Subgroups — C₁₃₉×#33334 #48027930

Order 35584.

1172 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	C₂²×#131×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#504×C₁₃₉	C₂	1	2	yes	maximal, complemented
3	#2113×C₁₃₉	C₂	1	6	yes	maximal, complemented
4	#2112×C₁₃₉	C₂	1	3	yes	maximal, complemented
5	C₂×C₂²⋊C₁₆×C₁₃₉	C₂	1	3	yes	maximal, complemented
6	C₂²×#41×C₁₃₉	C₄	1	1	yes	characteristic
7	C₂×#131×C₁₃₉	—	2	8	no	—
8	C₂×#131×C₁₃₉	C₄	1	1	yes	characteristic
9	C₂×#131×C₁₃₉	C₄	1	3	yes	—
10	C₂×#131×C₁₃₉	C₂²	1	3	yes	—
11	C₂×#131×C₁₃₉	C₂²	1	1	yes	characteristic
12	C₂²×M₁₆×C₁₃₉	C₄	1	3	yes	—
13	C₂³×C₈×C₁₃₉	C₂²	1	3	yes	—
14	#504×C₁₃₉	—	2	8	no	—
15	C₂×M₃₂×C₁₃₉	C₂²	1	6	yes	complemented
16	C₂²×C₁₆×C₁₃₉	C₂²	1	6	yes	complemented
17	C₂²⋊C₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
18	C₂²⋊C₁₆×C₁₃₉	C₂²	1	3	yes	—
19	C₂²⋊C₁₆×C₁₃₉	C₂²	1	6	yes	complemented
20	C₂²⋊C₁₆×C₁₃₉	C₂²	1	6	yes	complemented
21	C₂×#41×C₁₃₉	—	2	4	no	—
22	C₂×#41×C₁₃₉	—	2	4	no	—
23	C₂×#41×C₁₃₉	C₈	1	1	yes	characteristic
24	C₂×#41×C₁₃₉	C₈	1	3	yes	—
25	C₂×#41×C₁₃₉	C₂×C₄	1	3	yes	—
26	C₂×#41×C₁₃₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×Q₈×C₁₃₉	C₈	1	1	yes	characteristic
28	C₂²×D₄×C₁₃₉	C₈	1	3	yes	—
29	C₂³×C₄×C₁₃₉	C₂×C₄	1	3	yes	—
30	#131×C₁₃₉	—	2	4	no	—
31	#131×C₁₃₉	—	2	12	no	—
32	#131×C₁₃₉	—	2	12	no	—
33	#131×C₁₃₉	D₄	1	8	yes	—
34	C₂×M₁₆×C₁₃₉	—	2	12	no	—
35	C₂×M₁₆×C₁₃₉	C₂×C₄	1	3	yes	—
36	C₂×M₁₆×C₁₃₉	C₂×C₄	1	6	yes	—
37	C₂×M₁₆×C₁₃₉	C₂×C₄	1	3	yes	—
38	C₂²×C₈×C₁₃₉	—	2	12	no	—
39	C₂²×C₈×C₁₃₉	C₂×C₄	1	3	yes	—
40	C₂²×C₈×C₁₃₉	C₂×C₄	1	3	yes	—
41	C₂²×C₈×C₁₃₉	C₂×C₄	1	3	yes	—
42	C₂²×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₁₃₉	C₂³	1	3	yes	—
44	C₂²×C₈×C₁₃₉	C₂³	1	3	yes	—
45	M₃₂×C₁₃₉	—	2	12	no	—
46	C₂×C₁₆×C₁₃₉	—	2	12	no	—
47	C₂×C₁₆×C₁₃₉	C₂³	1	2	yes	—
48	C₂×C₁₆×C₁₃₉	C₂³	1	6	yes	—
49	C₂⁴×C₁₃₉	C₂×C₈	1	3	yes	—
50	#41×C₁₃₉	—	2	4	no	—
51	#41×C₁₃₉	—	2	12	no	—
52	#41×C₁₃₉	—	2	6	no	—
53	#41×C₁₃₉	—	2	6	no	—
54	#41×C₁₃₉	C₂²⋊C₄	1	4	yes	—
55	#41×C₁₃₉	M₁₆	1	4	yes	—
