Subgroups — C₁₂₇×#3062 #30636014

Order 16256.

504 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×AGL(1,ℤ/8)×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#131×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#537×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#535×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#469×C₁₂₇	C₂	1	2	yes	maximal, complemented
6	#468×C₁₂₇	C₂	1	2	yes	maximal, complemented
7	#466×C₁₂₇	C₂	1	2	yes	maximal, complemented
8	#465×C₁₂₇	C₂	1	2	yes	maximal, complemented
9	#428×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#416×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#102×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×#41×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
15	AGL(1,ℤ/8)×C₁₂₇	—	2	4	no	—
16	C₂×SD₁₆×C₁₂₇	C₂²	1	2	yes	—
17	C₂×D₈×C₁₂₇	C₂²	1	2	yes	—
18	#131×C₁₂₇	—	2	4	no	—
19	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	complemented
21	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
22	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	complemented
23	#127×C₁₂₇	—	2	1	no	—
24	#124×C₁₂₇	—	2	1	no	—
25	C₂²⋊Q₈×C₁₂₇	—	2	1	no	—
26	C₂²⋊Q₈×C₁₂₇	C₂²	1	2	yes	complemented
27	#121×C₁₂₇	—	2	1	no	—
28	#121×C₁₂₇	—	2	2	no	—
29	#121×C₁₂₇	C₂²	1	2	yes	—
30	#121×C₁₂₇	C₂²	1	2	yes	—
31	#121×C₁₂₇	C₂²	1	2	yes	complemented
32	#120×C₁₂₇	—	2	2	no	—
33	#117×C₁₂₇	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
35	#107×C₁₂₇	C₂²	1	2	yes	complemented
36	#106×C₁₂₇	C₂²	1	2	yes	complemented
37	#103×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
38	#103×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
39	#102×C₁₂₇	C₂²	1	2	yes	—
40	#102×C₁₂₇	C₂²	1	2	yes	—
41	#102×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
42	#102×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
43	C₂⁴×C₁₂₇	—	2	1	no	—
44	#41×C₁₂₇	—	2	2	no	—
45	#41×C₁₂₇	—	2	2	no	—
46	#41×C₁₂₇	—	2	2	no	—
47	#41×C₁₂₇	—	2	2	no	—
48	C₂×Q₈×C₁₂₇	D₄	1	1	yes	characteristic
49	C₂×Q₈×C₁₂₇	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₁₂₇	—	2	1	no	—
51	C₂×D₄×C₁₂₇	—	2	2	no	—
52	C₂×D₄×C₁₂₇	—	2	2	no	—
53	C₂×D₄×C₁₂₇	—	2	2	no	—
54	C₂×D₄×C₁₂₇	—	2	2	no	—
55	C₂×D₄×C₁₂₇	—	2	1	no	—
56	C₂×D₄×C₁₂₇	—	2	2	no	—
57	C₂×D₄×C₁₂₇	D₄	1	2	yes	—
58	C₂×D₄×C₁₂₇	D₄	1	1	yes	characteristic
59	C₂×D₄×C₁₂₇	C₂³	1	2	yes	—
60	C₂×D₄×C₁₂₇	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₂₇	—	2	1	no	—
62	C₂²×C₄×C₁₂₇	—	2	1	no	—
63	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
64	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
65	SD₁₆×C₁₂₇	—	2	4	no	—
66	D₈×C₁₂₇	—	2	4	no	—
67	M₁₆×C₁₂₇	—	2	4	no	—
68	M₁₆×C₁₂₇	D₄	1	4	yes	complemented
69	C₂×C₈×C₁₂₇	—	2	2	no	—
70	C₂×C₈×C₁₂₇	—	2	4	no	—
71	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
74	C₄⋊C₄×C₁₂₇	—	2	1	no	—
75	C₄⋊C₄×C₁₂₇	—	2	2	no	—
76	C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
