Subgroups — C₁₂₇×#12903 #44576576

Order 32512.

334 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#365×C₁₂₇	C₂	1	4	yes	maximal
2	C₄×M₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1848	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1844	C₂	1	1	yes	maximal, characteristic
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×#106×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	#384×C₁₂₇	—	2	1	no	—
13	#365×C₁₂₇	—	2	8	no	—
14	#346×C₁₂₇	C₄	1	2	yes	complemented
15	C₄×C₁₆×C₁₂₇	C₄	1	2	yes	complemented
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	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	M₃₂×C₁₂₇	D₄	1	4	yes	—
23	M₃₂×C₁₂₇	Q₈	1	4	yes	—
24	C₂×C₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
25	#108×C₁₂₇	—	2	4	no	—
26	#108×C₁₂₇	—	4	2	no	—
27	#106×C₁₂₇	—	2	4	no	—
28	#106×C₁₂₇	—	4	2	no	—
29	C₄⋊C₈×C₁₂₇	—	4	2	no	—
30	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
34	C₂²×C₄×C₁₂₇	—	4	2	no	—
35	C₂²×C₄×C₁₂₇	SD₁₆	1	2	yes	—
36	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
37	M₁₆×C₁₂₇	—	4	2	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	C₂×D₄	1	2	yes	—
46	C₂×C₈×C₁₂₇	C₂×Q₈	1	2	yes	—
47	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
48	C₄⋊C₄×C₁₂₇	—	4	2	no	—
49	C₄⋊C₄×C₁₂₇	—	4	2	no	—
50	C₄⋊C₄×C₁₂₇	—	8	4	no	—
51	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₁₂₇	SD₁₆	1	2	yes	—
54	C₁₆×C₁₂₇	C₄⋊C₄	1	8	yes	complemented
55	C₂³×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
56	C₂×C₄×C₁₂₇	—	2	4	no	—
57	C₂×C₄×C₁₂₇	—	8	4	no	—
58	C₂×C₄×C₁₂₇	—	8	2	no	—
59	C₂×C₄×C₁₂₇	#106	1	2	yes	—
60	C₂×C₄×C₁₂₇	#106	1	2	yes	—
61	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
63	C₂×C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
67	C₈×C₁₂₇	—	8	2	no	—
68	C₈×C₁₂₇	#106	1	2	yes	—
69	C₈×C₁₂₇	#106	1	2	yes	—
70	C₈×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
71	C₈×C₁₂₇	C₄⋊Q₈	1	2	yes	—
72	C₂²×C₁₂₇	—	2	2	no	—
73	C₂²×C₁₂₇	C₂×#106	1	1	yes	characteristic
74	C₂²×C₁₂₇	#492	1	1	yes	characteristic
75	C₂²×C₁₂₇	#498	1	1	yes	characteristic
76	C₄×C₁₂₇	—	2	2	no	—
77	C₄×C₁₂₇	—	16	2	no	—
78	C₄×C₁₂₇	#365	1	4	yes	complemented
79	C₄×C₁₂₇	#384	1	1	yes	characteristic
80	C₄×C₁₂₇	C₂×#106	1	1	yes	characteristic
81	C₄×C₁₂₇	#492	1	1	yes	characteristic
82	C₄×C₁₂₇	#498	1	1	yes	characteristic
83	#12903	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂×C₁₂₇	—	2	1	no	—
85	C₂×C₁₂₇	—	2	1	no	—
86	C₂×C₁₂₇	#1844	1	1	yes	characteristic
87	C₂×C₁₂₇	C₂×#365	1	2	yes	—
88	C₂×#365	C₂×C₁₂₇	1	4	yes	—
89	C₄×M₃₂	C₂×C₁₂₇	1	1	yes	characteristic
90	#1848	C₂×C₁₂₇	1	1	yes	characteristic
91	#1844	C₂×C₁₂₇	1	1	yes	characteristic
92	C₁₂₇	#12903	1	1	yes	characteristic, direct product factor, Sylow
93	C₂×M₃₂	C₂²×C₁₂₇	1	2	yes	—
94	C₂×#108	—	2	1	no	—
95	C₂×#108	C₂²×C₁₂₇	1	2	yes	—
96	C₂×#106	—	2	1	no	—
97	C₂×#106	C₂²×C₁₂₇	1	2	yes	—
98	#423	—	2	1	no	—
99	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
100	#384	—	2	1	no	—
101	#365	—	2	8	no	—
102	#346	C₄×C₁₂₇	1	2	yes	complemented
103	C₄×C₁₆	C₄×C₁₂₇	1	2	yes	complemented
104	C₂×M₁₆	—	2	2	no	—
105	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
106	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
