Subgroups — C₁₂₇×#11742 #44575415

Order 32512.

394 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#3169	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#2162	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#107×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₂₇×#1347	C₂	1	4	yes	maximal, complemented
5	#532×C₁₂₇	—	2	1	no	—
6	#500×C₁₂₇	C₄	1	1	yes	characteristic, complemented
7	#493×C₁₂₇	C₄	1	1	yes	characteristic, complemented
8	#486×C₁₂₇	—	2	1	no	—
9	#466×C₁₂₇	—	2	1	no	—
10	#466×C₁₂₇	C₂²	1	2	yes	—
11	C₄×D₈×C₁₂₇	—	2	1	no	—
12	C₄×D₈×C₁₂₇	C₄	1	2	yes	complemented
13	C₂×#107×C₁₂₇	—	2	1	no	—
14	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₁₂₇	—	2	1	no	—
16	C₄⋊C₁₆×C₁₂₇	—	2	1	no	—
17	C₂²⋊C₁₆×C₁₂₇	C₂²	1	2	yes	—
18	C₄×C₁₆×C₁₂₇	—	2	1	no	—
19	#340×C₁₂₇	C₂²	1	2	yes	—
20	C₂×#41×C₁₂₇	—	4	1	no	—
21	C₂×D₈×C₁₂₇	—	2	1	no	—
22	C₂×D₈×C₁₂₇	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
25	C₄⋊Q₈×C₁₂₇	—	2	1	no	—
26	#127×C₁₂₇	—	2	1	no	—
27	#124×C₁₂₇	—	4	1	no	—
28	#121×C₁₂₇	—	2	2	no	—
29	#121×C₁₂₇	—	4	1	no	—
30	C₄×D₄×C₁₂₇	—	2	2	no	—
31	C₄×D₄×C₁₂₇	—	4	1	no	—
32	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
33	C₂×C₄²×C₁₂₇	—	4	1	no	—
34	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₁₂₇	—	2	2	no	—
36	#107×C₁₂₇	—	2	4	no	—
37	#107×C₁₂₇	—	2	1	no	—
38	#107×C₁₂₇	C₂×C₄	1	2	yes	—
39	#102×C₁₂₇	—	2	2	no	—
40	#102×C₁₂₇	—	2	2	no	—
41	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
45	#95×C₁₂₇	—	4	1	no	—
46	#41×C₁₂₇	—	8	2	no	—
47	#41×C₁₂₇	—	8	2	no	—
48	C₂×Q₈×C₁₂₇	—	4	1	no	—
49	C₂×D₄×C₁₂₇	—	2	2	no	—
50	C₂×D₄×C₁₂₇	—	4	2	no	—
51	C₂×D₄×C₁₂₇	—	4	1	no	—
52	C₂×D₄×C₁₂₇	—	4	1	no	—
53	C₂²×C₄×C₁₂₇	—	4	2	no	—
54	C₂²×C₄×C₁₂₇	—	4	2	no	—
55	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
58	D₈×C₁₂₇	—	4	2	no	—
59	D₈×C₁₂₇	—	4	2	no	—
60	C₂×C₈×C₁₂₇	—	2	2	no	—
61	C₂×C₈×C₁₂₇	—	4	1	no	—
62	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
66	C₄⋊C₄×C₁₂₇	—	2	4	no	—
67	C₄⋊C₄×C₁₂₇	—	2	2	no	—
68	C₄⋊C₄×C₁₂₇	—	4	2	no	—
69	C₄⋊C₄×C₁₂₇	—	4	1	no	—
70	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
71	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
72	C₄²×C₁₂₇	—	8	2	no	—
73	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₁₂₇	SD₁₆	1	1	yes	characteristic
76	C₄²×C₁₂₇	SD₁₆	1	1	yes	characteristic
77	C₁₆×C₁₂₇	—	4	2	no	—
78	C₂³×C₁₂₇	—	4	2	no	—
79	C₂³×C₁₂₇	C₂×D₈	1	1	yes	characteristic
80	Q₈×C₁₂₇	—	8	2	no	—
81	D₄×C₁₂₇	—	4	2	no	—
82	D₄×C₁₂₇	—	4	2	no	—
83	D₄×C₁₂₇	—	8	2	no	—
84	D₄×C₁₂₇	—	8	2	no	—
85	D₄×C₁₂₇	—	8	2	no	—
86	D₄×C₁₂₇	—	8	2	no	—
87	C₂×C₄×C₁₂₇	—	2	1	no	—
88	C₂×C₄×C₁₂₇	—	2	1	no	—
89	C₂×C₄×C₁₂₇	—	4	1	no	—
90	C₂×C₄×C₁₂₇	—	4	4	no	—
91	C₂×C₄×C₁₂₇	—	4	2	no	—
92	C₂×C₄×C₁₂₇	—	8	2	no	—
93	C₂×C₄×C₁₂₇	—	8	2	no	—
94	C₂×C₄×C₁₂₇	—	8	4	no	—
95	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
96	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
97	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
