Subgroups — C₁₂₇×#20529 #44584202

Order 32512.

1170 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#407×C₁₂₇	C₂	1	2	yes	maximal
2	C₁₂₇×#2280	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1753	C₂	1	8	yes	maximal, complemented
4	C₁₂₇×#1750	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
6	C₂²×M₁₆×C₁₂₇	C₂²	1	4	yes	—
7	C₂×#117×C₁₂₇	C₂²	1	2	yes	—
8	#407×C₁₂₇	C₄	1	16	yes	complemented
9	C₂×C₂²⋊C₈×C₁₂₇	C₂²	1	4	yes	—
10	C₂×C₂²⋊C₈×C₁₂₇	C₂²	1	4	yes	—
11	C₂×#97×C₁₂₇	C₂²	1	4	yes	complemented
12	C₂×C₄×C₈×C₁₂₇	C₂²	1	4	yes	complemented
13	#381×C₁₂₇	C₄	1	2	yes	—
14	#380×C₁₂₇	C₄	1	4	yes	—
15	#379×C₁₂₇	C₄	1	2	yes	—
16	C₄×C₂²⋊C₄×C₁₂₇	C₂²	1	4	yes	—
17	#336×C₁₂₇	C₂²	1	8	yes	complemented
18	C₂⁵×C₁₂₇	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₁₂₇	—	4	4	no	—
20	C₂³×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₂₇	C₂×C₄	1	8	yes	—
23	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
24	C₂³×C₄×C₁₂₇	C₂³	1	2	yes	—
25	C₂×M₁₆×C₁₂₇	—	4	8	no	—
26	C₂×M₁₆×C₁₂₇	C₂×C₄	1	16	yes	—
27	C₂²×C₈×C₁₂₇	C₂³	1	8	yes	—
28	#117×C₁₂₇	—	4	4	no	—
29	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
30	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
31	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	8	yes	—
32	C₂×C₄²×C₁₂₇	C₂³	1	4	yes	—
33	C₂²⋊C₈×C₁₂₇	C₂×C₄	1	16	yes	—
34	C₂²⋊C₈×C₁₂₇	C₂×C₄	1	16	yes	—
35	#97×C₁₂₇	—	4	4	no	—
36	C₄×C₈×C₁₂₇	—	4	4	no	—
37	#95×C₁₂₇	C₂×C₄	1	8	yes	—
38	C₂⁴×C₁₂₇	—	4	2	no	—
39	C₂⁴×C₁₂₇	—	4	4	no	—
40	C₂⁴×C₁₂₇	C₄²	1	4	yes	—
41	C₂⁴×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
42	C₂⁴×C₁₂₇	C₂²×C₄	1	2	yes	—
43	C₂²×C₄×C₁₂₇	—	2	4	no	—
44	C₂²×C₄×C₁₂₇	—	2	8	no	—
45	C₂²×C₄×C₁₂₇	—	4	16	no	—
46	C₂²×C₄×C₁₂₇	—	4	4	no	—
47	C₂²×C₄×C₁₂₇	—	4	2	no	—
48	C₂²×C₄×C₁₂₇	C₄²	1	8	yes	—
49	C₂²×C₄×C₁₂₇	C₄²	1	4	yes	—
50	C₂²×C₄×C₁₂₇	C₂²×C₄	1	8	yes	—
51	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
54	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
55	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
57	M₁₆×C₁₂₇	—	4	16	no	—
58	C₂×C₈×C₁₂₇	—	2	8	no	—
59	C₂×C₈×C₁₂₇	—	2	8	no	—
60	C₂×C₈×C₁₂₇	C₂²×C₄	1	16	yes	—
61	C₄⋊C₄×C₁₂₇	—	4	4	no	—
62	C₂²⋊C₄×C₁₂₇	—	4	8	no	—
63	C₂²⋊C₄×C₁₂₇	—	4	4	no	—
64	C₄²×C₁₂₇	—	2	4	no	—
65	C₄²×C₁₂₇	—	2	4	no	—
66	C₂³×C₁₂₇	—	2	2	no	—
67	C₂³×C₁₂₇	—	2	4	no	—
68	C₂³×C₁₂₇	—	2	4	no	—
69	C₂³×C₁₂₇	—	2	4	no	—
70	C₂³×C₁₂₇	—	2	4	no	—
71	C₂³×C₁₂₇	—	4	4	no	—
72	C₂³×C₁₂₇	—	4	8	no	—
73	C₂³×C₁₂₇	—	4	16	no	—
74	C₂³×C₁₂₇	C₂×C₄²	1	2	yes	—
75	C₂³×C₁₂₇	C₂×C₄²	1	4	yes	—
76	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₂₇	—	2	2	no	—
78	C₂×C₄×C₁₂₇	—	2	4	no	—
79	C₂×C₄×C₁₂₇	—	2	8	no	—
80	C₂×C₄×C₁₂₇	—	2	8	no	—
81	C₂×C₄×C₁₂₇	—	2	8	no	—
