Subgroups — #31819604

Order 17792.

1540 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×D₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#22581436	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581434	C₂	1	3	yes	maximal, complemented
4	#22581433	C₂	1	3	yes	maximal, complemented
5	#22581429	C₂	1	2	yes	maximal, complemented
6	#22581425	C₂	1	6	yes	maximal, complemented
7	C₄×#14555786	C₂	1	2	yes	maximal, complemented
8	C₄×#14555784	C₂	1	6	yes	maximal, complemented
9	C₂×#17984800	C₂	1	3	yes	maximal, complemented
10	#117×D₁₃₉	C₂	1	3	yes	maximal, complemented
11	#520×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14555786	C₂²	1	1	yes	characteristic
13	C₂×#14555785	C₂²	1	3	yes	—
14	C₂×#14555784	C₂²	1	3	yes	—
15	#41×D₁₃₉	C₄	1	16	yes	complemented
16	C₂×Q₈×D₁₃₉	C₂²	1	1	yes	characteristic
17	C₂×D₄×D₁₃₉	C₂²	1	3	yes	—
18	C₂²×C₄×D₁₃₉	C₂²	1	3	yes	—
19	Q₈×Dic₁₃₉	C₂²	1	2	yes	complemented
20	C₄×Dic₂₇₈	C₂²	1	6	yes	complemented
21	#17984807	C₂²	1	6	yes	complemented
22	C₄×D₅₅₆	C₂²	1	6	yes	complemented
23	D₄×Dic₁₃₉	C₂²	1	6	yes	complemented
24	#17984804	C₂²	1	6	yes	complemented
25	C₄×C₂²⋊D₁₃₉	C₂²	1	12	yes	complemented
26	(C₂²⋊D₁₃₉)⋊C₄	C₂²	1	12	yes	complemented
27	#17984801	C₂²	1	3	yes	complemented
28	#17984800	C₂²	1	1	yes	characteristic
29	#17984800	C₂²	1	3	yes	—
30	#17984800	C₂²	1	6	yes	—
31	#17984800	C₂²	1	6	yes	complemented
32	#17984800	C₂²	1	6	yes	complemented
33	#17984799	C₂²	1	6	yes	—
34	#17984798	C₂²	1	6	yes	complemented
35	C₂×C₄⋊Dic₁₃₉	C₂²	1	3	yes	complemented
36	C₄⋊C₄×D₁₃₉	C₂²	1	3	yes	complemented
37	C₄⋊C₄×D₁₃₉	C₂²	1	3	yes	complemented
38	C₂×(C₂×D₁₃₉)⋊C₄	C₂²	1	6	yes	—
39	C₂²⋊C₄×D₁₃₉	C₂²	1	6	yes	—
40	C₂×C₄×Dic₁₃₉	C₂²	1	6	yes	complemented
41	C₄²×D₁₃₉	C₂²	1	6	yes	complemented
42	C₂×#41×C₁₃₉	C₂²	1	1	yes	characteristic
43	C₄×Q₈×C₁₃₉	C₂²	1	2	yes	complemented
44	C₄×D₄×C₁₃₉	C₂²	1	6	yes	complemented
45	#117×C₁₃₉	C₂²	1	3	yes	complemented
46	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	3	yes	complemented
47	C₂³×D₁₃₉	C₂³	1	3	yes	—
48	#14555786	C₂×C₄	1	8	yes	complemented
49	#14555785	C₂×C₄	1	24	yes	complemented
50	#14555784	C₂×C₄	1	24	yes	complemented
