Subgroups — C₁₂₇×#13211 #44576884

Order 32512.

494 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#3147	C₂	1	1	yes	maximal, characteristic
2	C₂×#360×C₁₂₇	C₂	1	4	yes	maximal, complemented
3	C₄×M₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₂₇×#1846	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#135×C₁₂₇	—	2	1	no	—
6	C₂×#135×C₁₂₇	C₂²	1	2	yes	—
7	#532×C₁₂₇	—	2	1	no	—
8	C₂×M₃₂×C₁₂₇	C₂²	1	2	yes	—
9	#495×C₁₂₇	—	2	1	no	—
10	#495×C₁₂₇	C₄	1	2	yes	complemented
11	#494×C₁₂₇	C₄	1	1	yes	characteristic, complemented
12	#493×C₁₂₇	C₄	1	1	yes	characteristic, complemented
13	#492×C₁₂₇	—	2	1	no	—
14	C₂×#107×C₁₂₇	—	2	1	no	—
15	C₂×#106×C₁₂₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
17	#382×C₁₂₇	—	2	1	no	—
18	#360×C₁₂₇	—	2	8	no	—
19	#346×C₁₂₇	—	2	1	no	—
20	C₄×C₁₆×C₁₂₇	—	2	1	no	—
21	C₂×#41×C₁₂₇	—	2	2	no	—
22	#135×C₁₂₇	—	2	4	no	—
23	#135×C₁₂₇	—	2	4	no	—
24	#135×C₁₂₇	—	4	2	no	—
25	#135×C₁₂₇	—	4	2	no	—
26	C₂×Q₁₆×C₁₂₇	—	2	1	no	—
27	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
28	C₂×SD₁₆×C₁₂₇	—	2	2	no	—
29	C₂×SD₁₆×C₁₂₇	—	2	2	no	—
30	C₂×D₈×C₁₂₇	—	2	1	no	—
31	C₂×D₈×C₁₂₇	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
34	C₄⋊Q₈×C₁₂₇	—	2	1	no	—
35	#127×C₁₂₇	—	2	1	no	—
36	#124×C₁₂₇	—	2	2	no	—
37	#121×C₁₂₇	—	4	2	no	—
38	C₄×D₄×C₁₂₇	—	4	2	no	—
39	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
40	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
41	M₃₂×C₁₂₇	—	2	2	no	—
42	M₃₂×C₁₂₇	D₄	1	4	yes	complemented
43	C₂×C₁₆×C₁₂₇	—	2	2	no	—
44	#107×C₁₂₇	—	4	2	no	—
45	#106×C₁₂₇	—	2	4	no	—
46	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
49	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
50	#95×C₁₂₇	—	4	2	no	—
51	#41×C₁₂₇	—	4	2	no	—
52	#41×C₁₂₇	—	4	2	no	—
53	#41×C₁₂₇	—	4	2	no	—
54	#41×C₁₂₇	—	4	2	no	—
55	C₂×Q₈×C₁₂₇	—	2	2	no	—
56	C₂×D₄×C₁₂₇	—	2	2	no	—
57	C₂×D₄×C₁₂₇	—	4	2	no	—
58	C₂×D₄×C₁₂₇	—	4	2	no	—
59	C₂²×C₄×C₁₂₇	—	4	2	no	—
60	C₂²×C₄×C₁₂₇	—	4	2	no	—
61	C₂²×C₄×C₁₂₇	D₈	1	2	yes	—
62	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
63	Q₁₆×C₁₂₇	—	2	4	no	—
64	Q₁₆×C₁₂₇	—	4	2	no	—
65	SD₁₆×C₁₂₇	—	4	4	no	—
66	SD₁₆×C₁₂₇	—	4	4	no	—
67	D₈×C₁₂₇	—	2	4	no	—
68	D₈×C₁₂₇	—	4	2	no	—
69	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₂₇	D₈	1	2	yes	—
72	C₂×C₈×C₁₂₇	SD₁₆	1	2	yes	—
73	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₂₇	#41	1	2	yes	—
77	C₄⋊C₄×C₁₂₇	—	4	2	no	—
78	C₄⋊C₄×C₁₂₇	—	4	2	no	—
79	C₄⋊C₄×C₁₂₇	—	4	2	no	—
80	C₂²⋊C₄×C₁₂₇	—	4	4	no	—
81	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₄²×C₁₂₇	SD₁₆	1	2	yes	—
84	C₁₆×C₁₂₇	—	2	4	no	—
85	C₂³×C₁₂₇	—	4	2	no	—
86	C₂³×C₁₂₇	C₂×D₈	1	1	yes	characteristic
87	Q₈×C₁₂₇	—	4	2	no	—
88	Q₈×C₁₂₇	—	4	2	no	—
89	D₄×C₁₂₇	—	4	2	no	—
90	D₄×C₁₂₇	—	4	2	no	—
91	D₄×C₁₂₇	—	8	4	no	—
92	D₄×C₁₂₇	—	8	2	no	—
93	D₄×C₁₂₇	—	8	2	no	—
94	C₂×C₄×C₁₂₇	—	4	2	no	—
95	C₂×C₄×C₁₂₇	—	4	2	no	—
96	C₂×C₄×C₁₂₇	—	8	2	no	—
97	C₂×C₄×C₁₂₇	—	8	2	no	—
98	C₂×C₄×C₁₂₇	—	8	4	no	—
99	C₂×C₄×C₁₂₇	—	8	2	no	—
100	C₂×C₄×C₁₂₇	#102	1	2	yes	—
101	C₂×C₄×C₁₂₇	#102	1	2	yes	—
