Subgroups — C₁₂₇×#11492 #44575165

Order 32512.

360 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#2931	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#2163	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₂₇×#2162	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₂₇×#1315	C₂	1	4	yes	maximal, complemented
5	C₂×#131×C₁₂₇	C₄	1	2	yes	—
6	#532×C₁₂₇	C₄	1	1	yes	characteristic
7	#436×C₁₂₇	—	2	1	no	—
8	#436×C₁₂₇	—	2	1	no	—
9	C₈×D₄×C₁₂₇	—	2	1	no	—
10	C₈×D₄×C₁₂₇	—	2	1	no	—
11	#423×C₁₂₇	C₄	1	1	yes	characteristic
12	#408×C₁₂₇	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₁₂₇	—	2	1	no	—
15	C₄⋊C₁₆×C₁₂₇	—	2	1	no	—
16	C₂²⋊C₁₆×C₁₂₇	C₂²	1	2	yes	—
17	C₂²⋊C₁₆×C₁₂₇	C₂²	1	2	yes	—
18	#346×C₁₂₇	—	2	1	no	—
19	C₄×C₁₆×C₁₂₇	—	2	1	no	—
20	C₂×#41×C₁₂₇	C₂×C₄	1	2	yes	—
21	#131×C₁₂₇	—	4	4	no	—
22	C₂×M₁₆×C₁₂₇	—	2	2	no	—
23	C₂×M₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₂₇	—	2	2	no	—
25	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
27	C₄⋊Q₈×C₁₂₇	C₈	1	1	yes	characteristic
28	#127×C₁₂₇	C₈	1	1	yes	characteristic
29	#124×C₁₂₇	C₈	1	2	yes	—
30	#121×C₁₂₇	—	2	1	no	—
31	#121×C₁₂₇	—	2	1	no	—
32	C₄×D₄×C₁₂₇	—	2	1	no	—
33	C₄×D₄×C₁₂₇	—	2	1	no	—
34	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₆×C₁₂₇	—	2	2	no	—
36	C₂×C₁₆×C₁₂₇	—	2	2	no	—
37	C₄⋊C₈×C₁₂₇	—	2	1	no	—
38	C₄⋊C₈×C₁₂₇	—	2	1	no	—
39	C₂²⋊C₈×C₁₂₇	—	2	2	no	—
40	C₂²⋊C₈×C₁₂₇	—	2	2	no	—
41	C₄×C₈×C₁₂₇	D₄	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	D₄	1	1	yes	characteristic
45	#41×C₁₂₇	—	4	2	no	—
46	#41×C₁₂₇	—	4	2	no	—
47	C₂×Q₈×C₁₂₇	C₂×C₈	1	2	yes	—
48	C₂×D₄×C₁₂₇	—	2	2	no	—
49	C₂×D₄×C₁₂₇	—	2	1	no	—
50	C₂×D₄×C₁₂₇	—	2	1	no	—
51	C₂×D₄×C₁₂₇	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₁₂₇	—	2	2	no	—
53	C₂²×C₄×C₁₂₇	M₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₁₂₇	M₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
56	M₁₆×C₁₂₇	—	4	4	no	—
57	M₁₆×C₁₂₇	—	4	2	no	—
58	C₂×C₈×C₁₂₇	—	2	2	no	—
59	C₂×C₈×C₁₂₇	—	2	2	no	—
60	C₂×C₈×C₁₂₇	—	4	1	no	—
61	C₂×C₈×C₁₂₇	—	4	4	no	—
62	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
66	C₄⋊C₄×C₁₂₇	—	2	1	no	—
67	C₄⋊C₄×C₁₂₇	—	2	1	no	—
68	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
69	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
70	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₁₂₇	C₂×C₈	1	1	yes	characteristic
73	C₄²×C₁₂₇	C₂×C₈	1	1	yes	characteristic
74	C₁₆×C₁₂₇	—	4	2	no	—
75	C₁₆×C₁₂₇	—	4	2	no	—
76	C₂³×C₁₂₇	—	2	2	no	—
77	C₂³×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
78	Q₈×C₁₂₇	—	4	2	no	—
79	D₄×C₁₂₇	—	4	2	no	—
80	D₄×C₁₂₇	—	4	2	no	—
81	D₄×C₁₂₇	—	4	4	no	—
82	D₄×C₁₂₇	—	4	2	no	—
83	C₂×C₄×C₁₂₇	—	2	2	no	—
84	C₂×C₄×C₁₂₇	—	2	1	no	—
85	C₂×C₄×C₁₂₇	—	2	1	no	—
86	C₂×C₄×C₁₂₇	—	2	2	no	—
87	C₂×C₄×C₁₂₇	—	4	1	no	—
