Subgroups — C₁₂₇×#21969 #44585642

Order 32512.

734 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#2920	C₂	1	2	yes	maximal
2	C₂²×C₄⋊C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1950	C₂	1	8	yes	maximal, complemented
4	C₁₂₇×#1810	C₂	1	4	yes	maximal
5	C₂³×C₈×C₁₂₇	C₂²	1	2	yes	—
6	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
7	#432×C₁₂₇	—	2	2	no	—
8	#432×C₁₂₇	—	2	2	no	—
9	#431×C₁₂₇	—	2	2	no	—
10	#431×C₁₂₇	—	2	2	no	—
11	C₂×#108×C₁₂₇	C₂²	1	8	yes	—
12	#423×C₁₂₇	C₂²	1	4	yes	—
13	C₂×C₄⋊C₈×C₁₂₇	C₄	1	4	yes	—
14	C₂×C₄⋊C₈×C₁₂₇	C₄	1	2	yes	—
15	C₂×C₄⋊C₈×C₁₂₇	C₄	1	2	yes	—
16	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	4	yes	—
17	#407×C₁₂₇	C₂²	1	4	yes	—
18	C₄×M₁₆×C₁₂₇	C₂²	1	4	yes	—
19	#341×C₁₂₇	C₂²	1	8	yes	—
20	C₂³×C₄×C₁₂₇	Q₈	1	1	yes	characteristic
21	C₂³×C₄×C₁₂₇	Q₈	1	1	yes	characteristic
22	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
23	C₂×M₁₆×C₁₂₇	C₂³	1	8	yes	—
24	C₂²×C₈×C₁₂₇	—	4	4	no	—
25	C₂²×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₁₂₇	C₂³	1	4	yes	—
28	C₂×C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₂₇	D₄	1	2	yes	—
33	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₂₇	Q₈	1	2	yes	—
35	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
36	#108×C₁₂₇	—	4	8	no	—
37	C₄⋊C₈×C₁₂₇	—	2	4	no	—
38	C₄⋊C₈×C₁₂₇	—	2	4	no	—
39	C₄⋊C₈×C₁₂₇	C₂×C₄	1	8	yes	—
40	C₄⋊C₈×C₁₂₇	C₂×C₄	1	8	yes	—
41	C₂²⋊C₈×C₁₂₇	—	2	4	no	—
42	C₂²⋊C₈×C₁₂₇	—	2	4	no	—
43	#97×C₁₂₇	—	2	4	no	—
44	C₄×C₈×C₁₂₇	—	2	4	no	—
45	C₂⁴×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₁₂₇	—	2	4	no	—
47	C₂²×C₄×C₁₂₇	—	2	2	no	—
48	C₂²×C₄×C₁₂₇	—	2	2	no	—
49	C₂²×C₄×C₁₂₇	—	4	2	no	—
50	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
64	M₁₆×C₁₂₇	—	2	8	no	—
65	M₁₆×C₁₂₇	—	2	8	no	—
66	C₂×C₈×C₁₂₇	—	2	8	no	—
67	C₂×C₈×C₁₂₇	—	2	8	no	—
68	C₂×C₈×C₁₂₇	—	4	4	no	—
69	C₂×C₈×C₁₂₇	—	4	4	no	—
70	C₂×C₈×C₁₂₇	C₂²×C₄	1	8	yes	—
71	C₄²×C₁₂₇	C₄⋊C₄	1	2	yes	—
72	C₄²×C₁₂₇	C₄⋊C₄	1	2	yes	—
73	C₄²×C₁₂₇	C₄⋊C₄	1	2	yes	—
74	C₄²×C₁₂₇	C₄⋊C₄	1	2	yes	—
75	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
76	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
77	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
78	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
79	C₂³×C₁₂₇	—	2	2	no	—
80	C₂³×C₁₂₇	—	2	2	no	—
81	C₂³×C₁₂₇	—	2	2	no	—
82	C₂³×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂³×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂³×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
85	C₂×C₄×C₁₂₇	—	2	2	no	—
86	C₂×C₄×C₁₂₇	—	2	2	no	—
87	C₂×C₄×C₁₂₇	—	2	4	no	—
88	C₂×C₄×C₁₂₇	—	2	1	no	—
89	C₂×C₄×C₁₂₇	—	2	1	no	—
90	C₂×C₄×C₁₂₇	—	2	1	no	—
91	C₂×C₄×C₁₂₇	—	2	1	no	—
92	C₂×C₄×C₁₂₇	—	2	2	no	—
93	C₂×C₄×C₁₂₇	—	2	2	no	—
94	C₂×C₄×C₁₂₇	—	2	4	no	—
95	C₂×C₄×C₁₂₇	—	2	4	no	—
96	C₂×C₄×C₁₂₇	—	4	1	no	—
97	C₂×C₄×C₁₂₇	—	4	1	no	—
98	C₂×C₄×C₁₂₇	—	4	2	no	—
99	C₂×C₄×C₁₂₇	#108	1	4	yes	complemented