102	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₂₇	#124	1	1	yes	characteristic
104	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
105	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
108	C₈×C₁₂₇	#102	1	2	yes	—
109	C₈×C₁₂₇	#102	1	2	yes	—
110	C₈×C₁₂₇	C₄×D₄	1	2	yes	—
111	C₈×C₁₂₇	#121	1	2	yes	—
112	C₂²×C₁₂₇	—	2	2	no	—
113	C₂²×C₁₂₇	—	8	4	no	—
114	C₂²×C₁₂₇	—	8	2	no	—
115	C₂²×C₁₂₇	C₂×#102	1	1	yes	characteristic
116	C₂²×C₁₂₇	#486	1	1	yes	characteristic
117	C₂²×C₁₂₇	#493	1	1	yes	characteristic
118	C₄×C₁₂₇	—	2	2	no	—
119	C₄×C₁₂₇	—	2	2	no	—
120	C₄×C₁₂₇	—	8	2	no	—
121	C₄×C₁₂₇	—	16	2	no	—
122	C₄×C₁₂₇	#390	1	1	yes	characteristic
123	C₄×C₁₂₇	C₂×#102	1	1	yes	characteristic
124	C₄×C₁₂₇	#486	1	1	yes	characteristic
125	C₄×C₁₂₇	#492	1	1	yes	characteristic
126	#13211	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂×C₁₂₇	—	2	1	no	—
128	C₂×C₁₂₇	—	2	1	no	—
129	C₂×C₁₂₇	—	16	2	no	—
130	C₂×C₁₂₇	#1957	1	1	yes	characteristic
131	C₂×C₁₂₇	#2145	1	2	yes	—
132	#3147	C₂×C₁₂₇	1	1	yes	characteristic
133	C₂×#360	C₂×C₁₂₇	1	4	yes	complemented
134	C₄×M₃₂	C₂×C₁₂₇	1	1	yes	characteristic, complemented
135	#1846	C₂×C₁₂₇	1	1	yes	characteristic, complemented
136	C₁₂₇	#13211	1	1	yes	characteristic, direct product factor, Sylow
137	C₂×#135	—	2	1	no	—
138	C₂×#135	C₂²×C₁₂₇	1	2	yes	—
139	#532	—	2	1	no	—
140	C₂×M₃₂	C₂²×C₁₂₇	1	2	yes	—
141	#495	—	2	1	no	—
142	#495	C₄×C₁₂₇	1	2	yes	complemented
143	#494	C₄×C₁₂₇	1	1	yes	characteristic, complemented
144	#493	C₄×C₁₂₇	1	1	yes	characteristic, complemented
145	#492	—	2	1	no	—
146	C₂×#107	—	2	1	no	—
147	C₂×#106	C₂²×C₁₂₇	1	2	yes	—
148	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
149	#382	—	2	1	no	—
150	#360	—	2	8	no	—
151	#346	—	2	1	no	—
152	C₄×C₁₆	—	2	1	no	—
153	C₂×#41	—	2	2	no	—
154	#135	—	2	4	no	—
155	#135	—	2	4	no	—
156	#135	—	4	2	no	—
157	#135	—	4	2	no	—
158	C₂×Q₁₆	—	2	1	no	—
159	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
160	C₂×SD₁₆	—	2	2	no	—
161	C₂×SD₁₆	—	2	2	no	—
162	C₂×D₈	—	2	1	no	—
163	C₂×D₈	C₂×C₄×C₁₂₇	1	2	yes	—
164	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
165	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
166	C₄⋊Q₈	—	2	1	no	—
167	#127	—	2	1	no	—
168	#124	—	2	2	no	—
169	#121	—	4	2	no	—
170	C₄×D₄	—	4	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
173	M₃₂	—	2	2	no	—
174	M₃₂	D₄×C₁₂₇	1	4	yes	complemented
175	C₂×C₁₆	—	2	2	no	—
176	#107	—	4	2	no	—
177	#106	—	2	4	no	—
178	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
179	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
180	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
181	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
182	#95	—	4	2	no	—
183	#41	—	4	2	no	—
184	#41	—	4	2	no	—
185	#41	—	4	2	no	—
186	#41	—	4	2	no	—
187	C₂×Q₈	—	2	2	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	4	2	no	—
190	C₂×D₄	—	4	2	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	D₈×C₁₂₇	1	2	yes	—
194	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
195	Q₁₆	—	2	4	no	—
196	Q₁₆	—	4	2	no	—
