Subgroups — C₁₂₇×#11646 #44575319

Order 32512.

314 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#3170	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#2162	C₂	1	1	yes	maximal, characteristic
3	C₂×#335×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1332	C₂	1	4	yes	maximal, complemented
5	#533×C₁₂₇	—	2	1	no	—
6	#500×C₁₂₇	C₄	1	1	yes	characteristic
7	#494×C₁₂₇	C₄	1	1	yes	characteristic
8	#487×C₁₂₇	—	2	1	no	—
9	#467×C₁₂₇	—	2	1	no	—
10	#467×C₁₂₇	C₄	1	2	yes	—
11	C₄×Q₁₆×C₁₂₇	—	2	1	no	—
12	C₄×Q₁₆×C₁₂₇	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₁₂₇	—	2	1	no	—
14	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₁₂₇	—	2	1	no	—
16	C₂²⋊C₁₆×C₁₂₇	—	2	1	no	—
17	C₄×C₁₆×C₁₂₇	C₂²	1	2	yes	—
18	#335×C₁₂₇	—	2	1	no	—
19	#335×C₁₂₇	C₂²	1	2	yes	—
20	C₂×Q₁₆×C₁₂₇	—	2	1	no	—
21	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₂₇	—	4	1	no	—
23	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
25	C₄⋊Q₈×C₁₂₇	—	2	1	no	—
26	C₄⋊Q₈×C₁₂₇	—	2	1	no	—
27	#125×C₁₂₇	—	4	1	no	—
28	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
29	C₂²⋊Q₈×C₁₂₇	—	4	1	no	—
30	C₄×Q₈×C₁₂₇	—	2	2	no	—
31	C₄×Q₈×C₁₂₇	—	4	1	no	—
32	#117×C₁₂₇	—	4	1	no	—
33	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
34	C₂×C₁₆×C₁₂₇	—	2	2	no	—
35	#107×C₁₂₇	—	2	1	no	—
36	#107×C₁₂₇	C₂×C₄	1	2	yes	—
37	C₄⋊C₈×C₁₂₇	—	2	2	no	—
38	C₄⋊C₈×C₁₂₇	—	4	1	no	—
39	#103×C₁₂₇	—	2	2	no	—
40	#103×C₁₂₇	—	2	2	no	—
41	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
43	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
45	C₂×Q₈×C₁₂₇	—	2	2	no	—
46	C₂×Q₈×C₁₂₇	—	4	1	no	—
47	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
50	Q₁₆×C₁₂₇	—	2	4	no	—
51	Q₁₆×C₁₂₇	—	4	2	no	—
52	C₂×C₈×C₁₂₇	—	2	2	no	—
53	C₂×C₈×C₁₂₇	—	4	1	no	—
54	C₂×C₈×C₁₂₇	—	4	2	no	—
55	C₂×C₈×C₁₂₇	—	8	2	no	—
56	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
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	—	4	1	no	—
62	C₄⋊C₄×C₁₂₇	—	4	1	no	—
63	C₄⋊C₄×C₁₂₇	—	4	2	no	—
64	C₄⋊C₄×C₁₂₇	—	4	1	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	—	4	2	no	—
75	Q₈×C₁₂₇	—	4	2	no	—
76	Q₈×C₁₂₇	—	8	2	no	—
77	C₂×C₄×C₁₂₇	—	2	1	no	—
78	C₂×C₄×C₁₂₇	—	2	1	no	—
79	C₂×C₄×C₁₂₇	—	4	1	no	—
80	C₂×C₄×C₁₂₇	—	4	2	no	—
81	C₂×C₄×C₁₂₇	—	4	2	no	—
82	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
83	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
84	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
85	C₂×C₄×C₁₂₇	SD₃₂	1	2	yes	—
86	C₂×C₄×C₁₂₇	Q₃₂	1	2	yes	—
87	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
90	C₈×C₁₂₇	—	2	1	no	—
91	C₈×C₁₂₇	—	2	1	no	—
92	C₈×C₁₂₇	—	2	1	no	—
93	C₈×C₁₂₇	—	8	2	no	—
94	C₈×C₁₂₇	#131	1	2	yes	—
95	C₂²×C₁₂₇	—	2	1	no	—
96	C₂²×C₁₂₇	—	2	1	no	—
97	C₂²×C₁₂₇	#358	1	2	yes	—
98	C₂²×C₁₂₇	C₂×#102	1	1	yes	characteristic
99	C₄×C₁₂₇	—	2	1	no	—
100	C₄×C₁₂₇	—	2	1	no	—
101	C₄×C₁₂₇	—	2	1	no	—
102	C₄×C₁₂₇	—	8	2	no	—
103	C₄×C₁₂₇	—	8	2	no	—
104	C₄×C₁₂₇	#358	1	2	yes	—
105	C₄×C₁₂₇	#408	1	1	yes	characteristic
106	C₄×C₁₂₇	#421	1	1	yes	characteristic
107	C₄×C₁₂₇	C₂×SD₃₂	1	1	yes	characteristic
108	C₄×C₁₂₇	C₂×Q₃₂	1	1	yes	characteristic
109	#11646	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂×C₁₂₇	—	2	2	no	—
111	C₂×C₁₂₇	#1480	1	1	yes	characteristic
112	C₂×C₁₂₇	C₂×#358	1	1	yes	characteristic
113	C₂×C₁₂₇	#2143	1	1	yes	characteristic
114	#3170	C₂×C₁₂₇	1	1	yes	characteristic
115	#2162	C₂×C₁₂₇	1	1	yes	characteristic
116	C₂×#335	C₂×C₁₂₇	1	1	yes	characteristic
117	#1332	C₂×C₁₂₇	1	4	yes	complemented
118	C₁₂₇	#11646	1	1	yes	characteristic, direct product factor, Sylow
119	#533	—	2	1	no	—
120	#500	C₄×C₁₂₇	1	1	yes	characteristic
121	#494	C₄×C₁₂₇	1	1	yes	characteristic
122	#487	—	2	1	no	—
123	#467	—	2	1	no	—
124	#467	C₄×C₁₂₇	1	2	yes	—
