Subgroups — C₁₃₉×#12738 #48007334

Order 35584.

926 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#135×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#362×C₁₃₉	C₂	1	2	yes	maximal, complemented
3	C₂×#361×C₁₃₉	C₂	1	2	yes	maximal, complemented
4	#2112×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#1810×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#41×C₁₃₉	—	2	1	no	—
7	C₂×#135×C₁₃₉	—	2	4	no	—
8	C₂×#135×C₁₃₉	—	2	4	no	—
9	C₂×#135×C₁₃₉	—	2	2	no	—
10	C₂×#135×C₁₃₉	C₄	1	4	yes	—
11	C₂²×Q₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
12	C₂²×SD₁₆×C₁₃₉	—	2	1	no	—
13	C₂²×D₈×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₂×M₃₂×C₁₃₉	C₂²	1	2	yes	—
16	C₂×#108×C₁₃₉	C₂²	1	2	yes	—
17	#407×C₁₃₉	—	2	1	no	—
18	#362×C₁₃₉	—	2	4	no	—
19	#361×C₁₃₉	—	2	4	no	—
20	C₂²⋊C₁₆×C₁₃₉	—	2	2	no	—
21	#341×C₁₃₉	—	2	1	no	—
22	C₂×#41×C₁₃₉	—	2	8	no	—
23	C₂×#41×C₁₃₉	—	2	4	no	—
24	C₂×#41×C₁₃₉	—	4	4	no	—
25	C₂×#41×C₁₃₉	—	4	2	no	—
26	C₂²×Q₈×C₁₃₉	—	2	1	no	—
27	C₂²×D₄×C₁₃₉	—	2	1	no	—
28	C₂²×D₄×C₁₃₉	—	4	1	no	—
29	C₂³×C₄×C₁₃₉	—	4	1	no	—
30	C₂³×C₄×C₁₃₉	D₄	1	1	yes	characteristic
31	#135×C₁₃₉	—	2	16	no	—
32	#135×C₁₃₉	—	2	8	no	—
33	#135×C₁₃₉	—	2	8	no	—
34	C₂×Q₁₆×C₁₃₉	—	2	2	no	—
35	C₂×Q₁₆×C₁₃₉	—	2	1	no	—
36	C₂×Q₁₆×C₁₃₉	C₂×C₄	1	2	yes	—
37	C₂×Q₁₆×C₁₃₉	D₄	1	4	yes	—
38	C₂×SD₁₆×C₁₃₉	—	2	8	no	—
39	C₂×SD₁₆×C₁₃₉	—	2	4	no	—
40	C₂×D₈×C₁₃₉	—	2	2	no	—
41	C₂×D₈×C₁₃₉	—	2	1	no	—
42	C₂×D₈×C₁₃₉	C₂×C₄	1	2	yes	—
43	C₂×D₈×C₁₃₉	D₄	1	4	yes	—
44	C₂×M₁₆×C₁₃₉	—	2	2	no	—
45	C₂²×C₈×C₁₃₉	—	2	2	no	—
46	C₂²×C₈×C₁₃₉	—	2	2	no	—
47	C₂²×C₈×C₁₃₉	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₁₃₉	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₁₃₉	D₄	1	2	yes	—
50	C₂²×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
51	M₃₂×C₁₃₉	—	2	2	no	—
52	M₃₂×C₁₃₉	—	2	2	no	—
53	C₂×C₁₆×C₁₃₉	—	2	2	no	—
54	#108×C₁₃₉	—	2	2	no	—
55	#108×C₁₃₉	—	2	2	no	—
56	C₂²⋊C₈×C₁₃₉	—	4	2	no	—
57	C₂⁴×C₁₃₉	—	4	1	no	—
58	C₂⁴×C₁₃₉	SD₁₆	1	1	yes	characteristic
59	#41×C₁₃₉	—	2	16	no	—
60	#41×C₁₃₉	—	4	8	no	—
61	#41×C₁₃₉	—	4	16	no	—
62	C₂×Q₈×C₁₃₉	—	2	2	no	—
63	C₂×Q₈×C₁₃₉	—	2	4	no	—
64	C₂×Q₈×C₁₃₉	—	4	1	no	—
65	C₂×Q₈×C₁₃₉	—	4	2	no	—
66	C₂×D₄×C₁₃₉	—	2	2	no	—
67	C₂×D₄×C₁₃₉	—	2	4	no	—
68	C₂×D₄×C₁₃₉	—	4	8	no	—
69	C₂×D₄×C₁₃₉	—	4	4	no	—
70	C₂×D₄×C₁₃₉	—	4	1	no	—
71	C₂×D₄×C₁₃₉	—	4	2	no	—
72	C₂²×C₄×C₁₃₉	—	2	2	no	—
73	C₂²×C₄×C₁₃₉	—	2	2	no	—
74	C₂²×C₄×C₁₃₉	—	4	8	no	—
75	C₂²×C₄×C₁₃₉	—	4	1	no	—
