Subgroups — #31818971

Order 17792.

348 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581653	C₂	1	1	yes	maximal, characteristic
2	#22581641	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581352	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581296	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581285	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581275	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581270	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581261	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581198	C₂	1	1	yes	maximal, characteristic, complemented
10	#22581198	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#14555773	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581030	C₂	1	1	yes	maximal, characteristic, complemented
13	#22580994	C₂	1	1	yes	maximal, characteristic, complemented
14	#22580992	C₂	1	1	yes	maximal, characteristic, complemented
15	#478×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555773	C₂²	1	1	yes	characteristic
17	C₂×#14555768	C₂²	1	1	yes	characteristic
18	C₂×#14555768	C₂²	1	1	yes	characteristic
19	#17984837	—	2	1	no	—
20	#17984835	—	2	1	no	—
21	#17984834	C₂²	1	1	yes	characteristic
22	#17984833	—	2	1	no	—
23	#17984827	—	2	1	no	—
24	#17984825	—	2	1	no	—
25	#17984822	C₂²	1	1	yes	characteristic
26	#17984822	C₂²	1	1	yes	characteristic
27	#17984815	—	2	1	no	—
28	#17984813	C₂²	1	1	yes	characteristic
29	#17984813	C₂²	1	1	yes	characteristic
30	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
31	C₄×D₅₅₆	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊D₁₃₉	—	2	1	no	—
33	#17984800	—	2	1	no	—
34	#17984800	C₂²	1	1	yes	characteristic
35	#17984800	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₁₃₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₄×D₁₃₉	C₂²	1	1	yes	characteristic
38	#17984776	C₂²	1	1	yes	characteristic
39	#17984775	C₂²	1	1	yes	characteristic
40	#17984775	C₂²	1	1	yes	characteristic
41	#17984774	C₂²	1	1	yes	characteristic
42	#17984773	C₂²	1	1	yes	characteristic
43	#17984773	C₂²	1	1	yes	characteristic
44	#17984773	C₂²	1	1	yes	characteristic
45	#17984767	C₂²	1	1	yes	characteristic
46	#17984767	C₂²	1	1	yes	characteristic
47	#17984765	C₂²	1	1	yes	characteristic
48	#17984763	C₂²	1	1	yes	characteristic
49	(C₂×D₁₃₉)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₁₃₉)⋊C₈	C₂²	1	1	yes	characteristic
51	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
52	#125×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₄×D₄×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	#102×C₁₃₉	C₂²	1	1	yes	characteristic
58	#102×C₁₃₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
60	C₂×D₅₅₆	D₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
62	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
63	C₂×C₄×D₁₃₉	—	2	1	no	—
64	C₂×C₄×D₁₃₉	—	2	1	no	—
65	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
67	#14555773	—	2	2	no	—
68	#14555768	—	2	2	no	—
69	#14555768	—	2	2	no	—
70	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₁₃₉	—	2	1	no	—
73	C₄⋊Dic₁₃₉	—	2	1	no	—
74	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
75	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
78	Dic₁₃₉⋊C₄	—	2	1	no	—
79	Dic₁₃₉⋊C₄	—	2	1	no	—
80	Dic₁₃₉⋊C₄	—	2	1	no	—
81	Dic₁₃₉⋊C₄	—	2	1	no	—
82	Dic₁₃₉⋊C₄	—	2	1	no	—
83	C₂²⋊Dic₁₃₉	—	2	1	no	—
84	C₂²⋊Dic₁₃₉	—	2	1	no	—
85	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
86	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
