Subgroups — C₁₃₉×#29283 #48023879

Order 35584.

618 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	#2331×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#2330×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	#2101×C₁₃₉	C₂	1	2	yes	maximal
4	#2057×C₁₃₉	C₂	1	2	yes	maximal
5	#1989×C₁₃₉	C₂	1	2	yes	maximal
6	#1947×C₁₃₉	C₂	1	2	yes	maximal
7	#1921×C₁₃₉	C₂	1	2	yes	maximal
8	#1918×C₁₃₉	C₂	1	2	yes	maximal
9	#1832×C₁₃₉	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
11	C₂×#125×C₁₃₉	C₂²	1	1	yes	characteristic
12	C₂×#107×C₁₃₉	C₂²	1	2	yes	—
13	C₂×#107×C₁₃₉	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
15	#391×C₁₃₉	—	2	2	no	—
16	#389×C₁₃₉	—	2	2	no	—
17	#389×C₁₃₉	C₂²	1	2	yes	—
18	#389×C₁₃₉	C₂²	1	2	yes	—
19	#387×C₁₃₉	—	2	2	no	—
20	#387×C₁₃₉	—	2	2	no	—
21	#384×C₁₃₉	C₂²	1	1	yes	characteristic
22	#382×C₁₃₉	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
25	#376×C₁₃₉	C₂²	1	1	yes	characteristic
26	#340×C₁₃₉	C₂²	1	2	yes	—
27	#340×C₁₃₉	C₂²	1	2	yes	—
28	#340×C₁₃₉	C₂²	1	2	yes	—
29	#340×C₁₃₉	C₂²	1	2	yes	—
30	#340×C₁₃₉	C₂²	1	2	yes	—
31	#340×C₁₃₉	C₂²	1	2	yes	—
32	#336×C₁₃₉	C₂²	1	2	yes	—
33	#336×C₁₃₉	C₂²	1	1	yes	characteristic
34	#336×C₁₃₉	C₂²	1	1	yes	characteristic
35	C₂²×Q₈×C₁₃₉	—	2	1	no	—
36	C₂²×C₈×C₁₃₉	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃₉	C₂³	1	2	yes	—
38	C₄⋊Q₈×C₁₃₉	—	2	4	no	—
39	#125×C₁₃₉	—	2	2	no	—
40	#125×C₁₃₉	Q₈	1	4	yes	complemented
41	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₃₉	—	2	1	no	—
43	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
45	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
46	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₃₉	—	2	2	no	—
51	C₂×C₄²×C₁₃₉	—	2	2	no	—
52	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
55	#107×C₁₃₉	—	2	4	no	—
56	#107×C₁₃₉	—	2	4	no	—
57	C₄⋊C₈×C₁₃₉	—	2	4	no	—
58	#95×C₁₃₉	—	2	2	no	—
59	#95×C₁₃₉	—	2	2	no	—
60	#95×C₁₃₉	—	2	1	no	—
61	#95×C₁₃₉	—	2	1	no	—
62	#95×C₁₃₉	—	2	2	no	—
63	#95×C₁₃₉	—	2	2	no	—
64	#95×C₁₃₉	D₄	1	2	yes	—
65	#95×C₁₃₉	D₄	1	2	yes	—
66	C₂×Q₈×C₁₃₉	—	2	4	no	—
67	C₂×Q₈×C₁₃₉	—	8	2	no	—
68	C₂²×C₄×C₁₃₉	—	2	2	no	—
69	C₂²×C₄×C₁₃₉	—	2	2	no	—
70	C₂²×C₄×C₁₃₉	—	2	2	no	—
71	C₂²×C₄×C₁₃₉	—	2	2	no	—
72	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₁₃₉	—	2	2	no	—
78	C₂×C₈×C₁₃₉	—	2	2	no	—
79	C₂×C₈×C₁₃₉	—	2	2	no	—
80	C₂×C₈×C₁₃₉	—	2	2	no	—
81	C₂×C₈×C₁₃₉	—	2	2	no	—
82	C₂×C₈×C₁₃₉	—	2	2	no	—
83	C₄⋊C₄×C₁₃₉	—	2	4	no	—
84	C₄⋊C₄×C₁₃₉	—	2	8	no	—
85	C₄⋊C₄×C₁₃₉	—	2	2	no	—
86	C₄⋊C₄×C₁₃₉	—	2	2	no	—
87	C₄⋊C₄×C₁₃₉	—	2	2	no	—
88	C₄⋊C₄×C₁₃₉	—	2	2	no	—
89	C₄⋊C₄×C₁₃₉	—	2	4	no	—
90	C₄⋊C₄×C₁₃₉	—	8	2	no	—
91	C₄⋊C₄×C₁₃₉	—	8	2	no	—
92	C₄⋊C₄×C₁₃₉	—	8	2	no	—
93	C₄⋊C₄×C₁₃₉	C₂×Q₈	1	4	yes	—
94	C₄⋊C₄×C₁₃₉	#41	1	4	yes	—
95	C₄²×C₁₃₉	—	2	2	no	—
96	C₄²×C₁₃₉	—	2	2	no	—
97	C₄²×C₁₃₉	—	8	2	no	—
98	C₄²×C₁₃₉	—	8	2	no	—
