Subgroups — #31816501

Order 17792.

444 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581507	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₄×D₁₃₉	C₂	1	1	yes	maximal, characteristic
3	#22581290	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581287	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581232	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581231	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581157	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581156	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581105	C₂	1	1	yes	maximal, characteristic
10	#22581102	C₂	1	1	yes	maximal, characteristic
11	#22581055	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581053	C₂	1	1	yes	maximal, characteristic, complemented
13	#22581051	C₂	1	1	yes	maximal, characteristic, complemented
14	#22581008	C₂	1	1	yes	maximal, characteristic, complemented
15	#440×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₁₃₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₃₉	—	2	1	no	—
18	C₂×#14555774	C₂²	1	1	yes	characteristic
19	C₂×#14555773	C₂²	1	1	yes	characteristic
20	C₂×#14555772	C₂²	1	1	yes	characteristic
21	#17984841	C₂²	1	1	yes	characteristic, complemented
22	#17984840	C₂²	1	1	yes	characteristic, complemented
23	#17984832	—	2	1	no	—
24	#17984822	C₂²	1	1	yes	characteristic
25	#17984820	—	2	1	no	—
26	#17984818	C₂²	1	1	yes	characteristic
27	C₄×Dic₂₇₈	—	2	1	no	—
28	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic, complemented
29	C₄×D₅₅₆	C₂²	1	1	yes	characteristic
30	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
31	#17984804	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊D₁₃₉	—	2	1	no	—
33	(C₂²⋊D₁₃₉)⋊C₄	—	2	1	no	—
34	#17984799	—	2	1	no	—
35	C₄⋊C₄×D₁₃₉	C₂²	1	1	yes	characteristic
36	C₂²⋊C₄×D₁₃₉	—	2	1	no	—
37	C₄²×D₁₃₉	C₂²	1	1	yes	characteristic
38	#17984776	C₂²	1	1	yes	characteristic, complemented
39	#17984775	C₂²	1	1	yes	characteristic, complemented
40	#17984775	C₂²	1	1	yes	characteristic, complemented
41	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic, complemented
42	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic, complemented
43	#17984767	C₂²	1	1	yes	characteristic, complemented
44	#17984766	C₂²	1	1	yes	characteristic, complemented
45	#17984765	C₂²	1	1	yes	characteristic, complemented
46	#17984764	C₂²	1	1	yes	characteristic
47	#17984763	C₂²	1	1	yes	characteristic
48	#17984762	C₂²	1	1	yes	characteristic
49	(C₂×D₁₃₉)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₁₃₉)⋊C₈	C₂²	1	1	yes	characteristic
51	#17984753	C₂²	1	1	yes	characteristic, complemented
52	C₂×SD₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
54	C₄×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
55	#107×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
56	#103×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
57	#102×C₁₃₉	C₂²	1	1	yes	characteristic
58	#97×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₁₃₉	—	2	1	no	—
60	C₂×Dic₂₇₈	—	2	1	no	—
61	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
62	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
