Subgroups — C₁₂₇×#11592 #44575265

Order 32512.

402 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#135×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₄×M₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#335×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₂₇×#1332	C₂	1	2	yes	maximal, complemented
5	C₁₂₇×#1331	C₂	1	2	yes	maximal, complemented
6	C₂×#135×C₁₂₇	C₄	1	1	yes	characteristic
7	C₄×#41×C₁₂₇	—	2	1	no	—
8	C₂×M₃₂×C₁₂₇	—	2	1	no	—
9	C₄×Q₁₆×C₁₂₇	C₄	1	1	yes	characteristic
10	C₄×Q₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₄×SD₁₆×C₁₂₇	—	2	1	no	—
12	C₄×SD₁₆×C₁₂₇	—	2	1	no	—
13	C₄×D₈×C₁₂₇	C₄	1	1	yes	characteristic
14	C₄×D₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	#427×C₁₂₇	C₄	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₁₂₇	—	2	1	no	—
17	#416×C₁₂₇	—	2	1	no	—
18	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
19	#346×C₁₂₇	—	2	1	no	—
20	C₄×C₁₆×C₁₂₇	C₂²	1	2	yes	—
21	#335×C₁₂₇	—	2	1	no	—
22	#335×C₁₂₇	C₂²	1	2	yes	—
23	C₂×#41×C₁₂₇	—	2	1	no	—
24	#135×C₁₂₇	—	2	2	no	—
25	#135×C₁₂₇	C₈	1	4	yes	complemented
26	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
27	C₂×SD₁₆×C₁₂₇	—	2	1	no	—
28	C₂×D₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₂₇	—	4	1	no	—
30	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
32	C₄×Q₈×C₁₂₇	—	2	1	no	—
33	C₄×Q₈×C₁₂₇	—	2	1	no	—
34	C₄×D₄×C₁₂₇	—	2	1	no	—
35	C₄×D₄×C₁₂₇	—	2	1	no	—
36	C₄×D₄×C₁₂₇	—	4	1	no	—
37	C₄×D₄×C₁₂₇	—	4	1	no	—
38	#117×C₁₂₇	—	2	1	no	—
39	#117×C₁₂₇	—	4	1	no	—
40	C₂×C₄²×C₁₂₇	—	4	1	no	—
41	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
42	M₃₂×C₁₂₇	—	2	4	no	—
43	C₂×C₁₆×C₁₂₇	—	2	2	no	—
44	#107×C₁₂₇	C₂×C₄	1	1	yes	characteristic
45	#107×C₁₂₇	C₂×C₄	1	1	yes	characteristic
46	#106×C₁₂₇	—	2	1	no	—
47	C₄⋊C₈×C₁₂₇	—	2	2	no	—
48	C₄⋊C₈×C₁₂₇	—	4	1	no	—
49	#103×C₁₂₇	—	2	1	no	—
50	#103×C₁₂₇	—	2	1	no	—
51	#102×C₁₂₇	—	2	1	no	—
52	#102×C₁₂₇	—	2	1	no	—
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	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
57	#41×C₁₂₇	—	2	4	no	—
58	#41×C₁₂₇	—	4	2	no	—
59	C₂×Q₈×C₁₂₇	—	2	1	no	—
60	C₂×D₄×C₁₂₇	—	2	1	no	—
61	C₂×D₄×C₁₂₇	—	4	1	no	—
62	C₂²×C₄×C₁₂₇	—	4	1	no	—
63	C₂²×C₄×C₁₂₇	—	4	1	no	—
64	C₂²×C₄×C₁₂₇	—	4	1	no	—
65	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
68	Q₁₆×C₁₂₇	—	2	1	no	—
69	Q₁₆×C₁₂₇	C₂×C₈	1	2	yes	complemented
70	SD₁₆×C₁₂₇	—	2	4	no	—
71	D₈×C₁₂₇	—	2	1	no	—
72	D₈×C₁₂₇	C₂×C₈	1	2	yes	complemented
73	C₂×C₈×C₁₂₇	—	2	2	no	—
74	C₂×C₈×C₁₂₇	—	4	2	no	—
75	C₂×C₈×C₁₂₇	—	8	2	no	—
76	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₂₇	C₂×C₈	1	2	yes	—
79	C₂×C₈×C₁₂₇	M₁₆	1	2	yes	—
80	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
