Subgroups — C₁₃₉×#35679 #48030275

Order 35584.

2470 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#124×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#519×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#2482×C₁₃₉	C₂	1	8	yes	maximal, complemented
4	#2448×C₁₃₉	C₂	1	16	yes	maximal, complemented
5	#2310×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#2309×C₁₃₉	C₂	1	2	yes	maximal
7	#2296×C₁₃₉	C₂	1	2	yes	maximal, complemented
8	C₂³×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
9	C₂³×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
10	C₂×#124×C₁₃₉	—	2	8	no	—
11	C₂×#124×C₁₃₉	C₂²	1	8	yes	—
12	#519×C₁₃₉	—	2	8	no	—
13	C₂×C₄×Q₈×C₁₃₉	C₂²	1	8	yes	complemented
14	C₂×#117×C₁₃₉	C₂²	1	2	yes	—
15	C₂×#117×C₁₃₉	C₂²	1	2	yes	—
16	C₂×#117×C₁₃₉	C₂²	1	4	yes	—
17	C₂²×C₂²⋊C₄×C₁₃₉	C₂²	1	4	yes	—
18	C₂²×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
19	#399×C₁₃₉	C₂²	1	16	yes	complemented
20	#397×C₁₃₉	C₂²	1	16	yes	complemented
21	#391×C₁₃₉	C₂²	1	8	yes	complemented
22	#390×C₁₃₉	C₂²	1	8	yes	—
23	#390×C₁₃₉	C₂²	1	8	yes	—
24	#389×C₁₃₉	C₂²	1	8	yes	complemented
25	#388×C₁₃₉	C₂²	1	32	yes	complemented
26	#385×C₁₃₉	C₂²	1	8	yes	—
27	#381×C₁₃₉	C₂²	1	4	yes	—
28	#379×C₁₃₉	C₂²	1	8	yes	—
29	C₄×C₄⋊C₄×C₁₃₉	C₂²	1	8	yes	complemented
30	C₂⁵×C₁₃₉	C₂³	1	2	yes	—
31	C₂²×Q₈×C₁₃₉	—	2	8	no	—
32	C₂²×Q₈×C₁₃₉	D₄	1	8	yes	—
33	C₂²×Q₈×C₁₃₉	C₂³	1	4	yes	—
34	C₂²×D₄×C₁₃₉	—	2	4	no	—
35	C₂²×D₄×C₁₃₉	—	2	8	no	—
36	C₂²×D₄×C₁₃₉	C₂³	1	4	yes	—
37	C₂³×C₄×C₁₃₉	C₂³	1	2	yes	—
38	C₂³×C₄×C₁₃₉	C₂³	1	2	yes	—
39	C₂³×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
40	#124×C₁₃₉	—	2	32	no	—
41	C₄×Q₈×C₁₃₉	—	2	32	no	—
42	#117×C₁₃₉	—	2	16	no	—
43	#117×C₁₃₉	—	2	8	no	—
44	#117×C₁₃₉	—	2	8	no	—
45	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	8	yes	—
46	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	4	yes	—
47	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	4	yes	—
48	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	16	yes	—
49	C₂×C₂²⋊C₄×C₁₃₉	—	2	16	no	—
