Subgroups — C₁₂₇×#11588 #44575261

Order 32512.

390 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#2931	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#2150	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#346×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₂₇×#1330	C₂	1	4	yes	maximal, complemented
5	C₂×#131×C₁₂₇	—	2	1	no	—
6	#532×C₁₂₇	C₄	1	1	yes	characteristic
7	C₂×M₃₂×C₁₂₇	—	2	1	no	—
8	C₂²×C₁₆×C₁₂₇	—	2	1	no	—
9	#436×C₁₂₇	—	2	1	no	—
10	#436×C₁₂₇	C₂²	1	2	yes	—
11	C₈×D₄×C₁₂₇	—	2	1	no	—
12	C₈×D₄×C₁₂₇	C₄	1	2	yes	—
13	#423×C₁₂₇	C₄	1	1	yes	characteristic
14	#408×C₁₂₇	—	2	1	no	—
15	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₁₆×C₁₂₇	—	2	1	no	—
17	C₄⋊C₁₆×C₁₂₇	C₂²	1	2	yes	—
18	#346×C₁₂₇	—	2	1	no	—
19	#346×C₁₂₇	C₂²	1	2	yes	—
20	C₂×#41×C₁₂₇	—	2	1	no	—
21	#131×C₁₂₇	—	4	2	no	—
22	#131×C₁₂₇	—	4	2	no	—
23	C₂×M₁₆×C₁₂₇	—	2	1	no	—
24	C₂×M₁₆×C₁₂₇	—	2	2	no	—
25	C₂²×C₈×C₁₂₇	—	2	2	no	—
26	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
28	C₄⋊Q₈×C₁₂₇	C₈	1	1	yes	characteristic
29	#127×C₁₂₇	C₈	1	1	yes	characteristic
30	#124×C₁₂₇	—	2	1	no	—
31	#121×C₁₂₇	—	2	1	no	—
32	#121×C₁₂₇	C₈	1	2	yes	—
33	C₄×D₄×C₁₂₇	—	2	1	no	—
34	C₄×D₄×C₁₂₇	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
36	M₃₂×C₁₂₇	—	4	2	no	—
37	C₂×C₁₆×C₁₂₇	—	2	4	no	—
38	C₂×C₁₆×C₁₂₇	—	2	2	no	—
39	C₂×C₁₆×C₁₂₇	—	2	2	no	—
40	C₄⋊C₈×C₁₂₇	—	2	1	no	—
41	C₄⋊C₈×C₁₂₇	C₂×C₄	1	2	yes	—
42	C₂²⋊C₈×C₁₂₇	—	2	2	no	—
43	C₂²⋊C₈×C₁₂₇	—	2	2	no	—
44	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
46	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
47	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
48	#41×C₁₂₇	—	4	2	no	—
49	#41×C₁₂₇	—	4	2	no	—
50	C₂×Q₈×C₁₂₇	—	2	1	no	—
51	C₂×D₄×C₁₂₇	—	2	2	no	—
52	C₂×D₄×C₁₂₇	—	2	1	no	—
53	C₂×D₄×C₁₂₇	—	2	1	no	—
54	C₂×D₄×C₁₂₇	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₁₂₇	—	2	2	no	—
56	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₂₇	M₁₆	1	1	yes	characteristic
59	M₁₆×C₁₂₇	—	4	2	no	—
60	M₁₆×C₁₂₇	—	4	2	no	—
61	M₁₆×C₁₂₇	—	4	2	no	—
62	C₂×C₈×C₁₂₇	—	2	2	no	—
63	C₂×C₈×C₁₂₇	—	2	2	no	—
64	C₂×C₈×C₁₂₇	—	4	4	no	—
65	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₂₇	D₈	1	2	yes	—
68	C₂×C₈×C₁₂₇	SD₁₆	1	2	yes	—
69	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
71	C₄⋊C₄×C₁₂₇	—	2	1	no	—
72	C₄⋊C₄×C₁₂₇	C₂×C₈	1	2	yes	—
73	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
74	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
75	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₁₂₇	C₂×C₈	1	1	yes	characteristic
77	C₄²×C₁₂₇	M₁₆	1	1	yes	characteristic
78	C₄²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
79	C₁₆×C₁₂₇	—	2	4	no	—
80	C₁₆×C₁₂₇	—	4	2	no	—
81	C₂³×C₁₂₇	—	2	2	no	—
82	C₂³×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
83	Q₈×C₁₂₇	—	4	2	no	—
84	D₄×C₁₂₇	—	4	2	no	—
85	D₄×C₁₂₇	—	4	2	no	—