56	C₂×Q₈×C₁₃₉	—	2	4	no	—
57	C₂×Q₈×C₁₃₉	C₂×C₈	1	3	yes	—
58	C₂×Q₈×C₁₃₉	C₂×C₈	1	1	yes	characteristic
59	C₂×D₄×C₁₃₉	—	2	12	no	—
60	C₂×D₄×C₁₃₉	C₂×C₈	1	6	yes	—
61	C₂×D₄×C₁₃₉	C₂×C₈	1	3	yes	—
62	C₂×D₄×C₁₃₉	C₂×C₈	1	3	yes	—
63	C₂²×C₄×C₁₃₉	—	2	6	no	—
64	C₂²×C₄×C₁₃₉	—	2	6	no	—
65	C₂²×C₄×C₁₃₉	C₂×C₈	1	3	yes	—
66	C₂²×C₄×C₁₃₉	C₂×C₈	1	3	yes	—
67	C₂²×C₄×C₁₃₉	C₂×C₈	1	3	yes	—
68	C₂²×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃₉	C₂²×C₄	1	3	yes	—
70	C₂²×C₄×C₁₃₉	C₂²×C₄	1	3	yes	—
71	M₁₆×C₁₃₉	—	2	6	no	—
72	M₁₆×C₁₃₉	—	2	12	no	—
73	M₁₆×C₁₃₉	C₂²⋊C₄	1	12	yes	—
74	C₂×C₈×C₁₃₉	—	2	6	no	—
75	C₂×C₈×C₁₃₉	—	2	6	no	—
76	C₂×C₈×C₁₃₉	—	2	6	no	—
77	C₂×C₈×C₁₃₉	—	2	2	no	—
78	C₂×C₈×C₁₃₉	—	2	6	no	—
79	C₂×C₈×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃₉	C₂²×C₄	1	3	yes	—
81	C₂×C₈×C₁₃₉	C₂²×C₄	1	3	yes	—
82	C₂×C₈×C₁₃₉	C₂×D₄	1	12	yes	—
83	C₂×C₈×C₁₃₉	C₂⁴	1	1	yes	characteristic
84	C₁₆×C₁₃₉	—	2	2	no	—
85	C₁₆×C₁₃₉	—	2	6	no	—
86	C₂³×C₁₃₉	—	2	6	no	—
87	C₂³×C₁₃₉	—	2	6	no	—
88	C₂³×C₁₃₉	—	4	3	no	—
89	C₂³×C₁₃₉	C₂²×C₈	1	3	yes	—
90	C₂³×C₁₃₉	C₂²×C₈	1	3	yes	—
91	C₂³×C₁₃₉	C₂²×C₈	1	1	yes	characteristic
92	Q₈×C₁₃₉	—	2	6	no	—
93	Q₈×C₁₃₉	C₂²⋊C₈	1	4	yes	—
94	D₄×C₁₃₉	—	2	12	no	—
95	D₄×C₁₃₉	—	2	6	no	—
96	D₄×C₁₃₉	C₂²⋊C₈	1	12	yes	—
97	C₂×C₄×C₁₃₉	—	2	6	no	—
98	C₂×C₄×C₁₃₉	—	2	1	no	—
99	C₂×C₄×C₁₃₉	—	2	3	no	—
100	C₂×C₄×C₁₃₉	—	2	1	no	—
101	C₂×C₄×C₁₃₉	—	2	3	no	—
102	C₂×C₄×C₁₃₉	—	2	6	no	—
103	C₂×C₄×C₁₃₉	—	2	6	no	—
104	C₂×C₄×C₁₃₉	C₂×C₂²⋊C₄	1	6	yes	—
105	C₂×C₄×C₁₃₉	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₁₃₉	C₂²×C₈	1	3	yes	—
107	C₂×C₄×C₁₃₉	C₂²×C₈	1	3	yes	—
108	C₂×C₄×C₁₃₉	C₂×M₁₆	1	6	yes	—
109	C₂×C₄×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
110	C₈×C₁₃₉	—	2	1	no	—
111	C₈×C₁₃₉	—	2	3	no	—
112	C₈×C₁₃₉	C₂×C₂²⋊C₄	1	6	yes	—
113	C₈×C₁₃₉	C₂²×D₄	1	2	yes	—
114	C₂²×C₁₃₉	—	2	1	no	—
115	C₂²×C₁₃₉	—	2	3	no	—
116	C₂²×C₁₃₉	—	2	1	no	—
117	C₂²×C₁₃₉	—	2	3	no	—
118	C₂²×C₁₃₉	—	2	3	no	—
119	C₂²×C₁₃₉	—	2	3	no	—
120	C₂²×C₁₃₉	—	2	3	no	—
121	C₂²×C₁₃₉	—	4	12	no	—
