Subgroups — C₁₃₉×#33457 #48028053

Order 35584.

682 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×SD₃₂×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#2950×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#2948×C₁₃₉	C₂	1	1	yes	maximal, characteristic
4	#2175×C₁₃₉	C₂	1	4	yes	maximal, complemented
5	C₄×SD₃₂×C₁₃₉	C₂	1	4	yes	maximal, complemented
6	#2158×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#2141×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#2140×C₁₃₉	C₂	1	1	yes	maximal, characteristic
9	#2108×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
11	C₂²×D₈×C₁₃₉	C₂²	1	1	yes	characteristic
12	#519×C₁₃₉	—	2	1	no	—
13	#518×C₁₃₉	—	2	1	no	—
14	C₂×SD₃₂×C₁₃₉	C₄	1	8	yes	complemented
15	C₂×SD₃₂×C₁₃₉	C₂²	1	2	yes	—
16	C₂×SD₃₂×C₁₃₉	C₂²	1	2	yes	—
17	C₂×M₃₂×C₁₃₉	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
19	#442×C₁₃₉	—	2	2	no	—
20	#441×C₁₃₉	—	2	2	no	—
21	C₄×Q₁₆×C₁₃₉	C₂²	1	4	yes	complemented
22	C₄×D₈×C₁₃₉	C₂²	1	4	yes	—
23	#427×C₁₃₉	C₂²	1	1	yes	characteristic
24	#427×C₁₃₉	C₂²	1	1	yes	characteristic
25	#419×C₁₃₉	—	2	1	no	—
26	#418×C₁₃₉	—	2	1	no	—
27	#405×C₁₃₉	C₂²	1	1	yes	characteristic
28	#367×C₁₃₉	C₂²	1	2	yes	complemented
29	#366×C₁₃₉	C₂²	1	2	yes	complemented
30	#358×C₁₃₉	C₂²	1	4	yes	complemented
31	#357×C₁₃₉	C₂²	1	4	yes	—
32	#346×C₁₃₉	C₂²	1	2	yes	complemented
33	C₄×C₁₆×C₁₃₉	C₂²	1	2	yes	complemented
34	C₂²×Q₈×C₁₃₉	—	2	1	no	—
35	C₂²×D₄×C₁₃₉	—	2	1	no	—
36	C₂×Q₁₆×C₁₃₉	—	2	1	no	—
37	C₂×Q₁₆×C₁₃₉	—	2	2	no	—
38	C₂×Q₁₆×C₁₃₉	C₂×C₄	1	4	yes	complemented
39	C₂×Q₁₆×C₁₃₉	C₂³	1	2	yes	—
40	C₂×D₈×C₁₃₉	—	2	1	no	—
41	C₂×D₈×C₁₃₉	—	2	2	no	—
42	C₂×D₈×C₁₃₉	C₂×C₄	1	4	yes	—
43	C₂×D₈×C₁₃₉	C₂³	1	2	yes	—
44	C₂×M₁₆×C₁₃₉	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
46	C₄×Q₈×C₁₃₉	—	2	4	no	—
47	C₄×Q₈×C₁₃₉	—	4	2	no	—
48	C₄×D₄×C₁₃₉	—	2	4	no	—
49	C₄×D₄×C₁₃₉	—	4	2	no	—
50	#117×C₁₃₉	—	2	1	no	—
51	#117×C₁₃₉	—	2	1	no	—
52	#117×C₁₃₉	—	4	2	no	—
53	#117×C₁₃₉	D₄	1	1	yes	characteristic
54	C₂×C₂²⋊C₄×C₁₃₉	—	4	2	no	—
55	SD₃₂×C₁₃₉	C₂×C₄	1	8	yes	complemented
56	SD₃₂×C₁₃₉	C₂×C₄	1	8	yes	complemented
57	M₃₂×C₁₃₉	—	2	2	no	—
58	C₂×C₁₆×C₁₃₉	C₂×C₄	1	4	yes	complemented
59	C₂×C₁₆×C₁₃₉	C₂³	1	2	yes	—
60	C₂×C₁₆×C₁₃₉	C₂³	1	1	yes	characteristic
