Subgroups — C₁₂₇×#11489 #44575162

Order 32512.

390 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#436×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#2163	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₂₇×#2162	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₂₇×#1314	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₁₂₇	C₄	1	2	yes	—
6	C₂×C₄×D₄×C₁₂₇	C₄	1	1	yes	characteristic
7	#436×C₁₂₇	—	2	1	no	—
8	#436×C₁₂₇	—	2	1	no	—
9	#436×C₁₂₇	—	2	1	no	—
10	#436×C₁₂₇	—	2	1	no	—
11	C₂×C₄⋊C₈×C₁₂₇	C₄	1	1	yes	characteristic
12	C₂×C₂²⋊C₈×C₁₂₇	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₁₂₇	—	2	1	no	—
15	C₄⋊C₁₆×C₁₂₇	—	2	1	no	—
16	C₂²⋊C₁₆×C₁₂₇	C₂²	1	2	yes	—
17	C₂²⋊C₁₆×C₁₂₇	C₂²	1	2	yes	—
18	#346×C₁₂₇	—	2	1	no	—
19	C₄×C₁₆×C₁₂₇	—	2	1	no	—
20	C₂²×D₄×C₁₂₇	C₈	1	1	yes	characteristic
21	C₂³×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
22	C₂×M₁₆×C₁₂₇	—	2	2	no	—
23	C₂×M₁₆×C₁₂₇	—	2	2	no	—
24	C₂×M₁₆×C₁₂₇	—	4	4	no	—
25	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
28	C₄×D₄×C₁₂₇	—	2	1	no	—
29	C₄×D₄×C₁₂₇	—	2	1	no	—
30	C₄×D₄×C₁₂₇	—	2	1	no	—
31	C₄×D₄×C₁₂₇	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₂₇	C₈	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₁₂₇	C₈	1	2	yes	—
34	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₆×C₁₂₇	—	2	2	no	—
36	C₂×C₁₆×C₁₂₇	—	2	2	no	—
37	C₄⋊C₈×C₁₂₇	—	2	1	no	—
38	C₄⋊C₈×C₁₂₇	—	2	1	no	—
39	C₂²⋊C₈×C₁₂₇	—	2	2	no	—
40	C₂²⋊C₈×C₁₂₇	—	2	2	no	—
41	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
42	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
43	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
45	C₂⁴×C₁₂₇	C₂×C₈	1	2	yes	—
46	C₂×D₄×C₁₂₇	—	2	1	no	—
47	C₂×D₄×C₁₂₇	—	2	1	no	—
48	C₂×D₄×C₁₂₇	—	4	2	no	—
49	C₂²×C₄×C₁₂₇	—	2	2	no	—
50	C₂²×C₄×C₁₂₇	—	2	2	no	—
51	C₂²×C₄×C₁₂₇	—	4	2	no	—
52	C₂²×C₄×C₁₂₇	—	4	2	no	—
53	C₂²×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₂₇	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
57	M₁₆×C₁₂₇	—	4	4	no	—
58	M₁₆×C₁₂₇	—	4	4	no	—
59	C₂×C₈×C₁₂₇	—	2	2	no	—
60	C₂×C₈×C₁₂₇	—	2	2	no	—
61	C₂×C₈×C₁₂₇	—	4	1	no	—
62	C₂×C₈×C₁₂₇	—	4	2	no	—
63	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
67	C₄⋊C₄×C₁₂₇	—	2	1	no	—
68	C₄⋊C₄×C₁₂₇	—	2	1	no	—
69	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
70	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
71	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₁₂₇	M₁₆	1	1	yes	characteristic
74	C₄²×C₁₂₇	M₁₆	1	1	yes	characteristic
75	C₁₆×C₁₂₇	—	4	2	no	—
76	C₁₆×C₁₂₇	—	4	2	no	—
77	C₂³×C₁₂₇	—	2	2	no	—
78	C₂³×C₁₂₇	—	2	4	no	—
79	C₂³×C₁₂₇	—	4	2	no	—
80	C₂³×C₁₂₇	—	4	2	no	—
81	C₂³×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
82	D₄×C₁₂₇	—	4	2	no	—
83	D₄×C₁₂₇	—	4	2	no	—
84	C₂×C₄×C₁₂₇	—	2	2	no	—
85	C₂×C₄×C₁₂₇	—	2	2	no	—
86	C₂×C₄×C₁₂₇	—	2	1	no	—
87	C₂×C₄×C₁₂₇	—	2	1	no	—
88	C₂×C₄×C₁₂₇	—	2	2	no	—
