Subgroups — #31816514

Order 17792.

392 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581489	C₂	1	1	yes	maximal, characteristic, complemented
2	#41×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581291	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581278	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581262	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581255	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581177	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581170	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581098	C₂	1	1	yes	maximal, characteristic
10	#22581062	C₂	1	1	yes	maximal, characteristic, complemented
11	#22581060	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581056	C₂	1	1	yes	maximal, characteristic, complemented
13	#22581054	C₂	1	1	yes	maximal, characteristic, complemented
14	#22581007	C₂	1	1	yes	maximal, characteristic, complemented
15	#452×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555772	C₂²	1	1	yes	characteristic
17	C₂×#14555770	C₂²	1	1	yes	characteristic
18	#17984837	—	2	1	no	—
19	#17984833	C₂²	1	1	yes	characteristic
20	#17984831	C₂²	1	1	yes	characteristic
21	#17984826	—	2	1	no	—
22	#17984816	C₂²	1	1	yes	characteristic
23	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
24	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
25	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
26	D₄×Dic₁₃₉	—	2	1	no	—
27	D₄×Dic₁₃₉	—	2	1	no	—
28	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
29	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
30	#17984804	C₂²	1	1	yes	characteristic
31	(C₂²⋊D₁₃₉)⋊C₄	—	2	1	no	—
32	#17984801	—	2	1	no	—
33	#17984801	C₂²	1	1	yes	characteristic
34	#17984800	—	2	1	no	—
35	C₂×C₄×Dic₁₃₉	—	2	1	no	—
36	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
37	#17984776	C₂²	1	1	yes	characteristic
38	#17984775	C₂²	1	1	yes	characteristic
39	#17984774	C₂²	1	1	yes	characteristic
40	#17984773	C₂²	1	1	yes	characteristic
41	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
42	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
43	#17984766	C₂²	1	1	yes	characteristic
44	#17984766	C₂²	1	1	yes	characteristic
45	#17984766	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	C₂²⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
50	#17984752	C₂²	1	1	yes	characteristic
51	#17984752	C₂²	1	1	yes	characteristic
52	C₂×#41×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
54	C₂×SD₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
56	#103×C₁₃₉	C₂²	1	1	yes	characteristic
57	#102×C₁₃₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
59	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
60	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
61	C₂²×Dic₁₃₉	—	2	1	no	—
62	C₂²×Dic₁₃₉	—	2	1	no	—
63	C₂²×Dic₁₃₉	D₄	1	1	yes	characteristic
64	C₂²×Dic₁₃₉	D₄	1	1	yes	characteristic
65	C₂×C₄×D₁₃₉	—	2	1	no	—
66	#14555772	—	2	2	no	—
67	#14555770	—	2	2	no	—
68	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₁₃₉	—	2	1	no	—
71	C₄⋊Dic₁₃₉	—	2	1	no	—
72	C₄⋊Dic₁₃₉	—	2	1	no	—
73	C₄⋊Dic₁₃₉	—	2	1	no	—
74	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
77	Dic₁₃₉⋊C₄	—	2	1	no	—
78	C₂²⋊Dic₁₃₉	—	2	1	no	—
79	C₂²⋊Dic₁₃₉	—	2	1	no	—
80	C₂²⋊Dic₁₃₉	—	2	1	no	—
81	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
82	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
83	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
84	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
85	C₄×Dic₁₃₉	—	2	1	no	—
86	C₄×Dic₁₃₉	—	2	1	no	—
87	C₄×Dic₁₃₉	—	2	1	no	—
88	C₄×Dic₁₃₉	—	2	1	no	—
89	C₄×Dic₁₃₉	D₄	1	1	yes	characteristic
90	C₄×Dic₁₃₉	D₄	1	1	yes	characteristic
91	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
