Subgroups — #31816968

Order 17792.

380 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581487	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581352	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581335	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581310	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581307	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581192	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581187	C₂	1	1	yes	maximal, characteristic
8	#22581184	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581180	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#14555775	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#14555772	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#14555773	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#14555770	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₄×C₁₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
15	#529×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555775	C₂²	1	1	yes	characteristic
17	C₂×#14555773	C₂²	1	1	yes	characteristic
18	C₂×#14555772	C₂²	1	1	yes	characteristic
19	C₂×#14555770	C₂²	1	1	yes	characteristic
20	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
21	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
22	#17984836	—	2	1	no	—
23	#17984834	C₂²	1	1	yes	characteristic
24	#17984830	C₂²	1	1	yes	characteristic
25	#17984825	—	2	1	no	—
26	#17984821	C₂²	1	1	yes	characteristic
27	#17984816	C₂²	1	1	yes	characteristic
28	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
29	C₄×D₅₅₆	C₂²	1	1	yes	characteristic
30	C₄×C₂²⋊D₁₃₉	—	2	1	no	—
31	#17984799	—	2	1	no	—
32	#17984775	C₂²	1	1	yes	characteristic
33	#17984775	C₂²	1	1	yes	characteristic
34	#17984773	C₂²	1	1	yes	characteristic
35	#17984773	C₂²	1	1	yes	characteristic
36	#17984766	C₂²	1	1	yes	characteristic
37	#17984766	C₂²	1	1	yes	characteristic
38	#17984765	C₂²	1	1	yes	characteristic
39	#17984765	C₂²	1	1	yes	characteristic
40	#17984763	C₂²	1	1	yes	characteristic
41	#17984763	C₂²	1	1	yes	characteristic
42	#17984762	C₂²	1	1	yes	characteristic
43	#17984762	C₂²	1	1	yes	characteristic
44	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
45	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
46	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
47	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
48	#126×C₁₃₉	—	2	1	no	—
49	#125×C₁₃₉	C₂²	1	1	yes	characteristic
50	#124×C₁₃₉	C₂²	1	1	yes	characteristic
51	#123×C₁₃₉	—	2	1	no	—
52	C₂²⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
53	#121×C₁₃₉	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₁₃₉	—	2	1	no	—
56	C₄×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
57	#117×C₁₃₉	—	2	1	no	—
58	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
60	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
62	C₂×C₄×D₁₃₉	—	2	1	no	—
63	#14555775	—	2	2	no	—
64	#14555773	—	2	2	no	—
65	#14555772	—	2	2	no	—
66	#14555770	—	2	2	no	—
67	C₂×C₁₃₉⋊C₈	—	2	2	no	—
68	C₂×C₁₃₉⋊C₈	—	2	2	no	—
69	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₁₃₉⋊C₈	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	C₂²⋊Dic₁₃₉	—	2	1	no	—
80	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
81	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
82	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
83	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
84	C₄×Dic₁₃₉	—	2	1	no	—
85	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
86	C₂×D₄×C₁₃₉	—	2	1	no	—
87	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
88	C₂²×C₄×C₁₃₉	—	2	1	no	—
89	C₂²×C₄×C₁₃₉	D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₃₉	D₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
92	C₄⋊C₄×C₁₃₉	—	2	1	no	—
93	C₄⋊C₄×C₁₃₉	—	2	1	no	—
