Subgroups — C₁₂₇×#11934 #44575607

Order 32512.

306 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#3160	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1564	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1363	C₂	1	4	yes	maximal, complemented
5	#533×C₁₂₇	—	2	1	no	—
6	C₂²×C₁₆×C₁₂₇	—	2	1	no	—
7	#500×C₁₂₇	C₄	1	1	yes	characteristic
8	#500×C₁₂₇	C₄	1	1	yes	characteristic
9	#499×C₁₂₇	—	2	1	no	—
10	#499×C₁₂₇	C₂²	1	2	yes	—
11	#498×C₁₂₇	—	2	1	no	—
12	#431×C₁₂₇	—	2	1	no	—
13	#427×C₁₂₇	—	2	1	no	—
14	#427×C₁₂₇	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₁₆×C₁₂₇	—	2	1	no	—
17	C₄⋊C₁₆×C₁₂₇	C₂²	1	2	yes	—
18	#335×C₁₂₇	—	2	1	no	—
19	#334×C₁₂₇	C₂²	1	2	yes	—
20	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
22	C₄⋊Q₈×C₁₂₇	—	2	1	no	—
23	C₄⋊Q₈×C₁₂₇	—	2	1	no	—
24	#125×C₁₂₇	—	2	2	no	—
25	C₂²⋊Q₈×C₁₂₇	—	4	1	no	—
26	C₂²⋊Q₈×C₁₂₇	—	4	1	no	—
27	C₄×Q₈×C₁₂₇	—	4	1	no	—
28	C₄×Q₈×C₁₂₇	—	4	1	no	—
29	#117×C₁₂₇	—	2	2	no	—
30	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₁₂₇	—	2	2	no	—
32	C₂×C₁₆×C₁₂₇	—	4	2	no	—
33	#107×C₁₂₇	—	2	1	no	—
34	#107×C₁₂₇	—	2	1	no	—
35	#107×C₁₂₇	C₂×C₄	1	2	yes	—
36	#107×C₁₂₇	C₂×C₄	1	2	yes	—
37	#106×C₁₂₇	—	2	2	no	—
38	#106×C₁₂₇	—	2	2	no	—
39	C₄⋊C₈×C₁₂₇	—	2	2	no	—
40	C₂²⋊C₈×C₁₂₇	—	4	1	no	—
41	C₄×C₈×C₁₂₇	—	4	1	no	—
42	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
45	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
46	C₂×Q₈×C₁₂₇	—	4	1	no	—
47	C₂×Q₈×C₁₂₇	—	4	1	no	—
48	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
51	C₂×C₈×C₁₂₇	—	2	2	no	—
52	C₂×C₈×C₁₂₇	—	4	2	no	—
53	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₂₇	SD₁₆	1	2	yes	—
56	C₂×C₈×C₁₂₇	Q₁₆	1	2	yes	—
57	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
59	C₄⋊C₄×C₁₂₇	—	2	2	no	—
60	C₄⋊C₄×C₁₂₇	—	2	2	no	—
61	C₄⋊C₄×C₁₂₇	—	4	1	no	—
62	C₄⋊C₄×C₁₂₇	—	4	1	no	—
63	C₄⋊C₄×C₁₂₇	—	4	2	no	—
64	C₄⋊C₄×C₁₂₇	—	4	2	no	—
65	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
66	C₄²×C₁₂₇	—	4	2	no	—
67	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²×C₁₂₇	D₈	1	1	yes	characteristic
69	C₄²×C₁₂₇	SD₁₆	1	1	yes	characteristic
70	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
71	C₁₆×C₁₂₇	—	4	2	no	—
72	C₂³×C₁₂₇	#102	1	1	yes	characteristic
73	Q₈×C₁₂₇	—	8	2	no	—
74	Q₈×C₁₂₇	—	8	2	no	—
75	C₂×C₄×C₁₂₇	—	2	2	no	—
76	C₂×C₄×C₁₂₇	—	4	1	no	—
77	C₂×C₄×C₁₂₇	—	4	2	no	—
78	C₂×C₄×C₁₂₇	—	4	2	no	—
79	C₂×C₄×C₁₂₇	#101	1	2	yes	—
80	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
81	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
82	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
83	C₂×C₄×C₁₂₇	#103	1	2	yes	—
84	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
87	C₈×C₁₂₇	—	2	1	no	—
88	C₈×C₁₂₇	—	2	1	no	—
89	C₈×C₁₂₇	—	8	2	no	—
90	C₈×C₁₂₇	#103	1	2	yes	—
91	C₈×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
92	C₈×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
93	C₂²×C₁₂₇	—	2	1	no	—
94	C₂²×C₁₂₇	—	2	1	no	—
95	C₂²×C₁₂₇	#332	1	2	yes	—
96	C₂²×C₁₂₇	C₂×#102	1	1	yes	characteristic
97	C₄×C₁₂₇	—	2	1	no	—
98	C₄×C₁₂₇	—	2	1	no	—
99	C₄×C₁₂₇	—	2	1	no	—
100	C₄×C₁₂₇	—	8	2	no	—
101	C₄×C₁₂₇	—	8	2	no	—
102	C₄×C₁₂₇	#332	1	2	yes	—
103	C₄×C₁₂₇	#359	1	2	yes	—
104	C₄×C₁₂₇	C₂×#101	1	1	yes	characteristic
105	C₄×C₁₂₇	C₂×#103	1	1	yes	characteristic
106	#11934	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂×C₁₂₇	—	2	2	no	—
108	C₂×C₁₂₇	C₂×#332	1	1	yes	characteristic
109	C₂×C₁₂₇	C₂×#359	1	1	yes	characteristic
110	C₂×C₁₂₇	#2145	1	1	yes	characteristic
111	#3160	C₂×C₁₂₇	1	1	yes	characteristic
112	C₂×C₄⋊C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
113	#1564	C₂×C₁₂₇	1	1	yes	characteristic
114	#1363	C₂×C₁₂₇	1	4	yes	complemented
115	C₁₂₇	#11934	1	1	yes	characteristic, direct product factor, Sylow
116	#533	—	2	1	no	—
117	C₂²×C₁₆	—	2	1	no	—
118	#500	C₄×C₁₂₇	1	1	yes	characteristic
119	#500	C₄×C₁₂₇	1	1	yes	characteristic
