Subgroups — #31816958

Order 17792.

412 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581505	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581498	C₂	1	1	yes	maximal, characteristic
3	#22581350	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581324	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581302	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581298	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581184	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581182	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581180	C₂	1	1	yes	maximal, characteristic, complemented
10	#22581044	C₂	1	1	yes	maximal, characteristic, complemented
11	SD₁₆×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581040	C₂	1	1	yes	maximal, characteristic, complemented
13	#22581038	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₈×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#465×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555784	—	2	1	no	—
17	C₂×#14555774	C₂²	1	1	yes	characteristic
18	C₂×#14555773	C₂²	1	1	yes	characteristic
19	C₂×#14555772	C₂²	1	1	yes	characteristic
20	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
21	#17984841	C₂²	1	1	yes	characteristic
22	#17984841	C₂²	1	1	yes	characteristic
23	#17984840	C₂²	1	1	yes	characteristic
24	#17984838	C₂²	1	1	yes	characteristic
25	#17984832	—	2	1	no	—
26	#17984828	—	2	1	no	—
27	#17984821	C₂²	1	1	yes	characteristic
28	#17984819	—	2	1	no	—
29	#17984816	C₂²	1	1	yes	characteristic
30	#17984814	—	2	1	no	—
31	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
32	#17984807	—	2	1	no	—
33	#17984807	C₂²	1	1	yes	characteristic
34	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
35	#17984804	C₂²	1	1	yes	characteristic
36	(C₂²⋊D₁₃₉)⋊C₄	—	2	1	no	—
37	#17984798	—	2	1	no	—
38	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
39	#17984774	C₂²	1	1	yes	characteristic
40	#17984773	C₂²	1	1	yes	characteristic
41	#17984773	C₂²	1	1	yes	characteristic
42	#17984767	C₂²	1	1	yes	characteristic
43	#17984766	C₂²	1	1	yes	characteristic
44	#17984765	C₂²	1	1	yes	characteristic
45	#17984764	C₂²	1	1	yes	characteristic
46	#17984763	C₂²	1	1	yes	characteristic
47	#17984762	C₂²	1	1	yes	characteristic
48	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
49	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
54	C₂²⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
55	#121×C₁₃₉	C₂²	1	1	yes	characteristic
56	#106×C₁₃₉	C₂²	1	1	yes	characteristic
57	#103×C₁₃₉	C₂²	1	1	yes	characteristic
58	#102×C₁₃₉	C₂²	1	1	yes	characteristic
59	#14555784	—	4	4	no	—
60	C₂×Dic₂₇₈	—	2	1	no	—
61	C₂×Dic₂₇₈	—	2	1	no	—
62	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
63	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
65	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
66	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
67	C₂²×Dic₁₃₉	—	2	1	no	—
68	C₂²×Dic₁₃₉	D₄	1	1	yes	characteristic
69	C₂²×Dic₁₃₉	D₄	1	1	yes	characteristic
70	C₂×C₄×D₁₃₉	—	2	1	no	—
71	#14555774	—	2	2	no	—
72	#14555773	—	2	2	no	—
73	#14555772	—	2	2	no	—
74	C₂×C₁₃₉⋊C₈	—	2	2	no	—
75	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
76	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₃₉	—	2	1	no	—
78	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
80	Dic₁₃₉⋊C₄	—	2	1	no	—
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	C₂²⋊Dic₁₃₉	—	2	1	no	—
