Subgroups — C₁₂₇×#12785 #44576458

Order 32512.

486 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#2150	C₂	1	2	yes	maximal, complemented
3	C₂×#349×C₁₂₇	C₂	1	2	yes	maximal
4	C₁₂₇×#1377	C₂	1	2	yes	maximal, complemented
5	C₂²×M₁₆×C₁₂₇	C₄	1	1	yes	characteristic
6	C₂²×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₁₂₇	C₄	1	1	yes	characteristic
8	C₂×M₃₂×C₁₂₇	C₂²	1	4	yes	—
9	C₄×M₁₆×C₁₂₇	—	2	4	no	—
10	C₂×#97×C₁₂₇	C₄	1	1	yes	characteristic
11	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₂₇	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₁₂₇	—	2	4	no	—
15	#349×C₁₂₇	—	2	4	no	—
16	C₂³×C₄×C₁₂₇	C₈	1	2	yes	—
17	C₂³×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
18	C₂×M₁₆×C₁₂₇	—	2	2	no	—
19	C₂×M₁₆×C₁₂₇	—	2	2	no	—
20	C₂×M₁₆×C₁₂₇	—	2	2	no	—
21	C₂×M₁₆×C₁₂₇	—	2	2	no	—
22	C₂×M₁₆×C₁₂₇	—	2	2	no	—
23	C₂×M₁₆×C₁₂₇	D₄	1	2	yes	—
24	C₂×M₁₆×C₁₂₇	Q₈	1	2	yes	—
25	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₁₂₇	—	2	2	no	—
30	C₂×C₄²×C₁₂₇	—	2	1	no	—
31	C₂×C₄²×C₁₂₇	—	2	1	no	—
32	C₂×C₄²×C₁₂₇	C₈	1	2	yes	—
33	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
35	M₃₂×C₁₂₇	—	2	4	no	—
36	M₃₂×C₁₂₇	—	2	4	no	—
37	C₂×C₁₆×C₁₂₇	—	2	4	no	—
38	#97×C₁₂₇	—	2	2	no	—
39	#97×C₁₂₇	—	2	2	no	—
40	C₄×C₈×C₁₂₇	—	2	2	no	—
41	C₄×C₈×C₁₂₇	D₄	1	4	yes	—
42	C₂⁴×C₁₂₇	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₂₇	—	2	1	no	—
44	C₂²×C₄×C₁₂₇	—	2	1	no	—
45	C₂²×C₄×C₁₂₇	—	2	1	no	—
46	C₂²×C₄×C₁₂₇	—	2	1	no	—
47	C₂²×C₄×C₁₂₇	—	2	2	no	—
48	C₂²×C₄×C₁₂₇	—	2	2	no	—
49	C₂²×C₄×C₁₂₇	—	4	4	no	—
50	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₁₂₇	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₁₂₇	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₁₂₇	M₁₆	1	2	yes	—
55	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
56	M₁₆×C₁₂₇	—	2	8	no	—
57	M₁₆×C₁₂₇	—	2	4	no	—
58	M₁₆×C₁₂₇	—	2	4	no	—
59	C₂×C₈×C₁₂₇	—	2	1	no	—
60	C₂×C₈×C₁₂₇	—	2	1	no	—
61	C₂×C₈×C₁₂₇	—	2	1	no	—
62	C₂×C₈×C₁₂₇	—	2	1	no	—
63	C₂×C₈×C₁₂₇	—	2	1	no	—
64	C₂×C₈×C₁₂₇	—	2	1	no	—
65	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
67	C₂×C₈×C₁₂₇	M₁₆	1	2	yes	—
68	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₈×C₁₂₇	#41	1	2	yes	—
72	C₄²×C₁₂₇	—	2	2	no	—
73	C₄²×C₁₂₇	—	2	2	no	—
74	C₄²×C₁₂₇	—	2	1	no	—
75	C₄²×C₁₂₇	—	2	1	no	—
76	C₄²×C₁₂₇	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₁₂₇	M₁₆	1	1	yes	characteristic
78	C₄²×C₁₂₇	M₁₆	1	1	yes	characteristic
79	C₁₆×C₁₂₇	—	4	4	no	—
80	C₂³×C₁₂₇	—	2	1	no	—
81	C₂³×C₁₂₇	—	2	1	no	—
82	C₂³×C₁₂₇	—	4	2	no	—
83	C₂³×C₁₂₇	C₄⋊C₈	1	2	yes	—
84	C₂³×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₁₂₇	—	2	1	no	—
86	C₂×C₄×C₁₂₇	—	2	1	no	—
87	C₂×C₄×C₁₂₇	—	2	4	no	—
88	C₂×C₄×C₁₂₇	—	2	2	no	—
89	C₂×C₄×C₁₂₇	—	2	2	no	—
90	C₂×C₄×C₁₂₇	—	2	2	no	—
91	C₂×C₄×C₁₂₇	—	2	2	no	—
92	C₂×C₄×C₁₂₇	—	2	2	no	—
93	C₂×C₄×C₁₂₇	—	4	4	no	—
94	C₂×C₄×C₁₂₇	—	4	4	no	—
95	C₂×C₄×C₁₂₇	—	4	2	no	—
