Subgroups — #31816524

Order 17792.

424 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581500	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581268	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581265	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581247	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581246	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581152	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581151	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581093	C₂	1	1	yes	maximal, characteristic, complemented
10	#22581090	C₂	1	1	yes	maximal, characteristic, complemented
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×#14555785	—	2	1	no	—
17	C₂×Q₈×D₁₃₉	C₂²	1	1	yes	characteristic
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
22	#17984838	C₂²	1	1	yes	characteristic
23	#17984828	—	2	1	no	—
24	#17984817	C₂²	1	1	yes	characteristic
25	#17984815	—	2	1	no	—
26	#17984813	C₂²	1	1	yes	characteristic
27	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
28	C₄×Dic₂₇₈	—	2	1	no	—
29	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
30	#17984807	—	2	1	no	—
31	#17984807	C₂²	1	1	yes	characteristic
32	C₄×D₅₅₆	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₁₃₉	—	2	1	no	—
34	C₄⋊C₄×D₁₃₉	—	2	1	no	—
35	C₄⋊C₄×D₁₃₉	C₂²	1	1	yes	characteristic
36	C₄²×D₁₃₉	—	2	1	no	—
37	C₄²×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	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
42	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
43	#17984767	C₂²	1	1	yes	characteristic
44	#17984766	C₂²	1	1	yes	characteristic
45	#17984765	C₂²	1	1	yes	characteristic
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
52	C₂×SD₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
55	#107×C₁₃₉	C₂²	1	1	yes	characteristic
56	#103×C₁₃₉	C₂²	1	1	yes	characteristic
57	#102×C₁₃₉	C₂²	1	1	yes	characteristic
58	#97×C₁₃₉	C₂²	1	1	yes	characteristic
59	#14555785	—	4	4	no	—
60	C₂×Dic₂₇₈	—	2	1	no	—
61	C₂×Dic₂₇₈	—	2	1	no	—
62	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
63	Q₈×D₁₃₉	—	2	2	no	—
64	Q₈×D₁₃₉	—	2	2	no	—
65	C₂×D₅₅₆	—	2	1	no	—
66	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
68	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
69	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
70	C₂×C₄×D₁₃₉	—	2	1	no	—
71	C₂×C₄×D₁₃₉	—	2	1	no	—
72	C₂×C₄×D₁₃₉	—	2	1	no	—
73	C₂×C₄×D₁₃₉	D₄	1	1	yes	characteristic
74	C₂×C₄×D₁₃₉	D₄	1	1	yes	characteristic
75	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
76	#14555774	—	2	2	no	—
77	#14555773	—	2	2	no	—
78	#14555772	—	2	2	no	—
79	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
80	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₁₃₉	—	2	1	no	—
82	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
83	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
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	C₄×Dic₁₃₉	—	2	1	no	—
92	C₄×Dic₁₃₉	—	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	1	no	—
110	C₂²×D₁₃₉	C₂×D₄	1	1	yes	characteristic
111	Dic₂₇₈	—	2	4	no	—
112	Dic₂₇₈	—	2	1	no	—
113	Dic₂₇₈	—	4	2	no	—
114	Dic₂₇₈	#41	1	2	yes	complemented
115	D₅₅₆	—	2	1	no	—
116	D₅₅₆	—	2	1	no	—
117	D₅₅₆	—	4	2	no	—
118	C₂²⋊D₁₃₉	—	4	2	no	—
119	C₂²⋊D₁₃₉	—	4	2	no	—
