Subgroups — C₁₂₇×#13166 #44576839

Order 32512.

326 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#367×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#367×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	#367×C₁₂₇	—	2	2	no	—
13	#367×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	2	yes	—
38	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₁₂₇	—	2	1	no	—
40	C₂×C₈×C₁₂₇	—	2	1	no	—
41	C₂×C₈×C₁₂₇	Q₁₆	1	2	yes	—
42	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
47	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
48	C₄⋊C₄×C₁₂₇	—	2	4	no	—
49	C₄⋊C₄×C₁₂₇	—	4	2	no	—
50	C₄⋊C₄×C₁₂₇	—	4	2	no	—
51	C₄⋊C₄×C₁₂₇	—	4	2	no	—
52	C₄⋊C₄×C₁₂₇	—	8	4	no	—
53	C₄²×C₁₂₇	—	2	1	no	—
54	C₄²×C₁₂₇	—	2	1	no	—
55	C₁₆×C₁₂₇	—	2	2	no	—
56	C₁₆×C₁₂₇	C₄⋊C₄	1	4	yes	complemented
57	C₂³×C₁₂₇	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₁₂₇	—	2	2	no	—
59	C₂×C₄×C₁₂₇	—	2	2	no	—
60	C₂×C₄×C₁₂₇	—	4	4	no	—
61	C₂×C₄×C₁₂₇	—	8	2	no	—
62	C₂×C₄×C₁₂₇	—	8	2	no	—
63	C₂×C₄×C₁₂₇	—	8	2	no	—
64	C₂×C₄×C₁₂₇	—	8	2	no	—
65	C₂×C₄×C₁₂₇	—	8	2	no	—
66	C₂×C₄×C₁₂₇	SD₃₂	1	4	yes	—
67	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₁₂₇	#121	1	1	yes	characteristic
69	C₂×C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×C₄×C₁₂₇	#135	1	1	yes	characteristic
72	C₂×C₄×C₁₂₇	#135	1	1	yes	characteristic
73	C₈×C₁₂₇	—	2	2	no	—
74	C₈×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₈×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
76	C₈×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
77	C₈×C₁₂₇	#125	1	1	yes	characteristic
78	C₂²×C₁₂₇	C₄×D₈	1	1	yes	characteristic
79	C₂²×C₁₂₇	#466	1	1	yes	characteristic
80	C₂²×C₁₂₇	#500	1	1	yes	characteristic
81	C₂²×C₁₂₇	C₂×SD₃₂	1	2	yes	—
82	C₂²×C₁₂₇	#508	1	2	yes	—
83	C₄×C₁₂₇	—	2	4	no	—
84	C₄×C₁₂₇	—	8	4	no	—
85	C₄×C₁₂₇	—	16	2	no	—
86	C₄×C₁₂₇	#389	1	1	yes	characteristic
87	C₄×C₁₂₇	C₄×Q₁₆	1	1	yes	characteristic
88	C₄×C₁₂₇	#467	1	1	yes	characteristic
89	C₄×C₁₂₇	#499	1	1	yes	characteristic
90	#13166	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂×C₁₂₇	#1942	1	1	yes	characteristic
92	C₂×C₁₂₇	C₄×SD₃₂	1	2	yes	—
93	C₂×C₁₂₇	#2217	1	2	yes	—
94	C₂×C₁₂₇	#2253	1	1	yes	characteristic
95	C₂×C₁₂₇	#2254	1	1	yes	characteristic
96	C₂×#367	C₂×C₁₂₇	1	1	yes	characteristic
97	C₂×#367	C₂×C₁₂₇	1	1	yes	characteristic
98	C₂×C₄×C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
99	#1942	C₂×C₁₂₇	1	2	yes	—
100	#1380	C₂×C₁₂₇	1	2	yes	—
101	C₁₂₇	#13166	1	1	yes	characteristic, direct product factor, Sylow
102	C₂²×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
103	C₂²×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
104	C₂×#107	C₂²×C₁₂₇	1	2	yes	—
105	C₂×#107	C₂²×C₁₂₇	1	2	yes	—
106	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
107	#389	—	2	2	no	—
108	#367	—	2	2	no	—
109	#367	C₄×C₁₂₇	1	4	yes	complemented
110	C₄×C₁₆	—	2	2	no	—
111	#340	—	2	2	no	—
112	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