50	C₂×C₂²⋊C₄×C₁₃₉	C₂³	1	16	yes	—
51	C₂×C₂²⋊C₄×C₁₃₉	C₂³	1	4	yes	—
52	C₂×C₂²⋊C₄×C₁₃₉	C₂³	1	4	yes	—
53	C₂×C₂²⋊C₄×C₁₃₉	C₂³	1	32	yes	—
54	C₂×C₄²×C₁₃₉	—	2	4	no	—
55	C₂×C₄²×C₁₃₉	C₂³	1	4	yes	—
56	C₂×C₄²×C₁₃₉	C₂³	1	4	yes	—
57	C₂×C₄²×C₁₃₉	C₂³	1	8	yes	—
58	C₂×C₄²×C₁₃₉	C₂³	1	4	yes	—
59	#95×C₁₃₉	C₂³	1	16	yes	—
60	#95×C₁₃₉	C₂³	1	16	yes	—
61	C₂⁴×C₁₃₉	—	2	4	no	—
62	C₂⁴×C₁₃₉	—	2	8	no	—
63	C₂⁴×C₁₃₉	—	4	8	no	—
64	C₂⁴×C₁₃₉	C₂⁴	1	4	yes	—
65	C₂⁴×C₁₃₉	C₂⁴	1	1	yes	characteristic
66	C₂×Q₈×C₁₃₉	—	2	32	no	—
67	C₂×Q₈×C₁₃₉	—	2	8	no	—
68	C₂×Q₈×C₁₃₉	—	2	8	no	—
69	C₂×Q₈×C₁₃₉	C₂×D₄	1	16	yes	—
70	C₂×D₄×C₁₃₉	—	2	8	no	—
71	C₂×D₄×C₁₃₉	—	2	8	no	—
72	C₂×D₄×C₁₃₉	—	2	8	no	—
73	C₂×D₄×C₁₃₉	—	4	16	no	—
74	C₂²×C₄×C₁₃₉	—	2	8	no	—
75	C₂²×C₄×C₁₃₉	—	2	4	no	—
76	C₂²×C₄×C₁₃₉	—	4	4	no	—
77	C₂²×C₄×C₁₃₉	C₂×D₄	1	8	yes	—
78	C₂²×C₄×C₁₃₉	C₂×D₄	1	4	yes	—
79	C₂²×C₄×C₁₃₉	#41	1	8	yes	—
80	C₂²×C₄×C₁₃₉	C₂⁴	1	8	yes	—
81	C₂²×C₄×C₁₃₉	C₂⁴	1	4	yes	—
82	C₂²×C₄×C₁₃₉	C₂⁴	1	8	yes	—
83	C₂²×C₄×C₁₃₉	C₂⁴	1	4	yes	—
84	C₂²×C₄×C₁₃₉	C₂⁴	1	2	yes	—
85	C₄⋊C₄×C₁₃₉	—	2	16	no	—
86	C₄⋊C₄×C₁₃₉	—	2	8	no	—
87	C₄⋊C₄×C₁₃₉	—	2	8	no	—
88	C₄⋊C₄×C₁₃₉	—	4	16	no	—
89	C₂²⋊C₄×C₁₃₉	—	2	8	no	—
90	C₂²⋊C₄×C₁₃₉	—	2	8	no	—
91	C₂²⋊C₄×C₁₃₉	—	2	32	no	—
92	C₂²⋊C₄×C₁₃₉	—	4	32	no	—
93	C₄²×C₁₃₉	—	2	8	no	—
94	C₄²×C₁₃₉	—	2	16	no	—
95	C₄²×C₁₃₉	—	2	8	no	—
96	C₄²×C₁₃₉	—	2	8	no	—
97	C₂³×C₁₃₉	—	2	4	no	—
98	C₂³×C₁₃₉	—	2	4	no	—
99	C₂³×C₁₃₉	—	2	8	no	—
100	C₂³×C₁₃₉	—	2	16	no	—
101	C₂³×C₁₃₉	—	4	16	no	—
102	C₂³×C₁₃₉	—	4	8	no	—
103	C₂³×C₁₃₉	—	8	16	no	—
104	C₂³×C₁₃₉	C₂²×D₄	1	2	yes	—
105	C₂³×C₁₃₉	C₂×#41	1	4	yes	—
106	C₂³×C₁₃₉	C₂⁵	1	1	yes	characteristic
107	Q₈×C₁₃₉	—	2	32	no	—
108	D₄×C₁₃₉	—	4	16	no	—
109	C₂×C₄×C₁₃₉	—	2	16	no	—
110	C₂×C₄×C₁₃₉	—	2	8	no	—
111	C₂×C₄×C₁₃₉	—	2	8	no	—
112	C₂×C₄×C₁₃₉	—	2	4	no	—
113	C₂×C₄×C₁₃₉	—	2	8	no	—
114	C₂×C₄×C₁₃₉	—	2	16	no	—
115	C₂×C₄×C₁₃₉	—	2	16	no	—
116	C₂×C₄×C₁₃₉	—	2	8	no	—
117	C₂×C₄×C₁₃₉	—	4	8	no	—
118	C₂×C₄×C₁₃₉	#123	1	16	yes	—
119	C₂×C₄×C₁₃₉	C₂²×D₄	1	8	yes	—
120	C₂×C₄×C₁₃₉	C₂²×D₄	1	4	yes	—
121	C₂×C₄×C₁₃₉	C₂×#41	1	4	yes	—