107	C₂×C₄⋊C₄	—	2	2	no	—
108	C₂×C₄⋊C₄	—	4	2	no	—
109	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
110	M₃₂	D₄×C₁₂₇	1	4	yes	—
111	M₃₂	Q₈×C₁₂₇	1	4	yes	—
112	C₂×C₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
113	#108	—	2	4	no	—
114	#108	—	4	2	no	—
115	#106	—	2	4	no	—
116	#106	—	4	2	no	—
117	C₄⋊C₈	—	4	2	no	—
118	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
119	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
120	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
121	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
122	C₂²×C₄	—	4	2	no	—
123	C₂²×C₄	SD₁₆×C₁₂₇	1	2	yes	—
124	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
125	M₁₆	—	4	2	no	—
126	M₁₆	—	4	2	no	—
127	C₂×C₈	—	4	2	no	—
128	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
129	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
130	C₂×C₈	SD₁₆×C₁₂₇	1	2	yes	—
131	C₂×C₈	SD₁₆×C₁₂₇	1	2	yes	—
132	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
133	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
134	C₂×C₈	C₂×Q₈×C₁₂₇	1	2	yes	—
135	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
136	C₄⋊C₄	—	4	2	no	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄⋊C₄	—	8	4	no	—
139	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
140	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
141	C₄²	SD₁₆×C₁₂₇	1	2	yes	—
142	C₁₆	C₄⋊C₄×C₁₂₇	1	8	yes	complemented
143	C₂³	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	8	4	no	—
146	C₂×C₄	—	8	2	no	—
147	C₂×C₄	#106×C₁₂₇	1	2	yes	—
148	C₂×C₄	#106×C₁₂₇	1	2	yes	—
149	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
150	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
151	C₂×C₄	C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
152	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
153	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
154	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
155	C₈	—	8	2	no	—
156	C₈	#106×C₁₂₇	1	2	yes	—
157	C₈	#106×C₁₂₇	1	2	yes	—
158	C₈	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
159	C₈	C₄⋊Q₈×C₁₂₇	1	2	yes	—
160	C₂²	—	2	2	no	—
161	C₂²	C₂×#106×C₁₂₇	1	1	yes	characteristic
162	C₂²	#492×C₁₂₇	1	1	yes	characteristic
163	C₂²	#498×C₁₂₇	1	1	yes	characteristic
164	C₄	—	2	2	no	—
165	C₄	—	16	2	no	—
166	C₄	#365×C₁₂₇	1	4	yes	complemented
167	C₄	#384×C₁₂₇	1	1	yes	characteristic
168	C₄	C₂×#106×C₁₂₇	1	1	yes	characteristic
169	C₄	#492×C₁₂₇	1	1	yes	characteristic
170	C₄	#498×C₁₂₇	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	C₁₂₇×#1844	1	1	yes	characteristic
174	C₂	C₂×#365×C₁₂₇	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
14. #346×C₁₂₇	C₄	semidirect
15. C₄×C₁₆×C₁₂₇	C₄	semidirect
54. C₁₆×C₁₂₇	C₄⋊C₄	semidirect
78. C₄×C₁₂₇	#365	semidirect
83. #12903	C₁₂₇	direct (N × H)
92. C₁₂₇	#12903	direct (N × H)
102. #346	C₄×C₁₂₇	semidirect
103. C₄×C₁₆	C₄×C₁₂₇	semidirect
142. C₁₆	C₄⋊C₄×C₁₂₇	semidirect
166. C₄	#365×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#365×C₁₂₇	2
C₄×M₃₂×C₁₂₇	2
C₁₂₇×#1848	2
C₁₂₇×#1844	2
#12903	127

Subgroups — C127×#12903 #44576576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#12903 #44576576