88	C₂×C₄×C₁₂₇	—	4	2	no	—
89	C₂×C₄×C₁₂₇	—	4	2	no	—
90	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
91	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
92	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
93	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
94	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
96	C₈×C₁₂₇	—	2	1	no	—
97	C₈×C₁₂₇	—	2	1	no	—
98	C₈×C₁₂₇	—	2	1	no	—
99	C₈×C₁₂₇	—	4	2	no	—
100	C₈×C₁₂₇	#100	1	2	yes	—
101	C₂²×C₁₂₇	—	2	1	no	—
102	C₂²×C₁₂₇	—	2	1	no	—
103	C₂²×C₁₂₇	—	2	1	no	—
104	C₂²×C₁₂₇	—	4	2	no	—
105	C₂²×C₁₂₇	—	4	2	no	—
106	C₂²×C₁₂₇	—	4	2	no	—
107	C₂²×C₁₂₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
108	C₄×C₁₂₇	—	2	1	no	—
109	C₄×C₁₂₇	—	2	1	no	—
110	C₄×C₁₂₇	—	2	1	no	—
111	C₄×C₁₂₇	—	2	1	no	—
112	C₄×C₁₂₇	—	4	2	no	—
113	C₄×C₁₂₇	#350	1	2	yes	—
114	C₄×C₁₂₇	#410	1	1	yes	characteristic
115	C₄×C₁₂₇	C₂×#100	1	1	yes	characteristic
116	#11492	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂×C₁₂₇	—	4	1	no	—
118	C₂×C₁₂₇	—	8	2	no	—
119	C₂×C₁₂₇	#1460	1	1	yes	characteristic
120	C₂×C₁₂₇	C₂×#350	1	1	yes	characteristic
121	C₂×C₁₂₇	#2117	1	1	yes	characteristic
122	#2931	C₂×C₁₂₇	1	1	yes	characteristic
123	#2163	C₂×C₁₂₇	1	1	yes	characteristic, complemented
124	#2162	C₂×C₁₂₇	1	1	yes	characteristic, complemented
125	#1315	C₂×C₁₂₇	1	4	yes	complemented
126	C₁₂₇	#11492	1	1	yes	characteristic, direct product factor, Sylow
127	C₂×#131	C₄×C₁₂₇	1	2	yes	—
128	#532	C₄×C₁₂₇	1	1	yes	characteristic
129	#436	—	2	1	no	—
130	#436	—	2	1	no	—
131	C₈×D₄	—	2	1	no	—
132	C₈×D₄	—	2	1	no	—
133	#423	C₄×C₁₂₇	1	1	yes	characteristic
134	#408	C₂²×C₁₂₇	1	2	yes	—
135	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
136	C₄⋊C₁₆	—	2	1	no	—
137	C₄⋊C₁₆	—	2	1	no	—
138	C₂²⋊C₁₆	C₂²×C₁₂₇	1	2	yes	—
139	C₂²⋊C₁₆	C₂²×C₁₂₇	1	2	yes	—
140	#346	—	2	1	no	—
141	C₄×C₁₆	—	2	1	no	—
142	C₂×#41	C₂×C₄×C₁₂₇	1	2	yes	—
143	#131	—	4	4	no	—
144	C₂×M₁₆	—	2	2	no	—
145	C₂×M₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
146	C₂²×C₈	—	2	2	no	—
147	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
148	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
149	C₄⋊Q₈	C₈×C₁₂₇	1	1	yes	characteristic
150	#127	C₈×C₁₂₇	1	1	yes	characteristic
151	#124	C₈×C₁₂₇	1	2	yes	—
152	#121	—	2	1	no	—
153	#121	—	2	1	no	—
154	C₄×D₄	—	2	1	no	—
155	C₄×D₄	—	2	1	no	—
156	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
157	C₂×C₁₆	—	2	2	no	—
158	C₂×C₁₆	—	2	2	no	—
159	C₄⋊C₈	—	2	1	no	—
160	C₄⋊C₈	—	2	1	no	—
161	C₂²⋊C₈	—	2	2	no	—
162	C₂²⋊C₈	—	2	2	no	—
163	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
164	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
165	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
166	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
