Subgroups — C₁₂₇×#2452 #30635404

Order 16256.

620 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊Q₈×C₁₂₇	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×D₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#396×C₁₂₇	C₂	1	2	yes	maximal, complemented
5	#393×C₁₂₇	C₂	1	2	yes	maximal, complemented
6	#390×C₁₂₇	C₂	1	1	yes	maximal, characteristic
7	#390×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	#389×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#389×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#385×C₁₂₇	C₂	1	2	yes	maximal, complemented
11	C₄×C₄⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₁₂₇	C₂²	1	2	yes	—
13	C₂²×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₁₂₇	C₂²	1	2	yes	—
15	C₄⋊Q₈×C₁₂₇	—	2	4	no	—
16	C₂²⋊Q₈×C₁₂₇	—	2	8	no	—
17	C₄×D₄×C₁₂₇	—	2	4	no	—
18	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
30	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
31	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
34	#95×C₁₂₇	C₂²	1	1	yes	characteristic
35	#95×C₁₂₇	C₂²	1	1	yes	characteristic
36	#95×C₁₂₇	C₂²	1	2	yes	—
37	#95×C₁₂₇	C₂²	1	2	yes	—
38	C₂⁴×C₁₂₇	C₂³	1	2	yes	—
39	C₂×Q₈×C₁₂₇	—	2	4	no	—
40	C₂×Q₈×C₁₂₇	—	2	4	no	—
41	C₂×Q₈×C₁₂₇	—	4	2	no	—
42	C₂×D₄×C₁₂₇	—	2	2	no	—
43	C₂×D₄×C₁₂₇	—	2	2	no	—
44	C₂×D₄×C₁₂₇	Q₈	1	4	yes	complemented
45	C₂²×C₄×C₁₂₇	—	2	4	no	—
46	C₂²×C₄×C₁₂₇	—	2	4	no	—
47	C₂²×C₄×C₁₂₇	—	4	2	no	—
48	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
51	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
52	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
55	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
58	C₄⋊C₄×C₁₂₇	—	2	4	no	—
59	C₄⋊C₄×C₁₂₇	—	2	4	no	—
60	C₄⋊C₄×C₁₂₇	—	2	4	no	—
61	C₄⋊C₄×C₁₂₇	—	2	2	no	—
62	C₄⋊C₄×C₁₂₇	—	2	2	no	—
63	C₄⋊C₄×C₁₂₇	—	2	2	no	—
64	C₄⋊C₄×C₁₂₇	D₄	1	4	yes	—
65	C₄⋊C₄×C₁₂₇	D₄	1	4	yes	complemented
66	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
67	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
68	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
69	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
70	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
71	C₄²×C₁₂₇	—	2	2	no	—
72	C₄²×C₁₂₇	—	2	2	no	—
73	C₄²×C₁₂₇	—	2	2	no	—
74	C₂³×C₁₂₇	—	2	2	no	—
75	C₂³×C₁₂₇	—	2	2	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	C₂×Q₈	1	4	yes	—
81	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
82	Q₈×C₁₂₇	—	4	8	no	—
83	D₄×C₁₂₇	—	2	8	no	—
84	C₂×C₄×C₁₂₇	—	2	1	no	—
85	C₂×C₄×C₁₂₇	—	2	1	no	—
86	C₂×C₄×C₁₂₇	—	2	1	no	—
87	C₂×C₄×C₁₂₇	—	2	2	no	—
88	C₂×C₄×C₁₂₇	—	2	1	no	—
89	C₂×C₄×C₁₂₇	—	2	1	no	—
90	C₂×C₄×C₁₂₇	—	2	1	no	—
91	C₂×C₄×C₁₂₇	—	2	2	no	—
92	C₂×C₄×C₁₂₇	—	2	1	no	—
93	C₂×C₄×C₁₂₇	—	2	1	no	—
94	C₂×C₄×C₁₂₇	—	2	1	no	—
95	C₂×C₄×C₁₂₇	—	2	2	no	—
96	C₂×C₄×C₁₂₇	—	2	2	no	—
97	C₂×C₄×C₁₂₇	—	2	2	no	—
98	C₂×C₄×C₁₂₇	—	2	1	no	—
99	C₂×C₄×C₁₂₇	—	2	2	no	—
100	C₂×C₄×C₁₂₇	—	2	2	no	—
101	C₂×C₄×C₁₂₇	—	2	1	no	—
102	C₂×C₄×C₁₂₇	—	2	1	no	—
103	C₂×C₄×C₁₂₇	—	2	2	no	—
104	C₂×C₄×C₁₂₇	—	2	1	no	—
105	C₂×C₄×C₁₂₇	—	2	1	no	—
106	C₂×C₄×C₁₂₇	—	4	8	no	—
107	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
108	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
110	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
111	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
112	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
