Subgroups — C₁₂₇×#12890 #44576563

Order 32512.

318 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	C₂×#346×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#1846	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1845	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1380	C₂	1	4	yes	maximal
5	C₂²×C₁₆×C₁₂₇	C₂²	1	2	yes	—
6	C₂×#107×C₁₂₇	—	2	1	no	—
7	C₂×#107×C₁₂₇	C₂²	1	2	yes	—
8	C₂×#107×C₁₂₇	C₂²	1	2	yes	—
9	C₂×#106×C₁₂₇	—	2	1	no	—
10	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
11	#384×C₁₂₇	—	2	1	no	—
12	#382×C₁₂₇	—	2	1	no	—
13	#346×C₁₂₇	C₄	1	4	yes	complemented
14	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
17	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
18	C₂×C₄⋊C₄×C₁₂₇	—	4	2	no	—
19	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
20	C₂×C₁₆×C₁₂₇	—	2	2	no	—
21	C₂×C₁₆×C₁₂₇	—	2	2	no	—
22	C₂×C₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
23	#107×C₁₂₇	—	2	4	no	—
24	#107×C₁₂₇	—	2	4	no	—
25	#107×C₁₂₇	—	4	2	no	—
26	#106×C₁₂₇	—	4	2	no	—
27	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₁₂₇	Q₈	1	2	yes	—
29	#95×C₁₂₇	—	4	2	no	—
30	C₂²×C₄×C₁₂₇	—	4	2	no	—
31	C₂²×C₄×C₁₂₇	—	4	2	no	—
32	C₂²×C₄×C₁₂₇	D₈	1	2	yes	—
33	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₁₂₇	SD₁₆	1	2	yes	—
36	C₂×C₈×C₁₂₇	SD₁₆	1	2	yes	—
37	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₁₂₇	#41	1	2	yes	—
40	C₂×C₈×C₁₂₇	#41	1	2	yes	—
41	C₄⋊C₄×C₁₂₇	—	4	2	no	—
42	C₄⋊C₄×C₁₂₇	—	4	2	no	—
43	C₄⋊C₄×C₁₂₇	—	4	2	no	—
44	C₄⋊C₄×C₁₂₇	—	4	2	no	—
45	C₄⋊C₄×C₁₂₇	—	8	4	no	—
46	C₄²×C₁₂₇	C₄⋊C₄	1	2	yes	—
47	C₄²×C₁₂₇	Q₁₆	1	2	yes	—
48	C₁₆×C₁₂₇	—	2	4	no	—
49	C₂³×C₁₂₇	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₁₂₇	—	2	2	no	—
51	C₂×C₄×C₁₂₇	—	2	2	no	—
52	C₂×C₄×C₁₂₇	—	8	2	no	—
53	C₂×C₄×C₁₂₇	—	8	2	no	—
54	C₂×C₄×C₁₂₇	—	8	4	no	—
55	C₂×C₄×C₁₂₇	—	8	2	no	—
56	C₂×C₄×C₁₂₇	—	8	2	no	—
57	C₂×C₄×C₁₂₇	#107	1	4	yes	—
58	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₂₇	#124	1	1	yes	characteristic
60	C₂×C₄×C₁₂₇	#124	1	1	yes	characteristic
61	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
64	C₈×C₁₂₇	#106	1	4	yes	—
65	C₈×C₁₂₇	#117	1	2	yes	—
66	C₈×C₁₂₇	#125	1	2	yes	—
67	C₂²×C₁₂₇	C₂×#107	1	1	yes	characteristic
68	C₂²×C₁₂₇	#486	1	1	yes	characteristic
69	C₂²×C₁₂₇	#486	1	1	yes	characteristic
70	C₂²×C₁₂₇	#509	1	2	yes	—
71	C₂²×C₁₂₇	#510	1	2	yes	—
72	C₄×C₁₂₇	—	2	4	no	—
73	C₄×C₁₂₇	—	16	2	no	—
74	C₄×C₁₂₇	—	16	2	no	—
75	C₄×C₁₂₇	#382	1	1	yes	characteristic
76	C₄×C₁₂₇	C₂×#106	1	1	yes	characteristic
77	C₄×C₁₂₇	#487	1	1	yes	characteristic
78	C₄×C₁₂₇	#487	1	1	yes	characteristic
79	#12890	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂×C₁₂₇	#1834	1	1	yes	characteristic
81	C₂×C₁₂₇	#2159	1	2	yes	—
82	C₂×C₁₂₇	#2243	1	2	yes	—
83	C₂×C₁₂₇	#2244	1	1	yes	characteristic
84	C₂×C₁₂₇	#2245	1	1	yes	characteristic
85	C₂×#346	C₂×C₁₂₇	1	1	yes	characteristic
86	#1846	C₂×C₁₂₇	1	1	yes	characteristic
87	#1845	C₂×C₁₂₇	1	1	yes	characteristic
88	#1380	C₂×C₁₂₇	1	4	yes	—
89	C₁₂₇	#12890	1	1	yes	characteristic, direct product factor, Sylow
