Subgroups — #31819054

Order 17792.

350 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581675	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581662	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581309	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581308	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581302	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581263	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581260	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581242	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581170	C₂	1	1	yes	maximal, characteristic
10	#22581168	C₂	1	1	yes	maximal, characteristic
11	#22581156	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈×Dic₂₇₈	C₂	1	1	yes	maximal, characteristic, complemented
13	#22581044	C₂	1	1	yes	maximal, characteristic, complemented
14	#22581033	C₂	1	1	yes	maximal, characteristic
15	C₄×SD₁₆×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555773	C₂²	1	1	yes	characteristic
17	C₂×#14555770	C₂²	1	1	yes	characteristic
18	#17984841	C₂²	1	1	yes	characteristic
19	#17984840	C₂²	1	1	yes	characteristic
20	#17984837	—	2	1	no	—
21	#17984836	—	2	1	no	—
22	#17984834	C₂²	1	1	yes	characteristic
23	#17984833	C₂²	1	1	yes	characteristic
24	#17984831	C₂²	1	1	yes	characteristic
25	#17984830	C₂²	1	1	yes	characteristic
26	#17984821	—	2	1	no	—
27	#17984820	—	2	1	no	—
28	#17984819	—	2	1	no	—
29	#17984819	—	2	1	no	—
30	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
31	#17984807	C₂²	1	1	yes	characteristic
32	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
33	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
34	#17984804	C₂²	1	1	yes	characteristic
35	#17984799	—	2	1	no	—
36	C₂×Dic₁₃₉⋊C₄	—	2	1	no	—
37	#17984775	C₂²	1	1	yes	characteristic
38	#17984773	C₂²	1	1	yes	characteristic
39	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
40	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
41	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
42	#17984767	C₂²	1	1	yes	characteristic
43	#17984767	C₂²	1	1	yes	characteristic
44	#17984766	C₂²	1	1	yes	characteristic
45	#17984766	C₂²	1	1	yes	characteristic
46	#17984765	C₂²	1	1	yes	characteristic
47	#17984764	C₂²	1	1	yes	characteristic
48	#17984764	C₂²	1	1	yes	characteristic
49	#17984762	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
55	#106×C₁₃₉	C₂²	1	1	yes	characteristic
56	#103×C₁₃₉	C₂²	1	1	yes	characteristic
57	#102×C₁₃₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₇₈	—	2	1	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	C₂²×Dic₁₃₉	—	2	1	no	—
64	C₂²×Dic₁₃₉	—	2	1	no	—
65	C₂×C₄×D₁₃₉	—	2	1	no	—
66	#14555773	—	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	D₄	1	1	yes	characteristic
72	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
74	Dic₁₃₉⋊C₄	—	2	1	no	—
75	Dic₁₃₉⋊C₄	—	2	1	no	—
76	Dic₁₃₉⋊C₄	—	2	1	no	—
77	Dic₁₃₉⋊C₄	—	2	1	no	—
78	Dic₁₃₉⋊C₄	—	2	1	no	—
79	Dic₁₃₉⋊C₄	—	2	1	no	—
80	Dic₁₃₉⋊C₄	—	2	1	no	—
81	Dic₁₃₉⋊C₄	D₄	1	1	yes	characteristic
82	Dic₁₃₉⋊C₄	D₄	1	1	yes	characteristic
83	C₂²⋊Dic₁₃₉	—	2	1	no	—
84	C₂²⋊Dic₁₃₉	—	2	1	no	—
85	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
86	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
87	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
88	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
89	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
90	C₄×Dic₁₃₉	—	2	1	no	—
91	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