120	#499	—	2	1	no	—
121	#499	C₂²×C₁₂₇	1	2	yes	—
122	#498	—	2	1	no	—
123	#431	—	2	1	no	—
124	#427	—	2	1	no	—
125	#427	C₄×C₁₂₇	1	2	yes	—
126	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
127	C₄⋊C₁₆	—	2	1	no	—
128	C₄⋊C₁₆	C₂²×C₁₂₇	1	2	yes	—
129	#335	—	2	1	no	—
130	#334	C₂²×C₁₂₇	1	2	yes	—
131	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
132	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
133	C₄⋊Q₈	—	2	1	no	—
134	C₄⋊Q₈	—	2	1	no	—
135	#125	—	2	2	no	—
136	C₂²⋊Q₈	—	4	1	no	—
137	C₂²⋊Q₈	—	4	1	no	—
138	C₄×Q₈	—	4	1	no	—
139	C₄×Q₈	—	4	1	no	—
140	#117	—	2	2	no	—
141	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
142	C₂×C₁₆	—	2	2	no	—
143	C₂×C₁₆	—	4	2	no	—
144	#107	—	2	1	no	—
145	#107	—	2	1	no	—
146	#107	C₂×C₄×C₁₂₇	1	2	yes	—
147	#107	C₂×C₄×C₁₂₇	1	2	yes	—
148	#106	—	2	2	no	—
149	#106	—	2	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₂²⋊C₈	—	4	1	no	—
152	C₄×C₈	—	4	1	no	—
153	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
154	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
155	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
156	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
157	C₂×Q₈	—	4	1	no	—
158	C₂×Q₈	—	4	1	no	—
159	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
160	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
161	C₂²×C₄	SD₁₆×C₁₂₇	1	1	yes	characteristic
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	4	2	no	—
164	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
165	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
166	C₂×C₈	SD₁₆×C₁₂₇	1	2	yes	—
167	C₂×C₈	Q₁₆×C₁₂₇	1	2	yes	—
168	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
169	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	4	1	no	—
173	C₄⋊C₄	—	4	1	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₂²⋊C₄	—	4	2	no	—
177	C₄²	—	4	2	no	—
178	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
179	C₄²	D₈×C₁₂₇	1	1	yes	characteristic
180	C₄²	SD₁₆×C₁₂₇	1	1	yes	characteristic
181	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
182	C₁₆	—	4	2	no	—
183	C₂³	#102×C₁₂₇	1	1	yes	characteristic
184	Q₈	—	8	2	no	—
185	Q₈	—	8	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	4	1	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	#101×C₁₂₇	1	2	yes	—
191	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
192	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
193	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
194	C₂×C₄	#103×C₁₂₇	1	2	yes	—
195	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
196	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
197	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
198	C₈	—	2	1	no	—
199	C₈	—	2	1	no	—
200	C₈	—	8	2	no	—
201	C₈	#103×C₁₂₇	1	2	yes	—
202	C₈	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
203	C₈	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	#332×C₁₂₇	1	2	yes	—
207	C₂²	C₂×#102×C₁₂₇	1	1	yes	characteristic
208	C₄	—	2	1	no	—
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	8	2	no	—
212	C₄	—	8	2	no	—
213	C₄	#332×C₁₂₇	1	2	yes	—
214	C₄	#359×C₁₂₇	1	2	yes	—
215	C₄	C₂×#101×C₁₂₇	1	1	yes	characteristic
216	C₄	C₂×#103×C₁₂₇	1	1	yes	characteristic
217	C₂	—	2	2	no	—
218	C₂	C₂×#332×C₁₂₇	1	1	yes	characteristic
219	C₂	C₂×#359×C₁₂₇	1	1	yes	characteristic
220	C₂	C₁₂₇×#2145	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₂₇	#102
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂×#102
prime radical	C₂³×C₁₂₇	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₂₇×#1363	C₂	semidirect
106. #11934	C₁₂₇	direct (N × H)
114. #1363	C₂×C₁₂₇	semidirect
115. C₁₂₇	#11934	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₂₇×#3160	2
C₂×C₄⋊C₁₆×C₁₂₇	2
C₁₂₇×#1564	2
C₁₂₇×#1363	2
#11934	127

Subgroups — C127×#11934 #44575607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#11934 #44575607