Subgroups — C₁₂₇×#20462 #44584135

Order 32512.

986 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	C₂×#431×C₁₂₇	C₂	1	2	yes	maximal
2	C₄×#117×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1768	C₂	1	4	yes	maximal, complemented
4	C₁₂₇×#1750	C₂	1	4	yes	maximal
5	C₁₂₇×#1744	C₂	1	2	yes	maximal, complemented
6	C₄×#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄²×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#117×C₁₂₇	C₂²	1	2	yes	—
9	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
10	#431×C₁₂₇	C₄	1	16	yes	complemented
11	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	4	yes	—
12	C₂×C₂²⋊C₈×C₁₂₇	C₂²	1	4	yes	—
13	C₂×#97×C₁₂₇	C₂²	1	4	yes	—
14	C₂×C₄×C₈×C₁₂₇	C₂²	1	4	yes	—
15	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
17	C₄×C₄⋊C₄×C₁₂₇	C₄	1	4	yes	—
18	C₄×C₂²⋊C₄×C₁₂₇	C₄	1	4	yes	—
19	#376×C₁₂₇	C₂²	1	2	yes	—
20	C₄³×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₄³×C₁₂₇	C₂²	1	1	yes	characteristic
22	#336×C₁₂₇	C₂²	1	8	yes	—
23	C₂³×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₁₂₇	C₂³	1	4	yes	—
26	C₂²×C₈×C₁₂₇	C₂³	1	4	yes	—
27	#117×C₁₂₇	—	2	8	no	—
28	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
29	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₂₇	C₂×C₄	1	8	yes	—
34	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
35	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
37	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₁₂₇	C₂×C₄	1	16	yes	—
39	C₂²⋊C₈×C₁₂₇	C₂×C₄	1	16	yes	—
40	#97×C₁₂₇	—	2	8	no	—
41	C₄×C₈×C₁₂₇	—	2	8	no	—
42	C₄×C₈×C₁₂₇	C₂×C₄	1	8	yes	—
43	C₄×C₈×C₁₂₇	C₂×C₄	1	8	yes	—
44	#95×C₁₂₇	C₂×C₄	1	8	yes	—
45	C₂⁴×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₂₇	—	2	8	no	—
47	C₂²×C₄×C₁₂₇	—	2	4	no	—
48	C₂²×C₄×C₁₂₇	C₄²	1	8	yes	—
49	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
50	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
53	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
54	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
56	C₂×C₈×C₁₂₇	—	2	8	no	—
57	C₂×C₈×C₁₂₇	—	2	8	no	—
58	C₂×C₈×C₁₂₇	C₂²×C₄	1	8	yes	—
59	C₂×C₈×C₁₂₇	C₂²×C₄	1	8	yes	—
60	C₄⋊C₄×C₁₂₇	—	2	8	no	—
61	C₂²⋊C₄×C₁₂₇	—	2	8	no	—
62	C₄²×C₁₂₇	—	2	8	no	—
63	C₄²×C₁₂₇	—	2	8	no	—
64	C₄²×C₁₂₇	—	2	4	no	—
65	C₄²×C₁₂₇	C₄²	1	8	yes	—
66	C₄²×C₁₂₇	C₂²×C₄	1	4	yes	—
67	C₄²×C₁₂₇	C₂²×C₄	1	4	yes	—
68	C₄²×C₁₂₇	#41	1	8	yes	—
69	C₂³×C₁₂₇	—	2	2	no	—
70	C₂³×C₁₂₇	—	2	4	no	—
71	C₂³×C₁₂₇	C₂×C₄²	1	2	yes	—
72	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₂₇	—	2	2	no	—
74	C₂×C₄×C₁₂₇	—	2	8	no	—
75	C₂×C₄×C₁₂₇	—	2	4	no	—
76	C₂×C₄×C₁₂₇	—	2	8	no	—
77	C₂×C₄×C₁₂₇	—	2	16	no	—
78	C₂×C₄×C₁₂₇	—	2	4	no	—
79	C₂×C₄×C₁₂₇	—	2	8	no	—