197	SD₁₆	—	4	4	no	—
198	SD₁₆	—	4	4	no	—
199	D₈	—	2	4	no	—
200	D₈	—	4	2	no	—
201	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
202	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
203	C₂×C₈	D₈×C₁₂₇	1	2	yes	—
204	C₂×C₈	SD₁₆×C₁₂₇	1	2	yes	—
205	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
206	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
207	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
208	C₂×C₈	#41×C₁₂₇	1	2	yes	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₂²⋊C₄	—	4	4	no	—
213	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
214	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
215	C₄²	SD₁₆×C₁₂₇	1	2	yes	—
216	C₁₆	—	2	4	no	—
217	C₂³	—	4	2	no	—
218	C₂³	C₂×D₈×C₁₂₇	1	1	yes	characteristic
219	Q₈	—	4	2	no	—
220	Q₈	—	4	2	no	—
221	D₄	—	4	2	no	—
222	D₄	—	4	2	no	—
223	D₄	—	8	4	no	—
224	D₄	—	8	2	no	—
225	D₄	—	8	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	8	2	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	—	8	4	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	#102×C₁₂₇	1	2	yes	—
233	C₂×C₄	#102×C₁₂₇	1	2	yes	—
234	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
235	C₂×C₄	#124×C₁₂₇	1	1	yes	characteristic
236	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
237	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
238	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
239	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
240	C₈	#102×C₁₂₇	1	2	yes	—
241	C₈	#102×C₁₂₇	1	2	yes	—
242	C₈	C₄×D₄×C₁₂₇	1	2	yes	—
243	C₈	#121×C₁₂₇	1	2	yes	—
244	C₂²	—	2	2	no	—
245	C₂²	—	8	4	no	—
246	C₂²	—	8	2	no	—
247	C₂²	C₂×#102×C₁₂₇	1	1	yes	characteristic
248	C₂²	#486×C₁₂₇	1	1	yes	characteristic
249	C₂²	#493×C₁₂₇	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	8	2	no	—
253	C₄	—	16	2	no	—
254	C₄	#390×C₁₂₇	1	1	yes	characteristic
255	C₄	C₂×#102×C₁₂₇	1	1	yes	characteristic
256	C₄	#486×C₁₂₇	1	1	yes	characteristic
257	C₄	#492×C₁₂₇	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	16	2	no	—
261	C₂	C₁₂₇×#1957	1	1	yes	characteristic
262	C₂	C₁₂₇×#2145	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₂₇	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
prime radical	C₁₂₇×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ 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
2. C₂×#360×C₁₂₇	C₂	semidirect
3. C₄×M₃₂×C₁₂₇	C₂	semidirect
4. C₁₂₇×#1846	C₂	semidirect
10. #495×C₁₂₇	C₄	semidirect
11. #494×C₁₂₇	C₄	semidirect
12. #493×C₁₂₇	C₄	semidirect
42. M₃₂×C₁₂₇	D₄	semidirect
126. #13211	C₁₂₇	direct (N × H)
133. C₂×#360	C₂×C₁₂₇	semidirect
134. C₄×M₃₂	C₂×C₁₂₇	semidirect
135. #1846	C₂×C₁₂₇	semidirect
136. C₁₂₇	#13211	direct (N × H)
142. #495	C₄×C₁₂₇	semidirect
143. #494	C₄×C₁₂₇	semidirect
144. #493	C₄×C₁₂₇	semidirect
174. M₃₂	D₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×#3147	2
C₂×#360×C₁₂₇	2
C₄×M₃₂×C₁₂₇	2
C₁₂₇×#1846	2
#13211	127

Subgroups — C127×#13211 #44576884

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#13211 #44576884