Subgroups — C₁₂₇×#20528 #44584201

Order 32512.

1250 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊C₈×C₁₂₇	C₂	1	2	yes	maximal
2	C₁₂₇×#2280	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1752	C₂	1	4	yes	maximal, complemented
4	C₈×C₂²⋊C₄×C₁₂₇	C₂	1	4	yes	maximal, complemented
5	C₁₂₇×#1750	C₂	1	4	yes	maximal, complemented
6	C₂⁴×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₂₇	C₂²	1	4	yes	—
8	C₂×#117×C₁₂₇	C₂²	1	2	yes	—
9	C₂×C₂²⋊C₈×C₁₂₇	C₄	1	16	yes	complemented
10	C₂×C₂²⋊C₈×C₁₂₇	C₂²	1	4	yes	—
11	C₂×C₂²⋊C₈×C₁₂₇	C₂²	1	2	yes	—
12	C₂×C₂²⋊C₈×C₁₂₇	C₂²	1	2	yes	—
13	C₂×#97×C₁₂₇	C₂²	1	4	yes	complemented
14	C₂×C₄×C₈×C₁₂₇	C₂²	1	4	yes	complemented
15	#381×C₁₂₇	C₄	1	2	yes	—
16	#380×C₁₂₇	C₄	1	4	yes	—
17	#379×C₁₂₇	C₄	1	2	yes	—
18	C₄×C₂²⋊C₄×C₁₂₇	C₂²	1	4	yes	—
19	#336×C₁₂₇	C₂²	1	8	yes	complemented
20	C₂⁵×C₁₂₇	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₂₇	—	4	4	no	—
22	C₂³×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₂₇	C₂×C₄	1	8	yes	—
25	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
26	C₂³×C₄×C₁₂₇	C₂³	1	2	yes	—
27	C₂²×C₈×C₁₂₇	—	4	8	no	—
28	C₂²×C₈×C₁₂₇	C₂×C₄	1	16	yes	—
29	C₂²×C₈×C₁₂₇	C₂³	1	4	yes	—
30	C₂²×C₈×C₁₂₇	C₂³	1	4	yes	—
31	#117×C₁₂₇	—	4	4	no	—
32	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
33	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	8	yes	—
34	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₁₂₇	C₂³	1	4	yes	—
36	C₂²⋊C₈×C₁₂₇	C₂×C₄	1	16	yes	—
37	C₂²⋊C₈×C₁₂₇	C₂×C₄	1	8	yes	—
38	C₂²⋊C₈×C₁₂₇	C₂×C₄	1	8	yes	—
39	#97×C₁₂₇	—	4	4	no	—
40	C₄×C₈×C₁₂₇	—	4	4	no	—
41	#95×C₁₂₇	C₂×C₄	1	8	yes	—
42	C₂⁴×C₁₂₇	—	2	8	no	—
43	C₂⁴×C₁₂₇	—	4	2	no	—
44	C₂⁴×C₁₂₇	C₄²	1	4	yes	—
45	C₂⁴×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
46	C₂⁴×C₁₂₇	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₁₂₇	—	2	4	no	—
48	C₂²×C₄×C₁₂₇	—	2	8	no	—
49	C₂²×C₄×C₁₂₇	—	4	16	no	—
50	C₂²×C₄×C₁₂₇	—	4	4	no	—
51	C₂²×C₄×C₁₂₇	—	4	2	no	—
52	C₂²×C₄×C₁₂₇	C₄²	1	4	yes	—
53	C₂²×C₄×C₁₂₇	C₄²	1	8	yes	—
54	C₂²×C₄×C₁₂₇	C₂²×C₄	1	8	yes	—
55	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
61	C₂×C₈×C₁₂₇	—	2	8	no	—
62	C₂×C₈×C₁₂₇	—	2	8	no	—
63	C₂×C₈×C₁₂₇	—	4	16	no	—
64	C₂×C₈×C₁₂₇	C₂²×C₄	1	8	yes	—
65	C₂×C₈×C₁₂₇	C₂²×C₄	1	8	yes	—
66	C₄⋊C₄×C₁₂₇	—	4	4	no	—
67	C₂²⋊C₄×C₁₂₇	—	2	16	no	—
68	C₂²⋊C₄×C₁₂₇	—	4	4	no	—
69	C₄²×C₁₂₇	—	2	4	no	—
70	C₄²×C₁₂₇	—	2	4	no	—
71	C₂³×C₁₂₇	—	2	4	no	—
72	C₂³×C₁₂₇	—	2	4	no	—
73	C₂³×C₁₂₇	—	2	2	no	—
74	C₂³×C₁₂₇	—	2	4	no	—
75	C₂³×C₁₂₇	—	2	16	no	—
76	C₂³×C₁₂₇	—	4	4	no	—
77	C₂³×C₁₂₇	—	4	16	no	—
78	C₂³×C₁₂₇	C₂×C₄²	1	2	yes	—
79	C₂³×C₁₂₇	C₂×C₄²	1	4	yes	—
80	C₂³×C₁₂₇	#131	1	8	yes	—
81	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₂₇	—	2	2	no	—
83	C₂×C₄×C₁₂₇	—	2	4	no	—
84	C₂×C₄×C₁₂₇	—	2	8	no	—
85	C₂×C₄×C₁₂₇	—	2	8	no	—
