Subgroups — C₂×#449×C₁₂₇ #30635952

Order 16256.

812 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×#41×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×SD₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×D₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#121×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#449×C₁₂₇	C₂	1	8	yes	maximal, direct product factor
6	C₂×#103×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#102×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂²⋊C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₁₂₇	—	2	4	no	—
10	C₂×#41×C₁₂₇	—	2	4	no	—
11	C₂×#41×C₁₂₇	—	2	2	no	—
12	C₂×#41×C₁₂₇	C₂²	1	4	yes	complemented
13	C₂²×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₁₂₇	—	2	1	no	—
15	C₂²×D₄×C₁₂₇	—	2	1	no	—
16	C₂²×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂²×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂³×C₄×C₁₂₇	—	2	1	no	—
19	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₂×SD₁₆×C₁₂₇	—	2	4	no	—
21	C₂×SD₁₆×C₁₂₇	C₂²	1	4	yes	complemented
22	C₂×D₈×C₁₂₇	—	2	4	no	—
23	C₂×D₈×C₁₂₇	C₂²	1	4	yes	complemented
24	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
25	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
26	#121×C₁₂₇	—	2	2	no	—
27	#121×C₁₂₇	C₂²	1	4	yes	complemented
28	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₁₂₇	—	2	1	no	—
30	#103×C₁₂₇	C₂²	1	4	yes	complemented
31	#102×C₁₂₇	C₂²	1	4	yes	complemented
32	C₂²⋊C₈×C₁₂₇	C₂²	1	4	yes	complemented
33	C₂⁴×C₁₂₇	—	2	1	no	—
34	C₂⁴×C₁₂₇	—	2	1	no	—
35	C₂⁴×C₁₂₇	D₄	1	1	yes	characteristic
36	#41×C₁₂₇	—	2	16	no	—
37	#41×C₁₂₇	—	2	8	no	—
38	#41×C₁₂₇	—	2	8	no	—
39	C₂×Q₈×C₁₂₇	—	2	2	no	—
40	C₂×Q₈×C₁₂₇	—	2	1	no	—
41	C₂×Q₈×C₁₂₇	D₄	1	4	yes	complemented
42	C₂×Q₈×C₁₂₇	C₂³	1	2	yes	—
43	C₂×D₄×C₁₂₇	—	2	8	no	—
44	C₂×D₄×C₁₂₇	—	2	4	no	—
45	C₂×D₄×C₁₂₇	—	2	4	no	—
46	C₂×D₄×C₁₂₇	—	2	1	no	—
47	C₂×D₄×C₁₂₇	—	2	2	no	—
48	C₂×D₄×C₁₂₇	—	2	2	no	—
49	C₂×D₄×C₁₂₇	—	2	2	no	—
50	C₂×D₄×C₁₂₇	—	2	1	no	—
51	C₂×D₄×C₁₂₇	—	4	4	no	—
52	C₂×D₄×C₁₂₇	D₄	1	4	yes	complemented
53	C₂×D₄×C₁₂₇	C₂³	1	2	yes	—
54	C₂×D₄×C₁₂₇	C₂³	1	2	yes	—
55	C₂²×C₄×C₁₂₇	—	2	8	no	—
56	C₂²×C₄×C₁₂₇	—	2	1	no	—
57	C₂²×C₄×C₁₂₇	—	2	4	no	—
58	C₂²×C₄×C₁₂₇	—	2	2	no	—
59	C₂²×C₄×C₁₂₇	—	2	2	no	—
60	C₂²×C₄×C₁₂₇	—	2	1	no	—
61	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
62	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
64	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
65	SD₁₆×C₁₂₇	—	2	8	no	—
66	D₈×C₁₂₇	—	2	8	no	—
67	C₂×C₈×C₁₂₇	—	2	2	no	—
68	C₂×C₈×C₁₂₇	—	2	2	no	—
69	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
70	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
