Subgroups — C₁₃₉×#21957 #48016553

Order 35584.

790 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#433×C₁₃₉	C₂	1	2	yes	maximal
2	#2296×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	#1940×C₁₃₉	C₂	1	4	yes	maximal, complemented
4	#1939×C₁₃₉	C₂	1	4	yes	maximal, complemented
5	#1798×C₁₃₉	C₂	1	2	yes	maximal
6	#1797×C₁₃₉	C₂	1	2	yes	maximal
7	C₂×#117×C₁₃₉	C₂²	1	1	yes	characteristic
8	C₂×#117×C₁₃₉	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
10	#433×C₁₃₉	—	2	8	no	—
11	C₂×C₄⋊C₈×C₁₃₉	C₂²	1	4	yes	—
12	C₂×C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₁₃₉	C₂²	1	4	yes	—
15	C₂×C₂²⋊C₈×C₁₃₉	C₂²	1	4	yes	—
16	C₂×#97×C₁₃₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₃₉	C₂²	1	2	yes	—
18	#389×C₁₃₉	C₄	1	2	yes	—
19	#389×C₁₃₉	C₄	1	2	yes	—
20	#385×C₁₃₉	C₄	1	4	yes	—
21	C₄×C₄⋊C₄×C₁₃₉	C₂²	1	4	yes	—
22	#336×C₁₃₉	C₂²	1	4	yes	—
23	#336×C₁₃₉	C₂²	1	2	yes	—
24	#336×C₁₃₉	C₂²	1	2	yes	—
25	C₂³×C₄×C₁₃₉	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₁₃₉	C₂³	1	4	yes	—
28	C₂²×C₈×C₁₃₉	C₂³	1	4	yes	—
29	#117×C₁₃₉	—	4	2	no	—
30	#117×C₁₃₉	D₄	1	4	yes	—
31	#117×C₁₃₉	Q₈	1	4	yes	—
32	C₂×C₄⋊C₄×C₁₃₉	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₁₃₉	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₁₃₉	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₁₃₉	C₂×C₄	1	2	yes	—
36	C₂×C₂²⋊C₄×C₁₃₉	C₂×C₄	1	2	yes	—
37	C₂×C₂²⋊C₄×C₁₃₉	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₁₃₉	—	4	2	no	—
39	C₂×C₄²×C₁₃₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₃₉	C₂³	1	2	yes	—
42	C₂×C₄²×C₁₃₉	C₂³	1	2	yes	—
43	C₂×C₄²×C₁₃₉	C₂³	1	2	yes	—
44	C₄⋊C₈×C₁₃₉	—	2	8	no	—
45	C₄⋊C₈×C₁₃₉	—	4	2	no	—
46	C₄⋊C₈×C₁₃₉	—	4	2	no	—
47	C₄⋊C₈×C₁₃₉	—	4	4	no	—
48	C₂²⋊C₈×C₁₃₉	—	2	8	no	—
49	#97×C₁₃₉	—	2	4	no	—
50	C₄×C₈×C₁₃₉	—	2	4	no	—
51	#95×C₁₃₉	C₂×C₄	1	4	yes	—
52	#95×C₁₃₉	C₂×C₄	1	4	yes	—
53	C₂⁴×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃₉	—	2	2	no	—
55	C₂²×C₄×C₁₃₉	—	2	2	no	—
56	C₂²×C₄×C₁₃₉	—	2	4	no	—
57	C₂²×C₄×C₁₃₉	—	4	2	no	—
58	C₂²×C₄×C₁₃₉	—	4	2	no	—
59	C₂²×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₃₉	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
67	C₂×C₈×C₁₃₉	—	2	4	no	—