61	C₂×C₁₆×C₁₃₉	C₂³	1	1	yes	characteristic
62	#107×C₁₃₉	C₂³	1	2	yes	—
63	#107×C₁₃₉	C₂³	1	2	yes	—
64	#106×C₁₃₉	—	2	1	no	—
65	#106×C₁₃₉	—	2	1	no	—
66	#103×C₁₃₉	—	2	4	no	—
67	#102×C₁₃₉	—	2	4	no	—
68	#97×C₁₃₉	D₄	1	2	yes	—
69	C₄×C₈×C₁₃₉	C₂³	1	2	yes	—
70	C₂⁴×C₁₃₉	—	4	1	no	—
71	C₂×Q₈×C₁₃₉	—	2	4	no	—
72	C₂×Q₈×C₁₃₉	—	2	2	no	—
73	C₂×Q₈×C₁₃₉	—	4	1	no	—
74	C₂×Q₈×C₁₃₉	—	4	2	no	—
75	C₂×D₄×C₁₃₉	—	2	4	no	—
76	C₂×D₄×C₁₃₉	—	2	2	no	—
77	C₂×D₄×C₁₃₉	—	4	1	no	—
78	C₂×D₄×C₁₃₉	—	4	2	no	—
79	C₂²×C₄×C₁₃₉	—	4	1	no	—
80	C₂²×C₄×C₁₃₉	—	4	4	no	—
81	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
82	Q₁₆×C₁₃₉	—	2	4	no	—
83	Q₁₆×C₁₃₉	—	2	2	no	—
84	Q₁₆×C₁₃₉	C₂²×C₄	1	4	yes	—
85	D₈×C₁₃₉	—	2	4	no	—
86	D₈×C₁₃₉	—	2	2	no	—
87	D₈×C₁₃₉	C₂²×C₄	1	4	yes	—
88	M₁₆×C₁₃₉	—	2	2	no	—
89	C₂×C₈×C₁₃₉	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₁₃₉	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₁₃₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃₉	#41	1	2	yes	—
93	C₂×C₈×C₁₃₉	C₂⁴	1	1	yes	characteristic
94	C₄⋊C₄×C₁₃₉	—	2	2	no	—
95	C₄⋊C₄×C₁₃₉	—	2	2	no	—
96	C₄⋊C₄×C₁₃₉	—	4	4	no	—
97	C₄⋊C₄×C₁₃₉	D₈	1	2	yes	—
98	C₂²⋊C₄×C₁₃₉	—	4	1	no	—
99	C₂²⋊C₄×C₁₃₉	—	4	1	no	—
100	C₂²⋊C₄×C₁₃₉	—	4	4	no	—
101	C₂²⋊C₄×C₁₃₉	D₈	1	2	yes	—
102	C₄²×C₁₃₉	—	4	1	no	—
103	C₄²×C₁₃₉	—	4	1	no	—
104	C₄²×C₁₃₉	—	4	4	no	—
105	C₄²×C₁₃₉	C₂×D₄	1	2	yes	—
106	C₁₆×C₁₃₉	—	2	2	no	—
107	C₁₆×C₁₃₉	C₂²×C₄	1	2	yes	—
108	C₁₆×C₁₃₉	C₂²×C₄	1	2	yes	—
109	C₂³×C₁₃₉	—	4	2	no	—
110	C₂³×C₁₃₉	—	4	4	no	—
111	C₂³×C₁₃₉	—	16	2	no	—
112	C₂³×C₁₃₉	C₂×D₈	1	1	yes	characteristic
113	Q₈×C₁₃₉	—	2	4	no	—
114	Q₈×C₁₃₉	—	4	2	no	—
115	Q₈×C₁₃₉	—	4	4	no	—
116	D₄×C₁₃₉	—	2	4	no	—
117	D₄×C₁₃₉	—	4	2	no	—
118	D₄×C₁₃₉	—	4	4	no	—
119	C₂×C₄×C₁₃₉	—	4	2	no	—
120	C₂×C₄×C₁₃₉	—	4	4	no	—
121	C₂×C₄×C₁₃₉	—	4	2	no	—
122	C₂×C₄×C₁₃₉	—	4	2	no	—
123	C₂×C₄×C₁₃₉	—	16	4	no	—
124	C₂×C₄×C₁₃₉	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₁₃₉	C₂×D₈	1	4	yes	—
126	C₂×C₄×C₁₃₉	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₁₃₉	#135	1	2	yes	—
128	C₂×C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
129	C₈×C₁₃₉	—	2	2	no	—
130	C₈×C₁₃₉	C₄×D₄	1	2	yes	—
131	C₈×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
132	C₈×C₁₃₉	C₂×#41	1	1	yes	characteristic
133	C₂²×C₁₃₉	—	2	2	no	—
134	C₂²×C₁₃₉	—	4	4	no	—
