Subgroups — C₁₂₇×#20366 #44584039

Order 32512.

1078 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₄×#117×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1746	C₂	1	4	yes	maximal, complemented
5	C₁₂₇×#1744	C₂	1	4	yes	maximal, complemented
6	C₁₂₇×#1729	C₂	1	4	yes	maximal
7	C₂²×M₁₆×C₁₂₇	C₂²	1	2	yes	—
8	C₂³×C₈×C₁₂₇	C₂²	1	2	yes	—
9	C₂×#117×C₁₂₇	C₂²	1	2	yes	—
10	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₄×M₁₆×C₁₂₇	C₄	1	8	yes	complemented
12	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₂₇	C₄	1	8	yes	complemented
14	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	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	2	yes	—
21	#336×C₁₂₇	C₂²	1	16	yes	—
22	C₂³×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
24	C₂×M₁₆×C₁₂₇	—	2	8	no	—
25	C₂×M₁₆×C₁₂₇	C₂×C₄	1	8	yes	—
26	C₂²×C₈×C₁₂₇	—	2	8	no	—
27	C₂²×C₈×C₁₂₇	C₂×C₄	1	8	yes	—
28	C₂²×C₈×C₁₂₇	C₂³	1	4	yes	—
29	C₂²×C₈×C₁₂₇	C₂³	1	4	yes	—
30	#117×C₁₂₇	—	2	8	no	—
31	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
32	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₁₂₇	C₂×C₄	1	8	yes	—
34	C₂×C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₁₂₇	C₂³	1	4	yes	—
38	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
40	#97×C₁₂₇	C₂×C₄	1	8	yes	—
41	C₄×C₈×C₁₂₇	C₂×C₄	1	8	yes	—
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	C₄²	1	8	yes	—
48	C₂²×C₄×C₁₂₇	C₂²×C₄	1	8	yes	—
49	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₁₂₇	#41	1	8	yes	—
55	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
56	M₁₆×C₁₂₇	—	2	16	no	—
57	C₂×C₈×C₁₂₇	—	2	16	no	—
58	C₂×C₈×C₁₂₇	—	2	8	no	—
59	C₂×C₈×C₁₂₇	—	2	8	no	—
60	C₂×C₈×C₁₂₇	C₂²×C₄	1	8	yes	—
61	C₂×C₈×C₁₂₇	C₂²×C₄	1	8	yes	—
62	C₄⋊C₄×C₁₂₇	—	2	8	no	—
63	C₂²⋊C₄×C₁₂₇	—	2	8	no	—
64	C₄²×C₁₂₇	—	2	16	no	—
65	C₄²×C₁₂₇	—	2	8	no	—
66	C₄²×C₁₂₇	C₄²	1	8	yes	—
67	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
68	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
69	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
70	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
71	C₂³×C₁₂₇	—	2	4	no	—
72	C₂³×C₁₂₇	C₂×C₄²	1	2	yes	—
73	C₂³×C₁₂₇	#117	1	4	yes	—
74	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₂₇	—	2	4	no	—
76	C₂×C₄×C₁₂₇	—	2	16	no	—
77	C₂×C₄×C₁₂₇	—	2	8	no	—
78	C₂×C₄×C₁₂₇	—	2	16	no	—
79	C₂×C₄×C₁₂₇	C₂×C₄²	1	4	yes	—
80	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
81	C₂×C₄×C₁₂₇	C₂×C₄²	1	4	yes	—
82	C₂×C₄×C₁₂₇	#117	1	8	yes	—
83	C₂×C₄×C₁₂₇	#117	1	8	yes	—
84	C₂×C₄×C₁₂₇	#117	1	4	yes	—
85	C₂×C₄×C₁₂₇	#131	1	8	yes	—
86	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₂₇	C₂×#41	1	4	yes	—