122	C₂²×C₁₃₉	C₂×C₂²⋊C₈	1	6	yes	—
123	C₂²×C₁₃₉	#504	1	2	yes	complemented
124	C₂²×C₁₃₉	C₂³×C₈	1	1	yes	characteristic
125	C₄×C₁₃₉	—	2	1	no	—
126	C₄×C₁₃₉	—	2	3	no	—
127	C₄×C₁₃₉	C₂×C₂²⋊C₈	1	6	yes	—
128	C₄×C₁₃₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
129	C₄×C₁₃₉	C₂²×M₁₆	1	1	yes	characteristic
130	C₂×C₁₃₉	—	2	2	no	—
131	C₂×C₁₃₉	—	2	6	no	—
132	C₂×C₁₃₉	—	4	3	no	—
133	C₂×C₁₃₉	C₂²×C₂²⋊C₈	1	1	yes	characteristic
134	C₂×C₁₃₉	C₂×#504	1	1	yes	characteristic
135	C₂×C₁₃₉	#3407	1	1	yes	characteristic
136	#33334	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₁₃₉	#33334	1	1	yes	characteristic, direct product factor, Sylow
138	C₂²×#131	C₂×C₁₃₉	1	1	yes	characteristic
139	C₂×#504	C₂×C₁₃₉	1	2	yes	complemented
140	#2113	C₂×C₁₃₉	1	6	yes	complemented
141	#2112	C₂×C₁₃₉	1	3	yes	complemented
142	C₂×C₂²⋊C₁₆	C₂×C₁₃₉	1	3	yes	complemented
143	C₂²×#41	C₄×C₁₃₉	1	1	yes	characteristic
144	C₂×#131	—	2	8	no	—
145	C₂×#131	C₄×C₁₃₉	1	1	yes	characteristic
146	C₂×#131	C₄×C₁₃₉	1	3	yes	—
147	C₂×#131	C₂²×C₁₃₉	1	3	yes	—
148	C₂×#131	C₂²×C₁₃₉	1	1	yes	characteristic
149	C₂²×M₁₆	C₄×C₁₃₉	1	3	yes	—
150	C₂³×C₈	C₂²×C₁₃₉	1	3	yes	—
151	#504	—	2	8	no	—
152	C₂×M₃₂	C₂²×C₁₃₉	1	6	yes	complemented
153	C₂²×C₁₆	C₂²×C₁₃₉	1	6	yes	complemented
154	C₂²⋊C₁₆	C₂²×C₁₃₉	1	1	yes	characteristic
155	C₂²⋊C₁₆	C₂²×C₁₃₉	1	3	yes	—
156	C₂²⋊C₁₆	C₂²×C₁₃₉	1	6	yes	complemented
157	C₂²⋊C₁₆	C₂²×C₁₃₉	1	6	yes	complemented
158	C₂×#41	—	2	4	no	—
159	C₂×#41	—	2	4	no	—
160	C₂×#41	C₈×C₁₃₉	1	1	yes	characteristic
161	C₂×#41	C₈×C₁₃₉	1	3	yes	—
162	C₂×#41	C₂×C₄×C₁₃₉	1	3	yes	—
163	C₂×#41	C₂×C₄×C₁₃₉	1	1	yes	characteristic
164	C₂²×Q₈	C₈×C₁₃₉	1	1	yes	characteristic
165	C₂²×D₄	C₈×C₁₃₉	1	3	yes	—
166	C₂³×C₄	C₂×C₄×C₁₃₉	1	3	yes	—
167	#131	—	2	4	no	—
168	#131	—	2	12	no	—
169	#131	—	2	12	no	—
170	#131	D₄×C₁₃₉	1	8	yes	—
171	C₂×M₁₆	—	2	12	no	—
172	C₂×M₁₆	C₂×C₄×C₁₃₉	1	3	yes	—
173	C₂×M₁₆	C₂×C₄×C₁₃₉	1	6	yes	—
174	C₂×M₁₆	C₂×C₄×C₁₃₉	1	3	yes	—
175	C₂²×C₈	—	2	12	no	—
176	C₂²×C₈	C₂×C₄×C₁₃₉	1	3	yes	—
