Subgroups — C₁₂₇×#20427 #44584100

Order 32512.

1118 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#424×C₁₂₇	C₂	1	2	yes	maximal
2	C₂×C₄×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₄×#117×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₁₂₇×#1768	C₂	1	4	yes	maximal, complemented
6	C₁₂₇×#1767	C₂	1	4	yes	maximal, complemented
7	C₁₂₇×#1729	C₂	1	2	yes	maximal
8	C₂²×M₁₆×C₁₂₇	C₂²	1	2	yes	—
9	C₂³×C₈×C₁₂₇	C₂²	1	2	yes	—
10	C₂×#117×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×#117×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
13	#424×C₁₂₇	C₄	1	16	yes	complemented
14	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	8	yes	—
15	C₄×M₁₆×C₁₂₇	—	2	4	no	—
16	C₂×#97×C₁₂₇	—	2	4	no	—
17	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
18	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
19	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
21	C₄×C₄⋊C₄×C₁₂₇	C₄	1	4	yes	—
22	C₄×C₂²⋊C₄×C₁₂₇	C₄	1	4	yes	—
23	#376×C₁₂₇	C₂²	1	2	yes	—
24	C₄³×C₁₂₇	C₂²	1	2	yes	—
25	#336×C₁₂₇	C₂²	1	8	yes	—
26	C₂³×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₁₂₇	—	2	4	no	—
29	C₂×M₁₆×C₁₂₇	—	2	4	no	—
30	C₂×M₁₆×C₁₂₇	C₂×C₄	1	8	yes	—
31	C₂²×C₈×C₁₂₇	—	2	4	no	—
32	C₂²×C₈×C₁₂₇	—	2	4	no	—
33	C₂²×C₈×C₁₂₇	C₂×C₄	1	8	yes	—
34	C₂²×C₈×C₁₂₇	C₂³	1	4	yes	—
35	C₂²×C₈×C₁₂₇	C₂³	1	4	yes	—
36	#117×C₁₂₇	—	2	4	no	—
37	#117×C₁₂₇	C₂×C₄	1	8	yes	—
38	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
40	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₁₂₇	C₂×C₄	1	8	yes	—
43	C₂×C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₁₂₇	D₄	1	4	yes	—
45	C₂×C₄²×C₁₂₇	Q₈	1	4	yes	—
46	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
47	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
48	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
50	C₄⋊C₈×C₁₂₇	C₂×C₄	1	32	yes	complemented
51	#97×C₁₂₇	—	2	4	no	—
52	#97×C₁₂₇	—	2	4	no	—
53	#97×C₁₂₇	—	2	4	no	—
54	C₄×C₈×C₁₂₇	—	2	4	no	—
55	#95×C₁₂₇	C₂×C₄	1	8	yes	—
56	C₂⁴×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₂₇	—	4	4	no	—
58	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	8	yes	—
59	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₁₂₇	#41	1	4	yes	—
68	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
69	M₁₆×C₁₂₇	—	2	16	no	—
70	C₂×C₈×C₁₂₇	—	2	16	no	—
71	C₂×C₈×C₁₂₇	—	2	4	no	—
72	C₂×C₈×C₁₂₇	—	2	4	no	—
73	C₂×C₈×C₁₂₇	—	2	4	no	—
74	C₂×C₈×C₁₂₇	—	2	4	no	—
75	C₂×C₈×C₁₂₇	C₂²×C₄	1	8	yes	—
76	C₂×C₈×C₁₂₇	C₂²×C₄	1	8	yes	—
77	C₄⋊C₄×C₁₂₇	—	2	4	no	—
78	C₄⋊C₄×C₁₂₇	C₄²	1	8	yes	—
79	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
80	C₂²⋊C₄×C₁₂₇	C₄²	1	8	yes	—
81	C₄²×C₁₂₇	—	2	4	no	—
82	C₄²×C₁₂₇	—	4	8	no	—
83	C₄²×C₁₂₇	C₄⋊C₄	1	8	yes	—
84	C₄²×C₁₂₇	C₂²×C₄	1	8	yes	—
85	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
86	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
87	C₄²×C₁₂₇	C₂×Q₈	1	2	yes	—
88	C₄²×C₁₂₇	C₂×Q₈	1	2	yes	—
89	C₂³×C₁₂₇	—	2	4	no	—
90	C₂³×C₁₂₇	C₂×C₄²	1	2	yes	—
91	C₂³×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
92	C₂³×C₁₂₇	#117	1	2	yes	—
93	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₂₇	—	2	4	no	—
95	C₂×C₄×C₁₂₇	—	2	8	no	—
96	C₂×C₄×C₁₂₇	—	2	4	no	—
97	C₂×C₄×C₁₂₇	—	2	4	no	—
98	C₂×C₄×C₁₂₇	—	2	8	no	—
99	C₂×C₄×C₁₂₇	—	4	8	no	—
100	C₂×C₄×C₁₂₇	C₂×C₄²	1	8	yes	—
