Subgroups — C₁₃₉×#14677 #48009273

Order 35584.

1182 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#382×C₁₃₉	C₂	1	2	yes	maximal
2	C₂×#382×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	#1444×C₁₃₉	C₂	1	4	yes	maximal
4	#1442×C₁₃₉	C₂	1	4	yes	maximal
5	#1441×C₁₃₉	C₂	1	4	yes	maximal
6	C₂²×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
7	C₂²×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
8	C₂²×C₄⋊C₄×C₁₃₉	C₂²	1	4	yes	—
9	C₂²×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
10	C₂²×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
11	#382×C₁₃₉	—	2	8	no	—
12	#382×C₁₃₉	C₄	1	8	yes	complemented
13	C₂×#95×C₁₃₉	C₂²	1	4	yes	—
14	C₂×#95×C₁₃₉	C₂²	1	4	yes	—
15	C₂×#95×C₁₃₉	C₂²	1	4	yes	—
16	C₂×#95×C₁₃₉	C₂²	1	8	yes	—
17	C₂×#95×C₁₃₉	C₂²	1	4	yes	—
18	C₂³×C₄×C₁₃₉	C₂³	1	4	yes	—
19	C₂³×C₄×C₁₃₉	C₂³	1	4	yes	—
20	C₂³×C₄×C₁₃₉	C₂³	1	4	yes	—
21	C₂³×C₄×C₁₃₉	C₂³	1	2	yes	—
22	C₂³×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₁₃₉	—	2	4	no	—
24	C₂×C₄⋊C₄×C₁₃₉	—	2	4	no	—
25	C₂×C₄⋊C₄×C₁₃₉	—	4	2	no	—
26	C₂×C₄⋊C₄×C₁₃₉	—	4	2	no	—
27	C₂×C₄⋊C₄×C₁₃₉	—	4	4	no	—
28	C₂×C₄⋊C₄×C₁₃₉	—	4	4	no	—
29	C₂×C₄⋊C₄×C₁₃₉	—	4	8	no	—
30	C₂×C₄⋊C₄×C₁₃₉	—	4	4	no	—
31	C₂×C₄⋊C₄×C₁₃₉	C₂×C₄	1	8	yes	—
32	C₂×C₄²×C₁₃₉	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₁₃₉	D₄	1	4	yes	—
34	C₂×C₄²×C₁₃₉	Q₈	1	4	yes	—
35	#95×C₁₃₉	—	2	8	no	—
36	#95×C₁₃₉	—	2	8	no	—
37	#95×C₁₃₉	—	4	4	no	—
38	#95×C₁₃₉	—	4	8	no	—
39	#95×C₁₃₉	C₂×C₄	1	16	yes	—
40	C₂⁴×C₁₃₉	C₂⁴	1	1	yes	characteristic
41	C₂²×C₄×C₁₃₉	—	2	1	no	—
42	C₂²×C₄×C₁₃₉	—	2	1	no	—
43	C₂²×C₄×C₁₃₉	—	2	4	no	—
44	C₂²×C₄×C₁₃₉	—	2	1	no	—
45	C₂²×C₄×C₁₃₉	—	2	1	no	—
46	C₂²×C₄×C₁₃₉	—	2	4	no	—
47	C₂²×C₄×C₁₃₉	—	2	4	no	—
48	C₂²×C₄×C₁₃₉	—	2	8	no	—
49	C₂²×C₄×C₁₃₉	—	2	4	no	—
50	C₂²×C₄×C₁₃₉	—	2	8	no	—
51	C₂²×C₄×C₁₃₉	—	2	4	no	—
52	C₂²×C₄×C₁₃₉	—	2	4	no	—
53	C₂²×C₄×C₁₃₉	—	2	4	no	—
54	C₂²×C₄×C₁₃₉	—	2	8	no	—
55	C₂²×C₄×C₁₃₉	—	2	8	no	—
56	C₂²×C₄×C₁₃₉	—	2	8	no	—
57	C₂²×C₄×C₁₃₉	—	2	4	no	—
58	C₂²×C₄×C₁₃₉	—	2	8	no	—
59	C₂²×C₄×C₁₃₉	—	2	4	no	—
60	C₂²×C₄×C₁₃₉	—	2	4	no	—
61	C₂²×C₄×C₁₃₉	C₂²×C₄	1	8	yes	—
62	C₂²×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₃₉	C₂×D₄	1	4	yes	—
66	C₂²×C₄×C₁₃₉	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₁₃₉	C₂×Q₈	1	4	yes	—
68	C₄⋊C₄×C₁₃₉	—	4	8	no	—
