Subgroups — C₁₂₇×#11408 #44575081

Order 32512.

290 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#2162	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₈×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1312	C₂	1	2	yes	maximal, complemented
5	C₁₂₇×#1311	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₆×C₁₂₇	C₄	1	2	yes	—
7	#431×C₁₂₇	C₄	1	1	yes	characteristic
8	C₄×M₁₆×C₁₂₇	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₄⋊C₁₆×C₁₂₇	C₄	1	2	yes	—
11	C₂²⋊C₁₆×C₁₂₇	C₄	1	2	yes	—
12	C₄×C₁₆×C₁₂₇	C₄	1	2	yes	—
13	C₄×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₁₂₇	C₂²	1	2	yes	—
15	C₄×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
16	#322×C₁₂₇	C₄	1	2	yes	—
17	C₈²×C₁₂₇	C₂²	1	2	yes	—
18	C₂×M₁₆×C₁₂₇	C₈	1	2	yes	—
19	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₁₂₇	—	2	4	no	—
23	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
24	C₂×C₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
25	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
26	C₄⋊C₈×C₁₂₇	C₈	1	2	yes	—
27	C₂²⋊C₈×C₁₂₇	C₈	1	2	yes	—
28	#97×C₁₂₇	C₈	1	2	yes	—
29	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₁₂₇	C₄²	1	1	yes	characteristic
36	C₂²×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
38	M₁₆×C₁₂₇	—	2	4	no	—
39	C₂×C₈×C₁₂₇	—	2	2	no	—
40	C₂×C₈×C₁₂₇	—	2	2	no	—
41	C₂×C₈×C₁₂₇	C₄²	1	1	yes	characteristic
42	C₂×C₈×C₁₂₇	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₁₂₇	C₂×C₈	1	4	yes	—
44	C₂×C₈×C₁₂₇	C₂×C₈	1	4	yes	—
45	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₁₂₇	C₄²	1	1	yes	characteristic
48	C₄²×C₁₂₇	C₂×C₈	1	1	yes	characteristic
49	C₄²×C₁₂₇	C₂×C₈	1	1	yes	characteristic
50	C₄²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
51	C₁₆×C₁₂₇	—	2	2	no	—
52	C₁₆×C₁₂₇	—	2	4	no	—
53	C₁₆×C₁₂₇	M₁₆	1	4	yes	complemented
54	C₂³×C₁₂₇	C₄×C₈	1	1	yes	characteristic
55	C₂×C₄×C₁₂₇	—	2	1	no	—
56	C₂×C₄×C₁₂₇	—	2	1	no	—
57	C₂×C₄×C₁₂₇	—	2	2	no	—
58	C₂×C₄×C₁₂₇	C₄×C₈	1	1	yes	characteristic
59	C₂×C₄×C₁₂₇	C₄×C₈	1	1	yes	characteristic
60	C₂×C₄×C₁₂₇	C₄×C₈	1	1	yes	characteristic
61	C₂×C₄×C₁₂₇	M₃₂	1	4	yes	—
62	C₂×C₄×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
65	C₈×C₁₂₇	—	2	1	no	—
66	C₈×C₁₂₇	—	2	1	no	—
67	C₈×C₁₂₇	—	2	4	no	—
68	C₈×C₁₂₇	—	2	4	no	—
69	C₈×C₁₂₇	C₂×M₁₆	1	2	yes	—
70	C₈×C₁₂₇	#131	1	2	yes	—
71	C₂²×C₁₂₇	—	2	1	no	—
72	C₂²×C₁₂₇	—	2	1	no	—
73	C₂²×C₁₂₇	#346	1	2	yes	—
74	C₂²×C₁₂₇	C₂×C₄×C₈	1	1	yes	characteristic
75	C₄×C₁₂₇	—	2	1	no	—
76	C₄×C₁₂₇	—	2	1	no	—
77	C₄×C₁₂₇	#346	1	2	yes	—
78	C₄×C₁₂₇	C₄×M₁₆	1	1	yes	characteristic
79	C₄×C₁₂₇	#405	1	1	yes	characteristic
80	C₄×C₁₂₇	C₂×M₃₂	1	2	yes	—
81	C₄×C₁₂₇	#504	1	2	yes	—
82	#11408	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂×C₁₂₇	—	2	2	no	—
84	C₂×C₁₂₇	C₈×M₁₆	1	1	yes	characteristic
85	C₂×C₁₂₇	C₂×#346	1	1	yes	characteristic
86	C₂×C₁₂₇	#2108	1	1	yes	characteristic
87	#2162	C₂×C₁₂₇	1	1	yes	characteristic
88	C₂×C₄×C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
89	C₈×M₁₆	C₂×C₁₂₇	1	1	yes	characteristic
90	#1312	C₂×C₁₂₇	1	2	yes	complemented
91	#1311	C₂×C₁₂₇	1	2	yes	complemented
92	C₁₂₇	#11408	1	1	yes	characteristic, direct product factor, Sylow
93	C₂²×C₁₆	C₄×C₁₂₇	1	2	yes	—
94	#431	C₄×C₁₂₇	1	1	yes	characteristic
95	C₄×M₁₆	C₄×C₁₂₇	1	1	yes	characteristic
96	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
97	C₄⋊C₁₆	C₄×C₁₂₇	1	2	yes	—
98	C₂²⋊C₁₆	C₄×C₁₂₇	1	2	yes	—
