Subgroups — C₁₂₇×#13165 #44576838

Order 32512.

326 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#366×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#366×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1942	C₂	1	2	yes	maximal
5	C₁₂₇×#1380	C₂	1	2	yes	maximal
6	C₂²×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂×#107×C₁₂₇	C₂²	1	2	yes	—
9	C₂×#107×C₁₂₇	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
11	#389×C₁₂₇	—	2	2	no	—
12	#366×C₁₂₇	—	2	2	no	—
13	#366×C₁₂₇	C₄	1	4	yes	complemented
14	C₄×C₁₆×C₁₂₇	—	2	2	no	—
15	#340×C₁₂₇	—	2	2	no	—
16	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
19	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
20	C₂×C₄⋊C₄×C₁₂₇	—	4	2	no	—
21	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₁₂₇	—	2	1	no	—
23	C₂×C₁₆×C₁₂₇	—	2	1	no	—
24	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
25	C₂×C₁₆×C₁₂₇	D₄	1	2	yes	—
26	C₂×C₁₆×C₁₂₇	Q₈	1	2	yes	—
27	C₂×C₁₆×C₁₂₇	Q₈	1	2	yes	—
28	#107×C₁₂₇	—	2	2	no	—
29	#107×C₁₂₇	—	2	2	no	—
30	#107×C₁₂₇	—	2	2	no	—
31	#107×C₁₂₇	C₂×C₄	1	4	yes	—
32	C₄×C₈×C₁₂₇	—	2	1	no	—
33	C₄×C₈×C₁₂₇	—	2	1	no	—
34	#95×C₁₂₇	—	4	2	no	—
35	C₂²×C₄×C₁₂₇	—	4	2	no	—
36	C₂²×C₄×C₁₂₇	—	4	2	no	—
37	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
38	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
39	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₁₂₇	—	2	1	no	—
41	C₂×C₈×C₁₂₇	—	2	1	no	—
42	C₂×C₈×C₁₂₇	Q₁₆	1	2	yes	—
43	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
48	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
49	C₄⋊C₄×C₁₂₇	—	2	4	no	—
50	C₄⋊C₄×C₁₂₇	—	4	2	no	—
51	C₄⋊C₄×C₁₂₇	—	4	2	no	—
52	C₄⋊C₄×C₁₂₇	—	4	2	no	—
53	C₄⋊C₄×C₁₂₇	—	8	4	no	—
54	C₄²×C₁₂₇	—	2	1	no	—
55	C₄²×C₁₂₇	—	2	1	no	—
56	C₁₆×C₁₂₇	—	2	2	no	—
57	C₁₆×C₁₂₇	C₄⋊C₄	1	4	yes	complemented
58	C₂³×C₁₂₇	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₁₂₇	—	2	1	no	—
60	C₂×C₄×C₁₂₇	—	2	1	no	—
61	C₂×C₄×C₁₂₇	—	2	1	no	—
62	C₂×C₄×C₁₂₇	—	2	1	no	—
63	C₂×C₄×C₁₂₇	—	4	4	no	—
64	C₂×C₄×C₁₂₇	—	8	2	no	—
65	C₂×C₄×C₁₂₇	—	8	2	no	—
66	C₂×C₄×C₁₂₇	—	8	2	no	—
67	C₂×C₄×C₁₂₇	—	8	2	no	—
68	C₂×C₄×C₁₂₇	—	8	2	no	—
69	C₂×C₄×C₁₂₇	D₁₆	1	2	yes	—
70	C₂×C₄×C₁₂₇	Q₃₂	1	2	yes	—
71	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₁₂₇	#121	1	1	yes	characteristic
73	C₂×C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×C₄×C₁₂₇	#135	1	1	yes	characteristic
76	C₂×C₄×C₁₂₇	#135	1	1	yes	characteristic
77	C₈×C₁₂₇	—	2	2	no	—
78	C₈×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₈×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
80	C₈×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
81	C₈×C₁₂₇	#125	1	1	yes	characteristic
82	C₂²×C₁₂₇	C₄×D₈	1	1	yes	characteristic
83	C₂²×C₁₂₇	#466	1	1	yes	characteristic
84	C₂²×C₁₂₇	#500	1	1	yes	characteristic
85	C₂²×C₁₂₇	C₂×D₁₆	1	1	yes	characteristic
86	C₂²×C₁₂₇	C₂×Q₃₂	1	1	yes	characteristic
87	C₂²×C₁₂₇	#508	1	1	yes	characteristic
88	C₂²×C₁₂₇	#508	1	1	yes	characteristic
89	C₄×C₁₂₇	—	2	2	no	—
90	C₄×C₁₂₇	—	2	2	no	—
91	C₄×C₁₂₇	—	8	4	no	—
92	C₄×C₁₂₇	—	16	2	no	—
93	C₄×C₁₂₇	#389	1	1	yes	characteristic
94	C₄×C₁₂₇	C₄×Q₁₆	1	1	yes	characteristic
95	C₄×C₁₂₇	#467	1	1	yes	characteristic
96	C₄×C₁₂₇	#499	1	1	yes	characteristic
97	#13165	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂×C₁₂₇	#1942	1	1	yes	characteristic
99	C₂×C₁₂₇	C₄×D₁₆	1	1	yes	characteristic
100	C₂×C₁₂₇	C₄×Q₃₂	1	1	yes	characteristic
101	C₂×C₁₂₇	#2215	1	1	yes	characteristic
102	C₂×C₁₂₇	#2216	1	1	yes	characteristic
103	C₂×C₁₂₇	#2252	1	1	yes	characteristic
104	C₂×C₁₂₇	#2253	1	1	yes	characteristic
105	C₂×#366	C₂×C₁₂₇	1	1	yes	characteristic
106	C₂×#366	C₂×C₁₂₇	1	1	yes	characteristic
107	C₂×C₄×C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
108	#1942	C₂×C₁₂₇	1	2	yes	—
109	#1380	C₂×C₁₂₇	1	2	yes	—