177	C₂²×C₈	C₂×C₄×C₁₃₉	1	3	yes	—
178	C₂²×C₈	C₂×C₄×C₁₃₉	1	3	yes	—
179	C₂²×C₈	C₂³×C₁₃₉	1	1	yes	characteristic
180	C₂²×C₈	C₂³×C₁₃₉	1	3	yes	—
181	C₂²×C₈	C₂³×C₁₃₉	1	3	yes	—
182	M₃₂	—	2	12	no	—
183	C₂×C₁₆	—	2	12	no	—
184	C₂×C₁₆	C₂³×C₁₃₉	1	2	yes	—
185	C₂×C₁₆	C₂³×C₁₃₉	1	6	yes	—
186	C₂⁴	C₂×C₈×C₁₃₉	1	3	yes	—
187	#41	—	2	4	no	—
188	#41	—	2	12	no	—
189	#41	—	2	6	no	—
190	#41	—	2	6	no	—
191	#41	C₂²⋊C₄×C₁₃₉	1	4	yes	—
192	#41	M₁₆×C₁₃₉	1	4	yes	—
193	C₂×Q₈	—	2	4	no	—
194	C₂×Q₈	C₂×C₈×C₁₃₉	1	3	yes	—
195	C₂×Q₈	C₂×C₈×C₁₃₉	1	1	yes	characteristic
196	C₂×D₄	—	2	12	no	—
197	C₂×D₄	C₂×C₈×C₁₃₉	1	6	yes	—
198	C₂×D₄	C₂×C₈×C₁₃₉	1	3	yes	—
199	C₂×D₄	C₂×C₈×C₁₃₉	1	3	yes	—
200	C₂²×C₄	—	2	6	no	—
201	C₂²×C₄	—	2	6	no	—
202	C₂²×C₄	C₂×C₈×C₁₃₉	1	3	yes	—
203	C₂²×C₄	C₂×C₈×C₁₃₉	1	3	yes	—
204	C₂²×C₄	C₂×C₈×C₁₃₉	1	3	yes	—
205	C₂²×C₄	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
206	C₂²×C₄	C₂²×C₄×C₁₃₉	1	3	yes	—
207	C₂²×C₄	C₂²×C₄×C₁₃₉	1	3	yes	—
208	M₁₆	—	2	6	no	—
209	M₁₆	—	2	12	no	—
210	M₁₆	C₂²⋊C₄×C₁₃₉	1	12	yes	—
211	C₂×C₈	—	2	6	no	—
212	C₂×C₈	—	2	6	no	—
213	C₂×C₈	—	2	6	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	6	no	—
216	C₂×C₈	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
217	C₂×C₈	C₂²×C₄×C₁₃₉	1	3	yes	—
218	C₂×C₈	C₂²×C₄×C₁₃₉	1	3	yes	—
219	C₂×C₈	C₂×D₄×C₁₃₉	1	12	yes	—
220	C₂×C₈	C₂⁴×C₁₃₉	1	1	yes	characteristic
221	C₁₆	—	2	2	no	—
222	C₁₆	—	2	6	no	—
223	C₂³	—	2	6	no	—
224	C₂³	—	2	6	no	—
225	C₂³	—	4	3	no	—
226	C₂³	C₂²×C₈×C₁₃₉	1	3	yes	—
227	C₂³	C₂²×C₈×C₁₃₉	1	3	yes	—
228	C₂³	C₂²×C₈×C₁₃₉	1	1	yes	characteristic
229	Q₈	—	2	6	no	—
230	Q₈	C₂²⋊C₈×C₁₃₉	1	4	yes	—
231	D₄	—	2	12	no	—
232	D₄	—	2	6	no	—
233	D₄	C₂²⋊C₈×C₁₃₉	1	12	yes	—
234	C₂×C₄	—	2	6	no	—
235	C₂×C₄	—	2	6	no	—
236	C₂×C₄	—	2	6	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	3	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	3	no	—
241	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₉	1	6	yes	—
242	C₂×C₄	C₂²×C₈×C₁₃₉	1	1	yes	characteristic
243	C₂×C₄	C₂²×C₈×C₁₃₉	1	3	yes	—