68	C₂×C₈×C₁₃₉	—	2	4	no	—
69	C₂×C₈×C₁₃₉	—	2	4	no	—
70	C₂×C₈×C₁₃₉	—	2	4	no	—
71	C₂×C₈×C₁₃₉	—	2	4	no	—
72	C₂×C₈×C₁₃₉	—	2	4	no	—
73	C₄⋊C₄×C₁₃₉	—	2	8	no	—
74	C₄⋊C₄×C₁₃₉	—	4	4	no	—
75	C₄⋊C₄×C₁₃₉	—	4	2	no	—
76	C₄⋊C₄×C₁₃₉	C₄⋊C₄	1	8	yes	—
77	C₂²⋊C₄×C₁₃₉	—	4	2	no	—
78	C₂²⋊C₄×C₁₃₉	C₄⋊C₄	1	8	yes	—
79	C₄²×C₁₃₉	—	2	2	no	—
80	C₄²×C₁₃₉	—	2	2	no	—
81	C₄²×C₁₃₉	—	2	4	no	—
82	C₄²×C₁₃₉	—	4	1	no	—
83	C₄²×C₁₃₉	—	4	1	no	—
84	C₄²×C₁₃₉	C₂×D₄	1	4	yes	—
85	C₄²×C₁₃₉	C₂×Q₈	1	4	yes	—
86	C₂³×C₁₃₉	—	2	1	no	—
87	C₂³×C₁₃₉	—	2	1	no	—
88	C₂³×C₁₃₉	—	2	2	no	—
89	C₂³×C₁₃₉	—	2	2	no	—
90	C₂³×C₁₃₉	C₂×C₄⋊C₄	1	2	yes	—
91	C₂³×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃₉	—	2	2	no	—
93	C₂×C₄×C₁₃₉	—	2	1	no	—
94	C₂×C₄×C₁₃₉	—	2	1	no	—
95	C₂×C₄×C₁₃₉	—	2	4	no	—
96	C₂×C₄×C₁₃₉	—	2	4	no	—
97	C₂×C₄×C₁₃₉	—	2	2	no	—
98	C₂×C₄×C₁₃₉	—	2	4	no	—
99	C₂×C₄×C₁₃₉	—	2	4	no	—
100	C₂×C₄×C₁₃₉	—	2	2	no	—
101	C₂×C₄×C₁₃₉	—	2	4	no	—
102	C₂×C₄×C₁₃₉	—	2	4	no	—
103	C₂×C₄×C₁₃₉	—	2	2	no	—
104	C₂×C₄×C₁₃₉	—	2	2	no	—
105	C₂×C₄×C₁₃₉	—	2	4	no	—
106	C₂×C₄×C₁₃₉	—	4	8	no	—
107	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	8	yes	—
108	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₁₃₉	#131	1	4	yes	—
110	C₂×C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₁₃₉	H(5,𝔽₂)	1	2	yes	—
113	C₂×C₄×C₁₃₉	2₋¹⁺⁴	1	2	yes	—
114	C₈×C₁₃₉	—	4	4	no	—
115	C₈×C₁₃₉	—	4	4	no	—
116	C₂²×C₁₃₉	—	2	2	no	—
117	C₂²×C₁₃₉	—	2	4	no	—
118	C₂²×C₁₃₉	—	4	2	no	—
119	C₂²×C₁₃₉	—	4	1	no	—
120	C₂²×C₁₃₉	—	4	1	no	—
121	C₂²×C₁₃₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
122	C₂²×C₁₃₉	#518	1	1	yes	characteristic
123	C₂²×C₁₃₉	#519	1	1	yes	characteristic
124	C₂²×C₁₃₉	C₂×#131	1	2	yes	—
125	C₂²×C₁₃₉	#568	1	2	yes	—
126	C₄×C₁₃₉	—	2	2	no	—
127	C₄×C₁₃₉	—	2	8	no	—
128	C₄×C₁₃₉	—	2	4	no	—
129	C₄×C₁₃₉	—	4	4	no	—
130	C₄×C₁₃₉	—	4	2	no	—
131	C₄×C₁₃₉	#537	1	2	yes	—
132	C₄×C₁₃₉	#544	1	2	yes	—
133	C₂×C₁₃₉	—	4	2	no	—
134	C₂×C₁₃₉	#2316	1	1	yes	characteristic
135	C₂×C₁₃₉	#2906	1	2	yes	—
136	C₂×C₁₃₉	#2976	1	2	yes	—
137	C₂×C₁₃₉	#2986	1	1	yes	characteristic
138	C₂×C₁₃₉	#2987	1	1	yes	characteristic
139	#21957	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₁₃₉	#21957	1	1	yes	characteristic, direct product factor, Sylow
