Subgroups — C₂×C₁₂₇×#1695 #44582809

Order 32512.

694 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#476×C₁₂₇	C₂	1	2	yes	maximal
3	C₂×#436×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1695	C₂	1	8	yes	maximal, direct product factor
5	C₂×#334×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#333×C₁₂₇	C₂	1	2	yes	maximal, complemented
7	C₂²×M₁₆×C₁₂₇	—	2	1	no	—
8	C₂×C₄⋊Q₈×C₁₂₇	C₂²	1	2	yes	—
9	C₂×C₄×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
10	#500×C₁₂₇	—	2	2	no	—
11	#500×C₁₂₇	C₂²	1	4	yes	complemented
12	#476×C₁₂₇	—	2	4	no	—
13	#476×C₁₂₇	C₂²	1	8	yes	—
14	#436×C₁₂₇	—	2	2	no	—
15	#436×C₁₂₇	C₂²	1	4	yes	—
16	C₂×#107×C₁₂₇	—	2	1	no	—
17	C₂×#107×C₁₂₇	—	2	1	no	—
18	C₂×#106×C₁₂₇	—	2	2	no	—
19	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₂×#102×C₁₂₇	—	2	2	no	—
22	C₂×C₂²⋊C₈×C₁₂₇	—	2	1	no	—
23	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
24	#334×C₁₂₇	C₂²	1	4	yes	complemented
25	#333×C₁₂₇	C₂²	1	8	yes	complemented
26	C₂²×Q₈×C₁₂₇	—	2	2	no	—
27	C₂²×D₄×C₁₂₇	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₂₇	—	2	1	no	—
29	C₂×M₁₆×C₁₂₇	—	2	4	no	—
30	C₂×M₁₆×C₁₂₇	—	4	4	no	—
31	C₂²×C₈×C₁₂₇	—	2	1	no	—
32	C₂²×C₈×C₁₂₇	—	2	2	no	—
33	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
35	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
36	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
37	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
38	C₄⋊Q₈×C₁₂₇	C₂³	1	4	yes	—
39	C₄×D₄×C₁₂₇	—	2	1	no	—
40	C₄×D₄×C₁₂₇	—	2	1	no	—
41	C₄×D₄×C₁₂₇	D₄	1	2	yes	—
42	C₄×D₄×C₁₂₇	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₁₂₇	D₄	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₁₂₇	—	2	1	no	—
47	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
48	#107×C₁₂₇	—	2	4	no	—
49	#107×C₁₂₇	—	2	4	no	—
50	#106×C₁₂₇	—	2	8	no	—
51	C₄⋊C₈×C₁₂₇	—	2	1	no	—
52	C₄⋊C₈×C₁₂₇	—	2	2	no	—
53	C₄⋊C₈×C₁₂₇	C₂³	1	2	yes	—
54	C₄⋊C₈×C₁₂₇	C₂³	1	4	yes	—
55	#102×C₁₂₇	—	2	8	no	—
56	C₂²⋊C₈×C₁₂₇	—	2	4	no	—
57	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
58	C₄×C₈×C₁₂₇	C₂³	1	2	yes	—
59	C₂⁴×C₁₂₇	—	2	1	no	—
60	C₂×Q₈×C₁₂₇	—	2	4	no	—
61	C₂×Q₈×C₁₂₇	—	4	2	no	—
62	C₂×Q₈×C₁₂₇	—	8	4	no	—
63	C₂×D₄×C₁₂₇	—	2	1	no	—
64	C₂×D₄×C₁₂₇	—	2	4	no	—
65	C₂×D₄×C₁₂₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₂₇	—	2	1	no	—
67	C₂²×C₄×C₁₂₇	—	2	2	no	—
68	C₂²×C₄×C₁₂₇	—	2	2	no	—
69	C₂²×C₄×C₁₂₇	—	4	2	no	—
70	C₂²×C₄×C₁₂₇	—	4	2	no	—
71	C₂²×C₄×C₁₂₇	—	4	2	no	—