101	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
102	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
103	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₁₂₇	#117	1	2	yes	—
106	C₂×C₄×C₁₂₇	C₄×D₄	1	8	yes	—
107	C₂×C₄×C₁₂₇	C₄×Q₈	1	8	yes	—
108	C₂×C₄×C₁₂₇	C₂³×C₄	1	2	yes	—
109	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₁₂₇	C₂×#41	1	2	yes	—
112	C₈×C₁₂₇	—	2	8	no	—
113	C₈×C₁₂₇	—	2	8	no	—
114	C₂²×C₁₂₇	—	2	4	no	—
115	C₂²×C₁₂₇	—	2	4	no	—
116	C₂²×C₁₂₇	—	2	4	no	—
117	C₂²×C₁₂₇	C₄×C₄⋊C₄	1	4	yes	—
118	C₂²×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
119	C₂²×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₁₂₇	C₂×#117	1	1	yes	characteristic
121	C₂²×C₁₂₇	#568	1	4	yes	—
122	C₄×C₁₂₇	—	2	8	no	—
123	C₄×C₁₂₇	—	2	4	no	—
124	C₄×C₁₂₇	—	2	4	no	—
125	C₄×C₁₂₇	—	4	4	no	—
126	C₄×C₁₂₇	C₄×C₄⋊C₄	1	4	yes	—
127	C₄×C₁₂₇	C₂×C₄×D₄	1	2	yes	—
128	C₄×C₁₂₇	C₂×C₄×Q₈	1	2	yes	—
129	#20427	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂×C₁₂₇	—	2	4	no	—
131	C₂×C₁₂₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₁₂₇	#2875	1	2	yes	—
133	C₂×C₁₂₇	#2905	1	2	yes	—
134	C₂×C₁₂₇	#2923	1	2	yes	—
135	C₂×#424	C₂×C₁₂₇	1	2	yes	—
136	C₂×C₄×M₁₆	C₂×C₁₂₇	1	1	yes	characteristic
137	C₂²×#97	C₂×C₁₂₇	1	1	yes	characteristic
138	C₄×#117	C₂×C₁₂₇	1	1	yes	characteristic
139	#1768	C₂×C₁₂₇	1	4	yes	complemented
140	#1767	C₂×C₁₂₇	1	4	yes	complemented
141	#1729	C₂×C₁₂₇	1	2	yes	—
142	C₁₂₇	#20427	1	1	yes	characteristic, direct product factor, Sylow
143	C₂²×M₁₆	C₂²×C₁₂₇	1	2	yes	—
144	C₂³×C₈	C₂²×C₁₂₇	1	2	yes	—
145	C₂×#117	C₂²×C₁₂₇	1	1	yes	characteristic
146	C₂×#117	C₂²×C₁₂₇	1	1	yes	characteristic
147	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
148	#424	C₄×C₁₂₇	1	16	yes	complemented
149	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	8	yes	—
150	C₄×M₁₆	—	2	4	no	—
151	C₂×#97	—	2	4	no	—
152	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
153	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
154	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
155	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
156	C₄×C₄⋊C₄	C₄×C₁₂₇	1	4	yes	—
157	C₄×C₂²⋊C₄	C₄×C₁₂₇	1	4	yes	—
158	#376	C₂²×C₁₂₇	1	2	yes	—
159	C₄³	C₂²×C₁₂₇	1	2	yes	—
160	#336	C₂²×C₁₂₇	1	8	yes	—
161	C₂³×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
162	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
163	C₂×M₁₆	—	2	4	no	—
164	C₂×M₁₆	—	2	4	no	—
165	C₂×M₁₆	C₂×C₄×C₁₂₇	1	8	yes	—
166	C₂²×C₈	—	2	4	no	—
167	C₂²×C₈	—	2	4	no	—
168	C₂²×C₈	C₂×C₄×C₁₂₇	1	8	yes	—
169	C₂²×C₈	C₂³×C₁₂₇	1	4	yes	—
170	C₂²×C₈	C₂³×C₁₂₇	1	4	yes	—
171	#117	—	2	4	no	—
172	#117	C₂×C₄×C₁₂₇	1	8	yes	—
173	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
174	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
175	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
176	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
177	C₂×C₄²	C₂×C₄×C₁₂₇	1	8	yes	—
178	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
179	C₂×C₄²	D₄×C₁₂₇	1	4	yes	—
180	C₂×C₄²	Q₈×C₁₂₇	1	4	yes	—
181	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
182	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
183	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
184	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