98	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
99	C₂×C₄×C₁₂₇	D₁₆	1	2	yes	—
100	C₂×C₄×C₁₂₇	SD₃₂	1	2	yes	—
101	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
103	C₈×C₁₂₇	—	2	1	no	—
104	C₈×C₁₂₇	—	2	1	no	—
105	C₈×C₁₂₇	—	2	1	no	—
106	C₈×C₁₂₇	#104	1	2	yes	—
107	C₂²×C₁₂₇	—	2	1	no	—
108	C₂²×C₁₂₇	—	2	1	no	—
109	C₂²×C₁₂₇	—	8	2	no	—
110	C₂²×C₁₂₇	—	8	2	no	—
111	C₂²×C₁₂₇	—	8	2	no	—
112	C₂²×C₁₂₇	C₂×#102	1	1	yes	characteristic
113	C₂²×C₁₂₇	C₂×D₁₆	1	1	yes	characteristic
114	C₂²×C₁₂₇	C₂×SD₃₂	1	1	yes	characteristic
115	C₄×C₁₂₇	—	2	1	no	—
116	C₄×C₁₂₇	—	2	1	no	—
117	C₄×C₁₂₇	—	2	1	no	—
118	C₄×C₁₂₇	—	8	4	no	—
119	C₄×C₁₂₇	—	8	2	no	—
120	C₄×C₁₂₇	#357	1	2	yes	—
121	C₄×C₁₂₇	#357	1	2	yes	—
122	C₄×C₁₂₇	#410	1	1	yes	characteristic
123	C₄×C₁₂₇	C₂×#104	1	1	yes	characteristic
124	#11742	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂×C₁₂₇	—	2	1	no	—
126	C₂×C₁₂₇	—	2	1	no	—
127	C₂×C₁₂₇	—	16	2	no	—
128	C₂×C₁₂₇	#1502	1	1	yes	characteristic
129	C₂×C₁₂₇	C₂×#357	1	1	yes	characteristic
130	C₂×C₁₂₇	#2145	1	1	yes	characteristic
131	#3169	C₂×C₁₂₇	1	1	yes	characteristic
132	#2162	C₂×C₁₂₇	1	1	yes	characteristic, complemented
133	C₄×#107	C₂×C₁₂₇	1	1	yes	characteristic, complemented
134	#1347	C₂×C₁₂₇	1	4	yes	complemented
135	C₁₂₇	#11742	1	1	yes	characteristic, direct product factor, Sylow
136	#532	—	2	1	no	—
137	#500	C₄×C₁₂₇	1	1	yes	characteristic, complemented
138	#493	C₄×C₁₂₇	1	1	yes	characteristic, complemented
139	#486	—	2	1	no	—
140	#466	—	2	1	no	—
141	#466	C₂²×C₁₂₇	1	2	yes	—
142	C₄×D₈	—	2	1	no	—
143	C₄×D₈	C₄×C₁₂₇	1	2	yes	complemented
144	C₂×#107	—	2	1	no	—
145	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
146	C₄×C₄⋊C₄	—	2	1	no	—
147	C₄⋊C₁₆	—	2	1	no	—
148	C₂²⋊C₁₆	C₂²×C₁₂₇	1	2	yes	—
149	C₄×C₁₆	—	2	1	no	—
150	#340	C₂²×C₁₂₇	1	2	yes	—
151	C₂×#41	—	4	1	no	—
152	C₂×D₈	—	2	1	no	—
153	C₂×D₈	C₂×C₄×C₁₂₇	1	2	yes	—
154	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
155	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
156	C₄⋊Q₈	—	2	1	no	—
157	#127	—	2	1	no	—
158	#124	—	4	1	no	—
159	#121	—	2	2	no	—
160	#121	—	4	1	no	—
161	C₄×D₄	—	2	2	no	—
162	C₄×D₄	—	4	1	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄²	—	4	1	no	—
165	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
166	C₂×C₁₆	—	2	2	no	—
167	#107	—	2	4	no	—
168	#107	—	2	1	no	—
169	#107	C₂×C₄×C₁₂₇	1	2	yes	—
170	#102	—	2	2	no	—
171	#102	—	2	2	no	—
172	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
173	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
174	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
175	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
176	#95	—	4	1	no	—
177	#41	—	8	2	no	—
178	#41	—	8	2	no	—
179	C₂×Q₈	—	4	1	no	—
180	C₂×D₄	—	2	2	no	—
181	C₂×D₄	—	4	2	no	—
182	C₂×D₄	—	4	1	no	—
183	C₂×D₄	—	4	1	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
187	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