51	C₂×Dic₂₇₈	C₂³	1	3	yes	—
52	Q₈×D₁₃₉	C₂×C₄	1	8	yes	complemented
53	C₂×D₅₅₆	C₂³	1	3	yes	—
54	C₂×C₂²⋊D₁₃₉	C₂³	1	6	yes	—
55	D₄×D₁₃₉	C₂×C₄	1	24	yes	complemented
56	C₂²×Dic₁₃₉	C₂³	1	6	yes	—
57	C₂²×Dic₁₃₉	C₂³	1	3	yes	—
58	C₂×C₄×D₁₃₉	C₂×C₄	1	24	yes	complemented
59	C₂×C₄×D₁₃₉	C₂³	1	6	yes	—
60	C₂×C₄×D₁₃₉	C₂³	1	6	yes	—
61	C₂×C₄×D₁₃₉	C₂³	1	3	yes	—
62	C₂×C₄×D₁₃₉	C₂³	1	3	yes	—
63	C₂×C₄×D₁₃₉	C₂³	1	3	yes	—
64	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
65	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
66	C₄⋊Dic₁₃₉	C₂³	1	3	yes	—
67	C₄⋊Dic₁₃₉	C₂³	1	6	yes	—
68	Dic₁₃₉⋊C₄	C₂³	1	6	yes	—
69	Dic₁₃₉⋊C₄	C₂³	1	6	yes	—
70	C₂²⋊Dic₁₃₉	C₂³	1	6	yes	—
71	(C₂×D₁₃₉)⋊C₄	C₂³	1	2	yes	—
72	(C₂×D₁₃₉)⋊C₄	C₂³	1	6	yes	—
73	(C₂×D₁₃₉)⋊C₄	C₂³	1	12	yes	—
74	C₄×Dic₁₃₉	C₂³	1	2	yes	—
75	C₄×Dic₁₃₉	C₂³	1	6	yes	—
76	C₄×Dic₁₃₉	C₂³	1	6	yes	—
77	C₄×Dic₁₃₉	C₂³	1	12	yes	—
78	#41×C₁₃₉	C₂×C₄	1	8	yes	complemented
79	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
80	C₂×D₄×C₁₃₉	C₂³	1	3	yes	—
81	C₂²×C₄×C₁₃₉	C₂³	1	6	yes	—
82	C₂²×C₄×C₁₃₉	C₂³	1	3	yes	—
83	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
84	C₄⋊C₄×C₁₃₉	C₂³	1	3	yes	—
85	C₄⋊C₄×C₁₃₉	C₂³	1	6	yes	—
86	C₂²⋊C₄×C₁₃₉	C₂³	1	6	yes	—
87	C₄²×C₁₃₉	C₂³	1	6	yes	—
88	C₂²×D₁₃₉	—	4	3	no	—
89	C₂²×D₁₃₉	—	4	3	no	—
90	C₂²×D₁₃₉	C₂²×C₄	1	12	yes	—
91	C₂²×D₁₃₉	C₂⁴	1	1	yes	characteristic
92	C₂²×D₁₃₉	C₂⁴	1	3	yes	—
93	Dic₂₇₈	C₂²×C₄	1	12	yes	—
94	D₅₅₆	C₂²×C₄	1	12	yes	—
95	C₂²⋊D₁₃₉	C₂²×C₄	1	24	yes	—
96	C₂×Dic₁₃₉	—	4	6	no	—
97	C₂×Dic₁₃₉	C₂²×C₄	1	12	yes	—
98	C₂×Dic₁₃₉	C₂⁴	1	6	yes	—
99	C₂×Dic₁₃₉	C₂⁴	1	2	yes	—
100	C₂×Dic₁₃₉	C₂⁴	1	1	yes	characteristic
101	C₂×Dic₁₃₉	C₂⁴	1	3	yes	—
102	C₄×D₁₃₉	—	4	6	no	—
103	C₄×D₁₃₉	—	4	6	no	—
104	C₄×D₁₃₉	C₂²×C₄	1	4	yes	—
105	C₄×D₁₃₉	C₂²×C₄	1	12	yes	—
106	C₄×D₁₃₉	C₂²×C₄	1	12	yes	—
107	C₄×D₁₃₉	C₂²×C₄	1	12	yes	—
108	C₂³×C₁₃₉	C₂⁴	1	3	yes	—
109	Q₈×C₁₃₉	C₂²×C₄	1	4	yes	—
110	D₄×C₁₃₉	C₂²×C₄	1	12	yes	—
111	C₂×C₄×C₁₃₉	—	4	6	no	—
112	C₂×C₄×C₁₃₉	C₂²×C₄	1	12	yes	—
113	C₂×C₄×C₁₃₉	C₂⁴	1	2	yes	—
114	C₂×C₄×C₁₃₉	C₂⁴	1	6	yes	—
115	C₂×C₄×C₁₃₉	C₂⁴	1	3	yes	—
116	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