82	C₄⋊C₄×C₁₂₇	—	2	1	no	—
83	C₄⋊C₄×C₁₂₇	—	2	1	no	—
84	C₄⋊C₄×C₁₂₇	—	4	1	no	—
85	C₄⋊C₄×C₁₂₇	—	4	1	no	—
86	C₂²⋊C₄×C₁₂₇	—	4	1	no	—
87	C₂²⋊C₄×C₁₂₇	—	4	1	no	—
88	C₂²⋊C₄×C₁₂₇	—	4	1	no	—
89	C₄²×C₁₂₇	—	4	1	no	—
90	C₄²×C₁₂₇	—	4	1	no	—
91	C₄²×C₁₂₇	—	4	1	no	—
92	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
93	C₄²×C₁₂₇	D₈	1	1	yes	characteristic
94	C₄²×C₁₂₇	SD₁₆	1	1	yes	characteristic
95	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
96	C₁₆×C₁₂₇	—	2	4	no	—
97	C₂³×C₁₂₇	—	4	1	no	—
98	C₂³×C₁₂₇	#102	1	1	yes	characteristic
99	Q₈×C₁₂₇	—	2	2	no	—
100	Q₈×C₁₂₇	—	4	1	no	—
101	D₄×C₁₂₇	—	2	2	no	—
102	D₄×C₁₂₇	—	4	4	no	—
103	D₄×C₁₂₇	—	4	1	no	—
104	C₂×C₄×C₁₂₇	—	2	1	no	—
105	C₂×C₄×C₁₂₇	—	2	1	no	—
106	C₂×C₄×C₁₂₇	—	4	1	no	—
107	C₂×C₄×C₁₂₇	—	4	1	no	—
108	C₂×C₄×C₁₂₇	—	4	4	no	—
109	C₂×C₄×C₁₂₇	—	4	1	no	—
110	C₂×C₄×C₁₂₇	—	4	1	no	—
111	C₂×C₄×C₁₂₇	—	8	2	no	—
112	C₂×C₄×C₁₂₇	—	8	1	no	—
113	C₂×C₄×C₁₂₇	—	8	1	no	—
114	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	2	yes	—
115	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
116	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
117	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
118	C₂×C₄×C₁₂₇	#104	1	2	yes	—
119	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
122	C₈×C₁₂₇	—	2	1	no	—
123	C₈×C₁₂₇	—	2	1	no	—
124	C₈×C₁₂₇	—	8	2	no	—
125	C₈×C₁₂₇	C₂²⋊C₈	1	2	yes	—
126	C₈×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
127	C₈×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
128	C₂²×C₁₂₇	—	2	1	no	—
129	C₂²×C₁₂₇	—	2	1	no	—
130	C₂²×C₁₂₇	—	4	2	no	—
131	C₂²×C₁₂₇	—	8	1	no	—
132	C₂²×C₁₂₇	—	8	1	no	—
133	C₂²×C₁₂₇	D₄⋊C₈	1	2	yes	—
134	C₂²×C₁₂₇	C₂×#102	1	1	yes	characteristic
135	C₄×C₁₂₇	—	2	1	no	—
136	C₄×C₁₂₇	—	2	1	no	—
137	C₄×C₁₂₇	—	2	1	no	—
138	C₄×C₁₂₇	—	4	2	no	—
139	C₄×C₁₂₇	—	8	1	no	—
140	C₄×C₁₂₇	—	8	1	no	—
141	C₄×C₁₂₇	D₄⋊C₈	1	2	yes	—
142	C₄×C₁₂₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
143	C₄×C₁₂₇	C₂×#104	1	1	yes	characteristic
144	C₄×C₁₂₇	#509	1	1	yes	characteristic
145	C₄×C₁₂₇	#510	1	1	yes	characteristic
146	#11592	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
147	C₂×C₁₂₇	—	2	2	no	—
148	C₂×C₁₂₇	—	8	2	no	—
149	C₂×C₁₂₇	C₂×D₄⋊C₈	1	1	yes	characteristic
150	C₂×C₁₂₇	#2139	1	1	yes	characteristic
151	C₂×C₁₂₇	#2143	1	1	yes	characteristic
152	C₄×#135	C₂×C₁₂₇	1	1	yes	characteristic
153	C₄×M₃₂	C₂×C₁₂₇	1	1	yes	characteristic, complemented
154	C₂×#335	C₂×C₁₂₇	1	1	yes	characteristic, complemented
155	#1332	C₂×C₁₂₇	1	2	yes	complemented
156	#1331	C₂×C₁₂₇	1	2	yes	complemented
157	C₁₂₇	#11592	1	1	yes	characteristic, direct product factor, Sylow
158	C₂×#135	C₄×C₁₂₇	1	1	yes	characteristic
159	C₄×#41	—	2	1	no	—