80	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
81	C₂×C₄×C₁₂₇	C₂×C₄²	1	4	yes	—
82	C₂×C₄×C₁₂₇	C₂×C₄²	1	4	yes	—
83	C₂×C₄×C₁₂₇	#117	1	8	yes	—
84	C₂×C₄×C₁₂₇	#131	1	8	yes	—
85	C₂×C₄×C₁₂₇	C₂³×C₄	1	2	yes	—
86	C₂×C₄×C₁₂₇	C₂×#41	1	4	yes	—
87	C₈×C₁₂₇	—	2	8	no	—
88	C₈×C₁₂₇	—	2	8	no	—
89	C₂²×C₁₂₇	—	2	4	no	—
90	C₂²×C₁₂₇	—	2	2	no	—
91	C₂²×C₁₂₇	—	4	4	no	—
92	C₂²×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
93	C₂²×C₁₂₇	C₂×#117	1	2	yes	—
94	C₂²×C₁₂₇	C₂×#131	1	4	yes	—
95	C₄×C₁₂₇	—	2	4	no	—
96	C₄×C₁₂₇	—	2	8	no	—
97	C₄×C₁₂₇	—	2	2	no	—
98	C₄×C₁₂₇	—	4	4	no	—
99	C₄×C₁₂₇	#405	1	8	yes	—
100	C₄×C₁₂₇	C₄×#41	1	4	yes	—
101	#20462	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂×C₁₂₇	—	4	2	no	—
103	C₂×C₁₂₇	C₄×#117	1	1	yes	characteristic
104	C₂×C₁₂₇	C₂×#405	1	2	yes	—
105	C₂×C₁₂₇	#2922	1	4	yes	—
106	C₂×#431	C₂×C₁₂₇	1	2	yes	—
107	C₄×#117	C₂×C₁₂₇	1	1	yes	characteristic
108	#1768	C₂×C₁₂₇	1	4	yes	complemented
109	#1750	C₂×C₁₂₇	1	4	yes	—
110	#1744	C₂×C₁₂₇	1	2	yes	complemented
111	C₄×#97	C₂×C₁₂₇	1	1	yes	characteristic, complemented
112	C₄²×C₈	C₂×C₁₂₇	1	1	yes	characteristic, complemented
113	C₁₂₇	#20462	1	1	yes	characteristic, direct product factor, Sylow
114	C₂×#117	C₂²×C₁₂₇	1	2	yes	—
115	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
116	#431	C₄×C₁₂₇	1	16	yes	complemented
117	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	4	yes	—
118	C₂×C₂²⋊C₈	C₂²×C₁₂₇	1	4	yes	—
119	C₂×#97	C₂²×C₁₂₇	1	4	yes	—
120	C₂×C₄×C₈	C₂²×C₁₂₇	1	4	yes	—
121	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
122	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
123	C₄×C₄⋊C₄	C₄×C₁₂₇	1	4	yes	—
124	C₄×C₂²⋊C₄	C₄×C₁₂₇	1	4	yes	—
125	#376	C₂²×C₁₂₇	1	2	yes	—
126	C₄³	C₂²×C₁₂₇	1	1	yes	characteristic
127	C₄³	C₂²×C₁₂₇	1	1	yes	characteristic
128	#336	C₂²×C₁₂₇	1	8	yes	—
129	C₂³×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
130	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
131	C₂²×C₈	C₂³×C₁₂₇	1	4	yes	—
132	C₂²×C₈	C₂³×C₁₂₇	1	4	yes	—
133	#117	—	2	8	no	—
134	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
135	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
136	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
137	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
138	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
139	C₂×C₄²	C₂×C₄×C₁₂₇	1	8	yes	—
140	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
141	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
142	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
143	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
144	C₄⋊C₈	C₂×C₄×C₁₂₇	1	16	yes	—
145	C₂²⋊C₈	C₂×C₄×C₁₂₇	1	16	yes	—
146	#97	—	2	8	no	—