99	C₄²×C₁₃₉	C₂×Q₈	1	2	yes	—
100	C₄²×C₁₃₉	#41	1	2	yes	—
101	C₂³×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
102	Q₈×C₁₃₉	—	8	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	—	2	1	no	—
107	C₂×C₄×C₁₃₉	—	2	1	no	—
108	C₂×C₄×C₁₃₉	—	2	4	no	—
109	C₂×C₄×C₁₃₉	—	2	2	no	—
110	C₂×C₄×C₁₃₉	—	2	2	no	—
111	C₂×C₄×C₁₃₉	—	2	2	no	—
112	C₂×C₄×C₁₃₉	—	2	2	no	—
113	C₂×C₄×C₁₃₉	—	2	2	no	—
114	C₂×C₄×C₁₃₉	—	4	2	no	—
115	C₂×C₄×C₁₃₉	—	4	2	no	—
116	C₂×C₄×C₁₃₉	—	4	2	no	—
117	C₂×C₄×C₁₃₉	—	4	2	no	—
118	C₂×C₄×C₁₃₉	—	4	2	no	—
119	C₂×C₄×C₁₃₉	—	4	2	no	—
120	C₂×C₄×C₁₃₉	—	4	2	no	—
121	C₂×C₄×C₁₃₉	—	4	2	no	—
122	C₂×C₄×C₁₃₉	—	4	2	no	—
123	C₂×C₄×C₁₃₉	—	4	2	no	—
124	C₂×C₄×C₁₃₉	C₂²⋊Q₈	1	4	yes	—
125	C₂×C₄×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₁₃₉	C₂×#41	1	2	yes	—
128	C₂×C₄×C₁₃₉	2₋¹⁺⁴	1	2	yes	—
129	C₈×C₁₃₉	—	4	2	no	—
130	C₈×C₁₃₉	—	4	2	no	—
131	C₂²×C₁₃₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
132	C₂²×C₁₃₉	#525	1	1	yes	characteristic
133	C₂²×C₁₃₉	#536	1	1	yes	characteristic
134	C₂²×C₁₃₉	#576	1	2	yes	—
135	C₂²×C₁₃₉	#578	1	2	yes	—
136	C₄×C₁₃₉	—	2	4	no	—
137	C₄×C₁₃₉	—	4	4	no	—
138	C₄×C₁₃₉	—	4	1	no	—
139	C₄×C₁₃₉	—	4	4	no	—
140	C₄×C₁₃₉	—	4	1	no	—
141	C₄×C₁₃₉	—	4	2	no	—
142	C₄×C₁₃₉	—	8	2	no	—
143	C₄×C₁₃₉	—	8	2	no	—
144	C₄×C₁₃₉	#556	1	2	yes	—
145	C₄×C₁₃₉	#557	1	2	yes	—
146	C₂×C₁₃₉	#2581	1	1	yes	characteristic
147	C₂×C₁₃₉	#3083	1	2	yes	—
148	C₂×C₁₃₉	#3122	1	1	yes	characteristic
149	C₂×C₁₃₉	#3127	1	1	yes	characteristic
150	C₂×C₁₃₉	#3243	1	2	yes	—
151	#29283	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₁₃₉	#29283	1	1	yes	characteristic, direct product factor, Sylow
153	#2331	C₂×C₁₃₉	1	1	yes	characteristic
154	#2330	C₂×C₁₃₉	1	1	yes	characteristic
155	#2101	C₂×C₁₃₉	1	2	yes	—
156	#2057	C₂×C₁₃₉	1	2	yes	—
157	#1989	C₂×C₁₃₉	1	2	yes	—
158	#1947	C₂×C₁₃₉	1	2	yes	—
159	#1921	C₂×C₁₃₉	1	2	yes	—
160	#1918	C₂×C₁₃₉	1	2	yes	—
161	#1832	C₂×C₁₃₉	1	1	yes	characteristic
162	C₂×C₄⋊Q₈	C₂²×C₁₃₉	1	1	yes	characteristic
163	C₂×#125	C₂²×C₁₃₉	1	1	yes	characteristic
164	C₂×#107	C₂²×C₁₃₉	1	2	yes	—
165	C₂×#107	C₂²×C₁₃₉	1	2	yes	—
166	C₂×C₄⋊C₈	C₂²×C₁₃₉	1	2	yes	—
167	#391	—	2	2	no	—
168	#389	—	2	2	no	—
169	#389	C₂²×C₁₃₉	1	2	yes	—
170	#389	C₂²×C₁₃₉	1	2	yes	—
171	#387	—	2	2	no	—
172	#387	—	2	2	no	—
173	#384	C₂²×C₁₃₉	1	1	yes	characteristic
174	#382	C₂²×C₁₃₉	1	1	yes	characteristic
175	C₄×C₄⋊C₄	C₂²×C₁₃₉	1	2	yes	—
176	C₄×C₄⋊C₄	C₂²×C₁₃₉	1	2	yes	—
177	#376	C₂²×C₁₃₉	1	1	yes	characteristic
178	#340	C₂²×C₁₃₉	1	2	yes	—
179	#340	C₂²×C₁₃₉	1	2	yes	—
180	#340	C₂²×C₁₃₉	1	2	yes	—
181	#340	C₂²×C₁₃₉	1	2	yes	—
182	#340	C₂²×C₁₃₉	1	2	yes	—
183	#340	C₂²×C₁₃₉	1	2	yes	—
184	#336	C₂²×C₁₃₉	1	2	yes	—
185	#336	C₂²×C₁₃₉	1	1	yes	characteristic
186	#336	C₂²×C₁₃₉	1	1	yes	characteristic
187	C₂²×Q₈	—	2	1	no	—
188	C₂²×C₈	C₂³×C₁₃₉	1	2	yes	—
189	C₂²×C₈	C₂³×C₁₃₉	1	2	yes	—