122	C₂×C₄×C₁₃₉	C₂×#41	1	4	yes	—
123	C₂²×C₁₃₉	—	2	4	no	—
124	C₂²×C₁₃₉	—	2	8	no	—
125	C₂²×C₁₃₉	—	4	4	no	—
126	C₂²×C₁₃₉	—	4	8	no	—
127	C₂²×C₁₃₉	—	4	16	no	—
128	C₂²×C₁₃₉	—	8	16	no	—
129	C₂²×C₁₃₉	C₂×#123	1	4	yes	—
130	C₂²×C₁₃₉	C₂³×D₄	1	1	yes	characteristic
131	C₂²×C₁₃₉	C₂²×#41	1	2	yes	—
132	C₂²×C₁₃₉	#585	1	4	yes	—
133	C₄×C₁₃₉	—	2	8	no	—
134	C₄×C₁₃₉	—	4	8	no	—
135	C₄×C₁₃₉	—	4	4	no	—
136	C₄×C₁₃₉	—	4	8	no	—
137	C₄×C₁₃₉	C₂×#123	1	8	yes	—
138	C₂×C₁₃₉	—	2	4	no	—
139	C₂×C₁₃₉	—	8	4	no	—
140	C₂×C₁₃₉	C₂²×#123	1	1	yes	characteristic
141	C₂×C₁₃₉	#3487	1	2	yes	—
142	C₂×C₁₃₉	#3510	1	4	yes	—
143	#35679	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₁₃₉	#35679	1	1	yes	characteristic, direct product factor, Sylow
145	C₂²×#124	C₂×C₁₃₉	1	1	yes	characteristic
146	C₂×#519	C₂×C₁₃₉	1	1	yes	characteristic, complemented
147	#2482	C₂×C₁₃₉	1	8	yes	complemented
148	#2448	C₂×C₁₃₉	1	16	yes	complemented
149	#2310	C₂×C₁₃₉	1	1	yes	characteristic, complemented
150	#2309	C₂×C₁₃₉	1	2	yes	—
151	#2296	C₂×C₁₃₉	1	2	yes	complemented
152	C₂³×Q₈	C₂²×C₁₃₉	1	1	yes	characteristic
153	C₂³×D₄	C₂²×C₁₃₉	1	1	yes	characteristic
154	C₂×#124	—	2	8	no	—
155	C₂×#124	C₂²×C₁₃₉	1	8	yes	—
156	#519	—	2	8	no	—
157	C₂×C₄×Q₈	C₂²×C₁₃₉	1	8	yes	complemented
158	C₂×#117	C₂²×C₁₃₉	1	2	yes	—
159	C₂×#117	C₂²×C₁₃₉	1	2	yes	—
160	C₂×#117	C₂²×C₁₃₉	1	4	yes	—
161	C₂²×C₂²⋊C₄	C₂²×C₁₃₉	1	4	yes	—
162	C₂²×C₄²	C₂²×C₁₃₉	1	1	yes	characteristic
163	#399	C₂²×C₁₃₉	1	16	yes	complemented
164	#397	C₂²×C₁₃₉	1	16	yes	complemented
165	#391	C₂²×C₁₃₉	1	8	yes	complemented
166	#390	C₂²×C₁₃₉	1	8	yes	—
167	#390	C₂²×C₁₃₉	1	8	yes	—
168	#389	C₂²×C₁₃₉	1	8	yes	complemented
169	#388	C₂²×C₁₃₉	1	32	yes	complemented
170	#385	C₂²×C₁₃₉	1	8	yes	—
171	#381	C₂²×C₁₃₉	1	4	yes	—
172	#379	C₂²×C₁₃₉	1	8	yes	—
173	C₄×C₄⋊C₄	C₂²×C₁₃₉	1	8	yes	complemented
174	C₂⁵	C₂³×C₁₃₉	1	2	yes	—
175	C₂²×Q₈	—	2	8	no	—
176	C₂²×Q₈	D₄×C₁₃₉	1	8	yes	—
177	C₂²×Q₈	C₂³×C₁₃₉	1	4	yes	—
178	C₂²×D₄	—	2	4	no	—
179	C₂²×D₄	—	2	8	no	—
180	C₂²×D₄	C₂³×C₁₃₉	1	4	yes	—
181	C₂³×C₄	C₂³×C₁₃₉	1	2	yes	—
