Subgroups — #31819489

Order 17792.

380 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581689	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581685	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581341	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581334	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581173	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581172	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581170	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581166	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581146	C₂	1	1	yes	maximal, characteristic, complemented
10	#22581145	C₂	1	1	yes	maximal, characteristic, complemented
11	#22581081	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581055	C₂	1	1	yes	maximal, characteristic, complemented
13	#22581054	C₂	1	1	yes	maximal, characteristic, complemented
14	D₈×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#496×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555775	C₂²	1	1	yes	characteristic
17	C₂×#14555774	C₂²	1	1	yes	characteristic
18	C₂×#14555773	C₂²	1	1	yes	characteristic
19	C₂×#14555770	C₂²	1	1	yes	characteristic
20	C₂×#14555770	C₂²	1	1	yes	characteristic
21	#17984838	C₂²	1	1	yes	characteristic
22	#17984831	C₂²	1	1	yes	characteristic
23	#17984829	—	2	1	no	—
24	#17984829	C₂²	1	1	yes	characteristic
25	#17984829	C₂²	1	1	yes	characteristic
26	#17984828	—	2	1	no	—
27	#17984826	—	2	1	no	—
28	#17984824	—	2	1	no	—
29	#17984823	—	2	1	no	—
30	#17984823	—	2	1	no	—
31	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
32	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
33	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
34	C₄×D₅₅₆	C₂²	1	1	yes	characteristic
35	D₄×Dic₁₃₉	—	2	1	no	—
36	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
37	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
38	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
39	(C₂²⋊D₁₃₉)⋊C₄	—	2	1	no	—
40	#17984776	C₂²	1	1	yes	characteristic
41	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
42	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
43	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
44	#17984767	C₂²	1	1	yes	characteristic
45	#17984766	C₂²	1	1	yes	characteristic
46	#17984764	C₂²	1	1	yes	characteristic
47	#17984763	C₂²	1	1	yes	characteristic
48	#17984763	C₂²	1	1	yes	characteristic
49	#17984763	C₂²	1	1	yes	characteristic
50	#17984752	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
54	C₂×D₈×C₁₃₉	C₂²	1	1	yes	characteristic
55	C₂×D₈×C₁₃₉	C₂²	1	1	yes	characteristic
56	#127×C₁₃₉	C₂²	1	1	yes	characteristic
57	#124×C₁₃₉	C₂²	1	1	yes	characteristic
58	#97×C₁₃₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₇₈	D₄	1	1	yes	characteristic
60	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
61	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
63	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
64	C₂²×Dic₁₃₉	—	2	1	no	—
65	C₂²×Dic₁₃₉	—	2	1	no	—
66	C₂²×Dic₁₃₉	—	2	1	no	—
67	C₂×C₄×D₁₃₉	—	2	1	no	—
68	#14555775	—	4	1	no	—
69	#14555774	—	2	2	no	—
70	#14555773	—	4	1	no	—
71	#14555770	—	4	1	no	—
72	#14555770	—	4	1	no	—
73	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₃₉	—	2	1	no	—
76	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
77	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
79	Dic₁₃₉⋊C₄	—	2	1	no	—
80	Dic₁₃₉⋊C₄	D₄	1	1	yes	characteristic
81	Dic₁₃₉⋊C₄	D₄	1	1	yes	characteristic
82	C₂²⋊Dic₁₃₉	—	2	1	no	—
83	C₂²⋊Dic₁₃₉	—	2	1	no	—
84	C₂²⋊Dic₁₃₉	—	2	1	no	—
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	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
90	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
