Subgroups — #31819449

Order 17792.

418 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581686	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581623	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581341	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581340	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581328	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581169	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581166	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581152	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581147	C₂	1	1	yes	maximal, characteristic, complemented
10	#22581146	C₂	1	1	yes	maximal, characteristic, complemented
11	#22581082	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581063	C₂	1	1	yes	maximal, characteristic, complemented
13	#22581062	C₂	1	1	yes	maximal, characteristic, complemented
14	#22581044	C₂	1	1	yes	maximal, characteristic, complemented
15	#459×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×#14555773	C₂²	1	1	yes	characteristic
20	C₂×D₁₁₁₂	C₂²	1	1	yes	characteristic
21	C₂×#14555770	C₂²	1	1	yes	characteristic
22	C₂×#14555770	C₂²	1	1	yes	characteristic
23	C₂×#14555770	C₂²	1	1	yes	characteristic
24	#17984838	C₂²	1	1	yes	characteristic
25	#17984838	C₂²	1	1	yes	characteristic
26	#17984838	C₂²	1	1	yes	characteristic
27	#17984830	C₂²	1	1	yes	characteristic
28	#17984829	C₂²	1	1	yes	characteristic
29	#17984828	—	2	1	no	—
30	#17984825	—	2	1	no	—
31	#17984824	—	2	1	no	—
32	#17984814	—	2	1	no	—
33	#17984814	—	2	1	no	—
34	#17984813	—	2	1	no	—
35	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
36	#17984807	C₂²	1	1	yes	characteristic
37	C₄×D₅₅₆	C₂²	1	1	yes	characteristic
38	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
39	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
40	#17984804	C₂²	1	1	yes	characteristic
41	C₄×C₂²⋊D₁₃₉	—	2	1	no	—
42	#17984773	C₂²	1	1	yes	characteristic
43	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
44	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
45	#17984767	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	#17984753	C₂²	1	1	yes	characteristic
50	#17984752	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₉	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	#14555784	—	4	4	no	—
60	C₂×Dic₂₇₈	D₄	1	1	yes	characteristic
61	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
62	C₂×D₅₅₆	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	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
68	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
69	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
70	C₂²×Dic₁₃₉	—	2	1	no	—
71	C₂²×Dic₁₃₉	—	2	1	no	—
72	C₂×C₄×D₁₃₉	—	2	1	no	—
73	C₂×C₄×D₁₃₉	—	2	1	no	—
74	#14555774	—	4	1	no	—
75	#14555773	—	2	2	no	—
76	#14555773	—	4	1	no	—
77	D₁₁₁₂	—	4	1	no	—
78	#14555770	—	2	2	no	—
79	#14555770	—	2	2	no	—
80	#14555770	—	4	1	no	—
81	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
82	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
83	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