89	C₂×C₄×C₁₂₇	—	4	4	no	—
90	C₂×C₄×C₁₂₇	—	4	2	no	—
91	C₂×C₄×C₁₂₇	—	4	2	no	—
92	C₂×C₄×C₁₂₇	—	4	2	no	—
93	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
94	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
95	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
96	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
97	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
99	C₈×C₁₂₇	—	2	1	no	—
100	C₈×C₁₂₇	—	2	1	no	—
101	C₈×C₁₂₇	—	2	1	no	—
102	C₈×C₁₂₇	—	4	2	no	—
103	C₈×C₁₂₇	#100	1	2	yes	—
104	C₂²×C₁₂₇	—	2	1	no	—
105	C₂²×C₁₂₇	—	2	1	no	—
106	C₂²×C₁₂₇	—	4	2	no	—
107	C₂²×C₁₂₇	—	4	2	no	—
108	C₂²×C₁₂₇	—	4	2	no	—
109	C₂²×C₁₂₇	—	8	4	no	—
110	C₂²×C₁₂₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
111	C₂²×C₁₂₇	#504	1	2	yes	—
112	C₄×C₁₂₇	—	2	1	no	—
113	C₄×C₁₂₇	—	2	1	no	—
114	C₄×C₁₂₇	—	2	1	no	—
115	C₄×C₁₂₇	—	2	1	no	—
116	C₄×C₁₂₇	—	4	2	no	—
117	C₄×C₁₂₇	#349	1	2	yes	—
118	C₄×C₁₂₇	#410	1	1	yes	characteristic
119	C₄×C₁₂₇	C₂×#100	1	1	yes	characteristic
120	#11489	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂×C₁₂₇	—	2	2	no	—
122	C₂×C₁₂₇	—	8	2	no	—
123	C₂×C₁₂₇	#1460	1	1	yes	characteristic
124	C₂×C₁₂₇	#2113	1	1	yes	characteristic
125	C₂×C₁₂₇	C₂×#349	1	1	yes	characteristic
126	C₂×#436	C₂×C₁₂₇	1	1	yes	characteristic
127	#2163	C₂×C₁₂₇	1	1	yes	characteristic, complemented
128	#2162	C₂×C₁₂₇	1	1	yes	characteristic, complemented
129	#1314	C₂×C₁₂₇	1	4	yes	complemented
130	C₁₂₇	#11489	1	1	yes	characteristic, direct product factor, Sylow
131	C₂²×M₁₆	C₄×C₁₂₇	1	2	yes	—
132	C₂×C₄×D₄	C₄×C₁₂₇	1	1	yes	characteristic
133	#436	—	2	1	no	—
134	#436	—	2	1	no	—
135	#436	—	2	1	no	—
136	#436	—	2	1	no	—
137	C₂×C₄⋊C₈	C₄×C₁₂₇	1	1	yes	characteristic
138	C₂×C₂²⋊C₈	C₂²×C₁₂₇	1	2	yes	—
139	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
140	C₄⋊C₁₆	—	2	1	no	—
141	C₄⋊C₁₆	—	2	1	no	—
142	C₂²⋊C₁₆	C₂²×C₁₂₇	1	2	yes	—
143	C₂²⋊C₁₆	C₂²×C₁₂₇	1	2	yes	—
144	#346	—	2	1	no	—
145	C₄×C₁₆	—	2	1	no	—
146	C₂²×D₄	C₈×C₁₂₇	1	1	yes	characteristic
147	C₂³×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
148	C₂×M₁₆	—	2	2	no	—
149	C₂×M₁₆	—	2	2	no	—
150	C₂×M₁₆	—	4	4	no	—
151	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
152	C₂²×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
153	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
154	C₄×D₄	—	2	1	no	—
155	C₄×D₄	—	2	1	no	—
156	C₄×D₄	—	2	1	no	—
157	C₄×D₄	—	2	1	no	—
158	C₂×C₄⋊C₄	C₈×C₁₂₇	1	1	yes	characteristic
159	C₂×C₂²⋊C₄	C₈×C₁₂₇	1	2	yes	—
160	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
161	C₂×C₁₆	—	2	2	no	—
162	C₂×C₁₆	—	2	2	no	—
163	C₄⋊C₈	—	2	1	no	—
164	C₄⋊C₈	—	2	1	no	—
165	C₂²⋊C₈	—	2	2	no	—
166	C₂²⋊C₈	—	2	2	no	—
167	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
168	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
169	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
