Subgroups — C₄×#498×C₁₂₇ #44585392

Order 32512.

878 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#498×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Q₈×C₁₂₇	C₂	1	2	yes	maximal
3	C₁₂₇×#1990	C₂	1	2	yes	maximal
4	C₁₂₇×#1941	C₂	1	4	yes	maximal
5	C₁₂₇×#1844	C₂	1	1	yes	maximal, characteristic
6	C₄×#106×C₁₂₇	C₂	1	4	yes	maximal
7	C₄²×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈×C₁₂₇	C₂²	1	2	yes	—
9	C₂×C₄×Q₈×C₁₂₇	—	2	2	no	—
10	#498×C₁₂₇	C₄	1	8	yes	direct product factor
11	C₂×#106×C₁₂₇	C₂²	1	4	yes	—
12	C₂×#106×C₁₂₇	C₂²	1	4	yes	—
13	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
17	#391×C₁₂₇	—	2	2	no	—
18	#389×C₁₂₇	—	2	2	no	—
19	#389×C₁₂₇	C₂²	1	4	yes	—
20	#384×C₁₂₇	C₂²	1	2	yes	—
21	C₄×C₄⋊C₄×C₁₂₇	—	2	2	no	—
22	C₄×C₄⋊C₄×C₁₂₇	C₂²	1	4	yes	—
23	C₄³×C₁₂₇	C₂²	1	1	yes	characteristic
24	#340×C₁₂₇	C₂²	1	8	yes	—
25	C₂²×Q₈×C₁₂₇	—	2	2	no	—
26	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
27	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
28	C₄⋊Q₈×C₁₂₇	—	2	4	no	—
29	C₄⋊Q₈×C₁₂₇	C₂×C₄	1	8	yes	—
30	C₄×Q₈×C₁₂₇	—	4	8	no	—
31	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
32	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
33	C₂×C₄⋊C₄×C₁₂₇	—	2	4	no	—
34	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
36	C₂×C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
37	C₂×C₄²×C₁₂₇	—	2	4	no	—
38	C₂×C₄²×C₁₂₇	D₄	1	2	yes	—
39	C₂×C₄²×C₁₂₇	D₄	1	2	yes	—
40	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
43	#106×C₁₂₇	—	2	8	no	—
44	#106×C₁₂₇	C₂×C₄	1	16	yes	—
45	C₄×C₈×C₁₂₇	—	2	4	no	—
46	C₄×C₈×C₁₂₇	—	2	2	no	—
47	C₄×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
48	C₄×C₈×C₁₂₇	Q₈	1	8	yes	complemented
49	#95×C₁₂₇	—	2	4	no	—
50	#95×C₁₂₇	—	2	4	no	—
51	C₂×Q₈×C₁₂₇	—	2	8	no	—
52	C₂×Q₈×C₁₂₇	—	4	4	no	—
53	C₂×Q₈×C₁₂₇	—	4	4	no	—
54	C₂²×C₄×C₁₂₇	—	2	4	no	—
55	C₂²×C₄×C₁₂₇	—	2	4	no	—
56	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
61	C₂×C₈×C₁₂₇	—	2	2	no	—
62	C₂×C₈×C₁₂₇	—	2	2	no	—
63	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₁₂₇	C₂×Q₈	1	4	yes	—
65	C₂×C₈×C₁₂₇	C₂×Q₈	1	4	yes	—
66	C₂×C₈×C₁₂₇	#41	1	4	yes	—
67	C₄⋊C₄×C₁₂₇	—	2	8	no	—
68	C₄⋊C₄×C₁₂₇	—	2	4	no	—
69	C₄⋊C₄×C₁₂₇	—	2	8	no	—
70	C₄⋊C₄×C₁₂₇	—	4	8	no	—
71	C₄⋊C₄×C₁₂₇	—	4	4	no	—
72	C₄⋊C₄×C₁₂₇	—	4	4	no	—
73	C₄⋊C₄×C₁₂₇	C₂²×C₄	1	8	yes	—
74	C₄²×C₁₂₇	—	2	4	no	—
75	C₄²×C₁₂₇	—	2	2	no	—
76	C₄²×C₁₂₇	—	4	8	no	—
77	C₄²×C₁₂₇	SD₁₆	1	8	yes	—
78	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
79	C₄²×C₁₂₇	C₂×Q₈	1	4	yes	—
80	C₄²×C₁₂₇	#41	1	2	yes	—
81	C₂³×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
82	Q₈×C₁₂₇	—	4	16	no	—
83	C₂×C₄×C₁₂₇	—	2	2	no	—
84	C₂×C₄×C₁₂₇	—	2	2	no	—
85	C₂×C₄×C₁₂₇	—	2	2	no	—
86	C₂×C₄×C₁₂₇	—	2	8	no	—
87	C₂×C₄×C₁₂₇	—	4	4	no	—
88	C₂×C₄×C₁₂₇	—	4	4	no	—
89	C₂×C₄×C₁₂₇	—	4	4	no	—
90	C₂×C₄×C₁₂₇	—	4	4	no	—
91	C₂×C₄×C₁₂₇	—	4	4	no	—
92	C₂×C₄×C₁₂₇	C₄×D₄	1	4	yes	—
93	C₂×C₄×C₁₂₇	C₄⋊Q₈	1	2	yes	—
94	C₂×C₄×C₁₂₇	C₄⋊Q₈	1	2	yes	—
95	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	4	yes	—
