Subgroups — C₂×C₁₂₇×#1381 #44576231

Order 32512.

574 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#108×C₁₂₇	C₂	1	2	yes	maximal
3	C₁₂₇×#1381	C₂	1	4	yes	maximal, direct product factor
4	C₂²×M₁₆×C₁₂₇	—	2	2	no	—
5	C₂³×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
6	C₂²×C₁₆×C₁₂₇	—	2	4	no	—
7	C₂²×C₁₆×C₁₂₇	C₄	1	2	yes	—
8	C₂²×C₁₆×C₁₂₇	C₄	1	2	yes	—
9	C₂²×C₁₆×C₁₂₇	C₂²	1	2	yes	—
10	C₂×#108×C₁₂₇	—	2	2	no	—
11	C₂×#108×C₁₂₇	—	2	4	no	—
12	C₂×#108×C₁₂₇	C₄	1	8	yes	—
13	C₂×#108×C₁₂₇	C₂²	1	4	yes	—
14	C₂³×C₄×C₁₂₇	D₄	1	1	yes	characteristic
15	C₂×M₁₆×C₁₂₇	—	2	8	no	—
16	C₂×M₁₆×C₁₂₇	—	2	4	no	—
17	C₂×M₁₆×C₁₂₇	—	4	2	no	—
18	C₂×M₁₆×C₁₂₇	—	4	4	no	—
19	C₂²×C₈×C₁₂₇	—	4	2	no	—
20	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₂₇	D₄	1	4	yes	—
25	C₂²×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
26	C₂²×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
27	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
28	C₂×C₁₆×C₁₂₇	—	2	4	no	—
29	C₂×C₁₆×C₁₂₇	—	2	4	no	—
30	C₂×C₁₆×C₁₂₇	—	2	4	no	—
31	C₂×C₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
32	#108×C₁₂₇	—	2	8	no	—
33	#108×C₁₂₇	—	2	4	no	—
34	#108×C₁₂₇	C₂×C₄	1	8	yes	—
35	C₂⁴×C₁₂₇	Q₁₆	1	1	yes	characteristic
36	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
37	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₂₇	SD₁₆	1	4	yes	—
42	C₂²×C₄×C₁₂₇	Q₁₆	1	1	yes	characteristic
43	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
44	M₁₆×C₁₂₇	—	2	8	no	—
45	M₁₆×C₁₂₇	—	4	8	no	—
46	M₁₆×C₁₂₇	—	4	4	no	—
47	C₂×C₈×C₁₂₇	—	4	8	no	—
48	C₂×C₈×C₁₂₇	—	4	4	no	—
49	C₂×C₈×C₁₂₇	C₄²	1	2	yes	—
50	C₂×C₈×C₁₂₇	C₄²	1	2	yes	—
51	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	4	yes	—
52	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	2	yes	—
53	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₂₇	C₄⋊C₄	1	4	yes	—
57	C₂×C₈×C₁₂₇	C₄⋊C₄	1	4	yes	—
58	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
62	C₁₆×C₁₂₇	—	2	4	no	—
63	C₂³×C₁₂₇	—	2	4	no	—
64	C₂³×C₁₂₇	#103	1	4	yes	—
65	C₂³×C₁₂₇	#107	1	1	yes	characteristic
66	C₂³×C₁₂₇	#107	1	1	yes	characteristic
67	C₂³×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
68	C₂×C₄×C₁₂₇	#95	1	2	yes	—
69	C₂×C₄×C₁₂₇	#95	1	2	yes	—
70	C₂×C₄×C₁₂₇	#102	1	4	yes	—
71	C₂×C₄×C₁₂₇	#102	1	4	yes	—
72	C₂×C₄×C₁₂₇	#103	1	4	yes	—
73	C₂×C₄×C₁₂₇	#106	1	2	yes	—
74	C₂×C₄×C₁₂₇	#106	1	2	yes	—
75	C₂×C₄×C₁₂₇	#107	1	1	yes	characteristic
76	C₂×C₄×C₁₂₇	#107	1	1	yes	characteristic
77	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
78	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	2	yes	—
81	C₈×C₁₂₇	—	4	8	no	—
82	C₈×C₁₂₇	#95	1	2	yes	—
83	C₈×C₁₂₇	#95	1	2	yes	—
84	C₈×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
85	C₈×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₈×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
