Subgroups — C₁₂₇×#12668 #44576341

Order 32512.

308 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#2155	C₂	1	2	yes	maximal
2	C₄×M₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1772	C₂	1	2	yes	maximal
4	C₁₂₇×#1385	C₂	1	1	yes	maximal, characteristic
5	C₁₂₇×#1384	C₂	1	1	yes	maximal, characteristic
6	C₂×M₃₂×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₂×M₃₂×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂×#108×C₁₂₇	C₂²	1	2	yes	—
9	#427×C₁₂₇	C₂²	1	2	yes	—
10	#405×C₁₂₇	—	2	2	no	—
11	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	#365×C₁₂₇	C₄	1	8	yes	complemented
13	#346×C₁₂₇	C₄	1	2	yes	complemented
14	C₄×C₁₆×C₁₂₇	C₄	1	2	yes	complemented
15	#339×C₁₂₇	—	2	2	no	—
16	C₂×M₁₆×C₁₂₇	—	2	2	no	—
17	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
19	#117×C₁₂₇	—	2	2	no	—
20	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
21	M₃₂×C₁₂₇	—	2	2	no	—
22	M₃₂×C₁₂₇	C₂×C₄	1	4	yes	—
23	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
24	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
25	#108×C₁₂₇	—	2	2	no	—
26	#108×C₁₂₇	C₂×C₄	1	4	yes	—
27	#107×C₁₂₇	—	2	2	no	—
28	#106×C₁₂₇	C₂×C₄	1	4	yes	—
29	#97×C₁₂₇	—	4	2	no	—
30	C₄×C₈×C₁₂₇	—	4	2	no	—
31	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
34	C₄×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
35	C₂²×C₄×C₁₂₇	—	2	1	no	—
36	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
37	M₁₆×C₁₂₇	—	2	4	no	—
38	M₁₆×C₁₂₇	—	4	2	no	—
39	C₂×C₈×C₁₂₇	—	4	2	no	—
40	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₂₇	SD₁₆	1	2	yes	—
43	C₂×C₈×C₁₂₇	SD₁₆	1	2	yes	—
44	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
46	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₁₂₇	#41	1	2	yes	—
48	C₄⋊C₄×C₁₂₇	—	2	4	no	—
49	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
50	C₄²×C₁₂₇	—	2	1	no	—
51	C₄²×C₁₂₇	—	4	2	no	—
52	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
54	C₁₆×C₁₂₇	—	2	2	no	—
55	C₁₆×C₁₂₇	C₄²	1	4	yes	—
56	C₂³×C₁₂₇	#135	1	1	yes	characteristic
57	C₂×C₄×C₁₂₇	—	2	1	no	—
58	C₂×C₄×C₁₂₇	—	2	1	no	—
59	C₂×C₄×C₁₂₇	—	4	2	no	—
60	C₂×C₄×C₁₂₇	—	4	4	no	—
61	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
63	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	2	yes	—
64	C₂×C₄×C₁₂₇	#135	1	1	yes	characteristic
65	C₈×C₁₂₇	—	4	4	no	—
66	C₈×C₁₂₇	#106	1	2	yes	—
67	C₈×C₁₂₇	#106	1	1	yes	characteristic
68	C₈×C₁₂₇	#106	1	1	yes	characteristic
69	C₈×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
70	C₈×C₁₂₇	#117	1	1	yes	characteristic
71	C₈×C₁₂₇	C₄×Q₈	1	2	yes	—
72	C₂²×C₁₂₇	—	4	1	no	—
73	C₂²×C₁₂₇	#427	1	1	yes	characteristic
74	C₂²×C₁₂₇	C₄×SD₁₆	1	2	yes	—
75	C₄×C₁₂₇	—	2	2	no	—
76	C₄×C₁₂₇	—	4	1	no	—
77	C₄×C₁₂₇	—	8	4	no	—
78	C₄×C₁₂₇	C₄×C₄⋊C₄	1	1	yes	characteristic
79	C₄×C₁₂₇	C₂×#106	1	1	yes	characteristic
80	C₄×C₁₂₇	C₄×SD₁₆	1	2	yes	—
81	#12668	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂×C₁₂₇	—	2	1	no	—
83	C₂×C₁₂₇	—	2	2	no	—
84	C₂×C₁₂₇	C₄×#106	1	1	yes	characteristic
85	#2155	C₂×C₁₂₇	1	2	yes	—
86	C₄×M₃₂	C₂×C₁₂₇	1	1	yes	characteristic
87	#1772	C₂×C₁₂₇	1	2	yes	—
88	#1385	C₂×C₁₂₇	1	1	yes	characteristic
89	#1384	C₂×C₁₂₇	1	1	yes	characteristic
90	C₁₂₇	#12668	1	1	yes	characteristic, direct product factor, Sylow
91	C₂×M₃₂	C₂²×C₁₂₇	1	1	yes	characteristic
92	C₂×M₃₂	C₂²×C₁₂₇	1	1	yes	characteristic
93	C₂×#108	C₂²×C₁₂₇	1	2	yes	—
94	#427	C₂²×C₁₂₇	1	2	yes	—
