Subgroups — C₁₂₇×#20554 #44584227

Order 32512.

850 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	C₂×#433×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#424×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#2298	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1771	C₂	1	2	yes	maximal, complemented
5	C₁₂₇×#1770	C₂	1	2	yes	maximal, complemented
6	C₁₂₇×#1768	C₂	1	2	yes	maximal, complemented
7	C₁₂₇×#1767	C₂	1	2	yes	maximal, complemented
8	C₁₂₇×#1753	C₂	1	2	yes	maximal
9	C₁₂₇×#1752	C₂	1	2	yes	maximal
10	C₂²×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂×#117×C₁₂₇	C₂²	1	1	yes	characteristic
13	#433×C₁₂₇	C₄	1	8	yes	complemented
14	#424×C₁₂₇	C₄	1	8	yes	complemented
15	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₈×C₁₂₇	C₂²	1	2	yes	—
19	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
20	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
21	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
22	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
25	#388×C₁₂₇	C₄	1	4	yes	—
26	#387×C₁₂₇	C₄	1	4	yes	—
27	C₄×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
28	C₄×C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
29	#376×C₁₂₇	C₂²	1	1	yes	characteristic
30	C₄³×C₁₂₇	C₂²	1	1	yes	characteristic
31	#336×C₁₂₇	C₂²	1	2	yes	—
32	#336×C₁₂₇	C₂²	1	2	yes	—
33	#336×C₁₂₇	C₂²	1	2	yes	—
34	#336×C₁₂₇	C₂²	1	2	yes	—
35	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₁₂₇	—	4	2	no	—
37	C₂×M₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
38	C₂²×C₈×C₁₂₇	—	4	2	no	—
39	C₂²×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
40	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
42	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
43	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
44	#117×C₁₂₇	C₂×C₄	1	4	yes	—
45	#117×C₁₂₇	C₂×C₄	1	4	yes	—
46	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
47	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
49	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
52	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
53	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₁₂₇	C₂×C₄	1	8	yes	—
56	C₄⋊C₈×C₁₂₇	C₂×C₄	1	8	yes	—
57	C₄⋊C₈×C₁₂₇	C₂×C₄	1	8	yes	—
58	C₂²⋊C₈×C₁₂₇	C₂×C₄	1	8	yes	—
59	#97×C₁₂₇	—	4	2	no	—
60	#97×C₁₂₇	—	4	2	no	—
61	#97×C₁₂₇	—	4	2	no	—
62	#97×C₁₂₇	C₂×C₄	1	4	yes	—
63	C₄×C₈×C₁₂₇	—	4	2	no	—
64	C₄×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
65	#95×C₁₂₇	C₂×C₄	1	4	yes	—
66	#95×C₁₂₇	C₂×C₄	1	4	yes	—
67	#95×C₁₂₇	C₂×C₄	1	4	yes	—
68	C₂⁴×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₂₇	—	2	2	no	—
70	C₂²×C₄×C₁₂₇	—	2	2	no	—
71	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
79	M₁₆×C₁₂₇	—	4	4	no	—
80	C₂×C₈×C₁₂₇	—	2	2	no	—
81	C₂×C₈×C₁₂₇	—	2	2	no	—
82	C₂×C₈×C₁₂₇	—	2	2	no	—
83	C₂×C₈×C₁₂₇	—	2	2	no	—
84	C₂×C₈×C₁₂₇	—	2	2	no	—
85	C₂×C₈×C₁₂₇	—	2	2	no	—
86	C₂×C₈×C₁₂₇	—	2	2	no	—
87	C₂×C₈×C₁₂₇	—	2	2	no	—
88	C₂×C₈×C₁₂₇	—	4	4	no	—
89	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
90	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
92	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
93	C₄⋊C₄×C₁₂₇	—	4	4	no	—
94	C₄⋊C₄×C₁₂₇	C₄²	1	8	yes	—
95	C₂²⋊C₄×C₁₂₇	—	4	4	no	—
96	C₂²⋊C₄×C₁₂₇	C₄²	1	8	yes	—
97	C₄²×C₁₂₇	—	2	2	no	—
98	C₄²×C₁₂₇	—	2	2	no	—
99	C₄²×C₁₂₇	—	2	2	no	—
100	C₄²×C₁₂₇	—	2	2	no	—
101	C₄²×C₁₂₇	—	4	4	no	—
102	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
103	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
104	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
105	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
106	C₂³×C₁₂₇	—	2	2	no	—
