Subgroups — #31818979

Order 17792.

336 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581638	C₂	1	1	yes	maximal, characteristic
2	#22581637	C₂	1	1	yes	maximal, characteristic
3	#22581354	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581295	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581293	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581290	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581259	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581258	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581210	C₂	1	1	yes	maximal, characteristic, complemented
10	#22581209	C₂	1	1	yes	maximal, characteristic, complemented
11	#22581045	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581028	C₂	1	1	yes	maximal, characteristic
13	#22580994	C₂	1	1	yes	maximal, characteristic, complemented
14	#22580993	C₂	1	1	yes	maximal, characteristic, complemented
15	#484×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555775	C₂²	1	1	yes	characteristic
17	C₂×#14555769	C₂²	1	1	yes	characteristic
18	C₂×#14555768	C₂²	1	1	yes	characteristic
19	#17984837	—	2	1	no	—
20	#17984835	—	2	1	no	—
21	#17984833	C₂²	1	1	yes	characteristic
22	#17984833	C₂²	1	1	yes	characteristic
23	#17984832	—	2	1	no	—
24	#17984832	—	2	1	no	—
25	#17984825	—	2	1	no	—
26	#17984824	—	2	1	no	—
27	#17984822	C₂²	1	1	yes	characteristic
28	#17984821	C₂²	1	1	yes	characteristic
29	#17984818	C₂²	1	1	yes	characteristic
30	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
31	#17984807	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊D₁₃₉	C₂²	1	1	yes	characteristic
33	#17984801	—	2	1	no	—
34	#17984801	C₂²	1	1	yes	characteristic
35	#17984800	—	2	1	no	—
36	#17984800	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	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
46	#17984767	C₂²	1	1	yes	characteristic
47	#17984767	C₂²	1	1	yes	characteristic
48	#17984766	C₂²	1	1	yes	characteristic
49	#17984765	C₂²	1	1	yes	characteristic
50	(C₂×D₁₃₉)⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
54	#107×C₁₃₉	C₂²	1	1	yes	characteristic
55	#107×C₁₃₉	C₂²	1	1	yes	characteristic
56	#103×C₁₃₉	C₂²	1	1	yes	characteristic
57	#103×C₁₃₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
60	C₂×C₂²⋊D₁₃₉	D₄	1	1	yes	characteristic
61	C₂×C₄×D₁₃₉	—	2	1	no	—
62	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
63	#14555775	—	2	2	no	—
64	#14555769	—	2	2	no	—
65	#14555768	—	2	2	no	—
66	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
67	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
68	C₄⋊Dic₁₃₉	—	2	1	no	—
69	C₄⋊Dic₁₃₉	—	2	1	no	—
70	C₄⋊Dic₁₃₉	—	2	1	no	—
71	C₄⋊Dic₁₃₉	—	2	1	no	—
72	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
75	Dic₁₃₉⋊C₄	—	2	1	no	—
76	Dic₁₃₉⋊C₄	—	2	1	no	—
77	Dic₁₃₉⋊C₄	—	2	1	no	—
78	Dic₁₃₉⋊C₄	—	2	1	no	—
79	Dic₁₃₉⋊C₄	—	2	1	no	—
80	Dic₁₃₉⋊C₄	—	2	1	no	—
81	Dic₁₃₉⋊C₄	D₄	1	1	yes	characteristic
82	C₂²⋊Dic₁₃₉	—	2	1	no	—
83	C₂²⋊Dic₁₃₉	—	2	1	no	—
84	C₂²⋊Dic₁₃₉	D₄	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
89	C₄×Dic₁₃₉	—	2	1	no	—
90	C₄×Dic₁₃₉	—	2	1	no	—
91	C₄×Dic₁₃₉	—	2	1	no	—
92	C₄×Dic₁₃₉	—	2	1	no	—
93	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
95	C₂²×C₄×C₁₃₉	—	2	1	no	—
96	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
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	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