71	C₄⋊C₄×C₁₂₇	—	2	1	no	—
72	C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
73	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
74	C₂³×C₁₂₇	—	2	2	no	—
75	C₂³×C₁₂₇	—	2	4	no	—
76	C₂³×C₁₂₇	—	2	2	no	—
77	C₂³×C₁₂₇	—	2	2	no	—
78	C₂³×C₁₂₇	—	2	2	no	—
79	C₂³×C₁₂₇	—	2	2	no	—
80	C₂³×C₁₂₇	—	4	2	no	—
81	C₂³×C₁₂₇	—	4	2	no	—
82	C₂³×C₁₂₇	—	4	2	no	—
83	C₂³×C₁₂₇	—	4	2	no	—
84	C₂³×C₁₂₇	—	4	2	no	—
85	C₂³×C₁₂₇	C₂×D₄	1	2	yes	—
86	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
87	Q₈×C₁₂₇	—	2	4	no	—
88	Q₈×C₁₂₇	—	2	2	no	—
89	Q₈×C₁₂₇	C₂×D₄	1	4	yes	complemented
90	D₄×C₁₂₇	—	2	16	no	—
91	D₄×C₁₂₇	—	2	4	no	—
92	D₄×C₁₂₇	—	2	2	no	—
93	D₄×C₁₂₇	—	2	2	no	—
94	D₄×C₁₂₇	—	2	2	no	—
95	D₄×C₁₂₇	—	4	8	no	—
96	D₄×C₁₂₇	—	4	2	no	—
97	D₄×C₁₂₇	C₂×D₄	1	4	yes	complemented
98	C₂×C₄×C₁₂₇	—	2	16	no	—
99	C₂×C₄×C₁₂₇	—	2	2	no	—
100	C₂×C₄×C₁₂₇	—	2	4	no	—
101	C₂×C₄×C₁₂₇	—	2	2	no	—
102	C₂×C₄×C₁₂₇	—	2	4	no	—
103	C₂×C₄×C₁₂₇	—	2	2	no	—
104	C₂×C₄×C₁₂₇	—	2	2	no	—
105	C₂×C₄×C₁₂₇	—	2	2	no	—
106	C₂×C₄×C₁₂₇	—	4	2	no	—
107	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
108	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
111	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
112	C₈×C₁₂₇	—	2	2	no	—
113	C₈×C₁₂₇	—	2	2	no	—
114	C₂²×C₁₂₇	—	2	2	no	—
115	C₂²×C₁₂₇	—	2	4	no	—
116	C₂²×C₁₂₇	—	2	2	no	—
117	C₂²×C₁₂₇	—	2	2	no	—
118	C₂²×C₁₂₇	—	4	2	no	—
119	C₂²×C₁₂₇	—	4	2	no	—
120	C₂²×C₁₂₇	—	4	4	no	—
121	C₂²×C₁₂₇	—	4	8	no	—
122	C₂²×C₁₂₇	—	4	2	no	—
123	C₂²×C₁₂₇	—	4	2	no	—
124	C₂²×C₁₂₇	—	4	2	no	—
125	C₂²×C₁₂₇	—	8	4	no	—
126	C₂²×C₁₂₇	#120	1	2	yes	—
127	C₂²×C₁₂₇	#135	1	2	yes	—
128	C₂²×C₁₂₇	AGL(1,ℤ/8)	1	2	yes	—
129	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
130	C₄×C₁₂₇	—	2	2	no	—
131	C₄×C₁₂₇	—	2	4	no	—
132	C₄×C₁₂₇	—	4	2	no	—
133	C₄×C₁₂₇	#120	1	2	yes	—
134	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
135	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₁₂₇	—	4	2	no	—
137	C₂×C₁₂₇	—	4	4	no	—
138	C₂×C₁₂₇	—	8	2	no	—
139	C₂×C₁₂₇	#449	1	4	yes	direct product factor
140	C₂×C₁₂₇	C₂×#120	1	1	yes	characteristic
141	C₂×C₁₂₇	C₂×#135	1	1	yes	characteristic
142	C₂×C₁₂₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×#449	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₁₂₇	C₂×#449	1	1	yes	characteristic, direct product factor, Sylow
145	C₂²×#41	C₂×C₁₂₇	1	1	yes	characteristic, complemented
146	C₂²×SD₁₆	C₂×C₁₂₇	1	1	yes	characteristic, complemented
147	C₂²×D₈	C₂×C₁₂₇	1	1	yes	characteristic, complemented
148	C₂×#121	C₂×C₁₂₇	1	1	yes	characteristic, complemented
149	#449	C₂×C₁₂₇	1	8	yes	direct product factor
150	C₂×#103	C₂×C₁₂₇	1	1	yes	characteristic, complemented
