Subgroups — C₁₃₉×#13008 #48007604

Order 35584.

516 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3054×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#2911×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#2145×C₁₃₉	C₂	1	2	yes	maximal, complemented
4	C₂×#349×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#1384×C₁₃₉	C₂	1	2	yes	maximal, complemented
6	C₂×#135×C₁₃₉	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
8	#525×C₁₃₉	—	2	1	no	—
9	C₂×#117×C₁₃₉	—	2	1	no	—
10	C₂×M₃₂×C₁₃₉	C₂²	1	2	yes	—
11	#470×C₁₃₉	C₄	1	2	yes	complemented
12	#469×C₁₃₉	C₄	1	2	yes	complemented
13	#468×C₁₃₉	—	2	2	no	—
14	#428×C₁₃₉	—	2	4	no	—
15	#427×C₁₃₉	—	2	1	no	—
16	#427×C₁₃₉	C₂²	1	1	yes	characteristic
17	#427×C₁₃₉	C₂²	1	2	yes	—
18	#416×C₁₃₉	—	2	1	no	—
19	#360×C₁₃₉	—	2	4	no	—
20	#349×C₁₃₉	—	2	2	no	—
21	C₂×#41×C₁₃₉	—	2	1	no	—
22	C₂³×C₄×C₁₃₉	D₄	1	1	yes	characteristic
23	#135×C₁₃₉	—	2	2	no	—
24	#135×C₁₃₉	—	2	2	no	—
25	C₂×Q₁₆×C₁₃₉	C₂×C₄	1	1	yes	characteristic, complemented
26	C₂×SD₁₆×C₁₃₉	—	2	1	no	—
27	C₂×D₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic, complemented
28	C₂×M₁₆×C₁₃₉	—	2	2	no	—
29	C₂×M₁₆×C₁₃₉	—	2	2	no	—
30	C₂×M₁₆×C₁₃₉	C₂×C₄	1	2	yes	—
31	C₂×M₁₆×C₁₃₉	D₄	1	2	yes	—
32	C₂²×C₈×C₁₃₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
34	#123×C₁₃₉	—	4	1	no	—
35	C₂²⋊Q₈×C₁₃₉	—	2	2	no	—
36	#121×C₁₃₉	—	2	2	no	—
37	#120×C₁₃₉	—	4	1	no	—
38	C₄×D₄×C₁₃₉	—	4	2	no	—
39	#117×C₁₃₉	—	2	2	no	—
40	#117×C₁₃₉	—	2	1	no	—
41	#117×C₁₃₉	—	2	2	no	—
42	#117×C₁₃₉	—	4	1	no	—
43	#117×C₁₃₉	—	4	1	no	—
44	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
45	C₂×C₂²⋊C₄×C₁₃₉	—	4	1	no	—
46	C₂×C₄²×C₁₃₉	—	4	1	no	—
47	M₃₂×C₁₃₉	—	2	2	no	—
48	M₃₂×C₁₃₉	—	2	2	no	—
49	C₂×C₁₆×C₁₃₉	—	2	2	no	—
50	#107×C₁₃₉	—	2	2	no	—
51	#107×C₁₃₉	—	2	2	no	—
52	#107×C₁₃₉	—	2	1	no	—
53	#106×C₁₃₉	—	2	2	no	—
54	#106×C₁₃₉	C₂×C₄	1	2	yes	complemented
55	#106×C₁₃₉	D₄	1	4	yes	complemented
56	#103×C₁₃₉	—	2	2	no	—
57	#102×C₁₃₉	—	2	2	no	—
58	C₂⁴×C₁₃₉	D₈	1	1	yes	characteristic
59	#41×C₁₃₉	—	4	1	no	—
60	#41×C₁₃₉	—	4	1	no	—
61	#41×C₁₃₉	—	4	1	no	—
62	#41×C₁₃₉	—	4	1	no	—
63	C₂×Q₈×C₁₃₉	—	2	1	no	—
64	C₂×D₄×C₁₃₉	—	2	1	no	—
65	C₂×D₄×C₁₃₉	—	4	2	no	—
66	C₂×D₄×C₁₃₉	—	4	1	no	—
67	C₂²×C₄×C₁₃₉	—	2	1	no	—
68	C₂²×C₄×C₁₃₉	—	2	1	no	—
69	C₂²×C₄×C₁₃₉	—	2	1	no	—
70	C₂²×C₄×C₁₃₉	—	2	1	no	—
71	C₂²×C₄×C₁₃₉	—	4	2	no	—
72	C₂²×C₄×C₁₃₉	—	4	1	no	—
73	C₂²×C₄×C₁₃₉	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₁₃₉	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₁₃₉	SD₁₆	1	2	yes	—
76	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
77	Q₁₆×C₁₃₉	—	2	2	no	—
78	SD₁₆×C₁₃₉	—	4	2	no	—
79	D₈×C₁₃₉	—	2	2	no	—