86	C₂×C₄×C₁₂₇	—	2	8	no	—
87	C₂×C₄×C₁₂₇	—	2	8	no	—
88	C₂×C₄×C₁₂₇	—	2	16	no	—
89	C₂×C₄×C₁₂₇	—	4	8	no	—
90	C₂×C₄×C₁₂₇	—	4	4	no	—
91	C₂×C₄×C₁₂₇	C₂×C₄²	1	4	yes	—
92	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
93	C₂×C₄×C₁₂₇	C₂³×C₄	1	2	yes	—
94	C₂×C₄×C₁₂₇	H(5,𝔽₂)	1	4	yes	—
95	C₈×C₁₂₇	—	2	8	no	—
96	C₈×C₁₂₇	—	2	8	no	—
97	C₂²×C₁₂₇	—	2	4	no	—
98	C₂²×C₁₂₇	—	2	16	no	—
99	C₂²×C₁₂₇	—	2	4	no	—
100	C₂²×C₁₂₇	—	2	4	no	—
101	C₂²×C₁₂₇	—	2	2	no	—
102	C₂²×C₁₂₇	—	4	16	no	—
103	C₂²×C₁₂₇	—	4	4	no	—
104	C₂²×C₁₂₇	#405	1	8	yes	—
105	C₂²×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
106	C₂²×C₁₂₇	#518	1	2	yes	—
107	C₂²×C₁₂₇	C₂×#131	1	4	yes	—
108	C₄×C₁₂₇	—	2	2	no	—
109	C₄×C₁₂₇	—	2	4	no	—
110	C₄×C₁₂₇	—	4	8	no	—
111	C₄×C₁₂₇	#518	1	4	yes	—
112	#20528	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂×C₁₂₇	—	2	8	no	—
114	C₂×C₁₂₇	—	4	2	no	—
115	C₂×C₁₂₇	#2273	1	1	yes	characteristic
116	C₂×C₁₂₇	C₂×#405	1	2	yes	—
117	C₂×C₁₂₇	#2977	1	4	yes	—
118	C₂²×C₂²⋊C₈	C₂×C₁₂₇	1	2	yes	—
119	#2280	C₂×C₁₂₇	1	1	yes	characteristic
120	#1752	C₂×C₁₂₇	1	4	yes	complemented
121	C₈×C₂²⋊C₄	C₂×C₁₂₇	1	4	yes	complemented
122	#1750	C₂×C₁₂₇	1	4	yes	complemented
123	C₁₂₇	#20528	1	1	yes	characteristic, direct product factor, Sylow
124	C₂⁴×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
125	C₂³×C₈	C₂²×C₁₂₇	1	4	yes	—
126	C₂×#117	C₂²×C₁₂₇	1	2	yes	—
127	C₂×C₂²⋊C₈	C₄×C₁₂₇	1	16	yes	complemented
128	C₂×C₂²⋊C₈	C₂²×C₁₂₇	1	4	yes	—
129	C₂×C₂²⋊C₈	C₂²×C₁₂₇	1	2	yes	—
130	C₂×C₂²⋊C₈	C₂²×C₁₂₇	1	2	yes	—
131	C₂×#97	C₂²×C₁₂₇	1	4	yes	complemented
132	C₂×C₄×C₈	C₂²×C₁₂₇	1	4	yes	complemented
133	#381	C₄×C₁₂₇	1	2	yes	—
134	#380	C₄×C₁₂₇	1	4	yes	—
135	#379	C₄×C₁₂₇	1	2	yes	—
136	C₄×C₂²⋊C₄	C₂²×C₁₂₇	1	4	yes	—
137	#336	C₂²×C₁₂₇	1	8	yes	complemented
138	C₂⁵	C₂×C₄×C₁₂₇	1	1	yes	characteristic
139	C₂³×C₄	—	4	4	no	—
140	C₂³×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
141	C₂³×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
142	C₂³×C₄	C₂×C₄×C₁₂₇	1	8	yes	—
143	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
144	C₂³×C₄	C₂³×C₁₂₇	1	2	yes	—
145	C₂²×C₈	—	4	8	no	—
146	C₂²×C₈	C₂×C₄×C₁₂₇	1	16	yes	—
147	C₂²×C₈	C₂³×C₁₂₇	1	4	yes	—
148	C₂²×C₈	C₂³×C₁₂₇	1	4	yes	—
149	#117	—	4	4	no	—
150	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
151	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	8	yes	—
152	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
153	C₂×C₄²	C₂³×C₁₂₇	1	4	yes	—
154	C₂²⋊C₈	C₂×C₄×C₁₂₇	1	16	yes	—
155	C₂²⋊C₈	C₂×C₄×C₁₂₇	1	8	yes	—
156	C₂²⋊C₈	C₂×C₄×C₁₂₇	1	8	yes	—
157	#97	—	4	4	no	—
158	C₄×C₈	—	4	4	no	—
159	#95	C₂×C₄×C₁₂₇	1	8	yes	—
160	C₂⁴	—	2	8	no	—
161	C₂⁴	—	4	2	no	—
162	C₂⁴	C₄²×C₁₂₇	1	4	yes	—
163	C₂⁴	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
164	C₂⁴	C₂²×C₄×C₁₂₇	1	2	yes	—