76	C₂²×C₄×C₁₃₉	—	4	4	no	—
77	C₂²×C₄×C₁₃₉	C₂²⋊C₄	1	2	yes	—
78	C₂²×C₄×C₁₃₉	D₈	1	2	yes	—
79	C₂²×C₄×C₁₃₉	SD₁₆	1	1	yes	characteristic
80	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
81	Q₁₆×C₁₃₉	—	2	2	no	—
82	Q₁₆×C₁₃₉	—	2	4	no	—
83	Q₁₆×C₁₃₉	C₂²⋊C₄	1	4	yes	—
84	SD₁₆×C₁₃₉	—	2	16	no	—
85	D₈×C₁₃₉	—	2	2	no	—
86	D₈×C₁₃₉	—	2	4	no	—
87	D₈×C₁₃₉	C₂²⋊C₄	1	4	yes	—
88	M₁₆×C₁₃₉	—	4	2	no	—
89	M₁₆×C₁₃₉	—	4	2	no	—
90	C₂×C₈×C₁₃₉	—	2	2	no	—
91	C₂×C₈×C₁₃₉	—	2	2	no	—
92	C₂×C₈×C₁₃₉	—	2	2	no	—
93	C₂×C₈×C₁₃₉	—	2	4	no	—
94	C₂×C₈×C₁₃₉	—	4	2	no	—
95	C₂×C₈×C₁₃₉	C₂²⋊C₄	1	2	yes	—
96	C₂×C₈×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃₉	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₁₃₉	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₁₃₉	C₂×D₄	1	1	yes	characteristic
100	C₁₆×C₁₃₉	—	4	2	no	—
101	C₂³×C₁₃₉	—	2	2	no	—
102	C₂³×C₁₃₉	—	4	2	no	—
103	C₂³×C₁₃₉	—	4	2	no	—
104	C₂³×C₁₃₉	—	4	4	no	—
105	C₂³×C₁₃₉	—	8	4	no	—
106	C₂³×C₁₃₉	#102	1	2	yes	—
107	C₂³×C₁₃₉	C₂×SD₁₆	1	1	yes	characteristic
108	Q₈×C₁₃₉	—	2	4	no	—
109	Q₈×C₁₃₉	—	4	2	no	—
110	Q₈×C₁₃₉	—	4	4	no	—
111	D₄×C₁₃₉	—	2	4	no	—
112	D₄×C₁₃₉	—	4	16	no	—
113	D₄×C₁₃₉	—	4	2	no	—
114	D₄×C₁₃₉	—	4	4	no	—
115	C₂×C₄×C₁₃₉	—	2	2	no	—
116	C₂×C₄×C₁₃₉	—	2	2	no	—
117	C₂×C₄×C₁₃₉	—	2	2	no	—
118	C₂×C₄×C₁₃₉	—	2	4	no	—
119	C₂×C₄×C₁₃₉	—	4	16	no	—
120	C₂×C₄×C₁₃₉	—	4	2	no	—
121	C₂×C₄×C₁₃₉	—	4	4	no	—
122	C₂×C₄×C₁₃₉	#102	1	2	yes	—
123	C₂×C₄×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃₉	#120	1	2	yes	—
125	C₂×C₄×C₁₃₉	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₁₃₉	AGL(1,ℤ/8)	1	2	yes	—
127	C₈×C₁₃₉	—	2	2	no	—
128	C₈×C₁₃₉	—	8	2	no	—
129	C₈×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₈×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₈×C₁₃₉	#120	1	2	yes	—
132	C₂²×C₁₃₉	—	2	2	no	—
133	C₂²×C₁₃₉	—	2	4	no	—
134	C₂²×C₁₃₉	—	2	2	no	—
135	C₂²×C₁₃₉	—	4	4	no	—
136	C₂²×C₁₃₉	—	4	4	no	—
137	C₂²×C₁₃₉	—	8	8	no	—
138	C₂²×C₁₃₉	—	8	4	no	—
139	C₂²×C₁₃₉	C₂×#102	1	1	yes	characteristic
140	C₂²×C₁₃₉	#450	1	2	yes	—
141	C₄×C₁₃₉	—	2	2	no	—
142	C₄×C₁₃₉	—	4	4	no	—
143	C₄×C₁₃₉	#379	1	1	yes	characteristic
144	C₄×C₁₃₉	#418	1	1	yes	characteristic
145	C₄×C₁₃₉	#447	1	2	yes	—
146	C₂×C₁₃₉	—	2	2	no	—
147	C₂×C₁₃₉	—	4	2	no	—
148	C₂×C₁₃₉	—	8	4	no	—
149	C₂×C₁₃₉	#1786	1	1	yes	characteristic
150	C₂×C₁₃₉	#2148	1	2	yes	—
151	#12738	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₁₃₉	#12738	1	1	yes	characteristic, direct product factor, Sylow
