Subgroups — C₁₂₇×#12915 #44576588

Order 32512.

374 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#3407	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#2151	C₂	1	2	yes	maximal, complemented
3	C₁₂₇×#2113	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1757	C₂	1	1	yes	maximal, characteristic
5	C₁₂₇×#1378	C₂	1	2	yes	maximal, complemented
6	C₂×#131×C₁₂₇	C₂²	1	1	yes	characteristic
7	#504×C₁₂₇	—	2	4	no	—
8	C₂×M₃₂×C₁₂₇	—	2	2	no	—
9	C₂×M₃₂×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂×M₃₂×C₁₂₇	C₂²	1	2	yes	—
11	C₂²×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
12	#413×C₁₂₇	C₄	1	1	yes	characteristic
13	#410×C₁₂₇	C₄	1	1	yes	characteristic
14	#405×C₁₂₇	C₂²	1	2	yes	—
15	C₄⋊C₁₆×C₁₂₇	—	2	4	no	—
16	C₂²⋊C₁₆×C₁₂₇	—	2	2	no	—
17	#344×C₁₂₇	C₄	1	2	yes	—
18	C₂×#41×C₁₂₇	C₂×C₄	1	1	yes	characteristic
19	#131×C₁₂₇	—	2	2	no	—
20	#131×C₁₂₇	—	2	2	no	—
21	C₂×M₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
22	C₂×M₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
23	C₂×M₁₆×C₁₂₇	Q₈	1	2	yes	—
24	C₂²×C₈×C₁₂₇	D₄	1	2	yes	—
25	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
26	#117×C₁₂₇	C₂×C₄	1	2	yes	—
27	M₃₂×C₁₂₇	—	2	4	no	—
28	M₃₂×C₁₂₇	—	2	2	no	—
29	M₃₂×C₁₂₇	—	2	2	no	—
30	M₃₂×C₁₂₇	—	2	2	no	—
31	C₂×C₁₆×C₁₂₇	—	2	4	no	—
32	C₂×C₁₆×C₁₂₇	—	2	1	no	—
33	C₂×C₁₆×C₁₂₇	—	2	2	no	—
34	C₂×C₁₆×C₁₂₇	—	2	1	no	—
35	C₂×C₁₆×C₁₂₇	D₄	1	4	yes	—
36	#101×C₁₂₇	—	2	1	no	—
37	#100×C₁₂₇	—	2	1	no	—
38	#99×C₁₂₇	C₈	1	4	yes	—
39	#97×C₁₂₇	—	2	2	no	—
40	C₄×C₈×C₁₂₇	—	2	2	no	—
41	#41×C₁₂₇	—	2	2	no	—
42	#41×C₁₂₇	—	2	2	no	—
43	C₂×Q₈×C₁₂₇	M₁₆	1	1	yes	characteristic
44	C₂×D₄×C₁₂₇	C₂×C₈	1	2	yes	—
45	C₂×D₄×C₁₂₇	M₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
48	M₁₆×C₁₂₇	—	2	2	no	—
49	M₁₆×C₁₂₇	—	2	4	no	—
50	M₁₆×C₁₂₇	—	2	2	no	—
51	M₁₆×C₁₂₇	—	2	2	no	—
52	C₂×C₈×C₁₂₇	—	2	2	no	—
53	C₂×C₈×C₁₂₇	—	2	2	no	—
54	C₂×C₈×C₁₂₇	—	2	2	no	—
55	C₂×C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
57	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₈×C₁₂₇	#41	1	2	yes	—
60	C₄⋊C₄×C₁₂₇	—	2	2	no	—
61	C₂²⋊C₄×C₁₂₇	C₂×C₈	1	4	yes	—
62	C₄²×C₁₂₇	—	2	2	no	—
63	C₁₆×C₁₂₇	—	2	4	no	—
64	C₁₆×C₁₂₇	—	4	1	no	—
65	C₁₆×C₁₂₇	—	4	1	no	—
66	C₂³×C₁₂₇	C₄⋊C₈	1	2	yes	—
67	C₂³×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
68	Q₈×C₁₂₇	—	2	2	no	—
69	D₄×C₁₂₇	—	2	4	no	—
70	D₄×C₁₂₇	—	2	2	no	—
71	C₂×C₄×C₁₂₇	—	2	2	no	—
72	C₂×C₄×C₁₂₇	—	2	2	no	—
73	C₂×C₄×C₁₂₇	—	2	4	no	—
74	C₂×C₄×C₁₂₇	C₄⋊C₈	1	2	yes	—
75	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
77	C₂×C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₂₇	#131	1	2	yes	—
79	C₈×C₁₂₇	—	2	2	no	—
80	C₈×C₁₂₇	—	4	2	no	—
81	C₈×C₁₂₇	#120	1	1	yes	characteristic
82	C₈×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
83	C₈×C₁₂₇	#123	1	1	yes	characteristic
84	C₂²×C₁₂₇	—	2	2	no	—
85	C₂²×C₁₂₇	—	4	1	no	—
86	C₂²×C₁₂₇	—	4	2	no	—
87	C₂²×C₁₂₇	C₂×C₄⋊C₈	1	1	yes	characteristic
88	C₂²×C₁₂₇	C₈×D₄	1	2	yes	—
89	C₄×C₁₂₇	—	2	2	no	—
90	C₄×C₁₂₇	—	4	4	no	—
91	C₄×C₁₂₇	#385	1	1	yes	characteristic
92	C₄×C₁₂₇	#424	1	1	yes	characteristic
93	C₄×C₁₂₇	#435	1	2	yes	—
94	#12915	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂×C₁₂₇	—	2	2	no	—
96	C₂×C₁₂₇	—	2	1	no	—
97	C₂×C₁₂₇	—	4	2	no	—
98	C₂×C₁₂₇	#1850	1	1	yes	characteristic
99	#3407	C₂×C₁₂₇	1	1	yes	characteristic
100	#2151	C₂×C₁₂₇	1	2	yes	complemented
101	#2113	C₂×C₁₂₇	1	1	yes	characteristic
102	#1757	C₂×C₁₂₇	1	1	yes	characteristic
103	#1378	C₂×C₁₂₇	1	2	yes	complemented
104	C₁₂₇	#12915	1	1	yes	characteristic, direct product factor, Sylow
105	C₂×#131	C₂²×C₁₂₇	1	1	yes	characteristic