96	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	2	yes	—
97	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	2	yes	—
98	C₂×C₄×C₁₂₇	C₄⋊C₈	1	2	yes	—
99	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₂₇	#117	1	1	yes	characteristic
102	C₂×C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
105	C₈×C₁₂₇	—	2	2	no	—
106	C₈×C₁₂₇	—	2	2	no	—
107	C₈×C₁₂₇	—	2	2	no	—
108	C₈×C₁₂₇	#121	1	2	yes	—
109	C₈×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
110	C₂²×C₁₂₇	—	2	2	no	—
111	C₂²×C₁₂₇	—	2	2	no	—
112	C₂²×C₁₂₇	—	4	2	no	—
113	C₂²×C₁₂₇	—	4	2	no	—
114	C₂²×C₁₂₇	—	4	2	no	—
115	C₂²×C₁₂₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
116	C₂²×C₁₂₇	C₂×C₄⋊C₈	1	1	yes	characteristic
117	C₂²×C₁₂₇	#431	1	1	yes	characteristic
118	C₄×C₁₂₇	—	2	2	no	—
119	C₄×C₁₂₇	—	2	2	no	—
120	C₄×C₁₂₇	—	4	2	no	—
121	C₄×C₁₂₇	#349	1	4	yes	complemented
122	C₄×C₁₂₇	#380	1	1	yes	characteristic
123	C₄×C₁₂₇	#407	1	1	yes	characteristic
124	C₄×C₁₂₇	#423	1	1	yes	characteristic
125	C₄×C₁₂₇	#432	1	1	yes	characteristic
126	#12785	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂×C₁₂₇	—	2	1	no	—
128	C₂×C₁₂₇	—	2	1	no	—
129	C₂×C₁₂₇	—	4	2	no	—
130	C₂×C₁₂₇	#1797	1	1	yes	characteristic
131	C₂×C₁₂₇	C₂×#349	1	2	yes	—
132	C₂×C₄×M₁₆	C₂×C₁₂₇	1	1	yes	characteristic
133	#2150	C₂×C₁₂₇	1	2	yes	complemented
134	C₂×#349	C₂×C₁₂₇	1	2	yes	—
135	#1377	C₂×C₁₂₇	1	2	yes	complemented
136	C₁₂₇	#12785	1	1	yes	characteristic, direct product factor, Sylow
137	C₂²×M₁₆	C₄×C₁₂₇	1	1	yes	characteristic
138	C₂²×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
139	C₂²×C₄²	C₄×C₁₂₇	1	1	yes	characteristic
140	C₂×M₃₂	C₂²×C₁₂₇	1	4	yes	—
141	C₄×M₁₆	—	2	4	no	—
142	C₂×#97	C₄×C₁₂₇	1	1	yes	characteristic
143	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
144	C₂×C₄×C₈	C₄×C₁₂₇	1	1	yes	characteristic
145	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
146	C₄⋊C₁₆	—	2	4	no	—
147	#349	—	2	4	no	—
148	C₂³×C₄	C₈×C₁₂₇	1	2	yes	—
149	C₂³×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
150	C₂×M₁₆	—	2	2	no	—
151	C₂×M₁₆	—	2	2	no	—
152	C₂×M₁₆	—	2	2	no	—
153	C₂×M₁₆	—	2	2	no	—
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	D₄×C₁₂₇	1	2	yes	—
156	C₂×M₁₆	Q₈×C₁₂₇	1	2	yes	—
157	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
158	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
159	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
160	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
161	C₂×C₄²	—	2	2	no	—
162	C₂×C₄²	—	2	1	no	—
163	C₂×C₄²	—	2	1	no	—
164	C₂×C₄²	C₈×C₁₂₇	1	2	yes	—
165	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
166	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
167	M₃₂	—	2	4	no	—
168	M₃₂	—	2	4	no	—
169	C₂×C₁₆	—	2	4	no	—
170	#97	—	2	2	no	—
171	#97	—	2	2	no	—
172	C₄×C₈	—	2	2	no	—
173	C₄×C₈	D₄×C₁₂₇	1	4	yes	—
174	C₂⁴	C₂×C₈×C₁₂₇	1	1	yes	characteristic