80	M₁₆×C₁₃₉	—	2	4	no	—
81	M₁₆×C₁₃₉	—	2	4	no	—
82	C₂×C₈×C₁₃₉	—	2	1	no	—
83	C₂×C₈×C₁₃₉	—	2	1	no	—
84	C₂×C₈×C₁₃₉	C₂²⋊C₄	1	2	yes	—
85	C₂×C₈×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃₉	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₁₃₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃₉	#41	1	2	yes	—
89	C₄⋊C₄×C₁₃₉	—	2	2	no	—
90	C₄⋊C₄×C₁₃₉	—	2	4	no	—
91	C₄⋊C₄×C₁₃₉	—	4	2	no	—
92	C₄⋊C₄×C₁₃₉	—	4	2	no	—
93	C₂²⋊C₄×C₁₃₉	—	4	2	no	—
94	C₂²⋊C₄×C₁₃₉	—	4	2	no	—
95	C₂²⋊C₄×C₁₃₉	—	4	2	no	—
96	C₂²⋊C₄×C₁₃₉	—	4	1	no	—
97	C₂²⋊C₄×C₁₃₉	—	4	2	no	—
98	C₄²×C₁₃₉	—	4	2	no	—
99	C₄²×C₁₃₉	—	4	1	no	—
100	C₄²×C₁₃₉	—	4	2	no	—
101	C₁₆×C₁₃₉	—	4	2	no	—
102	C₂³×C₁₃₉	—	2	1	no	—
103	C₂³×C₁₃₉	—	2	1	no	—
104	C₂³×C₁₃₉	—	4	1	no	—
105	C₂³×C₁₃₉	—	8	1	no	—
106	C₂³×C₁₃₉	#102	1	2	yes	—
107	C₂³×C₁₃₉	C₂×D₈	1	1	yes	characteristic
108	Q₈×C₁₃₉	—	4	1	no	—
109	Q₈×C₁₃₉	—	4	1	no	—
110	D₄×C₁₃₉	—	4	1	no	—
111	D₄×C₁₃₉	—	4	1	no	—
112	D₄×C₁₃₉	—	8	4	no	—
113	D₄×C₁₃₉	—	8	1	no	—
114	D₄×C₁₃₉	—	8	1	no	—
115	C₂×C₄×C₁₃₉	—	2	1	no	—
116	C₂×C₄×C₁₃₉	—	2	1	no	—
117	C₂×C₄×C₁₃₉	—	2	2	no	—
118	C₂×C₄×C₁₃₉	—	2	2	no	—
119	C₂×C₄×C₁₃₉	—	2	2	no	—
120	C₂×C₄×C₁₃₉	—	2	2	no	—
121	C₂×C₄×C₁₃₉	—	4	2	no	—
122	C₂×C₄×C₁₃₉	—	4	1	no	—
123	C₂×C₄×C₁₃₉	—	4	4	no	—
124	C₂×C₄×C₁₃₉	—	8	4	no	—
125	C₂×C₄×C₁₃₉	—	8	1	no	—
126	C₂×C₄×C₁₃₉	—	8	1	no	—
127	C₂×C₄×C₁₃₉	#102	1	2	yes	—
128	C₂×C₄×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃₉	#120	1	1	yes	characteristic
130	C₂×C₄×C₁₃₉	#123	1	1	yes	characteristic
131	C₂×C₄×C₁₃₉	C₂×SD₁₆	1	1	yes	characteristic
132	C₂×C₄×C₁₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₄×C₁₃₉	#137	1	1	yes	characteristic
134	C₈×C₁₃₉	—	2	2	no	—
135	C₈×C₁₃₉	C₄×D₄	1	2	yes	—
136	C₈×C₁₃₉	#121	1	2	yes	—
137	C₂²×C₁₃₉	—	2	2	no	—
138	C₂²×C₁₃₉	—	4	1	no	—
139	C₂²×C₁₃₉	—	4	2	no	—
140	C₂²×C₁₃₉	—	4	2	no	—
141	C₂²×C₁₃₉	—	8	1	no	—
142	C₂²×C₁₃₉	—	8	2	no	—
143	C₂²×C₁₃₉	—	8	1	no	—
144	C₂²×C₁₃₉	C₂×#102	1	1	yes	characteristic
145	C₂²×C₁₃₉	#447	1	1	yes	characteristic
146	C₂²×C₁₃₉	#480	1	1	yes	characteristic
147	C₄×C₁₃₉	—	2	2	no	—
148	C₄×C₁₃₉	—	8	4	no	—
149	C₄×C₁₃₉	—	8	2	no	—
150	C₄×C₁₃₉	—	8	1	no	—
151	C₄×C₁₃₉	#386	1	1	yes	characteristic
152	C₄×C₁₃₉	#417	1	1	yes	characteristic
153	C₄×C₁₃₉	#448	1	1	yes	characteristic
154	C₄×C₁₃₉	#481	1	1	yes	characteristic
155	C₂×C₁₃₉	—	2	1	no	—
156	C₂×C₁₃₉	—	2	1	no	—
157	C₂×C₁₃₉	—	2	1	no	—
158	C₂×C₁₃₉	—	4	2	no	—
159	C₂×C₁₃₉	—	16	1	no	—
160	C₂×C₁₃₉	#1874	1	1	yes	characteristic
161	#13008	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₁₃₉	#13008	1	1	yes	characteristic, direct product factor, Sylow
163	#3054	C₂×C₁₃₉	1	1	yes	characteristic
164	#2911	C₂×C₁₃₉	1	1	yes	characteristic, complemented
