Subgroups — C₂×C₁₂₇×#1993 #44584993

Order 32512.

1422 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#382×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1993	C₂	1	8	yes	maximal, direct product factor
5	C₂×#340×C₁₂₇	C₂	1	4	yes	maximal
6	C₂³×Q₈×C₁₂₇	—	2	1	no	—
7	C₂³×C₈×C₁₂₇	C₂²	1	2	yes	—
8	C₂×C₄⋊Q₈×C₁₂₇	—	2	2	no	—
9	C₂×C₄⋊Q₈×C₁₂₇	—	2	4	no	—
10	C₂×C₄⋊Q₈×C₁₂₇	C₄	1	8	yes	complemented
11	C₂×C₄⋊Q₈×C₁₂₇	C₂²	1	4	yes	—
12	C₂²×C₄⋊C₄×C₁₂₇	—	2	1	no	—
13	C₂²×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
14	C₂²×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×#97×C₁₂₇	—	2	4	no	—
17	C₂×#97×C₁₂₇	C₂²	1	4	yes	—
18	#382×C₁₂₇	—	2	2	no	—
19	#382×C₁₂₇	C₂²	1	4	yes	—
20	C₂×#95×C₁₂₇	—	2	1	no	—
21	C₂×#95×C₁₂₇	—	2	1	no	—
22	#340×C₁₂₇	C₂²	1	16	yes	—
23	C₂²×Q₈×C₁₂₇	—	2	8	no	—
24	C₂²×Q₈×C₁₂₇	—	2	4	no	—
25	C₂²×Q₈×C₁₂₇	—	4	8	no	—
26	C₂³×C₄×C₁₂₇	—	2	2	no	—
27	C₂³×C₄×C₁₂₇	—	2	2	no	—
28	C₂³×C₄×C₁₂₇	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₁₂₇	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₁₂₇	—	2	4	no	—
32	C₂²×C₈×C₁₂₇	—	2	4	no	—
33	C₂²×C₈×C₁₂₇	Q₈	1	8	yes	complemented
34	C₂²×C₈×C₁₂₇	C₂³	1	4	yes	—
35	C₄⋊Q₈×C₁₂₇	—	2	16	no	—
36	C₄⋊Q₈×C₁₂₇	—	2	8	no	—
37	C₄⋊Q₈×C₁₂₇	C₂×C₄	1	16	yes	complemented
38	C₂×C₄⋊C₄×C₁₂₇	—	2	8	no	—
39	C₂×C₄⋊C₄×C₁₂₇	—	2	4	no	—
40	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₁₂₇	—	2	4	no	—
42	C₂×C₄⋊C₄×C₁₂₇	—	2	8	no	—
43	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	8	yes	—
45	C₂×C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
47	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₁₂₇	D₄	1	2	yes	—
49	C₂×C₄²×C₁₂₇	D₄	1	4	yes	—
50	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
51	#97×C₁₂₇	—	2	8	no	—
52	#95×C₁₂₇	—	2	4	no	—
53	#95×C₁₂₇	—	2	4	no	—
54	C₂⁴×C₁₂₇	C₂×D₄	1	1	yes	characteristic
55	C₂×Q₈×C₁₂₇	—	2	16	no	—
56	C₂×Q₈×C₁₂₇	—	4	16	no	—
57	C₂×Q₈×C₁₂₇	—	4	32	no	—
58	C₂²×C₄×C₁₂₇	—	2	4	no	—
59	C₂²×C₄×C₁₂₇	—	2	4	no	—
60	C₂²×C₄×C₁₂₇	—	2	8	no	—
61	C₂²×C₄×C₁₂₇	—	2	4	no	—
62	C₂²×C₄×C₁₂₇	—	2	4	no	—
63	C₂²×C₄×C₁₂₇	—	4	8	no	—
64	C₂²×C₄×C₁₂₇	—	4	4	no	—
65	C₂²×C₄×C₁₂₇	—	4	4	no	—
66	C₂²×C₄×C₁₂₇	—	4	4	no	—
67	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
68	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
69	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
75	C₂²×C₄×C₁₂₇	#41	1	4	yes	—
76	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₁₂₇	—	2	16	no	—
78	C₂×C₈×C₁₂₇	—	2	4	no	—
79	C₂×C₈×C₁₂₇	—	2	4	no	—
80	C₂×C₈×C₁₂₇	C₂×Q₈	1	8	yes	complemented
81	C₄⋊C₄×C₁₂₇	—	2	16	no	—
82	C₄⋊C₄×C₁₂₇	—	2	8	no	—