84	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
85	Dic₁₃₉⋊C₄	—	2	1	no	—
86	Dic₁₃₉⋊C₄	D₄	1	1	yes	characteristic
87	Dic₁₃₉⋊C₄	D₄	1	1	yes	characteristic
88	C₂²⋊Dic₁₃₉	—	2	1	no	—
89	C₂²⋊Dic₁₃₉	—	2	1	no	—
90	C₂²⋊Dic₁₃₉	—	2	1	no	—
91	C₂²⋊Dic₁₃₉	—	2	1	no	—
92	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
93	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
94	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
95	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
96	C₄×Dic₁₃₉	—	2	1	no	—
97	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
98	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
99	C₂×D₄×C₁₃₉	—	2	1	no	—
100	C₂×D₄×C₁₃₉	—	2	1	no	—
101	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
102	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
103	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
104	C₂²×C₄×C₁₃₉	—	2	1	no	—
105	D₈×C₁₃₉	—	4	1	no	—
106	D₈×C₁₃₉	—	4	1	no	—
107	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
108	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
109	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
110	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
111	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
112	C₂²×D₁₃₉	—	2	1	no	—
113	C₂²×D₁₃₉	—	2	1	no	—
114	Dic₂₇₈	—	2	2	no	—
115	Dic₂₇₈	—	2	1	no	—
116	Dic₂₇₈	—	2	1	no	—
117	D₅₅₆	—	2	1	no	—
118	D₅₅₆	—	2	1	no	—
119	D₅₅₆	—	2	1	no	—
120	D₅₅₆	—	2	1	no	—
121	C₂²⋊D₁₃₉	—	4	2	no	—
122	C₂²⋊D₁₃₉	—	4	2	no	—
123	C₂²⋊D₁₃₉	—	4	2	no	—
124	C₂²⋊D₁₃₉	—	8	1	no	—
125	C₂²⋊D₁₃₉	—	8	1	no	—
126	C₂²⋊D₁₃₉	—	8	1	no	—
127	C₂×Dic₁₃₉	—	2	1	no	—
128	C₂×Dic₁₃₉	—	2	1	no	—
129	C₂×Dic₁₃₉	—	4	2	no	—
130	C₂×Dic₁₃₉	—	4	2	no	—
131	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
135	C₄×D₁₃₉	—	4	2	no	—
136	C₄×D₁₃₉	—	8	1	no	—
137	C₁₃₉⋊C₈	—	2	1	no	—
138	C₁₃₉⋊C₈	#41	1	2	yes	complemented
139	C₂³×C₁₃₉	—	2	1	no	—
140	C₂³×C₁₃₉	—	2	1	no	—
141	C₂³×C₁₃₉	—	2	1	no	—
142	D₄×C₁₃₉	—	2	1	no	—
143	D₄×C₁₃₉	—	2	1	no	—
144	D₄×C₁₃₉	—	2	1	no	—
145	D₄×C₁₃₉	—	2	1	no	—
146	D₄×C₁₃₉	—	2	1	no	—
147	D₄×C₁₃₉	—	2	1	no	—
148	D₄×C₁₃₉	—	4	2	no	—
149	D₄×C₁₃₉	—	8	1	no	—
150	C₂×C₄×C₁₃₉	—	2	1	no	—
151	C₂×C₄×C₁₃₉	—	8	1	no	—
152	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
153	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
154	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
155	C₈×C₁₃₉	—	2	1	no	—
156	C₈×C₁₃₉	—	2	1	no	—
157	C₂²×C₁₃₉	—	4	1	no	—
158	C₂²×C₁₃₉	—	4	1	no	—
159	C₂²×C₁₃₉	—	4	1	no	—
160	C₂²×C₁₃₉	—	4	1	no	—
161	C₂²×C₁₃₉	—	4	1	no	—
162	C₂²×C₁₃₉	—	4	1	no	—
163	C₂²×C₁₃₉	—	4	1	no	—
164	C₂²×C₁₃₉	—	4	1	no	—
165	C₂²×C₁₃₉	—	4	1	no	—
166	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
167	C₂×D₁₃₉	—	4	1	no	—
168	C₂×D₁₃₉	—	4	1	no	—
169	C₂×D₁₃₉	—	4	1	no	—
170	C₂×D₁₃₉	—	4	1	no	—
171	C₂×D₁₃₉	—	4	1	no	—
172	C₂×D₁₃₉	—	4	1	no	—
173	C₄×C₁₃₉	—	2	1	no	—
174	C₄×C₁₃₉	—	2	1	no	—
175	C₄×C₁₃₉	—	4	1	no	—
176	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
177	C₄×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
178	Dic₁₃₉	—	2	1	no	—