165	#2145	C₂×C₁₃₉	1	2	yes	complemented
166	C₂×#349	C₂×C₁₃₉	1	1	yes	characteristic, complemented
167	#1384	C₂×C₁₃₉	1	2	yes	complemented
168	C₂×#135	C₂²×C₁₃₉	1	1	yes	characteristic
169	C₂²×M₁₆	C₂²×C₁₃₉	1	1	yes	characteristic
170	#525	—	2	1	no	—
171	C₂×#117	—	2	1	no	—
172	C₂×M₃₂	C₂²×C₁₃₉	1	2	yes	—
173	#470	C₄×C₁₃₉	1	2	yes	complemented
174	#469	C₄×C₁₃₉	1	2	yes	complemented
175	#468	—	2	2	no	—
176	#428	—	2	4	no	—
177	#427	—	2	1	no	—
178	#427	C₂²×C₁₃₉	1	1	yes	characteristic
179	#427	C₂²×C₁₃₉	1	2	yes	—
180	#416	—	2	1	no	—
181	#360	—	2	4	no	—
182	#349	—	2	2	no	—
183	C₂×#41	—	2	1	no	—
184	C₂³×C₄	D₄×C₁₃₉	1	1	yes	characteristic
185	#135	—	2	2	no	—
186	#135	—	2	2	no	—
187	C₂×Q₁₆	C₂×C₄×C₁₃₉	1	1	yes	characteristic, complemented
188	C₂×SD₁₆	—	2	1	no	—
189	C₂×D₈	C₂×C₄×C₁₃₉	1	1	yes	characteristic, complemented
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	C₂×C₄×C₁₃₉	1	2	yes	—
193	C₂×M₁₆	D₄×C₁₃₉	1	2	yes	—
194	C₂²×C₈	D₄×C₁₃₉	1	1	yes	characteristic
195	C₂²×C₈	C₂³×C₁₃₉	1	1	yes	characteristic
196	#123	—	4	1	no	—
197	C₂²⋊Q₈	—	2	2	no	—
198	#121	—	2	2	no	—
199	#120	—	4	1	no	—
200	C₄×D₄	—	4	2	no	—
201	#117	—	2	2	no	—
202	#117	—	2	2	no	—
203	#117	—	2	1	no	—
204	#117	—	4	1	no	—
205	#117	—	4	1	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₂²⋊C₄	—	4	1	no	—
208	C₂×C₄²	—	4	1	no	—
209	M₃₂	—	2	2	no	—
210	M₃₂	—	2	2	no	—
211	C₂×C₁₆	—	2	2	no	—
212	#107	—	2	2	no	—
213	#107	—	2	2	no	—
214	#107	—	2	1	no	—
215	#106	—	2	2	no	—
216	#106	C₂×C₄×C₁₃₉	1	2	yes	complemented
217	#106	D₄×C₁₃₉	1	4	yes	complemented
218	#103	—	2	2	no	—
219	#102	—	2	2	no	—
220	C₂⁴	D₈×C₁₃₉	1	1	yes	characteristic
221	#41	—	4	1	no	—
222	#41	—	4	1	no	—
223	#41	—	4	1	no	—
224	#41	—	4	1	no	—
225	C₂×Q₈	—	2	1	no	—
226	C₂×D₄	—	2	1	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂×D₄	—	4	1	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	—	4	1	no	—
235	C₂²×C₄	C₂²⋊C₄×C₁₃₉	1	2	yes	—
236	C₂²×C₄	D₈×C₁₃₉	1	1	yes	characteristic
237	C₂²×C₄	SD₁₆×C₁₃₉	1	2	yes	—
238	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
239	Q₁₆	—	2	2	no	—
240	SD₁₆	—	4	2	no	—
241	D₈	—	2	2	no	—
242	M₁₆	—	2	4	no	—
243	M₁₆	—	2	4	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	C₂²⋊C₄×C₁₃₉	1	2	yes	—
247	C₂×C₈	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
248	C₂×C₈	C₂×D₄×C₁₃₉	1	2	yes	—
249	C₂×C₈	C₂×D₄×C₁₃₉	1	1	yes	characteristic
250	C₂×C₈	#41×C₁₃₉	1	2	yes	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₂²⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₂²⋊C₄	—	4	1	no	—
259	C₂²⋊C₄	—	4	2	no	—
260	C₄²	—	4	2	no	—
261	C₄²	—	4	1	no	—
262	C₄²	—	4	2	no	—
263	C₁₆	—	4	2	no	—
