Subgroups — #2564×C₁₃₉ #31803190

Order 17792.

584 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊Q₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₄×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	#400×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#399×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#395×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#393×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#387×C₁₃₉	C₂	1	2	yes	maximal, complemented
8	#386×C₁₃₉	C₂	1	2	yes	maximal
9	#385×C₁₃₉	C₂	1	2	yes	maximal, complemented
10	#380×C₁₃₉	C₂	1	2	yes	maximal, complemented
11	C₂×#95×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₁₃₉	C₂²	1	2	yes	—
15	C₂³×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
17	C₂²⋊Q₈×C₁₃₉	—	2	4	no	—
18	C₄×D₄×C₁₃₉	—	2	4	no	—
19	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₁₃₉	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₄×C₁₃₉	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₄×C₁₃₉	C₂²	1	2	yes	—
29	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
30	#95×C₁₃₉	C₂²	1	1	yes	characteristic
31	#95×C₁₃₉	C₂²	1	2	yes	—
32	#95×C₁₃₉	C₂²	1	1	yes	characteristic
33	#95×C₁₃₉	C₂²	1	2	yes	—
34	#95×C₁₃₉	C₂²	1	2	yes	—
35	#95×C₁₃₉	C₂²	1	2	yes	—
36	#95×C₁₃₉	C₂²	1	2	yes	—
37	C₂⁴×C₁₃₉	C₂³	1	1	yes	characteristic
38	C₂⁴×C₁₃₉	C₂³	1	1	yes	characteristic
39	C₂×Q₈×C₁₃₉	—	2	2	no	—
40	C₂×Q₈×C₁₃₉	—	4	2	no	—
41	C₂×D₄×C₁₃₉	—	2	2	no	—
42	C₂×D₄×C₁₃₉	—	2	2	no	—
43	C₂×D₄×C₁₃₉	Q₈	1	4	yes	complemented
44	C₂²×C₄×C₁₃₉	—	2	4	no	—
45	C₂²×C₄×C₁₃₉	—	2	2	no	—
46	C₂²×C₄×C₁₃₉	—	2	4	no	—
47	C₂²×C₄×C₁₃₉	—	4	2	no	—
48	C₂²×C₄×C₁₃₉	—	4	2	no	—
49	C₂²×C₄×C₁₃₉	—	4	2	no	—
50	C₂²×C₄×C₁₃₉	D₄	1	4	yes	—
51	C₂²×C₄×C₁₃₉	C₂³	1	2	yes	—
52	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₃₉	C₂³	1	2	yes	—
54	C₂²×C₄×C₁₃₉	C₂³	1	2	yes	—
55	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₁₃₉	C₂³	1	2	yes	—
60	C₄⋊C₄×C₁₃₉	—	2	2	no	—
61	C₄⋊C₄×C₁₃₉	—	2	2	no	—
62	C₄⋊C₄×C₁₃₉	—	2	4	no	—
63	C₄⋊C₄×C₁₃₉	—	2	4	no	—
64	C₄⋊C₄×C₁₃₉	—	4	2	no	—
65	C₂²⋊C₄×C₁₃₉	—	2	2	no	—
66	C₂²⋊C₄×C₁₃₉	—	2	4	no	—
67	C₂²⋊C₄×C₁₃₉	—	2	2	no	—
68	C₂²⋊C₄×C₁₃₉	—	2	4	no	—
69	C₂²⋊C₄×C₁₃₉	—	4	2	no	—
70	C₄²×C₁₃₉	—	2	2	no	—
71	C₂³×C₁₃₉	—	2	2	no	—