153	C₂²×#135	C₂×C₁₃₉	1	1	yes	characteristic
154	C₂×#362	C₂×C₁₃₉	1	2	yes	complemented
155	C₂×#361	C₂×C₁₃₉	1	2	yes	complemented
156	#2112	C₂×C₁₃₉	1	1	yes	characteristic, complemented
157	#1810	C₂×C₁₃₉	1	1	yes	characteristic, complemented
158	C₂²×#41	—	2	1	no	—
159	C₂×#135	—	2	4	no	—
160	C₂×#135	—	2	4	no	—
161	C₂×#135	—	2	2	no	—
162	C₂×#135	C₄×C₁₃₉	1	4	yes	—
163	C₂²×Q₁₆	C₂²×C₁₃₉	1	1	yes	characteristic
164	C₂²×SD₁₆	—	2	1	no	—
165	C₂²×D₈	C₂²×C₁₃₉	1	1	yes	characteristic
166	C₂³×C₈	C₂²×C₁₃₉	1	1	yes	characteristic
167	C₂×M₃₂	C₂²×C₁₃₉	1	2	yes	—
168	C₂×#108	C₂²×C₁₃₉	1	2	yes	—
169	#407	—	2	1	no	—
170	#362	—	2	4	no	—
171	#361	—	2	4	no	—
172	C₂²⋊C₁₆	—	2	2	no	—
173	#341	—	2	1	no	—
174	C₂×#41	—	2	8	no	—
175	C₂×#41	—	2	4	no	—
176	C₂×#41	—	4	4	no	—
177	C₂×#41	—	4	2	no	—
178	C₂²×Q₈	—	2	1	no	—
179	C₂²×D₄	—	2	1	no	—
180	C₂²×D₄	—	4	1	no	—
181	C₂³×C₄	—	4	1	no	—
182	C₂³×C₄	D₄×C₁₃₉	1	1	yes	characteristic
183	#135	—	2	16	no	—
184	#135	—	2	8	no	—
185	#135	—	2	8	no	—
186	C₂×Q₁₆	—	2	2	no	—
187	C₂×Q₁₆	—	2	1	no	—
188	C₂×Q₁₆	C₂×C₄×C₁₃₉	1	2	yes	—
189	C₂×Q₁₆	D₄×C₁₃₉	1	4	yes	—
190	C₂×SD₁₆	—	2	8	no	—
191	C₂×SD₁₆	—	2	4	no	—
192	C₂×D₈	—	2	2	no	—
193	C₂×D₈	—	2	1	no	—
194	C₂×D₈	C₂×C₄×C₁₃₉	1	2	yes	—
195	C₂×D₈	D₄×C₁₃₉	1	4	yes	—
196	C₂×M₁₆	—	2	2	no	—
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	C₂×C₄×C₁₃₉	1	2	yes	—
200	C₂²×C₈	D₄×C₁₃₉	1	1	yes	characteristic
201	C₂²×C₈	D₄×C₁₃₉	1	2	yes	—
202	C₂²×C₈	C₂³×C₁₃₉	1	1	yes	characteristic
203	M₃₂	—	2	2	no	—
204	M₃₂	—	2	2	no	—
205	C₂×C₁₆	—	2	2	no	—
206	#108	—	2	2	no	—
207	#108	—	2	2	no	—
208	C₂²⋊C₈	—	4	2	no	—
209	C₂⁴	—	4	1	no	—
210	C₂⁴	SD₁₆×C₁₃₉	1	1	yes	characteristic
211	#41	—	2	16	no	—
212	#41	—	4	8	no	—
213	#41	—	4	16	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	—	2	4	no	—
216	C₂×Q₈	—	4	1	no	—
217	C₂×Q₈	—	4	2	no	—
218	C₂×D₄	—	2	2	no	—
219	C₂×D₄	—	2	4	no	—
220	C₂×D₄	—	4	8	no	—
221	C₂×D₄	—	4	4	no	—
222	C₂×D₄	—	4	1	no	—
223	C₂×D₄	—	4	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	4	8	no	—
227	C₂²×C₄	—	4	1	no	—
228	C₂²×C₄	—	4	4	no	—
229	C₂²×C₄	C₂²⋊C₄×C₁₃₉	1	2	yes	—
230	C₂²×C₄	D₈×C₁₃₉	1	2	yes	—
231	C₂²×C₄	SD₁₆×C₁₃₉	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
233	Q₁₆	—	2	2	no	—
234	Q₁₆	—	2	4	no	—
235	Q₁₆	C₂²⋊C₄×C₁₃₉	1	4	yes	—
236	SD₁₆	—	2	16	no	—
237	D₈	—	2	2	no	—
238	D₈	—	2	4	no	—
239	D₈	C₂²⋊C₄×C₁₃₉	1	4	yes	—
240	M₁₆	—	4	2	no	—
