Subgroups — C₁₂₇×#12035 #44575708

Order 32512.

214 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	C₁₂₇×#2162	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#1568	C₂	1	1	yes	maximal, characteristic
3	C₂×#335×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1372	C₂	1	2	yes	maximal, complemented
5	C₁₂₇×#1371	C₂	1	2	yes	maximal, complemented
6	#432×C₁₂₇	—	2	1	no	—
7	C₂×C₄⋊C₈×C₁₂₇	—	2	1	no	—
8	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₄⋊C₁₆×C₁₂₇	C₄	1	2	yes	—
10	C₂²⋊C₁₆×C₁₂₇	C₄	1	2	yes	—
11	C₄×C₁₆×C₁₂₇	C₂²	1	2	yes	—
12	#335×C₁₂₇	—	2	1	no	—
13	#335×C₁₂₇	C₂²	1	2	yes	—
14	#335×C₁₂₇	C₂²	1	1	yes	characteristic
15	#335×C₁₂₇	C₂²	1	1	yes	characteristic
16	#334×C₁₂₇	—	2	1	no	—
17	C₂²×C₈×C₁₂₇	—	4	1	no	—
18	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
20	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
22	C₄⋊C₈×C₁₂₇	—	2	1	no	—
23	C₄⋊C₈×C₁₂₇	—	2	1	no	—
24	C₄⋊C₈×C₁₂₇	—	2	2	no	—
25	C₄⋊C₈×C₁₂₇	—	4	1	no	—
26	C₂²⋊C₈×C₁₂₇	—	4	1	no	—
27	#97×C₁₂₇	—	4	1	no	—
28	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
30	C₄×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
31	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
34	C₂²×C₄×C₁₂₇	Q₁₆	1	1	yes	characteristic
35	C₂×C₈×C₁₂₇	—	2	2	no	—
36	C₂×C₈×C₁₂₇	—	4	1	no	—
37	C₂×C₈×C₁₂₇	—	4	1	no	—
38	C₂×C₈×C₁₂₇	—	4	1	no	—
39	C₂×C₈×C₁₂₇	—	4	2	no	—
40	C₂×C₈×C₁₂₇	—	8	2	no	—
41	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
45	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₁₂₇	D₈	1	1	yes	characteristic
47	C₄²×C₁₂₇	Q₁₆	1	1	yes	characteristic
48	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
49	C₁₆×C₁₂₇	—	2	4	no	—
50	C₂³×C₁₂₇	#107	1	1	yes	characteristic
51	C₂×C₄×C₁₂₇	—	2	2	no	—
52	C₂×C₄×C₁₂₇	—	2	1	no	—
53	C₂×C₄×C₁₂₇	—	2	1	no	—
54	C₂×C₄×C₁₂₇	—	2	1	no	—
55	C₂×C₄×C₁₂₇	—	2	1	no	—
56	C₂×C₄×C₁₂₇	#107	1	1	yes	characteristic
57	C₂×C₄×C₁₂₇	#107	1	1	yes	characteristic
58	C₂×C₄×C₁₂₇	#107	1	1	yes	characteristic
59	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
62	C₈×C₁₂₇	—	2	1	no	—
63	C₈×C₁₂₇	—	2	1	no	—
64	C₈×C₁₂₇	—	2	1	no	—
65	C₈×C₁₂₇	—	8	1	no	—
66	C₈×C₁₂₇	—	8	2	no	—
67	C₈×C₁₂₇	—	8	1	no	—
68	C₈×C₁₂₇	#131	1	2	yes	—
69	C₂²×C₁₂₇	—	2	1	no	—
70	C₂²×C₁₂₇	—	2	1	no	—
71	C₂²×C₁₂₇	#368	1	2	yes	—
72	C₂²×C₁₂₇	C₂×#107	1	1	yes	characteristic
73	C₄×C₁₂₇	—	2	1	no	—
74	C₄×C₁₂₇	—	2	1	no	—
75	C₄×C₁₂₇	—	2	1	no	—
76	C₄×C₁₂₇	—	2	1	no	—
77	C₄×C₁₂₇	#424	1	1	yes	characteristic
78	C₄×C₁₂₇	#430	1	1	yes	characteristic
79	C₄×C₁₂₇	#509	1	1	yes	characteristic
80	C₄×C₁₂₇	#510	1	1	yes	characteristic
81	#12035	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂×C₁₂₇	—	2	2	no	—
83	C₂×C₁₂₇	#1571	1	1	yes	characteristic
84	C₂×C₁₂₇	#2159	1	1	yes	characteristic
85	C₂×C₁₂₇	C₂×#368	1	1	yes	characteristic
86	#2162	C₂×C₁₂₇	1	1	yes	characteristic
87	#1568	C₂×C₁₂₇	1	1	yes	characteristic
88	C₂×#335	C₂×C₁₂₇	1	1	yes	characteristic
89	#1372	C₂×C₁₂₇	1	2	yes	complemented
90	#1371	C₂×C₁₂₇	1	2	yes	complemented
91	C₁₂₇	#12035	1	1	yes	characteristic, direct product factor, Sylow
92	#432	—	2	1	no	—
93	C₂×C₄⋊C₈	—	2	1	no	—
94	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
95	C₄⋊C₁₆	C₄×C₁₂₇	1	2	yes	—
96	C₂²⋊C₁₆	C₄×C₁₂₇	1	2	yes	—
