Subgroups — #31816223

Order 17792.

384 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#125×D₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#14555786	C₂	1	1	yes	maximal, characteristic
3	#22581262	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581260	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581258	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581256	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581254	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581252	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581250	C₂	1	1	yes	maximal, characteristic, complemented
10	#22580994	C₂	1	1	yes	maximal, characteristic, complemented
11	#22580992	C₂	1	1	yes	maximal, characteristic, complemented
12	#22580975	C₂	1	1	yes	maximal, characteristic, complemented
13	#22580953	C₂	1	1	yes	maximal, characteristic
14	#22580953	C₂	1	1	yes	maximal, characteristic
15	#479×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555786	C₂²	1	1	yes	characteristic
17	C₂×#14555768	C₂²	1	1	yes	characteristic
18	C₂×#14555768	C₂²	1	1	yes	characteristic
19	#17984834	C₂²	1	1	yes	characteristic
20	#17984833	C₂²	1	1	yes	characteristic
21	#17984833	C₂²	1	1	yes	characteristic
22	#17984832	—	2	1	no	—
23	#17984832	—	2	1	no	—
24	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
25	C₄×D₅₅₆	—	2	1	no	—
26	C₄×D₅₅₆	C₂²	1	1	yes	characteristic
27	#17984804	—	2	1	no	—
28	#17984804	C₂²	1	1	yes	characteristic
29	#17984800	—	2	1	no	—
30	#17984800	C₂²	1	1	yes	characteristic
31	C₄⋊C₄×D₁₃₉	—	2	1	no	—
32	C₄⋊C₄×D₁₃₉	—	2	1	no	—
33	C₄⋊C₄×D₁₃₉	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₁₃₉	C₂²	1	1	yes	characteristic
35	C₄²×D₁₃₉	—	2	1	no	—
36	C₄²×D₁₃₉	C₂²	1	1	yes	characteristic
37	#17984776	C₂²	1	1	yes	characteristic
38	#17984775	C₂²	1	1	yes	characteristic
39	#17984775	C₂²	1	1	yes	characteristic
40	#17984774	C₂²	1	1	yes	characteristic
41	#17984773	C₂²	1	1	yes	characteristic
42	#17984773	C₂²	1	1	yes	characteristic
43	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
44	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
45	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
46	#17984766	C₂²	1	1	yes	characteristic
47	#17984766	C₂²	1	1	yes	characteristic
48	#17984764	C₂²	1	1	yes	characteristic
49	#17984764	C₂²	1	1	yes	characteristic
50	#17984762	C₂²	1	1	yes	characteristic
51	#17984762	C₂²	1	1	yes	characteristic
52	#125×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
54	#107×C₁₃₉	C₂²	1	1	yes	characteristic
55	#106×C₁₃₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
57	#103×C₁₃₉	C₂²	1	1	yes	characteristic
58	#103×C₁₃₉	C₂²	1	1	yes	characteristic
59	#14555786	—	2	2	no	—
