Subgroups — C₁₂₇×#11936 #44575609

Order 32512.

310 conjugacy classes of subgroups.

The subgroup lattice has 224 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	C₁₂₇×#2150	C₂	1	1	yes	maximal, characteristic
3	C₂×#334×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1363	C₂	1	4	yes	maximal, complemented
5	#533×C₁₂₇	—	2	1	no	—
6	C₂×M₃₂×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	#427×C₁₂₇	—	2	1	no	—
13	#427×C₁₂₇	C₄	1	2	yes	—
14	C₂×C₄⋊C₈×C₁₂₇	—	2	1	no	—
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	#334×C₁₂₇	—	2	1	no	—
19	#334×C₁₂₇	C₂²	1	2	yes	—
20	C₂²×C₈×C₁₂₇	—	4	1	no	—
21	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
23	C₄⋊Q₈×C₁₂₇	—	2	1	no	—
24	C₄⋊Q₈×C₁₂₇	—	2	1	no	—
25	#125×C₁₂₇	—	2	2	no	—
26	C₂²⋊Q₈×C₁₂₇	—	4	1	no	—
27	C₂²⋊Q₈×C₁₂₇	—	4	1	no	—
28	C₄×Q₈×C₁₂₇	—	4	1	no	—
29	C₄×Q₈×C₁₂₇	—	4	1	no	—
30	#117×C₁₂₇	—	2	2	no	—
31	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
32	M₃₂×C₁₂₇	—	4	2	no	—
33	C₂×C₁₆×C₁₂₇	—	2	2	no	—
34	#107×C₁₂₇	—	2	1	no	—
35	#107×C₁₂₇	—	2	1	no	—
36	#107×C₁₂₇	C₂×C₄	1	2	yes	—
37	#107×C₁₂₇	C₂×C₄	1	2	yes	—
38	#106×C₁₂₇	—	2	2	no	—
39	#106×C₁₂₇	—	2	2	no	—
40	C₄⋊C₈×C₁₂₇	—	2	2	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	—	8	2	no	—
54	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₂₇	SD₁₆	1	2	yes	—
57	C₂×C₈×C₁₂₇	Q₁₆	1	2	yes	—
58	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
60	C₄⋊C₄×C₁₂₇	—	2	2	no	—
61	C₄⋊C₄×C₁₂₇	—	2	2	no	—
62	C₄⋊C₄×C₁₂₇	—	4	1	no	—
63	C₄⋊C₄×C₁₂₇	—	4	1	no	—
64	C₄⋊C₄×C₁₂₇	—	4	2	no	—
65	C₄⋊C₄×C₁₂₇	—	4	2	no	—
66	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
67	C₄²×C₁₂₇	—	4	2	no	—
68	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₄²×C₁₂₇	D₈	1	1	yes	characteristic
70	C₄²×C₁₂₇	SD₁₆	1	1	yes	characteristic
71	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
72	C₁₆×C₁₂₇	—	4	2	no	—
73	C₂³×C₁₂₇	#102	1	1	yes	characteristic
74	Q₈×C₁₂₇	—	8	2	no	—
75	Q₈×C₁₂₇	—	8	2	no	—
76	C₂×C₄×C₁₂₇	—	2	2	no	—
77	C₂×C₄×C₁₂₇	—	4	1	no	—
78	C₂×C₄×C₁₂₇	—	4	2	no	—
79	C₂×C₄×C₁₂₇	—	4	2	no	—
80	C₂×C₄×C₁₂₇	#101	1	2	yes	—
81	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
82	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
83	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
84	C₂×C₄×C₁₂₇	#103	1	2	yes	—
85	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
88	C₈×C₁₂₇	—	2	1	no	—
89	C₈×C₁₂₇	—	2	1	no	—
90	C₈×C₁₂₇	—	8	2	no	—
91	C₈×C₁₂₇	#103	1	2	yes	—
92	C₈×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
93	C₈×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
94	C₂²×C₁₂₇	—	2	1	no	—
95	C₂²×C₁₂₇	—	2	1	no	—
96	C₂²×C₁₂₇	#332	1	2	yes	—
97	C₂²×C₁₂₇	C₂×#102	1	1	yes	characteristic
98	C₄×C₁₂₇	—	2	1	no	—
99	C₄×C₁₂₇	—	2	1	no	—
100	C₄×C₁₂₇	—	2	1	no	—
101	C₄×C₁₂₇	—	8	2	no	—
102	C₄×C₁₂₇	—	8	2	no	—
103	C₄×C₁₂₇	#332	1	2	yes	—
104	C₄×C₁₂₇	#360	1	2	yes	—
105	C₄×C₁₂₇	C₂×#101	1	1	yes	characteristic
106	C₄×C₁₂₇	C₂×#103	1	1	yes	characteristic
107	#11936	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂×C₁₂₇	—	2	2	no	—
109	C₂×C₁₂₇	C₂×#332	1	1	yes	characteristic
110	C₂×C₁₂₇	#2143	1	1	yes	characteristic
111	C₂×C₁₂₇	C₂×#360	1	1	yes	characteristic
112	#3160	C₂×C₁₂₇	1	1	yes	characteristic
113	#2150	C₂×C₁₂₇	1	1	yes	characteristic
114	C₂×#334	C₂×C₁₂₇	1	1	yes	characteristic
115	#1363	C₂×C₁₂₇	1	4	yes	complemented
116	C₁₂₇	#11936	1	1	yes	characteristic, direct product factor, Sylow
117	#533	—	2	1	no	—
118	C₂×M₃₂	—	2	1	no	—
119	#500	C₄×C₁₂₇	1	1	yes	characteristic
120	#500	C₄×C₁₂₇	1	1	yes	characteristic
121	#499	—	2	1	no	—
122	#499	C₂²×C₁₂₇	1	2	yes	—
123	#498	—	2	1	no	—
124	#427	—	2	1	no	—
125	#427	C₄×C₁₂₇	1	2	yes	—
126	C₂×C₄⋊C₈	—	2	1	no	—
