Subgroups — C₁₂₇×#11597 #44575270

Order 32512.

394 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	C₄×#135×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#346×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₂₇×#1564	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₂₇×#1334	C₂	1	2	yes	maximal, complemented
5	C₁₂₇×#1333	C₂	1	2	yes	maximal, complemented
6	C₂×#135×C₁₂₇	C₄	1	1	yes	characteristic
7	C₄×#41×C₁₂₇	—	2	1	no	—
8	C₂²×C₁₆×C₁₂₇	—	2	1	no	—
9	C₄×Q₁₆×C₁₂₇	C₄	1	1	yes	characteristic
10	C₄×Q₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₄×SD₁₆×C₁₂₇	—	2	1	no	—
12	C₄×SD₁₆×C₁₂₇	—	2	1	no	—
13	C₄×D₈×C₁₂₇	C₄	1	1	yes	characteristic
14	C₄×D₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	#431×C₁₂₇	—	2	1	no	—
16	#427×C₁₂₇	C₄	1	1	yes	characteristic
17	#416×C₁₂₇	—	2	1	no	—
18	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
19	#346×C₁₂₇	—	2	1	no	—
20	#346×C₁₂₇	C₂²	1	2	yes	—
21	#335×C₁₂₇	—	2	1	no	—
22	#334×C₁₂₇	C₂²	1	2	yes	—
23	C₂×#41×C₁₂₇	—	2	1	no	—
24	#135×C₁₂₇	—	2	2	no	—
25	#135×C₁₂₇	C₈	1	4	yes	complemented
26	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
27	C₂×SD₁₆×C₁₂₇	—	2	1	no	—
28	C₂×D₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
31	C₄×Q₈×C₁₂₇	—	2	1	no	—
32	C₄×Q₈×C₁₂₇	—	2	1	no	—
33	C₄×D₄×C₁₂₇	—	2	1	no	—
34	C₄×D₄×C₁₂₇	—	2	1	no	—
35	C₄×D₄×C₁₂₇	—	4	1	no	—
36	C₄×D₄×C₁₂₇	—	4	1	no	—
37	#117×C₁₂₇	—	2	1	no	—
38	#117×C₁₂₇	—	4	1	no	—
39	C₂×C₄²×C₁₂₇	—	4	1	no	—
40	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
41	C₂×C₁₆×C₁₂₇	—	2	4	no	—
42	C₂×C₁₆×C₁₂₇	—	2	2	no	—
43	#107×C₁₂₇	C₂×C₄	1	1	yes	characteristic
44	#107×C₁₂₇	C₂×C₄	1	1	yes	characteristic
45	#106×C₁₂₇	—	2	1	no	—
46	C₄⋊C₈×C₁₂₇	—	2	2	no	—
47	#103×C₁₂₇	—	2	1	no	—
48	#103×C₁₂₇	—	2	1	no	—
49	#102×C₁₂₇	—	2	1	no	—
50	#102×C₁₂₇	—	2	1	no	—
51	C₂²⋊C₈×C₁₂₇	—	4	1	no	—
52	C₄×C₈×C₁₂₇	—	4	1	no	—
53	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
55	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
56	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
57	#41×C₁₂₇	—	2	4	no	—
58	#41×C₁₂₇	—	4	2	no	—
59	C₂×Q₈×C₁₂₇	—	2	1	no	—
60	C₂×D₄×C₁₂₇	—	2	1	no	—
61	C₂×D₄×C₁₂₇	—	4	1	no	—
62	C₂²×C₄×C₁₂₇	—	4	1	no	—
63	C₂²×C₄×C₁₂₇	—	4	1	no	—
64	C₂²×C₄×C₁₂₇	—	4	1	no	—
65	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
68	Q₁₆×C₁₂₇	—	2	1	no	—
69	Q₁₆×C₁₂₇	C₂×C₈	1	2	yes	—
70	SD₁₆×C₁₂₇	—	2	4	no	—
71	D₈×C₁₂₇	—	2	1	no	—
72	D₈×C₁₂₇	C₂×C₈	1	2	yes	—
73	C₂×C₈×C₁₂₇	—	2	2	no	—
74	C₂×C₈×C₁₂₇	—	4	2	no	—
75	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₂₇	C₂×C₈	1	2	yes	—
78	C₂×C₈×C₁₂₇	M₁₆	1	2	yes	—
79	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
81	C₄⋊C₄×C₁₂₇	—	2	1	no	—
82	C₄⋊C₄×C₁₂₇	—	2	1	no	—
83	C₄⋊C₄×C₁₂₇	—	4	1	no	—
84	C₄⋊C₄×C₁₂₇	—	4	1	no	—
85	C₂²⋊C₄×C₁₂₇	—	4	1	no	—
86	C₂²⋊C₄×C₁₂₇	—	4	1	no	—
87	C₂²⋊C₄×C₁₂₇	—	4	1	no	—
88	C₄²×C₁₂₇	—	4	1	no	—
89	C₄²×C₁₂₇	—	4	1	no	—
90	C₄²×C₁₂₇	—	4	1	no	—
91	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
92	C₄²×C₁₂₇	D₈	1	1	yes	characteristic
93	C₄²×C₁₂₇	SD₁₆	1	1	yes	characteristic
94	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
95	C₁₆×C₁₂₇	—	2	4	no	—
96	C₂³×C₁₂₇	—	4	1	no	—
97	C₂³×C₁₂₇	#102	1	1	yes	characteristic
98	Q₈×C₁₂₇	—	2	2	no	—
99	Q₈×C₁₂₇	—	4	1	no	—
100	D₄×C₁₂₇	—	2	2	no	—
101	D₄×C₁₂₇	—	4	4	no	—
102	D₄×C₁₂₇	—	4	1	no	—
103	C₂×C₄×C₁₂₇	—	4	1	no	—
104	C₂×C₄×C₁₂₇	—	4	1	no	—