97	C₄×C₁₆	C₂²×C₁₂₇	1	2	yes	—
98	#335	—	2	1	no	—
99	#335	C₂²×C₁₂₇	1	2	yes	—
100	#335	C₂²×C₁₂₇	1	1	yes	characteristic
101	#335	C₂²×C₁₂₇	1	1	yes	characteristic
102	#334	—	2	1	no	—
103	C₂²×C₈	—	4	1	no	—
104	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
105	C₂²×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
106	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
107	C₂×C₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
108	C₄⋊C₈	—	2	1	no	—
109	C₄⋊C₈	—	2	1	no	—
110	C₄⋊C₈	—	2	2	no	—
111	C₄⋊C₈	—	4	1	no	—
112	C₂²⋊C₈	—	4	1	no	—
113	#97	—	4	1	no	—
114	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
115	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
116	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
117	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
118	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
119	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
120	C₂²×C₄	Q₁₆×C₁₂₇	1	1	yes	characteristic
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	4	1	no	—
123	C₂×C₈	—	4	1	no	—
124	C₂×C₈	—	4	1	no	—
125	C₂×C₈	—	4	2	no	—
126	C₂×C₈	—	8	2	no	—
127	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
128	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
129	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
130	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
131	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
132	C₄²	D₈×C₁₂₇	1	1	yes	characteristic
133	C₄²	Q₁₆×C₁₂₇	1	1	yes	characteristic
134	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
135	C₁₆	—	2	4	no	—
136	C₂³	#107×C₁₂₇	1	1	yes	characteristic
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	#107×C₁₂₇	1	1	yes	characteristic
143	C₂×C₄	#107×C₁₂₇	1	1	yes	characteristic
144	C₂×C₄	#107×C₁₂₇	1	1	yes	characteristic
145	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
146	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
147	C₂×C₄	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
148	C₈	—	2	1	no	—
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	—	8	2	no	—
152	C₈	—	8	1	no	—
153	C₈	—	8	1	no	—
154	C₈	#131×C₁₂₇	1	2	yes	—
155	C₂²	—	2	1	no	—
156	C₂²	—	2	1	no	—
157	C₂²	#368×C₁₂₇	1	2	yes	—
158	C₂²	C₂×#107×C₁₂₇	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	#424×C₁₂₇	1	1	yes	characteristic
164	C₄	#430×C₁₂₇	1	1	yes	characteristic
165	C₄	#509×C₁₂₇	1	1	yes	characteristic
166	C₄	#510×C₁₂₇	1	1	yes	characteristic
167	C₂	—	2	2	no	—
168	C₂	C₁₂₇×#1571	1	1	yes	characteristic
169	C₂	C₁₂₇×#2159	1	1	yes	characteristic
170	C₂	C₂×#368×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂×#107
prime radical	C₂³×C₁₂₇	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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
4. C₁₂₇×#1372	C₂	semidirect
5. C₁₂₇×#1371	C₂	semidirect
81. #12035	C₁₂₇	direct (N × H)
89. #1372	C₂×C₁₂₇	semidirect
90. #1371	C₂×C₁₂₇	semidirect
91. C₁₂₇	#12035	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₂₇×#2162	2
C₁₂₇×#1568	2
C₂×#335×C₁₂₇	2
C₁₂₇×#1372	2
C₁₂₇×#1371	2
#12035	127

Subgroups — C127×#12035 #44575708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#12035 #44575708