179	Dic₁₃₉	—	2	1	no	—
180	Dic₁₃₉	—	2	1	no	—
181	Dic₁₃₉	—	4	1	no	—
182	Dic₁₃₉	—	4	1	no	—
183	Dic₁₃₉	AGL(1,ℤ/8)	1	2	yes	—
184	C₂×C₁₃₉	—	8	1	no	—
185	C₂×C₁₃₉	—	8	1	no	—
186	C₂×C₁₃₉	—	8	1	no	—
187	C₂×C₁₃₉	#550	1	1	yes	characteristic
188	C₂×C₁₃₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
189	C₂×C₁₃₉	#576	1	1	yes	characteristic
190	D₁₃₉	—	8	1	no	—
191	D₁₃₉	—	8	1	no	—
192	C₁₃₉	#3399	1	1	yes	characteristic, complemented, Sylow
193	#3399	—	139	1	no	maximal, Sylow
194	#559	—	139	1	no	—
195	#550	—	139	1	no	—
196	#496	—	139	1	no	—
197	#496	—	139	1	no	—
198	#493	—	139	1	no	—
199	#463	—	139	1	no	—
200	#463	—	139	1	no	—
201	#460	—	139	1	no	—
202	#459	—	139	1	no	—
203	#459	—	139	1	no	—
204	#459	D₁₃₉	1	1	yes	characteristic, complemented
205	#446	—	139	1	no	—
206	#442	—	139	1	no	—
207	#442	—	139	1	no	—
208	C₄×SD₁₆	—	139	1	no	—
209	C₂×#41	—	278	1	no	—
210	C₂×SD₁₆	—	139	1	no	—
211	C₂×SD₁₆	—	139	1	no	—
212	C₂×SD₁₆	—	139	1	no	—
213	C₂×D₈	—	139	1	no	—
214	C₂×D₈	—	139	1	no	—
215	C₂×D₈	—	139	1	no	—
216	C₂×D₈	—	139	1	no	—
217	C₂×D₈	C₂×D₁₃₉	1	1	yes	characteristic
218	C₂×D₈	C₂×D₁₃₉	1	1	yes	characteristic
219	#127	—	139	1	no	—
220	#127	—	139	1	no	—
221	#127	—	139	1	no	—
222	#127	C₂×D₁₃₉	1	1	yes	characteristic
223	#124	—	278	1	no	—
224	#124	—	139	1	no	—
225	#124	—	139	1	no	—
226	#123	—	278	1	no	—
227	#123	—	278	1	no	—
228	#121	—	278	1	no	—
229	#121	—	278	1	no	—
230	#121	—	278	1	no	—
231	C₄×Q₈	—	139	1	no	—
232	C₄×Q₈	—	139	1	no	—
233	C₄×D₄	—	278	1	no	—
234	C₄×D₄	—	139	1	no	—
235	C₄×D₄	—	139	1	no	—
236	C₄×D₄	—	139	1	no	—
237	C₄×D₄	—	139	1	no	—
238	C₄×D₄	C₂×D₁₃₉	1	1	yes	characteristic
239	#106	—	139	1	no	—
240	C₄⋊C₈	—	139	1	no	—
241	C₄⋊C₈	—	139	1	no	—
242	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
243	#103	—	139	1	no	—
244	#102	—	139	1	no	—
245	#102	—	139	1	no	—
246	#102	—	139	1	no	—
247	#102	C₂×D₁₃₉	1	1	yes	characteristic
248	#102	C₂×D₁₃₉	1	1	yes	characteristic
249	#97	—	139	1	no	—
250	#97	—	139	1	no	—
251	C₄×C₈	—	139	1	no	—
252	#41	—	556	4	no	—
253	C₂×Q₈	—	139	1	no	—
254	C₂×Q₈	—	139	1	no	—
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	278	1	no	—
258	C₂×D₄	—	278	1	no	—
259	C₂×D₄	—	278	1	no	—
260	C₂×D₄	—	278	1	no	—
261	C₂×D₄	—	278	1	no	—
262	C₂×D₄	—	278	1	no	—
263	C₂×D₄	—	139	1	no	—
264	C₂×D₄	—	139	1	no	—
265	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
266	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
267	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
268	C₂²×C₄	—	2	1	no	—
269	C₂²×C₄	—	278	1	no	—
270	C₂²×C₄	—	278	1	no	—
271	C₂²×C₄	—	278	1	no	—
272	C₂²×C₄	—	278	1	no	—
273	SD₁₆	—	278	2	no	—
274	SD₁₆	—	556	1	no	—
275	SD₁₆	—	556	1	no	—
276	D₈	—	4	1	no	—
277	D₈	—	4	1	no	—
278	D₈	—	278	2	no	—
279	D₈	—	278	2	no	—
280	D₈	—	556	1	no	—
281	D₈	—	556	1	no	—
282	C₂×C₈	—	139	1	no	—
283	C₂×C₈	—	139	1	no	—
284	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
285	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
286	C₄⋊C₄	—	278	1	no	—
287	C₄⋊C₄	—	139	1	no	—
288	C₄⋊C₄	—	139	1	no	—
289	C₄⋊C₄	—	139	1	no	—
290	C₄⋊C₄	—	139	1	no	—
