Subgroups — C₁₂₇×#12887 #44576560

Order 32512.

286 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×M₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#1848	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1846	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1383	C₂	1	4	yes	maximal
5	C₂×M₃₂×C₁₂₇	C₂²	1	2	yes	—
6	C₂×#108×C₁₂₇	—	2	1	no	—
7	C₂×#108×C₁₂₇	C₂²	1	2	yes	—
8	C₂×#107×C₁₂₇	—	2	1	no	—
9	C₂×#106×C₁₂₇	C₂²	1	2	yes	—
10	#423×C₁₂₇	—	2	1	no	—
11	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	#382×C₁₂₇	—	2	1	no	—
13	#346×C₁₂₇	C₄	1	2	yes	complemented
14	C₄×C₁₆×C₁₂₇	C₄	1	2	yes	complemented
15	C₂×M₁₆×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×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
20	M₃₂×C₁₂₇	—	2	2	no	—
21	M₃₂×C₁₂₇	—	2	2	no	—
22	C₂×C₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
23	#108×C₁₂₇	—	2	4	no	—
24	#108×C₁₂₇	—	4	2	no	—
25	#107×C₁₂₇	—	4	2	no	—
26	#106×C₁₂₇	—	2	4	no	—
27	C₄⋊C₈×C₁₂₇	—	4	2	no	—
28	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
31	C₄×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
32	#95×C₁₂₇	—	4	2	no	—
33	C₂²×C₄×C₁₂₇	—	4	2	no	—
34	C₂²×C₄×C₁₂₇	SD₁₆	1	2	yes	—
35	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
36	M₁₆×C₁₂₇	—	4	2	no	—
37	M₁₆×C₁₂₇	—	4	2	no	—
38	C₂×C₈×C₁₂₇	—	4	2	no	—
39	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₂₇	D₈	1	2	yes	—
42	C₂×C₈×C₁₂₇	Q₁₆	1	2	yes	—
43	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₁₂₇	#41	1	2	yes	—
46	C₂×C₈×C₁₂₇	#41	1	2	yes	—
47	C₄⋊C₄×C₁₂₇	—	4	2	no	—
48	C₄⋊C₄×C₁₂₇	—	4	2	no	—
49	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₁₂₇	SD₁₆	1	2	yes	—
52	C₁₆×C₁₂₇	—	2	4	no	—
53	C₂³×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
54	C₂×C₄×C₁₂₇	—	4	2	no	—
55	C₂×C₄×C₁₂₇	—	8	4	no	—
56	C₂×C₄×C₁₂₇	—	8	2	no	—
57	C₂×C₄×C₁₂₇	#106	1	2	yes	—
58	C₂×C₄×C₁₂₇	#106	1	2	yes	—
59	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₂₇	#124	1	1	yes	characteristic
61	C₂×C₄×C₁₂₇	#124	1	1	yes	characteristic
62	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
65	C₈×C₁₂₇	—	8	2	no	—
66	C₈×C₁₂₇	#107	1	2	yes	—
67	C₈×C₁₂₇	#107	1	2	yes	—
68	C₈×C₁₂₇	#117	1	2	yes	—
69	C₈×C₁₂₇	#125	1	2	yes	—
70	C₂²×C₁₂₇	—	2	2	no	—
71	C₂²×C₁₂₇	C₂×#106	1	1	yes	characteristic
72	C₂²×C₁₂₇	#486	1	1	yes	characteristic
73	C₂²×C₁₂₇	#487	1	1	yes	characteristic
74	C₄×C₁₂₇	—	2	2	no	—
75	C₄×C₁₂₇	—	2	2	no	—
76	C₄×C₁₂₇	—	16	2	no	—
77	C₄×C₁₂₇	#382	1	1	yes	characteristic
78	C₄×C₁₂₇	C₂×#107	1	1	yes	characteristic
79	C₄×C₁₂₇	#486	1	1	yes	characteristic
80	C₄×C₁₂₇	#487	1	1	yes	characteristic
81	#12887	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂×C₁₂₇	—	2	1	no	—
83	C₂×C₁₂₇	—	2	1	no	—
84	C₂×C₁₂₇	#1834	1	1	yes	characteristic
85	C₂×C₁₂₇	#2155	1	2	yes	—
86	C₄×M₃₂	C₂×C₁₂₇	1	1	yes	characteristic
87	#1848	C₂×C₁₂₇	1	1	yes	characteristic
88	#1846	C₂×C₁₂₇	1	1	yes	characteristic
89	#1383	C₂×C₁₂₇	1	4	yes	—
90	C₁₂₇	#12887	1	1	yes	characteristic, direct product factor, Sylow
91	C₂×M₃₂	C₂²×C₁₂₇	1	2	yes	—
92	C₂×#108	—	2	1	no	—
93	C₂×#108	C₂²×C₁₂₇	1	2	yes	—
94	C₂×#107	—	2	1	no	—