92	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
93	#41×C₁₃₉	—	2	2	no	—
94	#41×C₁₃₉	—	2	2	no	—
95	C₂×Q₈×C₁₃₉	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	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₁₃₉	—	2	1	no	—
100	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
101	Q₁₆×C₁₃₉	—	2	2	no	—
102	SD₁₆×C₁₃₉	—	2	2	no	—
103	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₃₉	—	2	1	no	—
106	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
107	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
108	C₂²×D₁₃₉	—	2	1	no	—
109	D₅₅₆	—	2	1	no	—
110	D₅₅₆	—	2	1	no	—
111	C₂²⋊D₁₃₉	—	4	2	no	—
112	C₂×Dic₁₃₉	—	2	1	no	—
113	C₂×Dic₁₃₉	—	2	1	no	—
114	C₂×Dic₁₃₉	—	2	1	no	—
115	C₂×Dic₁₃₉	—	4	1	no	—
116	C₂×Dic₁₃₉	—	4	1	no	—
117	C₂×Dic₁₃₉	—	4	2	no	—
118	C₂×Dic₁₃₉	—	4	2	no	—
119	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
123	C₄×D₁₃₉	—	4	2	no	—
124	C₁₃₉⋊C₈	—	2	1	no	—
125	C₁₃₉⋊C₈	—	2	1	no	—
126	C₂³×C₁₃₉	—	2	1	no	—
127	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
128	Q₈×C₁₃₉	—	2	1	no	—
129	Q₈×C₁₃₉	—	2	1	no	—
130	Q₈×C₁₃₉	—	2	1	no	—
131	Q₈×C₁₃₉	#41	1	2	yes	—
132	D₄×C₁₃₉	—	2	4	no	—
133	D₄×C₁₃₉	—	2	1	no	—
134	D₄×C₁₃₉	#41	1	2	yes	—
135	C₂×C₄×C₁₃₉	—	2	4	no	—
136	C₂×C₄×C₁₃₉	—	2	1	no	—
137	C₂×C₄×C₁₃₉	—	2	1	no	—
138	C₂×C₄×C₁₃₉	—	2	1	no	—
139	C₂×C₄×C₁₃₉	—	2	1	no	—
140	C₂×C₄×C₁₃₉	—	2	1	no	—
141	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
143	C₈×C₁₃₉	—	2	1	no	—
144	C₈×C₁₃₉	—	2	1	no	—
145	C₂²×C₁₃₉	—	2	1	no	—
146	C₂²×C₁₃₉	—	2	2	no	—
147	C₂²×C₁₃₉	—	2	1	no	—
148	C₂²×C₁₃₉	—	2	1	no	—
149	C₂²×C₁₃₉	—	4	1	no	—
150	C₂²×C₁₃₉	—	4	1	no	—
151	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₁₃₉	—	4	1	no	—
153	C₂×D₁₃₉	—	4	1	no	—
154	C₂×D₁₃₉	—	4	1	no	—
155	C₄×C₁₃₉	—	2	1	no	—
156	C₄×C₁₃₉	—	2	2	no	—
157	C₄×C₁₃₉	—	4	1	no	—
158	C₄×C₁₃₉	—	4	1	no	—
159	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
160	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
161	Dic₁₃₉	—	2	1	no	—
162	Dic₁₃₉	—	2	1	no	—
163	Dic₁₃₉	—	2	1	no	—
164	Dic₁₃₉	—	2	1	no	—
165	Dic₁₃₉	—	4	1	no	—
166	Dic₁₃₉	—	4	1	no	—
167	Dic₁₃₉	—	4	1	no	—
168	C₂×C₁₃₉	—	4	1	no	—
169	C₂×C₁₃₉	—	4	2	no	—
170	C₂×C₁₃₉	#525	1	1	yes	characteristic
171	C₂×C₁₃₉	#575	1	1	yes	characteristic
172	C₂×C₁₃₉	#575	1	1	yes	characteristic
173	D₁₃₉	—	8	1	no	—
174	C₁₃₉	#3111	1	1	yes	characteristic, complemented, Sylow
175	#3111	—	139	1	no	maximal, Sylow
176	#529	—	139	1	no	—
177	C₄×#41	—	139	1	no	—
178	#485	—	139	1	no	—
179	#482	—	139	1	no	—
180	#479	—	139	1	no	—
181	#478	—	139	1	no	—
182	#464	—	139	1	no	—
183	#463	—	139	1	no	—
184	#452	D₁₃₉	1	1	yes	characteristic, complemented
185	#449	—	139	1	no	—
186	#442	—	139	1	no	—
187	#441	—	139	1	no	—
188	#440	—	139	1	no	—
189	#440	—	139	1	no	—
190	#432	—	139	1	no	—
191	C₂×#41	C₂×D₁₃₉	1	1	yes	characteristic
192	C₂×Q₁₆	C₂×D₁₃₉	1	1	yes	characteristic
193	C₂×SD₁₆	—	139	1	no	—
194	C₂×SD₁₆	C₂×D₁₃₉	1	1	yes	characteristic
195	C₂×D₈	—	139	1	no	—
196	#126	—	278	1	no	—
197	#125	—	139	1	no	—
198	#124	—	139	1	no	—
199	#123	—	278	1	no	—
200	C₂²⋊Q₈	C₂×D₁₃₉	1	1	yes	characteristic
201	#121	—	139	1	no	—
202	C₄×Q₈	—	139	1	no	—
203	C₄×Q₈	—	139	1	no	—
204	C₄×Q₈	—	139	1	no	—
205	C₄×D₄	—	278	1	no	—
206	C₄×D₄	—	278	1	no	—
207	C₄×D₄	—	278	1	no	—
208	C₄×D₄	—	139	1	no	—
209	C₄×D₄	—	139	1	no	—
210	C₄×D₄	—	139	1	no	—
211	#117	—	278	1	no	—
212	#117	—	278	1	no	—
213	#117	—	139	1	no	—
214	C₂×C₄²	—	278	1	no	—
215	C₂×C₄²	—	139	1	no	—
216	#107	—	139	1	no	—
217	#107	—	139	1	no	—
218	#106	—	139	1	no	—
219	#106	—	139	1	no	—
220	C₄⋊C₈	—	139	1	no	—
221	C₄⋊C₈	—	139	1	no	—
222	#103	—	139	1	no	—
223	#103	—	139	1	no	—
224	#103	—	139	1	no	—
225	#103	C₂×D₁₃₉	1	1	yes	characteristic
226	#102	—	139	1	no	—
227	#102	—	139	1	no	—
228	#102	—	139	1	no	—