291	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
292	C₂²⋊C₄	—	2	1	no	—
293	C₂²⋊C₄	—	278	1	no	—
294	C₂²⋊C₄	—	278	1	no	—
295	C₂²⋊C₄	—	278	1	no	—
296	C₂²⋊C₄	—	278	1	no	—
297	C₂²⋊C₄	—	278	1	no	—
298	C₂²⋊C₄	—	278	1	no	—
299	C₂²⋊C₄	—	278	1	no	—
300	C₂²⋊C₄	—	278	1	no	—
301	C₄²	—	278	1	no	—
302	C₄²	—	139	1	no	—
303	C₄²	—	139	1	no	—
304	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
305	C₂³	—	2	1	no	—
306	C₂³	—	2	1	no	—
307	C₂³	—	2	1	no	—
308	C₂³	—	278	1	no	—
309	C₂³	—	278	1	no	—
310	Q₈	—	278	2	no	—
311	Q₈	—	278	1	no	—
312	Q₈	—	278	1	no	—
313	D₄	—	2	1	no	—
314	D₄	—	2	1	no	—
315	D₄	—	2	1	no	—
316	D₄	—	2	1	no	—
317	D₄	—	2	1	no	—
318	D₄	—	2	1	no	—
319	D₄	—	4	2	no	—
320	D₄	—	8	1	no	—
321	D₄	—	278	1	no	—
322	D₄	—	278	1	no	—
323	D₄	—	278	1	no	—
324	D₄	—	278	1	no	—
325	D₄	—	556	2	no	—
326	D₄	—	556	2	no	—
327	D₄	—	556	2	no	—
328	D₄	—	1112	1	no	—
329	D₄	—	1112	1	no	—
330	D₄	—	1112	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	8	1	no	—
333	C₂×C₄	—	278	1	no	—
334	C₂×C₄	—	278	1	no	—
335	C₂×C₄	—	556	2	no	—
336	C₂×C₄	—	556	2	no	—
337	C₂×C₄	—	556	2	no	—
338	C₂×C₄	—	1112	1	no	—
339	C₂×C₄	—	139	1	no	—
340	C₂×C₄	—	139	1	no	—
341	C₂×C₄	—	139	1	no	—
342	C₂×C₄	—	139	1	no	—
343	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
344	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
345	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
346	C₈	—	2	1	no	—
347	C₈	—	2	1	no	—
348	C₈	—	278	1	no	—
349	C₈	—	139	2	no	—
350	C₂²	—	4	1	no	—
351	C₂²	—	4	1	no	—
352	C₂²	—	4	1	no	—
353	C₂²	—	4	1	no	—
354	C₂²	—	4	1	no	—
355	C₂²	—	4	1	no	—
356	C₂²	—	4	1	no	—
357	C₂²	—	4	1	no	—
358	C₂²	—	4	1	no	—
359	C₂²	—	556	1	no	—
360	C₂²	—	556	1	no	—
361	C₂²	—	556	1	no	—
362	C₂²	—	556	1	no	—
363	C₂²	—	556	1	no	—
364	C₂²	—	556	1	no	—
365	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	4	1	no	—
369	C₄	—	278	1	no	—
370	C₄	—	278	1	no	—
371	C₄	—	278	1	no	—
372	C₄	—	556	1	no	—
373	C₄	—	556	1	no	—
374	C₄	—	139	2	no	—
375	C₄	#41×D₁₃₉	1	1	yes	characteristic
376	C₄	#17984887	1	1	yes	characteristic
377	C₂	—	8	1	no	—
378	C₂	—	8	1	no	—
379	C₂	—	8	1	no	—
380	C₂	—	1112	1	no	—
381	C₂	—	1112	1	no	—
382	C₂	#22581621	1	1	yes	characteristic
383	C₂	AGL(1,ℤ/8)×D₁₃₉	1	1	yes	characteristic
384	C₂	#22581754	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	#3399
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. #22581686	C₂	semidirect
2. #22581623	C₂	semidirect
3. #22581341	C₂	semidirect
4. #22581340	C₂	semidirect
5. #22581328	C₂	semidirect
6. #22581169	C₂	semidirect
7. #22581166	C₂	semidirect
8. #22581152	C₂	semidirect
9. #22581147	C₂	semidirect
10. #22581146	C₂	semidirect
11. #22581082	C₂	semidirect
12. #22581063	C₂	semidirect
13. #22581062	C₂	semidirect
14. #22581044	C₂	semidirect
15. #459×C₁₃₉	C₂	semidirect
138. C₁₃₉⋊C₈	#41	semidirect
192. C₁₃₉	#3399	semidirect
204. #459	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581686	2
#22581623	2
#22581341	2
#22581340	2
#22581328	2
#22581169	2
#22581166	2
#22581152	2
#22581147	2
#22581146	2
#22581082	2
#22581063	2
#22581062	2
#22581044	2
#459×C₁₃₉	2
#3399	139