Subgroups — #31819136

Order 17792.

382 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581680	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581624	C₂	1	1	yes	maximal, characteristic
3	#22581319	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581316	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581298	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581252	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581232	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581154	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581152	C₂	1	1	yes	maximal, characteristic, complemented
10	#22581152	C₂	1	1	yes	maximal, characteristic, complemented
11	#22581082	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581053	C₂	1	1	yes	maximal, characteristic, complemented
13	#22581052	C₂	1	1	yes	maximal, characteristic, complemented
14	#22581040	C₂	1	1	yes	maximal, characteristic, complemented
15	#478×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555785	—	2	1	no	—
17	C₂×#14555774	C₂²	1	1	yes	characteristic
18	C₂×#14555774	C₂²	1	1	yes	characteristic
19	C₂×#14555773	C₂²	1	1	yes	characteristic
20	#17984841	—	2	1	no	—
21	#17984841	C₂²	1	1	yes	characteristic
22	#17984839	C₂²	1	1	yes	characteristic
23	#17984838	C₂²	1	1	yes	characteristic
24	#17984838	C₂²	1	1	yes	characteristic
25	#17984833	C₂²	1	1	yes	characteristic
26	#17984832	—	2	1	no	—
27	#17984832	—	2	1	no	—
28	#17984816	—	2	1	no	—
29	#17984815	—	2	1	no	—
30	#17984815	—	2	1	no	—
31	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
32	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
33	#17984807	—	2	1	no	—
34	#17984807	C₂²	1	1	yes	characteristic
35	#17984807	C₂²	1	1	yes	characteristic
36	#17984804	C₂²	1	1	yes	characteristic
37	#17984804	C₂²	1	1	yes	characteristic
38	#17984775	C₂²	1	1	yes	characteristic
39	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
40	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
41	#17984765	C₂²	1	1	yes	characteristic
42	#17984764	C₂²	1	1	yes	characteristic
43	#17984764	C₂²	1	1	yes	characteristic
44	#17984763	C₂²	1	1	yes	characteristic
45	#17984762	C₂²	1	1	yes	characteristic
46	#17984762	C₂²	1	1	yes	characteristic
47	#17984762	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	#125×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
54	#107×C₁₃₉	C₂²	1	1	yes	characteristic
55	#106×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	#14555785	—	4	4	no	—
60	C₂×Dic₂₇₈	—	2	1	no	—
61	C₂×Dic₂₇₈	D₄	1	1	yes	characteristic
62	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
63	C₂×D₅₅₆	—	2	1	no	—
64	C₂×D₅₅₆	—	2	1	no	—
65	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
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×C₂²⋊D₁₃₉	—	2	1	no	—
71	C₂×C₄×D₁₃₉	—	2	1	no	—
72	C₂×C₄×D₁₃₉	—	2	1	no	—
73	#14555774	—	2	2	no	—
74	#14555774	—	2	2	no	—
75	#14555773	—	2	2	no	—
76	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
77	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₁₃₉	—	2	1	no	—
79	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
80	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
81	Dic₁₃₉⋊C₄	—	2	1	no	—
82	Dic₁₃₉⋊C₄	—	2	1	no	—
83	Dic₁₃₉⋊C₄	—	2	1	no	—
84	Dic₁₃₉⋊C₄	—	2	1	no	—
85	Dic₁₃₉⋊C₄	—	2	1	no	—
86	Dic₁₃₉⋊C₄	D₄	1	1	yes	characteristic
87	Dic₁₃₉⋊C₄	D₄	1	1	yes	characteristic
88	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
89	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
90	C₄×Dic₁₃₉	—	2	1	no	—
91	C₄×Dic₁₃₉	—	2	1	no	—
92	C₄×Dic₁₃₉	—	2	1	no	—
93	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