77	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
79	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
80	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
81	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
82	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
83	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
84	C₄²×C₁₂₇	—	2	1	no	—
85	C₂³×C₁₂₇	—	2	1	no	—
86	C₂³×C₁₂₇	—	2	2	no	—
87	C₂³×C₁₂₇	—	4	2	no	—
88	C₂³×C₁₂₇	—	8	1	no	—
89	C₂³×C₁₂₇	—	8	1	no	—
90	C₂³×C₁₂₇	C₂×D₄	1	2	yes	—
91	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
92	Q₈×C₁₂₇	—	2	2	no	—
93	Q₈×C₁₂₇	—	2	1	no	—
94	Q₈×C₁₂₇	—	2	1	no	—
95	D₄×C₁₂₇	—	2	4	no	—
96	D₄×C₁₂₇	—	2	2	no	—
97	D₄×C₁₂₇	—	2	1	no	—
98	D₄×C₁₂₇	—	2	2	no	—
99	D₄×C₁₂₇	—	2	2	no	—
100	D₄×C₁₂₇	—	2	1	no	—
101	D₄×C₁₂₇	—	4	2	no	—
102	D₄×C₁₂₇	—	4	2	no	—
103	D₄×C₁₂₇	—	8	2	no	—
104	D₄×C₁₂₇	—	8	2	no	—
105	D₄×C₁₂₇	—	8	2	no	—
106	C₂×C₄×C₁₂₇	—	2	2	no	—
107	C₂×C₄×C₁₂₇	—	2	2	no	—
108	C₂×C₄×C₁₂₇	—	2	1	no	—
109	C₂×C₄×C₁₂₇	—	2	1	no	—
110	C₂×C₄×C₁₂₇	—	2	2	no	—
111	C₂×C₄×C₁₂₇	—	2	1	no	—
112	C₂×C₄×C₁₂₇	—	4	2	no	—
113	C₂×C₄×C₁₂₇	—	8	1	no	—
114	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
115	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
117	C₂×C₄×C₁₂₇	#41	1	2	yes	—
118	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
119	C₈×C₁₂₇	—	2	1	no	—
120	C₈×C₁₂₇	—	2	1	no	—
121	C₈×C₁₂₇	C₂×D₄	1	4	yes	—
122	C₂²×C₁₂₇	—	2	1	no	—
123	C₂²×C₁₂₇	—	2	1	no	—
124	C₂²×C₁₂₇	—	2	2	no	—
125	C₂²×C₁₂₇	—	2	2	no	—
126	C₂²×C₁₂₇	—	2	2	no	—
127	C₂²×C₁₂₇	—	4	1	no	—
128	C₂²×C₁₂₇	—	4	2	no	—
129	C₂²×C₁₂₇	—	4	1	no	—
130	C₂²×C₁₂₇	—	4	2	no	—
131	C₂²×C₁₂₇	—	4	2	no	—
132	C₂²×C₁₂₇	—	4	1	no	—
133	C₂²×C₁₂₇	—	8	4	no	—
134	C₂²×C₁₂₇	#121	1	2	yes	—
135	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
136	C₄×C₁₂₇	—	2	2	no	—
137	C₄×C₁₂₇	—	4	1	no	—
138	C₄×C₁₂₇	—	4	2	no	—
139	C₄×C₁₂₇	—	4	1	no	—
140	C₄×C₁₂₇	—	4	1	no	—
141	C₄×C₁₂₇	#121	1	2	yes	—
142	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
143	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
144	C₂×C₁₂₇	—	2	2	no	—
145	C₂×C₁₂₇	—	4	2	no	—
146	C₂×C₁₂₇	—	8	1	no	—
147	C₂×C₁₂₇	—	8	2	no	—
148	C₂×C₁₂₇	C₂×#121	1	1	yes	characteristic
149	C₂×C₁₂₇	#576	1	1	yes	characteristic
150	C₂×C₁₂₇	#577	1	1	yes	characteristic
151	#3062	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₁₂₇	#3062	1	1	yes	characteristic, direct product factor, Sylow
153	C₂×AGL(1,ℤ/8)	C₂×C₁₂₇	1	1	yes	characteristic, complemented
154	C₂×#131	C₂×C₁₂₇	1	1	yes	characteristic, complemented
155	#537	C₂×C₁₂₇	1	1	yes	characteristic, complemented
156	#535	C₂×C₁₂₇	1	1	yes	characteristic, complemented
157	#469	C₂×C₁₂₇	1	2	yes	complemented
158	#468	C₂×C₁₂₇	1	2	yes	complemented
159	#466	C₂×C₁₂₇	1	2	yes	complemented
160	#465	C₂×C₁₂₇	1	2	yes	complemented
161	#428	C₂×C₁₂₇	1	1	yes	characteristic, complemented