107	C₂³×C₁₂₇	—	2	4	no	—
108	C₂³×C₁₂₇	C₂×C₄²	1	2	yes	—
109	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₂₇	—	2	2	no	—
111	C₂×C₄×C₁₂₇	—	2	4	no	—
112	C₂×C₄×C₁₂₇	—	2	4	no	—
113	C₂×C₄×C₁₂₇	—	2	2	no	—
114	C₂×C₄×C₁₂₇	—	2	2	no	—
115	C₂×C₄×C₁₂₇	—	2	4	no	—
116	C₂×C₄×C₁₂₇	—	2	4	no	—
117	C₂×C₄×C₁₂₇	—	2	4	no	—
118	C₂×C₄×C₁₂₇	—	2	4	no	—
119	C₂×C₄×C₁₂₇	—	2	4	no	—
120	C₂×C₄×C₁₂₇	—	2	4	no	—
121	C₂×C₄×C₁₂₇	—	2	4	no	—
122	C₂×C₄×C₁₂₇	C₂×C₄²	1	4	yes	—
123	C₂×C₄×C₁₂₇	C₂×C₄²	1	4	yes	—
124	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
125	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₂₇	H(5,𝔽₂)	1	2	yes	—
128	C₂×C₄×C₁₂₇	2₋¹⁺⁴	1	2	yes	—
129	C₈×C₁₂₇	—	2	4	no	—
130	C₈×C₁₂₇	—	2	4	no	—
131	C₈×C₁₂₇	—	2	4	no	—
132	C₈×C₁₂₇	—	2	4	no	—
133	C₂²×C₁₂₇	—	2	4	no	—
134	C₂²×C₁₂₇	—	2	2	no	—
135	C₂²×C₁₂₇	—	4	4	no	—
136	C₂²×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
137	C₂²×C₁₂₇	#520	1	2	yes	—
138	C₂²×C₁₂₇	#568	1	4	yes	—
139	C₄×C₁₂₇	—	2	4	no	—
140	C₄×C₁₂₇	—	2	4	no	—
141	C₄×C₁₂₇	—	2	2	no	—
142	C₄×C₁₂₇	—	4	4	no	—
143	C₄×C₁₂₇	—	4	4	no	—
144	C₄×C₁₂₇	#518	1	1	yes	characteristic
145	C₄×C₁₂₇	#519	1	1	yes	characteristic
146	C₄×C₁₂₇	#520	1	2	yes	—
147	#20554	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂×C₁₂₇	—	4	2	no	—
149	C₂×C₁₂₇	#2274	1	1	yes	characteristic
150	C₂×C₁₂₇	#2875	1	2	yes	—
151	C₂×C₁₂₇	#2995	1	4	yes	—
152	C₂×#433	C₂×C₁₂₇	1	1	yes	characteristic
153	C₂×#424	C₂×C₁₂₇	1	1	yes	characteristic
154	#2298	C₂×C₁₂₇	1	1	yes	characteristic
155	#1771	C₂×C₁₂₇	1	2	yes	complemented
156	#1770	C₂×C₁₂₇	1	2	yes	complemented
157	#1768	C₂×C₁₂₇	1	2	yes	complemented
158	#1767	C₂×C₁₂₇	1	2	yes	complemented
159	#1753	C₂×C₁₂₇	1	2	yes	—
160	#1752	C₂×C₁₂₇	1	2	yes	—
161	C₁₂₇	#20554	1	1	yes	characteristic, direct product factor, Sylow
162	C₂²×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
163	C₂³×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
164	C₂×#117	C₂²×C₁₂₇	1	1	yes	characteristic
165	#433	C₄×C₁₂₇	1	8	yes	complemented
166	#424	C₄×C₁₂₇	1	8	yes	complemented
167	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
168	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
169	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
170	C₂×C₂²⋊C₈	C₂²×C₁₂₇	1	2	yes	—
171	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
172	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
173	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
174	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
175	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
176	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
177	#388	C₄×C₁₂₇	1	4	yes	—
178	#387	C₄×C₁₂₇	1	4	yes	—
179	C₄×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
180	C₄×C₂²⋊C₄	C₂²×C₁₂₇	1	2	yes	—
181	#376	C₂²×C₁₂₇	1	1	yes	characteristic
182	C₄³	C₂²×C₁₂₇	1	1	yes	characteristic
183	#336	C₂²×C₁₂₇	1	2	yes	—
184	#336	C₂²×C₁₂₇	1	2	yes	—
185	#336	C₂²×C₁₂₇	1	2	yes	—
186	#336	C₂²×C₁₂₇	1	2	yes	—
187	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
188	C₂×M₁₆	—	4	2	no	—
189	C₂×M₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
190	C₂²×C₈	—	4	2	no	—
191	C₂²×C₈	C₂×C₄×C₁₂₇	1	4	yes	—
192	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
193	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
194	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
195	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
196	#117	C₂×C₄×C₁₂₇	1	4	yes	—
197	#117	C₂×C₄×C₁₂₇	1	4	yes	—
198	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
199	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
200	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