170	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
171	C₂⁴	C₂×C₈×C₁₂₇	1	2	yes	—
172	C₂×D₄	—	2	1	no	—
173	C₂×D₄	—	2	1	no	—
174	C₂×D₄	—	4	2	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	C₂×C₈×C₁₂₇	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₈×C₁₂₇	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₈×C₁₂₇	1	2	yes	—
182	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
183	M₁₆	—	4	4	no	—
184	M₁₆	—	4	4	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	4	1	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
190	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
191	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
192	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₂²⋊C₄	—	2	2	no	—
196	C₂²⋊C₄	—	2	2	no	—
197	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
198	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
199	C₄²	M₁₆×C₁₂₇	1	1	yes	characteristic
200	C₄²	M₁₆×C₁₂₇	1	1	yes	characteristic
201	C₁₆	—	4	2	no	—
202	C₁₆	—	4	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	4	no	—
205	C₂³	—	4	2	no	—
206	C₂³	—	4	2	no	—
207	C₂³	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
208	D₄	—	4	2	no	—
209	D₄	—	4	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
220	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
221	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
222	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
223	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
224	C₂×C₄	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
225	C₈	—	2	1	no	—
226	C₈	—	2	1	no	—
227	C₈	—	2	1	no	—
228	C₈	—	4	2	no	—
229	C₈	#100×C₁₂₇	1	2	yes	—
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	4	2	no	—
233	C₂²	—	4	2	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	8	4	no	—
236	C₂²	C₂×C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
237	C₂²	#504×C₁₂₇	1	2	yes	—
238	C₄	—	2	1	no	—
239	C₄	—	2	1	no	—
240	C₄	—	2	1	no	—
241	C₄	—	2	1	no	—
242	C₄	—	4	2	no	—
243	C₄	#349×C₁₂₇	1	2	yes	—
244	C₄	#410×C₁₂₇	1	1	yes	characteristic
245	C₄	C₂×#100×C₁₂₇	1	1	yes	characteristic
246	C₂	—	2	2	no	—
247	C₂	—	8	2	no	—
248	C₂	C₁₂₇×#1460	1	1	yes	characteristic
249	C₂	C₁₂₇×#2113	1	1	yes	characteristic
250	C₂	C₂×#349×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂×C₂²⋊C₈
prime radical	C₂²×D₄×C₁₂₇	C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₂₇×#2163	C₂	semidirect
3. C₁₂₇×#2162	C₂	semidirect
4. C₁₂₇×#1314	C₂	semidirect
120. #11489	C₁₂₇	direct (N × H)
127. #2163	C₂×C₁₂₇	semidirect
128. #2162	C₂×C₁₂₇	semidirect
129. #1314	C₂×C₁₂₇	semidirect
130. C₁₂₇	#11489	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#436×C₁₂₇	2
C₁₂₇×#2163	2
C₁₂₇×#2162	2
C₁₂₇×#1314	2
#11489	127

Subgroups — C127×#11489 #44575162

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#11489 #44575162