94	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₁₃₉	—	2	1	no	—
97	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₁₃₉	—	2	1	no	—
100	C₄⋊C₄×C₁₃₉	—	2	1	no	—
101	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
104	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
105	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
106	C₂²×D₁₃₉	—	2	1	no	—
107	C₂²×D₁₃₉	—	2	1	no	—
108	Dic₂₇₈	—	2	1	no	—
109	Dic₂₇₈	—	2	2	no	—
110	Dic₂₇₈	—	4	2	no	—
111	Dic₂₇₈	#41	1	2	yes	complemented
112	D₅₅₆	—	2	1	no	—
113	D₅₅₆	—	2	1	no	—
114	D₅₅₆	—	2	1	no	—
115	D₅₅₆	—	2	1	no	—
116	D₅₅₆	—	4	2	no	—
117	D₅₅₆	—	8	1	no	—
118	C₂²⋊D₁₃₉	—	4	2	no	—
119	C₂²⋊D₁₃₉	—	4	2	no	—
120	C₂²⋊D₁₃₉	—	8	1	no	—
121	C₂²⋊D₁₃₉	—	8	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	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
128	C₄×D₁₃₉	—	4	2	no	—
129	C₄×D₁₃₉	—	4	2	no	—
130	C₁₃₉⋊C₈	—	2	1	no	—
131	C₁₃₉⋊C₈	—	2	1	no	—
132	C₂³×C₁₃₉	—	2	1	no	—
133	D₄×C₁₃₉	—	2	1	no	—
134	D₄×C₁₃₉	—	2	1	no	—
135	C₂×C₄×C₁₃₉	—	2	1	no	—
136	C₂×C₄×C₁₃₉	—	2	1	no	—
137	C₂×C₄×C₁₃₉	—	2	1	no	—
138	C₂×C₄×C₁₃₉	—	4	2	no	—
139	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
142	C₈×C₁₃₉	—	2	1	no	—
143	C₈×C₁₃₉	—	2	1	no	—
144	C₂²×C₁₃₉	—	4	1	no	—
145	C₂²×C₁₃₉	—	4	1	no	—
146	C₂²×C₁₃₉	—	4	1	no	—
147	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
148	C₂×D₁₃₉	—	4	1	no	—
149	C₂×D₁₃₉	—	4	1	no	—
150	C₂×D₁₃₉	—	4	1	no	—
151	C₂×D₁₃₉	—	4	1	no	—
152	C₂×D₁₃₉	—	4	1	no	—
153	C₂×D₁₃₉	—	4	1	no	—
154	C₄×C₁₃₉	—	2	1	no	—
155	C₄×C₁₃₉	—	2	1	no	—
156	C₄×C₁₃₉	—	4	1	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	2₋¹⁺⁴	1	1	yes	characteristic
161	Dic₁₃₉	—	2	2	no	—
162	Dic₁₃₉	—	2	1	no	—
163	Dic₁₃₉	—	2	1	no	—
164	Dic₁₃₉	—	2	1	no	—
165	Dic₁₃₉	—	4	1	no	—
166	Dic₁₃₉	AGL(1,ℤ/8)	1	2	yes	—
167	C₂×C₁₃₉	—	8	1	no	—
168	C₂×C₁₃₉	#548	1	1	yes	characteristic
169	C₂×C₁₃₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
170	C₂×C₁₃₉	#577	1	1	yes	characteristic
171	D₁₃₉	—	8	1	no	—
172	D₁₃₉	—	8	1	no	—
173	C₁₃₉	#3368	1	1	yes	characteristic, complemented, Sylow
174	#3368	—	139	1	no	maximal, Sylow
175	#557	—	139	1	no	—
176	#550	—	139	1	no	—
177	#490	—	139	1	no	—
178	#490	—	139	1	no	—
179	#486	—	139	1	no	—
180	#478	—	139	1	no	—
181	#478	D₁₃₉	1	1	yes	characteristic, complemented
182	#474	—	139	1	no	—
183	#460	—	139	1	no	—
184	#460	—	139	1	no	—
185	#460	—	139	1	no	—
186	#446	—	139	1	no	—
187	#440	—	139	1	no	—
188	#440	—	139	1	no	—
189	C₄×SD₁₆	—	139	1	no	—
190	C₂×#41	—	278	1	no	—
191	C₂×SD₁₆	—	139	1	no	—
192	C₂×SD₁₆	—	139	1	no	—
193	C₂×SD₁₆	—	139	1	no	—
194	C₄⋊Q₈	—	278	1	no	—
195	C₄⋊Q₈	—	139	1	no	—
196	#127	—	139	1	no	—
197	#127	—	139	1	no	—
198	#127	—	139	1	no	—
199	#125	—	278	1	no	—
200	#125	—	278	1	no	—
201	#125	—	139	1	no	—
202	#125	C₂×D₁₃₉	1	1	yes	characteristic
203	#121	—	278	1	no	—
204	#121	—	278	1	no	—
205	#121	—	278	1	no	—
206	C₄×Q₈	—	278	1	no	—
207	C₄×Q₈	—	139	1	no	—
208	C₄×Q₈	—	139	1	no	—
209	C₄×Q₈	—	139	1	no	—
210	C₄×Q₈	—	139	1	no	—
211	C₄×D₄	—	139	1	no	—
212	C₄×D₄	—	139	1	no	—
213	C₄×D₄	C₂×D₁₃₉	1	1	yes	characteristic
214	#107	—	139	1	no	—
215	#107	C₂×D₁₃₉	1	1	yes	characteristic
216	#106	C₂×D₁₃₉	1	1	yes	characteristic
217	C₄⋊C₈	—	139	1	no	—
218	C₄⋊C₈	—	139	1	no	—
219	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
220	#103	—	139	1	no	—
221	#102	—	139	1	no	—
222	#102	—	139	1	no	—
223	#102	—	139	1	no	—
224	#102	—	139	1	no	—
225	#102	—	139	1	no	—
226	#102	—	139	1	no	—
227	#102	—	139	1	no	—
228	#102	C₂×D₁₃₉	1	1	yes	characteristic
229	#102	C₂×D₁₃₉	1	1	yes	characteristic
230	#97	—	139	1	no	—
