Subgroups — C₁₂₇×#11612 #44575285

Order 32512.

322 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#2960	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#2150	C₂	1	1	yes	maximal, characteristic
3	C₂×#346×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1338	C₂	1	4	yes	maximal, complemented
5	#533×C₁₂₇	C₄	1	1	yes	characteristic
6	C₂×M₃₂×C₁₂₇	—	2	1	no	—
7	C₂²×C₁₆×C₁₂₇	—	2	1	no	—
8	#446×C₁₂₇	—	2	1	no	—
9	#446×C₁₂₇	C₂²	1	2	yes	—
10	C₈×Q₈×C₁₂₇	—	2	1	no	—
11	C₈×Q₈×C₁₂₇	C₄	1	2	yes	—
12	#424×C₁₂₇	—	2	1	no	—
13	#423×C₁₂₇	C₄	1	1	yes	characteristic
14	#405×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	#346×C₁₂₇	—	2	1	no	—
19	#346×C₁₂₇	C₂²	1	2	yes	—
20	C₂×M₁₆×C₁₂₇	—	2	1	no	—
21	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
23	C₄⋊Q₈×C₁₂₇	C₈	1	1	yes	characteristic
24	C₄⋊Q₈×C₁₂₇	C₈	1	1	yes	characteristic
25	#125×C₁₂₇	—	2	1	no	—
26	C₂²⋊Q₈×C₁₂₇	—	2	1	no	—
27	C₂²⋊Q₈×C₁₂₇	C₈	1	2	yes	—
28	C₄×Q₈×C₁₂₇	—	2	1	no	—
29	C₄×Q₈×C₁₂₇	C₂×C₄	1	2	yes	—
30	#117×C₁₂₇	—	2	1	no	—
31	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
32	M₃₂×C₁₂₇	—	4	2	no	—
33	C₂×C₁₆×C₁₂₇	—	2	4	no	—
34	C₂×C₁₆×C₁₂₇	—	2	2	no	—
35	C₂×C₁₆×C₁₂₇	—	2	2	no	—
36	C₄⋊C₈×C₁₂₇	—	2	2	no	—
37	C₄⋊C₈×C₁₂₇	—	2	2	no	—
38	C₄⋊C₈×C₁₂₇	—	2	1	no	—
39	C₄⋊C₈×C₁₂₇	C₂×C₄	1	2	yes	—
40	#97×C₁₂₇	—	2	2	no	—
41	C₄×C₈×C₁₂₇	—	2	2	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	—	2	1	no	—
47	C₂×Q₈×C₁₂₇	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₂₇	M₁₆	1	1	yes	characteristic
51	M₁₆×C₁₂₇	—	4	2	no	—
52	C₂×C₈×C₁₂₇	—	2	2	no	—
53	C₂×C₈×C₁₂₇	—	2	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	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
60	C₄⋊C₄×C₁₂₇	—	2	1	no	—
61	C₄⋊C₄×C₁₂₇	—	2	1	no	—
62	C₄⋊C₄×C₁₂₇	—	2	1	no	—
63	C₄⋊C₄×C₁₂₇	—	2	2	no	—
64	C₄⋊C₄×C₁₂₇	—	2	2	no	—
65	C₄⋊C₄×C₁₂₇	C₂×C₈	1	2	yes	—
66	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
67	C₄²×C₁₂₇	—	2	2	no	—
68	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₄²×C₁₂₇	C₂×C₈	1	1	yes	characteristic
70	C₄²×C₁₂₇	M₁₆	1	1	yes	characteristic
71	C₄²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
72	C₁₆×C₁₂₇	—	2	4	no	—
73	C₁₆×C₁₂₇	—	4	2	no	—
74	C₂³×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
75	Q₈×C₁₂₇	—	4	2	no	—
76	Q₈×C₁₂₇	—	4	2	no	—
77	C₂×C₄×C₁₂₇	—	2	2	no	—
78	C₂×C₄×C₁₂₇	—	2	2	no	—
79	C₂×C₄×C₁₂₇	—	4	1	no	—
80	C₂×C₄×C₁₂₇	—	4	1	no	—
81	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
82	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
83	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
84	C₂×C₄×C₁₂₇	#103	1	2	yes	—
85	C₂×C₄×C₁₂₇	#104	1	2	yes	—
86	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
89	C₈×C₁₂₇	—	2	1	no	—
90	C₈×C₁₂₇	—	2	1	no	—
91	C₈×C₁₂₇	—	4	2	no	—
92	C₈×C₁₂₇	#103	1	2	yes	—
93	C₈×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
94	C₈×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
95	C₂²×C₁₂₇	—	2	1	no	—
96	C₂²×C₁₂₇	—	2	1	no	—
97	C₂²×C₁₂₇	Q₈⋊C₈	1	2	yes	—
98	C₂²×C₁₂₇	C₂×C₂²⋊C₈	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	—	2	1	no	—
103	C₄×C₁₂₇	—	4	2	no	—
104	C₄×C₁₂₇	—	4	2	no	—
105	C₄×C₁₂₇	Q₈⋊C₈	1	2	yes	—
106	C₄×C₁₂₇	C₂×#103	1	1	yes	characteristic
107	C₄×C₁₂₇	C₂×#104	1	1	yes	characteristic
108	#11612	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂×C₁₂₇	—	2	2	no	—
110	C₂×C₁₂₇	C₂×Q₈⋊C₈	1	1	yes	characteristic
111	C₂×C₁₂₇	#2115	1	1	yes	characteristic
112	C₂×C₁₂₇	#2117	1	1	yes	characteristic
113	#2960	C₂×C₁₂₇	1	1	yes	characteristic
114	#2150	C₂×C₁₂₇	1	1	yes	characteristic
115	C₂×#346	C₂×C₁₂₇	1	1	yes	characteristic
116	#1338	C₂×C₁₂₇	1	4	yes	complemented