64	D₄×D₁₃₉	—	2	2	no	—
65	D₄×D₁₃₉	—	2	2	no	—
66	C₂²×Dic₁₃₉	—	2	1	no	—
67	C₂×C₄×D₁₃₉	—	2	1	no	—
68	C₂×C₄×D₁₃₉	—	2	1	no	—
69	C₂×C₄×D₁₃₉	—	2	1	no	—
70	C₂×C₄×D₁₃₉	—	4	4	no	—
71	C₂×C₄×D₁₃₉	D₄	1	1	yes	characteristic
72	C₂×C₄×D₁₃₉	D₄	1	1	yes	characteristic
73	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
74	#14555774	—	2	2	no	—
75	#14555773	—	2	2	no	—
76	#14555772	—	2	2	no	—
77	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
78	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₁₃₉	—	2	1	no	—
80	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
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	Dic₁₃₉⋊C₄	—	2	1	no	—
88	C₂²⋊Dic₁₃₉	—	2	1	no	—
89	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
90	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
91	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
92	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
93	C₄×Dic₁₃₉	—	2	1	no	—
94	C₄×Dic₁₃₉	D₄	1	1	yes	characteristic
95	C₄×Dic₁₃₉	D₄	1	1	yes	characteristic
96	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
98	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₁₃₉	—	2	1	no	—
100	SD₁₆×C₁₃₉	—	2	2	no	—
101	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₃₉	—	2	1	no	—
104	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
106	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
107	C₄²×C₁₃₉	—	2	1	no	—
108	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
109	C₂²×D₁₃₉	—	2	4	no	—
110	C₂²×D₁₃₉	—	2	1	no	—
111	C₂²×D₁₃₉	—	4	2	no	—
112	C₂²×D₁₃₉	—	4	2	no	—
113	C₂²×D₁₃₉	C₂×D₄	1	1	yes	characteristic
114	Dic₂₇₈	—	2	1	no	—
115	Dic₂₇₈	—	2	1	no	—
116	Dic₂₇₈	—	4	2	no	—
117	D₅₅₆	—	2	1	no	—
118	D₅₅₆	#41	1	2	yes	—
119	C₂²⋊D₁₃₉	—	2	4	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	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
127	C₄×D₁₃₉	—	2	2	no	—
128	C₄×D₁₃₉	—	2	1	no	—
129	C₄×D₁₃₉	—	2	1	no	—
130	C₄×D₁₃₉	—	2	2	no	—
131	C₄×D₁₃₉	—	4	2	no	—
132	C₄×D₁₃₉	—	4	2	no	—
133	C₄×D₁₃₉	—	4	2	no	—
134	C₁₃₉⋊C₈	—	2	1	no	—
135	C₁₃₉⋊C₈	—	2	1	no	—
136	C₂³×C₁₃₉	—	2	1	no	—
137	Q₈×C₁₃₉	—	2	1	no	—
138	Q₈×C₁₃₉	—	2	1	no	—
139	D₄×C₁₃₉	—	2	1	no	—
140	D₄×C₁₃₉	#41	1	2	yes	—
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	—	4	2	no	—
145	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
146	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
147	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
148	C₈×C₁₃₉	—	2	1	no	—
149	C₈×C₁₃₉	—	2	1	no	—
150	C₂²×C₁₃₉	—	2	2	no	—
151	C₂²×C₁₃₉	—	4	1	no	—
152	C₂²×C₁₃₉	—	4	1	no	—
153	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
154	C₂×D₁₃₉	—	2	1	no	—
155	C₂×D₁₃₉	—	2	1	no	—
156	C₂×D₁₃₉	—	2	2	no	—
157	C₂×D₁₃₉	—	4	8	no	—
158	C₂×D₁₃₉	—	4	1	no	—
159	C₂×D₁₃₉	—	4	1	no	—
160	C₂×D₁₃₉	#137	1	2	yes	—
161	C₄×C₁₃₉	—	2	1	no	—
162	C₄×C₁₃₉	—	2	1	no	—
163	C₄×C₁₃₉	—	4	1	no	—
164	C₄×C₁₃₉	—	4	1	no	—
165	C₄×C₁₃₉	—	4	1	no	—
166	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
167	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
168	Dic₁₃₉	—	2	1	no	—
169	Dic₁₃₉	—	2	1	no	—
170	Dic₁₃₉	—	4	1	no	—
171	Dic₁₃₉	—	4	1	no	—
172	Dic₁₃₉	—	4	1	no	—
173	Dic₁₃₉	AGL(1,ℤ/8)	1	2	yes	—
174	C₂×C₁₃₉	—	4	2	no	—