83	C₄⋊C₄×C₁₂₇	—	2	4	no	—
84	C₄⋊C₄×C₁₂₇	—	2	8	no	—
85	C₄⋊C₄×C₁₂₇	—	4	4	no	—
86	C₄⋊C₄×C₁₂₇	C₂²×C₄	1	8	yes	—
87	C₄²×C₁₂₇	C₂²⋊C₄	1	8	yes	—
88	C₄²×C₁₂₇	C₂²×C₄	1	4	yes	—
89	C₄²×C₁₂₇	C₂×D₄	1	4	yes	—
90	C₂³×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
91	C₂³×C₁₂₇	#124	1	2	yes	—
92	C₂³×C₁₂₇	C₄⋊Q₈	1	2	yes	—
93	C₂³×C₁₂₇	AGL(1,ℤ/8)	1	4	yes	—
94	C₂³×C₁₂₇	#137	1	4	yes	—
95	C₂³×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
96	Q₈×C₁₂₇	—	4	32	no	—
97	C₂×C₄×C₁₂₇	—	2	4	no	—
98	C₂×C₄×C₁₂₇	—	2	4	no	—
99	C₂×C₄×C₁₂₇	—	2	8	no	—
100	C₂×C₄×C₁₂₇	—	2	8	no	—
101	C₂×C₄×C₁₂₇	—	2	8	no	—
102	C₂×C₄×C₁₂₇	—	4	8	no	—
103	C₂×C₄×C₁₂₇	—	4	16	no	—
104	C₂×C₄×C₁₂₇	—	4	4	no	—
105	C₂×C₄×C₁₂₇	—	4	4	no	—
106	C₂×C₄×C₁₂₇	—	4	4	no	—
107	C₂×C₄×C₁₂₇	—	8	8	no	—
108	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
109	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
110	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
111	C₂×C₄×C₁₂₇	C₄×Q₈	1	8	yes	—
112	C₂×C₄×C₁₂₇	C₂²⋊Q₈	1	8	yes	—
113	C₂×C₄×C₁₂₇	#124	1	2	yes	—
114	C₂×C₄×C₁₂₇	C₄⋊Q₈	1	2	yes	—
115	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₂₇	C₂²×Q₈	1	2	yes	—
118	C₂×C₄×C₁₂₇	C₂×#41	1	2	yes	—
119	C₈×C₁₂₇	—	2	8	no	—
120	C₂²×C₁₂₇	#391	1	4	yes	—
121	C₂²×C₁₂₇	#418	1	4	yes	—
122	C₂²×C₁₂₇	#419	1	4	yes	—
123	C₂²×C₁₂₇	#489	1	8	yes	—
124	C₂²×C₁₂₇	#501	1	8	yes	—
125	C₂²×C₁₂₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
126	C₂²×C₁₂₇	C₂×#124	1	1	yes	characteristic
127	C₂²×C₁₂₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
128	C₂²×C₁₂₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
129	C₂²×C₁₂₇	C₂×#137	1	2	yes	—
130	C₄×C₁₂₇	—	2	4	no	—
131	C₄×C₁₂₇	—	2	4	no	—
132	C₄×C₁₂₇	—	4	8	no	—
133	C₄×C₁₂₇	—	8	4	no	—
134	C₄×C₁₂₇	#391	1	4	yes	—
135	C₄×C₁₂₇	C₂×C₄×Q₈	1	2	yes	—
136	C₄×C₁₂₇	C₂×C₂²⋊Q₈	1	2	yes	—
137	C₂×#1993	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₂×C₁₂₇	#1993	1	8	yes	direct product factor
139	C₂×C₁₂₇	C₂×#391	1	1	yes	characteristic
140	C₂×C₁₂₇	C₂×#418	1	1	yes	characteristic
141	C₂×C₁₂₇	C₂×#419	1	1	yes	characteristic
142	C₂×C₁₂₇	C₂×#489	1	2	yes	—
143	C₂×C₁₂₇	C₂×#501	1	2	yes	—
144	C₂²×C₄⋊Q₈	C₂×C₁₂₇	1	1	yes	characteristic
145	C₂²×#97	C₂×C₁₂₇	1	1	yes	characteristic
146	C₂×#382	C₂×C₁₂₇	1	1	yes	characteristic
147	#1993	C₂×C₁₂₇	1	8	yes	direct product factor
148	C₂×#340	C₂×C₁₂₇	1	4	yes	—
149	C₁₂₇	C₂×#1993	1	1	yes	characteristic, direct product factor, Sylow
150	C₂³×Q₈	—	2	1	no	—
151	C₂³×C₈	C₂²×C₁₂₇	1	2	yes	—
152	C₂×C₄⋊Q₈	—	2	2	no	—
153	C₂×C₄⋊Q₈	—	2	4	no	—
154	C₂×C₄⋊Q₈	C₄×C₁₂₇	1	8	yes	complemented