82	C₂×C₄×C₁₂₇	—	2	8	no	—
83	C₂×C₄×C₁₂₇	—	2	16	no	—
84	C₂×C₄×C₁₂₇	—	4	8	no	—
85	C₂×C₄×C₁₂₇	—	4	4	no	—
86	C₂×C₄×C₁₂₇	C₂×C₄²	1	4	yes	—
87	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
88	C₂×C₄×C₁₂₇	C₂³×C₄	1	2	yes	—
89	C₂×C₄×C₁₂₇	H(5,𝔽₂)	1	4	yes	—
90	C₈×C₁₂₇	—	2	16	no	—
91	C₂²×C₁₂₇	—	2	4	no	—
92	C₂²×C₁₂₇	—	2	4	no	—
93	C₂²×C₁₂₇	—	2	4	no	—
94	C₂²×C₁₂₇	—	2	4	no	—
95	C₂²×C₁₂₇	—	2	2	no	—
96	C₂²×C₁₂₇	—	4	16	no	—
97	C₂²×C₁₂₇	—	4	4	no	—
98	C₂²×C₁₂₇	—	4	8	no	—
99	C₂²×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
100	C₂²×C₁₂₇	#518	1	2	yes	—
101	C₂²×C₁₂₇	#568	1	4	yes	—
102	C₄×C₁₂₇	—	2	2	no	—
103	C₄×C₁₂₇	—	2	4	no	—
104	C₄×C₁₂₇	—	4	8	no	—
105	C₄×C₁₂₇	#518	1	4	yes	—
106	#20529	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂×C₁₂₇	—	4	4	no	—
108	C₂×C₁₂₇	—	4	2	no	—
109	C₂×C₁₂₇	#2273	1	1	yes	characteristic
110	C₂×C₁₂₇	#2873	1	2	yes	—
111	C₂×C₁₂₇	#2978	1	4	yes	—
112	C₂×#407	C₂×C₁₂₇	1	2	yes	—
113	#2280	C₂×C₁₂₇	1	1	yes	characteristic
114	#1753	C₂×C₁₂₇	1	8	yes	complemented
115	#1750	C₂×C₁₂₇	1	4	yes	complemented
116	C₁₂₇	#20529	1	1	yes	characteristic, direct product factor, Sylow
117	C₂⁴×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
118	C₂²×M₁₆	C₂²×C₁₂₇	1	4	yes	—
119	C₂×#117	C₂²×C₁₂₇	1	2	yes	—
120	#407	C₄×C₁₂₇	1	16	yes	complemented
121	C₂×C₂²⋊C₈	C₂²×C₁₂₇	1	4	yes	—
122	C₂×C₂²⋊C₈	C₂²×C₁₂₇	1	4	yes	—
123	C₂×#97	C₂²×C₁₂₇	1	4	yes	complemented
124	C₂×C₄×C₈	C₂²×C₁₂₇	1	4	yes	complemented
125	#381	C₄×C₁₂₇	1	2	yes	—
126	#380	C₄×C₁₂₇	1	4	yes	—
127	#379	C₄×C₁₂₇	1	2	yes	—
128	C₄×C₂²⋊C₄	C₂²×C₁₂₇	1	4	yes	—
129	#336	C₂²×C₁₂₇	1	8	yes	complemented
130	C₂⁵	C₂×C₄×C₁₂₇	1	1	yes	characteristic
131	C₂³×C₄	—	4	4	no	—
132	C₂³×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
133	C₂³×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
134	C₂³×C₄	C₂×C₄×C₁₂₇	1	8	yes	—
135	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
136	C₂³×C₄	C₂³×C₁₂₇	1	2	yes	—
137	C₂×M₁₆	—	4	8	no	—
138	C₂×M₁₆	C₂×C₄×C₁₂₇	1	16	yes	—
139	C₂²×C₈	C₂³×C₁₂₇	1	8	yes	—
140	#117	—	4	4	no	—
141	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
142	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
143	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	8	yes	—
144	C₂×C₄²	C₂³×C₁₂₇	1	4	yes	—
145	C₂²⋊C₈	C₂×C₄×C₁₂₇	1	16	yes	—
146	C₂²⋊C₈	C₂×C₄×C₁₂₇	1	16	yes	—
147	#97	—	4	4	no	—
148	C₄×C₈	—	4	4	no	—
149	#95	C₂×C₄×C₁₂₇	1	8	yes	—
150	C₂⁴	—	4	2	no	—
151	C₂⁴	—	4	4	no	—
152	C₂⁴	C₄²×C₁₂₇	1	4	yes	—
153	C₂⁴	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
154	C₂⁴	C₂²×C₄×C₁₂₇	1	2	yes	—
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	2	8	no	—
157	C₂²×C₄	—	4	16	no	—