87	C₂²⋊Dic₁₃₉	—	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	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₁₃₉	—	2	1	no	—
98	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
100	SD₁₆×C₁₃₉	—	2	2	no	—
101	C₂×C₈×C₁₃₉	—	2	2	no	—
102	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
103	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₃₉	—	2	1	no	—
105	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
106	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
107	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
108	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
109	C₂²×D₁₃₉	—	2	1	no	—
110	Dic₂₇₈	—	2	1	no	—
111	Dic₂₇₈	—	2	1	no	—
112	Dic₂₇₈	—	4	2	no	—
113	Dic₂₇₈	—	4	2	no	—
114	D₅₅₆	—	2	1	no	—
115	D₅₅₆	—	2	1	no	—
116	C₂²⋊D₁₃₉	—	4	2	no	—
117	C₂²⋊D₁₃₉	—	4	2	no	—
118	C₂²⋊D₁₃₉	—	4	2	no	—
119	C₂×Dic₁₃₉	—	2	2	no	—
120	C₂×Dic₁₃₉	—	2	1	no	—
121	C₂×Dic₁₃₉	—	2	1	no	—
122	C₂×Dic₁₃₉	—	2	1	no	—
123	C₂×Dic₁₃₉	—	4	2	no	—
124	C₂×Dic₁₃₉	SD₁₆	1	4	yes	complemented
125	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
129	C₄×D₁₃₉	—	4	2	no	—
130	C₁₃₉⋊C₈	—	2	1	no	—
131	C₁₃₉⋊C₈	#41	1	2	yes	complemented
132	C₂³×C₁₃₉	—	2	1	no	—
133	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
134	Q₈×C₁₃₉	—	2	1	no	—
135	Q₈×C₁₃₉	—	2	1	no	—
136	D₄×C₁₃₉	—	2	1	no	—
137	D₄×C₁₃₉	—	2	1	no	—
138	D₄×C₁₃₉	—	4	2	no	—
139	C₂×C₄×C₁₃₉	—	2	1	no	—
140	C₂×C₄×C₁₃₉	—	2	1	no	—
141	C₂×C₄×C₁₃₉	—	2	1	no	—
142	C₂×C₄×C₁₃₉	—	2	1	no	—
143	C₂×C₄×C₁₃₉	—	2	1	no	—
144	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
146	C₈×C₁₃₉	—	2	1	no	—
147	C₈×C₁₃₉	#41	1	2	yes	complemented
148	C₂²×C₁₃₉	—	2	1	no	—
149	C₂²×C₁₃₉	—	2	1	no	—
150	C₂²×C₁₃₉	—	4	1	no	—
151	C₂²×C₁₃₉	—	4	1	no	—
152	C₂²×C₁₃₉	—	4	1	no	—
153	C₂²×C₁₃₉	C₂×SD₁₆	1	2	yes	—
154	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
155	C₂×D₁₃₉	—	4	1	no	—
156	C₂×D₁₃₉	—	4	1	no	—
157	C₂×D₁₃₉	—	4	1	no	—
158	C₄×C₁₃₉	—	2	1	no	—
159	C₄×C₁₃₉	—	4	1	no	—
160	C₄×C₁₃₉	—	4	1	no	—
161	C₄×C₁₃₉	—	4	1	no	—
162	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
163	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
164	Dic₁₃₉	—	2	1	no	—
165	Dic₁₃₉	—	4	1	no	—
166	Dic₁₃₉	—	4	1	no	—
167	Dic₁₃₉	—	4	1	no	—
168	Dic₁₃₉	C₂×SD₁₆	1	2	yes	—
169	Dic₁₃₉	C₂×SD₁₆	1	2	yes	—
170	Dic₁₃₉	#135	1	2	yes	—
171	C₂×C₁₃₉	—	2	2	no	—
172	C₂×C₁₃₉	—	8	1	no	—
173	C₂×C₁₃₉	#532	1	1	yes	characteristic
174	C₂×C₁₃₉	C₂²×SD₁₆	1	1	yes	characteristic
175	C₂×C₁₃₉	C₂×#135	1	1	yes	characteristic
176	D₁₃₉	—	8	1	no	—
177	C₁₃₉	#3167	1	1	yes	characteristic, complemented, Sylow
178	#3167	—	139	1	no	maximal, Sylow
179	#533	—	139	1	no	—
180	#532	—	139	1	no	—
181	#498	—	139	1	no	—
182	#492	—	139	1	no	—
183	#487	—	139	1	no	—
184	#486	—	139	1	no	—
185	#465	—	139	1	no	—
186	#465	—	139	1	no	—
187	#465	—	139	1	no	—
188	#465	D₁₃₉	1	1	yes	characteristic, complemented
189	C₄×SD₁₆	—	139	1	no	—
190	C₄×SD₁₆	—	139	1	no	—
191	C₄×SD₁₆	—	139	1	no	—
192	C₄×SD₁₆	—	139	1	no	—
193	C₂×C₄×C₈	—	139	1	no	—
194	C₂×#41	—	278	1	no	—
195	C₂×SD₁₆	—	139	1	no	—
196	C₂×SD₁₆	—	139	1	no	—
197	C₂×SD₁₆	—	139	1	no	—
198	C₂×SD₁₆	C₂×D₁₃₉	1	1	yes	characteristic
199	C₂²×C₈	—	139	1	no	—
200	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
201	C₄⋊Q₈	—	139	1	no	—
202	C₄⋊Q₈	—	139	1	no	—
203	C₄⋊Q₈	—	139	1	no	—
204	#127	—	139	1	no	—
205	#125	—	278	1	no	—
206	#124	—	278	1	no	—
207	C₂²⋊Q₈	—	278	1	no	—
208	C₂²⋊Q₈	—	139	1	no	—
209	C₂²⋊Q₈	C₂×D₁₃₉	1	1	yes	characteristic
210	#121	—	278	1	no	—
211	#121	—	139	1	no	—
212	#121	C₂×D₁₃₉	1	1	yes	characteristic
213	C₄×Q₈	—	278	1	no	—