113	C₂×C₄×C₁₂₇	#41	1	2	yes	—
114	C₂×C₄×C₁₂₇	#41	1	2	yes	—
115	C₂²×C₁₂₇	—	2	4	no	—
116	C₂²×C₁₂₇	—	2	4	no	—
117	C₂²×C₁₂₇	—	2	4	no	—
118	C₂²×C₁₂₇	—	4	2	no	—
119	C₂²×C₁₂₇	—	4	2	no	—
120	C₂²×C₁₂₇	—	4	2	no	—
121	C₂²×C₁₂₇	—	4	2	no	—
122	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
123	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
124	C₂²×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
125	C₂²×C₁₂₇	C₂×#41	1	1	yes	characteristic
126	C₂²×C₁₂₇	C₂×#41	1	1	yes	characteristic
127	C₂²×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
128	C₂²×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
129	C₄×C₁₂₇	—	2	2	no	—
130	C₄×C₁₂₇	—	2	2	no	—
131	C₄×C₁₂₇	—	2	4	no	—
132	C₄×C₁₂₇	—	2	2	no	—
133	C₄×C₁₂₇	—	2	2	no	—
134	C₄×C₁₂₇	—	4	2	no	—
135	C₄×C₁₂₇	—	4	1	no	—
136	C₄×C₁₂₇	—	4	2	no	—
137	C₄×C₁₂₇	—	4	1	no	—
138	C₄×C₁₂₇	C₂²⋊Q₈	1	4	yes	complemented
139	C₂×C₁₂₇	—	4	4	no	—
140	C₂×C₁₂₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
141	C₂×C₁₂₇	#532	1	1	yes	characteristic
142	C₂×C₁₂₇	#542	1	1	yes	characteristic
143	C₂×C₁₂₇	D₄²	1	1	yes	characteristic
144	C₂×C₁₂₇	#548	1	1	yes	characteristic
145	C₂×C₁₂₇	D₄×Q₈	1	1	yes	characteristic
146	C₂×C₁₂₇	#554	1	1	yes	characteristic
147	#2452	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₁₂₇	#2452	1	1	yes	characteristic, direct product factor, Sylow
149	C₂×C₄⋊Q₈	C₂×C₁₂₇	1	1	yes	characteristic, complemented
150	C₂×C₂²⋊Q₈	C₂×C₁₂₇	1	2	yes	complemented
151	C₂×C₄×D₄	C₂×C₁₂₇	1	1	yes	characteristic
152	#396	C₂×C₁₂₇	1	2	yes	complemented
153	#393	C₂×C₁₂₇	1	2	yes	complemented
154	#390	C₂×C₁₂₇	1	1	yes	characteristic
155	#390	C₂×C₁₂₇	1	1	yes	characteristic
156	#389	C₂×C₁₂₇	1	1	yes	characteristic, complemented
157	#389	C₂×C₁₂₇	1	1	yes	characteristic, complemented
158	#385	C₂×C₁₂₇	1	2	yes	complemented
159	C₄×C₄⋊C₄	C₂×C₁₂₇	1	1	yes	characteristic, complemented
160	C₂²×Q₈	C₂²×C₁₂₇	1	2	yes	—
161	C₂²×D₄	C₂²×C₁₂₇	1	1	yes	characteristic
162	C₂³×C₄	C₂²×C₁₂₇	1	2	yes	—
163	C₄⋊Q₈	—	2	4	no	—
164	C₂²⋊Q₈	—	2	8	no	—
165	C₄×D₄	—	2	4	no	—
166	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
167	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
168	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
169	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
170	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
171	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
172	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
173	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
174	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	2	yes	—
175	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	2	yes	—
176	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	2	yes	—
177	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	2	yes	—
178	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	2	yes	—
179	C₂×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
180	C₂×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
181	C₂×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
182	#95	C₂²×C₁₂₇	1	1	yes	characteristic
183	#95	C₂²×C₁₂₇	1	1	yes	characteristic
184	#95	C₂²×C₁₂₇	1	2	yes	—
185	#95	C₂²×C₁₂₇	1	2	yes	—
186	C₂⁴	C₂³×C₁₂₇	1	2	yes	—
187	C₂×Q₈	—	2	4	no	—
188	C₂×Q₈	—	2	4	no	—
189	C₂×Q₈	—	4	2	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	Q₈×C₁₂₇	1	4	yes	complemented