185	C₄⋊C₈	C₂×C₄×C₁₂₇	1	32	yes	complemented
186	#97	—	2	4	no	—
187	#97	—	2	4	no	—
188	#97	—	2	4	no	—
189	C₄×C₈	—	2	4	no	—
190	#95	C₂×C₄×C₁₂₇	1	8	yes	—
191	C₂⁴	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
192	C₂²×C₄	—	4	4	no	—
193	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	8	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	1	yes	characteristic
199	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
200	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
201	C₂²×C₄	C₂×Q₈×C₁₂₇	1	2	yes	—
202	C₂²×C₄	#41×C₁₂₇	1	4	yes	—
203	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
204	M₁₆	—	2	16	no	—
205	C₂×C₈	—	2	16	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	C₂²×C₄×C₁₂₇	1	8	yes	—
211	C₂×C₈	C₂²×C₄×C₁₂₇	1	8	yes	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	C₄²×C₁₂₇	1	8	yes	—
214	C₂²⋊C₄	—	2	4	no	—
215	C₂²⋊C₄	C₄²×C₁₂₇	1	8	yes	—
216	C₄²	—	2	4	no	—
217	C₄²	—	4	8	no	—
218	C₄²	C₄⋊C₄×C₁₂₇	1	8	yes	—
219	C₄²	C₂²×C₄×C₁₂₇	1	8	yes	—
220	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
221	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
222	C₄²	C₂×Q₈×C₁₂₇	1	2	yes	—
223	C₄²	C₂×Q₈×C₁₂₇	1	2	yes	—
224	C₂³	—	2	4	no	—
225	C₂³	C₂×C₄²×C₁₂₇	1	2	yes	—
226	C₂³	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
227	C₂³	#117×C₁₂₇	1	2	yes	—
228	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	4	8	no	—
235	C₂×C₄	C₂×C₄²×C₁₂₇	1	8	yes	—
236	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
237	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	4	yes	—
238	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	4	yes	—
239	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
240	C₂×C₄	#117×C₁₂₇	1	2	yes	—
241	C₂×C₄	C₄×D₄×C₁₂₇	1	8	yes	—
242	C₂×C₄	C₄×Q₈×C₁₂₇	1	8	yes	—
243	C₂×C₄	C₂³×C₄×C₁₂₇	1	2	yes	—
244	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
245	C₂×C₄	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
246	C₂×C₄	C₂×#41×C₁₂₇	1	2	yes	—
247	C₈	—	2	8	no	—
248	C₈	—	2	8	no	—
249	C₂²	—	2	4	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	4	no	—
252	C₂²	C₄×C₄⋊C₄×C₁₂₇	1	4	yes	—
253	C₂²	C₂²×C₄²×C₁₂₇	1	1	yes	characteristic
254	C₂²	C₂²×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
255	C₂²	C₂×#117×C₁₂₇	1	1	yes	characteristic
256	C₂²	#568×C₁₂₇	1	4	yes	—
257	C₄	—	2	8	no	—
258	C₄	—	2	4	no	—
259	C₄	—	2	4	no	—
260	C₄	—	4	4	no	—
261	C₄	C₄×C₄⋊C₄×C₁₂₇	1	4	yes	—
262	C₄	C₂×C₄×D₄×C₁₂₇	1	2	yes	—
263	C₄	C₂×C₄×Q₈×C₁₂₇	1	2	yes	—
264	C₂	—	2	4	no	—
265	C₂	C₂×C₄×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
266	C₂	C₁₂₇×#2875	1	2	yes	—
267	C₂	C₁₂₇×#2905	1	2	yes	—
268	C₂	C₁₂₇×#2923	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
5. C₁₂₇×#1768	C₂	semidirect
6. C₁₂₇×#1767	C₂	semidirect
13. #424×C₁₂₇	C₄	semidirect
50. C₄⋊C₈×C₁₂₇	C₂×C₄	semidirect
129. #20427	C₁₂₇	direct (N × H)
139. #1768	C₂×C₁₂₇	semidirect
140. #1767	C₂×C₁₂₇	semidirect
142. C₁₂₇	#20427	direct (N × H)
148. #424	C₄×C₁₂₇	semidirect
185. C₄⋊C₈	C₂×C₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#424×C₁₂₇	2
C₂×C₄×M₁₆×C₁₂₇	2
C₂²×#97×C₁₂₇	2
C₄×#117×C₁₂₇	2
C₁₂₇×#1768	2
C₁₂₇×#1767	2
C₁₂₇×#1729	2
#20427	127

Subgroups — C127×#20427 #44584100

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#20427 #44584100