87	C₂²×C₁₂₇	—	2	8	no	—
88	C₂²×C₁₂₇	—	2	2	no	—
89	C₂²×C₁₂₇	—	2	2	no	—
90	C₂²×C₁₂₇	#340	1	2	yes	—
91	C₂²×C₁₂₇	#340	1	2	yes	—
92	C₂²×C₁₂₇	#368	1	4	yes	—
93	C₂²×C₁₂₇	C₂×#103	1	2	yes	—
94	C₂²×C₁₂₇	C₂×#107	1	1	yes	characteristic
95	C₄×C₁₂₇	#340	1	2	yes	—
96	C₄×C₁₂₇	#340	1	2	yes	—
97	C₄×C₁₂₇	C₂×#95	1	1	yes	characteristic
98	C₄×C₁₂₇	C₂×#102	1	2	yes	—
99	C₄×C₁₂₇	C₂×#106	1	1	yes	characteristic
100	C₂×#1381	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂×C₁₂₇	—	2	2	no	—
102	C₂×C₁₂₇	—	2	2	no	—
103	C₂×C₁₂₇	#1381	1	4	yes	direct product factor
104	C₂×C₁₂₇	C₂×#340	1	1	yes	characteristic
105	C₂×C₁₂₇	C₂×#368	1	2	yes	—
106	C₂³×C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
107	C₂²×#108	C₂×C₁₂₇	1	2	yes	—
108	#1381	C₂×C₁₂₇	1	4	yes	direct product factor
109	C₁₂₇	C₂×#1381	1	1	yes	characteristic, direct product factor, Sylow
110	C₂²×M₁₆	—	2	2	no	—
111	C₂³×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
112	C₂²×C₁₆	—	2	4	no	—
113	C₂²×C₁₆	C₄×C₁₂₇	1	2	yes	—
114	C₂²×C₁₆	C₄×C₁₂₇	1	2	yes	—
115	C₂²×C₁₆	C₂²×C₁₂₇	1	2	yes	—
116	C₂×#108	—	2	2	no	—
117	C₂×#108	—	2	4	no	—
118	C₂×#108	C₄×C₁₂₇	1	8	yes	—
119	C₂×#108	C₂²×C₁₂₇	1	4	yes	—
120	C₂³×C₄	D₄×C₁₂₇	1	1	yes	characteristic
121	C₂×M₁₆	—	2	8	no	—
122	C₂×M₁₆	—	2	4	no	—
123	C₂×M₁₆	—	4	2	no	—
124	C₂×M₁₆	—	4	4	no	—
125	C₂²×C₈	—	4	2	no	—
126	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
127	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
128	C₂²×C₈	C₂×C₄×C₁₂₇	1	4	yes	—
129	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
130	C₂²×C₈	D₄×C₁₂₇	1	4	yes	—
131	C₂²×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
132	C₂²×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
133	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
134	C₂×C₁₆	—	2	4	no	—
135	C₂×C₁₆	—	2	4	no	—
136	C₂×C₁₆	—	2	4	no	—
137	C₂×C₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
138	#108	—	2	8	no	—
139	#108	—	2	4	no	—
140	#108	C₂×C₄×C₁₂₇	1	8	yes	—
141	C₂⁴	Q₁₆×C₁₂₇	1	1	yes	characteristic
142	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	4	yes	—
143	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
144	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
145	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
146	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
147	C₂²×C₄	SD₁₆×C₁₂₇	1	4	yes	—
148	C₂²×C₄	Q₁₆×C₁₂₇	1	1	yes	characteristic
149	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
150	M₁₆	—	2	8	no	—
151	M₁₆	—	4	8	no	—
152	M₁₆	—	4	4	no	—
153	C₂×C₈	—	4	8	no	—
154	C₂×C₈	—	4	4	no	—
155	C₂×C₈	C₄²×C₁₂₇	1	2	yes	—
156	C₂×C₈	C₄²×C₁₂₇	1	2	yes	—
157	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	4	yes	—
158	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	2	yes	—