60	#14555786	Q₈	1	4	yes	—
61	C₂×D₅₅₆	—	2	1	no	—
62	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₁₃₉	—	2	1	no	—
64	C₂×C₄×D₁₃₉	—	2	1	no	—
65	C₂×C₄×D₁₃₉	—	2	1	no	—
66	C₂×C₄×D₁₃₉	—	2	1	no	—
67	C₂×C₄×D₁₃₉	—	2	1	no	—
68	C₂×C₄×D₁₃₉	D₄	1	1	yes	characteristic
69	C₂×C₄×D₁₃₉	D₄	1	1	yes	characteristic
70	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
71	#14555768	—	2	2	no	—
72	#14555768	—	2	2	no	—
73	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₃₉	—	2	1	no	—
76	C₄⋊Dic₁₃₉	—	2	1	no	—
77	C₄⋊Dic₁₃₉	—	2	1	no	—
78	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
81	Dic₁₃₉⋊C₄	—	2	1	no	—
82	Dic₁₃₉⋊C₄	—	2	1	no	—
83	Dic₁₃₉⋊C₄	—	2	1	no	—
84	Dic₁₃₉⋊C₄	—	2	1	no	—
85	Dic₁₃₉⋊C₄	—	2	1	no	—
86	Dic₁₃₉⋊C₄	—	2	1	no	—
87	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
88	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
89	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
90	C₄×Dic₁₃₉	—	2	1	no	—
91	C₄×Dic₁₃₉	—	2	1	no	—
92	C₄×Dic₁₃₉	D₄	1	1	yes	characteristic
93	C₄×Dic₁₃₉	D₄	1	1	yes	characteristic
94	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
96	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₁₃₉	—	2	1	no	—
99	C₄⋊C₄×C₁₃₉	—	2	1	no	—
100	C₄⋊C₄×C₁₃₉	—	2	1	no	—
101	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
104	C₄²×C₁₃₉	—	2	1	no	—
105	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
106	C₂²×D₁₃₉	—	2	1	no	—
107	C₂²×D₁₃₉	C₂×D₄	1	1	yes	characteristic
108	D₅₅₆	—	2	4	no	—
109	D₅₅₆	—	2	1	no	—
110	D₅₅₆	C₂×Q₈	1	2	yes	—
111	C₂×Dic₁₃₉	—	2	1	no	—
112	C₂×Dic₁₃₉	—	2	1	no	—
113	C₂×Dic₁₃₉	—	2	1	no	—
114	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
115	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
116	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
117	C₄×D₁₃₉	—	2	4	no	—
118	C₄×D₁₃₉	—	4	1	no	—
119	C₄×D₁₃₉	—	4	1	no	—
120	C₄×D₁₃₉	—	4	2	no	—
121	C₄×D₁₃₉	—	4	2	no	—
122	C₄×D₁₃₉	—	4	2	no	—
123	C₄×D₁₃₉	—	4	2	no	—
124	C₄×D₁₃₉	C₂×Q₈	1	2	yes	—
125	C₄×D₁₃₉	#41	1	2	yes	—
126	C₁₃₉⋊C₈	—	2	1	no	—
127	C₁₃₉⋊C₈	—	2	1	no	—
128	Q₈×C₁₃₉	—	2	1	no	—
129	Q₈×C₁₃₉	C₂×Q₈	1	2	yes	—
130	C₂×C₄×C₁₃₉	—	2	1	no	—
131	C₂×C₄×C₁₃₉	—	2	1	no	—
132	C₂×C₄×C₁₃₉	—	2	1	no	—
133	C₂×C₄×C₁₃₉	—	2	1	no	—
134	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
137	C₈×C₁₃₉	—	2	1	no	—
138	C₈×C₁₃₉	—	2	1	no	—
139	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
140	C₂×D₁₃₉	—	2	1	no	—
141	C₂×D₁₃₉	—	2	1	no	—
142	C₂×D₁₃₉	—	2	2	no	—
143	C₂×D₁₃₉	—	4	1	no	—
144	C₂×D₁₃₉	—	4	1	no	—
145	C₂×D₁₃₉	C₂²⋊Q₈	1	2	yes	—
146	C₄×C₁₃₉	—	2	2	no	—
147	C₄×C₁₃₉	—	2	1	no	—
148	C₄×C₁₃₉	—	2	1	no	—
149	C₄×C₁₃₉	—	4	1	no	—
150	C₄×C₁₃₉	—	4	1	no	—
151	C₄×C₁₃₉	—	4	1	no	—
152	C₄×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
