Subgroups — C₁₂₇×#20682 #44584355

Order 32512.

2022 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄²×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#407×C₁₂₇	C₂	1	2	yes	maximal
3	C₁₂₇×#1817	C₂	1	8	yes	maximal, complemented
4	C₁₂₇×#1728	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₁₂₇	C₄	1	2	yes	—
6	C₂⁴×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₁₂₇	C₂²	1	4	yes	—
8	C₂²×C₄²×C₁₂₇	—	2	8	no	—
9	C₂²×C₄²×C₁₂₇	C₄	1	2	yes	—
10	C₂²×C₄²×C₁₂₇	C₄	1	4	yes	—
11	C₂²×C₄²×C₁₂₇	C₂²	1	4	yes	—
12	C₂²×C₄²×C₁₂₇	C₂²	1	2	yes	—
13	#407×C₁₂₇	—	2	8	no	—
14	C₂×C₂²⋊C₈×C₁₂₇	C₂²	1	8	yes	—
15	#336×C₁₂₇	C₂²	1	16	yes	complemented
16	C₂⁵×C₁₂₇	C₂×C₄	1	1	yes	characteristic
17	C₂³×C₄×C₁₂₇	—	2	8	no	—
18	C₂³×C₄×C₁₂₇	—	2	16	no	—
19	C₂³×C₄×C₁₂₇	—	2	8	no	—
20	C₂³×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
21	C₂³×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
22	C₂³×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
25	C₂³×C₄×C₁₂₇	D₄	1	8	yes	—
26	C₂³×C₄×C₁₂₇	C₂³	1	2	yes	—
27	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₁₂₇	—	2	8	no	—
29	C₂×M₁₆×C₁₂₇	—	2	16	no	—
30	C₂²×C₈×C₁₂₇	C₂³	1	8	yes	—
31	C₂×C₄²×C₁₂₇	—	2	8	no	—
32	C₂×C₄²×C₁₂₇	—	2	16	no	—
33	C₂×C₄²×C₁₂₇	—	2	16	no	—
34	C₂×C₄²×C₁₂₇	—	2	8	no	—
35	C₂×C₄²×C₁₂₇	C₂×C₄	1	8	yes	—
36	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₁₂₇	C₂³	1	4	yes	—
38	C₂²⋊C₈×C₁₂₇	—	2	16	no	—
39	C₂⁴×C₁₂₇	—	2	4	no	—
40	C₂⁴×C₁₂₇	—	2	8	no	—
41	C₂⁴×C₁₂₇	C₂²⋊C₄	1	4	yes	—
42	C₂⁴×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
43	C₂⁴×C₁₂₇	C₂²×C₄	1	2	yes	—
44	C₂²×C₄×C₁₂₇	—	2	64	no	—
45	C₂²×C₄×C₁₂₇	—	2	8	no	—
46	C₂²×C₄×C₁₂₇	—	2	8	no	—
47	C₂²×C₄×C₁₂₇	—	2	8	no	—
48	C₂²×C₄×C₁₂₇	—	2	4	no	—
49	C₂²×C₄×C₁₂₇	—	2	8	no	—
50	C₂²×C₄×C₁₂₇	—	2	8	no	—
51	C₂²×C₄×C₁₂₇	—	2	32	no	—
52	C₂²×C₄×C₁₂₇	—	2	16	no	—
53	C₂²×C₄×C₁₂₇	—	2	16	no	—
54	C₂²×C₄×C₁₂₇	—	2	8	no	—
55	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
56	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	8	yes	—
57	C₂²×C₄×C₁₂₇	M₁₆	1	16	yes	complemented
58	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₁₂₇	C₂²×C₄	1	8	yes	—
60	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₂₇	C₂×D₄	1	8	yes	—
62	C₂²×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
63	C₂²×C₄×C₁₂₇	#41	1	8	yes	—
64	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
65	M₁₆×C₁₂₇	—	4	16	no	—
66	C₂×C₈×C₁₂₇	—	2	8	no	—
67	C₂×C₈×C₁₂₇	—	2	16	no	—
68	C₄²×C₁₂₇	—	2	16	no	—
69	C₄²×C₁₂₇	—	2	8	no	—
70	C₄²×C₁₂₇	—	2	8	no	—
71	C₂³×C₁₂₇	—	2	32	no	—
72	C₂³×C₁₂₇	—	2	4	no	—
73	C₂³×C₁₂₇	—	2	4	no	—
74	C₂³×C₁₂₇	—	2	4	no	—
75	C₂³×C₁₂₇	—	2	8	no	—
76	C₂³×C₁₂₇	—	2	16	no	—
77	C₂³×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
78	C₂³×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
79	C₂³×C₁₂₇	#117	1	4	yes	—
80	C₂³×C₁₂₇	C₂×M₁₆	1	8	yes	—
81	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₂₇	—	2	32	no	—