141	C₂×#433	C₂×C₁₃₉	1	2	yes	—
142	#2296	C₂×C₁₃₉	1	1	yes	characteristic
143	#1940	C₂×C₁₃₉	1	4	yes	complemented
144	#1939	C₂×C₁₃₉	1	4	yes	complemented
145	#1798	C₂×C₁₃₉	1	2	yes	—
146	#1797	C₂×C₁₃₉	1	2	yes	—
147	C₂×#117	C₂²×C₁₃₉	1	1	yes	characteristic
148	C₂×#117	C₂²×C₁₃₉	1	1	yes	characteristic
149	C₂²×C₄²	C₂²×C₁₃₉	1	1	yes	characteristic
150	#433	—	2	8	no	—
151	C₂×C₄⋊C₈	C₂²×C₁₃₉	1	4	yes	—
152	C₂×C₄⋊C₈	C₂²×C₁₃₉	1	2	yes	—
153	C₂×C₄⋊C₈	C₂²×C₁₃₉	1	2	yes	—
154	C₂×C₄⋊C₈	C₂²×C₁₃₉	1	4	yes	—
155	C₂×C₂²⋊C₈	C₂²×C₁₃₉	1	4	yes	—
156	C₂×#97	C₂²×C₁₃₉	1	2	yes	—
157	C₂×C₄×C₈	C₂²×C₁₃₉	1	2	yes	—
158	#389	C₄×C₁₃₉	1	2	yes	—
159	#389	C₄×C₁₃₉	1	2	yes	—
160	#385	C₄×C₁₃₉	1	4	yes	—
161	C₄×C₄⋊C₄	C₂²×C₁₃₉	1	4	yes	—
162	#336	C₂²×C₁₃₉	1	4	yes	—
163	#336	C₂²×C₁₃₉	1	2	yes	—
164	#336	C₂²×C₁₃₉	1	2	yes	—
165	C₂³×C₄	C₂×C₄×C₁₃₉	1	2	yes	—
166	C₂³×C₄	C₂³×C₁₃₉	1	1	yes	characteristic
167	C₂²×C₈	C₂³×C₁₃₉	1	4	yes	—
168	C₂²×C₈	C₂³×C₁₃₉	1	4	yes	—
169	#117	—	4	2	no	—
170	#117	D₄×C₁₃₉	1	4	yes	—
171	#117	Q₈×C₁₃₉	1	4	yes	—
172	C₂×C₄⋊C₄	C₂×C₄×C₁₃₉	1	4	yes	—
173	C₂×C₄⋊C₄	C₂×C₄×C₁₃₉	1	2	yes	—
174	C₂×C₄⋊C₄	C₂×C₄×C₁₃₉	1	4	yes	—
175	C₂×C₄⋊C₄	C₂×C₄×C₁₃₉	1	2	yes	—
176	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₉	1	2	yes	—
177	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₉	1	2	yes	—
178	C₂×C₄²	—	4	2	no	—
179	C₂×C₄²	C₂×C₄×C₁₃₉	1	1	yes	characteristic
180	C₂×C₄²	C₂×C₄×C₁₃₉	1	1	yes	characteristic
181	C₂×C₄²	C₂³×C₁₃₉	1	2	yes	—
182	C₂×C₄²	C₂³×C₁₃₉	1	2	yes	—
183	C₂×C₄²	C₂³×C₁₃₉	1	2	yes	—
184	C₄⋊C₈	—	2	8	no	—
185	C₄⋊C₈	—	4	2	no	—
186	C₄⋊C₈	—	4	2	no	—
187	C₄⋊C₈	—	4	4	no	—
188	C₂²⋊C₈	—	2	8	no	—
189	#97	—	2	4	no	—
190	C₄×C₈	—	2	4	no	—
191	#95	C₂×C₄×C₁₃₉	1	4	yes	—
192	#95	C₂×C₄×C₁₃₉	1	4	yes	—
193	C₂⁴	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	C₂²×C₄×C₁₃₉	1	2	yes	—
200	C₂²×C₄	C₂²×C₄×C₁₃₉	1	4	yes	—
201	C₂²×C₄	C₂²×C₄×C₁₃₉	1	4	yes	—
202	C₂²×C₄	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
203	C₂²×C₄	C₂²×C₄×C₁₃₉	1	2	yes	—
204	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
205	C₂²×C₄	C₂×Q₈×C₁₃₉	1	2	yes	—
206	C₂²×C₄	C₂⁴×C₁₃₉	1	1	yes	characteristic