167	#41	—	4	2	no	—
168	#41	—	4	2	no	—
169	C₂×Q₈	C₂×C₈×C₁₂₇	1	2	yes	—
170	C₂×D₄	—	2	2	no	—
171	C₂×D₄	—	2	1	no	—
172	C₂×D₄	—	2	1	no	—
173	C₂×D₄	C₂×C₈×C₁₂₇	1	2	yes	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	M₁₆×C₁₂₇	1	1	yes	characteristic
176	C₂²×C₄	M₁₆×C₁₂₇	1	1	yes	characteristic
177	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
178	M₁₆	—	4	4	no	—
179	M₁₆	—	4	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	4	1	no	—
183	C₂×C₈	—	4	4	no	—
184	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
185	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
186	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
187	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
188	C₄⋊C₄	—	2	1	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	2	2	no	—
192	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
193	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
194	C₄²	C₂×C₈×C₁₂₇	1	1	yes	characteristic
195	C₄²	C₂×C₈×C₁₂₇	1	1	yes	characteristic
196	C₁₆	—	4	2	no	—
197	C₁₆	—	4	2	no	—
198	C₂³	—	2	2	no	—
199	C₂³	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
200	Q₈	—	4	2	no	—
201	D₄	—	4	2	no	—
202	D₄	—	4	2	no	—
203	D₄	—	4	4	no	—
204	D₄	—	4	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
213	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
214	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
215	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
216	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
217	C₂×C₄	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
218	C₈	—	2	1	no	—
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₈	—	4	2	no	—
222	C₈	#100×C₁₂₇	1	2	yes	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	4	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	C₂×C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	4	2	no	—
235	C₄	#350×C₁₂₇	1	2	yes	—
236	C₄	#410×C₁₂₇	1	1	yes	characteristic
237	C₄	C₂×#100×C₁₂₇	1	1	yes	characteristic
238	C₂	—	4	1	no	—
239	C₂	—	8	2	no	—
240	C₂	C₁₂₇×#1460	1	1	yes	characteristic
241	C₂	C₂×#350×C₁₂₇	1	1	yes	characteristic
242	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	C₂²×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂×C₂²⋊C₈
prime radical	#532×C₁₂₇	C₄
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
2. C₁₂₇×#2163	C₂	semidirect
3. C₁₂₇×#2162	C₂	semidirect
4. C₁₂₇×#1315	C₂	semidirect
116. #11492	C₁₂₇	direct (N × H)
123. #2163	C₂×C₁₂₇	semidirect
124. #2162	C₂×C₁₂₇	semidirect
125. #1315	C₂×C₁₂₇	semidirect
126. C₁₂₇	#11492	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×#2931	2
C₁₂₇×#2163	2
C₁₂₇×#2162	2
C₁₂₇×#1315	2
#11492	127

Subgroups — C127×#11492 #44575165

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#11492 #44575165