175	C₂×C₁₃₉	#525	1	1	yes	characteristic
176	C₂×C₁₃₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
177	C₂×C₁₃₉	C₂×#137	1	1	yes	characteristic
178	D₁₃₉	—	2	2	no	—
179	D₁₃₉	—	4	2	no	—
180	C₁₃₉	#3110	1	1	yes	characteristic, complemented, Sylow
181	#3110	—	139	1	no	maximal, Sylow
182	#533	—	139	1	no	—
183	C₂×C₄×D₄	—	139	1	no	—
184	#484	—	139	1	no	—
185	#484	—	139	1	no	—
186	#474	—	139	1	no	—
187	#474	—	139	1	no	—
188	#461	—	139	1	no	—
189	#461	—	139	1	no	—
190	#450	—	139	1	no	—
191	#450	—	139	1	no	—
192	#440	—	139	1	no	—
193	#440	—	139	1	no	—
194	#440	—	139	1	no	—
195	#440	D₁₃₉	1	1	yes	characteristic, complemented
196	#432	—	139	1	no	—
197	C₂²×D₄	—	139	1	no	—
198	C₂³×C₄	—	278	1	no	—
199	C₂×SD₁₆	—	139	1	no	—
200	C₂×SD₁₆	—	139	1	no	—
201	C₂×SD₁₆	—	139	1	no	—
202	C₂×SD₁₆	C₂×D₁₃₉	1	1	yes	characteristic
203	C₄⋊Q₈	—	139	1	no	—
204	C₄⋊Q₈	—	139	1	no	—
205	#125	—	278	1	no	—
206	C₂²⋊Q₈	—	278	1	no	—
207	C₂²⋊Q₈	—	139	1	no	—
208	C₂²⋊Q₈	—	139	1	no	—
209	C₄×Q₈	—	278	1	no	—
210	C₄×Q₈	—	139	1	no	—
211	C₄×Q₈	C₂×D₁₃₉	1	1	yes	characteristic, complemented
212	C₄×D₄	—	278	1	no	—
213	C₄×D₄	—	278	1	no	—
214	C₄×D₄	—	139	1	no	—
215	C₄×D₄	—	139	1	no	—
216	C₄×D₄	—	139	1	no	—
217	C₄×D₄	C₂×D₁₃₉	1	1	yes	characteristic
218	#117	—	278	1	no	—
219	C₂×C₄⋊C₄	—	139	1	no	—
220	C₂×C₂²⋊C₄	—	278	1	no	—
221	C₂×C₄²	—	139	1	no	—
222	#107	—	139	1	no	—
223	#107	—	139	1	no	—
224	#107	—	139	1	no	—
225	#107	C₂×D₁₃₉	1	1	yes	characteristic, complemented
226	C₄⋊C₈	—	139	1	no	—
227	C₄⋊C₈	—	139	1	no	—
228	#103	—	139	1	no	—
229	#103	—	139	1	no	—
230	#103	—	139	1	no	—
231	#103	C₂×D₁₃₉	1	1	yes	characteristic, complemented
232	#102	—	139	1	no	—
233	#102	—	139	1	no	—
234	#102	—	139	1	no	—
235	#102	C₂×D₁₃₉	1	1	yes	characteristic
236	C₂²⋊C₈	—	139	1	no	—
237	C₂²⋊C₈	—	139	1	no	—
238	#97	—	139	1	no	—
239	#97	C₂×D₁₃₉	1	1	yes	characteristic, complemented
240	C₂⁴	—	278	1	no	—
241	C₂×Q₈	—	278	1	no	—
242	C₂×Q₈	—	139	1	no	—
243	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
244	C₂×D₄	—	278	2	no	—
245	C₂×D₄	—	278	2	no	—
246	C₂×D₄	—	278	1	no	—
247	C₂×D₄	—	139	1	no	—
248	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	278	1	no	—
251	C₂²×C₄	—	278	1	no	—
252	C₂²×C₄	—	278	1	no	—
253	C₂²×C₄	—	278	1	no	—
254	C₂²×C₄	—	556	4	no	—
255	C₂²×C₄	—	139	1	no	—
256	C₂²×C₄	—	139	1	no	—
257	C₂²×C₄	—	139	1	no	—
258	SD₁₆	—	2	2	no	—
259	SD₁₆	—	278	2	no	—
260	SD₁₆	—	278	2	no	—
261	SD₁₆	—	278	2	no	—
262	C₂×C₈	—	139	1	no	—
263	C₂×C₈	—	139	1	no	—
264	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
265	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
266	C₄⋊C₄	—	2	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	—	278	1	no	—
272	C₄⋊C₄	—	278	1	no	—
273	C₄⋊C₄	—	278	1	no	—
274	C₄⋊C₄	—	139	1	no	—
275	C₄⋊C₄	—	139	1	no	—
276	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
277	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	278	1	no	—
280	C₂²⋊C₄	—	278	1	no	—
281	C₂²⋊C₄	—	278	1	no	—
282	C₂²⋊C₄	—	278	1	no	—
283	C₂²⋊C₄	—	278	1	no	—
284	C₄²	—	2	1	no	—
285	C₄²	—	278	1	no	—
286	C₄²	—	139	1	no	—
287	C₄²	—	139	1	no	—
288	C₄²	—	139	1	no	—
289	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