244	C₂×C₄	C₂²×C₈×C₁₃₉	1	3	yes	—
245	C₂×C₄	C₂×M₁₆×C₁₃₉	1	6	yes	—
246	C₂×C₄	C₂³×C₄×C₁₃₉	1	1	yes	characteristic
247	C₈	—	2	1	no	—
248	C₈	—	2	3	no	—
249	C₈	C₂×C₂²⋊C₄×C₁₃₉	1	6	yes	—
250	C₈	C₂²×D₄×C₁₃₉	1	2	yes	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	3	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	3	no	—
255	C₂²	—	2	3	no	—
256	C₂²	—	2	3	no	—
257	C₂²	—	2	3	no	—
258	C₂²	—	4	12	no	—
259	C₂²	C₂×C₂²⋊C₈×C₁₃₉	1	6	yes	—
260	C₂²	#504×C₁₃₉	1	2	yes	complemented
261	C₂²	C₂³×C₈×C₁₃₉	1	1	yes	characteristic
262	C₄	—	2	1	no	—
263	C₄	—	2	3	no	—
264	C₄	C₂×C₂²⋊C₈×C₁₃₉	1	6	yes	—
265	C₄	C₂²×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
266	C₄	C₂²×M₁₆×C₁₃₉	1	1	yes	characteristic
267	C₂	—	2	2	no	—
268	C₂	—	2	6	no	—
269	C₂	—	4	3	no	—
270	C₂	C₂²×C₂²⋊C₈×C₁₃₉	1	1	yes	characteristic
271	C₂	C₂×#504×C₁₃₉	1	1	yes	characteristic
272	C₂	#3407×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₁₃₉	C₂⁴
commutator	C₂²	C₂³×C₈×C₁₃₉
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₉
socle	C₂²×C₁₃₉	C₂³×C₈
prime radical	C₂²×#41×C₁₃₉	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#504×C₁₃₉	C₂	semidirect
3. #2113×C₁₃₉	C₂	semidirect
4. #2112×C₁₃₉	C₂	semidirect
5. C₂×C₂²⋊C₁₆×C₁₃₉	C₂	semidirect
15. C₂×M₃₂×C₁₃₉	C₂²	semidirect
16. C₂²×C₁₆×C₁₃₉	C₂²	semidirect
19. C₂²⋊C₁₆×C₁₃₉	C₂²	semidirect
20. C₂²⋊C₁₆×C₁₃₉	C₂²	semidirect
123. C₂²×C₁₃₉	#504	semidirect
136. #33334	C₁₃₉	direct (N × H)
137. C₁₃₉	#33334	direct (N × H)
139. C₂×#504	C₂×C₁₃₉	semidirect
140. #2113	C₂×C₁₃₉	semidirect
141. #2112	C₂×C₁₃₉	semidirect
142. C₂×C₂²⋊C₁₆	C₂×C₁₃₉	semidirect
152. C₂×M₃₂	C₂²×C₁₃₉	semidirect
153. C₂²×C₁₆	C₂²×C₁₃₉	semidirect
156. C₂²⋊C₁₆	C₂²×C₁₃₉	semidirect
157. C₂²⋊C₁₆	C₂²×C₁₃₉	semidirect
260. C₂²	#504×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#131×C₁₃₉	2
C₂×#504×C₁₃₉	2
#2113×C₁₃₉	2
#2112×C₁₃₉	2
C₂×C₂²⋊C₁₆×C₁₃₉	2
#33334	139

Subgroups — C139×#33334 #48027930

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#33334 #48027930