89	C₈×C₁₂₇	—	2	8	no	—
90	C₈×C₁₂₇	—	2	8	no	—
91	C₂²×C₁₂₇	—	2	4	no	—
92	C₂²×C₁₂₇	—	2	8	no	—
93	C₂²×C₁₂₇	#376	1	4	yes	—
94	C₂²×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
95	C₂²×C₁₂₇	C₂×#117	1	2	yes	—
96	C₂²×C₁₂₇	C₂×#131	1	4	yes	—
97	C₄×C₁₂₇	—	2	4	no	—
98	C₄×C₁₂₇	—	4	8	no	—
99	C₄×C₁₂₇	#376	1	4	yes	—
100	C₄×C₁₂₇	#405	1	8	yes	—
101	C₄×C₁₂₇	C₂×#117	1	2	yes	—
102	C₄×C₁₂₇	C₂×#117	1	2	yes	—
103	#20366	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂×C₁₂₇	—	2	4	no	—
105	C₂×C₁₂₇	C₂×#376	1	1	yes	characteristic
106	C₂×C₁₂₇	C₂×#405	1	2	yes	—
107	C₂×C₁₂₇	#2922	1	4	yes	—
108	C₂×C₄×M₁₆	C₂×C₁₂₇	1	1	yes	characteristic
109	C₂²×C₄×C₈	C₂×C₁₂₇	1	1	yes	characteristic
110	C₄×#117	C₂×C₁₂₇	1	1	yes	characteristic
111	#1746	C₂×C₁₂₇	1	4	yes	complemented
112	#1744	C₂×C₁₂₇	1	4	yes	complemented
113	#1729	C₂×C₁₂₇	1	4	yes	—
114	C₁₂₇	#20366	1	1	yes	characteristic, direct product factor, Sylow
115	C₂²×M₁₆	C₂²×C₁₂₇	1	2	yes	—
116	C₂³×C₈	C₂²×C₁₂₇	1	2	yes	—
117	C₂×#117	C₂²×C₁₂₇	1	2	yes	—
118	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
119	C₄×M₁₆	C₄×C₁₂₇	1	8	yes	complemented
120	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
121	C₂×C₄×C₈	C₄×C₁₂₇	1	8	yes	complemented
122	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
123	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
124	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
125	C₄×C₄⋊C₄	C₄×C₁₂₇	1	4	yes	—
126	C₄×C₂²⋊C₄	C₄×C₁₂₇	1	4	yes	—
127	#376	C₂²×C₁₂₇	1	2	yes	—
128	C₄³	C₂²×C₁₂₇	1	2	yes	—
129	#336	C₂²×C₁₂₇	1	16	yes	—
130	C₂³×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
131	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
132	C₂×M₁₆	—	2	8	no	—
133	C₂×M₁₆	C₂×C₄×C₁₂₇	1	8	yes	—
134	C₂²×C₈	—	2	8	no	—
135	C₂²×C₈	C₂×C₄×C₁₂₇	1	8	yes	—
136	C₂²×C₈	C₂³×C₁₂₇	1	4	yes	—
137	C₂²×C₈	C₂³×C₁₂₇	1	4	yes	—
138	#117	—	2	8	no	—
139	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
140	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
141	C₂×C₄²	C₂×C₄×C₁₂₇	1	8	yes	—
142	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
143	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
144	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
145	C₂×C₄²	C₂³×C₁₂₇	1	4	yes	—
146	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
147	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
148	#97	C₂×C₄×C₁₂₇	1	8	yes	—
149	C₄×C₈	C₂×C₄×C₁₂₇	1	8	yes	—
150	C₄×C₈	C₂×C₄×C₁₂₇	1	8	yes	—
151	C₄×C₈	C₂×C₄×C₁₂₇	1	8	yes	—
152	#95	C₂×C₄×C₁₂₇	1	8	yes	—
153	C₂⁴	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
154	C₂²×C₄	—	2	8	no	—
155	C₂²×C₄	C₄²×C₁₂₇	1	8	yes	—
