Subgroups — C₁₂₇×#12866 #44576539

Order 32512.

470 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#107×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1808	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1380	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
6	C₂²×C₄⋊C₄×C₁₂₇	—	2	1	no	—
7	C₂²×C₁₆×C₁₂₇	C₂²	1	2	yes	—
8	C₂×#107×C₁₂₇	—	2	1	no	—
9	C₂×#107×C₁₂₇	—	2	2	no	—
10	C₂×#107×C₁₂₇	C₄	1	4	yes	complemented
11	C₂×#107×C₁₂₇	C₂²	1	2	yes	—
12	C₂×#107×C₁₂₇	C₂²	1	2	yes	—
13	#380×C₁₂₇	—	2	1	no	—
14	C₂²⋊C₁₆×C₁₂₇	—	2	2	no	—
15	#340×C₁₂₇	—	2	1	no	—
16	C₂³×C₄×C₁₂₇	—	4	1	no	—
17	C₂³×C₄×C₁₂₇	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₂₇	—	2	4	no	—
19	C₂²×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₂₇	D₄	1	2	yes	—
22	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₁₂₇	—	2	4	no	—
24	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
25	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
26	C₂×C₄⋊C₄×C₁₂₇	—	4	1	no	—
27	C₂×C₄⋊C₄×C₁₂₇	—	4	2	no	—
28	C₂×C₂²⋊C₄×C₁₂₇	—	4	1	no	—
29	C₂×C₁₆×C₁₂₇	—	2	2	no	—
30	C₂×C₁₆×C₁₂₇	—	2	4	no	—
31	#107×C₁₂₇	—	2	4	no	—
32	#107×C₁₂₇	—	2	2	no	—
33	#107×C₁₂₇	—	2	4	no	—
34	#107×C₁₂₇	C₂×C₄	1	4	yes	—
35	#95×C₁₂₇	—	4	1	no	—
36	C₂⁴×C₁₂₇	D₈	1	1	yes	characteristic
37	C₂²×C₄×C₁₂₇	—	2	2	no	—
38	C₂²×C₄×C₁₂₇	—	2	2	no	—
39	C₂²×C₄×C₁₂₇	—	4	4	no	—
40	C₂²×C₄×C₁₂₇	—	4	2	no	—
41	C₂²×C₄×C₁₂₇	—	4	2	no	—
42	C₂²×C₄×C₁₂₇	—	8	2	no	—
43	C₂²×C₄×C₁₂₇	—	8	2	no	—
44	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
46	C₂²×C₄×C₁₂₇	SD₁₆	1	2	yes	—
47	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₁₂₇	—	2	2	no	—
49	C₂×C₈×C₁₂₇	—	2	4	no	—
50	C₂×C₈×C₁₂₇	—	2	4	no	—
51	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	2	yes	—
52	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₁₂₇	#41	1	4	yes	—
55	C₄⋊C₄×C₁₂₇	—	2	4	no	—
56	C₄⋊C₄×C₁₂₇	—	4	4	no	—
57	C₄⋊C₄×C₁₂₇	—	4	2	no	—
58	C₄⋊C₄×C₁₂₇	—	4	2	no	—
59	C₄⋊C₄×C₁₂₇	—	4	2	no	—
60	C₂²⋊C₄×C₁₂₇	—	8	2	no	—
61	C₁₆×C₁₂₇	—	4	2	no	—
62	C₂³×C₁₂₇	—	2	2	no	—
63	C₂³×C₁₂₇	—	2	2	no	—
64	C₂³×C₁₂₇	#102	1	2	yes	—
65	C₂³×C₁₂₇	D₁₆	1	2	yes	—
66	C₂³×C₁₂₇	SD₃₂	1	2	yes	—
67	C₂³×C₁₂₇	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₁₂₇	—	2	2	no	—
69	C₂×C₄×C₁₂₇	—	2	2	no	—
70	C₂×C₄×C₁₂₇	—	2	2	no	—
71	C₂×C₄×C₁₂₇	—	2	4	no	—
72	C₂×C₄×C₁₂₇	—	4	4	no	—
73	C₂×C₄×C₁₂₇	—	8	8	no	—
74	C₂×C₄×C₁₂₇	—	8	2	no	—
75	C₂×C₄×C₁₂₇	—	8	2	no	—
