Subgroups — #31817841

Order 17792.

370 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581557	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581554	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581255	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581254	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581253	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581251	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581205	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581203	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581188	C₂	1	1	yes	maximal, characteristic, complemented
10	#22581186	C₂	1	1	yes	maximal, characteristic, complemented
11	#22581146	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581146	C₂	1	1	yes	maximal, characteristic, complemented
13	#22581145	C₂	1	1	yes	maximal, characteristic, complemented
14	#22580978	C₂	1	1	yes	maximal, characteristic, complemented
15	#459×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555770	C₂²	1	1	yes	characteristic
17	C₂×#14555770	C₂²	1	1	yes	characteristic
18	C₂×#14555768	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₃₉	C₂²	1	1	yes	characteristic
20	#17984838	C₂²	1	1	yes	characteristic
21	#17984834	C₂²	1	1	yes	characteristic
22	#17984833	C₂²	1	1	yes	characteristic
23	#17984825	—	2	1	no	—
24	#17984824	—	2	1	no	—
25	#17984823	—	2	1	no	—
26	#17984823	—	2	1	no	—
27	#17984817	C₂²	1	1	yes	characteristic
28	#17984816	—	2	1	no	—
29	#17984815	—	2	1	no	—
30	#17984814	—	2	1	no	—
31	#17984814	—	2	1	no	—
32	#17984813	C₂²	1	1	yes	characteristic
33	#17984813	C₂²	1	1	yes	characteristic
34	#17984813	C₂²	1	1	yes	characteristic
35	C₄×D₅₅₆	C₂²	1	1	yes	characteristic
36	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
37	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
38	#17984799	C₂²	1	1	yes	characteristic
39	#17984776	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	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
44	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
45	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
46	#17984764	C₂²	1	1	yes	characteristic
47	#17984764	C₂²	1	1	yes	characteristic
48	#17984763	C₂²	1	1	yes	characteristic
49	#17984763	C₂²	1	1	yes	characteristic
50	#17984763	C₂²	1	1	yes	characteristic
51	#17984763	C₂²	1	1	yes	characteristic
52	C₂×D₈×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₂×D₈×C₁₃₉	C₂²	1	1	yes	characteristic
54	#127×C₁₃₉	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
57	#102×C₁₃₉	C₂²	1	1	yes	characteristic
58	#102×C₁₃₉	C₂²	1	1	yes	characteristic
59	C₂×D₅₅₆	—	2	1	no	—
60	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
62	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
63	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
64	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
65	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