214	C₄×Q₈	—	139	1	no	—
215	C₄×Q₈	—	139	1	no	—
216	C₄×D₄	—	278	1	no	—
217	C₄×D₄	—	139	1	no	—
218	C₄×D₄	—	139	1	no	—
219	#117	—	278	1	no	—
220	C₂×C₄²	—	139	1	no	—
221	#106	—	139	1	no	—
222	#106	—	139	1	no	—
223	#106	—	139	1	no	—
224	#106	C₂×D₁₃₉	1	1	yes	characteristic
225	#103	—	139	1	no	—
226	#103	—	139	1	no	—
227	#103	—	139	1	no	—
228	#103	C₂×D₁₃₉	1	1	yes	characteristic
229	#102	—	139	1	no	—
230	#102	—	139	1	no	—
231	#102	—	139	1	no	—
232	#102	C₂×D₁₃₉	1	1	yes	characteristic
233	C₄×C₈	—	139	1	no	—
234	C₄×C₈	—	139	1	no	—
235	C₄×C₈	—	139	1	no	—
236	C₄×C₈	—	139	1	no	—
237	#41	—	556	4	no	—
238	C₂×Q₈	—	278	1	no	—
239	C₂×Q₈	—	278	1	no	—
240	C₂×Q₈	—	139	1	no	—
241	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	—	278	1	no	—
244	C₂×D₄	—	278	1	no	—
245	C₂×D₄	—	278	1	no	—
246	C₂×D₄	—	139	1	no	—
247	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
248	C₂²×C₄	—	278	1	no	—
249	C₂²×C₄	—	278	1	no	—
250	C₂²×C₄	—	139	1	no	—
251	C₂²×C₄	—	139	1	no	—
252	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
253	SD₁₆	—	2	2	no	—
254	SD₁₆	—	278	2	no	—
255	SD₁₆	—	278	2	no	—
256	SD₁₆	—	278	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	278	2	no	—
259	C₂×C₈	—	139	1	no	—
260	C₂×C₈	—	139	1	no	—
261	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	278	1	no	—
265	C₄⋊C₄	—	278	1	no	—
266	C₄⋊C₄	—	278	1	no	—
267	C₄⋊C₄	—	278	1	no	—
268	C₄⋊C₄	—	278	1	no	—
269	C₄⋊C₄	—	278	1	no	—
270	C₄⋊C₄	—	278	1	no	—
271	C₄⋊C₄	—	139	1	no	—
272	C₄⋊C₄	—	139	1	no	—
273	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	278	1	no	—
278	C₂²⋊C₄	—	278	1	no	—
279	C₂²⋊C₄	—	278	1	no	—
280	C₂²⋊C₄	—	278	1	no	—
281	C₄²	—	278	1	no	—
282	C₄²	—	278	1	no	—
283	C₄²	—	139	1	no	—
284	C₄²	—	139	1	no	—
285	C₄²	—	139	1	no	—
286	C₄²	—	139	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	278	1	no	—
289	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
290	Q₈	—	2	1	no	—
291	Q₈	—	2	1	no	—
292	Q₈	—	278	1	no	—
293	Q₈	—	278	1	no	—
294	Q₈	—	556	2	no	—
295	Q₈	—	556	2	no	—
296	D₄	—	2	1	no	—
297	D₄	—	2	1	no	—
298	D₄	—	4	2	no	—
299	D₄	—	278	1	no	—
300	D₄	—	278	1	no	—
301	D₄	—	556	2	no	—
302	D₄	—	556	2	no	—
303	D₄	—	556	2	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	278	2	no	—
310	C₂×C₄	—	278	1	no	—
311	C₂×C₄	—	278	1	no	—
312	C₂×C₄	—	278	1	no	—
313	C₂×C₄	—	556	2	no	—
314	C₂×C₄	—	556	2	no	—
315	C₂×C₄	—	139	1	no	—
316	C₂×C₄	—	139	4	no	—
317	C₂×C₄	—	139	1	no	—
318	C₂×C₄	—	139	1	no	—
319	C₂×C₄	—	139	1	no	—
320	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
321	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
322	C₈	—	2	1	no	—
323	C₈	—	278	1	no	—
324	C₈	—	139	2	no	—
325	C₈	#14555784	1	2	yes	complemented
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	556	1	no	—
332	C₂²	—	556	1	no	—
333	C₂²	—	556	1	no	—
334	C₂²	SD₁₆×D₁₃₉	1	2	yes	—
335	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	278	1	no	—
341	C₄	—	556	1	no	—
342	C₄	—	556	1	no	—
343	C₄	—	556	1	no	—
344	C₄	—	139	2	no	—
345	C₄	—	139	2	no	—
346	C₄	—	139	2	no	—
347	C₄	#41×D₁₃₉	1	1	yes	characteristic
348	C₄	C₂×#14555784	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	—	8	1	no	—
351	C₂	—	1112	1	no	—
352	C₂	#22581498	1	1	yes	characteristic
353	C₂	C₂×SD₁₆×D₁₃₉	1	1	yes	characteristic
354	C₂	#135×D₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#465×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	#22581498	C₂
hypercenter	#465	D₁₃₉
hypocenter	C₁₃₉	#3167
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #465