151	C₂×#102	C₂×C₁₂₇	1	1	yes	characteristic, complemented
152	C₂×C₂²⋊C₈	C₂×C₁₂₇	1	1	yes	characteristic, complemented
153	C₂×#41	—	2	4	no	—
154	C₂×#41	—	2	4	no	—
155	C₂×#41	—	2	2	no	—
156	C₂×#41	C₂²×C₁₂₇	1	4	yes	complemented
157	C₂²×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
158	C₂²×D₄	—	2	1	no	—
159	C₂²×D₄	—	2	1	no	—
160	C₂²×D₄	C₂²×C₁₂₇	1	1	yes	characteristic
161	C₂²×D₄	C₂²×C₁₂₇	1	1	yes	characteristic
162	C₂³×C₄	—	2	1	no	—
163	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
164	C₂×SD₁₆	—	2	4	no	—
165	C₂×SD₁₆	C₂²×C₁₂₇	1	4	yes	complemented
166	C₂×D₈	—	2	4	no	—
167	C₂×D₈	C₂²×C₁₂₇	1	4	yes	complemented
168	C₂²×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
169	C₂²×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
170	#121	—	2	2	no	—
171	#121	C₂²×C₁₂₇	1	4	yes	complemented
172	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
173	C₂×C₂²⋊C₄	—	2	1	no	—
174	#103	C₂²×C₁₂₇	1	4	yes	complemented
175	#102	C₂²×C₁₂₇	1	4	yes	complemented
176	C₂²⋊C₈	C₂²×C₁₂₇	1	4	yes	complemented
177	C₂⁴	—	2	1	no	—
178	C₂⁴	—	2	1	no	—
179	C₂⁴	D₄×C₁₂₇	1	1	yes	characteristic
180	#41	—	2	16	no	—
181	#41	—	2	8	no	—
182	#41	—	2	8	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×Q₈	—	2	1	no	—
185	C₂×Q₈	D₄×C₁₂₇	1	4	yes	complemented
186	C₂×Q₈	C₂³×C₁₂₇	1	2	yes	—
187	C₂×D₄	—	2	8	no	—
188	C₂×D₄	—	2	4	no	—
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	2	1	no	—
195	C₂×D₄	—	4	4	no	—
196	C₂×D₄	D₄×C₁₂₇	1	4	yes	complemented
197	C₂×D₄	C₂³×C₁₂₇	1	2	yes	—
198	C₂×D₄	C₂³×C₁₂₇	1	2	yes	—
199	C₂²×C₄	—	2	8	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	D₄×C₁₂₇	1	1	yes	characteristic
206	C₂²×C₄	D₄×C₁₂₇	1	2	yes	—
207	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
208	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
209	SD₁₆	—	2	8	no	—
210	D₈	—	2	8	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	C₂³×C₁₂₇	1	2	yes	—
214	C₂×C₈	C₂³×C₁₂₇	1	2	yes	—
215	C₄⋊C₄	—	2	1	no	—
216	C₄⋊C₄	C₂³×C₁₂₇	1	2	yes	—
217	C₂²⋊C₄	—	2	4	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	4	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	4	2	no	—
225	C₂³	—	4	2	no	—
226	C₂³	—	4	2	no	—
227	C₂³	—	4	2	no	—
228	C₂³	—	4	2	no	—
229	C₂³	C₂×D₄×C₁₂₇	1	2	yes	—
230	C₂³	C₂×D₄×C₁₂₇	1	1	yes	characteristic
231	Q₈	—	2	4	no	—
232	Q₈	—	2	2	no	—
233	Q₈	C₂×D₄×C₁₂₇	1	4	yes	complemented
234	D₄	—	2	16	no	—
235	D₄	—	2	4	no	—
236	D₄	—	2	2	no	—
237	D₄	—	2	2	no	—
238	D₄	—	2	2	no	—
239	D₄	—	4	8	no	—
240	D₄	—	4	2	no	—
241	D₄	C₂×D₄×C₁₂₇	1	4	yes	complemented
242	C₂×C₄	—	2	16	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
252	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
