Subgroups — C₁₂₇×#21321 #44584994

Order 32512.

1166 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#533×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#2285	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1993	C₂	1	4	yes	maximal, complemented
5	C₁₂₇×#1992	C₂	1	4	yes	maximal, complemented
6	C₁₂₇×#1735	C₂	1	4	yes	maximal
7	C₂³×C₈×C₁₂₇	C₂²	1	2	yes	—
8	#533×C₁₂₇	—	2	4	no	—
9	#533×C₁₂₇	C₄	1	8	yes	complemented
10	C₂×C₄⋊Q₈×C₁₂₇	C₂²	1	2	yes	—
11	C₂×#125×C₁₂₇	C₂²	1	2	yes	—
12	C₂×C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
13	C₂×C₄×Q₈×C₁₂₇	—	2	2	no	—
14	C₂×#117×C₁₂₇	C₂²	1	2	yes	—
15	C₂×#117×C₁₂₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂×#97×C₁₂₇	—	2	4	no	—
18	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
19	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
20	#384×C₁₂₇	C₂²	1	2	yes	—
21	#382×C₁₂₇	C₂²	1	2	yes	—
22	#380×C₁₂₇	—	2	2	no	—
23	C₄×C₄⋊C₄×C₁₂₇	—	2	2	no	—
24	#340×C₁₂₇	C₂²	1	16	yes	—
25	C₂²×Q₈×C₁₂₇	—	2	2	no	—
26	C₂³×C₄×C₁₂₇	D₄	1	2	yes	—
27	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₁₂₇	—	2	4	no	—
29	C₂²×C₈×C₁₂₇	—	2	4	no	—
30	C₂²×C₈×C₁₂₇	Q₈	1	8	yes	complemented
31	C₂²×C₈×C₁₂₇	C₂³	1	4	yes	—
32	C₄⋊Q₈×C₁₂₇	—	2	4	no	—
33	C₄⋊Q₈×C₁₂₇	C₂×C₄	1	8	yes	—
34	#125×C₁₂₇	—	2	4	no	—
35	#125×C₁₂₇	C₂×C₄	1	8	yes	—
36	C₂²⋊Q₈×C₁₂₇	—	2	16	no	—
37	C₄×Q₈×C₁₂₇	—	2	16	no	—
38	#117×C₁₂₇	—	2	4	no	—
39	#117×C₁₂₇	—	2	8	no	—
40	#117×C₁₂₇	C₂×C₄	1	8	yes	—
41	C₂×C₄⋊C₄×C₁₂₇	—	2	4	no	—
42	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₂₇	—	2	4	no	—
44	C₂×C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
45	C₂×C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
46	C₂×C₂²⋊C₄×C₁₂₇	—	2	2	no	—
47	C₂×C₂²⋊C₄×C₁₂₇	—	2	2	no	—
48	C₂×C₄²×C₁₂₇	—	2	2	no	—
49	C₂×C₄²×C₁₂₇	—	2	2	no	—
50	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₁₂₇	D₄	1	2	yes	—
52	C₂×C₄²×C₁₂₇	D₄	1	4	yes	—
53	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
55	#97×C₁₂₇	—	2	4	no	—
56	#97×C₁₂₇	—	2	4	no	—
57	#95×C₁₂₇	—	2	4	no	—
58	C₂⁴×C₁₂₇	C₂×D₄	1	1	yes	characteristic
59	C₂×Q₈×C₁₂₇	—	2	8	no	—
60	C₂×Q₈×C₁₂₇	—	4	8	no	—
61	C₂²×C₄×C₁₂₇	—	2	4	no	—
62	C₂²×C₄×C₁₂₇	—	2	4	no	—
63	C₂²×C₄×C₁₂₇	—	4	4	no	—
64	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	8	yes	—
65	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
71	C₂²×C₄×C₁₂₇	#41	1	4	yes	—
72	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
73	C₂×C₈×C₁₂₇	—	2	16	no	—
74	C₂×C₈×C₁₂₇	—	2	4	no	—
75	C₂×C₈×C₁₂₇	—	2	4	no	—
76	C₂×C₈×C₁₂₇	C₂×Q₈	1	8	yes	—
77	C₄⋊C₄×C₁₂₇	—	2	16	no	—
78	C₄⋊C₄×C₁₂₇	—	2	8	no	—
79	C₄⋊C₄×C₁₂₇	—	2	4	no	—
80	C₄⋊C₄×C₁₂₇	—	2	8	no	—
81	C₄⋊C₄×C₁₂₇	—	4	8	no	—
82	C₄⋊C₄×C₁₂₇	C₂²×C₄	1	8	yes	—
83	C₂²⋊C₄×C₁₂₇	—	2	8	no	—
84	C₂²⋊C₄×C₁₂₇	—	4	4	no	—
85	C₄²×C₁₂₇	—	2	8	no	—
86	C₄²×C₁₂₇	—	4	4	no	—
87	C₄²×C₁₂₇	C₂²⋊C₄	1	8	yes	—
88	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
89	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
90	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
91	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
92	C₂³×C₁₂₇	—	2	4	no	—
93	C₂³×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
94	C₂³×C₁₂₇	#124	1	2	yes	—
95	C₂³×C₁₂₇	C₄⋊Q₈	1	2	yes	—
96	C₂³×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