135	C₂²×C₁₃₉	—	8	4	no	—
136	C₂²×C₁₃₉	—	16	4	no	—
137	C₂²×C₁₃₉	C₄×D₈	1	2	yes	—
138	C₂²×C₁₃₉	C₂²×D₈	1	1	yes	characteristic
139	C₄×C₁₃₉	—	2	4	no	—
140	C₄×C₁₃₉	—	4	4	no	—
141	C₄×C₁₃₉	—	8	2	no	—
142	C₄×C₁₃₉	—	8	2	no	—
143	C₄×C₁₃₉	C₄×D₈	1	2	yes	—
144	C₄×C₁₃₉	C₂×C₄×D₄	1	1	yes	characteristic
145	C₄×C₁₃₉	C₂×#135	1	1	yes	characteristic
146	C₂×C₁₃₉	—	2	2	no	—
147	C₂×C₁₃₉	—	8	4	no	—
148	C₂×C₁₃₉	C₂×C₄×D₈	1	1	yes	characteristic
149	C₂×C₁₃₉	#3416	1	2	yes	—
150	#33457	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₁₃₉	#33457	1	1	yes	characteristic, direct product factor, Sylow
152	C₂²×SD₃₂	C₂×C₁₃₉	1	1	yes	characteristic
153	#2950	C₂×C₁₃₉	1	1	yes	characteristic, complemented
154	#2948	C₂×C₁₃₉	1	1	yes	characteristic
155	#2175	C₂×C₁₃₉	1	4	yes	complemented
156	C₄×SD₃₂	C₂×C₁₃₉	1	4	yes	complemented
157	#2158	C₂×C₁₃₉	1	1	yes	characteristic, complemented
158	#2141	C₂×C₁₃₉	1	1	yes	characteristic, complemented
159	#2140	C₂×C₁₃₉	1	1	yes	characteristic
160	#2108	C₂×C₁₃₉	1	1	yes	characteristic, complemented
161	C₂²×Q₁₆	C₂²×C₁₃₉	1	1	yes	characteristic
162	C₂²×D₈	C₂²×C₁₃₉	1	1	yes	characteristic
163	#519	—	2	1	no	—
164	#518	—	2	1	no	—
165	C₂×SD₃₂	C₄×C₁₃₉	1	8	yes	complemented
166	C₂×SD₃₂	C₂²×C₁₃₉	1	2	yes	—
167	C₂×SD₃₂	C₂²×C₁₃₉	1	2	yes	—
168	C₂×M₃₂	C₂²×C₁₃₉	1	1	yes	characteristic
169	C₂²×C₁₆	C₂²×C₁₃₉	1	1	yes	characteristic
170	#442	—	2	2	no	—
171	#441	—	2	2	no	—
172	C₄×Q₁₆	C₂²×C₁₃₉	1	4	yes	complemented
173	C₄×D₈	C₂²×C₁₃₉	1	4	yes	—
174	#427	C₂²×C₁₃₉	1	1	yes	characteristic
175	#427	C₂²×C₁₃₉	1	1	yes	characteristic
176	#419	—	2	1	no	—
177	#418	—	2	1	no	—
178	#405	C₂²×C₁₃₉	1	1	yes	characteristic
179	#367	C₂²×C₁₃₉	1	2	yes	complemented
180	#366	C₂²×C₁₃₉	1	2	yes	complemented
181	#358	C₂²×C₁₃₉	1	4	yes	complemented
182	#357	C₂²×C₁₃₉	1	4	yes	—
183	#346	C₂²×C₁₃₉	1	2	yes	complemented
184	C₄×C₁₆	C₂²×C₁₃₉	1	2	yes	complemented
185	C₂²×Q₈	—	2	1	no	—
186	C₂²×D₄	—	2	1	no	—
187	C₂×Q₁₆	—	2	1	no	—
188	C₂×Q₁₆	—	2	2	no	—
189	C₂×Q₁₆	C₂×C₄×C₁₃₉	1	4	yes	complemented
190	C₂×Q₁₆	C₂³×C₁₃₉	1	2	yes	—
191	C₂×D₈	—	2	1	no	—
192	C₂×D₈	—	2	2	no	—
193	C₂×D₈	C₂×C₄×C₁₃₉	1	4	yes	—
194	C₂×D₈	C₂³×C₁₃₉	1	2	yes	—
195	C₂×M₁₆	D₄×C₁₃₉	1	1	yes	characteristic
196	C₂²×C₈	C₂³×C₁₃₉	1	1	yes	characteristic