91	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
92	C₄×Dic₁₃₉	—	2	1	no	—
93	C₄×Dic₁₃₉	—	2	1	no	—
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	—	2	1	no	—
99	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
100	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
101	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
102	SD₁₆×C₁₃₉	—	2	2	no	—
103	SD₁₆×C₁₃₉	—	4	1	no	—
104	D₈×C₁₃₉	—	2	2	no	—
105	D₈×C₁₃₉	—	4	1	no	—
106	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
107	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
108	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
109	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
110	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
111	C₂²×D₁₃₉	—	2	1	no	—
112	Dic₂₇₈	—	2	1	no	—
113	Dic₂₇₈	—	2	1	no	—
114	Dic₂₇₈	—	4	1	no	—
115	D₅₅₆	—	2	1	no	—
116	D₅₅₆	—	2	1	no	—
117	C₂²⋊D₁₃₉	—	4	2	no	—
118	C₂²⋊D₁₃₉	—	8	1	no	—
119	C₂×Dic₁₃₉	—	2	1	no	—
120	C₂×Dic₁₃₉	—	2	1	no	—
121	C₂×Dic₁₃₉	—	2	1	no	—
122	C₂×Dic₁₃₉	—	8	1	no	—
123	C₂×Dic₁₃₉	—	8	1	no	—
124	C₂×Dic₁₃₉	—	8	1	no	—
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	—	8	1	no	—
130	C₁₃₉⋊C₈	—	2	1	no	—
131	C₁₃₉⋊C₈	—	2	1	no	—
132	C₂³×C₁₃₉	—	2	1	no	—
133	C₂³×C₁₃₉	—	2	1	no	—
134	C₂³×C₁₃₉	—	2	1	no	—
135	Q₈×C₁₃₉	—	2	1	no	—
136	Q₈×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	—	2	1	no	—
141	D₄×C₁₃₉	—	2	1	no	—
142	D₄×C₁₃₉	—	2	1	no	—
143	D₄×C₁₃₉	—	4	2	no	—
144	D₄×C₁₃₉	—	8	1	no	—
145	C₂×C₄×C₁₃₉	—	2	1	no	—
146	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
147	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
148	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
149	C₈×C₁₃₉	—	2	1	no	—
150	C₈×C₁₃₉	#41	1	2	yes	—
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	—	4	1	no	—
159	C₂²×C₁₃₉	—	4	1	no	—
160	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
161	C₂×D₁₃₉	—	4	1	no	—
162	C₂×D₁₃₉	—	4	1	no	—
163	C₂×D₁₃₉	—	4	1	no	—
164	C₄×C₁₃₉	—	2	1	no	—
165	C₄×C₁₃₉	—	2	1	no	—
166	C₄×C₁₃₉	—	4	1	no	—
167	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
168	C₄×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
169	Dic₁₃₉	—	2	1	no	—
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	#550	1	1	yes	characteristic
180	C₂×C₁₃₉	#575	1	1	yes	characteristic
181	C₂×C₁₃₉	#576	1	1	yes	characteristic
182	D₁₃₉	—	8	1	no	—
183	C₁₃₉	#3403	1	1	yes	characteristic, complemented, Sylow
184	#3403	—	139	1	no	maximal, Sylow
185	#559	—	139	1	no	—
186	#559	—	139	1	no	—
187	#496	—	139	1	no	—
188	#496	D₁₃₉	1	1	yes	characteristic, complemented
189	#495	—	139	1	no	—
190	#464	—	139	1	no	—
191	#464	—	139	1	no	—
192	#463	—	139	1	no	—
193	#463	—	139	1	no	—
194	#459	—	139	1	no	—
195	#459	—	139	1	no	—
196	#446	—	139	1	no	—
197	#440	—	139	1	no	—
198	#440	—	139	1	no	—
199	C₄×D₈	—	139	1	no	—
200	C₂×Q₁₆	—	139	1	no	—
201	C₂×SD₁₆	—	139	1	no	—
202	C₂×SD₁₆	—	139	1	no	—
203	C₂×SD₁₆	C₂×D₁₃₉	1	1	yes	characteristic
204	C₂×SD₁₆	C₂×D₁₃₉	1	1	yes	characteristic
205	C₂×D₈	—	139	1	no	—
206	C₂×D₈	—	139	1	no	—
207	C₂×D₈	C₂×D₁₃₉	1	1	yes	characteristic
208	C₂×D₈	C₂×D₁₃₉	1	1	yes	characteristic
209	#127	—	139	1	no	—
210	#127	C₂×D₁₃₉	1	1	yes	characteristic
211	#124	—	278	1	no	—
212	#124	—	278	1	no	—
213	#124	—	139	1	no	—
214	#124	—	139	1	no	—
215	#124	—	139	1	no	—
216	#124	C₂×D₁₃₉	1	1	yes	characteristic
217	#123	—	278	1	no	—
218	#123	—	278	1	no	—
219	#123	—	278	1	no	—
220	#123	—	278	1	no	—
221	C₄×Q₈	—	139	1	no	—
222	C₄×Q₈	—	139	1	no	—
223	C₄×Q₈	—	139	1	no	—
224	C₄×D₄	—	278	1	no	—
225	C₄×D₄	—	278	1	no	—
226	C₄×D₄	—	139	1	no	—
227	C₄×D₄	—	139	1	no	—
228	C₄×D₄	—	139	1	no	—
229	C₄×D₄	—	139	1	no	—
230	#107	—	139	1	no	—
231	C₄⋊C₈	—	139	1	no	—
232	C₄⋊C₈	—	139	1	no	—
233	C₄⋊C₈	—	139	1	no	—