87	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
88	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
89	(C₂×D₁₃₉)⋊C₄	D₄	1	1	yes	characteristic
90	(C₂×D₁₃₉)⋊C₄	D₄	1	1	yes	characteristic
91	C₄×Dic₁₃₉	—	2	1	no	—
92	C₄×Dic₁₃₉	—	2	1	no	—
93	C₄×Dic₁₃₉	—	2	1	no	—
94	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
95	C₂²×C₄×C₁₃₉	—	2	1	no	—
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	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
103	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
104	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
105	C₂²×D₁₃₉	C₂×D₄	1	1	yes	characteristic
106	C₂²×D₁₃₉	C₂×D₄	1	1	yes	characteristic
107	Dic₂₇₈	—	2	1	no	—
108	Dic₂₇₈	—	2	1	no	—
109	D₅₅₆	—	4	1	no	—
110	C₂²⋊D₁₃₉	—	4	2	no	—
111	C₂²⋊D₁₃₉	—	8	1	no	—
112	C₂×Dic₁₃₉	—	2	1	no	—
113	C₂×Dic₁₃₉	—	2	1	no	—
114	C₂×Dic₁₃₉	—	2	1	no	—
115	C₂×Dic₁₃₉	—	2	1	no	—
116	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
117	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
118	C₄×D₁₃₉	—	2	1	no	—
119	C₄×D₁₃₉	—	2	1	no	—
120	C₄×D₁₃₉	—	2	1	no	—
121	C₄×D₁₃₉	—	2	1	no	—
122	C₄×D₁₃₉	—	4	2	no	—
123	C₄×D₁₃₉	—	8	1	no	—
124	C₁₃₉⋊C₈	—	2	1	no	—
125	C₁₃₉⋊C₈	#41	1	2	yes	—
126	C₂³×C₁₃₉	—	2	1	no	—
127	D₄×C₁₃₉	—	2	1	no	—
128	D₄×C₁₃₉	—	2	1	no	—
129	C₂×C₄×C₁₃₉	—	2	1	no	—
130	C₂×C₄×C₁₃₉	—	2	1	no	—
131	C₂×C₄×C₁₃₉	—	2	1	no	—
132	C₂×C₄×C₁₃₉	—	8	1	no	—
133	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
136	C₈×C₁₃₉	—	2	1	no	—
137	C₈×C₁₃₉	—	2	1	no	—
138	C₂²×C₁₃₉	—	4	1	no	—
139	C₂²×C₁₃₉	—	4	1	no	—
140	C₂²×C₁₃₉	—	4	1	no	—
141	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
142	C₂×D₁₃₉	—	2	1	no	—
143	C₂×D₁₃₉	—	2	1	no	—
144	C₂×D₁₃₉	—	2	1	no	—
145	C₂×D₁₃₉	—	2	1	no	—
146	C₂×D₁₃₉	—	2	1	no	—
147	C₂×D₁₃₉	—	2	1	no	—
148	C₄×C₁₃₉	—	2	1	no	—
149	C₄×C₁₃₉	—	2	1	no	—
150	C₄×C₁₃₉	—	4	1	no	—
151	C₄×C₁₃₉	—	4	1	no	—
152	C₄×C₁₃₉	—	4	1	no	—
153	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
154	C₄×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
155	Dic₁₃₉	—	2	1	no	—
156	Dic₁₃₉	—	2	1	no	—
157	Dic₁₃₉	—	4	1	no	—
158	Dic₁₃₉	—	4	1	no	—
159	Dic₁₃₉	—	4	1	no	—
160	Dic₁₃₉	—	4	1	no	—
161	C₂×C₁₃₉	—	8	1	no	—
162	C₂×C₁₃₉	#547	1	1	yes	characteristic
163	C₂×C₁₃₉	#575	1	1	yes	characteristic
164	C₂×C₁₃₉	#577	1	1	yes	characteristic
165	D₁₃₉	—	4	1	no	—
166	D₁₃₉	—	4	1	no	—
167	C₁₃₉	#3351	1	1	yes	characteristic, complemented, Sylow
168	#3351	—	139	1	no	maximal, Sylow
169	#553	—	139	1	no	—
170	#552	—	139	1	no	—
171	#499	—	139	1	no	—
172	#485	—	139	1	no	—
173	#483	—	139	1	no	—
174	#482	—	139	1	no	—
175	#481	—	139	1	no	—
176	#479	—	139	1	no	—
177	#478	D₁₃₉	1	1	yes	characteristic, complemented
178	#468	—	139	1	no	—
179	#468	—	139	1	no	—
180	C₄×SD₁₆	—	139	1	no	—
181	#436	—	139	1	no	—
182	#428	—	139	1	no	—
183	#428	—	139	1	no	—
184	C₂×SD₁₆	—	139	1	no	—
185	C₂×M₁₆	—	139	1	no	—
186	C₂×M₁₆	—	139	1	no	—
187	#126	—	278	1	no	—
188	#126	—	278	1	no	—
189	#125	—	278	1	no	—
190	#125	—	139	1	no	—
191	#125	C₂×D₁₃₉	1	1	yes	characteristic
192	#123	—	278	1	no	—
193	#123	—	278	1	no	—
194	C₂²⋊Q₈	—	139	1	no	—
195	C₂²⋊Q₈	—	139	1	no	—
196	#121	—	278	1	no	—
197	#121	—	139	1	no	—
198	#121	—	139	1	no	—
199	C₄×Q₈	—	139	1	no	—
200	C₄×D₄	—	278	1	no	—
201	C₄×D₄	—	139	1	no	—
202	C₄×D₄	C₂×D₁₃₉	1	1	yes	characteristic
203	#117	—	278	1	no	—
204	#117	—	139	1	no	—
205	#117	—	139	1	no	—
206	C₂×C₄⋊C₄	—	139	1	no	—
207	C₂×C₄⋊C₄	—	139	1	no	—
208	#107	—	139	1	no	—
209	#107	—	139	1	no	—
210	#107	—	139	1	no	—
211	#107	C₂×D₁₃₉	1	1	yes	characteristic
212	#106	—	139	1	no	—
213	#106	—	139	1	no	—
214	#106	—	139	1	no	—