188	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
189	D₈	—	4	2	no	—
190	D₈	—	4	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	4	1	no	—
193	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
194	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
195	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
196	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₄⋊C₄	—	4	1	no	—
201	C₂²⋊C₄	—	4	2	no	—
202	C₂²⋊C₄	—	4	2	no	—
203	C₄²	—	8	2	no	—
204	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
205	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
206	C₄²	SD₁₆×C₁₂₇	1	1	yes	characteristic
207	C₄²	SD₁₆×C₁₂₇	1	1	yes	characteristic
208	C₁₆	—	4	2	no	—
209	C₂³	—	4	2	no	—
210	C₂³	C₂×D₈×C₁₂₇	1	1	yes	characteristic
211	Q₈	—	8	2	no	—
212	D₄	—	4	2	no	—
213	D₄	—	4	2	no	—
214	D₄	—	8	2	no	—
215	D₄	—	8	2	no	—
216	D₄	—	8	2	no	—
217	D₄	—	8	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	8	2	no	—
224	C₂×C₄	—	8	2	no	—
225	C₂×C₄	—	8	4	no	—
226	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
227	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
228	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
229	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
230	C₂×C₄	D₁₆×C₁₂₇	1	2	yes	—
231	C₂×C₄	SD₃₂×C₁₂₇	1	2	yes	—
232	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
233	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	#104×C₁₂₇	1	2	yes	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	8	2	no	—
241	C₂²	—	8	2	no	—
242	C₂²	—	8	2	no	—
243	C₂²	C₂×#102×C₁₂₇	1	1	yes	characteristic
244	C₂²	C₂×D₁₆×C₁₂₇	1	1	yes	characteristic
245	C₂²	C₂×SD₃₂×C₁₂₇	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	8	4	no	—
250	C₄	—	8	2	no	—
251	C₄	#357×C₁₂₇	1	2	yes	—
252	C₄	#357×C₁₂₇	1	2	yes	—
253	C₄	#410×C₁₂₇	1	1	yes	characteristic
254	C₄	C₂×#104×C₁₂₇	1	1	yes	characteristic
255	C₂	—	2	1	no	—
256	C₂	—	2	1	no	—
257	C₂	—	16	2	no	—
258	C₂	C₁₂₇×#1502	1	1	yes	characteristic
259	C₂	C₂×#357×C₁₂₇	1	1	yes	characteristic
260	C₂	C₁₂₇×#2145	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₂₇	#102
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂×#102
prime radical	C₁₂₇×#3169	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×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. C₁₂₇×#2162	C₂	semidirect
3. C₄×#107×C₁₂₇	C₂	semidirect
4. C₁₂₇×#1347	C₂	semidirect
6. #500×C₁₂₇	C₄	semidirect
7. #493×C₁₂₇	C₄	semidirect
12. C₄×D₈×C₁₂₇	C₄	semidirect
124. #11742	C₁₂₇	direct (N × H)
132. #2162	C₂×C₁₂₇	semidirect
133. C₄×#107	C₂×C₁₂₇	semidirect
134. #1347	C₂×C₁₂₇	semidirect
135. C₁₂₇	#11742	direct (N × H)
137. #500	C₄×C₁₂₇	semidirect
138. #493	C₄×C₁₂₇	semidirect
143. C₄×D₈	C₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×#3169	2
C₁₂₇×#2162	2
C₄×#107×C₁₂₇	2
C₁₂₇×#1347	2
#11742	127

Subgroups — C127×#11742 #44575415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#11742 #44575415