Subgroups — C₁₂₇×#11929 #44575602

Order 32512.

414 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#3147	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₂₇×#1564	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₂₇×#1362	C₂	1	2	yes	maximal, complemented
5	C₁₂₇×#1361	C₂	1	2	yes	maximal, complemented
6	C₂×#135×C₁₂₇	—	2	1	no	—
7	C₂×#135×C₁₂₇	C₄	1	2	yes	—
8	#532×C₁₂₇	—	2	1	no	—
9	C₂²×C₁₆×C₁₂₇	—	2	1	no	—
10	#495×C₁₂₇	—	2	1	no	—
11	#495×C₁₂₇	C₄	1	2	yes	—
12	#494×C₁₂₇	C₂²	1	1	yes	characteristic
13	#493×C₁₂₇	C₂²	1	1	yes	characteristic
14	#492×C₁₂₇	—	2	1	no	—
15	#431×C₁₂₇	—	2	1	no	—
16	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₁₆×C₁₂₇	—	2	1	no	—
18	C₄⋊C₁₆×C₁₂₇	C₂²	1	2	yes	—
19	#335×C₁₂₇	C₂²	1	2	yes	—
20	#334×C₁₂₇	—	2	1	no	—
21	C₂×#41×C₁₂₇	—	2	2	no	—
22	#135×C₁₂₇	—	4	4	no	—
23	#135×C₁₂₇	—	4	2	no	—
24	#135×C₁₂₇	—	4	2	no	—
25	C₂×Q₁₆×C₁₂₇	—	2	1	no	—
26	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
27	C₂×SD₁₆×C₁₂₇	—	2	2	no	—
28	C₂×SD₁₆×C₁₂₇	—	2	2	no	—
29	C₂×D₈×C₁₂₇	—	2	1	no	—
30	C₂×D₈×C₁₂₇	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
33	C₄⋊Q₈×C₁₂₇	—	2	1	no	—
34	#127×C₁₂₇	—	2	1	no	—
35	#124×C₁₂₇	—	2	2	no	—
36	#121×C₁₂₇	—	4	1	no	—
37	#121×C₁₂₇	—	4	1	no	—
38	C₄×D₄×C₁₂₇	—	4	1	no	—
39	C₄×D₄×C₁₂₇	—	4	1	no	—
40	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
41	C₂×C₁₆×C₁₂₇	—	2	2	no	—
42	C₂×C₁₆×C₁₂₇	—	4	2	no	—
43	C₄⋊C₈×C₁₂₇	—	2	2	no	—
44	C₂²⋊C₈×C₁₂₇	—	4	1	no	—
45	C₄×C₈×C₁₂₇	—	4	1	no	—
46	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
48	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
49	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
50	#41×C₁₂₇	—	4	4	no	—
51	#41×C₁₂₇	—	4	4	no	—
52	C₂×Q₈×C₁₂₇	—	2	2	no	—
53	C₂×D₄×C₁₂₇	—	2	2	no	—
54	C₂×D₄×C₁₂₇	—	4	2	no	—
55	C₂×D₄×C₁₂₇	—	4	1	no	—
56	C₂×D₄×C₁₂₇	—	4	1	no	—
57	C₂²×C₄×C₁₂₇	—	4	2	no	—
58	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
61	Q₁₆×C₁₂₇	—	4	2	no	—
62	Q₁₆×C₁₂₇	—	4	1	no	—
63	Q₁₆×C₁₂₇	—	4	1	no	—
64	SD₁₆×C₁₂₇	—	4	4	no	—
65	SD₁₆×C₁₂₇	—	4	4	no	—
66	D₈×C₁₂₇	—	4	2	no	—
67	D₈×C₁₂₇	—	4	1	no	—
68	D₈×C₁₂₇	—	4	1	no	—
69	C₂×C₈×C₁₂₇	—	2	2	no	—
70	C₂×C₈×C₁₂₇	—	4	2	no	—
71	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₂₇	D₈	1	2	yes	—
74	C₂×C₈×C₁₂₇	SD₁₆	1	2	yes	—
75	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
77	C₄⋊C₄×C₁₂₇	—	4	1	no	—
78	C₄⋊C₄×C₁₂₇	—	4	1	no	—
79	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
80	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
81	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₄²×C₁₂₇	D₈	1	1	yes	characteristic
83	C₄²×C₁₂₇	SD₁₆	1	1	yes	characteristic
84	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
85	C₁₆×C₁₂₇	—	4	2	no	—
86	C₂³×C₁₂₇	—	4	2	no	—
87	C₂³×C₁₂₇	#102	1	1	yes	characteristic
88	Q₈×C₁₂₇	—	4	2	no	—
89	Q₈×C₁₂₇	—	4	2	no	—
90	D₄×C₁₂₇	—	4	2	no	—
91	D₄×C₁₂₇	—	4	2	no	—
92	D₄×C₁₂₇	—	8	2	no	—
93	D₄×C₁₂₇	—	8	4	no	—
94	D₄×C₁₂₇	—	8	2	no	—
95	C₂×C₄×C₁₂₇	—	2	2	no	—
96	C₂×C₄×C₁₂₇	—	4	1	no	—