160	C₂×M₃₂	—	2	1	no	—
161	C₄×Q₁₆	C₄×C₁₂₇	1	1	yes	characteristic
162	C₄×Q₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
163	C₄×SD₁₆	—	2	1	no	—
164	C₄×SD₁₆	—	2	1	no	—
165	C₄×D₈	C₄×C₁₂₇	1	1	yes	characteristic
166	C₄×D₈	C₂²×C₁₂₇	1	1	yes	characteristic
167	#427	C₄×C₁₂₇	1	1	yes	characteristic
168	C₂×C₄⋊C₈	—	2	1	no	—
169	#416	—	2	1	no	—
170	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
171	#346	—	2	1	no	—
172	C₄×C₁₆	C₂²×C₁₂₇	1	2	yes	—
173	#335	—	2	1	no	—
174	#335	C₂²×C₁₂₇	1	2	yes	—
175	C₂×#41	—	2	1	no	—
176	#135	—	2	2	no	—
177	#135	C₈×C₁₂₇	1	4	yes	complemented
178	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
179	C₂×SD₁₆	—	2	1	no	—
180	C₂×D₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
181	C₂²×C₈	—	4	1	no	—
182	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
183	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
184	C₄×Q₈	—	2	1	no	—
185	C₄×Q₈	—	2	1	no	—
186	C₄×D₄	—	2	1	no	—
187	C₄×D₄	—	2	1	no	—
188	C₄×D₄	—	4	1	no	—
189	C₄×D₄	—	4	1	no	—
190	#117	—	2	1	no	—
191	#117	—	4	1	no	—
192	C₂×C₄²	—	4	1	no	—
193	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
194	M₃₂	—	2	4	no	—
195	C₂×C₁₆	—	2	2	no	—
196	#107	C₂×C₄×C₁₂₇	1	1	yes	characteristic
197	#107	C₂×C₄×C₁₂₇	1	1	yes	characteristic
198	#106	—	2	1	no	—
199	C₄⋊C₈	—	2	2	no	—
200	C₄⋊C₈	—	4	1	no	—
201	#103	—	2	1	no	—
202	#103	—	2	1	no	—
203	#102	—	2	1	no	—
204	#102	—	2	1	no	—
205	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
206	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
207	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
208	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
209	#41	—	2	4	no	—
210	#41	—	4	2	no	—
211	C₂×Q₈	—	2	1	no	—
212	C₂×D₄	—	2	1	no	—
213	C₂×D₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	1	no	—
217	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
218	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
219	C₂²×C₄	SD₁₆×C₁₂₇	1	1	yes	characteristic
220	Q₁₆	—	2	1	no	—
221	Q₁₆	C₂×C₈×C₁₂₇	1	2	yes	complemented
222	SD₁₆	—	2	4	no	—
223	D₈	—	2	1	no	—
224	D₈	C₂×C₈×C₁₂₇	1	2	yes	complemented
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	8	2	no	—
228	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
229	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
230	C₂×C₈	C₂×C₈×C₁₂₇	1	2	yes	—
231	C₂×C₈	M₁₆×C₁₂₇	1	2	yes	—
232	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
233	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	4	1	no	—
237	C₄⋊C₄	—	4	1	no	—
238	C₂²⋊C₄	—	4	1	no	—
239	C₂²⋊C₄	—	4	1	no	—
240	C₂²⋊C₄	—	4	1	no	—
241	C₄²	—	4	1	no	—
242	C₄²	—	4	1	no	—
243	C₄²	—	4	1	no	—
244	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
