Subgroups — C₁₂₇×#20668 #44584341

Order 32512.

1322 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#131×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#108×C₁₂₇	C₂	1	2	yes	maximal, complemented
3	C₁₂₇×#2879	C₂	1	2	yes	maximal
4	C₁₂₇×#1811	C₂	1	4	yes	maximal, complemented
5	C₁₂₇×#1810	C₂	1	4	yes	maximal, complemented
6	C₁₂₇×#1738	C₂	1	2	yes	maximal, complemented
7	C₂²×#41×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂×#131×C₁₂₇	—	2	8	no	—
9	C₂×#131×C₁₂₇	C₄	1	8	yes	—
10	C₂²×M₁₆×C₁₂₇	C₂²	1	4	yes	—
11	C₂²×M₁₆×C₁₂₇	C₂²	1	2	yes	—
12	C₂²×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₁₂₇	C₂²	1	2	yes	—
14	C₂³×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×#108×C₁₂₇	—	2	8	no	—
16	C₂×#108×C₁₂₇	C₂²	1	4	yes	—
17	C₂×#108×C₁₂₇	C₂²	1	4	yes	—
18	#408×C₁₂₇	—	2	8	no	—
19	#407×C₁₂₇	C₂²	1	8	yes	—
20	#341×C₁₂₇	C₂²	1	8	yes	—
21	C₂×#41×C₁₂₇	—	2	4	no	—
22	C₂×#41×C₁₂₇	—	2	4	no	—
23	C₂×#41×C₁₂₇	C₂×C₄	1	4	yes	—
24	C₂×#41×C₁₂₇	D₄	1	4	yes	—
25	C₂²×Q₈×C₁₂₇	Q₈	1	1	yes	characteristic
26	C₂²×D₄×C₁₂₇	Q₈	1	2	yes	—
27	C₂²×D₄×C₁₂₇	Q₈	1	1	yes	characteristic
28	C₂³×C₄×C₁₂₇	C₂³	1	2	yes	—
29	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
30	#131×C₁₂₇	—	2	32	no	—
31	C₂×M₁₆×C₁₂₇	—	2	4	no	—
32	C₂×M₁₆×C₁₂₇	—	2	4	no	—
33	C₂×M₁₆×C₁₂₇	—	2	8	no	—
34	C₂×M₁₆×C₁₂₇	—	2	8	no	—
35	C₂×M₁₆×C₁₂₇	—	2	8	no	—
36	C₂×M₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
37	C₂×M₁₆×C₁₂₇	C₂×C₄	1	8	yes	—
38	C₂×M₁₆×C₁₂₇	C₂³	1	8	yes	—
39	C₂²×C₈×C₁₂₇	—	2	8	no	—
40	C₂²×C₈×C₁₂₇	—	2	4	no	—
41	C₂²×C₈×C₁₂₇	C₂×C₄	1	8	yes	—
42	C₂²×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
43	C₂²×C₈×C₁₂₇	D₄	1	8	yes	—
44	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
45	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
47	#108×C₁₂₇	—	2	8	no	—
48	#108×C₁₂₇	—	2	8	no	—
49	C₂²⋊C₈×C₁₂₇	—	2	16	no	—
50	C₂⁴×C₁₂₇	C₂×Q₈	1	2	yes	—
51	C₂⁴×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
52	#41×C₁₂₇	—	2	8	no	—
53	#41×C₁₂₇	—	2	8	no	—
54	#41×C₁₂₇	—	2	8	no	—
55	#41×C₁₂₇	—	2	8	no	—
56	C₂×Q₈×C₁₂₇	—	2	4	no	—
57	C₂×Q₈×C₁₂₇	C₄⋊C₄	1	4	yes	—
58	C₂×D₄×C₁₂₇	—	2	8	no	—
59	C₂×D₄×C₁₂₇	—	2	4	no	—
60	C₂×D₄×C₁₂₇	C₄⋊C₄	1	8	yes	—
61	C₂×D₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₁₂₇	—	2	4	no	—
63	C₂²×C₄×C₁₂₇	—	2	4	no	—
64	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
67	C₂²×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
68	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄×C₁₂₇	#41	1	4	yes	—
72	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
73	M₁₆×C₁₂₇	—	2	8	no	—
74	M₁₆×C₁₂₇	—	2	16	no	—
75	M₁₆×C₁₂₇	—	2	8	no	—
76	M₁₆×C₁₂₇	—	2	8	no	—
77	M₁₆×C₁₂₇	—	4	8	no	—
78	C₂×C₈×C₁₂₇	—	2	4	no	—
79	C₂×C₈×C₁₂₇	—	2	16	no	—
80	C₂×C₈×C₁₂₇	—	2	8	no	—
81	C₂×C₈×C₁₂₇	—	4	8	no	—
82	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	16	yes	—
83	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
84	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
87	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
88	C₂³×C₁₂₇	—	2	4	no	—
89	C₂³×C₁₂₇	—	4	2	no	—
90	C₂³×C₁₂₇	—	4	4	no	—
91	C₂³×C₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
92	C₂³×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
93	C₂³×C₁₂₇	C₂²⋊Q₈	1	4	yes	—
94	C₂³×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
95	Q₈×C₁₂₇	—	2	8	no	—
96	D₄×C₁₂₇	—	2	16	no	—
97	D₄×C₁₂₇	—	2	8	no	—
