Subgroups — #31821174

Order 17792.

1424 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581790	C₂	1	2	yes	maximal, complemented
2	#22581766	C₂	1	3	yes	maximal, complemented
3	#22581762	C₂	1	6	yes	maximal, complemented
4	#22581761	C₂	1	2	yes	maximal, complemented
5	#22581758	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581755	C₂	1	3	yes	maximal, complemented
7	#22581752	C₂	1	6	yes	maximal, complemented
8	C₂×#17984865	C₂	1	3	yes	maximal, complemented
9	#135×D₁₃₉	C₂	1	3	yes	maximal, complemented
10	#131×D₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
11	#578×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#17984890	—	2	2	no	—
13	#17984890	C₂²	1	6	yes	complemented
14	#17984888	—	2	6	no	—
15	#17984888	C₂²	1	2	yes	complemented
16	#17984888	C₂²	1	6	yes	complemented
17	C₂×#14555786	—	2	2	no	—
18	C₂×#14555786	C₂²	1	6	yes	complemented
19	#41×D₁₃₉	—	2	6	no	—
20	#41×D₁₃₉	C₂²	1	1	yes	characteristic
21	#41×D₁₃₉	C₂²	1	6	yes	complemented
22	#17984874	C₂²	1	12	yes	complemented
23	#17984873	C₂²	1	6	yes	complemented
24	#17984870	C₂²	1	6	yes	complemented
25	#17984869	C₂²	1	12	yes	complemented
26	#17984867	C₂²	1	6	yes	complemented
27	#17984865	C₂²	1	6	yes	complemented
28	#17984865	C₂²	1	6	yes	complemented
29	#17984865	C₂²	1	3	yes	complemented
30	#17984865	C₂²	1	6	yes	complemented
31	#17984864	C₂²	1	3	yes	complemented
32	#17984863	C₂²	1	6	yes	complemented
33	#17984862	C₂²	1	6	yes	complemented
34	#17984861	C₂²	1	1	yes	characteristic, complemented
35	Q₁₆×D₁₃₉	C₂²	1	3	yes	complemented
36	C₂×#14555772	C₂²	1	6	yes	complemented
37	SD₁₆×D₁₃₉	C₂²	1	6	yes	complemented
38	C₂×D₁₁₁₂	C₂²	1	3	yes	complemented
39	D₈×D₁₃₉	C₂²	1	3	yes	complemented
40	#17984850	C₂²	1	3	yes	complemented
41	#17984849	C₂²	1	1	yes	characteristic, complemented
42	#17984848	C₂²	1	3	yes	complemented
43	M₁₆×D₁₃₉	C₂²	1	3	yes	complemented
44	C₂×C₈×D₁₃₉	C₂²	1	3	yes	complemented
45	2₋¹⁺⁴×C₁₃₉	C₂²	1	2	yes	complemented
46	#137×C₁₃₉	C₂²	1	6	yes	complemented
47	#135×C₁₃₉	C₂²	1	3	yes	complemented
48	C₂×Q₁₆×C₁₃₉	C₂²	1	3	yes	complemented
49	#131×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
50	#14555786	—	2	6	no	—
51	#14555786	—	2	6	no	—
52	#14555786	—	2	6	no	—
53	#14555786	—	2	2	no	—
54	#14555786	D₄	1	1	yes	characteristic