97	Q₈×C₁₂₇	—	4	16	no	—
98	C₂×C₄×C₁₂₇	—	2	4	no	—
99	C₂×C₄×C₁₂₇	—	2	4	no	—
100	C₂×C₄×C₁₂₇	—	2	8	no	—
101	C₂×C₄×C₁₂₇	—	4	8	no	—
102	C₂×C₄×C₁₂₇	—	4	8	no	—
103	C₂×C₄×C₁₂₇	—	4	4	no	—
104	C₂×C₄×C₁₂₇	—	4	8	no	—
105	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
106	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
107	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
108	C₂×C₄×C₁₂₇	C₄×Q₈	1	8	yes	—
109	C₂×C₄×C₁₂₇	C₂²⋊Q₈	1	8	yes	—
110	C₂×C₄×C₁₂₇	#124	1	2	yes	—
111	C₂×C₄×C₁₂₇	C₄⋊Q₈	1	2	yes	—
112	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₂₇	C₂²×Q₈	1	2	yes	—
115	C₂×C₄×C₁₂₇	C₂×#41	1	2	yes	—
116	C₈×C₁₂₇	—	2	8	no	—
117	C₂²×C₁₂₇	—	2	4	no	—
118	C₂²×C₁₂₇	—	2	4	no	—
119	C₂²×C₁₂₇	—	2	4	no	—
120	C₂²×C₁₂₇	#391	1	4	yes	—
121	C₂²×C₁₂₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
122	C₂²×C₁₂₇	C₂×#124	1	1	yes	characteristic
123	C₂²×C₁₂₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
124	C₂²×C₁₂₇	#575	1	4	yes	—
125	C₄×C₁₂₇	—	2	4	no	—
126	C₄×C₁₂₇	—	2	4	no	—
127	C₄×C₁₂₇	—	4	8	no	—
128	C₄×C₁₂₇	—	8	4	no	—
129	C₄×C₁₂₇	#391	1	4	yes	—
130	C₄×C₁₂₇	C₂×C₄×Q₈	1	2	yes	—
131	C₄×C₁₂₇	C₂×C₂²⋊Q₈	1	2	yes	—
132	#21321	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂×C₁₂₇	—	2	4	no	—
134	C₂×C₁₂₇	C₂×#391	1	1	yes	characteristic
135	C₂×C₁₂₇	#2896	1	2	yes	—
136	C₂×C₁₂₇	#3135	1	2	yes	—
137	C₂×C₁₂₇	#3162	1	2	yes	—
138	C₂×#533	C₂×C₁₂₇	1	1	yes	characteristic
139	C₂²×#97	C₂×C₁₂₇	1	1	yes	characteristic
140	#2285	C₂×C₁₂₇	1	1	yes	characteristic
141	#1993	C₂×C₁₂₇	1	4	yes	complemented
142	#1992	C₂×C₁₂₇	1	4	yes	complemented
143	#1735	C₂×C₁₂₇	1	4	yes	—
144	C₁₂₇	#21321	1	1	yes	characteristic, direct product factor, Sylow
145	C₂³×C₈	C₂²×C₁₂₇	1	2	yes	—
146	#533	—	2	4	no	—
147	#533	C₄×C₁₂₇	1	8	yes	complemented
148	C₂×C₄⋊Q₈	C₂²×C₁₂₇	1	2	yes	—
149	C₂×#125	C₂²×C₁₂₇	1	2	yes	—
150	C₂×C₂²⋊Q₈	—	2	2	no	—
151	C₂×C₄×Q₈	—	2	2	no	—
152	C₂×#117	C₂²×C₁₂₇	1	2	yes	—
153	C₂×#117	C₂²×C₁₂₇	1	2	yes	—
154	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
155	C₂×#97	—	2	4	no	—
156	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
157	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
158	#384	C₂²×C₁₂₇	1	2	yes	—
159	#382	C₂²×C₁₂₇	1	2	yes	—
160	#380	—	2	2	no	—
161	C₄×C₄⋊C₄	—	2	2	no	—
162	#340	C₂²×C₁₂₇	1	16	yes	—
163	C₂²×Q₈	—	2	2	no	—
164	C₂³×C₄	D₄×C₁₂₇	1	2	yes	—
165	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
166	C₂²×C₈	—	2	4	no	—
167	C₂²×C₈	—	2	4	no	—
168	C₂²×C₈	Q₈×C₁₂₇	1	8	yes	complemented
169	C₂²×C₈	C₂³×C₁₂₇	1	4	yes	—
170	C₄⋊Q₈	—	2	4	no	—
171	C₄⋊Q₈	C₂×C₄×C₁₂₇	1	8	yes	—
172	#125	—	2	4	no	—
173	#125	C₂×C₄×C₁₂₇	1	8	yes	—
174	C₂²⋊Q₈	—	2	16	no	—
175	C₄×Q₈	—	2	16	no	—
176	#117	—	2	4	no	—
177	#117	—	2	8	no	—
178	#117	C₂×C₄×C₁₂₇	1	8	yes	—
179	C₂×C₄⋊C₄	—	2	4	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	4	no	—
182	C₂×C₄⋊C₄	C₂³×C₁₂₇	1	4	yes	—
183	C₂×C₄⋊C₄	C₂³×C₁₂₇	1	4	yes	—
184	C₂×C₂²⋊C₄	—	2	2	no	—
185	C₂×C₂²⋊C₄	—	2	2	no	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
189	C₂×C₄²	D₄×C₁₂₇	1	2	yes	—
190	C₂×C₄²	D₄×C₁₂₇	1	4	yes	—
191	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
192	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
193	#97	—	2	4	no	—
194	#97	—	2	4	no	—
195	#95	—	2	4	no	—
196	C₂⁴	C₂×D₄×C₁₂₇	1	1	yes	characteristic