72	C₂³×C₁₃₉	—	2	2	no	—
73	C₂³×C₁₃₉	—	2	1	no	—
74	C₂³×C₁₃₉	—	2	2	no	—
75	C₂³×C₁₃₉	—	2	1	no	—
76	C₂³×C₁₃₉	—	2	2	no	—
77	C₂³×C₁₃₉	C₂×D₄	1	2	yes	—
78	C₂³×C₁₃₉	C₂×Q₈	1	2	yes	—
79	C₂³×C₁₃₉	C₂×Q₈	1	2	yes	—
80	C₂³×C₁₃₉	#41	1	2	yes	—
81	C₂³×C₁₃₉	C₂⁴	1	1	yes	characteristic
82	Q₈×C₁₃₉	—	4	4	no	—
83	D₄×C₁₃₉	—	2	8	no	—
84	C₂×C₄×C₁₃₉	—	2	1	no	—
85	C₂×C₄×C₁₃₉	—	2	2	no	—
86	C₂×C₄×C₁₃₉	—	2	1	no	—
87	C₂×C₄×C₁₃₉	—	2	2	no	—
88	C₂×C₄×C₁₃₉	—	2	2	no	—
89	C₂×C₄×C₁₃₉	—	2	2	no	—
90	C₂×C₄×C₁₃₉	—	2	1	no	—
91	C₂×C₄×C₁₃₉	—	2	2	no	—
92	C₂×C₄×C₁₃₉	—	2	1	no	—
93	C₂×C₄×C₁₃₉	—	2	2	no	—
94	C₂×C₄×C₁₃₉	—	2	2	no	—
95	C₂×C₄×C₁₃₉	—	2	2	no	—
96	C₂×C₄×C₁₃₉	—	2	2	no	—
97	C₂×C₄×C₁₃₉	—	2	2	no	—
98	C₂×C₄×C₁₃₉	—	2	2	no	—
99	C₂×C₄×C₁₃₉	—	2	2	no	—
100	C₂×C₄×C₁₃₉	—	2	2	no	—
101	C₂×C₄×C₁₃₉	—	2	2	no	—
102	C₂×C₄×C₁₃₉	—	2	2	no	—
103	C₂×C₄×C₁₃₉	—	4	4	no	—
104	C₂×C₄×C₁₃₉	—	4	8	no	—
105	C₂×C₄×C₁₃₉	—	4	4	no	—
106	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
107	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
108	C₂×C₄×C₁₃₉	C₂×Q₈	1	2	yes	—
109	C₂×C₄×C₁₃₉	#41	1	4	yes	—
110	C₂²×C₁₃₉	—	2	2	no	—
111	C₂²×C₁₃₉	—	2	2	no	—
112	C₂²×C₁₃₉	—	2	4	no	—
113	C₂²×C₁₃₉	—	2	4	no	—
114	C₂²×C₁₃₉	—	2	2	no	—
115	C₂²×C₁₃₉	—	2	4	no	—
116	C₂²×C₁₃₉	—	4	1	no	—
117	C₂²×C₁₃₉	—	4	1	no	—
118	C₂²×C₁₃₉	—	4	2	no	—
119	C₂²×C₁₃₉	C₂²⋊Q₈	1	4	yes	—
120	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
121	C₂²×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
122	C₂²×C₁₃₉	C₂×#41	1	1	yes	characteristic
123	C₂²×C₁₃₉	C₂×#41	1	2	yes	—
124	C₂²×C₁₃₉	H(5,𝔽₂)	1	2	yes	—
125	C₄×C₁₃₉	—	2	2	no	—
126	C₄×C₁₃₉	—	2	2	no	—
127	C₄×C₁₃₉	—	2	2	no	—
128	C₄×C₁₃₉	—	2	4	no	—
129	C₄×C₁₃₉	—	4	1	no	—
130	C₄×C₁₃₉	—	4	2	no	—
131	C₄×C₁₃₉	—	4	1	no	—
132	C₄×C₁₃₉	—	4	2	no	—
133	C₄×C₁₃₉	—	4	2	no	—
134	C₂×C₁₃₉	—	2	4	no	—
135	C₂×C₁₃₉	—	4	2	no	—
136	C₂×C₁₃₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
137	C₂×C₁₃₉	#532	1	1	yes	characteristic
138	C₂×C₁₃₉	#534	1	1	yes	characteristic
139	C₂×C₁₃₉	#551	1	2	yes	—
140	C₂×C₁₃₉	#554	1	2	yes	—
141	C₁₃₉	#2564	1	1	yes	characteristic, direct product factor, Sylow
142	#2564	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂×C₂²⋊Q₈	C₂×C₁₃₉	1	1	yes	characteristic, complemented