253	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
254	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
255	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
256	C₈	—	2	2	no	—
257	C₈	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	4	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	4	no	—
265	C₂²	—	4	8	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	8	4	no	—
270	C₂²	#120×C₁₂₇	1	2	yes	—
271	C₂²	#135×C₁₂₇	1	2	yes	—
272	C₂²	AGL(1,ℤ/8)×C₁₂₇	1	2	yes	—
273	C₂²	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	4	no	—
276	C₄	—	4	2	no	—
277	C₄	#120×C₁₂₇	1	2	yes	—
278	C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
279	C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
280	C₂	—	4	2	no	—
281	C₂	—	4	4	no	—
282	C₂	—	8	2	no	—
283	C₂	#449×C₁₂₇	1	4	yes	direct product factor
284	C₂	C₂×#120×C₁₂₇	1	1	yes	characteristic
285	C₂	C₂×#135×C₁₂₇	1	1	yes	characteristic
286	C₂	C₂×AGL(1,ℤ/8)×C₁₂₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂²×#41×C₁₂₇	C₂	semidirect
2. C₂²×SD₁₆×C₁₂₇	C₂	semidirect
3. C₂²×D₈×C₁₂₇	C₂	semidirect
4. C₂×#121×C₁₂₇	C₂	semidirect
5. #449×C₁₂₇	C₂	direct (N × H)
6. C₂×#103×C₁₂₇	C₂	semidirect
7. C₂×#102×C₁₂₇	C₂	semidirect
8. C₂×C₂²⋊C₈×C₁₂₇	C₂	semidirect
12. C₂×#41×C₁₂₇	C₂²	semidirect
21. C₂×SD₁₆×C₁₂₇	C₂²	semidirect
23. C₂×D₈×C₁₂₇	C₂²	semidirect
27. #121×C₁₂₇	C₂²	semidirect
30. #103×C₁₂₇	C₂²	semidirect
31. #102×C₁₂₇	C₂²	semidirect
32. C₂²⋊C₈×C₁₂₇	C₂²	semidirect
41. C₂×Q₈×C₁₂₇	D₄	semidirect
52. C₂×D₄×C₁₂₇	D₄	semidirect
89. Q₈×C₁₂₇	C₂×D₄	semidirect
97. D₄×C₁₂₇	C₂×D₄	semidirect
139. C₂×C₁₂₇	#449	direct (N × H)
143. C₂×#449	C₁₂₇	direct (N × H)
144. C₁₂₇	C₂×#449	direct (N × H)
145. C₂²×#41	C₂×C₁₂₇	semidirect
146. C₂²×SD₁₆	C₂×C₁₂₇	semidirect
147. C₂²×D₈	C₂×C₁₂₇	semidirect
148. C₂×#121	C₂×C₁₂₇	semidirect
149. #449	C₂×C₁₂₇	direct (N × H)
150. C₂×#103	C₂×C₁₂₇	semidirect
151. C₂×#102	C₂×C₁₂₇	semidirect
152. C₂×C₂²⋊C₈	C₂×C₁₂₇	semidirect
156. C₂×#41	C₂²×C₁₂₇	semidirect
165. C₂×SD₁₆	C₂²×C₁₂₇	semidirect
167. C₂×D₈	C₂²×C₁₂₇	semidirect
171. #121	C₂²×C₁₂₇	semidirect
174. #103	C₂²×C₁₂₇	semidirect
175. #102	C₂²×C₁₂₇	semidirect
176. C₂²⋊C₈	C₂²×C₁₂₇	semidirect
185. C₂×Q₈	D₄×C₁₂₇	semidirect
196. C₂×D₄	D₄×C₁₂₇	semidirect
233. Q₈	C₂×D₄×C₁₂₇	semidirect
241. D₄	C₂×D₄×C₁₂₇	semidirect
283. C₂	#449×C₁₂₇	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×#41×C₁₂₇	2
C₂²×SD₁₆×C₁₂₇	2
C₂²×D₈×C₁₂₇	2
C₂×#121×C₁₂₇	2
#449×C₁₂₇	2
C₂×#103×C₁₂₇	2
C₂×#102×C₁₂₇	2
C₂×C₂²⋊C₈×C₁₂₇	2
C₂×#449	127

Subgroups — C2×#449×C127 #30635952

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#449×C₁₂₇ #30635952