Complements

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

N	H ≅ G/N	type
1. #22581505	C₂	semidirect
3. #22581350	C₂	semidirect
4. #22581324	C₂	semidirect
5. #22581302	C₂	semidirect
6. #22581298	C₂	semidirect
7. #22581184	C₂	semidirect
8. #22581182	C₂	semidirect
9. #22581180	C₂	semidirect
10. #22581044	C₂	semidirect
11. SD₁₆×Dic₁₃₉	C₂	semidirect
12. #22581040	C₂	semidirect
13. #22581038	C₂	semidirect
14. C₂×C₈×Dic₁₃₉	C₂	semidirect
15. #465×C₁₃₉	C₂	semidirect
124. C₂×Dic₁₃₉	SD₁₆	semidirect
131. C₁₃₉⋊C₈	#41	semidirect
147. C₈×C₁₃₉	#41	semidirect
177. C₁₃₉	#3167	semidirect
188. #465	D₁₃₉	semidirect
325. C₈	#14555784	semidirect

Maximal subgroups

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

subgroup	index
#22581505	2
#22581498	2
#22581350	2
#22581324	2
#22581302	2
#22581298	2
#22581184	2
#22581182	2
#22581180	2
#22581044	2
SD₁₆×Dic₁₃₉	2
#22581040	2
#22581038	2
C₂×C₈×Dic₁₃₉	2
#465×C₁₃₉	2
#3167	139