159	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	2	yes	—
160	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
161	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
162	C₂×C₈	C₄⋊C₄×C₁₂₇	1	4	yes	—
163	C₂×C₈	C₄⋊C₄×C₁₂₇	1	4	yes	—
164	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
165	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
166	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
167	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
168	C₁₆	—	2	4	no	—
169	C₂³	—	2	4	no	—
170	C₂³	#103×C₁₂₇	1	4	yes	—
171	C₂³	#107×C₁₂₇	1	1	yes	characteristic
172	C₂³	#107×C₁₂₇	1	1	yes	characteristic
173	C₂³	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
174	C₂×C₄	#95×C₁₂₇	1	2	yes	—
175	C₂×C₄	#95×C₁₂₇	1	2	yes	—
176	C₂×C₄	#102×C₁₂₇	1	4	yes	—
177	C₂×C₄	#102×C₁₂₇	1	4	yes	—
178	C₂×C₄	#103×C₁₂₇	1	4	yes	—
179	C₂×C₄	#106×C₁₂₇	1	2	yes	—
180	C₂×C₄	#106×C₁₂₇	1	2	yes	—
181	C₂×C₄	#107×C₁₂₇	1	1	yes	characteristic
182	C₂×C₄	#107×C₁₂₇	1	1	yes	characteristic
183	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
184	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
185	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
186	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	2	yes	—
187	C₈	—	4	8	no	—
188	C₈	#95×C₁₂₇	1	2	yes	—
189	C₈	#95×C₁₂₇	1	2	yes	—
190	C₈	C₂×C₄²×C₁₂₇	1	1	yes	characteristic
191	C₈	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
192	C₈	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
193	C₂²	—	2	8	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	2	2	no	—
196	C₂²	#340×C₁₂₇	1	2	yes	—
197	C₂²	#340×C₁₂₇	1	2	yes	—
198	C₂²	#368×C₁₂₇	1	4	yes	—
199	C₂²	C₂×#103×C₁₂₇	1	2	yes	—
200	C₂²	C₂×#107×C₁₂₇	1	1	yes	characteristic
201	C₄	#340×C₁₂₇	1	2	yes	—
202	C₄	#340×C₁₂₇	1	2	yes	—
203	C₄	C₂×#95×C₁₂₇	1	1	yes	characteristic
204	C₄	C₂×#102×C₁₂₇	1	2	yes	—
205	C₄	C₂×#106×C₁₂₇	1	1	yes	characteristic
206	C₂	—	2	2	no	—
207	C₂	—	2	2	no	—
208	C₂	C₁₂₇×#1381	1	4	yes	direct product factor
209	C₂	C₂×#340×C₁₂₇	1	1	yes	characteristic
210	C₂	C₂×#368×C₁₂₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₂₇	D₈
commutator	C₈	C₂×C₄²×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂³×C₁₂₇	#107
prime radical	C₂⁴×C₁₂₇	Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₂₇ ◁ C₂³×C₄×C₁₂₇ ◁ C₂³×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₂₇×#1381	C₂	direct (N × H)
100. C₂×#1381	C₁₂₇	direct (N × H)
103. C₂×C₁₂₇	#1381	direct (N × H)
108. #1381	C₂×C₁₂₇	direct (N × H)
109. C₁₂₇	C₂×#1381	direct (N × H)
208. C₂	C₁₂₇×#1381	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₁₆×C₁₂₇	2
C₂²×#108×C₁₂₇	2
C₁₂₇×#1381	2
C₂×#1381	127

Subgroups — C2×C127×#1381 #44576231

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₂₇×#1381 #44576231