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	4	no	—
213	C₄⋊C₄	—	2	8	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	C₄⋊C₄×C₁₃₉	1	8	yes	—
217	C₂²⋊C₄	—	4	2	no	—
218	C₂²⋊C₄	C₄⋊C₄×C₁₃₉	1	8	yes	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	4	no	—
222	C₄²	—	4	1	no	—
223	C₄²	—	4	1	no	—
224	C₄²	C₂×D₄×C₁₃₉	1	4	yes	—
225	C₄²	C₂×Q₈×C₁₃₉	1	4	yes	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	C₂×C₄⋊C₄×C₁₃₉	1	2	yes	—
231	C₂³	C₂³×C₄×C₁₃₉	1	1	yes	characteristic
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	4	8	no	—
247	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	8	yes	—
248	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	2	yes	—
249	C₂×C₄	#131×C₁₃₉	1	4	yes	—
250	C₂×C₄	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
251	C₂×C₄	C₂²×Q₈×C₁₃₉	1	1	yes	characteristic
252	C₂×C₄	H(5,𝔽₂)×C₁₃₉	1	2	yes	—
253	C₂×C₄	2₋¹⁺⁴×C₁₃₉	1	2	yes	—
254	C₈	—	4	4	no	—
255	C₈	—	4	4	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	C₂²×C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
262	C₂²	#518×C₁₃₉	1	1	yes	characteristic
263	C₂²	#519×C₁₃₉	1	1	yes	characteristic
264	C₂²	C₂×#131×C₁₃₉	1	2	yes	—
265	C₂²	#568×C₁₃₉	1	2	yes	—
266	C₄	—	2	2	no	—
267	C₄	—	2	8	no	—
268	C₄	—	2	4	no	—
269	C₄	—	4	2	no	—
270	C₄	—	4	4	no	—
271	C₄	#537×C₁₃₉	1	2	yes	—
272	C₄	#544×C₁₃₉	1	2	yes	—
273	C₂	—	4	2	no	—
274	C₂	#2316×C₁₃₉	1	1	yes	characteristic
275	C₂	#2906×C₁₃₉	1	2	yes	—
276	C₂	#2976×C₁₃₉	1	2	yes	—
277	C₂	#2986×C₁₃₉	1	1	yes	characteristic
278	C₂	#2987×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₃₉	C₂⁴
commutator	C₂³	C₂³×C₄×C₁₃₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₉
socle	C₂³×C₁₃₉	C₂³×C₄
prime radical	C₂⁴×C₁₃₉	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. #1940×C₁₃₉	C₂	semidirect
4. #1939×C₁₃₉	C₂	semidirect
139. #21957	C₁₃₉	direct (N × H)
140. C₁₃₉	#21957	direct (N × H)
143. #1940	C₂×C₁₃₉	semidirect
144. #1939	C₂×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#433×C₁₃₉	2
#2296×C₁₃₉	2
#1940×C₁₃₉	2
#1939×C₁₃₉	2
#1798×C₁₃₉	2
#1797×C₁₃₉	2
#21957	139

Subgroups — C139×#21957 #48016553

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#21957 #48016553