69	C₄⋊C₄×C₁₃₉	—	4	8	no	—
70	C₄⋊C₄×C₁₃₉	—	4	8	no	—
71	C₄⋊C₄×C₁₃₉	—	4	16	no	—
72	C₄²×C₁₃₉	C₄⋊C₄	1	16	yes	complemented
73	C₂³×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
74	C₂³×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
75	C₂³×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
76	C₂³×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
77	C₂³×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
78	C₂³×C₁₃₉	H(5,𝔽₂)	1	4	yes	—
79	C₂³×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
80	C₂³×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
81	C₂³×C₁₃₉	2₋¹⁺⁴	1	4	yes	—
82	C₂×C₄×C₁₃₉	—	2	2	no	—
83	C₂×C₄×C₁₃₉	—	2	2	no	—
84	C₂×C₄×C₁₃₉	—	2	8	no	—
85	C₂×C₄×C₁₃₉	—	2	4	no	—
86	C₂×C₄×C₁₃₉	—	2	4	no	—
87	C₂×C₄×C₁₃₉	—	2	2	no	—
88	C₂×C₄×C₁₃₉	—	2	2	no	—
89	C₂×C₄×C₁₃₉	—	2	2	no	—
90	C₂×C₄×C₁₃₉	—	2	2	no	—
91	C₂×C₄×C₁₃₉	—	2	8	no	—
92	C₂×C₄×C₁₃₉	—	2	8	no	—
93	C₂×C₄×C₁₃₉	—	2	16	no	—
94	C₂×C₄×C₁₃₉	—	4	4	no	—
95	C₂×C₄×C₁₃₉	—	4	8	no	—
96	C₂×C₄×C₁₃₉	—	4	4	no	—
97	C₂×C₄×C₁₃₉	—	4	4	no	—
98	C₂×C₄×C₁₃₉	—	4	4	no	—
99	C₂×C₄×C₁₃₉	—	4	4	no	—
100	C₂×C₄×C₁₃₉	—	4	4	no	—
101	C₂×C₄×C₁₃₉	—	4	4	no	—
102	C₂×C₄×C₁₃₉	—	4	4	no	—
103	C₂×C₄×C₁₃₉	—	4	8	no	—
104	C₂×C₄×C₁₃₉	—	4	8	no	—
105	C₂×C₄×C₁₃₉	—	4	8	no	—
106	C₂×C₄×C₁₃₉	—	4	8	no	—
107	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	4	yes	—
108	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	8	yes	—
109	C₂²×C₁₃₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
110	C₂²×C₁₃₉	#518	1	1	yes	characteristic
111	C₂²×C₁₃₉	#519	1	1	yes	characteristic
112	C₂²×C₁₃₉	#520	1	4	yes	—
113	C₂²×C₁₃₉	#535	1	2	yes	—
114	C₂²×C₁₃₉	#536	1	2	yes	—
115	C₂²×C₁₃₉	#537	1	1	yes	characteristic
116	C₂²×C₁₃₉	#537	1	1	yes	characteristic
117	C₂²×C₁₃₉	#544	1	1	yes	characteristic
118	C₂²×C₁₃₉	#544	1	1	yes	characteristic
119	C₂²×C₁₃₉	#544	1	4	yes	—
120	C₂²×C₁₃₉	H(3,𝔽₄)	1	2	yes	—
121	C₂²×C₁₃₉	#562	1	2	yes	—
122	C₂²×C₁₃₉	#562	1	4	yes	—
123	C₂²×C₁₃₉	#563	1	4	yes	—
124	C₂²×C₁₃₉	#563	1	2	yes	—
125	C₂²×C₁₃₉	#564	1	2	yes	—
126	C₄×C₁₃₉	—	2	4	no	—
127	C₄×C₁₃₉	—	2	8	no	—
128	C₄×C₁₃₉	—	4	8	no	—
129	C₄×C₁₃₉	—	8	4	no	—
130	C₄×C₁₃₉	—	8	4	no	—
131	C₂×C₁₃₉	#2316	1	1	yes	characteristic
132	C₂×C₁₃₉	#2317	1	2	yes	—
133	C₂×C₁₃₉	#2379	1	2	yes	—
134	C₂×C₁₃₉	#2382	1	2	yes	—