113	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	4	2	no	—
117	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
118	C₂×C₁₆	—	2	1	no	—
119	C₂×C₁₆	—	2	1	no	—
120	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
121	C₂×C₁₆	D₄×C₁₂₇	1	2	yes	—
122	C₂×C₁₆	Q₈×C₁₂₇	1	2	yes	—
123	C₂×C₁₆	Q₈×C₁₂₇	1	2	yes	—
124	#107	—	2	2	no	—
125	#107	—	2	2	no	—
126	#107	—	2	2	no	—
127	#107	C₂×C₄×C₁₂₇	1	4	yes	—
128	C₄×C₈	—	2	1	no	—
129	C₄×C₈	—	2	1	no	—
130	#95	—	4	2	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	4	2	no	—
133	C₂²×C₄	D₈×C₁₂₇	1	2	yes	—
134	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	Q₁₆×C₁₂₇	1	2	yes	—
138	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
139	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
140	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
141	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
142	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
143	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
144	C₄⋊C₄	—	2	4	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	8	4	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	2	1	no	—
151	C₁₆	—	2	2	no	—
152	C₁₆	C₄⋊C₄×C₁₂₇	1	4	yes	complemented
153	C₂³	C₂×D₈×C₁₂₇	1	1	yes	characteristic
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	4	4	no	—
157	C₂×C₄	—	8	2	no	—
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	SD₃₂×C₁₂₇	1	4	yes	—
163	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
164	C₂×C₄	#121×C₁₂₇	1	1	yes	characteristic
165	C₂×C₄	C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
166	C₂×C₄	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
167	C₂×C₄	#135×C₁₂₇	1	1	yes	characteristic
168	C₂×C₄	#135×C₁₂₇	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
171	C₈	C₄×Q₈×C₁₂₇	1	1	yes	characteristic
172	C₈	C₂²⋊Q₈×C₁₂₇	1	1	yes	characteristic
173	C₈	#125×C₁₂₇	1	1	yes	characteristic
174	C₂²	C₄×D₈×C₁₂₇	1	1	yes	characteristic
175	C₂²	#466×C₁₂₇	1	1	yes	characteristic
176	C₂²	#500×C₁₂₇	1	1	yes	characteristic
177	C₂²	C₂×SD₃₂×C₁₂₇	1	2	yes	—
178	C₂²	#508×C₁₂₇	1	2	yes	—
179	C₄	—	2	4	no	—
180	C₄	—	8	4	no	—
181	C₄	—	16	2	no	—
182	C₄	#389×C₁₂₇	1	1	yes	characteristic
183	C₄	C₄×Q₁₆×C₁₂₇	1	1	yes	characteristic
184	C₄	#467×C₁₂₇	1	1	yes	characteristic
185	C₄	#499×C₁₂₇	1	1	yes	characteristic
186	C₂	C₁₂₇×#1942	1	1	yes	characteristic
187	C₂	C₄×SD₃₂×C₁₂₇	1	2	yes	—
188	C₂	C₁₂₇×#2217	1	2	yes	—
189	C₂	C₁₂₇×#2253	1	1	yes	characteristic
190	C₂	C₁₂₇×#2254	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. #367×C₁₂₇	C₄	semidirect
56. C₁₆×C₁₂₇	C₄⋊C₄	semidirect
90. #13166	C₁₂₇	direct (N × H)
101. C₁₂₇	#13166	direct (N × H)
109. #367	C₄×C₁₂₇	semidirect
152. 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×#367×C₁₂₇	2
C₂×C₄×C₁₆×C₁₂₇	2
C₁₂₇×#1942	2
C₁₂₇×#1380	2
#13166	127

Subgroups — C127×#13166 #44576839

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#13166 #44576839