264	C₂³	—	2	1	no	—
265	C₂³	—	2	1	no	—
266	C₂³	—	4	1	no	—
267	C₂³	—	8	1	no	—
268	C₂³	#102×C₁₃₉	1	2	yes	—
269	C₂³	C₂×D₈×C₁₃₉	1	1	yes	characteristic
270	Q₈	—	4	1	no	—
271	Q₈	—	4	1	no	—
272	D₄	—	4	1	no	—
273	D₄	—	4	1	no	—
274	D₄	—	8	4	no	—
275	D₄	—	8	1	no	—
276	D₄	—	8	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	8	4	no	—
287	C₂×C₄	—	8	1	no	—
288	C₂×C₄	—	8	1	no	—
289	C₂×C₄	#102×C₁₃₉	1	2	yes	—
290	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
291	C₂×C₄	#120×C₁₃₉	1	1	yes	characteristic
292	C₂×C₄	#123×C₁₃₉	1	1	yes	characteristic
293	C₂×C₄	C₂×SD₁₆×C₁₃₉	1	1	yes	characteristic
294	C₂×C₄	AGL(1,ℤ/8)×C₁₃₉	1	1	yes	characteristic
295	C₂×C₄	#137×C₁₃₉	1	1	yes	characteristic
296	C₈	—	2	2	no	—
297	C₈	C₄×D₄×C₁₃₉	1	2	yes	—
298	C₈	#121×C₁₃₉	1	2	yes	—
299	C₂²	—	2	2	no	—
300	C₂²	—	4	1	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	—	8	1	no	—
304	C₂²	—	8	2	no	—
305	C₂²	—	8	1	no	—
306	C₂²	C₂×#102×C₁₃₉	1	1	yes	characteristic
307	C₂²	#447×C₁₃₉	1	1	yes	characteristic
308	C₂²	#480×C₁₃₉	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	8	4	no	—
311	C₄	—	8	2	no	—
312	C₄	—	8	1	no	—
313	C₄	#386×C₁₃₉	1	1	yes	characteristic
314	C₄	#417×C₁₃₉	1	1	yes	characteristic
315	C₄	#448×C₁₃₉	1	1	yes	characteristic
316	C₄	#481×C₁₃₉	1	1	yes	characteristic
317	C₂	—	2	1	no	—
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	—	4	2	no	—
321	C₂	—	16	1	no	—
322	C₂	#1874×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃₉	#481
commutator	C₂×C₈	C₂²×C₄×C₁₃₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₉
socle	C₂×C₁₃₉	#1874
prime radical	#3054×C₁₃₉	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃₉ ◁ C₂²×C₄×C₁₃₉ ◁ C₂²×M₁₆×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2911×C₁₃₉	C₂	semidirect
3. #2145×C₁₃₉	C₂	semidirect
4. C₂×#349×C₁₃₉	C₂	semidirect
5. #1384×C₁₃₉	C₂	semidirect
11. #470×C₁₃₉	C₄	semidirect
12. #469×C₁₃₉	C₄	semidirect
25. C₂×Q₁₆×C₁₃₉	C₂×C₄	semidirect
27. C₂×D₈×C₁₃₉	C₂×C₄	semidirect
54. #106×C₁₃₉	C₂×C₄	semidirect
55. #106×C₁₃₉	D₄	semidirect
161. #13008	C₁₃₉	direct (N × H)
162. C₁₃₉	#13008	direct (N × H)
164. #2911	C₂×C₁₃₉	semidirect
165. #2145	C₂×C₁₃₉	semidirect
166. C₂×#349	C₂×C₁₃₉	semidirect
167. #1384	C₂×C₁₃₉	semidirect
173. #470	C₄×C₁₃₉	semidirect
174. #469	C₄×C₁₃₉	semidirect
187. C₂×Q₁₆	C₂×C₄×C₁₃₉	semidirect
189. C₂×D₈	C₂×C₄×C₁₃₉	semidirect
216. #106	C₂×C₄×C₁₃₉	semidirect
217. #106	D₄×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#3054×C₁₃₉	2
#2911×C₁₃₉	2
#2145×C₁₃₉	2
C₂×#349×C₁₃₉	2
#1384×C₁₃₉	2
#13008	139

Subgroups — C139×#13008 #48007604

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#13008 #48007604