135	C₂×C₁₃₉	#2819	1	1	yes	characteristic
136	C₂×C₁₃₉	#2820	1	2	yes	—
137	C₂×C₁₃₉	#2822	1	1	yes	characteristic
138	C₂×C₁₃₉	#2844	1	2	yes	—
139	C₂×C₁₃₉	#2845	1	2	yes	—
140	#14677	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₁₃₉	#14677	1	1	yes	characteristic, direct product factor, Sylow
142	C₂×#382	C₂×C₁₃₉	1	2	yes	—
143	C₂×#382	C₂×C₁₃₉	1	1	yes	characteristic
144	#1444	C₂×C₁₃₉	1	4	yes	—
145	#1442	C₂×C₁₃₉	1	4	yes	—
146	#1441	C₂×C₁₃₉	1	4	yes	—
147	C₂²×C₄⋊C₄	C₂²×C₁₃₉	1	2	yes	—
148	C₂²×C₄⋊C₄	C₂²×C₁₃₉	1	2	yes	—
149	C₂²×C₄⋊C₄	C₂²×C₁₃₉	1	4	yes	—
150	C₂²×C₄⋊C₄	C₂²×C₁₃₉	1	2	yes	—
151	C₂²×C₄²	C₂²×C₁₃₉	1	1	yes	characteristic
152	#382	—	2	8	no	—
153	#382	C₄×C₁₃₉	1	8	yes	complemented
154	C₂×#95	C₂²×C₁₃₉	1	4	yes	—
155	C₂×#95	C₂²×C₁₃₉	1	4	yes	—
156	C₂×#95	C₂²×C₁₃₉	1	4	yes	—
157	C₂×#95	C₂²×C₁₃₉	1	8	yes	—
158	C₂×#95	C₂²×C₁₃₉	1	4	yes	—
159	C₂³×C₄	C₂³×C₁₃₉	1	4	yes	—
160	C₂³×C₄	C₂³×C₁₃₉	1	4	yes	—
161	C₂³×C₄	C₂³×C₁₃₉	1	4	yes	—
162	C₂³×C₄	C₂³×C₁₃₉	1	2	yes	—
163	C₂³×C₄	C₂³×C₁₃₉	1	1	yes	characteristic
164	C₂×C₄⋊C₄	—	2	4	no	—
165	C₂×C₄⋊C₄	—	2	4	no	—
166	C₂×C₄⋊C₄	—	4	2	no	—
167	C₂×C₄⋊C₄	—	4	2	no	—
168	C₂×C₄⋊C₄	—	4	4	no	—
169	C₂×C₄⋊C₄	—	4	4	no	—
170	C₂×C₄⋊C₄	—	4	8	no	—
171	C₂×C₄⋊C₄	—	4	4	no	—
172	C₂×C₄⋊C₄	C₂×C₄×C₁₃₉	1	8	yes	—
173	C₂×C₄²	C₂×C₄×C₁₃₉	1	4	yes	—
174	C₂×C₄²	D₄×C₁₃₉	1	4	yes	—
175	C₂×C₄²	Q₈×C₁₃₉	1	4	yes	—
176	#95	—	2	8	no	—
177	#95	—	2	8	no	—
178	#95	—	4	4	no	—
179	#95	—	4	8	no	—
180	#95	C₂×C₄×C₁₃₉	1	16	yes	—
181	C₂⁴	C₂⁴×C₁₃₉	1	1	yes	characteristic
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	8	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	2	8	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	8	no	—
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	—	2	8	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	8	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	C₂²×C₄×C₁₃₉	1	8	yes	—
203	C₂²×C₄	C₂²×C₄×C₁₃₉	1	4	yes	—
204	C₂²×C₄	C₂²×C₄×C₁₃₉	1	2	yes	—
205	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
206	C₂²×C₄	C₂×D₄×C₁₃₉	1	4	yes	—
207	C₂²×C₄	C₂×Q₈×C₁₃₉	1	2	yes	—
208	C₂²×C₄	C₂×Q₈×C₁₃₉	1	4	yes	—
209	C₄⋊C₄	—	4	8	no	—
210	C₄⋊C₄	—	4	8	no	—
211	C₄⋊C₄	—	4	8	no	—
212	C₄⋊C₄	—	4	16	no	—
213	C₄²	C₄⋊C₄×C₁₃₉	1	16	yes	complemented
214	C₂³	C₂³×C₄×C₁₃₉	1	1	yes	characteristic