96	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	4	yes	—
97	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	4	yes	—
98	C₂×C₄×C₁₂₇	#135	1	4	yes	—
99	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₂₇	C₂²×Q₈	1	2	yes	—
101	C₂×C₄×C₁₂₇	C₂×#41	1	2	yes	—
102	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
103	C₈×C₁₂₇	—	2	4	no	—
104	C₈×C₁₂₇	C₄×Q₈	1	8	yes	complemented
105	C₂²×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
106	C₂²×C₁₂₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
107	C₂²×C₁₂₇	#532	1	1	yes	characteristic
108	C₂²×C₁₂₇	C₂²×SD₁₆	1	2	yes	—
109	C₂²×C₁₂₇	C₂×#135	1	2	yes	—
110	C₄×C₁₂₇	—	2	2	no	—
111	C₄×C₁₂₇	—	2	4	no	—
112	C₄×C₁₂₇	—	4	8	no	—
113	C₄×C₁₂₇	—	8	4	no	—
114	C₄×C₁₂₇	C₄×SD₁₆	1	8	yes	—
115	C₄×C₁₂₇	#498	1	4	yes	direct product factor
116	C₄×C₁₂₇	C₂×C₄×Q₈	1	2	yes	—
117	C₄×C₁₂₇	#533	1	2	yes	—
118	C₄×#498	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂×C₁₂₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
120	C₂×C₁₂₇	C₂×C₄×SD₁₆	1	2	yes	—
121	C₂×C₁₂₇	C₂×#498	1	1	yes	characteristic
122	C₂×C₁₂₇	#3160	1	1	yes	characteristic
123	C₂×C₁₂₇	#3167	1	2	yes	—
124	C₂×#498	C₂×C₁₂₇	1	1	yes	characteristic
125	C₄×C₄⋊Q₈	C₂×C₁₂₇	1	2	yes	—
126	#1990	C₂×C₁₂₇	1	2	yes	—
127	#1941	C₂×C₁₂₇	1	4	yes	—
128	#1844	C₂×C₁₂₇	1	1	yes	characteristic
129	C₄×#106	C₂×C₁₂₇	1	4	yes	—
130	C₄²×C₈	C₂×C₁₂₇	1	1	yes	characteristic
131	C₁₂₇	C₄×#498	1	1	yes	characteristic, direct product factor, Sylow
132	C₂×C₄⋊Q₈	C₂²×C₁₂₇	1	2	yes	—
133	C₂×C₄×Q₈	—	2	2	no	—
134	#498	C₄×C₁₂₇	1	8	yes	direct product factor
135	C₂×#106	C₂²×C₁₂₇	1	4	yes	—
136	C₂×#106	C₂²×C₁₂₇	1	4	yes	—
137	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
138	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
139	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
140	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
141	#391	—	2	2	no	—
142	#389	—	2	2	no	—
143	#389	C₂²×C₁₂₇	1	4	yes	—
144	#384	C₂²×C₁₂₇	1	2	yes	—
145	C₄×C₄⋊C₄	—	2	2	no	—
146	C₄×C₄⋊C₄	C₂²×C₁₂₇	1	4	yes	—
147	C₄³	C₂²×C₁₂₇	1	1	yes	characteristic
148	#340	C₂²×C₁₂₇	1	8	yes	—
149	C₂²×Q₈	—	2	2	no	—
150	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
151	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
152	C₄⋊Q₈	—	2	4	no	—
153	C₄⋊Q₈	C₂×C₄×C₁₂₇	1	8	yes	—
154	C₄×Q₈	—	4	8	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	—	2	4	no	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄⋊C₄	C₂³×C₁₂₇	1	4	yes	—
160	C₂×C₄⋊C₄	C₂³×C₁₂₇	1	4	yes	—
161	C₂×C₄²	—	2	4	no	—
162	C₂×C₄²	D₄×C₁₂₇	1	2	yes	—
163	C₂×C₄²	D₄×C₁₂₇	1	2	yes	—
164	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
165	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
166	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
167	#106	—	2	8	no	—
168	#106	C₂×C₄×C₁₂₇	1	16	yes	—
169	C₄×C₈	—	2	4	no	—
170	C₄×C₈	—	2	2	no	—
171	C₄×C₈	C₂×C₄×C₁₂₇	1	4	yes	—
172	C₄×C₈	Q₈×C₁₂₇	1	8	yes	complemented
173	#95	—	2	4	no	—
174	#95	—	2	4	no	—
175	C₂×Q₈	—	2	8	no	—
176	C₂×Q₈	—	4	4	no	—
177	C₂×Q₈	—	4	4	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