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
197	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
198	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
199	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
200	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
201	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
202	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
203	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
204	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
205	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	D₄×C₁₂₇	1	4	yes	—
213	C₄⋊C₄	D₄×C₁₂₇	1	4	yes	complemented
214	C₂²⋊C₄	—	2	4	no	—
215	C₂²⋊C₄	—	2	4	no	—
216	C₂²⋊C₄	—	2	4	no	—
217	C₂²⋊C₄	—	2	4	no	—
218	C₂²⋊C₄	—	4	2	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	2	2	no	—
228	C₂³	C₂×Q₈×C₁₂₇	1	4	yes	—
229	C₂³	C₂⁴×C₁₂₇	1	1	yes	characteristic
230	Q₈	—	4	8	no	—
231	D₄	—	2	8	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	4	8	no	—
255	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
256	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
257	C₂×C₄	C₂×D₄×C₁₂₇	1	4	yes	—
258	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
259	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
260	C₂×C₄	C₂×Q₈×C₁₂₇	1	2	yes	—
261	C₂×C₄	#41×C₁₂₇	1	2	yes	—
262	C₂×C₄	#41×C₁₂₇	1	2	yes	—
263	C₂²	—	2	4	no	—
264	C₂²	—	2	4	no	—
265	C₂²	—	2	4	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
271	C₂²	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
272	C₂²	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
273	C₂²	C₂×#41×C₁₂₇	1	1	yes	characteristic
274	C₂²	C₂×#41×C₁₂₇	1	1	yes	characteristic
275	C₂²	H(5,𝔽₂)×C₁₂₇	1	1	yes	characteristic
276	C₂²	2₋¹⁺⁴×C₁₂₇	1	1	yes	characteristic
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	4	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	4	2	no	—
283	C₄	—	4	1	no	—
284	C₄	—	4	2	no	—
285	C₄	—	4	1	no	—
286	C₄	C₂²⋊Q₈×C₁₂₇	1	4	yes	complemented
287	C₂	—	4	4	no	—
288	C₂	C₂×C₂²⋊Q₈×C₁₂₇	1	1	yes	characteristic
289	C₂	#532×C₁₂₇	1	1	yes	characteristic
290	C₂	#542×C₁₂₇	1	1	yes	characteristic
291	C₂	D₄²×C₁₂₇	1	1	yes	characteristic
292	C₂	#548×C₁₂₇	1	1	yes	characteristic
293	C₂	D₄×Q₈×C₁₂₇	1	1	yes	characteristic
294	C₂	#554×C₁₂₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊Q₈×C₁₂₇	C₂	semidirect
2. C₂×C₂²⋊Q₈×C₁₂₇	C₂	semidirect
4. #396×C₁₂₇	C₂	semidirect
5. #393×C₁₂₇	C₂	semidirect
8. #389×C₁₂₇	C₂	semidirect
9. #389×C₁₂₇	C₂	semidirect
10. #385×C₁₂₇	C₂	semidirect
11. C₄×C₄⋊C₄×C₁₂₇	C₂	semidirect
44. C₂×D₄×C₁₂₇	Q₈	semidirect
65. C₄⋊C₄×C₁₂₇	D₄	semidirect
138. C₄×C₁₂₇	C₂²⋊Q₈	semidirect
147. #2452	C₁₂₇	direct (N × H)
148. C₁₂₇	#2452	direct (N × H)
149. C₂×C₄⋊Q₈	C₂×C₁₂₇	semidirect
150. C₂×C₂²⋊Q₈	C₂×C₁₂₇	semidirect
152. #396	C₂×C₁₂₇	semidirect
153. #393	C₂×C₁₂₇	semidirect
156. #389	C₂×C₁₂₇	semidirect
157. #389	C₂×C₁₂₇	semidirect
158. #385	C₂×C₁₂₇	semidirect
159. C₄×C₄⋊C₄	C₂×C₁₂₇	semidirect
192. C₂×D₄	Q₈×C₁₂₇	semidirect
213. C₄⋊C₄	D₄×C₁₂₇	semidirect
286. C₄	C₂²⋊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×C₄⋊Q₈×C₁₂₇	2
C₂×C₂²⋊Q₈×C₁₂₇	2
C₂×C₄×D₄×C₁₂₇	2
#396×C₁₂₇	2
#393×C₁₂₇	2
#390×C₁₂₇	2
#389×C₁₂₇	2
#385×C₁₂₇	2
C₄×C₄⋊C₄×C₁₂₇	2
#2452	127

Subgroups — C127×#2452 #30635404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#2452 #30635404