95	C₂×#106	C₂²×C₁₂₇	1	2	yes	—
96	#423	—	2	1	no	—
97	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
98	#382	—	2	1	no	—
99	#346	C₄×C₁₂₇	1	2	yes	complemented
100	C₄×C₁₆	C₄×C₁₂₇	1	2	yes	complemented
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	C₂×C₄⋊C₄	—	2	2	no	—
105	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
106	M₃₂	—	2	2	no	—
107	M₃₂	—	2	2	no	—
108	C₂×C₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
109	#108	—	2	4	no	—
110	#108	—	4	2	no	—
111	#107	—	4	2	no	—
112	#106	—	2	4	no	—
113	C₄⋊C₈	—	4	2	no	—
114	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
115	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
116	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
117	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
118	#95	—	4	2	no	—
119	C₂²×C₄	—	4	2	no	—
120	C₂²×C₄	SD₁₆×C₁₂₇	1	2	yes	—
121	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
122	M₁₆	—	4	2	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	D₈×C₁₂₇	1	2	yes	—
128	C₂×C₈	Q₁₆×C₁₂₇	1	2	yes	—
129	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
130	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
131	C₂×C₈	#41×C₁₂₇	1	2	yes	—
132	C₂×C₈	#41×C₁₂₇	1	2	yes	—
133	C₄⋊C₄	—	4	2	no	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
136	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
137	C₄²	SD₁₆×C₁₂₇	1	2	yes	—
138	C₁₆	—	2	4	no	—
139	C₂³	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	8	4	no	—
142	C₂×C₄	—	8	2	no	—
143	C₂×C₄	#106×C₁₂₇	1	2	yes	—
144	C₂×C₄	#106×C₁₂₇	1	2	yes	—
145	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
146	C₂×C₄	#124×C₁₂₇	1	1	yes	characteristic
147	C₂×C₄	#124×C₁₂₇	1	1	yes	characteristic
148	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
149	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
150	C₂×C₄	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
151	C₈	—	8	2	no	—
152	C₈	#107×C₁₂₇	1	2	yes	—
153	C₈	#107×C₁₂₇	1	2	yes	—
154	C₈	#117×C₁₂₇	1	2	yes	—
155	C₈	#125×C₁₂₇	1	2	yes	—
156	C₂²	—	2	2	no	—
157	C₂²	C₂×#106×C₁₂₇	1	1	yes	characteristic
158	C₂²	#486×C₁₂₇	1	1	yes	characteristic
159	C₂²	#487×C₁₂₇	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	16	2	no	—
163	C₄	#382×C₁₂₇	1	1	yes	characteristic
164	C₄	C₂×#107×C₁₂₇	1	1	yes	characteristic
165	C₄	#486×C₁₂₇	1	1	yes	characteristic
166	C₄	#487×C₁₂₇	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	C₁₂₇×#1834	1	1	yes	characteristic
170	C₂	C₁₂₇×#2155	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₂₇	C₂×#106
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
prime radical	C₂³×C₁₂₇	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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. #346×C₁₂₇	C₄	semidirect
14. C₄×C₁₆×C₁₂₇	C₄	semidirect
81. #12887	C₁₂₇	direct (N × H)
90. C₁₂₇	#12887	direct (N × H)
99. #346	C₄×C₁₂₇	semidirect
100. 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
C₄×M₃₂×C₁₂₇	2
C₁₂₇×#1848	2
C₁₂₇×#1846	2
C₁₂₇×#1383	2
#12887	127

Subgroups — C127×#12887 #44576560

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#12887 #44576560