162	#416	C₂×C₁₂₇	1	1	yes	characteristic, complemented
163	C₂×#102	C₂×C₁₂₇	1	1	yes	characteristic, complemented
164	C₂×#41	C₂²×C₁₂₇	1	1	yes	characteristic
165	C₂×#41	C₂²×C₁₂₇	1	1	yes	characteristic
166	C₂²×D₄	C₂²×C₁₂₇	1	1	yes	characteristic
167	AGL(1,ℤ/8)	—	2	4	no	—
168	C₂×SD₁₆	C₂²×C₁₂₇	1	2	yes	—
169	C₂×D₈	C₂²×C₁₂₇	1	2	yes	—
170	#131	—	2	4	no	—
171	C₂×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
172	C₂×M₁₆	C₂²×C₁₂₇	1	2	yes	complemented
173	C₂²×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
174	C₂²×C₈	C₂²×C₁₂₇	1	2	yes	complemented
175	#127	—	2	1	no	—
176	#124	—	2	1	no	—
177	C₂²⋊Q₈	—	2	1	no	—
178	C₂²⋊Q₈	C₂²×C₁₂₇	1	2	yes	complemented
179	#121	—	2	1	no	—
180	#121	—	2	2	no	—
181	#121	C₂²×C₁₂₇	1	2	yes	—
182	#121	C₂²×C₁₂₇	1	2	yes	—
183	#121	C₂²×C₁₂₇	1	2	yes	complemented
184	#120	—	2	2	no	—
185	#117	C₂²×C₁₂₇	1	1	yes	characteristic
186	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
187	#107	C₂²×C₁₂₇	1	2	yes	complemented
188	#106	C₂²×C₁₂₇	1	2	yes	complemented
189	#103	C₂²×C₁₂₇	1	1	yes	characteristic, complemented
190	#103	C₂²×C₁₂₇	1	1	yes	characteristic, complemented
191	#102	C₂²×C₁₂₇	1	2	yes	—
192	#102	C₂²×C₁₂₇	1	2	yes	—
193	#102	C₂²×C₁₂₇	1	1	yes	characteristic, complemented
194	#102	C₂²×C₁₂₇	1	1	yes	characteristic, complemented
195	C₂⁴	—	2	1	no	—
196	#41	—	2	2	no	—
197	#41	—	2	2	no	—
198	#41	—	2	2	no	—
199	#41	—	2	2	no	—
200	C₂×Q₈	D₄×C₁₂₇	1	1	yes	characteristic
201	C₂×Q₈	C₂³×C₁₂₇	1	1	yes	characteristic
202	C₂×D₄	—	2	1	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	2	1	no	—
208	C₂×D₄	—	2	2	no	—
209	C₂×D₄	D₄×C₁₂₇	1	2	yes	—
210	C₂×D₄	D₄×C₁₂₇	1	1	yes	characteristic
211	C₂×D₄	C₂³×C₁₂₇	1	2	yes	—
212	C₂×D₄	C₂³×C₁₂₇	1	1	yes	characteristic
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
216	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
217	SD₁₆	—	2	4	no	—
218	D₈	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	D₄×C₁₂₇	1	4	yes	complemented
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
224	C₂×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
225	C₂×C₈	C₂³×C₁₂₇	1	2	yes	—
226	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	C₂³×C₁₂₇	1	2	yes	—
229	C₄⋊C₄	C₂³×C₁₂₇	1	1	yes	characteristic
230	C₄⋊C₄	C₂³×C₁₂₇	1	1	yes	characteristic
231	C₂²⋊C₄	—	2	1	no	—
232	C₂²⋊C₄	—	2	2	no	—
233	C₂²⋊C₄	—	2	2	no	—
234	C₂²⋊C₄	—	2	2	no	—
235	C₂²⋊C₄	—	2	2	no	—
236	C₄²	—	2	1	no	—
237	C₂³	—	2	1	no	—
238	C₂³	—	2	2	no	—
239	C₂³	—	4	2	no	—
240	C₂³	—	8	1	no	—
241	C₂³	—	8	1	no	—
242	C₂³	C₂×D₄×C₁₂₇	1	2	yes	—
243	C₂³	C₂×D₄×C₁₂₇	1	1	yes	characteristic
244	Q₈	—	2	2	no	—
245	Q₈	—	2	1	no	—
246	Q₈	—	2	1	no	—
247	D₄	—	2	4	no	—
248	D₄	—	2	2	no	—
249	D₄	—	2	1	no	—
250	D₄	—	2	2	no	—
251	D₄	—	2	2	no	—
252	D₄	—	2	1	no	—
253	D₄	—	4	2	no	—
254	D₄	—	4	2	no	—
255	D₄	—	8	2	no	—
256	D₄	—	8	2	no	—
257	D₄	—	8	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	8	1	no	—