86	D₄×C₁₂₇	—	4	4	no	—
87	D₄×C₁₂₇	—	4	2	no	—
88	C₂×C₄×C₁₂₇	—	2	2	no	—
89	C₂×C₄×C₁₂₇	—	4	1	no	—
90	C₂×C₄×C₁₂₇	—	4	1	no	—
91	C₂×C₄×C₁₂₇	—	4	2	no	—
92	C₂×C₄×C₁₂₇	—	4	2	no	—
93	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
94	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
95	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
96	C₂×C₄×C₁₂₇	#102	1	2	yes	—
97	C₂×C₄×C₁₂₇	#104	1	2	yes	—
98	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
101	C₈×C₁₂₇	—	2	1	no	—
102	C₈×C₁₂₇	—	2	1	no	—
103	C₈×C₁₂₇	—	4	2	no	—
104	C₈×C₁₂₇	#102	1	2	yes	—
105	C₈×C₁₂₇	C₂×D₈	1	1	yes	characteristic
106	C₈×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
107	C₂²×C₁₂₇	—	2	1	no	—
108	C₂²×C₁₂₇	—	2	1	no	—
109	C₂²×C₁₂₇	—	4	2	no	—
110	C₂²×C₁₂₇	—	4	2	no	—
111	C₂²×C₁₂₇	—	4	2	no	—
112	C₂²×C₁₂₇	D₄⋊C₈	1	2	yes	—
113	C₂²×C₁₂₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
114	C₄×C₁₂₇	—	2	1	no	—
115	C₄×C₁₂₇	—	2	1	no	—
116	C₄×C₁₂₇	—	2	1	no	—
117	C₄×C₁₂₇	—	2	1	no	—
118	C₄×C₁₂₇	—	4	2	no	—
119	C₄×C₁₂₇	D₄⋊C₈	1	2	yes	—
120	C₄×C₁₂₇	C₂×#102	1	1	yes	characteristic
121	C₄×C₁₂₇	C₂×#104	1	1	yes	characteristic
122	#11588	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂×C₁₂₇	—	2	2	no	—
124	C₂×C₁₂₇	—	8	2	no	—
125	C₂×C₁₂₇	C₂×D₄⋊C₈	1	1	yes	characteristic
126	C₂×C₁₂₇	#2115	1	1	yes	characteristic
127	C₂×C₁₂₇	#2117	1	1	yes	characteristic
128	#2931	C₂×C₁₂₇	1	1	yes	characteristic
129	#2150	C₂×C₁₂₇	1	1	yes	characteristic, complemented
130	C₂×#346	C₂×C₁₂₇	1	1	yes	characteristic, complemented
131	#1330	C₂×C₁₂₇	1	4	yes	complemented
132	C₁₂₇	#11588	1	1	yes	characteristic, direct product factor, Sylow
133	C₂×#131	—	2	1	no	—
134	#532	C₄×C₁₂₇	1	1	yes	characteristic
135	C₂×M₃₂	—	2	1	no	—
136	C₂²×C₁₆	—	2	1	no	—
137	#436	—	2	1	no	—
138	#436	C₂²×C₁₂₇	1	2	yes	—
139	C₈×D₄	—	2	1	no	—
140	C₈×D₄	C₄×C₁₂₇	1	2	yes	—
141	#423	C₄×C₁₂₇	1	1	yes	characteristic
142	#408	—	2	1	no	—
143	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
144	C₄⋊C₁₆	—	2	1	no	—
145	C₄⋊C₁₆	C₂²×C₁₂₇	1	2	yes	—
146	#346	—	2	1	no	—
147	#346	C₂²×C₁₂₇	1	2	yes	—
148	C₂×#41	—	2	1	no	—
149	#131	—	4	2	no	—
150	#131	—	4	2	no	—
151	C₂×M₁₆	—	2	1	no	—
152	C₂×M₁₆	—	2	2	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
155	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
156	C₄⋊Q₈	C₈×C₁₂₇	1	1	yes	characteristic
157	#127	C₈×C₁₂₇	1	1	yes	characteristic
158	#124	—	2	1	no	—
159	#121	—	2	1	no	—
160	#121	C₈×C₁₂₇	1	2	yes	—
161	C₄×D₄	—	2	1	no	—
162	C₄×D₄	C₂×C₄×C₁₂₇	1	2	yes	—
163	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
164	M₃₂	—	4	2	no	—
165	C₂×C₁₆	—	2	4	no	—
166	C₂×C₁₆	—	2	2	no	—
167	C₂×C₁₆	—	2	2	no	—
168	C₄⋊C₈	—	2	1	no	—
169	C₄⋊C₈	C₂×C₄×C₁₂₇	1	2	yes	—
170	C₂²⋊C₈	—	2	2	no	—
171	C₂²⋊C₈	—	2	2	no	—
172	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
173	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