190	C₄⋊Q₈	—	2	4	no	—
191	#125	—	2	2	no	—
192	#125	Q₈×C₁₃₉	1	4	yes	complemented
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	2	yes	—
199	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	2	yes	—
200	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	2	yes	—
201	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	2	yes	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
205	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
206	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
207	#107	—	2	4	no	—
208	#107	—	2	4	no	—
209	C₄⋊C₈	—	2	4	no	—
210	#95	—	2	2	no	—
211	#95	—	2	2	no	—
212	#95	—	2	1	no	—
213	#95	—	2	1	no	—
214	#95	—	2	2	no	—
215	#95	—	2	2	no	—
216	#95	D₄×C₁₃₉	1	2	yes	—
217	#95	D₄×C₁₃₉	1	2	yes	—
218	C₂×Q₈	—	2	4	no	—
219	C₂×Q₈	—	8	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
225	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
226	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
228	C₂²×C₄	C₂⁴×C₁₃₉	1	1	yes	characteristic
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	8	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	8	2	no	—
243	C₄⋊C₄	—	8	2	no	—
244	C₄⋊C₄	—	8	2	no	—
245	C₄⋊C₄	C₂×Q₈×C₁₃₉	1	4	yes	—
246	C₄⋊C₄	#41×C₁₃₉	1	4	yes	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	8	2	no	—
250	C₄²	—	8	2	no	—
251	C₄²	C₂×Q₈×C₁₃₉	1	2	yes	—
252	C₄²	#41×C₁₃₉	1	2	yes	—
253	C₂³	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
254	Q₈	—	8	4	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	C₂²⋊Q₈×C₁₃₉	1	4	yes	—
277	C₂×C₄	C₂²×Q₈×C₁₃₉	1	1	yes	characteristic
278	C₂×C₄	C₂×#41×C₁₃₉	1	1	yes	characteristic
279	C₂×C₄	C₂×#41×C₁₃₉	1	2	yes	—
280	C₂×C₄	2₋¹⁺⁴×C₁₃₉	1	2	yes	—
281	C₈	—	4	2	no	—
282	C₈	—	4	2	no	—
283	C₂²	C₂×C₂²⋊Q₈×C₁₃₉	1	1	yes	characteristic
284	C₂²	#525×C₁₃₉	1	1	yes	characteristic
285	C₂²	#536×C₁₃₉	1	1	yes	characteristic
286	C₂²	#576×C₁₃₉	1	2	yes	—
287	C₂²	#578×C₁₃₉	1	2	yes	—
288	C₄	—	2	4	no	—
289	C₄	—	4	4	no	—
290	C₄	—	4	1	no	—
291	C₄	—	4	4	no	—
292	C₄	—	4	1	no	—
293	C₄	—	4	2	no	—
294	C₄	—	8	2	no	—
295	C₄	—	8	2	no	—
296	C₄	#556×C₁₃₉	1	2	yes	—
297	C₄	#557×C₁₃₉	1	2	yes	—
298	C₂	#2581×C₁₃₉	1	1	yes	characteristic
299	C₂	#3083×C₁₃₉	1	2	yes	—
300	C₂	#3122×C₁₃₉	1	1	yes	characteristic
301	C₂	#3127×C₁₃₉	1	1	yes	characteristic
302	C₂	#3243×C₁₃₉	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₉ ◁ #376×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
40. #125×C₁₃₉	Q₈	semidirect
151. #29283	C₁₃₉	direct (N × H)
152. C₁₃₉	#29283	direct (N × H)
192. #125	Q₈×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#2331×C₁₃₉	2
#2330×C₁₃₉	2
#2101×C₁₃₉	2
#2057×C₁₃₉	2
#1989×C₁₃₉	2
#1947×C₁₃₉	2
#1921×C₁₃₉	2
#1918×C₁₃₉	2
#1832×C₁₃₉	2
#29283	139

Subgroups — C139×#29283 #48023879

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#29283 #48023879