98	C₂×C₄×C₁₂₇	—	2	4	no	—
99	C₂×C₄×C₁₂₇	—	2	4	no	—
100	C₂×C₄×C₁₂₇	—	2	4	no	—
101	C₂×C₄×C₁₂₇	—	2	8	no	—
102	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
103	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₁₂₇	#117	1	4	yes	—
106	C₂×C₄×C₁₂₇	#121	1	4	yes	—
107	C₂×C₄×C₁₂₇	#121	1	4	yes	—
108	C₂×C₄×C₁₂₇	C₂²⋊Q₈	1	4	yes	—
109	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
111	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₂₇	C₂×#41	1	2	yes	—
113	C₈×C₁₂₇	—	2	4	no	—
114	C₈×C₁₂₇	—	4	8	no	—
115	C₈×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
116	C₈×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
117	C₂²×C₁₂₇	—	2	2	no	—
118	C₂²×C₁₂₇	—	2	4	no	—
119	C₂²×C₁₂₇	—	2	8	no	—
120	C₂²×C₁₂₇	—	2	4	no	—
121	C₂²×C₁₂₇	—	4	4	no	—
122	C₂²×C₁₂₇	—	4	8	no	—
123	C₂²×C₁₂₇	#380	1	4	yes	—
124	C₂²×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
125	C₂²×C₁₂₇	C₂×C₂²⋊Q₈	1	2	yes	—
126	C₄×C₁₂₇	—	2	4	no	—
127	C₄×C₁₂₇	#380	1	4	yes	—
128	C₄×C₁₂₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
129	C₄×C₁₂₇	C₂×#117	1	1	yes	characteristic
130	C₄×C₁₂₇	C₂×#121	1	2	yes	—
131	#20668	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂×C₁₂₇	—	2	2	no	—
133	C₂×C₁₂₇	—	2	4	no	—
134	C₂×C₁₂₇	—	4	4	no	—
135	C₂×C₁₂₇	C₂×#380	1	1	yes	characteristic
136	C₂×C₁₂₇	#2916	1	2	yes	—
137	C₂²×#131	C₂×C₁₂₇	1	1	yes	characteristic
138	C₂²×#108	C₂×C₁₂₇	1	2	yes	complemented
139	#2879	C₂×C₁₂₇	1	2	yes	—
140	#1811	C₂×C₁₂₇	1	4	yes	complemented
141	#1810	C₂×C₁₂₇	1	4	yes	complemented
142	#1738	C₂×C₁₂₇	1	2	yes	complemented
143	C₁₂₇	#20668	1	1	yes	characteristic, direct product factor, Sylow
144	C₂²×#41	C₂²×C₁₂₇	1	1	yes	characteristic
145	C₂×#131	—	2	8	no	—
146	C₂×#131	C₄×C₁₂₇	1	8	yes	—
147	C₂²×M₁₆	C₂²×C₁₂₇	1	4	yes	—
148	C₂²×M₁₆	C₂²×C₁₂₇	1	2	yes	—
149	C₂²×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
150	C₂³×C₈	C₂²×C₁₂₇	1	2	yes	—
151	C₂³×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
152	C₂×#108	—	2	8	no	—
153	C₂×#108	C₂²×C₁₂₇	1	4	yes	—
154	C₂×#108	C₂²×C₁₂₇	1	4	yes	—
155	#408	—	2	8	no	—
156	#407	C₂²×C₁₂₇	1	8	yes	—
157	#341	C₂²×C₁₂₇	1	8	yes	—
158	C₂×#41	—	2	4	no	—
159	C₂×#41	—	2	4	no	—
160	C₂×#41	C₂×C₄×C₁₂₇	1	4	yes	—
161	C₂×#41	D₄×C₁₂₇	1	4	yes	—
162	C₂²×Q₈	Q₈×C₁₂₇	1	1	yes	characteristic
163	C₂²×D₄	Q₈×C₁₂₇	1	2	yes	—
164	C₂²×D₄	Q₈×C₁₂₇	1	1	yes	characteristic
165	C₂³×C₄	C₂³×C₁₂₇	1	2	yes	—
166	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
167	#131	—	2	32	no	—
168	C₂×M₁₆	—	2	4	no	—
169	C₂×M₁₆	—	2	4	no	—
170	C₂×M₁₆	—	2	8	no	—
171	C₂×M₁₆	—	2	8	no	—
172	C₂×M₁₆	—	2	8	no	—
173	C₂×M₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
174	C₂×M₁₆	C₂×C₄×C₁₂₇	1	8	yes	—
175	C₂×M₁₆	C₂³×C₁₂₇	1	8	yes	—
176	C₂²×C₈	—	2	8	no	—
177	C₂²×C₈	—	2	4	no	—
178	C₂²×C₈	C₂×C₄×C₁₂₇	1	8	yes	—
179	C₂²×C₈	C₂×C₄×C₁₂₇	1	4	yes	—
180	C₂²×C₈	D₄×C₁₂₇	1	8	yes	—
181	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
182	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
183	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
184	#108	—	2	8	no	—
185	#108	—	2	8	no	—
186	C₂²⋊C₈	—	2	16	no	—
187	C₂⁴	C₂×Q₈×C₁₂₇	1	2	yes	—
188	C₂⁴	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
189	#41	—	2	8	no	—
190	#41	—	2	8	no	—
191	#41	—	2	8	no	—
192	#41	—	2	8	no	—
193	C₂×Q₈	—	2	4	no	—
194	C₂×Q₈	C₄⋊C₄×C₁₂₇	1	4	yes	—
195	C₂×D₄	—	2	8	no	—
196	C₂×D₄	—	2	4	no	—
197	C₂×D₄	C₄⋊C₄×C₁₂₇	1	8	yes	—
