Subgroups — C₁₂₇×#12732 #44576405

Order 32512.

934 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#358×C₁₂₇	C₂	1	4	yes	maximal, complemented
3	C₂×C₂²⋊C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1808	C₂	1	1	yes	maximal, characteristic
5	C₂³×Q₈×C₁₂₇	—	2	1	no	—
6	C₂²×Q₁₆×C₁₂₇	—	2	8	no	—
7	C₂²×Q₁₆×C₁₂₇	—	2	1	no	—
8	C₂²×Q₁₆×C₁₂₇	—	2	2	no	—
9	C₂²×Q₁₆×C₁₂₇	C₄	1	4	yes	complemented
10	C₂²×Q₁₆×C₁₂₇	C₂²	1	2	yes	—
11	C₂³×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₁₂₇	C₂²	1	2	yes	—
13	C₂×#107×C₁₂₇	C₂²	1	2	yes	—
14	#380×C₁₂₇	—	2	1	no	—
15	#358×C₁₂₇	—	2	8	no	—
16	C₂²⋊C₁₆×C₁₂₇	—	2	2	no	—
17	#340×C₁₂₇	—	2	1	no	—
18	C₂²×Q₈×C₁₂₇	—	2	8	no	—
19	C₂²×Q₈×C₁₂₇	—	2	4	no	—
20	C₂²×Q₈×C₁₂₇	—	2	2	no	—
21	C₂²×Q₈×C₁₂₇	—	4	4	no	—
22	C₂²×Q₈×C₁₂₇	—	4	1	no	—
23	C₂²×Q₈×C₁₂₇	—	4	2	no	—
24	C₂³×C₄×C₁₂₇	—	4	1	no	—
25	C₂³×C₄×C₁₂₇	D₄	1	1	yes	characteristic
26	C₂×Q₁₆×C₁₂₇	—	2	8	no	—
27	C₂×Q₁₆×C₁₂₇	—	2	16	no	—
28	C₂×Q₁₆×C₁₂₇	—	2	16	no	—
29	C₂×Q₁₆×C₁₂₇	—	2	4	no	—
30	C₂×Q₁₆×C₁₂₇	—	2	4	no	—
31	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
32	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
33	C₂×Q₁₆×C₁₂₇	D₄	1	8	yes	—
34	C₂²×C₈×C₁₂₇	—	2	4	no	—
35	C₂²×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₂₇	D₄	1	2	yes	—
38	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
40	C₂×C₂²⋊C₄×C₁₂₇	—	4	1	no	—
41	C₂×C₁₆×C₁₂₇	—	2	2	no	—
42	C₂×C₁₆×C₁₂₇	—	2	4	no	—
43	#107×C₁₂₇	—	2	4	no	—
44	#95×C₁₂₇	—	4	1	no	—
45	C₂⁴×C₁₂₇	D₈	1	1	yes	characteristic
46	C₂×Q₈×C₁₂₇	—	2	16	no	—
47	C₂×Q₈×C₁₂₇	—	2	4	no	—
48	C₂×Q₈×C₁₂₇	—	2	8	no	—
49	C₂×Q₈×C₁₂₇	—	4	8	no	—
50	C₂×Q₈×C₁₂₇	—	4	8	no	—
51	C₂×Q₈×C₁₂₇	—	4	16	no	—
52	C₂×Q₈×C₁₂₇	—	4	4	no	—
53	C₂×Q₈×C₁₂₇	—	4	2	no	—
54	C₂×Q₈×C₁₂₇	—	4	4	no	—
55	C₂²×C₄×C₁₂₇	—	2	2	no	—
56	C₂²×C₄×C₁₂₇	—	2	2	no	—
57	C₂²×C₄×C₁₂₇	—	4	8	no	—
58	C₂²×C₄×C₁₂₇	—	4	4	no	—
59	C₂²×C₄×C₁₂₇	—	4	2	no	—
60	C₂²×C₄×C₁₂₇	—	4	2	no	—
61	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₂₇	SD₁₆	1	2	yes	—
64	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
65	Q₁₆×C₁₂₇	—	2	16	no	—
66	Q₁₆×C₁₂₇	—	2	4	no	—
67	Q₁₆×C₁₂₇	—	2	8	no	—
68	Q₁₆×C₁₂₇	C₂²⋊C₄	1	8	yes	complemented
69	C₂×C₈×C₁₂₇	—	2	2	no	—
70	C₂×C₈×C₁₂₇	—	2	4	no	—
71	C₂×C₈×C₁₂₇	—	2	4	no	—
72	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
75	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
76	C₄⋊C₄×C₁₂₇	—	4	2	no	—
77	C₄⋊C₄×C₁₂₇	—	4	2	no	—
78	C₂²⋊C₄×C₁₂₇	—	8	2	no	—
79	C₁₆×C₁₂₇	—	4	2	no	—
80	C₂³×C₁₂₇	—	2	2	no	—
81	C₂³×C₁₂₇	—	2	2	no	—
82	C₂³×C₁₂₇	#102	1	2	yes	—
83	C₂³×C₁₂₇	SD₃₂	1	2	yes	—
84	C₂³×C₁₂₇	Q₃₂	1	2	yes	—
85	C₂³×C₁₂₇	C₂×D₈	1	1	yes	characteristic
86	Q₈×C₁₂₇	—	2	8	no	—
87	Q₈×C₁₂₇	—	4	16	no	—
88	Q₈×C₁₂₇	—	4	4	no	—
89	Q₈×C₁₂₇	—	4	8	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	—	2	4	no	—
94	C₂×C₄×C₁₂₇	—	4	16	no	—
95	C₂×C₄×C₁₂₇	—	4	4	no	—
96	C₂×C₄×C₁₂₇	—	4	8	no	—