182	C₂³×C₄	C₂³×C₁₃₉	1	2	yes	—
183	C₂³×C₄	C₂³×C₁₃₉	1	1	yes	characteristic
184	#124	—	2	32	no	—
185	C₄×Q₈	—	2	32	no	—
186	#117	—	2	16	no	—
187	#117	—	2	8	no	—
188	#117	—	2	8	no	—
189	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	4	yes	—
190	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	8	yes	—
191	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	4	yes	—
192	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	16	yes	—
193	C₂×C₂²⋊C₄	—	2	16	no	—
194	C₂×C₂²⋊C₄	C₂³×C₁₃₉	1	16	yes	—
195	C₂×C₂²⋊C₄	C₂³×C₁₃₉	1	4	yes	—
196	C₂×C₂²⋊C₄	C₂³×C₁₃₉	1	4	yes	—
197	C₂×C₂²⋊C₄	C₂³×C₁₃₉	1	32	yes	—
198	C₂×C₄²	—	2	4	no	—
199	C₂×C₄²	C₂³×C₁₃₉	1	4	yes	—
200	C₂×C₄²	C₂³×C₁₃₉	1	8	yes	—
201	C₂×C₄²	C₂³×C₁₃₉	1	4	yes	—
202	C₂×C₄²	C₂³×C₁₃₉	1	4	yes	—
203	#95	C₂³×C₁₃₉	1	16	yes	—
204	#95	C₂³×C₁₃₉	1	16	yes	—
205	C₂⁴	—	2	4	no	—
206	C₂⁴	—	2	8	no	—
207	C₂⁴	—	4	8	no	—
208	C₂⁴	C₂⁴×C₁₃₉	1	4	yes	—
209	C₂⁴	C₂⁴×C₁₃₉	1	1	yes	characteristic
210	C₂×Q₈	—	2	32	no	—
211	C₂×Q₈	—	2	8	no	—
212	C₂×Q₈	—	2	8	no	—
213	C₂×Q₈	C₂×D₄×C₁₃₉	1	16	yes	—
214	C₂×D₄	—	2	8	no	—
215	C₂×D₄	—	2	8	no	—
216	C₂×D₄	—	2	8	no	—
217	C₂×D₄	—	4	16	no	—
218	C₂²×C₄	—	2	8	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	4	4	no	—
221	C₂²×C₄	C₂×D₄×C₁₃₉	1	8	yes	—
222	C₂²×C₄	C₂×D₄×C₁₃₉	1	4	yes	—
223	C₂²×C₄	#41×C₁₃₉	1	8	yes	—
224	C₂²×C₄	C₂⁴×C₁₃₉	1	8	yes	—
225	C₂²×C₄	C₂⁴×C₁₃₉	1	4	yes	—
226	C₂²×C₄	C₂⁴×C₁₃₉	1	8	yes	—
227	C₂²×C₄	C₂⁴×C₁₃₉	1	4	yes	—
228	C₂²×C₄	C₂⁴×C₁₃₉	1	2	yes	—
229	C₄⋊C₄	—	2	16	no	—
230	C₄⋊C₄	—	2	8	no	—
231	C₄⋊C₄	—	2	8	no	—
232	C₄⋊C₄	—	4	16	no	—
233	C₂²⋊C₄	—	2	8	no	—
234	C₂²⋊C₄	—	2	8	no	—
235	C₂²⋊C₄	—	2	32	no	—
236	C₂²⋊C₄	—	4	32	no	—
237	C₄²	—	2	8	no	—
238	C₄²	—	2	16	no	—
239	C₄²	—	2	8	no	—
240	C₄²	—	2	8	no	—
241	C₂³	—	2	4	no	—
242	C₂³	—	2	4	no	—
243	C₂³	—	2	8	no	—
244	C₂³	—	2	16	no	—
245	C₂³	—	4	16	no	—
246	C₂³	—	4	8	no	—
247	C₂³	—	8	16	no	—
248	C₂³	C₂²×D₄×C₁₃₉	1	2	yes	—
249	C₂³	C₂×#41×C₁₃₉	1	4	yes	—
250	C₂³	C₂⁵×C₁₃₉	1	1	yes	characteristic
251	Q₈	—	2	32	no	—
252	D₄	—	4	16	no	—
