Subgroups — C₁₃₉×#23824 #48018420

Order 35584.

882 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#127×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	C₄×#125×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#2099×C₁₃₉	C₂	1	2	yes	maximal, complemented
4	#2073×C₁₃₉	C₂	1	2	yes	maximal, complemented
5	#2041×C₁₃₉	C₂	1	2	yes	maximal
6	#2025×C₁₃₉	C₂	1	2	yes	maximal, complemented
7	#1762×C₁₃₉	C₂	1	4	yes	maximal, complemented
8	#1745×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#127×C₁₃₉	C₂²	1	1	yes	characteristic
10	C₂×#125×C₁₃₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₄×C₁₃₉	—	2	2	no	—
12	C₂×C₄×D₄×C₁₃₉	C₂²	1	2	yes	—
13	C₂×#102×C₁₃₉	C₂²	1	4	yes	—
14	C₂×#102×C₁₃₉	C₂²	1	4	yes	—
15	C₂×#97×C₁₃₉	C₂²	1	2	yes	—
16	#390×C₁₃₉	—	2	2	no	—
17	#390×C₁₃₉	C₂²	1	2	yes	—
18	#389×C₁₃₉	C₂²	1	2	yes	—
19	#387×C₁₃₉	—	2	2	no	—
20	#384×C₁₃₉	C₂²	1	1	yes	characteristic
21	#382×C₁₃₉	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₁₃₉	—	2	2	no	—
23	C₄×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
24	C₄³×C₁₃₉	C₂²	1	1	yes	characteristic
25	#340×C₁₃₉	C₂²	1	4	yes	—
26	#340×C₁₃₉	C₂²	1	4	yes	—
27	#336×C₁₃₉	C₂²	1	2	yes	—
28	#336×C₁₃₉	C₂²	1	2	yes	—
29	C₂²×D₄×C₁₃₉	—	2	2	no	—
30	C₂²×D₄×C₁₃₉	C₂³	1	2	yes	—
31	C₂³×C₄×C₁₃₉	—	2	2	no	—
32	C₂²×C₈×C₁₃₉	C₂³	1	4	yes	—
33	#127×C₁₃₉	—	2	4	no	—
34	#125×C₁₃₉	—	2	4	no	—
35	C₄×D₄×C₁₃₉	—	2	8	no	—
36	C₄×D₄×C₁₃₉	—	4	8	no	—
37	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
38	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
40	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	2	yes	—
41	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	2	yes	—
43	C₂×C₂²⋊C₄×C₁₃₉	—	2	4	no	—
44	C₂×C₂²⋊C₄×C₁₃₉	—	2	2	no	—
45	C₂×C₄²×C₁₃₉	—	2	2	no	—
46	C₂×C₄²×C₁₃₉	D₄	1	2	yes	—
47	C₂×C₄²×C₁₃₉	D₄	1	2	yes	—
48	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
51	#102×C₁₃₉	—	2	8	no	—
52	#102×C₁₃₉	—	2	8	no	—
53	#97×C₁₃₉	—	2	4	no	—
54	#95×C₁₃₉	—	2	2	no	—
55	#95×C₁₃₉	—	2	2	no	—
56	#95×C₁₃₉	—	2	2	no	—
57	C₂⁴×C₁₃₉	—	2	2	no	—
58	C₂×D₄×C₁₃₉	—	2	8	no	—
59	C₂×D₄×C₁₃₉	—	2	4	no	—
60	C₂×D₄×C₁₃₉	—	2	4	no	—
61	C₂×D₄×C₁₃₉	—	2	2	no	—
62	C₂×D₄×C₁₃₉	—	4	4	no	—
63	C₂×D₄×C₁₃₉	—	4	4	no	—
64	C₂×D₄×C₁₃₉	#41	1	4	yes	—
65	C₂²×C₄×C₁₃₉	—	2	2	no	—
66	C₂²×C₄×C₁₃₉	—	2	2	no	—
67	C₂²×C₄×C₁₃₉	—	2	2	no	—
68	C₂²×C₄×C₁₃₉	—	4	8	no	—