155	C₂×C₄⋊Q₈	C₂²×C₁₂₇	1	4	yes	—
156	C₂²×C₄⋊C₄	—	2	1	no	—
157	C₂²×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
158	C₂²×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
159	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
160	C₂×#97	—	2	4	no	—
161	C₂×#97	C₂²×C₁₂₇	1	4	yes	—
162	#382	—	2	2	no	—
163	#382	C₂²×C₁₂₇	1	4	yes	—
164	C₂×#95	—	2	1	no	—
165	C₂×#95	—	2	1	no	—
166	#340	C₂²×C₁₂₇	1	16	yes	—
167	C₂²×Q₈	—	2	8	no	—
168	C₂²×Q₈	—	2	4	no	—
169	C₂²×Q₈	—	4	8	no	—
170	C₂³×C₄	—	2	2	no	—
171	C₂³×C₄	—	2	2	no	—
172	C₂³×C₄	D₄×C₁₂₇	1	1	yes	characteristic
173	C₂³×C₄	D₄×C₁₂₇	1	1	yes	characteristic
174	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
175	C₂²×C₈	—	2	4	no	—
176	C₂²×C₈	—	2	4	no	—
177	C₂²×C₈	Q₈×C₁₂₇	1	8	yes	complemented
178	C₂²×C₈	C₂³×C₁₂₇	1	4	yes	—
179	C₄⋊Q₈	—	2	16	no	—
180	C₄⋊Q₈	—	2	8	no	—
181	C₄⋊Q₈	C₂×C₄×C₁₂₇	1	16	yes	complemented
182	C₂×C₄⋊C₄	—	2	8	no	—
183	C₂×C₄⋊C₄	—	2	4	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	—	2	4	no	—
186	C₂×C₄⋊C₄	—	2	8	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	8	yes	—
189	C₂×C₄⋊C₄	C₂³×C₁₂₇	1	4	yes	—
190	C₂×C₄⋊C₄	C₂³×C₁₂₇	1	4	yes	—
191	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
192	C₂×C₄²	D₄×C₁₂₇	1	2	yes	—
193	C₂×C₄²	D₄×C₁₂₇	1	4	yes	—
194	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
195	#97	—	2	8	no	—
196	#95	—	2	4	no	—
197	#95	—	2	4	no	—
198	C₂⁴	C₂×D₄×C₁₂₇	1	1	yes	characteristic
199	C₂×Q₈	—	2	16	no	—
200	C₂×Q₈	—	4	16	no	—
201	C₂×Q₈	—	4	32	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	2	8	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	4	8	no	—
208	C₂²×C₄	—	4	4	no	—
209	C₂²×C₄	—	4	4	no	—
210	C₂²×C₄	—	4	4	no	—
211	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	4	yes	—
212	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	4	yes	—
213	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
214	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
215	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
216	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
218	C₂²×C₄	C₂×Q₈×C₁₂₇	1	4	yes	—
219	C₂²×C₄	#41×C₁₂₇	1	4	yes	—
220	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
221	C₂×C₈	—	2	16	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	C₂×Q₈×C₁₂₇	1	8	yes	complemented
225	C₄⋊C₄	—	2	16	no	—
226	C₄⋊C₄	—	2	8	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	2	8	no	—
229	C₄⋊C₄	—	4	4	no	—
230	C₄⋊C₄	C₂²×C₄×C₁₂₇	1	8	yes	—
231	C₄²	C₂²⋊C₄×C₁₂₇	1	8	yes	—
232	C₄²	C₂²×C₄×C₁₂₇	1	4	yes	—
233	C₄²	C₂×D₄×C₁₂₇	1	4	yes	—
234	C₂³	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