97	C₂×C₄×C₁₂₇	—	8	2	no	—
98	C₂×C₄×C₁₂₇	—	8	2	no	—
99	C₂×C₄×C₁₂₇	—	8	2	no	—
100	C₂×C₄×C₁₂₇	#102	1	2	yes	—
101	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₂₇	#120	1	2	yes	—
103	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
104	C₂×C₄×C₁₂₇	AGL(1,ℤ/8)	1	2	yes	—
105	C₈×C₁₂₇	—	2	2	no	—
106	C₈×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
107	C₈×C₁₂₇	#120	1	2	yes	—
108	C₂²×C₁₂₇	—	2	2	no	—
109	C₂²×C₁₂₇	—	2	4	no	—
110	C₂²×C₁₂₇	—	2	4	no	—
111	C₂²×C₁₂₇	—	2	2	no	—
112	C₂²×C₁₂₇	#358	1	4	yes	complemented
113	C₂²×C₁₂₇	C₂×#102	1	1	yes	characteristic
114	C₂²×C₁₂₇	#447	1	2	yes	—
115	C₂²×C₁₂₇	C₂×SD₃₂	1	1	yes	characteristic
116	C₂²×C₁₂₇	C₂×Q₃₂	1	1	yes	characteristic
117	C₂²×C₁₂₇	#510	1	2	yes	—
118	C₄×C₁₂₇	—	2	2	no	—
119	C₄×C₁₂₇	—	4	8	no	—
120	C₄×C₁₂₇	—	16	2	no	—
121	C₄×C₁₂₇	#379	1	1	yes	characteristic
122	C₄×C₁₂₇	#418	1	1	yes	characteristic
123	C₄×C₁₂₇	#450	1	2	yes	—
124	#12732	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂×C₁₂₇	—	2	4	no	—
126	C₂×C₁₂₇	#1786	1	1	yes	characteristic
127	C₂×C₁₂₇	C₂×#358	1	1	yes	characteristic
128	C₂×C₁₂₇	#2141	1	1	yes	characteristic
129	C₂×C₁₂₇	#2185	1	2	yes	—
130	C₂×C₁₂₇	#2188	1	2	yes	—
131	C₂³×Q₁₆	C₂×C₁₂₇	1	1	yes	characteristic
132	C₂×#358	C₂×C₁₂₇	1	4	yes	complemented
133	C₂×C₂²⋊C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
134	#1808	C₂×C₁₂₇	1	1	yes	characteristic
135	C₁₂₇	#12732	1	1	yes	characteristic, direct product factor, Sylow
136	C₂³×Q₈	—	2	1	no	—
137	C₂²×Q₁₆	—	2	8	no	—
138	C₂²×Q₁₆	—	2	1	no	—
139	C₂²×Q₁₆	—	2	2	no	—
140	C₂²×Q₁₆	C₄×C₁₂₇	1	4	yes	complemented
141	C₂²×Q₁₆	C₂²×C₁₂₇	1	2	yes	—
142	C₂³×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
143	C₂²×C₁₆	C₂²×C₁₂₇	1	2	yes	—
144	C₂×#107	C₂²×C₁₂₇	1	2	yes	—
145	#380	—	2	1	no	—
146	#358	—	2	8	no	—
147	C₂²⋊C₁₆	—	2	2	no	—
148	#340	—	2	1	no	—
149	C₂²×Q₈	—	2	8	no	—
150	C₂²×Q₈	—	2	4	no	—
151	C₂²×Q₈	—	2	2	no	—
152	C₂²×Q₈	—	4	4	no	—
153	C₂²×Q₈	—	4	1	no	—
154	C₂²×Q₈	—	4	2	no	—
155	C₂³×C₄	—	4	1	no	—
156	C₂³×C₄	D₄×C₁₂₇	1	1	yes	characteristic
157	C₂×Q₁₆	—	2	8	no	—
158	C₂×Q₁₆	—	2	16	no	—
159	C₂×Q₁₆	—	2	16	no	—
160	C₂×Q₁₆	—	2	4	no	—
161	C₂×Q₁₆	—	2	4	no	—
162	C₂×Q₁₆	—	2	2	no	—
163	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
164	C₂×Q₁₆	D₄×C₁₂₇	1	8	yes	—
165	C₂²×C₈	—	2	4	no	—
166	C₂²×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
167	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
168	C₂²×C₈	D₄×C₁₂₇	1	2	yes	—
169	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₂²⋊C₄	—	4	1	no	—
172	C₂×C₁₆	—	2	2	no	—
173	C₂×C₁₆	—	2	4	no	—
174	#107	—	2	4	no	—
175	#95	—	4	1	no	—
176	C₂⁴	D₈×C₁₂₇	1	1	yes	characteristic
177	C₂×Q₈	—	2	16	no	—
178	C₂×Q₈	—	2	4	no	—
179	C₂×Q₈	—	2	8	no	—
180	C₂×Q₈	—	4	8	no	—
181	C₂×Q₈	—	4	8	no	—
182	C₂×Q₈	—	4	16	no	—
183	C₂×Q₈	—	4	4	no	—
184	C₂×Q₈	—	4	2	no	—
185	C₂×Q₈	—	4	4	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	4	8	no	—