92	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
93	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
94	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
95	C₂²×C₄×C₁₃₉	—	2	1	no	—
96	SD₁₆×C₁₃₉	—	2	2	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	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
102	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
103	C₄²×C₁₃₉	—	2	1	no	—
104	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
105	C₂²×D₁₃₉	—	2	1	no	—
106	Dic₂₇₈	—	2	2	no	—
107	Dic₂₇₈	—	2	1	no	—
108	Dic₂₇₈	—	2	1	no	—
109	Dic₂₇₈	—	8	1	no	—
110	D₅₅₆	—	2	1	no	—
111	D₅₅₆	—	2	1	no	—
112	C₂×Dic₁₃₉	—	2	1	no	—
113	C₂×Dic₁₃₉	—	2	1	no	—
114	C₂×Dic₁₃₉	—	2	1	no	—
115	C₂×Dic₁₃₉	—	4	2	no	—
116	C₂×Dic₁₃₉	—	8	1	no	—
117	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
118	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
121	C₄×D₁₃₉	—	4	2	no	—
122	C₁₃₉⋊C₈	—	2	1	no	—
123	C₁₃₉⋊C₈	—	2	1	no	—
124	C₂³×C₁₃₉	—	2	1	no	—
125	Q₈×C₁₃₉	—	2	1	no	—
126	Q₈×C₁₃₉	—	2	1	no	—
127	D₄×C₁₃₉	—	2	1	no	—
128	D₄×C₁₃₉	#41	1	2	yes	—
129	C₂×C₄×C₁₃₉	—	2	1	no	—
130	C₂×C₄×C₁₃₉	—	2	1	no	—
131	C₂×C₄×C₁₃₉	—	2	1	no	—
132	C₂×C₄×C₁₃₉	—	8	1	no	—
133	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
136	C₈×C₁₃₉	—	2	1	no	—
137	C₈×C₁₃₉	—	2	1	no	—
138	C₂²×C₁₃₉	—	2	2	no	—
139	C₂²×C₁₃₉	—	4	1	no	—
140	C₂²×C₁₃₉	—	4	1	no	—
141	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
142	C₂×D₁₃₉	—	4	1	no	—
143	C₂×D₁₃₉	—	4	1	no	—
144	C₂×D₁₃₉	—	4	1	no	—
145	C₄×C₁₃₉	—	2	1	no	—
146	C₄×C₁₃₉	—	2	1	no	—
147	C₄×C₁₃₉	—	4	1	no	—
148	C₄×C₁₃₉	—	4	1	no	—
149	C₄×C₁₃₉	—	4	1	no	—
150	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
151	C₄×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
152	Dic₁₃₉	—	2	1	no	—
153	Dic₁₃₉	—	2	1	no	—
154	Dic₁₃₉	—	2	1	no	—
155	Dic₁₃₉	—	4	1	no	—
156	Dic₁₃₉	—	4	1	no	—
157	Dic₁₃₉	—	4	1	no	—
158	Dic₁₃₉	#135	1	2	yes	—
159	C₂×C₁₃₉	—	4	2	no	—
160	C₂×C₁₃₉	#548	1	1	yes	characteristic
161	C₂×C₁₃₉	C₂×#135	1	1	yes	characteristic
162	C₂×C₁₃₉	#577	1	1	yes	characteristic
163	D₁₃₉	—	8	1	no	—
164	C₁₃₉	#3360	1	1	yes	characteristic, complemented, Sylow
165	#3360	—	139	1	no	maximal, Sylow
166	#556	—	139	1	no	—
167	#554	—	139	1	no	—
168	#488	—	139	1	no	—
169	#488	—	139	1	no	—
170	#487	—	139	1	no	—
171	#479	—	139	1	no	—
172	#479	—	139	1	no	—
173	#476	—	139	1	no	—
174	#463	—	139	1	no	—
175	#463	—	139	1	no	—
176	#461	—	139	1	no	—
177	C₈×Q₈	—	139	1	no	—
178	C₄×SD₁₆	—	139	1	no	—
179	C₄×SD₁₆	D₁₃₉	1	1	yes	characteristic, complemented
180	C₄×D₈	—	139	1	no	—
181	C₂×SD₁₆	—	139	1	no	—
182	C₂×SD₁₆	C₂×D₁₃₉	1	1	yes	characteristic
183	C₂×D₈	—	139	1	no	—
184	C₄⋊Q₈	—	139	1	no	—
185	C₄⋊Q₈	—	139	1	no	—
186	#126	—	278	1	no	—
187	#126	—	278	1	no	—
188	#125	—	139	1	no	—
189	#125	—	139	1	no	—
190	#124	—	139	1	no	—
191	#124	—	139	1	no	—
192	C₂²⋊Q₈	—	278	1	no	—
193	C₂²⋊Q₈	—	278	1	no	—
194	C₂²⋊Q₈	—	278	1	no	—
195	C₂²⋊Q₈	—	278	1	no	—
196	C₄×Q₈	—	139	1	no	—
197	C₄×Q₈	—	139	1	no	—
198	C₄×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
199	C₄×D₄	—	139	1	no	—
200	C₄×D₄	—	139	1	no	—
201	C₄×D₄	—	139	1	no	—
202	C₄×D₄	C₂×D₁₃₉	1	1	yes	characteristic
203	#117	—	278	1	no	—
204	C₂×C₄⋊C₄	—	278	1	no	—
205	#107	—	139	1	no	—
206	#106	—	139	1	no	—
207	#106	C₂×D₁₃₉	1	1	yes	characteristic
208	C₄⋊C₈	—	139	1	no	—
209	C₄⋊C₈	—	139	1	no	—
210	C₄⋊C₈	—	139	1	no	—
211	#103	—	139	1	no	—
212	#103	—	139	1	no	—
213	#103	—	139	1	no	—
214	#103	—	139	1	no	—
215	#103	—	139	1	no	—
216	#103	C₂×D₁₃₉	1	1	yes	characteristic
