—
179	C₂×D₈	C₈×C₁₂₇	1	1	yes	characteristic
180	#131	—	2	2	no	—
181	#131	—	2	2	no	—
182	#131	—	2	2	no	—
183	#131	—	2	2	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
188	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
189	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
190	M₃₂	—	2	1	no	—
191	M₃₂	—	2	1	no	—
192	M₃₂	—	2	2	no	—
193	M₃₂	—	2	2	no	—
194	M₃₂	D₄×C₁₂₇	1	2	yes	—
195	M₃₂	D₄×C₁₂₇	1	2	yes	—
196	C₂×C₁₆	—	2	1	no	—
197	C₂×C₁₆	—	2	1	no	—
198	C₂×C₁₆	—	2	2	no	—
199	#108	—	2	1	no	—
200	#108	—	2	1	no	—
201	#104	—	2	2	no	—
202	#104	—	2	2	no	—
203	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
204	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
205	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
206	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
207	#41	—	2	2	no	—
208	#41	—	2	2	no	—
209	#41	—	2	2	no	—
210	#41	—	2	2	no	—
211	C₂×Q₈	C₂×C₈×C₁₂₇	1	2	yes	—
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	C₂×C₈×C₁₂₇	1	2	yes	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
217	Q₁₆	—	2	1	no	—
218	Q₁₆	—	2	1	no	—
219	SD₁₆	—	2	2	no	—
220	SD₁₆	—	2	2	no	—
221	D₈	—	2	1	no	—
222	D₈	—	2	1	no	—
223	M₁₆	—	2	2	no	—
224	M₁₆	—	2	2	no	—
225	M₁₆	—	4	2	no	—
226	M₁₆	—	4	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
231	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
232	C₂×C₈	C₂×C₈×C₁₂₇	1	1	yes	characteristic
233	C₂×C₈	C₂×C₈×C₁₂₇	1	1	yes	characteristic
234	C₂×C₈	M₁₆×C₁₂₇	1	1	yes	characteristic
235	C₂×C₈	M₁₆×C₁₂₇	1	1	yes	characteristic
236	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
237	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
238	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
239	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
240	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
241	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
242	C₄²	—	2	1	no	—
243	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
244	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
245	C₁₆	—	2	2	no	—
246	C₁₆	—	2	2	no	—
247	C₁₆	—	4	2	no	—
248	C₂³	—	2	2	no	—
249	C₂³	#131×C₁₂₇	1	1	yes	characteristic
250	Q₈	—	2	2	no	—
251	Q₈	—	2	2	no	—
252	D₄	—	2	2	no	—
253	D₄	—	2	2	no	—
254	D₄	—	4	2	no	—
255	D₄	—	4	2	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
263	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
264	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
265	C₂×C₄	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
266	C₂×C₄	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
267	C₂×C₄	#131×C₁₂₇	1	1	yes	characteristic
268	C₈	—	4	2	no	—
269	C₈	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
270	C₈	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
271	C₈	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
272	C₈	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
273	C₈	#121×C₁₂₇	1	1	yes	characteristic
274	C₈	#121×C₁₂₇	1	1	yes	characteristic
275	C₈	#124×C₁₂₇	1	1	yes	characteristic
276	C₈	#127×C₁₂₇	1	1	yes	characteristic
277	C₂²	—	4	1	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	8	2	no	—
280	C₂²	#408×C₁₂₇	1	1	yes	characteristic
281	C₂²	C₈×D₄×C₁₂₇	1	1	yes	characteristic
282	C₂²	#436×C₁₂₇	1	1	yes	characteristic
283	C₄	—	2	2	no	—
284	C₄	—	4	1	no	—
285	C₄	—	4	2	no	—
286	C₄	#390×C₁₂₇	1	1	yes	characteristic
287	C₄	C₂×C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
288	C₄	C₈×D₄×C₁₂₇	1	1	yes	characteristic
289	C₄	#436×C₁₂₇	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	2	1	no	—
293	C₂	—	8	2	no	—
294	C₂	C₁₂₇×#1954	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₂₇	#121
commutator	C₂×C₄	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂×C₁₂₇	#1954
prime radical	C₂×#135×C₁₂₇	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₂₇ ◁ C₂²×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#364×C₁₂₇	C₂	semidirect
3. C₁₂₇×#2117	C₂	semidirect
4. C₁₂₇×#2115	C₂	semidirect
5. C₄×M₃₂×C₁₂₇	C₂	semidirect
142. #13197	C₁₂₇	direct (N × H)
149. C₂×#364	C₂×C₁₂₇	semidirect
150. #2117	C₂×C₁₂₇	semidirect
151. #2115	C₂×C₁₂₇	semidirect
152. C₄×M₃₂	C₂×C₁₂₇	semidirect
153. C₁₂₇	#13197	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×#443×C₁₂₇	2
C₂×#364×C₁₂₇	2
C₁₂₇×#2117	2
C₁₂₇×#2115	2
C₄×M₃₂×C₁₂₇	2
#13197	127

Subgroups — C127×#13197 #44576870

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#13197 #44576870