189	C₂²×C₄	—	4	4	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	2	yes	—
193	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
194	C₂²×C₄	SD₁₆×C₁₂₇	1	2	yes	—
195	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
196	Q₁₆	—	2	16	no	—
197	Q₁₆	—	2	4	no	—
198	Q₁₆	—	2	8	no	—
199	Q₁₆	C₂²⋊C₄×C₁₂₇	1	8	yes	complemented
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	2	yes	—
204	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
205	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
206	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₂²⋊C₄	—	8	2	no	—
210	C₁₆	—	4	2	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	#102×C₁₂₇	1	2	yes	—
214	C₂³	SD₃₂×C₁₂₇	1	2	yes	—
215	C₂³	Q₃₂×C₁₂₇	1	2	yes	—
216	C₂³	C₂×D₈×C₁₂₇	1	1	yes	characteristic
217	Q₈	—	2	8	no	—
218	Q₈	—	4	16	no	—
219	Q₈	—	4	4	no	—
220	Q₈	—	4	8	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	4	16	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	8	no	—
228	C₂×C₄	—	8	2	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	#102×C₁₂₇	1	2	yes	—
232	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
233	C₂×C₄	#120×C₁₂₇	1	2	yes	—
234	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
235	C₂×C₄	AGL(1,ℤ/8)×C₁₂₇	1	2	yes	—
236	C₈	—	2	2	no	—
237	C₈	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
238	C₈	#120×C₁₂₇	1	2	yes	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	2	2	no	—
243	C₂²	#358×C₁₂₇	1	4	yes	complemented
244	C₂²	C₂×#102×C₁₂₇	1	1	yes	characteristic
245	C₂²	#447×C₁₂₇	1	2	yes	—
246	C₂²	C₂×SD₃₂×C₁₂₇	1	1	yes	characteristic
247	C₂²	C₂×Q₃₂×C₁₂₇	1	1	yes	characteristic
248	C₂²	#510×C₁₂₇	1	2	yes	—
249	C₄	—	2	2	no	—
250	C₄	—	4	8	no	—
251	C₄	—	16	2	no	—
252	C₄	#379×C₁₂₇	1	1	yes	characteristic
253	C₄	#418×C₁₂₇	1	1	yes	characteristic
254	C₄	#450×C₁₂₇	1	2	yes	—
255	C₂	—	2	4	no	—
256	C₂	C₁₂₇×#1786	1	1	yes	characteristic
257	C₂	C₂×#358×C₁₂₇	1	1	yes	characteristic
258	C₂	C₁₂₇×#2141	1	1	yes	characteristic
259	C₂	C₁₂₇×#2185	1	2	yes	—
260	C₂	C₁₂₇×#2188	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₂₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
prime radical	C₂⁴×C₁₂₇	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ C₂³×C₄×C₁₂₇ ◁ C₂³×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#358×C₁₂₇	C₂	semidirect
9. C₂²×Q₁₆×C₁₂₇	C₄	semidirect
68. Q₁₆×C₁₂₇	C₂²⋊C₄	semidirect
112. C₂²×C₁₂₇	#358	semidirect
124. #12732	C₁₂₇	direct (N × H)
132. C₂×#358	C₂×C₁₂₇	semidirect
135. C₁₂₇	#12732	direct (N × H)
140. C₂²×Q₁₆	C₄×C₁₂₇	semidirect
199. Q₁₆	C₂²⋊C₄×C₁₂₇	semidirect
243. C₂²	#358×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂³×Q₁₆×C₁₂₇	2
C₂×#358×C₁₂₇	2
C₂×C₂²⋊C₁₆×C₁₂₇	2
C₁₂₇×#1808	2
#12732	127

Subgroups — C127×#12732 #44576405

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#12732 #44576405