110	C₁₂₇	#13165	1	1	yes	characteristic, direct product factor, Sylow
111	C₂²×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
112	C₂²×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
113	C₂×#107	C₂²×C₁₂₇	1	2	yes	—
114	C₂×#107	C₂²×C₁₂₇	1	2	yes	—
115	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
116	#389	—	2	2	no	—
117	#366	—	2	2	no	—
118	#366	C₄×C₁₂₇	1	4	yes	complemented
119	C₄×C₁₆	—	2	2	no	—
120	#340	—	2	2	no	—
121	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
122	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄⋊C₄	—	4	2	no	—
126	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
127	C₂×C₁₆	—	2	1	no	—
128	C₂×C₁₆	—	2	1	no	—
129	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
130	C₂×C₁₆	D₄×C₁₂₇	1	2	yes	—
131	C₂×C₁₆	Q₈×C₁₂₇	1	2	yes	—
132	C₂×C₁₆	Q₈×C₁₂₇	1	2	yes	—
133	#107	—	2	2	no	—
134	#107	—	2	2	no	—
135	#107	—	2	2	no	—
136	#107	C₂×C₄×C₁₂₇	1	4	yes	—
137	C₄×C₈	—	2	1	no	—
138	C₄×C₈	—	2	1	no	—
139	#95	—	4	2	no	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	—	4	2	no	—
142	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
143	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
144	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	Q₁₆×C₁₂₇	1	2	yes	—
148	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
149	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
150	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
151	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
152	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
153	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	8	4	no	—
159	C₄²	—	2	1	no	—
160	C₄²	—	2	1	no	—
161	C₁₆	—	2	2	no	—
162	C₁₆	C₄⋊C₄×C₁₂₇	1	4	yes	complemented
163	C₂³	C₂×D₈×C₁₂₇	1	1	yes	characteristic
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	—	8	2	no	—
170	C₂×C₄	—	8	2	no	—
171	C₂×C₄	—	8	2	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	D₁₆×C₁₂₇	1	2	yes	—
175	C₂×C₄	Q₃₂×C₁₂₇	1	2	yes	—
176	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
177	C₂×C₄	#121×C₁₂₇	1	1	yes	characteristic
178	C₂×C₄	C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
179	C₂×C₄	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
180	C₂×C₄	#135×C₁₂₇	1	1	yes	characteristic
181	C₂×C₄	#135×C₁₂₇	1	1	yes	characteristic
182	C₈	—	2	2	no	—
183	C₈	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
184	C₈	C₄×Q₈×C₁₂₇	1	1	yes	characteristic
185	C₈	C₂²⋊Q₈×C₁₂₇	1	1	yes	characteristic
186	C₈	#125×C₁₂₇	1	1	yes	characteristic
187	C₂²	C₄×D₈×C₁₂₇	1	1	yes	characteristic
188	C₂²	#466×C₁₂₇	1	1	yes	characteristic
189	C₂²	#500×C₁₂₇	1	1	yes	characteristic
190	C₂²	C₂×D₁₆×C₁₂₇	1	1	yes	characteristic
191	C₂²	C₂×Q₃₂×C₁₂₇	1	1	yes	characteristic
192	C₂²	#508×C₁₂₇	1	1	yes	characteristic
193	C₂²	#508×C₁₂₇	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	—	8	4	no	—
197	C₄	—	16	2	no	—
198	C₄	#389×C₁₂₇	1	1	yes	characteristic
199	C₄	C₄×Q₁₆×C₁₂₇	1	1	yes	characteristic
200	C₄	#467×C₁₂₇	1	1	yes	characteristic
201	C₄	#499×C₁₂₇	1	1	yes	characteristic
202	C₂	C₁₂₇×#1942	1	1	yes	characteristic
203	C₂	C₄×D₁₆×C₁₂₇	1	1	yes	characteristic
204	C₂	C₄×Q₃₂×C₁₂₇	1	1	yes	characteristic
205	C₂	C₁₂₇×#2215	1	1	yes	characteristic
206	C₂	C₁₂₇×#2216	1	1	yes	characteristic
207	C₂	C₁₂₇×#2252	1	1	yes	characteristic
208	C₂	C₁₂₇×#2253	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₂₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
13. #366×C₁₂₇	C₄	semidirect
57. C₁₆×C₁₂₇	C₄⋊C₄	semidirect
97. #13165	C₁₂₇	direct (N × H)
110. C₁₂₇	#13165	direct (N × H)
118. #366	C₄×C₁₂₇	semidirect
162. C₁₆	C₄⋊C₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#366×C₁₂₇	2
C₂×C₄×C₁₆×C₁₂₇	2
C₁₂₇×#1942	2
C₁₂₇×#1380	2
#13165	127

Subgroups — C127×#13165 #44576838

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#13165 #44576838