229	#102	C₂×D₁₃₉	1	1	yes	characteristic
230	C₂²⋊C₈	—	139	1	no	—
231	C₂²⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
232	#97	—	139	1	no	—
233	#97	—	139	1	no	—
234	#41	—	2	2	no	—
235	#41	—	2	2	no	—
236	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
237	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	278	1	no	—
240	C₂×D₄	—	139	1	no	—
241	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	278	1	no	—
244	C₂²×C₄	—	278	1	no	—
245	C₂²×C₄	—	278	1	no	—
246	C₂²×C₄	—	139	1	no	—
247	C₂²×C₄	—	139	1	no	—
248	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
249	Q₁₆	—	2	2	no	—
250	SD₁₆	—	2	2	no	—
251	SD₁₆	—	278	2	no	—
252	D₈	—	278	2	no	—
253	C₂×C₈	—	139	1	no	—
254	C₂×C₈	—	139	1	no	—
255	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
256	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	278	1	no	—
259	C₄⋊C₄	—	278	1	no	—
260	C₄⋊C₄	—	278	1	no	—
261	C₄⋊C₄	—	278	1	no	—
262	C₄⋊C₄	—	278	1	no	—
263	C₄⋊C₄	—	139	1	no	—
264	C₄⋊C₄	—	139	1	no	—
265	C₄⋊C₄	—	139	1	no	—
266	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	278	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	C₂²⋊C₄	—	278	1	no	—
274	C₂²⋊C₄	—	278	1	no	—
275	C₄²	—	278	1	no	—
276	C₄²	—	278	1	no	—
277	C₄²	—	278	1	no	—
278	C₄²	—	278	1	no	—
279	C₄²	—	139	1	no	—
280	C₄²	—	139	1	no	—
281	C₄²	—	139	1	no	—
282	C₄²	—	139	1	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	278	1	no	—
285	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
286	Q₈	—	2	1	no	—
287	Q₈	—	2	1	no	—
288	Q₈	—	2	1	no	—
289	Q₈	#14555784	1	2	yes	—
290	D₄	—	2	4	no	—
291	D₄	—	2	1	no	—
292	D₄	—	278	1	no	—
293	D₄	—	278	1	no	—
294	D₄	—	556	2	no	—
295	D₄	#14555784	1	2	yes	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	278	1	no	—
303	C₂×C₄	—	278	1	no	—
304	C₂×C₄	—	278	1	no	—
305	C₂×C₄	—	556	2	no	—
306	C₂×C₄	—	556	1	no	—
307	C₂×C₄	—	556	1	no	—
308	C₂×C₄	—	556	2	no	—
309	C₂×C₄	—	556	2	no	—
310	C₂×C₄	—	139	1	no	—
311	C₂×C₄	—	139	1	no	—
312	C₂×C₄	—	139	1	no	—
313	C₂×C₄	—	139	1	no	—
314	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
315	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	278	1	no	—
319	C₈	—	278	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	556	1	no	—
327	C₂²	—	556	1	no	—
328	C₂²	—	556	1	no	—
329	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	2	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	278	1	no	—
335	C₄	—	278	1	no	—
336	C₄	—	278	1	no	—
337	C₄	—	278	1	no	—
338	C₄	—	556	1	no	—
339	C₄	—	556	1	no	—
340	C₄	—	556	1	no	—
341	C₄	C₂×#14555784	1	1	yes	characteristic
342	C₄	C₂×#14555784	1	1	yes	characteristic
343	C₂	—	4	1	no	—
344	C₂	—	4	2	no	—
345	C₂	—	1112	1	no	—
346	C₂	#22581459	1	1	yes	characteristic
347	C₂	#22581750	1	1	yes	characteristic
348	C₂	#22581752	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#452×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	#22581098	C₂
hypercenter	#452	D₁₃₉
hypocenter	C₁₃₉	#3111
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #452

Complements

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

N	H ≅ G/N	type
1. #22581489	C₂	semidirect
2. #41×Dic₁₃₉	C₂	semidirect
3. #22581291	C₂	semidirect
4. #22581278	C₂	semidirect
5. #22581262	C₂	semidirect
6. #22581255	C₂	semidirect
7. #22581177	C₂	semidirect
8. #22581170	C₂	semidirect
10. #22581062	C₂	semidirect
11. #22581060	C₂	semidirect
12. #22581056	C₂	semidirect
13. #22581054	C₂	semidirect
14. #22581007	C₂	semidirect
15. #452×C₁₃₉	C₂	semidirect
174. C₁₃₉	#3111	semidirect
184. #452	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581489	2
#41×Dic₁₃₉	2
#22581291	2
#22581278	2
#22581262	2
#22581255	2
#22581177	2
#22581170	2
#22581098	2
#22581062	2
#22581060	2
#22581056	2
#22581054	2
#22581007	2
#452×C₁₃₉	2
#3111	139