158	C₂²×C₄	—	4	4	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	C₄²×C₁₂₇	1	8	yes	—
161	C₂²×C₄	C₄²×C₁₂₇	1	4	yes	—
162	C₂²×C₄	C₂²×C₄×C₁₂₇	1	8	yes	—
163	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
164	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
165	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
166	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
167	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
168	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
169	M₁₆	—	4	16	no	—
170	C₂×C₈	—	2	8	no	—
171	C₂×C₈	—	2	8	no	—
172	C₂×C₈	C₂²×C₄×C₁₂₇	1	16	yes	—
173	C₄⋊C₄	—	4	4	no	—
174	C₂²⋊C₄	—	4	8	no	—
175	C₂²⋊C₄	—	4	4	no	—
176	C₄²	—	2	4	no	—
177	C₄²	—	2	4	no	—
178	C₂³	—	2	2	no	—
179	C₂³	—	2	4	no	—
180	C₂³	—	2	4	no	—
181	C₂³	—	2	4	no	—
182	C₂³	—	2	4	no	—
183	C₂³	—	4	4	no	—
184	C₂³	—	4	8	no	—
185	C₂³	—	4	16	no	—
186	C₂³	C₂×C₄²×C₁₂₇	1	2	yes	—
187	C₂³	C₂×C₄²×C₁₂₇	1	4	yes	—
188	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	—	2	8	no	—
193	C₂×C₄	—	2	8	no	—
194	C₂×C₄	—	2	8	no	—
195	C₂×C₄	—	2	16	no	—
196	C₂×C₄	—	4	8	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	C₂×C₄²×C₁₂₇	1	4	yes	—
199	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
200	C₂×C₄	C₂³×C₄×C₁₂₇	1	2	yes	—
201	C₂×C₄	H(5,𝔽₂)×C₁₂₇	1	4	yes	—
202	C₈	—	2	16	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	4	no	—
205	C₂²	—	2	4	no	—
206	C₂²	—	2	4	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	4	16	no	—
209	C₂²	—	4	4	no	—
210	C₂²	—	4	8	no	—
211	C₂²	C₂²×C₄²×C₁₂₇	1	1	yes	characteristic
212	C₂²	#518×C₁₂₇	1	2	yes	—
213	C₂²	#568×C₁₂₇	1	4	yes	—
214	C₄	—	2	2	no	—
215	C₄	—	2	4	no	—
216	C₄	—	4	8	no	—
217	C₄	#518×C₁₂₇	1	4	yes	—
218	C₂	—	4	4	no	—
219	C₂	—	4	2	no	—
220	C₂	C₁₂₇×#2273	1	1	yes	characteristic
221	C₂	C₁₂₇×#2873	1	2	yes	—
222	C₂	C₁₂₇×#2978	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₂₇	C₂⁴
commutator	C₂²	C₂²×C₄²×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle	C₂³×C₁₂₇	C₂³×C₄
prime radical	C₂⁵×C₁₂₇	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₂₇×#1753	C₂	semidirect
4. C₁₂₇×#1750	C₂	semidirect
8. #407×C₁₂₇	C₄	semidirect
11. C₂×#97×C₁₂₇	C₂²	semidirect
12. C₂×C₄×C₈×C₁₂₇	C₂²	semidirect
17. #336×C₁₂₇	C₂²	semidirect
106. #20529	C₁₂₇	direct (N × H)
114. #1753	C₂×C₁₂₇	semidirect
115. #1750	C₂×C₁₂₇	semidirect
116. C₁₂₇	#20529	direct (N × H)
120. #407	C₄×C₁₂₇	semidirect
123. C₂×#97	C₂²×C₁₂₇	semidirect
124. C₂×C₄×C₈	C₂²×C₁₂₇	semidirect
129. #336	C₂²×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#407×C₁₂₇	2
C₁₂₇×#2280	2
C₁₂₇×#1753	2
C₁₂₇×#1750	2
#20529	127

Subgroups — C127×#20529 #44584202

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#20529 #44584202