Subgroups — C₂×C₁₂₇×#1817 #44584353

Order 32512.

2110 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄²×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂²⋊C₈×C₁₂₇	C₂	1	2	yes	maximal
3	C₁₂₇×#1817	C₂	1	8	yes	maximal, direct product factor
4	C₂×#336×C₁₂₇	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₁₂₇	C₄	1	1	yes	characteristic
6	C₂⁴×C₄×C₁₂₇	C₄	1	1	yes	characteristic
7	C₂⁴×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₁₂₇	C₂²	1	4	yes	—
9	C₂²×C₄²×C₁₂₇	—	2	8	no	—
10	C₂²×C₄²×C₁₂₇	C₄	1	4	yes	—
11	C₂²×C₄²×C₁₂₇	C₄	1	2	yes	—
12	C₂²×C₄²×C₁₂₇	C₂²	1	4	yes	—
13	C₂²×C₄²×C₁₂₇	C₂²	1	2	yes	—
14	C₂×C₂²⋊C₈×C₁₂₇	—	2	8	no	—
15	C₂×C₂²⋊C₈×C₁₂₇	C₂²	1	8	yes	—
16	#336×C₁₂₇	C₂²	1	16	yes	complemented
17	C₂⁵×C₁₂₇	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₁₂₇	—	2	4	no	—
19	C₂³×C₄×C₁₂₇	—	2	4	no	—
20	C₂³×C₄×C₁₂₇	—	2	4	no	—
21	C₂³×C₄×C₁₂₇	—	2	8	no	—
22	C₂³×C₄×C₁₂₇	—	2	8	no	—
23	C₂³×C₄×C₁₂₇	C₈	1	8	yes	complemented
24	C₂³×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
25	C₂³×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
29	C₂³×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₁₂₇	D₄	1	8	yes	—
31	C₂³×C₄×C₁₂₇	C₂³	1	2	yes	—
32	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₂₇	—	2	8	no	—
34	C₂²×C₈×C₁₂₇	—	2	16	no	—
35	C₂²×C₈×C₁₂₇	C₂³	1	8	yes	—
36	C₂×C₄²×C₁₂₇	—	2	8	no	—
37	C₂×C₄²×C₁₂₇	—	2	16	no	—
38	C₂×C₄²×C₁₂₇	—	2	16	no	—
39	C₂×C₄²×C₁₂₇	—	2	8	no	—
40	C₂×C₄²×C₁₂₇	C₂×C₄	1	8	yes	—
41	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₁₂₇	C₂³	1	4	yes	—
43	C₂²⋊C₈×C₁₂₇	—	2	16	no	—
44	C₂⁴×C₁₂₇	—	2	4	no	—
45	C₂⁴×C₁₂₇	—	2	4	no	—
46	C₂⁴×C₁₂₇	C₂²⋊C₄	1	4	yes	—
47	C₂⁴×C₁₂₇	C₂×C₈	1	4	yes	—
48	C₂⁴×C₁₂₇	M₁₆	1	4	yes	—
49	C₂⁴×C₁₂₇	C₂²×C₄	1	2	yes	—
50	C₂⁴×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₂₇	—	2	32	no	—
52	C₂²×C₄×C₁₂₇	—	2	32	no	—
53	C₂²×C₄×C₁₂₇	—	2	8	no	—
54	C₂²×C₄×C₁₂₇	—	2	4	no	—
55	C₂²×C₄×C₁₂₇	—	2	8	no	—
56	C₂²×C₄×C₁₂₇	—	2	4	no	—
57	C₂²×C₄×C₁₂₇	—	2	4	no	—
58	C₂²×C₄×C₁₂₇	—	2	4	no	—
59	C₂²×C₄×C₁₂₇	—	2	8	no	—
60	C₂²×C₄×C₁₂₇	—	2	4	no	—
61	C₂²×C₄×C₁₂₇	—	2	8	no	—
62	C₂²×C₄×C₁₂₇	—	2	32	no	—
63	C₂²×C₄×C₁₂₇	—	2	16	no	—
64	C₂²×C₄×C₁₂₇	—	2	8	no	—
65	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
66	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	8	yes	—
67	C₂²×C₄×C₁₂₇	C₂×C₈	1	8	yes	—
68	C₂²×C₄×C₁₂₇	C₂×C₈	1	16	yes	complemented
69	C₂²×C₄×C₁₂₇	M₁₆	1	8	yes	—
70	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
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	C₂×D₄	1	8	yes	—
75	C₂²×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
76	C₂²×C₄×C₁₂₇	#41	1	8	yes	—
77	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
78	C₂×C₈×C₁₂₇	—	2	8	no	—
