Subgroups — C₁₂₇×#13207 #44576880

Order 32512.

486 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#494×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#358×C₁₂₇	C₂	1	4	yes	maximal
3	C₂×#346×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1846	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₁₆×C₁₂₇	—	2	1	no	—
6	C₂²×Q₁₆×C₁₂₇	C₂²	1	2	yes	—
7	C₂×C₄⋊Q₈×C₁₂₇	—	2	1	no	—
8	C₂²×C₁₆×C₁₂₇	C₂²	1	2	yes	—
9	#494×C₁₂₇	—	2	2	no	—
10	#494×C₁₂₇	C₄	1	4	yes	complemented
11	C₂×#107×C₁₂₇	C₂²	1	2	yes	—
12	C₂×#106×C₁₂₇	—	2	1	no	—
13	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	#382×C₁₂₇	—	2	1	no	—
15	#358×C₁₂₇	—	2	8	no	—
16	#346×C₁₂₇	—	2	2	no	—
17	C₂²×Q₈×C₁₂₇	—	2	2	no	—
18	C₂×Q₁₆×C₁₂₇	—	2	4	no	—
19	C₂×Q₁₆×C₁₂₇	—	2	4	no	—
20	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
21	C₂×Q₁₆×C₁₂₇	—	2	4	no	—
22	C₂×Q₁₆×C₁₂₇	—	4	2	no	—
23	C₂×Q₁₆×C₁₂₇	—	4	2	no	—
24	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
27	C₄⋊Q₈×C₁₂₇	—	2	4	no	—
28	C₄⋊Q₈×C₁₂₇	—	4	2	no	—
29	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₂₇	—	4	2	no	—
31	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
32	C₂×C₁₆×C₁₂₇	—	2	2	no	—
33	C₂×C₁₆×C₁₂₇	—	2	2	no	—
34	C₂×C₁₆×C₁₂₇	D₄	1	4	yes	—
35	#107×C₁₂₇	—	2	4	no	—
36	#106×C₁₂₇	—	4	2	no	—
37	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
39	#95×C₁₂₇	—	4	2	no	—
40	C₂×Q₈×C₁₂₇	—	2	4	no	—
41	C₂×Q₈×C₁₂₇	—	4	2	no	—
42	C₂×Q₈×C₁₂₇	—	4	2	no	—
43	C₂×Q₈×C₁₂₇	—	4	2	no	—
44	C₂×Q₈×C₁₂₇	—	4	2	no	—
45	C₂×Q₈×C₁₂₇	—	4	2	no	—
46	C₂²×C₄×C₁₂₇	—	4	2	no	—
47	C₂²×C₄×C₁₂₇	—	4	2	no	—
48	C₂²×C₄×C₁₂₇	D₈	1	2	yes	—
49	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
50	Q₁₆×C₁₂₇	—	2	8	no	—
51	Q₁₆×C₁₂₇	—	4	8	no	—
52	Q₁₆×C₁₂₇	—	4	4	no	—
53	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₁₂₇	D₈	1	2	yes	—
55	C₂×C₈×C₁₂₇	SD₁₆	1	2	yes	—
56	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₁₂₇	#41	1	2	yes	—
60	C₄⋊C₄×C₁₂₇	—	4	8	no	—
61	C₄⋊C₄×C₁₂₇	—	4	2	no	—
62	C₄⋊C₄×C₁₂₇	—	4	2	no	—
63	C₄²×C₁₂₇	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₁₂₇	SD₁₆	1	2	yes	—
65	C₁₆×C₁₂₇	—	2	4	no	—
66	C₂³×C₁₂₇	C₂×D₈	1	1	yes	characteristic
67	Q₈×C₁₂₇	—	4	4	no	—
68	Q₈×C₁₂₇	—	4	4	no	—
69	Q₈×C₁₂₇	—	8	2	no	—
70	Q₈×C₁₂₇	—	8	2	no	—
71	C₂×C₄×C₁₂₇	—	2	4	no	—
72	C₂×C₄×C₁₂₇	—	4	4	no	—
73	C₂×C₄×C₁₂₇	—	8	2	no	—
74	C₂×C₄×C₁₂₇	—	8	2	no	—
75	C₂×C₄×C₁₂₇	—	8	2	no	—
76	C₂×C₄×C₁₂₇	—	8	4	no	—
77	C₂×C₄×C₁₂₇	#102	1	4	yes	—
78	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₂₇	#124	1	1	yes	characteristic
80	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
81	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
83	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
84	C₈×C₁₂₇	#102	1	4	yes	—
85	C₈×C₁₂₇	C₄×D₄	1	2	yes	—
86	C₈×C₁₂₇	#121	1	2	yes	—
87	C₂²×C₁₂₇	C₂×#102	1	1	yes	characteristic
88	C₂²×C₁₂₇	#486	1	1	yes	characteristic
89	C₂²×C₁₂₇	#493	1	1	yes	characteristic
90	C₂²×C₁₂₇	#510	1	4	yes	—
91	C₄×C₁₂₇	—	2	4	no	—
92	C₄×C₁₂₇	—	8	4	no	—
93	C₄×C₁₂₇	—	16	2	no	—
94	C₄×C₁₂₇	#390	1	1	yes	characteristic
95	C₄×C₁₂₇	C₂×#102	1	1	yes	characteristic
96	C₄×C₁₂₇	#486	1	1	yes	characteristic
97	C₄×C₁₂₇	#492	1	1	yes	characteristic
98	#13207	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂×C₁₂₇	#1957	1	1	yes	characteristic
100	C₂×C₁₂₇	#2141	1	2	yes	—
101	C₂×C₁₂₇	#2245	1	2	yes	—
102	C₂×C₁₂₇	#2251	1	2	yes	—
103	C₂×#494	C₂×C₁₂₇	1	1	yes	characteristic
104	C₂×#358	C₂×C₁₂₇	1	4	yes	—
105	C₂×#346	C₂×C₁₂₇	1	1	yes	characteristic
106	#1846	C₂×C₁₂₇	1	1	yes	characteristic
107	C₁₂₇	#13207	1	1	yes	characteristic, direct product factor, Sylow