55	#14555786	C₂³	1	6	yes	—
56	#14555786	C₂³	1	12	yes	—
57	#14555785	—	2	6	no	—
58	#14555785	—	2	12	no	—
59	#14555785	—	2	6	no	—
60	#14555785	C₂³	1	6	yes	—
61	#14555785	C₂³	1	3	yes	—
62	#14555784	—	2	6	no	—
63	#14555784	D₄	1	3	yes	—
64	#14555784	C₂³	1	2	yes	—
65	Q₈×D₁₃₉	—	2	6	no	—
66	Q₈×D₁₃₉	D₄	1	1	yes	characteristic
67	Q₈×D₁₃₉	C₂³	1	6	yes	—
68	C₂×D₅₅₆	—	2	6	no	—
69	C₂×D₅₅₆	—	2	6	no	—
70	C₂×D₅₅₆	C₂³	1	6	yes	—
71	D₄×D₁₃₉	—	2	6	no	—
72	D₄×D₁₃₉	—	2	12	no	—
73	D₄×D₁₃₉	—	2	6	no	—
74	D₄×D₁₃₉	D₄	1	3	yes	—
75	D₄×D₁₃₉	C₂³	1	6	yes	—
76	C₂×C₄×D₁₃₉	—	2	6	no	—
77	C₂×C₄×D₁₃₉	—	2	6	no	—
78	C₂×C₄×D₁₃₉	C₂³	1	3	yes	—
79	Dic₅₅₆	C₂³	1	1	yes	characteristic
80	#14555775	C₂³	1	6	yes	—
81	#14555774	C₂³	1	6	yes	—
82	#14555773	C₂³	1	2	yes	—
83	#14555772	C₂³	1	6	yes	—
84	#14555772	C₂³	1	12	yes	—
85	D₁₁₁₂	C₂³	1	3	yes	—
86	D₁₁₁₂	C₂³	1	6	yes	—
87	#14555770	C₂³	1	6	yes	—
88	#14555769	C₂³	1	3	yes	—
89	#14555768	C₂³	1	6	yes	—
90	C₂×C₁₃₉⋊C₈	C₂³	1	3	yes	—
91	C₈×D₁₃₉	C₂³	1	1	yes	characteristic
92	C₈×D₁₃₉	C₂³	1	3	yes	—
93	C₈×D₁₃₉	C₂³	1	3	yes	—
94	C₈×D₁₃₉	C₂³	1	3	yes	—
95	#41×C₁₃₉	—	2	6	no	—
96	#41×C₁₃₉	C₂³	1	6	yes	—
97	#41×C₁₃₉	C₂³	1	1	yes	characteristic
98	C₂×Q₈×C₁₃₉	—	2	2	no	—
99	C₂×Q₈×C₁₃₉	C₂³	1	6	yes	—
100	Q₁₆×C₁₃₉	C₂³	1	3	yes	—
101	Q₁₆×C₁₃₉	C₂³	1	6	yes	—
102	SD₁₆×C₁₃₉	C₂³	1	6	yes	—
103	D₈×C₁₃₉	C₂³	1	1	yes	characteristic
104	M₁₆×C₁₃₉	C₂³	1	3	yes	—
105	C₂×C₈×C₁₃₉	C₂³	1	3	yes	—
106	C₂²×D₁₃₉	—	2	6	no	—
107	C₂²×D₁₃₉	—	2	6	no	—
108	C₂²×D₁₃₉	C₂×D₄	1	3	yes	—
109	Dic₂₇₈	—	2	6	no	—
110	Dic₂₇₈	C₂×D₄	1	3	yes	—
111	Dic₂₇₈	C₂⁴	1	2	yes	—
112	D₅₅₆	—	2	12	no	—
113	D₅₅₆	—	2	6	no	—
114	D₅₅₆	—	2	6	no	—
115	D₅₅₆	C₂×D₄	1	3	yes	—
116	D₅₅₆	C₂⁴	1	6	yes	—
117	C₂²⋊D₁₃₉	—	2	6	no	—
118	C₂²⋊D₁₃₉	—	2	2	no	—
119	C₂²⋊D₁₃₉	—	2	6	no	—
120	C₂²⋊D₁₃₉	C₂×D₄	1	6	yes	—
121	C₂×Dic₁₃₉	—	2	2	no	—
122	C₂×Dic₁₃₉	C₂×D₄	1	3	yes	—
123	C₄×D₁₃₉	—	2	6	no	—
124	C₄×D₁₃₉	—	2	6	no	—
125	C₄×D₁₃₉	—	2	6	no	—
126	C₄×D₁₃₉	—	2	12	no	—
127	C₄×D₁₃₉	C₂×D₄	1	3	yes	—
128	C₄×D₁₃₉	C₂×D₄	1	3	yes	—
129	C₄×D₁₃₉	C₂⁴	1	1	yes	characteristic
130	C₄×D₁₃₉	C₂⁴	1	3	yes	—
131	C₁₃₉⋊C₈	C₂⁴	1	1	yes	characteristic
132	C₁₃₉⋊C₈	C₂⁴	1	3	yes	—
133	Q₈×C₁₃₉	—	2	6	no	—
134	Q₈×C₁₃₉	C₂×D₄	1	1	yes	characteristic
135	Q₈×C₁₃₉	C₂⁴	1	6	yes	—
136	D₄×C₁₃₉	—	2	6	no	—
137	D₄×C₁₃₉	C₂×D₄	1	3	yes	—
138	D₄×C₁₃₉	C₂⁴	1	2	yes	—