79	C₂×C₈×C₁₂₇	—	2	16	no	—
80	C₂×C₈×C₁₂₇	—	4	16	no	—
81	C₄²×C₁₂₇	—	2	16	no	—
82	C₄²×C₁₂₇	—	2	8	no	—
83	C₄²×C₁₂₇	—	2	8	no	—
84	C₂³×C₁₂₇	—	2	4	no	—
85	C₂³×C₁₂₇	—	2	4	no	—
86	C₂³×C₁₂₇	—	2	8	no	—
87	C₂³×C₁₂₇	—	2	8	no	—
88	C₂³×C₁₂₇	—	2	16	no	—
89	C₂³×C₁₂₇	—	2	16	no	—
90	C₂³×C₁₂₇	C₂²⋊C₈	1	16	yes	—
91	C₂³×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
92	C₂³×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
93	C₂³×C₁₂₇	#117	1	4	yes	—
94	C₂³×C₁₂₇	C₂²×C₈	1	4	yes	—
95	C₂³×C₁₂₇	C₂²×C₈	1	2	yes	—
96	C₂³×C₁₂₇	C₂×M₁₆	1	4	yes	—
97	C₂³×C₁₂₇	C₂×M₁₆	1	2	yes	—
98	C₂³×C₁₂₇	C₂×M₁₆	1	4	yes	—
99	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₂₇	—	2	32	no	—
101	C₂×C₄×C₁₂₇	—	2	4	no	—
102	C₂×C₄×C₁₂₇	—	2	4	no	—
103	C₂×C₄×C₁₂₇	—	2	4	no	—
104	C₂×C₄×C₁₂₇	—	2	4	no	—
105	C₂×C₄×C₁₂₇	—	2	4	no	—
106	C₂×C₄×C₁₂₇	—	2	8	no	—
107	C₂×C₄×C₁₂₇	—	2	16	no	—
108	C₂×C₄×C₁₂₇	—	2	32	no	—
109	C₂×C₄×C₁₂₇	—	2	16	no	—
110	C₂×C₄×C₁₂₇	—	2	16	no	—
111	C₂×C₄×C₁₂₇	—	2	16	no	—
112	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
113	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
114	C₂×C₄×C₁₂₇	#117	1	4	yes	—
115	C₂×C₄×C₁₂₇	#123	1	16	yes	—
116	C₂×C₄×C₁₂₇	C₂²×C₈	1	8	yes	—
117	C₂×C₄×C₁₂₇	C₂×M₁₆	1	8	yes	—
118	C₂×C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
119	C₂×C₄×C₁₂₇	C₂×#41	1	4	yes	—
120	C₈×C₁₂₇	—	4	8	no	—
121	C₂²×C₁₂₇	—	2	32	no	—
122	C₂²×C₁₂₇	—	2	4	no	—
123	C₂²×C₁₂₇	—	2	8	no	—
124	C₂²×C₁₂₇	—	2	8	no	—
125	C₂²×C₁₂₇	—	2	4	no	—
126	C₂²×C₁₂₇	#381	1	4	yes	—
127	C₂²×C₁₂₇	C₂×C₂²⋊C₈	1	8	yes	—
128	C₂²×C₁₂₇	C₂×C₂²⋊C₈	1	4	yes	—
129	C₂²×C₁₂₇	#407	1	4	yes	—
130	C₂²×C₁₂₇	#431	1	8	yes	—
131	C₂²×C₁₂₇	#432	1	8	yes	—
132	C₂²×C₁₂₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₁₂₇	C₂×#117	1	2	yes	—
134	C₂²×C₁₂₇	C₂³×C₈	1	1	yes	characteristic
135	C₂²×C₁₂₇	C₂²×M₁₆	1	1	yes	characteristic
136	C₂²×C₁₂₇	C₂²×M₁₆	1	2	yes	—
137	C₄×C₁₂₇	—	2	8	no	—
138	C₄×C₁₂₇	—	2	8	no	—
139	C₄×C₁₂₇	—	2	4	no	—
140	C₄×C₁₂₇	#381	1	4	yes	—
141	C₄×C₁₂₇	C₂×#123	1	4	yes	—
142	C₂×#1817	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂×C₁₂₇	—	2	8	no	—
144	C₂×C₁₂₇	#1817	1	8	yes	direct product factor
145	C₂×C₁₂₇	C₂×#381	1	1	yes	characteristic
146	C₂×C₁₂₇	C₂²×C₂²⋊C₈	1	1	yes	characteristic
147	C₂×C₁₂₇	C₂×#407	1	1	yes	characteristic
148	C₂×C₁₂₇	C₂×#431	1	2	yes	—
149	C₂×C₁₂₇	C₂×#432	1	2	yes	—
150	C₂³×C₄²	C₂×C₁₂₇	1	1	yes	characteristic
151	C₂²×C₂²⋊C₈	C₂×C₁₂₇	1	2	yes	—
152	#1817	C₂×C₁₂₇	1	8	yes	direct product factor
153	C₂×#336	C₂×C₁₂₇	1	4	yes	complemented
154	C₁₂₇	C₂×#1817	1	1	yes	characteristic, direct product factor, Sylow
155	C₂⁴×C₄	C₄×C₁₂₇	1	1	yes	characteristic
156	C₂⁴×C₄	C₄×C₁₂₇	1	1	yes	characteristic
157	C₂⁴×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