117	C₂²×C₁₃₉	—	2	3	no	—
118	C₂²×C₁₃₉	—	2	3	no	—
119	C₂²×C₁₃₉	C₂³×C₄	1	6	yes	—
120	C₂²×C₁₃₉	C₂⁵	1	1	yes	characteristic
121	C₂×D₁₃₉	—	2	3	no	—
122	C₂×D₁₃₉	—	2	3	no	—
123	C₂×D₁₃₉	—	2	1	no	—
124	C₂×D₁₃₉	—	2	1	no	—
125	C₂×D₁₃₉	—	4	12	no	—
126	C₂×D₁₃₉	C₂³×C₄	1	2	yes	—
127	C₂×D₁₃₉	C₂³×C₄	1	6	yes	—
128	C₄×C₁₃₉	—	2	6	no	—
129	C₄×C₁₃₉	—	2	2	no	—
130	C₄×C₁₃₉	C₂³×C₄	1	2	yes	—
131	C₄×C₁₃₉	C₂³×C₄	1	6	yes	—
132	Dic₁₃₉	—	2	2	no	—
133	Dic₁₃₉	—	2	6	no	—
134	Dic₁₃₉	C₂³×C₄	1	2	yes	—
135	Dic₁₃₉	C₂³×C₄	1	6	yes	—
136	C₂×C₁₃₉	—	2	6	no	—
137	C₂×C₁₃₉	C₂⁴×C₄	1	1	yes	characteristic
138	C₂×C₁₃₉	#585	1	1	yes	characteristic
139	C₂×C₁₃₉	#585	1	1	yes	characteristic
140	D₁₃₉	—	2	2	no	—
141	D₁₃₉	—	2	6	no	—
142	C₁₃₉	#3428	1	1	yes	characteristic, complemented, Sylow
143	#3428	—	139	1	no	maximal, Sylow
144	C₂²×#41	—	139	1	no	—
145	#520	—	139	1	no	—
146	#520	—	139	3	no	—
147	#520	—	139	3	no	—
148	#520	D₁₃₉	1	1	yes	characteristic, complemented
149	#519	—	139	2	no	—
150	#518	—	139	6	no	—
151	C₄×#41	—	139	2	no	—
152	C₄×#41	—	139	6	no	—
153	C₂×#117	—	139	3	no	—
154	C₂×#117	—	139	3	no	—
155	C₂×#41	—	139	16	no	—
156	C₂×#41	—	139	1	no	—
157	C₂×#41	—	139	3	no	—
158	C₂×#41	—	139	3	no	—
159	C₂×#41	C₂×D₁₃₉	1	1	yes	characteristic
160	C₂²×Q₈	—	139	1	no	—
161	C₂²×D₄	—	139	3	no	—
162	C₂³×C₄	—	139	3	no	—
163	C₄×Q₈	—	139	2	no	—
164	C₄×Q₈	—	139	6	no	—
165	C₄×Q₈	—	139	6	no	—
166	C₄×Q₈	C₂×D₁₃₉	1	2	yes	complemented
167	C₄×D₄	—	139	6	no	—
168	C₄×D₄	—	139	12	no	—
169	C₄×D₄	—	139	6	no	—
170	C₄×D₄	—	139	12	no	—
171	C₄×D₄	—	139	6	no	—
172	C₄×D₄	C₂×D₁₃₉	1	6	yes	complemented
173	#117	—	139	3	no	—
174	#117	—	139	1	no	—
175	#117	—	139	3	no	—
176	#117	—	139	6	no	—
177	#117	—	139	6	no	—
178	#117	—	139	6	no	—
179	#117	—	139	6	no	—
180	#117	—	139	6	no	—
181	#117	C₂×D₁₃₉	1	3	yes	complemented
182	C₂×C₄⋊C₄	—	139	3	no	—
183	C₂×C₄⋊C₄	—	139	3	no	—
184	C₂×C₄⋊C₄	—	139	3	no	—
185	C₂×C₄⋊C₄	C₂×D₁₃₉	1	3	yes	complemented
186	C₂×C₂²⋊C₄	—	139	6	no	—
187	C₂×C₂²⋊C₄	—	139	6	no	—
188	C₂×C₄²	—	139	6	no	—
189	C₂×C₄²	—	139	6	no	—
190	C₂⁴	—	139	3	no	—
191	#41	—	139	8	no	—
192	#41	—	139	24	no	—
193	#41	—	139	24	no	—
194	#41	C₄×D₁₃₉	1	8	yes	complemented