245	C₄²	D₈×C₁₂₇	1	1	yes	characteristic
246	C₄²	SD₁₆×C₁₂₇	1	1	yes	characteristic
247	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
248	C₁₆	—	2	4	no	—
249	C₂³	—	4	1	no	—
250	C₂³	#102×C₁₂₇	1	1	yes	characteristic
251	Q₈	—	2	2	no	—
252	Q₈	—	4	1	no	—
253	D₄	—	2	2	no	—
254	D₄	—	4	4	no	—
255	D₄	—	4	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	8	2	no	—
264	C₂×C₄	—	8	1	no	—
265	C₂×C₄	—	8	1	no	—
266	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	2	yes	—
267	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
268	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
269	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
270	C₂×C₄	#104×C₁₂₇	1	2	yes	—
271	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
272	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
273	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
274	C₈	—	2	1	no	—
275	C₈	—	2	1	no	—
276	C₈	—	8	2	no	—
277	C₈	C₂²⋊C₈×C₁₂₇	1	2	yes	—
278	C₈	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
279	C₈	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	8	1	no	—
284	C₂²	—	8	1	no	—
285	C₂²	D₄⋊C₈×C₁₂₇	1	2	yes	—
286	C₂²	C₂×#102×C₁₂₇	1	1	yes	characteristic
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	4	2	no	—
291	C₄	—	8	1	no	—
292	C₄	—	8	1	no	—
293	C₄	D₄⋊C₈×C₁₂₇	1	2	yes	—
294	C₄	C₂×C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
295	C₄	C₂×#104×C₁₂₇	1	1	yes	characteristic
296	C₄	#509×C₁₂₇	1	1	yes	characteristic
297	C₄	#510×C₁₂₇	1	1	yes	characteristic
298	C₂	—	2	2	no	—
299	C₂	—	8	2	no	—
300	C₂	C₂×D₄⋊C₈×C₁₂₇	1	1	yes	characteristic
301	C₂	C₁₂₇×#2139	1	1	yes	characteristic
302	C₂	C₁₂₇×#2143	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂×D₈
commutator	C₈	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂×#102
prime radical	C₂×#135×C₁₂₇	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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. C₄×M₃₂×C₁₂₇	C₂	semidirect
3. C₂×#335×C₁₂₇	C₂	semidirect
4. C₁₂₇×#1332	C₂	semidirect
5. C₁₂₇×#1331	C₂	semidirect
25. #135×C₁₂₇	C₈	semidirect
69. Q₁₆×C₁₂₇	C₂×C₈	semidirect
72. D₈×C₁₂₇	C₂×C₈	semidirect
146. #11592	C₁₂₇	direct (N × H)
153. C₄×M₃₂	C₂×C₁₂₇	semidirect
154. C₂×#335	C₂×C₁₂₇	semidirect
155. #1332	C₂×C₁₂₇	semidirect
156. #1331	C₂×C₁₂₇	semidirect
157. C₁₂₇	#11592	direct (N × H)
177. #135	C₈×C₁₂₇	semidirect
221. Q₁₆	C₂×C₈×C₁₂₇	semidirect
224. D₈	C₂×C₈×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×#135×C₁₂₇	2
C₄×M₃₂×C₁₂₇	2
C₂×#335×C₁₂₇	2
C₁₂₇×#1332	2
C₁₂₇×#1331	2
#11592	127

Subgroups — C127×#11592 #44575265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#11592 #44575265