139	C₂×C₄×C₁₃₉	—	2	6	no	—
140	C₂×C₄×C₁₃₉	—	2	6	no	—
141	C₂×C₄×C₁₃₉	C₂⁴	1	3	yes	—
142	C₈×C₁₃₉	C₂⁴	1	1	yes	characteristic
143	C₈×C₁₃₉	C₂⁴	1	3	yes	—
144	C₂²×C₁₃₉	—	2	2	no	—
145	C₂²×C₁₃₉	C₂²×D₄	1	3	yes	—
146	C₂×D₁₃₉	—	2	6	no	—
147	C₂×D₁₃₉	—	4	3	no	—
148	C₂×D₁₃₉	—	8	6	no	—
149	C₂×D₁₃₉	—	8	6	no	—
150	C₂×D₁₃₉	C₂²×D₄	1	3	yes	—
151	C₂×D₁₃₉	C₂²×D₄	1	1	yes	characteristic
152	C₄×C₁₃₉	—	2	6	no	—
153	C₄×C₁₃₉	C₂²×D₄	1	3	yes	—
154	C₄×C₁₃₉	C₂⁵	1	1	yes	characteristic
155	Dic₁₃₉	—	2	2	no	—
156	Dic₁₃₉	C₂²×D₄	1	1	yes	characteristic
157	Dic₁₃₉	C₂²×D₄	1	3	yes	—
158	C₂×C₁₃₉	—	2	3	no	—
159	C₂×C₁₃₉	—	4	2	no	—
160	C₂×C₁₃₉	C₂³×D₄	1	1	yes	characteristic
161	D₁₃₉	—	2	1	no	—
162	D₁₃₉	—	2	3	no	—
163	D₁₃₉	—	4	6	no	—
164	C₁₃₉	#3584	1	1	yes	characteristic, complemented, Sylow
165	#3584	—	139	1	no	maximal, Sylow
166	#585	—	139	2	no	—
167	#578	—	139	3	no	—
168	#578	D₁₃₉	1	1	yes	characteristic, complemented
169	#577	—	139	2	no	—
170	#577	—	139	6	no	—
171	#576	—	139	3	no	—
172	#576	—	139	1	no	—
173	#575	—	139	6	no	—
174	C₂×#135	—	139	3	no	—
175	C₂×#135	—	139	3	no	—
176	C₂×#131	—	139	1	no	—
177	2₋¹⁺⁴	—	278	2	no	—
178	2₋¹⁺⁴	—	139	6	no	—
179	2₋¹⁺⁴	C₂×D₁₃₉	1	2	yes	complemented
180	H(5,𝔽₂)	—	278	6	no	—
181	H(5,𝔽₂)	—	139	6	no	—
182	H(5,𝔽₂)	—	139	2	no	—
183	C₂×#41	—	278	2	no	—
184	C₂×#41	—	278	6	no	—
185	C₂×#41	—	139	6	no	—
186	C₂×#41	—	139	6	no	—
187	C₂×#41	—	139	1	no	—
188	#137	—	139	6	no	—
189	#137	—	139	12	no	—
190	#137	C₂×D₁₃₉	1	6	yes	complemented
191	AGL(1,ℤ/8)	—	139	6	no	—
192	AGL(1,ℤ/8)	—	139	12	no	—
193	AGL(1,ℤ/8)	—	139	6	no	—
194	#135	—	139	6	no	—
195	#135	—	139	3	no	—
196	#135	—	139	6	no	—
197	#135	—	139	6	no	—
198	#135	—	139	6	no	—
199	#135	—	139	6	no	—
200	#135	—	139	1	no	—
201	#135	—	139	3	no	—
202	#135	C₂×D₁₃₉	1	3	yes	complemented
203	C₂×Q₁₆	—	139	3	no	—
204	C₂×Q₁₆	C₂×D₁₃₉	1	3	yes	complemented
205	C₂×SD₁₆	—	139	6	no	—
206	C₂×SD₁₆	—	139	6	no	—
207	C₂×D₈	—	139	3	no	—
208	C₂×D₈	—	139	3	no	—
209	#131	—	139	1	no	—
210	#131	—	139	3	no	—
211	#131	—	139	3	no	—
212	#131	C₂×D₁₃₉	1	1	yes	characteristic, complemented
213	C₂×M₁₆	—	139	3	no	—
214	C₂²×C₈	—	139	3	no	—
215	#41	—	2	6	no	—
216	#41	—	278	6	no	—
217	#41	—	278	12	no	—
218	#41	—	278	6	no	—
219	#41	—	278	2	no	—
220	#41	—	278	6	no	—
221	#41	—	278	6	no	—
222	#41	—	278	6	no	—