117	C₁₂₇	#11612	1	1	yes	characteristic, direct product factor, Sylow
118	#533	C₄×C₁₂₇	1	1	yes	characteristic
119	C₂×M₃₂	—	2	1	no	—
120	C₂²×C₁₆	—	2	1	no	—
121	#446	—	2	1	no	—
122	#446	C₂²×C₁₂₇	1	2	yes	—
123	C₈×Q₈	—	2	1	no	—
124	C₈×Q₈	C₄×C₁₂₇	1	2	yes	—
125	#424	—	2	1	no	—
126	#423	C₄×C₁₂₇	1	1	yes	characteristic
127	#405	—	2	1	no	—
128	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
129	C₄⋊C₁₆	—	2	1	no	—
130	C₄⋊C₁₆	C₂²×C₁₂₇	1	2	yes	—
131	#346	—	2	1	no	—
132	#346	C₂²×C₁₂₇	1	2	yes	—
133	C₂×M₁₆	—	2	1	no	—
134	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
135	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
136	C₄⋊Q₈	C₈×C₁₂₇	1	1	yes	characteristic
137	C₄⋊Q₈	C₈×C₁₂₇	1	1	yes	characteristic
138	#125	—	2	1	no	—
139	C₂²⋊Q₈	—	2	1	no	—
140	C₂²⋊Q₈	C₈×C₁₂₇	1	2	yes	—
141	C₄×Q₈	—	2	1	no	—
142	C₄×Q₈	C₂×C₄×C₁₂₇	1	2	yes	—
143	#117	—	2	1	no	—
144	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
145	M₃₂	—	4	2	no	—
146	C₂×C₁₆	—	2	4	no	—
147	C₂×C₁₆	—	2	2	no	—
148	C₂×C₁₆	—	2	2	no	—
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	2	1	no	—
152	C₄⋊C₈	C₂×C₄×C₁₂₇	1	2	yes	—
153	#97	—	2	2	no	—
154	C₄×C₈	—	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	1	no	—
160	C₂×Q₈	C₂×C₈×C₁₂₇	1	2	yes	—
161	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
162	C₂²×C₄	C₂×C₈×C₁₂₇	1	1	yes	characteristic
163	C₂²×C₄	M₁₆×C₁₂₇	1	1	yes	characteristic
164	M₁₆	—	4	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
168	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
169	C₂×C₈	SD₁₆×C₁₂₇	1	2	yes	—
170	C₂×C₈	Q₁₆×C₁₂₇	1	2	yes	—
171	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
172	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	—	2	1	no	—
175	C₄⋊C₄	—	2	1	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	C₂×C₈×C₁₂₇	1	2	yes	—
179	C₂²⋊C₄	—	2	2	no	—
180	C₄²	—	2	2	no	—
181	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
182	C₄²	C₂×C₈×C₁₂₇	1	1	yes	characteristic
183	C₄²	M₁₆×C₁₂₇	1	1	yes	characteristic
184	C₄²	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
185	C₁₆	—	2	4	no	—
186	C₁₆	—	4	2	no	—
187	C₂³	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
188	Q₈	—	4	2	no	—
189	Q₈	—	4	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
195	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
196	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
197	C₂×C₄	#103×C₁₂₇	1	2	yes	—
198	C₂×C₄	#104×C₁₂₇	1	2	yes	—
199	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
200	C₂×C₄	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
201	C₂×C₄	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
202	C₈	—	2	1	no	—
203	C₈	—	2	1	no	—
204	C₈	—	4	2	no	—
205	C₈	#103×C₁₂₇	1	2	yes	—
206	C₈	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
207	C₈	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	Q₈⋊C₈×C₁₂₇	1	2	yes	—
211	C₂²	C₂×C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	4	2	no	—
217	C₄	—	4	2	no	—
218	C₄	Q₈⋊C₈×C₁₂₇	1	2	yes	—
219	C₄	C₂×#103×C₁₂₇	1	1	yes	characteristic
220	C₄	C₂×#104×C₁₂₇	1	1	yes	characteristic
221	C₂	—	2	2	no	—
222	C₂	C₂×Q₈⋊C₈×C₁₂₇	1	1	yes	characteristic
223	C₂	C₁₂₇×#2115	1	1	yes	characteristic
224	C₂	C₁₂₇×#2117	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₁₂₇×#1338	C₂	semidirect
108. #11612	C₁₂₇	direct (N × H)
116. #1338	C₂×C₁₂₇	semidirect
117. C₁₂₇	#11612	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×#2960	2
C₁₂₇×#2150	2
C₂×#346×C₁₂₇	2
C₁₂₇×#1338	2
#11612	127

Subgroups — C127×#11612 #44575285

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#11612 #44575285