198	C₂×D₄	C₄⋊C₄×C₁₂₇	1	4	yes	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
202	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
203	C₂²×C₄	C₂×D₄×C₁₂₇	1	4	yes	—
204	C₂²×C₄	C₂×D₄×C₁₂₇	1	4	yes	—
205	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
206	C₂²×C₄	C₂×Q₈×C₁₂₇	1	2	yes	—
207	C₂²×C₄	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
208	C₂²×C₄	#41×C₁₂₇	1	4	yes	—
209	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
210	M₁₆	—	2	8	no	—
211	M₁₆	—	2	16	no	—
212	M₁₆	—	2	8	no	—
213	M₁₆	—	2	8	no	—
214	M₁₆	—	4	8	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	16	no	—
217	C₂×C₈	—	2	8	no	—
218	C₂×C₈	—	4	8	no	—
219	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	16	yes	—
220	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
221	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
222	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
223	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
224	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
225	C₂³	—	2	4	no	—
226	C₂³	—	4	2	no	—
227	C₂³	—	4	4	no	—
228	C₂³	C₂×C₄⋊C₄×C₁₂₇	1	4	yes	—
229	C₂³	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
230	C₂³	C₂²⋊Q₈×C₁₂₇	1	4	yes	—
231	C₂³	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
232	Q₈	—	2	8	no	—
233	D₄	—	2	16	no	—
234	D₄	—	2	8	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	8	no	—
239	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
240	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	4	yes	—
241	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
242	C₂×C₄	#117×C₁₂₇	1	4	yes	—
243	C₂×C₄	#121×C₁₂₇	1	4	yes	—
244	C₂×C₄	#121×C₁₂₇	1	4	yes	—
245	C₂×C₄	C₂²⋊Q₈×C₁₂₇	1	4	yes	—
246	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
247	C₂×C₄	C₂²×D₄×C₁₂₇	1	2	yes	—
248	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
249	C₂×C₄	C₂×#41×C₁₂₇	1	2	yes	—
250	C₈	—	2	4	no	—
251	C₈	—	4	8	no	—
252	C₈	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
253	C₈	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	4	no	—
256	C₂²	—	2	8	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	4	4	no	—
259	C₂²	—	4	8	no	—
260	C₂²	#380×C₁₂₇	1	4	yes	—
261	C₂²	C₂²×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
262	C₂²	C₂×C₂²⋊Q₈×C₁₂₇	1	2	yes	—
263	C₄	—	2	4	no	—
264	C₄	#380×C₁₂₇	1	4	yes	—
265	C₄	C₂²×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
266	C₄	C₂×#117×C₁₂₇	1	1	yes	characteristic
267	C₄	C₂×#121×C₁₂₇	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	2	4	no	—
270	C₂	—	4	4	no	—
271	C₂	C₂×#380×C₁₂₇	1	1	yes	characteristic
272	C₂	C₁₂₇×#2916	1	2	yes	—

	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	C₂²×#41×C₁₂₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂²×#41×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×#108×C₁₂₇	C₂	semidirect
4. C₁₂₇×#1811	C₂	semidirect
5. C₁₂₇×#1810	C₂	semidirect
6. C₁₂₇×#1738	C₂	semidirect
131. #20668	C₁₂₇	direct (N × H)
138. C₂²×#108	C₂×C₁₂₇	semidirect
140. #1811	C₂×C₁₂₇	semidirect
141. #1810	C₂×C₁₂₇	semidirect
142. #1738	C₂×C₁₂₇	semidirect
143. C₁₂₇	#20668	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#131×C₁₂₇	2
C₂²×#108×C₁₂₇	2
C₁₂₇×#2879	2
C₁₂₇×#1811	2
C₁₂₇×#1810	2
C₁₂₇×#1738	2
#20668	127

Subgroups — C127×#20668 #44584341

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#20668 #44584341