72	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₂₇	#41	1	1	yes	characteristic
75	M₁₆×C₁₂₇	—	4	8	no	—
76	C₂×C₈×C₁₂₇	—	2	2	no	—
77	C₂×C₈×C₁₂₇	—	2	2	no	—
78	C₂×C₈×C₁₂₇	—	2	4	no	—
79	C₂×C₈×C₁₂₇	—	4	2	no	—
80	C₂×C₈×C₁₂₇	—	4	4	no	—
81	C₂×C₈×C₁₂₇	SD₁₆	1	4	yes	—
82	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
84	C₄⋊C₄×C₁₂₇	—	2	1	no	—
85	C₄⋊C₄×C₁₂₇	—	2	4	no	—
86	C₄⋊C₄×C₁₂₇	—	2	4	no	—
87	C₄⋊C₄×C₁₂₇	—	4	2	no	—
88	C₄⋊C₄×C₁₂₇	—	4	2	no	—
89	C₄⋊C₄×C₁₂₇	C₂×D₄	1	2	yes	—
90	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
91	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
92	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₁₂₇	#41	1	1	yes	characteristic
95	C₄²×C₁₂₇	C₂⁴	1	1	yes	characteristic
96	C₂³×C₁₂₇	—	2	2	no	—
97	C₂³×C₁₂₇	—	4	2	no	—
98	C₂³×C₁₂₇	—	4	2	no	—
99	C₂³×C₁₂₇	—	4	2	no	—
100	C₂³×C₁₂₇	#121	1	1	yes	characteristic
101	Q₈×C₁₂₇	—	8	4	no	—
102	Q₈×C₁₂₇	—	8	4	no	—
103	D₄×C₁₂₇	—	2	4	no	—
104	D₄×C₁₂₇	—	4	2	no	—
105	C₂×C₄×C₁₂₇	—	2	2	no	—
106	C₂×C₄×C₁₂₇	—	2	2	no	—
107	C₂×C₄×C₁₂₇	—	4	2	no	—
108	C₂×C₄×C₁₂₇	—	4	2	no	—
109	C₂×C₄×C₁₂₇	—	4	2	no	—
110	C₂×C₄×C₁₂₇	—	4	4	no	—
111	C₂×C₄×C₁₂₇	—	4	4	no	—
112	C₂×C₄×C₁₂₇	—	8	4	no	—
113	C₂×C₄×C₁₂₇	#121	1	1	yes	characteristic
114	C₂×C₄×C₁₂₇	#121	1	1	yes	characteristic
115	C₂×C₄×C₁₂₇	#121	1	1	yes	characteristic
116	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	2	yes	—
117	C₂×C₄×C₁₂₇	#137	1	2	yes	—
118	C₂×C₄×C₁₂₇	#137	1	4	yes	—
119	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
122	C₈×C₁₂₇	—	2	2	no	—
123	C₈×C₁₂₇	—	4	2	no	—
124	C₈×C₁₂₇	—	4	4	no	—
125	C₈×C₁₂₇	C₂×SD₁₆	1	4	yes	—
126	C₂²×C₁₂₇	—	4	2	no	—
127	C₂²×C₁₂₇	—	4	2	no	—
128	C₂²×C₁₂₇	—	4	2	no	—
129	C₂²×C₁₂₇	—	8	4	no	—
130	C₂²×C₁₂₇	#461	1	2	yes	—
131	C₂²×C₁₂₇	#470	1	2	yes	—
132	C₂²×C₁₂₇	#472	1	2	yes	—
133	C₂²×C₁₂₇	C₂×#121	1	1	yes	characteristic
134	C₄×C₁₂₇	—	2	1	no	—
135	C₄×C₁₂₇	—	2	1	no	—
136	C₄×C₁₂₇	—	4	2	no	—
137	C₄×C₁₂₇	—	8	4	no	—
138	C₄×C₁₂₇	#461	1	2	yes	—
139	C₄×C₁₂₇	#470	1	2	yes	—
140	C₄×C₁₂₇	C₂²×SD₁₆	1	1	yes	characteristic
141	C₄×C₁₂₇	C₂×#137	1	1	yes	characteristic
142	C₄×C₁₂₇	C₂×#137	1	2	yes	—
143	C₂×#1695	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂×C₁₂₇	—	8	2	no	—
145	C₂×C₁₂₇	#1695	1	4	yes	direct product factor
146	C₂×C₁₂₇	C₂×#461	1	1	yes	characteristic
147	C₂×C₁₂₇	C₂×#470	1	1	yes	characteristic
148	C₂×C₁₂₇	C₂×#472	1	1	yes	characteristic
149	C₂×#500	C₂×C₁₂₇	1	1	yes	characteristic, complemented
150	C₂×#476	C₂×C₁₂₇	1	2	yes	—
151	C₂×#436	C₂×C₁₂₇	1	1	yes	characteristic