106	#504	—	2	4	no	—
107	C₂×M₃₂	—	2	2	no	—
108	C₂×M₃₂	C₂²×C₁₂₇	1	1	yes	characteristic
109	C₂×M₃₂	C₂²×C₁₂₇	1	2	yes	—
110	C₂²×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
111	#413	C₄×C₁₂₇	1	1	yes	characteristic
112	#410	C₄×C₁₂₇	1	1	yes	characteristic
113	#405	C₂²×C₁₂₇	1	2	yes	—
114	C₄⋊C₁₆	—	2	4	no	—
115	C₂²⋊C₁₆	—	2	2	no	—
116	#344	C₄×C₁₂₇	1	2	yes	—
117	C₂×#41	C₂×C₄×C₁₂₇	1	1	yes	characteristic
118	#131	—	2	2	no	—
119	#131	—	2	2	no	—
120	C₂×M₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
121	C₂×M₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
122	C₂×M₁₆	Q₈×C₁₂₇	1	2	yes	—
123	C₂²×C₈	D₄×C₁₂₇	1	2	yes	—
124	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
125	#117	C₂×C₄×C₁₂₇	1	2	yes	—
126	M₃₂	—	2	4	no	—
127	M₃₂	—	2	2	no	—
128	M₃₂	—	2	2	no	—
129	M₃₂	—	2	2	no	—
130	C₂×C₁₆	—	2	4	no	—
131	C₂×C₁₆	—	2	1	no	—
132	C₂×C₁₆	—	2	1	no	—
133	C₂×C₁₆	—	2	2	no	—
134	C₂×C₁₆	D₄×C₁₂₇	1	4	yes	—
135	#101	—	2	1	no	—
136	#100	—	2	1	no	—
137	#99	C₈×C₁₂₇	1	4	yes	—
138	#97	—	2	2	no	—
139	C₄×C₈	—	2	2	no	—
140	#41	—	2	2	no	—
141	#41	—	2	2	no	—
142	C₂×Q₈	M₁₆×C₁₂₇	1	1	yes	characteristic
143	C₂×D₄	C₂×C₈×C₁₂₇	1	2	yes	—
144	C₂×D₄	M₁₆×C₁₂₇	1	1	yes	characteristic
145	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	2	yes	—
146	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
147	M₁₆	—	2	2	no	—
148	M₁₆	—	2	4	no	—
149	M₁₆	—	2	2	no	—
150	M₁₆	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	C₄⋊C₄×C₁₂₇	1	2	yes	—
155	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
156	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
157	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
158	C₂×C₈	#41×C₁₂₇	1	2	yes	—
159	C₄⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	C₂×C₈×C₁₂₇	1	4	yes	—
161	C₄²	—	2	2	no	—
162	C₁₆	—	2	4	no	—
163	C₁₆	—	4	1	no	—
164	C₁₆	—	4	1	no	—
165	C₂³	C₄⋊C₈×C₁₂₇	1	2	yes	—
166	C₂³	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
167	Q₈	—	2	2	no	—
168	D₄	—	2	4	no	—
169	D₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	C₄⋊C₈×C₁₂₇	1	2	yes	—
174	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
175	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
176	C₂×C₄	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
177	C₂×C₄	#131×C₁₂₇	1	2	yes	—
178	C₈	—	2	2	no	—
179	C₈	—	4	2	no	—
180	C₈	#120×C₁₂₇	1	1	yes	characteristic
181	C₈	C₂²⋊Q₈×C₁₂₇	1	2	yes	—
182	C₈	#123×C₁₂₇	1	1	yes	characteristic
183	C₂²	—	2	2	no	—
184	C₂²	—	4	1	no	—
185	C₂²	—	4	2	no	—
186	C₂²	C₂×C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
187	C₂²	C₈×D₄×C₁₂₇	1	2	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	4	4	no	—
190	C₄	#385×C₁₂₇	1	1	yes	characteristic
191	C₄	#424×C₁₂₇	1	1	yes	characteristic
192	C₄	#435×C₁₂₇	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	—	2	1	no	—
195	C₂	—	4	2	no	—
196	C₂	C₁₂₇×#1850	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₂₇	#120
commutator	C₂³	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂×C₁₂₇	#1850
prime radical	C₂×#41×C₁₂₇	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₂₇ ◁ C₂²×C₈×C₁₂₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₁₂₇×#2151	C₂	semidirect
5. C₁₂₇×#1378	C₂	semidirect
94. #12915	C₁₂₇	direct (N × H)
100. #2151	C₂×C₁₂₇	semidirect
103. #1378	C₂×C₁₂₇	semidirect
104. C₁₂₇	#12915	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×#3407	2
C₁₂₇×#2151	2
C₁₂₇×#2113	2
C₁₂₇×#1757	2
C₁₂₇×#1378	2
#12915	127

Subgroups — C127×#12915 #44576588

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#12915 #44576588