223	#41	—	278	6	no	—
224	#41	—	139	6	no	—
225	#41	—	139	12	no	—
226	#41	—	139	2	no	—
227	#41	—	139	6	no	—
228	#41	—	139	1	no	—
229	#41	—	139	3	no	—
230	#41	—	139	3	no	—
231	#41	C₂²×D₁₃₉	1	6	yes	—
232	#41	C₂²×D₁₃₉	1	1	yes	characteristic
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	278	6	no	—
235	C₂×Q₈	—	139	6	no	—
236	C₂×Q₈	—	139	1	no	—
237	C₂×Q₈	C₂²×D₁₃₉	1	6	yes	—
238	C₂×D₄	—	278	6	no	—
239	C₂×D₄	—	278	6	no	—
240	C₂×D₄	—	278	6	no	—
241	C₂×D₄	—	278	12	no	—
242	C₂×D₄	—	278	6	no	—
243	C₂×D₄	—	139	6	no	—
244	C₂×D₄	—	139	3	no	—
245	C₂×D₄	—	139	6	no	—
246	C₂²×C₄	—	278	6	no	—
247	C₂²×C₄	—	278	6	no	—
248	C₂²×C₄	—	139	3	no	—
249	Q₁₆	—	139	6	no	—
250	Q₁₆	—	139	1	no	—
251	Q₁₆	C₂²×D₁₃₉	1	3	yes	—
252	Q₁₆	C₂²×D₁₃₉	1	6	yes	—
253	SD₁₆	—	139	6	no	—
254	SD₁₆	—	139	12	no	—
255	SD₁₆	—	139	2	no	—
256	SD₁₆	—	139	6	no	—
257	SD₁₆	C₂²×D₁₃₉	1	6	yes	—
258	D₈	—	139	6	no	—
259	D₈	—	139	3	no	—
260	D₈	—	139	6	no	—
261	D₈	C₂²×D₁₃₉	1	1	yes	characteristic
262	M₁₆	—	139	3	no	—
263	M₁₆	—	139	6	no	—
264	M₁₆	C₂²×D₁₃₉	1	3	yes	—
265	C₂×C₈	—	139	3	no	—
266	C₂×C₈	—	139	1	no	—
267	C₂×C₈	—	139	3	no	—
268	C₂×C₈	—	139	3	no	—
269	C₂×C₈	—	139	3	no	—
270	C₂×C₈	C₂²×D₁₃₉	1	3	yes	—
271	C₂³	—	278	6	no	—
272	C₂³	—	278	6	no	—
273	C₂³	—	139	3	no	—
274	Q₈	—	2	6	no	—
275	Q₈	—	278	6	no	—
276	Q₈	—	139	2	no	—
277	Q₈	—	139	3	no	—
278	Q₈	D₄×D₁₃₉	1	1	yes	characteristic
279	Q₈	C₂³×D₁₃₉	1	6	yes	—
280	D₄	—	2	6	no	—
281	D₄	—	278	12	no	—
282	D₄	—	278	6	no	—
283	D₄	—	278	2	no	—
284	D₄	—	278	6	no	—
285	D₄	—	278	6	no	—
286	D₄	—	278	6	no	—
287	D₄	—	139	6	no	—
288	D₄	—	139	3	no	—
289	D₄	—	139	6	no	—
290	D₄	D₄×D₁₃₉	1	3	yes	—
291	D₄	C₂³×D₁₃₉	1	2	yes	—
292	C₂×C₄	—	2	6	no	—
293	C₂×C₄	—	2	6	no	—
294	C₂×C₄	—	278	6	no	—
295	C₂×C₄	—	278	6	no	—
296	C₂×C₄	—	278	12	no	—
297	C₂×C₄	—	278	2	no	—
298	C₂×C₄	—	278	6	no	—
299	C₂×C₄	—	139	3	no	—
300	C₂×C₄	—	139	3	no	—
301	C₂×C₄	—	139	3	no	—
302	C₂×C₄	—	139	1	no	—
303	C₂×C₄	—	139	3	no	—
304	C₂×C₄	C₂³×D₁₃₉	1	3	yes	—
305	C₈	—	139	1	no	—
306	C₈	—	139	3	no	—
307	C₈	C₂³×D₁₃₉	1	1	yes	characteristic
308	C₈	C₂³×D₁₃₉	1	3	yes	—
309	C₂²	—	2	2	no	—
310	C₂²	—	278	6	no	—
311	C₂²	—	556	3	no	—
312	C₂²	—	1112	6	no	—
313	C₂²	—	1112	6	no	—
314	C₂²	—	139	3	no	—
315	C₂²	—	139	1	no	—
316	C₂²	C₂×D₄×D₁₃₉	1	3	yes	—
317	C₄	—	2	6	no	—
318	C₄	—	278	2	no	—