152	#1695	C₂×C₁₂₇	1	8	yes	direct product factor
153	C₂×#334	C₂×C₁₂₇	1	1	yes	characteristic, complemented
154	C₂×#333	C₂×C₁₂₇	1	2	yes	complemented
155	C₁₂₇	C₂×#1695	1	1	yes	characteristic, direct product factor, Sylow
156	C₂²×M₁₆	—	2	1	no	—
157	C₂×C₄⋊Q₈	C₂²×C₁₂₇	1	2	yes	—
158	C₂×C₄×D₄	C₂²×C₁₂₇	1	1	yes	characteristic
159	#500	—	2	2	no	—
160	#500	C₂²×C₁₂₇	1	4	yes	complemented
161	#476	—	2	4	no	—
162	#476	C₂²×C₁₂₇	1	8	yes	—
163	#436	—	2	2	no	—
164	#436	C₂²×C₁₂₇	1	4	yes	—
165	C₂×#107	—	2	1	no	—
166	C₂×#107	—	2	1	no	—
167	C₂×#106	—	2	2	no	—
168	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	1	yes	characteristic
169	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
170	C₂×#102	—	2	2	no	—
171	C₂×C₂²⋊C₈	—	2	1	no	—
172	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
173	#334	C₂²×C₁₂₇	1	4	yes	complemented
174	#333	C₂²×C₁₂₇	1	8	yes	complemented
175	C₂²×Q₈	—	2	2	no	—
176	C₂²×D₄	D₄×C₁₂₇	1	1	yes	characteristic
177	C₂³×C₄	—	2	1	no	—
178	C₂×M₁₆	—	2	4	no	—
179	C₂×M₁₆	—	4	4	no	—
180	C₂²×C₈	—	2	1	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
183	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
184	C₄⋊Q₈	—	2	2	no	—
185	C₄⋊Q₈	—	2	2	no	—
186	C₄⋊Q₈	—	2	2	no	—
187	C₄⋊Q₈	C₂³×C₁₂₇	1	4	yes	—
188	C₄×D₄	—	2	1	no	—
189	C₄×D₄	—	2	1	no	—
190	C₄×D₄	D₄×C₁₂₇	1	2	yes	—
191	C₄×D₄	C₂³×C₁₂₇	1	2	yes	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	D₄×C₁₂₇	1	1	yes	characteristic
195	C₂×C₂²⋊C₄	—	2	1	no	—
196	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
197	#107	—	2	4	no	—
198	#107	—	2	4	no	—
199	#106	—	2	8	no	—
200	C₄⋊C₈	—	2	1	no	—
201	C₄⋊C₈	—	2	2	no	—
202	C₄⋊C₈	C₂³×C₁₂₇	1	2	yes	—
203	C₄⋊C₈	C₂³×C₁₂₇	1	4	yes	—
204	#102	—	2	8	no	—
205	C₂²⋊C₈	—	2	4	no	—
206	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
207	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
208	C₂⁴	—	2	1	no	—
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	4	2	no	—
211	C₂×Q₈	—	8	4	no	—
212	C₂×D₄	—	2	1	no	—
213	C₂×D₄	—	2	4	no	—
214	C₂×D₄	C₂×D₄×C₁₂₇	1	2	yes	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
223	C₂²×C₄	#41×C₁₂₇	1	1	yes	characteristic
224	M₁₆	—	4	8	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	4	no	—
230	C₂×C₈	SD₁₆×C₁₂₇	1	4	yes	—
231	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
232	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	C₂×D₄×C₁₂₇	1	2	yes	—
239	C₂²⋊C₄	—	2	2	no	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
242	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