105	C₂²×D₁₃₉	C₂×D₄	1	1	yes	characteristic
106	Dic₂₇₈	—	2	1	no	—
107	Dic₂₇₈	—	2	1	no	—
108	C₂²⋊D₁₃₉	—	4	1	no	—
109	C₂×Dic₁₃₉	—	2	1	no	—
110	C₂×Dic₁₃₉	—	2	1	no	—
111	C₂×Dic₁₃₉	—	2	1	no	—
112	C₂×Dic₁₃₉	—	2	1	no	—
113	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
114	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
115	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
116	C₄×D₁₃₉	—	2	1	no	—
117	C₄×D₁₃₉	—	2	1	no	—
118	C₄×D₁₃₉	—	4	2	no	—
119	C₁₃₉⋊C₈	—	2	1	no	—
120	C₁₃₉⋊C₈	#41	1	2	yes	—
121	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
122	Q₈×C₁₃₉	—	2	1	no	—
123	Q₈×C₁₃₉	—	2	1	no	—
124	C₂×C₄×C₁₃₉	—	2	1	no	—
125	C₂×C₄×C₁₃₉	—	2	1	no	—
126	C₂×C₄×C₁₃₉	—	2	1	no	—
127	C₂×C₄×C₁₃₉	—	2	1	no	—
128	C₂×C₄×C₁₃₉	—	2	1	no	—
129	C₂×C₄×C₁₃₉	—	2	1	no	—
130	C₂×C₄×C₁₃₉	—	4	2	no	—
131	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
133	C₈×C₁₃₉	—	2	1	no	—
134	C₈×C₁₃₉	—	2	1	no	—
135	C₂²×C₁₃₉	—	2	1	no	—
136	C₂²×C₁₃₉	—	2	1	no	—
137	C₂²×C₁₃₉	—	2	1	no	—
138	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
139	C₂×D₁₃₉	—	2	1	no	—
140	C₂×D₁₃₉	—	2	1	no	—
141	C₂×D₁₃₉	—	2	1	no	—
142	C₄×C₁₃₉	—	2	1	no	—
143	C₄×C₁₃₉	—	4	1	no	—
144	C₄×C₁₃₉	—	4	1	no	—
145	C₄×C₁₃₉	—	4	1	no	—
146	C₄×C₁₃₉	—	4	1	no	—
147	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
148	C₄×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
149	Dic₁₃₉	—	2	1	no	—
150	Dic₁₃₉	—	2	1	no	—
151	Dic₁₃₉	—	2	1	no	—
152	Dic₁₃₉	—	4	1	no	—
153	Dic₁₃₉	—	4	1	no	—
154	Dic₁₃₉	—	4	1	no	—
155	Dic₁₃₉	—	4	1	no	—
156	C₂×C₁₃₉	—	4	1	no	—
157	C₂×C₁₃₉	#547	1	1	yes	characteristic
158	C₂×C₁₃₉	#575	1	1	yes	characteristic
159	C₂×C₁₃₉	#578	1	1	yes	characteristic
160	D₁₃₉	—	4	1	no	—
161	C₁₃₉	#3352	1	1	yes	characteristic, complemented, Sylow
162	#3352	—	139	1	no	maximal, Sylow
163	#552	—	139	1	no	—
164	#552	—	139	1	no	—
165	#499	—	139	1	no	—
166	#485	—	139	1	no	—
167	#485	—	139	1	no	—
168	#484	—	139	1	no	—
169	#484	D₁₃₉	1	1	yes	characteristic, complemented
170	#479	—	139	1	no	—
171	#479	—	139	1	no	—
172	#470	—	139	1	no	—
173	#470	—	139	1	no	—
174	C₄×Q₁₆	—	139	1	no	—
175	#436	—	139	1	no	—
176	#428	—	139	1	no	—
177	#428	—	139	1	no	—
178	C₂×Q₁₆	—	139	1	no	—
179	C₂×M₁₆	—	139	1	no	—
180	C₂×M₁₆	—	139	1	no	—
181	#126	—	278	1	no	—
182	#126	—	278	1	no	—
183	#125	—	278	1	no	—
184	#125	—	278	1	no	—
185	#125	—	139	1	no	—
186	#125	—	139	1	no	—
187	#123	—	278	1	no	—
188	#123	—	278	1	no	—
189	C₂²⋊Q₈	—	139	1	no	—
190	C₂²⋊Q₈	—	139	1	no	—
191	C₂²⋊Q₈	—	139	1	no	—
192	C₂²⋊Q₈	C₂×D₁₃₉	1	1	yes	characteristic
193	C₄×Q₈	—	139	1	no	—
194	C₄×Q₈	—	139	1	no	—
195	C₄×D₄	—	139	1	no	—
196	#117	—	278	1	no	—
197	#117	—	278	1	no	—
198	#117	—	139	1	no	—
199	#117	—	139	1	no	—
200	C₂×C₄⋊C₄	—	139	1	no	—
201	C₂×C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
202	#107	—	139	1	no	—
203	#107	—	139	1	no	—
204	#107	—	139	1	no	—
205	#107	C₂×D₁₃₉	1	1	yes	characteristic
206	#107	C₂×D₁₃₉	1	1	yes	characteristic
207	#106	—	139	1	no	—
208	#106	—	139	1	no	—
209	#106	—	139	1	no	—
210	#106	—	139	1	no	—
211	C₄⋊C₈	—	139	1	no	—
212	#103	—	139	1	no	—
213	#103	—	139	1	no	—
214	#103	—	139	1	no	—
215	#103	—	139	1	no	—
216	#103	C₂×D₁₃₉	1	1	yes	characteristic
217	#103	C₂×D₁₃₉	1	1	yes	characteristic
218	C₂²⋊C₈	—	139	1	no	—
219	C₂²⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
220	C₄×C₈	—	139	1	no	—