290	C₂³	—	2	1	no	—
291	C₂³	—	278	4	no	—
292	C₂³	—	278	1	no	—
293	C₂³	—	556	2	no	—
294	C₂³	—	556	2	no	—
295	C₂³	—	139	1	no	—
296	Q₈	—	2	1	no	—
297	Q₈	—	2	1	no	—
298	Q₈	—	278	1	no	—
299	Q₈	—	278	1	no	—
300	Q₈	—	556	2	no	—
301	D₄	—	2	1	no	—
302	D₄	—	278	4	no	—
303	D₄	—	278	1	no	—
304	D₄	—	139	2	no	—
305	D₄	#14555785	1	2	yes	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	278	2	no	—
311	C₂×C₄	—	278	2	no	—
312	C₂×C₄	—	278	1	no	—
313	C₂×C₄	—	278	1	no	—
314	C₂×C₄	—	278	1	no	—
315	C₂×C₄	—	278	1	no	—
316	C₂×C₄	—	278	1	no	—
317	C₂×C₄	—	556	2	no	—
318	C₂×C₄	—	556	2	no	—
319	C₂×C₄	—	556	2	no	—
320	C₂×C₄	—	556	2	no	—
321	C₂×C₄	—	139	1	no	—
322	C₂×C₄	—	139	1	no	—
323	C₂×C₄	—	139	1	no	—
324	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
325	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
326	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
327	C₈	—	2	1	no	—
328	C₈	—	2	1	no	—
329	C₈	—	278	1	no	—
330	C₈	—	278	1	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	278	1	no	—
335	C₂²	—	278	1	no	—
336	C₂²	—	278	2	no	—
337	C₂²	—	556	8	no	—
338	C₂²	—	556	1	no	—
339	C₂²	—	556	1	no	—
340	C₂²	—	139	2	no	—
341	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	4	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	278	1	no	—
348	C₄	—	278	1	no	—
349	C₄	—	556	1	no	—
350	C₄	—	556	1	no	—
351	C₄	—	556	1	no	—
352	C₄	—	139	2	no	—
353	C₄	#41×D₁₃₉	1	1	yes	characteristic
354	C₄	C₂×#14555785	1	1	yes	characteristic
355	C₂	—	4	2	no	—
356	C₂	—	278	2	no	—
357	C₂	—	556	2	no	—
358	C₂	#22581462	1	1	yes	characteristic
359	C₂	AGL(1,ℤ/8)×D₁₃₉	1	1	yes	characteristic
360	C₂	#137×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	#440×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂×D₄×D₁₃₉	C₂²
hypercenter	#440	D₁₃₉
hypocenter	C₁₃₉	#3110
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

Complements

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

N	H ≅ G/N	type
1. #22581507	C₂	semidirect
3. #22581290	C₂	semidirect
4. #22581287	C₂	semidirect
5. #22581232	C₂	semidirect
6. #22581231	C₂	semidirect
7. #22581157	C₂	semidirect
8. #22581156	C₂	semidirect
11. #22581055	C₂	semidirect
12. #22581053	C₂	semidirect
13. #22581051	C₂	semidirect
14. #22581008	C₂	semidirect
15. #440×C₁₃₉	C₂	semidirect
21. #17984841	C₂²	semidirect
22. #17984840	C₂²	semidirect
28. C₄×Dic₂₇₈	C₂²	semidirect
38. #17984776	C₂²	semidirect
39. #17984775	C₂²	semidirect
40. #17984775	C₂²	semidirect
41. Dic₁₃₉⋊C₈	C₂²	semidirect
42. Dic₁₃₉⋊C₈	C₂²	semidirect
43. #17984767	C₂²	semidirect
44. #17984766	C₂²	semidirect
45. #17984765	C₂²	semidirect
51. #17984753	C₂²	semidirect
53. C₄×Q₈×C₁₃₉	C₂²	semidirect
55. #107×C₁₃₉	C₂²	semidirect
56. #103×C₁₃₉	C₂²	semidirect
58. #97×C₁₃₉	C₂²	semidirect
180. C₁₃₉	#3110	semidirect
195. #440	D₁₃₉	semidirect
211. C₄×Q₈	C₂×D₁₃₉	semidirect
225. #107	C₂×D₁₃₉	semidirect
231. #103	C₂×D₁₃₉	semidirect
239. #97	C₂×D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581507	2
C₄×D₄×D₁₃₉	2
#22581290	2
#22581287	2
#22581232	2
#22581231	2
#22581157	2
#22581156	2
#22581105	2
#22581102	2
#22581055	2
#22581053	2
#22581051	2
#22581008	2
#440×C₁₃₉	2
#3110	139