97	C₂×C₄×C₁₂₇	—	4	2	no	—
98	C₂×C₄×C₁₂₇	—	8	4	no	—
99	C₂×C₄×C₁₂₇	#100	1	2	yes	—
100	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
101	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
102	C₂×C₄×C₁₂₇	#102	1	2	yes	—
103	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
104	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
107	C₈×C₁₂₇	—	2	1	no	—
108	C₈×C₁₂₇	—	2	1	no	—
109	C₈×C₁₂₇	—	8	2	no	—
110	C₈×C₁₂₇	#102	1	2	yes	—
111	C₈×C₁₂₇	C₂×D₈	1	1	yes	characteristic
112	C₈×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
113	C₂²×C₁₂₇	—	2	1	no	—
114	C₂²×C₁₂₇	—	2	1	no	—
115	C₂²×C₁₂₇	—	8	2	no	—
116	C₂²×C₁₂₇	—	8	2	no	—
117	C₂²×C₁₂₇	—	8	2	no	—
118	C₂²×C₁₂₇	#331	1	2	yes	—
119	C₂²×C₁₂₇	C₂×#102	1	1	yes	characteristic
120	C₄×C₁₂₇	—	2	1	no	—
121	C₄×C₁₂₇	—	2	1	no	—
122	C₄×C₁₂₇	—	2	1	no	—
123	C₄×C₁₂₇	—	8	2	no	—
124	C₄×C₁₂₇	#331	1	2	yes	—
125	C₄×C₁₂₇	C₂×#100	1	1	yes	characteristic
126	C₄×C₁₂₇	C₂×#102	1	1	yes	characteristic
127	C₄×C₁₂₇	#508	1	2	yes	—
128	#11929	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂×C₁₂₇	—	2	2	no	—
130	C₂×C₁₂₇	—	16	2	no	—
131	C₂×C₁₂₇	C₂×#331	1	1	yes	characteristic
132	C₂×C₁₂₇	#2138	1	1	yes	characteristic
133	C₂×C₁₂₇	#2148	1	1	yes	characteristic
134	#3147	C₂×C₁₂₇	1	1	yes	characteristic
135	C₂×C₄⋊C₁₆	C₂×C₁₂₇	1	1	yes	characteristic, complemented
136	#1564	C₂×C₁₂₇	1	1	yes	characteristic, complemented
137	#1362	C₂×C₁₂₇	1	2	yes	complemented
138	#1361	C₂×C₁₂₇	1	2	yes	complemented
139	C₁₂₇	#11929	1	1	yes	characteristic, direct product factor, Sylow
140	C₂×#135	—	2	1	no	—
141	C₂×#135	C₄×C₁₂₇	1	2	yes	—
142	#532	—	2	1	no	—
143	C₂²×C₁₆	—	2	1	no	—
144	#495	—	2	1	no	—
145	#495	C₄×C₁₂₇	1	2	yes	—
146	#494	C₂²×C₁₂₇	1	1	yes	characteristic
147	#493	C₂²×C₁₂₇	1	1	yes	characteristic
148	#492	—	2	1	no	—
149	#431	—	2	1	no	—
150	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
151	C₄⋊C₁₆	—	2	1	no	—
152	C₄⋊C₁₆	C₂²×C₁₂₇	1	2	yes	—
153	#335	C₂²×C₁₂₇	1	2	yes	—
154	#334	—	2	1	no	—
155	C₂×#41	—	2	2	no	—
156	#135	—	4	4	no	—
157	#135	—	4	2	no	—
158	#135	—	4	2	no	—
159	C₂×Q₁₆	—	2	1	no	—
160	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
161	C₂×SD₁₆	—	2	2	no	—
162	C₂×SD₁₆	—	2	2	no	—
163	C₂×D₈	—	2	1	no	—
164	C₂×D₈	C₂×C₄×C₁₂₇	1	2	yes	—
165	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
166	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
167	C₄⋊Q₈	—	2	1	no	—
168	#127	—	2	1	no	—
169	#124	—	2	2	no	—
170	#121	—	4	1	no	—
171	#121	—	4	1	no	—
172	C₄×D₄	—	4	1	no	—
173	C₄×D₄	—	4	1	no	—
174	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
175	C₂×C₁₆	—	2	2	no	—
176	C₂×C₁₆	—	4	2	no	—
177	C₄⋊C₈	—	2	2	no	—
178	C₂²⋊C₈	—	4	1	no	—
179	C₄×C₈	—	4	1	no	—
180	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
181	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
182	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
183	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
184	#41	—	4	4	no	—
185	#41	—	4	4	no	—