158	C₂³×C₈	C₂²×C₁₂₇	1	4	yes	—
159	C₂²×C₄²	—	2	8	no	—
160	C₂²×C₄²	C₄×C₁₂₇	1	4	yes	—
161	C₂²×C₄²	C₄×C₁₂₇	1	2	yes	—
162	C₂²×C₄²	C₂²×C₁₂₇	1	4	yes	—
163	C₂²×C₄²	C₂²×C₁₂₇	1	2	yes	—
164	C₂×C₂²⋊C₈	—	2	8	no	—
165	C₂×C₂²⋊C₈	C₂²×C₁₂₇	1	8	yes	—
166	#336	C₂²×C₁₂₇	1	16	yes	complemented
167	C₂⁵	C₂×C₄×C₁₂₇	1	1	yes	characteristic
168	C₂³×C₄	—	2	4	no	—
169	C₂³×C₄	—	2	4	no	—
170	C₂³×C₄	—	2	4	no	—
171	C₂³×C₄	—	2	8	no	—
172	C₂³×C₄	—	2	8	no	—
173	C₂³×C₄	C₈×C₁₂₇	1	8	yes	complemented
174	C₂³×C₄	C₂×C₄×C₁₂₇	1	4	yes	—
175	C₂³×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
176	C₂³×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
177	C₂³×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
178	C₂³×C₄	C₂×C₄×C₁₂₇	1	4	yes	—
179	C₂³×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
180	C₂³×C₄	D₄×C₁₂₇	1	8	yes	—
181	C₂³×C₄	C₂³×C₁₂₇	1	2	yes	—
182	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
183	C₂²×C₈	—	2	8	no	—
184	C₂²×C₈	—	2	16	no	—
185	C₂²×C₈	C₂³×C₁₂₇	1	8	yes	—
186	C₂×C₄²	—	2	8	no	—
187	C₂×C₄²	—	2	16	no	—
188	C₂×C₄²	—	2	16	no	—
189	C₂×C₄²	—	2	8	no	—
190	C₂×C₄²	C₂×C₄×C₁₂₇	1	8	yes	—
191	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
192	C₂×C₄²	C₂³×C₁₂₇	1	4	yes	—
193	C₂²⋊C₈	—	2	16	no	—
194	C₂⁴	—	2	4	no	—
195	C₂⁴	—	2	4	no	—
196	C₂⁴	C₂²⋊C₄×C₁₂₇	1	4	yes	—
197	C₂⁴	C₂×C₈×C₁₂₇	1	4	yes	—
198	C₂⁴	M₁₆×C₁₂₇	1	4	yes	—
199	C₂⁴	C₂²×C₄×C₁₂₇	1	2	yes	—
200	C₂⁴	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
201	C₂²×C₄	—	2	32	no	—
202	C₂²×C₄	—	2	32	no	—
203	C₂²×C₄	—	2	8	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	8	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	8	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	8	no	—
212	C₂²×C₄	—	2	32	no	—
213	C₂²×C₄	—	2	16	no	—
214	C₂²×C₄	—	2	8	no	—
215	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	4	yes	—
216	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	8	yes	—
217	C₂²×C₄	C₂×C₈×C₁₂₇	1	8	yes	—
218	C₂²×C₄	C₂×C₈×C₁₂₇	1	16	yes	complemented
219	C₂²×C₄	M₁₆×C₁₂₇	1	8	yes	—
220	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
221	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
222	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
223	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₁₂₇	1	8	yes	—
225	C₂²×C₄	C₂×D₄×C₁₂₇	1	4	yes	—
226	C₂²×C₄	#41×C₁₂₇	1	8	yes	—
227	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
228	C₂×C₈	—	2	8	no	—
229	C₂×C₈	—	2	16	no	—
230	C₂×C₈	—	4	16	no	—
231	C₄²	—	2	16	no	—
232	C₄²	—	2	8	no	—
233	C₄²	—	2	8	no	—
234	C₂³	—	2	4	no	—
235	C₂³	—	2	4	no	—
236	C₂³	—	2	8	no	—
237	C₂³	—	2	8	no	—
238	C₂³	—	2	16	no	—
239	C₂³	—	2	16	no	—
240	C₂³	C₂²⋊C₈×C₁₂₇	1	16	yes	—