201	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
202	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
203	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
204	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
205	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
206	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
207	C₄⋊C₈	C₂×C₄×C₁₂₇	1	8	yes	—
208	C₄⋊C₈	C₂×C₄×C₁₂₇	1	8	yes	—
209	C₄⋊C₈	C₂×C₄×C₁₂₇	1	8	yes	—
210	C₂²⋊C₈	C₂×C₄×C₁₂₇	1	8	yes	—
211	#97	—	4	2	no	—
212	#97	—	4	2	no	—
213	#97	—	4	2	no	—
214	#97	C₂×C₄×C₁₂₇	1	4	yes	—
215	C₄×C₈	—	4	2	no	—
216	C₄×C₈	C₂×C₄×C₁₂₇	1	4	yes	—
217	#95	C₂×C₄×C₁₂₇	1	4	yes	—
218	#95	C₂×C₄×C₁₂₇	1	4	yes	—
219	#95	C₂×C₄×C₁₂₇	1	4	yes	—
220	C₂⁴	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
224	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
225	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
226	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
227	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
228	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
229	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
230	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
231	M₁₆	—	4	4	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	4	4	no	—
241	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
242	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
243	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
244	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
245	C₄⋊C₄	—	4	4	no	—
246	C₄⋊C₄	C₄²×C₁₂₇	1	8	yes	—
247	C₂²⋊C₄	—	4	4	no	—
248	C₂²⋊C₄	C₄²×C₁₂₇	1	8	yes	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	4	4	no	—
254	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
255	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
256	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
257	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	4	no	—
260	C₂³	C₂×C₄²×C₁₂₇	1	2	yes	—
261	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	C₂×C₄²×C₁₂₇	1	4	yes	—
275	C₂×C₄	C₂×C₄²×C₁₂₇	1	4	yes	—
276	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
277	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
278	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
279	C₂×C₄	H(5,𝔽₂)×C₁₂₇	1	2	yes	—
280	C₂×C₄	2₋¹⁺⁴×C₁₂₇	1	2	yes	—
281	C₈	—	2	4	no	—
282	C₈	—	2	4	no	—
283	C₈	—	2	4	no	—
284	C₈	—	2	4	no	—
285	C₂²	—	2	4	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	4	no	—
288	C₂²	C₂²×C₄²×C₁₂₇	1	1	yes	characteristic
289	C₂²	#520×C₁₂₇	1	2	yes	—
290	C₂²	#568×C₁₂₇	1	4	yes	—
291	C₄	—	2	4	no	—
292	C₄	—	2	4	no	—
293	C₄	—	2	2	no	—
294	C₄	—	4	4	no	—
295	C₄	—	4	4	no	—
296	C₄	#518×C₁₂₇	1	1	yes	characteristic
297	C₄	#519×C₁₂₇	1	1	yes	characteristic
298	C₄	#520×C₁₂₇	1	2	yes	—
299	C₂	—	4	2	no	—
300	C₂	C₁₂₇×#2274	1	1	yes	characteristic
301	C₂	C₁₂₇×#2875	1	2	yes	—
302	C₂	C₁₂₇×#2995	1	4	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
4. C₁₂₇×#1771	C₂	semidirect
5. C₁₂₇×#1770	C₂	semidirect
6. C₁₂₇×#1768	C₂	semidirect
7. C₁₂₇×#1767	C₂	semidirect
13. #433×C₁₂₇	C₄	semidirect
14. #424×C₁₂₇	C₄	semidirect
147. #20554	C₁₂₇	direct (N × H)
155. #1771	C₂×C₁₂₇	semidirect
156. #1770	C₂×C₁₂₇	semidirect
157. #1768	C₂×C₁₂₇	semidirect
158. #1767	C₂×C₁₂₇	semidirect
161. C₁₂₇	#20554	direct (N × H)
165. #433	C₄×C₁₂₇	semidirect
166. #424	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×#433×C₁₂₇	2
C₂×#424×C₁₂₇	2
C₁₂₇×#2298	2
C₁₂₇×#1771	2
C₁₂₇×#1770	2
C₁₂₇×#1768	2
C₁₂₇×#1767	2
C₁₂₇×#1753	2
C₁₂₇×#1752	2
#20554	127

Subgroups — C127×#20554 #44584227

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#20554 #44584227