99	C₄×C₁₆	C₄×C₁₂₇	1	2	yes	—
100	C₄×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
101	C₄×C₁₆	C₂²×C₁₂₇	1	2	yes	—
102	C₄×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
103	#322	C₄×C₁₂₇	1	2	yes	—
104	C₈²	C₂²×C₁₂₇	1	2	yes	—
105	C₂×M₁₆	C₈×C₁₂₇	1	2	yes	—
106	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
107	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
108	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
109	C₂×C₁₆	—	2	4	no	—
110	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
111	C₂×C₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
112	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
113	C₄⋊C₈	C₈×C₁₂₇	1	2	yes	—
114	C₂²⋊C₈	C₈×C₁₂₇	1	2	yes	—
115	#97	C₈×C₁₂₇	1	2	yes	—
116	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
117	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
118	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
119	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
120	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
121	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
122	C₂²×C₄	C₄²×C₁₂₇	1	1	yes	characteristic
123	C₂²×C₄	C₂×C₈×C₁₂₇	1	1	yes	characteristic
124	C₂²×C₄	C₂×C₈×C₁₂₇	1	1	yes	characteristic
125	M₁₆	—	2	4	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	C₄²×C₁₂₇	1	1	yes	characteristic
129	C₂×C₈	C₄²×C₁₂₇	1	1	yes	characteristic
130	C₂×C₈	C₂×C₈×C₁₂₇	1	4	yes	—
131	C₂×C₈	C₂×C₈×C₁₂₇	1	4	yes	—
132	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
133	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
134	C₄²	C₄²×C₁₂₇	1	1	yes	characteristic
135	C₄²	C₂×C₈×C₁₂₇	1	1	yes	characteristic
136	C₄²	C₂×C₈×C₁₂₇	1	1	yes	characteristic
137	C₄²	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
138	C₁₆	—	2	2	no	—
139	C₁₆	—	2	4	no	—
140	C₁₆	M₁₆×C₁₂₇	1	4	yes	complemented
141	C₂³	C₄×C₈×C₁₂₇	1	1	yes	characteristic
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	C₄×C₈×C₁₂₇	1	1	yes	characteristic
146	C₂×C₄	C₄×C₈×C₁₂₇	1	1	yes	characteristic
147	C₂×C₄	C₄×C₈×C₁₂₇	1	1	yes	characteristic
148	C₂×C₄	M₃₂×C₁₂₇	1	4	yes	—
149	C₂×C₄	C₂×C₄²×C₁₂₇	1	1	yes	characteristic
150	C₂×C₄	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
151	C₂×C₄	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
152	C₈	—	2	4	no	—
153	C₈	—	2	4	no	—
154	C₈	—	2	1	no	—
155	C₈	—	2	1	no	—
156	C₈	C₂×M₁₆×C₁₂₇	1	2	yes	—
157	C₈	#131×C₁₂₇	1	2	yes	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	#346×C₁₂₇	1	2	yes	—
161	C₂²	C₂×C₄×C₈×C₁₂₇	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	#346×C₁₂₇	1	2	yes	—
165	C₄	C₄×M₁₆×C₁₂₇	1	1	yes	characteristic
166	C₄	#405×C₁₂₇	1	1	yes	characteristic
167	C₄	C₂×M₃₂×C₁₂₇	1	2	yes	—
168	C₄	#504×C₁₂₇	1	2	yes	—
169	C₂	—	2	2	no	—
170	C₂	C₈×M₁₆×C₁₂₇	1	1	yes	characteristic
171	C₂	C₂×#346×C₁₂₇	1	1	yes	characteristic
172	C₂	C₁₂₇×#2108	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₁₂₇×#1312	C₂	semidirect
5. C₁₂₇×#1311	C₂	semidirect
53. C₁₆×C₁₂₇	M₁₆	semidirect
82. #11408	C₁₂₇	direct (N × H)
90. #1312	C₂×C₁₂₇	semidirect
91. #1311	C₂×C₁₂₇	semidirect
92. C₁₂₇	#11408	direct (N × H)
140. C₁₆	M₁₆×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×#2162	2
C₂×C₄×C₁₆×C₁₂₇	2
C₈×M₁₆×C₁₂₇	2
C₁₂₇×#1312	2
C₁₂₇×#1311	2
#11408	127

Subgroups — C127×#11408 #44575081

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#11408 #44575081