69	C₂²×C₄×C₁₃₉	—	4	4	no	—
70	C₂²×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
75	C₂×C₈×C₁₃₉	—	2	4	no	—
76	C₂×C₈×C₁₃₉	—	2	4	no	—
77	C₂×C₈×C₁₃₉	—	2	4	no	—
78	C₄⋊C₄×C₁₃₉	—	2	8	no	—
79	C₄⋊C₄×C₁₃₉	—	2	2	no	—
80	C₄⋊C₄×C₁₃₉	—	2	4	no	—
81	C₄⋊C₄×C₁₃₉	—	2	4	no	—
82	C₄⋊C₄×C₁₃₉	—	4	4	no	—
83	C₄⋊C₄×C₁₃₉	—	4	4	no	—
84	C₄⋊C₄×C₁₃₉	#41	1	4	yes	—
85	C₂²⋊C₄×C₁₃₉	—	4	8	no	—
86	C₂²⋊C₄×C₁₃₉	—	4	4	no	—
87	C₄²×C₁₃₉	—	2	4	no	—
88	C₄²×C₁₃₉	—	2	2	no	—
89	C₄²×C₁₃₉	—	2	2	no	—
90	C₄²×C₁₃₉	—	4	2	no	—
91	C₄²×C₁₃₉	—	4	4	no	—
92	C₄²×C₁₃₉	#41	1	4	yes	—
93	C₂³×C₁₃₉	—	2	4	no	—
94	C₂³×C₁₃₉	—	4	2	no	—
95	C₂³×C₁₃₉	—	4	2	no	—
96	C₂³×C₁₃₉	—	4	4	no	—
97	C₂³×C₁₃₉	—	4	4	no	—
98	C₂³×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
99	D₄×C₁₃₉	—	2	8	no	—
100	D₄×C₁₃₉	—	4	16	no	—
101	D₄×C₁₃₉	—	4	4	no	—
102	C₂×C₄×C₁₃₉	—	2	2	no	—
103	C₂×C₄×C₁₃₉	—	2	2	no	—
104	C₂×C₄×C₁₃₉	—	2	2	no	—
105	C₂×C₄×C₁₃₉	—	2	2	no	—
106	C₂×C₄×C₁₃₉	—	2	2	no	—
107	C₂×C₄×C₁₃₉	—	2	4	no	—
108	C₂×C₄×C₁₃₉	—	2	2	no	—
109	C₂×C₄×C₁₃₉	—	2	2	no	—
110	C₂×C₄×C₁₃₉	—	4	2	no	—
111	C₂×C₄×C₁₃₉	—	4	2	no	—
112	C₂×C₄×C₁₃₉	—	4	2	no	—
113	C₂×C₄×C₁₃₉	—	4	2	no	—
114	C₂×C₄×C₁₃₉	—	4	4	no	—
115	C₂×C₄×C₁₃₉	—	4	2	no	—
116	C₂×C₄×C₁₃₉	—	4	2	no	—
117	C₂×C₄×C₁₃₉	—	8	8	no	—
118	C₂×C₄×C₁₃₉	#124	1	2	yes	—
119	C₂×C₄×C₁₃₉	#124	1	2	yes	—
120	C₂×C₄×C₁₃₉	AGL(1,ℤ/8)	1	4	yes	—
121	C₂×C₄×C₁₃₉	C₂×#41	1	2	yes	—
122	C₂×C₄×C₁₃₉	C₂×#41	1	2	yes	—
123	C₂×C₄×C₁₃₉	C₂×#41	1	2	yes	—
124	C₈×C₁₃₉	—	4	4	no	—
125	C₂²×C₁₃₉	—	4	8	no	—
126	C₂²×C₁₃₉	—	4	4	no	—
127	C₂²×C₁₃₉	—	8	4	no	—
128	C₂²×C₁₃₉	—	8	2	no	—
129	C₂²×C₁₃₉	—	8	2	no	—
130	C₂²×C₁₃₉	#525	1	1	yes	characteristic
131	C₂²×C₁₃₉	#525	1	1	yes	characteristic
132	C₂²×C₁₃₉	C₂×#124	1	1	yes	characteristic
133	C₂²×C₁₃₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
134	C₂²×C₁₃₉	#575	1	2	yes	—
135	C₄×C₁₃₉	—	2	4	no	—
136	C₄×C₁₃₉	—	2	2	no	—
137	C₄×C₁₃₉	—	2	2	no	—
138	C₄×C₁₃₉	—	4	2	no	—
139	C₄×C₁₃₉	—	4	4	no	—
140	C₄×C₁₃₉	—	8	2	no	—
141	C₄×C₁₃₉	—	8	2	no	—
142	C₄×C₁₃₉	#529	1	4	yes	—
143	C₂×C₁₃₉	—	8	4	no	—
144	C₂×C₁₃₉	#2398	1	1	yes	characteristic
145	C₂×C₁₃₉	#3114	1	2	yes	—
146	C₂×C₁₃₉	#3115	1	2	yes	—
147	C₂×C₁₃₉	#3138	1	1	yes	characteristic
148	C₂×C₁₃₉	#3142	1	1	yes	characteristic
149	#23824	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₁₃₉	#23824	1	1	yes	characteristic, direct product factor, Sylow