94	C₄⋊C₄×C₁₃₉	—	2	1	no	—
95	C₄⋊C₄×C₁₃₉	—	2	1	no	—
96	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
97	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
98	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
99	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
100	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
101	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
102	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
103	C₄²×C₁₃₉	—	2	1	no	—
104	C₄²×C₁₃₉	D₄	1	1	yes	characteristic
105	C₄²×C₁₃₉	D₄	1	1	yes	characteristic
106	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
107	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
108	C₂²×D₁₃₉	—	2	1	no	—
109	Dic₂₇₈	—	2	1	no	—
110	Dic₂₇₈	—	2	1	no	—
111	D₅₅₆	—	2	1	no	—
112	D₅₅₆	—	2	1	no	—
113	C₂²⋊D₁₃₉	—	4	2	no	—
114	C₂×Dic₁₃₉	—	2	1	no	—
115	C₂×Dic₁₃₉	—	2	1	no	—
116	C₂×Dic₁₃₉	—	2	1	no	—
117	C₄×D₁₃₉	—	4	2	no	—
118	C₁₃₉⋊C₈	—	2	1	no	—
119	C₁₃₉⋊C₈	—	2	1	no	—
120	C₁₃₉⋊C₈	#41	1	2	yes	—
121	C₁₃₉⋊C₈	#41	1	2	yes	—
122	C₂³×C₁₃₉	—	2	1	no	—
123	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
124	Q₈×C₁₃₉	—	2	1	no	—
125	Q₈×C₁₃₉	—	2	1	no	—
126	D₄×C₁₃₉	—	2	1	no	—
127	D₄×C₁₃₉	—	2	1	no	—
128	D₄×C₁₃₉	—	4	2	no	—
129	C₂×C₄×C₁₃₉	—	2	2	no	—
130	C₂×C₄×C₁₃₉	—	2	2	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	—	2	1	no	—
135	C₂×C₄×C₁₃₉	—	2	1	no	—
136	C₂×C₄×C₁₃₉	—	4	2	no	—
137	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
143	C₂²×C₁₃₉	—	2	1	no	—
144	C₂²×C₁₃₉	—	2	1	no	—
145	C₂²×C₁₃₉	—	4	1	no	—
146	C₂²×C₁₃₉	—	4	1	no	—
147	C₂²×C₁₃₉	—	4	1	no	—
148	C₂²×C₁₃₉	#135	1	2	yes	—
149	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
150	C₂×D₁₃₉	—	4	1	no	—
151	C₂×D₁₃₉	—	4	1	no	—
152	C₂×D₁₃₉	—	4	1	no	—
153	C₄×C₁₃₉	—	2	1	no	—
154	C₄×C₁₃₉	—	2	1	no	—
155	C₄×C₁₃₉	—	2	1	no	—
156	C₄×C₁₃₉	—	2	1	no	—
157	C₄×C₁₃₉	—	4	1	no	—
158	C₄×C₁₃₉	—	4	1	no	—
159	C₄×C₁₃₉	—	4	1	no	—
160	C₄×C₁₃₉	#135	1	2	yes	—
161	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
162	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
163	Dic₁₃₉	—	4	1	no	—
164	Dic₁₃₉	—	4	1	no	—
165	Dic₁₃₉	—	4	1	no	—
166	C₂×C₁₃₉	—	2	2	no	—
167	C₂×C₁₃₉	—	8	1	no	—
168	C₂×C₁₃₉	#532	1	1	yes	characteristic
169	C₂×C₁₃₉	C₂×#135	1	1	yes	characteristic
170	C₂×C₁₃₉	C₂×#135	1	1	yes	characteristic
171	D₁₃₉	—	8	1	no	—
172	C₁₃₉	#3168	1	1	yes	characteristic, complemented, Sylow
173	#3168	—	139	1	no	maximal, Sylow
174	#529	—	139	1	no	—
175	#529	D₁₃₉	1	1	yes	characteristic, complemented
176	#499	—	139	1	no	—
177	#495	—	139	1	no	—
178	#488	—	139	1	no	—
179	#488	—	139	1	no	—
180	#467	—	139	1	no	—
181	#466	—	139	1	no	—
182	#465	—	139	1	no	—
183	#465	—	139	1	no	—
184	C₄×Q₁₆	—	139	1	no	—
185	C₄×SD₁₆	—	139	1	no	—
186	C₄×SD₁₆	—	139	1	no	—
187	C₄×D₈	—	139	1	no	—
188	C₂×C₄×C₈	—	139	1	no	—
189	C₂×Q₁₆	—	139	1	no	—
190	C₂×SD₁₆	—	139	1	no	—
191	C₂×SD₁₆	—	139	1	no	—
192	C₂×D₈	—	139	1	no	—
193	C₂²×C₈	—	139	1	no	—
194	C₂²×C₈	—	139	1	no	—
195	#126	—	2	1	no	—
196	#126	—	278	1	no	—
197	#125	—	139	1	no	—
198	#125	C₂×D₁₃₉	1	1	yes	characteristic
199	#124	—	139	1	no	—
200	#124	C₂×D₁₃₉	1	1	yes	characteristic
201	#123	—	2	1	no	—
202	#123	—	278	1	no	—
203	C₂²⋊Q₈	—	139	1	no	—
204	C₂²⋊Q₈	C₂×D₁₃₉	1	1	yes	characteristic
205	#121	—	139	1	no	—
206	#121	C₂×D₁₃₉	1	1	yes	characteristic
207	C₄×Q₈	—	139	1	no	—
208	C₄×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
209	C₄×D₄	—	2	1	no	—
210	C₄×D₄	—	278	1	no	—
211	C₄×D₄	—	139	1	no	—
212	C₄×D₄	C₂×D₁₃₉	1	1	yes	characteristic
213	#117	—	2	1	no	—
214	#117	—	278	1	no	—
215	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
216	#107	—	139	1	no	—
217	#107	—	139	1	no	—
218	#106	—	139	1	no	—
219	#106	—	139	1	no	—
220	#103	—	139	1	no	—
221	#103	—	139	1	no	—
222	#103	—	139	1	no	—
223	#103	—	139	1	no	—
224	#102	—	139	1	no	—
225	#102	—	139	1	no	—
