Subgroups — C₄×#361×C₁₂₇ #44576317

Order 32512.

498 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#361×C₁₂₇	C₂	1	2	yes	maximal
3	C₄×M₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#108×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₂₇×#1387	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₄×C₁₂₇	—	2	1	no	—
8	C₂×M₃₂×C₁₂₇	C₂²	1	2	yes	—
9	C₄×D₈×C₁₂₇	—	2	1	no	—
10	C₄×D₈×C₁₂₇	—	2	1	no	—
11	C₄×D₈×C₁₂₇	C₄	1	2	yes	—
12	C₄×D₈×C₁₂₇	C₄	1	2	yes	—
13	C₂×#108×C₁₂₇	C₂²	1	2	yes	—
14	C₂×#107×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×#102×C₁₂₇	—	2	1	no	—
16	C₄×M₁₆×C₁₂₇	—	2	1	no	—
17	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
18	#361×C₁₂₇	C₄	1	8	yes	direct product factor
19	#346×C₁₂₇	—	2	1	no	—
20	C₄×C₁₆×C₁₂₇	—	2	1	no	—
21	#341×C₁₂₇	—	2	1	no	—
22	C₂²×D₄×C₁₂₇	—	2	1	no	—
23	C₂³×C₄×C₁₂₇	—	4	1	no	—
24	C₂×D₈×C₁₂₇	—	2	1	no	—
25	C₂×D₈×C₁₂₇	—	2	2	no	—
26	C₂×D₈×C₁₂₇	C₂×C₄	1	4	yes	—
27	C₂×D₈×C₁₂₇	C₂×C₄	1	2	yes	—
28	C₂×M₁₆×C₁₂₇	—	2	2	no	—
29	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
31	C₄×D₄×C₁₂₇	—	2	2	no	—
32	C₄×D₄×C₁₂₇	—	2	2	no	—
33	C₄×D₄×C₁₂₇	—	4	1	no	—
34	C₄×D₄×C₁₂₇	—	4	1	no	—
35	C₂×C₄⋊C₄×C₁₂₇	—	2	1	no	—
36	C₂×C₂²⋊C₄×C₁₂₇	—	4	1	no	—
37	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
38	M₃₂×C₁₂₇	—	2	2	no	—
39	M₃₂×C₁₂₇	C₂×C₄	1	4	yes	complemented
40	C₂×C₁₆×C₁₂₇	—	2	2	no	—
41	#108×C₁₂₇	—	2	2	no	—
42	#108×C₁₂₇	C₂×C₄	1	4	yes	complemented
43	#107×C₁₂₇	—	2	1	no	—
44	#107×C₁₂₇	C₂×C₄	1	2	yes	—
45	#102×C₁₂₇	—	2	2	no	—
46	#102×C₁₂₇	—	2	2	no	—
47	#97×C₁₂₇	—	4	1	no	—
48	C₄×C₈×C₁₂₇	—	4	1	no	—
49	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
52	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
53	C₂⁴×C₁₂₇	—	4	1	no	—
54	C₂×D₄×C₁₂₇	—	2	2	no	—
55	C₂×D₄×C₁₂₇	—	2	4	no	—
56	C₂×D₄×C₁₂₇	—	4	1	no	—
57	C₂×D₄×C₁₂₇	—	4	2	no	—
58	C₂²×C₄×C₁₂₇	—	4	1	no	—
59	C₂²×C₄×C₁₂₇	—	4	2	no	—
60	C₂²×C₄×C₁₂₇	—	4	2	no	—
61	C₂²×C₄×C₁₂₇	—	8	4	no	—
62	C₂²×C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
65	D₈×C₁₂₇	—	2	2	no	—
66	D₈×C₁₂₇	—	2	4	no	—
67	D₈×C₁₂₇	C₄²	1	4	yes	—
68	M₁₆×C₁₂₇	—	2	4	no	—
69	M₁₆×C₁₂₇	—	4	2	no	—
70	C₂×C₈×C₁₂₇	—	2	2	no	—
71	C₂×C₈×C₁₂₇	—	4	2	no	—
72	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₂₇	#41	1	2	yes	—
78	C₄⋊C₄×C₁₂₇	—	2	2	no	—
79	C₄⋊C₄×C₁₂₇	—	4	1	no	—
80	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
81	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
82	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
84	C₄²×C₁₂₇	D₈	1	1	yes	characteristic
85	C₄²×C₁₂₇	D₈	1	1	yes	characteristic
86	C₁₆×C₁₂₇	—	2	4	no	—
87	C₂³×C₁₂₇	—	4	2	no	—
88	C₂³×C₁₂₇	—	4	4	no	—
89	C₂³×C₁₂₇	—	8	2	no	—
90	C₂³×C₁₂₇	—	8	2	no	—
91	C₂³×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
92	D₄×C₁₂₇	—	2	4	no	—
93	D₄×C₁₂₇	—	4	2	no	—
94	D₄×C₁₂₇	—	4	4	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	—	8	2	no	—
100	C₂×C₄×C₁₂₇	—	8	4	no	—
101	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
102	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
103	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
104	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
105	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₁₂₇	#135	1	2	yes	—