195	C₂×Q₈	—	139	8	no	—
196	C₂×Q₈	—	139	3	no	—
197	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
198	C₂×D₄	—	139	6	no	—
199	C₂×D₄	—	139	24	no	—
200	C₂×D₄	—	139	3	no	—
201	C₂×D₄	C₂²×D₁₃₉	1	3	yes	—
202	C₂²×C₄	—	139	6	no	—
203	C₂²×C₄	—	139	6	no	—
204	C₂²×C₄	—	139	6	no	—
205	C₂²×C₄	—	139	3	no	—
206	C₂²×C₄	—	139	24	no	—
207	C₂²×C₄	—	139	3	no	—
208	C₂²×C₄	—	139	3	no	—
209	C₂²×C₄	—	139	1	no	—
210	C₂²×C₄	—	139	3	no	—
211	C₂²×C₄	C₂²×D₁₃₉	1	6	yes	—
212	C₂²×C₄	C₂²×D₁₃₉	1	3	yes	—
213	C₄⋊C₄	—	139	3	no	—
214	C₄⋊C₄	—	139	1	no	—
215	C₄⋊C₄	—	139	6	no	—
216	C₄⋊C₄	—	139	6	no	—
217	C₄⋊C₄	—	139	6	no	—
218	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
219	C₄⋊C₄	C₂²×D₁₃₉	1	3	yes	—
220	C₄⋊C₄	C₂²×D₁₃₉	1	6	yes	—
221	C₂²⋊C₄	—	139	6	no	—
222	C₂²⋊C₄	—	139	6	no	—
223	C₂²⋊C₄	—	139	2	no	—
224	C₂²⋊C₄	—	139	12	no	—
225	C₂²⋊C₄	C₂²×D₁₃₉	1	6	yes	—
226	C₄²	—	139	6	no	—
227	C₄²	—	139	2	no	—
228	C₄²	—	139	6	no	—
229	C₄²	—	139	12	no	—
230	C₄²	C₂²×D₁₃₉	1	6	yes	—
231	C₂³	—	556	3	no	—
232	C₂³	—	556	3	no	—
233	C₂³	—	139	1	no	—
234	C₂³	—	139	3	no	—
235	C₂³	—	139	12	no	—
236	C₂³	C₂³×D₁₃₉	1	3	yes	—
237	Q₈	—	139	12	no	—
238	Q₈	C₂×C₄×D₁₃₉	1	4	yes	—
239	D₄	—	139	24	no	—
240	D₄	—	139	12	no	—
241	D₄	C₂×C₄×D₁₃₉	1	12	yes	—
242	C₂×C₄	—	4	6	no	—
243	C₂×C₄	—	556	6	no	—
244	C₂×C₄	—	556	6	no	—
245	C₂×C₄	—	556	6	no	—
246	C₂×C₄	—	139	2	no	—
247	C₂×C₄	—	139	6	no	—
248	C₂×C₄	—	139	12	no	—
249	C₂×C₄	—	139	1	no	—
250	C₂×C₄	—	139	3	no	—
251	C₂×C₄	—	139	4	no	—
252	C₂×C₄	—	139	12	no	—
253	C₂×C₄	—	139	12	no	—
254	C₂×C₄	—	139	12	no	—
255	C₂×C₄	C₂×C₄×D₁₃₉	1	12	yes	—
256	C₂×C₄	C₂³×D₁₃₉	1	6	yes	—
257	C₂×C₄	C₂³×D₁₃₉	1	2	yes	—
258	C₂×C₄	C₂³×D₁₃₉	1	3	yes	—
259	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
260	C₂²	—	2	3	no	—
261	C₂²	—	2	3	no	—
262	C₂²	—	278	3	no	—
263	C₂²	—	278	1	no	—
264	C₂²	—	278	3	no	—
265	C₂²	—	278	1	no	—
266	C₂²	—	556	12	no	—
267	C₂²	—	139	2	no	—
268	C₂²	—	139	6	no	—
269	C₂²	C₂²×C₄×D₁₃₉	1	6	yes	—
270	C₂²	C₂⁴×D₁₃₉	1	1	yes	characteristic
271	C₄	—	2	6	no	—
272	C₄	—	2	2	no	—
273	C₄	—	278	2	no	—
274	C₄	—	278	6	no	—
275	C₄	—	139	2	no	—
276	C₄	—	139	6	no	—
277	C₄	C₂²×C₄×D₁₃₉	1	2	yes	—