215	#106	—	139	1	no	—
216	#106	C₂×D₁₃₉	1	1	yes	characteristic
217	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
218	#103	—	139	1	no	—
219	#103	—	139	1	no	—
220	#103	—	139	1	no	—
221	#102	—	139	1	no	—
222	#102	C₂×D₁₃₉	1	1	yes	characteristic
223	#102	C₂×D₁₃₉	1	1	yes	characteristic
224	C₂²⋊C₈	—	139	1	no	—
225	C₂²⋊C₈	—	139	1	no	—
226	C₄×C₈	—	139	1	no	—
227	C₂×Q₈	—	139	1	no	—
228	C₂×D₄	—	278	1	no	—
229	C₂×D₄	—	278	1	no	—
230	C₂×D₄	—	139	1	no	—
231	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
232	C₂²×C₄	—	2	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	SD₁₆	—	278	2	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	—	278	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	—	139	1	no	—
254	C₄⋊C₄	—	139	1	no	—
255	C₄⋊C₄	—	139	1	no	—
256	C₄⋊C₄	—	139	1	no	—
257	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
258	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
259	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	278	1	no	—
262	C₂²⋊C₄	—	278	1	no	—
263	C₂²⋊C₄	—	278	1	no	—
264	C₂²⋊C₄	—	278	1	no	—
265	C₂²⋊C₄	—	278	1	no	—
266	C₂²⋊C₄	—	278	1	no	—
267	C₂²⋊C₄	—	139	1	no	—
268	C₂²⋊C₄	—	139	1	no	—
269	C₄²	—	278	1	no	—
270	C₄²	—	278	1	no	—
271	C₄²	—	278	1	no	—
272	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
273	C₂³	—	2	1	no	—
274	C₂³	—	139	1	no	—
275	C₂³	—	139	1	no	—
276	Q₈	—	278	1	no	—
277	Q₈	—	278	1	no	—
278	D₄	—	2	1	no	—
279	D₄	—	2	1	no	—
280	D₄	—	556	1	no	—
281	D₄	—	556	2	no	—
282	D₄	—	1112	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	8	1	no	—
287	C₂×C₄	—	278	1	no	—
288	C₂×C₄	—	278	1	no	—
289	C₂×C₄	—	278	1	no	—
290	C₂×C₄	—	278	1	no	—
291	C₂×C₄	—	278	1	no	—
292	C₂×C₄	—	278	1	no	—
293	C₂×C₄	—	278	1	no	—
294	C₂×C₄	—	278	1	no	—
295	C₂×C₄	—	556	2	no	—
296	C₂×C₄	—	1112	1	no	—
297	C₂×C₄	—	139	1	no	—
298	C₂×C₄	—	139	1	no	—
299	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
300	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
301	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	—	278	1	no	—
305	C₈	—	139	2	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	278	1	no	—
310	C₂²	—	278	1	no	—
311	C₂²	—	278	1	no	—
312	C₂²	—	278	1	no	—
313	C₂²	—	278	1	no	—
314	C₂²	—	278	1	no	—
315	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	278	1	no	—
322	C₄	—	278	1	no	—
323	C₄	—	556	1	no	—
324	C₄	—	556	1	no	—
325	C₄	—	556	1	no	—
326	C₄	—	556	1	no	—
327	C₄	#41×D₁₃₉	1	1	yes	characteristic
328	C₄	#17984890	1	1	yes	characteristic
329	C₂	—	8	1	no	—
330	C₂	—	556	1	no	—
331	C₂	—	556	1	no	—
332	C₂	#22581605	1	1	yes	characteristic
333	C₂	#22581750	1	1	yes	characteristic
334	C₂	#22581763	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#478×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	#22581653	C₂
hypercenter	#478	D₁₃₉
hypocenter	C₁₃₉	#3351
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
2. #22581641	C₂	semidirect
3. #22581352	C₂	semidirect
4. #22581296	C₂	semidirect
5. #22581285	C₂	semidirect
6. #22581275	C₂	semidirect
7. #22581270	C₂	semidirect
8. #22581261	C₂	semidirect
9. #22581198	C₂	semidirect
10. #22581198	C₂	semidirect
11. C₄×#14555773	C₂	semidirect
12. #22581030	C₂	semidirect
13. #22580994	C₂	semidirect
14. #22580992	C₂	semidirect
15. #478×C₁₃₉	C₂	semidirect
167. C₁₃₉	#3351	semidirect
177. #478	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581653	2
#22581641	2
#22581352	2
#22581296	2
#22581285	2
#22581275	2
#22581270	2
#22581261	2
#22581198	2
C₄×#14555773	2
#22581030	2
#22580994	2
#22580992	2
#478×C₁₃₉	2
#3351	139