76	C₂×C₄×C₁₂₇	—	8	2	no	—
77	C₂×C₄×C₁₂₇	—	8	2	no	—
78	C₂×C₄×C₁₂₇	—	8	2	no	—
79	C₂×C₄×C₁₂₇	#102	1	2	yes	—
80	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₂₇	#123	1	2	yes	—
82	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
83	C₂×C₄×C₁₂₇	#137	1	2	yes	—
84	C₈×C₁₂₇	—	2	2	no	—
85	C₈×C₁₂₇	#117	1	2	yes	—
86	C₈×C₁₂₇	#123	1	2	yes	—
87	C₂²×C₁₂₇	—	2	2	no	—
88	C₂²×C₁₂₇	—	2	2	no	—
89	C₂²×C₁₂₇	—	2	4	no	—
90	C₂²×C₁₂₇	—	2	4	no	—
91	C₂²×C₁₂₇	#357	1	4	yes	—
92	C₂²×C₁₂₇	C₂×#102	1	1	yes	characteristic
93	C₂²×C₁₂₇	#480	1	2	yes	—
94	C₂²×C₁₂₇	C₂×D₁₆	1	1	yes	characteristic
95	C₂²×C₁₂₇	C₂×SD₃₂	1	1	yes	characteristic
96	C₂²×C₁₂₇	#510	1	2	yes	—
97	C₄×C₁₂₇	—	2	2	no	—
98	C₄×C₁₂₇	—	8	4	no	—
99	C₄×C₁₂₇	—	16	2	no	—
100	C₄×C₁₂₇	#381	1	1	yes	characteristic
101	C₄×C₁₂₇	#419	1	1	yes	characteristic
102	C₄×C₁₂₇	#483	1	2	yes	—
103	#12866	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂×C₁₂₇	—	2	4	no	—
105	C₂×C₁₂₇	#1823	1	1	yes	characteristic
106	C₂×C₁₂₇	C₂×#357	1	1	yes	characteristic
107	C₂×C₁₂₇	#2141	1	1	yes	characteristic
108	C₂×C₁₂₇	#2232	1	2	yes	—
109	C₂×C₁₂₇	#2235	1	2	yes	—
110	C₂²×#107	C₂×C₁₂₇	1	1	yes	characteristic
111	C₂×C₂²⋊C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
112	#1808	C₂×C₁₂₇	1	1	yes	characteristic
113	#1380	C₂×C₁₂₇	1	4	yes	complemented
114	C₁₂₇	#12866	1	1	yes	characteristic, direct product factor, Sylow
115	C₂³×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
116	C₂²×C₄⋊C₄	—	2	1	no	—
117	C₂²×C₁₆	C₂²×C₁₂₇	1	2	yes	—
118	C₂×#107	—	2	1	no	—
119	C₂×#107	—	2	2	no	—
120	C₂×#107	C₄×C₁₂₇	1	4	yes	complemented
121	C₂×#107	C₂²×C₁₂₇	1	2	yes	—
122	C₂×#107	C₂²×C₁₂₇	1	2	yes	—
123	#380	—	2	1	no	—
124	C₂²⋊C₁₆	—	2	2	no	—
125	#340	—	2	1	no	—
126	C₂³×C₄	—	4	1	no	—
127	C₂³×C₄	D₄×C₁₂₇	1	1	yes	characteristic
128	C₂²×C₈	—	2	4	no	—
129	C₂²×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
130	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
131	C₂²×C₈	D₄×C₁₂₇	1	2	yes	—
132	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
133	C₂×C₄⋊C₄	—	2	4	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	4	1	no	—
137	C₂×C₄⋊C₄	—	4	2	no	—
138	C₂×C₂²⋊C₄	—	4	1	no	—
139	C₂×C₁₆	—	2	2	no	—
140	C₂×C₁₆	—	2	4	no	—
141	#107	—	2	4	no	—
142	#107	—	2	2	no	—
143	#107	—	2	4	no	—
144	#107	C₂×C₄×C₁₂₇	1	4	yes	—
145	#95	—	4	1	no	—
146	C₂⁴	D₈×C₁₂₇	1	1	yes	characteristic
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	4	4	no	—