153	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
154	Dic₁₃₉	—	2	1	no	—
155	Dic₁₃₉	—	2	1	no	—
156	Dic₁₃₉	—	4	1	no	—
157	Dic₁₃₉	—	4	1	no	—
158	Dic₁₃₉	—	4	1	no	—
159	Dic₁₃₉	C₂²⋊Q₈	1	2	yes	—
160	C₂×C₁₃₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
161	C₂×C₁₃₉	#575	1	1	yes	characteristic
162	C₂×C₁₃₉	#575	1	1	yes	characteristic
163	D₁₃₉	—	2	2	no	—
164	D₁₃₉	—	4	2	no	—
165	C₁₃₉	#3080	1	1	yes	characteristic, complemented, Sylow
166	#3080	—	139	1	no	maximal, Sylow
167	C₂×#125	—	139	1	no	—
168	C₄×#41	—	139	1	no	—
169	#479	—	139	1	no	—
170	#479	—	139	1	no	—
171	#479	—	139	1	no	—
172	#479	D₁₃₉	1	1	yes	characteristic, complemented
173	#478	—	139	1	no	—
174	#478	—	139	1	no	—
175	#478	—	139	1	no	—
176	#478	—	139	1	no	—
177	#428	—	139	1	no	—
178	#428	—	139	1	no	—
179	#423	—	139	1	no	—
180	#417	—	139	1	no	—
181	#417	—	139	1	no	—
182	C₂×#41	—	139	1	no	—
183	C₂×M₁₆	—	139	1	no	—
184	C₂×M₁₆	—	139	1	no	—
185	#125	—	278	1	no	—
186	#125	—	278	1	no	—
187	#125	—	139	1	no	—
188	#125	—	139	1	no	—
189	#125	—	139	1	no	—
190	#125	C₂×D₁₃₉	1	1	yes	characteristic
191	C₄×Q₈	—	139	1	no	—
192	C₄×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
193	C₄×D₄	—	278	1	no	—
194	C₄×D₄	—	278	1	no	—
195	C₄×D₄	—	139	1	no	—
196	C₄×D₄	—	139	1	no	—
197	#117	—	278	1	no	—
198	#117	—	139	1	no	—
199	C₂×C₄⋊C₄	—	278	1	no	—
200	C₂×C₄⋊C₄	—	278	1	no	—
201	C₂×C₄⋊C₄	—	139	1	no	—
202	C₂×C₄⋊C₄	—	139	1	no	—
203	C₂×C₄²	—	278	1	no	—
204	C₂×C₄²	—	139	1	no	—
205	#107	—	139	1	no	—
206	#107	—	139	1	no	—
207	#107	—	139	1	no	—
208	#107	C₂×D₁₃₉	1	1	yes	characteristic
209	#106	—	139	1	no	—
210	#106	—	139	1	no	—
211	#106	—	139	1	no	—
212	#106	C₂×D₁₃₉	1	1	yes	characteristic
213	C₄⋊C₈	—	139	1	no	—
214	C₄⋊C₈	—	139	1	no	—
215	C₄⋊C₈	—	139	1	no	—
216	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
217	#103	—	139	1	no	—
218	#103	—	139	1	no	—
219	#103	C₂×D₁₃₉	1	1	yes	characteristic
220	#103	C₂×D₁₃₉	1	1	yes	characteristic
221	#102	—	139	1	no	—
222	#102	—	139	1	no	—
223	#102	—	139	1	no	—
224	#102	—	139	1	no	—
225	#41	—	278	2	no	—
226	#41	—	139	4	no	—
227	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
228	C₂×D₄	—	278	1	no	—
229	C₂×D₄	—	139	1	no	—
230	C₂²×C₄	—	278	1	no	—
231	C₂²×C₄	—	278	1	no	—
232	C₂²×C₄	—	278	1	no	—
233	C₂²×C₄	—	278	1	no	—
234	C₂²×C₄	—	278	1	no	—
235	C₂²×C₄	—	139	1	no	—
236	C₂²×C₄	—	139	1	no	—
237	C₂²×C₄	—	139	1	no	—
238	M₁₆	—	278	2	no	—
239	M₁₆	—	278	2	no	—
240	C₂×C₈	—	139	1	no	—
241	C₂×C₈	—	139	1	no	—
242	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	278	1	no	—
248	C₄⋊C₄	—	278	1	no	—
249	C₄⋊C₄	—	278	1	no	—
250	C₄⋊C₄	—	278	1	no	—
251	C₄⋊C₄	—	278	1	no	—
252	C₄⋊C₄	—	278	1	no	—
253	C₄⋊C₄	—	278	1	no	—
254	C₄⋊C₄	—	278	1	no	—