156	C₂²×C₄	C₂²×C₄×C₁₂₇	1	8	yes	—
157	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
158	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
159	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
160	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
161	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
162	C₂²×C₄	#41×C₁₂₇	1	8	yes	—
163	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
164	M₁₆	—	2	16	no	—
165	C₂×C₈	—	2	16	no	—
166	C₂×C₈	—	2	8	no	—
167	C₂×C₈	—	2	8	no	—
168	C₂×C₈	C₂²×C₄×C₁₂₇	1	8	yes	—
169	C₂×C₈	C₂²×C₄×C₁₂₇	1	8	yes	—
170	C₄⋊C₄	—	2	8	no	—
171	C₂²⋊C₄	—	2	8	no	—
172	C₄²	—	2	16	no	—
173	C₄²	—	2	8	no	—
174	C₄²	C₄²×C₁₂₇	1	8	yes	—
175	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
176	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
177	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
178	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
179	C₂³	—	2	4	no	—
180	C₂³	C₂×C₄²×C₁₂₇	1	2	yes	—
181	C₂³	#117×C₁₂₇	1	4	yes	—
182	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	16	no	—
185	C₂×C₄	—	2	8	no	—
186	C₂×C₄	—	2	16	no	—
187	C₂×C₄	C₂×C₄²×C₁₂₇	1	4	yes	—
188	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
189	C₂×C₄	C₂×C₄²×C₁₂₇	1	4	yes	—
190	C₂×C₄	#117×C₁₂₇	1	8	yes	—
191	C₂×C₄	#117×C₁₂₇	1	8	yes	—
192	C₂×C₄	#117×C₁₂₇	1	4	yes	—
193	C₂×C₄	#131×C₁₂₇	1	8	yes	—
194	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
195	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
196	C₂×C₄	C₂×#41×C₁₂₇	1	4	yes	—
197	C₈	—	2	8	no	—
198	C₈	—	2	8	no	—
199	C₂²	—	2	4	no	—
200	C₂²	—	2	8	no	—
201	C₂²	#376×C₁₂₇	1	4	yes	—
202	C₂²	C₂²×C₄²×C₁₂₇	1	1	yes	characteristic
203	C₂²	C₂×#117×C₁₂₇	1	2	yes	—
204	C₂²	C₂×#131×C₁₂₇	1	4	yes	—
205	C₄	—	2	4	no	—
206	C₄	—	4	8	no	—
207	C₄	#376×C₁₂₇	1	4	yes	—
208	C₄	#405×C₁₂₇	1	8	yes	—
209	C₄	C₂×#117×C₁₂₇	1	2	yes	—
210	C₄	C₂×#117×C₁₂₇	1	2	yes	—
211	C₂	—	2	4	no	—
212	C₂	C₂×#376×C₁₂₇	1	1	yes	characteristic
213	C₂	C₂×#405×C₁₂₇	1	2	yes	—
214	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
4. C₁₂₇×#1746	C₂	semidirect
5. C₁₂₇×#1744	C₂	semidirect
11. C₄×M₁₆×C₁₂₇	C₄	semidirect
13. C₂×C₄×C₈×C₁₂₇	C₄	semidirect
103. #20366	C₁₂₇	direct (N × H)
111. #1746	C₂×C₁₂₇	semidirect
112. #1744	C₂×C₁₂₇	semidirect
114. C₁₂₇	#20366	direct (N × H)
119. C₄×M₁₆	C₄×C₁₂₇	semidirect
121. C₂×C₄×C₈	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×C₄×M₁₆×C₁₂₇	2
C₂²×C₄×C₈×C₁₂₇	2
C₄×#117×C₁₂₇	2
C₁₂₇×#1746	2
C₁₂₇×#1744	2
C₁₂₇×#1729	2
#20366	127

Subgroups — C127×#20366 #44584039

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#20366 #44584039