243	C₄²	#41×C₁₂₇	1	1	yes	characteristic
244	C₄²	C₂⁴×C₁₂₇	1	1	yes	characteristic
245	C₂³	—	2	2	no	—
246	C₂³	—	4	2	no	—
247	C₂³	—	4	2	no	—
248	C₂³	—	4	2	no	—
249	C₂³	#121×C₁₂₇	1	1	yes	characteristic
250	Q₈	—	8	4	no	—
251	Q₈	—	8	4	no	—
252	D₄	—	2	4	no	—
253	D₄	—	4	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	8	4	no	—
262	C₂×C₄	#121×C₁₂₇	1	1	yes	characteristic
263	C₂×C₄	#121×C₁₂₇	1	1	yes	characteristic
264	C₂×C₄	#121×C₁₂₇	1	1	yes	characteristic
265	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	2	yes	—
266	C₂×C₄	#137×C₁₂₇	1	2	yes	—
267	C₂×C₄	#137×C₁₂₇	1	4	yes	—
268	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
269	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
270	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	—	4	2	no	—
273	C₈	—	4	4	no	—
274	C₈	C₂×SD₁₆×C₁₂₇	1	4	yes	—
275	C₂²	—	4	2	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	8	4	no	—
279	C₂²	#461×C₁₂₇	1	2	yes	—
280	C₂²	#470×C₁₂₇	1	2	yes	—
281	C₂²	#472×C₁₂₇	1	2	yes	—
282	C₂²	C₂×#121×C₁₂₇	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	4	2	no	—
286	C₄	—	8	4	no	—
287	C₄	#461×C₁₂₇	1	2	yes	—
288	C₄	#470×C₁₂₇	1	2	yes	—
289	C₄	C₂²×SD₁₆×C₁₂₇	1	1	yes	characteristic
290	C₄	C₂×#137×C₁₂₇	1	1	yes	characteristic
291	C₄	C₂×#137×C₁₂₇	1	2	yes	—
292	C₂	—	8	2	no	—
293	C₂	C₁₂₇×#1695	1	4	yes	direct product factor
294	C₂	C₂×#461×C₁₂₇	1	1	yes	characteristic
295	C₂	C₂×#470×C₁₂₇	1	1	yes	characteristic
296	C₂	C₂×#472×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₂₇	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₂₇
prime radical	C₂²×D₄×C₁₂₇	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#500×C₁₂₇	C₂	semidirect
4. C₁₂₇×#1695	C₂	direct (N × H)
5. C₂×#334×C₁₂₇	C₂	semidirect
6. C₂×#333×C₁₂₇	C₂	semidirect
11. #500×C₁₂₇	C₂²	semidirect
24. #334×C₁₂₇	C₂²	semidirect
25. #333×C₁₂₇	C₂²	semidirect
143. C₂×#1695	C₁₂₇	direct (N × H)
145. C₂×C₁₂₇	#1695	direct (N × H)
149. C₂×#500	C₂×C₁₂₇	semidirect
152. #1695	C₂×C₁₂₇	direct (N × H)
153. C₂×#334	C₂×C₁₂₇	semidirect
154. C₂×#333	C₂×C₁₂₇	semidirect
155. C₁₂₇	C₂×#1695	direct (N × H)
160. #500	C₂²×C₁₂₇	semidirect
173. #334	C₂²×C₁₂₇	semidirect
174. #333	C₂²×C₁₂₇	semidirect
293. C₂	C₁₂₇×#1695	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#500×C₁₂₇	2
C₂×#476×C₁₂₇	2
C₂×#436×C₁₂₇	2
C₁₂₇×#1695	2
C₂×#334×C₁₂₇	2
C₂×#333×C₁₂₇	2
C₂×#1695	127

Subgroups — C2×C127×#1695 #44582809

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₂₇×#1695 #44582809