226	#102	—	139	1	no	—
227	#102	—	139	1	no	—
228	C₄×C₈	—	139	1	no	—
229	C₄×C₈	—	139	1	no	—
230	C₄×C₈	—	139	1	no	—
231	C₄×C₈	—	139	1	no	—
232	C₂×Q₈	—	139	1	no	—
233	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	—	278	1	no	—
236	C₂×D₄	—	139	1	no	—
237	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	278	1	no	—
240	C₂²×C₄	C₂²⋊D₁₃₉	1	1	yes	characteristic
241	C₂²×C₄	C₂²⋊D₁₃₉	1	1	yes	characteristic
242	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
243	Q₁₆	—	278	2	no	—
244	SD₁₆	—	278	2	no	—
245	SD₁₆	—	278	2	no	—
246	D₈	—	278	2	no	—
247	C₂×C₈	—	278	2	no	—
248	C₂×C₈	—	278	2	no	—
249	C₂×C₈	—	139	1	no	—
250	C₂×C₈	—	139	1	no	—
251	C₂×C₈	—	139	1	no	—
252	C₂×C₈	—	139	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	278	1	no	—
258	C₄⋊C₄	—	278	1	no	—
259	C₄⋊C₄	—	278	1	no	—
260	C₄⋊C₄	—	278	1	no	—
261	C₄⋊C₄	—	139	1	no	—
262	C₄⋊C₄	—	139	1	no	—
263	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	2	1	no	—
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	278	1	no	—
271	C₂²⋊C₄	—	278	1	no	—
272	C₂²⋊C₄	—	278	1	no	—
273	C₂²⋊C₄	—	278	1	no	—
274	C₂²⋊C₄	—	278	1	no	—
275	C₄²	—	2	1	no	—
276	C₄²	—	278	1	no	—
277	C₄²	C₂²⋊D₁₃₉	1	1	yes	characteristic
278	C₄²	C₂²⋊D₁₃₉	1	1	yes	characteristic
279	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
280	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
281	C₂³	—	2	1	no	—
282	C₂³	—	278	1	no	—
283	C₂³	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
284	Q₈	—	2	1	no	—
285	Q₈	—	2	1	no	—
286	Q₈	—	278	1	no	—
287	Q₈	—	278	1	no	—
288	D₄	—	2	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	4	2	no	—
291	D₄	—	278	1	no	—
292	D₄	—	278	1	no	—
293	D₄	—	556	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	278	1	no	—
303	C₂×C₄	—	278	1	no	—
304	C₂×C₄	—	278	1	no	—
305	C₂×C₄	—	556	2	no	—
306	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
307	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
308	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
309	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
310	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
311	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
312	C₈	—	278	1	no	—
313	C₈	—	278	1	no	—
314	C₈	—	139	2	no	—
315	C₈	—	139	2	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	556	1	no	—
322	C₂²	—	556	1	no	—
323	C₂²	—	556	1	no	—
324	C₂²	#17984862	1	2	yes	—
325	C₂²	C₂²×C₂²⋊D₁₃₉	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	556	1	no	—
334	C₄	—	556	1	no	—
335	C₄	—	556	1	no	—
336	C₄	#17984862	1	2	yes	—
337	C₄	#41×D₁₃₉	1	1	yes	characteristic
338	C₄	#41×D₁₃₉	1	1	yes	characteristic
339	C₂	—	2	2	no	—
340	C₂	—	8	1	no	—
341	C₂	—	1112	1	no	—
342	C₂	#22581499	1	1	yes	characteristic
343	C₂	C₂×#17984862	1	1	yes	characteristic
344	C₂	C₂×#17984862	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#529×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	#22581187	C₂
hypercenter	#529	D₁₃₉
hypocenter	C₁₃₉	#3168
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #529

Complements

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

N	H ≅ G/N	type
1. #22581487	C₂	semidirect
2. #22581352	C₂	semidirect
3. #22581335	C₂	semidirect
4. #22581310	C₂	semidirect
5. #22581307	C₂	semidirect
6. #22581192	C₂	semidirect
8. #22581184	C₂	semidirect
9. #22581180	C₂	semidirect
10. C₄×#14555775	C₂	semidirect
11. C₄×#14555772	C₂	semidirect
12. C₄×#14555773	C₂	semidirect
13. C₄×#14555770	C₂	semidirect
14. C₂×C₄×C₁₃₉⋊C₈	C₂	semidirect
15. #529×C₁₃₉	C₂	semidirect
172. C₁₃₉	#3168	semidirect
175. #529	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581487	2
#22581352	2
#22581335	2
#22581310	2
#22581307	2
#22581192	2
#22581187	2
#22581184	2
#22581180	2
C₄×#14555775	2
C₄×#14555772	2
C₄×#14555773	2
C₄×#14555770	2
C₂×C₄×C₁₃₉⋊C₈	2
#529×C₁₃₉	2
#3168	139