234	#103	—	139	1	no	—
235	#103	—	139	1	no	—
236	#102	—	139	1	no	—
237	#102	—	139	1	no	—
238	#102	—	139	1	no	—
239	#102	—	139	1	no	—
240	#97	—	139	1	no	—
241	#97	C₂×D₁₃₉	1	1	yes	characteristic
242	C₄×C₈	—	139	1	no	—
243	C₂×Q₈	—	139	1	no	—
244	C₂×Q₈	—	139	1	no	—
245	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	2	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	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
253	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
254	C₂²×C₄	—	278	1	no	—
255	C₂²×C₄	—	278	1	no	—
256	C₂²×C₄	—	278	1	no	—
257	C₂²×C₄	—	278	1	no	—
258	Q₁₆	—	556	1	no	—
259	SD₁₆	—	2	2	no	—
260	SD₁₆	—	4	1	no	—
261	SD₁₆	—	278	2	no	—
262	SD₁₆	—	556	1	no	—
263	D₈	—	2	2	no	—
264	D₈	—	4	1	no	—
265	D₈	—	556	1	no	—
266	D₈	—	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	—	139	1	no	—
274	C₄⋊C₄	—	139	1	no	—
275	C₄⋊C₄	—	139	1	no	—
276	C₄⋊C₄	—	139	1	no	—
277	C₄⋊C₄	—	139	1	no	—
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	2	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	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	—	278	1	no	—
290	C₄²	—	278	1	no	—
291	C₄²	—	278	1	no	—
292	C₄²	—	139	1	no	—
293	C₄²	—	139	1	no	—
294	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
295	C₂³	—	2	1	no	—
296	C₂³	—	2	1	no	—
297	C₂³	—	2	1	no	—
298	C₂³	—	278	1	no	—
299	Q₈	—	2	1	no	—
300	Q₈	—	2	1	no	—
301	Q₈	—	278	1	no	—
302	Q₈	—	278	1	no	—
303	Q₈	—	556	1	no	—
304	D₄	—	2	1	no	—
305	D₄	—	2	1	no	—
306	D₄	—	2	1	no	—
307	D₄	—	2	1	no	—
308	D₄	—	2	1	no	—
309	D₄	—	2	1	no	—
310	D₄	—	4	2	no	—
311	D₄	—	8	1	no	—
312	D₄	—	278	1	no	—
313	D₄	—	278	1	no	—
314	D₄	—	556	2	no	—
315	D₄	—	1112	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	278	1	no	—
318	C₂×C₄	—	278	1	no	—
319	C₂×C₄	—	278	1	no	—
320	C₂×C₄	—	1112	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	—	139	1	no	—
328	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
329	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
330	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
331	C₈	—	2	1	no	—
332	C₈	—	278	1	no	—
333	C₈	—	278	1	no	—
334	C₈	#14555784	1	2	yes	—
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	—	4	1	no	—
344	C₂²	—	556	1	no	—
345	C₂²	—	556	1	no	—
346	C₂²	—	556	1	no	—
347	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	4	1	no	—
351	C₄	—	278	1	no	—
352	C₄	—	278	1	no	—
353	C₄	—	278	1	no	—
354	C₄	—	278	1	no	—
355	C₄	—	556	1	no	—
356	C₄	—	556	1	no	—
357	C₄	—	556	1	no	—
358	C₄	C₂×#14555784	1	1	yes	characteristic
359	C₄	#17984887	1	1	yes	characteristic
360	C₂	—	8	1	no	—
361	C₂	—	8	1	no	—
362	C₂	—	8	1	no	—
363	C₂	—	1112	1	no	—
364	C₂	#22581623	1	1	yes	characteristic
365	C₂	#22581749	1	1	yes	characteristic
366	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	#496×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
hypercenter	#496	D₁₃₉
hypocenter	C₁₃₉	#3403
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #496

Complements

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

N	H ≅ G/N	type
1. #22581689	C₂	semidirect
2. #22581685	C₂	semidirect
3. #22581341	C₂	semidirect
4. #22581334	C₂	semidirect
5. #22581173	C₂	semidirect
6. #22581172	C₂	semidirect
7. #22581170	C₂	semidirect
8. #22581166	C₂	semidirect
9. #22581146	C₂	semidirect
10. #22581145	C₂	semidirect
11. #22581081	C₂	semidirect
12. #22581055	C₂	semidirect
13. #22581054	C₂	semidirect
14. D₈×Dic₁₃₉	C₂	semidirect
15. #496×C₁₃₉	C₂	semidirect
183. C₁₃₉	#3403	semidirect
188. #496	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581689	2
#22581685	2
#22581341	2
#22581334	2
#22581173	2
#22581172	2
#22581170	2
#22581166	2
#22581146	2
#22581145	2
#22581081	2
#22581055	2
#22581054	2
D₈×Dic₁₃₉	2
#496×C₁₃₉	2
#3403	139