151	C₄×#127	C₂×C₁₃₉	1	1	yes	characteristic
152	C₄×#125	C₂×C₁₃₉	1	1	yes	characteristic, complemented
153	#2099	C₂×C₁₃₉	1	2	yes	complemented
154	#2073	C₂×C₁₃₉	1	2	yes	complemented
155	#2041	C₂×C₁₃₉	1	2	yes	—
156	#2025	C₂×C₁₃₉	1	2	yes	complemented
157	#1762	C₂×C₁₃₉	1	4	yes	complemented
158	#1745	C₂×C₁₃₉	1	1	yes	characteristic, complemented
159	C₂×#127	C₂²×C₁₃₉	1	1	yes	characteristic
160	C₂×#125	C₂²×C₁₃₉	1	1	yes	characteristic
161	C₂×C₄×D₄	—	2	2	no	—
162	C₂×C₄×D₄	C₂²×C₁₃₉	1	2	yes	—
163	C₂×#102	C₂²×C₁₃₉	1	4	yes	—
164	C₂×#102	C₂²×C₁₃₉	1	4	yes	—
165	C₂×#97	C₂²×C₁₃₉	1	2	yes	—
166	#390	—	2	2	no	—
167	#390	C₂²×C₁₃₉	1	2	yes	—
168	#389	C₂²×C₁₃₉	1	2	yes	—
169	#387	—	2	2	no	—
170	#384	C₂²×C₁₃₉	1	1	yes	characteristic
171	#382	C₂²×C₁₃₉	1	1	yes	characteristic
172	C₄×C₄⋊C₄	—	2	2	no	—
173	C₄×C₄⋊C₄	C₂²×C₁₃₉	1	2	yes	—
174	C₄³	C₂²×C₁₃₉	1	1	yes	characteristic
175	#340	C₂²×C₁₃₉	1	4	yes	—
176	#340	C₂²×C₁₃₉	1	4	yes	—
177	#336	C₂²×C₁₃₉	1	2	yes	—
178	#336	C₂²×C₁₃₉	1	2	yes	—
179	C₂²×D₄	—	2	2	no	—
180	C₂²×D₄	C₂³×C₁₃₉	1	2	yes	—
181	C₂³×C₄	—	2	2	no	—
182	C₂²×C₈	C₂³×C₁₃₉	1	4	yes	—
183	#127	—	2	4	no	—
184	#125	—	2	4	no	—
185	C₄×D₄	—	2	8	no	—
186	C₄×D₄	—	4	8	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	2	yes	—
191	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	2	yes	—
192	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	2	yes	—
193	C₂×C₂²⋊C₄	—	2	4	no	—
194	C₂×C₂²⋊C₄	—	2	2	no	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	D₄×C₁₃₉	1	2	yes	—
197	C₂×C₄²	D₄×C₁₃₉	1	2	yes	—
198	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
199	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
200	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
201	#102	—	2	8	no	—
202	#102	—	2	8	no	—
203	#97	—	2	4	no	—
204	#95	—	2	2	no	—
205	#95	—	2	2	no	—
206	#95	—	2	2	no	—
207	C₂⁴	—	2	2	no	—
208	C₂×D₄	—	2	8	no	—
209	C₂×D₄	—	2	4	no	—
210	C₂×D₄	—	2	4	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	—	4	4	no	—
213	C₂×D₄	—	4	4	no	—
214	C₂×D₄	#41×C₁₃₉	1	4	yes	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	4	8	no	—
219	C₂²×C₄	—	4	4	no	—
220	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
221	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
224	C₂²×C₄	C₂⁴×C₁₃₉	1	1	yes	characteristic