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	—	2	8	no	—
167	C₂²×C₄	—	4	16	no	—
168	C₂²×C₄	—	4	4	no	—
169	C₂²×C₄	—	4	2	no	—
170	C₂²×C₄	C₄²×C₁₂₇	1	4	yes	—
171	C₂²×C₄	C₄²×C₁₂₇	1	8	yes	—
172	C₂²×C₄	C₂²×C₄×C₁₂₇	1	8	yes	—
173	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
174	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
175	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
176	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
177	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
178	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
179	C₂×C₈	—	2	8	no	—
180	C₂×C₈	—	2	8	no	—
181	C₂×C₈	—	4	16	no	—
182	C₂×C₈	C₂²×C₄×C₁₂₇	1	8	yes	—
183	C₂×C₈	C₂²×C₄×C₁₂₇	1	8	yes	—
184	C₄⋊C₄	—	4	4	no	—
185	C₂²⋊C₄	—	2	16	no	—
186	C₂²⋊C₄	—	4	4	no	—
187	C₄²	—	2	4	no	—
188	C₄²	—	2	4	no	—
189	C₂³	—	2	4	no	—
190	C₂³	—	2	4	no	—
191	C₂³	—	2	2	no	—
192	C₂³	—	2	4	no	—
193	C₂³	—	2	16	no	—
194	C₂³	—	4	4	no	—
195	C₂³	—	4	16	no	—
196	C₂³	C₂×C₄²×C₁₂₇	1	2	yes	—
197	C₂³	C₂×C₄²×C₁₂₇	1	4	yes	—
198	C₂³	#131×C₁₂₇	1	8	yes	—
199	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	8	no	—
203	C₂×C₄	—	2	8	no	—
204	C₂×C₄	—	2	8	no	—
205	C₂×C₄	—	2	8	no	—
206	C₂×C₄	—	2	16	no	—
207	C₂×C₄	—	4	8	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	C₂×C₄²×C₁₂₇	1	4	yes	—
210	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
211	C₂×C₄	C₂³×C₄×C₁₂₇	1	2	yes	—
212	C₂×C₄	H(5,𝔽₂)×C₁₂₇	1	4	yes	—
213	C₈	—	2	8	no	—
214	C₈	—	2	8	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	2	16	no	—
217	C₂²	—	2	4	no	—
218	C₂²	—	2	4	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	4	16	no	—
221	C₂²	—	4	4	no	—
222	C₂²	#405×C₁₂₇	1	8	yes	—
223	C₂²	C₂²×C₄²×C₁₂₇	1	1	yes	characteristic
224	C₂²	#518×C₁₂₇	1	2	yes	—
225	C₂²	C₂×#131×C₁₂₇	1	4	yes	—
226	C₄	—	2	2	no	—
227	C₄	—	2	4	no	—
228	C₄	—	4	8	no	—
229	C₄	#518×C₁₂₇	1	4	yes	—
230	C₂	—	2	8	no	—
231	C₂	—	4	2	no	—
232	C₂	C₁₂₇×#2273	1	1	yes	characteristic
233	C₂	C₂×#405×C₁₂₇	1	2	yes	—
234	C₂	C₁₂₇×#2977	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×#1752	C₂	semidirect
4. C₈×C₂²⋊C₄×C₁₂₇	C₂	semidirect
5. C₁₂₇×#1750	C₂	semidirect
9. C₂×C₂²⋊C₈×C₁₂₇	C₄	semidirect
13. C₂×#97×C₁₂₇	C₂²	semidirect
14. C₂×C₄×C₈×C₁₂₇	C₂²	semidirect
19. #336×C₁₂₇	C₂²	semidirect
112. #20528	C₁₂₇	direct (N × H)
120. #1752	C₂×C₁₂₇	semidirect
121. C₈×C₂²⋊C₄	C₂×C₁₂₇	semidirect
122. #1750	C₂×C₁₂₇	semidirect
123. C₁₂₇	#20528	direct (N × H)
127. C₂×C₂²⋊C₈	C₄×C₁₂₇	semidirect
131. C₂×#97	C₂²×C₁₂₇	semidirect
132. C₂×C₄×C₈	C₂²×C₁₂₇	semidirect
137. #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
C₂²×C₂²⋊C₈×C₁₂₇	2
C₁₂₇×#2280	2
C₁₂₇×#1752	2
C₈×C₂²⋊C₄×C₁₂₇	2
C₁₂₇×#1750	2
#20528	127

Subgroups — C127×#20528 #44584201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#20528 #44584201