66	C₂²×Dic₁₃₉	—	2	1	no	—
67	C₂²×Dic₁₃₉	—	2	1	no	—
68	C₂×C₄×D₁₃₉	—	2	1	no	—
69	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
70	#14555770	—	4	1	no	—
71	#14555770	—	4	1	no	—
72	#14555768	—	4	1	no	—
73	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
75	C₈×D₁₃₉	—	4	1	no	—
76	C₄⋊Dic₁₃₉	—	2	1	no	—
77	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
84	Dic₁₃₉⋊C₄	D₄	1	1	yes	characteristic
85	C₂²⋊Dic₁₃₉	—	2	1	no	—
86	C₂²⋊Dic₁₃₉	—	2	1	no	—
87	C₂²⋊Dic₁₃₉	—	2	1	no	—
88	C₂²⋊Dic₁₃₉	—	2	1	no	—
89	C₂²⋊Dic₁₃₉	—	2	1	no	—
90	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
91	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
92	(C₂×D₁₃₉)⋊C₄	D₄	1	1	yes	characteristic
93	(C₂×D₁₃₉)⋊C₄	D₄	1	1	yes	characteristic
94	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₁₃₉	—	2	1	no	—
96	C₂×D₄×C₁₃₉	—	2	1	no	—
97	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
98	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
99	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
100	C₂²×C₄×C₁₃₉	—	2	1	no	—
101	D₈×C₁₃₉	—	4	1	no	—
102	D₈×C₁₃₉	—	4	1	no	—
103	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
104	C₂×C₈×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	C₂³	1	1	yes	characteristic
108	C₂²×D₁₃₉	—	2	1	no	—
109	C₂²×D₁₃₉	C₂×D₄	1	1	yes	characteristic
110	D₅₅₆	—	2	1	no	—
111	D₅₅₆	—	2	1	no	—
112	D₅₅₆	—	8	1	no	—
113	C₂²⋊D₁₃₉	—	8	1	no	—
114	C₂²⋊D₁₃₉	—	8	1	no	—
115	C₂²⋊D₁₃₉	—	8	1	no	—
116	C₂²⋊D₁₃₉	—	8	1	no	—
117	C₂²⋊D₁₃₉	—	8	1	no	—
118	C₂×Dic₁₃₉	—	2	1	no	—
119	C₂×Dic₁₃₉	—	2	1	no	—
120	C₂×Dic₁₃₉	—	2	1	no	—
121	C₂×Dic₁₃₉	—	8	1	no	—
122	C₂×Dic₁₃₉	—	8	1	no	—
123	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
126	C₄×D₁₃₉	—	2	1	no	—
127	C₄×D₁₃₉	—	2	1	no	—
128	C₄×D₁₃₉	—	8	1	no	—
129	C₁₃₉⋊C₈	—	2	1	no	—
130	C₁₃₉⋊C₈	—	2	1	no	—
131	C₂³×C₁₃₉	—	2	1	no	—
132	C₂³×C₁₃₉	—	2	1	no	—
133	C₂³×C₁₃₉	—	2	1	no	—
134	D₄×C₁₃₉	—	2	1	no	—
135	D₄×C₁₃₉	—	2	1	no	—
136	D₄×C₁₃₉	—	2	1	no	—
137	D₄×C₁₃₉	—	2	1	no	—
138	D₄×C₁₃₉	—	2	1	no	—
139	D₄×C₁₃₉	—	2	1	no	—
140	D₄×C₁₃₉	—	8	1	no	—
141	D₄×C₁₃₉	—	8	1	no	—
142	C₂×C₄×C₁₃₉	—	2	1	no	—
143	C₂×C₄×C₁₃₉	—	8	1	no	—
144	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
146	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
147	C₈×C₁₃₉	—	2	1	no	—
148	C₈×C₁₃₉	—	2	1	no	—
149	C₂²×C₁₃₉	—	4	1	no	—
150	C₂²×C₁₃₉	—	4	1	no	—
151	C₂²×C₁₃₉	—	4	1	no	—
152	C₂²×C₁₃₉	—	4	1	no	—
153	C₂²×C₁₃₉	—	4	1	no	—
154	C₂²×C₁₃₉	—	4	1	no	—
155	C₂²×C₁₃₉	—	4	1	no	—
156	C₂²×C₁₃₉	—	4	1	no	—
157	C₂²×C₁₃₉	—	4	1	no	—
158	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
159	C₂×D₁₃₉	—	2	1	no	—
160	C₂×D₁₃₉	—	2	1	no	—
161	C₂×D₁₃₉	—	2	1	no	—
162	C₂×D₁₃₉	—	4	1	no	—
163	C₂×D₁₃₉	—	4	1	no	—
164	C₂×D₁₃₉	—	4	1	no	—
165	C₄×C₁₃₉	—	2	1	no	—
166	C₄×C₁₃₉	—	2	1	no	—
167	C₄×C₁₃₉	—	4	1	no	—
168	C₄×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
169	C₄×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
170	Dic₁₃₉	—	2	1	no	—
171	Dic₁₃₉	—	2	1	no	—
