Subgroups — C₁₂₇×#12033 #44575706

Order 32512.

222 conjugacy classes of subgroups.

The subgroup lattice has 164 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×#1567	C₂	1	1	yes	maximal, characteristic
3	C₂×#335×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1372	C₂	1	4	yes	maximal, complemented
5	#423×C₁₂₇	—	2	1	no	—
6	C₂×C₄⋊C₈×C₁₂₇	—	2	1	no	—
7	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₄⋊C₁₆×C₁₂₇	C₄	1	2	yes	—
9	C₂²⋊C₁₆×C₁₂₇	C₄	1	2	yes	—
10	C₄×C₁₆×C₁₂₇	C₂²	1	2	yes	—
11	#335×C₁₂₇	—	2	1	no	—
12	#335×C₁₂₇	—	2	1	no	—
13	#335×C₁₂₇	C₂²	1	2	yes	—
14	#335×C₁₂₇	C₂²	1	2	yes	—
15	C₂×M₁₆×C₁₂₇	—	4	1	no	—
16	C₂²×C₈×C₁₂₇	—	4	1	no	—
17	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
19	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
20	C₂×C₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
21	C₄⋊C₈×C₁₂₇	—	2	2	no	—
22	C₄⋊C₈×C₁₂₇	—	2	2	no	—
23	C₄⋊C₈×C₁₂₇	—	4	1	no	—
24	C₄⋊C₈×C₁₂₇	—	4	1	no	—
25	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
27	C₄×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
28	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
31	C₂²×C₄×C₁₂₇	Q₁₆	1	1	yes	characteristic
32	M₁₆×C₁₂₇	—	8	2	no	—
33	C₂×C₈×C₁₂₇	—	2	2	no	—
34	C₂×C₈×C₁₂₇	—	4	1	no	—
35	C₂×C₈×C₁₂₇	—	4	2	no	—
36	C₂×C₈×C₁₂₇	—	4	2	no	—
37	C₂×C₈×C₁₂₇	—	8	2	no	—
38	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
42	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₁₂₇	D₈	1	1	yes	characteristic
44	C₄²×C₁₂₇	Q₁₆	1	1	yes	characteristic
45	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
46	C₁₆×C₁₂₇	—	2	4	no	—
47	C₂³×C₁₂₇	#107	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	—	2	1	no	—
49	C₂×C₄×C₁₂₇	—	2	1	no	—
50	C₂×C₄×C₁₂₇	—	4	1	no	—
51	C₂×C₄×C₁₂₇	#107	1	1	yes	characteristic
52	C₂×C₄×C₁₂₇	#107	1	1	yes	characteristic
53	C₂×C₄×C₁₂₇	#107	1	1	yes	characteristic
54	C₂×C₄×C₁₂₇	SD₃₂	1	2	yes	—
55	C₂×C₄×C₁₂₇	SD₃₂	1	2	yes	—
56	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
59	C₈×C₁₂₇	—	2	1	no	—
60	C₈×C₁₂₇	—	2	1	no	—
61	C₈×C₁₂₇	—	2	1	no	—
62	C₈×C₁₂₇	—	8	2	no	—
63	C₈×C₁₂₇	—	8	2	no	—
64	C₈×C₁₂₇	#131	1	2	yes	—
65	C₂²×C₁₂₇	—	2	1	no	—
66	C₂²×C₁₂₇	—	2	1	no	—
67	C₂²×C₁₂₇	#367	1	2	yes	—
68	C₂²×C₁₂₇	C₂×#107	1	1	yes	characteristic
69	C₄×C₁₂₇	—	2	1	no	—
70	C₄×C₁₂₇	—	2	1	no	—
71	C₄×C₁₂₇	—	2	1	no	—
72	C₄×C₁₂₇	#367	1	2	yes	—
73	C₄×C₁₂₇	#424	1	1	yes	characteristic
74	C₄×C₁₂₇	#430	1	1	yes	characteristic
75	C₄×C₁₂₇	C₂×SD₃₂	1	1	yes	characteristic
76	C₄×C₁₂₇	C₂×SD₃₂	1	1	yes	characteristic
77	#12033	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂×C₁₂₇	—	2	2	no	—
79	C₂×C₁₂₇	#1571	1	1	yes	characteristic
80	C₂×C₁₂₇	C₂×#367	1	1	yes	characteristic
81	C₂×C₁₂₇	#2161	1	1	yes	characteristic
82	#2162	C₂×C₁₂₇	1	1	yes	characteristic
83	#1567	C₂×C₁₂₇	1	1	yes	characteristic
84	C₂×#335	C₂×C₁₂₇	1	1	yes	characteristic
85	#1372	C₂×C₁₂₇	1	4	yes	complemented
86	C₁₂₇	#12033	1	1	yes	characteristic, direct product factor, Sylow
87	#423	—	2	1	no	—
88	C₂×C₄⋊C₈	—	2	1	no	—
