Subgroups — C₁₂₇×#13755 #44577428

Order 32512.

1774 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₄⋊C₄×C₁₂₇	C₂	1	2	yes	maximal
2	C₂×#376×C₁₂₇	C₂	1	2	yes	maximal
3	C₂×C₄³×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×C₄⋊#95	C₂	1	4	yes	maximal
5	C₁₂₇×#1434	C₂	1	4	yes	maximal
6	C₄×#95×C₁₂₇	C₂	1	2	yes	maximal
7	C₂²×C₄⋊C₄×C₁₂₇	C₂²	1	4	yes	—
8	C₂²×C₄²×C₁₂₇	C₂²	1	2	yes	—
9	C₂²×C₄²×C₁₂₇	C₂²	1	2	yes	—
10	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₁₂₇	C₂²	1	4	yes	—
12	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊C₄×C₁₂₇	C₄	1	16	yes	complemented
15	#376×C₁₂₇	—	2	8	no	—
16	C₂×#95×C₁₂₇	C₂²	1	8	yes	—
17	C₂×#95×C₁₂₇	C₂²	1	8	yes	—
18	C₂×#95×C₁₂₇	C₂²	1	4	yes	—
19	C₄³×C₁₂₇	C₄	1	8	yes	complemented
20	C₂³×C₄×C₁₂₇	C₂³	1	8	yes	—
21	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
22	C₂³×C₄×C₁₂₇	C₂³	1	2	yes	—
23	C₂³×C₄×C₁₂₇	C₂³	1	2	yes	—
24	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
25	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₁₂₇	—	2	16	no	—
27	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	16	yes	—
28	C₂×C₄²×C₁₂₇	—	2	8	no	—
29	C₂×C₄²×C₁₂₇	—	2	8	no	—
30	C₂×C₄²×C₁₂₇	—	2	8	no	—
31	C₂×C₄²×C₁₂₇	—	2	4	no	—
32	C₂×C₄²×C₁₂₇	—	2	8	no	—
33	C₂×C₄²×C₁₂₇	C₂×C₄	1	8	yes	—
34	C₂×C₄²×C₁₂₇	C₂×C₄	1	8	yes	—
35	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₁₂₇	C₂×C₄	1	16	yes	—
37	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₁₂₇	C₂×C₄	1	8	yes	—
39	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₁₂₇	D₄	1	4	yes	—
41	C₂×C₄²×C₁₂₇	Q₈	1	4	yes	—
42	#95×C₁₂₇	—	2	16	no	—
43	#95×C₁₂₇	—	2	16	no	—
44	#95×C₁₂₇	C₂×C₄	1	16	yes	—
45	C₂⁴×C₁₂₇	C₂⁴	1	1	yes	characteristic
46	C₂²×C₄×C₁₂₇	—	2	4	no	—
47	C₂²×C₄×C₁₂₇	—	2	4	no	—
48	C₂²×C₄×C₁₂₇	—	2	4	no	—
49	C₂²×C₄×C₁₂₇	—	2	4	no	—
50	C₂²×C₄×C₁₂₇	—	2	4	no	—
51	C₂²×C₄×C₁₂₇	—	2	8	no	—
52	C₂²×C₄×C₁₂₇	—	2	16	no	—
53	C₂²×C₄×C₁₂₇	—	2	8	no	—
54	C₂²×C₄×C₁₂₇	—	2	16	no	—
55	C₂²×C₄×C₁₂₇	C₂²×C₄	1	16	yes	—
56	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
65	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
67	C₂²×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₁₂₇	#41	1	8	yes	—
69	C₂²×C₄×C₁₂₇	#41	1	8	yes	—
70	C₂²×C₄×C₁₂₇	#41	1	4	yes	—
71	C₄⋊C₄×C₁₂₇	—	2	32	no	—
72	C₄²×C₁₂₇	—	2	16	no	—
73	C₄²×C₁₂₇	—	2	16	no	—
74	C₄²×C₁₂₇	—	2	8	no	—
75	C₄²×C₁₂₇	—	2	32	no	—
76	C₄²×C₁₂₇	C₄²	1	16	yes	—
77	C₄²×C₁₂₇	C₄⋊C₄	1	16	yes	complemented
78	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