172	Dic₁₃₉	—	2	1	no	—
173	Dic₁₃₉	—	4	1	no	—
174	Dic₁₃₉	—	4	1	no	—
175	Dic₁₃₉	—	4	1	no	—
176	C₂×C₁₃₉	—	8	1	no	—
177	C₂×C₁₃₉	—	8	1	no	—
178	C₂×C₁₃₉	—	8	1	no	—
179	C₂×C₁₃₉	#537	1	1	yes	characteristic
180	C₂×C₁₃₉	#576	1	1	yes	characteristic
181	C₂×C₁₃₉	#576	1	1	yes	characteristic
182	D₁₃₉	—	4	1	no	—
183	D₁₃₉	—	8	1	no	—
184	C₁₃₉	#3259	1	1	yes	characteristic, complemented, Sylow
185	#3259	—	139	1	no	maximal, Sylow
186	#540	—	139	1	no	—
187	#540	—	139	1	no	—
188	#478	—	139	1	no	—
189	#478	—	139	1	no	—
190	#478	—	139	1	no	—
191	#478	—	139	1	no	—
192	#469	—	139	1	no	—
193	#469	—	139	1	no	—
194	#466	—	139	1	no	—
195	#466	—	139	1	no	—
196	#459	—	139	1	no	—
197	#459	—	139	1	no	—
198	#459	—	139	1	no	—
199	#459	D₁₃₉	1	1	yes	characteristic, complemented
200	#424	—	139	1	no	—
201	C₂×D₈	—	139	1	no	—
202	C₂×D₈	—	139	1	no	—
203	C₂×D₈	C₂×D₁₃₉	1	1	yes	characteristic
204	C₂×D₈	C₂×D₁₃₉	1	1	yes	characteristic
205	C₂×M₁₆	—	139	1	no	—
206	C₂²×C₈	—	139	1	no	—
207	#127	—	139	1	no	—
208	#127	C₂×D₁₃₉	1	1	yes	characteristic
209	#125	—	139	1	no	—
210	#125	—	139	1	no	—
211	#123	—	278	1	no	—
212	#123	—	278	1	no	—
213	#123	—	278	1	no	—
214	#123	—	278	1	no	—
215	#121	—	278	1	no	—
216	#121	—	278	1	no	—
217	#121	—	278	1	no	—
218	#121	—	278	1	no	—
219	#121	—	139	1	no	—
220	#121	—	139	1	no	—
221	#121	—	139	1	no	—
222	#121	—	139	1	no	—
223	C₄×D₄	—	139	1	no	—
224	C₄×D₄	—	139	1	no	—
225	C₄×D₄	—	139	1	no	—
226	C₄×D₄	C₂×D₁₃₉	1	1	yes	characteristic
227	#117	—	139	1	no	—
228	#107	—	139	1	no	—
229	#107	—	139	1	no	—
230	#106	—	139	1	no	—
231	#106	—	139	1	no	—
232	C₄⋊C₈	—	139	1	no	—
233	C₄⋊C₈	—	139	1	no	—
234	C₄⋊C₈	—	139	1	no	—
235	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
236	#102	—	139	1	no	—
237	#102	—	139	1	no	—
238	#102	—	139	1	no	—
239	#102	—	139	1	no	—
240	#102	—	139	1	no	—
241	#102	—	139	1	no	—
242	#102	C₂×D₁₃₉	1	1	yes	characteristic
243	#102	C₂×D₁₃₉	1	1	yes	characteristic
244	C₂×D₄	—	2	1	no	—
245	C₂×D₄	—	2	1	no	—
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	—	278	1	no	—
251	C₂×D₄	—	278	1	no	—
252	C₂×D₄	—	139	1	no	—
253	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
254	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
255	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
256	C₂²×C₄	—	2	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	D₈	—	4	1	no	—
262	D₈	—	4	1	no	—
263	D₈	—	556	1	no	—
264	D₈	—	556	1	no	—
265	M₁₆	—	556	1	no	—
266	C₂×C₈	—	556	1	no	—
267	C₂×C₈	—	139	1	no	—
268	C₂×C₈	—	139	1	no	—
269	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
270	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
271	C₄⋊C₄	—	278	1	no	—
272	C₄⋊C₄	—	278	1	no	—
273	C₄⋊C₄	—	278	1	no	—
274	C₄⋊C₄	—	278	1	no	—
275	C₄⋊C₄	—	139	1	no	—
276	C₄⋊C₄	—	139	1	no	—
277	C₄⋊C₄	—	139	1	no	—
278	C₄⋊C₄	—	139	1	no	—
279	C₄⋊C₄	—	139	1	no	—
280	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
281	C₂²⋊C₄	—	2	1	no	—
282	C₂²⋊C₄	—	278	1	no	—
283	C₂²⋊C₄	—	278	1	no	—