90	C₂²×C₁₆	C₂²×C₁₂₇	1	2	yes	—
91	C₂×#107	—	2	1	no	—
92	C₂×#107	C₂²×C₁₂₇	1	2	yes	—
93	C₂×#107	C₂²×C₁₂₇	1	2	yes	—
94	C₂×#106	—	2	1	no	—
95	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
96	#384	—	2	1	no	—
97	#382	—	2	1	no	—
98	#346	C₄×C₁₂₇	1	4	yes	complemented
99	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
100	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
101	C₂×C₄⋊C₄	—	2	2	no	—
102	C₂×C₄⋊C₄	—	2	2	no	—
103	C₂×C₄⋊C₄	—	4	2	no	—
104	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
105	C₂×C₁₆	—	2	2	no	—
106	C₂×C₁₆	—	2	2	no	—
107	C₂×C₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
108	#107	—	2	4	no	—
109	#107	—	2	4	no	—
110	#107	—	4	2	no	—
111	#106	—	4	2	no	—
112	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
113	C₄×C₈	Q₈×C₁₂₇	1	2	yes	—
114	#95	—	4	2	no	—
115	C₂²×C₄	—	4	2	no	—
116	C₂²×C₄	—	4	2	no	—
117	C₂²×C₄	D₈×C₁₂₇	1	2	yes	—
118	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
119	C₂×C₈	C₄⋊C₄×C₁₂₇	1	2	yes	—
120	C₂×C₈	SD₁₆×C₁₂₇	1	2	yes	—
121	C₂×C₈	SD₁₆×C₁₂₇	1	2	yes	—
122	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
123	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
124	C₂×C₈	#41×C₁₂₇	1	2	yes	—
125	C₂×C₈	#41×C₁₂₇	1	2	yes	—
126	C₄⋊C₄	—	4	2	no	—
127	C₄⋊C₄	—	4	2	no	—
128	C₄⋊C₄	—	4	2	no	—
129	C₄⋊C₄	—	4	2	no	—
130	C₄⋊C₄	—	8	4	no	—
131	C₄²	C₄⋊C₄×C₁₂₇	1	2	yes	—
132	C₄²	Q₁₆×C₁₂₇	1	2	yes	—
133	C₁₆	—	2	4	no	—
134	C₂³	C₂×D₈×C₁₂₇	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	8	4	no	—
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	—	8	2	no	—
140	C₂×C₄	—	8	2	no	—
141	C₂×C₄	—	8	2	no	—
142	C₂×C₄	#107×C₁₂₇	1	4	yes	—
143	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
144	C₂×C₄	#124×C₁₂₇	1	1	yes	characteristic
145	C₂×C₄	#124×C₁₂₇	1	1	yes	characteristic
146	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
147	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
148	C₂×C₄	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
149	C₈	#106×C₁₂₇	1	4	yes	—
150	C₈	#117×C₁₂₇	1	2	yes	—
151	C₈	#125×C₁₂₇	1	2	yes	—
152	C₂²	C₂×#107×C₁₂₇	1	1	yes	characteristic
153	C₂²	#486×C₁₂₇	1	1	yes	characteristic
154	C₂²	#486×C₁₂₇	1	1	yes	characteristic
155	C₂²	#509×C₁₂₇	1	2	yes	—
156	C₂²	#510×C₁₂₇	1	2	yes	—
157	C₄	—	2	4	no	—
158	C₄	—	16	2	no	—
159	C₄	—	16	2	no	—
160	C₄	#382×C₁₂₇	1	1	yes	characteristic
161	C₄	C₂×#106×C₁₂₇	1	1	yes	characteristic
162	C₄	#487×C₁₂₇	1	1	yes	characteristic
163	C₄	#487×C₁₂₇	1	1	yes	characteristic
164	C₂	C₁₂₇×#1834	1	1	yes	characteristic
165	C₂	C₁₂₇×#2159	1	2	yes	—
166	C₂	C₁₂₇×#2243	1	2	yes	—
167	C₂	C₁₂₇×#2244	1	1	yes	characteristic
168	C₂	C₁₂₇×#2245	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. #346×C₁₂₇	C₄	semidirect
79. #12890	C₁₂₇	direct (N × H)
89. C₁₂₇	#12890	direct (N × H)
98. #346	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×#346×C₁₂₇	2
C₁₂₇×#1846	2
C₁₂₇×#1845	2
C₁₂₇×#1380	2
#12890	127

Subgroups — C127×#12890 #44576563

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#12890 #44576563