79	C₂³×C₁₂₇	C₂³×C₄	1	2	yes	—
80	C₂³×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
81	C₂³×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
82	C₂³×C₁₂₇	C₂×#41	1	4	yes	—
83	C₂³×C₁₂₇	C₂×#41	1	2	yes	—
84	C₂³×C₁₂₇	C₂×#41	1	4	yes	—
85	C₂×C₄×C₁₂₇	—	2	4	no	—
86	C₂×C₄×C₁₂₇	—	2	4	no	—
87	C₂×C₄×C₁₂₇	—	2	4	no	—
88	C₂×C₄×C₁₂₇	—	2	4	no	—
89	C₂×C₄×C₁₂₇	—	2	4	no	—
90	C₂×C₄×C₁₂₇	—	2	8	no	—
91	C₂×C₄×C₁₂₇	—	2	16	no	—
92	C₂×C₄×C₁₂₇	—	2	8	no	—
93	C₂×C₄×C₁₂₇	—	2	8	no	—
94	C₂×C₄×C₁₂₇	—	2	16	no	—
95	C₂×C₄×C₁₂₇	—	2	16	no	—
96	C₂×C₄×C₁₂₇	—	2	16	no	—
97	C₂×C₄×C₁₂₇	—	4	16	no	—
98	C₂×C₄×C₁₂₇	—	4	16	no	—
99	C₂×C₄×C₁₂₇	C₂×C₄²	1	4	yes	—
100	C₂×C₄×C₁₂₇	C₂×C₄²	1	4	yes	—
101	C₂×C₄×C₁₂₇	C₂×C₄²	1	4	yes	—
102	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	8	yes	—
103	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄×C₁₂₇	#117	1	8	yes	—
105	C₂×C₄×C₁₂₇	#117	1	8	yes	—
106	C₂×C₄×C₁₂₇	#117	1	16	yes	—
107	C₂×C₄×C₁₂₇	#117	1	4	yes	—
108	C₂×C₄×C₁₂₇	C₄×D₄	1	8	yes	—
109	C₂×C₄×C₁₂₇	C₄×Q₈	1	8	yes	—
110	C₂²×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
111	C₂²×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
112	C₂²×C₁₂₇	C₂×#117	1	2	yes	—
113	C₂²×C₁₂₇	C₂×#117	1	4	yes	—
114	C₂²×C₁₂₇	C₂×#117	1	1	yes	characteristic
115	C₂²×C₁₂₇	C₂×#117	1	2	yes	—
116	C₂²×C₁₂₇	C₂×C₄×D₄	1	2	yes	—
117	C₂²×C₁₂₇	C₂×C₄×Q₈	1	2	yes	—
118	C₂²×C₁₂₇	C₄×#41	1	4	yes	—
119	C₂²×C₁₂₇	#529	1	8	yes	—
120	C₂²×C₁₂₇	#532	1	4	yes	—
121	C₂²×C₁₂₇	#533	1	4	yes	—
122	C₄×C₁₂₇	—	2	4	no	—
123	C₄×C₁₂₇	—	2	8	no	—
124	C₄×C₁₂₇	—	2	8	no	—
125	C₄×C₁₂₇	—	4	16	no	—
126	C₄×C₁₂₇	#376	1	8	yes	complemented
127	C₄×C₁₂₇	C₄×C₄⋊C₄	1	8	yes	—
128	#13755	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂×C₁₂₇	C₂×#376	1	1	yes	characteristic
130	C₂×C₁₂₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₁₂₇	C₄×#117	1	1	yes	characteristic
132	C₂×C₁₂₇	#2283	1	2	yes	—
133	C₂×C₁₂₇	#2285	1	2	yes	—
134	C₂×C₁₂₇	#2293	1	4	yes	—
135	C₂×C₁₂₇	#2295	1	4	yes	—
136	C₂×C₄×C₄⋊C₄	C₂×C₁₂₇	1	2	yes	—
137	C₂×#376	C₂×C₁₂₇	1	2	yes	—
138	C₂×C₄³	C₂×C₁₂₇	1	1	yes	characteristic
139	C₄⋊#95	C₂×C₁₂₇	1	4	yes	—
140	#1434	C₂×C₁₂₇	1	4	yes	—
141	C₄×#95	C₂×C₁₂₇	1	2	yes	—
142	C₁₂₇	#13755	1	1	yes	characteristic, direct product factor, Sylow
143	C₂²×C₄⋊C₄	C₂²×C₁₂₇	1	4	yes	—
144	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
145	C₂²×C₄²	C₂²×C₁₂₇	1	4	yes	—
146	C₂²×C₄²	C₂²×C₁₂₇	1	2	yes	—
147	C₂²×C₄²	C₂²×C₁₂₇	1	2	yes	—
148	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