217	#102	—	139	1	no	—
218	#102	—	139	1	no	—
219	#102	—	139	1	no	—
220	#102	C₂×D₁₃₉	1	1	yes	characteristic
221	C₄×C₈	—	139	1	no	—
222	C₄×C₈	—	139	1	no	—
223	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
224	C₂×Q₈	—	278	1	no	—
225	C₂×Q₈	—	139	1	no	—
226	C₂×Q₈	—	139	1	no	—
227	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
228	C₂×D₄	—	139	1	no	—
229	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	278	1	no	—
232	C₂²×C₄	—	278	1	no	—
233	C₂²×C₄	—	278	1	no	—
234	SD₁₆	—	2	2	no	—
235	SD₁₆	—	278	2	no	—
236	D₈	—	278	2	no	—
237	C₂×C₈	—	139	1	no	—
238	C₂×C₈	—	139	1	no	—
239	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	278	1	no	—
243	C₄⋊C₄	—	278	1	no	—
244	C₄⋊C₄	—	278	1	no	—
245	C₄⋊C₄	—	278	1	no	—
246	C₄⋊C₄	—	278	1	no	—
247	C₄⋊C₄	—	278	1	no	—
248	C₄⋊C₄	—	278	1	no	—
249	C₄⋊C₄	—	278	1	no	—
250	C₄⋊C₄	—	139	1	no	—
251	C₄⋊C₄	—	139	1	no	—
252	C₄⋊C₄	—	139	1	no	—
253	C₄⋊C₄	—	139	1	no	—
254	C₄⋊C₄	—	139	1	no	—
255	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
257	C₂²⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	278	1	no	—
259	C₂²⋊C₄	—	278	1	no	—
260	C₂²⋊C₄	—	278	1	no	—
261	C₂²⋊C₄	—	278	1	no	—
262	C₂²⋊C₄	—	278	1	no	—
263	C₂²⋊C₄	—	278	1	no	—
264	C₂²⋊C₄	—	278	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	278	1	no	—
267	C₄²	—	139	1	no	—
268	C₄²	—	139	1	no	—
269	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
270	C₂³	—	2	1	no	—
271	C₂³	—	278	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	278	2	no	—
275	Q₈	—	278	1	no	—
276	Q₈	—	278	1	no	—
277	Q₈	—	1112	1	no	—
278	D₄	—	2	1	no	—
279	D₄	—	278	1	no	—
280	D₄	—	278	1	no	—
281	D₄	#14555785	1	2	yes	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	8	1	no	—
286	C₂×C₄	—	278	1	no	—
287	C₂×C₄	—	278	1	no	—
288	C₂×C₄	—	278	1	no	—
289	C₂×C₄	—	556	2	no	—
290	C₂×C₄	—	556	2	no	—
291	C₂×C₄	—	1112	1	no	—
292	C₂×C₄	—	139	1	no	—
293	C₂×C₄	—	139	1	no	—
294	C₂×C₄	—	139	1	no	—
295	C₂×C₄	—	139	1	no	—
296	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
297	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	278	1	no	—
302	C₈	—	278	1	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	556	1	no	—
307	C₂²	—	556	1	no	—
308	C₂²	—	556	1	no	—
309	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	278	1	no	—
316	C₄	—	278	1	no	—
317	C₄	—	278	1	no	—
318	C₄	—	556	1	no	—
319	C₄	—	556	1	no	—
320	C₄	—	556	1	no	—
321	C₄	—	139	2	no	—
322	C₄	C₂×#14555785	1	1	yes	characteristic
323	C₄	#17984889	1	1	yes	characteristic
324	C₂	—	4	2	no	—
325	C₂	—	1112	1	no	—
326	C₂	#22581612	1	1	yes	characteristic
327	C₂	#135×D₁₃₉	1	1	yes	characteristic
328	C₂	#22581759	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	C₄×SD₁₆×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂×#14555770	C₂²
hypercenter	C₄×SD₁₆	D₁₃₉
hypocenter	C₁₃₉	#3360
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #22581675	C₂	semidirect
2. #22581662	C₂	semidirect
3. #22581309	C₂	semidirect
4. #22581308	C₂	semidirect
5. #22581302	C₂	semidirect
6. #22581263	C₂	semidirect
7. #22581260	C₂	semidirect
8. #22581242	C₂	semidirect
11. #22581156	C₂	semidirect
12. C₈×Dic₂₇₈	C₂	semidirect
13. #22581044	C₂	semidirect
15. C₄×SD₁₆×C₁₃₉	C₂	semidirect
164. C₁₃₉	#3360	semidirect
179. C₄×SD₁₆	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581675	2
#22581662	2
#22581309	2
#22581308	2
#22581302	2
#22581263	2
#22581260	2
#22581242	2
#22581170	2
#22581168	2
#22581156	2
C₈×Dic₂₇₈	2
#22581044	2
#22581033	2
C₄×SD₁₆×C₁₃₉	2
#3360	139