186	C₂×Q₈	—	2	2	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂×D₄	—	4	1	no	—
190	C₂×D₄	—	4	1	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
193	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
194	C₂²×C₄	SD₁₆×C₁₂₇	1	1	yes	characteristic
195	Q₁₆	—	4	2	no	—
196	Q₁₆	—	4	1	no	—
197	Q₁₆	—	4	1	no	—
198	SD₁₆	—	4	4	no	—
199	SD₁₆	—	4	4	no	—
200	D₈	—	4	2	no	—
201	D₈	—	4	1	no	—
202	D₈	—	4	1	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
206	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
207	C₂×C₈	D₈×C₁₂₇	1	2	yes	—
208	C₂×C₈	SD₁₆×C₁₂₇	1	2	yes	—
209	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
210	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
211	C₄⋊C₄	—	4	1	no	—
212	C₄⋊C₄	—	4	1	no	—
213	C₂²⋊C₄	—	4	2	no	—
214	C₂²⋊C₄	—	4	2	no	—
215	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
216	C₄²	D₈×C₁₂₇	1	1	yes	characteristic
217	C₄²	SD₁₆×C₁₂₇	1	1	yes	characteristic
218	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
219	C₁₆	—	4	2	no	—
220	C₂³	—	4	2	no	—
221	C₂³	#102×C₁₂₇	1	1	yes	characteristic
222	Q₈	—	4	2	no	—
223	Q₈	—	4	2	no	—
224	D₄	—	4	2	no	—
225	D₄	—	4	2	no	—
226	D₄	—	8	2	no	—
227	D₄	—	8	4	no	—
228	D₄	—	8	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	8	4	no	—
233	C₂×C₄	#100×C₁₂₇	1	2	yes	—
234	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
235	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
236	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
237	C₂×C₄	#102×C₁₂₇	1	2	yes	—
238	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
239	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
240	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	8	2	no	—
244	C₈	#102×C₁₂₇	1	2	yes	—
245	C₈	C₂×D₈×C₁₂₇	1	1	yes	characteristic
246	C₈	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	8	2	no	—
250	C₂²	—	8	2	no	—
251	C₂²	—	8	2	no	—
252	C₂²	#331×C₁₂₇	1	2	yes	—
253	C₂²	C₂×#102×C₁₂₇	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	8	2	no	—
258	C₄	#331×C₁₂₇	1	2	yes	—
259	C₄	C₂×#100×C₁₂₇	1	1	yes	characteristic
260	C₄	C₂×#102×C₁₂₇	1	1	yes	characteristic
261	C₄	#508×C₁₂₇	1	2	yes	—
262	C₂	—	2	2	no	—
263	C₂	—	16	2	no	—
264	C₂	C₂×#331×C₁₂₇	1	1	yes	characteristic
265	C₂	C₁₂₇×#2138	1	1	yes	characteristic
266	C₂	C₁₂₇×#2148	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₂₇	#102
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂×#102
prime radical	C₁₂₇×#3147	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. C₂×C₄⋊C₁₆×C₁₂₇	C₂	semidirect
3. C₁₂₇×#1564	C₂	semidirect
4. C₁₂₇×#1362	C₂	semidirect
5. C₁₂₇×#1361	C₂	semidirect
128. #11929	C₁₂₇	direct (N × H)
135. C₂×C₄⋊C₁₆	C₂×C₁₂₇	semidirect
136. #1564	C₂×C₁₂₇	semidirect
137. #1362	C₂×C₁₂₇	semidirect
138. #1361	C₂×C₁₂₇	semidirect
139. C₁₂₇	#11929	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×#3147	2
C₂×C₄⋊C₁₆×C₁₂₇	2
C₁₂₇×#1564	2
C₁₂₇×#1362	2
C₁₂₇×#1361	2
#11929	127

Subgroups — C127×#11929 #44575602

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#11929 #44575602