197	C₄×Q₈	—	2	4	no	—
198	C₄×Q₈	—	4	2	no	—
199	C₄×D₄	—	2	4	no	—
200	C₄×D₄	—	4	2	no	—
201	#117	—	2	1	no	—
202	#117	—	2	1	no	—
203	#117	—	4	2	no	—
204	#117	D₄×C₁₃₉	1	1	yes	characteristic
205	C₂×C₂²⋊C₄	—	4	2	no	—
206	SD₃₂	C₂×C₄×C₁₃₉	1	8	yes	complemented
207	SD₃₂	C₂×C₄×C₁₃₉	1	8	yes	complemented
208	M₃₂	—	2	2	no	—
209	C₂×C₁₆	C₂×C₄×C₁₃₉	1	4	yes	complemented
210	C₂×C₁₆	C₂³×C₁₃₉	1	2	yes	—
211	C₂×C₁₆	C₂³×C₁₃₉	1	1	yes	characteristic
212	C₂×C₁₆	C₂³×C₁₃₉	1	1	yes	characteristic
213	#107	C₂³×C₁₃₉	1	2	yes	—
214	#107	C₂³×C₁₃₉	1	2	yes	—
215	#106	—	2	1	no	—
216	#106	—	2	1	no	—
217	#103	—	2	4	no	—
218	#102	—	2	4	no	—
219	#97	D₄×C₁₃₉	1	2	yes	—
220	C₄×C₈	C₂³×C₁₃₉	1	2	yes	—
221	C₂⁴	—	4	1	no	—
222	C₂×Q₈	—	2	4	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	4	1	no	—
225	C₂×Q₈	—	4	2	no	—
226	C₂×D₄	—	2	4	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	4	1	no	—
229	C₂×D₄	—	4	2	no	—
230	C₂²×C₄	—	4	1	no	—
231	C₂²×C₄	—	4	4	no	—
232	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
233	Q₁₆	—	2	4	no	—
234	Q₁₆	—	2	2	no	—
235	Q₁₆	C₂²×C₄×C₁₃₉	1	4	yes	—
236	D₈	—	2	4	no	—
237	D₈	—	2	2	no	—
238	D₈	C₂²×C₄×C₁₃₉	1	4	yes	—
239	M₁₆	—	2	2	no	—
240	C₂×C₈	C₂²×C₄×C₁₃₉	1	2	yes	—
241	C₂×C₈	C₂×D₄×C₁₃₉	1	2	yes	—
242	C₂×C₈	C₂×D₄×C₁₃₉	1	1	yes	characteristic
243	C₂×C₈	#41×C₁₃₉	1	2	yes	—
244	C₂×C₈	C₂⁴×C₁₃₉	1	1	yes	characteristic
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	4	4	no	—
248	C₄⋊C₄	D₈×C₁₃₉	1	2	yes	—
249	C₂²⋊C₄	—	4	1	no	—
250	C₂²⋊C₄	—	4	1	no	—
251	C₂²⋊C₄	—	4	4	no	—
252	C₂²⋊C₄	D₈×C₁₃₉	1	2	yes	—
253	C₄²	—	4	1	no	—
254	C₄²	—	4	1	no	—
255	C₄²	—	4	4	no	—
256	C₄²	C₂×D₄×C₁₃₉	1	2	yes	—
257	C₁₆	—	2	2	no	—
258	C₁₆	C₂²×C₄×C₁₃₉	1	2	yes	—
259	C₁₆	C₂²×C₄×C₁₃₉	1	2	yes	—
260	C₂³	—	4	2	no	—
261	C₂³	—	4	4	no	—
262	C₂³	—	16	2	no	—
263	C₂³	C₂×D₈×C₁₃₉	1	1	yes	characteristic
264	Q₈	—	2	4	no	—
265	Q₈	—	4	4	no	—
266	Q₈	—	4	2	no	—
267	D₄	—	2	4	no	—
268	D₄	—	4	4	no	—
269	D₄	—	4	2	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	16	4	no	—
275	C₂×C₄	C₄×D₄×C₁₃₉	1	2	yes	—
276	C₂×C₄	C₂×D₈×C₁₃₉	1	4	yes	—
277	C₂×C₄	C₂×D₈×C₁₃₉	1	1	yes	characteristic
278	C₂×C₄	#135×C₁₃₉	1	2	yes	—
279	C₂×C₄	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
280	C₈	—	2	2	no	—
281	C₈	C₄×D₄×C₁₃₉	1	2	yes	—