149	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
150	C₄×C₄⋊C₄	C₄×C₁₂₇	1	16	yes	complemented
151	#376	—	2	8	no	—
152	C₂×#95	C₂²×C₁₂₇	1	8	yes	—
153	C₂×#95	C₂²×C₁₂₇	1	8	yes	—
154	C₂×#95	C₂²×C₁₂₇	1	4	yes	—
155	C₄³	C₄×C₁₂₇	1	8	yes	complemented
156	C₂³×C₄	C₂³×C₁₂₇	1	8	yes	—
157	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
158	C₂³×C₄	C₂³×C₁₂₇	1	2	yes	—
159	C₂³×C₄	C₂³×C₁₂₇	1	2	yes	—
160	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
161	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
162	C₂×C₄⋊C₄	—	2	16	no	—
163	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	16	yes	—
164	C₂×C₄²	—	2	8	no	—
165	C₂×C₄²	—	2	8	no	—
166	C₂×C₄²	—	2	8	no	—
167	C₂×C₄²	—	2	4	no	—
168	C₂×C₄²	—	2	8	no	—
169	C₂×C₄²	C₂×C₄×C₁₂₇	1	8	yes	—
170	C₂×C₄²	C₂×C₄×C₁₂₇	1	8	yes	—
171	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
172	C₂×C₄²	C₂×C₄×C₁₂₇	1	16	yes	—
173	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
174	C₂×C₄²	C₂×C₄×C₁₂₇	1	8	yes	—
175	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
176	C₂×C₄²	D₄×C₁₂₇	1	4	yes	—
177	C₂×C₄²	Q₈×C₁₂₇	1	4	yes	—
178	#95	—	2	16	no	—
179	#95	—	2	16	no	—
180	#95	C₂×C₄×C₁₂₇	1	16	yes	—
181	C₂⁴	C₂⁴×C₁₂₇	1	1	yes	characteristic
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	2	8	no	—
188	C₂²×C₄	—	2	16	no	—
189	C₂²×C₄	—	2	8	no	—
190	C₂²×C₄	—	2	16	no	—
191	C₂²×C₄	C₂²×C₄×C₁₂₇	1	16	yes	—
192	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
193	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
194	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
195	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
196	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
197	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
198	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
199	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
200	C₂²×C₄	C₂×D₄×C₁₂₇	1	4	yes	—
201	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
202	C₂²×C₄	C₂×Q₈×C₁₂₇	1	4	yes	—
203	C₂²×C₄	C₂×Q₈×C₁₂₇	1	2	yes	—
204	C₂²×C₄	#41×C₁₂₇	1	8	yes	—
205	C₂²×C₄	#41×C₁₂₇	1	8	yes	—
206	C₂²×C₄	#41×C₁₂₇	1	4	yes	—
207	C₄⋊C₄	—	2	32	no	—
208	C₄²	—	2	16	no	—
209	C₄²	—	2	16	no	—
210	C₄²	—	2	8	no	—
211	C₄²	—	2	32	no	—
212	C₄²	C₄²×C₁₂₇	1	16	yes	—
213	C₄²	C₄⋊C₄×C₁₂₇	1	16	yes	complemented
214	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
215	C₂³	C₂³×C₄×C₁₂₇	1	2	yes	—
216	C₂³	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