120	C₂²⋊D₁₃₉	—	4	2	no	—
121	C₂×Dic₁₃₉	—	2	1	no	—
122	C₂×Dic₁₃₉	—	2	1	no	—
123	C₂×Dic₁₃₉	—	2	1	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	4	no	—
128	C₄×D₁₃₉	—	2	2	no	—
129	C₄×D₁₃₉	—	2	1	no	—
130	C₄×D₁₃₉	—	2	1	no	—
131	C₄×D₁₃₉	—	2	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	#41	1	2	yes	complemented
139	D₄×C₁₃₉	—	2	1	no	—
140	D₄×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	—	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	—	4	1	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	—	4	1	no	—
157	C₂×D₁₃₉	—	4	1	no	—
158	C₂×D₁₃₉	—	4	1	no	—
159	C₂×D₁₃₉	AGL(1,ℤ/8)	1	2	yes	—
160	C₄×C₁₃₉	—	2	2	no	—
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	C₂×#41	1	1	yes	characteristic
166	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
167	Dic₁₃₉	—	2	2	no	—
168	Dic₁₃₉	—	2	1	no	—
169	Dic₁₃₉	—	2	1	no	—
170	Dic₁₃₉	—	4	1	no	—
171	Dic₁₃₉	—	4	1	no	—
172	Dic₁₃₉	#137	1	2	yes	—
173	C₂×C₁₃₉	—	8	1	no	—
174	C₂×C₁₃₉	#525	1	1	yes	characteristic
175	C₂×C₁₃₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
176	C₂×C₁₃₉	C₂×#137	1	1	yes	characteristic
177	D₁₃₉	—	2	2	no	—
178	D₁₃₉	—	8	1	no	—
179	C₁₃₉	#3112	1	1	yes	characteristic, complemented, Sylow
180	#3112	—	139	1	no	maximal, Sylow
181	#532	—	139	1	no	—
182	C₂×C₄×Q₈	—	139	1	no	—
183	#480	—	139	1	no	—
184	#480	—	139	1	no	—
185	#477	—	139	1	no	—
186	#477	—	139	1	no	—
187	#460	—	139	1	no	—
188	#460	—	139	1	no	—
189	#448	—	139	1	no	—
190	#448	—	139	1	no	—
191	#440	—	139	1	no	—
192	#440	—	139	1	no	—
193	#440	—	139	1	no	—
194	#440	D₁₃₉	1	1	yes	characteristic, complemented
195	#432	—	139	1	no	—
196	C₂×#41	—	278	1	no	—
197	C₂²×Q₈	—	139	1	no	—
198	C₂×SD₁₆	—	139	1	no	—
199	C₂×SD₁₆	—	139	1	no	—
200	C₂×SD₁₆	—	139	1	no	—
201	C₂×SD₁₆	C₂×D₁₃₉	1	1	yes	characteristic
202	C₄⋊Q₈	—	139	1	no	—
203	#127	—	139	1	no	—
204	#124	—	278	1	no	—
205	#121	—	278	1	no	—
206	#121	—	139	1	no	—
207	#121	—	139	1	no	—
208	C₄×Q₈	—	278	1	no	—
209	C₄×Q₈	—	278	1	no	—
210	C₄×Q₈	—	139	1	no	—
211	C₄×Q₈	—	139	1	no	—
212	C₄×Q₈	—	139	1	no	—
213	C₄×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
214	C₄×D₄	—	278	1	no	—
215	C₄×D₄	—	139	1	no	—
216	C₄×D₄	C₂×D₁₃₉	1	1	yes	characteristic
217	C₂×C₄⋊C₄	—	278	1	no	—
218	C₂×C₄⋊C₄	—	139	1	no	—
219	C₂×C₄²	—	278	1	no	—
220	C₂×C₄²	—	139	1	no	—
221	#107	—	139	1	no	—
222	#107	—	139	1	no	—
223	#107	—	139	1	no	—
224	#107	C₂×D₁₃₉	1	1	yes	characteristic
225	C₄⋊C₈	—	139	1	no	—
226	C₄⋊C₈	—	139	1	no	—
227	#103	—	139	1	no	—
228	#103	—	139	1	no	—
229	#103	—	139	1	no	—
230	#103	C₂×D₁₃₉	1	1	yes	characteristic
231	#102	—	139	1	no	—
232	#102	—	139	1	no	—
233	#102	—	139	1	no	—
234	#102	C₂×D₁₃₉	1	1	yes	characteristic
235	C₂²⋊C₈	—	139	1	no	—
236	C₂²⋊C₈	—	139	1	no	—
237	#97	—	139	1	no	—
238	#97	C₂×D₁₃₉	1	1	yes	characteristic
239	#41	—	556	4	no	—
240	C₂×Q₈	—	278	1	no	—
241	C₂×Q₈	—	278	1	no	—
242	C₂×Q₈	—	278	2	no	—
243	C₂×Q₈	—	278	2	no	—
244	C₂×Q₈	—	139	1	no	—
245	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
246	C₂×D₄	—	278	1	no	—
247	C₂×D₄	—	278	1	no	—
248	C₂×D₄	—	278	1	no	—
249	C₂×D₄	—	278	1	no	—
250	C₂×D₄	—	139	1	no	—
251	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	278	1	no	—
254	C₂²×C₄	—	278	1	no	—
255	C₂²×C₄	—	278	1	no	—
256	C₂²×C₄	—	139	1	no	—
257	C₂²×C₄	—	139	1	no	—
258	C₂²×C₄	—	139	1	no	—