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₄⋊C₄	—	2	8	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	4	4	no	—
233	C₄⋊C₄	—	4	4	no	—
234	C₄⋊C₄	#41×C₁₃₉	1	4	yes	—
235	C₂²⋊C₄	—	4	8	no	—
236	C₂²⋊C₄	—	4	4	no	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	4	2	no	—
241	C₄²	—	4	4	no	—
242	C₄²	#41×C₁₃₉	1	4	yes	—
243	C₂³	—	2	4	no	—
244	C₂³	—	4	2	no	—
245	C₂³	—	4	2	no	—
246	C₂³	—	4	4	no	—
247	C₂³	—	4	4	no	—
248	C₂³	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
249	D₄	—	2	8	no	—
250	D₄	—	4	16	no	—
251	D₄	—	4	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	8	8	no	—
268	C₂×C₄	#124×C₁₃₉	1	2	yes	—
269	C₂×C₄	#124×C₁₃₉	1	2	yes	—
270	C₂×C₄	AGL(1,ℤ/8)×C₁₃₉	1	4	yes	—
271	C₂×C₄	C₂×#41×C₁₃₉	1	2	yes	—
272	C₂×C₄	C₂×#41×C₁₃₉	1	2	yes	—
273	C₂×C₄	C₂×#41×C₁₃₉	1	2	yes	—
274	C₈	—	4	4	no	—
275	C₂²	—	4	8	no	—
276	C₂²	—	4	4	no	—
277	C₂²	—	8	4	no	—
278	C₂²	—	8	2	no	—
279	C₂²	—	8	2	no	—
280	C₂²	#525×C₁₃₉	1	1	yes	characteristic
281	C₂²	#525×C₁₃₉	1	1	yes	characteristic
282	C₂²	C₂×#124×C₁₃₉	1	1	yes	characteristic
283	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₉	1	2	yes	—
284	C₂²	#575×C₁₃₉	1	2	yes	—
285	C₄	—	2	4	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	4	2	no	—
289	C₄	—	4	4	no	—
290	C₄	—	8	2	no	—
291	C₄	—	8	2	no	—
292	C₄	#529×C₁₃₉	1	4	yes	—
293	C₂	—	8	4	no	—
294	C₂	#2398×C₁₃₉	1	1	yes	characteristic
295	C₂	#3114×C₁₃₉	1	2	yes	—
296	C₂	#3115×C₁₃₉	1	2	yes	—
297	C₂	#3138×C₁₃₉	1	1	yes	characteristic
298	C₂	#3142×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₉
prime radical	C₂×#127×C₁₃₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₉ ◁ C₄³×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×#125×C₁₃₉	C₂	semidirect
3. #2099×C₁₃₉	C₂	semidirect
4. #2073×C₁₃₉	C₂	semidirect
6. #2025×C₁₃₉	C₂	semidirect
7. #1762×C₁₃₉	C₂	semidirect
8. #1745×C₁₃₉	C₂	semidirect
149. #23824	C₁₃₉	direct (N × H)
150. C₁₃₉	#23824	direct (N × H)
152. C₄×#125	C₂×C₁₃₉	semidirect
153. #2099	C₂×C₁₃₉	semidirect
154. #2073	C₂×C₁₃₉	semidirect
156. #2025	C₂×C₁₃₉	semidirect
157. #1762	C₂×C₁₃₉	semidirect
158. #1745	C₂×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×#127×C₁₃₉	2
C₄×#125×C₁₃₉	2
#2099×C₁₃₉	2
#2073×C₁₃₉	2
#2041×C₁₃₉	2
#2025×C₁₃₉	2
#1762×C₁₃₉	2
#1745×C₁₃₉	2
#23824	139

Subgroups — C139×#23824 #48018420

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#23824 #48018420