241	C₂³	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
242	C₂³	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
243	C₂³	#117×C₁₂₇	1	4	yes	—
244	C₂³	C₂²×C₈×C₁₂₇	1	4	yes	—
245	C₂³	C₂²×C₈×C₁₂₇	1	2	yes	—
246	C₂³	C₂×M₁₆×C₁₂₇	1	4	yes	—
247	C₂³	C₂×M₁₆×C₁₂₇	1	2	yes	—
248	C₂³	C₂×M₁₆×C₁₂₇	1	4	yes	—
249	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
250	C₂×C₄	—	2	32	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	2	16	no	—
258	C₂×C₄	—	2	32	no	—
259	C₂×C₄	—	2	16	no	—
260	C₂×C₄	—	2	16	no	—
261	C₂×C₄	—	2	16	no	—
262	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
263	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
264	C₂×C₄	#117×C₁₂₇	1	4	yes	—
265	C₂×C₄	#123×C₁₂₇	1	16	yes	—
266	C₂×C₄	C₂²×C₈×C₁₂₇	1	8	yes	—
267	C₂×C₄	C₂×M₁₆×C₁₂₇	1	8	yes	—
268	C₂×C₄	C₂²×D₄×C₁₂₇	1	2	yes	—
269	C₂×C₄	C₂×#41×C₁₂₇	1	4	yes	—
270	C₈	—	4	8	no	—
271	C₂²	—	2	32	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	8	no	—
274	C₂²	—	2	8	no	—
275	C₂²	—	2	4	no	—
276	C₂²	#381×C₁₂₇	1	4	yes	—
277	C₂²	C₂×C₂²⋊C₈×C₁₂₇	1	8	yes	—
278	C₂²	C₂×C₂²⋊C₈×C₁₂₇	1	4	yes	—
279	C₂²	#407×C₁₂₇	1	4	yes	—
280	C₂²	#431×C₁₂₇	1	8	yes	—
281	C₂²	#432×C₁₂₇	1	8	yes	—
282	C₂²	C₂²×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
283	C₂²	C₂×#117×C₁₂₇	1	2	yes	—
284	C₂²	C₂³×C₈×C₁₂₇	1	1	yes	characteristic
285	C₂²	C₂²×M₁₆×C₁₂₇	1	1	yes	characteristic
286	C₂²	C₂²×M₁₆×C₁₂₇	1	2	yes	—
287	C₄	—	2	8	no	—
288	C₄	—	2	8	no	—
289	C₄	—	2	4	no	—
290	C₄	#381×C₁₂₇	1	4	yes	—
291	C₄	C₂×#123×C₁₂₇	1	4	yes	—
292	C₂	—	2	8	no	—
293	C₂	C₁₂₇×#1817	1	8	yes	direct product factor
294	C₂	C₂×#381×C₁₂₇	1	1	yes	characteristic
295	C₂	C₂²×C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
296	C₂	C₂×#407×C₁₂₇	1	1	yes	characteristic
297	C₂	C₂×#431×C₁₂₇	1	2	yes	—
298	C₂	C₂×#432×C₁₂₇	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₁₂₇	C₂³
commutator	C₂²	C₂³×C₈×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle	C₂⁴×C₁₂₇	C₂²×C₄
prime radical	C₂⁵×C₁₂₇	C₂×C₄
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
3. C₁₂₇×#1817	C₂	direct (N × H)
4. C₂×#336×C₁₂₇	C₂	semidirect
16. #336×C₁₂₇	C₂²	semidirect
23. C₂³×C₄×C₁₂₇	C₈	semidirect
68. C₂²×C₄×C₁₂₇	C₂×C₈	semidirect
142. C₂×#1817	C₁₂₇	direct (N × H)
144. C₂×C₁₂₇	#1817	direct (N × H)
152. #1817	C₂×C₁₂₇	direct (N × H)
153. C₂×#336	C₂×C₁₂₇	semidirect
154. C₁₂₇	C₂×#1817	direct (N × H)
166. #336	C₂²×C₁₂₇	semidirect
173. C₂³×C₄	C₈×C₁₂₇	semidirect
218. C₂²×C₄	C₂×C₈×C₁₂₇	semidirect
293. C₂	C₁₂₇×#1817	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄²×C₁₂₇	2
C₂²×C₂²⋊C₈×C₁₂₇	2
C₁₂₇×#1817	2
C₂×#336×C₁₂₇	2
C₂×#1817	127

Subgroups — C2×C127×#1817 #44584353

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₂₇×#1817 #44584353