181	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
182	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
183	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
184	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
188	C₂×C₈	C₂×Q₈×C₁₂₇	1	4	yes	—
189	C₂×C₈	C₂×Q₈×C₁₂₇	1	4	yes	—
190	C₂×C₈	#41×C₁₂₇	1	4	yes	—
191	C₄⋊C₄	—	2	8	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	2	8	no	—
194	C₄⋊C₄	—	4	8	no	—
195	C₄⋊C₄	—	4	4	no	—
196	C₄⋊C₄	—	4	4	no	—
197	C₄⋊C₄	C₂²×C₄×C₁₂₇	1	8	yes	—
198	C₄²	—	2	4	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	4	8	no	—
201	C₄²	SD₁₆×C₁₂₇	1	8	yes	—
202	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
203	C₄²	C₂×Q₈×C₁₂₇	1	4	yes	—
204	C₄²	#41×C₁₂₇	1	2	yes	—
205	C₂³	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
206	Q₈	—	4	16	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	8	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	C₄×D₄×C₁₂₇	1	4	yes	—
217	C₂×C₄	C₄⋊Q₈×C₁₂₇	1	2	yes	—
218	C₂×C₄	C₄⋊Q₈×C₁₂₇	1	2	yes	—
219	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	4	yes	—
220	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	4	yes	—
221	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	4	yes	—
222	C₂×C₄	#135×C₁₂₇	1	4	yes	—
223	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
224	C₂×C₄	C₂²×Q₈×C₁₂₇	1	2	yes	—
225	C₂×C₄	C₂×#41×C₁₂₇	1	2	yes	—
226	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
227	C₈	—	2	4	no	—
228	C₈	C₄×Q₈×C₁₂₇	1	8	yes	complemented
229	C₂²	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
230	C₂²	C₂×C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
231	C₂²	#532×C₁₂₇	1	1	yes	characteristic
232	C₂²	C₂²×SD₁₆×C₁₂₇	1	2	yes	—
233	C₂²	C₂×#135×C₁₂₇	1	2	yes	—
234	C₄	—	2	2	no	—
235	C₄	—	2	4	no	—
236	C₄	—	4	8	no	—
237	C₄	—	8	4	no	—
238	C₄	C₄×SD₁₆×C₁₂₇	1	8	yes	—
239	C₄	#498×C₁₂₇	1	4	yes	direct product factor
240	C₄	C₂×C₄×Q₈×C₁₂₇	1	2	yes	—
241	C₄	#533×C₁₂₇	1	2	yes	—
242	C₂	C₄×C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
243	C₂	C₂×C₄×SD₁₆×C₁₂₇	1	2	yes	—
244	C₂	C₂×#498×C₁₂₇	1	1	yes	characteristic
245	C₂	C₁₂₇×#3160	1	1	yes	characteristic
246	C₂	C₁₂₇×#3167	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₂₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle, prime radical	C₂³×C₁₂₇	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₂₇ ◁ C₄³×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
10. #498×C₁₂₇	C₄	direct (N × H)
48. C₄×C₈×C₁₂₇	Q₈	semidirect
104. C₈×C₁₂₇	C₄×Q₈	semidirect
115. C₄×C₁₂₇	#498	direct (N × H)
118. C₄×#498	C₁₂₇	direct (N × H)
131. C₁₂₇	C₄×#498	direct (N × H)
134. #498	C₄×C₁₂₇	direct (N × H)
172. C₄×C₈	Q₈×C₁₂₇	semidirect
228. C₈	C₄×Q₈×C₁₂₇	semidirect
239. C₄	#498×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×#498×C₁₂₇	2
C₄×C₄⋊Q₈×C₁₂₇	2
C₁₂₇×#1990	2
C₁₂₇×#1941	2
C₁₂₇×#1844	2
C₄×#106×C₁₂₇	2
C₄²×C₈×C₁₂₇	2
C₄×#498	127

Subgroups — C4×#498×C127 #44585392

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#498×C₁₂₇ #44585392