105	C₂×C₄×C₁₂₇	—	4	1	no	—
106	C₂×C₄×C₁₂₇	—	4	4	no	—
107	C₂×C₄×C₁₂₇	—	4	1	no	—
108	C₂×C₄×C₁₂₇	—	4	1	no	—
109	C₂×C₄×C₁₂₇	—	8	2	no	—
110	C₂×C₄×C₁₂₇	—	8	2	no	—
111	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	2	yes	—
112	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
113	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
114	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
115	C₂×C₄×C₁₂₇	#104	1	2	yes	—
116	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
119	C₈×C₁₂₇	—	2	1	no	—
120	C₈×C₁₂₇	—	2	1	no	—
121	C₈×C₁₂₇	—	8	2	no	—
122	C₈×C₁₂₇	C₂²⋊C₈	1	2	yes	—
123	C₈×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
124	C₈×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
125	C₂²×C₁₂₇	—	2	1	no	—
126	C₂²×C₁₂₇	—	2	1	no	—
127	C₂²×C₁₂₇	—	4	2	no	—
128	C₂²×C₁₂₇	—	8	1	no	—
129	C₂²×C₁₂₇	—	8	1	no	—
130	C₂²×C₁₂₇	D₄⋊C₈	1	2	yes	—
131	C₂²×C₁₂₇	C₂×#102	1	1	yes	characteristic
132	C₄×C₁₂₇	—	2	1	no	—
133	C₄×C₁₂₇	—	2	1	no	—
134	C₄×C₁₂₇	—	2	1	no	—
135	C₄×C₁₂₇	—	2	1	no	—
136	C₄×C₁₂₇	—	4	2	no	—
137	C₄×C₁₂₇	—	8	1	no	—
138	C₄×C₁₂₇	—	8	1	no	—
139	C₄×C₁₂₇	D₄⋊C₈	1	2	yes	—
140	C₄×C₁₂₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
141	C₄×C₁₂₇	C₂×#104	1	1	yes	characteristic
142	#11597	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂×C₁₂₇	—	2	2	no	—
144	C₂×C₁₂₇	—	8	2	no	—
145	C₂×C₁₂₇	C₂×D₄⋊C₈	1	1	yes	characteristic
146	C₂×C₁₂₇	#2145	1	1	yes	characteristic
147	C₂×C₁₂₇	#2148	1	1	yes	characteristic
148	C₄×#135	C₂×C₁₂₇	1	1	yes	characteristic
149	C₂×#346	C₂×C₁₂₇	1	1	yes	characteristic, complemented
150	#1564	C₂×C₁₂₇	1	1	yes	characteristic, complemented
151	#1334	C₂×C₁₂₇	1	2	yes	complemented
152	#1333	C₂×C₁₂₇	1	2	yes	complemented
153	C₁₂₇	#11597	1	1	yes	characteristic, direct product factor, Sylow
154	C₂×#135	C₄×C₁₂₇	1	1	yes	characteristic
155	C₄×#41	—	2	1	no	—
156	C₂²×C₁₆	—	2	1	no	—
157	C₄×Q₁₆	C₄×C₁₂₇	1	1	yes	characteristic
158	C₄×Q₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
159	C₄×SD₁₆	—	2	1	no	—
160	C₄×SD₁₆	—	2	1	no	—
161	C₄×D₈	C₄×C₁₂₇	1	1	yes	characteristic
162	C₄×D₈	C₂²×C₁₂₇	1	1	yes	characteristic
163	#431	—	2	1	no	—
164	#427	C₄×C₁₂₇	1	1	yes	characteristic
165	#416	—	2	1	no	—
166	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
167	#346	—	2	1	no	—
168	#346	C₂²×C₁₂₇	1	2	yes	—
169	#335	—	2	1	no	—
170	#334	C₂²×C₁₂₇	1	2	yes	—
171	C₂×#41	—	2	1	no	—
172	#135	—	2	2	no	—
173	#135	C₈×C₁₂₇	1	4	yes	complemented
174	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
175	C₂×SD₁₆	—	2	1	no	—
176	C₂×D₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
177	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
178	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
179	C₄×Q₈	—	2	1	no	—
180	C₄×Q₈	—	2	1	no	—
181	C₄×D₄	—	2	1	no	—
182	C₄×D₄	—	2	1	no	—
183	C₄×D₄	—	4	1	no	—
184	C₄×D₄	—	4	1	no	—
185	#117	—	2	1	no	—
186	#117	—	4	1	no	—
187	C₂×C₄²	—	4	1	no	—
188	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
189	C₂×C₁₆	—	2	4	no	—
190	C₂×C₁₆	—	2	2	no	—
191	#107	C₂×C₄×C₁₂₇	1	1	yes	characteristic
192	#107	C₂×C₄×C₁₂₇	1	1	yes	characteristic
193	#106	—	2	1	no	—
194	C₄⋊C₈	—	2	2	no	—
195	#103	—	2	1	no	—
196	#103	—	2	1	no	—
197	#102	—	2	1	no	—
198	#102	—	2	1	no	—
199	C₂²⋊C₈	—	4	1	no	—
200	C₄×C₈	—	4	1	no	—
201	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
202	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