253	C₂×C₄	—	2	16	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	16	no	—
259	C₂×C₄	—	2	16	no	—
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	4	8	no	—
262	C₂×C₄	#123×C₁₃₉	1	16	yes	—
263	C₂×C₄	C₂²×D₄×C₁₃₉	1	8	yes	—
264	C₂×C₄	C₂²×D₄×C₁₃₉	1	4	yes	—
265	C₂×C₄	C₂×#41×C₁₃₉	1	4	yes	—
266	C₂×C₄	C₂×#41×C₁₃₉	1	4	yes	—
267	C₂²	—	2	4	no	—
268	C₂²	—	2	8	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	4	8	no	—
271	C₂²	—	4	16	no	—
272	C₂²	—	8	16	no	—
273	C₂²	C₂×#123×C₁₃₉	1	4	yes	—
274	C₂²	C₂³×D₄×C₁₃₉	1	1	yes	characteristic
275	C₂²	C₂²×#41×C₁₃₉	1	2	yes	—
276	C₂²	#585×C₁₃₉	1	4	yes	—
277	C₄	—	2	8	no	—
278	C₄	—	4	4	no	—
279	C₄	—	4	8	no	—
280	C₄	—	4	8	no	—
281	C₄	C₂×#123×C₁₃₉	1	8	yes	—
282	C₂	—	2	4	no	—
283	C₂	—	8	4	no	—
284	C₂	C₂²×#123×C₁₃₉	1	1	yes	characteristic
285	C₂	#3487×C₁₃₉	1	2	yes	—
286	C₂	#3510×C₁₃₉	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₃₉
prime radical	C₂³×D₄×C₁₃₉	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#519×C₁₃₉	C₂	semidirect
3. #2482×C₁₃₉	C₂	semidirect
4. #2448×C₁₃₉	C₂	semidirect
5. #2310×C₁₃₉	C₂	semidirect
7. #2296×C₁₃₉	C₂	semidirect
13. C₂×C₄×Q₈×C₁₃₉	C₂²	semidirect
19. #399×C₁₃₉	C₂²	semidirect
20. #397×C₁₃₉	C₂²	semidirect
21. #391×C₁₃₉	C₂²	semidirect
24. #389×C₁₃₉	C₂²	semidirect
25. #388×C₁₃₉	C₂²	semidirect
29. C₄×C₄⋊C₄×C₁₃₉	C₂²	semidirect
143. #35679	C₁₃₉	direct (N × H)
144. C₁₃₉	#35679	direct (N × H)
146. C₂×#519	C₂×C₁₃₉	semidirect
147. #2482	C₂×C₁₃₉	semidirect
148. #2448	C₂×C₁₃₉	semidirect
149. #2310	C₂×C₁₃₉	semidirect
151. #2296	C₂×C₁₃₉	semidirect
157. C₂×C₄×Q₈	C₂²×C₁₃₉	semidirect
163. #399	C₂²×C₁₃₉	semidirect
164. #397	C₂²×C₁₃₉	semidirect
165. #391	C₂²×C₁₃₉	semidirect
168. #389	C₂²×C₁₃₉	semidirect
169. #388	C₂²×C₁₃₉	semidirect
173. C₄×C₄⋊C₄	C₂²×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#124×C₁₃₉	2
C₂×#519×C₁₃₉	2
#2482×C₁₃₉	2
#2448×C₁₃₉	2
#2310×C₁₃₉	2
#2309×C₁₃₉	2
#2296×C₁₃₉	2
#35679	139

Subgroups — C139×#35679 #48030275

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#35679 #48030275