255	C₄⋊C₄	—	278	1	no	—
256	C₄⋊C₄	—	139	1	no	—
257	C₄⋊C₄	—	139	1	no	—
258	C₄⋊C₄	—	139	1	no	—
259	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
260	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
261	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
262	C₂²⋊C₄	—	278	1	no	—
263	C₂²⋊C₄	—	278	1	no	—
264	C₂²⋊C₄	—	278	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	278	1	no	—
267	C₄²	—	278	1	no	—
268	C₄²	—	139	1	no	—
269	C₄²	—	139	1	no	—
270	C₄²	—	139	1	no	—
271	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
272	C₂³	—	278	1	no	—
273	C₂³	—	139	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	Q₈×D₁₃₉	1	2	yes	—
276	D₄	—	278	4	no	—
277	D₄	—	278	1	no	—
278	D₄	—	139	2	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	278	4	no	—
284	C₂×C₄	—	278	1	no	—
285	C₂×C₄	—	278	1	no	—
286	C₂×C₄	—	278	1	no	—
287	C₂×C₄	—	556	1	no	—
288	C₂×C₄	—	556	1	no	—
289	C₂×C₄	—	556	2	no	—
290	C₂×C₄	—	556	2	no	—
291	C₂×C₄	—	556	2	no	—
292	C₂×C₄	—	556	2	no	—
293	C₂×C₄	—	139	1	no	—
294	C₂×C₄	—	139	1	no	—
295	C₂×C₄	—	139	1	no	—
296	C₂×C₄	—	139	2	no	—
297	C₂×C₄	—	139	2	no	—
298	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
299	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
300	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	278	1	no	—
304	C₈	—	278	1	no	—
305	C₂²	—	278	1	no	—
306	C₂²	—	278	1	no	—
307	C₂²	—	278	2	no	—
308	C₂²	—	556	1	no	—
309	C₂²	—	556	1	no	—
310	C₂²	—	139	2	no	—
311	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	278	1	no	—
319	C₄	—	278	1	no	—
320	C₄	—	556	1	no	—
321	C₄	—	556	1	no	—
322	C₄	—	556	1	no	—
323	C₄	—	139	2	no	—
324	C₄	C₂×Q₈×D₁₃₉	1	1	yes	characteristic
325	C₄	#41×D₁₃₉	1	1	yes	characteristic
326	C₂	—	278	2	no	—
327	C₂	—	556	2	no	—
328	C₂	C₂²⋊Q₈×D₁₃₉	1	1	yes	characteristic
329	C₂	#22581750	1	1	yes	characteristic
330	C₂	#22581752	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#479×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂×#14555786	C₂²
hypercenter	#479	D₁₃₉
hypocenter	C₁₃₉	#3080
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
1. #125×D₁₃₉	C₂	semidirect
3. #22581262	C₂	semidirect
4. #22581260	C₂	semidirect
5. #22581258	C₂	semidirect
6. #22581256	C₂	semidirect
7. #22581254	C₂	semidirect
8. #22581252	C₂	semidirect
9. #22581250	C₂	semidirect
10. #22580994	C₂	semidirect
11. #22580992	C₂	semidirect
12. #22580975	C₂	semidirect
15. #479×C₁₃₉	C₂	semidirect
165. C₁₃₉	#3080	semidirect
172. #479	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#125×D₁₃₉	2
C₄×#14555786	2
#22581262	2
#22581260	2
#22581258	2
#22581256	2
#22581254	2
#22581252	2
#22581250	2
#22580994	2
#22580992	2
#22580975	2
#22580953	2
#479×C₁₃₉	2
#3080	139