203	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
204	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
205	#41	—	2	4	no	—
206	#41	—	4	2	no	—
207	C₂×Q₈	—	2	1	no	—
208	C₂×D₄	—	2	1	no	—
209	C₂×D₄	—	4	1	no	—
210	C₂²×C₄	—	4	1	no	—
211	C₂²×C₄	—	4	1	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
214	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
215	C₂²×C₄	SD₁₆×C₁₂₇	1	1	yes	characteristic
216	Q₁₆	—	2	1	no	—
217	Q₁₆	C₂×C₈×C₁₂₇	1	2	yes	—
218	SD₁₆	—	2	4	no	—
219	D₈	—	2	1	no	—
220	D₈	C₂×C₈×C₁₂₇	1	2	yes	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
224	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
225	C₂×C₈	C₂×C₈×C₁₂₇	1	2	yes	—
226	C₂×C₈	M₁₆×C₁₂₇	1	2	yes	—
227	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
228	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	4	1	no	—
232	C₄⋊C₄	—	4	1	no	—
233	C₂²⋊C₄	—	4	1	no	—
234	C₂²⋊C₄	—	4	1	no	—
235	C₂²⋊C₄	—	4	1	no	—
236	C₄²	—	4	1	no	—
237	C₄²	—	4	1	no	—
238	C₄²	—	4	1	no	—
239	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
240	C₄²	D₈×C₁₂₇	1	1	yes	characteristic
241	C₄²	SD₁₆×C₁₂₇	1	1	yes	characteristic
242	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
243	C₁₆	—	2	4	no	—
244	C₂³	—	4	1	no	—
245	C₂³	#102×C₁₂₇	1	1	yes	characteristic
246	Q₈	—	2	2	no	—
247	Q₈	—	4	1	no	—
248	D₄	—	2	2	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	8	2	no	—
258	C₂×C₄	—	8	2	no	—
259	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	2	yes	—
260	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
261	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
262	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
263	C₂×C₄	#104×C₁₂₇	1	2	yes	—
264	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
265	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
266	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	8	2	no	—
270	C₈	C₂²⋊C₈×C₁₂₇	1	2	yes	—
271	C₈	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
272	C₈	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	4	2	no	—
276	C₂²	—	8	1	no	—
277	C₂²	—	8	1	no	—
278	C₂²	D₄⋊C₈×C₁₂₇	1	2	yes	—
279	C₂²	C₂×#102×C₁₂₇	1	1	yes	characteristic
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	4	2	no	—
285	C₄	—	8	1	no	—
286	C₄	—	8	1	no	—
287	C₄	D₄⋊C₈×C₁₂₇	1	2	yes	—
288	C₄	C₂×C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
289	C₄	C₂×#104×C₁₂₇	1	1	yes	characteristic
290	C₂	—	2	2	no	—
291	C₂	—	8	2	no	—
292	C₂	C₂×D₄⋊C₈×C₁₂₇	1	1	yes	characteristic
293	C₂	C₁₂₇×#2145	1	1	yes	characteristic
294	C₂	C₁₂₇×#2148	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₂₇	#102
commutator	C₈	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂×#102
prime radical	C₂×#135×C₁₂₇	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#346×C₁₂₇	C₂	semidirect
3. C₁₂₇×#1564	C₂	semidirect
4. C₁₂₇×#1334	C₂	semidirect
5. C₁₂₇×#1333	C₂	semidirect
25. #135×C₁₂₇	C₈	semidirect
142. #11597	C₁₂₇	direct (N × H)
149. C₂×#346	C₂×C₁₂₇	semidirect
150. #1564	C₂×C₁₂₇	semidirect
151. #1334	C₂×C₁₂₇	semidirect
152. #1333	C₂×C₁₂₇	semidirect
153. C₁₂₇	#11597	direct (N × H)
173. #135	C₈×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×#135×C₁₂₇	2
C₂×#346×C₁₂₇	2
C₁₂₇×#1564	2
C₁₂₇×#1334	2
C₁₂₇×#1333	2
#11597	127

Subgroups — C127×#11597 #44575270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#11597 #44575270