282	C₈	C₂³×C₄×C₁₃₉	1	1	yes	characteristic
283	C₈	C₂×#41×C₁₃₉	1	1	yes	characteristic
284	C₂²	—	2	2	no	—
285	C₂²	—	4	4	no	—
286	C₂²	—	8	4	no	—
287	C₂²	—	16	4	no	—
288	C₂²	C₄×D₈×C₁₃₉	1	2	yes	—
289	C₂²	C₂²×D₈×C₁₃₉	1	1	yes	characteristic
290	C₄	—	2	4	no	—
291	C₄	—	4	4	no	—
292	C₄	—	8	2	no	—
293	C₄	—	8	2	no	—
294	C₄	C₄×D₈×C₁₃₉	1	2	yes	—
295	C₄	C₂×C₄×D₄×C₁₃₉	1	1	yes	characteristic
296	C₄	C₂×#135×C₁₃₉	1	1	yes	characteristic
297	C₂	—	2	2	no	—
298	C₂	—	8	4	no	—
299	C₂	C₂×C₄×D₈×C₁₃₉	1	1	yes	characteristic
300	C₂	#3416×C₁₃₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₉	C₂²×D₈
commutator	C₈	C₂³×C₄×C₁₃₉
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₉
prime radical	C₂²×D₈×C₁₃₉	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₉ ◁ #117×C₁₃₉ ◁ #405×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2950×C₁₃₉	C₂	semidirect
4. #2175×C₁₃₉	C₂	semidirect
5. C₄×SD₃₂×C₁₃₉	C₂	semidirect
6. #2158×C₁₃₉	C₂	semidirect
7. #2141×C₁₃₉	C₂	semidirect
9. #2108×C₁₃₉	C₂	semidirect
14. C₂×SD₃₂×C₁₃₉	C₄	semidirect
21. C₄×Q₁₆×C₁₃₉	C₂²	semidirect
28. #367×C₁₃₉	C₂²	semidirect
29. #366×C₁₃₉	C₂²	semidirect
30. #358×C₁₃₉	C₂²	semidirect
32. #346×C₁₃₉	C₂²	semidirect
33. C₄×C₁₆×C₁₃₉	C₂²	semidirect
38. C₂×Q₁₆×C₁₃₉	C₂×C₄	semidirect
55. SD₃₂×C₁₃₉	C₂×C₄	semidirect
56. SD₃₂×C₁₃₉	C₂×C₄	semidirect
58. C₂×C₁₆×C₁₃₉	C₂×C₄	semidirect
150. #33457	C₁₃₉	direct (N × H)
151. C₁₃₉	#33457	direct (N × H)
153. #2950	C₂×C₁₃₉	semidirect
155. #2175	C₂×C₁₃₉	semidirect
156. C₄×SD₃₂	C₂×C₁₃₉	semidirect
157. #2158	C₂×C₁₃₉	semidirect
158. #2141	C₂×C₁₃₉	semidirect
160. #2108	C₂×C₁₃₉	semidirect
165. C₂×SD₃₂	C₄×C₁₃₉	semidirect
172. C₄×Q₁₆	C₂²×C₁₃₉	semidirect
179. #367	C₂²×C₁₃₉	semidirect
180. #366	C₂²×C₁₃₉	semidirect
181. #358	C₂²×C₁₃₉	semidirect
183. #346	C₂²×C₁₃₉	semidirect
184. C₄×C₁₆	C₂²×C₁₃₉	semidirect
189. C₂×Q₁₆	C₂×C₄×C₁₃₉	semidirect
206. SD₃₂	C₂×C₄×C₁₃₉	semidirect
207. SD₃₂	C₂×C₄×C₁₃₉	semidirect
209. C₂×C₁₆	C₂×C₄×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×SD₃₂×C₁₃₉	2
#2950×C₁₃₉	2
#2948×C₁₃₉	2
#2175×C₁₃₉	2
C₄×SD₃₂×C₁₃₉	2
#2158×C₁₃₉	2
#2141×C₁₃₉	2
#2140×C₁₃₉	2
#2108×C₁₃₉	2
#33457	139

Subgroups — C139×#33457 #48028053

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#33457 #48028053