95	#405	—	2	2	no	—
96	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
97	#365	C₄×C₁₂₇	1	8	yes	complemented
98	#346	C₄×C₁₂₇	1	2	yes	complemented
99	C₄×C₁₆	C₄×C₁₂₇	1	2	yes	complemented
100	#339	—	2	2	no	—
101	C₂×M₁₆	—	2	2	no	—
102	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
103	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
104	#117	—	2	2	no	—
105	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
106	M₃₂	—	2	2	no	—
107	M₃₂	C₂×C₄×C₁₂₇	1	4	yes	—
108	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
109	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
110	#108	—	2	2	no	—
111	#108	C₂×C₄×C₁₂₇	1	4	yes	—
112	#107	—	2	2	no	—
113	#106	C₂×C₄×C₁₂₇	1	4	yes	—
114	#97	—	4	2	no	—
115	C₄×C₈	—	4	2	no	—
116	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
117	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
118	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
119	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
122	M₁₆	—	2	4	no	—
123	M₁₆	—	4	2	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
126	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
127	C₂×C₈	SD₁₆×C₁₂₇	1	2	yes	—
128	C₂×C₈	SD₁₆×C₁₂₇	1	2	yes	—
129	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
130	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
131	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
132	C₂×C₈	#41×C₁₂₇	1	2	yes	—
133	C₄⋊C₄	—	2	4	no	—
134	C₂²⋊C₄	—	4	2	no	—
135	C₄²	—	2	1	no	—
136	C₄²	—	4	2	no	—
137	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
138	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
139	C₁₆	—	2	2	no	—
140	C₁₆	C₄²×C₁₂₇	1	4	yes	—
141	C₂³	#135×C₁₂₇	1	1	yes	characteristic
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	4	4	no	—
146	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
147	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
148	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	2	yes	—
149	C₂×C₄	#135×C₁₂₇	1	1	yes	characteristic
150	C₈	—	4	4	no	—
151	C₈	#106×C₁₂₇	1	2	yes	—
152	C₈	#106×C₁₂₇	1	1	yes	characteristic
153	C₈	#106×C₁₂₇	1	1	yes	characteristic
154	C₈	C₂×C₄²×C₁₂₇	1	1	yes	characteristic
155	C₈	#117×C₁₂₇	1	1	yes	characteristic
156	C₈	C₄×Q₈×C₁₂₇	1	2	yes	—
157	C₂²	—	4	1	no	—
158	C₂²	#427×C₁₂₇	1	1	yes	characteristic
159	C₂²	C₄×SD₁₆×C₁₂₇	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	4	1	no	—
162	C₄	—	8	4	no	—
163	C₄	C₄×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
164	C₄	C₂×#106×C₁₂₇	1	1	yes	characteristic
165	C₄	C₄×SD₁₆×C₁₂₇	1	2	yes	—
166	C₂	—	2	1	no	—
167	C₂	—	2	2	no	—
168	C₂	C₄×#106×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₂₇	#106
commutator	C₈	C₂×C₄²×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂×C₁₂₇	C₄×#106
prime radical	C₂³×C₁₂₇	#135
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₂₇ ◁ C₂²×C₈×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
12. #365×C₁₂₇	C₄	semidirect
13. #346×C₁₂₇	C₄	semidirect
14. C₄×C₁₆×C₁₂₇	C₄	semidirect
81. #12668	C₁₂₇	direct (N × H)
90. C₁₂₇	#12668	direct (N × H)
97. #365	C₄×C₁₂₇	semidirect
98. #346	C₄×C₁₂₇	semidirect
99. C₄×C₁₆	C₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×#2155	2
C₄×M₃₂×C₁₂₇	2
C₁₂₇×#1772	2
C₁₂₇×#1385	2
C₁₂₇×#1384	2
#12668	127

Subgroups — C127×#12668 #44576341

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#12668 #44576341