147	C₄×C₈	—	2	8	no	—
148	C₄×C₈	C₂×C₄×C₁₂₇	1	8	yes	—
149	C₄×C₈	C₂×C₄×C₁₂₇	1	8	yes	—
150	#95	C₂×C₄×C₁₂₇	1	8	yes	—
151	C₂⁴	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
152	C₂²×C₄	—	2	8	no	—
153	C₂²×C₄	—	2	4	no	—
154	C₂²×C₄	C₄²×C₁₂₇	1	8	yes	—
155	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
156	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
157	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
158	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
159	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
160	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
161	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
162	C₂×C₈	—	2	8	no	—
163	C₂×C₈	—	2	8	no	—
164	C₂×C₈	C₂²×C₄×C₁₂₇	1	8	yes	—
165	C₂×C₈	C₂²×C₄×C₁₂₇	1	8	yes	—
166	C₄⋊C₄	—	2	8	no	—
167	C₂²⋊C₄	—	2	8	no	—
168	C₄²	—	2	8	no	—
169	C₄²	—	2	8	no	—
170	C₄²	—	2	4	no	—
171	C₄²	C₄²×C₁₂₇	1	8	yes	—
172	C₄²	C₂²×C₄×C₁₂₇	1	4	yes	—
173	C₄²	C₂²×C₄×C₁₂₇	1	4	yes	—
174	C₄²	#41×C₁₂₇	1	8	yes	—
175	C₂³	—	2	2	no	—
176	C₂³	—	2	4	no	—
177	C₂³	C₂×C₄²×C₁₂₇	1	2	yes	—
178	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	8	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	8	no	—
183	C₂×C₄	—	2	16	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	8	no	—
186	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
187	C₂×C₄	C₂×C₄²×C₁₂₇	1	4	yes	—
188	C₂×C₄	C₂×C₄²×C₁₂₇	1	4	yes	—
189	C₂×C₄	#117×C₁₂₇	1	8	yes	—
190	C₂×C₄	#131×C₁₂₇	1	8	yes	—
191	C₂×C₄	C₂³×C₄×C₁₂₇	1	2	yes	—
192	C₂×C₄	C₂×#41×C₁₂₇	1	4	yes	—
193	C₈	—	2	8	no	—
194	C₈	—	2	8	no	—
195	C₂²	—	2	4	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	4	4	no	—
198	C₂²	C₂²×C₄²×C₁₂₇	1	1	yes	characteristic
199	C₂²	C₂×#117×C₁₂₇	1	2	yes	—
200	C₂²	C₂×#131×C₁₂₇	1	4	yes	—
201	C₄	—	2	4	no	—
202	C₄	—	2	8	no	—
203	C₄	—	2	2	no	—
204	C₄	—	4	4	no	—
205	C₄	#405×C₁₂₇	1	8	yes	—
206	C₄	C₄×#41×C₁₂₇	1	4	yes	—
207	C₂	—	4	2	no	—
208	C₂	C₄×#117×C₁₂₇	1	1	yes	characteristic
209	C₂	C₂×#405×C₁₂₇	1	2	yes	—
210	C₂	C₁₂₇×#2922	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₁₂₇×#1768	C₂	semidirect
5. C₁₂₇×#1744	C₂	semidirect
6. C₄×#97×C₁₂₇	C₂	semidirect
7. C₄²×C₈×C₁₂₇	C₂	semidirect
10. #431×C₁₂₇	C₄	semidirect
101. #20462	C₁₂₇	direct (N × H)
108. #1768	C₂×C₁₂₇	semidirect
110. #1744	C₂×C₁₂₇	semidirect
111. C₄×#97	C₂×C₁₂₇	semidirect
112. C₄²×C₈	C₂×C₁₂₇	semidirect
113. C₁₂₇	#20462	direct (N × H)
116. #431	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×#431×C₁₂₇	2
C₄×#117×C₁₂₇	2
C₁₂₇×#1768	2
C₁₂₇×#1750	2
C₁₂₇×#1744	2
C₄×#97×C₁₂₇	2
C₄²×C₈×C₁₂₇	2
#20462	127

Subgroups — C127×#20462 #44584135

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#20462 #44584135