83	C₂×C₄×C₁₂₇	—	2	32	no	—
84	C₂×C₄×C₁₂₇	—	2	8	no	—
85	C₂×C₄×C₁₂₇	—	2	4	no	—
86	C₂×C₄×C₁₂₇	—	2	8	no	—
87	C₂×C₄×C₁₂₇	—	2	8	no	—
88	C₂×C₄×C₁₂₇	—	2	32	no	—
89	C₂×C₄×C₁₂₇	—	2	16	no	—
90	C₂×C₄×C₁₂₇	—	2	16	no	—
91	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
92	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
93	C₂×C₄×C₁₂₇	#117	1	4	yes	—
94	C₂×C₄×C₁₂₇	#123	1	16	yes	—
95	C₂×C₄×C₁₂₇	C₂×M₁₆	1	16	yes	—
96	C₂×C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
97	C₂×C₄×C₁₂₇	C₂×#41	1	4	yes	—
98	C₈×C₁₂₇	—	4	8	no	—
99	C₂²×C₁₂₇	—	2	32	no	—
100	C₂²×C₁₂₇	—	2	4	no	—
101	C₂²×C₁₂₇	—	2	4	no	—
102	C₂²×C₁₂₇	—	2	8	no	—
103	C₂²×C₁₂₇	—	2	16	no	—
104	C₂²×C₁₂₇	—	2	4	no	—
105	C₂²×C₁₂₇	#381	1	4	yes	—
106	C₂²×C₁₂₇	#407	1	8	yes	—
107	C₂²×C₁₂₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
108	C₂²×C₁₂₇	C₂×#117	1	2	yes	—
109	C₂²×C₁₂₇	C₂²×M₁₆	1	4	yes	—
110	C₄×C₁₂₇	—	2	16	no	—
111	C₄×C₁₂₇	—	2	4	no	—
112	C₄×C₁₂₇	#381	1	4	yes	—
113	C₄×C₁₂₇	C₂×#123	1	4	yes	—
114	#20682	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂×C₁₂₇	—	2	4	no	—
116	C₂×C₁₂₇	—	2	8	no	—
117	C₂×C₁₂₇	C₂×#381	1	1	yes	characteristic
118	C₂×C₁₂₇	C₂×#407	1	2	yes	—
119	C₂×C₁₂₇	#2920	1	4	yes	—
120	C₂³×C₄²	C₂×C₁₂₇	1	1	yes	characteristic
121	C₂×#407	C₂×C₁₂₇	1	2	yes	—
122	#1817	C₂×C₁₂₇	1	8	yes	complemented
123	#1728	C₂×C₁₂₇	1	4	yes	complemented
124	C₁₂₇	#20682	1	1	yes	characteristic, direct product factor, Sylow
125	C₂⁴×C₄	C₄×C₁₂₇	1	2	yes	—
126	C₂⁴×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
127	C₂²×M₁₆	C₂²×C₁₂₇	1	4	yes	—
128	C₂²×C₄²	—	2	8	no	—
129	C₂²×C₄²	C₄×C₁₂₇	1	2	yes	—
130	C₂²×C₄²	C₄×C₁₂₇	1	4	yes	—
131	C₂²×C₄²	C₂²×C₁₂₇	1	4	yes	—
132	C₂²×C₄²	C₂²×C₁₂₇	1	2	yes	—
133	#407	—	2	8	no	—
134	C₂×C₂²⋊C₈	C₂²×C₁₂₇	1	8	yes	—
135	#336	C₂²×C₁₂₇	1	16	yes	complemented
136	C₂⁵	C₂×C₄×C₁₂₇	1	1	yes	characteristic
137	C₂³×C₄	—	2	8	no	—
138	C₂³×C₄	—	2	16	no	—
139	C₂³×C₄	—	2	8	no	—
140	C₂³×C₄	C₂×C₄×C₁₂₇	1	4	yes	—
141	C₂³×C₄	C₂×C₄×C₁₂₇	1	4	yes	—
142	C₂³×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
143	C₂³×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
144	C₂³×C₄	C₂×C₄×C₁₂₇	1	4	yes	—
145	C₂³×C₄	D₄×C₁₂₇	1	8	yes	—
146	C₂³×C₄	C₂³×C₁₂₇	1	2	yes	—
147	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
148	C₂×M₁₆	—	2	8	no	—
149	C₂×M₁₆	—	2	16	no	—
150	C₂²×C₈	C₂³×C₁₂₇	1	8	yes	—
151	C₂×C₄²	—	2	8	no	—
152	C₂×C₄²	—	2	16	no	—
153	C₂×C₄²	—	2	16	no	—
154	C₂×C₄²	—	2	8	no	—
155	C₂×C₄²	C₂×C₄×C₁₂₇	1	8	yes	—
156	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
157	C₂×C₄²	C₂³×C₁₂₇	1	4	yes	—
158	C₂²⋊C₈	—	2	16	no	—
159	C₂⁴	—	2	4	no	—
160	C₂⁴	—	2	8	no	—
161	C₂⁴	C₂²⋊C₄×C₁₂₇	1	4	yes	—
162	C₂⁴	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
163	C₂⁴	C₂²×C₄×C₁₂₇	1	2	yes	—
164	C₂²×C₄	—	2	64	no	—
165	C₂²×C₄	—	2	8	no	—
166	C₂²×C₄	—	2	8	no	—
167	C₂²×C₄	—	2	8	no	—
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	—	2	8	no	—