109	C₈×C₁₂₇	—	2	1	no	—
110	C₈×C₁₂₇	—	2	1	no	—
111	C₈×C₁₂₇	—	4	4	no	—
112	C₈×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
113	C₈×C₁₂₇	#117	1	1	yes	characteristic
114	C₈×C₁₂₇	C₄×D₄	1	2	yes	—
115	C₂²×C₁₂₇	—	2	1	no	—
116	C₂²×C₁₂₇	—	2	1	no	—
117	C₂²×C₁₂₇	—	4	4	no	—
118	C₂²×C₁₂₇	—	8	8	no	—
119	C₂²×C₁₂₇	—	8	2	no	—
120	C₂²×C₁₂₇	—	8	2	no	—
121	C₂²×C₁₂₇	C₂×#102	1	1	yes	characteristic
122	C₂²×C₁₂₇	C₄×SD₁₆	1	2	yes	—
123	C₄×C₁₂₇	—	2	1	no	—
124	C₄×C₁₂₇	—	2	1	no	—
125	C₄×C₁₂₇	—	2	1	no	—
126	C₄×C₁₂₇	—	8	2	no	—
127	C₄×C₁₂₇	#361	1	2	yes	direct product factor
128	C₄×C₁₂₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
129	C₄×C₁₂₇	#416	1	1	yes	characteristic
130	C₄×C₁₂₇	C₄×D₈	1	2	yes	—
131	C₄×#361	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂×C₁₂₇	—	2	2	no	—
133	C₂×C₁₂₇	—	8	4	no	—
134	C₂×C₁₂₇	C₄×#102	1	1	yes	characteristic
135	C₂×C₁₂₇	C₂×#361	1	1	yes	characteristic
136	C₂×C₁₂₇	#2148	1	1	yes	characteristic
137	C₂×C₄×D₈	C₂×C₁₂₇	1	1	yes	characteristic
138	C₂×#361	C₂×C₁₂₇	1	2	yes	—
139	C₄×M₃₂	C₂×C₁₂₇	1	1	yes	characteristic, complemented
140	C₄×#108	C₂×C₁₂₇	1	1	yes	characteristic, complemented
141	#1387	C₂×C₁₂₇	1	2	yes	complemented
142	C₁₂₇	C₄×#361	1	1	yes	characteristic, direct product factor, Sylow
143	C₂²×D₈	C₂²×C₁₂₇	1	1	yes	characteristic
144	C₂×C₄×D₄	—	2	1	no	—
145	C₂×M₃₂	C₂²×C₁₂₇	1	2	yes	—
146	C₄×D₈	—	2	1	no	—
147	C₄×D₈	—	2	1	no	—
148	C₄×D₈	C₄×C₁₂₇	1	2	yes	—
149	C₄×D₈	C₄×C₁₂₇	1	2	yes	—
150	C₂×#108	C₂²×C₁₂₇	1	2	yes	—
151	C₂×#107	C₂²×C₁₂₇	1	1	yes	characteristic
152	C₂×#102	—	2	1	no	—
153	C₄×M₁₆	—	2	1	no	—
154	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
155	#361	C₄×C₁₂₇	1	8	yes	direct product factor
156	#346	—	2	1	no	—
157	C₄×C₁₆	—	2	1	no	—
158	#341	—	2	1	no	—
159	C₂²×D₄	—	2	1	no	—
160	C₂³×C₄	—	4	1	no	—
161	C₂×D₈	—	2	1	no	—
162	C₂×D₈	—	2	2	no	—
163	C₂×D₈	C₂×C₄×C₁₂₇	1	4	yes	—
164	C₂×D₈	C₂×C₄×C₁₂₇	1	2	yes	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
167	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
168	C₄×D₄	—	2	2	no	—
169	C₄×D₄	—	2	2	no	—
170	C₄×D₄	—	4	1	no	—
171	C₄×D₄	—	4	1	no	—
172	C₂×C₄⋊C₄	—	2	1	no	—
173	C₂×C₂²⋊C₄	—	4	1	no	—
174	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
175	M₃₂	—	2	2	no	—
176	M₃₂	C₂×C₄×C₁₂₇	1	4	yes	complemented
177	C₂×C₁₆	—	2	2	no	—
178	#108	—	2	2	no	—
179	#108	C₂×C₄×C₁₂₇	1	4	yes	complemented
180	#107	—	2	1	no	—
181	#107	C₂×C₄×C₁₂₇	1	2	yes	—
182	#102	—	2	2	no	—
183	#102	—	2	2	no	—
184	#97	—	4	1	no	—
185	C₄×C₈	—	4	1	no	—
186	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
187	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
188	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
189	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
190	C₂⁴	—	4	1	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	2	4	no	—
193	C₂×D₄	—	4	1	no	—
194	C₂×D₄	—	4	2	no	—
195	C₂²×C₄	—	4	1	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	—	8	4	no	—
199	C₂²×C₄	SD₁₆×C₁₂₇	1	1	yes	characteristic
200	C₂²×C₄	SD₁₆×C₁₂₇	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
202	D₈	—	2	2	no	—
203	D₈	—	2	4	no	—
204	D₈	C₄²×C₁₂₇	1	4	yes	—
205	M₁₆	—	2	4	no	—