235	C₂³	#124×C₁₂₇	1	2	yes	—
236	C₂³	C₄⋊Q₈×C₁₂₇	1	2	yes	—
237	C₂³	AGL(1,ℤ/8)×C₁₂₇	1	4	yes	—
238	C₂³	#137×C₁₂₇	1	4	yes	—
239	C₂³	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
240	Q₈	—	4	32	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	2	8	no	—
246	C₂×C₄	—	4	8	no	—
247	C₂×C₄	—	4	16	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	8	8	no	—
252	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
253	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
254	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
255	C₂×C₄	C₄×Q₈×C₁₂₇	1	8	yes	—
256	C₂×C₄	C₂²⋊Q₈×C₁₂₇	1	8	yes	—
257	C₂×C₄	#124×C₁₂₇	1	2	yes	—
258	C₂×C₄	C₄⋊Q₈×C₁₂₇	1	2	yes	—
259	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
260	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
261	C₂×C₄	C₂²×Q₈×C₁₂₇	1	2	yes	—
262	C₂×C₄	C₂×#41×C₁₂₇	1	2	yes	—
263	C₈	—	2	8	no	—
264	C₂²	#391×C₁₂₇	1	4	yes	—
265	C₂²	#418×C₁₂₇	1	4	yes	—
266	C₂²	#419×C₁₂₇	1	4	yes	—
267	C₂²	#489×C₁₂₇	1	8	yes	—
268	C₂²	#501×C₁₂₇	1	8	yes	—
269	C₂²	C₂²×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
270	C₂²	C₂×#124×C₁₂₇	1	1	yes	characteristic
271	C₂²	C₂×C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
272	C₂²	C₂×AGL(1,ℤ/8)×C₁₂₇	1	2	yes	—
273	C₂²	C₂×#137×C₁₂₇	1	2	yes	—
274	C₄	—	2	4	no	—
275	C₄	—	2	4	no	—
276	C₄	—	4	8	no	—
277	C₄	—	8	4	no	—
278	C₄	#391×C₁₂₇	1	4	yes	—
279	C₄	C₂×C₄×Q₈×C₁₂₇	1	2	yes	—
280	C₄	C₂×C₂²⋊Q₈×C₁₂₇	1	2	yes	—
281	C₂	C₁₂₇×#1993	1	8	yes	direct product factor
282	C₂	C₂×#391×C₁₂₇	1	1	yes	characteristic
283	C₂	C₂×#418×C₁₂₇	1	1	yes	characteristic
284	C₂	C₂×#419×C₁₂₇	1	1	yes	characteristic
285	C₂	C₂×#489×C₁₂₇	1	2	yes	—
286	C₂	C₂×#501×C₁₂₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₂₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×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	direct (N × H)
10. C₂×C₄⋊Q₈×C₁₂₇	C₄	semidirect
33. C₂²×C₈×C₁₂₇	Q₈	semidirect
37. C₄⋊Q₈×C₁₂₇	C₂×C₄	semidirect
80. C₂×C₈×C₁₂₇	C₂×Q₈	semidirect
137. C₂×#1993	C₁₂₇	direct (N × H)
138. C₂×C₁₂₇	#1993	direct (N × H)
147. #1993	C₂×C₁₂₇	direct (N × H)
149. C₁₂₇	C₂×#1993	direct (N × H)
154. C₂×C₄⋊Q₈	C₄×C₁₂₇	semidirect
177. C₂²×C₈	Q₈×C₁₂₇	semidirect
181. C₄⋊Q₈	C₂×C₄×C₁₂₇	semidirect
224. C₂×C₈	C₂×Q₈×C₁₂₇	semidirect
281. C₂	C₁₂₇×#1993	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×C₄⋊Q₈×C₁₂₇	2
C₂²×#97×C₁₂₇	2
C₂×#382×C₁₂₇	2
C₁₂₇×#1993	2
C₂×#340×C₁₂₇	2
C₂×#1993	127

Subgroups — C2×C127×#1993 #44584993

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₂₇×#1993 #44584993