144	C₂×C₄×D₄	C₂×C₁₃₉	1	1	yes	characteristic
145	#400	C₂×C₁₃₉	1	1	yes	characteristic, complemented
146	#399	C₂×C₁₃₉	1	1	yes	characteristic, complemented
147	#395	C₂×C₁₃₉	1	1	yes	characteristic, complemented
148	#393	C₂×C₁₃₉	1	1	yes	characteristic, complemented
149	#387	C₂×C₁₃₉	1	2	yes	complemented
150	#386	C₂×C₁₃₉	1	2	yes	—
151	#385	C₂×C₁₃₉	1	2	yes	complemented
152	#380	C₂×C₁₃₉	1	2	yes	complemented
153	C₂×#95	C₂×C₁₃₉	1	1	yes	characteristic, complemented
154	C₂²×Q₈	C₂²×C₁₃₉	1	1	yes	characteristic
155	C₂²×D₄	C₂²×C₁₃₉	1	1	yes	characteristic
156	C₂³×C₄	C₂²×C₁₃₉	1	2	yes	—
157	C₂³×C₄	C₂²×C₁₃₉	1	1	yes	characteristic
158	C₂³×C₄	C₂²×C₁₃₉	1	1	yes	characteristic
159	C₂²⋊Q₈	—	2	4	no	—
160	C₄×D₄	—	2	4	no	—
161	C₂×C₄⋊C₄	C₂²×C₁₃₉	1	1	yes	characteristic
162	C₂×C₄⋊C₄	C₂²×C₁₃₉	1	2	yes	—
163	C₂×C₄⋊C₄	C₂²×C₁₃₉	1	1	yes	characteristic
164	C₂×C₄⋊C₄	C₂²×C₁₃₉	1	2	yes	—
165	C₂×C₄⋊C₄	C₂²×C₁₃₉	1	2	yes	—
166	C₂×C₂²⋊C₄	C₂²×C₁₃₉	1	1	yes	characteristic
167	C₂×C₂²⋊C₄	C₂²×C₁₃₉	1	1	yes	characteristic
168	C₂×C₂²⋊C₄	C₂²×C₁₃₉	1	2	yes	—
169	C₂×C₂²⋊C₄	C₂²×C₁₃₉	1	2	yes	—
170	C₂×C₂²⋊C₄	C₂²×C₁₃₉	1	2	yes	—
171	C₂×C₄²	C₂²×C₁₃₉	1	1	yes	characteristic
172	#95	C₂²×C₁₃₉	1	2	yes	—
173	#95	C₂²×C₁₃₉	1	1	yes	characteristic
174	#95	C₂²×C₁₃₉	1	1	yes	characteristic
175	#95	C₂²×C₁₃₉	1	2	yes	—
176	#95	C₂²×C₁₃₉	1	2	yes	—
177	#95	C₂²×C₁₃₉	1	2	yes	—
178	#95	C₂²×C₁₃₉	1	2	yes	—
179	C₂⁴	C₂³×C₁₃₉	1	1	yes	characteristic
180	C₂⁴	C₂³×C₁₃₉	1	1	yes	characteristic
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	4	2	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	Q₈×C₁₃₉	1	4	yes	complemented
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	D₄×C₁₃₉	1	4	yes	—
193	C₂²×C₄	C₂³×C₁₃₉	1	2	yes	—
194	C₂²×C₄	C₂³×C₁₃₉	1	1	yes	characteristic
195	C₂²×C₄	C₂³×C₁₃₉	1	2	yes	—
196	C₂²×C₄	C₂³×C₁₃₉	1	2	yes	—
197	C₂²×C₄	C₂³×C₁₃₉	1	1	yes	characteristic
198	C₂²×C₄	C₂³×C₁₃₉	1	1	yes	characteristic
199	C₂²×C₄	C₂³×C₁₃₉	1	1	yes	characteristic
200	C₂²×C₄	C₂³×C₁₃₉	1	1	yes	characteristic
201	C₂²×C₄	C₂³×C₁₃₉	1	2	yes	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	2	4	no	—
209	C₂²⋊C₄	—	2	2	no	—
210	C₂²⋊C₄	—	2	4	no	—
211	C₂²⋊C₄	—	4	2	no	—
212	C₄²	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	2	1	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	2	2	no	—