100	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₂₇	H(5,𝔽₂)	1	2	yes	—
109	C₂×C₄×C₁₂₇	2₋¹⁺⁴	1	2	yes	—
110	C₈×C₁₂₇	—	2	8	no	—
111	C₈×C₁₂₇	—	4	8	no	—
112	C₂²×C₁₂₇	—	2	8	no	—
113	C₂²×C₁₂₇	—	2	1	no	—
114	C₂²×C₁₂₇	—	2	1	no	—
115	C₂²×C₁₂₇	—	2	1	no	—
116	C₂²×C₁₂₇	—	2	1	no	—
117	C₂²×C₁₂₇	—	2	1	no	—
118	C₂²×C₁₂₇	—	2	1	no	—
119	C₂²×C₁₂₇	C₂×#108	1	2	yes	—
120	C₂²×C₁₂₇	#430	1	2	yes	—
121	C₂²×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
122	C₂²×C₁₂₇	#544	1	2	yes	—
123	C₄×C₁₂₇	—	2	1	no	—
124	C₄×C₁₂₇	—	2	1	no	—
125	C₄×C₁₂₇	—	2	2	no	—
126	C₄×C₁₂₇	—	2	2	no	—
127	C₄×C₁₂₇	—	2	2	no	—
128	C₄×C₁₂₇	C₂×#108	1	4	yes	—
129	C₄×C₁₂₇	#518	1	1	yes	characteristic
130	C₄×C₁₂₇	#519	1	1	yes	characteristic
131	C₄×C₁₂₇	#537	1	2	yes	—
132	#21969	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂×C₁₂₇	—	2	2	no	—
134	C₂×C₁₂₇	—	2	2	no	—
135	C₂×C₁₂₇	—	2	2	no	—
136	C₂×C₁₂₇	#2316	1	1	yes	characteristic
137	C₂×C₁₂₇	C₂²×#108	1	1	yes	characteristic
138	C₂×C₁₂₇	C₂×#430	1	1	yes	characteristic
139	#2920	C₂×C₁₂₇	1	2	yes	—
140	C₂²×C₄⋊C₈	C₂×C₁₂₇	1	1	yes	characteristic
141	#1950	C₂×C₁₂₇	1	8	yes	complemented
142	#1810	C₂×C₁₂₇	1	4	yes	—
143	C₁₂₇	#21969	1	1	yes	characteristic, direct product factor, Sylow
144	C₂³×C₈	C₂²×C₁₂₇	1	2	yes	—
145	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
146	#432	—	2	2	no	—
147	#432	—	2	2	no	—
148	#431	—	2	2	no	—
149	#431	—	2	2	no	—
150	C₂×#108	C₂²×C₁₂₇	1	8	yes	—
151	#423	C₂²×C₁₂₇	1	4	yes	—
152	C₂×C₄⋊C₈	C₄×C₁₂₇	1	4	yes	—
153	C₂×C₄⋊C₈	C₄×C₁₂₇	1	2	yes	—
154	C₂×C₄⋊C₈	C₄×C₁₂₇	1	2	yes	—
155	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	4	yes	—
156	#407	C₂²×C₁₂₇	1	4	yes	—
157	C₄×M₁₆	C₂²×C₁₂₇	1	4	yes	—
158	#341	C₂²×C₁₂₇	1	8	yes	—
159	C₂³×C₄	Q₈×C₁₂₇	1	1	yes	characteristic
160	C₂³×C₄	Q₈×C₁₂₇	1	1	yes	characteristic
161	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
162	C₂×M₁₆	C₂³×C₁₂₇	1	8	yes	—
163	C₂²×C₈	—	4	4	no	—
164	C₂²×C₈	C₂×C₄×C₁₂₇	1	4	yes	—
165	C₂²×C₈	C₂×C₄×C₁₂₇	1	4	yes	—
166	C₂²×C₈	C₂³×C₁₂₇	1	4	yes	—
167	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
168	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
169	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
170	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
171	C₂×C₄²	D₄×C₁₂₇	1	2	yes	—
172	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
173	C₂×C₄²	Q₈×C₁₂₇	1	2	yes	—
174	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
175	#108	—	4	8	no	—
176	C₄⋊C₈	—	2	4	no	—
177	C₄⋊C₈	—	2	4	no	—
178	C₄⋊C₈	C₂×C₄×C₁₂₇	1	8	yes	—
179	C₄⋊C₈	C₂×C₄×C₁₂₇	1	8	yes	—
180	C₂²⋊C₈	—	2	4	no	—
181	C₂²⋊C₈	—	2	4	no	—
182	#97	—	2	4	no	—
183	C₄×C₈	—	2	4	no	—
184	C₂⁴	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
190	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	2	yes	—
191	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	2	yes	—
192	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