125	C₄×Q₁₆	—	2	1	no	—
126	C₄×Q₁₆	C₂²×C₁₂₇	1	2	yes	—
127	C₂×C₄⋊C₈	—	2	1	no	—
128	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
129	C₄⋊C₁₆	—	2	1	no	—
130	C₂²⋊C₁₆	—	2	1	no	—
131	C₄×C₁₆	C₂²×C₁₂₇	1	2	yes	—
132	#335	—	2	1	no	—
133	#335	C₂²×C₁₂₇	1	2	yes	—
134	C₂×Q₁₆	—	2	1	no	—
135	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
136	C₂²×C₈	—	4	1	no	—
137	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
138	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
139	C₄⋊Q₈	—	2	1	no	—
140	C₄⋊Q₈	—	2	1	no	—
141	#125	—	4	1	no	—
142	C₂²⋊Q₈	—	2	2	no	—
143	C₂²⋊Q₈	—	4	1	no	—
144	C₄×Q₈	—	2	2	no	—
145	C₄×Q₈	—	4	1	no	—
146	#117	—	4	1	no	—
147	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
148	C₂×C₁₆	—	2	2	no	—
149	#107	—	2	1	no	—
150	#107	C₂×C₄×C₁₂₇	1	2	yes	—
151	C₄⋊C₈	—	2	2	no	—
152	C₄⋊C₈	—	4	1	no	—
153	#103	—	2	2	no	—
154	#103	—	2	2	no	—
155	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
156	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
157	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
158	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	—	4	1	no	—
161	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
162	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
163	C₂²×C₄	SD₁₆×C₁₂₇	1	1	yes	characteristic
164	Q₁₆	—	2	4	no	—
165	Q₁₆	—	4	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	4	1	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	8	2	no	—
170	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
171	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
172	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
173	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	4	1	no	—
176	C₄⋊C₄	—	4	1	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	1	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₂²⋊C₄	—	4	2	no	—
181	C₄²	—	4	2	no	—
182	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
183	C₄²	D₈×C₁₂₇	1	1	yes	characteristic
184	C₄²	SD₁₆×C₁₂₇	1	1	yes	characteristic
185	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
186	C₁₆	—	4	2	no	—
187	C₂³	#102×C₁₂₇	1	1	yes	characteristic
188	Q₈	—	4	2	no	—
189	Q₈	—	4	2	no	—
190	Q₈	—	8	2	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
197	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
198	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
199	C₂×C₄	SD₃₂×C₁₂₇	1	2	yes	—
200	C₂×C₄	Q₃₂×C₁₂₇	1	2	yes	—
201	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
202	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
203	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
204	C₈	—	2	1	no	—
205	C₈	—	2	1	no	—
206	C₈	—	2	1	no	—
207	C₈	—	8	2	no	—
208	C₈	#131×C₁₂₇	1	2	yes	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	#358×C₁₂₇	1	2	yes	—
212	C₂²	C₂×#102×C₁₂₇	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	8	2	no	—
217	C₄	—	8	2	no	—
218	C₄	#358×C₁₂₇	1	2	yes	—
219	C₄	#408×C₁₂₇	1	1	yes	characteristic
220	C₄	#421×C₁₂₇	1	1	yes	characteristic
221	C₄	C₂×SD₃₂×C₁₂₇	1	1	yes	characteristic
222	C₄	C₂×Q₃₂×C₁₂₇	1	1	yes	characteristic
223	C₂	—	2	2	no	—
224	C₂	C₁₂₇×#1480	1	1	yes	characteristic
225	C₂	C₂×#358×C₁₂₇	1	1	yes	characteristic
226	C₂	C₁₂₇×#2143	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂×D₈
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 ▷ 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×#1332	C₂	semidirect
109. #11646	C₁₂₇	direct (N × H)
117. #1332	C₂×C₁₂₇	semidirect
118. C₁₂₇	#11646	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×#3170	2
C₁₂₇×#2162	2
C₂×#335×C₁₂₇	2
C₁₂₇×#1332	2
#11646	127

Subgroups — C127×#11646 #44575319

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#11646 #44575319