206	M₁₆	—	4	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
210	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
211	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
212	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
213	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
214	C₂×C₈	#41×C₁₂₇	1	2	yes	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	4	1	no	—
217	C₂²⋊C₄	—	4	2	no	—
218	C₂²⋊C₄	—	4	2	no	—
219	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
220	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
221	C₄²	D₈×C₁₂₇	1	1	yes	characteristic
222	C₄²	D₈×C₁₂₇	1	1	yes	characteristic
223	C₁₆	—	2	4	no	—
224	C₂³	—	4	2	no	—
225	C₂³	—	4	4	no	—
226	C₂³	—	8	2	no	—
227	C₂³	—	8	2	no	—
228	C₂³	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
229	D₄	—	2	4	no	—
230	D₄	—	4	2	no	—
231	D₄	—	4	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	8	4	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	8	4	no	—
238	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
239	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
240	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
241	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
242	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
243	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
244	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
245	C₂×C₄	#135×C₁₂₇	1	2	yes	—
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	—	4	4	no	—
249	C₈	C₂×C₄²×C₁₂₇	1	1	yes	characteristic
250	C₈	#117×C₁₂₇	1	1	yes	characteristic
251	C₈	C₄×D₄×C₁₂₇	1	2	yes	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	4	4	no	—
255	C₂²	—	8	8	no	—
256	C₂²	—	8	2	no	—
257	C₂²	—	8	2	no	—
258	C₂²	C₂×#102×C₁₂₇	1	1	yes	characteristic
259	C₂²	C₄×SD₁₆×C₁₂₇	1	2	yes	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	8	2	no	—
264	C₄	#361×C₁₂₇	1	2	yes	direct product factor
265	C₄	C₄×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
266	C₄	#416×C₁₂₇	1	1	yes	characteristic
267	C₄	C₄×D₈×C₁₂₇	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	8	4	no	—
270	C₂	C₄×#102×C₁₂₇	1	1	yes	characteristic
271	C₂	C₂×#361×C₁₂₇	1	1	yes	characteristic
272	C₂	C₁₂₇×#2148	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₂₇	#102
commutator	C₈	C₂×C₄²×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂×#102
prime radical	C₂²×D₈×C₁₂₇	C₂²
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
3. C₄×M₃₂×C₁₂₇	C₂	semidirect
4. C₄×#108×C₁₂₇	C₂	semidirect
5. C₁₂₇×#1387	C₂	semidirect
18. #361×C₁₂₇	C₄	direct (N × H)
39. M₃₂×C₁₂₇	C₂×C₄	semidirect
42. #108×C₁₂₇	C₂×C₄	semidirect
127. C₄×C₁₂₇	#361	direct (N × H)
131. C₄×#361	C₁₂₇	direct (N × H)
139. C₄×M₃₂	C₂×C₁₂₇	semidirect
140. C₄×#108	C₂×C₁₂₇	semidirect
141. #1387	C₂×C₁₂₇	semidirect
142. C₁₂₇	C₄×#361	direct (N × H)
155. #361	C₄×C₁₂₇	direct (N × H)
176. M₃₂	C₂×C₄×C₁₂₇	semidirect
179. #108	C₂×C₄×C₁₂₇	semidirect
264. C₄	#361×C₁₂₇	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×C₄×D₈×C₁₂₇	2
C₂×#361×C₁₂₇	2
C₄×M₃₂×C₁₂₇	2
C₄×#108×C₁₂₇	2
C₁₂₇×#1387	2
C₄×#361	127

Subgroups — C4×#361×C127 #44576317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#361×C₁₂₇ #44576317