217	C₂³	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
218	C₂³	C₂×#41×C₁₂₇	1	4	yes	—
219	C₂³	C₂×#41×C₁₂₇	1	2	yes	—
220	C₂³	C₂×#41×C₁₂₇	1	4	yes	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	—	2	16	no	—
228	C₂×C₄	—	2	8	no	—
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	2	16	no	—
231	C₂×C₄	—	2	16	no	—
232	C₂×C₄	—	2	16	no	—
233	C₂×C₄	—	4	16	no	—
234	C₂×C₄	—	4	16	no	—
235	C₂×C₄	C₂×C₄²×C₁₂₇	1	4	yes	—
236	C₂×C₄	C₂×C₄²×C₁₂₇	1	4	yes	—
237	C₂×C₄	C₂×C₄²×C₁₂₇	1	4	yes	—
238	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	8	yes	—
239	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	4	yes	—
240	C₂×C₄	#117×C₁₂₇	1	8	yes	—
241	C₂×C₄	#117×C₁₂₇	1	8	yes	—
242	C₂×C₄	#117×C₁₂₇	1	16	yes	—
243	C₂×C₄	#117×C₁₂₇	1	4	yes	—
244	C₂×C₄	C₄×D₄×C₁₂₇	1	8	yes	—
245	C₂×C₄	C₄×Q₈×C₁₂₇	1	8	yes	—
246	C₂²	C₂²×C₄²×C₁₂₇	1	1	yes	characteristic
247	C₂²	C₂²×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
248	C₂²	C₂×#117×C₁₂₇	1	2	yes	—
249	C₂²	C₂×#117×C₁₂₇	1	4	yes	—
250	C₂²	C₂×#117×C₁₂₇	1	1	yes	characteristic
251	C₂²	C₂×#117×C₁₂₇	1	2	yes	—
252	C₂²	C₂×C₄×D₄×C₁₂₇	1	2	yes	—
253	C₂²	C₂×C₄×Q₈×C₁₂₇	1	2	yes	—
254	C₂²	C₄×#41×C₁₂₇	1	4	yes	—
255	C₂²	#529×C₁₂₇	1	8	yes	—
256	C₂²	#532×C₁₂₇	1	4	yes	—
257	C₂²	#533×C₁₂₇	1	4	yes	—
258	C₄	—	2	4	no	—
259	C₄	—	2	8	no	—
260	C₄	—	2	8	no	—
261	C₄	—	4	16	no	—
262	C₄	#376×C₁₂₇	1	8	yes	complemented
263	C₄	C₄×C₄⋊C₄×C₁₂₇	1	8	yes	—
264	C₂	C₂×#376×C₁₂₇	1	1	yes	characteristic
265	C₂	C₂×C₄×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
266	C₂	C₄×#117×C₁₂₇	1	1	yes	characteristic
267	C₂	C₁₂₇×#2283	1	2	yes	—
268	C₂	C₁₂₇×#2285	1	2	yes	—
269	C₂	C₁₂₇×#2293	1	4	yes	—
270	C₂	C₁₂₇×#2295	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄×C₁₂₇	C₂³
commutator	C₂²	C₂²×C₄²×C₁₂₇
Frattini, cosocle	C₂⁴	C₂⁴×C₁₂₇
socle, prime radical	C₂⁴×C₁₂₇	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
14. C₄×C₄⋊C₄×C₁₂₇	C₄	semidirect
19. C₄³×C₁₂₇	C₄	semidirect
77. C₄²×C₁₂₇	C₄⋊C₄	semidirect
126. C₄×C₁₂₇	#376	semidirect
128. #13755	C₁₂₇	direct (N × H)
142. C₁₂₇	#13755	direct (N × H)
150. C₄×C₄⋊C₄	C₄×C₁₂₇	semidirect
155. C₄³	C₄×C₁₂₇	semidirect
213. C₄²	C₄⋊C₄×C₁₂₇	semidirect
262. C₄	#376×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₄⋊C₄×C₁₂₇	2
C₂×#376×C₁₂₇	2
C₂×C₄³×C₁₂₇	2
C₁₂₇×C₄⋊#95	2
C₁₂₇×#1434	2
C₄×#95×C₁₂₇	2
#13755	127

Subgroups — C127×#13755 #44577428

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#13755 #44577428