174	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
175	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
176	#41	—	4	2	no	—
177	#41	—	4	2	no	—
178	C₂×Q₈	—	2	1	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	2	1	no	—
181	C₂×D₄	—	2	1	no	—
182	C₂×D₄	C₂×C₈×C₁₂₇	1	2	yes	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₈×C₁₂₇	1	1	yes	characteristic
186	C₂²×C₄	M₁₆×C₁₂₇	1	1	yes	characteristic
187	M₁₆	—	4	2	no	—
188	M₁₆	—	4	2	no	—
189	M₁₆	—	4	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	4	4	no	—
193	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
194	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
195	C₂×C₈	D₈×C₁₂₇	1	2	yes	—
196	C₂×C₈	SD₁₆×C₁₂₇	1	2	yes	—
197	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
198	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	C₂×C₈×C₁₂₇	1	2	yes	—
201	C₂²⋊C₄	—	2	2	no	—
202	C₂²⋊C₄	—	2	2	no	—
203	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
204	C₄²	C₂×C₈×C₁₂₇	1	1	yes	characteristic
205	C₄²	M₁₆×C₁₂₇	1	1	yes	characteristic
206	C₄²	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
207	C₁₆	—	2	4	no	—
208	C₁₆	—	4	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
211	Q₈	—	4	2	no	—
212	D₄	—	4	2	no	—
213	D₄	—	4	2	no	—
214	D₄	—	4	4	no	—
215	D₄	—	4	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
222	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
223	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
224	C₂×C₄	#102×C₁₂₇	1	2	yes	—
225	C₂×C₄	#104×C₁₂₇	1	2	yes	—
226	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
227	C₂×C₄	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
228	C₂×C₄	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
229	C₈	—	2	1	no	—
230	C₈	—	2	1	no	—
231	C₈	—	4	2	no	—
232	C₈	#102×C₁₂₇	1	2	yes	—
233	C₈	C₂×D₈×C₁₂₇	1	1	yes	characteristic
234	C₈	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	D₄⋊C₈×C₁₂₇	1	2	yes	—
241	C₂²	C₂×C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	4	2	no	—
247	C₄	D₄⋊C₈×C₁₂₇	1	2	yes	—
248	C₄	C₂×#102×C₁₂₇	1	1	yes	characteristic
249	C₄	C₂×#104×C₁₂₇	1	1	yes	characteristic
250	C₂	—	2	2	no	—
251	C₂	—	8	2	no	—
252	C₂	C₂×D₄⋊C₈×C₁₂₇	1	1	yes	characteristic
253	C₂	C₁₂₇×#2115	1	1	yes	characteristic
254	C₂	C₁₂₇×#2117	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂×C₂²⋊C₈
prime radical	#532×C₁₂₇	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₂₇×#2150	C₂	semidirect
3. C₂×#346×C₁₂₇	C₂	semidirect
4. C₁₂₇×#1330	C₂	semidirect
122. #11588	C₁₂₇	direct (N × H)
129. #2150	C₂×C₁₂₇	semidirect
130. C₂×#346	C₂×C₁₂₇	semidirect
131. #1330	C₂×C₁₂₇	semidirect
132. C₁₂₇	#11588	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₂₇×#2931	2
C₁₂₇×#2150	2
C₂×#346×C₁₂₇	2
C₁₂₇×#1330	2
#11588	127

Subgroups — C127×#11588 #44575261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#11588 #44575261