219	C₂³	C₂×D₄×C₁₃₉	1	2	yes	—
220	C₂³	C₂×Q₈×C₁₃₉	1	2	yes	—
221	C₂³	C₂×Q₈×C₁₃₉	1	2	yes	—
222	C₂³	#41×C₁₃₉	1	2	yes	—
223	C₂³	C₂⁴×C₁₃₉	1	1	yes	characteristic
224	Q₈	—	4	4	no	—
225	D₄	—	2	8	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	8	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
249	C₂×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
250	C₂×C₄	C₂×Q₈×C₁₃₉	1	2	yes	—
251	C₂×C₄	#41×C₁₃₉	1	4	yes	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	2	4	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	2	no	—
261	C₂²	C₂²⋊Q₈×C₁₃₉	1	4	yes	—
262	C₂²	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
263	C₂²	C₂²×Q₈×C₁₃₉	1	1	yes	characteristic
264	C₂²	C₂×#41×C₁₃₉	1	1	yes	characteristic
265	C₂²	C₂×#41×C₁₃₉	1	2	yes	—
266	C₂²	H(5,𝔽₂)×C₁₃₉	1	2	yes	—
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	4	no	—
271	C₄	—	4	1	no	—
272	C₄	—	4	2	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	2	no	—
275	C₄	—	4	2	no	—
276	C₂	—	2	4	no	—
277	C₂	—	4	2	no	—
278	C₂	C₂×C₂²⋊Q₈×C₁₃₉	1	1	yes	characteristic
279	C₂	#532×C₁₃₉	1	1	yes	characteristic
280	C₂	#534×C₁₃₉	1	1	yes	characteristic
281	C₂	#551×C₁₃₉	1	2	yes	—
282	C₂	#554×C₁₃₉	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₂²⋊Q₈×C₁₃₉	C₂	semidirect
3. #400×C₁₃₉	C₂	semidirect
4. #399×C₁₃₉	C₂	semidirect
5. #395×C₁₃₉	C₂	semidirect
6. #393×C₁₃₉	C₂	semidirect
7. #387×C₁₃₉	C₂	semidirect
9. #385×C₁₃₉	C₂	semidirect
10. #380×C₁₃₉	C₂	semidirect
11. C₂×#95×C₁₃₉	C₂	semidirect
43. C₂×D₄×C₁₃₉	Q₈	semidirect
141. C₁₃₉	#2564	direct (N × H)
142. #2564	C₁₃₉	direct (N × H)
143. C₂×C₂²⋊Q₈	C₂×C₁₃₉	semidirect
145. #400	C₂×C₁₃₉	semidirect
146. #399	C₂×C₁₃₉	semidirect
147. #395	C₂×C₁₃₉	semidirect
148. #393	C₂×C₁₃₉	semidirect
149. #387	C₂×C₁₃₉	semidirect
151. #385	C₂×C₁₃₉	semidirect
152. #380	C₂×C₁₃₉	semidirect
153. C₂×#95	C₂×C₁₃₉	semidirect
185. C₂×D₄	Q₈×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊Q₈×C₁₃₉	2
C₂×C₄×D₄×C₁₃₉	2
#400×C₁₃₉	2
#399×C₁₃₉	2
#395×C₁₃₉	2
#393×C₁₃₉	2
#387×C₁₃₉	2
#386×C₁₃₉	2
#385×C₁₃₉	2
#380×C₁₃₉	2
C₂×#95×C₁₃₉	2
#2564	139

Subgroups — #2564×C139 #31803190

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2564×C₁₃₉ #31803190