221	C₂×Q₈	—	139	1	no	—
222	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
223	C₂×D₄	—	139	1	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	278	1	no	—
226	C₂²×C₄	—	139	1	no	—
227	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
228	Q₁₆	—	278	2	no	—
229	M₁₆	—	278	2	no	—
230	M₁₆	—	278	2	no	—
231	C₂×C₈	—	139	1	no	—
232	C₂×C₈	—	139	1	no	—
233	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
234	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	278	1	no	—
238	C₄⋊C₄	—	278	1	no	—
239	C₄⋊C₄	—	278	1	no	—
240	C₄⋊C₄	—	278	1	no	—
241	C₄⋊C₄	—	278	1	no	—
242	C₄⋊C₄	—	278	1	no	—
243	C₄⋊C₄	—	278	1	no	—
244	C₄⋊C₄	—	278	1	no	—
245	C₄⋊C₄	—	278	1	no	—
246	C₄⋊C₄	—	278	1	no	—
247	C₄⋊C₄	—	139	1	no	—
248	C₄⋊C₄	—	139	1	no	—
249	C₄⋊C₄	—	139	1	no	—
250	C₄⋊C₄	—	139	1	no	—
251	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
252	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
253	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
254	C₂²⋊C₄	—	2	1	no	—
255	C₂²⋊C₄	—	278	1	no	—
256	C₂²⋊C₄	—	278	1	no	—
257	C₂²⋊C₄	—	278	1	no	—
258	C₂²⋊C₄	—	278	1	no	—
259	C₂²⋊C₄	—	278	1	no	—
260	C₂²⋊C₄	—	139	1	no	—
261	C₂²⋊C₄	—	139	1	no	—
262	C₄²	—	278	1	no	—
263	C₄²	—	278	1	no	—
264	C₄²	—	278	1	no	—
265	C₄²	—	278	1	no	—
266	C₄²	—	139	1	no	—
267	C₂³	—	139	1	no	—
268	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
269	Q₈	—	2	1	no	—
270	Q₈	—	2	1	no	—
271	Q₈	—	278	1	no	—
272	Q₈	—	278	1	no	—
273	D₄	—	556	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	278	1	no	—
282	C₂×C₄	—	278	1	no	—
283	C₂×C₄	—	278	1	no	—
284	C₂×C₄	—	278	1	no	—
285	C₂×C₄	—	278	1	no	—
286	C₂×C₄	—	278	1	no	—
287	C₂×C₄	—	556	2	no	—
288	C₂×C₄	—	139	1	no	—
289	C₂×C₄	—	139	1	no	—
290	C₂×C₄	—	139	1	no	—
291	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
292	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	—	278	1	no	—
296	C₈	—	139	2	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	278	1	no	—
301	C₂²	—	278	1	no	—
302	C₂²	—	278	1	no	—
303	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	278	1	no	—
310	C₄	—	278	1	no	—
311	C₄	—	278	1	no	—
312	C₄	—	556	1	no	—
313	C₄	—	556	1	no	—
314	C₄	—	556	1	no	—
315	C₄	—	556	1	no	—
316	C₄	#41×D₁₃₉	1	1	yes	characteristic
317	C₄	#17984889	1	1	yes	characteristic
318	C₂	—	4	1	no	—
319	C₂	—	556	1	no	—
320	C₂	#22581601	1	1	yes	characteristic
321	C₂	#22581749	1	1	yes	characteristic
322	C₂	#22581766	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#484×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂×C₂²⋊D₁₃₉	D₄
hypercenter	#484	D₁₃₉
hypocenter	C₁₃₉	#3352
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #484

Complements

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

N	H ≅ G/N	type
3. #22581354	C₂	semidirect
4. #22581295	C₂	semidirect
5. #22581293	C₂	semidirect
6. #22581290	C₂	semidirect
7. #22581259	C₂	semidirect
8. #22581258	C₂	semidirect
9. #22581210	C₂	semidirect
10. #22581209	C₂	semidirect
11. #22581045	C₂	semidirect
13. #22580994	C₂	semidirect
14. #22580993	C₂	semidirect
15. #484×C₁₃₉	C₂	semidirect
161. C₁₃₉	#3352	semidirect
169. #484	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581638	2
#22581637	2
#22581354	2
#22581295	2
#22581293	2
#22581290	2
#22581259	2
#22581258	2
#22581210	2
#22581209	2
#22581045	2
#22581028	2
#22580994	2
#22580993	2
#484×C₁₃₉	2
#3352	139