284	C₂²⋊C₄	—	278	1	no	—
285	C₂²⋊C₄	—	278	1	no	—
286	C₂²⋊C₄	—	278	1	no	—
287	C₂²⋊C₄	—	278	1	no	—
288	C₂²⋊C₄	—	278	1	no	—
289	C₂²⋊C₄	—	139	1	no	—
290	C₂²⋊C₄	—	139	1	no	—
291	C₄²	—	139	1	no	—
292	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
293	C₂³	—	2	1	no	—
294	C₂³	—	2	1	no	—
295	C₂³	—	2	1	no	—
296	C₂³	—	278	1	no	—
297	C₂³	—	139	1	no	—
298	D₄	—	2	1	no	—
299	D₄	—	2	1	no	—
300	D₄	—	2	1	no	—
301	D₄	—	2	1	no	—
302	D₄	—	2	1	no	—
303	D₄	—	2	1	no	—
304	D₄	—	8	1	no	—
305	D₄	—	8	1	no	—
306	D₄	—	278	1	no	—
307	D₄	—	278	1	no	—
308	D₄	—	1112	1	no	—
309	D₄	—	1112	1	no	—
310	D₄	—	1112	1	no	—
311	D₄	—	1112	1	no	—
312	D₄	—	1112	1	no	—
313	D₄	—	1112	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	8	1	no	—
316	C₂×C₄	—	278	1	no	—
317	C₂×C₄	—	278	1	no	—
318	C₂×C₄	—	278	1	no	—
319	C₂×C₄	—	278	1	no	—
320	C₂×C₄	—	278	1	no	—
321	C₂×C₄	—	1112	1	no	—
322	C₂×C₄	—	1112	1	no	—
323	C₂×C₄	—	1112	1	no	—
324	C₂×C₄	—	139	1	no	—
325	C₂×C₄	—	139	1	no	—
326	C₂×C₄	—	139	1	no	—
327	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
328	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
329	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
330	C₈	—	2	1	no	—
331	C₈	—	2	1	no	—
332	C₈	—	278	1	no	—
333	C₈	—	278	1	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	4	1	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	278	1	no	—
344	C₂²	—	278	1	no	—
345	C₂²	—	278	1	no	—
346	C₂²	—	556	1	no	—
347	C₂²	—	556	1	no	—
348	C₂²	—	556	1	no	—
349	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	4	1	no	—
353	C₄	—	278	1	no	—
354	C₄	—	278	1	no	—
355	C₄	—	278	1	no	—
356	C₄	—	556	1	no	—
357	C₄	—	556	1	no	—
358	C₄	—	556	1	no	—
359	C₄	#17984887	1	1	yes	characteristic
360	C₄	#17984889	1	1	yes	characteristic
361	C₂	—	8	1	no	—
362	C₂	—	8	1	no	—
363	C₂	—	8	1	no	—
364	C₂	—	556	1	no	—
365	C₂	—	1112	1	no	—
366	C₂	#22581532	1	1	yes	characteristic
367	C₂	#22581754	1	1	yes	characteristic
368	C₂	#22581757	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#459×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
hypercenter	#459	D₁₃₉
hypocenter	C₁₃₉	#3259
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #459

Complements

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

N	H ≅ G/N	type
1. #22581557	C₂	semidirect
2. #22581554	C₂	semidirect
3. #22581255	C₂	semidirect
4. #22581254	C₂	semidirect
5. #22581253	C₂	semidirect
6. #22581251	C₂	semidirect
7. #22581205	C₂	semidirect
8. #22581203	C₂	semidirect
9. #22581188	C₂	semidirect
10. #22581186	C₂	semidirect
11. #22581146	C₂	semidirect
12. #22581146	C₂	semidirect
13. #22581145	C₂	semidirect
14. #22580978	C₂	semidirect
15. #459×C₁₃₉	C₂	semidirect
184. C₁₃₉	#3259	semidirect
199. #459	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581557	2
#22581554	2
#22581255	2
#22581254	2
#22581253	2
#22581251	2
#22581205	2
#22581203	2
#22581188	2
#22581186	2
#22581146	2
#22581145	2
#22580978	2
#459×C₁₃₉	2
#3259	139