108	C₂²×Q₁₆	—	2	1	no	—
109	C₂²×Q₁₆	C₂²×C₁₂₇	1	2	yes	—
110	C₂×C₄⋊Q₈	—	2	1	no	—
111	C₂²×C₁₆	C₂²×C₁₂₇	1	2	yes	—
112	#494	—	2	2	no	—
113	#494	C₄×C₁₂₇	1	4	yes	complemented
114	C₂×#107	C₂²×C₁₂₇	1	2	yes	—
115	C₂×#106	—	2	1	no	—
116	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
117	#382	—	2	1	no	—
118	#358	—	2	8	no	—
119	#346	—	2	2	no	—
120	C₂²×Q₈	—	2	2	no	—
121	C₂×Q₁₆	—	2	4	no	—
122	C₂×Q₁₆	—	2	4	no	—
123	C₂×Q₁₆	—	2	2	no	—
124	C₂×Q₁₆	—	2	4	no	—
125	C₂×Q₁₆	—	4	2	no	—
126	C₂×Q₁₆	—	4	2	no	—
127	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
128	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
129	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
130	C₄⋊Q₈	—	2	4	no	—
131	C₄⋊Q₈	—	4	2	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	—	4	2	no	—
134	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	—	2	2	no	—
137	C₂×C₁₆	D₄×C₁₂₇	1	4	yes	—
138	#107	—	2	4	no	—
139	#106	—	4	2	no	—
140	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
141	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
142	#95	—	4	2	no	—
143	C₂×Q₈	—	2	4	no	—
144	C₂×Q₈	—	4	2	no	—
145	C₂×Q₈	—	4	2	no	—
146	C₂×Q₈	—	4	2	no	—
147	C₂×Q₈	—	4	2	no	—
148	C₂×Q₈	—	4	2	no	—
149	C₂²×C₄	—	4	2	no	—
150	C₂²×C₄	—	4	2	no	—
151	C₂²×C₄	D₈×C₁₂₇	1	2	yes	—
152	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
153	Q₁₆	—	2	8	no	—
154	Q₁₆	—	4	8	no	—
155	Q₁₆	—	4	4	no	—
156	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	2	yes	—
157	C₂×C₈	D₈×C₁₂₇	1	2	yes	—
158	C₂×C₈	SD₁₆×C₁₂₇	1	2	yes	—
159	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
160	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
161	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
162	C₂×C₈	#41×C₁₂₇	1	2	yes	—
163	C₄⋊C₄	—	4	8	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄²	C₂²⋊C₄×C₁₂₇	1	2	yes	—
167	C₄²	SD₁₆×C₁₂₇	1	2	yes	—
168	C₁₆	—	2	4	no	—
169	C₂³	C₂×D₈×C₁₂₇	1	1	yes	characteristic
170	Q₈	—	4	4	no	—
171	Q₈	—	4	4	no	—
172	Q₈	—	8	2	no	—
173	Q₈	—	8	2	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	8	2	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	8	4	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	#102×C₁₂₇	1	4	yes	—
181	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
182	C₂×C₄	#124×C₁₂₇	1	1	yes	characteristic
183	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
184	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
185	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
186	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
187	C₈	#102×C₁₂₇	1	4	yes	—
188	C₈	C₄×D₄×C₁₂₇	1	2	yes	—
189	C₈	#121×C₁₂₇	1	2	yes	—
190	C₂²	C₂×#102×C₁₂₇	1	1	yes	characteristic
191	C₂²	#486×C₁₂₇	1	1	yes	characteristic
192	C₂²	#493×C₁₂₇	1	1	yes	characteristic
193	C₂²	#510×C₁₂₇	1	4	yes	—
194	C₄	—	2	4	no	—
195	C₄	—	8	4	no	—
196	C₄	—	16	2	no	—
197	C₄	#390×C₁₂₇	1	1	yes	characteristic
198	C₄	C₂×#102×C₁₂₇	1	1	yes	characteristic
199	C₄	#486×C₁₂₇	1	1	yes	characteristic
200	C₄	#492×C₁₂₇	1	1	yes	characteristic
201	C₂	C₁₂₇×#1957	1	1	yes	characteristic
202	C₂	C₁₂₇×#2141	1	2	yes	—
203	C₂	C₁₂₇×#2245	1	2	yes	—
204	C₂	C₁₂₇×#2251	1	2	yes	—

	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
10. #494×C₁₂₇	C₄	semidirect
98. #13207	C₁₂₇	direct (N × H)
107. C₁₂₇	#13207	direct (N × H)
113. #494	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×#494×C₁₂₇	2
C₂×#358×C₁₂₇	2
C₂×#346×C₁₂₇	2
C₁₂₇×#1846	2
#13207	127

Subgroups — C127×#13207 #44576880

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#13207 #44576880