193	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
194	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
195	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
196	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
199	C₂²×C₄	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
200	C₂²×C₄	C₂×Q₈×C₁₂₇	1	2	yes	—
201	C₂²×C₄	C₂×Q₈×C₁₂₇	1	2	yes	—
202	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
203	M₁₆	—	2	8	no	—
204	M₁₆	—	2	8	no	—
205	C₂×C₈	—	2	8	no	—
206	C₂×C₈	—	2	8	no	—
207	C₂×C₈	—	4	4	no	—
208	C₂×C₈	—	4	4	no	—
209	C₂×C₈	C₂²×C₄×C₁₂₇	1	8	yes	—
210	C₄²	C₄⋊C₄×C₁₂₇	1	2	yes	—
211	C₄²	C₄⋊C₄×C₁₂₇	1	2	yes	—
212	C₄²	C₄⋊C₄×C₁₂₇	1	2	yes	—
213	C₄²	C₄⋊C₄×C₁₂₇	1	2	yes	—
214	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
215	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
216	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
217	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
222	C₂³	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
223	C₂³	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	#108×C₁₂₇	1	4	yes	complemented
239	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
240	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
241	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
242	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
243	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
244	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
245	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
246	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
247	C₂×C₄	H(5,𝔽₂)×C₁₂₇	1	2	yes	—
248	C₂×C₄	2₋¹⁺⁴×C₁₂₇	1	2	yes	—
249	C₈	—	2	8	no	—
250	C₈	—	4	8	no	—
251	C₂²	—	2	8	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	C₂×#108×C₁₂₇	1	2	yes	—
259	C₂²	#430×C₁₂₇	1	2	yes	—
260	C₂²	C₂²×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
261	C₂²	#544×C₁₂₇	1	2	yes	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	C₂×#108×C₁₂₇	1	4	yes	—
268	C₄	#518×C₁₂₇	1	1	yes	characteristic
269	C₄	#519×C₁₂₇	1	1	yes	characteristic
270	C₄	#537×C₁₂₇	1	2	yes	—
271	C₂	—	2	2	no	—
272	C₂	—	2	2	no	—
273	C₂	—	2	2	no	—
274	C₂	C₁₂₇×#2316	1	1	yes	characteristic
275	C₂	C₂²×#108×C₁₂₇	1	1	yes	characteristic
276	C₂	C₂×#430×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle	C₂²×C₁₂₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₁₂₇	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂²×C₄²×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₂₇×#1950	C₂	semidirect
99. C₂×C₄×C₁₂₇	#108	semidirect
132. #21969	C₁₂₇	direct (N × H)
141. #1950	C₂×C₁₂₇	semidirect
143. C₁₂₇	#21969	direct (N × H)
238. C₂×C₄	#108×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×#2920	2
C₂²×C₄⋊C₈×C₁₂₇	2
C₁₂₇×#1950	2
C₁₂₇×#1810	2
#21969	127

Subgroups — C127×#21969 #44585642

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#21969 #44585642