215	C₂³	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
216	C₂³	C₂²×Q₈×C₁₃₉	1	1	yes	characteristic
217	C₂³	H(5,𝔽₂)×C₁₃₉	1	1	yes	characteristic
218	C₂³	H(5,𝔽₂)×C₁₃₉	1	1	yes	characteristic
219	C₂³	H(5,𝔽₂)×C₁₃₉	1	4	yes	—
220	C₂³	2₋¹⁺⁴×C₁₃₉	1	1	yes	characteristic
221	C₂³	2₋¹⁺⁴×C₁₃₉	1	1	yes	characteristic
222	C₂³	2₋¹⁺⁴×C₁₃₉	1	4	yes	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	2	16	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	8	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	8	no	—
245	C₂×C₄	—	4	8	no	—
246	C₂×C₄	—	4	8	no	—
247	C₂×C₄	—	4	8	no	—
248	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	4	yes	—
249	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	8	yes	—
250	C₂²	C₂²×C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
251	C₂²	#518×C₁₃₉	1	1	yes	characteristic
252	C₂²	#519×C₁₃₉	1	1	yes	characteristic
253	C₂²	#520×C₁₃₉	1	4	yes	—
254	C₂²	#535×C₁₃₉	1	2	yes	—
255	C₂²	#536×C₁₃₉	1	2	yes	—
256	C₂²	#537×C₁₃₉	1	1	yes	characteristic
257	C₂²	#537×C₁₃₉	1	1	yes	characteristic
258	C₂²	#544×C₁₃₉	1	1	yes	characteristic
259	C₂²	#544×C₁₃₉	1	1	yes	characteristic
260	C₂²	#544×C₁₃₉	1	4	yes	—
261	C₂²	H(3,𝔽₄)×C₁₃₉	1	2	yes	—
262	C₂²	#562×C₁₃₉	1	2	yes	—
263	C₂²	#562×C₁₃₉	1	4	yes	—
264	C₂²	#563×C₁₃₉	1	4	yes	—
265	C₂²	#563×C₁₃₉	1	2	yes	—
266	C₂²	#564×C₁₃₉	1	2	yes	—
267	C₄	—	2	4	no	—
268	C₄	—	2	8	no	—
269	C₄	—	4	8	no	—
270	C₄	—	8	4	no	—
271	C₄	—	8	4	no	—
272	C₂	#2316×C₁₃₉	1	1	yes	characteristic
273	C₂	#2317×C₁₃₉	1	2	yes	—
274	C₂	#2379×C₁₃₉	1	2	yes	—
275	C₂	#2382×C₁₃₉	1	2	yes	—
276	C₂	#2819×C₁₃₉	1	1	yes	characteristic
277	C₂	#2820×C₁₃₉	1	2	yes	—
278	C₂	#2822×C₁₃₉	1	1	yes	characteristic
279	C₂	#2844×C₁₃₉	1	2	yes	—
280	C₂	#2845×C₁₃₉	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₃₉	C₂⁴
commutator	C₂³	C₂³×C₄×C₁₃₉
Frattini, cosocle	C₂⁴	C₂⁴×C₁₃₉
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
12. #382×C₁₃₉	C₄	semidirect
72. C₄²×C₁₃₉	C₄⋊C₄	semidirect
140. #14677	C₁₃₉	direct (N × H)
141. C₁₃₉	#14677	direct (N × H)
153. #382	C₄×C₁₃₉	semidirect
213. C₄²	C₄⋊C₄×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#382×C₁₃₉	2
#1444×C₁₃₉	2
#1442×C₁₃₉	2
#1441×C₁₃₉	2
#14677	139

Subgroups — C139×#14677 #48009273

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#14677 #48009273