89	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
90	C₄⋊C₁₆	C₄×C₁₂₇	1	2	yes	—
91	C₂²⋊C₁₆	C₄×C₁₂₇	1	2	yes	—
92	C₄×C₁₆	C₂²×C₁₂₇	1	2	yes	—
93	#335	—	2	1	no	—
94	#335	—	2	1	no	—
95	#335	C₂²×C₁₂₇	1	2	yes	—
96	#335	C₂²×C₁₂₇	1	2	yes	—
97	C₂×M₁₆	—	4	1	no	—
98	C₂²×C₈	—	4	1	no	—
99	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
100	C₂²×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
101	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
102	C₂×C₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
103	C₄⋊C₈	—	2	2	no	—
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	4	1	no	—
106	C₄⋊C₈	—	4	1	no	—
107	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
108	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
109	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
110	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
111	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
112	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
113	C₂²×C₄	Q₁₆×C₁₂₇	1	1	yes	characteristic
114	M₁₆	—	8	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	4	1	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	8	2	no	—
120	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
121	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
122	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
123	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
124	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
125	C₄²	D₈×C₁₂₇	1	1	yes	characteristic
126	C₄²	Q₁₆×C₁₂₇	1	1	yes	characteristic
127	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
128	C₁₆	—	2	4	no	—
129	C₂³	#107×C₁₂₇	1	1	yes	characteristic
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	#107×C₁₂₇	1	1	yes	characteristic
134	C₂×C₄	#107×C₁₂₇	1	1	yes	characteristic
135	C₂×C₄	#107×C₁₂₇	1	1	yes	characteristic
136	C₂×C₄	SD₃₂×C₁₂₇	1	2	yes	—
137	C₂×C₄	SD₃₂×C₁₂₇	1	2	yes	—
138	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
139	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
140	C₂×C₄	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	8	2	no	—
145	C₈	—	8	2	no	—
146	C₈	#131×C₁₂₇	1	2	yes	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	#367×C₁₂₇	1	2	yes	—
150	C₂²	C₂×#107×C₁₂₇	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	#367×C₁₂₇	1	2	yes	—
155	C₄	#424×C₁₂₇	1	1	yes	characteristic
156	C₄	#430×C₁₂₇	1	1	yes	characteristic
157	C₄	C₂×SD₃₂×C₁₂₇	1	1	yes	characteristic
158	C₄	C₂×SD₃₂×C₁₂₇	1	1	yes	characteristic
159	C₂	—	2	2	no	—
160	C₂	C₁₂₇×#1571	1	1	yes	characteristic
161	C₂	C₂×#367×C₁₂₇	1	1	yes	characteristic
162	C₂	C₁₂₇×#2161	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
77. #12033	C₁₂₇	direct (N × H)
85. #1372	C₂×C₁₂₇	semidirect
86. C₁₂₇	#12033	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×#1567	2
C₂×#335×C₁₂₇	2
C₁₂₇×#1372	2
#12033	127

Subgroups — C127×#12033 #44575706

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#12033 #44575706