266	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
267	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
268	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
269	C₂×C₄	#41×C₁₂₇	1	2	yes	—
270	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
271	C₈	—	2	1	no	—
272	C₈	—	2	1	no	—
273	C₈	C₂×D₄×C₁₂₇	1	4	yes	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	8	4	no	—
286	C₂²	#121×C₁₂₇	1	2	yes	—
287	C₂²	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
288	C₄	—	2	2	no	—
289	C₄	—	4	1	no	—
290	C₄	—	4	2	no	—
291	C₄	—	4	1	no	—
292	C₄	—	4	1	no	—
293	C₄	#121×C₁₂₇	1	2	yes	—
294	C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
295	C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
296	C₂	—	2	2	no	—
297	C₂	—	4	2	no	—
298	C₂	—	8	2	no	—
299	C₂	—	8	1	no	—
300	C₂	C₂×#121×C₁₂₇	1	1	yes	characteristic
301	C₂	#576×C₁₂₇	1	1	yes	characteristic
302	C₂	#577×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₂₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₂₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂×#41×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×AGL(1,ℤ/8)×C₁₂₇	C₂	semidirect
2. C₂×#131×C₁₂₇	C₂	semidirect
3. #537×C₁₂₇	C₂	semidirect
4. #535×C₁₂₇	C₂	semidirect
5. #469×C₁₂₇	C₂	semidirect
6. #468×C₁₂₇	C₂	semidirect
7. #466×C₁₂₇	C₂	semidirect
8. #465×C₁₂₇	C₂	semidirect
9. #428×C₁₂₇	C₂	semidirect
10. #416×C₁₂₇	C₂	semidirect
11. C₂×#102×C₁₂₇	C₂	semidirect
20. C₂×M₁₆×C₁₂₇	C₂²	semidirect
22. C₂²×C₈×C₁₂₇	C₂²	semidirect
26. C₂²⋊Q₈×C₁₂₇	C₂²	semidirect
31. #121×C₁₂₇	C₂²	semidirect
35. #107×C₁₂₇	C₂²	semidirect
36. #106×C₁₂₇	C₂²	semidirect
37. #103×C₁₂₇	C₂²	semidirect
38. #103×C₁₂₇	C₂²	semidirect
41. #102×C₁₂₇	C₂²	semidirect
42. #102×C₁₂₇	C₂²	semidirect
68. M₁₆×C₁₂₇	D₄	semidirect
151. #3062	C₁₂₇	direct (N × H)
152. C₁₂₇	#3062	direct (N × H)
153. C₂×AGL(1,ℤ/8)	C₂×C₁₂₇	semidirect
154. C₂×#131	C₂×C₁₂₇	semidirect
155. #537	C₂×C₁₂₇	semidirect
156. #535	C₂×C₁₂₇	semidirect
157. #469	C₂×C₁₂₇	semidirect
158. #468	C₂×C₁₂₇	semidirect
159. #466	C₂×C₁₂₇	semidirect
160. #465	C₂×C₁₂₇	semidirect
161. #428	C₂×C₁₂₇	semidirect
162. #416	C₂×C₁₂₇	semidirect
163. C₂×#102	C₂×C₁₂₇	semidirect
172. C₂×M₁₆	C₂²×C₁₂₇	semidirect
174. C₂²×C₈	C₂²×C₁₂₇	semidirect
178. C₂²⋊Q₈	C₂²×C₁₂₇	semidirect
183. #121	C₂²×C₁₂₇	semidirect
187. #107	C₂²×C₁₂₇	semidirect
188. #106	C₂²×C₁₂₇	semidirect
189. #103	C₂²×C₁₂₇	semidirect
190. #103	C₂²×C₁₂₇	semidirect
193. #102	C₂²×C₁₂₇	semidirect
194. #102	C₂²×C₁₂₇	semidirect
220. M₁₆	D₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×AGL(1,ℤ/8)×C₁₂₇	2
C₂×#131×C₁₂₇	2
#537×C₁₂₇	2
#535×C₁₂₇	2
#469×C₁₂₇	2
#468×C₁₂₇	2
#466×C₁₂₇	2
#465×C₁₂₇	2
#428×C₁₂₇	2
#416×C₁₂₇	2
C₂×#102×C₁₂₇	2
#3062	127

Subgroups — C127×#3062 #30636014

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#3062 #30636014