259	SD₁₆	—	2	2	no	—
260	SD₁₆	—	278	2	no	—
261	SD₁₆	—	278	2	no	—
262	SD₁₆	—	278	2	no	—
263	C₂×C₈	—	139	1	no	—
264	C₂×C₈	—	139	1	no	—
265	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
266	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
267	C₄⋊C₄	—	2	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	—	139	1	no	—
274	C₄⋊C₄	—	139	1	no	—
275	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	278	1	no	—
279	C₂²⋊C₄	—	278	1	no	—
280	C₂²⋊C₄	—	278	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	278	1	no	—
283	C₄²	—	278	1	no	—
284	C₄²	—	278	1	no	—
285	C₄²	—	139	1	no	—
286	C₄²	—	139	1	no	—
287	C₄²	—	139	1	no	—
288	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
289	C₂³	—	2	1	no	—
290	C₂³	—	278	1	no	—
291	C₂³	—	139	1	no	—
292	Q₈	—	2	1	no	—
293	Q₈	—	278	4	no	—
294	Q₈	—	278	1	no	—
295	Q₈	—	556	2	no	—
296	Q₈	—	139	2	no	—
297	Q₈	#14555785	1	2	yes	complemented
298	D₄	—	2	1	no	—
299	D₄	—	2	1	no	—
300	D₄	—	278	1	no	—
301	D₄	—	278	1	no	—
302	D₄	—	556	2	no	—
303	D₄	—	556	2	no	—
304	D₄	—	556	2	no	—
305	D₄	—	556	2	no	—
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	4	no	—
311	C₂×C₄	—	278	2	no	—
312	C₂×C₄	—	278	2	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	—	278	1	no	—
318	C₂×C₄	—	556	2	no	—
319	C₂×C₄	—	556	2	no	—
320	C₂×C₄	—	139	1	no	—
321	C₂×C₄	—	139	1	no	—
322	C₂×C₄	—	139	1	no	—
323	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
324	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
325	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
326	C₈	—	2	1	no	—
327	C₈	—	2	1	no	—
328	C₈	—	278	1	no	—
329	C₈	—	278	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	278	1	no	—
334	C₂²	—	278	1	no	—
335	C₂²	—	556	1	no	—
336	C₂²	—	556	1	no	—
337	C₂²	—	556	1	no	—
338	C₂²	—	139	2	no	—
339	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
340	C₄	—	2	2	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	4	1	no	—
344	C₄	—	4	1	no	—
345	C₄	—	278	2	no	—
346	C₄	—	278	1	no	—
347	C₄	—	278	1	no	—
348	C₄	—	556	1	no	—
349	C₄	—	556	1	no	—
350	C₄	—	139	2	no	—
351	C₄	#41×D₁₃₉	1	1	yes	characteristic
352	C₄	C₂×#14555785	1	1	yes	characteristic
353	C₂	—	8	1	no	—
354	C₂	—	278	2	no	—
355	C₂	—	1112	1	no	—
356	C₂	#22581462	1	1	yes	characteristic
357	C₂	AGL(1,ℤ/8)×D₁₃₉	1	1	yes	characteristic
358	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	#22581500	C₂
hypercenter	#440	D₁₃₉
hypocenter	C₁₃₉	#3112
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
2. C₄×Q₈×D₁₃₉	C₂	semidirect
3. #22581268	C₂	semidirect
4. #22581265	C₂	semidirect
5. #22581247	C₂	semidirect
6. #22581246	C₂	semidirect
7. #22581152	C₂	semidirect
8. #22581151	C₂	semidirect
9. #22581093	C₂	semidirect
10. #22581090	C₂	semidirect
11. #22581055	C₂	semidirect
12. #22581053	C₂	semidirect
13. #22581051	C₂	semidirect
14. #22581008	C₂	semidirect
15. #440×C₁₃₉	C₂	semidirect
114. Dic₂₇₈	#41	semidirect
138. Q₈×C₁₃₉	#41	semidirect
179. C₁₃₉	#3112	semidirect
194. #440	D₁₃₉	semidirect
297. Q₈	#14555785	semidirect

Maximal subgroups

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

subgroup	index
#22581500	2
C₄×Q₈×D₁₃₉	2
#22581268	2
#22581265	2
#22581247	2
#22581246	2
#22581152	2
#22581151	2
#22581093	2
#22581090	2
#22581055	2
#22581053	2
#22581051	2
#22581008	2
#440×C₁₃₉	2
#3112	139