127	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
128	C₄⋊C₁₆	—	2	1	no	—
129	C₄⋊C₁₆	C₂²×C₁₂₇	1	2	yes	—
130	#334	—	2	1	no	—
131	#334	C₂²×C₁₂₇	1	2	yes	—
132	C₂²×C₈	—	4	1	no	—
133	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
134	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
135	C₄⋊Q₈	—	2	1	no	—
136	C₄⋊Q₈	—	2	1	no	—
137	#125	—	2	2	no	—
138	C₂²⋊Q₈	—	4	1	no	—
139	C₂²⋊Q₈	—	4	1	no	—
140	C₄×Q₈	—	4	1	no	—
141	C₄×Q₈	—	4	1	no	—
142	#117	—	2	2	no	—
143	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
144	M₃₂	—	4	2	no	—
145	C₂×C₁₆	—	2	2	no	—
146	#107	—	2	1	no	—
147	#107	—	2	1	no	—
148	#107	C₂×C₄×C₁₂₇	1	2	yes	—
149	#107	C₂×C₄×C₁₂₇	1	2	yes	—
150	#106	—	2	2	no	—
151	#106	—	2	2	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	4	1	no	—
154	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
155	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
156	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
157	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
158	C₂×Q₈	—	4	1	no	—
159	C₂×Q₈	—	4	1	no	—
160	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
161	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
162	C₂²×C₄	SD₁₆×C₁₂₇	1	1	yes	characteristic
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	4	2	no	—
165	C₂×C₈	—	8	2	no	—
166	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
167	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
168	C₂×C₈	SD₁₆×C₁₂₇	1	2	yes	—
169	C₂×C₈	Q₁₆×C₁₂₇	1	2	yes	—
170	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
171	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	4	1	no	—
175	C₄⋊C₄	—	4	1	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₂²⋊C₄	—	4	2	no	—
179	C₄²	—	4	2	no	—
180	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
181	C₄²	D₈×C₁₂₇	1	1	yes	characteristic
182	C₄²	SD₁₆×C₁₂₇	1	1	yes	characteristic
183	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
184	C₁₆	—	4	2	no	—
185	C₂³	#102×C₁₂₇	1	1	yes	characteristic
186	Q₈	—	8	2	no	—
187	Q₈	—	8	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	4	1	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	#101×C₁₂₇	1	2	yes	—
193	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
194	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
195	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
196	C₂×C₄	#103×C₁₂₇	1	2	yes	—
197	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
198	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
199	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
200	C₈	—	2	1	no	—
201	C₈	—	2	1	no	—
202	C₈	—	8	2	no	—
203	C₈	#103×C₁₂₇	1	2	yes	—
204	C₈	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
205	C₈	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#332×C₁₂₇	1	2	yes	—
209	C₂²	C₂×#102×C₁₂₇	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	8	2	no	—
214	C₄	—	8	2	no	—
215	C₄	#332×C₁₂₇	1	2	yes	—
216	C₄	#360×C₁₂₇	1	2	yes	—
217	C₄	C₂×#101×C₁₂₇	1	1	yes	characteristic
218	C₄	C₂×#103×C₁₂₇	1	1	yes	characteristic
219	C₂	—	2	2	no	—
220	C₂	C₂×#332×C₁₂₇	1	1	yes	characteristic
221	C₂	C₁₂₇×#2143	1	1	yes	characteristic
222	C₂	C₂×#360×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₂₇	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
prime radical	C₂³×C₁₂₇	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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
107. #11936	C₁₂₇	direct (N × H)
115. #1363	C₂×C₁₂₇	semidirect
116. C₁₂₇	#11936	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
C₁₂₇×#2150	2
C₂×#334×C₁₂₇	2
C₁₂₇×#1363	2
#11936	127

Subgroups — C127×#11936 #44575609

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#11936 #44575609