241	M₁₆	—	4	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	C₂²⋊C₄×C₁₃₉	1	2	yes	—
248	C₂×C₈	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
249	C₂×C₈	C₂×D₄×C₁₃₉	1	2	yes	—
250	C₂×C₈	C₂×D₄×C₁₃₉	1	2	yes	—
251	C₂×C₈	C₂×D₄×C₁₃₉	1	1	yes	characteristic
252	C₁₆	—	4	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	—	4	2	no	—
255	C₂³	—	4	2	no	—
256	C₂³	—	4	4	no	—
257	C₂³	—	8	4	no	—
258	C₂³	#102×C₁₃₉	1	2	yes	—
259	C₂³	C₂×SD₁₆×C₁₃₉	1	1	yes	characteristic
260	Q₈	—	2	4	no	—
261	Q₈	—	4	2	no	—
262	Q₈	—	4	4	no	—
263	D₄	—	2	4	no	—
264	D₄	—	4	16	no	—
265	D₄	—	4	2	no	—
266	D₄	—	4	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	4	16	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	#102×C₁₃₉	1	2	yes	—
275	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
276	C₂×C₄	#120×C₁₃₉	1	2	yes	—
277	C₂×C₄	C₂×D₈×C₁₃₉	1	1	yes	characteristic
278	C₂×C₄	AGL(1,ℤ/8)×C₁₃₉	1	2	yes	—
279	C₈	—	2	2	no	—
280	C₈	—	8	2	no	—
281	C₈	C₂×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
282	C₈	C₂×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
283	C₈	#120×C₁₃₉	1	2	yes	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	4	4	no	—
288	C₂²	—	4	4	no	—
289	C₂²	—	8	8	no	—
290	C₂²	—	8	4	no	—
291	C₂²	C₂×#102×C₁₃₉	1	1	yes	characteristic
292	C₂²	#450×C₁₃₉	1	2	yes	—
293	C₄	—	2	2	no	—
294	C₄	—	4	4	no	—
295	C₄	#379×C₁₃₉	1	1	yes	characteristic
296	C₄	#418×C₁₃₉	1	1	yes	characteristic
297	C₄	#447×C₁₃₉	1	2	yes	—
298	C₂	—	2	2	no	—
299	C₂	—	4	2	no	—
300	C₂	—	8	4	no	—
301	C₂	#1786×C₁₃₉	1	1	yes	characteristic
302	C₂	#2148×C₁₃₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₉	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₁₃₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₉
prime radical	C₂²×#135×C₁₃₉	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₉ ◁ C₂³×C₄×C₁₃₉ ◁ C₂³×C₈×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#362×C₁₃₉	C₂	semidirect
3. C₂×#361×C₁₃₉	C₂	semidirect
4. #2112×C₁₃₉	C₂	semidirect
5. #1810×C₁₃₉	C₂	semidirect
151. #12738	C₁₃₉	direct (N × H)
152. C₁₃₉	#12738	direct (N × H)
154. C₂×#362	C₂×C₁₃₉	semidirect
155. C₂×#361	C₂×C₁₃₉	semidirect
156. #2112	C₂×C₁₃₉	semidirect
157. #1810	C₂×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#135×C₁₃₉	2
C₂×#362×C₁₃₉	2
C₂×#361×C₁₃₉	2
#2112×C₁₃₉	2
#1810×C₁₃₉	2
#12738	139

Subgroups — C139×#12738 #48007334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#12738 #48007334