197	C₂×Q₈	—	2	8	no	—
198	C₂×Q₈	—	4	8	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	4	4	no	—
202	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	8	yes	—
203	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
204	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
205	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
206	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
208	C₂²×C₄	C₂×Q₈×C₁₂₇	1	4	yes	—
209	C₂²×C₄	#41×C₁₂₇	1	4	yes	—
210	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
211	C₂×C₈	—	2	16	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	C₂×Q₈×C₁₂₇	1	8	yes	—
215	C₄⋊C₄	—	2	16	no	—
216	C₄⋊C₄	—	2	8	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	2	8	no	—
219	C₄⋊C₄	—	4	8	no	—
220	C₄⋊C₄	C₂²×C₄×C₁₂₇	1	8	yes	—
221	C₂²⋊C₄	—	2	8	no	—
222	C₂²⋊C₄	—	4	4	no	—
223	C₄²	—	2	8	no	—
224	C₄²	—	4	4	no	—
225	C₄²	C₂²⋊C₄×C₁₂₇	1	8	yes	—
226	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
227	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
228	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
229	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
230	C₂³	—	2	4	no	—
231	C₂³	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
232	C₂³	#124×C₁₂₇	1	2	yes	—
233	C₂³	C₄⋊Q₈×C₁₂₇	1	2	yes	—
234	C₂³	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
235	Q₈	—	4	16	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	8	no	—
239	C₂×C₄	—	4	8	no	—
240	C₂×C₄	—	4	8	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	8	no	—
243	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
244	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
245	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
246	C₂×C₄	C₄×Q₈×C₁₂₇	1	8	yes	—
247	C₂×C₄	C₂²⋊Q₈×C₁₂₇	1	8	yes	—
248	C₂×C₄	#124×C₁₂₇	1	2	yes	—
249	C₂×C₄	C₄⋊Q₈×C₁₂₇	1	2	yes	—
250	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
251	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
252	C₂×C₄	C₂²×Q₈×C₁₂₇	1	2	yes	—
253	C₂×C₄	C₂×#41×C₁₂₇	1	2	yes	—
254	C₈	—	2	8	no	—
255	C₂²	—	2	4	no	—
256	C₂²	—	2	4	no	—
257	C₂²	—	2	4	no	—
258	C₂²	#391×C₁₂₇	1	4	yes	—
259	C₂²	C₂²×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
260	C₂²	C₂×#124×C₁₂₇	1	1	yes	characteristic
261	C₂²	C₂×C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
262	C₂²	#575×C₁₂₇	1	4	yes	—
263	C₄	—	2	4	no	—
264	C₄	—	2	4	no	—
265	C₄	—	4	8	no	—
266	C₄	—	8	4	no	—
267	C₄	#391×C₁₂₇	1	4	yes	—
268	C₄	C₂×C₄×Q₈×C₁₂₇	1	2	yes	—
269	C₄	C₂×C₂²⋊Q₈×C₁₂₇	1	2	yes	—
270	C₂	—	2	4	no	—
271	C₂	C₂×#391×C₁₂₇	1	1	yes	characteristic
272	C₂	C₁₂₇×#2896	1	2	yes	—
273	C₂	C₁₂₇×#3135	1	2	yes	—
274	C₂	C₁₂₇×#3162	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₁₂₇×#1993	C₂	semidirect
5. C₁₂₇×#1992	C₂	semidirect
9. #533×C₁₂₇	C₄	semidirect
30. C₂²×C₈×C₁₂₇	Q₈	semidirect
132. #21321	C₁₂₇	direct (N × H)
141. #1993	C₂×C₁₂₇	semidirect
142. #1992	C₂×C₁₂₇	semidirect
144. C₁₂₇	#21321	direct (N × H)
147. #533	C₄×C₁₂₇	semidirect
168. C₂²×C₈	Q₈×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#533×C₁₂₇	2
C₂²×#97×C₁₂₇	2
C₁₂₇×#2285	2
C₁₂₇×#1993	2
C₁₂₇×#1992	2
C₁₂₇×#1735	2
#21321	127

Subgroups — C127×#21321 #44584994

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#21321 #44584994