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	4	4	no	—
182	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	2	yes	—
183	C₂²×C₄	C₂×C₈×C₁₂₇	1	2	yes	—
184	C₂²×C₄	C₂×C₈×C₁₂₇	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₈×C₁₂₇	1	2	yes	—
186	C₂²×C₄	M₁₆×C₁₂₇	1	2	yes	—
187	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
188	M₁₆	—	2	8	no	—
189	M₁₆	—	2	4	no	—
190	M₁₆	—	2	4	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	2	yes	—
198	C₂×C₈	C₄⋊C₄×C₁₂₇	1	2	yes	—
199	C₂×C₈	M₁₆×C₁₂₇	1	2	yes	—
200	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
201	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
202	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
203	C₂×C₈	#41×C₁₂₇	1	2	yes	—
204	C₄²	—	2	2	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	2	1	no	—
208	C₄²	C₂²⋊C₄×C₁₂₇	1	2	yes	—
209	C₄²	M₁₆×C₁₂₇	1	1	yes	characteristic
210	C₄²	M₁₆×C₁₂₇	1	1	yes	characteristic
211	C₁₆	—	4	4	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	2	1	no	—
214	C₂³	—	4	2	no	—
215	C₂³	C₄⋊C₈×C₁₂₇	1	2	yes	—
216	C₂³	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	2	yes	—
229	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	2	yes	—
230	C₂×C₄	C₄⋊C₈×C₁₂₇	1	2	yes	—
231	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
232	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
233	C₂×C₄	#117×C₁₂₇	1	1	yes	characteristic
234	C₂×C₄	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
235	C₂×C₄	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
236	C₂×C₄	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	—	2	2	no	—
239	C₈	—	2	2	no	—
240	C₈	#121×C₁₂₇	1	2	yes	—
241	C₈	C₂²⋊Q₈×C₁₂₇	1	2	yes	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	C₂×C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
248	C₂²	C₂×C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
249	C₂²	#431×C₁₂₇	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	4	2	no	—
253	C₄	#349×C₁₂₇	1	4	yes	complemented
254	C₄	#380×C₁₂₇	1	1	yes	characteristic
255	C₄	#407×C₁₂₇	1	1	yes	characteristic
256	C₄	#423×C₁₂₇	1	1	yes	characteristic
257	C₄	#432×C₁₂₇	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	4	2	no	—
261	C₂	C₁₂₇×#1797	1	1	yes	characteristic
262	C₂	C₂×#349×C₁₂₇	1	2	yes	—

	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	C₂⁴×C₁₂₇	C₂×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×#2150	C₂	semidirect
4. C₁₂₇×#1377	C₂	semidirect
121. C₄×C₁₂₇	#349	semidirect
126. #12785	C₁₂₇	direct (N × H)
133. #2150	C₂×C₁₂₇	semidirect
135. #1377	C₂×C₁₂₇	semidirect
136. C₁₂₇	#12785	direct (N × H)
253. C₄	#349×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×M₁₆×C₁₂₇	2
C₁₂₇×#2150	2
C₂×#349×C₁₂₇	2
C₁₂₇×#1377	2
#12785	127

Subgroups — C127×#12785 #44576458

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#12785 #44576458