231	#97	—	139	1	no	—
232	C₄×C₈	—	139	1	no	—
233	#41	—	556	4	no	—
234	C₂×Q₈	—	278	1	no	—
235	C₂×Q₈	—	139	1	no	—
236	C₂×Q₈	—	139	1	no	—
237	C₂×D₄	—	278	1	no	—
238	C₂×D₄	—	278	1	no	—
239	C₂×D₄	—	278	1	no	—
240	C₂×D₄	—	278	1	no	—
241	C₂×D₄	—	278	1	no	—
242	C₂×D₄	—	278	1	no	—
243	C₂×D₄	—	139	1	no	—
244	C₂×D₄	—	139	1	no	—
245	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	278	1	no	—
248	C₂²×C₄	—	278	1	no	—
249	SD₁₆	—	278	2	no	—
250	SD₁₆	—	278	2	no	—
251	SD₁₆	—	278	2	no	—
252	C₂×C₈	—	139	1	no	—
253	C₂×C₈	—	139	1	no	—
254	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
255	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
256	C₄⋊C₄	—	2	1	no	—
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	—	278	1	no	—
264	C₄⋊C₄	—	139	1	no	—
265	C₄⋊C₄	—	139	1	no	—
266	C₄⋊C₄	—	139	1	no	—
267	C₄⋊C₄	—	139	1	no	—
268	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
269	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
270	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
271	C₂²⋊C₄	—	2	1	no	—
272	C₂²⋊C₄	—	278	1	no	—
273	C₂²⋊C₄	—	278	1	no	—
274	C₄²	—	278	1	no	—
275	C₄²	—	278	1	no	—
276	C₄²	—	278	1	no	—
277	C₄²	—	139	1	no	—
278	C₄²	—	139	1	no	—
279	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
280	C₂³	—	2	1	no	—
281	C₂³	—	278	1	no	—
282	C₂³	—	278	1	no	—
283	Q₈	—	278	2	no	—
284	Q₈	—	278	1	no	—
285	Q₈	—	556	2	no	—
286	Q₈	—	139	2	no	—
287	D₄	—	2	1	no	—
288	D₄	—	2	1	no	—
289	D₄	—	278	1	no	—
290	D₄	—	278	1	no	—
291	D₄	—	278	1	no	—
292	D₄	—	278	1	no	—
293	D₄	—	556	2	no	—
294	D₄	—	556	2	no	—
295	D₄	—	556	2	no	—
296	D₄	—	1112	1	no	—
297	D₄	—	1112	1	no	—
298	D₄	—	1112	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	278	1	no	—
304	C₂×C₄	—	278	1	no	—
305	C₂×C₄	—	556	2	no	—
306	C₂×C₄	—	556	2	no	—
307	C₂×C₄	—	139	1	no	—
308	C₂×C₄	—	139	1	no	—
309	C₂×C₄	—	139	1	no	—
310	C₂×C₄	—	139	1	no	—
311	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
312	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
313	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	278	1	no	—
317	C₈	—	278	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	556	1	no	—
322	C₂²	—	556	1	no	—
323	C₂²	—	556	1	no	—
324	C₂²	—	556	1	no	—
325	C₂²	—	556	1	no	—
326	C₂²	—	556	1	no	—
327	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	278	1	no	—
334	C₄	—	278	1	no	—
335	C₄	—	278	2	no	—
336	C₄	—	278	1	no	—
337	C₄	—	556	1	no	—
338	C₄	—	139	2	no	—
339	C₄	#41×D₁₃₉	1	1	yes	characteristic
340	C₄	#17984890	1	1	yes	characteristic
341	C₂	—	8	1	no	—
342	C₂	—	1112	1	no	—
343	C₂	—	1112	1	no	—
344	C₂	#22581613	1	1	yes	characteristic
345	C₂	AGL(1,ℤ/8)×D₁₃₉	1	1	yes	characteristic
346	C₂	#22581760	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#478×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	#22581624	C₂
hypercenter	#478	D₁₃₉
hypocenter	C₁₃₉	#3368
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #22581680	C₂	semidirect
3. #22581319	C₂	semidirect
4. #22581316	C₂	semidirect
5. #22581298	C₂	semidirect
6. #22581252	C₂	semidirect
7. #22581232	C₂	semidirect
8. #22581154	C₂	semidirect
9. #22581152	C₂	semidirect
10. #22581152	C₂	semidirect
11. #22581082	C₂	semidirect
12. #22581053	C₂	semidirect
13. #22581052	C₂	semidirect
14. #22581040	C₂	semidirect
15. #478×C₁₃₉	C₂	semidirect
111. Dic₂₇₈	#41	semidirect
173. C₁₃₉	#3368	semidirect
181. #478	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581680	2
#22581624	2
#22581319	2
#22581316	2
#22581298	2
#22581252	2
#22581232	2
#22581154	2
#22581152	2
#22581082	2
#22581053	2
#22581052	2
#22581040	2
#478×C₁₃₉	2
#3368	139