150	C₂²×C₄	—	4	2	no	—
151	C₂²×C₄	—	4	2	no	—
152	C₂²×C₄	—	8	2	no	—
153	C₂²×C₄	—	8	2	no	—
154	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	2	yes	—
155	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
156	C₂²×C₄	SD₁₆×C₁₂₇	1	2	yes	—
157	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	2	yes	—
162	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
163	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
164	C₂×C₈	#41×C₁₂₇	1	4	yes	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	4	4	no	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₂²⋊C₄	—	8	2	no	—
171	C₁₆	—	4	2	no	—
172	C₂³	—	2	2	no	—
173	C₂³	—	2	2	no	—
174	C₂³	#102×C₁₂₇	1	2	yes	—
175	C₂³	D₁₆×C₁₂₇	1	2	yes	—
176	C₂³	SD₃₂×C₁₂₇	1	2	yes	—
177	C₂³	C₂×D₈×C₁₂₇	1	1	yes	characteristic
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	8	2	no	—
184	C₂×C₄	—	8	2	no	—
185	C₂×C₄	—	8	8	no	—
186	C₂×C₄	—	8	2	no	—
187	C₂×C₄	—	8	2	no	—
188	C₂×C₄	—	8	2	no	—
189	C₂×C₄	#102×C₁₂₇	1	2	yes	—
190	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
191	C₂×C₄	#123×C₁₂₇	1	2	yes	—
192	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
193	C₂×C₄	#137×C₁₂₇	1	2	yes	—
194	C₈	—	2	2	no	—
195	C₈	#117×C₁₂₇	1	2	yes	—
196	C₈	#123×C₁₂₇	1	2	yes	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	2	no	—
199	C₂²	—	2	4	no	—
200	C₂²	—	2	4	no	—
201	C₂²	#357×C₁₂₇	1	4	yes	—
202	C₂²	C₂×#102×C₁₂₇	1	1	yes	characteristic
203	C₂²	#480×C₁₂₇	1	2	yes	—
204	C₂²	C₂×D₁₆×C₁₂₇	1	1	yes	characteristic
205	C₂²	C₂×SD₃₂×C₁₂₇	1	1	yes	characteristic
206	C₂²	#510×C₁₂₇	1	2	yes	—
207	C₄	—	2	2	no	—
208	C₄	—	8	4	no	—
209	C₄	—	16	2	no	—
210	C₄	#381×C₁₂₇	1	1	yes	characteristic
211	C₄	#419×C₁₂₇	1	1	yes	characteristic
212	C₄	#483×C₁₂₇	1	2	yes	—
213	C₂	—	2	4	no	—
214	C₂	C₁₂₇×#1823	1	1	yes	characteristic
215	C₂	C₂×#357×C₁₂₇	1	1	yes	characteristic
216	C₂	C₁₂₇×#2141	1	1	yes	characteristic
217	C₂	C₁₂₇×#2232	1	2	yes	—
218	C₂	C₁₂₇×#2235	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
4. C₁₂₇×#1380	C₂	semidirect
10. C₂×#107×C₁₂₇	C₄	semidirect
103. #12866	C₁₂₇	direct (N × H)
113. #1380	C₂×C₁₂₇	semidirect
114. C₁₂₇	#12866	direct (N × H)
120. C₂×#107	C₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#107×C₁₂₇	2
C₂×C₂²⋊C₁₆×C₁₂₇	2
C₁₂₇×#1808	2
C₁₂₇×#1380	2
#12866	127

Subgroups — C127×#12866 #44576539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#12866 #44576539