319	C₄	—	139	1	no	—
320	C₄	—	139	3	no	—
321	C₄	C₂×D₄×D₁₃₉	1	3	yes	—
322	C₄	C₂⁴×D₁₃₉	1	1	yes	characteristic
323	C₂	—	2	3	no	—
324	C₂	—	4	2	no	—
325	C₂	—	278	1	no	—
326	C₂	—	278	3	no	—
327	C₂	—	556	6	no	—
328	C₂	C₂²×D₄×D₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₄×D₁₃₉
commutator, cosocle	C₄×C₁₃₉	C₂⁵
Frattini	C₄	C₂⁴×D₁₃₉
Fitting	#578×C₁₃₉	C₂
socle	C₂×C₁₃₉	C₂³×D₄
hypercenter	#578	D₁₃₉
hypocenter	C₁₃₉	#3584
LCS	G ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂ ◁ #41 ◁ #578

Complements

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

N	H ≅ G/N	type
1. #22581790	C₂	semidirect
2. #22581766	C₂	semidirect
3. #22581762	C₂	semidirect
4. #22581761	C₂	semidirect
5. #22581758	C₂	semidirect
6. #22581755	C₂	semidirect
7. #22581752	C₂	semidirect
8. C₂×#17984865	C₂	semidirect
9. #135×D₁₃₉	C₂	semidirect
10. #131×D₁₃₉	C₂	semidirect
11. #578×C₁₃₉	C₂	semidirect
13. #17984890	C₂²	semidirect
15. #17984888	C₂²	semidirect
16. #17984888	C₂²	semidirect
18. C₂×#14555786	C₂²	semidirect
21. #41×D₁₃₉	C₂²	semidirect
22. #17984874	C₂²	semidirect
23. #17984873	C₂²	semidirect
24. #17984870	C₂²	semidirect
25. #17984869	C₂²	semidirect
26. #17984867	C₂²	semidirect
27. #17984865	C₂²	semidirect
28. #17984865	C₂²	semidirect
29. #17984865	C₂²	semidirect
30. #17984865	C₂²	semidirect
31. #17984864	C₂²	semidirect
32. #17984863	C₂²	semidirect
33. #17984862	C₂²	semidirect
34. #17984861	C₂²	semidirect
35. Q₁₆×D₁₃₉	C₂²	semidirect
36. C₂×#14555772	C₂²	semidirect
37. SD₁₆×D₁₃₉	C₂²	semidirect
38. C₂×D₁₁₁₂	C₂²	semidirect
39. D₈×D₁₃₉	C₂²	semidirect
40. #17984850	C₂²	semidirect
41. #17984849	C₂²	semidirect
42. #17984848	C₂²	semidirect
43. M₁₆×D₁₃₉	C₂²	semidirect
44. C₂×C₈×D₁₃₉	C₂²	semidirect
45. 2₋¹⁺⁴×C₁₃₉	C₂²	semidirect
46. #137×C₁₃₉	C₂²	semidirect
47. #135×C₁₃₉	C₂²	semidirect
48. C₂×Q₁₆×C₁₃₉	C₂²	semidirect
49. #131×C₁₃₉	C₂²	semidirect
164. C₁₃₉	#3584	semidirect
168. #578	D₁₃₉	semidirect
179. 2₋¹⁺⁴	C₂×D₁₃₉	semidirect
190. #137	C₂×D₁₃₉	semidirect
202. #135	C₂×D₁₃₉	semidirect
204. C₂×Q₁₆	C₂×D₁₃₉	semidirect
212. #131	C₂×D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581790	2
#22581766	2
#22581762	2
#22581761	2
#22581758	2
#22581755	2
#22581752	2
C₂×#17984865	2
#135×D₁₃₉	2
#131×D₁₃₉	2
#578×C₁₃₉	2
#3584	139