278	C₄	C₂²×C₄×D₁₃₉	1	6	yes	—
279	C₂	—	2	6	no	—
280	C₂	—	278	2	no	—
281	C₂	—	278	6	no	—
282	C₂	C₂³×C₄×D₁₃₉	1	1	yes	characteristic
283	C₂	#22581788	1	1	yes	characteristic
284	C₂	#22581790	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃₉
commutator	C₂×C₁₃₉	C₂⁴×C₄
Fitting	#520×C₁₃₉	C₂
socle, cosocle	C₂²×C₁₃₉	C₂⁵
prime radical	C₂×#41×D₁₃₉	C₂
hypercenter	#520	D₁₃₉
hypocenter	C₁₃₉	#3428
LCS	G ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ #520

Complements

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

N	H ≅ G/N	type
2. #22581436	C₂	semidirect
3. #22581434	C₂	semidirect
4. #22581433	C₂	semidirect
5. #22581429	C₂	semidirect
6. #22581425	C₂	semidirect
7. C₄×#14555786	C₂	semidirect
8. C₄×#14555784	C₂	semidirect
9. C₂×#17984800	C₂	semidirect
10. #117×D₁₃₉	C₂	semidirect
11. #520×C₁₃₉	C₂	semidirect
15. #41×D₁₃₉	C₄	semidirect
19. Q₈×Dic₁₃₉	C₂²	semidirect
20. C₄×Dic₂₇₈	C₂²	semidirect
21. #17984807	C₂²	semidirect
22. C₄×D₅₅₆	C₂²	semidirect
23. D₄×Dic₁₃₉	C₂²	semidirect
24. #17984804	C₂²	semidirect
25. C₄×C₂²⋊D₁₃₉	C₂²	semidirect
26. (C₂²⋊D₁₃₉)⋊C₄	C₂²	semidirect
27. #17984801	C₂²	semidirect
31. #17984800	C₂²	semidirect
32. #17984800	C₂²	semidirect
34. #17984798	C₂²	semidirect
35. C₂×C₄⋊Dic₁₃₉	C₂²	semidirect
36. C₄⋊C₄×D₁₃₉	C₂²	semidirect
37. C₄⋊C₄×D₁₃₉	C₂²	semidirect
40. C₂×C₄×Dic₁₃₉	C₂²	semidirect
41. C₄²×D₁₃₉	C₂²	semidirect
43. C₄×Q₈×C₁₃₉	C₂²	semidirect
44. C₄×D₄×C₁₃₉	C₂²	semidirect
45. #117×C₁₃₉	C₂²	semidirect
46. C₂×C₄⋊C₄×C₁₃₉	C₂²	semidirect
48. #14555786	C₂×C₄	semidirect
49. #14555785	C₂×C₄	semidirect
50. #14555784	C₂×C₄	semidirect
52. Q₈×D₁₃₉	C₂×C₄	semidirect
55. D₄×D₁₃₉	C₂×C₄	semidirect
58. C₂×C₄×D₁₃₉	C₂×C₄	semidirect
78. #41×C₁₃₉	C₂×C₄	semidirect
142. C₁₃₉	#3428	semidirect
148. #520	D₁₃₉	semidirect
166. C₄×Q₈	C₂×D₁₃₉	semidirect
172. C₄×D₄	C₂×D₁₃₉	semidirect
181. #117	C₂×D₁₃₉	semidirect
185. C₂×C₄⋊C₄	C₂×D₁₃₉	semidirect
194. #41	C₄×D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41×D₁₃₉	2
#22581436	2
#22581434	2
#22581433	2
#22581429	2
#22581425	2
C₄×#14555786	2
C₄×#14555784	2
C₂×#17984800	2
#117×D₁₃₉	2
#520×C₁₃₉	2
#3428	139