170	C₂²×C₄	—	2	8	no	—
171	C₂²×C₄	—	2	32	no	—
172	C₂²×C₄	—	2	16	no	—
173	C₂²×C₄	—	2	16	no	—
174	C₂²×C₄	—	2	8	no	—
175	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	4	yes	—
176	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	8	yes	—
177	C₂²×C₄	M₁₆×C₁₂₇	1	16	yes	complemented
178	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
179	C₂²×C₄	C₂²×C₄×C₁₂₇	1	8	yes	—
180	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
181	C₂²×C₄	C₂×D₄×C₁₂₇	1	8	yes	—
182	C₂²×C₄	C₂×D₄×C₁₂₇	1	4	yes	—
183	C₂²×C₄	#41×C₁₂₇	1	8	yes	—
184	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
185	M₁₆	—	4	16	no	—
186	C₂×C₈	—	2	8	no	—
187	C₂×C₈	—	2	16	no	—
188	C₄²	—	2	16	no	—
189	C₄²	—	2	8	no	—
190	C₄²	—	2	8	no	—
191	C₂³	—	2	32	no	—
192	C₂³	—	2	4	no	—
193	C₂³	—	2	4	no	—
194	C₂³	—	2	4	no	—
195	C₂³	—	2	8	no	—
196	C₂³	—	2	16	no	—
197	C₂³	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
198	C₂³	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
199	C₂³	#117×C₁₂₇	1	4	yes	—
200	C₂³	C₂×M₁₆×C₁₂₇	1	8	yes	—
201	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
202	C₂×C₄	—	2	32	no	—
203	C₂×C₄	—	2	32	no	—
204	C₂×C₄	—	2	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	8	no	—
207	C₂×C₄	—	2	8	no	—
208	C₂×C₄	—	2	32	no	—
209	C₂×C₄	—	2	16	no	—
210	C₂×C₄	—	2	16	no	—
211	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
212	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
213	C₂×C₄	#117×C₁₂₇	1	4	yes	—
214	C₂×C₄	#123×C₁₂₇	1	16	yes	—
215	C₂×C₄	C₂×M₁₆×C₁₂₇	1	16	yes	—
216	C₂×C₄	C₂²×D₄×C₁₂₇	1	2	yes	—
217	C₂×C₄	C₂×#41×C₁₂₇	1	4	yes	—
218	C₈	—	4	8	no	—
219	C₂²	—	2	32	no	—
220	C₂²	—	2	4	no	—
221	C₂²	—	2	4	no	—
222	C₂²	—	2	8	no	—
223	C₂²	—	2	16	no	—
224	C₂²	—	2	4	no	—
225	C₂²	#381×C₁₂₇	1	4	yes	—
226	C₂²	#407×C₁₂₇	1	8	yes	—
227	C₂²	C₂²×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
228	C₂²	C₂×#117×C₁₂₇	1	2	yes	—
229	C₂²	C₂²×M₁₆×C₁₂₇	1	4	yes	—
230	C₄	—	2	16	no	—
231	C₄	—	2	4	no	—
232	C₄	#381×C₁₂₇	1	4	yes	—
233	C₄	C₂×#123×C₁₂₇	1	4	yes	—
234	C₂	—	2	4	no	—
235	C₂	—	2	8	no	—
236	C₂	C₂×#381×C₁₂₇	1	1	yes	characteristic
237	C₂	C₂×#407×C₁₂₇	1	2	yes	—
238	C₂	C₁₂₇×#2920	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₂₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle	C₂³×C₁₂₇	C₂³×C₄
prime radical	C₂⁵×C₁₂₇	C₂×C₄
LCS	G ▷ C₂³ ▷ 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×#1817	C₂	semidirect
4. C₁₂₇×#1728	C₂	semidirect
15. #336×C₁₂₇	C₂²	semidirect
57. C₂²×C₄×C₁₂₇	M₁₆	semidirect
114. #20682	C₁₂₇	direct (N × H)
122. #1817	C₂×C₁₂₇	semidirect
123. #1728	C₂×C₁₂₇	semidirect
124. C₁₂₇	#20682	direct (N × H)
135. #336	C₂²×C₁₂₇	semidirect
177. C₂²×C₄	M₁₆×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₄²×C₁₂₇	2
C₂×#407×C₁₂₇	2
C₁₂₇×#1817	2
C₁₂₇×#1728	2
#20682	127

Subgroups — C127×#20682 #44584355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#20682 #44584355